Computational Fluid-Structure Interaction: Methods, Models, and Applications [1 ed.] 0128147709, 9780128147702

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Computational Fluid-Structure Interaction

Computational Fluid-Structure Interaction Methods, Models, and Applications

Yong Zhao Professor, Nazarbayev University, Astana, Kazakhstan

Xiaohui Su Professor, Dalian University of Technology, Liaoning, China

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright r 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-814770-2 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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Preface Modern Computational Science and Engineering (CSE) is developing rapidly and coming out of age to become more and more sophisticated in order to tackle numerous real-life problems, such as multiphysics and multiscale ones. Fluid structure interaction (FSI) is one of these challenging problems, which are found in nature and numerous engineering applications. Despite recent intensive research, FSI is still not fully understood and well studied numerically and experimentally. There is an obvious lack of reliable, robust, and efficient computational methods and related specialized books and textbooks for studying 3D complex FSI problems. Over the last decade a number of novel, high efficient numerical techniques, and solution methods have come to the fore in computational fluid dynamics (CFD) for the study of FSI. Our research group have developed many such numerical methods specifically for FSI simulation over the past two and a half decades. The solvers and FSI simulation methods developed have been applied in industrial research and designs, such as the design and analysis of artificial heart valves, the study of instabilities of rotating disks in data storage devices, as well as the design and analysis of flapping-wing micro air vehicles (MAVs). The results have been widely published and the methods have been well received. As such we are motivated to write this book to provide the much-needed reference materials and introduce the latest FSI methods in detail for the benefit of researchers, practicing engineers, and future students. We acknowledge the contributions from our existing and former group members, whose works appear in this book and many of the results presented here in this book mainly come from their Masters and Ph.D. dissertations. The main contributors of this book include: 1. Dr. Xia Guohua, who contributes to the development of the immersed membrane method (IMM), the finite volume structural dynamic solver and their FSI applications in artificial tissue heart valves. 2. Dr. Jonathon Tai, who has made major contributions to the development of the finite volume unstructured multigrid parallel 2D/3D incompressible flow solvers, the immersed object method (IOM) and their FSI applications in artificial mechanical heart valves. 3. Dr. Lv Xin, who helps to develop the preconditioned finite volume multigrid parallel compressible 2D/3D solvers, the immersed membrane xi

xii

4. 5. 6. 7. 8.

Preface

method (IMM), finite volume structural dynamic solver and their FSI applications in the studies of instabilities of rotating disks in data storage devices and transonic flow flutter of the M6 wing. Mr. Huang Hai, Daniel, who contributes to the development of the early version of finite volume unstructured grid 2D compressible flow solver. Mr. Barnhard Bals, who develops the parallel arbitrary Lagrange Euler (ALE) technique for its FSI application in the study of incompressible flow in a collapsible channel. Mr. Yao Cao, who works on the development of the arbitrary Lagrange Euler (ALE) technique for its FSI applications in the study of MAVs. Mr. Yuanwei Cao, who contributes to the development of the arbitrary Lagrange Euler (ALE) technique and its FSI applications in the study of rigid wing MAVs. Mr. Zhen Yin, who helps to develop the arbitrary Lagrange Euler (ALE) technique and works on its FSI applications in the study of deformable wing MAVs.

Yong Zhao and Xiaohui Su Professor, Nazarbayev University, Astana, Kazakhstan, Professor, Dalian University of Technology, Liaoning, China

Chapter 1

Introduction 1.1 BACKGROUND The interaction of a flexible structure with a flowing fluid, in which it is submersed or by which it is surrounded, gives rise to a rich variety of physical phenomena with applications in many fields of engineering, for example, the stability and response of aircraft wings, the flow of blood through arteries, the response of bridges and tall buildings to winds, the vibration of turbine and compressor blades or hard disks for computer data storage, the opening and closing of heart valves, and the oscillation of heat exchangers. To understand these phenomena we need to model both the structure and the fluid and simulate their interaction to gain insights into the rich and complicated physics involved. However, in keeping with the overall theme of this volume, the emphasis here is on numerical methods, physical modeling, and their engineering applications. Furthermore, the applications are largely drawn from mechanical and aerospace engineering, although the methods and fundamental physical phenomena have much wider applications. In this book we emphasize recent developments and future challenges. The emphasis is on the enhanced physical understanding and the ease of simulating highly complex fluid structure interaction (FSI) with accuracy, made possible by the numerical methods developed.

1.2 COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD), usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows, heat transfer, and associated phenomena such as chemical reactions. The fundamental basis of almost all CFD problems is the Navier Stokes equations. Numerical methods developed for CFD computations were first proposed by Richardson in his book Weather Prediction by Numerical Process [1], which forms the basis for modern CFD and numerical meteorology. Modern CFD has its origins dating back to the 1960s and 1970s, when universities and companies started to experiment with more and more sophisticated methods to solve more complicated fluid flow problems for engineering applications, such as Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00001-5 © 2019 Elsevier Inc. All rights reserved.

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aircraft designs, to take advantage of the development in computer technologies. During the last decades significant progress has been made in CFD as a result of the development of powerful algorithms and fast supercomputers. In the meantime CFD has firmly established itself as the third approach in the field of fluid dynamics, equal to pure experiment and theory. Nowadays CFD has been used in a wide range of applications in many industries. Here are a few examples: G G G G G

G G G G G G G G G

Aerospace/aeronautics Automotive engineering Astrophysics and cosmology Atmospheric sciences Building and construction: HVAC (heating, ventilation, and air conditioning) and wind engineering Biomedical engineering Chemical/petrochemical engineering Energy/power generation Earth science Environmental engineering Manufacturing/process engineering Naval architecture and marine/offshore engineering Oil and gas industry Product design and optimization

CFD cannot completely replace experiments, both of which are complementary and have to be applied together to obtain optimal results. Phenomena that are too complex to be predicted by theory and too expensive or dangerous to be reproduced in the laboratory can now be simulated on supercomputers. Computational results that can be arbitrarily detailed and digitally processed for more comprehensive visualization are the advantages of CFD over experimental methods. Changes in the environment and parameters of the simulation can be made more accurate and faster in a computer simulation than in an experimental construction. CFD needs much less infrastructure and space, usually even less energy than laboratory experiments. These considerations mean that computational simulations are less expensive, especially because the same supercomputer can be used for many completely different simulations. The availability of computational results and data files together with the internet makes CFD portable all over the world and increases the effectiveness of research and development work. Nevertheless practical experiments are still necessary to create a basis for the validation of a numerical solver for each investigated problem. A complete CFD program contains three main elements, a preprocessor, a solver, and a postprocessor. Preprocessing consists of a few software modules for the definition of the physical and chemical phenomena to be

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modeled, the generation of a computational grid, as well as the fluid properties and the specification of appropriate boundary conditions. The flow simulation takes place in the solver of a CFD program. The governing analytic equations are discretized and solved for the node locations by iteration methods. Usually one of the three discretization approaches, finite difference method (FDM), finite elements method (FEM), or finite volume method (FVM) is used. An FV Runge Kutta time-stepping algorithm is used in this book and is described in later chapters. One of the main advantages of CFD mentioned previously is the detailed visualization of the results. This is accomplished by the postprocessing section of a CFD package. TECPLOT, one of the several commercial visualization tools, is mainly used in this book. Some examples for visualization options are domain geometry and grid visualization, vector plots, flow contour plots, or particle tracking. The output can be almost arbitrarily detailed and easily exported into other file formats to make the results portable.

1.3 COMPUTATIONAL STRUCTURAL DYNAMICS Similar to CFD, computational structural dynamics, usually abbreviated as CSD, is a branch of structural dynamics that uses numerical methods and algorithms to solve and analyze solid dynamics problems that involve single or multibody dynamics, including rigid or flexible, linear or nonlinear behaviors. The fundamental basis of CSD simulations are the Cauchy’s equations. Over the last five decades a wide variety of numerical methods have been proposed for the numerical solution of partial differential equations of solid dynamics. Among these methods, the finite element method (FEM) has firmly established itself as the standard approach for problems in computational solid mechanics (CSM), especially with regard to deformation problems involving nonlinear material analysis. As it is contemporary the FVM is developed from early finite difference techniques and has similarly established itself within the field of CFD. Both classes of methods integrate governing equations over predefined control volumes, which are associated with the elements making up the domain of interest. Furthermore, both approaches can be classified as weighted residual methods where they differ in the weighting functions adopted. In many engineering applications there is an emerging need to model multiphysics problems in a coupled manner. In principle, because of their local conservation properties the FVMs should be in a good position to solve such problems effectively. Over the last decade a number of researchers have applied FVMs to solve problems in CSM and it is now possible to classify these methods into two approaches, cell-centered and vertex-based ones. Subsequently such techniques have been applied to CSM problems using structured and unstructured meshes. With regard to these techniques it should

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be noted that when solid bodies undergo deformation the application of mechanical boundary conditions is the most effective if they can be imposed directly on the physical boundary. Obviously the cell-centered approximation may have difficulty in prescribing the boundary conditions, when complex geometries are considered and where displacements at the boundary are not prescribed directly and in a straightforward manner. The second approach is based on some basic ideas of traditional FE methods, which employ shape functions to describe the variation of an independent variable, such as displacement, over an element and is therefore well suited to complex geometries. The approach can be roughly classified as a cell-vertex FVM. Both the above-mentioned FV approaches apply strict conservation laws over a control volume and have demonstrated superiority over traditional FE methods with regard to accuracy. Some researchers have attributed this to the local conservation of independent variables as enforced by the controlvolume methods employed and others to the enforced continuity of the derivatives of the independent variables across cell boundaries. In this book we solve both NS and Cauchy’s equations using a novel FVM scheme in which such treatment has expressed significant merit in the solver development of two-way FSI.

1.4 MOVING BOUNDARY Moving boundary problems have important engineering applications in a variety of engineering fields such as solid and fluid dynamics, combustion, heat transfer, material sciences. Currently many techniques are devoted to solve the moving boundary problems, and these techniques are classified into three major groups, namely Eulerian method, arbitrary Lagrangian Eulerian (ALE) method, and hybrid method, based on the property of computational grids used in the calculations. The following subsections give only a review of the methods used to tackle moving boundary problems.

1.4.1 Eulerian Method In the Eulerian method the mesh is stationary and does not move or deform. Therefore fluid motion can be conveniently described with respect to this Eulerian reference frame. The Eulerian method includes the fictitious domain method (FDM), immersed boundary method (IBM), immersed object method (IOM), immersed membrane method (IMM), ghost cell method (GCM), ghost fluid method (GFM), and volume of fluid (VOF) method, etc. IOM and IMM will be reviewed in the following subsections.

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1.4.1.1 Immersed Object Method A novel and efficient IOM is proposed by the authors’ research group for numerical simulation of blood flows in prosthetic heart valves under physiological conditions and blood leaflet interaction. The IOM is similar to the IBM in the sense that they both use a combination of Eulerian approach for the fluid and Lagrangian for the structure. Peskin et al. [2 4] proposed IBM to simulate the natural heart and heart valves combining Eulerian flow equations and a Lagrangian description of heart walls and valves. The philosophy of the IBM is to treat the elastic material as a part of the fluid in which additional forces (arising from the elastic stresses) are applied. The fluid equations are solved on a regular mesh grid, the structure of which is not modified in any way by the presence of the immersed elastic bodies and the geometry of which may be quite complicated. The elastic material is tracked in Lagrangian fashion, by following a collection of representative material points. The spatial configuration of these points is used to compute elastic forces, which are applied to the nearby grid points of the fluid. The fluid velocity is updated under the influence of these forces, and the new velocity is then interpolated at the elastic material points, which are moved at the interpolated velocity to complete the time step. The IBM has been applied to a wide range of problems, mostly in biofluid dynamics, including blood flow in the heart [4], platelet aggregation during blood clotting [5], fluid dynamics of the inner ear [6], flow in arterioles [7], and simulating the motion of flexible pulp fibers [8]. Later Peskin and coinvestigators developed the method further and published an adaptive version [9] and a second-order accurate [10] one to enhance its accuracy. The IOM involves assuming the solid body to be filled with fluid and specifying the body force term in such a way as to simulate the motion of the solid body and presence of its boundary within the computational domain that does not coincide with the computational grid and without altering the computational grid [11]. A direct forcing method is employed to drive the fluid velocity within the object to the moving object velocity, which will be detailed in Chapter 7, The Immersed Object Method With Overlapping Grids. To our knowledge there are still no existing publications in the literature dealing with flows using IOM with overlapping grid (OG) incorporating the unstructured parallel-MG method. The developed solver is used to compute a 29-mm St. Jude Medical (SJM) aortic bileaflet mechanical heart valve (MHV) under physiological conditions and blood leaflet interaction for both opening and closing phases. 1.4.1.2 Immersed Membrane Method The IMM is proposed to solve moving boundary problems with thin structures [12,13]. In the IMM a stationary Eulerian mesh is used for the computation of the fluid domain, while the domain occupied by the moving and/or

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deforming solid structures are treated as “vacuum,” whose domain is defined by the fluid structure interface, which may be varying with time. To obtain accurate flow conditions in the vicinity of the interface, without changing the topology and geometry of the fluid mesh, the concept of ghost nodes with ghost values is introduced to account for discontinuities in pressures and fluid stresses across the interface. Velocity continuity at the fluid structure interface is directly enforced by a newly developed thirdorder MUSCL-like one-sided extrapolation scheme for fluid velocity, with the ghost nodes and the extrapolated velocities being treated as boundary conditions for the fluid domain. The fluid structure interaction is calculated in a staggered and iterative manner. First the governing equations for the fluid domain are solved, after which fluid forces are calculated along the fluid structure interface. Then the structure dynamics is calculated under the influence of fluid forces and the structure is moved to a new position with a new configuration. Lastly the IMM is employed to obtain appropriate boundary conditions at the fluid structure interface for the fluid domain with the newly updated structure configuration. The previous procedure is repeated up to a desired time instant. The proposed IMM is suitable for various kinds of fluid structure interaction problems including compressible and incompressible flows. It can solve FSI phenomena involving both rigid and flexible bodies. It can also be extended to solve the FSI problems for objects with arbitrary shapes, where surfaces of the immersed objects are treated as immersed membranes and the insides of the objects do not have fluids.

1.4.2 Arbitrary Lagrangian Eulerian (ALE) Method The arbitrary Lagrangian Eulerian (ALE) method, first proposed by Donea et al. [14], involves a continuous adaptation of the mesh without modifying the mesh topology in solving the fluid structure interaction and moving boundary problem. The mathematical formulation of the equation of motion for a fluid is most conveniently described with respect to an Eulerian reference frame. However, this is incompatible with the Lagrangian formulation, which is more appropriate to describe a solid phase. The ALE methodology combines the best of features of these two different formulations. The essence of the ALE is that the mesh motion can be chosen arbitrarily [15]. It includes three phases: an explicit Lagrangian update, an implicit iteration of the momentum equation with the equation of state, and a rezone/map phase [16]. The Lagrangian phase provides an explicit update of the equation of motion. When the material velocities are much smaller than the fluid sound speed, the optional implicit phase allows sound waves to move many cells per cycle, thereby significantly improving computational efficiency. Finally a rezone algorithm may prescribe mesh velocities relative to the fluid, thus necessitating remap phase in which the solution from the end of phase 2 is

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mapped onto the new mesh. A particularly important property of ALE is that it provides a means of minimizing advection errors [17]. ALE techniques have gained popularity for transient, high speed, and small deformation problems [18]. They are most commonly used in the aerospace engineering, where solid structures are subjected to complex airflows and flutter, for example, see Ref. [19]. The ALE method is developed here for the numerical study of flexible pipe wall [20] and lift force generation by insect wings [21,22]. The detailed treatment and derivation of ALE method are introduced in Chapter 9, Arbitrary Lagrangian Eulerian (ALE) Method and Fluid Structure Interaction.

1.4.3 Hybrid Method Hybrid method combines the Lagrangian and Eulerian approaches. The Chimera mesh method is a hybrid method. When the boundary moves without changing its geometry, Chimera mesh might be a suitable method for large displacement cases. In this method there are two sets of meshes, Eulerian and Lagrangian. The Lagrangian mesh surrounds the moving boundary and moves with it. The Eulerian background mesh occupies other regions of the flow field. The two meshes have an overlapping region or a transition region. When the moving boundary moves to a new position, the transition region is regenerated or the overlapping region is redefined. Flow variables on the Eulerian mesh are computed in an Eulerian manner and those on the Lagrangian mesh are evaluated in a Lagrangian manner. Kiris et al. [23] calculated blood flow through artificial heart devices with Chimera mesh method. In their work structured Chimera mesh was used around the heart valves in mitral and aorta positions. The Chimera mesh and the background mesh exchanged flow information through an overlapping zone. Comparison between numerical results and experimental measurements showed that they correlated well. For a moving boundary with fixed geometry the Chimera mesh method could be a good candidate. It is computationally less expensive than the mesh regeneration method, and could effectively deal with large displacements. The shortcoming is the introduction of errors during information exchange between the two meshes and difficulty in implementing the digging of holes in the Eulerian mesh while the structure is moving. Here in this book the overlapping grid (OG) method is developed in company with IOM for numerical simulation of blood flows in prosthetic heart valves under physiological conditions and blood leaflet interactions. Since the proposed OG method is coupled and implemented with IOM, we give the detailed introduction in the same chapter where IOM is described.

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1.5 MOTIVATION OF THIS BOOK Over the last decade a number of novel and highly efficient numerical techniques and solution methods have been developed in CFD for the study of FSI. The authors’ group has developed many such numerical methods specifically for FSI simulation over the past two and a half decades. The solvers and FSI simulation methods developed have been applied in industrial research and designs, such as the design and analysis of artificial heart valves, the study of instabilities of rotating disks in data storage devices, as well as the design and analysis of flapping-wing micro air vehicles (MAVs). The results have been widely published and the methods have been well received. As such we are motivated to write this book to provide the much needed reference materials and introduce the latest FSI methods in detail for the benefit of researchers, practicing engineers, and future students.

1.6 STRUCTURE OF THIS BOOK The book is divided into two parts. First we introduce the solvers and various numerical methods for FSI in Part I and the validation of the methods described and their engineering applications in Part II. Part I describes the basic ideas, methodologies, and equations of fluid and structural dynamics and FSI, mainly covered in nine chapters. Chapter 1, Introduction, gives a literature review of FSI and the relevant numerical methods developed for studying FSI. Chapter 2, Mathematical Formulation for Preconditioned Compressible Flow Solver, is concerned with the mathematical models for unsteady compressible fluid flows and the numerical approaches employed in the flow solver. Chapter 3, Mathematical Formulation for Incompressible Flow Solver, presents the mathematical models for unsteady incompressible flows and the related numerical methods used in the flow solver. Chapter 4, Mathematical Formulation for Computational Structure Dynamics, presents the linear-elastic Cauthy equations and the numerical methods used for modeling structural dynamics. Chapter 5, The Multigrid Method, provides a full description of the mathematical formulation of the multigrid method. Chapter 6, Parallel Computation, describes the implementation of parallel computation. Chapter 7, The Immersed Object Method With Overlapping Grids, deals with the implementation details of the immersed object method with OGs and how it can be used for coupling the structure and fluid solvers. Chapter 8, The Immersed Membrane Method and Fluid Structure Interaction, presents another moving boundary treatment method, the IMM. Chapter 9, Arbitrary Lagrangian Eulerian (ALE) Method and Fluid Structure Interaction, details the implementation of the recently developed moving boundary treatment method, ALE. After the descriptions of the basic equations and numerical methods in the FSI solvers, model validation work and some FSI applications using the

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proposed methods are then introduced in Part II, which has four chapters. Chapter 10, IOM FSI Model Validations and Applications, focuses on the validation and application of FSI by using IOM with OG. Chapter 11, IMM FSI Model Validations and Applications for Incompressible Flows, covers the validation and application of FSI by using IMM for incompressible flows, while Chapter 12, IMM FSI Model Validations and Applications for Compressible Flows, deals with the validation and applications of the IMM for compressible flows. In the last chapter (Chapter 13: ALE FSI Model Validations and Applications) model validation work and FSI applications using ALE methods are introduced.

REFERENCES [1] Hunt, Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict, Ann. Rev. Fluid Mech. 30 (1997). [2] S.C. Peskin, D.M. McQueen, Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys. 37 (1980) 113 132. [3] S.C. Peskin, D.M. McQueen, A three-dimensional computational method for the blood flow in the heart, J. Comput. Phys. 81 (1989) 372 405. [4] M.F. McCracken, S.C. Peskin, A vortex method for blood flow through heart valves, J. Comput. Phys. 35 (1980) 183 205. [5] A.L. Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys. 56 (1984) 111 134. [6] R.P. Beyer, A computational model of the cochlea using the immersed boundary method, J. Comput. Phys. 98 (1992) 145 162. [7] K.M. Arthurs, L.C. Moore, S.C. Peskin, E.B. Pitman, H.E. Layton, Modeling arteriolar flow and mass transport using the immersed boundary method, J. Comput. Phys. 147 (1998) 402 440. [8] J.M. Stockie, S.I. Green, Simulating the motion of flexible pulp fibres using the Immersed boundary method, J. Comput. Phys. 147 (1998) 147 165. [9] A.M. Roma, S.C. Peskin, M.J. Berger, An adaptive version of the Immersed boundary method, J. Comput. Phys. 153 (1999) 509 534. [10] M.-C. Lai, S.C. Peskin, An Immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705 719. [11] C.H. Tai, Y. Zhao, K.M. Liew, Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids, J. Comput. Phys. 207 (1) (2005) 151 172. [12] X. Lv, Y. Zhao, X.Y. Huang, G.H. Xia, Z.J. Wang, An efficient parallel/unstructuredmultigrid preconditioned implicit method for simulating 3D unsteady compressible flows with arbitrarily moving objects, J. Comput. Phys. 215 (2) (2006) 661 690. [13] G.H. Xia, Y. Zhao, J.H. Yeo, Parallel unstructured multigrid simulation of 3D unsteady flows and fluid-structure interaction using Immersed membrane method, Comp. Fluids 38 (1) (2009) 71 79. [14] J. Donea, S. Giuliani, J.P. Halleux, An Arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comp. Methods Appl. Mech. Eng. 33 (1982) 689 723.

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[15] L.G. Margolin, Introduction to an arbitrary Lagrangian Eulerian computing method for all flow speeds, J. Comput. Phys. 135 (1997) 198 202. [16] C.W. Hirt, A.A. Amsden, J.L. Cook, An Arbitrary Lagrangian Eulerian computing method for all flow speeds, J. Comput. Phys. 135 (1997) 203 216. [17] J.U. Brackbill, W.E. Pracht, An implicit, Almost-Lagrangian algorithm for magnetohydrogynamics, J. Comput. Phys. 13 (1973) 455. [18] D.J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng. 99 (1992) 235 394. [19] C. Farhat, M. Lesoinne, P. LeTallec, Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity, Comput. Methods Appl. Mech. Eng. 157 (1998) 95 114. [20] Y. Zhao, A. Ahmed Forhard, General method for simulation of fluid flows with moving and compliant boundaries using unstructured grids, Comp. Methods Appl. Mech. Eng. (United States) 192 (39 40) (2003) 4439 4466. [21] X.H. Su, Y.W. Cao, Y. Zhao, An unstructured mesh Arbitrary Lagrangian-Eulerian unsteady incompressible flow solver and its application to insect flight aerodynamics, Phys. Fluids 28 (6) (2016) 061901. [22] X.H. Su, Y. Zhen, Y.W. Cao, Y. Zhao, Numerical investigations on aerodynamic forces of deformable foils in hovering motions, Phys. Fluids 29 (4) (2017) 041902. [23] C. Kiris, D. Kwak, S. Rogers, I.-D. Chang, Computational approach for probing the flow through artificial heart devices, J. Biomech. Eng. 119 (1997) 452 460.

Chapter 2

Mathematical Formulation for Preconditioned Compressible Flow Solver 2.1 BASIC GOVERNING EQUATIONS FOR FLUID FLOWS The 3D unsteady Navier
Stokes equations may be written in a number of forms. One common form is as follows: Continuity: @ρ @ðρuÞ @ðρvÞ @ðρwÞ 1 1 1 50 @t @x @y @z

ð2:1Þ

u-Momentum: @ðρuÞ @ðρu2 1 pÞ @ðρuvÞ @ðρuwÞ @τ xx @τ xy @τ xz 1 1 1 5 1 1 @t @x @y @z @x @y @z

ð2:2Þ

v-Momentum: @ðρvÞ @ðρuvÞ @ðρv2 1 pÞ @ðρvwÞ @τ yx @τ yy @τ yz 1 1 1 5 1 1 @t @x @y @z @x @y @z

ð2:3Þ

w-Momentum: @ðρwÞ @ðρuwÞ @ðρvwÞ @ðρw2 1 pÞ @τ zx @τ zy @τ zz 1 1 1 5 1 1 @t @x @y @z @x @y @z

ð2:4Þ

Energy equation: @ðρet Þ @ðρet uÞ @ðρet vÞ ðρet wÞ 1 1 1 @t @x @y @z 5

@ðκð@T=@xÞÞ @ðκð@T=@yÞÞ @ðκð@T=@zÞÞ @ðupÞ @ðvpÞ @ðwpÞ 1 1 2 2 2 @x @y @z @x @y @z

1

@ðuτ xx 1 vτ xy 1 wτ xz Þ @ðuτ yx 1 vτ yy 1 wτ yz Þ @ðuτ zx 1 vτ zy 1 wτ zz Þ 1 1 @x @y @z ð2:5Þ

Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00002-7 © 2019 Elsevier Inc. All rights reserved.

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where the volumetric heating, such as absorption or emission of radiation and body source terms, such as gravity terms are not included. ρ is density; u, v, and w are the Cartesian components of velocity along x-, y-, and z-axes; p is pressure; and T is temperature. They can be related by the state equation of perfect gas p 5 ρRT, where R is the gas constant, which is 287 m2 =ðs2 KÞ for air at standard conditions. Furthermore, et is the total energy defined as et 5 e 1 ð1=2Þðu2 1 v2 1 w2 Þ, where e is internal energy per unit mass of the fluid and e 5 Cv T, where Cv is the specific heat at constant volume and Cv 5 R=ðγ 2 1Þ. γ is the ratio of specific heats and γ 5 1:4. If we assume that the fluid considered is a calorically perfect gas, then the Navier
Stokes equations are closed by the state equation for perfect gas: p 5 ðγ 2 1Þρe T5

ðγ 2 1Þe R

H 5e1

p 1 1 ðu2 1 v2 1 w2 Þ ρ 2

Finally, the viscous stresses are related to relations:
 @u 2 @u @v @w 1 1 τ xx 5 2μ 2 μ ; @x 3 @x @y @z
 @v 2 @u @v @w 1 1 τ yy 5 2μ 2 μ ; @y 3 @x @y @z
 @w 2 @u @v @w 2 μ 1 1 τ zz 5 2μ ; @z 3 @x @y @z

the velocity field by the Stokes
 @v @u 1 τ xy 5 τ yx 5 μ @x @y
 @u @w 1 τ xz 5 τ zx 5 μ @z @x
 @w @v 1 τ yz 5 τ zy 5 μ @y @z

The dynamic viscosity μ and conductivity κ are properties of the fluid and are functions of temperature. These two quantities are related by the Prandtl number: Pr 5

μCp κ

For air, the Prandtl number is around 0.72 at room temperature. Equations of fluid motion may be nondimensionalized to achieve certain objectives. First, it would provide conditions upon which dynamic and energetic similarity may be obtained for geometrically similar situations. Second, the solution of such equations would usually provide values within limits between zero and one, which can enhance accuracy for numerical computation.

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The nondimensional variables used in this study are defined as follows:
 x y z  u v w ðx ; y ; z Þ 5 ; ; ; ; ðu ; v ; w Þ 5 L L L UN UN UN t 5 ρ 5

t p 2 pin et H  ; p 5 ; et 5 ; H 5 2 2 L=U N ρN ðU N Þ ðU N Þ ðU N Þ2

ρ T μ ρN U N L  γμ   Cp  r T ; γ5 ; T5 ; μ 5 ; Re 5 ; q 52 N 2 ρN μN μN Pr Cv ðU N Þ =Cv

where pin is the inlet or reference pressure; L is the characteristic length of the computed model; U N is the inflow velocity; and μ is the dynamic viscosity of the fluid. The variables with a superscript asterisk sign ( ) are nondimensional parameters and the asterisk sign will be dropped in subsequent equations for sake of convenience. It should be noted that we subtract a constant value (the reference pressure) from the pressure term to control the round-off errors for low-speed flows, which is found to be critical in controlling computational errors in the momentum equations for low-speed compressible flows. The use of gauge pressure is a common treatment for incompressible solvers because only pressure gradients are needed for all calculations. For compressible solvers, the absolute value of pressure must be used when dealing with the energy equation and state equation of gases. But when we use nondimensional absolute pressure at low Mach numbers, it becomes extremely large although the pressure gradients in momentum equations are small. The use of gauge pressure can avoid performing the addition and subtraction operations between two extraordinarily large values of nondimensional absolute pressures. In our experience, the result obtained with gauge pressure is more accurate than without using it. And the relative error can be up to 4%
5% of the result. It should be noted that the original absolute pressure (designated by p0 hereafter) should be used instead of gauge pressure whenever the state equation of gas is involved. The nondimensional equations of fluid motion may be expressed in vector form as @W c @Ei @Fi @Gi @Ev @Fv @Gv 1 1 1 5 1 1 @t @x @y @z @x @y @z 2 3 ρ 6 ρu 7 6 7 7 Wc 5 6 6 ρv 7 4 ρw 5 ρet

ð2:6Þ

ð2:7Þ

14

Computational Fluid-Structure Interaction

2

3 ρu 6 ρu2 1 p 7 6 7 7 Ei 5 6 6 ρuv 7 4 ρuw 5 ðρet 1 pÞu 3 2 0 7 6 τ xx 7 1 6 7 6 τ xy Ev 5 7 6 ReN 4 τ 5  xz  uτ xx 1 vτ xy 1 wτ xz 2 qx 2 3 ρv 6 ρvu 7 6 2 7 7 Fi 5 6 ρv 1 p 6 7 4 ρvw 5 ðρet 1 pÞv 3 2 0 7 6 τ yx 7 1 6 7 6 τ yy Fv 5 7 6 ReN 4 τ 5 yz uτ yx 1 vτ yy 1 wτ yz 2 qy 3 2 ρw 7 6 ρwu 7 6 7 6 Gi 5 6 ρwv 7 4 ρw2 1 p 5 ðρet 1 pÞw 2 3 0 6 τ zx 7 7 1 6 6 7 τ zy Gv 5 7 ReN 6 4 τ zz 5 uτ zx 1 vτ zy 1 wτ zz 2 qz

ð2:8Þ

ð2:9Þ

ð2:10Þ

ð2:11Þ

ð2:12Þ

ð2:13Þ

2.2 LOW-SPEED PRECONDITIONING FORMULATION One difficulty with compressible Navier
Stokes solvers is their slow convergence rates and even unstable solutions for low-Mach-number flows. This difficulty can be traced to a disparity between the acoustic and convective speeds [1
10], and can be addressed by a preconditioning algorithm. Previous work in this area has been reported by Venkateswaran and

Mathematical Formulation Chapter | 2

15

Merkle [4
7], Turkel [8], Van Leer et al. [9], and Weiss and Smith [10]. The applications of the preconditioning methods have been found in the computation of steady flows without considering arbitrarily moving objects in the flow field. The preconditioned Navier
Stokes equations for 3D compressible unsteady flows can be given in vector form, explicitly expressing the conservation laws of mass, momentum, and energy. We also introduce, in the equations, pseudo time terms to provide pseudo time marching for their numerical solutions: Γ1 where

@W p @W c 1 1 rUFi 5 rUFv @τ @t 3 ρ 6 ρu 7 6 7 7 Wc 5 6 6 ρv 7 4 ρw 5 ρet 2 3 p 6u 7 6 7 7 Wp 5 6 6v 7 4w 5 T0 3 2 ρU 6 ρuU 1 pi 7 7 6 7 Fi 5 6 6 ρvU 1 pj 7 4 ρwU 1 pk 5 ρHU 2 3 0 6 τx 7 7 1 6 6 τy 7 Fv 5 6 7 ReN 4 τ 5 z   τ UU 2 q

ð2:14Þ

2

ð2:15Þ

ð2:16Þ

ð2:17Þ

ð2:18Þ

τ is the pseudo time and Γ1 is the preconditioning matrix in the pseudo time terms for low-Mach-number flows, which is defined in the appendix. W c and W p are the vectors of conservative and primitive dependent variables, respectively; Fi and Fv are the inviscid convective flux and viscous flux vectors. Furthermore, we have the following formulas: U 5 ui 1 vj 1 wk

16

Computational Fluid-Structure Interaction

τ 5 τ xi 1 τ yj 1 τ zk τ i 5 τ ix i 1 τ iy j 1 τ iz k
 @ui @uj 2 τ ij 5 μ 1 2 δij rUU 3 @xj @xi q 5 qx i 1 qy j 1 qz k T 0 5 p0 =ρ 5 c2 =γ i, j, and k are the three unit vectors in three Cartesian directions; τ ix , τ iy , and τ iz are the viscous stresses.

2.3 DISCRETIZATION METHODS The 3D equations (2.14) is transformed into an integral form and discretized on an unstructured grid. A cell-vertex finite-volume scheme is adopted here. For every vertex, a control volume is constructed using the median duals of the tetrahedral cells by connecting all the neighboring centroids of edges, surfaces, and tetrahedra. As shown in Fig. 2.1 is a part of the control volume associated with vertex P within tetrahedron PABC. Spatial discretization is performed by using the integral form of the conservation equations over the control volume surrounding node P: ððð ððð ððð ððð @Q0 1 @W c dV 1 dV 1 rUFi dV 2 rUFv dV 5 0 ð2:19Þ cv @τ cv @t cv cv

FIG. 2.1 Construction of control volume within a tetrahedron for a node P.

Mathematical Formulation Chapter | 2

17

Noted that a new variable Q0 1 has arisen as @Q0 1 =ð@τÞ 5 Γ1 @W p =ð@τÞ, and the Jacobian Γ1 5 @Q0 1 =ð@W p Þ. So that we have: @Q0 1 @Q0 1 @W p @W p 5 5 Γ1 @τ @W p @τ @τ

2.3.1 Edge-Based Method and Cell-Based Method The convective term in Eq. (2.19) is transformed into a summation: ððð ð nbseg X rUFi dV 5 Fi UndS 5 ½ðFi Þij UnΔSk cv

Scv

ð2:20Þ

k51

where nbseg is the number of the edges associated with node P; ðFi Þkij is the inviscid flux through the part of control volume surface associated with edge k; and n is the unit normal vector of the control volume surface. Finally, ΔSk is a part of the control volume surface associated with edge k (as shown in Fig. 2.1, if k is edge PC, then 1O2c is ΔSk ). Therefore all the fluxes are calculated for the edges and then collected at the two ends of each edge for updating of flow variables in time marching, which is the so-called edgebased data structure for fast inviscid flux computation and efficient data storage and retrieval. On the other hand, the viscous term is calculated using a cell-based method for the same reason of high computational efficiency: ððð ð ncell X ½Fv UnΔSc i rUFv dV 5 Fv UndS 5 ð2:21Þ cv

Scv

i51

where ncell is the number of tetrahedral elements associated with node P and ΔSci is the part of control volume surface in cell i. By using the following relation: ð dS 5 0: Scv

The total vector surface of the control volume in a cell i becomes nΔSci 5 ð1=3ÞðnΔSpi Þ. Thus the calculation of viscous terms can be simplified as ncell X i51

½Fv UnΔSc i 5

ncell 1X ½Fv UnΔSp i 3 i51

ð2:22Þ

where nΔSpi is the surface vector of the face opposite node P of the tetrahedron under consideration, where the ðFv Þi is calculated at the center of the tetrahedron with a node P, and can be obtained by using the Green’s theorem based on the variables at the four vertices of the tetrahedron. Similar to the

18

Computational Fluid-Structure Interaction

Galerkin type of formulation, the gradient of a flow variable φ at the center of a tetrahedron is evaluated as follows: P4 P 1 4i51 φi Si i51 φi 9Si 52 ð2:23Þ gradφc 5 2 3 27V V where φi is the flow variable at a vertex i of the tetrahedron; Si is the surface area that is opposite to node i; and V is the volume of the tetrahedron. Gradients at the vertices are obtained by a volume averaging of the gradients at the centers of cells associated with the vertex under consideration.

2.3.2 Roe’s TVD Scheme for Inviscid Flux in Edge-Based Method The Riemann problem is an initial problem with piecewise constant initial data. Its exact solution represents the real physical characteristics of a flow with several families of waves and their propagation. When the wave is a shock, the equations for this initial problem cannot be explicitly solved and an iterative solution has to be used. In order to reduce the amount of calculation, some approximate treatments have been employed in numerical solutions. In this work, a high-order Roe’s TVD scheme for compressible flow for arbitrary unstructured 3D grids has been adopted. In Roe’s approach [11], averaged values of density, velocity, and enthalpy at the interface between the two initial states at two neighboring points are calculated by using a special averaging procedure, called Roe’s averaging: pffiffiffiffiffipffiffiffiffiffi ρ 5 ρL ρR pffiffiffiffiffi pffiffiffiffiffi ðu; v; wÞL ρL 1 ðu; v; wÞR ρR pffiffiffiffiffi pffiffiffiffiffi ðu; v; wÞ 5 ð2:24Þ ρL 1 ρR pffiffiffiffiffi pffiffiffiffiffi H L ρL 1 H R ρR p p ffiffiffiffiffi ffiffiffiffiffi H5 ð2:25Þ ρL 1 ρR where L and R represent the two neighboring points on an edge, and H denotes total enthalpy which has the following relation: H5e1

 p0 1 1 u2 1 v 2 1 w 2 2 ρ

The speed of sound can be derived as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1 2 2 2 c5 H 2 ðu 1 v 1 w Þ ðγ 2 1Þ 2

ð2:26Þ

ð2:27Þ

Mathematical Formulation Chapter | 2

19

The inviscid flux ðFi Þij is evaluated based on Roe’s approximate Riemann solver (or the flux difference splitting scheme):  1

ðFi Þij 5 FLi 1 FRi 2 A ðW R 2 W L Þ 2  1 L 5 Fi 1 FRi 2 jΔFj ð2:28Þ 2 5 FLi 1 ΔF2 5 FRi 2 ΔF1 where

  FLi 5 Fi W Lc   FRi 5 Fi W Rc

and W Lc and W Rc are the left and right conserved state vectors on the two neighboring points of an edge. The flux difference is: ð2:29Þ jΔFj 5 ΔF1 2 ΔF2 8 3 39 2 2 1 0 > > > > > > > 7 7> 6 6 > > u ΔU Δu2n > > 7 7 6 6
 x n = < 7 7 6 6 Δp 6 6 7 7 6 6 1ρ ΔF 5λ1 Δρ2 2 6 v Δv2n ΔU y n 7 7> 6 > ðcÞ 6 > 7 7> 6 > > > 5 5> 4w 4 Δw2nz ΔUn > > > > ; : 2 ðU n Þ =2 uΔu1vΔv1wΔw2U n ΔUn 3 2 1 7 6
6 u1nx c 7 7 6 Δp1ρcΔU n 6 v1ny c 7 1λ46 7 6 2 2ðcÞ 7 6 4 w1nz c 5 H 1U n c 3 2 1 7 6
6 u6nx c 7 7 6 Δp2ρcΔUn 6 1λ56 v6ny c 7 2 7 6 2ðcÞ 7 6 4 w6nz c 5 H 6U n c ð2:30Þ where ðU n Þ2 5 u2 1 v2 1 w2 ΔUn 5 Δunx 1 Δvny 1 Δwnz U n 5 unx 1 vny 1 wnz

20

Computational Fluid-Structure Interaction

Furthermore, the eigenvalues of A are λ1 5 U n , λ4;5 5 U n 6 c. Because of the introduction of preconditioning matrix Γ1 , the inviscid fluxes, ðFi Þijk , through the face k is now reformulated as 1 1 @Fi 0 k ðFi Þij  ððFi Þi 1ðFi Þj Þk 2 @Q0 ðδQ1 Þk 2 2 1 k ! ð2:31Þ 1 1 @Fi @W p @Q01 5 ððFi Þi 1ðFi Þj Þk 2 @W p @Q0 @W p ððW p Þj 2ðW p Þi Þk 2 2 1 k

k

Note that we have retained the variable, Q0 1 , in computing this flux. Defining the Jacobian in the normal direction as
 @Fi ðHp Þk 5 @W p k And using the previously defined Jacobian Γ1 5 @Q0 1 =@W p , then the above expression becomes 1 1 ðFi Þkij  ððFi Þi 1ðFi Þj Þk 2 Hp Γ21 1 k Γ1k ððW p Þj 2ðW p Þi Þk 2 2

Drop the subscript k in the flux vector and the Jacobian with the assumption that the fluxes and Jacobians all correspond to conditions in the normal direction on the given control volume surface. And after some simple algebraic derivations, we have 1 1 ðFi Þij  ððFi Þi 1 ðFi Þj Þ 2 Γ1 Γ21 1 Hp ððW p Þj 2 ðW p Þi Þ 2 2

ð2:32Þ

2.3.3 Upwind-Biased Interpolation Combined with the third-order Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) interpolation, it can produce accurate and stable solution on unstructured grids. The left and right state vectors W L and W R at a control volume surface are evaluated using a nominally third-order upwind-biased interpolation scheme:  1

1 ð1 2 κÞΔ2 i 1 ð1 1 κÞΔi 4 i 1h 2 W R 5 W j 2 ð1 2 κÞΔ1 j 1 ð1 1 κÞΔj 4 WL 5 Wi 1

where 2 Δ1 i 5 Δj 5 W j 2 W i 1 Δ2 i 5 W i 2 W i21 5 2ijUrW i 2 ðW j 2 W i Þ 5 2ijUrW i 2 Δi 2 Δ1 j 5 W j11 2 W j 5 2ijUrW j 2 ðW j 2 W i Þ 5 2ijUrW j 2 Δj

ð2:33aÞ ð2:33bÞ

Mathematical Formulation Chapter | 2

21

Therefore substituting the above equations into Eqs. (2.33a) and (2.33b), the final equations based on upwind-biased interpolation scheme is shown as follows:  1

ð1 2 κÞijUrW i 1 κΔ1 i 2 i 1h W R 5 W j 2 ð1 2 κÞijUrW j 1 κΔ2 j 2 WL 5 Wi 1

ð2:34aÞ ð2:34bÞ

where κ is set to 1/3, which corresponds to a nominally third-order accuracy. ij is the vector representing the edge, which points from node P to its neighboring node under consideration. The gradients of W at i and j are calculated by volume averaging the gradients of the cells that surround i and j. Finally, for a given node P, the spatially discretized Eq. (2.19) form a system of coupled ordinary differential equations. This can be reformulated as: @Q0 1 @W c ΔVcv 1 ΔVcv @τ @t 8 9 ncell vx >  < = p1z : vy 5 ðp1x vx 1p1y vy 1p1z vz Þ1 > : > ; vz 8 9 > vx >  < = p3z : vy 5 ðp3x vx 1p3y vy 1p3z vz Þ3 > : > ; vz 8 9  < vx =  2 p2z : vy 5 p2x vx 1p2y vy 1p2z vz : ; 8 vz 9  < vx =  3 p3z : vy 5 p3x vx 1p3y vy 1p3z vz : ; vz

8 9  < vx = p1z : vy 5 ðp1x vx 1p1y vy 1p1z vz Þ1 : ; 8 vz 9  < vx = p2z : vy 5 ðp2x vx 1p2y vy 1p2z vz Þ2 : ; vz

.

0

,

0

ð5:9aÞ .

0

,

0

ð5:9bÞ .

0

,

0

ð5:9cÞ

,

0

.

0

ð5:10aÞ

The Multigrid Method Chapter | 5



ðpv Þ1 5 fpg1 : fvg 5 p1x

p1y

 ðpv Þ3 5 fpg3 : fvg 5 p3x

p3y

 ðpv Þ2 5 fpg2 : fvg 5 p2x

p2y

 ðpv Þ3 5 fpg3 : fvg 5 p3x

p3y

8 9  < vx = p1z : vy 5 ðp1x vx 1p1y vy 1p1z vz Þ1 : ; 8 vz 9  < vx = p3z : vy 5 ðp3x vx 1p3y vy 1p3z vz Þ3 : ; vz

75

,

0

.

0

ð5:10bÞ 8 9  < vx = p2z : vy 5 ðp2x vx 1p2y vy 1p2z vz Þ2 , 0 : ; 8 vz 9  < vx = p3z : vy 5 ðp3x vx 1 p3y vy 1 p3z vz Þ 3 . 0 : ; vz ð5:10cÞ

The purpose of this second criterion is mainly to ensure that the nodes being tested is within the boundary face instead of testing other node belonging to the neighboring faces, as shown in Fig. 5.6C for 2D. This figure shows that the sign for these two dot products are the same, which both the values of dot product are less than zero (i.e., p1 .v , 0 and p2 .v , 0, having the same sign). If the node being tested fulfills these two criteria, then the node is projected downward onto the boundary edge (2D) and the projected lengths between this projected node and the two edge nodes are computed for the transfer operator alogorithms, as depicted in Figure 5.7. This is to ensure that the residual transfer is conservative. The above-described algorithm can be summarized in the flowchart as shown in Fig. 5.8 and the pseudo code for the algorithm is given after the flowchart.

FIGURE 5.6C Node belonging to the neigboring edge (i.e., ~ p 1 U~ v , 0 and ~ p 2 U~ v , 0).

76

Computational Fluid-Structure Interaction FIGURE 5.7 Projected lengths for the transfer operator algorithms (2D).

! ---------------------------------------------------------------------------------------------------! Pseudo-code for finding nodes enclose within cell using dot product of two vectors ! ---------------------------------------------------------------------------------------------------subroutine containabc variables definition initialize all the arrays do loop for number of multigrid levels (loop 1) initialize flag as 0 for all the nodes do loop for number of zones (loop 2) loop for number of cells within the zone (loop 3) establish inward normal vectors ~ n for the 4 faces of the cell enclose cell within an imaginary bounding box do loop for number of nodes within the zone (loop 4) if (flag node.eq.0 .and. within the bounding box) then establish vectors ~ p for the 4 faces of the cell compute ~ p .~ n for the 4 faces if (~ p .~ n $ 0 for the 4 faces) then flag node 5 1 call subroutine to calculate volumes for the 4 sub-tetrahedral & volume of the cell go back to loop 4 go back to loop 3 go back to loop 2 if (geometry got curvature) call subroutine to test for nodes within boundary faces go back to loop 1 return end

77

The Multigrid Method Chapter | 5

5.5 MESH-TO-MESH TRANSFER OPERATORS Following the two-dimensional approach presented in Ref. [7] for data transfer within the domain and incorporate the new technique developed here for a curved boundary, there are two classes of mesh-to-mesh transfer operators being implemented in the method for three-dimensions, which are restriction operators and prolongation operators. List of nodes and cells within the specified zones for three-grid levels

Fine-to-coarse grids (1--2 grid level – Group 1) Coarse-to-coarsest grids (2--3 grid level – Group 2)

Initialize the flag as 0 for all the nodes

List of cells in the array for the zone Go to the next group of grid levels (eg., 2--3 grid level for three-grid levels)

Establish the inward normal vectors, n, for the 4 faces of the cells

Enclose the cell within an imaginary bounding box

Is the node flag 0 and within the bounding box?

No

Yes Establish the vectors, p, for the 4 faces of the cell

Compute p .n for the 4 faces

Is p .n ≥ 0 for the 4

No

Yes

Go to the next zone

Go to the next in the list for the zone

Go to the next cell in the list for the zone

List of nodes in the array for the zone

(1) Compute the volumes for the 4 subtetrahedral within the cell and the volume of the cell. (2) Flag the node as 1

A

B

C

D

E

FIGURE 5.8 Flowchart for interconnectivity relationships between meshes algorithm for threegrid levels.

78

Computational Fluid-Structure Interaction

A

B

C

D

End of the list of nodes for the zone?

E

No

Yes No

End of the list of cells for the zone?

Yes End of zones specified by user?

No

Yes No Curvature in the geometry?

Yes Call subroutine to test for nodes that contain within the boundary faces using algorithm depicted in Fig. 5.6 Yes No

End of three-grid levels?

Yes End of algorithm

FIGURE 5.8 (Continued).

5.5.1 Flow-Field Variables Transfer Operators Once the interconnectivity relationship between meshes has been determined, the restriction transfer operator, Thh11 , that transfers the flow-field values from the fine grid to the coarse grid will be as follows: W1 5

Va Wa 1 Vb Wb 1 Vc Wc 1 Vd Wd Va 1 Vb 1 Vc 1 V d

ð5:11Þ

Fig. 5.9A depicts the operation of this transfer operator for those nodes within a cell. Lower case letters denote fine-grid nodes and arabic numbers

The Multigrid Method Chapter | 5

79

FIGURE 5.9A Transfer of flow-field values from the fine mesh to the coarse mesh using restriction transfer operator.

FIGURE 5.9B Transfer of variables from fine nodes to the coarse node at the boundary.

denote coarse-grid nodes for all the figures in this section. The flow-field values at the coarse node 1, which is contained in the fine cell formed by nodes a, b, c, and d, is a weighted average of the values at those nodes. Va is the volume of the sub tetrahedron cell with vertices b, 1, c, and d. Vb is the volume of the sub tetrahedron cell with vertices a, 1, c, and d. Vc is the sub tetrahedron cell with vertices a, 1, d, and b. Vd is the subtetrahedron cell with vertices a, 1, b, and c. The volume of the corresponding tetrahedron cell is the one opposite to the node. According to Eq. (5.11), if node 1 coincides with node a, then W1 will be equal to Wa. At the boundary with curvature, the flow-field values are transferred from the fine nodes of the triangle formed by vertices a, b, and c onto the coarse node 1 using an area-weighted contribution calculation, as illustrated in Fig. 5.9B. The restriction transfer operator, Thh11 , that transfers the flow field values is as follows: W1 5

Aa W a 1 A b W b 1 Ac W c Aa 1 Ab 1 Ac

ð5:12Þ

Fig. 5.9B illustrates the operation of this transfer operator for those nodes within the boundary triangles. The flow-field values at the coarse node 1,

80

Computational Fluid-Structure Interaction FIGURE 5.10 An illustration of transfer of residuals using prolongation operator will not be conserved.

which is projected downward onto the fine triangle face formed by nodes a, b, and c, is a weighted average of the values at those nodes. Aa is the area of the triangle with vertices b, 1, and c. Ab is the area of the triangle with vertices c, 1, and a. Ac is the area of the triangle with vertices a, 1, and b. The area of the corresponding triangle is the one opposite to the node. Similarly, according to Eq. (5.12), if node 1 coincides with node a, then W1 will be equal to Wa.

5.5.2 Residual Transfer Operators In the current MG method, the restriction transfer operator is used for transferring residuals so that the transferred residuals will be conserved. Fig. 5.10 illustrates the reason why using prolongation operators to transfer residuals will not be conserved. For illustration purposes, Fig. 5.10 is illustrated with a 2D mesh. The coarse-grid nodes 1, 2, and 3 are super-positioned onto the three fine cells as shown in the figure. The first fine cell is formed by node a, b, and f, the second fine cell is formed by node i, j, and n, and the third fine cell is formed by node l, m, and q. If the residuals are transferred using prolongation operators, then the residuals at these nine fine nodes will be transferred to the three coarse nodes, while the residual at fine node k will not be transferred to any of the coarse nodes. It is possible to encounter a situation whereby the residuals of these nine fine nodes are zero and the residual of fine node k is nonzero. A negative transfer will result in the three coarse nodes to have zero residual if using prolongation operators. To ensure that the overall convergence will be conserved and to avoid negative transferring between meshes, the selection of the class of operators to be used in transferring the residuals is very important to ensure that the sum of coarsegrid residuals after the transfer are always conserved, that is equal to the total of the fine-grid residuals. where the transfer of residuals from the fine grid to the coarse grid is based on a weighted average of the residual from the fine node to the four coarse nodes that form the coarse tetrahedral cell. As depicted in Fig. 5.11A,

The Multigrid Method Chapter | 5

81

FIGURE 5.11A Transfer of residuals from the fine mesh to the coarse mesh using restriction transfer operator.

the residual from node a is transferred to coarse nodes 1, 2, 3, and 4 by the transfer operator, Q h11 h . The transferred residual to a particular coarse node is the summation of all the transferred weighted-average residuals from other fine nodes in those tetrahedra surrounding this coarse node, as shown in the following equations, where the contribution from fine-grid node a is calculated and added to the residual of coarse-grid node 1: ~~ 5 R ~~ 1 R 1 1 ~~ 5 R ~~ 1 R 3 3

V1 R~~ a ; V1 1 V2 1 V3 1 V4

V3 R~~ a ; V 1 1 V 2 1 V 3 1 V4

~~ 1 R~~ 2 5 R 2 R~~ 4 5 R~~ 4 1

~~ V2 R a ; V1 1 V2 1 V 3 1 V 4

~~ V4 R a V1 1 V2 1 V3 1 V4

ð5:13Þ

where V1 is the volume of the subtetrahedron cell with vertices 2, a, 3, and 4. V2 is the volume of the subtetrahedron cell with vertices 1, a, 3, and 4. V3 is the volume of the subtetrahedron cell with vertices 1, a, 2, and 4. V4 is the volume of the sub tetrahedron cell with vertices 1, a, 2, and 3. The volume of the corresponding tetrahedron cell is the one opposite to node. It is easy to show that this transfer is conservative in the sense that the total fine mesh residuals are equal to the coarse mesh ones. The residual for a find-grid node on the boundary face of the curvature geometry is transferred to the coarse nodes of the triangle formed by vertices 1, 2, and 3 using an area-weighted contribution calculation as shown in Fig. 5.11B. The transfer operator, Q h11 h , that transfers the residuals from the fine grid to the three coarse-grid nodes is as follows: ~~ 5 R ~~ 1 R 1 1

A1 R~~ a A 1 1 A 2 1 A3

82

Computational Fluid-Structure Interaction FIGURE 5.11B Transfer of residuals from fine node to the coarse nodes at the boundary.

R~~ 2 5 R~~ 2 1

~~ A2 R a ; A1 1 A2 1 A3

~~ 5 R ~~ 1 R 3 3

A3 R~~ a A1 1 A2 1 A3

ð5:14Þ

Fig. 5.11B depicts the operation of this transfer operator typically for those nodes on the boundary faces. The residual at the fine node a, which is projected downward onto the coarse triangle formed by nodes 1, 2, and 3, is a weighted average of the value at this fine node. The transferred residual is the summation of all the transferred residuals from other fine nodes to these coarse nodes. This also ensures that the residual transfer is conservative on the boundary. It has been proven that the total fine-grid residuals are equal to the coarse grid ones and this is achieved by performing a summation of the residuals on both fine and coarse grids and the totals are equal to each other [7]. This proves that this restriction transfer operator is conservative.

5.5.3 Correction Transfer Operators Prolongation operators are used to transfer corrections of the flow-field variables from the coarse mesh to the fine mesh, as depicted in Figs. 5.12A and 5.12B. As shown in Eq. (5.15), the correction, dWh11 , is the difference 1 between the newly computed value on the coarse grid, Wh11 , and the initial ð0Þ value that was transferred from the fine grid, Wh11 . ð0Þ 1 dWh11 5 Wh11 2 Wh11

ð5:15Þ

Corrections are transferred to the fine mesh by the prolongation operator, I

h h11 : h dWh11 vh 5 Ih11

and

Wh1 5 Wh 1 vh

According to Fig. 5.12A, the correction of the flow-field variables transferred from the coarse nodes 1, 2, 3, and 4 to the fine node a is a weighted

The Multigrid Method Chapter | 5

83

FIGURE 5.12A Transfer of corrections from the coarse mesh to the fine mesh using prolongation transfer operator.

FIGURE 5.12B Transfer of corrections from the coarse nodes to the fine node at the boundary.

average of the corrections at these nodes and the expression for the transferred correction is ðva Þh 5

V1 ðdW1 Þh11 1 V2 ðdW2 Þh11 1 V3 ðdW3 Þh11 1 V4 ðdW4 Þh11 V1 1 V2 1 V3 1 V4

ð5:16Þ

where V1 is the volume of the subtetrahedron cell with vertices 2, a, 3, and 5. V2 is the volume of the subtetrahedron cell with vertices 1, a, 3, and 5. V3 is the volume of the subtetrahedron cell with vertices 1, a, 2, and 5. V4 is the volume of the subtetrahedron cell with vertices 1, a, 2, and 3. The volume of the corresponding tetrahedron cell is the one opposite to the node. The corrections of the flow-field variables at the boundary are transferred from the coarse nodes of the triangle formed by vertices 1, 2, and 3 onto the fine node a. The transfer operator, I hh11 , that transfers the corrections is ðva Þh 5

A1 ðdW1 Þh11 1 A2 ðdW2 Þh11 1 A3 ðdW3 Þh11 A 1 1 A2 1 A3

ð5:17Þ

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Computational Fluid-Structure Interaction

Fig. 5.12B shows the operation of this transfer operator for those nodes on the boundary faces. The correction at the fine node a, which is projected downward onto the coarse face formed by nodes 1, 2, and 3, is a weighted average of the values at those nodes. A1 is the area of the triangle with vertices 2, a, and 3. A2 is the area of the triangle with vertices 1, a, and 3. A3 is the area of the triangle with vertices 1, a, and 2. The area of the corresponding triangle is the one opposite to the node. Similarly, according to Eq. (5.17), if node a coincides with node 1, then ðva Þh will be equal to ðdW1 Þh11 .

5.6 THE MULTIGRID METHOD FOR STRUCTURAL DYNAMIC SOLVER In this section, an efficient unstructured MG scheme developed for the equation of dynamic equilibrium is introduced. The MG algorithm is described as follows. The discretized dynamic equation can be expressed by equation (4.25), which is repeated here for convenience: ΔU ~~ n11;m 5R Δτ

ð5:18Þ

where U is the velocity vector used in CSD solver. This equation is solved iteratively by a dual-time-stepping scheme. An pseudo code is given below to illustrate the basic procedures of the MG scheme. The outer cycles are based on the physical time t, which is numbered from 1 to ktmax, whereas the inner cycles are based on the pseudo time τ, which is numbered from 1 to max_itr_sub. The grid levels range from 1 to nmg, where 1 is the finest level and nmg is the coarsest level. Phh11 is the prolongation operator from level h 1 1 to h, and Qh11 and Thh11 are the residual transfer and restriction h operators from level h to h 1 1. ALGORITHM START call setupinterconnect !---. build up inter-connectivity relationship between levels DO kt 5 1, ktmax !---. start of physical time step DO mg 5 1, nmg call cvvol(mg) !---. calculate the volumes for control volume (CV) and cell call clhaut(mg) !---. calculate CV characteristic length for determination !---. of Δτ and Δt computation ENDDO call dtsize(mg 5 1) !---. computation of the time-step size for the finest level DO itersub 5 1, max_itr_sub !---. start of sub-iteration

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DO ialpha 5 1, 5 !---. 5-stage Runge-Kutta time integration process kt call fvsolver(mg 5 1) !---. solve equation (3.1) for ~ U1 following the way !--- . described in forgoing sections ENDDO DO mg 5 2, nmg call transfrsol(mg-1) !---. h to h 1 1 ð0Þ U h ). !---. (~ U h11 5 Thh11 ~ call transfresd(mg-1) !---. h to h 1 1 ~  ~ ð0Þ R~ U !---. (R~ h11 5 Q h11 h h ).

restrict

solution

from

restrict

residual

from

IF (itersub.eq.1) THEN call dtsize(mg) !---. calculate the time-step size for the current level ENDIF DO ialpha 5 1, 5 kt call fvsolver(mg) !---. solve equation (3.1) for ~ U mg based on the initial ð0Þ ~ ð0Þ !---. solution ~ U h11 and initial residual R~ h11 ENDDO ENDDO DO mg 5 nmg, 2, -1 call transfrcorct(mg) !---. prolongate correction from h 1 1 to h  1 1 ð0Þ  1 !---. (~ Uh 5 ~ U h 1 Ihh11 ~ U h11 ). Here ~ U h11 2 ~ U h is !---. the updated solution for the finer grids ENDDO ENDDO !---. end of sub-iteration DO mg 5 2, nmg call transfrsol(mg-1) !---. restrict solution from h to h 1 1 ENDDO DO mg 5 1, nmg call updgrid(mg) !---. finally update the solid mesh ENDDO ENDDO !---. end of physical time step ALGORITHM END

One should note that we always use the initial intermesh connectivity, which is built up before the first time-step starts, to perform the mesh-tomesh transfer operations. In order to enable the solver to tackle more geometrically complex structures, the search for nodes within the boundary faces (edges for 2D) will continue after the search for nodes within the cells are performed if the user specifies that the physical wall of the geometry is

86

Computational Fluid-Structure Interaction Coarser-grid node

Coarser-grid boundary

Finer-grid node

Finer-grid boundary

Projection node for P

FIGURE 5.13 Schematic of boundary node projection algorithm, Ω is the computational domain.

Node P p1 1

θ1 θ 2

p2 2

V Ω

curved (e.g., see Fig. 5.13 ), a normal procedure cannot find a corresponding finer grid cell for the coarser grid node P. It means that the node P cannot get a proper solution vector from the finer grid, which can pose a serious problem while solving the dynamic equation (4.28). Following the boundary node projection algorithm proposed in Section 5.6, we resort to finding a projection node for node P in the nearest finer mesh cell face and then use this projected node to perform the necessary MG operations.

REFERENCES [1] R.P. Fedorenko, The speed of convergence of one iterative process, U.S.S.R., Comput. Math. Math. Phys. 4 (3) (1964) 227235. [2] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput. 31 (1977) 333390. [3] A. Jameson, Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Appl. Math.Comput 13 (1983) 327355. [4] Jameson A., Time Dependent Calculations Using Multigrid, with Applications to Unsteady flows Past Airfoils and Wings, AIAA Paper 91-1596, June 1991. [5] E. Turkel, Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations, J. Comput. Physics 72 (1987). [6] Mavriplis D.J., Large-scale Parallel Viscous Flow Computations Using An Unstructured Multigrid Algorithm, NASA/CR-1999-209724, ICASE Report No. 99-44, Nov 1999. [7] C.H. Tai, Y. Zhao, A Finite Volume Unstructured Multigrid Method for Efficient Computation of Unsteady Incompressible Viscous Flows, Int. J. Num. Meth. Fluids 46 (1) (2004) 5984.

Chapter 6

Parallel Computation 6.1 PARALLELIZATION STRATEGY In this chapter the methods are introduced to parallelize the serial single grid (SG) and MG codes, using a concept known as single program multiple data (SPMD) based on geometric domain decomposition. This technique involves decomposing the problem domain into a set of S partitions (subdomains) that may be distributed over P processors in a parallel machine. Each processor runs the same program and operates only on its part of the problem with the boundary node variables and gradients being exchanged with other processors using the message-passing interface (MPI) [1]. The purpose of this message passing between the processors is to maintain consistency with the original serial solver. MPI is based on independent processes, which do not share any memory. An overlapping mesh-partitioning technique is employed so as to build the control volumes for the finite-volume formulation. Typically a fixed number of MPI processes are used, corresponding to the number of subdomains. The flowchart presented in Fig. 6.1 depicts the geometric domain decomposition parallelization strategy employing MPI.

6.2 DOMAIN DECOMPOSITION Domain decomposition is a generic name given to a variety of computational activities, which involves the division of a problem space into two or more parts that can be operated on separately to some advantage. Domain decomposition of a mesh into a set of S subdomains that may be allocated to a set of P processors involves finding a partition of the mesh so that the amount of computational time on each processor is almost equal. This method has the advantage of flexibility to allow variations in the decomposition strategy to be used to minimize the communication and maximize processor utilization. This must be done with care so that the amount of computation and/or message passing associated with subdomain boundary interfaces does not grow rapidly with the number of processors. For structured meshes the mesh partitioning is quite straightforward. However, for unstructured meshes, the mesh partitioning becomes a nontrivial problem. Following the 2D approach presented in Ref. [2], domain decomposition constitutes the most important Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00006-4 © 2019 Elsevier Inc. All rights reserved.

87

88

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Start Mesh partitioning and processing of ghost nodes and overlapping elements for SG or MG kt = kt +1 or t = t +Δt Coarse grids Transfer solutions and residuals Exchange flow field variables and forcing term

Compute local time steps Compute nodal gradients Compute diffusive fluxes Exchange of nodal gradients Compute convective fluxes Exchange of flow field variables

SG or MG?

Fine grid Compute local time steps

MG

Compute nodal gradients Compute diffusive fluxes Exchange of nodal gradients Compute convective fluxes Exchange of flow field variables

SG No

kt = ktmax or t = Yes

Transfer corrections Exchange of flow field variables ktmax denotes maximum time steps; tmax denotes maximum time for running.

End

FIGURE 6.1 Flowchart depicting geometric domain decomposition parallelization strategy using message-passing interface (MPI).

part of this parallel implementation and new algorithms are developed for 3D computations [3
5], which are outlined in the following subsections.

6.2.1 METIS METIS [6] is the computer code used to produce the partitions. It is a software package developed by Department of Computer Science, University of Minnesota, United States, for partitioning unstructured meshes, large irregular graphs, and computing fill-reducing orderings of sparse matrices. It provides two programs partnmesh and partdmesh for partitioning meshes into k equal-size parts. These programs take the element node array of the mesh as input and compute a partitioning for both its element and nodes. The difference between these two programs is that partnmesh converts the mesh into a nodal graph (i.e., each node of the mesh becomes a vertex of the graph), whereas partdmesh converts the mesh into a dual graph (i.e., each element becomes a vertex of the graph). In the case of partnmesh the partitioning of the nodal graph is used to derive partitions of the elements. In the case of partdmesh the partitioning of the dual graph is used to derive a partitioning of the nodes. Partnmesh is chosen over partdmesh here as numerical

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experiments show that the run time for simulation is much faster for partitioning file obtained from partnmesh. This may be due to different type of algorithms used to partition the mesh for both partnmesh and partdmesh. Since partnmesh takes the element node array of the mesh as the input, the unstructured tetrahedral grid file generated is yet to be used as the input for partnmesh. In view of this a program named prepmesh3D is developed to output a new graph file that is written according to the element node array of the mesh. A script file, prepgrid3D, is written to integrate both the programs prepmesh3D and partnmesh. On successful execution of prepgrid3D the actual partitioning is stored in two files MeshFile.npart.Nparts, which stores the partitioning of the nodes, and MeshFile.epart.Nparts, which stores the partitioning of the elements. MeshFile is the name of the file that stores the mesh and Nparts is the number of required partitions. MeshFile.npart.Nparts is the only file required for the identification of ghost nodes and overlapping elements. In this context a core node is defined as the real and physical node of the partitioned mesh, whereas a ghost node is defined as the nonphysical node of the partitioned mesh that belongs to the neighboring subdomains and no computation is being performed on it. Overlapping elements in this context is defined as those elements that overlap each other along the interprocessor boundaries and the nodes that formed these elements are a mixture of core and ghost nodes. The developed algorithm is written in a program named Partition and is fully integrated into the 3D Navier
Stokes solver so as to have little intervention from the user. Basically, it consists of four essential steps: 1. Identify ghost nodes and overlapping elements between the neighboring subdomains. 2. Generate individual grid files with local numbering for every partition. 3. Establish data structure for communication between core and ghost nodes. 4. Generate a script file, combine, to integrate all the result files into one after the parallel simulation has been performed. The flowchart shown in Fig. 6.2 depicts the previously described algorithm.

6.2.2 Identification of Ghost Nodes and Overlapping Elements After having obtained a partition of the mesh into P parts using METIS, new algorithms are developed to identify the ghost nodes and overlapping elements and to write the individual grid files with local numbering for each partition. The program also includes establishing the data structure for communication and generating a script file, combine, to integrate all the result files into one as each processor writes their own result based on local numbering. The nodes and elements that are allocated uniquely to a processor are

90

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Start

Read global grid file and partition file for nodes generated for partnmesh

Search for core and overlapping elements with element nodes having the same and different partition number, respectively for all subdomains

Establish an array for core and ghost nodes for all subdomains using the correlation establish in Section 6.2.2

Write local partitioned grid files including ghost nodes and overlapping elements for each subdomain with local numbering for every partition

Write data structure to grid files for communication between core and ghost nodes

Generate script file, combine, to integrate all result files into one

End FIGURE 6.2 Flowchart depicting algorithm developed in program Partition.

referred to as core mesh components here and each processor calculates the flow-field variables and nodal gradients for it. Each subdomain is enclosed with a layer of ghost nodes and overlapping elements, which overlap the subdomains along the interprocessor boundaries as depicted in Fig. 6.3. Fig. 6.3 illustrates a 2D mesh being partitioned. The developed algorithms for 3D mesh are much more complex than the illustrated 2D mesh. Fig. 6.3 shows that the irregular partition lines bisect the 2D mesh whereas irregular partition faces bisect the 3D mesh and the mesh elements are tetrahedral instead of triangles. In general the ideas behind the algorithms for both 2D and 3D models are the same. Fig. 6.4 shows a 3D cavity mesh being partitioned into various subdomains generated by this partitioning algorithm. These ghost nodes contain flow-field variables and nodal gradients from neighboring subdomains that are required for the solution of variables within the subdomain so as to maintain a solution consistent with the original serial code and no computation is being performed on it. Communication between these core and ghost nodes is based on MPI and the data flow direction is always from core nodes to ghost nodes. The main concept of this algorithm is that those elements along the interprocessor boundaries and with nodes having different partition numbers are considered as overlapping elements, which are cut through by partition lines. And those nodes that formed these elements are a mixture of core and ghost nodes. Basically the ghost node of

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91

FIGURE 6.3 Mesh decomposed into four partitions showing each subdomain extended with a layer of ghost nodes and overlapping elements.

92

Computational Fluid-Structure Interaction

(A)

(B)

(D)

(C)

(E)

(F)

FIGURE 6.4 Cavity mesh decomposed into various partitions (A) original mesh; (B) 2 partitions; (C) 4 partitions; (D) 8 partitions; (E) 16 partitions; and (F) 32 partitions.

this partition is the mirror image of the core node of the other neighboring partition. For example, as shown in Fig. 6.3, the ghost node number 11 of partition 3 is the mirror image of core node number 10 of partition 1. Therefore making use of this correlation between the core and the ghost nodes, the rest of the ghost nodes can be identified so as to build the control volume for the core node.

6.2.3 Grid Files With Local Numbering Since SPMD programming paradigm is employed here for geometric domain decomposition, which distributes the problem domain over P processors and each processor runs the same program individually. It is a wise strategy to rewrite the global grid file into the respective local partitioned grid files with local numbering. In this way, each processor will only read and write their respective grid file and output files individually, without having any master processor to control the I/O operations. This avoids any synchronization in

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93

the event of reading and writing and it behaves like a serial program, each performs its own computation. The only communication is done during the passing of nodal gradients and flow variables. Thus a substantial amount of computational time and memory allocation is saved. Another advantage is the solver only reads the respective partitioned grid file for the next simulation run without both accessing the global grid file and generating the mesh decomposition algorithm again.

6.2.4 Data Structure for the Exchange of Ghost Nodes Variables and Gradients The communication between the core and ghost nodes requires a data structure for each subdomain, which holds the nodes and processor number to be sent and received. Each processor reflects its renumbered subdomain as a complete mesh consisting of ne elements and np nodes, where ne and np are the local number of elements and nodes, respectively. In the methods used here the communication and outputting of variables to files are based on local numbering rather than global numbering, and therefore no translation back from local to global numbering is necessary. With reference to the grid decomposed in Fig. 6.3 the data structure established for communication between the core and ghost nodes is shown in Table 6.1 and it only shows the data structure for Partition 1 and the data structure for the rest of the partitions will follow exactly the same fashion. The data structure is appended to the end of each individual grid file and the respective processor will only read its grid file. Part of the code for reading the data structure is shown as follows for a particular processor, myid: read (nunit, ) nsenrev do id 5 1,nsenrev read(nunit, ) igver,nprtn,icore,npart,ighst if (nprtn.eq.myid) then nsend(npart,mg) 5 nsend(npart,mg) 1 1 sendvert(npart,nsend(npart,mg),mg) 5 icore else nrecv(nprtn,mg) 5 nrecv(nprtn,mg) 1 1 recevert(nprtn,nrecv(nprtn,mg),mg) 5 ighst endif enddo

where nsenrev is the total number of nodes to be sent and received; sendvert is a three-dimensional array that contains the locally numbered core nodes, icore, which will be sent to the partition number, npart, for each level of multigrid, mg; and recevert is similar to the array sendvert except that it contains the locally numbered ghost nodes, ighst, which will be received from the partition number, nprtn.

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Computational Fluid-Structure Interaction

TABLE 6.1 Communication Between Core and Ghost Nodes for Partition 1 as Depicted in Fig. 6.3. Global Node Number

Send

Receive

Partition Number

Local Node Number (core node)

Partition Number

Local Node Number (ghost node)

igver

nprtn

icore

npart

ighst

11

1

5

2

17

31

8

16

27

9

15

26

10

14

26

10

23

7

10

6

3

9

26

10

4

12

5

1

17

10

2

3

11

32

9

16

29

10

15

24

8

13

22

7

12

7

3

11

8

14

28

3

4

On most parallel computers, moving data from one processor to another, takes more time than moving or manipulating data within a single process. To keep a program from being slowed down, many parallel computers allow users to start sending or receiving several messages and to proceed with other operations. Message passing is performed asynchronously using the buffered-send mpi_bsend( ) routine here. This allows the messages to be placed in the buffer until it is delivered. Once all the flow-field variables or nodal gradients or forcing terms have been sent, the processes are ready to receive all these from the respective processes. Messages are received by the standard blocking receive mpi_recv( ) routine. To reduce the communication time and the latency overheads by creating fewer larger messages, the flow-field variables (ua( )), nodal gradients, and

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forcing terms are packed into a single array (sendsol( ) for flow-field variables only) before sending them to the respective partitions, as depicted in part of the code shown later. The flow-field variables, nodal gradients, or forcing terms are unpacked from the single array (recvsol( ) for flow-field variables only) after receiving from the respective partitions. do npart 5 0,numprocs-1 ..... iq 5 0 do iv 5 1,nsend(npart,mg) ip 5 sendvert(npart,iv,mg) is 5 ip 1 nsdex(mg) - 1 do ivar 5 1,nvar iq 5 iq 1 1 sendsol(iq) 5 ua(ivar,is) enddo enddo call mpi_bsend(sendsol,(nsend(npart,mg) nvar), & mpi_real,npart,myid,mpi_comm_world,ierr) call mpi_recv(recvsol,(nrecv(npart,mg) nvar), & mpi_real,npart,npart,mpi_comm_world,istatus,ierr) id 5 0 do iv 5 1,nrecv(npart,mg) ip 5 recevert(npart,iv,mg) is 5 ip 1 nsdex(mg) - 1 do ivar 5 1,nvar id 5 id 1 1 ua(ivar,is) 5 recvsol(id) enddo enddo ..... enddo

The first argument in mpi_bsend( ) is the address of the variables to be sent, the second argument is the number of variables to be sent, the third argument is the MPI-derived data type, mpi_real, the fourth argument is the destination where the variables are to be sent, and the fifth argument is called the tag, which is an arbitrary nonnegative integer to restrict receipt of the message. Finally, a communicator, mpi_comm_world, is specified and ierr, in which to return an error code. The mpi_recv( ) routine is a blocking receive, in which the control is not returned to the program until the message is received. The first argument is the address of the buffer into which the received message is to be put, the second argument is the number of variables to be received, the third argument is the same as mpi_bsend, the fourth argument is the source of the variables to be received from, and the fifth argument is the tag.

96

Computational Fluid-Structure Interaction

6.3 MULTIGRID PARALLELIZATION In Mavriplis’ works [2,7,8] the domain decomposition combined with multigrid (DD-MG) approach is adopted where an agglomeration-based MG algorithm was parallelized based on independent partitioning of each level of mesh in the MG hierarchy and this results in an unrelated coarse and fine grid partitions. The disadvantage is that the partitions of coarser grids may bear no relation to the partition of the corresponding finer grids and results in increase in intergrid communication. In the present work the need for communication between processors for different grid levels during transferring of variables and residuals and interpolation of corrections are eliminated. The multigrid domain decomposition (MG-DD) approach is adopted for the MG parallelization in this study [3,9]. This means that the nonnested MGs are independently generated first. Then domain decomposition of the finest grid is performed, which is followed by decomposition of the various coarse levels of grids guided by the finer grid partitions. This is achieved by using the fine grid partitions to infer the coarse level partitions (i.e., the coarse grid is to inherit its partition from that of the corresponding fine grid) and load balancing in the coarse mesh is reasonably well ensured. The only communication needed to perform once on the coarse grids is after transferring of flow-field variables and residuals and on the fine grid after transferring of corrections, as shown in Fig. 6.1. This communication is to ensure that the ghost nodes will have updated solution from the respective partitions after the transferring of variables from the fine grid and transferring of corrections from the coarse grid. In the MG process the flow-field variables and residuals of the coarse grid nodes are obtained directly from their corresponding fine grid nodes. A two-level MG and two subdomains are used to describe the procedure of partitioning the coarse grid using the fine grid. The main idea about this MG parallelization algorithm is that the fine grid is partitioned into two subdomains according to the algorithm developed for SG depicted in Section 6.2. And both the maximum and minimum values in the x- and y-direction (xmin, xmax, ymin, and ymax) of each partition for the fine grid are noted. With these dimensions an imaginary bounding box enclosing the subdomains is formed as shown in Fig. 6.6. The main purpose of this bounding box is to identify the coarse nodes that fall within this box according to the fine grid partitioned including those nodes beyond the subdomains boundary. Next, a search is done for those actual coarse nodes that fall within the subdomain and filter out those nodes that fall onto the shaded portion as shown in Fig. 6.5 using the algorithm depicted in Section 4.6. After classifying the respective coarse nodes according to which partition they belong to, and then the ghost nodes and overlapping elements are identified using the algorithm depicted in Section 6.2.2. The individual grid file for the partitioned coarse grid and data structure for communication is written to the

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97

FIGURE 6.5 Fine grid partitions to infer the coarse grid partitions.

Start

Domain decomposition using METIS and identify ghost nodes and overlapping elements for the fine grid using “Partition”

Using algorithm depicted in Section 4.5 to detect coarse node that fall onto the core elements instead of overlapping elements

Classified coarse nods according to the maximum and minimum dimensions of the partitioned fine grid

Using “Partition” to identify ghost nodes and overlapping elements for coarse grid

End FIGURE 6.6 Flowchart depicting the multigrid (MG) parallelization processes.

respective partition. The flowchart presented in Fig. 6.6 shows the whole procedure of using the fine grid partition to infer the coarse level partitions.

6.4 MEASURING PERFORMANCE Both speedup and efficiency are commonly used to measure the performance of a parallel code. The run time of the original serial code is used as a measure of the run time on one processor. In this study the run time or total simulation time starts from the moment before mesh partitioning for either SG or MG, identifying ghost nodes and writing local grid files for different partitions to the time after writing all the results to the respective files. Here both CPU time and wall-clock time are used to record the total simulation time. The main difference is that CPU time is the recorded time

98

Computational Fluid-Structure Interaction

when only the processor performs a calculation whereas clock time is almost similar to the CPU time but it includes idling time when the processor idles while waiting for other processors to communicate. The wall-clock time is used to represent the total simulation time in the current method since it includes the idling time, computation time, and communication time, which is the true representation of the total simulation time. Computation time is the time spent on computing, and communication time is the total time spent on exchange of flow-field variables, nodal gradients, and forcing terms.

6.4.1 Speedup The parallel speedup Sp is the ratio of the run time on one processor t1 to the run time on P processors, tp [10]. Sp 5

t1 tp

ð6:1Þ

If the parallelization is 100% efficient, then Sp 5 P, but this is rarely the case as both communication and load balancing adversely affect the performance of parallel computation. A theoretical model has been formulated for the prediction of the parallel computational and communication times [3,9]. The total simulation wall-clock time, tp, is used as the run time to compute the speedup and it is the maximum wall-clock time among all the processors defined as follows: j j Total simulation wall-clock time; tp 5 tcpu 1 tidle ;

for all j

ð6:2Þ

where j is the processor number; tjcpu and tjidle are the CPU time and idling time for processor j, respectively. CPU time for processor j is defined as follows: j j j CPU time; tcpu 5 tcomp 1 tcomm

ð6:3Þ

j j where tcomp and tcomm is the computation and communication time for processor j, respectively. The overall communication time is defined as the maximum communication time among all the processors:
j NG FT Communication time; tcomm 5 tFFV ð6:4Þ comm 1tcomm 1tcomm j NG FT where tFFV comm , tcomm , and tcomm are the communication time for exchanging of flow-field variables, nodal gradients, and forcing terms, respectively. And the representative computation time of a problem is defined as follows: j j j Computation time; tcomp 5 tp 2 tcomm 2 tidle

ð6:5Þ

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99

6.4.2 Parallel Efficiency The parallel efficiency is sometimes used as the performance parameter for a parallel code. Parallel efficiency Ep is simply the ratio of the parallel speedup Sp to the number of processors P [10]. Ep 5

Sp P

ð6:6Þ

where 0 # Ep # 1.

REFERENCES [1] W. Gropp, E. Lusk, A. Skjellum, Using MPI: Portable Parallel Programming With the Message-Passing Interface, The MIT Press, Cambridge, Massachusetts, 1994. [2] D.J. Mavriplis, Large-Scale Parallel Viscous Flow Computations Using an Unstructured Multigrid Algorithm, NASA/CR-1999-209724, ICASE Report No. 99-44, Nov 1999. [3] C.H. Tai, Y. Zhao, Parallel unsteady incompressible viscous flow simulation using an unstructured multigrid method, J. Comput. Phys. 192 (1) (2003) 277
311. [4] C.H. Tai, Y. Zhao, K.M. Liew, Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, The Second M.I.T. Conference on Computational Fluid and Solid Mechanics, 17
20 June, 2003, MIT, Cambridge, MA 02139, United States, 2003. [5] C.H. Tai, Y. Zhao, K.M. Liew, Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, in: G.A. Gravvanis, R.W. Lewis (Eds.) Special issue on “preconditioning methods: algorithms, applications and software environments”, Comp. Struct. 82 (28) (2004), 2361
2452. [6] G. Karypis, V. Kumar, METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, Version 4.0, University of Minnesota, Department of Computer Science, Sep 1998. [7] D.J. Mavriplis, Parallel Unstructured Mesh Analysis of High-Lift Configurations, AIAA Paper 2000-0923, 38th AIAA Aerospace Sciences Meeting, January 2000. [8] D.J. Mavriplis, Multigrid strategies for viscous flow solvers on anisotrpic unstructured meshes, AIAA Paper 97-1952-CP, June 1997, Proceedings of the 13th AIAA CFD Conference, Snowmass, CO, 1997 pp. 659
675. [9] X. Lv, Y. Zhao, et al., An efficient parallel/unstructured-multigrid preconditioned implicit method for simulating 3D unsteady compressible flows with moving objects, J. Comput. Phys. 215 (2) (2006) 661
690. [10] R.T. Fenner, Engineering Elasticity: Applications of Numerical and Analytical Techniques, Ellis Horwood, 1986.

Chapter 7

The Immersed Object Method With Overlapping Grids 7.1 INTRODUCTION The simulation of arbitrarily moving bodies with complex geometries is one of the challenges faced by researchers in the field of computational fluid dynamics (CFD). The development of accurate, robust, and efficient methods that can tackle this problem would be very useful for many practical applications, for example, the biomedical application. In almost all practical CFD applications, one needs to account for solid boundaries with complex geometries. There are several approaches for dealing with stationary boundaries; for example, one may use structured body-fitted grids or unstructured grids to be able to describe the required boundary geometry. The main advantage of such approaches is that the boundary will be well defined and specifying of boundary conditions is relatively straightforward. For moving boundary problems, one will have to regenerate the grid if the boundary is moved over a large distance, which will lead to increased computational effort. Attempts are made to develop a novel, simple, and efficient immersed object method (IOM) with overlapping grids (OGs) for simulating unsteady incompressible viscous flows around moving rigid bodies. Instead of using the direct forcing method [1] along the object boundary, the fluid velocity within the whole object is set to the corresponding local object velocity by using a forcing term in the momentum equations. Compared with the other existing methods, this direct approach has the advantage that one does not need to perform elaborate interpolation to impose the boundary condition. The solid body or object is represented by an internal mesh, used only to do quadtree or octree search to identify the Eulerian mesh nodes inside the object for adding the forcing term. For high Reynolds number flows where the boundary layer is thin and has to be accurately captured, an OG surrounding the object can be added and the Eulerian solution can be transferred to the OG to perform further calculation to obtain a fined solution and the lift and drag coefficients. Compared with the Chimera method, this approach is relatively simpler to implement in computer codes because no hole cutting is required in the Eulerian mesh, and this is especially advantageous when the solid body is moving arbitrarily or when there are multiple moving objects. Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00007-6 © 2019 Elsevier Inc. All rights reserved.

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7.2 IMMERSED OBJECT METHOD IOM involves specifying the body force term in such a way as to simulate the presence of a flow boundary within the computational domain that does not coincide with the computational grid and without altering the computational grid [2]. The advantage of this is that bodies of almost arbitrary shape can be added without grid restructuring, often time-consuming and computationally expensive procedure. Furthermore, multiple bodies may be simulated, and relative motion of those bodies may be accomplished at reasonable computational cost. This method is quite similar to the fictitious domain method in some sense, except that the fictitious domain method requires a coupling between the actual boundary discretization and the grid used in the auxiliary domain. Whereas, the IOM developed in this study does not require specifying any boundary conditions to the stationary or moving object, nor does not require the complex information about how the 3D surface of the object intersects with the 3D unstructured mesh, because no elaborate interpolation at the interface is needed at each time step. And a direct forcing method is employed to drive the fluid velocity within the object to the moving object velocity, instead of applying it only on the boundary as in its original form [1]. A body force term F is introduced such that a desired velocity distribution Vo can be assigned within and over an object boundary Ω. The body force can be added to the incompressible Navier
Stokes equations and solved for U @U rp 1 UUr U 5 2 1 μUr2 UU 1 F @t ρ

ð7:1Þ

and the continuity equation rUU 5 0

ð7:2Þ

In principle, there are no restrictions for the object velocity distribution Vo and for the shape and motion of Ω; therefore a wide variety of boundary conditions can be imposed. The main advantage of this approach is that F can be prescribed on a regular mesh so that accuracy and efficiency of the solution procedure on simple grids are maintained. The body force F of Eq. (7.1) is prescribed at each time step to establish the desired velocity Vo on an arbitrary surface that need not coincide with the grid. The time-discretized equation with the introduction of the pseudo time τ of Eq. (7.1) can be written as follows:
 n11;m ΔVcv U n11;m11 2 U n11;m 5 R~~ 1 Fn11;m ð7:3Þ Δτ In order to drive the velocity Un11 to the desired value Vo, within every physical time-step, a number of subiteration are performed on the discrete

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Navier
Stokes equation (7.3) using a five-stage Runge
Kutta timeintegration algorithm to give the velocity update equation as follows:   n11;m Δτ R~~ U n11;m11 5 U n11;m 1 αRK 1 Fn11;m ð7:4Þ ΔVcv ~~ is defined in Eq. (3.57) for the matrix-free, implicit, dual timewhere R stepping computation, which includes the convective and viscous terms and the pressure gradient. αRK is the stage coefficients for the five-stage Runge
Kutta scheme. At the end of the subiteration, the original Navier
Stokes equations are recovered as follows: R~~ 5 0

ð7:5Þ

To impose the fluid velocity within the object to be the object velocity (i.e., Un11 5 Vo), the body force F must be stated as follows: 8 < 2 R~~ n11;m 1 1 ΔV cv
V 2 U n11;m ; on and within Ω o n11;m αRK Δτ 5 ð7:6Þ F : 0; elsewhere This forcing is direct in the sense that the desired value of object velocity is imposed directly on and within the boundary of the solid body without any dynamical processes, and no boundary conditions are explicitly enforced, which make the method simple, robust, and efficient. In addition the forcing of Eq. (7.6) does not require additional CPU time since it does not involve the computation of extra terms. Here there are two methods to define the physical boundary of the immersed object. The first method is to specify the geometric equation of the immersed object within the computational domain. The second method is to generate an object grid and embed it into the computational domain. For complex immersed objects, the latter is found to be more robust and accurate than the former. The body force F of Eq. (7.6) is imposed on those fluid nodes that fall on and within the immersed object. The first method is quite easy and straightforward for simple object. The second method is robust and accurate for more complex geometries. For ease of clarification and illustration, all the figures in this chapter will be in 2D. Fig. 7.1 shows a circular disk grid immersed into a square cavity domain where it is used to classify those fluid nodes in the computational domain as solid nodes. From the figure, it can be seen that the computational mesh that is covered by the immersed object needs to be refined so that the physical boundary can be defined as accurate as possible. The search algorithm for fluid nodes that are covered by the immersed object is based on the concept of dot product of two vectors: the unit normal vector oriented inward from an edge, n, and the vector ~ p that points from the center of the edge to a node under

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FIGURE 7.1 Immersed object grid into computational domain.

1

1 Fluid node, p p3

Immersed object cell p3 p1 θ1 n1

p2 n2

θ3 n3

p2

3

p1

θ2

Immersed object cell θ3 θ1

n3 n1

θ2

3

n2

Fluid node, p Bounding box 2

Bounding box 2

(A)

(B)

FIGURE 7.2 (A) Fluid node falls within an immersed object cell using dot product; (B) fluid node does not fall within an immersed object cell.

consideration. The dot product of these two vectors, (pn)nedg, for the three edges, is as follows:    nx
nedg nedg 5 fpgUfng 5 px py U $ 0; nedg 5 1 to 3 ðpn Þ 5 px nx 1py ny ny ð7:7Þ where the superscript nedg is the edge number of the cell. (pn)nedg must be positive for all the three edges if the fluid node under consideration falls within the cell of the immersed object as depicted in Fig. 7.2 and this fluid node is classified as a solid node. To reduce searching time, a quadtree search method is employed. Before the search, both the computational domains covered by the immersed object and the immersed object grid are decomposed into a number of square zones, in which searching for a particular fluid node is only done within the related square zones, instead of searching the entire flow field. The main advantage is that physical boundary of any arbitrary geometry can be defined within the computational domain. The disadvantage is that the physical boundary will not

The Immersed Object Method With Overlapping Grids Chapter | 7

105

FIGURE 7.3 Immersed object grid embedded in every processor.

be as smooth as those defined using the conventional methods and this disadvantage can be overcome by using OG. For parallel implementation, every processor has a copy of the immersed object grid, as shown in Fig. 7.3, and a search for solid nodes is done within its own processor using the search algorithm described earlier. For moving body problems, the search algorithm is performed at the beginning of each new time-step once the immersed object has been moved to a new position (i.e., translate, rotate, or both) and the speed of searching has a great influence on the overall performance. There are two kinds of meshes being constructed in the IOM. The first mesh is the Eulerian or background that remains stationary throughout the computation, and the stationary and/or moving object is defined by using the IOM, and the second mesh is a Lagrangian or OG that contains the physical object and moves with it. The methodology of OG will be explained in detail in Section 7.2. The parallel-MG method only applies to the background mesh whereas the parallel-SG computation applies to the OG. The reason for using parallel-SG computation for the OG rather than parallel-MG method is given in Section 7.2. The methodology to define the physical boundary of the immersed object on the coarse grid levels is exactly the same as those described in the previous paragraph for the fine grid. The search for the immersed object begins at the start of every time-step for all grid levels and IOM with direct forcing is enforced within the object at the coarse grid as well as on the fine grid levels. In this way, the convergence rate and computational time are accelerated for the IOM on the background grid.

7.3 OVERLAPPING GRIDS The overlapping grid (OG), which does not need to cover the entire background mesh, is constructed with very dense nodes around the object. This helps to fully resolve the physical boundary and enables accurate resolution of the boundary layer. The computation begins by calculating the solution on

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the background mesh with the IOM. The solution on the background mesh is then interpolated to the OG. The flow field around the object in the OG is then recalculated to obtain an accurate boundary layer of the object. Here the calculation of the convective flux on the OG requires the difference between the fluid velocity (Uf) and the grid velocity (Ug) in the velocity vector (U) as shown in convective matrix (Fc) after Eq. (3.17). This accounts the movement of the OG with reference to the background grid motion. Next, the solution on the OG is interpolated back to the background mesh and is reevaluated with a more accurate solution from the OG. In this way, the mesh density on the background mesh can be reduced to save computational effort. Since the main purpose of OG is to enhance the resolution of the boundary layer, parallel-MG method is not used to accelerate the convergence rate in the OG and only parallel-SG computation is employed here. To determine the solution on the OG, the solution on the background mesh is interpolated onto the OG as boundary conditions and this requires some form of interpolation in a transfer operator. Before the interpolation of solution takes place, it is necessary to determine each overlapping node in the background cell is located and vice versa. The search algorithm developed for defining the physical boundary of the immersed object is adopted here, as shown in Eq. (7.7) and Fig. 7.2. To reduce searching time, a quadtree search method is employed. Following the interpolation algorithm developed for MG in [3], the flow field variables for the background grid are interpolated onto the overlapping boundary according to the following equation: W OG 5

ABG1 W BG1 1 ABG2 W BG2 1 ABG3 W BG3 ABG1 1 ABG2 1 ABG3

ð7:8Þ

where W OG is the solution for the overlapping boundary node; W BG1 ; W BG2 ; and W BG3 are the solutions for the background mesh. ABG1 ; ABG2 ; and ABG3 are the areas of the corresponding triangles opposite to the background cell nodes. The flow field values at the overlapping boundary node, which is contained in the background cell formed by nodes BG1, BG2, and BG3, are a weighted average of the values at those nodes as shown in Fig. 7.4. The flow field variables for the OG are interpolated back onto the background node according to the following equation: AOG1 W OG1 1 AOG2 W OG2 1 AOG3 W OG3 ð7:9Þ AOG1 1 AOG2 1 AOG3 where W BG is the solution for the background node; W OG1 ; W OG2 ; and W OG3 are the solutions for the OG. AOG1 ; AOG2 ; and AOG3 are the areas of the corresponding triangles opposite to the overlapping cell nodes. The flow field values at the background node are a weighted average of the values at those nodes formed by nodes OG1, OG2, and OG3, as shown in Fig. 7.5. W BG 5

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BG1

ABG3 ABG2 OG ABG1

BG2

BG3 FIGURE 7.4 Transfer of flow field values from the background mesh to the overlapping grid.

OG1

AOG3 AOG2

BG AOG1

OG2

OG3 FIGURE 7.5 Transfer of flow field values back to the background mesh from the overlapping grid.

For parallel implementation, the OG is decomposed into a number of subdomains for different processors, which are independent from the computation for the background mesh. For example, as shown in Fig. 7.6, both the background mesh and OG are decomposed into four and two subdomains, respectively. The subdomains of the background mesh are distributed to processors 1, 2, 3, and 4 whereas the subdomains of the OG are distributed to processors 5 and 7. From the figure, it can be seen that the OG for processor 5 covers processors 1, 2, 3, and 4, and communications will take place between these processors. Likewise for the OG for processor 6, communications will take place between processor 6 and processors 1, 2, and 4. Every processor for the background mesh has a copy of the OG data, just like the immersed object grid shown in Fig. 7.3. The coordinates of the OG will be updated accordingly at every time-step in the background mesh processor for moving body problem, and in this way it does not require the OG processors to send the updated coordinates to the background mesh processors. The parallel computation on the OG is exactly the same as that employed for the background mesh, except interpolation is needed at the beginning of each new time-step. For moving body problems, the search algorithm is to determine that each overlapping node in the background cell is located and vice versa will

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FIGURE 7.6 Mesh partitions and communications between background and overlapping grids.

be performed at the beginning of each new time-step once the OG has been moved to a new position (i.e., translate, rotate, or both). The performance of this part has a large influence on the overall performance and the search algorithm is found to be quite efficient.

REFERENCES [1] J. Mohd-Yusof, Combined Immersed-Boundary/B-spline Methods for Simulations of Flow in Complex Geometries, CTR Annual Research Briefs, NASA Ames/Standford University, 1997. [2] C.H. Tai, Y. Zhao, K.M. Liew, Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids, J. Comput. Phys. 207 (1) (2005) 151
172. [3] C.H. Tai, Y. Zhao, Parallel unsteady incompressible viscous flow simulation using an unstructured multigrid method, J. Comput. Phys. 192 (1) (2003) 277
311.

Chapter 8

The Immersed Membrane Method and Fluid-Structure Interaction 8.1 DEFINITION OF GHOST FLUID NODES The immersed membrane method was initially developed to simulate the interaction between a fluid and its immersed thin membranes. When a thin structure is immersed in a flow field, it causes discontinuous flow conditions across the structure. Although velocity is continuous, the gradient of velocity, pressure, the gradient of pressure and fluid stresses are quite different, even having sharp changes over both sides of the thin structure. See Fig. 8.1 for a 2D example of thin membrane immersed into a flow field. In order to account for the discontinuous fluid conditions across the membrane, a set of ghost fluid nodes are introduced. Specifically, in cell 267, edges 26 and 27 are cut by the membrane, I1 and I2 are the intersection points respectively. Now we use the velocities at node 2 and point I1 to extrapolate the velocity at node 6, and assign this extrapolated velocity to the ghost node g26, which locates at the same position as node 6. Furthermore, we can use the pressure/ density and pressure/density gradients at node 2 to extrapolate the ghost node pressure/density at g26 as well. In a similar way, we can use the corresponding conditions at node 6 and I1 to extrapolate the ghost values at g62, which locates at the same location as node 2. While calculating the fluxes along edge 26, ghost values at g26 will be used in place of the real-node conditions of node 6 when the residual for node 2 is accumulated. Likewise, ghost values of g62 will be used in place of real-node conditions of node 2 when cumulating the residual for node 6. There is the possibility that a node may hold multiple ghost-node values, node 2 (g62 and g72) in this case. All of these possible ghost values should be stored and used accordingly, while calculating the respective fluxes along different edges. One should note that the ghost-node values should be only used while computing the fluxes along the edges cut by the membrane. For the normal edges, the real-node conditions should be used instead. For example, flow conditions at node 6 and 7 should be used while computing the fluxes along edge 68.

Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00008-8 © 2019 Elsevier Inc. All rights reserved.

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3

2 (g62, g72)

4

7 (g27)

I2

8

I1 1

6 (g26)

5 FIGURE 8.1 The concept of Ghost Nodes and Ghost Values.

The IMM is not only limited to the studies of immersed membrane structures; it can also be extended to study the fluid-structure interaction for arbitrary-shaped objects immersed in a fluid. For an arbitrary object immersed in a fluid as shown in Fig. 8.2, it occupies a part of the fluid domain. Some fluid nodes will fall within the object domain, which are represented by hollow circles in the figure. For these fluid nodes, special treatment should be performed to ensure that they will not take part in the fluid calculation since physically there are no fluids inside the object domain for which a pre-processing subroutine is designed. It will identify those fluid nodes which fall in the domain of the object. Before the identifying process begins, it would be beneficial if we can narrow down the search region of the fluid mesh that might contain the immersed object mesh. A bounding box is defined around the object by the extrema of the object mesh coordinates. Only those fluid nodes fall into this box will be checked and they are stored into an array named “suspected array”. This simple treatment can greatly save the search time according to our experiences. Next, what we need to do is just performing a do-loop based on the number of the cells of the immersed object mesh. During each cycle of this loop, we test the “suspected array” nodes one by one to see if they fall into a certain cell of the object mesh. If yes, we can identify it as a ghost node. The concept of bounding box is also applied in this testing process to improve the efficiency of the algorithm. The “suspected array” nodes are first tested against this box before the full test algorithm is performed. Basically, the algorithm for inter-connectivity relationship between the suspected nodes and object cell is based on the concept of dot product between

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

111

B B1 A1

B3

B2

A A2 A3

Real node

Ghost node

FIGURE 8.2 Treatment of the fluid nodes inside the immersed body; for ghost-node A, it can possess up to 3 different ghost values corresponding to fluid nodes A1, A2 and A3; for fluid node B, 3 ghost nodes B1, B2, B3 contribute to its flux computations.

two vectors: i) The unit normal vectors oriented inward from the triangular face, n; ii) and the vector p pointing from the centre of the triangular face of the cell to the node to be tested. The dot product between these two vectors, (pn)nface, must be greater than or equal to zero if the angle, θ, subtended between them is less than 90
and less than zero if the angle is greater than 90
, as depicted in the following equation [1,2]: 8 9 > nx >
< =  nface
 nface ðpn Þ 5 p :fng 5 px py pz : ny 5 px nx1py ny1pz nz ; nface5 1 to 4; > > : ; nz ð8:1Þ where the superscript nface is the triangular face number of the tetrahedral cell. For ease of clarification and illustration, all the figures in this section will be in 2D. With reference to Fig. 8.3A, the dot product between these two vectors must be positive for the three edges for 2D (four faces for 3D) if the suspected node is to fall within the structure cell. If one of the dot products between these two vectors is less than zero, then the node being tested is not within the cell, as depicted in Fig. 8.3B. The following equation shows that the criterion for the node to be within the tetrahedral cell is
 ðpn Þ1 :and:ðpn Þ2 :and:ðpn Þ3 :and:ðpn Þ4 $ 0: ð8:2Þ

112

Computational Fluid-Structure Interaction 1 Immersed object cell

Suspected node, p

p3 p1

θ3 p2

n1

n2

3

n3

θ1 θ2

Bounding box 2 FIGURE 8.3A Suspected node falls within a 2D immersed object triangle cell using dot product.

1 Suspected node, p

Immersed object cell

p3 p2 p1

θ3

n3

θ1

3

n2 n1

θ2

Bounding box enclosing cell 2

FIGURE 8.3B Suspected node does not fall within a 2D immersed object triangle cell.

In order to reduce the pre-processing CPU time further, those nodes that fall within a particular cell are marked with flag 1 and they will not be tested for the rest cells anymore. After all the suspected nodes with a flag of 0 in the box are tested against this immersed object cell, then the next object cell in the list will be tested with those fluid nodes having a flag of 0 again and those having a flag of 1 are skipped. This process will stop if all the suspected nodes are tested and marked as 1 or if all the object cells are tested whichever one is completed earlier. Subsequently, those fluid nodes with a flag 1 will not be involved in flow calculations. Only those ghost nodes having a connecting edge, that passes through the fluid-structure interface will be assigned for ghost-node

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

113

values and used to impose the immersed boundary conditions. For ease of clarification and illustration, we ignore those ghost-nodes which are not adjacent to the interface from now on. And when we mention ghost-nodes, we actually refer to those which contribute to flow computation.

8.2 INTERACTIONS BETWEEN FLUID AND STRUCTURE MESHES As described in forgoing chapters, in the fluid solver the convection fluxes are computed based on edges of fluid mesh. As a result, every node can become multiple ghost nodes with their corresponding ghost values since a node can be connected by multiple edges, which is especially true for 3D unstructured grids. The selection of a particular ghost value depends on which edge and node the computation involves. This is one of the novel features of IMM. Hence, it is very crucial to find all of the edges of the fluid mesh cut by the fluid-structure interface in an accurate and efficient manner. Our solution algorithm will be described in the following paragraphs. Because the object is fully immersed in the fluid, there should be no exception that every node of the object mesh is contained by a certain cell of the fluid mesh. As the first step of our algorithm, the relationship between the surface nodes of the object mesh and their containing cells of the fluid mesh will be established. Note that the same searching procedure determining the ghost nodes in the former section may apply here. After all of the surface nodes of the object mesh have been associated with their corresponding containing fluid cells, our algorithm starts by performing a do-loop over the number of the surface triangles of the object mesh. The basic operations involved in every iteration of the loop are outlined as follows: 1. If all of the three vertexes of the triangle locate in the same fluid cell, the iteration exit because there is definitely no edge of the fluid mesh intersecting with the current surface triangle; 2. If the two of the vertices locate in the same fluid cell while the third one does not, any one of the six edges of the fluid cell containing the third vertex might pass through the fluid-structure interface and the possible intersection points (if exists) will be searched using a separate subroutine, which will be given later; 3. If the three vertices reside in three different fluid cells, this becomes more complicated: a. The three cells have two common nodes. Then it can be made sure that among all of the edges of these three fluid cells, there is only one which passes through the fluid-structure interface, which is the edge connecting these two nodes.

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Computational Fluid-Structure Interaction 1

2 Intersected fluid cell

3 I1

I2

I3 Surface triangle of object mesh

4

FIGURE 8.4 Fluid cell that doesn’t contain any of the surface nodes of the object mesh.

b. The three cells have one common node. Then all of the edges of these three cells originate from such a common node might pass through the fluid-structure interface, and they are checked one by one using a separate subroutine (explained in following pages). c. The three cells have no common nodes. Then all of the edges of these three cells are likely to pass through the fluid-structure interface. Hence they are also checked one by one using a separate subroutine (explained in following pages). When the do-loop finishes its execution, most of the fluid edges cut by the interface are identified and marked with flag 1. But there could be some edges that are missed by the given scheme. As shown in Fig. 8.4, the fluid cell 1234 contains no surface node of the object mesh at all, but three of its edges pass through the interface. Obviously, they will be omitted by the given searching scheme. Therefore, a remedy procedure is designed to rescue such edges in the end, in which all of the fluid cells cut by the interface are examined to see whether any of their edges are cut by the interface, while those edges with flag 1 will be skipped. The following subroutine will find the intersection point (if it exists) between a line segment and a planar 3 vertex facet. The mathematics and solution can also be used to find the intersection between a plane and line, which is a simpler problem. The intersection between more complex polygons can be found by first triangulating them into multiple 3 vertex facets. Source code will be provided at the end, it illustrates the solution more than being written for efficiency. The labeling and naming conventions for the line segment and the facet are shown in the following diagram (Fig. 8.5)

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115

Pb n =(A, B, C)

Pa P

× Ax + By + Cz + D = 0

P = P1 + μ (P2 – P1)

P2 PC

P1 FIGURE 8.5 Determination of the intersection between a line segment and a planar facet with three vertices.

The procedure will be implemented given the line segment defined by its two end points and the facet bounded by its three vertices. The solution involves the following steps G G G G

Check that the line and plane are not parallel; Find the intersection of the line with the plane containing the facet; Check that the intersection lies along the line segment; Check that the intersection lies within the facet;

The intersection P is found by substituting the equation for the line P 5 P1 1 μðP2 2 P1 Þ into the equation for the plane Ax 1 By 1 Cz 1 D 5 0. Note that the values of A, B and C are the components of the normal unit vector to the plane which can be found by taking the cross product of any two normalized edge vectors, for example: ðA; B; C Þ 5 

ðPb 2 Pa Þ 3 ðPc 2 Pa Þ  ðPb 2 Pa Þ 3 ðPc 2 Pa Þ

ð8:3Þ

Then D is found by substituting one vertex into the equation for the plane: APax 1 BPay 1 CPaz 5 2 D

ð8:4Þ

This gives an expression for μ from which the point of intersection P can be found using the equation of the line.       μ 5 D 1 AP1x 1 BP1y 1 CP1z = AðP1x 2 P2x Þ 1 B P1y 2 P2y 1 C ðP1z 2 P2z Þ ð8:5Þ If the denominator above is 0 then the line is parallel to the plane and they don’t intersect. For the intersection point to lie on the line segment, μ must be between 0 and 1.

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Computational Fluid-Structure Interaction Pb

P S3 a1

a3

S1

a2 S2

Pa

Pc FIGURE 8.6 The sum of the internal angles of a point on the interior of a triangle is 2π.

Lastly, we need to check whether or not the intersection point lies within the planar facet bounded by Pa, Pb and Pc. The method used here relies on the fact that the sum of the internal angles of a point on the interior of a triangle is 2π, points outside the triangular facet will have lower angle sums. As shown in Fig. 8.6. If we form the unit vectors Pa1 , Pa2 and Pa3 as follows (P is the point being tested to see if it is in the interior) ð Pa 2 PÞ  Pa1 5  ð Pa 2 PÞ  ð Pb 2 PÞ Pa2 5    P ~b 2 P ~

ð8:6Þ

ð Pc 2 PÞ  Pa3 5  ð Pc 2 PÞ  the angles are a1 5 acosðPa1 UPa2 Þ a2 5 acosðPa2 UPa3 Þ a3 5 acosðPa3 UPa1 Þ The source code is as follows: /



/

Determine whether or not the line segment p1, p2 Intersects the 3 vertex facet bounded by pa, pb, pc Return true/false and the intersection point P The equation of the line is p 5 p1 1 μ(p2 2 p1) The equation of the plane is ax 1 by 1 cz 1 d 5 0 nxx 1 nyy 1 nzz 1 d 5 0

ð8:7Þ

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

117

int LineFacet(p1,p2,pa,pb,pc,p) XYZ p1,p2,pa,pb,pc, p; { double d; double a1,a2,a3; double total,denom,mu; XYZ n,pa1,pa2,pa3; / Calculate the parameters for the plane  / n.x 5 (pb.y - pa.y) (pc.z - pa.z) - (pb.z - pa.z) (pc.y - pa.y); n.y 5 (pb.z - pa.z) (pc.x - pa.x) - (pb.x - pa.x) (pc.z - pa.z); n.z 5 (pb.x - pa.x) (pc.y - pa.y) - (pb.y - pa.y) (pc.x - pa.x); Normalise(&n); d 5 2 n.x  pa.x - n.y  pa.y - n.z  pa.z; / Calculate the position on the line that intersects the plane  / denom 5 n.x  (p2.x - p1.x) 1 n.y  (p2.y - p1.y) 1 n.z  (p2.z - p1.z); if (ABS(denom) , EPS) / Line and plane don’t intersect  / return(FALSE); mu 5 - (d 1 n.x  p1.x 1 n.y  p1.y 1 n.z  p1.z) / denom; p- . x 5 p1.x 1 mu  (p2.x - p1.x); p- . y 5 p1.y 1 mu  (p2.y - p1.y); p- . z 5 p1.z 1 mu  (p2.z - p1.z); if (mu , 0 || mu . 1) / Intersection not along line segment  / return(FALSE); / Determine whether or not the intersection point is bounded by pa,pb,pc  / pa1.x 5 pa.x - p- . x; pa1.y 5 pa.y - p- . y; pa1.z 5 pa.z - p- . z; Normalise(&pa1); pa2.x 5 pb.x - p- . x; pa2.y 5 pb.y - p- . y; pa2.z 5 pb.z - p- . z; Normalise(&pa2); pa3.x 5 pc.x - p- . x; pa3.y 5 pc.y - p- . y; pa3.z 5 pc.z - p- . z; Normalise(&pa3); a1 5 pa1.x pa2.x 1 pa1.y pa2.y 1 pa1.z pa2.z; a2 5 pa2.x pa3.x 1 pa2.y pa3.y 1 pa2.z pa3.z; a3 5 pa3.x pa1.x 1 pa3.y pa1.y 1 pa3.z pa1.z; total 5 (acos(a1) 1 acos(a2) 1 acos(a3))  RTOD; if (ABS(total - 360) . EPS) return(FALSE); return(TRUE); }

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Computational Fluid-Structure Interaction

As it has been shown that the intersection will be returned if it exists. With the help of this information, the areas of the three sub-triangles (S1, S2 and S3 in Fig. 8.6) with the common vertex P will be calculated and stored for late use.

8.3 ONE-SIDED MUSCL-LIKE EXTRAPOLATION In the beginning of every physical time step, the fluid solver (TETRAKE) is executed to solve the flow equations in the the fluid domain. And during this stage, IMM is used to impose the boundary conditions across the fluidstructure interface. In this method, those fluid cells crossing the interface need special treatment. Take Fig. 8.7 for an example, where a fluid cell 1234 is cut by the interface. Nodes 1, 2 and 3 lie in the fluid domain while node 4 lies in the structural domain. As described in former chapters, convection fluxes are computed based on mesh edges. In the computation of the convection flux along edge 14, the flow conditions at node 1 and ghost conditions at node 4 (g14) will be involved. The ghost conditions of node 4 are determined as follows: 1. Identify the intersection point I1 between the edge 14 and the interface; 2. Identify the surface triangle in which intersection point I1 lies; 3. Determine the velocity u I of intersection point I1 using a area weighted scheme with the knowledge of velocities of node a, b and c; 4. Extrapolate the ghost velocity of node 4 using the flow condition of node 1 and the velocity of the intersection point I1. 5. Extrapolate the ghost pressure and density of node 4 using the known pressure, density and gradient of node 1.

1

2 Intersected fluid cell N

g

3

a b I1

h

d

c

I2

i I3

e

f

(g14, g24, g34)

4

Fluid – structure interface

FIGURE 8.7 Schematic of Immersed Membrane Method and one-sided extrapolation.

119

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8 (A)

(B) ur

u1

u2

u1

u1

uI ug14

I

ug14

4 (g14) r1I

II

I'

r4I

I

2 r1r

for 0 ≤⏐r4I⏐≤⏐r1I⏐

I1 r1I

4 (g14)

r4I

for⏐r1I⏐ > > u 2 u 5 2 jr j for 0 # jr4I j # jr1I j < 1 I 1I ð8:8Þ u 2 u jr g14 I 4I j > > j j j j for r 5 2 , r > 1I 4I : u2 2 uI jr2I j Therefore, 8 jr4I j > > u 52 Uðu1 2 uI Þ 1 uI > < g14 jr j

for 0 # jr4I j # jr1I j

1I

jr4I j > > > : ug14 5 2 jr2I j Uðu2 2 uI Þ 1 uI

for jr1I j , jr4I j

ð8:9Þ

where uI , u1 and ug14 are the velocity vectors at intersection point I1, node 1, ghost node g14. In Fig. 8.8(B), we have jr2I j 5 jr4I j and jr110 j 5 jr14 j. u2 is evaluated as following: u2 5

ðjr110 j 2 jr12 jÞu1 1 ðjr2I j 2 jr1I jÞu10 jr14 j

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Computational Fluid-Structure Interaction

where u10 5 u1 1 r41 Uru1 : In this work, the structure and fluid domains are coupled by enforcing the velocity continuity condition at the interface, us 5 uf

ð8:10Þ

Where us is the velocity vector of a structure surface node, uf is the neighbouring fluid velocity at the same location. And Equation (8.9) is used to extrapolate fluid velocity to its corresponding ghost nodes. The extrapolation of ghost-node pressure is illustrated in Fig. 8.8(C). Taking node 1 as real node, the ghost-node pressure and density of node 4 is obtained by pg14 5 p1 1 r14 Urp1 ρg14 5 ρ1 1 r14 Urρ1

ð8:11Þ

where pg14 and ρg14 are the ghost-node pressure and density at node 4, rp1 and rρ1 are the pressure and density gradients at node 1, r14 is the distance vector pointing from node 1 to node 4. Now all of the necessary ghost conditions for determining the convection flux along edge 14 are computed. Suppose edge 24 is the next edge along which the convection flux will be computed. We need to determine the ghost conditions for node 4 (g24) again with the knowledges of node 2 and intersection node I2. As a result, every node can hold multiple ghost nodes and thus multiple ghost values since a node can be connected by multiple edges, which is especially true for 3D unstructured grids. The selection of a particular ghost value depends on which edge and node the computation is for. This is a novel feature of the current implementation of unstructured IMM. This linear interpolation results in a second-order accurate scheme. It is noted that higher-order MUSCL interpolation can also be applied during the flux computations if higher order accuracy is required. As depicted in Fig. 8.9, edge 12 is cut by fluid-structure interface and I is the intersection point. C is the centre of edge 12. Local 3rd-order accuracy can be achieved by introducing uL and uR for the two sides of C:  1 1 ð1 2 κÞΔ2 1 1 ð1 1 κÞΔ1 4   uR 5 u1 1 ug12 =2

uL 5 u1 1

ð8:12Þ

where Δ1 1 5 ug12 2 u1 1 Δ2 1 5 2U12Uru1 2 ðug12 2 u1 Þ 5 2U12Uru1 2 Δ1

ð8:13Þ

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

uL = u1 +

121

1 [(1 – k)Δ–1 + (1 – k)Δ+1] 4 uR = (u1 + ug12 ) /2 u1

1

C

u1

ug12

1

2

FIGURE 8.9 Introduction of higher-order MUSCL like interpolation.

And ug12 is the ghost velocity vector of node 2, ru1 then velocity gradient at node 1. 12 is the vector representing the edge. When κ is set to 1/3, which corresponds to a nominally third-order accuracy. Pressure and density can be interpolated in a similar way. The viscous fluxes and gradients are computed based on fluid mesh cells. In the cell N in Fig. 8.7, when the viscous flux is computed for node 1, the flow conditions at nodes 1, 2 and 3 will be needed, as well as the ghost-node conditions at node 4 (It is g14 in this case. But it would be g24 if the viscous flux for node 2 is computed).

8.4 IMM BASED FLUID-STRUCTURE INTERACTION In previous chapters we have already developed a novel parallel unstructured multi-grid preconditioned compressible Navier-Stokes solver TETRAKE. Here we aim to extend it for fluid-structure interaction simulation. The biggest challenge here is how to couple the two modules and how to synchronize them. Our solution algorithm is shown in Fig. 8.10, from which it may be seen that different time stepping size between the fluid and the structure solvers are allowed. In the beginning of every physical time step, the fluid solver (TETRAKE) is executed to calculate flow in the fluid domain. And during this stage, IMM is used to impose the boundary conditions across the fluid-structure interface using the techniques introduced in the foregoing sections. After the computation of fluid domain, fluid forces acting on the structure are calculated along the fluid-structure interface and they are applied to advance the movement of the structure. In the structural domain, equation (4.1) is solved by the techniques described in former sections with the fluid forces exerted by ambient fluids as boundary conditions. Boundary conditions on the surface Γs of the structural domain Ωs are described in terms of prescribed traction ~ t P on the boundary Γt and prescribed displacement d~P on the boundary of the structure Γd . Just as described in Equation (4.8). At the fluid-structure interface the structure experiences a surface traction due to the fluid; hence the

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Computational Fluid-Structure Interaction

Start

Yes

Fluid step = Solid step?

No

Mesh partitioning & preprocessing for MG Compute fluid geometry

Initialize the velocities for fluid & structure domain

Compute structure geometry and build up the inter-connection between fluid and solid meshes

Compute local time step size for fluid solvers

Fluid solver sub-iteration

Update the solid mesh and Compute structure geometry Rebuild the inter-connection between fluid and solid meshes Yes No Converged ?

Compute displacement for current step

Compute nodal gradients Compute diffusive fluxes Exchange of nodal gradients Compute convective fluxes Exchange of flow field variables

Compute stress & strain Update the solid mesh and Compute structure geometry

Converged ? No

Treatment for freshly cleared fluid nodes

Structure dynamics solver sub-iteration

Feedback solid position & velocity

Read initial fluid and structure geometry

Yes

Fluid solver Computational fluid dynamics (CFD)

Calculate traction boundary condition for the solid module

Structure solver Computational structure dynamics (CSD)

FIGURE 8.10 Schematic of our FSI solution.

appropriate boundary condition for the structure at the fluid-structure interface is expressed as:

@ui @uj 2 1 tp 5 2 pδij 1 μ 2 μrUU f δij on Γt ; ði; j 5 1; 2; 3Þ ð8:14Þ 3 @xj @xi where μ is the dynamic viscosity, p the pressure, and ui the Cartesian component of fluid velocity U f corresponding to direction xi. In IMM, the loosely coupling relationship between the fluid domain and the structure domain is depicted in Fig. 8.11. The fluid forces include pressure p, shear stress σt and normal stress σn , Since the fluid mesh and structural are non-confirmatory, the fluid pressure needs to be extrapolated to the fluid-structure interface for the computation of fluid forces. And again this extrapolation is edge-based. In Fig. 8.7, I3 is the intersection point between edge 34 and the interface, and we

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

Fluid Domain

NS equations CFD

Fluid forces (pressure, shear and normal stress)

Velocity continuity ux = uf

123

Structure Domain Cauchy’s equations CSM

FIGURE 8.11 Relation between the structure domain and the fluid domain in fluid-structure interaction.

need to extrapolate the pressure of this point to calculate the fluid force. This extrapolation procedure can be expressed as following: pI3 5 p3 1 r3I Urp3

ð8:15Þ

where pI3 is the fluid pressure on the fluid-structure interface on the side of vertex 3, rp3 are the pressure gradient at vertex 3, r3I is the distance vector pointing from node 3 to point I3. This linear extrapolation leads to a second order accuracy. Note that the higher-order MUSCL scheme similar to equation (8.12 and 8.13) can also apply here. And then the resultant pressure will be distributed to the structure nodes d, e and f based on an area weighted scheme. For Newtonian fluids, the fluid stress tensor is given as follows: 2 3 σxx σxy σxz ð8:16Þ t 5 4 σyx σyy σyz 5 σzx σzy σzz and





@u 2 @u @v @w @v @u σxx 5 2μ 2 μ 1 1 1 ; σxy 5 σyx 5 μ @x 3 @x @y @z @x @y



@v 2 @u @v @w @u @w 1 1 1 σyy 5 2μ 2 μ ; σxz 5 σzx 5 μ @y 3 @x @y @z @z @x



@w 2 @u @v @w @w @v 2 μ 1 1 1 σzz 5 2μ ; σyz 5 σzy 5 μ @z 3 @x @y @z @y @z

ð8:17Þ

where the molecular viscosity μ is property of the fluid and is function of temperature. So the fluid stress σ can be calculated as the following: 2 3 2 3 σxx σxy σxz nx σ 5 tUn 5 4 σyx σyy σyz 5U4 ny 5 ð8:18Þ σzx σzy σzz nz

124

Computational Fluid-Structure Interaction

where n is the unit normal vector of the structure element. In the fluid solver, the velocity gradients are assumed to be constant within one fluid cell. It means that on the fluid-structure interface the gradients of fluid velocities are equal to those at the cell centroids. But for a cell intersected by the immersed structure, the gradients are only uniform and continuous for centroids on the same side of the structure. So the computations of fluid stresses on the interface should use the right gradients of velocity. Use edge 34 in Fig. 8.7 for example, the fluid stresses at intersection point I3 should be computed using the gradients of vertex 3, i.e.: 2 3 2 3 σ3;xx σ3;xy σ3;xz nx σ3 5 t3 Un 5 4 σ3;yx σ3;yy σ3;yz 5U4 ny 5 ð8:19Þ σ3;zx σ3;zy σ3;zz nz As can be seen from Fig. 8.10, there exists a special step at the end of each time step of structure computation, or at the beginning of each time step of fluid computation, namely ‘treatment of freshly cleared fluid nodes’. In some Immersed Boundary Methods where the immersed boundaries are treated as sharp interfaces, such as those presented by Bayyuk et al. [3], Yang [4] and Udaykumar et al. [5], the issue of change of material needs to be addressed. This arises when a computational point (as in Fig. 8.12), which was in the solid during one time step, emerges into the fluid at the next time step. In [4], because the fractional time advancement scheme is used, the evaluation of the RHS of the momentum equation at step k 1 1 requires physical values of the velocity vector and pressure, as well as their derivatives from step k at all fluid points. Due to the fact that the interface changes locations, it is possible that some of the required values from step k are not physical. To avoid forbiddingly complex special treatments of the computation of these derivatives every time such a change of flag is detected during the time advancement procedure, they propose a field-extension procedure, in which the velocity and pressure fields are “extended” in the solid phase at the end of each step. Practically, with this procedure the velocity and pressure fields are extrapolated at the Eulerian nodes in the solid that have at least one neighbor in the fluid phase. In the cut-cell formulation of Udaykumar et al. [5] this issue of freshly cleared cells was addressed using a cell-merging formulation in conjunction with quadratic interpolation among neighboring grid nodes in the fluid. In the current method, an approach similar to the field-extension procedure used in the finite-difference method of Yang [4] is employed. It is comprised of several basic substeps as described as follows. In the first substep, all of the computational nodes which locate inside the solid domain are successively ‘leveled’. The level tagging process is depicted in Fig. 8.12. All ghost nodes are tagged as ‘level 1’ nodes, which are grid points in the solid phase that have one or more neighboring points in the fluid phase (neighboring here means connected by an edge). ‘Level 2’

The Immersed Membrane Method and Fluid-Structure Interaction Chapter | 8

125

Interface at time k Interface at time k + 1 Neighboring fluid nodes Level 1 nodes Level 2 nodes Level 3 nodes Field extension direction

FIGURE 8.12 A two-dimensional example that demonstrates the possible changes of flags of the grid nodes near a moving interface. The classification of the various nodes is according to the notation introduced in the text and is based on the location of the body at time k. In this special case, all of the ‘level 1’ nodes (ghost nodes) will emerge into the fluid at time k 1 1, and all of the ‘level 2’ nodes will become ‘level 1’ nodes at time k 1 1 and so on.

nodes are grid points in the solid that have one or more ghost nodes as their neighboring points. ‘Level 3’ nodes are grid points in the solid that have one or more ‘Level 2’ nodes as their neighboring points (except ‘level 1’ points). The rest may be deduced by analogy. In the second substep of our approach, for every ‘level 1’ point, the labels of the neighboring points in the fluid phase are stored, as well as the lengths of the edges connecting the current point and the neighboring points in the fluid phase. For every ‘level 2’ point, the labels of the neighboring ‘level 1’ points are stored, as well as the lengths of the edges connecting the current point and the neighboring ‘level 1’ points. The same treatment will be performed for every point of successively levels. Finally, the velocity and pressure fields are “extended” in the solid through the levels. The inverse distance weighted method proposed by [6] is adopted in this study to perform the “field extension” task. This method has the property of preserving local maxima and producing smooth reconstruction. The interpolation at a certain location (x, y, z) is

126

Computational Fluid-Structure Interaction

φðx; y; zÞ 5

n X wm φm =q m51



p R2hm wm 5 Rhm

p n X R2hl q5 Rhl l51

ð8:20Þ

where φm represents the solution at a certain location, wm the weight, and hm the distance between the location (x, y, z) and the location of φm . R represents the maximum of all hm; p is a constant and normally set to 2. The extension process can be described as follows. The velocity and pressure fields of the fluid nodes neighboring to the solid interface are interpolated to the “Level 1” points using Equations (8.20). And then the velocity and pressure fields of the “Level 1” points are interpolated to the “Level 2” points using Equations (8.20) again and so on for successively levels. Generally, 3level extension is enough, provided that the CFL number has not been set to an extraordinarily large value. As a result, not only the velocity and pressure at the newly emerging points, but also their derivatives will have physical values, eliminating problems in the computation of the RHS in the next time step for these points.

REFERENCES [1] G.H. Xia, A three-dimensional computation of fluid-structure interaction in bioprosthetic heart valves using immersed membrane method, PhD Thesis, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 2005. [2] Y. Zhao, C.H. Tai, Parallel Computation of Unsteady Incompressible Viscous Flows using an Unstructured Multigrid Method, ASME Forum on Parallel Computing Methods, Paper No. IMECE2002-FE-34388, Proceedings of IMECE’02, 2002, ASME International Mechanical Engineering Congress & Exposition, New Orleans, Louisiana, November 1722, 2002. [3] S.A. Bayyuk, K.G. Powell, B. van Leer, A Simulation Technique for 2-D Unsteady Inviscid Flows around Arbitrarily Moving and Deforming Bodies of Arbitrary Geometry, Technical Paper 93-3391-CP, AIAA Press, Washington DC, 1993. [4] J. Yang, E. Balaras, An Embedded-Boundary Formulation for Large-Eddy Simulation of Turbulent Flows Interacting with Moving Boundaries, J. Comput. Phys. 215 (1) (2006) 1240. [5] H.S. Udaykumar, R. Mittal, P. Rampunggoon, A. Khanna, A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. Comput. Phys. 174 (2001) 345380. [6] R. Franke, Scattered data interpolation: Tests of some methods, Math. Computer 38 (1982) 181200.

Chapter 9

Arbitrary Lagrangian
Eulerian (ALE) Method and Fluid-Structure Interaction 9.1 GOVERNING EQUATIONS The arbitrary Lagrangian
Eulerian (ALE) method, first proposed by Donea et al. [1], involves a continuous adaptation of the mesh without modifying the mesh topology in solving the fluid
structure interaction and moving boundary problem. The essence of the ALE is that the mesh motion can be chosen arbitrarily [2]. It includes three phases: an explicit Lagrangian update, an implicit iteration of the momentum equation, and a rezone/map phase [3]. A particularly important property of ALE is that it provides a means of minimizing advection errors [4]. ALE techniques have gained popularity for transient, high speed, small deformation problems [5]. The incompressible unsteady Navier
Stokes governing equations, modified by the artificial compressibility method (ACM) dual-time-steps and ALE in nondimensional integral vector form, are as follows: ð ð I @ @ C W dV 1 K W dV 1 rUðFc 2 Fv Þ dS 5 0 ð9:1Þ @τ cv @t cv scv where

3 0 7 6 1 6 UrUu 7 7 6 Re 7 6 7 6 7 6 1 6 Fv 5 6 UrUv 7 7 7 6 Re 7 6 7 6 7 6 1 4 UrUw 5 Re 2

2 3 p 6u7 6 W54 7 ; v5 w

2

U

6 6 uU 1 pδij Fc 5 6 6 vU 1 pδ ij 4 wU 1 pδij

3 7 7 7; 7 5

Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00009-X © 2019 Elsevier Inc. All rights reserved.

127

128

Computational Fluid-Structure Interaction

2

0 0 6 60 1 K56 60 0 4

0

0 0

0

0 1

0

3

7 07 7; 07 5 1

3

2

1 6β 6 6 C56 60 6 40 0

0

0

0

1

0

0

1

7 7 7 07 7 7 05

0

0

1

where W is the vector of dependent variables and U is the velocity vector; Fc and Fv are the convective flux and viscous flux vectors, respectively; β is the artificial compressibility parameter as above and K is the unit matrix, except that its first element is zero; and C is a preconditioning matrix. It is noted that Eq. (9.1) are the same as Eq. (3.18), the governing equations for incompressible flow given in Chapter 3, Mathematical Formulation for Incompressible Flow  T Solver, but the velocity vector, U 5 Uf 2 U g , where Uf 5 u v w and U g are the fluid velocity and control volume surface velocity, respectively. If U g 5 0, Eq. (9.1) are Eulerian. But when U 5 U f , Eq. (9.1) becomes Lagrangian. In order to appreciate the difference between Eqs. (9.1) and (3.18), Eq. (9.1) are rearranged as follows: ð ð I @ @ W dV 1 K W dV 1 rUðFCC 2 FCG 2 FV ÞdS 5 0 ð9:2Þ C @τ cv @t cv scv 3 2 Ug 7 6 6 uU g 7 7 6 FCG 5 6 7 4 vU g 5 wUg where the term of convective flux Fc in Eq. (9.1) is expanded into two components, Fcc Fcg in which the effect of mesh movement is described as Fcg . Eq. (9.1) without Fcg are the Eulerian equations and are exactly the same as Eq. (3.18).

9.2 DISCRETIZATION METHODS Eq. (9.2) could be considered as Eq. (3.18) with one extra term added. Therefore, the discretization methods for Eq. (3.18), which are introduced in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, could be used again and only the discretization method for the term FCG and mesh movement algorithm are introduced in this section.

Arbitrary Lagrangian
Eulerian (ALE) Method Chapter | 9

129

9.2.1 Edge-Based Method for Term F CG The extra term FCG in Eq. (9.2) is transformed into a summation: I I nbseg X  rUFCG dS 5 FCG UndS 5 ðFCG Þij UnΔS k Scv

Scv

ð9:3Þ

k51

where nbseg is the number of the edges associated with node P, ðFCG Þkij is the new flux related to mesh movement through the part of control volume surface associated with edge k, and n is the unit normal vector of the control volume surface. ΔSk is a part of the control volume surface associated with edge k (as shown in Fig. 3.1, if k is edge PC, then 1O2c is ΔSk ). Therefore, all the fluxes are calculated for the edges and then collected at the two ends of each edge for updating of flow variables in time marching, which is the so-called edge-based data structure for fast inviscid flux computation and efficient data storage and retrieval. The new flux term FCG is linear because U g in each physical and pseudo time-step is known and kept constant. Together with the inviscid flux discretization method, the implicit scheme is applied for the discretization of the flux term FCG in which the flow variable appearing in the FCG is U n11;m11 .

9.2.2 Moving Mesh Algorithm in ALE After the wall boundary is moved, the mesh is deformed. The solution of the governing Navier
Stokes equations requires a new grid with updated nodal coordinates and the corresponding nodal speed at every time-step. Regeneration of the grid is not economical and too time consuming. For small displacement problems, the coordinates of the deformed mesh can be updated by algebraic methods while keeping the connectivity of the nodes unchanged. In this section, a novel mesh movement algorithm [6] is introduced in detail. In the present solver, the shortest distance d(is) of every inner node (is) to solid walls is calculated and the closest wall node (iswall) is identified. A maximum of all d(is) is determined for nondimensionalization (d max). Then the displacement δ of the node is obtained as the product of a distance function and the displacement δ of the associated wall node Fig. 9.1 shows the flow chart for the moving mesh algorithm. δrðisÞ 5 f ðisÞδrðiswallÞ

ð9:4Þ

A general distance function is constructed: f ðisÞ 5

ly2 ðisÞ lx2 ðisÞ 1 ly2 ðisÞ

ð9:5Þ

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Computational Fluid-Structure Interaction

Find the neighbors of each wall node Calculate the forces on each moving wall node

Calculate the displacement speed and distance of each moving wall node Smooth the displacement speed and distance of each moving wall node

Calculate the displacements of each inner node according to the nearest wall node Update the cell shapes using the spring analogy method FIGURE 9.1 Flowchart for the moving mesh algorithm.

By using two exponential damping functions: lxðisÞ 5

1 2 e2ðdðisÞ=dmaxÞ ðe 2 1Þ=e

and

lyðisÞ 5

1 2 e12ðdðisÞ=dmaxÞ 12e

ð9:6Þ

The function f(is) tends to 1 when d tends to 0 and it tends to 0 when d tends to d max. This makes the grid near the wall and far away from the wall very rigid, while those parts in between are elastic and easy to be deformed.

9.2.3 Spring Analogy Method In order to further improve the robustness of this method for large mesh deformation simulation, a smoothing procedure is implemented in the present solver to completely eliminate skewed or overlapping cells. The displacement of a node P is smoothed using the displacements of its surrounding nodes ({{{Fig. 9.2}}}). numedgeðPÞ P

δrP 5

Pi δri

i51 numedgeðPÞ P i51

ð9:7Þ Pi

Arbitrary Lagrangian
Eulerian (ALE) Method Chapter | 9

131

δr11 1

2

δr12

d1

δr1 =

P δrP 3

Pi = 5

1 2

(δr11 + δr12)

1 di

4 FIGURE 9.2 Smoothing of node displacement.

where, numedge(P) is the number of edges which surround node P. Pi is the inverse of the distance from node P to edge i and the denominator in the above expression is the sum of the Pi for node P. δri is the average displacement of edge i which can be considered as average of the displacements of the two nodes at its ends.

9.3 PARALLELIZATION OF THE ALE CODE WITH MOVING BOUNDARY The parallelization of the ALE solver consists of three steps. The first step is to calculate the displacements and velocities of the boundary nodes, considering the forces illustrated in Fig. 9.3 and smoothing their displacements by taking into account the geometric continuity of the neighboring parts of the solid. The second step is to obtain the inner node displacements relatively to the boundary movement as described in Section 9.2.2. Finally, the inner mesh is restructured in order to prevent excessive deformations of the cells for faster convergence. Therefore the main efforts in parallelization are to create a consistent movement of the boundary, so that the inner mesh will not be excessively distorted as a consequence.

9.3.1 Moving Boundary 9.3.1.1 Calculation of Forces on the Wall Boundary The forces on each wall node consist of the forces exerted by the fluid and those by the wall boundary. The latter are obtained by the determination of the tension between the present wall node and its left and right neighbors. The deviation of the tension from the state before the movement of the wall is calculated by comparing the current boundary normal vector of each node with its initial boundary normal vector before wall deformation. Therefore

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Computational Fluid-Structure Interaction

Find the neighbors of each wall node Find the neighboring wall ghost nodes at the partition boundaries Calculate the forces on each moving wall node Calculate the displacement speed and distance of each moving wall node

Subroutine smoothmov.f

Calculate the displaements of each inner node according to the nearest wall node

{

Gather the displacement speeds and distances of the moving nodes in a master process

Smooth the displacement speed and distance of each moving wall node Distribute the smoothened displacement speeds distances to the partitions

Update the cell shapes using the spring analogy method FIGURE 9.3 Flowchart for the parallel moving mesh algorithm.

the boundary normal vectors have to be exchanged between the core and ghost nodes at the partition boundaries.

9.3.1.2 Movement of Partition Boundaries Up to this point, the parallel computation of the flow field and boundary movement depends on the successful exchange of fluid solution vectors, nodal gradients, and boundary normal vectors, as well as the synchronous calculation of deformation speeds in each partition. In order to smooth this movement, a loop over all boundary nodes has to be run in each partition. It is necessary to know the smoothed deformation of the previous node in the loop. Thus, a discontinuity occurs at the partition borders as shown in Fig. 9.4 because it is impossible to enforce this condition in a concurrent

Arbitrary Lagrangian
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FIGURE 9.4 Problem: discontinuity of geometry during smoothing process.

parallel simulation. During smoothing, this discontinuity disappears but the overall movement is different from the serial solution. There are two possible options to solve this problem. Increasing the number of subiterations definitely removes this discontinuity because the moving boundary algorithm is convergent; the main disadvantage is that the convergence speed is relatively slow and the parallel speedup is decreased dramatically. The second option is transferring the initial deformation of the wall nodes to a so-called master processor that runs the smoothing algorithm on all of the moving wall nodes. Actually, this specific parallel problem is reduced to a serial problem using this method. The disadvantage here is that CPU time is wasted for all the used CPUs except the master processor during this time period. Nevertheless the latter solution is chosen in this work due to less time loss and better consistency compared to the first solution. In order to create an easy and efficient storage system for the values that have to be transmitted to the master processor, the wall nodes are globally numbered in the subroutine “addghost.f.” The size of the arrays storing the values in the master process is obtained by adding the local numbers of wall nodes. On the one hand, this method creates lots of empty spaces in the arrays and wastes memory space, since normally only a small percentage of wall nodes are actually moved. On the other hand, sorting the moving wall nodes into their global order in order to achieve best conformity to the serial code will slow down the program even more. The variables required to be passed are the global wall node number, the displacement and speed of the node, and the global wall node number of its left and right neighbors.

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TABLE 9.1 Information Stored in the Master Process mlogbndry

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Appendix 9.6 shows the source code of the algorithm that was developed for parallel moving boundary smoothing. The number of moving nodes of each partition (nfirst) and the partitions containing moving nodes (array masterwho) are determined during the calculation of the forces and initial movement in another subroutine and stored in an array. The master processor is chosen in a loop over the combined array masterhe by chosing the first available process without moving boundary nodes; if every partition contains moving boundary nodes, the first process is chosen. Table 9.1 shows the information for each moving wall node that is stored in the master process in order to perform the grid smoothing. After smoothing, the new deformation speeds and distances are transmitted to the wall ghost nodes.

9.3.2 Moving Grid After calculating the displacement speeds of the boundaries within each subiteration, at the end of each time-step, the total displacement for the boundary nodes is calculated and the cell shapes are updated according to the spring analogy method. Before moving the whole mesh, the displacement of the boundary nodes has to be synchronized with the ghost nodes of the neighboring partitions to avoid intersections of elements’ edges.

REFERENCES [1] J. Donea, S. Giuliani, J.P. Halleux, An arbitrary Lagrangian
Eulerian finite element method for transient dynamic fluid
structure interactions, Comput. Methods Appl. Mech. Eng. 33 (1982) 689
723. [2] L.G. Margolin, Introduction to an arbitrary Lagrangian
Eulerian computing method for all flow speeds, J. Comput. Phys. 135 (1997) 198
202. [3] C.W. Hirt, A.A. Amsden, J.L. Cook, An arbitrary Lagrangian
Eulerian computing method for all flow speeds, J. Comput. Phys. 135 (1997) 203
216. [4] J.U. Brackbill, W.E. Pracht, An implicit, almost-Lagrangian algorithm for magnetohydrogynamics, J. Comput. Phys. 13 (1973) 455. [5] D.J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng. 99 (1992) 235
394. [6] Y. Zhao, A. Ahmed Forhard, General method for simulation of fluid flows with moving and compliant boundaries using unstructured grids, Comput. Methods Appl. Mech. Eng. (United States) 192 (39
40) (2003) 4439
4466.

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APPENDIX Appendices. Fortran 90 Source Codes A9.1 Code for Creating Overlapping Elements (addghost.f)

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A9.2 Code for Creating Ghost Nodes (addghost.f)

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A9.4 Code for Creating Communication Data (addghost.f)

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A9.5 Code for Example Communication Module (communicate.f)

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A9.6 Code for Parallel boundary movement smoothing (smoothmov.f)

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Chapter 10

IOM FSI Model Validations and Applications 10.1 SERIAL/PARALLEL SINGLE GRID AND MULTIGRID COMPUTATIONS 10.1.1 3D Lid-Driven Cavity Steady Flow The lid-driven cavity flow, a classic recirculating flow, is an idealization of many environmental, geophysical, and industrial flows. It is a typical benchmark problem for solvers of the incompressible Navier
Stokes equations and has been widely used for validation of numerical schemes and algorithms. This problem is used to validate the steady computation for serial and parallel incompressible solvers using both SG and MG, introduced in this book. The choice of this problem is prompted by numerous experimental observations by Koseff and Street [1
3] and numerical computations by Jiang et al. [4]. In the open literature, most of the researchers have claimed that steady-state solutions could still be obtained and no Taylor-Go¨rtler-like (TGL) vortices for Re , 2000 were found. The number and location of these vortices were functions of Reynolds number, spanwise-aspect-ratio, and time. Fujima et al. [5] observed a pair of TGL vortices at Re 5 2000, but Jiang et al. [4] found corner and TGL vortices at the bottom region of the cavity at Re 5 1000 and the flow in the cavity was not stable as the appearance of the TGL vortices was consistent with this instability. In this work, the flow is confined in a unity cubic domain with the upper wall (z 5 1) moving at a constant speed of u 5 1. A grid with 97,336 nodes and 455,625 tetrahedral elements is used with grid clustering near the five cavity walls, which is shown in Fig. 10.1. The velocity components u 5 1, v 5 0, and w 5 0 in the x, y, and z direction, respectively, are specified on the top driven surface (z 5 1) and u 5 v 5 w 5 0 on all solid walls. The initial velocity in the flow field domain is set to zero everywhere except on the lid surface. The Reynolds numbers specified in the computations are 400 and 1000. The pseudo-compressibility coefficient β is set to 1.0 for faster convergence, CFL is set to 0.8 and the physical time-step is set to 0.02 and 0.01 for Re 5 400 and 1000, respectively. The third-order characteristics upwind scheme is used in all computations. The dual-time-stepping is used so as to take advantage of Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00010-6 © 2019 Elsevier Inc. All rights reserved.

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FIGURE 10.1 Geometry for the lid-driven cavity flow.

the faster convergence rate and the number of pseudo subiterations per timestep is set to 20. Since one pseudo subiteration is equivalent to 1 MG cycle, this leads to 20 MG cycles per time-step. The computational results are compared with the numerical results of Jiang et al. [4].

10.1.1.1 Grid Convergence Study The accuracy of numerical simulation is greatly dependent on the grid size employed, and therefore grid convergence study is one of the essential studies in any of the computational fluid dynamics (CFD) simulations. The main purpose of grid convergence study is to obtain an optimum grid size so that the accuracy of the solution is independent of the grid. The grid size and computational time are both related to each other; therefore increasing the mesh density to give better results will increase computational time. Hence, to minimize computational time but not at the expense of the accuracy of the solution, grid convergence study is a necessary test. In this study, three different grid densities of unstructured tetrahedral grids are generated, which have 32,768, 97,336, and 125,000 nodes, respectively. All the parameters for the three simulation runs are kept exactly the same and run until it converges for the three cases. The velocity profiles of u component on the vertical centerline and w component on the horizontal centerline of the symmetry plane y 5 0.5 for the three different grid densities are shown in Fig. 10.2. From the figure, it can be seen that both the u and w component velocity profiles for 32,768 nodes deviate from the other two finer grids marginally. The reason of this deviation is that the mesh nodes along both the vertical and horizontal centerlines are not dense enough to well capture the trend of the velocity profiles along these axes centerline. The u and w component velocity profiles for both grid size of 97,336 and 125,000 nodes coincide with each other as shown in Fig. 10.2 and it shows that the numerical solution will not change even for a mesh denser than

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125,000 nodes. A converged solution with a reasonable accuracy can be obtained using a grid size of 97,336 nodes and hence, the computational time is optimized in the present work.

10.1.1.2 Serial Single Grid and Multigrid Computations The benchmark lid-driven cavity flow problem is used for validation of the 3D serial SG and MG methods developed in this work. In this work, the three-level MG method makes use of a sequence of independently generated unstructured tetrahedral grids, as shown in Fig. 10.3. The fine grid consists of 97,336 nodes and 455,625 elements. Whereas, the coarse grid consists of 24,389 nodes and 109,760 elements, and the coarsest grid consists of 5832 nodes and 24,565 elements. The developed algorithm for inter-connectivity relationship between meshes is validated by Tai and Zhao [6
8] for 2D MG method.

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FIGURE 10.3 A sequence of independently generated unstructured tetrahedral grids for threelevel MG computations. (A) 97,336 nodes and 455,625 elements; (B) 24,389 nodes and 109,760 elements; (C) 5832 nodes and 24,565 elements.

The velocity profiles of u component on the vertical centerline and w component on the horizontal centerline of the symmetry plane y 5 0.5 for Re 5 400 and 1000 are shown in Figs. 10.4 and 10.5, respectively. The computed velocity profiles are compared with the numerical solutions obtained by Jiang et al. [4]. From the figures, it can be seen that the velocity profiles obtained by the MG solver produce quite a good agreement with the profiles of Jiang et al., as well as yielding excellent agreement with the SG. This shows that the MG method will not change the single grids’ (SGs) results, but only accelerate the convergence of the solution. Figs. 10.6 and 10.7 show the steady velocity vectors on three mid-planes: (A) x 5 0.5, (B) y 5 0.5, and (C) z 5 0.5 for Re 5 400 and 1000, respectively, using the MG method. The velocity vectors obtained for both Reynolds numbers are consistent and in good agreement with the results of Jiang et al. and other researchers [5
10]. It can be observed that at the mid-planes y 5 0.5, as shown in Fig. 10.6B and 10.7B, the secondary vortices appear in the two lower corners and the primary vortex moves toward the center of the cube as the Reynolds numbers increases. This phenomenon is similar to that in the 2D lid-driven square cavity, but there does not exist a secondary vortex near the left upper corner as reported by Fujima et al. [5] and Yang et al. [10]. In Fig. 10.6A and C, it can be observed that a pair of primary contra-rotating

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FIGURE 10.4 Velocity profiles of u on the vertical centerline and w on the horizontal centerline of the symmetry plane y 5 0.5 for Re 5 400.

FIGURE 10.5 Velocity profiles of u on the vertical centerline and w on the horizontal centerline of the symmetry plane y 5 0.5 for Re 5 1000.

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FIGURE 10.6 Velocity vectors for Re 5 400 on the mid-planes (A) x 5 0.5, (B) y 5 0.5, and (C) z 5 0.5.

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vortices near the upstream wall and near the bottom wall, respectively. Those pairs of primary and secondary vortices strengthen with increasing Reynolds number and become more distinctive at Re 5 1000, as shown in Fig. 10.7A and C. Since the main purpose of the MG method is to accelerate the convergence rate, a more realistic way of comparing the results obtained is to make a quantitative comparison between the SG and the MG computations in term of the order of magnitude reduction in residual versus number of time-steps as shown in Figs. 10.8 and 10.9 for both Re 5 400 and 1000, respectively. The figures show the convergence history plots for both SG and three-level MG (V- and W-cycles) computations. Fig. 10.8 shows that the flow with Re 5 400 takes 1200 time-steps for SG to reduce the residual by an order magnitude of five, 700 and 400 time-steps for the three-level MG V- and Wcycles, respectively, to converge to the same level of magnitude. Similarly, for Re 5 1000, it takes 2000 time-steps for SG to reduce the residual by an order magnitude of 5, 1000, and 700 time-steps for the three-level MG Vand W-cycles, respectively, to converge to the same level of magnitude. It

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indicates that the SG takes more physical time-steps to reduce the residual to the same order of magnitude. It is seen that there is a significant reduction of time-step iterations required for the MG method to compute the same flow problem as compared to the SG. The MG method shows a reduction of

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42% and 67% for Re 5 400, and 50% and 65% for Re 5 1000 in physical time-steps for both V- and W-cycles, respectively, over the SG. It can be seen that the MG method did accelerate the convergence rate quite significantly in this work. From the residual plots, the initial convergence rates for the MG method are very fast as compared to the SG, but the convergence rates start to decelerate after a certain period of time-steps. The main reason that causes this phenomenon is that the simulations result in a complex recirculating flow consisting of 3D vortex structures and the low-frequency errors are not easily damped out. This phenomenon is consistent with observation reported by Montero et al. [9] as follows: “The driven-cavity problem is a rotating flow for which standard multigrid schemes might have difficulties converging.” Figs. 10.10 and 10.11 show convergence history plots in term of CPU time for both SG and the MG method for different Reynolds numbers. It can be seen that there is a significant reduction in CPU time using the MG method. It takes 3500 minutes and 6000 minutes of CPU time for the SG of Re 5 400 and 1000, respectively, to reduce the residual by an order of five. As for the three-level MG method with Re 5 400, it takes 2500 minutes of CPU time for V-cycle and 1450 minutes of CPU time for W-cycle to converge to the same order of magnitude as the SG. Similarly, for Re 5 1000, it takes 3600 minutes of CPU time for V-cycle and 2580 minutes for W-cycle. The computational time obtained for the MG method shows that a reduction of 29% and 59% for Re 5 400, and 40% and 57% for Re 5 1000 in CPU

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time for both V- and W-cycles, respectively, over the SG. The percentage reduction in term of CPU time for the MG method is not that significant as compared to the SG, as it might be due to the 3D vortex structures phenomenon in the recirculating flow which limits the convergence rate of the MG method. The CPU time of W-cycle is found to be less than that of the Vcycle on a three-level of MG, because W-cycle performs lesser operations on the coarser grid. The acceleration gains from much iteration on coarse grids in the W-cycle are actually worthwhile because of the penalty from the additional operations for V-cycle becomes significantly longer. It can be concluded that the MG method is a fast and efficient technique to increase the converge rate than the SG. In a purely sequential environment, V- and W-cycles require approximately 75% and 90%, respectively, more CPU time than an SG. However, both MG strategies provide close to an order of magnitude increase in convergence rate, thus greatly outweighing their increased cost per cycle.

10.1.1.3 Parallel Single Grid and Multigrid Computations The lid-driven cavity flow problem as stated in Section 10.1 is used to validate the 3D parallel-SG and parallel-MG solvers based on geometric domain decomposition technique using the Single Program Multiple Data programming paradigm. The computational results are compared with those obtained in Section 10.1.2. The cavity grids shown in Fig. 10.3 are decomposed accordingly to the required partitions for computations, as reflected in

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Fig. 5.4. The number of nodes per partition is quite well spread across all the processors after performing mesh partitioning using the developed algorithms depicted in Section 5.2 for the SG and Section 5.3 for the MG method. In this work, the fine grid partition is used to infer the coarse level partitions for the MG method. This partitioning process eliminates the need for communication between processors for different grid levels during transferring of variables and residuals and interpolation of corrections. The only communication needed to perform once on the coarse grids is after transferring of flow field variables and residuals. The number of nodes per partition for the coarse and coarsest levels of grids is reasonably well balanced. This will result in a more efficient communication between the grid above the coarse and coarsest grids. The performance of different number of partitions for both parallel SG and MG is based on the speedup characteristics, efficiency of parallelization, comparison between percentage computation, and communication time, as shown in Figs. 10.12
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FIGURE 10.13 Total simulation wall-clock time for different number of processors: (A) SG method and (B) MG method (Re 5 400).

and MG, and a high degree of parallelism is achieved. In this work, both CPU time and wall-clock time are used to record the total simulation time. The main difference is that CPU time is the recorded time when only the processor performs a calculation, whereas wall-clock time includes idling

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FIGURE 10.14 Comparison between percentage computation and communication time for (A) Re 5 400 and (B) Re 5 1000.

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time when the processor idles while waiting for other processors to communicate. The wall-clock time is used to represent the total simulation time since it includes the idling time, computation time, and communication time, which is the true representation of the simulation time. In this work, communication time is the time spent on exchange of flow field variables, nodal gradients, and forcing terms. The speedup and efficiency shown in the figures are computed according to Eqs. (5.1) and (5.6), respectively. A perfect speedup is when the speedup is equal to the specified number of processors required to accomplish a simulation and all the speedups obtained in this work are compared to this perfect speedup. The speedup plots as presented in Fig. 10.12 show that the speedup increases almost linearly up to eight processors and after which; it starts to deviate away from the perfect speedup. This is to be expected, since the relative ratio of communication to computation is higher as the number of processors increases and inevitably the speedup declines. But the simulation time decreases significantly as the number of processors increases and the time taken to complete the simulation decreases, as shown in Fig. 10.13 for both SG and MG methods. From the speedup plots, it can be seen that the speedup is not that significant for the MG method as compared to the SG. This is mainly due to the additional overheads of transferring variables to the coarser grids, which increases the communication time. Although the speedup is lower for the MG method as compared to the SG due to additional overheads, the overall simulation time required to compute the same problem by parallel MG is much less than parallel SG. For instance, the simulation time using eight processors needed by the MG method is 393.88 minutes as compared to 501.37 minutes required by the SG, as shown in Fig. 10.13 for Re 5 400. From Fig. 10.12, it can be seen that the computation is scalable up to 32 processors since the gradient of the graph of speedup against the number of processors is positive. That is, if more processors are used then the run time will reduce. For a fixed problem, as the number of processors increases, the computation time on each processor decreases while the communication time increases. It is mainly due to this change in the ratio of calculation to communication that leads to the drop in speedup as the number of processors increases. The efficiency declines steadily as the number of processors increases, as shown in Fig. 10.12B and D. This is to be expected, since the speedup deviates away from the perfect speedup as the number of processors increase, as shown in Fig. 10.12A and C. Although the computation time has reduced significantly running on 32 processors, it is observed that the number of processors to achieve an optimum speedup and efficiency is 16 processors for both SG and MG methods. This is because the number of users and their load varied during the simulation and by using fewer processors may even produce better efficiency out of the machine. One of the reasons why speedup and efficiency deteriorate as the number of processors increases is the increase in communication time.

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Fig. 10.14 shows the comparison between percentage computation and communication time for the respective number of processors. The lines representing both the percentage computation time and communication time are moving toward each other as the number of processors increases. This is to be expected, since computation time and communication time will decrease and increase, respectively, inversely proportional to each other as the number of processors increases. The figure shows that the percentage communication time is much lower for parallel SG as compared to parallel MG running, because no additional coarse grid sweeps are performed on the SG code. This also explains why SG computation has much higher parallel efficiency. The communication time is extremely high for the MG method running on 32 processors which accounts for almost 80% of the total simulation time and the rest is spent on computation for Re 5 400, as indicated in Fig. 10.14A. This increase in communication overhead is mainly due to the increasing number of adjacent subdomains, which increases the number of messages that requires transmission. Similarly, increasing the number of partitions will definitely decrease the computation time due to the fact that the load on each processor is much smaller. A good mesh partition is one, which divides the computational load equally among the subdomains and minimizes the amount of communication required between subdomains. To achieve this, the perimeter interfaces for messages transmission between subdomains should be as short as possible, thus reducing the communication overhead. It is also important to have some degrees of balance in the communication. Especially there should not be one subdomain interface that is unduly larger than the average. And any exceptionally larger interface will delay the overall parallel execution. An important consideration in load balancing is that it is not so much essential to achieve a totally uniform balance of load but rather that no one processor should have significantly more than average load. Any processor with an exceptional workload will cause all other processors to incur idle time, which results in poor parallel performance. Fig. 10.15 shows the convergence history plot for the various numbers of partitions for both SG and MG simulations. It can be seen that the residuals for the various numbers of partitions coincide with each other and the reduction of residuals has reduced to the same order of magnitude. Fig. 10.16 shows the velocity profiles for Re 5 400 obtained using different number of processors for both SG and MG. The velocity profiles yield a good agreement with Jiang’s results [4] and those of serial codes. From the figures, it can be seen that the parallel strategy adopted in this work is effective and will not affect the convergence rate and the accuracy of the numerical solution regardless of the numbers of partitions used and reduces the total simulation time significantly.

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10.1.2 Viscous Unsteady Flow Past a Circular Cylinder The viscous flow past a circular cylinder is a classical benchmark problem, which has been the subject of many theoretical, experimental, and computational investigations due to its simple geometry and its representative behavior of general bluff body wakes. This unsteady problem is computed as a test case for validating and assessing the capability of the serial and parallel incompressible solvers using both SG and MG to perform time-dependent calculations for high Reynolds number unsteady computations. When the Reynolds number based on the free-stream velocity (U) and the cylinder diameter (D) is less than or equal to 200, the flow will undergo separations but still maintain laminar. The vortices induced by separation and shear is then shed from upper and lower surfaces alternatively and, thus forming the famous von Ka´rma´n vortex street phenomenon. At this stage, the contribution of pressure drag to the total drag is about three-quarters. The shedding of each vortex produces circulation, and hence gives rise to a lateral force on the cylinder. Since these forces are periodically following the frequency (f) of vortex shedding, the cylinder may be subjected to a forced vibration. The frequency of such forced vibration, sometimes called self-induced vibration, may be analyzed based on a nondimensional parameter called Strouhal number (St 5 f  D/U). The Reynolds number specified in this unsteady computation is 200 since the numerical and experimental observations of Liu et al. [11], Williamson [12], and Wille [13] are available. A cylinder with a diameter of unity with its axis in the z-direction is placed in a uniform x-directional flow. The computational domain is a rectangular box of length 20D, where D is the diameter of the cylinder, with the downstream boundary located at a distance of 16D from the center of the cylinder. The top and bottom walls are located at a distance of 4D from the center of the cylinder, while the side walls are 0.5D apart. A grid size of 138,299 nodes and 745,638 tetrahedral elements is used for all computations with the wake region further refined to accurately capture the fine details of the vortex shedding phenomenon as shown in Fig. 10.17. The flow is started from stationary conditions and the simulation is run until periodic shedding of vortices occurred. The inflow velocity to the domain is assumed to be uniform and all the side walls are assumed to be slip walls. The cylinder surface is assumed to be solid and no-slip conditions are applied. The pseudocompressibility coefficient β is set to 1.0 for faster convergence, CFL is set to 1.2 and the physical time-step is set to 0.1 for better temporal resolution. The third-order characteristics upwind scheme is used in all the computations together with the implicit dual-time-stepping scheme and the second-order temporal discretization. The number of pseudo subiterations per time-step is set to 200 for SG, 50 V-cycles per time-step and 25 W-cycles per time-step. In this section, the results for parallel computations are presented only since the parallel results are exactly the same as those of serial computations as validated in Section 10.1.1.3 for steady flow computation.

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FIGURE 10.17 Mesh with grid refinement in the wake region for unsteady flow past a circular cylinder.

10.1.2.1 Grid Convergence Study As mentioned in Section 10.1.1.1, grid convergence study is a necessary test in any of the CFD simulations. In this study, three different grid densities of unstructured tetrahedral grids are generated, which have 104,004, 138,299, and 173,126 nodes, respectively. All the parameters for the three simulation runs are kept exactly the same and run until nondimensional time, t 5 120.0 for Re 5 200. The computed value of lift coefficient, Cl is used as a criterion for convergence as shown in Fig. 10.18. Comparing the results shown in Fig. 10.18, the maximum value of Cl obtained using 104,004 nodes is 0.60 as compared to 0.64 obtained by both grids of 138,299 and 173,126 nodes. And similarly, comparing the results obtained by both 138,299 and 173,126 nodes, the value of Cl does not deviate from each other significantly. Since there is not much difference in results when the nodes density is increased from 138,299 to 173,126 nodes and to minimize computational time, a grid size of 138,299 nodes is being employed in the present work. 10.1.2.2 Parallel Unsteady Flow Computation The test case considered for high Reynolds number unsteady flow is the viscous flow past a 3D circular cylinder. The parameters and schemes for both the parallel SG and MG computations are stated in Section 10.1.2. A threelevel MG is used to compute the flow. There are 138,299 nodes and 745,638 tetrahedral elements in the fine grid, 34,983 nodes and 175,881 tetrahedral elements in the coarse grid, and 9094 nodes and 42,150 tetrahedral elements in the coarsest grid, of which the fine grid is shown in Fig. 10.17 and coarse level grids are shown in Fig. 10.19. The grids are decomposed accordingly

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FIGURE 10.18 Grid convergence study for unsteady flow past a circular cylinder.

FIGURE 10.19 Coarse level tetrahedral grids for three-level MG computations.

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FIGURE 10.20 Lift and drag coefficients versus time for flow over a circular cylinder. (Parallel SG, number of subiterations 5 200, three-level parallel MG, 50 V-cycles and 25 Wcycles per time-step, Re 5 200.)

to the required partitions for computations. The purpose of this section is to validate and assess the capability of the parallel incompressible Navier
Stokes solvers based on the artificial compressibility approach and a higher-order characteristics-based finite-volume to perform time-dependent calculation using an implicit dual-time-stepping scheme. The computed results are compared with numerical results obtained by other researchers, and with available experimental results. Fig. 10.20 shows the lift and drag coefficients on the circular cylinder versus nondimensional time for both SG and MG. A pronounced asymmetric wake began to appear at nondimensional time of 30 in both SG and MG runs. And the flow became completely periodic at a time of 60. Although both of the flows took the same nondimensional time to become fully periodic, the number of subiterations for MG is much less than the SG. This signifies that the MG method takes a shorter time than the SG to produce the vortex shedding phenomenon, thus less CPU time is needed for the flow to become fully periodic. The computed coefficients from the MG method are yielding excellent agreement with the SG. This shows that the MG method will not change the SGs’ results, but only accelerate the convergence of the solution. Table 10.1 tabulates the values of computed lift and drag coefficients and Strouhal number for this flow and these values are compared with both numerical and experimental studies obtained by other researchers [11
13]. In this table, Cl is the lift coefficients, Cd is the drag coefficients, and St is the Strouhal number. In comparison with other researcher’s results, the

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TABLE 10.1 Lift Coefficient, Drag Coefficient, and Strouhal Number for Unsteady Flow Over a 3D Circular Cylinder (Re 5 200) Reference

Cl

Cd

St

Present

6 0.64

1.39 6 0.040

0.201

Liu et al. [11]

6 0.69

1.31 6 0.049

0.192

Williamson (Expt.) [12]







0.197

Wille (Expt.) [13]




1.30




1.31 6 0.041

0.195

Tai and Zhao (2D) [6]

6 0.64

obtained value of 6 0.64 for Cl and the average value of 1.39 for Cd produce quite a good agreement with the numerical results by Liu et al. [11] and the experimental results obtained by Wille [13]. The computed St obtained yields a good agreement with the experimental data obtained by Williamson [12]. The coefficients obtained are compared with those obtained from 2D simulation [14] and presented in Table 10.1. The agreement between 2D and 3D simulations is good since the 3D features are weak. Fig. 10.21 shows the streamlines in sequence for one complete cycle of the von Ka´rma´n vortex shedding (Re 5 200). The series of streamlines plots shown for the vortex shedding will repeat exactly itself for every period of T 5 4.910. From the figures, it can be seen that the attached vortices become asymmetric and are shed alternately at a well-defined Strouhal frequency. As seen from Fig. 10.22, it indicates that the 3D features are weak with an unsteady wake since there is not much distortion of the vortices in the z-direction and the vortex axes are nearly aligned with the cylinder axis in the near wake. The center of the primary vortex is x 5 4.43, y 5 3.58 and secondary vortex is x 5 5.27, y 5 3.98 for all planes in the z-direction as shown in Fig. 10.22. The 3D features will get stronger as the Reynolds number increases and this distinct feature will be shown for Re 5 800 toward the end of this section. Fig. 10.23 shows the contour of vorticity at an instantaneous time for z 5 0.25, which basically outlines the von Ka´rma´n vortex street phenomenon. Fig. 10.24 shows the experimental results for Ka´rma´n vortex street behind a circular cylinder at Re 5 200 [15]. Both figures shows that the vortices induced by separation are shed from upper and lower surfaces alternatively and, thus forming the von Ka´rma´n vortex street phenomenon. Visualization of the vorticity magnitude reveals the presence of columnar vortices with their axis aligned with that of the cylinder as shown in Fig. 10.25. In the near-wake region, the columnar vortices exhibit sinusoidal variations in the axial direction, which appear to be fairly regular. Further downstream, diffusion causes the vortices to mingle and lose their individual nature.

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FIGURE 10.21 Streamlines patterns showing one cycle of the von Ka´rma´n vortex shedding for Re 5 200 at z 5 0.25.

Since the 3D features get stronger as the Reynolds number increases, viscous flow past a circular cylinder with Re 5 800 is computed to assess the capability of the parallel incompressible Navier
Stokes time-dependent solver to capture this distinct features. In order to accurately capture these fine 3D distinct features, the wake region is further refined and a grid size of 351,137 nodes and 1,943,446 tetrahedral elements is used for this computation. Fig. 10.26 shows the lift and drag coefficients on the circular cylinder versus nondimensional time. The figure shows the coefficients are no longer uniform as those measured at Re 5 200 and the vortex shedding phenomenon is no longer 2D. The lift and drag coefficients are compared with those obtained from 2D simulation as shown in Fig. 10.27. Since at higher Reynolds numbers, the boundary layer is sharper, the velocity gradients are larger, and resulting in the release of stronger vortices. As a result, the amplitudes of the coefficients for the 2D simulation are larger than those at

FIGURE 10.22 Instantaneous streamlines plots for all planes in the z-direction.

FIGURE 10.23 Vorticity contours plot for Re 5 200 at z 5 0.25.

FIGURE 10.24 Ka´rma´n vortex street behind a circular cylinder at Re 5 200. Experimental data from Van Dyke, D. Milton, An Album of Fluid Motion, Parabolic Press, Stanford, CA, USA, 1982. Photograph by Prof. Sadatoshi Taneda. Used with permission from Hiroshi Taneda

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FIGURE 10.25 Vorticity iso-surfaces for Re 5 200 at t 5 120.

FIGURE 10.26 Lift and drag coefficients versus time for flow over a 3D circular cylinder at Re 5 800 (3D simulation).

Re 5 200 [6]. However, this is not seen in the 3D simulation since the vortices are distorted and possess components besides that in the spanwise direction. Fig. 10.28 shows the instantaneous streamlines patterns at different planes along the z-direction. From the figure, it can be seen that the size of the vortices is different at different planes. The center of the vortices for all planes deviates with each other as observed in Fig. 10.28 and all these phenomena show that the 3D features are strong for Re 5 800, unlike the flow

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FIGURE 10.27 Lift and drag coefficients versus time for flow over a 2D circular cylinder at Re 5 800 (2D simulation).

for Re 5 200, which shows the 3D feature is weak. Fig. 10.29 shows the vorticity iso-surfaces plot at t 5 300 and Fig. 10.30 shows the iso-surfaces for two different vorticity values of 0.5 and 2.0. There is a breakdown in the spanwise vorticity and distinct strands of streamwise vorticity at jωj 5 0:5 can be observed in Fig. 10.30A. In Fig. 10.30B, it can be seen that the distortion of spanwise vortex cores to form loops. The performance results for unsteady flow of Re 5 200 using both parallel SG and MG is based on the speedup characteristics, efficiency of parallelization, and comparison between percentage computation and communication time, as shown in Fig. 10.31. The plot shown in Fig. 10.31A reveals that the speedup increases almost linearly up to four processors and after which, it starts to deviate away from the perfect speedup. The speedup performance starts to deteriorate as the number of processors increases, but the simulation time decreases significantly. This is to be expected, since the relative ratio of communication to computation is higher as the number of processors increases and inevitably the speedup declines. The speedups for both parallel SG and MG show a similar trend, but the computational time required to compute the same problem by parallel MG is much less than parallel SG. From Fig. 10.31A, it can be seen that the computation is scalable up to 32 processors. The efficiency plot shown in Fig. 10.31B is low for this test case using the MG method running on 32 processors, just slightly above 50% as shown in the figure. Therefore it is more efficient to use 16 processors to simulate this problem, rather than 32 processors for the MG method.

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FIGURE 10.28 Streamlines plots for different planes in the z-direction at Re 5 800.

On the other hand, parallel SG proved to be quite efficient running on 32 processors, indicating an efficiency of 67%. Fig. 10.31C shows the percentage computation time and communication time required in various computations. The lines representing both the percentage computation time and communication time are moving toward each other as the number of processors increases. This is to be expected, since

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FIGURE 10.29 Vorticity iso-surfaces for Re 5 800 at t 5 300.

FIGURE 10.30 Vorticity iso-surfaces for (A) jωj 5 0:5 and (B) jωj 5 2:0 at Re 5 800.

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(A) 32 Re = 200

28

Ideal Single grid Multigrid

Speedup

24

21.45

20 16

18.84 12.52

12 10.89

8 4 0

6.64 6.32

3.42 1.73

3.31 1.74

0

4

8 12 16 20 24 Number of processors (P)

(B)

32

(C) 1.0

0.87

0.86 0.83 0.83

0.8

0.79

Re = 200 Single grid Multigrid

0.79

0.7

0.67 0.58

0.6

0.5

0.53

0

4

8

12

16

20

24

Number of processors (P)

28

32

Percentage of total simulation time (%)

100

0.9

Efficiency

28

Re = 200

80

60

Single grid Multigrid Compatation time Communication time

40

20

0 0

4

8

12

16

20

24

28

32

Number of processors (P)

FIGURE 10.31 Performance for unsteady flow past a circular cylinder of Re 5 200; (A) speedup, (B) efficiency, and (C) comparison between percentage computation and communication time.

computation time and communication time will decrease and increase, respectively, inversely proportional to each other as the number of processor increases. The figure shows that the optimum number of processors to compute this problem using the MG method in order to have a balance of both communication time and computation time is 24 processors. Beyond this optimum number of processors, the communication time is extremely high for the MG method running on 32 processors, which accounts for almost 73% of the total simulation time as indicated in Fig. 10.31C. The percentage communication time is much lower for parallel SG as compared to parallel MG running on 32 processors, because no additional coarse grid sweeps are performed on the SG code.

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10.2 COMPUTATIONS WITH THE IMMERSED OBJECT METHOD WITH OVERLAPPING GRIDS 10.2.1 Viscous Flow Past a Stationary Circular Cylinder The first test case used for validating the immersed object method (IOM) with overlapping grids (OGs) for stationary object is the viscous flow past a stationary circular cylinder. Both low and high Reynolds numbers are used to demonstrate and examine the performance, accuracy, and robustness of the method for steady and unsteady flows. The computational domain for the Eulerian or background mesh has the same dimensions as those depicted in Section 10.1.2, except that it is without the physical cylinder in the computational domain. A 3D circular cylinder mesh as shown in Fig. 10.32 is immersed into the background mesh to define the physical boundary of the cylinder. The search algorithm depicted in Section 6.1 is used to find those fluid nodes within the immersed cylinder and classify it as solid nodes. From Fig. 10.32, it can be seen that the background mesh that is covered by the

FIGURE 10.32 Background mesh, OG, and 3D immersed circular cylinder mesh for viscous flow past a stationary circular cylinder.

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immersed cylinder needs to be refined so that the physical boundary can be defined as accurate as possible. The OG with a circular cylinder of unity diameter has a dimension of x 5 110.0, y 5 4.0, and z 5 0.49 with the origin located at (1.0, 2.0, 0.005). The background mesh has a grid size of 40,287 nodes and 206,017 tetrahedral elements and it is partitioned into four subdomains for parallel computation, whereas the OG consists of 188,603 nodes and 1,029,685 tetrahedral elements and partitioned into 12 subdomains, all of which are shown in Fig. 10.32. A three-level MG method for the background mesh is used to compute the flow. There are 10,601 nodes and 49,247 tetrahedral elements in the coarse grid, and 2585 nodes and 13,234 tetrahedral elements in the coarsest grid. All these meshes are used for both steady and unsteady flow computations and with the further refined wake region to accurately capture the fine details of the vortex shedding phenomenon for unsteady flow. The grid convergence studies were performed for all the meshes and an optimum grid size for both background mesh and OG are obtained. The inflow velocity to the domain is assumed to be uniform and all the side walls are assumed to be slip walls. The cylinder surface is assumed to be solid and no-slip conditions are applied. The third-order characteristicsbased upwind scheme is used in all computations together with the implicit dual-time-stepping scheme and the second-order temporal discretization. CFL is set to 1.2 and 0.6 for steady and unsteady flows, respectively. The physical time-step is set to 0.1 for better temporal resolution. The number of pseudo subiterations per time-step for the MG method is 100 W-cycles and 60 W-cycles for steady and unsteady flows, respectively. Since the cylinder is at stationary, the fluid velocity within the cylinder is set to an object velocity, Vo, of zero for every physical time-step.

10.2.1.1 Steady Viscous Flow (Re 5 41) The low Re specified in this computation is 41.0 and since the Re is low, the flow can be considered as steady flow. The flow is started from stationary conditions and the simulation is run until the flow has fully convergence as shown in the convergence history plot using the IOM with parallel-MG method in Fig. 10.33 and the normalized residual is reduced to an order of five. For a given cylinder of a known diameter immersed in a fluid, the Re is directly proportional to the velocity, and therefore the variation with Reynolds number would affect the steadiness of the flow. At very small values of Re, the inertia effects are negligible and the flow pattern is very similar to that for an ideal flow, the pressure recovery being nearly complete. Thus pressure drag is negligible and the total drag is nearly all due to skin friction. As the Re is increased, separation of the boundary layer occurs as separation point and the fluid inertia becomes more important. At the separation point, the fluid cannot flow along the curved path of the cylinder to the

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FIGURE 10.33 Convergence history plot for residual versus physical time-steps using IOM with parallel-MG method on flow past a circular cylinder at Re 5 41.

rear of the body. The region ahead of the cylinder in which viscous effects are important become smaller, with the viscous region extending only a short distance ahead of the cylinder. The viscous effects are convected downstream and the flow loses its symmetry. Two symmetrical eddies known as separation bubble, rotating in the opposite directions, are formed behind the cylinder in which some of the fluid is actually flowing upstream, against the direction of the upstream flow. Fig. 10.34A shows the photograph of a streamlines plot taken from an experiment for a flow over a circular cylinder for Re 5 41.0, which is the approximate upper limit for steady flow [15]. Fig. 10.34B and C show the streamlines plot obtained from the developed IOM for the background mesh and OG, respectively. Based on the qualitative comparison between the experimental results of Dyke et al. [15], the wake formed behind the cylinder agrees well with each other, where the separation point occurs at the same location, the separation bubble is of the same size and the reattachment point is at the same location. A quantitative comparison of the aspect ratio (separation bubble length, S, or cylinder diameter, d) with the experimental results obtained by Nishioka and Sato [16] is attempted. Fig. 10.35 shows the aspect ratio versus Re and with a Re of 41.0, an aspect ratio of 2.40 is obtained. Fig. 10.34B and C show that the aspect ratio obtained using the IOM with an OG is 2.33, and comparing this value with the aspect ratio obtained in Fig. 10.35, the aspect ratio agrees with each other.

10.2.1.2 Unsteady Viscous Flow (Re 5 200) The high Re specified in this computation is 200. The flow is started from stationary conditions and the simulation is run until periodic shedding of

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FIGURE 10.34 Streamlines plot for flow over a 3D stationary circular cylinder at Re 5 41 for (A) experimental measurement [15], (B) background mesh (z 5 0.25), and (C) OG (z 5 0.25). Photograph by Prof. Sadatoshi Taneda. Used with permission from Hiroshi Taneda

7 6 5

S/d

d

s

4 3

S/d ≈ 2.40

2 1 Re = 41 0

20

40

60

80

100

120

Re FIGURE 10.35 Length of separation bubbles behind cylinder versus Re [16].

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FIGURE 10.36 Lift and drag coefficients versus time for flow over a circular cylinder using IOM with OG and parallel MG at Re 5 200.

vortices occurred. The results obtained using the IOM with OG and parallelMG method on the background mesh are compared with the results obtained in Section 10.1.2.2. Fig. 10.36 shows the lift and drag coefficients on the cylinder versus nondimensional time obtained from the OG. A pronounced asymmetric wake began to appear at nondimensional time of 20 and the flow became completely periodic at a time of 55. The lift and drag coefficients obtained using the IOM is 6 0.645 and 1.39 6 0.046. The Strouhal number obtained is 0.202. In comparison with the results tabulated in Table 10.1, all the values obtained using the IOM with OG yield a good agreement with the numerical values in Section 10.1.2.2 and those of the experimental data [12,13]. Fig. 10.37 shows the streamlines plot in sequence for one complete cycle of the von Ka´rma´n vortex shedding. The series of streamlines plots shown for the vortex shedding will repeat exactly itself for every period of T 5 4.95. Visualization of the vorticity iso-surfaces is shown in Fig. 10.38 and it reveals the presence of columnar vortices with their axis aligned with that of the cylinder as those shown in Fig. 10.25.

10.2.2 Rotating Square Cylinder The second test case considered for validating the IOM with OG for moving object is viscous flow over a rotating square cylinder of unity at Re 5 41. The background mesh consists of 27,489 nodes and 54,595 elements and it is partitioned into eight subdomains for parallel computation, whereas the OG consists of 15,129 nodes and 29,658 elements and partitioned into four

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FIGURE 10.37 Streamlines patterns showing one cycle of the von Ka´rma´n vortex shedding using the IOM with OG at Re 5 200 at z 5 0.25.

FIGURE 10.38 Vorticity iso-surfaces for Re 5 200 using IOM (background mesh).

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15

15

10

10

y

y

20

5

5 0 –40

–10 –20 x Background mesh (eight sub-domains) (27,489 nodes and 54,595 elements) –30

0

0 –40

–10 –20 x Overlapping grid (four sub-domains) (15,129 nodes and 29,658 elements) –30

0

FIGURE 10.39 Partitioned background mesh and OG for rotating square cylinder flow. From Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids, Journal of Computational Physics, Volume 207, Issue 1, 20 July 2005, Pages 151-172.

subdomains, as shown in Fig. 10.39. A three-level MG method for the background mesh is used in this study. There are 6872 nodes and 13,652 elements in the coarse grid, and 1843 nodes and 3685 elements in the coarsest grid. The third-order characteristics-based scheme is used in the computation together with the matrix-free, implicit, dual-time-stepping scheme and the second-order temporal discretization. The number of pseudo subiterations per time-step is set to 100 W-cycles, CFL is set to 0.4 and the nondimensional physical time-step is set to 0.02. A square cylinder mesh of unity is immersed into the background mesh to define the physical boundary of the cylinder. The angle of rotation per time-step for the cylinder is set to 0.2 degrees or 3.5 3 1023 rad. The square cylinder is rotating counter-clockwise at an angular velocity of 0.175 rad/s for each time-step. The flow field is started from rest with the square cylinder rotates at the set angular velocity and the simulation is run until periodic shedding of vortices occurred. The computed lift and drag coefficients on the rotating square cylinder versus nondimensional time computed using the IOM from the OG are shown in Fig. 10.40. From Fig. 10.40, it can be seen that there are two peaks and one trough within a cycle. Fig. 10.41 shows the vorticity contour plots for the rotating square cylinder at different nondimensional physical time and angle of rotation, which outlines the vortex shedding phenomenon. These phenomena are different from the Ka´rma´n vortex street phenomenon. The series of vorticity contour plots shown for the vortex shedding will repeat exactly itself for every period of T 5 36 and thus it takes a nondimensional time of 36 for the square cylinder to complete one cycle.

10.2.3 Rotating and Translating Circular Disk The third test case for validation is the IOM with OG for moving object is the viscous flow over a rotating and translating circular disk of diameter

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FIGURE 10.40 Lift and drag coefficients versus time for flow over a rotating square cylinder computed using the IOM with OG.

d 5 0.25 at Re 5 25, based on kinematics viscosity ν 5 1 3 1022. The disk is moving within a computational domain of x 5 20.35 to 0.9 and y 5 20.5 to 0.5 and the center of the disk translates along a prescribed trajectory as follows:

πt


πt xðtÞ 5 0:25 1 2 cos ; yðtÞ 5 2 0:1 sin π 1 2 cos : ð10:1Þ 2 2 The computational domain, prescribed trajectory, and several different positions of the disk are shown in Fig. 10.42. The four sides of the computational domain are solid wall with zero velocity. While its center is moving along the trajectory, the disk is rotating counter-clockwise at an angular velocity of π and the fluid velocity within the disk is set to an object velocity, Vo, of π. The thirdorder characteristics-based scheme is used in the computation together with the matrix-free implicit dual-time-stepping scheme. The mesh shown in Fig. 10.42 consists of 27,323 nodes and 54,284 elements and partitioned into eight subdomains for parallel computation. A three-level MG method for the background mesh is used where the coarse grid consists of 7023 nodes and 13,953 elements, and 1802 nodes and 3725 elements in the coarsest grid. The number of pseudo subiterations per time-step is set to 200 W-cycles, CFL is set to 1.5 and the nondimensional physical time-step is set to 0.01. The flow is started with the fluid at rest and the disk is rotating at an angular velocity of π and run until the flow becomes periodic. The period of the periodic motion is 4. Fig. 10.43 shows the contour plots for both u and v velocities and streamline plots at different time instances between t 5 4.0 and 10.0. From the figure, it can be seen that the rotating disk causes the flow field within the confined domain to be in a rotational motion in the counter-clockwise direction.

FIGURE 10.41 Vorticity contour plots for rotating square cylinder at different time and angle of rotation using the IOM with OG.

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FIGURE 10.42 Moving disk along a prescribed trajectory.

At the same position of the rotating disk, the flow phenomenon is completely different at different time instances. For example, the flow patterns at t 5 5.5 and 6.5 are different from each other. This is mainly due to the rotating disk at t 5 5.5 is translating in the forward direction, whereas at t 5 6.5, it is translating back to the initial position at (0,0).

10.3 COMPUTATION OF ST. JUDE MEDICAL AORTIC BI-LEAFLET MECHANICAL HEART VALVE 10.3.1 Introduction The successful development of prosthetic heart valves requires not only the knowledge of the dynamic behaviors of the valves, but also the associated blood flow field around them. Many in vitro experimental studies provided useful field information on different heart valves; however, the physical and cost limitation of experimental investigations make it difficult to give a full and detailed description of the complex transvalvular flow phenomena. With the advent of high-speed parallel computers and the development of CFD analysis algorithms, bio-fluid dynamic simulation of the valvular function provides a viable alternative in the detailed analysis of the complex flow dynamics during the opening and closing phases. These simulations can be used to analyze the details of the fluid dynamic stresses in the regions of interest with the mechanical valve prostheses, where detailed experimental measurements are impractical. Prosthetic heart valve researchers have identified four phases of the valve within a cardiac cycle, which are the opening phase, the fully opened phase, the closing phase, and the fully closed phase.

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FIGURE 10.43 Streamlines and velocities contour plots for one period of the moving disk.

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Among these four phases, the fully opened and fully closed phases are relatively simple to study due to the fixed flow field structure, and many researchers have intensively and extensively studied these two phases [17
25]. For the opening and closing phases, the flow phenomenon is complex and the limitations of computing resources also contributed to the slow progress. Lai et al. [26] have developed a numerical model to simulate the leaflet motion during the closing phase in order to investigate the closure fluid dynamics. Shi et al. [24,27] have studied the opening process in a bileaflet mechanical heart valve under pulsatile flow condition. One of the important requirements of an accurate simulation of valve function is the development of a fluid
structure interaction (FSI) algorithm for the prediction of the leaflet motion resulting from the hemodynamic forces acting on the leaflets. In this present work involving FSI, the computation is first carried out by solving the Navier
Stokes equations for the fluid flow field using the numerical method described in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and the IOM with the leaflets immersed into the fluid computational domain and then the hemodynamic forces acting on the leaflets are available from the OGs for the structure dynamics equation. These hemodynamic forces are translated into rotating moments in the structure dynamics equation. The governing equation of motion for the rigid leaflets is solved using an implicit dual-time-stepping scheme. The new angular position and velocity of the leaflets are available once the structure equation for the leaflet is solved. The immersed leaflets boundary in the fluid domain is reconstructed with the new angular position and a new set of boundary conditions is applied to it. And with this new immersed leaflets position, the fluid flow field is recalculated again. Section 1 presents the numerical model properties for a 29 mm St. Jude Medical (SJM) aortic bi-leaflet MHV, the properties for the fluid computational domain and the physiological conditions applied to this work. Then the subsequent sections will define the mathematical model for the FSI algorithm developed in this work.

10.3.2 Model Properties In the present study, a 29 mm SJM aortic bi-leaflet MHV [28] is studied. In this aortic bi-leaflet valve model, the annulus diameter is 29 mm, whereas the diameter of the internal flow channel is 25 mm, and the thickness of the valve leaflets is 0.8 mm. The estimated orifice area of this valve model is 490.87 mm2. In the fully opened and closed position, the leaflets make an angle of 5 degrees and 60 degrees, respectively, with the x
z plane as illustrated in Figs. 10.44 and 10.45. This implies that each leaflet opens to 5 degrees with a travel arc of 55 degrees. The SJM standard valve is designed and manufactured of pyrolytic carbon and is tungsten impregnated. The density of it ranges from 1.5 3 103 to 2.0 3 103 kg/m3 [29]. In this study, the density of the leaflet is set to 1.8 3 103 kg/m3, thus the moment of inertia (I) of the leaflet is 3.6 3 1028 kg m2.

FIGURE 10.44 A 29 mm SJM aortic bi-leaflet MHV at fully opened position, 5 degrees.

FIGURE 10.45 A 29 mm SJM aortic bi-leaflet MHV at fully opened position, 60 degrees.

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As the non-Newtonian nature affects the flow field only when the diameter of the flow channel is comparable with the size of the blood cell [30
33], the human blood is treated as a Newtonian fluid in this computation, with a density of 1.06 3 103 kg/m3 and a dynamic viscosity of 3.5 cPoise ð3:5 3 1023 N s=m2 Þ. The initial velocity in the flow field domain is set to zero everywhere except at the inlet. The time-dependent inlet velocity for both opening and closing phases is specified according to the physiological mass flow rate of a normal human based on the requirements for pulsatile flow testing of heart valve substitutes published by British Standard organization [33], as shown in Fig. 10.46. The shape of the velocity profile is typical for the aortic valve system as illustrated in Fig. 10.47. In this study, a set of time-dependent inlet velocity equation for both opening and closing phases is devised according to the profile shown in Fig. 10.46 and is given as follows: 8 0:0 0 for 0 , t # 0:075T 1 > > > > > 5:0πðt 2 0:075TÞA > > for 0:075T , t # 0:175T U0 Usin@ > > T < 0 1 Uin ðm=sÞ 5 > > 1:98πðt 2 0:175TÞ > A for 0:175T , t # 0:44T > U0 Ucos@ > > > T > > : 0:0 for 0:44T , t # T ð10:2Þ

FIGURE 10.46 Time-dependent inlet velocity profile for opening and closing phases. (A) Aortic valve closed, 0.0 , t # 0.06 s; (B) aortic valve opening and closing, 0.06 , t # 0.35 s; (C) aortic valve closed, 0.35 , t # 0.8 s).

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FIGURE 10.47 Plot of the aortic and left ventricular pressure curves and the associated flow rate curve during a cardiac cycle.

where U0 5 6 0.5 m/s (positive for opening phase and negative for closing phase), and T is the period for one cardiac cycle which is 0.83 s and this corresponds to the normal heartbeat of 72 beats/min. The nondimensional form of Uin 5 Uin/UN. The corresponding maximum flow reaches 250 mL/s or 0.5 m/s, which is approximately half of the maximum physiological flow. A higher flow would investigate numerical instabilities. Within a systolic interval as shown in Fig. 10.46, the aortic heart valves undergo three phases of position, as labeled in the figure as A, B, and C. In phase A, the aortic valves are closed from 0.0 s to 0.06 s. In phase B, the aortic valves open from 0.06 s to 0.14 s and close from 0.14 s to 0.35 s. The maximum flow takes place at t 5 0.14 s. In phase C, the aortic valves are closed from 0.35 s to 0.8 s. For incompressible flow in this work, the pressure is treated as relative to the pressure value at the outflow plane and a fixed pressure of 10 kPa is assigned to the outflow plane in this study. The Reynolds number based on the free-stream velocity, the diameter of the flow channel, and the density and dynamic viscosity of blood is set to 1893 in this study. Flow in a typical MHV under the pulsatile flow condition in a cardiac cycle would normally exhibit transition from unsteady laminar to turbulent flow. The need for turbulence modeling arises from the local Reynolds number of the blood flow past MHV that may easily exceed 7000. However,

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under pulsatile flow conditions the turbulence is intermittent, peaking only during the deceleration phase following peak systole [34]. Currently there is still no suitable turbulence model that could successfully describe both the laminar and turbulent flows at the same time and also take into account of their transition with affordable computing resources [35]. Large eddy simulation (LES) turbulence model could be a choice to overcome this problem, but it is very time-consuming and is beyond the capability of most research institutes. From the open literature, numerical study of pulsatile flow in MHV is currently limited to unsteady laminar flow only. In this work, it is beyond the work scope to develop the LES turbulence model to compute the flow in the MHV. As the initial step toward the computation of pulsatile flow through the MHV, the turbulence effect of the flow field is not taken into account and the entire flow field is considered as an unsteady laminar flow in the all computations.

10.3.3 Fluid
Structure Interaction In this analysis, the FSI is considered between the fluid domain (blood) and the immersed structures, that is, the valve leaflets, while assuming the aortic valve leaflets to be perfectly rigid, undergoing no deformation. The main point is that the boundary motions that drive the flow are not prescribed, but are intrinsically coupled to the fluid motion. This shows how relatively simple analysis coupling fluids and structures can provide vital information on the dynamics of the MHV. The algorithm for FSI, the mathematical model for the structure domain and coupling of the fluid and structure domains are described in the subsequent subsections.

10.3.3.1 Fluid Domain The fluid domain is considered as blood flow and the physical parameters are set according to human physiological conditions. The blood flow is considered to be isothermal, incompressible, and Newtonian. The fluid domain is solved using the numerical methods described in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and the leaflets are immersed into the fluid domain and solved using the IOM with OGs. The solutions for the current timestep are calculated till it converges, so the pressure and velocity of the flow field are available for the structure dynamics equation. The wall shear stress on the leaflets surface is calculated based on the velocity gradients. 10.3.3.2 Structure Domain In this work, the bi-leaflet are considered as rigid body where each particle on it rotates with the same angular velocity and the dynamic model of the leaflets motion is a single degree of freedom only. The angular deviation of the leaflets is from some arbitrary starting position, say ~ θ. Hence, if the

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hemodynamics forces exerted on the leaflets surface are known from the previous pseudo time-step, then the structure dynamics equation can be solved. Based on Newton’s second law of motion in rotational form, the governing equation of the leaflets rotation can be modeled using the relationships between angular acceleration and moments due to the pressure and the leaflets wall shear stress acting on the leaflets. Due to the symmetric geometry of the bi-leaflet valve, the governing equation for one of the two leaflets is given as, d2~ d~ θ θ ~p 1 M ~τ 1 γU~ θ 5M 1 δU ð10:3Þ dt dt2 where I is the rotating moment of inertia of the leaflet, ~ θ is the rotating angle of the leaflet, δ is the friction coefficient for the hinges, γ is the elastic coefficient for the leaflet, M p and M τ are the moments of the leaflet about the pivotal axis generated by the pressure and the leaflet wall shear stress respectively, d2~ θ=dt are the angular acceleration and velocity, respecθ=dt2 and d~ tively, of the leaflet. In principle, the moment due to the weight of the leaflet should be included in governing equation, but this moment was found to be much smaller than the pressure moment at all angles [36]. The terms in Eq. (10.3) are nondimensionalized as follows: IU

I~ 5

I ρN

L5

M~ 5

;

M 2 L3 ρN U N

ð10:4Þ

For simplicity, the tilde for all nondimensional term is dropped for all equations.

10.3.3.2.1

Hemodynamics Forces and Moments

Considering the forces acting on a differential element on the surface of the leaflet as shown in Fig. 10.48, the leaflet will rotate about the pivot axis AA0 under the action of the moments generated by both the pressure and wall shear stress. The moments generated by both the pressure and wall shear stress are given as follows: ðð ~p 5 r 3 ðpU n ÞUdA ð10:5Þ M ~τ 5 M

ðð

Γ


 r 3 τU n UdA

Γ

ð10:6Þ

where Γ is the surface area of the leaflet, ~ r is the distance vector which points from the pivot axis to the center of the control area and is perpendicular to the pivot axis, p is the pressure on the control area, τ is the shear stress tensor on the small control area, ~ n is the normal vector of the control area,

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FIGURE 10.48 Hemodynamics forces acting on the surface of the valve leaflet.

and dA is the area of the small control area. In this work, counter-clockwise rotation of the leaflet is considered to be positive. 10.3.3.2.2

Discretization of Structure Equation

The governing equation for the leaflets is spatially discretized on every element of the OG that forms the leaflet. A matrix-free implicit dual-timestepping scheme similar to the method for solving the Navier
Stokes equations is used to solve the structure equation. In Eq. (10.3), the angular acceleration, α 5 d 2~ ω =dt, where ~ ω 5 d~ θ=dt is the angular velocity of the θ=dt2 5 d~ leaflet and ignoring the elastic coefficient for rigid body rotation, the equation is reformulated into nondimensional form and is given as IU

d~ ω ~p 1 M ~τ 5 M ~ tot 1 δU~ ω 5M dt

ð10:7Þ

Rearranging the above equation, ~ tot 2 δU~ d~ ω n11 ω n11 M 5 dt I

ð10:8Þ ,

The superscript (n 1 1) denotes the time level ðn 1 1ÞΔt and ω is evaluated at this time level. In this work, @=@t is discretized as a second-order accurate backward difference, so that Eq. (10.8) is reformulated as follows, ! ~ tot 2 δU~ M 1:5 ~ ω n11 2 2:0 ~ ω n 1 0:5 ~ ω n21 ω n11 5 RHSn11 ð10:9Þ 2 Δt I

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Computational Fluid-Structure Interaction

The derivative with respect to pseudo time,τ, is added to the above equation, d~ ω n11;m 5 2 RHSn11;m dτ

ð10:10Þ

Reformulate the above equation, ~ ω n11;m ω n11;m11 2 ~ 5 2 RHSn11;m dτ

ð10:11Þ

Here m denotes the pseudo time level mΔτ. ~ ω is sought by marching to a pseudo steady state in τ. Once the artificial steady state is reached, the derivative of ~ ω with respect to τ becomes zero, and RHS approaches to zero, which will recover the original structure equation in Eq. (10.3). Eq. (10.11) is integrated in pseudo time by an explicit five-stage Runge
Kutta scheme and the stage coefficients are the same as those shown in Chapter 3, Mathematical Formulation for Incompressible Flow Solver. However, the pseudo time-step size may be severely restricted if the physical time-step size is very small. Hence, following the matrix-free implicit dual-timestepping in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and modify it to suit the structure equation here, Eq. (10.11) is given as, ~ ω n11;m ω n11;m11 2 ~ 5 RHSn;m11 ð10:12Þ dτ 21 21 where RHSn;m11 5 A~ ð2 RHSn11;m Þ and A~ 5 ð11ð1:5Δτ=ΔtÞÞ21 . Pseudo time-stepping is then performed in Eq. (10.12) using a five-stage Runge
Kutta scheme as follows: ~ ω n11;m 2 αRK Δτ RHSn;m11 ω n11;m11 5 ~

ð10:13Þ

After obtaining the angular velocity, ~ ω , at pseudo time m 1 1, this serves as an input to obtain the angular displacement, ~ θ. Following the same second-order of discretization for ~ ω and the derivative with respect to pseudo time, τ, is added to the equation for ~ ω 5 d~ θ=dt and the equation becomes, n11;m11

n11;m n11;m d~ θ d~ θ 1 5~ ω n11;m11 dτ dt

ð10:14Þ

Reformulating the above equation, n11;m11 n11;m n11;m n;m n21;m ! ~ 2~ θ 1:5 ~ θ 2 2:0 ~ θ 1 0:5 ~ θ θ n11;m11 5~ ω 2 5 RHSn11;m dτ Δt

ð10:15Þ

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Using the same implicit five-stage Runge
Kutta scheme described above to obtain the angular velocity, the new angular position is obtained as follows: n11;m11 n11;m ~ 5~ θ 2 αRK Δτ RHSn;m11 θ

where RHS

n;m11

21

5 A~ RHS

n11;m

21

and A~

ð10:16Þ 21

5 ð11ð1:5Δτ=ΔtÞÞ .

10.3.4 Coupling of Fluid and Structure Domains In this work, a loose coupling of the fluid and structure domains is adopted. In loose-coupled methods, the fluid and structure equations are considered individually during one discrete time-step rather than considering the total system. These methods are also referred to as partitioned or staggered procedures [37
39], which are most frequently applied to solve three field problems, such as the fluid, structure, and moving (fluid) grid problems presented in this thesis. In this approach, the governing equations associated with each subsystem are solved using the most recently computed solutions of the other subsystems. As a result, well-established but distinct numerical solvers can be adopted for each subproblem, which are best suited for that particular subproblem. This makes the staggered approach very appealing in FSI problems. The loose coupling of fluid and structure domains in this work is coupled together by first solving the Navier
Stokes equations (Eq. (3.17)) for the fluid flow field using the numerical method described in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and the IOM with the leaflets immersed into the fluid computational domain and then the fluid pressure and wall shear stress acting on the leaflets, as shown in Fig. 10.48, are available from the OGs for the structure dynamics equation. These hemodynamic forces are translated into rotating moments in the structure dynamics equation (Eq. (10.3)). The governing equation of motion for the rigid leaflets is solved using a matrix-free implicit dual-time-stepping scheme. The new angular position and angular velocity of the leaflets are available once the structure equation for the leaflet is solved. Both the immersed leaflets and the OG are rotated to the new position along the leaflet pivotal axis. A grid velocity, U g , based on the angular velocity and the perpendicular distance vector, which points from the pivotal axis to the center of the control area, is computed. The movement of the OG following the heart valve motion is taken into consideration by adding the grid velocity, U g , to the convective term in Eq. (3.17). The immersed leaflets boundary in the fluid domain is reconstructed with the new angular position and a new set of boundary conditions is applied to it. And with this new immersed leaflets position, the fluid flow field is recalculated again. The whole procedure for the FSI algorithm is repeated again till the fully opened or fully closed position of the leaflets is reached. The procedure for the coupling of the fluid and structure domains is schematically depicted in Fig. 10.49.

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Computational Fluid-Structure Interaction

Start

Initialize variables in all domains

t = t + Δt and θ = θ ± Δθ

Define physiological velocity according to profile in fig. 10.46 Rotate immersed leaflet and overlapping grids with new θ Define boundary of immersed leaflet in background grid and grid velocity (Ug) in overlapping grids

Fluid domain

Determine node in cell between background and overlapping grids interpolate solutions Structure domain Solve fluid dynamic equations in Eq. (3.17) with numerical methods in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and IOM

No

Translate fluid forces into rotational moments

Evaluate pressure and wall shear stress in overlapping grid

Solve structure dynamics equation in Eq. (10.2) for new angular position (θ) and angular velocity (ω)

θ = 5degree or

Evaluate grid velocity (Ug) Moving grid

Yes End

FIGURE 10.49 Flowchart depicting the coupling of the fluid and structure domains.

10.3.5 Opening Phase The aortic MHV exists for the purpose of allowing blood to flow in one direction from the left ventricle to the aorta. It must serve this purpose without causing significant resistance to flow so as not to put an additional

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demand on the heart. It must work with almost 100% efficiency. Its opening and closing mechanisms must also not consume a significant amount of energy. For these reasons, it is important to understand the fluid dynamics of the aortic MHV. The motions of a 29 mm SJM aortic bi-leaflet MHV for both opening and closing phases and the blood flow through it are computed. The results are presented in terms of streamlines around the leaflets, computational blood velocities across the leaflets, rotating moment due to pressure and leaflet shear stresses, leaflet angular velocity, leaflet angular displacement, and vorticity iso-surfaces around the leaflets. The model properties, such as geometrical and material properties, and the appropriate boundary conditions for the flow system were given in Section 10.3.2. The presented aortic bi-leaflet MHV model is computed based on the IOM with OG and the parallel-MG method. The key features of this model are 3D, fully coupled blood
leaflet interaction, physiologically realistic flow conditions, avoidance of mesh update strategies, and parallel-MG computation. The friction coefficient of the hinges in the structure dynamics equation in Eq. (10.3) serves to prevent the leaflets from opening or closing. Since little is known of this behavior, the friction coefficient is set to zero as a first approximation, which implies that its effect on the joint is treated as a simple pinned hinge. A symmetric geometry is computationally used for simplicity due to the symmetric structure of the bi-leaflet valve. The systolic phase of the cardiac cycle is analyzed by applying a time-dependent flow profile shown in Fig. 10.46 and given in Eq. (10.2). No-slip boundary condition is specified on the rigid aorta walls and the leaflet surfaces and slip boundary condition is applied for the symmetry plane. Special boundary conditions are imposed on the OG to suppress the out of plane motion for the fluid, and the no-slip boundary conditions are enforced at the out of the plane fluid domain. In the present study, attention is not focused on the determination of an optimal artificial valve, rather on the flow fields downstream of the aorta immediately after the MHV. The computational grids for both the background mesh and the OG are shown in Figs. 10.50 and 10.51. Fig. 10.52 shows a perspective view of the combination of Figs. 10.50 and 10.51. Fig. 10.53 shows the definition of the various terms used in the present computation. Although the mechanical aortic valve in a human diseased heart valve is positioned right after the left ventricle, the configuration shown in Fig. 10.50 is to eliminate both the upstream and downstream effects. To eliminate the downstream effect, the distance from the valve to the outlet should be at least 10 times the diameter of the flow channel. In this study, the diameter of the flow channel is 25 mm and the length of the channel is 350 mm. The valve is located at 75 mm away from the inlet and 275 mm away from the outlet. The leaflet is rotated about the pivot axis located along the y-axis at x 5 72.71 mm and z 5 3.03 mm. The computational background mesh has a grid size of 193,344 nodes and 1,094,673 tetrahedral elements and it is partitioned into 16 subdomains for parallel computation, whereas

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Computational Fluid-Structure Interaction

FIGURE 10.50 Background mesh, immersed leaflet, and valve housing for opening phase.

FIGURE 10.51 OG with aortic valve leaflet at the fully closed position for the opening phase.

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FIGURE 10.52 Combination of background mesh and OG with leaflet.

FIGURE 10.53 Definition of terms used in the opening phase for x
z plane.

the OG consists of 85,899 nodes and 452,028 tetrahedral elements and partitioned into four subdomains. The grid convergence studies have performed for these meshes and an optimum grid size is obtained. A three-level MG is employed for the background mesh to compute the flow. There are 48,948

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Computational Fluid-Structure Interaction

nodes and 277,135 tetrahedral elements in the coarse grid, and 12,881 nodes and 72,930 tetrahedral elements in the coarsest grid. The valve leaflet shown in Fig. 10.44 is immersed into the background flow domain in the fully closed position (60 degrees) to define the physical boundary of the leaflet for IOM. The search algorithm depicted in Section 6.1 is used to find those fluid nodes within the immersed leaflet and classify it as solid nodes. In order to track the motion of the leaflet as accurate as possible and to accurately capture the fine details of the vortex shedding phenomena, the background mesh is refined in two regions from 65 mm upstream to 90 mm downstream, and 40 mm upstream to 150 mm downstream of the valve. The reason for generating two refined regions is to ensure that a smooth transition of grid density. For simplicity, the flow field variables in the simple pinned hinge and the valve housing shown in the figure are set to zero at all times. The OG shown in Fig. 10.45 will rotate about the pivot axis according to the blood
leaflet interaction from the fully closed position. The flow is initiated with the leaflet in the fully closed position and due to some leakages from the gap of the leaflets; a steady flow solution is obtained and used it as the initial condition for the moving leaflets computation. For this pulsatile flow, the CFL is set to 1.2 and the physical time-step is set to 0.001 s for better temporal resolution. The number of pseudo subiterations per time-step is set to 200 W-cycles in order for the computation to achieve a divergence free condition. The solution efficiency of the timeintegration algorithm for a moving boundary flow problem becomes more important than for a steady problem because the unsteady residual should be driven close to zero at each time-step. In the present study, the opening phase occurred from t 5 0.06 s to t 5 0.14 s as shown in Fig. 10.54. The opening phase can be subdivided into two phases, which consist of the opening phase from t 5 0.06 s to t 5 0.112 s and the fully opened phase from t 5 0.112 s to t 5 0.14 s. In this work, it can be seen that little time is spent in the opening phase and it took only 0.052 s out of 0.8 s. During a cardiac cycle, most of the time is spent in filling the heart when the valve is closed and ejecting the blood when the valve is open. The valve is open when the blood is ejected from the left ventricle that pushes the leaflets to open. Fig. 10.54 shows the nondimensionalized rotating moment derived from Eqs. (10.5) and (10.6) due to the forces generated by the pressure and shear stresses in the x- and z-planes. The figure also shows the summation of these moments. In this study, it is observed that the moment due to pressure force contributed to most part of the total moment. The moment due to the leaflet shear stresses might be under-estimated due to the lack of a suitable turbulence model in this study and turbulent shear stresses should be considered in the future work in order to have a more accurate contribution of the moment due to leaflet shear stresses. From Fig. 10.54, it can be seen that from t 5 0.0 s to t 5 0.06 s, there is no moment occurred due to the specified inlet velocity is zero. Thus there

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FIGURE 10.54 Rotating moment acting on the leaflet for the opening phase.

are no hemodynamic forces acting on the leaflets during this period. It is observed from t 5 0.06 s to t 5 0.075 s, there is an impetuous onset of moment due to the transition from fluid at rest to a sudden gain in momentum in the flow field. From t 5 0.075 s to t 5 0.094 s, the moment increased with a steep gradient within this short period and reached a maximum nondimensionalized moment of 5.3. It showed a deceleration of rotating moment after t 5 0.094 s and a small spike appeared at around t 5 0.11 s. The occurrence of this spike might be due to the slight fluctuation of the leaflets toward the end of the opening phase. After reaching the peak of the spike, there is a slight increase of moment, which might be due to a jerk occurred at the end of the opening phase. After t 5 0.112 s, the moment remained at zero value till the end of the opening phase. After obtaining the rotating moments generated by the hemodynamic forces shown in Fig. 10.54, the angular velocity (ω) and angular position (θ) for the opening phase are obtained by solving the structure dynamics equation (Eq. (10.3)). Fig. 10.55 shows both the change of leaflet angular velocity and position from the fully closed to the fully opened position. It should be noted that the negative moments and angular velocity shown in the respective figures indicating that the leaflet is rotating in the clockwise direction as shown in the sign convention stated in Fig. 10.48. From the figure, it is observed that the leaflet started to gain some angular velocity at the beginning of the systole from t 5 0.0 s to t 5 0.06 s, but it is too small to cause

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Computational Fluid-Structure Interaction

(A)

(B) 0.0

1.1

60

1.0 Angular position θ (radi)

–1.0 –1.5 –2.0 –2.5 –3.0 –3.5 –4.0

0.9 0.7

0.4

–5.0

0.1

Time (s)

0.1

0.12 0.14

30

0.5

–4.5

0.02 0.04 0.06 0.08

40

0.6

0.3 0.2

0

50

0.8

0.0

20

Angular position, θ (degree)

–0.5 Angular velocity, ω (rad/s)

70

1.2

0 0

0.02 0.04 0.06 0.08

0.1

0 0.12 0.14

Time (s)

FIGURE 10.55 Change of leaflet (A) angular velocity (ω) and (B) angular position (θ) from the fully closed to the fully opened position for the opening phase.

any significant movement in the leaflet due to the hemodynamic forces acting on both sides of the leaflet are almost equal. The valve opened gradually during this period. From t 5 0.06 s to t 5 0.085 s, the valve opened rapidly with a sudden increase of angular velocity from almost 0.0 rad/s to 1.0 rad/s. This is due to the fluid in the domain accelerated from almost at rest to a specified inlet velocity profile as shown in Fig. 10.54. This velocity propagated to the fluid around the leaflet and this caused an imbalance of hemodynamic forces acting on both sides of the leaflet. Thus the leaflet will rotate along the pivotal axis with an angular velocity. Once the propagation of velocity has reached to the rest of the fluid domain, it is observed that the valve will open gradually with a steep gradient from t 5 0.085 s to t 5 0.112 s. During the period toward the end of the opening phase, the valve opening speed will decelerate as the rotating moment from t 5 0.094 s to t 5 0.11 s has decreased as shown in Fig. 10.54. This phenomenon signifies that the valve will have a cushioning effect and it will not be opened with a sudden impact created on the rotating hinge. In this way, the valve will last longer and the blood platelets will not be sheared away too much. It can be seen that from t 5 0.11 s to t 5 0.112 s, the leaflet is fluctuating with a spike shown in the rotating moment and this action might be too minor to be noted in Fig. 10.55B. Fluttering of the leaflet may cause a momentarily immovable movement in the leaflet and would have an implication on the flow field immediately at the trailing edge of the leaflets. The leaflets will reach the fully opened position of 5 degrees with a peak angular velocity of 4.8 rad/s at t 5 0.112 s and maximum flow takes place. The leaflets will remain at the fully opened position till t 5 0.14 s. Fig. 10.56 shows a series of pressure contour plot for the opening phase at different time instant and angular positions on the mid-plane of y 5 12.5.

FIGURE 10.56 Pressure contour plots for the opening phase at different time instants and angular positions on the mid-plane of y 5 12.5. (A) t 5 0.06 s, θ 5 60.0 degrees; (B) t 5 0.09 s, θ 5 51.15 degrees; (C) t 5 0.094 s, θ 5 45.83 degrees; (D) t 5 0.098 s, θ 5 310.61 degrees; (E) t 5 0.102 s, θ 5 210.84 degrees; (F) t 5 0.106 s, θ 5 20.04 degrees; (G) t 5 0.109 s, θ 5 12.21 degrees; (H) t 5 0.112 s, θ 5 5 degrees.

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Computational Fluid-Structure Interaction

FIGURE 10.56 (Continued).

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FIGURE 10.56 (Continued).

Since a symmetric geometry for bi-leaflet valves with symmetric structure is simulated in this study, the computed results shown in the figures are mirrored along the x
y plane. Fig. 10.57 shows the pressure variation on the inner and outer surfaces of the leaflet near the leading and trailing edges during the opening phase. From Fig. 10.56, it can be seen that the pressure on the left ventricle is higher than the aorta for different time instants and angular positions of the leaflets. This pressure difference across the valve will cause a rotating moment of the leaflets and gradually affects the flow field around it. It is observed that there is a drastic pressure difference of about 13 kPa and 10.5 kPa between the leading edge and trailing edge, respectively, of the outer and inner surfaces of the leaflet at the beginning of the opening phase at t 5 0.06 s for θ 5 60.0 as seen in Fig. 10.57. This caused by an impetuous onset of moment as observed in Fig. 10.48 and the leaflets

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Computational Fluid-Structure Interaction

FIGURE 10.57 Pressure on the outer and inner leaflet surfaces during the opening phase.

began to accelerate with an angular velocity toward the fully opened position as seen in Fig. 10.54. As observed in Figs. 10.56 and 10.57, there are some locations in the immediate vicinity of the MHV experiencing a negative pressure at the inner surface of the leaflets, which indicates a recirculation zone occurring in that region. This phenomenon will be explained in details in the later part of this section. From t 5 0.09 s onwards, the pressure difference across the valve has gradually reduced and finally reached an equilibrium state at t 5 0.112 s for the fully opened position, where the ventricular pressure equals the aorta pressure. One consideration for an ideal MHV would be one where the pressure drop across the valve orifice is minimum. The effective orifice area of the device determines the value of this pressure drop. A smooth transition between left ventricle and ascending aorta will depend particularly on the geometry. The viscous effects, in this case, are secondary in the sense that the energy dissipated into the bulk fluid by viscosity is small compared with the work done by the pressure gradient. However, viscous effects such as separation can affect this pressure gradient quite considerably where the resulting wake, increases as a function of the downstream coordinate, decreases the effective orifice area. A series of streamline and velocity vector plots on the mid-plane of y 5 12.5, volume ribbons streamtrace plot, velocity and strength of vorticity

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FIGURE 10.58 Velocity profile extracted at different x position for the opening phase.

profiles extracted along the z-axis in different x position as indicated in Fig. 10.58, and vorticity iso-surfaces for the opening phase at different time instants and angular positions are shown in Figs. 10.59
10.66. Since a symmetric geometry for bi-leaflet valves with symmetric structure is simulated in this study, the computed results shown in the figures are mirrored along the x
y plane. From the figures, it is observed that the flow past the valve during the opening phase is characterized by the periodic vortex shedding in the wake of the valve leaflets. The typical three-entrainment-jet flow generated through bi-leaflet MHV orifices is formed; one jet is formed past the central orifice of the valve between the valve leaflets, and the other two jets are formed between the outer surfaces of the leaflets and the valve-housing wall. In this study, it is observed that vortices are formed immediately at the two outer jets when the leaflets open; the central jet appears to gain strength as it develops. It can be observed that there exist four well-organized counter-rotating vortices in the immediate vicinity of the MHV as seen in the x
z plane. This indicates that these structures survive throughout the duration of a heartbeat, which illustrates the presence of vortical structures that do not diffuse and induce a low-pressure region, which in turn enhances regurgitation. In Fig. 10.58, the positions X1 and X4 are along the pivot axis at x 5 72 mm. The positions X2 and X5 are at x 5 77 mm, and X3 and X6 are at x 5 82 mm. The positions at the left ventricle side located immediately before the leaflets are indicated by X7 and X8 at x 5 63 mm and x 5 67 mm, respectively, which are 4 mm apart. The positions at the aorta located immediately after the leaflets are indicated by X9 and X10 at x 5 86 mm and x 5 90 mm, respectively, which are 4 mm apart. Some of the velocity profiles may not be extracted in some of the x position due to the continuous motion of the leaflets.

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FIGURE 10.59 Various plots for the opening phase at t 5 0.06 s and θ 5 60.0 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.06 s and θ 5 60.0 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.06 s and θ 5 60.0 degrees. (C) Velocity profile on the mid-plane of y-axis at t 5 0.06 s and θ 5 60.0 degrees.

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205

FIGURE 10.59 (Continued).

At the instance when the leaflets start to move from the fully closed position at t 5 0.06 s, the flow in the central jet has a velocity range from 0.0 m/s to 0.8 m/s and registered a maximum value of around 3.1 m/s at the center of the jet at the X2 position as indicated in Fig. 10.59C. At the outer jet, it registered a maximum velocity of 1.46 m/s at z 5 11.9 mm and the X5 position. In the vicinity of the valve at both the left ventricle and aorta sides, the average velocity is 0.44 m/s at X7
X10 positions. There is no vortex formation seen in this angular position and the flow streamlines are quite uniform at the aorta. When the leaflets rotated to θ 5 51.15 degrees at t 5 0.09 s under the blood
leaflet interaction effect, there are two small counter-rotating vortices formed at the leading edge of the inner surface having a center at x 5 76.03 mm and z 5 6 1.18 mm, and two bigger ones formed at the trailing edge having a center at x 5 82.89 mm and z 5 6 10.93 mm as seen in Fig. 10.60A and B. The formation of these vortices is a result of the outer edge of the jet rolling up. At the center of the channel, the jet is a straight plane jet whereas on the outer two trailing edges of the leaflets, the jet is a

FIGURE 10.60 Various plots for the opening phase at t 5 0.09 s and θ 5 51.15 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.09 s and θ 5 51.15 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.09 s and θ 5 51.15 degrees. (C) Volume ribbons streamtrace at t 5 0.09 s and θ 5 51.15 degrees. (D) velocity profile on the mid-plane of y-axis at t 5 0.09 s and θ 5 51.15 degrees. (e) Vorticity profile on the mid-plane of y-axis at t 5 0.09 s and θ 5 51.15 degrees. (f) Vorticity iso-surfaces at t 5 0.09 s and θ 5 51.15 degrees.

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207

FIGURE 10.60 (Continued).

curved plane jet. It is evident from the bi-peaked velocity profile at X9 position, shown in Fig. 10.60D, that the center of the vortices experiences a velocity value of 1.16 m/s and a bell-shaped velocity profile at the central jet. These circulation zones create a negative pressure region as shown in Fig. 10.50B. The velocity is at its maximum at the outer jet as seen from the contour plot. From Fig. 10.54D, the central jet registered a maximum velocity of 3.6 m/s at X1 position and reduced to 2.4 m/s at X2 position. This high velocity at X1 is to be expected since the central orifice acted like an impinging jet in this position. The peak velocity at X5 position is almost twofolds higher than that of X4 position due to the converging passage at the outer jet. The leaflets rotated to a new angular position of θ 5 45.83 degrees at t 5 0.094 s under the hemodynamic forces acting on them. It can be seen that

208

Computational Fluid-Structure Interaction

FIGURE 10.60 (Continued).

the three jets form a central diverging channel and two sides converging channels. The clearances in the three jets are still quite narrow showing a strong tendency of the blood rushing toward the aorta with an increase in the inlet velocity. Due to the strong influence of the blood
leaflet interaction, viscous effect and the surge in velocity of the three jets, the four vortices have grown in size and moved away from the leaflets toward downstream having the centers of the vortices at x 5 710.21 mm, z 5 6 1.39 mm and x 5 85.22 mm, z 5 6 10.66 mm. Due to the recirculation zone, the velocity at X2 position is lower than that of X1 position having a value of 2.8 m/s for the former and 3.6 m/s for the latter as shown in Fig. 10.61D. The difference between the peak velocity at X4 and X5 positions is smaller in this angular position as compared to that of θ 5 51.15 degrees. This is mainly due to the widening of the gap between the leaflets and the valve-housing wall. A positive and negative bipeaked velocity profile is seen at X9 position with a peaked positive value of 2.7 m/s and peaked negative value of 0.3 m/s. This negative velocity

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209

FIGURE 10.60 (Continued).

characterized a recirculation zones exist in that region. The flow is accelerated and the velocity gradient became steeper. It is evident by a twofolds increased in the peak velocity from the previous angular position. At X10 position, further downstream at the aorta, a round bi-peaked velocity of 1.0 m/s with a bell-shaped velocity profile is obtained at this location. At t 5 0.098 s, the leaflets opened to an angular position of θ 5 310.61 degrees. Although the decreased in velocity to 2.4 m/s at X1 position as the gap at the central jet between the two leaflets widened, the blood
leaflet interaction caused the size of the vortices to increase further and moved further downstream to x 5 80.91 mm, z 5 6 1.81 mm and x 5 87.42 mm, z 5 6 7.32 mm as observed in Fig. 10.62A and B. The peak velocity at X5

FIGURE 10.61 Various plots for the opening phase at t 5 0.094 s and θ 5 45.83 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.094 s and θ 5 45.83 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.094 s and θ 5 45.83 degrees. (C) Volume ribbons streamtrace at t 5 0.094 s and θ 5 45.83 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.094 s and θ 5 45.83 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.094 s and θ 5 45.83 degrees. (F) Vorticity iso-surfaces at t 5 0.094 s and θ 5 45.83 degrees.

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211

FIGURE 10.61 (Continued).

position has gradually reduced to 1.1 m/s as the gap of the outer jet continued to widen. The positive bi-peaked velocity at X9 position has reduced to 2.35 m/s with the widening of the jets. On the other hand, the negative bipeaked velocity has increased to 2
0.55 m/s, which signifies a stronger recirculation zone as it is evident from Fig. 10.62A and B that the vortices have grown in size. At X10 position, the round bi-peaked velocity profile at θ 5 45.83 degrees has changed to a sharp pointed bi-peaked velocity profile having a velocity of 1.85 m/s as the vortex columns are being shed toward downstream. At t 5 0.102 s, the leaflets opened to an angular position of θ 5 210.84 degrees. From this position onwards, the leaflets are opened wide enough to allow the measurement of velocity at X3 and X6 positions as indicated in Fig. 10.58. The two vortices in the center have grown in size and moved further downstream at the aorta to x 5 81.55 mm, z 5 6 2.73 mm as shown in Fig. 10.63A and B. These vortices are being attached to the inner surfaces at the trailing edges of the leaflets and they will move downstream any further.

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Computational Fluid-Structure Interaction

FIGURE 10.61 (Continued).

In addition, they will start growing smaller due to the narrowing of the central channel. Whereas the other two vortices located farthest downstream at the aorta have shrunk in size and weaker in magnitude. This is to be expected since the vortices will lose its momentum as it moved further downstream away from the leaflets to a new location having the center of vortices at x 5 90.67 mm and z 5 6 5.84 mm. The peak velocity at the central jet has gradually decreased at X1 and X2 positions to 1.8 m/s and 1.7 m/s, respectively, as shown in Fig. 10.57D. There is a negative peak velocity of 0.4 m/s at X3 position, which indicates a recirculation zone as evidently shown by the two bigger vortices existed along X3 in Fig. 10.63A and B. At X6 position where the blood flows from the left ventricle to the aorta at the outer jets registered a peak velocity of 1.48 m/s and the peak velocity has reduced to 0.94 m/s at X5 position. A tri-peaked velocity profile is obtained in the X9 position having a velocity of 1.5 m/s at the two outer-peaked profile and 0.95 m/s at the center-peaked profile occurred at the center of the channel. At X10 position, a positive tri-peaked of 1.6 m/s and negative

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213

FIGURE 10.61 (Continued).

bi-peaked of 0.2 m/s velocity profile is obtained. Since the widening of the gap in the central orifice should reduce the velocity at the center of the channel, this flow phenomenon is not seen in this angular position and also the next few positions. This might be due to the increased in the inlet velocity and the angular velocity of the leaflets. The leaflets opened to an angular position of θ 5 20.04 degrees at t 5 0.106 s. The two vortices attached to the inner surface of the leaflets have shrunk in size and moved to x 5 83.03 mm and z 5 6 3.79 mm. There are two very small vortices observed at x 5 83.31 mm and z 5 6 6.69 mm and it is at the tip of the leaflets as shown in Fig. 10.64A and B. The

FIGURE 10.62 Various plots for the opening phase at t 5 0.098 s and θ 5 310.61 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.098 s and θ 5 310.61 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.098 s and θ 5 310.61 degrees. (C) Volume ribbons streamtrace at t 5 0.098 s and θ 5 310.61 degrees. (D) Velocity profile on the mid-plane of yaxis at t 5 0.098 s and θ 5 310.61 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.098 s and θ 5 310.61 degrees. (F) Vorticity iso-surfaces at t 5 0.098 s and θ 5 310.61 degrees.

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215

FIGURE 10.62 (Continued).

existence of these small vortices might be due to the vortices at the trailing edge have detached itself away from the edge formed by the inner surface and the tip of the leaflets. The two vortices existed farthest downstream at θ 5 210.84 degrees as shown in Fig. 10.63A and B have propagated out of the channel in this time instant with the increased in the inlet velocity. The continuous widening of the central orifice has come to an angular position where the velocity profiles at X1; X2 and X3 positions have the same peaked velocity of 1.32 m/s as shown in Fig. 10.64D. The peak velocities at X5 and X6 positions have reduced to 0.8 m/s and 1.02 m/s, respectively and it has a bigger U-shaped velocity profile with the widening of the outer jet. A tripeaked velocity profile is obtained at X9 position having a magnitude of 1.0 m/s for the outer bi-peaked profile and 1.2 m/s for the center-peaked profile. It is observed that at this time instant, the peaked velocity for the outer bi-peaked profile is smaller than the center-peaked profile due to the

216

Computational Fluid-Structure Interaction

FIGURE 10.62 (Continued).

continuous increase of momentum in the central jet. Similarly, a tri-peaked velocity profile is obtained at X10 position having an outer bi-peaked velocity of 1.0 m/s and a smaller center-peaked velocity of 0.58 m/s. At t 5 0.109 s, the leaflets opened to an angular position of θ 5 12.21 degrees. From Fig. 10.65A and B, it is observed that the two vortices at θ 5 20.04 degrees have shrunk even smaller in size and moved to the edge of the leaflets. These vortices have moved to x 5 83.63 mm and z 5 6 3.83 mm. This is due to the widening of the central jet that weakens the strength of the vortices. At X1, X2, and X3 positions, they have almost the same velocity of 1.1 m/s and this value is reduced from the previous angular position. The peak velocities at X5 and X6 positions have reduced to 0.74 m/s and 0.84 m/s, respectively. A tri-peaked velocity profile, smaller in magnitude, is obtained in X9 position having the same velocity of 0.8 m/s for the three peaked. A bi-peaked velocity profile of 0.82 m/s with a humpshaped profile of 0.4 m/s at the central jet is obtained at X10 position. It is

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217

FIGURE 10.62 (Continued).

observed that the flow has lost its momentum at this position as compared to the previous position. Finally, the leaflets have moved to the fully opened position of θ 5 5 degrees at t 5 0.112 s. From Fig. 10.66A, it is observed that all the vortices have disappeared in the channel with the outer and central jets at its maximum flow. The effect of the leaflets movement on the flow seems to have weakened with a reduced velocity profile obtained in X1, X2, and X3 positions as observed in Fig. 10.66C and the deceleration effect could have resulted from the diverging cross section between the leaflets in their fully opened position. The peaked velocity at X1 and X2 positions is 0.88 m/s and 0.76 m/s at X3 position. The velocity profiles at X4, X5, and

FIGURE 10.63 Various plots for the opening phase at t 5 0.102 s and θ 5 210.84 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.102 s and θ 5 210.84 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.102 s and θ 5 210.84 degrees. (C) Volume ribbons streamtrace at t 5 0.102 s and θ 5 210.841 degrees. (D) Velocity profile on the mid-plane of yaxis at t 5 0.102 s and θ 5 210.84 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.102 s and θ 5 210.84 degrees. (F) Vorticity iso-surfaces at t 5 0.102 s and θ 5 210.84 degrees.

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219

FIGURE 10.63 (Continued).

X6 positions almost coincided with each other having an average velocity of 0.68 m/s. This is to be expected since the passage at the outer jet did not converge too much and this allowed an almost uniform flow to take place. Two circumferential-shaped like velocity profile with a bell-shaped profile at the center is obtained at X9 position with a peak velocity of 0.67 m/s and 0.52 m/s for the former and latter profiles, respectively. The same shape like velocity profile as X9 is obtained at X10 position, except with a lower peak velocity of 0.63 m/s for the two circumferential-shaped like profile and 0.41 m/s for the bell-shaped profile. At this fully opened position, the leaflets might experience a slight fluttering due to the continuous inflow of inlet velocity till it reached the maximum velocity of 0.5 m/s at t 5 0.14 s as shown in Fig. 10.46. A series of interesting vorticity iso-surfaces plots at different time instants and angular positions are obtained in this study as shown in

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Computational Fluid-Structure Interaction

FIGURE 10.63 (Continued).

Figs. 10.60(F)
10.65(F). Fig. 10.67 shows the plan view of Figs. 10.60(F)
10.65(F) for different angular positions. Figs. 10.60(F)
10.65(F) and 10.59 represent the MHV leaflets. Figs. 10.60(E)
10.65(E) present the strength of vorticity profiles extracted along the z-axis in different x position as indicated in Fig. 10.58. Some of the velocity profiles may not be extracted in some of the x position due to the continuous motion of the leaflets. All the plots for vorticity iso-surfaces depict different contour surfaces represent different levels of vorticity or shear intensity. On the same surface, the magnitude of shear is constant. Vorticity surfaces also represent the 3D topology of the shear layers, vortex instability, and the flow structure. At t 5 0.09 s, when the leaflets opened to an angular position of θ 5 51.15 degrees, the wake of the valve revealed an intricate pattern of shed layers that roll up on top of each other at the outer jets. The formation of all these vortex columns might be due to the sharp edges of the leaflets. The vorticity strength at X1 position is quite high and registered a nondimensional value of 10.0 as

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221

FIGURE 10.63 (Continued).

indicated in Fig. 10.60E. This high vorticity strength is due to the smallconverged gap at the central jet. Similarly, the high vorticity strength at X5 position is due the converging passage at the outer jets. The vorticity strength is zero for most part along the z-axis for X7
X10 positions; expect at the outer jet near the wall where the shear layers roll-ups existed. As the leaflets continued to open to θ 5 45.83 degrees at t 5 0.094 s, the vorticity strength at the central jet has weakened for both X1 and X2 positions. It is observed that there are several semi-circular layers of vorticity iso-surfaces of different intensity formed at the outer jets near the channel wall. The shear intensity near the wall is high as compared to the shear intensity away from the wall. The shed layers that roll up on top of each other at the outer jets as observed

FIGURE 10.64 Various plots for the opening phase at t 5 0.106 s and θ 5 20.04 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.106 s and θ 5 20.04 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.106 s and θ 5 20.04 degrees. (C) Volume ribbons streamtrace at t 5 0.106 s and θ 5 20.04 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.106 s and θ 5 20.04 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.106 s and θ 5 20.04 degrees. (F) Vorticity iso-surfaces at t 5 0.106 s and θ 5 20.04 degrees.

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223

FIGURE 10.64 (Continued).

from the previous time instant is still being observed in this time instant with an elongated jet as seen in Fig. 10.67B. A positive and negative bi-peaked vorticity profile of 2.8 and 20.3, respectively is obtained at X9 position. It is noteworthy that the positive vorticity strength at X9 position considered in this time instant is the highest as compared to the rest considered at a later time instants, and as well as the negative vorticity strength is only obtained in this time instant. And the vorticity strength at X7 and X8 positions has a value of 0.5 for most part along the z-axis as compared to a value of 0.0 for the rest of the angular positions. The shed layers that roll up on top of each other at the outer jets have become less prominent for θ 5 310.61 degrees at t 5 0.098 s as observed in Fig. 10.62F and it rolled up to form a conical structures along the y-axis. The shear layers around the central jet started to roll up and bulge out with the center concave in, resembling the shape of a red blood cell from the y
z plane. This represents a vortex ring with a jet inside the ring at the central

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Computational Fluid-Structure Interaction

FIGURE 10.64 (Continued).

orifice and it becomes prominent as the leaflets continue to open as observed in the next few plots of vorticity iso-surfaces. Both the vorticity layers for the center and outer jets are elongated further downstream as seen in Fig. 10.63C. The vorticity profiles at X9 and X10 positions show four perks at the outer jets and zero vorticity along the central jet. The “mushroomheaded” like vorticity structure at the central jet viewing from the x
z plane in Fig. 10.63D for θ 5 210.84 degrees at t 5 0.102 s has grown in size and extended further downstream from the previous angular position. The shear layers at the outer jets are caving in toward the center of the channel and the roll up on top of each other is not so prominent as compared to the previous time instant. A bi-peaked vorticity profile of 1.6 is obtained at X1 position and a double “hump-shaped” like vorticity profiles of 1.05 and 0.85 are obtained at X2 and X3 positions, respectively. There are two peaks of 0.3 at the central jet, another two peaks of 0.85 at the outer jets and reached a maximum value of 1.7 near the channel wall for X9 position. And there are four peaks of 0.8 and 0.9 at the outer jets for X10 position.

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225

FIGURE 10.64 (Continued).

The shed layers, roll up at the outer jets, started to thin out or break up into smaller structures with the disappearance of the vortex pair for θ 5 20.04 degrees at t 5 0.106 s as observed in Fig. 10.64F and extended further downstream. The “mushroom-headed” like vorticity structure at the central jet has grown bigger in size with the outer shear layer rolled up inwards leaving a hollow at the center of the structure looked like an “O-ringed” shape, which represents the shape of the jet itself. The bi-peaked vorticity profile at X1 position from the previous position has reduced to 1.15 and the double “hump-shaped” like vorticity profiles X2 and X3 positions have

FIGURE 10.65 Various plots for the opening phase at t 5 0.109 s and θ 5 12.21 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.109 s and θ 5 12.21 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.109 s and θ 5 12.21 degrees. (C) Volume ribbons streamtrace at t 5 0.109 s and θ 5 12.21 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.109 s and θ 5 12.21 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.109 s and θ 5 12.21 degrees. (F) Vorticity iso-surfaces at t 5 0.109 s and θ 5 12.21 degrees.

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227

FIGURE 10.65 (Continued).

reduced to 0.7 and 0.5, respectively. It is observed in Fig. 10.65F that only one layer of vorticity of the same low intensity formed at the outer jets near the channel wall for θ 5 12.21 degrees at t 5 0.109 s. A smooth semi-circular layer of vorticity is seen at the outer jets with the roll up on top of each other and smaller structures for the vorticity disappeared. The hollow vorticity structure at the central jet looked like an “oval-ringed” with the center concave in. This hollow vorticity structure at the central jet explains why some part of the profiles at X9 and X10 positions are zero and it represents the core of the jet.

10.3.6 Closing Phase The closing phase of a 29 mm SJM aortic bi-leaflet MHV based on the parameters listed in Section 10.3.2 is computed. Since the flow field

228

Computational Fluid-Structure Interaction

FIGURE 10.65 (Continued).

configuration does not change for a cardiac cycle, the general flow field structure in the closing phase is the same as the opening phase in this study. However, due to the flow reversal in the closing phase, the upstream and downstream boundary conditions in the opening phase are totally reversed in the current closing phase. In the present study, attention is focused on the flow field downstream of the left ventricle immediately after the MHV. The computational grids for both the background mesh and the OG are shown in Figs. 10.68 and 10.69. Fig. 10.70 shows a perspective view of the combination of Figs. 10.68 and 10.69. Fig. 10.71 shows the definition of the various terms used in the present computation. There is a slight difference between the present configuration in Fig. 10.68 and the configuration for the opening phase in Fig. 10.50. The velocity boundary

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229

FIGURE 10.65 (Continued).

condition can be applied to the inlet without problem; however, the outlet is too near to where the valve is located, it is evident that the flow will not be fully developed in the outlet. This causes the problem of the pressure boundary not being readily applied to the outlet. It is improper to extend the outlet to a farther position since this is not the real situation in the human body and most of the heart valve test rigs. It is also impractical to apply a special developing pressure profile at the outlet, as it is difficult to devise it. To circumvent this effect, a rectangular box of 60 mm 3 80 mm 3 40 mm is added to the existing outlet, which has the

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Computational Fluid-Structure Interaction

FIGURE 10.66 Various plots for the opening phase at t 5 0.112 s and θ 5 5.0 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.112 s and θ 5 5.0 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.112 s and θ 5 5.0 degrees. (C) Velocity profile on the mid-plane of y-axis at t 5 0.112 s and θ 5 5.0 degrees.

same configuration as the heart valve test rig proposed by Shi [24]. As the rectangular cross-sectional area of 60 mm 3 40 mm is much bigger than that of π 3 12:5 mm 3 12:5 mm in the cylindrical flow channel, the pressure boundary condition can be readily applied to it, without worrying about the setting in the original outlet condition. To facilitate the computation for the closing phase, the velocity equation as specified in Eq. (10.2) is applied to

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231

FIGURE 10.66 (Continued).

the inlet, with the velocity set as backward flow (i.e., Uo 5 20.5 m/s), and a fixed pressure is applied to the outlet. It is noteworthy that the computation of closing phase is much more difficulty than the opening phase due to some contributing factors in the flow field. First, the turbulent effect is more prominent in this phase [35,36] and the historical data of the opening phase did not take account into the present study due to the change in the inlet and outlet positions. Second, the SJM valves close with a prominent backflow and the closing of this valve is an accelerating process, which in addition to the backflow, it acts as a source of water-hammer effect [30]. Next, at the instant when the valve is about to reach the fully closed position, a phenomenon known as cavitation and squeeze flow might take place, although the study on this phenomenon is not the focus of the present study. Cavitation is the rapid formation of vapor bubbles caused by a transient reduction in local pressure field. The

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Computational Fluid-Structure Interaction

FIGURE 10.67 Plan view for the vorticity iso-surfaces shown in Figs. 10.60(F)
10.65(F). (A) θ 5 51.15 degrees; (B) θ 5 45.83 degrees; (C) θ 5 310.61 degrees; (D) θ 5 210.84 degrees; (E) θ 5 20.04 degrees; (F) θ 5 12.21 degrees.

collapse of the cavitation bubbles subsequent to pressure recovery generates high-speed water jets with considerable energy that may damage the valve surface (erosion marks) or blood hemolysis [31]. The cavitation event is initiated by valve closure and its duration has been shown to be on the order of millisecond [32]. One of the mechanisms that contribute to cavitation formation is the appearance of high squeeze flow velocity that is

IOM FSI Model Validations and Applications Chapter | 10

FIGURE 10.68 Background mesh, immersed leaflet and valve housing for closing phase.

FIGURE 10.69 OG with aortic valve leaflet at fully opened position for the closing phase.

233

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Computational Fluid-Structure Interaction

FIGURE 10.70 Combination of background mesh and OG with leaflet.

FIGURE 10.71 Definition of terms used in the closing phase for x
z plane.

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235

characterized by large pressure drops. During valve closure, maximum velocities of up to 2
5 m/s have been experimentally documented as the flow is squeezed into a narrow passage while being subjected to the high closing velocity of the valve itself [33]. Despite all these difficulties faced, it is still worthwhile to carry out the current research on the closing phase as a first attempt in this field and results obtained might deviate from the real human conditions. The computational background mesh shown in Fig. 10.68 has a grid size of 219,326 nodes and 1,231,479 tetrahedral elements and it is partitioned into 16 subdomains for parallel computation, whereas the OG consists of 85,899 nodes and 452,028 tetrahedral elements and partitioned into four subdomains. The grid convergence studies are performed for all the meshes and an optimum grid size for both background mesh and OG are obtained. A three-level MG is employed for the background mesh to compute the flow. There are 54,834 nodes and 307,896 tetrahedral elements in the coarse grid, and 14,432 nodes and 81,032 tetrahedral elements in the coarsest grid. The valve leaflet shown in Fig. 10.68 is immersed into the background flow domain in the fully opened position (5 degrees) to define the physical boundary of the leaflet for IOM. The search algorithm depicted in Section 7.1 is used to find those fluid nodes within the immersed leaflet and classify it as solid nodes. In order to track the motion of the leaflet as accurate as possible and to accurately capture the fine details of the vortex shedding phenomena, the background mesh is refined in the same regions as the background mesh for the opening phase. For simplicity, the flow field variables in the simple pinned hinge and the valve housing shown in the figure are set to zero at all times. The flow is initiated with the leaflet in the fully opened position and a steady flow solution is obtained at the end of the fully opened phase. It is used as the initial condition for the moving leaflet closing phase computation. The OG shown in Fig. 10.69 will rotate about the pivot axis according to the blood
leaflet interaction from the fully opened position. For the pulsatile flow inlet condition specified in Eq. (10.2), the CFL is set to 1.2 and the physical timestep is set to 0.001 s for better temporal resolution. The number of pseudo subiterations per time-step is set to 200 W-cycles in order for the computation to achieve a divergence free condition. In the present study, the closing phase occurred from t 5 0.14 s to t 5 0.35 s as shown in Fig. 10.72. The closing phase can be subdivided into two phases, which consists of the closing phase from t 5 0.14 s to t 5 0.323 s; and fully opened phase from t 5 0.323 s to t 5 0.35 s. Fig. 10.72 shows the nondimensionalized rotating moment derived from Eqs. (10.5) and (10.6) due to the forces generated by the pressure and shear stresses in the x-plane and z-plane. The figure also shows the summation of these moments. Similar to the opening phase, the moment due to pressure force contributed to most part of the total moment. The moment due to the leaflet shear

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FIGURE 10.72 Rotating moment acting on the leaflet for the closing phase.

stresses might be under-estimated due to the lack of a suitable turbulence model in this study and turbulent shear stresses should be considered in the future work in order to have a more accurate contribution of the moment due to leaflet shear stresses. Although the shear stresses are small compared with the pressure, it can still be important to the red blood cells. From Fig. 10.66, it is observed that from t 5 0.14 s to t 5 0.175 s, the total moment is almost zero. From t 5 0.175 s, there is an impetuous onset of moment due to the gain in momentum in the flow field. From t 5 0.175 s to t 5 0.275 s, the magnitude of the total rotating moment increased in an exponential manner and reached a peak nondimensionalized moment of 0.96. It showed a deceleration of rotating moment after t 5 0.275 s and reached a low rotating moment of 0.44 at around t 5 0.295 s. This deceleration might be caused by the flow separation behind the leaflets, which causes a drop in lift, similar to an aerofoil when the angle of attack is large enough. It is observed that after t 5 0.295 s, the total rotating moment developed in a fluctuating manner till the end of the closing phase. The occurrence of this fluctuation might be due to the valve closure with a prominent regurgitated flow, which results in a water-hammer effect. After t 5 0.323 s, the valve has reached the fully closed position and the moment remained at zero value till the end of the closing phase.

IOM FSI Model Validations and Applications Chapter | 10 (A)

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FIGURE 10.73 Change of leaflet (A) angular velocity (ω) and (B) angular position (θ) from the fully opened to the fully closed position for the closing phase.

After obtaining the rotating moments generated by the hemodynamic forces shown in Fig. 10.72, the angular velocity (ω) and angular position (θ) for the closing phase are obtained by solving the structure dynamics equation (Eq. (10.3)). Fig. 10.73 shows both the change of leaflet angular velocity and position from the fully opened to the fully closed position. It should be noted that the positive rotating moments and angular velocity shown in the respective figures indicating that the leaflet is rotating in the counter-clockwise direction as shown in the sign convention as stated in Fig. 10.48. From the figure, it is observed that at the start of the closing phase at t 5 0.14 s to t 5 0.17 s, the leaflets remained stationary with zero angular velocity. From t 5 0.17 s onwards, the leaflets started to gain some angular velocity, but it is too small to cause any significant movement in the leaflet till t 5 0.21 s. This is due to the hemodynamic forces acting on both sides of the leaflets are almost identical. From t 5 0.21 s to t 5 0.26 s, the valve closed rapidly with an exponential increase of angular velocity from almost 0.0 rad/s to 0.45 rad/ s. The imbalance of hemodynamic forces acting on both sides of the leaflet caused it to rotate around the pivotal axis. It is observed that the valve will close gradually with a steep gradient from t 5 0.26 s to t 5 0.323 s. The leaflets will experience a decrease in angular velocity during the period from t 5 0.275 s to t 5 0.295 s due to the decreased in total rotating moment as observed from Fig. 10.72. The leaflets will reach the fully closed position of 60 degrees with a peak angular velocity of 2.2 rad/s at t 5 0.323 s. The leaflets will remain at the fully closed position till t 5 0.35 s and the next cycle of opening phase will take place again. Fig. 10.74 shows a series of pressure contour plot for the closing phase at different time instants and angular positions on the mid-plane of y 5 12.5.

FIGURE 10.74 Pressure contour plots for the closing phase at different time instants and angular positions on the mid-plane of y 5 12.5. (a) t 5 0.14 s, θ 5 5 degrees; (b) t 5 0.27 s, θ 5 12.86 degrees; (c) t 5 0.282 s, θ 5 20.06 degrees; (d) t 5 0.294 s, θ 5 210.74 degrees; (e) t 5 0.304 s, θ 5 310.07 degrees; (f) t 5 0.310 s, θ 5 45.19 degrees; (g) t 5 0.316 s, θ 5 51.77 degrees; (h) t 5 0.323 s, θ 5 60.0 degrees.

IOM FSI Model Validations and Applications Chapter | 10

FIGURE 10.74 (Continued).

239

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FIGURE 10.74 (Continued).

Since a symmetric geometry for bi-leaflet valve with symmetric structure is simulated in this study, the computed results shown in the figures are mirrored along the x
y plane. Fig. 10.75 shows the pressure variation on the inner and outer surfaces of the leaflet near the leading and trailing edges during the closing phase. From Fig. 10.74, it is observed that the pressure on the aorta is higher than the left ventricle for different time instants and angular positions of the leaflets. This pressure difference across the valve will cause a rotating moment of the leaflets and gradually affects the flow field around it. From Fig. 10.75, it can be

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Time (sec) FIGURE 10.75 Pressure on the outer and inner leaflet surfaces during closing phase.

seen that the pressure difference across the valve for both the leading and trailing edges at t 5 0.14 s is almost equal and this explains why the leaflets remained at the fully opened position as shown in Fig. 10.73B. The pressure difference across the valve has increased gradually and caused a noticeable rotating movement of the leaflets at t 5 0.21 s. The leaflets continued to rotate toward the fully closed position with an increasing pressure difference across the valve. The pressure difference is more prominent toward the end of the closed position as observed from Fig. 10.75. This huge pressure difference across the valve caused the leaflets to close tightly so that no backflow of blood is allowed from the left ventricle into the aorta. A series of streamline and velocity vector plots on the mid-plane of y 5 12.5, volume ribbons streamtrace plot, velocity and strength of vorticity profiles extracted along the z-axis in different x position as indicated in Fig. 10.76, and vorticity iso-surfaces for the closing phase at different time instants and angular positions are shown in Figs. 10.77
10.84. Since a

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FIGURE 10.76 Velocity profile extracted at different x position for the closing phase.

symmetric geometry for bi-leaflet valve with symmetric structure is simulated in this study, the computed results shown in the figures are mirrored along the x
y plane. The typical three-entrainment-jet flow as observed in the opening phase is also perceived in this phase. The positions in Fig. 10.76 are exactly the same as those in the opening phase and reiterated in the following sentence. The positions X1 and X4 are along the pivot axis at x 5 72 mm. The positions X2 and X5 are at x 5 77 mm, and X3 and X6 are at x 5 82 mm. The positions at the left ventricle side located immediately after the leaflets are indicated by X7 and X8 at x 5 63 mm and x 5 67 mm, respectively, which are 4 mm apart. The positions at the aorta located immediately before the leaflets are indicated by X9 and X10 at x 5 86 mm and x 5 90 mm, respectively, which are 4 mm apart. Some of the velocity profiles may not be extracted in some of the x position due to the continuous motion of the leaflets. The negative velocity profiles represent a backflow flow velocity from the aorta to the left ventricle and it should be taken as a velocity against the flow convention for the opening phase. There is no vortex formation seen in the fully opened position, θ 5 5 degrees, at the starting of the closing phase, t 5 0.14 s and the flow is quite smooth from the aorta to the left ventricle as observed from Fig. 10.77A and B. The flow in the central jet has a peak velocity of 20.68 m/s, 20.6 m/s, and 20.48 m/s at X1, X2, and X3 positions, respectively. At the outer jets, the velocity profiles for X4 and X5 positions are almost the same since the gap formed by the outer surface of the leaflets and the channel wall is almost equal. It registered an average maximum velocity of 20.65 m/s. In the vicinity near the valve leaflets at the left ventricle side, the profiles are almost the same with a bi-peaked velocity of 20.38 m/s and 20.26 m/s, and 20.62 m/s

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243

FIGURE 10.77 Various plots for the closing phase at t 5 0.14 s and θ 5 5.0 degrees. (a) Streamlines plot on the mid-plane of y-axis at t 5 0.14 s and θ 5 5.0 degrees. (b) Velocity vector plot on the mid-plane of y-axis at t 5 0.14 s and θ 5 5.0 degrees. (c) Velocity profile on the midplane of y-axis at t 5 0.14 s and θ 5 5.0 degrees.

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FIGURE 10.77 (Continued).

and 20.7 m/s at the center of the channel for X7 and X8 positions, respectively. The leaflets remained at the fully opened position for quite some time till t 5 0.21 s and at t 5 0.27 s, the leaflets have rotated to θ 5 12.86 degrees under the influence of the blood
leaflet interaction. At this position, there are two small counter-rotating vortices formed at x 5 65.52 mm and z 5 6 1.75 mm, in the immediate vicinity after the leading edge of the leaflets. And two smaller vortices circulating at the outer surface of the trailing edge, located at x 5 80.29 mm and z 5 6 6.07 mm. These vortices and recirculation might result in accumulation of harmful materials (fatty deposits) and blood clots on the leaflet surfaces. It is observed from Fig. 10.78A and B that there are two elongated vortices on both the sides of the channel wall, which might be due to the blood
leaflet interaction and viscous effects. The two small “hump-shaped” like velocity profiles extracted from X1 position have a velocity ranged from 20.15 m/s to 0.18 m/s as observed in Fig. 10.78D. At X2 and X3 positions, the bell-shaped velocity profiles have

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245

FIGURE 10.78 Various plots for the closing phase at t 5 0.27 s and θ 5 12.86 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.27 s and θ 5 12.86 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.27 s and θ 5 12.86 degrees. (C) Volume ribbons streamtrace at t 5 0.27 s and θ 5 12.86 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.27 s and θ 5 12.86 degrees.

246

Computational Fluid-Structure Interaction

FIGURE 10.78 (Continued).

an average maximum velocity of 20.08 m/s. The maximum velocity of 20.58 m/s at X8 position is higher than that of X7 position, having a value of 20.365 m/s. This is expected since the velocity in the immediate vicinity after the leaflets usually will be higher than those farther downstream. At t 5 0.282 s, the leaflets have rotated to a new angular position of θ 5 20.06 degrees under the hemodynamic forces acting on it. The two vortices in the immediate vicinity after the leading edge of the leaflets have grown in size and moved further downstream to x 5 63.9 mm and z 5 6 1.67 mm as observed in Fig. 10.79A and B. The two smaller vortices circulating at the outer surface of the trailing edge still existed in this angular position and they are located at x 5 710.9 mm and z 5 6 7.0 mm. It is observed that two small, tiny vortices circulating at the tip of the leading

FIGURE 10.79 Various plots for the closing phase at t 5 0.282 s and θ 5 20.06 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.282 s and θ 5 20.06 degrees. (B)Velocity vector plot on the mid-plane of y-axis at t 5 0.282 s and θ 5 20.06 degrees. (C) Volume ribbons streamtrace at t 5 0.282 s and θ 5 20.06 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.282 s and θ 5 20.06 degrees.

248

Computational Fluid-Structure Interaction

FIGURE 10.79 (Continued).

edge at x 5 610.69 mm and z 5 6 1.21 mm. The two elongated vortices on both sides of the channel wall are still circulating around in that area. The velocity profiles and maximum velocity values for X1
X3 positions are more or less the same as the previous angular position. In this position, the maximum velocity of 20.44 m/s at X8 position is lower than that of X7 position, having a value of 21.24 m/s, which is a completely opposite from the previous position. This is due to the narrowing of the gap between the leaflets in the central jet, which caused by the surge in the velocity along the center of the channel. The leaflets closed to an angular position of θ 5 210.74 degrees at t 5 0.294 s. The number of vortices remained the same as the previous position and has yet to propagate out of the channel as seen in Fig. 10.80A and B. The only difference is that they have propagated further downstream and grown bigger in size. Due to the narrowing of the gap

FIGURE 10.80 Various plots for the closing phase at t 5 0.294 s and θ 5 210.74 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.294 s and θ 5 210.74 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.294 s and θ 5 210.74 degrees. (C) Volume ribbons streamtrace at t 5 0.294 s and θ 5 210.74 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.294 s and θ 5 210.74 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.294 s and θ 5 210.74 degrees. (F) Vorticity iso-surfaces at t 5 0.294 s and θ 5 210.74 degrees.

250

Computational Fluid-Structure Interaction

FIGURE 10.80 (Continued).

in the central jet, the two bell-shaped velocity profiles at X2 and X3 positions started to show some velocity difference between them, having a maximum velocity of 20.02 m/s for the former and 20.09 m/s for the latter. The velocity profiles for X7 and X8 positions are almost the same as the previous position, except having a slightly higher value of 21.44 m/s at X7 position as observed in Fig. 10.80D. At t 5 0.304 s, the leaflets have closed to an angular position of θ 5 310.07 degrees. It can be seen that the three jets form a central converging channel at the leading edges and two side converging channels at the trailing edges. The clearances in the three jets are narrowed in this position, showing a strong tendency of the blood rushing toward the left ventricle side. The surging of blood at the converged jets with an impetuous onset of velocity can be shown by the velocity profiles at X4, X5, and X8 positions in Fig. 10.81D, which show a maximum value of 22.0 m/s, 21.9 m/s, and a

IOM FSI Model Validations and Applications Chapter | 10

251

FIGURE 10.80 (Continued).

bi-peaked of 21.2 m/s, respectively. From this position onwards, the velocity profiles at X3 and X6 positions cannot be extracted due to the continuous motion of the leaflets. As the flow progress, the two vortices farther downstream have propagated to a new location of x 5 510.02 mm and z 5 6 1.59 mm. There are two small new vortices appeared at the trailing edge of leaflets followed by the two existing vortices which have moved to x 5 76.27 mm and z 5 6 10.47 mm. Based on Fig. 10.81A and B, the two elongated vortices on both the sides of the channel wall show a strong circulation effect and the high-speed jet might cause a shearing effect on the channel wall. The leaflets continued to close under the influence of the hemodynamic forces acting on it and at t 5 0.31 s, it has reached to an angular position of θ 5 45.19 degrees. It is observed that most of the vortices seemed to have disappeared due to the stretching of the fluid caused by the sudden closing of the valve and the resulting water-hammer effect. There are

252

Computational Fluid-Structure Interaction

FIGURE 10.80 (Continued).

two small vortices circulating at the outer surface of the leaflets as seen in Fig. 10.82A and B. Although the effect of the leaflets movement on the flow seems to have weakened, the narrow gaps formed at the central and outer jets give rise to a high velocity at X7 and X8 positions. The bi-peaked velocities of 22.83 m/s and 22.44 m/s are measured at X7 and X8 positions as seen in Fig. 10.82D. The leaflets have closed to an angular position of θ 5 51.77 degrees at t 5 0.316 s. From Fig. 10.83A and B, it can be seen that the two vortices circulating at the outer surface of the leaflets at the previous position have

FIGURE 10.81 Various plots for the closing phase at t 5 0.304 ss and θ 5 310.07 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.304 s and θ 5 310.07 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.304 s and θ 5 310.07 degrees. (C) Volume ribbons streamtrace at t 5 0.304 s and θ 5 310.07 degrees. (D) Velocity profile on the mid-plane of yaxis at t 5 0.304 s and θ 5 310.07 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.304 s and θ 5 310.07 degrees. (F) Vorticity iso-surfaces at t 5 0.304 s and θ 5 310.07 degrees.

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Computational Fluid-Structure Interaction

FIGURE 10.81 (Continued).

propagated to x 5 61.42 mm and z 5 6 7.91 mm. In this angular position, the leaflets are almost in the closed position and the tip of it is quite near to the channel wall. The gaps at the central and outer jets are quite narrowed which caused a surge in the blood’s velocity at these locations. It is evident from Fig. 10.83D that the velocity profile at X5 position has a maximum value of 22.2 m/s and a bi-peaked velocity profile at X7 position having a maximum value of 22.3 m/s. Finally, the leaflets have rotated to the fully closed position of θ 5 60.0 degrees at t 5 0.323 s. From Fig. 10.84A and B, it is observed that there is no more vortices existed in the channel. There is blood flow observed in the left ventricle side and it might be caused by some leakages occurred at the central and outer jets through the tiny narrow gaps. The flow in the immediate vicinity after the leaflets indicated a high velocity region where the velocity profiles at X7 and X8 positions showed a

IOM FSI Model Validations and Applications Chapter | 10

255

FIGURE 10.81 (Continued).

maximum value of 22.1 m/s and 22.8 m/s, respectively, at the center of the channel as seen in Fig. 10.84C. At the instance before the leaflets reached the fully position, cavitation might occur and the valve closed with a backflow, which leads to the water-hammer effect. A series of vorticity iso-surfaces plots at different time instants and angular positions are obtained in this study as shown in Figs. 10.80(F)
10.83(F). Fig. 10.85 shows the plan view of Figs. 10.80(F)
10.83(F) for different angular positions. The vorticity iso-surfaces plot for θ 5 5 degrees, θ 5 12.86 degrees, θ 5 20.06 degrees and θ 5 60.0 degrees are omitted as the vorticity iso-surfaces are not prominent in these angular positions. Figs. 10.80(E)
10.83(E) present the strength of vorticity profiles extracted along the z-axis in different x position as indicated in Fig. 10.76. Some of the velocity profiles may not be extracted in some of the x position due to the continuous

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FIGURE 10.81 (Continued).

motion of the leaflets. All the plots for vorticity iso-surfaces depict different contour surfaces represent different levels of vorticity or shear intensity. On the same surface, the magnitude of shear is constant. Vorticity surfaces also represent the 3D topology of the shear layers, vortex instability, and the flow structure. At t 5 0.294 s, when the leaflets have closed to an angular position of θ 5 210.74 degrees, the wake of the valve revealed four ball-shaped like shear layers roll-ups at the outer jets after the leaflets edges. The shear layers around the central jet started to roll up and bulge out with the center concave in indicating a jet inside it and the shape of it resembled the shape of a red

FIGURE 10.82 Various plots for the closing phase at t 5 0.310 s and θ 5 45.19 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.310 s and θ 5 45.19 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.310 s and θ 5 45.19 degrees. (C) Volume ribbons streamtrace at t 5 0.310 s and θ 5 45.19 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.310 s and θ 5 45.19 degrees. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.310 s and θ 5 45.19 degrees. (F) Vorticity iso-surfaces at t 5 0.310 s and θ 5 45.19 degrees.

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FIGURE 10.82 (Continued).

blood cell. The vorticity strength at X6 position is quite high and registered a nondimensional value of 1.8 as indicated in Fig. 10.80E. The vorticity strength is zero for most part along the z-axis for X7
X10 positions, expect at the central jet where the shear layers roll-ups existed and it registered maximum vorticity strength of 0.9 at X7 position. As the leaflets continued to close to θ 5 310.07 degrees at t 5 0.304 s, the vorticity strength at the central jet has weakened for both X7 and X8 positions as seen in Fig. 10.81E. This is due the shear layers roll-ups at the central jet has propagated further downstream as observed in the plan view as shown in Fig. 10.85B. On the other hand, the vorticity values at X4 and X5 positions have gained strength as compared to the previous position. The shear layers have rolled up like a hand-fist shaped at the outer jets and it existed at the center of the leaflets as observed from Fig. 10.81F. These shear

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259

FIGURE 10.82 (Continued).

layers have propagated further downstream at t 5 0.310 s for θ 5 45.19 degrees as observed in Fig. 10.82F and assembling like a ball-shaped surfaces. The shear layers roll-ups at the central jet has grown in size and moved further downstream as seen in Fig. 10.85C. At the same time, it has gained substantial vorticity strength at X7 and X8 positions as compared to the previous angular position. This is observed in Fig. 10.82E where a bipeaked vorticity strength profile is obtained at X7 position having a nondimensional value of 2.10. The wake at the center of the leaflets revealed a complicated pattern of shear layers that roll up side-by-side and on top of each other at θ 5 51.77 degrees for t 5 0.316 s as observed in Fig. 10.83F. The shear layers roll-ups at the central jet has shrunk in size as seen in Fig. 10.85D and the vorticity strength at X7 position has reduced significantly as seen in Fig. 10.83E. On the other hand, the vorticity strength at X5

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FIGURE 10.82 (Continued).

and X8 positions has increased and this is due to the narrow gap at the central and outer jets.

10.3.7 Remarks The motions of a 29 mm SJM aortic bi-leaflet MHV for both opening and closing phases and the blood flow through it are computed in the present study. The general flow patterns downstream of the aorta and the left ventricle for the opening and closing phases, respectively, immediately after

FIGURE 10.83 Various plots for the closing phase at t 5 0.316 s and θ 5 51.77 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.316 s and θ 5 51.77 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.316 s and θ 5 51.77 degrees. (C) Volume ribbons streamtrace at t 5 0.316 s and θ 5 51.77 degrees. (D) Velocity profile on the mid-plane of y-axis at t 5 0.316 s and θ 5 51.77 degree. (E) Vorticity profile on the mid-plane of y-axis at t 5 0.316 s and θ 5 51.77 degrees. (F) Vorticity iso-surfaces at t 5 0.316 s and θ 5 51.77 degrees.

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FIGURE 10.83 (Continued).

the MHV showed very complicated and inherently 3D flow patterns as presented in the figures in the previous sections. The computed results are compared quantitatively with the flow visualization obtained by Krafczyk et al. [40]. It is observed that there are certain similarities between all these results. Krafczyk et al. [40] have reported the results of the simulation of 3D transient physiological flows in fixed geometries similar to a CarboMedics bi-leaflet heart valve at different opening angles. In their work, a LatticeBoltzmann method is used in their mathematical approach and blood
leaflet interaction is not considered. The geometry of the simulated flow problem employed in their work is shown in Fig. 10.86. The diameter of the tube is 20 mm and the length 40 mm. The peak Reynolds number used in

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263

FIGURE 10.83 (Continued).

their work is 1818 with upeak 5 300 mm/s and ν 5 3.3 mm2/s. They simplified the problem by assuming the leaflets to be fixed and simulated one test case with an angle of 40 degrees. The computational domain consists of 280 3 140 3 140 grid nodes. The flow enters the tube along the x-axis with a fully developed parabolic velocity profile of a time-dependent maximum amplitude as shown in Fig. 10.87. Fig. 10.88 shows a velocity vector plot of a mid-plane of y-axis at t 5 0.15 s with an angle of 40 degrees obtained in their work. In this time instant, they reported that the flow starts to become 3D. The velocity vector plot obtained in the present work at a valve angle of θ 5 310.61 degrees as shown in Fig. 10.62B is shown again in Fig. 10.89 for quantitative visual comparison with Krafczyk’s velocity vector plot [40]. Based on flow visualization comparison in Fig. 10.86, the general flow

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FIGURE 10.83 (Continued).

patterns for both display certain similarities. There are two small counterrotating vortices circulating at the inner surface of leaflets that are caused by the central jet for both figures. And two bigger ones formed immediately after the MHV that are caused by the outer jets. The difference in some part of the flow regime for both figures is caused by the blood
leaflet interaction effect in the present work, whereas the flow obtained in Ref. [40] is of a

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265

FIGURE 10.84 Various plots for the closing phase at t 5 0.323 s and θ 5 60.0 degrees. (A) Streamlines plot on the mid-plane of y-axis at t 5 0.323 s and θ 5 60.0 degrees. (B) Velocity vector plot on the mid-plane of y-axis at t 5 0.323 s and θ 5 60.0 degrees. (C) Velocity profile on the mid-plane of y-axis at t 5 0.323 s and θ 5 60.0 degrees.

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FIGURE 10.84 (Continued).

fixed geometry. The other difference is the inlet velocity profile used in their work and the present work. Finally, the present work shows a great improvement as compared to Shi’s work [24]. In his work, no clear and distinct vortices and vortex shedding is observed in all time instants. Fig. 10.90 shows one of the velocity vector plots at an angular angle of about 40 degrees at t 5 0.036 s for the opening phase obtained by Shi [24]. Based on flow visualization comparison between Figs. 10.89 and 10.90, there is a clear and distinct vortices observed in the present work which is not seen in Fig. 10.84. In summary, the proposed IOM with OG incorporating the parallel-MG computation is very viable, efficient and robust in solving FSI problems,

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267

FIGURE 10.85 Plan view for the vorticity iso-surfaces shown in Figs. 10.80(F) to 10.83(F). (A) θ 5 210.74 degrees; (B) θ 5 310.07 degrees; (C) θ 5 45.19 degrees; (D) θ 5 51.77 degrees.

such as blood
leaflet interaction. In this study, the numerical models are found to give a reasonable representation of the dominant flow patterns of the bi-leaflet heart valves and it shows great promise and potential as an effective computational tool to analyze FSI and different designs of engineering products with FSI, such as prosthetic heart valves under physiological conditions and blood
leaflet interaction.

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FIGURE 10.86 Geometry of valve used in Krafczyk’s simulation [40].

320 260

u(t) (mm/s)

200 140 80 20 –40 –100 0.00

0.20

0.40

0.60

0.80

1.00

t (s) FIGURE 10.87 Time-dependent inlet velocity used in Krafczyk’s simulation [40].

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FIGURE 10.88 Velocity vector plot at t 5 0.15 s obtained by Krafczyk et al. [40].

U: –0.515 –0.310 –0.105 0.100

0.305 0.510 0.714 0.919 1.124 1.329 1.534 1.739 1.944 2.148 2.353

FIGURE 10.89 Velocity vector plot at t 5 0.098 s and θ 5 310.61 degrees for opening phase.

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FIGURE 10.90 Velocity vector plot at t 5 0.036 s and θ  40 degrees obtained by Shi [24].

REFERENCES [1] J.R. Koseff, R.L. Street, Visualization studies of a shear driven three-dimensional recirculating flow, J. Fluids Eng. 106 (1984) 21
29. [2] J.R. Koseff, R.L. Street, On end wall effects in a lid-driven cavity flow, J. Fluids Eng. 106 (1984) 385
389. [3] J.R. Koseff, R.L. Street, The lid-driven cavity flow: a synthesis of qualitative and quantitative observations, J. Fluids Eng. 106 (1984) 390
398. [4] B. Jiang, T.L. Lin, L.A. Povinelli, Large-scale computation of incompressible viscous flow by least-squares finite element method, Comput. Methods Appl. Mech. Eng. 114 (1994) 213
231. [5] S. Fujima, M. Tabata, Y. Fukasawa, Extension to three-dimensional problems of the upwind finite element scheme based on the choice of up- and downwind points, Comput. Methods Appl. Mech. Eng. 112 (1994) 109
131. [6] C.H. Tai, Y. Zhao, A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows, Int. J. Numer. Methods Fluids 46 (1) (2004) 59
84. [7] Y. Zhao, C.H. Tai, Parallel unsteady incompressible viscous flow computations using an unstructured multigrid method, Parallel CFD 2002: 13th International Conference, JAERIKRE, Kansai Science City, Japan, May 20
22, 2002. [8] Zhao Y., and Tai C.H., Parallel computation of unsteady incompressible viscous flows using an unstructured multigrid method, ASME Forum on Parallel Computing Methods, Paper No. IMECE2002-FE-34388, Proceedings of IMECE’02, 2002 ASME International Mechanical Engineering Congress & Exposition New Orleans, Louisiana, November 17
22, 2002.

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[9] R.S. Montero, I.M. Llorente, M.D. Salas, Robust multigrid algorithms for the Navier
Stokes equations, J. Comput. Phys. 173 (2001) 412
432. [10] J.Y. Yang, S.C. Yang, Y.N. Chen, C.A. Hsu, Implicit weighted ENO schemes for the three-dimensional incompressible Navier
Stokes equations, J. Comput. Phys. 146 (1998) 464
487. [11] C. Liu, X. Zheng, C.H. Sung, Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys. 139 (1998) 35
57. [12] C.H.K. Williamson, Defining a universal and continuous Strouhal
Reynolds number relationship for the laminar vortex shedding of a circular cylinder, Phys. Fluids 31 (1988) 2742
2744. [13] R. Wille, Karman vortex streets, Advances in Applied Mechanics, vol. 6, Academic, New York, 1960, pp. 273
287. [14] Kim Jae Wook and Lee Duck Joo, Generalized formulation and application of characteristic boundary conditions, AIAA Paper, 98
222. [15] Van Dyke, D. Milton, An Album of Fluid Motion, Parabolic Press, Stanford, CA, USA, 1982. [16] M. Nishioka, H. Sato, Mechanism of determination of the shedding frequency of vortices behind a cylinder at low Reynolds numbers, J. Fluid Mech. 89 (1978) 49
60. [17] C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, 1997. [18] F.N. Underwood Jr, A Numerical Study of the Steady, Axisymmetric Flow Through a Disk Type Prosthetic Heart Valve (Ph.D. thesis), Graduate School of University of Notre Dame, 1975. [19] E.B. Shim, K.S. Chang, Numerical analysis of three-dimensional Bjo¨rk-Shiley valvular flow in an aorta, J. Biomech. Eng. 119 (1997) 45
51. [20] W.L. Lim, Y.T. Chew, T.C. Chew, H.T. Low, Steady flow dynamics of prosthetic aortic heart valves: a comparative evaluation with PIV techniques, J. Biomech. 31 (1998) 411
421. [21] S.C. Peskin, D.M. McQueen, Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys. 37 (1980) 113
132. [22] S.C. Peskin, D.M. McQueen, A three-dimensional computational method for the blood flow in the Heart, J. Comput. Phys. 81 (1989) 372
405. [23] M.F. McCracken, S.C. Peskin, A vortex method for blood flow through heart valves, J. Comput. Phys. 35 (1980) 183
205. [24] Y.B. Shi, Numerical and Experimental Study of Pulsatile Flow in Bi-leaflet Mechanical Heart Valves (Ph.D. thesis), Nanyang Technological University, 2002. [25] R.D. Blevins, Flow Induced Vibration, Van Nostrand Rheinholt, 1977. [26] Y.G. Lai, K.B. Chandran, J. Lemmon, A numerical simulation of mechanical heart valve closure fluid dynamics, J. Biomech. 35 (2002) 881
892. [27] Y.B. Shi, Y. Zhao, J.H. Yeo, N.H.C. Hwang, Numerical simulation of opening process in a bi-leaflet mechanical heart valve under pulsatile flow condition, J. Heart Valve Dis. 12 (2) (2003) 245
256. [28] St. Jude Medical Artificial Heart Valve, Cardiac Peacemakers and Other Medical Devices Home Page, ,http://www.sjm.com/..

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[29] O. Ritchie Robert, Fatigue and fracture of pyrolytic carbon: a damage-tolerant approach to structural integrity and life prediction in “ceramic” heart valve prostheses, J. Heart Valve Dis. 5 (Suppl. I) (1996) S9
S31. [30] Y.C. Fung, Biomechanics
Motion, Flow, Stress, and Growth, Springer-Verlag, 1990. [31] W.W. Nicholes, M.F. O’Rourke, McDonald’s Blood Flow in Arteries, Theoretical, Experimental and Clinical Principles, 3rd ed., Lea and Feibiger, 1990. [32] X.Y. Xu, M.W. Collins, Studies of blood flow in arterial bifurcations using computational fluid dynamics, Proc. Inst. Mech. Eng. H: J. Eng. Med. 208 (1994) 163
175. [33] British Standard BS EN 12006-1:1999, Non-Active Surgical Implants
Particular Requirement for Cardiac and Vascular Implants, Part 1: Heart Valve Substitutes, 1999. [34] D. Bluestein, S. Einav, Techniques in the analysis of stability of pulsatile flow through heart valves, in: C.T. Leondes (Ed.), Biomechanics Systems Techniques and Applications: Cardiovascular Techniques, Vol. II, CRC Press LLC, Boca Raton, FL, 2001. [35] G. Tzabiras, Calculation of Complex Turbulent Flows, WIT Press, 2000. [36] M.R. Myers, J.M. Porter, Impulsive-motion model for computing the closing motion of mechanical heart-valve leaflets, Ann. Biomed. Eng. 31 (2003) 1031
1039. [37] C. Farhat, M. Lesoinne, P. LeTallec, Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity, Comput. Methods Appl. Mech. Eng. 157 (1998) 95
114. [38] C.A. Felippa, K.C. Park, C. Farhat, Partitioned analysis of coupled systems, Computational Mechanics, in: E. On˜ate, S. Idelsohn (Eds.), Proceedings of the WCCM IV Conference, CIMNE, Barcelona, 1998. [39] W.A. Wall, R. Ekkehard, Fluid
Structure Interaction Based Upon A Stabilized (ALE) Finite Element Method, Computational Mechanics, in: E. On˜ate, S. Idelsohn (Eds.), Proceedings of the WCCM IV Conference, CIMNE, Barcelona, 1998. [40] M. Krafczyk, M. Cerrolaza, M. Schulz, E. Rank, Analysis of 3D transient blood flow passing through an artificial aortic valve by Lattice-Boltzmann methods, J. Biomech. 31 (1998) 453
462.

Chapter 11

IMM FSI Model Validations and Applications for Incompressible Flows 11.1 TWO-DIMENSIONAL IMMERSED MEMBRANE COMPUTATIONS The proposed immersed membrane method (IMM) is first applied to simulate two-dimensional (2D) flows with an immersed thin structure. At the beginning, the grid convergence study is carried out to demonstrate that the solution is grid independent. Then the 2D steady flow results are compared with those obtained with baseline solver to validate the newly proposed method. Results of moving rigid membrane and moving flexible membrane are presented at last.

11.1.1 Grid Convergence Study The main purpose of grid convergence study is to obtain an optimum grid size, so that the accuracy of solution is independent of the computational grid. The number of nodes and computational time are both related to each other; therefore increasing the grid density to give better results will increase computational time. Hence, to minimize computational time but not at the expense of accuracy of the solution, grid convergence study is a necessary test. The examination of the grid convergence of a simulation is a straightforward method for determining the discretization error in a computational fluid dynamics (CFD) simulation. The method involves performing the simulation on two or more successively finer grids. The term grid convergence study is equivalent to the commonly used term grid refinement study. As the grid is refined (grid cells become smaller and the number of cells in the flow domain increases) and the time-step is refined (reduced), the spatial and temporal discretization errors, respectively, should asymptotically approach zero, excluding computer round-off error. When the grid gets finer and finer, there will be a slight or no change in the result. The model used for the grid convergence study is a 2D channel with aspect ratio of 6:1, as depicted in Fig. 11.1. The width of the channel is 1 Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00011-8 © 2019 Elsevier Inc. All rights reserved.

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FIGURE 11.1 Geometry of the 2D channel model with an immersed membrane.

FIGURE 11.2 Flow field of grid (I) with 2000 nodes.

FIGURE 11.3 Flow field of grid (II) with 18,000 nodes.

FIGURE 11.4 Flow field of grid (III) with 32,000 nodes.

and the length is 6. A membrane with length of 0.5 is fixed at the middle of the channel. The membrane is vertical to the wall of the channel. The left side of the channel is inlet and the right side is outlet. Three grids of different sizes are used to simulate the steady flow for the 2D channel. Grid (I) has 2000 nodes, grid (II) has 18,000 nodes, and grid (III) has 32,000 nodes. The Reynolds number of the flow is 100 and the inflow velocity is 1.0 m/s. All the fluid and geometrical parameters for the three simulations are kept the same. And the solver runs until it achieves the convergence rate of 1024. The grid convergence of the solver is verified by comparing the numerical results of the flow field obtained from the three grids. With a slight change in result as the computational grid gets finer and finer, it can be demonstrated that the simulation results are independent of the computational grid and the solver is grid convergent. Fig. 11.2 shows the flow field obtained from grid (I). It is the coarsest grid used in the study. In this flow field, only one primary vortex appears. Figs. 11.3 and 11.4 show the results obtained from grids (II) and (III). In these two results, two vortices appear in the flow

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TABLE 11.1 Properties of the Vortices of the Three Grids Grids

Secondary Vortex

Primary Vortex

Vortex Center

Vorticity

Length of Vortex

Vortex Center

Vorticity

Length of Vortex

(I) 2000 nodes










(0.75, 0.31)

26.69

1.70

(II) 18,000 nodes

(20.06, 0.06)

20.23

0.17

(1.34, 0.31)

211.44

2.57

(III) 36,000 nodes

(20.07, 0.06)

20.26

0.17

(1.50, 0.31)

211.30

2.62

fields. One is the primary vortex behind the membrane and the other is a small secondary vortex before the membrane. This shows that the grid (I) with 2000 nodes is not fine enough to capture all properties of the flow field. The comparison of the flow fields is shown in Table 11.1. The results of grids (II) and (III) are quite similar. The centers of vortices are located in the similar locations with similar vorticities, and the velocity profiles at same location have little difference as shown in Fig. 11.5A, while the velocity profile of grid (I) at the same location is entirely different to those of the other two grids. The convergence history shown in Fig. 11.5B demonstrates that the solver developed based on the IMM converges fast and well. From the earlier analysis, it is concluded that the solver is grid convergent since the solution of the simulation will not change much if the grid is finer than 18,000 grid nodes.

11.1.2 Validation of Steady Flow To validate the proposed scheme and test its performance, a stationary rigid membrane is employed for the IMM. The model examined in this section is a 2D channel with a sinus cavity in the middle of the bottom wall, as sketched in Fig. 11.6. A rigid membrane is attached to the channel just before the sinus cavity. The channel has a length of 6L and a width of 1L (L is equal to 20 mm). The radius of the sinus cavity is 0.5L. The immersed membrane has a length of 0.5L and thickness of 0.5L/100, and it is attached to the bottom wall at an angle of α 5 42.5 degrees. The 2D channel is meshed into triangular cells. In the region where the membrane will span over the mesh is further refined to catch detailed fluid characteristics during

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FIGURE 11.5 (A) X-component velocity profiles at X 5 1; (B) convergence history plot for three different mesh densities.

the movement. The immersed membrane is discretized into 10 elements. Close-up view of the mesh and the membrane is shown in Fig. 11.7A. Since the membrane is considered rigid, the Young’s modulus is set to a very high value. The inflow velocity V is 1.0 m/s and the Reynolds number is 100.

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FIGURE 11.6 Computational model for validation of steady flow (L 5 20 mm).

FIGURE 11.7 (A) Close-up view of immersed membrane and surrounding mesh; (B) close-up view of internal boundary and surrounding mesh.

The results are compared with those of an internal boundary calculated by a well-validated 2D Navier
Stokes solver DELTINKE [1]. The internal boundary has the same geometry as the immersed membrane which is shown in Fig. 11.7B and it is under the same flow conditions as the immersed

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FIGURE 11.8 Convergence history of immersed membrane and internal boundary.

FIGURE 11.9 (A) Flow field with an immersed membrane computed by the immersed membrane method; (B) flow field with an internal boundary computed by solver DELTINKE.

membrane. Fig. 11.8 shows the convergence history of the immersed membrane and the internal boundary. The new solver based on the immersed membrane scheme converges faster and better than the baseline solver. Fig. 11.9 shows that the two flow fields have the same number of vortices with similar shapes, and the velocity profiles at same locations agree well with each other as shown in Fig. 11.10. Table 11.2 compares the two flow fields quantitatively. It demonstrates that the results obtained by the IMM agree well with the results obtained by standard Navier
Stokes solver using the internal boundary.

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FIGURE 11.10 (A) Comparison of X-component velocity profiles at X 5 0; (b) comparison of X-component velocity profiles at X 5 1.

11.1.3 Moving Rigid Membrane The rigid membrane in the developed steady flow as shown in Fig. 11.9A is then released and moved by the fluid forces. The immersed membrane is

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TABLE 11.2 Comparison of Flow Field With Immersed Membrane and Flow Field With Internal Boundary (L 5 20 mm) Immersed Membrane

Internal Boundary

Primary Vortex

Secondary Vortex

Primary Vortex

Secondary Vortex

Center

(0.926L, 0.270L)

(0.077L, 20.035L)

(0.925L, 0.275L)

(0.046L, 20.026L)

Vorticity

24.992

0.116

24.995

0.117

hinged with bottom wall at one end and it is allowed to rotate around this end. Fig. 11.11 shows a sequence of computed flow fields with corresponding membrane positions at different nondimensional time instants. Streamlines are shown to illustrate the evolution of the flow field. Starting from the steady position, the membrane is pushed forward around the joint point until it reaches the rest position almost parallel to the inflow velocity. The small vortex in the sinus cavity disappears soon after the membrane starts moving at t 5 0.05, while the primary vortex is pushed downstream along the bottom wall and its size decreases due to viscous dissipation until it disappears at approximately the end of the downstream field. The Central Processing Unit (CPU) time for this simulation is 216 minutes. The whole process is divided into 50 time-steps. Within each time-step, the number of pseudo time subiterations is set to 100 in order for the computation to achieve the divergence-free conditions. The order of the residual is 1026 and a second order of reduction of residual in every time-step is ensured.

11.1.4 Moving Flexible Membrane In this section, a flexible membrane is studied by the IMM. The computational model examined in this section is slightly different. The radius of the sinus cavity is 0.8L, and two small rounds are added to where the sinus cavity joins the bottom wall. The membrane is fully clamped by the wall at the lower end, while the other one is a free end. The flexible membrane begins to move in the developed steady flow as shown in Fig. 11.12, in which the membrane is stationary and the Young’s modulus is set to a very high value to harden the membrane to obtain the steady flow. The inflow velocity V is 1.0 m/s and the Reynolds number is 100. Table 11.3 lists geometrical and material parameters of the flexible membrane and the flow field. Fig. 11.13 shows the motion of the immersed flexible membrane. Starting from the steady flow, the flexible membrane bends and moves toward the sinus cavity. It is found that the cause of the bending of

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FIGURE 11.11 Motion of a rigid membrane, thickness of 0.5L/100, driven by a constant inflow at Re 5 100.

membrane at its center is due to the fact that the pressure difference at the center is the greatest, which result in the most significant bending there. The vortex decreases in size and moves downstream due to the motion of the membrane. Finally it disappears at the corner where the sinus cavity joins

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FIGURE 11.12 Steady flow with an immersed flexible membrane.

TABLE 11.3 Parameters of a Flexible Membrane Fluid Parameters

Geometric Parameters of Membrane

Material Parameters of Membrane

ρf (kg/m3)

1.0 3 103

Length (mm)

0.5L

ρm (kg/m3)

1.0 3 103

Re (2)

100

Thickness (mm)

0.5L/50

E (N/m2)

1.5 3 106

V (m/s)

1.0

α

45

ν

0.49

the bottom wall. The membrane continues moving toward the sinus cavity until it rests in the sinus cavity slightly declining inward the sinus. The CPU time for this simulation is 307 minutes. The whole process is divided into 50 time-steps. Within each time-step, the number of pseudo time subiterations is 100. The order of the residual is 1026 and the other order of reduction in residual in every time-step is ensured.

11.1.5 Remarks The earlier two examples demonstrate the capabilities of the IMM to analyze the large deformation, complex and time-dependent motions of the immersed thin structures driven by viscous flow, without changing the stationary background mesh.

11.2 SERIAL COMPUTATION OF BI-LEAFLET MECHANICAL HEART VALVE In this section, the proposed IMM method is applied to compute the flow fields around a St. Jude 29 bi-leaflet aortic mechanical heart valve for opening and closing phases under physiological conditions with a serial solver. Only one of the two valve leaflets is computed due to the symmetry of the two leaflets, and the results are mirrored along the plane x
z to obtain a full view of the flow field. The results of flow field are shown in the form of velocity vector plots, streamline distributions, and pressure contours with

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FIGURE 11.13 Motion of an immersed flexible membrane, driven by constant viscous inflow at Re 5 100. (A) t 5 0.2; (B) t 5 0.4; (C) t 5 0.6; (D) t 5 1.0; (E) t 5 1.6; (F) t 5 2.0; (G) t 5 2.6; (H) t 5 3.0.

corresponding valve positions. Detailed stress and pressure distributions on the leaflet surfaces are presented for analysis of loading variations of the valve. Comparison of computed results with experimental results conforms that the method can accurately simulate the opening/closing phases of the mechanical heart valves (MHVs).

11.2.1 Model Properties of St. Jude Bi-Leaflet Mechanical Heart Valve In this MHV model, the annulus diameter is 29 mm, the diameter of the internal flow channel is 25 mm, and the thickness of the valve leaflet is 0.8 mm [2]. At fully open and closed positions, the valve leaflets are 5 and 60 degrees, respectively, away from the x
z plane as illustrated in Fig. 11.14. Thus the valve leaflet moves a total angle of 55 degrees during its opening and closing phases. The density of the valve leaflets ranges from

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FIGURE 11.13 (Continued).

1.5 3 103 to 2.0 3 103 kg/m3 [2]. In this work, the density is set to be 1.8 3 103 kg/m3. The density of human blood is 1.06 3 103 kg/m3 and its dynamic viscosity is η 5 3.5 3 1023 N  s/m2. The pulsatile inflow velocity profile is specified according to the physiological mass flow rate of a normal human based on the requirements for pulsatile flow testing of heart valve substitutes published by British Standard organization [3]. In this study, a set of timedependent inlet velocity equation for both the opening and closing phases are devised according to the profile shown in Fig. 11.15 and is given as follows [2,4], 8 0:0 for 0 , t # 0:075T > > > > < U 0 Usin
5:0πðt 2 0:075TÞ=T  for 0:075T , t # 0:175T
 U in ðm=sÞ 5 > for 0:175T , t # 0:44T > U 0 Ucos 1:98πðt 2 0:175TÞ=T > > : 0:0 for 0:44T , t # T ð11:1Þ

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FIGURE 11.14 (A) Fully closed configuration of St. Jude bi-leaflet MHV; (B) fully open configuration of St. Jude bi-leaflet MHV.

FIGURE 11.15 Time-dependent inlet velocity profile for opening and closing phases. (A) Aortic valve closed, 0.0 s ,t # 0.06 s; (B) aortic valve opening and closing, 0.06 s ,t # 0.35 s; (C) aortic valve closed, 0.35 s , t # 0.8 s) [2].

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FIGURE 11.16 (A) Geometry of the vessel for simulation of St. Jude 29 mm bi-leaflet MHV; (B) geometry and position of the valve leaflet.

where Uo 5 6 0.5 m/s (positive for opening phase and negative for closing phase), and T is the period for one cardiac cycle which is 0.83 seconds and this corresponds to the normal heartbeat of 72 beats/minute. Thus the Reynolds number, defined as Re 5 ρUN D=η, reaching a value around 4000 at peak mainstream velocity at the moment of maximum systolic flow, will result in turbulent flow. However, the application of this Reynolds number is not feasible due to the limited computing resources and a lack of suitable model to simulate both the turbulent and laminar flows simultaneously. Under the pulsatile inflow condition, the presence of turbulence is a very brief time within a cardiac cycle. So in this work, the Reynolds number is set to a moderate value of 1800 to limit the flow condition to unsteady laminar flow. The MHV is immersed in a circular vessel with the same internal diameter of 25 mm. The length of the vessel is 100 mm, and the valve is located at 35 mm away from the inlet as shown in Fig. 11.16. A grid convergence study was carried out and a mesh size of 50,096 nodes and 251,547 tetrahedral elements was chosen for the fluid domain, while the mesh size of the valve leaflet is of 4200 nodes and 8151 triangular elements. Initially, velocities at all nodes are set to be zero except at the inlet the pulsatile inflow velocity is imposed. And a fixed pressure of 10 kPa is assigned at the outlet. No-slip boundary condition is specified on the rigid vessel walls and the leaflet surfaces. Slip boundary condition is applied for the symmetry plane.

11.2.2 Results for the Opening Phase The opening phase occurs from t 5 0.0 to t 5 0.14 seconds according to the pulsatile inflow velocity profile depicted in Fig. 11.15. From t 5 0.0 to t 5 0.06 seconds the velocity of the valve leaflet is set to be zero since the inflow velocity is zero. Only a very small amount of fluid flows in to push

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the leaflets as shown in Fig. 11.18A. The valve leaflets start moving from t 5 0.06 seconds when the inlet velocity increases. A series of successive pressure contours of x
y plane at z 5 0 with the corresponding leaflet positions during valve opening phase are shown in Fig. 11.17. At t 5 0.105 seconds, the leaflets reach their fully open positions, and from t 5 0.105 to t 5 0.14 seconds they remain fully open until the start of the leaflet closing phase. The opening of the valve leaflets takes about 0.045 seconds, which agrees well with the experimental and numerical results of Shi [2].

FIGURE 11.17 Pressure contours of the x
y plane at z 5 0 during opening phase at different time instants. (A) angle 5 60 degrees at t 5 0.06 s; (B) angle 5 50 degrees at t 5 0.093 s; (C) angle 5 30 degrees at t 5 0.098 s; (D) angle 5 22 degrees at t 5 0.100 s; (E) angle 5 10 degrees at t 5 0.102 s; (f) angle 5 5 degrees at t 5 0.105 s.

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FIGURE 11.17 (Continued).

This opening process is a very brief period compared with the whole opening phase of 0.14 seconds. It agrees with the fact that during the opening phase the leaflets open very fast and leave much of the time of the opening phase for blood to go through aortic valve. Pressure difference between the upstream and downstream of the valve leaflets keeps decreasing as the valve opens. Finally pressures at upstream and downstream of the leaflets are almost the same when the valve leaflets are resting at their fully open positions (angle 5 5 degrees). Fig. 11.18 shows six successive valve positions

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with corresponding velocity vector fields and volume streamlines around the valve leaflets of the opening phase. In Fig. 11.18B, three jets form when the leaflets start moving at t 5 0.06 seconds. One jet is at the central orifice of the two leaflets. The other two are at the small gaps between the leaflets and the solid housing wall. In the vector plots, it can be observed that near the region of the leaflets, fluid velocity is vertical to the leaflet surface. It implies that the leaflets follow the motion of the fluid or vice versa, which is an intrinsic phenomenon of dynamic fluid
structure interaction (FSI), because surface velocity on the rotating leaflets is always perpendicular to the surfaces. Fig. 11.19 shows the pressure and fluid shear-stress distributions on two surfaces of the valve leaflet. Fig. 11.19A1
F1 is the pressure distributions on the leaflet surface of the upstream side, and

FIGURE 11.18 Velocity vectors of the x
y plane at z 5 0 and volume stream around the valve leaflet during opening phase at different time instants. (A) angle 5 60 degrees at t 5 0.06 s; (B) angle 5 50 degrees at t 5 0.033 s; (C) angle 5 30 degrees at t 5 0.038 s; (D) angle 5 22 degrees at t 5 0.040 s; (E) angle 5 10 degrees at t 5 0.042 s; (F) angle 5 5 degrees at t 5 0.045 s.

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FIGURE 11.18 (Continued).

Fig. 11.19A2
F2 is of the downstream side. On the upstream leaflet surface, high pressures appear near the trailing edge of the leaflet. On the downstream leaflet surface, high pressures are at the central region. It is found that pressures on both surfaces across the leaflet have very different values and distribution patterns. This demonstrates that the IMM can consider the discontinuous fluid properties across the immersed structure. Fig. 11.19A3
F3 is the fluid shear-stress distributions on the upstream surface and Fig. 11.19A4
F4 is on the downstream surface. The shear stress also has radical changes on both sides of the valve leaflet. It is observed that the fluid shear stress is about 100 times smaller than the fluid pressure, so the pressure difference is the major cause of the valve motion. On both the upstream and downstream surfaces, the higher shear stress is near the trailing edge of the leaflet. The physical time-step size is set to 0.0005 seconds to track the transient movement of the valve leaflet. In each time-step, 400 subiterations are used

FIGURE 11.19 Pressure and fluid shear-stress distributions on leaflet surface for opening phase. (1) Pressure distribution on leaflet surface of the upstream side. (2) Pressure distribution on leaflet surface of the downstream side. (3) Fluid shear-stress distribution on leaflet surface of the upstream side. (4) Fluid shear-stress distribution on leaflet surface of the downstream side. (A) angle 5 60 degrees at t 5 0.06 s; (B) angle 5 50 degrees at t 5 0.033 s; (C) angle 5 30 degrees at t 5 0.038 s; (D) angle 5 22 degrees at t 5 0.040 s; (E) angle 5 10 degrees at t 5 0.042 s; (F) angle 5 5 degrees at t 5 0.045 s.

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to achieve the divergence-free condition. The order of the residual is 1026. The CPU time of the simulation is 780 minutes on one node of a SGI O3400 server.

11.2.3 Results for the Closing Phase The closing phase of the 29 mm bi-leaflet MHV is computed using the same model as that of the opening phase. During the closing phase, since the flow velocity direction is reversed, the inlet side is downstream to the valve while the previously downstream outlet is upstream to the valve now. To eliminate the downstream effects, the distance between inlet and the position where the valve is seated is elongated to 65 mm, and the distance between the outlet and the valve is shortened to 35 mm to maintain the total length of the model to be 100 mm. The pulsatile inflow velocity profile for closing phase from t 5 0.14 to t 5 0.83 seconds is applied at the new inlet and a fixed pressure of 10 kPa is assigned to the new outlet. The flow field of closing phase is far more complex than that of the opening phase because of the reversal of the flow velocity. Initially a steady flow shown in Fig. 11.20A and 11.21A is developed for the closing phase by fixing the leaflets at their fully open positions. The leaflets start moving in the developed steady flow. A series of pressure contours of x
y plane at z 5 0 with corresponding valve positions are shown in Fig. 11.20. It shows that pressure of the upstream side of the leaflets is larger than the pressure of the upstream side. The pressure difference will push the leaflets toward their closing positions. At the end of the opening phase, pressures at both the leading and trailing edges of the leaflets are almost the same, so the leaflets remain at their fully open positions for about 0.035 seconds from t 5 0.105 to t 5 0.14 seconds. When pulsatile inflow inverses its direction after t 5 0.14 seconds, the pressure difference gradually increases. In Fig. 11.20B, it is observed that the pressure increases at the upstream side, and the leaflets are pushed to an angle of 11 degrees. The leaflets continue to rotate toward the fully closed position with an increasing pressure difference across the valve as shown in Fig. 11.20C
F. The pressure difference is more prominent at the end of the closed position as observed in Fig. 11.20G. This huge transvalular pressure difference pushes the leaflets to close tightly so that no backflow of blood is allowed from the left ventricle (LV) into the aorta. A series of consecutive velocity vector plots and streamline plots with corresponding leaflet positions are shown in Fig. 11.21. In the closing phase, the flow field still has three channels formed as in the opening phase. At early stages of the closing phase, the flow field is quite smooth and no vertex appears as shown in Fig. 11.21B
D. As the leaflets further approach fully closing positions, the flow field becomes more and more obstructed by the leaflets as shown in Fig. 11.21E
G. The irregular velocity streamlines indicate that the flow field is disturbed by the closing leaflets. In

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FIGURE 11.20 Pressure contours of the x
y plane at z 5 0 around valve leaflets at different time instants of closing phase. (A) t 5 0.14 s angle 5 5 degrees (steady flow); (B) t 5 0.251 s angle 5 11 degrees; (C) t 5 0.264 s angle 5 18 degrees; (D) t 5 0.301 s angle 5 34 degrees; (E) t 5 0.308 angle 5 43 degrees; (F) t 5 0.319 s angle 5 56 degrees; (G) t 5 0.325 s angle 5 60 degrees.

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FIGURE 11.20 (Continued).

Fig. 11.21, it is observed that a jet is formed in the central channel due to the converging flow obstructed by the leaflets. Velocity of the jet increases as the leaflets approaching the fully closed positions. Near the fully closed positions, squeeze flow appears in the side channels. Fig. 11.22 shows the pressure and fluid shear-stress distributions on the two surfaces across valve leaflets. Both the pressure and the shear stress have radical changes across the valve leaflets. At early stages of the closing phase, pressures on both upstream and downstream surfaces have small difference, and pressures at the leading edges are higher than pressures at the trailing edges as shown in Figs. 11.22A1
C1 and 11.22A2
C2. As the leaflets move toward their closed positions, the pressure difference on both the upstream and

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FIGURE 11.21 Velocity vector of the x
y plane at z 5 0 and volume stream around the valve leaflet during closing phase at different time instants. (A) t 5 0.14 s angle 5 5 degrees (steady flow); (B) t 5 0.251 s angle 5 11 degrees; (C) t 5 0.264 s angle 5 18 degrees; (D) t 5 0.301 s angle 5 34 degrees; (E) t 5 0.308 angle 5 43 degrees; (F) t 5 0.319 s angle 5 56 degrees; (G) t 5 0.325 s angle 5 60 degrees.

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FIGURE 11.21 (Continued).

downstream surfaces increases as shown in Figs. 11.22D1
G1 and 11.22D2
G2. The pressure is much higher than the shear stress as in the opening phase. The physical time-step size is set to 0.0005 seconds. And 300 subiterations are used to achieve the divergence-free condition. The CPU time of the simulation is about 670 minutes on the SGI O3400 server.

11.2.4 Remarks An IMM is proposed in this work to compute the FSI problem in an MHV. In this method a stationary Eulerian background mesh is used for the computation of the fluid domain, thus avoiding the update of background mesh. By introducing fluid ghost nodes in the computations involving mesh nodes

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across the structure, a set of ghost values are linearly extrapolated from their corresponding real nodes to replace the real nodes in the cross-structure fluid computations. These one-sided extrapolations can account for radical

FIGURE 11.22 Pressure and fluid shear-stress distributions on leaflet surface for closing phase. (1) Pressure distribution on leaflet surface of the upstream side; (2) pressure distribution on leaflet surface of the downstream side; (3) fluid shear-stress distribution on leaflet surface of the upstream side; (4) fluid shear-stress distribution on leaflet surface of the downstream side. (A) t 5 0.14 s angle 5 5 degrees (steady flow); (B) t 5 0.251 s angle 5 11 degrees; (C) t 5 0.264 s angle 5 18 degrees; (D) t 5 0.301 s angle 5 34 degrees; (E) t 5 0.308 angle 5 43 degrees; (F) t 5 0.319 s angle 5 56 degrees ; (G) t 5 0.325 s angle 5 60 degrees.

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FIGURE 11.22 (Continued).

changes in fluid conditions across the immersed thin structure and provide a promising way to improve the accuracy of the FSI simulations. After computation of the fluid domain, the structure is driven directly by the fluid forces according to the Newton’s second law of motion, thus inertial effects can be considered in the computation. The flow field in a real St. Jude 29 mm was studied by the IMM using a 3D Navier
Stokes solver. The results of the opening and closing processes agree well with other available numerical results and experimental results.

11.3 GRID CONVERGENCE STUDY FOR THE PARALLEL SOLVER 11.3.1 Grid Convergence Study The grid convergence study is carried out first to demonstrate grid independence of the parallel computation solver. This study is performed for a case where a solid oscillating sphere is immersed in a fluid enclosed within a domain with solid no-slip walls, similar to the study cases of Gilmanov [5]

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FIGURE 11.23 Computational domain for oscillating sphere.

and Udaykumar [6]. The flow situation is depicted in Fig. 11.23. A rigid sphere of diameter D 5 1 unit is initially placed at the center of the cubic box with edge L 5 2 units within an incompressible and viscous fluid, which is initially at rest. Flow is induced by oscillating the immersed sphere back and forth horizontally along the x-axis. The oscillating velocity and location of the sphere are prescribed as follows: Vx 5 0:25πsinð2πtÞ; Vy 5 Vz 5 0; 0 # t # 1 xn ðtÞ 5 x0n 1 A½1 2 cosð2πtÞ; yn ðtÞ 5 y0n ; zn ðtÞ 5 z0n ; A 5 0:25D

ð11:2Þ ð11:3Þ

where xn , yn , and zn are the position of the n th node of the surface of the sphere. The motion is impulsively initiated at t 5 0 with an amplitude of A 5 0.25D and a nondimensional time period of 1. The Reynolds number for this flow is defined as Vx D=υ and it is set to be 50 for the grid convergence study. Four uniformly spacing and successively finer grids are used for this study, with 113, 213, 413, and 813 grid nodes, respectively. The meshes are constructed by two grid-generation programs: FORGEN and ST2UNST [7]. First in FORGEN, a structured grid is generated by specifying the length and the number of intervals for each edge. The grid nodes are placed along each edge with a uniform spacing distance decided by the edge length and interval numbers. Taking the grid of 113 nodes, for example, the length of cubic edge is 2 and 10 intervals are specified for each edge, so the interval distance hI is 0.2 and 11 nodes are placed along x, y, and z direction. With this spacing distance, a total number of 113 nodes can be placed in the cube, and the structured grid is finally generated by connecting all the nodes into an assembly of hexahedron elements. In construction of the next consecutively finer grid of 213 nodes, the current intervals of hI 5 0.2 are further divided into two smaller intervals of hII 5 0.1, thus increasing the number of nodes in each direction to 21, and the total number of nodes to 213. The next two grids are constructed in the same manner as intervals of hII 5 0.1 are divided into hIII 5 0.05 with 41 nodes in each direction and the total number of nodes of 413, and intervals of hIII 5 0.05 are divided into hIV 5 0.025, resulting in 81

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FIGURE 11.24 Grid generation for consecutively finer meshes.

nodes in each direction and the total number of nodes of 813. The grids are constructed in this way in order to identify the same nodes in the four grids. By consecutively dividing the intervals, only new nodes are added into the finer grids and old nodes are kept in the finer grids but with different numberings. A relation of the numberings can be established to identify the same nodes, which will be used for the error analysis for the four grids. In the program ST2UNST, the structured grids are transformed to unstructured grids by splitting the hexahedron elements into tetrahedron elements, with simply connecting the diagonal grid nodes of the structured hexahedron elements. The construction process for consecutively finer meshes is illustrated in Fig. 11.24 with two simple 2D meshes. To implement single program multiple data parallel computation, the four grids are partitioned into subdomains. Since eight processes are used in the parallel computing, all the grids are partitioned into eight subdomains as shown in Fig. 11.25. Although the first two grids of 113 and 213 have small amount of nodes, they are still partitioned into eight subdomains in order to keep consistency in the grid convergence study regarding parallel computation. Surface of the rigid sphere is discretized into an unstructured mesh of 7308 nodes and of 14,612 triangular elements. The simulations are run for one complete time period of 1. A small timestep size of Δt 5 0.005 is chosen for all the simulations in order to minimize the effect of temporal errors on solutions. So the simulation for one oscillation period is divided into 200 time-steps, and in each time-step, 400 subiterations are used for pseudo time-stepping in order to ensure divergencefree conditions. At the end of the oscillation period, flow fields obtained on the four grids are recorded and compared for grid convergence study. Fig. 11.26 shows the streamline and vector plots for the mid planes of y 5 0 of the four grids. Because the sphere oscillates back and forth in an

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FIGURE 11.25 Mesh partition for grid convergence study. (A) Grid of 113 nodes; (B) grid of 213 nodes; (C) grid of 413 nodes; (D) grid of 813 nodes.

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FIGURE 11.25 (Continued).

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incompressible fluid confined in the closed cube with nonslip boundaries, at the end of the oscillation period, a recirculating flow is induced with streamlines starting from the leading side (
x side) and ending at the trailing side (1x side) of the sphere. It is observed that two primary vortices appear symmetrically about the equatorial planes in all the four flow fields at similar positions with similar vorticities. However, for the grid of 113 nodes, it is too coarse to capture the small vortices near the corners of the cube, and for the grid of 213 nodes, it shows some signs of the development of the small vortices, though not apparent. In the grid of 413 nodes, the two small vortices develop well near the corners, and their positions and vorticities are in good agreement with the results obtained on the grid of 813 nodes. So the grid of 413 nodes are fine enough to obtain accurate solutions, and the solutions are grid independent since the results change little when the grid is finer than that of 413 nodes.

FIGURE 11.26 Flow fields at the moment of the end of one oscillation time period (pictures on the left are plots of streamline and x-component velocity contour for y 5 0 planes. Pictures on the right are vector plots for y 5 0 planes). (A) Grid of 113 nodes; (B) grid of 213 nodes; (C) grid of 413 nodes; (D) grid of 813 nodes.

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FIGURE 11.26 (Continued).

11.3.2 Error Analysis To demonstrate the second order of accuracy of the proposed IMM for moving boundaries and FSIs, an error analysis is carried out on the solutions obtained on the four consecutively finer meshes in grid convergence study. Without analytical solution to this flow problem, the solution on the finest grid used in the study, the grid of 813 nodes, is taken to be the “exact” solution, which is used as basis in calculating errors of the coarser grids. The errors are calculated for x-component velocities of all grid nodes under consideration at the end of the simulation period. For a grid of N3 nodes, the kth norm of x-component velocities are calculated as follows, " # k 1=k N3  1 X  ðN 3 Þ ð813 Þ  k εN 5 3 1 # t # N3 ð11:4Þ ðVx Þj 2ðVx Þj  N j and the infinity error norm of x-component velocities is calculated as,

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  3   ðN 3 Þ εN 2 ðVx Þjð81 Þ  N 5 max 3 ðVx Þj j 5 1;N

where ðVx ÞjðN

3

Þ

1 # t # N3

305

ð11:5Þ

is the x-component velocity of the jth node of the grid N3

nodes, ðVx Þjð81 Þ is the x-component velocity of the same node of the finest grid of 813 nodes as the jth node in the grid of N3 nodes. The relationship between numberings of one node in different grids is established when constructing the grids as described in the previous section (section 11.3.1). The convergence history of L1, L2, and LN error norms for the three coarser grids with respect to the finest grid are shown in Fig. 11.27 against with grid spacing distance h in a log-log manner. Slope 2 and slope 1 are also shown in the plot for comparison. In the plot, both L1 and L2 are very close to slope 2, which suggest that the current method converges at a rate close to the second order. The overall slope for LN is about 1.6, a small deviation from the second-order convergence rate. This may be caused by the fact of treating solution on the finest grid as “exact” solution while not using the exact analytical solution since it is not available for the governing equations. The results of error analysis confirm that the proposed IMM is second-order accurate, which is stated in the numerical formulation in Chapter 6, Parallel Computation. 3

FIGURE 11.27 Convergence history of L1, L2, and LN error norms of x-component velocity for the three coarser grids with respect to the finest grid.

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11.4 VALIDATION OF THE PARALLEL SOLVER The parallel solver with IMM for FSI problems is validated by performing simulations for the axially symmetrical steady flows induced by a sphere rotating at constant angular velocities about a diameter in an incompressible fluid which is rest at large distance from it under moderate Reynolds numbers. In this study, radius of the sphere is R and the fluid domain is a cube with edge of 10R. The sphere is placed at the center of the cube. It rotates about z-axis at a constant angular velocity of ω as shown in Fig. 11.28. The Reynolds number is defined as ωR2 =υ. For Reynolds numbers between 1 and 100, Dennis et al. [8] have obtained analytical solutions by solving the governing Navier
Stokes equations with a vorticity-stream function formulation in spherical polar coordinate systems. And Gilmonov et al. [5] reported their numerical simulation results for this problem at the Reynolds numbers of 20, 50, and 100. Under moderate and low Reynolds numbers, the resultant flow fields are steady and axisymmetric. However, intermediate unsteady simulation results obtained in the process from the sphere impulsively starts rotating from rest to the final steady status are also presented in this study. The simulation of intermediate unsteady flows induced by the rotating sphere provides a sound validation for the accuracy of the parallel solver and infuses some confidence about the correct implementation of the numerical infrastructure developed in this work for solving fluid domains with immersed moving structures. The computational domain is a closed cube of edge equal to 10R and with nonslip boundary conditions applied on all the six surfaces. Such a large fluid domain is chosen in order to minimize the boundary effects on fluid flows. The fluid domain is discretized into a uniformly spacing grid of 423 nodes, and for parallel computation it is partitioned into eight subdomains as shown in Fig. 11.29. Surface of the sphere is discretized into an unstructured mesh of 7306 nodes and 14,612 triangular elements. It starts rotating impulsively at t 5 0 around z-axis with a constant angular velocity of ω in the

FIGURE 11.28 Computational domain for rotating sphere.

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FIGURE 11.29 Mesh partition for steady flows induced by a rotating sphere.

clockwise direction, so the velocities of all the structure nodes are prescribed as follows: Vxi 5 2 ωR cosφi sin θi ; Vyi 5 ωR cosφi cosθi ; Vzi 5 0 if xi # 0 yi # 0 Vxi 5 ωR cosφi sin θi ; Vyi 5 2 ωR cosφi cosθi ; Vzi 5 0 if xi $ 0 yi $ 0 Vxi 5 ωR cosφi sin θi ; Vyi 5 ωR cosφi cosθi ; Vzi 5 0 if xi # 0 yi $ 0 Vxi 5 2 ωR cosφi sin θi ; Vyi 5 2 ωR cosφi cosθi ; Vzi 5 0 if xi $ 0 yi # 0 ð11:6Þ where Vxi , Vyi , and Vzi are velocity components of the ith structure node, θi is the angle between the structure node and the plane of x 5 0, φi is the azimuth angle of the structure node. Simulations are carried out for Reynolds numbers of 10, 20, 50, and 100. In all the simulations, a physical time-step of Δt 5 0.01T is used, where T 5 2π=ω is the rotation period of the sphere. Therefore simulation for one rotation period is divided into 100 physical time-steps, and in each time-steps 200 subiterations are used for pseudo time-stepping in order to obtain fairly accurate intermediate unsteady results.

11.4.1 Unsteady Flows at Reynolds Number of 100 The intermediate unsteady flows of Re 5 100 are presented first to demonstrate capacity of the parallel solver to capture accurate transient results. The unsteady flows are recorded at the time instants of t 5 0.1T, t 5 0.5T, t 5 1.0T, t 5 2.0T, and the flow field reaches steady status around t 5 3.0T. The instantaneous plots of streamlines with contour for x-component velocity and vector plots of the unsteady flows at y 5 0 plane are shown in

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Fig. 11.30, together with the steady flow recorded at the time instant of t 5 3.0T. These plots document the evolution history of the flow field due to a rotating sphere at the Reynolds number of 100. At early stage of the impulsive rotation, two small vortices form near both southern and northern poles of the sphere as shown in Fig. 11.30A. But subsequently the four

FIGURE 11.30 Evolution history for flow field due to a rotating sphere at Re 5 100. (A) t 5 0.1T; (B) t 5 0.5T; (C) t 5 1.0T; (D) t 5 2.0T; (E) t 5 3.0T.

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FIGURE 11.30 (Continued).

symmetrical vortices strengthen themselves as they move toward the equatorial plane as shown in Fig. 11.30B and C. And near the equatorial plane, the vortices change their moving direction to the radial direction and move away the sphere as shown in Fig. 11.30C
E. Finally the vortices stabilize near the equatorial plane with some distance from the sphere, and the flow field reaches its final axisymmetric steady state as shown in Fig. 11.30E. It is worth noting that the flow remains perfectly axisymmetrical during the whole evolution process. This flow phenomenon is caused by the imbalance between the component of the centrifugal force in the tangential direction to the sphere direction and the pressure gradient, which is directed in the direction normal to the sphere. The induced force imbalance will direct the fluids abound the sphere flowing from the poles to equator. And the two converging fluids from both poles collide at the equatorial plane and form a jet flow which flows away from the sphere in radial direction. To satisfy the mass conservation, the streamlines have to close and form two toroidal vortices located symmetrically with respect to the equatorial plane. Numerical results of Gilmanov [5] are shown in Fig. 11.31 for qualitative comparison, and the same flow patterns were also observed by the researcher.

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FIGURE 11.31 Unsteady results reported by Gilmonov [5] for flow due to a rotating sphere at Re 5 100.

11.4.2 Steady Flows Due to a Rotating Sphere Steady flows at low and moderate Reynolds numbers are also simulated in this work for quantitative comparison. The steady flow patterns for Reynolds numbers of 10, 20, 50, and 100 are shown for the plane of y 5 0 in Fig. 11.32. The main feature of these steady flows is that the polar inflow is balanced by the outflow at the equator. The inflow changes to an outflow at a critical angle which depends on Reynolds number. This inflow angle θinflow is the angle between z-axis and the radius that passes through vortex center as depicted in Fig. 11.32A. The inflow angles are 56.7, 61.2, 71.8, and 75.9 degrees for Reynolds numbers 10, 20, 50, and 100, respectively. It is observed that the inflow angle increases as the Reynolds number increases, which indicates that the polar inflow gets slower and expands a larger region as the Reynolds number increases, and the equatorial jet flow gets faster and narrower. It agrees with the fact that with a larger Reynolds number the

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sphere rotates at a larger angular velocity and induces a greater influence on the surrounding fluids. The values of θinflow at different Reynolds numbers are tabulated in Table 11.4 with the analytical results obtained by Dennis [8] and the numerical results obtained by Gilmonov [5] for quantitative comparison. The flow fields obtained by Gilmanov are also shown in Fig. 11.33. The numerical results in this work agree excellently with the previous works. It is proved that the proposed IMM is capable to solve the dynamic FSI problems accurately.

11.5 PARALLEL COMPUTATION OF BIOPROSTHETIC HEART VALVE The bioprosthetic heart valve (BHV) replaces diseased natural aortic valve and serves its function to allow the blood flow only in one direction from the LV to the aorta. A desirable BHV is expected to cause as little resistance

FIGURE 11.32 Steady flow due to rotating sphere at different Reynolds numbers. (A) Re 5 10; (B) Re 5 20; (C) Re 5 50; (D) Re 5 100.

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FIGURE 11.32 (Continued).

TABLE 11.4 Polar Inflow Angles at Different Reynolds Numbers Compared With [5] and [8] Re

Dennis ( ) [8]

Gilmonov ( ) [5]

Current ( )

ε % (Compared with [8])

10

58.0




56.7

2.2

20

62.6

61.6

61.2

2.2

50

69.4

68.2

71.8

3.4

100

73.8

72.1

74.9

1.5

to blood flow as possible, and it should seal firmly to prevent any regurgitation when the valve closes. The BHVs have better hemodynamic performance compared with MHVs, but their long-term performance is disappointing. Most of the degeneration of BHVs can be attributed to calcification and tearing of leaflets. It is believed that the calcification and tearing

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FIGURE 11.33 Numerical results by Gilmonov [5] of steady flow due to rotating sphere at different Reynolds numbers. (A) Re 5 20; (B) Re 5 50; (C) Re 5 100.

of valve leaflets are related to the distribution of surrounding fluid stress and internal structure stress. So it is important to understand the flow dynamics and stress distribution of the valve leaflet under physiological conditions. With advent of high-speed computers and the development of CFD and

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Computational Structural Dynamics (CSD) analysis algorithms, biofluid and structural dynamic simulation of valvular function provides a promising alternative in the detailed analysis of the complex flow dynamics during the opening and closing phases. These numerical simulations can be used to analyze the detailed fluid dynamic stress distribution in the region of interest with the valve prostheses and structure response of the valve prostheses to the fluid stresses. Researchers have identified four phases of the valve within one cardiac cycle, which are the opening phase, the fully opened phase, the closing phase, and the fully closed phase. Among these four phases, the fully open and fully closed phases are relatively simple due to the fixed flow-field structure, and many researchers have intensively and extensively studied these two phases. For the opening and closing phases, the flow phenomenon is far more complex and the limitations of computing resources also contribute to the slow progress. One of the important requirements of an accurate simulation of valve function is the development of fluid
structure algorithm for the prediction of the leaflet motion resulting from the hemodynamic forces acting on the leaflets. In the present work involving FSI, the computation is first carried out by solving the Navier
Stokes equations for the fluid flow field using the numerical method described in Chapter 3, Mathematical Formulation for Incompressible Flow Solver, and the IMM method with the valve leaflets immersed into the fluid computational domain. Then the hemodynamic forces are computed on the valve leaflets for solving the structure dynamics equation. The governing equation for structure domain is solved by using the numerical methods developed in Chapter 4, Mathematical Formulation for CSD. The new velocity and spatial configuration are available once the structure equation is solved. And with this new velocity and position, the fluid flow is calculated for a new time-step. In this numerical simulation, attention is not focused on the determination of an optimal artificial valve, rather on investigation of the flow fields around the BHV and the response of the valve leaflets. And the study is also focused on the complex features and the fluid dynamics of the moving leaflets due to the blood
leaflet interaction. In this section, the motions of a stented BHV for both opening and closing phases and the blood flow through it are computed. The results are presented in terms of fluid pressure contours, fluid stress distributions on leaflet surfaces, streamlines, velocity vectors, and extracted velocity profiles.

11.5.1 The Opening Phase 11.5.1.1 Computational Model Although in reality the three leaflets of a BHV are not precisely identical, it is assumed in this work that they are similar enough that the BHV can be treated as an axisymmetrical structure. Hence, for sake of limited computing resources, only one-third of the vessel with one leaflet is considered in the

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model as depicted in Fig. 11.34. The radius R of the model is 12.5 mm and the total length L is 150 mm. The inlet side is LV and the outlet side is aorta. The BHV is positioned right after the LV in reality, in order to minimize the inlet and outlet effects on the flow field, the sinus and valve is located 50 mm (4R) away from the inlet, and the outlet is 100 mm (8R) away from the valve as show in Fig. 11.34. The height of sinus is 20 mm and the depth is 6 mm. The computational domain is discretized into an unstructured mesh of tetrahedral elements. And in the region from 45 to 80 mm away from the inlet, where the valve is located, the mesh is further refined in order to capture detailed FSI phenomenon. A grid convergence study was performed on the model and the grid size of 173,885 nodes and 974,752 elements was chosen for the simulation. For parallel computation, the mesh is partitioned into eight subdomains (see Fig. 11.37).

FIGURE 11.34 Computational domain for opening phase. (A) Perspective view of the computational model; (B) geometry of the computational model.

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FIGURE 11.35 Configuration of a closed and unloaded porcine aortic valve leaflet.

TABLE 11.5 Geometrical and Material Properties of the Flow and Valve Leaflet Fluid Parameters

Geometric Parameters of Valve

Material Parameters of Valve

ρf (kg/m3)

1.06 3 103

Height (mm)

15

ρm (kg/m3)

1.06 3 103

Re

1500

Thickness (mm)

0.04

E (kPa)

100

Uin

Pulsatile inflow

Height of sinus (mm)

20

ν

0.30

11.5.1.2 Material and Geometrical Properties of Valve Leaflet The aortic valve consists of three highly flexible leaflets, which are attached to the aortic root from one commissural point along the aortic ring toward a second commissural point. Behind each leaflet the aortic root bulbs into a sinus cavity to form the beginning of the ascending aorta. The shape of aortic heart valve leaflet in closed and unloaded phase is assumed to be an elliptic paraboloid [9], see Fig. 11.35. In his PhD work, De Hart [10] used an isotropic aortic heart valve for simulations. Following De Hart, the BHV used in this work is modified based on his model. The valve has a length of 15 mm and a uniform thickness of 0.04 mm. It is assumed in this work that the valve is isotropic for simplicity, although the real valve is not the case. The Young’s modulus of the valve leaflets is 100 kPa and the Poisson’s ratio is 0.3. The valve leaflets are assumed to be neutrally buoyant in the blood flow, so density of the leaflet is taken to be equal to the density of blood, 1.06 3 103 kg/m3. The properties are tabulated into Table 11.5. The valve

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FIGURE 11.36 Mesh for a bioprosthetic aortic valve leaflet.

leaflet is discretized into an unstructured mesh of 4225 nodes and 8216 triangular shell elements, see Fig. 11.36. The valve leaflet mesh is immersed in the fluid mesh as show in Fig. 11.37.

11.5.1.3 Initial and Boundary Conditions As the non-Newtonian nature affects the flow field only when the diameter of the flow channel is comparable with the size of the blood cell [11
13], the human blood is treated as a Newtonian fluid in this computation, with a density of 1.06 3 103 kg/m3 and a dynamic viscosity of η 5 3.5 3 1023 Ns/ m2. The initial velocity in the flow field domain is set to zero everywhere except at the inlet. The time-dependent inlet velocity profile for both the opening and closing phases is depicted in Fig. 11.15, which is specified in Eq. (11.1) and repeated here, 8 0:0 for 0 , t # 0:075T > >
 < U 0 Usin
5:0πðt 2 0:075TÞ=T  for 0:075T , t # 0:175T U in ðm=sÞ 5 U Ucos 1:98πðt 2 0:175TÞ=T for 0:175T , t # 0:44T > > : 0 0:0 for 0:44T , t # T where Uo 5 6 0.5 m/s (positive for opening phase and negative for closing phase), and T is the period for one cardiac cycle which is 0.83 seconds, the normal heartbeat of 72 beats/minute. The nondimensional form of Uin 5 Uin/UN. The corresponding maximum flow reaches 250 mL/second or 0.5 m/s, which is approximately half of the maximum physiological flow. A higher flow would instigate numerical instabilities [2]. Within a systolic interval as shown in Fig. 11.15, the aortic heart valves undergo three phases of position, as labeled asphase A, phase B, and phase C. In phase A, the aortic valves are closed from 0.0 to 0.06 seconds. In phase B, the aortic valves open from 0.06 to 0.14 seconds and close from 0.14 to 0.35 seconds. The maximum flow takes place at t 5 0.14 seconds. In phase C, the aortic valves remain closed from

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0.35 to 0.8 seconds. For incompressible flow in this work, the pressure is treated as relative to the pressure value at the outflow plane and a fixed pressure of 10 kPa is assigned to the outflow plane in this study. The Reynolds number, defined as Re 5 ρU N R=η based on the free-stream velocity, the radius of the flow channel and the density and dynamic viscosity of blood is set to 1500 in this study. Flow in a typical BHV under the pulsatile flow condition in a cardiac cycle would normally exhibit transition from unsteady laminar to turbulent flow. The need for turbulence modeling arises from the local Reynolds number of the blood flow past the BHV that may easily exceed 7000. However,

FIGURE 11.37 (A) Fluid mesh, immersed leaflet mesh for opening phase; (B) fluid mesh extracted from the center of y-axis (x
z plane) with the immersed leaflet mesh.

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under pulsatile flow conditions, the turbulence is intermittent, peaking only during the deceleration phase following peak systole [14]. Currently there is still no suitable turbulence model that could successfully describe both the laminar and turbulent flows at the same time and also take into account of their transition with affordable computing resources [15]. Large eddy simulation (LES) turbulence model could be a choice to overcome this problem, but it is very time-consuming and is beyond the capability of most research institutes. From the open literature, numerical study of pulsatile flow in prosthetic heart valves is currently limited to unsteady laminar flow only. In this work, it is beyond the research scope to develop the LES turbulence model to compute the flow in the prosthetic heart valves. As the initial step toward the computation of pulsatile flow through the BHVs, the turbulence effect of the flow field is not taken into account and the entire flow field is considered as an unsteady laminar flow in the all computations. The aorta walls are assumed to be rigid to mimic a stented BHV, and the valve leaflet is attached to the rigid walls using fully clamped boundary conditions imposed to suppress relative motion of leaflet on rigid wall. No-slip boundary condition is specified on the rigid aorta walls and blood
leaflet interface. Slip boundary condition is applied for the symmetry planes. At the inlet the pulsatile velocity profile is applied as described by Eq. (11.1) and at the outflow plane a fixed pressure of 10 kPa is assigned for the incompressible flow.

11.5.1.4 Results The flow is initiated with the leaflet in the fully closed position. A steady flow solution is obtained and used as the initial condition for the moving leaflets computation. For the unsteady computation, Courant
Friedrichs
Lewy (CFL) is set to 1.0. And a small physical timestep size of 5 3 1024 seconds is used to obtain accurate unsteady results, so the opening phase with a duration of 0.14 seconds is divided into 280 steps. In each time-step, 400 subiterations are used for pseudo time-stepping in order to achieve a divergence-free condition. The solution efficiency of the time-integration algorithm for a moving boundary flow problem becomes more important for a steady flow problem because the unsteady residual should be driven close to zero at each time-step. All the simulations were run on an SGI Origin 3400 machine with 32 processors based on silicon graphics scalable distributed-shared-memory multiprocessing architecture. It has a system bandwidth of 44.8 GB/second, 32 GB onboard memory, 800 MFlops/CPU, eight nodes and each node has four MIPs 64 Bit R12000 400 MHz/8MB RISC CPUs. According to the pulsatile inflow velocity depicted in Fig. 11.15, the opening phase for the valve leaflets occurs from 0.06 (see Fig. 11.40A) to 0.14 seconds (see Fig. 11.40G). The results show that the opening phase can

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be subdivided into two phases regarding the motion status of the leaflet. Starting from 0.06 seconds, the valve leaflet opens quickly and around 0.12 seconds it reaches its maximum opening position (see Fig. 11.40F). The valve leaflet remains at its maximum opening position until the end of opening phase around t 5 0.14 seconds (see Fig. 11.40G). It is observed that the motion of the valve leaflet is very fast and little time spent on opening phase, which takes only 0.06 seconds out of the whole cardiac cycle of 0.8 seconds. Most of the time one cardiac cycle is spent on ejecting the blood to aorta when the valve is open and filling in the heart when the valve is closed. Fig. 11.38 shows the fluid pressure and stress distribution on the surfaces of the valve leaflet. Fig. 11.38(i) is the fluid pressure distribution on the surface of the valve leaflet on the LV side, and Fig. 11.38(ii) is the fluid pressure distribution on the surface of the valve leaflet on the aorta side. Fig. 11.38(iii) is the fluid stress distribution on the surface of the valve leaflet on the LV side, and Fig. 11.38(iv) is the fluid stress distribution on the surface of the valve leaflet on the aorta side. It is observed that the fluid pressure is usually 100 times of the fluid stress, which indicates that the fluid pressure is the major cause for valve motions. At t 5 0.06 seconds, the valve leaflet is at its initial fully closed configuration. An average transvalular pressure difference of 12 kPa is observed at this moment. The pressure difference will push the valve leaflet to move toward its fully open position. The fluid stresses on both sides are far lower than the fluid pressure and of similar values because that the fluid velocity is zero on both sides of the valve leaflet. As the inflow velocity increases, the valve leaflet is pushed open by the forward flows. At t 5 0.07 seconds, the valve opens a little and forms a small orifice for the blood flow to be ejected into the aorta. The transvalular pressure is similar to the pressure difference at its fully closed position. High fluid stress is observed on the leaflet surface near to the free edge, because the forward blood flow gain velocity and passes through the free edge at relatively high speed due to the small valve orifice at this moment. The existence of high fluid stresses near the free edge of valve leaflet indicates that these areas are high-risk regions for leaflet tearing. At t 5 0.09 seconds, the valve opens with a larger orifice with an area of about 70% of the maximum orifice area when the valve achieves its fully opened position. At this moment, the transvalular pressure difference is very small. The high stress near the free edges disappears, because the velocity of forward blood flow is of similar values over the valve leaflet surface due to a large valve orifice. From t 5 0.12 seconds onward, the valve reaches its fully open position, the pressure on the LV side equals the pressure on the aorta side. And the valve will no longer move toward the sinus cavity. It remains at this position and fluctuate till the end of the opening phase of t 5 0.14 seconds. Fig. 11.39 shows a series of pressure contour plots of y 5 0 plane in vicinity of the immersed valve leaflet for the opening phase at different time instants. It presents the evolution of the pressure distribution of the flow field

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FIGURE 11.38 Fluid pressure and stress distribution on valve leaflet at different time instants for opening phase. (i) Pressure distribution on the LV side; (ii) pressure distribution on the aorta side; (iii) fluid stress distribution on the LV side; (iv) fluid stress distribution on the aorta side. (A) t 5 0.06 s; (B) t 5 0.07 s; (C) t 5 0.08 s; (D) t 5 0.09 s; (E) t 5 0.10 s; (F) t 5 0.12 s; (G) t 5 0.14 s.

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FIGURE 11.38 (Continued).

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FIGURE 11.38 (Continued).

323

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FIGURE 11.38 (Continued).

during the opening phase with the motion of the valve leaflet. Fig. 11.39A shows the pressure contour at t 5 0.06 seconds when the valve leaflet is about to move. It can be seen that at this moment the average pressure on the LV side is higher than the average pressure at the aorta side. A transvalular pressure difference of 12 kPa is measured across the leaflet at this time instant. This pressure difference on the two sides of the valve provides a driving force to push the valve moving forward and toward the sinus cavity until it reaches its maximum opening position. From t 5 0.09 seconds onward, the pressure difference has gradually reduced and finally reached an equilibrium state at t 5 0.12 seconds for the fully opened position. One consideration for an ideal prosthetic heart valve would be one where the pressure drop across the valve orifice is minimum. The effective orifice area of the device determines the value of this pressure drop. A smooth transition between LV and the ascending aorta will depend particularly on the geometry. The viscous effects, in this case, are secondary because that the energy dissipated into the bulk fluid by viscosity is small compared with the work done by this pressure gradient. However, viscous effects such as separation can affect this pressure gradient quite considerably where the resulting wake, increasing as a function of the downstream coordinate, decrease the effective orifice area. The vector fields and the streamline distributions near the valve leaflet are given in Fig. 11.40 for the opening phase at different time instants with

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the corresponding positions of the valve leaflet. Fig. 11.40(i) shows the vector plots of plane y 5 0 and corresponding configurations of the valve leaflet. Fig. 11.40(ii) shows the velocity vector plots of the flow field near the valve leaflet. Fig. 11.40(iii) shows the velocity vector plots extracted at x 5 50 mm (Plane I), x 5 60 mm (Plane II), x 5 70 mm (Plane III), x 5 80 mm (Plane

FIGURE 11.39 Pressure contours of y 5 0 plane at different time instants for opening phase. (A) t 5 0.06 s; (B) t 5 0.07 s; (C) t 5 0.08 s; (D) t 5 0.09 s; (E) t 5 0.10 s; (F) t 5 0.12 s; (G) t 5 0.14 s.

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FIGURE 11.39 (Continued).

IV), and x 5 90 mm (Plane V). And Fig. 11.40(iv) shows the streamline distribution of the flow field around the valve leaflet. From the figure, it is observed that the opening phase is characterized by an acceleration central flow in the mainstream, and little flow is observed in the sinus cavity. The valve opens rapidly, which takes only 0.06 seconds out of the whole cardiac cycle of 0.8 seconds. As the leaflet moves into the sinus cavity, vortices form between the leaflets and the sinus walls. And the leaflet projects

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FIGURE 11.39 (Continued).

slightly into the sinus cavity, see Fig. 11.40G. The flow enters the sinus cavity at the leaflet free edge, curls back along the sinus wall, along the valve leaflet and then flows out into the mainstream. The same flow patter was observed by Bellhouse et al. [16]. At t 5 0.06 seconds as shown in Fig. 11.40A, the valve leaflet starts to move from its initial configuration, corresponding to a fully closed position. Little flow is observed at this moment since velocity of the pulsatile inflow

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FIGURE 11.39 (Continued).

FIGURE 11.40 Velocity vector plots at different time instants for opening phase. (i) Velocity vector plot for y 5 0 plane with valve leaflet; (ii) velocity vector plot of the flow field near the valve leaflet; (iii) velocity vector plots extracted at x 5 50 mm (Plane I), x 5 60 mm (Plane II), x 5 70 mm (Plane III), x 5 80 mm (Plane IV) and x 5 90 mm (Plane V); (iv) streamline distribution of the flow field around the valve leaflet. (A) t 5 0.06 s; (B) t 5 0.07 s; (C) t 5 0.08 s; (D) t 5 0.09 s; (E) t 5 0.10 s; (F) t 5 0.12 s; (G) t 5 0.14 s.

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FIGURE 11.40 (Continued).

is zero. At t 5 0.07 seconds, the valve leaflet is pushed open by the transvalular pressure difference. A forward flow is observed at every node of the fluid domain as shown in Fig. 11.40B, including in the sinus cavity, that is, the wash-out effect. At this moment, the valve opens with a small orifice of a triangular shape as shown in Fig. 11.41A. Five velocity profiles are extracted from plane y 5 0 at x 5 50 mm (Plane I), x 5 60 mm (Plane II), x 5 70 mm (Plane III), x 5 80 mm (Plane IV), and x 5 90 mm (Plane

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FIGURE 11.40 (Continued).

V) as shown in Fig. 11.42. At Plane I just in front of the valve leaflet, the velocity profile is fairly flat and the average velocity is about 0.2 m/s as shown in Fig. 11.43A. At Pane II, the average velocity inside the sinus cavity is about 0.1 m/s. And the average velocity within the mainstream is about 0.4 m/s due to the small valve orifice at this moment. At Plane III, the average velocity inside the sinus cavity is about 0.1 m/s, and the average velocity

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FIGURE 11.40 (Continued).

within the mainstream is about 0.6 m/s. The velocity inside the sinus cavity is much lower than the velocity in the mainstream, whose maximum value is about 10%
15% of the maximum velocity of the mainstream. van Steenhoven et al. [17,18] reported similar findings in their 2D experiments of the aortic valve model that the velocities in the sinus are about 15% of those in the mainstream. It is also observed that in the sinus the velocities at Planes II and III are of similar values, while in the mainstream the velocity

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FIGURE 11.40 (Continued).

at Plane III is about 0.2 m/s higher than that of Plane II, which indicates that the rate of change of velocity is higher in the mainstream than that in the sinus cavity. It results in a higher fluid stress on the valve leaflet surface at the mainstream side (LV side) than the fluid stress on the leaflet surface at the sinus side (aorta side), which are shown in frames (iii) and (iv) of Fig. 11.38B. In frame (iii), the average fluid stress on leaflet surface is about 150 Pa and in frame (iv) the average fluid stress on the leaflet surface is

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FIGURE 11.40 (Continued).

about 50 Pa. At Planes IV and V after the valve leaflet, the maximum velocity is 1.1 and 0.9 m/s, respectively. Both velocity profiles present parabolic shapes. The valve opens with a narrow orifice area under the hemodynamic forces at the time instant of 0.08 seconds as shown in Fig. 11.41B. At this moment, the orifice area is still quite narrow showing a strong tendency of the blood rushing toward the aorta with an increase in the inlet velocity as

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FIGURE 11.40 (Continued).

shown in Fig. 11.40C. At Plane I, the average velocity is 0.2 and the velocity profile does not change much compared with that of the time instant of 0.07 seconds as shown in Fig. 11.43B. At Planes II and III, the average velocities inside the sinus cavity are 0.1 m/s, which indicates that little flow is observed in the sinus cavity and no vortex is formed in the sinus. In the mainstream, the average velocity at plane is about 0.4 m/s at Plane II and the velocity profile is more flat due to a larger opening orifice. The average

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FIGURE 11.41 Valve orifice area of opening phase (view from aorta side to LV). (A) t 5 0.06 s; (B) t 5 0.07 s; (C) t 5 0.08 s; (D) t 5 0.09 s; (E) t 5 0.10 s; (F) t 5 0.12 s; (G) t 5 0.14 s.

FIGURE 11.42 Extracted positions of velocity profiles.

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velocity at Plane III in the mainstream increases to 1.0 m/s. With the high rate of change of velocity in the mainstream, the average fluid stress is about 120 Pa on the leaflet surface at the mainstream side as shown in Fig. 11.38C. And the higher fluid stress distributes mainly in the region near the free edge of the leaflet because the rate of change of velocity is higher at the free edge

FIGURE 11.43 Velocity profiles extracted for opening phase. (A) t 5 0.07 s; (B) t 5 0.08 s; (C) t 5 0.09 s; (D) t 5 0.10 s; (E) t 5 0.12 s; (F) t 5 0.14 s.

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FIGURE 11.43 (Continued).

than that at the central region of the leaflet. At the sinus cavity side, the average fluid stress is of a low value of around 10 Pa due to the low velocity value and the low rate of change of velocity. At Planes IV and V, the maximum velocities increase to 1.2 and 0.95 m/s, respectively, due to the acceleration of the pulsatile inflow. And the velocity profiles are more central compared those of the time instant of 0.07 seconds. In the region between r/

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R0 5 20.6 and r/R0 5 0.0, the velocity is larger than 0.2 m/s. While at t 5 0.06 seconds, this region is between r/R0 5 20.4 and r/R0 5 0.0 as shown in Fig. 11.43A. At t 5 0.09 seconds, the valve opens with a moderate valve orifice area as shown in Fig. 11.41C. The flow field remains accelerating till this time instant. At Plane I before the leaflet, the average velocity increases to 0.4 m/ s. At Planes IV and V after the leaflet, the maximum velocities increase to 1.4 and 1.2 m/s, respectively, with the region of the central flow extending to r/R0 5 20.75 as shown in Fig. 11.43C. At Plane II, the average velocity in the mainstream increases to 0.6 m/s. And at region between r/R0 5 20.60 and r/R0 5 21.0, a negative velocity of 20.1 m/s is observed near the valve leaflet surface in the sinus cavity. Near the sinus wall at r/R0 5 21.6, a forward flow of 0.2 m/s is present. This suggests that the mainstream flow enters the sinus at the center of the free edge. It moves along the leaflet surface and along the sinus wall and finally leaves the sinus near the commissures. The same flow pattern was also observed by Peskin et al. [19] in the 3D flow field with sinus cavity. At Plane III, the average velocity in the mainstream increases to 1.2 m/s. The rate of change of velocity in the mainstream between Planes II and III is 0.6 m/s, and in the sinus cavity the rate of change of the velocity near leaflet is 0.3 m/s. So the fluid stress on the leaflet surface at the mainstream side is about 130 Pa and on the sinus side the fluid stress is about 80 Pa as shown in Fig. 11.38D. The valve opens to a large orifice at t 5 0.10 seconds as shown in Fig. 11.41D. The orifice area is about 80% of the maximum orifice area. At this moment, a large region of forward flow is observed in the mainstream as shown in Fig. 11.40E due to the large valve orifice. The region of central flow in ascending aorta extends to r/R0 5 20.85 as shown in the velocity profiles at Planes IV and V of Fig. 11.43D. The maximum velocity at Plane IV registers a value of 1.8 m/s and at Plane V the maximum velocity is 1.0 m/s. In the mainstream, the average velocities at Planes III and II are 1.1 and 0.8 m/s, respectively, resulting a velocity change of 0.3 m/s. In the sinus cavity, the average velocities at Planes III and II are 0.15 and 0.1 m/s, respectively, with a velocity gradient of 0.05 m/s. So the fluid stress on the leaflet surface at the mainstream side is about 80 Pa and at the sinus side it is about 20 Pa. From t 5 0.12 seconds till the end of the opening phase, the valve reaches its fully open position with a maximum valve orifice as shown in Fig. 11.41D and E. The flow is more central and little flow is observed in the sinus cavity as shown in Fig. 11.40F and G. The leaflet projects slightly into the sinus cavity. It remains at the fully open position and fluctuates as the blood flow passing through the valve. At t 5 0.12 seconds as shown in Fig. 11.43E, the average velocities at the Planes I, IV, and V are 0.2, 0.6, and 0.8 m/s, respectively. At Plane II, the mainstream average velocity is 0.6 m/s and at Plane III it is 0.8 m/s. In the sinus cavity, the average

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velocities at both planes are about 0.1 m/s. This results in an average fluid stress of 110 Pa on the leaflet surface at the mainstream side and 10 Pa at the sinus cavity side as shown in Fig. 11.38F. At t 5 0.14 seconds, the average velocities at the Planes I, IV, and V are 0.1, 0.3, and 0.4 m/s, respectively, as shown in Fig. 11.43F. At Planes II and III, the mainstream average velocities are 0.3 m/s and at Plane III it is 0.7 m/s. By comparing the velocity profiles at t 5 0.12 seconds and t 5 0.14 seconds, it is observed that at the end of the opening phase the flow has started to decelerate already. Similar results were found by van Steenhoven et al. [17,18] for in vivo experiments and by De Hart [10] in the numerical simulation of a stented aortic heart valve.

11.5.2 The Closing Phase 11.5.2.1 Computational Model The closing phase of a stented BHV based on the parameters listed in Table 11.5 is computed under the physiological conditions specified by Eq. (11.1). The flow-field configuration should not change over the pulsatile period for simulation, so the general flow field structure in the closing phase remains the same as the opening phase. However, during the closing phase, the flow reverses its direction, which results in that the former upstream and downstream in the opening phase are totally reversed for the current closing phase. In order to keep consistency with the opening phase, the LV side of the BHV is still treated as upstream, and the opposite aorta side is treated as downstream. But in the computation, the inlet boundary conditions are applied to the downstream plane, while the outlet boundary conditions are applied to the upstream plane as shown in Fig. 11.44. The same mesh as shown in Fig. 11.34 is used for the computation of the closing phase. No-slip boundary conditions are applied to solid walls and blood
leaflet interface. Slip boundary conditions are applied to symmetry planes. And the valve leaflet is fully clamped into the solid walls.

FIGURE 11.44 Perspective view of the computational model for closing phase.

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The computation of the closing phase is far more difficult compared with the computation of the opening phase due to some factors. First, the turbulent effects are more prominent in the closing phase [20,21], because the closing leaflet blocks the flow and instigate much turbulence. Second, the historical development of the flow field affects its current status greatly. However, in this work, the historical development of the flow field is not taken into account due to the swapping of the outlet and inlet positions. Without the consideration of the historical development, the current computation might deviate from the real situation. Despite all these difficulties faced, it is still worthy to carry out the computation of the closing phase as a first attempt to obtain some knowledge of the flow field for sake of possible future research works in this field.

11.5.2.2 Results According to the pulsatile flow velocity profile depicted in Fig. 11.15, the closing phase occurs from 0.14 to 0.35 seconds. For the unsteady computation, the CFL is set to 1.0. And a small physical time-step size of 5.0 3 1024 seconds is used to obtain accurate unsteady results, so the opening phase with a duration of 0.21 seconds is divided into 420 steps. In each time-step, 400 subiterations are used for pseudo time-stepping in order to achieve a divergence-free condition. The results for the closing phase are presented in Figs. 11.45
11.48 in terms of pressure contour, fluid pressure, and stress distribution on leaflet surface, velocity vector, and streamlines, and velocity profiles extracted at several locations. From these figures, it is observed that the valve closure is initiated by the pressure reversal of the fluid domain. The negative transvalular pressure pushes the valve from the fully open position toward its fully closed position. This agrees with van Steenhoven’s theory [17] that the valve closure is induced mainly by the adverse pressure gradient. Fig. 11.45 presents the pressure contour of plane y 5 0 with corresponding valve positions at different time instants of the closing phase. From the figures, it is observed that the pressure on the aorta side is higher than the pressure on the LV side during the closing phase. At t 5 0.14 seconds as shown in Fig. 11.45A, the pressure difference between the aorta and LV is about 1 kPa. This pressure provides a driving force to push the valve away from it fully opened position toward the fully closed position and initiates the closing phase. This pressure difference across the valve has increased gradually as the valve closes and caused a noticeable motion of the leaflet at t 5 0.19 seconds, see Fig. 11.45B. And the pressure difference is more prominent toward the end of the closed position as observed in Fig. 11.45C and D. Finally the valve is sealed off with a large pressure difference at t 5 0.34 seconds, see Fig. 11.45E.

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A series of velocity vector plots and streamline distribution plots for the closing phase with corresponding valve configurations are presented in Fig. 11.41. Velocity profiles for plane y 5 0 are extracted at x 5 40 mm (Plane I), x 5 50 mm (Plane II), x 5 60 mm (Plane III), x 5 70 mm (Plane IV), and x 5 80 mm (Plane V) as shown in Fig. 11.48. At t 5 0.14 seconds, as shown in Fig. 11.47A, the flow is quite smooth from the aorta to the LV.

FIGURE 11.45 Pressure contours of y 5 0 plane at different time instants for the closing phase. (A) t 5 0.14 s; (B) t 5 0.19 s; (C) t 5 0.27 s; (D) t 5 0.32 s; (E) t 5 0.34 s.

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FIGURE 11.45 (Continued).

The maximum velocities at Planes I, II, and V are 20.53, 20.52, and 20.23 m/s, respectively, as shown in Fig. 11.49A. At Plane III, the average velocity inside the sinus is about 20.01 m/s. It is much smaller than the velocity in the mainstream of 20.5 m/s. For Plane IV just before the valve leaflet, the average velocity inside the sinus is about 20.02 m/s, and the average mainstream velocity is about 20.3 m/s. It can be seen that there is little flow in the sinus cavity at this moment. Both the velocity and the rate

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FIGURE 11.45 (Continued).

of change of the velocity in the mainstream is much larger than those in the sinus, thus resulting a larger fluid stress on the leaflet surface at the mainstream side than that on the sinus side. The average fluid stress on the leaflet surface is about 90 Pa and the average fluid stress on the valve leaflet is about 20 Pa on the sinus side as shown in Fig. 11.46A. The motion of the valve leaflet is relatively small at the early stages of the closing phase due to the small pressure differences. The valve leaflet moves to a new position at t 5 0.19 seconds under the influence of blood
leaflet interaction as shown in Fig. 11.47B. At this moment, the flow field is still smooth because the valve orifice is large for the reversing flow. In the central flow, the maximum velocities at Planes I, II, and V are 20.68 m/s, 20.67, and 20.24 m/s, respectively. At Planes III and IV, the average velocities inside the sinus are 0.06 and 0.08 m/s, while in the mainstream the average velocities are 20.5 and 20.36 m/s, respectively. The maximum velocity at Plane III is 20.72 m/s. This is expected because the presence of the leaflet limits the orifice area and increases the mainstream velocity. The larger rate of change of velocity in the mainstream results in a larger fluid stress of 60 Pa on the leaflet surface. On the leaflet surface at the sinus side, the fluid stress is about 8 Pa as shown in Figs. 11.46B and 11.50. At t 5 0.27 seconds, the valve moves to a new position under the hemodynamic forces acting on it as shown in Fig. 11.47C. At this moment, the flow is obstructed a little by the closing valve leaflet. The maximum velocities at Planes I, II, and V are 20.5, 20.5, and 20.11 m/s as shown in Fig. 11.49C. The velocities at Planes I and II are larger than that at Plane V,

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FIGURE 11.46 Fluid pressure and stress distribution on valve leaflet at different time instants for opening phase. (i) Pressure distribution on the LV side; (ii) pressure distribution on the aorta side; (iii) fluid stress distribution on the LV side; (iv) fluid stress distribution on the aorta side. (A) t 5 0.14 s; (B) t 5 0.19 s; (C) t 5 0.27 s; (D) t 5 0.32 s.

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FIGURE 11.46 (Continued).

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FIGURE 11.47 Velocity vector plots at different time instants for opening phase. (i) Velocity vector plot for y 5 0 plane with valve leaflet; (ii) velocity vector plot of the flow field near the valve leaflet; (iii) velocity vector plots extracted at x 5 40 mm (Plane I), x 5 50 mm (Plane II), x 5 60 mm (Plane III), x 5 70 mm (Plane IV), and x 5 80 mm (Plane V); (iv) streamline distribution of the flow field around the valve leaflet. (A) t 5 0.14 s; (B) t 5 0.19 s; (C) t 5 0.27 s; (D) t 5 0.32 s; (E) t 5 0.34 s.

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FIGURE 11.47 (Continued).

because the closing valve obstructs the blood flow and reduces the passage for flow after the valve leaflet, thus increasing the velocity after the leaflet. At Plane III, the maximum velocity in the mainstream is about 20.58 m/s due the influence of the blood
leaflet interaction. At Plane IV, the maximum velocity is about 20.2 m/s. Inside the sinus cavity, the average velocities are 0.01 m/s, which are much lower than the mainstream velocities. The fluid stress on the leaflet surface at the mainstream side is 150 Pa. And a

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FIGURE 11.47 (Continued).

high fluid stress of about 300 Pa is observed in the region near the free edge due to the small valve orifice area. On the surface at the sinus side, the average fluid stress is about 30 Pa as shown in Fig. 11.46C. The valve is about to close to at t 5 0.32 seconds as shown in Fig. 11.47D. The blood flow is obstructed a lot by the valve leaflet. The maximum velocities at Planes I, II, and V are 20.52, 20.38, and 20.14 m/s, respectively, as shown in Fig. 11.49D. These values increase because the

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FIGURE 11.47 (Continued).

valve orifice area is much smaller at this moment and increases the maximum velocity of the central flow. At Plane III, the maximum velocity in the mainstream is 20.57 m/s as the valve almost closes. At Plane IV, the maximum velocity in the mainstream is 20.18 m/s as the inflow slows down. And in the sinus cavity, the average velocities at the two planes are near zero, which indicates the flow inside the sinus cavity is almost stagnant at this moment. Thus the average fluid stress on the leaflet surface at the sinus

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FIGURE 11.47 (Continued).

FIGURE 11.48 Positions for velocity extraction of plane y 5 0.

side is of a very low value of 5 Pa, while at the mainstream side, the average fluid stress is about 150 Pa. The high stress region still exists near the free edge of the leaflet due the large rate of change of the velocity in the mainstream. The valve finally snaps to the fully closed position at t 5 0.34 seconds and little flow is observed in the sinus and the mainstream at this moment as shown in Fig. 11.47E.

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FIGURE 11.49 Velocity profiles extracted for opening phase. (A) t 5 0.14 s; (B) t 5 0.19 s; (C) t 5 0.27 s; (D) t 5 0.31 s.

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FIGURE 11.50 Valve orifice area for the closing phase (view from aorta side to LV). (A) t 5 0.14 s; (B) t 5 0.19 s; (C) t 5 0.27 s; (D) t 5 0.32 s; (E) t 5 0.34 s.

11.5.3 Remarks The motions of a stented BHV for both opening and closing phases under physiological conditions and the blood flows through it are simulated by the developed parallel solver with the IMM for blood
leaflet interaction. And the results are compared with experimental results and numerical results. Good agreements are observed between current numerical results and other researchers’ results. The general motions of the valve can be pictured as the follows: the valve leaflet opens and closes very fast. Most of the time of one cardiac cycle is spent on ejecting the blood to aorta when the valve is open and filling in the heart when the valve is closed. The transvalular pressure difference is the major cause for the motions of the valve leaflet. During the opening phase, the pressure at the LV side is higher than the pressure at the aorta side, and the positive pressure difference pushes the leaflet toward its fully opened position. While during the closing phase, the pressure at the LV side is lower than the pressure at the aorta side, and the adverse pressure difference pushes the leaflet toward its fully closed position. The opening phase starts from t 5 0.06 seconds, it reaches its maximum opening position around

IMM FSI Model Validations and Applications Chapter | 11

353

t 5 0.12 seconds. The valve leaflet remains at its maximum opening position and fluctuates until the end of the opening phase around t 5 0.14 seconds. The central flow enters the sinus cavity at the leaflet free edge, curls back along the sinus wall, along the valve leaflet and then flows out into the mainstream. For both the opening and closing phases, the velocity inside the sinus cavity is much lower than the velocity in the mainstream, whose maximum value is about 10%
15% of the maximum velocity of the mainstream. Both the velocity and the rate of change of velocity in the mainstream are higher than those in the sinus cavity, so the fluid stresses on the leaflet surfaces at the mainstream side are higher than those at the sinus side. And high fluid stresses exist in the region near the free edge of the leaflet, which indicates that these areas are high-risk regions for leaflet tearing. At the moment of complete valve opening, the blood flow has started to decelerate already. The negative transvalular pressure pushes the valve from the fully open position toward its fully closed position. And the motion of the valve leaflet is relatively small at the early stages of the closing phase due to the small pressure differences. The valve leaflet fluctuate at its fully opened position for quite some time till t 5 0.19 seconds. At the end of the systolic phase, the valve snaps to the fully closed position.

REFERENCES [1] Y. Zhao, C.H. Tai, Higher-order characteristics-based methods for incompressible flow computation on unstructured grids, AIAA J. 39 (7) (2001) 1280
1287. [2] Y.B. Shi, Numerical and Experimental Study of Pulsatile Flow in Bi-leaflet Mechanical Heart Valves, Ph.D. Thesis, Nanyang Technological University, 2002. [3] British Standard BS EN 12006-1:1999, Non-Active Surgical Implants
Particular Requirement for Cardiac and Vascular Implants, Part 1: Heart Valve Substitutes, 1999. [4] C.H. Tai, Parallel-multigrid Computation of Prosthetic Heart Valves Under Physiological Conditions Using an Immersed Object Method With Overlapping Grids, Ph.D. Thesis, Mechanical and Production Engineering, Nanyang Technological University, 2004. [5] A. Gilmanov, et al., A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. Comput. Phys. 207 (2) (2005) 457
492. [6] H.S. Udaykumar, et al., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. Comput. Phys. 174 (2001) 345
380. [7] Y. Zhao, Grid Generation Solver: FORGEN and ST2UNST, Research Report, School of MAE, Nanyang Technological University, 1999. [8] S.C.R. Dennis, et al., The steady flow due to a rotating sphere at low and moderate Reynolds numbers, J. Fluid Mech. 101 (1980) 257
259. [9] M.S. Hamid, H.N. Sabbah, P.D. Stein, Influence of stent height upon stresses on the cusps of closed bioprosthetic valves, J. Biomech. 19 (1986) 759
769. [10] J. De Hart, Fluid-Structure Interaction in the Aortic Heart Valve: A Three-Dimensional Computational Analysis, Ph.D. Thesis, Eindhoven University of Technology, 2003. [11] Y.C. Fung, Biomechanics
Motion, Flow, Stress, and Growth, Springer-Verlag, New York, 1990.

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[12] W. Nicholes Wilmwe, F. O’Rourke Michael, McDonald’s Blood Flow in Arteries, Theoretical, Experimental and Clinical Principles, third ed., Lea & Feibiger, 1990. [13] X.Y. Xu, M.W. Collins, Studies of blood flow in arterial bifurcations using computational fluid dynamics, Proc. Inst. Mech. Eng. Part H: J. Eng. Med. 208 (1994) 163
175. [14] D. Bluestein, S. Einav, Techniques in the analysis of stability of pulsatile flow through heart valves, in: C.T. Leondes (Ed.), Biomechanics Systems Techniques and Applications: Cardiovascular Techniques, Vol. II, CRC Press LLC, Boca Raton, FL, 2001. [15] G. Tzabiras, Calculation of Complex Turbulent Flows, WIT Press, Boston, 2000. [16] B.J. Bellhouse, L. Talbot, The fluid mechanics of the aortic valve, J. Fluid Mech. 35 (1969) 721. [17] A.A. van Steenhoven, M.E.H. van Dongen, Model studies of the closing behaviour of the aortic valve, J. Fluid Mech. 90 (1979) 21. [18] A.A. van Steenhoven, P.C. Veenstra, The effect of some hemodynamic factors on the behaviour of the aortic valve, J. Biomech. 15 (1982) 941. [19] C.S. Peskin, Flow Patterns Around Heart Valves: A Digital Computer Method for Solving the Equation of Motion, Ph.D. Thesis, Albert Einstein College of Medicine, Yeshiva University, 1972. [20] D. Bluestein, S. Einav, Transition to turbulence in pulsatile flow through heart valves
a modified stability approach, J. Biomech. Eng. 116 (1994) 477
487. [21] J.S. Liu, P.C. Lu, S.H. Chu, Turbulence characteristics downstream of bileaflet aortic valve prostheses, J. Biomech. Eng. 122 (2000) 118
124.

Chapter 12

IMM FSI Model Validations and Applications for Compressible Flows 12.1 VALIDATION OF BASELINE COMPRESSIBLE FLOW SOLVER WITH IMM 12.1.1 Steady Flow Induced by a Rotating Sphere in Quiescent Air—Order of Accuracy To determine the overall accuracy of the proposed methods, first we carry out a grid convergence study for a test problem, which is a 3D analog of the problem used by Gilmanov and Sotiropoulos [1]. In this case, we simulate flow induced by a sphere of radius R0 , rotating at constant angular velocity Ω about the z-axis in a nearly incompressible, viscous fluid with kinematic viscosity ν. The Reynolds number for this flow is defined as Re 5 ΩR20 =ν. For Reynolds numbers in the range Re 5 1 2 100, benchmark solutions for this problem have been reported by Dennis et al. [2] who solved numerically the steady, axisymmetric Navier
Stokes equations in polar coordinates using a vorticity-stream function formulation. In our studies, the Reynolds number is set to Re 5 100, and we solve the full 3D and unsteady flow problem with the sphere starting to rotate impulsively from quiescence relative to the stationary Cartesian grid. The computational domain is a ð10R0 Þ3 cube with its center located at x 5 5.0R0, y 5 5.0R0, and z 5 5.0R0. Four uniformly spaced and successively finer mesh sizes with 203, 403, 803, and 1603 grid points are used for error analysis, and the finest-mesh solution is considered to be the “exact” solution. The surface of the sphere is discretized with an unstructured triangular mesh consisting of 14,612 elements. The sphere is placed at the center of the cubical domain and starts to rotate impulsively at t 5 0 with constant rotational velocity Ω about the z-axis. On all the grids, the same physical time-step (Δt 5 0.01T) is employed in order to concentrate on the spatial resolution of the method, as done in Ref. [1]. For all the grids, the simulation is

Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00012-X © 2019 Elsevier Inc. All rights reserved.

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Computational Fluid-Structure Interaction

continued for one complete period, at the end of which the LN and Lq norms of the u-velocity errors are calculated as follows: " #


1=q N3
1 X
ðNÞ
ðNÞ e
q q N e
εN 5 max
ui 2 ui
; εN 5 3 ð12:1Þ
u 2ui
N i51 i i 5 1;N 3 ðNÞ q is the u-velocity where εN N and εN are the infinity and qth error norms, ui 3 e component at the ith node of the N mesh, and ui is the “exact” velocity field calculated on the 1603 grid. The results of the grid convergence study are summarized in Fig. 12.1, which shows the variation of the LN, L1, and L2 norms of errors with grid spacing in logarithmic coordinates. The lines with Slopes 1 and 2 are also given as reference. It is evident from Fig. 12.1 that the method is second-order accurate. To further demonstrate the accuracy of our method, we also use the Richardson estimation procedure to study the accuracy of the solver as in Ref. [1]. Let f N denote the numerical solution on the N3 mesh. Assume that the discrete solution is an γ-order approximation to its value f exact , and the flow field is continuous and has no singularity points, then we have   log :f N 2 f N=2 :=:f N=2 2 f N=4 : ð12:2Þ γ5 log 2

0 –0.2

Grid l : 20**3 nodes Grid ll : 40**3 nodes Grid lll : 80**3 nodes “Exact grid” : 160**3 nodes

Grid l

–0.4

Log ⎜Error ⎜

–0.6 –0.8 –1

Grid ll

–1.2 L2 error L1 error Linfinity error Slope 1 Slope 2

–1.4 –1.6

Grid lll

–1.8 –1.3

–1.2

–1.1

–1

–0.9 –0.8 Log ⎜h ⎜

–0.7

–0.6

–0.5

FIGURE 12.1 Convergence of the LN , L1 , and L2 error norms for the velocity field induced by a rotating sphere. Slopes 1 and 2 are the reference lines for first-order and second-order accuracy, respectively.

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IMM FSI Model Validations and Applications Chapter | 12

TABLE 12.1 Rate of Convergence γ Calculated for Different Error Norms Norm

Grids 403, 803, 1603

LN

1.82

L1

2.84

L2

2.15

where jj jj denotes an error norm (LN, L1, or L2). If γ  2 the solution is second-order accurate. We apply the above procedure for N 5 160 (using solutions obtained on meshes 403, 803, and 1603) to calculate γ for successively refined meshes. We use all three norms to compute the error and the results are summarized in Table 12.1, which strongly supports our assertion about the second-order accuracy of our method. Fig. 12.2 shows several snapshots of instantaneous streamlines on the plane at y 5 2.5. To compare the result quantitatively with those in Refs. [1,2], we compute the angle θs between the z-axis and the line that originates from the center of the sphere and passes through the center of the toroidal vortex ring (see Fig. 12.3 for definition). In Table 12.2, we compare our θs with those reported in Refs. [1,2] and the agreement is very good.

12.1.2 Flow Induced by an Oscillating Sphere in a Closed Cavity The grid convergence study is also performed for the case where a solid oscillating sphere is immersed in a fluid enclosed within a cube with solid walls. This is to demonstrate the capability of the method in handling objects moving with large displacements. The sphere of diameter D 5 1.0 units is placed initially at the center of the cube of dimension 2 3 2 units and oscillates horizontally with a nondimensional time period of 1.0 and an amplitude of 0.25D. The oscillation is effected by moving the sphere as a rigid body with velocity given by u 5 0:25πsinð2πtÞ;

v5w50

ð12:3Þ

The Reynolds number (based on the sphere diameter and maximum velocity) has been set to 20. The following sequence of grid sizes is employed in performing the error analysis: 203, 403, 803, and 1603. The result on the 1603 mesh is taken to be the “exact” solution for this case. A small time-step of Δt 5 0.005 is chosen for all these simulations in order to minimize the effect of temporal errors on the solution. The simulations are carried out for one oscillation period and the velocity components at each

t = 0.3 T

t = 0.6 T

z

z y

y

x

t = 1.0 T

x

t = 3.0 T

z y

z x

y

x

FIGURE 12.2 Instantaneous snapshots of streamlines at the y 5 2.5 plane depicting the early stages of flow evolution toward steady state for Re 5 100. Time is measured from the start of the impulsive rotation and T is the rotation period of the sphere.

Z

Y

X

θs = 74.3°

FIGURE 12.3 The angle θs between the z-axis and the radius that passes through the center of the toroidal vortex ring.

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359

TABLE 12.2 Comparison of the Measured [1,2] and Calculated Angle θs Reynolds number

θDennis

θGilmanov

θcalc

εð%Þ

100

73.8

72.1

74.3

0.68 (with Ref. [2]), 3.05 (with Ref. [1])

Z

Y

X

FIGURE 12.4 Instantaneous streamlines at t 5 1.0T for flow induced by an oscillating sphere in a closed cube filled with compressible and viscous fluid.

grid point are recorded for all the meshes under consideration at the end of the period. The instantaneous streamlines at the end of an oscillation cycle (t 5 1.0T) are shown in Fig. 12.4. Fig. 12.5 presents the error norms for the three meshes (203, 403, and 803). As can be noted, the convergence rates of errors in the simulations are close to Slope 2 reference line, indicating second-order accuracy.

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–0.2 –0.4

Grid l : 20**3 nodes Grid ll : 40**3 nodes Grid lll : 80**3 nodes “Exact grid” : 160**3 nodes

Grid l

Log ⎜Error ⎜

–0.6 –0.8 –1 Grid ll

–1.2 –1.4 –1.6

L2 error L1 error Linfinity error Slope 1 Slope 2

Grid lll

–1.8 –1.3

–1.2

–1.1

–1 –0.9 Log ⎜h ⎜

–0.8

–0.7

–0.6

–0.5

FIGURE 12.5 Convergence of the LN , L1 , and L2 error norms for the velocity field of oscillating sphere in a closed cube. Slopes 1 and 2 are the reference lines for first-order and secondorder accuracies, respectively.

12.1.3 Steady and Unsteady Flows Past a Circular Cylinder Flow past a circular cylinder is a classical benchmark problem, which has been the subject of many theoretical, experimental, and computational investigations due to its simple geometry and its representative behavior of general bluff body flows. Both low and high Reynolds numbers are used to demonstrate and examine the performance, accuracy, and robustness of the parallel multigrid (MG) compressible solver with the implementation of immersed membrane method (IMM) algorithm for steady and unsteady flows. The computational domain and the immersed cylinder are shown in Fig. 12.6. The parallel calculations are performed on the SGI Origin 3400 machine. Three different grids are generated, which have 84,054, 122,202, and 182,536 nodes and the immersed cylinder surface is discretized using 33,578 triangular elements. All the parameters for the three simulation runs are kept exactly the same and the simulation is run until nondimensional time, t 5 150.0 for Re 5 200 with an inflow Mach number of 0.2. The computed value of lift coefficient, Cl, is used as a criterion for convergence as shown in Fig. 12.7. Comparing the results shown in Fig. 12.7, the peak value of Cl obtained using 84,054 nodes is 0.56 as compared to 0.64 obtained by both grids of

IMM FSI Model Validations and Applications Chapter | 12

3D circular cylinder mesh with 33,578 triangular elements

3D circular cylinder mesh limmersed into background fluid mesh The wake region behind the cylinder is further refined

361

X Y

Z

FIGURE 12.6 The computational domain and the immersed cylinder for viscous flow past a circular cylinder.

84,054 nodes 122,202 nodes 182,536 nodes

1.4 1.2 1 Lift coefficient Cl

0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 50

100

150

Time

FIGURE 12.7 Lift coefficients versus physical time using three different grids for flow over a circular cylinder (parallel MG, preconditioned, Re 5 200, inflow Mach 5 0.2).

122,202 and 182,536 nodes. Similarly, comparing the results obtained by both 122,202 and 182,536 nodes, the value of Cl does not deviate from each other significantly. Since there is not much difference in the results when the nodes density is increased from 122,202 to 182,536 nodes and to minimize computational time, a grid size of 122,202 nodes is employed in the present work.

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Computational Fluid-Structure Interaction

12.1.3.1 Steady Flow The low Reynolds number specified in this computation is 40.0 and the inflow Mach number is set to be a very small value. Thus the flow can be considered as steady 2D, which is started from quiescent initial condition in the simulation. It is run using the low-speed preconditioning with the parallel MG method as shown in the convergence history plot in Fig. 12.8. It is noted that for the same CPU time there is significant improvement in residual reduction using the preconditioning, that is, improvement in real convergence rate. Fig. 12.9A and B shows the streamline plots in the wake region obtained with and without the preconditioning method, respectively. Based on the qualitative comparison with the experimental result of Van Dyke and Milton [3], the wake formed behind the cylinder predicted by the preconditioning method has better agreement with Dyke’s data than the nonpreconditioning one. A quantitative comparison of the aspect ratio (separation bubble length, S/cylinder diameter, d) with the experimental results obtained by Nishioka and Sato [4] is also carried out. Fig. 12.10 shows the aspect ratio versus Re. With an Re of 40.0, an aspect ratio of 2.35 is obtained, which agrees well with our preconditioned result (as shown in Fig. 12.9A). As can be observed, the low-speed preconditioning method not only gives better convergence rate, it also helps to improve the quality of the numerical results when the 100

Re = 40.0 M = 0.01 Parallel-MG Parallel-MG with Prec

10–1

R/R0

10–2

10–3

10–4

10–5 0

1000

2000 3000 CPU time

4000

5000

FIGURE 12.8 Convergence histories for steady flow over a circular cylinder.

IMM FSI Model Validations and Applications Chapter | 12 (A)

Parallel-MG with Prec Re = 40.0, inflow M = 0.01

(B)

363

Parallel-MG without Prec Re = 40.0, inflow M = 0.01

Y Z

X

d = 1.0

S = 2.35

d = 1.0

S = 1.76

FIGURE 12.9 Streamline plots (z 5 0.25) for flow over a 3D stationary circular cylinder at Re 5 40 and inflow Mach 5 0.01 using (A) parallel MG and preconditioning and (B) parallel MG without preconditioning.

6 5 d

S

S/d

4 3 S/d = 2.35

2 1

20

40 60 Reynold number

80

100

FIGURE 12.10 Length of separation bubbles behind cylinder versus Re [4].

flow speed is extremely low. This favorable characteristic is very crucial for compressible unsteady flow computation, especially for the cases where high-speed flow field is embedded in low-speed flow region.

12.1.3.2 Unsteady Flow The purpose here is to validate and assess the capability of the current parallel Navier
Stokes solver utilizing multigrid and low-speed preconditioning method as the basic convergence acceleration techniques. The computed results are compared with numerical ones obtained by other researchers, and

364

Computational Fluid-Structure Interaction

with available experimental results. The computed lift and drag coefficients on the cylinder versus nondimensional time with parallel MG for an eightpartition mesh are shown in Fig. 12.11. A pronounced asymmetric wake begins to appear at a nondimensional time of 30. The flow becomes completely periodic at a time of 63. It is found that the number of subiterations for MG is much less than that required by single-grid (SG) computation, which signifies that the MG method takes a shorter time than the SG to produce the vortex shedding phenomenon, thus less CPU time is needed for the flow to become fully periodic. Figs. 12.12 and 12.13 present the instant streamlines and Mach-number contours in one cycle of the von Ka´rma´n vortex shedding. Fig. 12.14 shows the convergence history in terms of numbers of subiterations in each physical time-step for SG, MG, and MG with preconditioning method for a Reynolds number equal to 200. It is obvious that the combination of preconditioning and MG contributes significantly to the improvement in convergence within each physical time-step, which is very useful for efficiently computing 3D unsteady flows. The lift coefficient, Cl, drag coefficient, Cd, and Strouhal number, St, obtained in this work are 6 0.65, 1.38 6 0.046, and 0.196, respectively, and they agree well with numerical solutions obtained by other researchers as well as with experimental measurements [5
8]. These results are tabulated in Table 12.3. The performance of the parallel solver for simulating unsteady flow over a cylinder with Re 5 200 using both parallel SG and MG is estimated based

Lift and drag coefficients

2 Cd

1.5 1

Cl 0.5 0 –0.5 –1 20

40

60

80 Time

100

120

140

FIGURE 12.11 Lift and drag coefficients versus physical time for flow over a circular cylinder using three-level parallel MG and preconditioning (number of subiterations 5 60 V-cycles, Re 5 200, inflow Mach 5 0.2).

IMM FSI Model Validations and Applications Chapter | 12

365

(A)

(B)

(C)

(D)

(E)

FIGURE 12.12 Streamline patterns showing one cycle of vortex shedding using three-level parallel MG and preconditioning (z 5 0.25, number of subiterations 5 60 V cycles, Re 5 200, inflow Mach 5 0.2).

on the speedup characteristics, efficiency of parallelization, and comparison between percentage computation and communication time, as shown in Table 12.4. The speedup and parallel efficiency of the proposed method is found to be reasonably good. Although parallel MG has slightly low speedup compared with parallel SG, the MG computation is still much more efficient because it requires much less number of subiterations for every physical time-step.

12.1.4 Flow Over an Immersed Fixed Membrane This case is to validate and assess the capability and accuracy of the developed method for thin structure problems. The model examined in this case is a 3D channel flow with a sinus cavity in the middle of the bottom wall. A rigid membrane is attached to the rigid channel just before the sinus cavity. The channel has a length of 10L and a width of 1L (L is equal to 20 mm). The radius of the sinus cavity is 0.5L. The immersed membrane has a length of 0.5L and a thickness of 0.5L/100, and it is attached to the bottom wall at an angle of α 5 42.5 degrees. In the region near the

366

Computational Fluid-Structure Interaction

(A)

(B)

(C)

(D)

(E)

FIGURE 12.13 Mach number contours showing one cycle of vortex shedding using three-level parallel MG and preconditioning (z 5 0.25, number of subiterations 5 60 V cycles, Re 5 200, inflow Mach 5 0.2).

IMM FSI Model Validations and Applications Chapter | 12

367

Single grid, Non Prec 3-level MG Non Prec 3-level MG Prec

100

Residual drop

10–1 10–2 10–3 10–4 10–5 21,600

21,700

21,800 No. of iter

21,900

22,000

FIGURE 12.14 Unsteady flow convergence history plot (MG used “V” cycle, number of subiterations 5 60, Re 5 200, inflow Mach 5 0.2).

TABLE 12.3 Lift and Drag Coefficients and Strouhal Number for Unsteady Flow Over a 3D Circular Cylinder (Re 5 200) Researchers

Cl

Cd

St

Present (parallel MG)

6 0.65

1.38 6 0.046

0.196

Tai and Zhao (parallel MG, incompressible solvers) [5]

6 0.64

1.31 6 0.041

0.195

Liu et al. [6]

6 0.69

1.31 6 0.049

0.192

Williamson (Expt.) [7]







0.197

Wille (Expt.) [8]




1.30




membrane, the mesh is further refined in order to capture the fine details of the flow. The immersed membrane is discretized into triangular cells. A closeup view of the mesh and the immersed membrane is shown in Fig. 12.15. The inflow Mach number is 0.3 and the Reynolds number is 100. The results are then compared with those using an internal boundary calculated by the baseline preconditioned compressible parallel

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Computational Fluid-Structure Interaction

TABLE 12.4 Performance for Parallel Computation of Unsteady Flow Past a Circular Cylinder (Re 5 200, M 5 0.3 with Preconditioning) Performance Measuring Techniques 2

SG

MG

Number of CPU

Number of CPU

4

8

Speedup

1.77

3.48

7.02

Efficiency

0.89

0.88

0.85

Computation time and communication time (% to total simulation wall-clock time)

99.3

95.2

16

2

13.4 0.81

87.7

79.3

4

8

16

1.72

3.29

6.39

10.18

0.88

0.83

0.79

0.72

97.2

91.1

82.4

70.1

3D membrane meshed with 6178 triangular elements

3D tube meshed with 116,865 nodes and 706,130 elements In the region which the membrane will span over the mesh is further 1 0.5 Y

0 –0.5 –0.5 0 0.5 Z

0

2

6

4

10

8 Y

X X Z

FIGURE 12.15 Geometry of the computational domain for the flow over an immersed fixed membrane (1 unit length 5 20 mm).

MG solver. The internal boundary has the same geometry as the immersed membrane, and it is under the same flow conditions as used by the IMM. The convergence history of the simulation given in Fig. 12.16 shows that the new solver based on the IMM actually converges faster and better than the baseline solver. Fig. 12.17 confirms that the two flow fields have the same number of vortices with similar shapes, and the u-velocity profiles at

IMM FSI Model Validations and Applications Chapter | 12 0

20,000

369

40,000

100

100 Immersed boundary

R/R0

Internal wall

10–1

10–1

10–2

10–2

10–3

10–3

10–4

10–4

0

20,000

40,000 kt

FIGURE 12.16 Convergence histories with immersed membrane and internal boundary.

1

1

0.5

0.5

0

0

0

1

2

–1

3

0

1

1

0.5

0.5

0

0 0

1

2 X

3

1

2

3

X

X

–1

0

1

2

3

X

FIGURE 12.17 (A) Streamlines on the channel central plane with the immersed membrane represented by the IMM; (B) flow field with an internal boundary computed by baseline solver.

the same locations agree well with each other as found in Fig. 12.18. Table 12.5 compares the two flow fields quantitatively, which demonstrates that the results obtained by the IMM agree well with those obtained by the baseline Navier
Stokes solver using an internal boundary.

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Computational Fluid-Structure Interaction 0

0.5

1

1.5

2

–0.5 1

1

0

0.5

1

1.5

2

2.5 1

1

0.8

0.8 0.8

0.8 Immersed boundary Internal wall

Immersed boundary Internal wall

0.6 0.6 Y

Y

0.6

0.6

0.4

0.4 0.4

0.4

0.2

0.2 0.2

0.2

0

0

0.5

1

1.5

2

0

0

–0.5

0

0.5

U

1 U

1.5

2

2.5

0

FIGURE 12.18 (left) Comparison of velocity profiles at X 5 0, Z 5 0.5; (right) comparison of velocity profiles at X 5 1, Z 5 0.5.

TABLE 12.5 Properties of Vortices of Flow Field with Immersed Membrane and Flow Field with Internal Boundary Immersed Membrane Center of the Primary Vortex

Center of the Secondary Vortex

Internal Boundary Center of the Primary Vortex

Center of the Secondary Vortex

X (mm)

0.932L

0.067L

0.923L

0.047L

Y (mm)

0.272L

2 0.031L

0.273L

2 0.027L

Vorticity

2 5.013

0.126

2 4.997

0.117

12.2 VALIDATION OF THE COMPUTAIONAL STRUCTURAL DYNAMICS SOLVER 12.2.1 Deformation of Point-Loaded Fixed-Free Cantilever Structures 12.2.1.1 Two-Dimensional Cantilever A standard problem in structural mechanics is that of a fixed-free cantilever supporting an applied load at the free end [9
11]. The fixed-free cantilever is shown in Fig. 12.19, where b 5 2.0 is the breadth, L 5 20.0 the length of the cantilever, and F the applied load. It is assumed that the depth d 5 1.0. The static solution to this problem given by Timoshenko and Goodier [12] allows a slight distortion at the fixed end of the cantilever, whereas the

371

IMM FSI Model Validations and Applications Chapter | 12

Load

Fixed end Breadth, b

Y Z

Length, L

X

Free end

Depth, d

FIGURE 12.19 Schematic of flexural (bending) deformation test of fixed-free cantilever.

TABLE 12.6 Computational Parameters for the 2D Cantilever 2D Fixed-Free Cantilever Computational Parameters

Value

Load, F

200 N

Length, L

20.0 m

Breadth, b

2.0 m

Density, ρ

2600.0 kg/m3

Young’s modulus, E

10 MPa

Poisson’s ratio, ν

0.0

solution given by Fenner [13] allows no such phenomenon. This test case requires no such displacement or distortion at the fixed end of the cantilever, hence at x 5 L the y-displacement at the free end of the cantilever according to Fenner [13] is given by dy 5 2

4FL3 Edb3

ð12:4Þ

where E is Young’s modulus and d is the depth of the cantilever. The fixedfree cantilever has a load of 200 N at the free end, as depicted in Fig. 12.19. Note that the gravity effect is not considered in this study. The static solution given in Eq. (12.4) is independent of Poisson’s ratio and is applicable to a cantilever undergoing pure flexure, that is, no axial loads are supported and the out-of-plane load on the cantilever is zero. Thus, for comparison with the analytic solution, a zero Poisson’s ratio is assumed. With the parameters as given in Table 12.6, Eq. (12.4) gives the static displacement in y-direction at the tip of the cantilever as 20.08 m. Before embarking on a dynamic problem, a grid convergence study for the static displacement problem is carried out. The domain is meshed using triangular

372

Computational Fluid-Structure Interaction

elements. The following five meshes are studied for increasingly higher mesh density: G G G G G

Coarse mesh, 20 3 2 elements and 33 nodes First refinement, 40 3 4 elements and 105 nodes Second refinement, 80 3 8 elements and 369 nodes Third refinement, 100 3 10 elements and 561 nodes Final refinement, 200 3 20 elements and 2121 nodes

For all of the grids, the simulation is initiated by exerting the same load on the right tip of the cantilever until the solution is converged. The percentage errors in the y-displacement are shown in Fig. 12.20, where percentage errors of less than 1.5% for the second mesh and less than 0.1% for the final mesh are observed. Thus the second mesh is considered to be sufficiently accurate and is used for the rest of the analysis. Fig. 12.21 shows the stress distribution in the equilibrium state. In order to test the capability of this method for predicting dynamics of solid structures, we also perform a dynamic bending simulation for the fixed-free cantilever. We use a dynamic load, which is a sinusoidal function of time t: F 5 200 sinð0:05tÞ

30

ð12:5Þ

29.37 33 nodes, 20*2 cells

Percentage error (%)

25 20 15 10

6.16 105 nodes, 40*4 cells

5

0.07 2121 nodes 200*20 cells

1.34 369 nodes, 80*8 cells

0

0.69 561 nodes 100*10 cells

20*2

40*4

80*8 Mesh size

FIGURE 12.20 Mesh refinements versus percentage error.

100*10

200*20

373

IMM FSI Model Validations and Applications Chapter | 12 tip displacement: 0.0800712 max x normal stress: 6740 min x normal stress: –6740 4942.67

–4942.67

4044

3145.33

–4044

2246.67

–3145.33

1348

–2246.67

449.333

–1348

Direct x stress σxx (Pa) –6740

tip displacement: 0.0800712 max y normal stress: 2887 min y normal stress: –1553

–4044

–1348

1348

4044

6740

1131.65

–107.419

–4.16279

99.093

202.349

265.634

–1139.98

Direct y stress σyy (Pa) –1553

–830.209 –107.419

512.116

1234.91

1957.7

2680.49

tip displacement: 0.0800712 max xy shear stress: 288 min xy shear stress: –1480 –648 288

28 –180

–128

–76

–42.4646

–596

80

Shear xy stress σxy (Pa) –1480

–1220

–960

–700

–440

–180

28

288

FIGURE 12.21 Direct and shear stress distribution in the equilibrium state for the static pointloaded free-fixed 2D cantilever.

With the other parameters as given in Table 12.6 and depth d 5 1.0 m, Eq. (12.4) also gives the maximum displacement in y-direction at the tip of the cantilever as 0.08 m. The simulation is kept running for a total of 2315 seconds. The cantilever tip displacement
time history in y-direction is then plotted in Fig. 12.22, which has a maximum displacement of 0.081 m in good agreement with the analytic solution.

12.2.1.2 Three-Dimensional Cantilever The 2D problem studied in the previous section is then extended to three dimensions and all of the computational parameters are given in Table 12.7. Thus the static y-displacement at the neutral axis of the tip given by Eq. (12.4) is about 20.1124 m, which provides an upper bound to the amplitude. Similar to the 2D simulation, the 3D statically loaded cantilever is first

374

Computational Fluid-Structure Interaction 0.1 Y max = 0.066898 Y min = –0.081088

Tip displacement, m

0.05

0

–0.05

–0.1

0

500

1000 Time, s

1500

2000

FIGURE 12.22 Displacement history of the tip in y-direction for dynamic point loading on free-fixed 3D cantilever.

TABLE 12.7 Computational Parameters for the 3D Cantilever 3D Fixed-Free Cantilever Computational Parameters

Value

Load, F

1.0 MN

Length, L

20.0 m

Breadth, b

2.0 m

Depth, d

2.0 m

Density, ρ

2600.0 kg/m3

Young’s modulus, E

17.8 GPa

Poisson’s ratio, ν

0.3

studied. The following three meshes are tested for grid convergence and MG studies: G G G

Coarse mesh, 2400 tetrahedral elements First refinement, 3200 tetrahedral elements Final refinement, 10,800 tetrahedral elements

IMM FSI Model Validations and Applications Chapter | 12

100

0

50 First level: Second level: Third level:

10–2

150

2989 nodes, 10,800 elements 1025 nodes, 3200 elements 775 nodes, 2400 elements

100

10–2

1. SG with 20 0 subiteration per time step 2. 3-level MG with 20 0 subiterations per time step 3: SG with 40 0 subiterations per time step 4: 3-level MG with 40 0 subiterations per time step

10–4 Residual drop

100

375

10–6

10–4

10–6 1

10–8

10–8

10–10

2

4

10–10 3

0

50

100 CPU time (min)

150

FIGURE 12.23 Convergence rate acceleration given by three-level MG method.

According to our results, the solutions for SG and three-level MG are identical. From the numerical solution for the final mesh, an amplitude error of 1.36% is observed. The tip displacement for this mesh turns out to be 20.11365 m. Fig. 12.23 shows the residual history and the convergence acceleration due to the three-level MG method, which makes the convergence at least three times faster than SG solver in terms of CPU time. Next, we will study the performance of the scheme for simulating dynamic loading. The natural frequency of a fixed-free cantilever [10] is given by sffiffiffiffiffiffiffiffi 3:516 EI f5 ð12:6Þ 2πL2 ρdb where I is the moment of inertia in the plane or the second moment of area of the cantilever about the neutral axis [12,13] and is given by ð 1b 2 db3 ð12:7Þ z2 dA 5 I5 12 212b Using the same parameters in Table 12.7, from Eqs. (12.6) and (12.7), the 3D fixed-free cantilever has a fundamental natural frequency of

376

Computational Fluid-Structure Interaction

0.1

0

20

40

60

80

1

120

140

160

0.1

Tip displacement in Z direction (m)

95.8

0.05

96

96.2

96.4

0.05 l = 0.4893 s

0

Z = 0.0

0

–0.05

–0.05

–0.1

–0.1

0.05

0.05

0

Z = –0.113265 –0.15

–0.15

–0.2

0

–0.05

–0.05 95.8

96

96.2 Time (s)

96.4

–0.2 0

20

40

60

80 100 Time (s)

120

140

160

FIGURE 12.24 Tip displacement history in z-direction for the point-loaded fixed-free 3D cantilever, three-level MG method was employed.

2.1134 Hz with a period of oscillation of 0.4732 seconds. To capture the sinusoidal motion of the structure accurately, the time-step for the solver is set to 0.05 seconds. The load is exerted on the initially undeformed 3D cantilever and then kept until the static equilibrium state of the cantilever is reached, after which the load is suddenly removed. The calculated tip displacement history is depicted in Fig. 12.24. The computed period is 0.4893 seconds, which again agrees well with the theoretical one. As pointed out by Slone et al. in Ref. [10], for accuracy considerations, the updating of the cantilever grid in this study is based on the initial position and the total displacement method. In order to accelerate the convergence rate, the MG method is used for all of the 3D problems.

12.2.2 Three-Dimensional Fixed-Free Cantilever Immersed in Fluid Flow The basic configurations of the problem are shown in Fig. 12.25 and the boundary conditions include an inflow boundary on the top of the domain, outflow boundary at the bottom, and a nonslip wall boundary to which the cantilever is clamped. A symmetry plane is also used to divide the field into two halves because of its geometric symmetry. The fluid domain is meshed using 1,307,075 tetrahedral elements and the overall dimensions are: depth d 5 2.0 m, breadth b 5 2.0 m, and length l 5 20.0 m. The cantilever is meshed using 12,196 tetrahedral elements. The Reynolds number and inflow Mach number are set to 112.7 105 and 0.05, respectively, for this case. The material properties of the cantilever are Young’s modulus 21.0 GPa, Poisson’s ratio 0.3, and density 2600 kg/m3. According to Eq. (5.3), the

IMM FSI Model Validations and Applications Chapter | 12

377

Characteristic inflow

Non-slip wall Fixed-free cantilever Clamped end

Free end

Breadth Symmetry planes

Characteristic outflow

Y Z

Length

X

Depth

FIGURE 12.25 Schematic 3D free-fixed cantilever in fluid flow.

cantilever has a period of oscillation of 0.4356 seconds. The simulation timestep is taken as 0.025 seconds. The gravity force is also ignored in the simulation. The stress distribution in the cantilever in its final equilibrium state is shown in Fig. 12.26 and the tip vertical displacement history is depicted in Fig. 12.27, while the flow field in the final steady state is shown in Figs. 12.28
12.30, respectively. They show that while the beam is oscillating, a complex flow field is also developing under the beam and the wake region fluctuates with the beam accordingly. A grid convergence study on this case was also performed. Four successively finer mesh sizes with 434,572, 921,586, 1,307,075, and 1,908,265 tetrahedron cells are used for error analysis, and the finest-mesh solution is considered to be the “exact” solution. On all the grids, the same physical time-step (Δt 5 0.01T) is employed in order to concentrate on the spatial resolution of the method. For all the grids, the LN and Lq norms of the u-velocity errors are calculated as follows: " #1=q N



1X q N e
e
q

εj 5 max ui 2 ui ; εj 5 ui 2ui i 5 1;N ð12:8Þ N i51 ðj 5 1; 4 j 5 1is the coarsest gridÞ where εN are the infinity and qth error norms on the jth grid, ui is j and the u-velocity component at the ith node of the current mesh, uei is the interpolated “exact” velocity field from the results calculated on the finest grid, and N is the number of grid nodes of current grid. The results of the grid convergence study are summarized in Fig. 12.31, which shows the variation εqj

Direct x stress σxx –2.2E+09 –1.6E+09 –1E+09 –4E+08

2E+08

8E+08 1.4E+09

2E+09

Direct y stress σyy –1.5E+08

190813

791923

7.74396E+06

2.5E+08

Shear xy stress σxy –2.2E+08

–1.4E+08

–7.24445E+07

–6E+07

2E+07

FIGURE 12.26 Direct and shear stress distribution in the equilibrium state for the fixed-free cantilever immersed in fluid.

Tip displacement history

0

–0.05

–0.1

–0.15 0

2

4

6 Time, s

8

10

FIGURE 12.27 Immersed cantilever tip vertical displacement history during the FSI process.

379

IMM FSI Model Validations and Applications Chapter | 12

v –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –0.963828 –0.97105 –0.989766 –1 –1.01523 –1.02654 –1.0504 –1.1 –1.12478 –1.2

Y

Z

0

X

20 X

30

20

10

0

40

FIGURE 12.28 V velocity contours in XY plane, Re 5 112.7 3 105.

V

30

20 Y

10 Y

Z

0.05 0 –0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4 –0.45 –0.5 –0.55 –0.6 –0.65 –0.7 –0.75 –0.8 –0.85 –0.9 –0.95 –1 –1.05 –1.1 –1.15 –1.2 –1.25

Y

Z

X

X

0

FIGURE 12.29 V velocity contours and streamlines in YZ plane, Re 5 112.7 3 105.

of the LN, L1, and L2 norms of errors with grid spacing in logarithmic coordinates. The lines with Slopes 1 and 2 are also given as reference. It is evident from the figure that the method is second-order accurate. To further demonstrate the accuracy of our method, we also use the Richardson estimation procedure to study the accuracy of the solver. Let f j denote the numerical solution on the jth mesh. Assume that the discrete solution is a γ-order approximation to its value f exact , and the flow field is continuous and has no singularity points, then we have

380

Computational Fluid-Structure Interaction

Y

Z

X

FIGURE 12.30 Volume streamlines pattern depict the separation under the immersed cantilever, Re 5 112.7 3 105.

–0.2 –0.4

Grid l : 434,572 cells Grid ll : 921,586 cells Grid lll : 1,307,075 cells “Exact grid” : 1,908,265 cells

Grid l

Log ⎜Error ⎜

–0.6 –0.8 –1 Grid ll

–1.2 –1.4 –1.6

L2 error L1 error Linfinity error Slope 1 Slope 2

Grid lll

–1.8 –1.3

–1.2

–1.1

–1

–0.9

–0.8

–0.7

–0.6

–0.5

Log ⎜h ⎜ FIGURE 12.31 Convergence of the LN , L1 , and L2 error norms for the velocity field with a immersed fixed-free cantilever. Slopes 1 and 2 are the reference lines for first-order and secondorder accuracy, respectively.

381

IMM FSI Model Validations and Applications Chapter | 12

TABLE 12.8 Rate of Convergence γ Calculated for Different Error Norms Norm

Grids j 5 2, 3, 4

LN

1.93

L1

2.18

L2

2.15

  log :f ðjÞ 2 f ðj21Þ :=:f ðj21Þ 2 f ðj22Þ : γ5 log 2

ð12:9Þ

where jj jj denotes an error norm (LN, L1, or L2). If γ  2 the solution is second-order accurate. We apply the above procedure for j 5 4 (involving interpolation of solutions from j 5 2, 3, to 4) to calculate γ for successively refined meshes. We use all three norms to compute the error and the results are summarized in Table 12.8, which strongly supports our assertion about the second-order accuracy of our method. To further demonstrate and evaluate the performance of the proposed basic structural solver, both the solver and the commercial solver (ANSYS 9.0) are used to simulate the equilibrium state of the point-loaded 3D fixedfree cantilever as used in the above FSI calculation, with a load of 1.0 MN. The geometry and material properties of the cantilever remain the same. Using the given parameters, the theoretical value of the vertical displacement of the tip is 0.0952381 m. Four meshes are tested here: G G G G

Grid Grid Grid Grid

1, 2, 3, 4,

3950 tetrahedral elements 6412 tetrahedral elements 12,196 tetrahedral elements 19,183 tetrahedral elements

Fig. 12.32 shows a comparison of the computational times (CPU times on an SGI O3400 workstation) used by the proposed basic structural solver and ANSYS 9.0. It is found that the two solvers use almost the same amount of CPU time to make the residual drop to the same level (R/R0 5 1025), with the current solver being slightly more efficient when the mesh density is increased. Fig. 12.33 shows a comparison of grid convergence for the proposed solver and ANSYS 9.0. It can be observed that the current solver can produce faster convergence toward the exact solution than ANSYS. From the above comparisons, we can conclude that the proposed basic solver is comparable to established solvers in terms of efficiency and accuracy. With the use of MG, the efficiency will be significantly improved, as demonstrated in Section 12.2.1.2.

382

Computational Fluid-Structure Interaction

35

30

Result by Ansys Result by proposed solver

CPU time (s)

25

20

15

10 Grid 1

Grid 2

Grid 3

Grid 4

FIGURE 12.32 Comparison of CPU times used by the two solvers.

Tip displacement (m)

0.094 0.092

Result by Ansys Result by proposed solver Theoretical value

0.09 0.088 0.086 0.084 0.082 Grid 1

Grid 2

FIGURE 12.33 Grid convergences of the two solvers.

Grid 3

Grid 4

IMM FSI Model Validations and Applications Chapter | 12

383

12.3 SUMMARY As presented in the previous two sections, a new method combining preconditioning, parallel unstructured MG method and a novel treatment of moving objects in fluids for efficient simulation of 3D unsteady compressible flows has been successfully developed and validated. The convergence of numerical solutions is found to be significantly improved with a combination of the preconditioning and unstructured MG methods. The use of gauge pressure in pressure gradient terms is found to be important to eliminate roundoff errors while the flow speed is very low. The parallel speedup and efficiency of the method for both steady and unsteady flows are found to be reasonably good. The newly developed IMM is shown to work well with very large displacements of immersed moving objects. Compared with the immersed boundary method, this method allows sharp changes of fluid conditions across immersed thin structures without complicated and time-consuming interpolation and extrapolation. A grid convergence study for the flow generated by a steadily rotating sphere is carried out, which shows that the method is second-order accurate. The method is also found to have similar order of accuracy in another study, where a sphere immersed in a cube filled with fluid oscillates with large amplitude. Results from a flow over a circular cylinder computed by the proposed method are found to agree well with existing numerical and experimental ones. Finally, flow over an immersed membrane is calculated and results are compared with those of the membrane erected as a wall, and satisfactory agreement is observed. These studies demonstrate that the method proposed is an effective tool to solve 3D unsteady low-Mach-number compressible flows with arbitrarily moving objects. A novel 3D matrix-free implicit unstructured MG structural dynamic finite-volume solver has been successfully developed and validated. The convergence of numerical solutions is found to be significantly improved with the help of the implicit unstructured MG method. The efficiency and accuracy of the solver are fully validated using a point-loaded fixed-free cantilever, for which both 2D and 3D static and dynamic cases are thoroughly tested. Through the case involving a fixed-free cantilever immersed in fluid flow, it is found that the current finite volume (FV) structural dynamic solver works well with our unstructured grid FV compressible fluid solver TETRAKE as well as the IMM [14]. These studies demonstrate the potential capability of the proposed method for large-scale complex fluid
structure interaction simulation.

12.4 STEADY AND UNSTEADY AIR FLOWS BETWEEN TWO RIGID DISKS COROTATING IN A FIXED CYLINDRICAL ENCLOSURE The flow field configuration of interest is shown in Fig. 12.34. It consists of a pair of disks clamped coaxially on a central hub that rotates in a stationary

384

Computational Fluid-Structure Interaction

Immersed rotating disk u=0 v=rΩ w=0

θ, v

Z, W

Ω

Stationary wall u=v=w=0 Periodicity or symmetry

r, u Rotating disk Fixed enclosure wall

Rotating hub

IV H

III

V IV Rotating disk

b

Il

R1 R2

l a

FIGURE 12.34 Schematic of the corotating disks flow configuration indicating the toroidal vortices that develop in the cross-stream plane near the fixed enclosure wall. The basic flow regions (I
V) induced in the interdisk space are sketched in accordance with previous studies [15
19]: (I) enclosure wall boundary layer; (II) disk Ekman layers; (III) the region in solidbody rotation; (IV) core potential flow; and (V) detached shear layer. In the bulk of the calculations, R1 5 56.4 mm, R2 5 105.7 mm, H 5 9.53 mm, and b 5 0. Selective calculations have been performed with R1 5 56.4 mm, R2 5 105.0 mm, H 5 9.53 mm, a 5 2.7 mm, and b 5 1.91 mm. The latter values correspond exactly to an experimental configuration used in other studies [15,18,20
22].

cylindrical enclosure. Small aspect ratios are considered here since they are especially relevant to industrial and technical applications such as hard disk drives. The configuration of coaxial disks corotating in a cylindrical enclosure provides a useful model for investigating flows in the hard disk drives used as data storage devices in computers. A better understanding of the complex unsteady flows that arise in disk storage devices is essential for their improved design and repeatable operation. A disk storage system consists of a stack of equidistant, centrally clamped disks corotating in a nonaxisymmetric enclosure. Electronic data are distributed along micron-sized circular tracks on the disk surfaces. Data transfer to and from the disks is accomplished by means of magnetic heads suspended at submicron distances from the rotating disk surfaces by rigid supports.

IMM FSI Model Validations and Applications Chapter | 12

385

For the configuration with Γ 5 H=ðR2 2 R1 1 aÞ  0:186, past studies show that the transition from axisymmetric 2D steady flow to nonaxisymmetric 3D unsteady flow occurs approximately at Re 5 23,150, where Re is the Reynolds number based on the disk radius, the tip speed of the disks, and the kinematic viscosity of the fluid. Below the critical Reynolds number, the steady-flow solutions are characterized by a symmetrical pair of counterrotating toroidal vortices in the cross-stream (r
z) plane. This secondary motion is driven by the radial imbalance between the outward-directed centrifugal force and the inward-directed pressure gradient force. Above this critical value, the motion is unsteady and periodic, while the features of the crossstream flow pattern are broadly preserved. The symmetry of the motion about the mid-plane is broken by alternating periodic crossings of the toroidal vortices. This instability is maintained through an interaction that arises between outward-directed fluid in the disk Ekman layers and inward directed fluid in the return core flow. 3D calculations presented in [20] at Re 5 22,200 and 44,400 show that the toroidal vortices acquire a time-varying sinuous shape in the circumferential direction. These calculations reveal circumferentially periodic reversals of the axial velocity component in the cross-stream plane, including the detached shear layer separating the region of motion in solidbody rotation near the hub from the potential core, in agreement with the flow visualization observations of Humphrey and Gor [21]. The wavelength of this oscillation is shown to be twice that of the circumferential velocity component which is responsible for the nodal distribution of axial vorticity. When plotted on the interdisk mid-plane, the axial component of vorticity manifests itself as an even integer number, 2n (n 5 1, 2, . . .), of circumferentially periodic foci. In this study, the steady and unsteady unobstructed laminar flow of air between the two corotating disks is simulated using proposed numerical procedure. The flow configurations with aspect ratio Γ  0:186 and Reynolds number ranging from 4400 to 44,400 are considered, corresponding to rotation rates ranging from 60 to 600 rpm. In the bulk of the calculations, the thickness of the disks is ignored (b 5 0). Nevertheless, a case corresponding to the experimental values of b 5 1.91 is also examined to evaluate the influence of such simplification. For the configurations of interest here, the grid nodes are distributed nonuniformly throughout the calculation domain in order to resolve the strongly sheared regions of the flow. These regions are: the Ekman layers along the respective disk surfaces, the shear layer along the fixed cylindrical enclosure wall, and the detached shear layer lying between the region of flow in solid body rotation and the fully 3D potential core. Estimates of the location and size of these regions provided in Refs. [15,20,21] were used to construct the present grids. This approach permits an effective distribution of nodes while avoiding unnecessary refinement where velocity gradients are weak, for example, in the solid body rotation region. Four meshes with different grid sizes are tested with 802,546, 1,316,248, 1,798,149, and 2,348,149 tetrahedron cells, respectively.

386

Computational Fluid-Structure Interaction

Due to limited computer time and storage, a grid convergence study was performed to determine the most appropriate mesh for the final test. Calculations are first performed in the configuration of Fig. 12.34 for conditions of the experiment of Schuler [18]. Radial profiles of the circumferential velocity component (tangent velocity component in the u, v plane) calculated on the interdisk mid-plane for Re 5 4440, 7400, and 14,800 (Ω 5 60, 100, and 200 rpm), are compared in Fig. 12.35 with the experimental data of Schuler. The calculations are started from a zero initial velocity condition, and allowed to evolve in time until the steady-state solution is reached. As can be seen from the figure, the agreement between the measurements and the calculations is very good, with only very slight differences arising in the near wall region. The results of the last two meshes virtually coincide with each other. Notwithstanding, the remaining calculations reported here correspond to the third grid with 1,798,149 elements.

12.4.1 Three-Dimensional Steady Flow Fig. 12.36 shows contours of the velocity components, Mach number, and temperature in the cross-stream (r
z) plane for the speeds of rotation of Re 5 7400 (Ω 5 100 rpm). In the figure, the hub is the right boundary, the enclosure wall is the left boundary, and the disks are the top and bottom boundaries. In this simulation, we ignored the thickness of the disks. The structure of the flow is remarkably similar to that predicted analytically by Schuler [18]. An Ekman layer develops along the surface of each disk, starting at the radial location where the circumferential velocity profile first deviates from solid-body rotation (see Fig. 12.36). Analysis shows that the thickness an  of1=2 Ekman layer, marked by an arrow in Fig. 12.36B, is of order δE 5 υ=Ω and uniform. The fluid forced radially outward in the Ekman layers is redirected in the axial direction along the enclosure wall, and then radially inward into the core of the flow about the interdisk mid-plane, thus creating a pair of symmetrical, toroidally shaped, counter-rotating vortices. A thin shear layer with thickness also of order δE develops very near the fixed enclosure wall. The numerically calculated Ekman layer thicknesses are nearly uniform along with r, and in agreement with the above estimate for δE , obtained by Schuler et al. The basic structure of the flow is relatively independent of the Reynolds number, also in agreement with the predictions of the theoretical analysis presented in that reference. Fig. 12.37 shows the streamline patterns on the same plane. From which, a pair of symmetrical, toroidally shaped, counter-rotating vortices can be easily identified. Results of current simulation in the form of contours of various flow variables on the interdisk mid-plane are also shown in Fig. 12.38.

IMM FSI Model Validations and Applications Chapter | 12 (A)

387

0.8

Circumferential velocity

0.7 0.6 0.5 0.4 Third mesh Second mesh First mesh Schuler’s measurement Last mesh

0.3 0.2 0.1 0

(B)

0.5

0.6

0.7

r

0.8

0.9

1

0.8

Circumferential velocity

0.7 0.6 0.5 0.4 Third mesh Second mesh First mesh Schuler’s measurement Last mesh

0.3 0.2 0.1 0

0.5

0.6

0.7

0.8

0.9

1

r

(C)

0.8

Circumferential velocity

0.7 0.6 0.5 0.4 Third mesh Second mesh First mesh Schuler’s measurement Last mesh

0.3 0.2 0.1 0

0.5

0.6

0.7

0.8

0.9

1

r

FIGURE 12.35 Comparison between measurements and calculations of the circumferential velocity component along the interdisk mid-plane for (A) Re 5 4440 (60 rpm); (B) 7400 (100 rpm); (C) 14,400 (200 rpm). The results for the last two grids virtually coincide.

Z

X

Y

u –0.9 –0.7 –0.6 –0.5 –0.4 –0.2 0

0.2 0.4 0.6 0.8

(A)

v –0.16 –0.12 –0.08 –0.04

0

0.04 0.08 0.12 0.16

(B)

w –0.1

–0.0184991

0.0189641

0.09

(C)

Mach number 0.005 0.02 0.035 0.05

0.07 0.085

(D)

Temperature 176

176.3 176.6 176.9 177.2 177.5 177.8

(E)

FIGURE 12.36 Iso-contours of the velocity components, Mach number, and temperature on the r
z plane. (A) Obtained from 3D calculations Re 5 7400; (B) arrow marks the thickness  for1=2 of an Ekman layer, estimated according to δE 5 υ=Ω .

FIGURE 12.37 Streamline patterns on the r
z plane. A pair of symmetrical, toroidally shaped, counter-rotating vortices can be easily identified.

IMM FSI Model Validations and Applications Chapter | 12

u

y

z

x

0.6 0.5 0.353561 0.2 0 –0.2 –0.4 –0.5 –0.6

y z

Circumferential velocity 0.6 0.4 0.2 x 0 –0.2 –0.4 –0.6

T

Pressure y

z

x

71.14 71.12 71.1 71.08 71.06 71.04 71.02 71

389

y z

x

177.95 177.85 177.75 177.65 177.55 177.45 177.35

FIGURE 12.38 Contours of u and circumferential velocity components, pressure, and temperature on the interdisk mid-plane.

12.4.2 Three-Dimensional Unsteady Flow 12.4.2.1 Case 1: Re 5 22,200 In contrast to steady flow, the axisymmetric flow at Re 5 22,200 oscillates periodically about the interdisk mid-plane. Fig. 12.39 shows the instantaneous flow field on the r
z plane at an instant in time. The oscillation develops without the imposition of a perturbation. Fig. 12.40 shows the time variation of the dimensional circumferential, radial and axial velocity components at a monitoring point located on the interdisk mid-plane. Due to the periodic change in sign of the axial velocity component with respect to the mid-plane, its frequency is half that of the other two velocity components and pressure which do not change sign. Power spectra of these time series (calculated using data for nondimensional time t . 100 only) reveal a pair of dominant frequencies in this flow, namely 2πf =Ω 5 10:42 for radial and circumferential components, and 2πf =Ω 5 5:21 for axial component.

(A)

(B)

(C)

(D)

FIGURE 12.39 (A) Circumferential velocity components; (B) radial velocity components; (C) axial velocity components; and (D) streamline patterns in the r
z plane. Obtained from 3D calculations for Re 5 22,200 (Ω 5 300 rpm) and t 5 103 s. 0.1 0

3 Radial velocity (m/s)

Circumferential velocity (m/s)

3.5

2.5 2 1.5 1

–0.2 –0.3 –0.4

0.5 0

–0.1

–0.5 0

20

40 60 Flow time (s)

80

20

40 60 Flow time (s)

80

0

20

40 60 Flow time (s)

80

12

0.1

Pressure coefficient

Axial velocity (m/s)

0.2

0

0

–0.1

10 8 6 4 2

–0.2

0 0

20

40 60 Flow time (s)

80

FIGURE 12.40 Time history of the dimensional circumferential, radial and axial velocity components, as well as the nondimensional parameter, pressure coefficient (Cp), at a monitoring point located on the interdisk mid-plane (r/R2 5 1.0). Obtained from 3D calculations for Re 5 22,200 (Ω 5 300 rpm).

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Fig. 12.41 compares unsteady 3D calculations of the mean and rms (rootmean-square) circumferential velocity components on the interdisk midplane at θ 5 0, with corresponding measurements for Re 5 22,200. The mean velocity profiles are in reasonable agreement. However, although qualitatively similar, the calculated rms profile is significantly smaller in magnitude than the experimental profile. The under-prediction of the velocity rms by the current numerical procedure, and the higher predicted value of the

Circumferential component

(A)

0.8

0.6

0.4 Results by proposed solver Results by measurement

0.2

0

0.6

0.7

0.8 r

0.9

1

(B) 12 11

Circumferential component

10 9 8

Results by proposed solver Results by measurement

7 6 5 4 3 2 1 0

0.6

0.7

0.8 r

0.9

1

FIGURE 12.41 Comparison between measurements (circles) and calculations (line with triangles) of the nondimensional circumferential velocity component (A) mean and (B) rms along the interdisk mid-plane for Re 5 22,200 (Ω 5 300 rpm).

392

Computational Fluid-Structure Interaction

Reynolds number for the transition to unsteady flow, can be attributed to several factors as discussed in [20]. First, the thickness of the disks is ignored in the calculations. The tip of a disk represents an additional source of instability for the flow ejected radially outward in the Ekman layer. Second, small but unavoidable imperfections in the disks, in the curvature of the enclosure wall, and in the relative orientation of various parts of the test section can induce perturbations in the experiment that result in higher rms and lower critical speed of rotation for transition to unsteady flow than calculated numerically. Finally, numerical diffusion error tends to stabilize the calculated flow, making necessary the use of extraordinarily refined grids or higher-order low-diffusion schemes to counteract this effect.

12.4.2.2 Case 2: Re 5 44,400 3D calculations are performed for a configuration very similar to that investigated numerically by Humphrey et al. [20], in which the effects of the disk rimenclosure wall gap are considered but the thickness of the disks is neglected. The configuration is defined by setting R1 5 56.4 mm, R2 5 105 mm, a 5 2.7 mm, H 5 9.53 mm and b 5 0 in Fig. 12.34, and fixing a symmetry boundary condition in the gap. The configuration has been calculated in this study for the same aspect ratio but different rotational speed (600 rpm, in this case corresponding to Re 5 44,400). Humphrey et al. predicted six foci in a converged calculation with Re 5 44,400. The calculation of this flow was performed using the previously calculated flow field shown in Fig. 12.39 as the initial condition. With no perturbation applied to this initial flow field, the result after 0.1 seconds (one-third of a disk revolution) shows little difference with the solution provided initially. Thus, a nonaxisymmetric perturbation (wdisk) was applied in the form of a simulated wobble of the disks. The perturbation was imposed for 0.1 second and was circumferentially sinusoidal, with amplitude 5% of the local disk speed: wdisk 5 0.05Ωr sin θ. This axial velocity was applied in phase to both disks in such a way that global continuity was preserved. The result was an irregular (nonperiodic) oscillatory flow field. The calculation was repeated with a quiescent fluid as an initial condition, and a similar irregular oscillatory flow was obtained. This ensured that the initial condition was not a factor in determining the fully developed flow. Results of the present converged calculation in the form of instantaneous contours of the z-vorticity component, ωz, on the interdisk mid-plane are shown in Fig. 12.42. (Here by converged, we mean that the present flow acquired a steady periodic state with primary oscillation frequencies corresponding to 2πf =Ω 5 10:42 for the radial and circumferential velocity components and 2πf =Ω 5 5:21 for the axial.) The contour levels plotted are specifically selected to display the circumferentially periodic foci of axial vorticity described by Humphrey et al. [20]. Fig. 12.43 shows the 3D iso-surface of axial velocity component, which is color-contoured by Mach number. In this figure, the nonaxisymmetric 3D flow structure is fully revealed.

FIGURE 12.42 Iso-contours of the axial vorticity component, ωz, on the interdisk mid-plane from 3D calculations for Re 5 44,400 (600 rpm). Six positive foci (colored by red) can be clearly identified.

Mach number

Z

Y

X

0.03 0.026 0.022 0.018 0.014 0.01 0.006 0.002

FIGURE 12.43 Iso-surface of the axial velocity component, uz, from 3D calculations for Re 5 44,400.

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Computational Fluid-Structure Interaction

Z

(A)

X

Y

Z

(B)

X

Y

Z

(C)

X

Y

Z

(D)

X

Y

Z

(E)

X

Y

Z

(F) X

Y

FIGURE 12.44 Instantaneous plot of the (A) cross-stream flow field from 3D calculation for Re 5 44,400 (600 rpm); (B) circumferential velocity components; (C) radial velocity components; (D) axial velocity components; (E) Mach number; (F) pressure.

Plots of the r
z cut-plane mesh and cross-stream flow field for an instant of time are shown in Fig. 12.44 for the flow at Re 5 44,400.

12.4.2.3 Effects of Disk-Enclosure Gap and Disk Thickness Numerical calculations were also performed for the configuration as shown in Fig. 12.34, with R1 5 56.4 mm, R2 5 105 mm, H 5 9.53 mm, a 5 2.7 mm, and b 5 1.91 mm, in order to assess the effects of the disk thickness and rimenclosure wall gap on the flow. These dimensions yield the same aspect ratio, Γ  0:186, as the previous configuration with a 5 0 and R2 5 107.7 mm. Fig. 12.45 shows the instantaneous flow field on r
z plane at an instant in time. No significant discrepancies were found between these results and the former results shown above without considering the disk thickness, except that the critical value of the Reynolds number for the flow transitioning to the unsteady region changes to about 15,000.

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Immersed rotating disk

(A)

(B)

(C)

(D)

FIGURE 12.45 (A) Circumferential velocity components; (B) radial velocity components; (C) axial velocity components; (D) streamline patterns on the r
z plane. Obtained from 3D calculations for Re 5 15,000.

12.4.2.4 Remarks 3D calculations performed for the unsteady laminar flow between a pair of disks corotating in a fixed cylindrical enclosure provide new insight for the interpretation of experimental observations and the improved understanding of this class of flows. The calculation results show good overall qualitative agreement with the experimental results available and reveal the instability phenomenon in the region near the curved enclosure wall, where the disk Ekman layers turn to collide against each other.

12.5 LARGE-EDDY SIMULATION OF AEROELASTIC FLUTTER FOR THE ONERA M6 WING IN THE TRANSONIC FLOW In this case, we study the flow over an ONERA M6 wing. It was tested in a wind tunnel at transonic Mach numbers (0.7, 0.84, 0.88, and 0.92) and various angles of attack up to 6 degrees by Schmitt and Charpin [23].

396

Computational Fluid-Structure Interaction

The Reynolds numbers were about 12 million based on the mean aerodynamic chord. The wind tunnel tests are documented in the AGARD Report AR-138 published in 1979 [23]. The ONERA M6 wing is a classic CFD validation case for external flows because of its simple geometry combined with complexities of transonic flow (i.e., local supersonic flow, shocks, and turbulent boundary layers separation). It has almost become a standard for CFD codes because of its inclusion as a validation case in numerous CFD papers over the years. To evaluate the accuracy of the steady aerodynamic solution, calculations for this wing are first carried out and compared to available experimental data. In this step, the wing body will be considered as a rigid solid and the structural domain will be ignored during the calculations. In Ref. [23], the ONERA M6 wing was tested thoroughly, and the pressures at various spanwise locations were recorded, which will be served as a main source of validation for the current study. Next, the former aerodynamic result is employed as the initial state in the full fluid
structure interaction simulation and the wing flutter will be predicted using the proposed method. The transonic flutter of the M6 wing has been extensively studied by many researchers [24
26] and numerous results are available for validation purpose.

12.5.1 Geometry and Calculation Parameters The ONERA M6 wing, which is a swept, semispan wing with no twist, has a leading-edge sweep angle of 30 degrees, an aspect ratio of 3.8, and a taper ratio of 0.562. It uses a symmetric airfoil using the ONERA D section, which is a 10% maximum thickness-to-chord ratio conventional section. Fig. 12.46 shows the geometric layout and some of the geometric properties of the wing. The values of the parameters for the steady aerodynamic and static aeroelastic calculations are as follows: Angle of attack, α Angle of sideslip, β Freesteam Mach number, MN Density of air, ρN Freesteam velocity, UN Root chord, cτ Reynolds number, Re Freesteam temperature, TN Freesteam pressure, PN

5 3.06 degrees 5 3.06 degrees 5 0.8395 5 1.223 kg/m3 5 285 m/s 5 0.8059 m 5 11.72 3 106 5 460.0 degrees Rankine 5 45.82899 psia

The computational domain is depicted in Fig. 12.47. Three successively refined grids are used for the grid convergence study purpose: Grid 1 Grid 2 Grid 3

1,769,472 elements, 306,577 nodes 3,566,552 elements, 604,583 nodes 5,308,416 elements, 909,521 nodes

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Swept wing M6 Aspect ratio

A = 3.8

Taper ratio

λ = 0.56

Sweep angle

∧25%= 26.7°

Rows of pressure taps N 1 2 3 4 5 6 7

z/b 0.20 0.44 0.65 0.80 0.90 0.95 0.99

upper 23 23 23 23 31 31 31

under 11 11 11 11 14 14 14

2734.5 mm

6 5 4 3

ONERA ‘D’ Wing section

2

30°

b = 1196.3 mm

1 End plate 805.9 mm 1578.7 mm

FIGURE 12.46 Geometric layout of the ONERA M6 wing.

Fig. 12.48 shows the grid about the tip of the wing. The grid is clustered about the wing surface to resolve the turbulent boundary layer.

12.5.2 Large Eddy Simulation (LES) of Steady Aerodynamic Computation of M6 Wing The computation is performed using the time-marching capabilities of TETRAKE to march to a convergent solution starting from a uniform initial solution tabulated above. Local time-stepping is used at each subiteration to enhance the convergence rate. The flow is assumed to be in transitional

398

Computational Fluid-Structure Interaction

Y

Symmetric plane

X Z

Immersed M6 wing (adiabatic non-slip wall) Refined area to capture Boundary layer & shock wave

Leading edge position = (0, 0, 0)

6.373 m

Height h = 12.746 m

Depth d = 6.373 m

Y X Z

Far field characteristic boundary

Width w = 13.7837 m

FIGURE 12.47 Schematic of computational domain of the ONERA M6 wing.

Y X Z

FIGURE 12.48 Computational grids near the tip of the ONERA M6 wing.

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0.012 0.011 Finest grid Intermediate grid Coarse grid 0.04

0.02

Cd (drag coefficient)

Cd (drag coefficient)

0.06

Only the history for medium grid is plotted

0.01 0.009 0.008 0.007 0.006

0

0

20

40

60 Time

80

100

0.005

100

120

105

110

115

120

125

Time

FIGURE 12.49 Time history of the drag coefficient of the ONERA M6 wing.

0.15 0.148

Only the history for medium grid is plotted

0.146

0.1 Finest grid Intermediate grid Coarse grid

Cl (lift coefficient)

CI (lift coefficient)

0.15

0.144 0.142 0.14

0.05 0.138

0.136

0 0

20

40

60 Time

80

100

100

120

105

110

115

120

125

Time

FIGURE 12.50 Time history of the lift coefficient of the ONERA M6 wing.

0.02 Finest grid Intermediate grid Coarse grid

0.025

0.0195 Cm (moment coefficient)

Cm (moment coefficient)

0.03

0.02 0.015 0.01

Only the history for medium grid is plotted

0.019

0.185

0.018

0.005 0 0

20

40

60 Time

80

100

120

0.175

100

105

110

115 Time

FIGURE 12.51 Time history of the moment coefficient of the ONERA M6 wing.

120

125

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Computational Fluid-Structure Interaction

regime. Figs. 12.49
12.51 show the convergence histories of the lift coefficients Cl and the drag coefficients Cd, as well as the moment coefficient Cm along z-axis, respectively. A mean lift coefficient Cl and a mean drag coefficient Cd are derived from the limited time samples. They are listed in Table 12.9, and show good agreements with other published results. Results for three different grids are tabulated for ease of comparison. Due to the fact that the computation on the intermediate grid (with 3,566,552 elements and 604,583 nodes) already gives satisfactory result, and in order to minimize the necessary computational time, this grid will be exclusively used in the following computation. All of the shown results hereafter are all corresponding to this mesh unless specified. The time-averaged Mach number distribution on the symmetric plane and upper wing surface is shown in Fig. 12.52. It shows that on the inboard region of the wing, the main shock is located slightly behind mid-chord. A leading edge shock at around 0.1c (c is local chord length) is also observed. For a wing calculation, another piece of important information is the pressure coefficient on the wing surface. The chordwise time-averaged

TABLE 12.9 Predicted Mean Lift and Drag Coefficients Mean Lift Coefficient

Mean Drag Coefficient

Finest grid

0.1428

0.0084704

Intermediate grid

0.1420

0.0084663

Coarse grid

0.1347

0.0073866

FIGURE 12.52 Time-averaged Mach number distribution on the symmetric plane and upper surface of the ONERA M6 wing.

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pressure coefficient distributions on the wing surface at seven spanwise locations are presented in Fig. 12.53. The collected sample time mean pressure coefficients data at the same locations of the wing from referenced experiment [23] are also identified in the figures for ease of comparison. These figures show that the predicted pressure coefficient agrees well with the experimental result, which in turn exhibits the accuracy of proposed IMM as well as the LES modeling. The distributions of time-averaged pressure coefficient, temperature, and density on the symmetric plane and upper wing surface are shown in Fig. 12.54. Fig. 12.55 shows the contours of the ratio between the test filtering width and grid filtering width on the symmetric plane. Fig. 12.56 shows the timeaveraged streamline ribbons around the wing surface, where the symmetric plane and the ribbons are contoured by velocity magnitude.

(A)

(B) Schmitt. {EXP} WIND {NASA} TETRAKE

1

WIND {NASA} Schmitt. {EXP} TETRAKE

1

0.5

Cp

Cp

0.5

0

0 z/b = 0.20

–0.5

–1 0

z/b = 0.44

–0.5

0.2

0.4

0.6

0.8

–1 0

1

0.2

0.4

x/c

(C)

x/c

0.6

0.8

1

(D) WIND {NASA} Schmitt. {EXP} TETRAKE

1

WIND {NASA} Schmitt. {EXP} TETRAKE

1

0.5

Cp

Cp

0.5

0

0 z/b = 0.65

–0.5

–1 0

z/b = 0.80

–0.5

0.2

0.4

0.6

x/c

0.8

1

–1 0

0.2

0.6

0.4

0.8

1

x/c

FIGURE 12.53 Time-averaged pressure coefficients distributions on seven selected locations on ONERA M6 wing.

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Computational Fluid-Structure Interaction

(F)

(E) WIND {NASA} Schmitt. {EXP} TETRAKE

1

WIND {NASA} Schmitt. {EXP} TETRAKE

1

0.5

Cp

Cp

0.5

0

0 z/b = 0.9

z/b = 0.95

–0.5

–1

–0.5

0

0.2

0.4

x/c

0.6

0.8

–1

1

0

0.2

0.4

x/c

0.6

0.8

1

(G) WIND {NASA} Schmitt. {EXP} TETRAKE

1

Cp

0.5

0 z/b = 0.95

–0.5

–1 0

0.2

0.4

x/c

0.6

0.8

1

FIGURE 12.53 (Continued)

12.5.3 LES of ONERA M6 Wing Flutter The M6 wing flutter computation is based on the same flow parameters listed in the previous section and the intermediate grid. The immersed structural domain (represented by the ONERA M6 wing body) is meshed by 42,150 tetrahedron elements and 9094 nodes. The wing surface is meshed with 7868 boundary triangles. The structure material parameters are as follows: Young’s modulus, E Material density, ρ Poisson’s ratio, ν

5 7.102 3 1010 Pa 5 2770.0 kg/m3 5 0.32

The simulation begins by employing the converged solution from the former computation as the initial solution for the flow domain and the initial wing velocity is zero. A fixed time-step (Δt 5 0.1 s) is employed in the current simulation. Fig. 12.57 shows the flutter responses of the ONERA M6 wing at MN 5 0:8395 and 3.06 degrees angle of attack.

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FIGURE 12.54 Distributions of time-averaged pressure coefficient, temperature, and density on symmetric plane and wing surface.

404

Computational Fluid-Structure Interaction

FIGURE 12.55 Distributions of ratio between test filtering width and grid filtering width on symmetric plane.

FIGURE 12.56 Time-averaged streamline ribbons around the M6 wing surface (contoured by dimensional velocity magnitude).

IMM FSI Model Validations and Applications Chapter | 12 (A)

(B)

405

0.15 Tip leading edge displacement Tip trailing edge displacement

0.008 0.1

Wing tip displacement

0.006

Energy

0.004 Aerodynamic works (We) Total energy (Etot) Etot-We

0.002 0

0

–0.05

–0.1

–0.002 –0.004

0.05

0

5

10 Time (S)

–0.15

15

0

5

10 Time (S)

15

FIGURE 12.57 Flutter responses for ONERA M6 wing using intermediate grid at MN 5 0:8395 and 3.06 degrees angle of attack (displacement in y-direction). (A)

(B)

0.0086

0.15

0.0085 Drag coefficient

Lift coefficient

0.145

0.14

0.135

0.13 0

0.0084

0.0083

5

10 Time (S)

15

20

0.0082

0

5

10 Time (S)

15

20

FIGURE 12.58 Time history of lift and drag coefficients for M6 wing fluttering using intermediate grid at MN 5 0:8395 and 3.06 degrees angle of attack.

While Fig. 12.58 shows the time variation of lift and drag coefficients for the ONERA M6 wing during the flutter process. As can be seen from the figures, the constant energy difference (Etot
WE) is equal to the initial energy. This verifies that the energy exchange between the structure and the fluid satisfies the global conservation law for the total energy. We also find that the maximum tip displacement at leading edge is about 87.6% of that at trailing edge, which means that the flutter of M6 wing is a composition motion of the first bending and first torsion mode while dominated by the former. Fig. 12.59 shows the Mach number distributions on the upper surface of the M6 wing at three different moments (t1, t2, and t3) during the flutter. At t 5 t3, the tip y-displacement of the wing reaches the maximum value.

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Computational Fluid-Structure Interaction

FIGURE 12.59 Instantaneous Mach number distributions at three different moments in one cycle of the wing flutter.

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REFERENCES [1] A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies, J. Comput. Phys. 207 (2) (2005) 457
492. [2] S.C.R. Dennis, S.N. Singh, D.B. Ingham, The steady flow due to a rotating sphere at low and moderate Reynolds numbers, Phys. Fluids 101 (1980) 257
279. [3] Van Dyke, D. Milton, An Album of Fluid Motion, Parabolic Press, Stanford, California, USA, 1982. [4] M. Nishioka, H. Sato, Mechanism of determination of the shedding frequency of vortices behind a cylinder at low Reynolds numbers, J. Fluid Mech. 89 (1978) 49
60. [5] C.H. Tai, Y. Zhao, Parallel unsteady incompressible viscous flow simulation using an unstructured multigrid method, J. Comput. Phys. 192 (1) (2003) 277
311. [6] C. Liu, X. Zheng, C.H. Sung, Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys. 139 (1998) 35
57. [7] C.H.K. Williamson, Defining a universal and continuous Strouhal
Reynolds number relationship for the laminar vortex shedding of a circular cylinder, Phys. Fluids 31 (1988) 2742
2744. [8] R. Wille, Karman vortex streets, Advances in Applied Mechanics, 6, Academic, New York, 1960, pp. 273
287. [9] J.H. Hattel, P.N. Hansen, A control volume-based finite difference method for solving the equilibrium equations in terms of displacements, Appl. Math. Model. 19 (1995) 210
243. [10] A.K. Slone, C. Bailey, M. Cross, Dynamic solid mechanics using finite volume methods, Appl. Math. Model. 27 (2003) 69
87. [11] A.K. Slone, K. Pericleous, C. Bailey, M. Cross, C. Bennett, A finite volume unstructured mesh approach to dynamic fluid
structure interaction: An assessment of the challenge of predicting the onset of flutter, Appl. Math. Model. 28 (2004) 211
239. [12] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1982. [13] R.T. Fenner, Engineering Elasticity: Applications of Numerical and Analytical Techniques, Ellis Horwood, 1986. [14] X. Lv, Y. Zhao, X.Y. Huang, G.H. Xia, X.H. Su, A matrix-free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid-structure interaction, J. Comput. Phys. 225 (2007) 120
1442. [15] C.A. Schuler, W.R. Usry, B. Weber, J.A.C. Humphrey, R. Greif, On the flow in the unobstructed space between shrouded corotating disks, Phys. Fluids A 2 (1990) 1760. [16] C.J. Chang, C.A. Schuler, J.A.C. Humphrey, R. Greif, Flow and heat transfer in the space between two corotating disks in an axisymmetric enclosure, J. Heat Transfer 111 (1989) 625. [17] C.J. Chang, J.A.C. Humphrey, R. Greif, Calculation of turbulent convection between corotating disks in axisymmetric enclosures, Int. J. Heat Mass Transfer 33 (1990) 2701. [18] C.A. Schuler, Investigation of the Flow Between Rotating Disks in an Enclosure, Ph.D. thesis, University of California, Berkeley, 1990. [19] I. Iglesias, J.A.C. Humphrey, Two- and three-dimensional laminar flows between disks co-rotating in a fixed cylindrical enclosure, Int. J. Numer. Meth. Fluids 26 (1998) 581
603. [20] J.A.C. Humphrey, C.A. Schuler, D.R. Webster, Unsteady laminar flow between a pair of disks corotating in a fixed cylindrical enclosure, Phys. Fluids A 7 (1995) 1225
1240.

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[21] J.A.C. Humphrey, D. Gor, Experimental observations of an unsteady detached shear layer in enclosed corotating disk flow, Phys. Fluids A 5 (1993) 2438
2442. [22] A.Z. Szeri, S.J. Schneider, F. Labbe, H.N. Kaulman, Flow between rotating disks; Part I, basic flow, J. Fluid Mech. 134 (1983) 103
131. [23] Schmitt, V. and F. Charpin, Pressure distributions on the ONERA-M6-Wing at transonic Mach Numbers, experimental data base for computer program assessment, Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138, May 1979. [24] G. Hwang, Parallel Finite Element Solutions of Nonlinear Aeroelastic and Aeroservoelastic Problems in Three-Dimensional Transonic Flows, Ph.D. Dissertation, Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA, 1997. [25] O.O. Bendiksen, Hwang, G.Y., Transonic flutter suppression using dynamic twist control, AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference and Exhibit, 37th, Salt Lake City, UT, Apr. 15
17, 1996, Technical Papers. Pt. 4 (A96-26801 06-39), Reston, VA, American Institute of Aeronautics and Astronautics, pp. 2670-2684, 1996. [26] Guclu Seber and Oddvar O. bendiksen, nonlinear flutter Calculations Using Finite Elements in a Large Deformation Direct Eulerian
Lagrangian FORMULATION, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference Austin, Texas, 2005.

Chapter 13

ALE FSI Model Validations and Applications 13.1 VALIDATION OF INCOMPRESSIBLE ARBITRARY LAGRANGE EULERIAN SOLVERS 13.1.1 Steady Flow in a Two-Sided, Lid-Driven Cavity 13.1.1.1 Introduction The lid-driven cavity flow is a fundamental and well-researched problem in fluid dynamics. Therefore this problem is often used as a test case to validate new CFD codes. Examples can be found in the few papers, which are published within the last 10 years by Wan et al. [1], Wright et al. [2], Ding et al. [3], and Danesh [4]. A more advanced problem is flow in a two-sided, lid-driven cavity. This case has been studied not only experimentally, but also numerically by Kuhlmann et al. [5
8] since 1997. They analyzed the critical 2D and 3D states for different geometries, including symmetric and asymmetric driven flow conditions. Here, the focus is not to validate the code—this has already been done before for the serial code—but to check the parallel performance and the consistency of the parallel version, no effort has been made to validate the existence of the critical states with different Reynolds numbers for several geometries as in Ref. [5]. The geometry used is shown in Fig. 13.1, which has the same aspect ratio as used by Kuhlman et al. [5], which is used for comparison with the results of the arbitrary Lagrange Eulerian (ALE) code. The steady solutions were obtained with the third-order characteristicsbased upwind scheme with a grid of 7381 nodes (121 3 61 nodes with 14,400 elements) for two different Reynolds numbers: 240 and 800. The flows were simulated for both Reynolds numbers with 100 subiterations for each time-step, with a time-step size of 0.05 seconds and a total of 2000 time-steps for a duration of 100 nondimensional time units. Although convergence is obtained after 500 steps for Re 5 240 and after 1600 steps for Re 5 800, this setting was chosen to have comparable simulation times for the parallel performance study, which was the main objective of this chapter. Another reason is that the solutions are more likely to be affected by initial Computational Fluid-Structure Interaction. DOI: https://doi.org/10.1016/B978-0-12-814770-2.00013-1 © 2019 Elsevier Inc. All rights reserved.

409

410

Computational Fluid-Structure Interaction

FIGURE 13.1 Domain geometry of the two-sided, lid-driven cavity.

startup randomness if too short simulation times are used. The artificial compressibility parameter β was set to 1 for fast convergence. The following boundary conditions were used: for x 5 2 0:98:

u 5 0; v 5 2 1:0

for x 51 0:98:

u 5 0; v 51 1:0

for rigid walls:

u 5 0; v 5 0

The initial condition for the inner nodes was: u 5 0; v 5 0 The computations were performed on an SGI Origin 2000 machine with 32 MPIs 64-Bit R10000 250 MHz RISC CPUs with 500 MFLOPS each. This system was provided by Center of Advanced Numerical Engineering Simulation CANES at Nanyang Technological University in Singapore.

13.1.1.2 Grid Dependency Study Grid dependency study was performed by comparing the convergence histories of a 4186-node (91 3 46) grid with a 7381-node (121 3 60) for both the Reynolds numbers Re 5 240 and 800. The residual was normalized and chosen as a tool for comparing the results of the two grids. Other parameters used for comparison are the vertical and horizontal velocities v and u along the centerline of the cavity. For Re 5 240 (Fig. 13.2A), the residual stops changing after 420-timesteps for both grids. The residual of the smaller 4186-node grid decreases faster than that of the residual of the 7381-node grid after 150 time-steps, and both residuals become the same after 400-time-steps. Therefore it can be concluded that there is no significant difference in convergence for the two grids, but the smaller grid is more advantageous for time-critical simulations

ALE FSI Model Validations and Applications Chapter | 13

411

FIGURE 13.2 History of normalized residual versus number of time-steps for (A) Re 5 230; (B) Re 5 800.

if accuracy is not adversely affected by the smaller grid size. Fig. 13.2B shows the history of the residual at Re 5 800. Up to 300-time-steps, there is no significant difference. Between 300 and 650 time-steps, the transition from two-vortex flow to one-vortex flow can be observed. The residual of the steady state after 1100 time-steps for the 4186-node grid is higher than for the 7381-node one but it is reached faster. For both Reynolds numbers, there is no significant change in the velocity profile v along the centerline in x-direction (Fig. 13.3). In order to conduct a parallel performance study, the larger grid is chosen.

13.1.1.3 Results and Discussion The main goal of this test case is to prove the consistency of the parallel results with the serial ones. It is shown that the results for 1, 2, 4, 8, and 16 partitions are exactly the same. Figs. 13.4 and 13.5 show the contour plots for the y-component of the velocity, v, for 1, 4, and 16 (Re 5 240), respectively, for 1, 8, and 16 (Re 5 800) partitions. Fig. 13.6 plots the velocity v over the x-axis (y 5 0). For both Reynolds numbers, a good consistency between the different parallel solutions and the serial solution can be

412

Computational Fluid-Structure Interaction

FIGURE 13.3 Velocity v along x at y 5 0 for Re 5 230 after 500 time-steps and Re 5 800 after 1600 time-steps.

observed. Thus it can be stated that the parallelization was successful for steady flows. According to Ref. [5], it has been shown that basically two different states occur, depending on the Reynolds number. Both states are point symmetric because both walls are moved with the same speed. For low Reynolds numbers there is a two-vortex flow and for high Reynolds numbers a onevortex flow. During the transition, a cat-eye shaped flow exists and which was found in this work. Since the solver is 2D, the latter state does not occur at the Reynolds number at which it has been located experimentally by Kuhlmann et al. [5]. First, the flow with a Reynolds number of 240 is investigated. After 100 time units, a steady state (Residual  5.7210) is obtained and the flow consists of two main vortexes, each one driven by the moving wall on its side. There are two minor vortexes near the y-axis at the upper and lower wall. The streamlines within the cavity are illustrated in Fig. 13.7 for one (A) and four (B) partitions, and compared with the solution of Kuhlmann et al. [5] (C). Location and size of the two minor vortexes near the y-axis are not found to be exactly the same. Increasing the Reynolds number produces better compliance. The reason for this is assumed to be the absence of 3D effects in the 2D solver used in this work opposite to the 3D solver used by Kuhlmann et al. [5]. A parameter variation study for the Reynolds number was conducted to find the transition between the two-vortex state and the one-vortex state in this section. This transition was expected to start at a Reynolds number between 234.3 [5,6] and 257 [5], forming a so-called cat-eye flow. The Reynolds number was increased by 5 for each variation and it was found out that the steady state was reobtained after a simulation time of 4.5 seconds. When the streamline pattern started to change significantly, the Reynolds

ALE FSI Model Validations and Applications Chapter | 13

413

FIGURE 13.4 Contour plot for vertical velocity v, Re 5 240.

number increment was decreased to 1. Due to the negligence of the 3D effects that cause this specific flow state it occurred at a Reynolds number of 388. Well after this transition, at a Reynolds number of 800, the flow is simulated for 100 nondimensional time units to obtain a steady state

414

Computational Fluid-Structure Interaction

V

0.4

0.90 0.80 0.70 0.60 0.50 0.41 0.31 0.21 0.11 0.01 – 0.09 – 0.19 – 0.29 – 0.39 – 0.49 – 0.58 – 0.68 – 0.78 – 0.88 – 0.98

y

0.2

0

–0.2

–0.4 –1

(A) 1 partition

–0.5

0 x

0.5

1

V

0.4

0.90 0.80 0.70 0.60 0.50 0.41 0.31 0.21 0.11 0.01 – 0.09 – 0.19 – 0.29 – 0.39 – 0.49 – 0.58 – 0.68 – 0.78 – 0.88 – 0.98

y

0.2

0

–0.2

–0.4 –1

(B) 4 partitions

–0.5

0 x

0.5

1

V

0.4

0.90 0.80 0.70 0.60 0.50 0.41 0.31 0.21 0.11 0.01 – 0.09 – 0.19 – 0.29 – 0.39 – 0.49 – 0.58 – 0.68 – 0.78 – 0.88 – 0.98

y

0.2

0

–0.2

–0.4 –1

–0.5

(C) 16 partitions

0 x

0.5

1

FIGURE 13.5 Contour plot for vertical velocity v, Re 5 240.

(Residual  5.0210). The flow now consists of one large vortex in the center of the cavity and two minor point symmetric vertices at the upper and lower walls located at x  6 0.7. Fig. 13.8 compares the streamline patterns for one (A) and two (B) partitions with the result of Kuhlmann et al. Location

ALE FSI Model Validations and Applications Chapter | 13

415

FIGURE 13.6 Velocity v over x at y 5 0 for (A) Re 5 230 and (B) 800.

and pattern of the streamlines belonging to the main vortex show a very good correspondence; location and size of the minor vortexes are not matching, but a better agreement was obtained by increasing the Reynolds number. It is assumed that, as mentioned above, the 2D nature of the solver is the main reason for this disagreement (Fig. 13.9). Thus it is shown that 3D effects for flows in two-sided, lid-driven cavities with the Reynolds numbers far below and far above the transition state are negligible, whereas the cat-eye state cannot be exactly located without considering the 3D effects.

13.1.1.4 Parallel Performance In this chapter, wall clock time was used for measurement of the total simulation time. The communication time is categorized into the time needed to pass the flow variables, the nodal gradients, and all other information, such as synchronizing the time-step size or computation of the residual. The latter part of the communication time was always below 2 seconds, so it will not be considered in detail for this performance study. Tables 13.1 and 13.2 summarize the performance data for calculations for both Reynolds numbers, i.e., Re 5 240 and 800. Since both simulations lasted for about the same wall clock time, comparing the times given allows us to judge the quality of this statistics. Roughly, the computation for Re 5 800 is always fractional faster than Re 5 240. The development of the communication time with rising number of partitions does agree quite accurate. Small oscillations are due to different loads of the parallel computer during the simulations. Thus we can state that the parallel data in Tables 13.1 and 13.2 is representative for concluding the performance of the parallel code developed in this chapter. The information of Tables 13.1 and 13.2 is visualized in Figs. 13.10 and 13.11. Analyzing Fig. 13.10, we find the parallel efficiency decreasing almost linear with the number of processors beginning with two processors.

416

Computational Fluid-Structure Interaction

FIGURE 13.7 Streamlines for Re 5 240.

417

ALE FSI Model Validations and Applications Chapter | 13

0.4

y

0.2

0

–0.2

–0.4 –1

–0.8

–0.6

–0.4

–0.2

0 x

0.2

0.4

0.6

0.8

1

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

(A) 1 partition 0.4

y

0.2

0

–0.2

–0.4 –1

–0.8

–0.6

(B) 2 partitions

0.4

y

0.2

0

–0.2

–0.4 –0.4

–0.2

0

(C) [22] x-axis is scaled in units of width FIGURE 13.8 Streamlines for Re 5 800.

0.2

0.4

x

418

Computational Fluid-Structure Interaction (A) 0.4

y

0.2

0

–0.2

–0.4 –1

–0.8

–0.6

–0.4

–0.2

0 x

(B)

0.2

0.4

0.6

0.8

1

0.4

y

0.2

0

–0.2

–0.4 –0.4

0

–0.2

0.4

0.2

x

FIGURE 13.9 Cat-eye flow state. 2D numerical solution from this work, (A) Re 5 388, 3D numerical solution [22], (B) Re 5 257.

TABLE 13.1 Cavity: Performance Table for Re 5 240 (t in min) Number of Processors

tcomp

NG tcom

FV tcom

tP

EP

SP

1

912.72







912.72







2

459.43

3.86

2.27

465.56

0.98

1.96

4

228.65

8.80

10.54

247.99

0.92

3.68

16.66

15.95

142.27

0.80

6.42

18.41

17.14

94.26

0.61

9.68

8 16

113,66 60.71

419

ALE FSI Model Validations and Applications Chapter | 13

TABLE 13.2 Cavity: Performance Table for Re 5 800 (t in min) NG tcom

FV tcom

908.76




2

457.44

4

Number of Processors

tcomp

tP

EP

SP

1




908.76







3.61

1.56

462.61

0.98

1.96

228.23

10.32

13.19

251.74

0.90

3.61

8

114.31

11.81

15.13

141.25

0.80

6.43

16

58.16

17.76

16.34

92.26

0.62

9.85

FIGURE 13.10 Cavity: Speedup and efficiency diagram for Re 5 240 (left) and 800 (right).

Re = 240 Re = 240 Re = 800 Re = 800

90

Gradient of speedup

80

0.9 0.8 0.7 0.6 0.5 0.4 0.3

70

60

1

0.2

Share of computing time

0

2

4

6

8

10

Number of processors, P

12

Speedup gradient

Computing time (%)

100

0.1 14

0 16

FIGURE 13.11 Cavity: Share of computing time and speedup gradient.

The parallel speedup behaves approximately linear up to 8 processors; then, there is a break in the graph and the angle against the speedup axis increases. Fig. 13.11 shows the gradient of speedup that documents a break after eight partitions. At 2 and 4 processors, the step size is almost equal but twice as high at 8 processors. Effectively means that while the loss in speedup between 4 and 8 processors is about 30% for both Re 5 240 and

420

Computational Fluid-Structure Interaction

800, it is already 60% between 8 and 16 processors. A possible reason for that can be found in the hardware of the Origin 2000; 4 processors each are arranged in clusters that share one memory module. So, with the amount of processes, the number of clusters to communicate between rises. A black dashed straight line in Fig. 13.10 shows the desired proportional speedup; with rising number of processors, the distance between this optimal speedup and the obtained speedup enlarges. The main reason for this is very obvious in Tables 13.1 and 13.2. While the computation time tcomp decreases almost proportionally, the communication time is increasing. Fig. 13.11 visualizes this phenomenon by a linear decrease of the share of computation time tcomp in the actual simulation time tP. All these observations obtained by analyzing the above diagrams (Figs 13.10 and 13.11) have been sensed subjectively by performing the computations. It seems that for the hard- and software setup used in this work, a number of 8 processors is quite insensitive against interferences such as intercluster communication or computer overload.

13.1.2 Flow in a Two-Dimensional Collapsible Channel 13.1.2.1 Introduction Channel flow with collapsible walls is a very basic problem in biofluid dynamics. The solution for this problem is necessary to simulate blood flow in the human body, for example, in the heart or in veins. First numerical simulations have been done by Rast [9], Lowe and Pedley [10], and Luo and Pedley [11] in 1994 and 1995, studying fluid flow through a channel with moving indentation without considering wall inertia. In 1998 Luo and Pedley [12] extended this work by considering wall inertia. They used model with equal material tension in each location of the membrane. Considering different tensions at each point in the membrane, Luo [13] and Cai and Luo [14] implemented a fluid beam model, replacing the membrane by an elastic beam. A different way was chosen by Forhad and Zhao [15], who derived the equation of motion for the membrane considering all the fluid and structure forces and so providing the most complete simulation of the fluid flow in a 2D collapsible channel. This study parallelized the ALE solver and used it to study fluid flow and the related FSI in a collapsible channel. Validation of the parallel implementation was carried out by comparing the parallel results with those of the original serial code in a channel with a membrane. The following nondimensional values were used for the geometry parameters and the external pressure according to Fig. 13.12: Lu 5 5;

Le 5 5;

Ld 5 30;

D 5 1;

pe 2 pd 5 1:953

ALE FSI Model Validations and Applications Chapter | 13 Lu

421

Ld

Le pe

Uin

pd

D

FIGURE 13.12 Geometry of the collapsible channel.

For the flow domain, the following nondimensional values are set as initial and boundary conditions: for x 5 0:

u 5 6yð1 2 yÞ; for x 5 40:

0#y#1

and

v50

p 5 1:0

The rigid wall boundary conditions are u 5 v 5 0 for rigid walls and uðtÞ 5 uw ðtÞ and vðtÞ 5 vw ðtÞ for moving walls. The following nondimensional parameters were used to describe the behavior of the membrane: T0 5 178:916=b;

b 5 30:0;

m 5 0:01;

E 5 0:1

where T0 is the initial tension in the membrane, b $ 1 is a parameter to control the magnitude of tension, m is a parameter to control the inertia of the membrane and E is the nondimensional Young’s modulus of elasticity, the used value is according to a rubber-like material. The simulation was computed with a Reynolds number of 300. The artificial compressibility parameter was set to 1, the time-step size was set to 0.9, with 250 subiterations per time-step. An unstructured grid with 42,448 nodes was used. Before computing the movement of the membrane, it was considered as a rigid body until a simulation time of t 5 10. The computations were performed on the NTU SGI Origin 2000 machine with 32 MPIs 64-Bit R10000 250 MHz RISC CPUs with 500 MFLOPS each.

13.1.2.2 Developing Flow Without Moving Grid Before the moving grid simulation is started, the channel flow is simulated for an interval of Dt 5 10 for all partitions. This is necessary since the velocity profile chosen as the initial condition does not consider the deformation in the channel. After Dt 5 10, the flow is so far developed that reasonable moving boundary simulation is possible. As seen in Fig. 13.13 the flow contour has developed exactly the same for the serial, 2, 4, 8, and 16 processors simulation. This flow state will be used as common starting point for all moving boundary simulations in this work. Another criterion to compare the consistency of the solution is the history of the lift and drag coefficient over the simulation time. Fig. 13.14 visualizes perfect synchrony for all configurations.

422

Computational Fluid-Structure Interaction

FIGURE 13.13 Result at t 5 0 after a simulation time of t 5 10.

13.1.2.3 Moving Grid The main objective of this test case is to validate the parallel consistency and performance for moving boundary simulation. Thus the boundary was moved for six different time intervals, Δt 5 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0.

ALE FSI Model Validations and Applications Chapter | 13

423

FIGURE 13.14 History of lift and drag coefficient over simulation time.

For each interval, the results were compared and the performance was measured. The main criterion for the consistency here is the deformation of the membrane. Exemplarily the deformation at t 5 1.0 is compared in Fig. 13.15. Visually, there is no difference between the deformations of the membrane in the different simulations. The overlay of the membrane for all numbers of partitions computed also shows a good match. Thus it can be stated that the consistency of the method developed in this work is validated for this case. Second objective is the analysis of the channel flow influenced by the membrane. Since the movement of the membrane is mainly influenced by pressure differences between the inside and the outside of the channel, the development of the pressure contour plot (Fig. 13.16) is important to investigate. The principle of the movement can be tracked by comparing the pressure distribution of each observed time with the change of deformation of the membrane at the next point in time. It can also be observed that the different tension value for each point of the membrane affects the movement by tracking the wave that moves the membrane in the positive x-direction between x 5 6 and 5 7 at t 5 1.5, 2.0, and 2.5. This wave is stopped and starts returning at t 5 3.0. Fig. 13.17 plots the development of the velocity contour in the observed time intervals.

13.1.2.4 Parallel Performance Similar to the previous case, wall clock time was used for measurement of the total simulation time for rigid and moving boundaries. It includes the computing time and the communication time. The communication time is categorized into the time needed to pass the flow variables and the nodal gradients. For moving boundary simulations, the time needed for combining the moving boundary data in the master processor is the time taken for proceeding and redistributing, it is considered as communication time for moving

424

Computational Fluid-Structure Interaction

FIGURE 13.15 Deformation of the membrane at t 5 1.0 for different partition numbers.

ALE FSI Model Validations and Applications Chapter | 13

FIGURE 13.16 Pressure contour plots for t 5 0, 0.5, 1.0 1.5, 2.0, 2.5, and 3.0.

425

426

Computational Fluid-Structure Interaction

FIGURE 13.17 Velocity contour plot between t 5 0 and 3.

427

ALE FSI Model Validations and Applications Chapter | 13

MB boundary, tcom . The time necessary to pass all other information, such as synchronizing the time-step size or computation of the residual was always below 2 seconds in this chapter, so it will not be considered in detail for this performance study. Due to heavy load on the parallel computer used for this work results for more than 16 processors could not be computed for this test case. Table 13.3 summarizes the performance data for the part of the simulation without moving boundary. The performance data for the moving boundary simulation is stated in Tables 13.4, 13.5, and 13.6. The communication MB time for the moving boundary part of the code, tcom , increases with a number of partitions as expected and observed in the performance of the above study. The other communication times behave similarly for 4, 8, and 16 processors. For 2 processors, the communication times for the nodal gradients and flow variables are the same in all cases, but the Δt 5 3.0 interval higher than for 4 processors. The speedup calculated with those values is visualized in

TABLE 13.3 Collapsible Channel: Performance Table for Fixed Boundary (t in min) FV tcom

Number of Processors

tcomp

NG tcom

1

1186.73




2

431.36

6.12

10.21

447.69

4

198.65

2.34

4.51

8

94.79

6.27

16

45.34

5.31

tP

EP (%)

1186.73







1.33

2.65

205.50

1.44

5.77

8.60

109.66

1.35

10.82

7.55

58.20

1.27

20.39




SP

TABLE 13.4 Collapsible Channel: Performance Table for Movement Δt 5 1.0 (t in min) Number of Processors

tcomp

FV tcom

NG tcom

MB tcom

tP

EP (%)

1

142.60










142.60







2

52.60

1.99

0.79

0.42

55.80

1.28

2.56

4

22.76

0.23

0.17

0.46

23.62

1.51

6.04

8

11.13

0.97

0.54

0.86

13.50

1.32

10.56

16

5.43

1.02

0.66

1.10

8.21

1.09

17.37

SP

428

Computational Fluid-Structure Interaction

TABLE 13.5 Collapsible Channel: Performance Table for Movement Δt 5 2.0 (t in min) Number of Processors

tcomp

FV tcom

NG tcom

MB tcom

tP

EP (%)

1

267.06










267.06







2

104.25

1.43

0.71

0.78

107.17

1.25

2.49

4

43.63

0.83

0.40

0.87

45.73

1.46

5.84

8

21.21

2.56

1.37

1.39

26.53

1.26

10.07

16

10.12

1.91

1.21

2.09

15.33

1.09

17.42

SP

TABLE 13.6 Collapsible Channel: Performance Table for Movement Δt 5 3.0 (t in min) Number of Processors

tcomp

FV tcom

NG tcom

MB tcom

1

392.44







2

154.47

0.25

0.15

4

64.54

4.70

8

30.94

16

14.69

tP

EP (%)

SP




492.44







1.10

155.97

1.58

3.16

2.26

1.39

72.89

1.69

6.76

2.53

1.30

1.91

36.68

1.68

13.43

2.21

1.44

2.94

21.28

1.45

23.14

Fig. 13.18. An eye-catching phenomenon is the unusual value for the parallel efficiency, for all configurations higher than one, what is practically impossible (Fig. 13.19). The performance measurement was repeated three times— taking special care that the Origin 2000 was not overloaded during the simulation time—and always the same result was obtained, the interference causing this abnormality has not been localized. Looking at the results for the parallel study from another point of view, the result fits better with previous studies about parallel performance. Only the results for 4, 8, and 16 processors will be considered, and the speedup and efficiency were calculated relative to 4 processors. Table 13.7 shows the result and compares it with the result of the two-sided, lid-driven cavity; the total computation time of that case is approximately the same so it fits quite well as a benchmark. For the calculation without moving grid, normalized speedup and efficiency for 8 processors are minimal better than for the twosided, lid-driven cavity, but are significantly higher for 16 processors.

ALE FSI Model Validations and Applications Chapter | 13

429

FIGURE 13.18 Measured parallel speedup.

FIGURE 13.19 Measured parallel efficiency.

TABLE 13.7 Efficiency and Speedup Normalized by 4 Processors Number of Partitions

4

8

16

E4/S4 fixed grid

1

0.94/1.87

0.88/3.53

E4/S4 Δt 5 1.0

1

0.87/1.75

0.72/2.87

E4/S4 Δt 5 2.0

1

0.86/1.72

0.76/2.98

E4/S4 Δt 5 3.0

1

0.99/1.99

0.86/3.43

Two-sided Cavity, Re 5 240 8

16

0.87/1.74

0.66/2.63

430

Computational Fluid-Structure Interaction

Speedup and efficiency for 8 processors and moving grid for t 5 1.0, t 5 2.0, respectively are about the same as the benchmark case, for 16 processors, there is a significant rise as well. A conclusion out of this is different from that of the smooth and steady cavity flow, for more complex unsteady flow, the advantage for larger amounts of processors rises.

13.1.3 Flow Over a Two-Dimensional Oscillating Circular Cylinder 13.1.3.1 Introduction The motion of harmonic oscillating circular cylinders in a fluid is especially of interest in the study of FSI, not only because the presence of many engineering applications in offshore and civil engineering, or in heat exchanger design, but also extensive literatures on the topic supported by plenty of experimental and numerical investigations. The special flows were investigated in many studies, for example by Keulegan and Carpenter [16], Sarpkaya [17], Williamson [18], Obasaju et al. [19], Tatsuno and Bearman [20], Justesen [21], Sarpkaya [22], Anagnostopoulos et al. [23], Lin et al. [24], Du¨tsch et al. [25], Guilmineau and Queutey [26], Lam and Dai [27], Elston et al. [28,29], Anh-Hung Pham et al. [30], Lam and Hu [30], and Ghozlani et al. [31]. Such flows are mainly characterized by complex vortex
vortex and vortex
structure interaction phenomena, involving stability, accuracy, and bifurcation problems. Hence, time-dependent computations have to be carried out carefully at specified Reynolds numbers, which requires the numerical models used to be more stable and accurate. It is well known that a bluff body is oscillating in a still fluid, secondary streaming is generated around the body owing to the influence of nonlinear effects. In general, there are two important parameters, which mainly control flow structure around the body. One is Keulegan
Carpenter number, KC, where KC 5 2πA0/d and A0 is the amplitude of oscillation motion and d is the cylinder diameter and the other one could be one of three candidates, Reynolds number, Re used in Ref. [25]; Stokes number, fd2/v, where f is the frequency of body oscillation and v is the kinetic viscosity of the fluid, used in Ref. [17]; and the symbol β used in Ref. [25], which is defined by the ratio of Re to KC. Here the latter three parameters mentioned mainly provide the frequency information of body oscillations and could be mutually transformed into each other. Tatsuno and Bearman [20] classified flow patterns around oscillating cylinder into eight regimes based on the values of KC and β and gave a general description for each regime. Those descriptions are frequently referenced by researchers who want to verify their new numerical models. In this section, validation work of simulating flow around oscillating cylinder in three cases, in which the cases are in A, E, and F regimes shown in Fig. 13.20 based on the definition in Ref. [20].

ALE FSI Model Validations and Applications Chapter | 13

12

431

F G

10

KC

8 D

6

A

E

C 4 2

0

β=

35

100

B

A°

200

300

400

500

Re FIGURE 13.20 Flow regimes defined by Tatsuno and Bearman [20].

13.1.3.2 Model Validation: For Re 5 100 and KC 5 5 The proposed ALE model is validated by the test case of flow around a vertically oscillating circular cylinder in a quiescent fluid. The diameter of oscillating cylinder is 1 in dimensionless scale while the far field boundary is taken to be circular in shape and is located at a distance of 25 from the center of the cylinder (see Fig. 13.21A). Mesh grids are shown in Fig. 13.21B. A harmonic oscillation is forced in the y-direction with the dimensionless displacement of the cylinder, given by
 KC 2π sin t ð13:1Þ yðt Þ 5 2 2π KC where KC 5 5 is the KC number, while the flow conditions are taken to correspond to a Reynolds number, Re 5 100. Eq. (13.1) is chosen as that in Ref. [24] for the comparison purpose. Fig. 13.22 shows mesh grids at different phase positions by using the proposed moving mesh algorithm. The mesh around the moving boundary is indeed keeping rigid, which would be good at calculating the flow field around moving boundary. A sequence of flow patterns is shown in Fig. 13.23, which is in good agreement with flow visualization results observed by experiment [24] and other published numerical results [24,25]. The computation results show that the flow structure around oscillating cylinder was characterized by stable, symmetric, and periodic vortex shedding. It can be observed that, as the cylinder moving down, the flow at the front of

432

Computational Fluid-Structure Interaction

FIGURE 13.21 Viscous flow around an oscillating cylinder. (A) Geometry of flow around an oscillating cylinder; (B) mesh grids used in the simulating case, 8544 elements and 4325 nodes.

FIGURE 13.22 Mesh grids at the different phase positions. (A) 0, (B) 36, (C) 96, (D) 108, (E) 192, (F) 216, (G) 288, and (H) 324 degrees.

cylinder is split and flow around cylinder develops and produces two counter-rotating vortices of the same size and shape. When the maximum downward position is reached, the production of the first pair of vortices is completed. Then, a new pair of eddies is generated by the upward movement of the cylinder, rotating in the opposite direction and they are shed when the cylinder reaches its maximum upward position. Unlike the postprocessing done in Refs. [24,25], in which the simulating results were obtained by phase

ALE FSI Model Validations and Applications Chapter | 13

433

FIGURE 13.23 Isolines of pressure (left) and vorticity (right) at the different phase positions. (A) 0, (B) 36, (C) 96, (D) 108, (E) 192, (F) 216, (G) 288, and (H) 324 degrees.

averaging, here in this chapter, the instantaneous velocity profile in the 51st period is plotted to compare with the experimental data. Fig. 13.24 shows the comparison between these values and the results of the experiment at four cross sections, with constant y-value, for three different phase angles of the oscillating motion. The comparison is considered to be satisfactory in quality with experimental results reported in Ref. [24]. The good agreement between the experimental and numerical results is also indicated by the comparisons provided in Fig. 13.25 of measured and predicted velocity fields for three angles of the cylinder motion.

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FIGURE 13.24 Comparison of velocity x-component u (left column) and velocity y-component v (right column) profiles of the simulation results with those of the experimental results [25] at three phases: (A) 180, (B) 210, and (C) 330 degrees.

13.1.3.3 Model Validation: For Re 5 200 and KC 5 10 The present ALE model is extended to investigate the parameter combination Re 5 200 and KC 5 10, which refers to regime F in Fig. 13.20 [20]. Visual study of the mentioned case in Ref. [20] revealed a different behavior of the vortex motion induced by the cylinder oscillation from those results of case

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FIGURE 13.25 Comparison of velocity vector fields and streamline between experimental results [25] (left column) and simulating results (right column) at three phases: (A) 180, (B) 210, and (C) 330 degrees.

Re 5 100 and KC 5 5. The visualization shows that symmetric vortex shedding occurred first and in roughly the same way as described for the Re 5 100 and KC 5 5 case as the cylinder starting to move. However, after a few cycles of oscillations, the symmetric flow pattern began to change. The symmetric vortex formation became unstable, resulting in the occurrence of a stronger vortex on one side of the cylinder and a weaker on the other side. The difference in the magnitude of the two vortices resulted in a motion of the vortices away from the cylinder at an angle of approximately 27 degrees with respect to the axis along which the cylinder oscillated. The flow around the oscillating cylinder convects diagonally and both vortices, which were formed do not cross this axis but moved away from it. Flow computations are performed for Re 5 200 and KC 5 10 by using the proposed ALE model to verify the above-described asymmetric vortex motion. The results of these computations are presented in Fig. 13.26, showing the same vortex motion as obtained in the experiments. Fig. 13.26 clearly shows that two-vortex pairs form within each oscillation cycle and this

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Computational Fluid-Structure Interaction

FIGURE 13.26 Vorticity isolines at Re 5 200 and KC 5 10.

formation repeats periodically. In contrast to the vortex formation for Re 5 100 and KC 5 5, each pair of vortices consisted of two structures of different age and different initial strength. A strong inclined vortex street extending in two opposite directions is created by the numerical predictions, which is quite similar to those observed in the experiments. Fig. 13.27 shows Cd and Cl history at Re 5 200 and KC 5 6, which indicates flow pattern around cylinder moves periodically after six periods. Slightly different numerical results from Ref. [25], the present model gives perfect periodical Cl profile in which there are small and regular oscillations occurred along with axis, not shaking with large deviation as given in Ref. [25]. Moreover, in Cd profile, there exist small oscillations, which could not be found out in Ref. [25]. Fig. 13.28 shows Cl versus Cd plot at Re 5 200 and KC 5 10. It is noted that the first five periods are omitted for the better demonstration in Fig. 13.28. The plot shows strong cycle-to-cycle variations of total force vector. The forces exerting on cylinder are changed periodically and diagonally per cycle and the direction of total force vector deviates about 27 degrees from axis Cd 5 0, which agrees with those discovers in experiments.

13.1.3.4 Model Validation: For Re 5 210 and KC 5 6 In the visual study of the flow around an oscillating circular cylinder [20], the induced flow patterns are classified into eight regimes in the range of 1.6 , KC , 15 and 5 , β , 160, in which β is defined as the ratio of Re to KC. The case of Re 5 210 and KC 5 6 is falling into E regime and located on the line β 5 35 (see Fig. 13.20). The principal feature description of

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FIGURE 13.27 Cd and Cl history at Re 5 200 and KC 5 10. (A) Du¨tsch’s results [25] at Re 5 200 and KC 5 10 (F1: drag force and F2: lift force); (B) the present ALE model results.

regime E (see Fig. 13.20) states irregular switching of flow convection direction and representing temporarily stable V-type vortex streets in Ref. [20]. The flow which convects to one side of the axis of oscillation, however, intermittently changes its direction to the other side. This switching of the flow occurs at irregular intervals and is presumably triggered by small

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FIGURE 13.28 Cl versus Cd for 30 cycles at Re 5 200 and KC 5 10.

disturbances. The present ALE model then is used to verify the abovementioned features. Fig. 13.29 gives Cd and Cl history at Re 5 210 and KC 5 6. The Cd and Cl profiles show that there appear disturbances from period number 26 to 50. During those periods, the magnitude of Cd decreases to zero and then increases to temporarily stable value while the magnitude of Cl does change too much except for small oscillations. The above-mentioned flow pattern switching phenomenon is believed to happened during this time. Fig. 13.30 shows Cd versus Cl profile at Re 5 210 and KC 5 6. The plot also shows strong cycle-to-cycle variations of the total force vector. But, the total force vector profile is quite different from the results of case Re 5 200 and KC 5 10 mentioned before, much disturbances existing during force development. Fig. 13.31 gives vorticity isolines at different phases for three specified periods. These three periods are flow switching, during switching, and after flow switching, respectively. The vortex shedding and motion, and flow switching are clearly represented in the figures. Before the time period 35, the V-type vortex streets appear in the lower region (see Fig. 13.31 (left column)). Then, there are quite long-time periods for forming opposite V-type vortex streets in the above region (see in Fig. 13.31 (right column)). In Ref. [25], the author considered that there was absence of flow switching due to absence of induced artificial disturbances. In this chapter, we double check

2

Cl

1

0

–1

–2 2

Cd

1

0

–1

–2 0

10

20

30

40

50

t/T FIGURE 13.29 Cl and Cd history at Re 5 210 and KC 5 6.

FIGURE 13.30 Cl versus Cd for 70 cycles at Re 5 210 and KC 5 6.

60

70

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FIGURE 13.31 Vorticity isolines at different phases for different periods at Re 5 210 and KC 5 6. (Phase positions: from top to bottom: 0, 54, 108, 162, 216, 270, and 324 degrees) (left column: 30th period; right column: 70th period).

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that with another case, Re 5 200 and KC 5 6. The flow switching phenomenons are still obtained. Yet, we could not find any real reason for that. However, we noticed the simulated time period is about 20 in Ref. [25]. The simulating time may be too short for making conclusion of flow switching absence.

13.1.4 Remarks An existing and validated 2D incompressible finite volume solver was parallelized in this chapter using an overlapping grid elements technique at the partition borders for setting up a communication structure and message passing interface for synchronizing the solution between neighboring partitions. Parallel consistency has been validated successfully for steady flows by comparing the serial and parallel result of a two-sided, lid-driven cavity flow for one- and two-vortex flow states at two different Reynolds numbers, Re 5 240 and 800. In addition, the previous results of Kuhlmann et al. could also be validated demonstrating that for the occurrence of a cat-eye flow state, the Reynolds number is affected by 3D influences. The parallel performance was analyzed and it was concluded that the most effective parallel computing—under conditions similar to this work, especially domain size and hardware—can be executed with 8 processors. Also for moving boundary simulation in a collapsible channel, the parallel implementation was validated successfully. The results for serial and parallel computation were demonstrated to be the same. The parallel performance study for this case generated two main results. First, an unknown interference extended the computing time for the serial and two processor simulations, and this phenomenon seems not to be random since the performance measurements were executed three times. Second, different from the steady case, the performance loss for 16 processors compared to 4 processors seems to be less pronounced. It is concluded that for more complex, for example, unsteady and even more for moving boundary cases, the optimal number of processors is larger than that of simple steady cases. The proposed ALE model is also applied to simulate flow patterns around oscillating cylinders for three typical cases, which belong to A, E, and F regimes, respectively, classified by Tatsuno and Bearman [20]. Good agreements are observed between measurements and numerical simulations. The phenomenon of flow switching is captured for Re 5 200 and KC 5 6 and reported for the first time. By using the proposed model, flow patterns around oscillating cylinders and force information including Cl and Cd are investigated at Re 5 100 and KC 5 5. Irrespective of the flow patterns around the oscillating cylinders, at the same Re and KC numbers the flow around them develops asymmetrically.

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13.2 AERODYNAMIC FORCES OF RIGID FOILS IN HOVERING MOTION 13.2.1 Introduction In recent years, micro air vehicles (MAVs) have been attracting more attention for their potentials in civilian and military applications [32
34]. Several MAV concepts have been proposed, which includes fixed wings [35], rotary wings [36], and flapping wings [37]. Considering the size of MAVs and their Reynolds numbers, conventional air vehicles, and MAVs have remarkable differences in aerodynamic characteristics, such as lift and drag. Fixed wing concept encounters challenges in lift generation and flight control for the design of MAVs. Because of the common features shared by MAVs and insects, scientists study the aerodynamics of insect flight for clues of better design. The understanding of insect flight aerodynamics becomes important for the design of MAVs. Hovering flight is a kind of flight mode where the body is assumed to be fixed in space and the freestream velocity is zero [32,34,38]. It is the most energetically expensive form of flight [39] and exceptionally fine control is also needed to remain stationary [33]. The power produced by the flapping of insect wings should sustain the insect itself in the air. To design and build MAVs with the capability of hovering flight, the mechanism of aerodynamic lift force generation of hovering flight is important to consider and worthy of detailed investigation. Numerous researches were carried out to understand the high-lift mechanism of insect flight. The traditional quasi-steady aerodynamic theory does not explain how an insect could support its weight during flight [40]. Because of this discovery, the study of insect flight has been focused on its unsteady mechanisms. Until now, several unsteady mechanisms are found to help generate the high lift. Clap and fling were the earliest unsteady mechanisms found by Weis-Fogh [41]. After this study, Lehmann et al. [42], Sun and Yu [43], Liu and Aono [38] conducted experimental and numerical research on this mechanism, which could enhance lift generation at low Reynolds numbers. Another high-lift mechanism found is rapid pitch by Shyy [39], who discovered that vorticity around the wing increases due to rapid pitch, which resulted in the augmentation of lift generation. In addition, wake capture mechanism, which means wing
wake interaction, leads to additional aerodynamic lift peak, demonstrated by Lehmann et al. [42] and Birch and Dickinson [44], who found that wake capture is dependent on temporal changes in the distribution and magnitude of vorticity during stroke reversal. Shyy et al. [45] further clarified the relationship between wake capture and lift augmentation of the instantaneous lift peak using 2D numerical simulations. Another mechanism is the delayed stall of the leading-edge vortex (LEV), which is deemed as the major source of lift generation [38,46]. It is a common flow feature for flapping wing aerodynamics at Reynolds

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number of approximately 100. In addition to those mentioned above, tip vortex (TiV) is also an important mechanism. It affects the forces exerted on the wing. Shyy [45] concluded that TiVs could either promote or make little impact on the aerodynamics of a low-aspect-ratio flapping wing by varying the kinematic motions. In addition, induced jet has been observed experimentally by Freymuth [47] and numerically by Trizila et al. [48,49], which can decrease the lift of a wing. As we all know, insect wing’s flexible and passive pitching motion is an effective way to reduce energy consumption. Ishihara et al. [50] found that LEV attached to the flexible wing longer than on a rigid wing both with passive pitching motion. Furthermore, the aerodynamics is also affected by flapping kinematics, flapping frequency, and flight speed. Although some progress has been made in understanding the unsteady flight mechanisms, it is still a challenge to manipulate unsteady flow features to enhance lift performance. Until now, sizes of MAVs cannot come close to those of insects and MAVs cannot fly as swiftly as insects do. In addition, very few numerical studies have reported on the low Reynolds number unsteady phenomena, such as wake capture, delayed stall of LEV in details relating their role in clap and fling motion, which are important physical mechanisms for successful flight of insects and hence provides useful design guidelines for MAVs. The difference of induced jet between single wing and wing pair (clap and fling) hovering motion has never been studied. In the past two decades, researchers have conducted many numerical research on 2D and 3D flapping wing motion to solve the insect flight problems. 2D simulation is considered more fundamental to understand the high-lift mechanisms, although 2D flow patterns are less realistic for insect flight. The reasons can be clarified as follows. The 2D elliptical wing simulation [51
53] is found to approximate experimental insect flight reasonably well and the 2D elliptical flapping wing is suitable for the study of insect flight at low Reynolds number [53]. Similarly, it might be sufficient to explain high-lift mechanisms of the hovering motion. In spite of these important works, there are still exit puzzling and unsatisfactory explanations about the high-lift mechanisms of the hovering motion even for 2D flow. For those above-mentioned reasons, 2D flapping wing motion is selected for the study to help explain the high-lift mechanisms of the hovering motion. In this section, aerodynamic forces of foils in hovering motions are numerically investigated. The lift force generation mechanisms and energy consumptions for rigid foils in hovering motions are studied.

13.2.2 Grid Dependency Study Air flow over a 2D elliptical wing at low Reynolds number has been simulated [53]. The 2D wing model is simplified from a 3D wing. Birch & Dickinson [44] found that the strongest vortex appears at a spanwise location

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of 0.65 R from the wing root (R is the wing span). Wang et al. [51] derived their 2D model with this distance and that model is used in the following simulation cases. The shape equation of 2D ellipse wing geometry is given as follows: x2 y2 1 2 51 2 a b

ð13:2Þ

where a and b correspond to the major and minor axis of the ellipse. The elliptical major axis, also the chord of wing model is set to 1, which is considered as the characteristic length. In the present study, the motion function in Wang [52] is adopted. The translation and rotation motions are controlled by xðtÞand αðtÞ, respectively. The function of translation motion and maximum velocity are shown below: xðtÞ 5

A0 ½cosð2πf0 tÞ 2 1 2

Umax 5 πf0 A0 The rotation function is expressed as: π αðtÞ 5 1 αsinð2πf0 t 1 ϕÞ 2

ð13:3Þ ð13:4Þ

ð13:5Þ

where A0 is translational amplitude, f0 is flapping frequency, Umax is maximum velocity, α is angular amplitude, and ϕ is the phase difference between translation and rotation motions. The corresponding Reynolds number can be derived based on velocity expression derived from Eq. (13.2): Re 5

Umax c πf0 A0 c 5 υ υ

ð13:6Þ

where υ is kinematic viscosity and c is elliptical chord. According to Dickinson et al. [39], the flow with a Re value ranging from 100 to 1000 is classified as laminar flow. In the following simulations, all Re values are less than 100. Hence, the flow could be considered as laminar flow. The track of the wing motion is shown in Fig. 13.32.

FIGURE 13.32 The trajectory of wing hovering motion.

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The outside boundary is chosen as circular shape and its radius is set to 30 times the chord length, while the ratio of major axis to minor one is set to 10:1. Mesh convergence study is conducted in advance for hovering motion. The constant parameters are as follows: A0 =c 5 3:185, f0 5 0:1 Hz, ϕ 5 0, Re 5 100, α 5 30 degrees, ϕ 5 45 degrees. Five-grid systems are tested here for mesh convergence analysis. We mainly change the mesh resolution around the wing boundary, from grid-size number 1:8 (one side on the wing boundary) 3 64 (on the outer boundary); grid-size number 2:16 3 64; gridsize number 3:32 3 64; grid-size number 4:64 3 64 to grid-size number 5:96 3 64. The mesh information used in mesh convergence has been summarized in Table 13.8. The pressure values of three nodes on wing boundary are collected and shown in Fig. 13.33 for mesh convergence analysis.

TABLE 13.8 The Grid Information for the Grid Dependence Analysis Grid System

Total Node Number

Total Element Number

1

615

1182

2

3017

5906

3

4020

7848

4

5753

11,186

5

7306

14,164

FIGURE 13.33 Grid dependence analysis.

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Fig. 13.33 shows that pressure value would remain unchanged for mesh 3, 4, and 5. In consideration of the computation cost, mesh 3 is chosen for the following simulation.

13.2.3 Lift Force Generation Mechanisms for Rigid Wings in Hovering Motion 13.2.3.1 Single Rigid Wing in Hovering Motion According to the phase difference in the models, there are three kinds of hovering motion styles for single rigid wing, advanced rotation style with positive phase angle, symmetrical rotation style with zero-phase angle, and delayed rotation style with negative phase angle, respectively. The constant parameters are as follows: A0 =c 5 3:185, f0 5 0:1 Hz, ϕ 5 0, Re 5 100, α 5 30 degrees . For advanced rotation, ϕϕ 5 45 degrees; for symmetrical rotation, ϕ 5 0 degrees; for delayed rotation, ϕ 5 2 45 degrees. The phase difference between translation and rotation is studied in this section. The lift and drag coefficients of three mentioned rotation styles are plotted in Fig. 13.34. Fig. 13.34 shows four peak Cl values in advanced and symmetrical rotation. However, there exists phase differences for peak values between advanced and symmetrical cases. At the same time, the delayed rotation case produces only three peaks in Cl values, which indicates that the lift generation mechanism may be different from the other two. The vortex and pressure contours are shown in Fig. 13.35 for the study of lift generation mechanism in symmetrical rotation hovering motion. The range of the dimensionless vorticity, vertical velocity, and pressure are set as [ 2 5, 5], [ 2 1, 1], and [ 2 1, 1], respectively. At t/T 5 0.1, the clockwise (A)

(B)

3

1.5

2.5

ϕ 45 ϕ –45 ϕ0

2 1.5

ϕ 45 ϕ –45 ϕ0

1

1 CI

Cd

0.5 0

0.5

–0.5 –1 –1.5

0

–2 –2.5 –3

0

0.2

0.4

0.6 t/T

0.8

1

–0.5

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.34 The distributions of lift and drag coefficients for advanced, delayed, and symmetrical rotation in hovering motion.

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FIGURE 13.35 Vorticity, pressure, and vertical velocity contours for symmetrical rotation hovering motion. Red: counter-clockwise for vortex, positive for pressure, and vertical velocity; blue: clockwise for vortex, negative for pressure, and negative velocity.

vortex (CWV) and counter-clockwise vortex (CCWV) referred to in Nakata et al. [54] appear clearly. At this time instance, the new LEV and trailingedge vortex (TEV) interact with old LEV and TEV. Because the directions of new LEV and TEV rotation are opposite to those of old ones’, it is believed that the lift force is mainly generated due to vortex
vortex interaction. There is a peak Cl value, about 0.55, at t/T 5 0.1, which confirms the theory that additional aerodynamic force peak is generated due to wake capture. As time increases to t/T 5 0.2, there exists a minimum Cl value, about 0.23 shown in Fig. 13.34. Induced jet is considered as the cause of the minimum in Cl. In order to illustrate the phenomena clearly, the vertical velocity contour is plotted in Fig. 13.35B. The wing encounters persistent jet, which accelerates the flow along its underside. The increase of velocity along the wing’s underside will decrease pressure; hence, a minimum Cl value is generated. At t/T 5 0.3, the new lift force generated is believed to be due to the

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delayed stall of a new LEV. The attached LEV creating a region of negative pressure results in the second lift peak. The second Cl peak, about 0.72, is larger than the first Cl peak, 0.55, which indicates that the delayed stall of LEV provides even larger Cl than that obtained from wake capture mechanics. At t/T 5 0.46, the second minimum lift appeared, which is below zero, about 20.08. That is believed to be caused by rapid pitch mechanism. The wing is rotating in a clockwise direction at a large angular velocity (biggest at t/T 5 0.5 according to the motion equation). The angular velocity results in the increase of the flow velocity in the left side of the wing near the trailing edge and decrease of the flow velocity on the right side of the wing near the leading edge. The pressure decreases on the left side of the wing near the trailing edge, but increases in the right side of the wing near the leading edge. So another minimal lift is generated. At t/T 5 0.5, the wing arrives in the leftmost position and at the start of rightward motion. Compared with results of the upper half cycle, almost asymmetrical Cd and similar Cl profiles in the next half cycle are obtained. The above results are reasonable, considering that the trajectory of the wing motion in the next half cycle is asymmetrical to that in the first half cycle. The drag force exerting on the wing is opposite to that in the first half cycle. At the same time, a similar Cl profile in the next half cycle shows that the wing experiences wake capture and delayed stall again. During one hovering motion cycle, positive lift force is provided all the time, which could be used for supporting the body weight. The pressure and vorticity contours for advanced rotation at different time instants are shown in Fig. 13.36 to illustrate the lift generation mechanism. At t/T 5 0, the lift peak results from rapid pitch mechanism. The wing flips before the stroke end in advanced rotation style. The flow is still keeping in rightward motion because of inertia force. When the wing rotates counter-clockwise, the fluid on the right side of the wing will accelerate and the fluid on the left will decelerate. The lift force will decrease so that a minimal lift is generated. At t/T 5 0.16, there appears a minimum lift force, about 20.06. At this moment, a CCWV generated in the last stroke begins to come into the TEV generated in the present stroke. This may be the reason for the generation of the minimum lift. That is to say, when the CCWV generated in the last stroke come into contact with the LEV generated in the present stroke, lift will be enhanced; but when it comes into contact with the TEV, lift will decrease. At t/T 5 0.28, the delayed stall of LEV can explain why there is a peak in lift at this instant. At t/T 5 0.39, the major axis of the wing is almost vertical to the x-axis so another minimum lift is generated. The above explanation is also suitable for the lift generation in the next half cycle. The pressure, vertical, and vorticity contours in delayed rotation style are shown in Fig. 13.37 to study the lift generation mechanism. At t/T 5 0.08, the minimum lift results from CCWV generated in the last stroke coming into contact with the TEV. The vortex
vortex interaction near the trailing

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FIGURE 13.36 Vorticity and pressure contours for advanced rotation.

edge generates a negative pressure area. At t/T 5 0.28, the delayed stall of LEV leads to the peak lift. At t/T 5 0.5, the minimum lift is caused by the clockwise rotation of the wing, which is similar to that for the symmetrical rotation. At t/T 5 0.68, wake capture causes the second peak in lift. At t/ T 5 0.78, the induced jet results in the minimum lift. At t/T 5 0.88, the delayed stall of LEV causes the peak lift. At t/T 5 0.96, the rapid pitch mechanism gives rise to the minimum lift again. According to Dickinson et al. [39], in delayed rotations, if a wing rotates back after the stroke reversal, then when the wing starts to accelerate it pitches down, resulting in reduced lift.

13.2.3.2 Two Rigid Wings in Hovering Motion In the cited works, only Gillebaart [55] used ALE method to simulate clap and fling mechanism, but the mesh topology in this method changed. To our knowledge, very few articles study the complete cyclic clap and fling motion. Here, complete cyclic clap and fling motion finally studied using the present ALE method, with mesh topology unchanged. The kinematics of the

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FIGURE 13.37 Pressure, vorticity, and vertical velocity contours for delayed rotation. (A) Vorticity contours, t/T 5 0.08; (B) vorticity contours, t/T 5 0.28; (C) pressure contours, t/T 5 0.5; (D) vorticity contours, t/T 5 0.68; (E) vertical velocity contours, t/T 5 0.78; (F) vorticity contours, t/T 5 0.88; (G) pressure contours, t/T 5 0.96.

left wing are described as follows: A0 =c 5 3:185, f0 5 0:1 Hz, ϕ 5 0, Re 5 100, α 5 453 . The right wing is the mirror image of the left wing at all time during their motions. The translational and rotational equations of the left wing are the same as Eqs. (13.3) and (13.5). Thick ellipse of 1/10c is used for wings in the present study. The motions of the wings are sketched in Fig. 13.38. Initially, the distance between these two wings is 1/2c, with their chord lines in parallel along the vertical direction and the wings start to move away from each other in the fling stroke first, which is then followed by the clap stroke. The drag and lift coefficients are shown in Fig. 13.39. The lift coefficient has three peak values, which is different from that of a single wing. And

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FIGURE 13.38 The trajectories of wings during clap and fling motions.

(A)

(B)

1.5

1.5

Wing pair Single wing

Wing pair Single wing

1 1

0.5

0

CI

Cd

0.5

–0.5 0 –1 –1.5

0

0.2

0.4

t/T

0.6

0.8

1

–0.5

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.39 The distributions of drag and lift coefficients in clap and fling motion.

there is no minimal lift between two lift peaks in one-half stroke. The vorticity contours at different time instants are shown in Fig. 13.40. At t/T 5 0.1, the first peak corresponds to the wake capture. But the wake capture is weakened because of the clap phase in the last stroke. At t/T 5 0.3, the second peak corresponds to the delayed stall of the LEV. But the second peak is larger than that of a single wing. At this instance, the wings encounter the TEVs generated in the last stroke, which have opposite signs and are not dissipated yet. At t/T 5 0.46, the minimum lift is due to the rapid pitch mechanism, which is similar to that of single wing hovering motion. In the clap stroke, the wings start to approach. At t/T 5 0.6, the wake capture and following delayed stall of LEV result in the third Cl peak. The induced jet phenomenon does not take place because of the impact of the two wings. But the wings meet no TEV generated in the last stroke anymore, so the peak in lift at t/T 5 0.8 is smaller than that at t/T 5 0.3. At t/T 5 0.96, the minimum lift is caused by the induced jet (Fig. 13.40E), which is located at the middle of the two wings. This is different from that for single wing hovering motion. Fig. 13.39 shows the amplitude of the Cl due to delayed stall mechanism in clap and fling motion is larger than that of the single wing in hovering motion while the situation reverses for the rest. Here, a quantitative statistics on the force and energy expenditure is done to further clarify this

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(A) t/T = 0.1

(B) t/T = 0.3

(C) t/T = 0.6

(D) t/T = 0.8

(E) t/T = 0.96 FIGURE 13.40 Vorticity contours for clap and fling at different time instants.

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TABLE 13.9 The Comparisons of Cl and Energy Expenditure Between Single Wing Hovering Motion and Clap and Fling Motion Hovering Case

Cl

P

Cl/P

Single wing

0.31

0.96

0.32

Clap and fling

0.28

0.33

0.83

appearance. The period-averaged results are calculated and summarized in Table 13.9. The results show that both period-averaged Cl and energy expenditure produced in clap and fling motion are smaller than those generated in single wing motion although there exists one larger peak Cl in clap and fling motion. However, the ratio of Cl to energy expenditure increases by 159%, which means that clap and fling motion would generate more lift supposed consuming the same energy used in single wing hovering motion.

13.3 AERODYNAMIC FORCES OF DEFORMABLE FOILS IN HOVERING MOTION 13.3.1 Introduction In the last decade, attention has been increasingly paid to the aerodynamics of deformable wings [56
65] due to the rising popularity of MAVs. On the one hand, based on solid research work on rigid wings [38,43,44,46,66
69], lift force generation mechanisms have been identified and studied continually, such as clapping and flying [38,43,67,68], dynamic stall (delayed stall) [38], rapid pitch [46], wake capture [44,45,69] and TiV [45], as well as induced jet [47
49]. Researchers have gradually obtained systematic and deep insights into unsteady aerodynamic forces on rigid wings. With those insights obtained, further research on flexible wings found in nature and their lift generation mechanisms is expected. On the other hand, with the development of high-speed and high-pixel camera and object-tracing technologies, researchers are now able to investigate microstructures and morphing of insect wings [70
78]. However, the structures and materials of flexible wings are found to be so complex that so far one could not yet find simple and accurate expressions to represent major wing features by measurement data or regression fit values, such as wing modulus of elasticity [70
73] and constitutive relationship [74
78]. It is the reason why there is very little research work on fully coupled two-way FSI found in flexible wing flapping motion. An alternative way to study the effect of flexible and morphing wings is the prescribed deformation method, which is made possible by point-tracing technologies used for free or fixed flying wings. These can be considered as

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semi-rigid wing models because wing deformation is prescribed with simplified harmonic equations during wing flapping based on experimental data. These models can be useful for MAV designers to obtain the best lift performance.

13.3.2 Descriptions of Deformation Wing Models in Hovering Motion The hovering motion of a 2D rigid wing can be described by three parameters, which are angle of attack [52,53,79,80], stroke amplitude [69,81,82], and wingbeat frequency [83
85], respectively. As for deformable wings, the hovering motion is considered as a combination of three components: translation, rotation, and deformation, respectively. Thus two extra parameters, the angle of incidence specified due to deformation and the camber [41,57
59,64,86
94], should be added and used to represent the timevarying twist and camber of the deformable wing. Research on deformable wings has revealed that the performance of a rigid flapping wing can be significantly improved by adding some level of flexibility to the wing surface [91]. The unsteady aerodynamics of a flapping wing is not only sensitive to variations in the wing kinematics but also the wing morphing [58,92]. The deformation of wing surface may potentially provide new aerodynamic force generating mechanisms [89,90,95]. In order to understand the effect of wing flexibility on the aerodynamics of flapping motion, three types of deformable foil models can be used for the purpose of the experiment and numerical investigation. They are full deformation foil models [59,96,97], partial deformation ones [98], and foils consisting of two hinge connected rigid plates [64,94,99
101], respectively.

13.3.2.1 Full Deformation Foil Model In this book, all three 2D foil models mentioned above are used to investigate the effect of foil deformation on lift force generation mechanisms. Except for the hinged plate model, the shape equations of full and partial deformation foil models are described as follows: x2 y2 1 2 51 2 a b

ð13:7Þ

where a and b correspond to the major and minor axes of ellipse. x and y mean horizontal and vertical coordinate, respectively. In this section, the ratio of long axis to short one is set to 30:1. Translation and rotation motions, as well as wingbeat frequency are set to be the same as our previous work for rigid wing investigation [85] for the purpose of comparison. The translation and rotation motions are controlled by xðtÞ and αðtÞ, and the expressions are given below:

ALE FSI Model Validations and Applications Chapter | 13

A0 ½cosð2πf0 tÞ 2 1 2 π αðtÞ 5 1 αsinð2πf0 t 1 ϕÞ 2 xðtÞ 5

455

ð13:8Þ ð13:9Þ

where A0 means the amplitude of translation, f0 means the frequency of translation and rotation, α means the amplitude of rotation, and ϕ means the phase difference between translation and rotation. The deformation of foils during hovering motion is mainly due to the camber deformation, which is a function of time and is called chord-wise deformation [41,99]. There is an extra angle of incidence of the foil here, which is also caused by the change of camber. As for full deformation foil, the deformation of camber is introduced to the central line of the foil firstly, then the whole foil deforms accordingly (see Fig. 13.41). In general, two quadratic functions are adopted to represent the camber shape of the central line of the foil for the full deformation foil model as follows: 8 0 1 > > > > yc 5 mðtÞ ð2px 2 x2 Þ @ 0 # x # pA > > 2 c2 > p c c < ð13:10Þ 0 1 > >   > yc mðtÞ x > > 5 ð1 2 2pÞ 1 2px 2 x2 @p , # 1A > > :c c ð12pÞ2 c2 where x means the abscissa of a point on the chord, yc is the deviation of the mean line, and m is the maximum of the deviation, which is called the maximum camber. Here, p is set to 0.5, and mðtÞ is described as follows: mðtÞ 5 m0 sinð2πf0 tÞ

ð13:11Þ

where m0 is the amplitude of the camber in time series. m0 is set from 0 (rigid case) to 0.2 3 chord in the simulations [57,59], which is plotted in Fig. 13.41. Y

m(t)

0

pc

FIGURE 13.41 The sketch of full deformation foil model.

c

X

456

Computational Fluid-Structure Interaction

FIGURE 13.42 The sketch of partial deformation foil model.

13.3.2.2 Partial Deformation Foil Model As for the partial deformation foil model, which is one-quarter rigid and the rest flexible, only one quadratic function is adopted to represent the camber shape of the mean line as follows:   yc mðtÞ  2 x 2 , 1 ð13:12Þ 5 ðx 2 2px 1 p p , c c ð12pÞ2 c2 where pc is the value of the camber position along the foil, and again mðtÞ is described as follows: mðtÞ 5 m0 sinð2πf0 tÞ

ð13:13Þ

where m0 is set from 0 (rigid) to 0.2 3 chord in the simulations, which is plotted in Fig. 13.42.

13.3.2.3 Hinge Connected Two-Rigid-Plate Model As for the hinge connected two-rigid-plate model, its two plates are moving separately and connected at the hinge point [64,92]. Fig. 13.43A shows the deformation sketch of the hinge connected rigid plate model in hovering motion, whose parameters are given as below: θL ðtÞ 5 β L sinð2πf0 tÞ

ð13:14Þ

θT ðtÞ 5 β T sinð2 2πf0 t 1 ϕÞ

ð13:15Þ

where θL is the orientation of the main plate relative to the vertical direction, f0 is flapping frequency and β L is the amplitude of rotation. The deflection angle θT of trailing-edge flap (TEF) is defined, based on the equation used by Li [94]. The deflection phase difference from main plate ϕ controls the TEF deflection timing, and thus forms a different camber patterns (see Fig. 13.43B).

ALE FSI Model Validations and Applications Chapter | 13 (A)

(B) Main plate

βL

θL

457

Main plate

TEF

0 –βL

HL TEF

2βT

θT ϕ 0

0.5

1

1.5

2

t/T

FIGURE 13.43 The sketch of hinge connected rigid plate model and its angle variations.

FIGURE 13.44 The trajectory of hovering motion for the full deformation model. (left: upstroke; right: downstroke).

13.3.3 Lift Force Generation Mechanisms for Deformable Wings in Hovering Motion 13.3.3.1 Results for Full Deformation Foil Model In this subsection, the differences of aerodynamic forces in hovering motion of the deformable and rigid foils are to be studied. The kinematic parameters of the movement of the hovering foils are set as:A0 =c 5 3:185, f0 5 0:1, ϕ 5 0, Re 5 100, and camber amplitude, m0 , varies from 0 to 0.2 3 chord with an interval of 0.05 3 chord for the deformable foils. The case of zero camber amplitude is equivalent to the rigid condition. The rotational center for the deformable foil model is located at the center of the foil. Based on the kinetic parameters mentioned above, the trajectory of hovering motion for the full deformation model is plotted in Fig. 13.44. Fig. 13.45 gives the time histories of lift force coefficient (Cl) and drag force coefficient (Cd) of the fully deformable foil in one hovering motion cycle for different camber amplitudes. One can observe clearly that no matter how much camber amplitude is used, there are four crests and troughs for the Cl and Cd curves. Moreover, the first and the third crests and troughs produce negative effects, while the second and fourth ones generate positive ones on

458

Computational Fluid-Structure Interaction

(A)

(B)

2

CAMBER = 0.00 CAMBER = 0.05 CAMBER = 0.10 CAMBER = 0.15 CAMBER = 0.20

1.5

1.5

CAMBER = 0.20 CAMBER = 0.05 CAMBER = 0.10 CAMBER = 0.15 CAMBER = 0.20

1

CD

CL

0.5 0.5

0

–0.5 –1 –1.5

–0.5 0

0.2

0.4

0.6

t/T

0.8

1

–2

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.45 The time histories of lift and drag coefficients for various camber amplitudes in the full deformation model. (A) The time histories of lift coefficients (Cl) and (B) the time histories of drag coefficients (Cd).

the Cl and Cd curves along with the increase of camber amplitude. The first and the third crests appearing in Fig. 13.45 are generated by the wake capture mechanism based on the understanding of the lift force generation mechanism for rigid wing in hovering motion, while the other two crests are generated by the delayed stall mechanism. At the same time, the first and the third troughs appearing in Fig. 13.45 are generated by induced jet mechanism, while the other two troughs are generated by rapid pitch mechanism. In the last section, the above-mentioned lift force generation mechanisms for rigid foils in hovering motion are explained in detailed. Here, in this section, the research focus is on how lift generation mechanisms are influenced by the deformation of foils. According to the results shown in Fig. 13.45, one could first observe that all the lift force generation mechanisms appearing in rigid foils, such as wake capture, delayed stall, induced jet, and rapid pitch, are still appearing in wing deformation cases. Second, with the influence of wing deformation, the occurrence of wake capture, induced jet, and rapid pitch mechanisms happen sooner, compared with those in the rigid wing cases. The larger the camber amplitude, the sooner are their occurrences. The only exception is the delayed stall mechanism. Thus the delayed stall would be delayed in the deformation case, compared with the rigid one. In rigid wings, the delayed stall mechanism contributes a larger part of the averaged lift force. Therefore it is believed that a longer delay in stall appearing in deformable wings should produce even larger lift force. Thus the effect of deformation should play a positive role during hovering motion of wings. In order to perform quantitative comparison and analysis, the differences of Cl and Cd between the two cases for every camber amplitude are summarized in Table 13.10 and plotted in Fig. 13.46. The differences in Cl between the two cases with different camber amplitudes show that the effect of deformation weakens the wake capture mechanism and promotes delayed stall mechanism. The tendency for Cd is just the

TABLE 13.10 Summary of Differences in Cd and Cl Between the Full Deformation and Rigid Cases Upstrokea

Camber Amplitude (1) Difference of Cl

Difference of Cd

(2)

(3)

Downstrokea (4)

(1)

(3)

(4)

0.05

2 0.18

0.06

0.19

0.07

2 0.14

0.02

0.14

0.07

0.1

2 0.26

0.06

0.33

0.11

2 0.20

0.032

0.26

0.10

0.15

2 0.33

0.07

0.42

0.13

2 0.25

2 0.01

0.33

0.11

0.2

2 0.38

0.03

0.57

0.14

2 0.30

2 0.05

0.41

0.14

0.27

2 0.06

2 0.14

2 0.13

2 0.23

0.02

0.09

0.15

0.05 0.1

0.39

2 0.09

2 0.25

2 0.18

2 0.30

0.07

0.19

0.22

0.15

0.46

2 0.17

2 0.33

2 0.18

2 0.36

0.10

0.25

0.23

0.2

0.50

2 0.24

2 0.47

2 0.18

2 0.40

0.18

0.35

0.27

Note: (1): Wake capture (t/T 5 0.1); (2): Induce Jet (t/T 5 0.2); (3): Delayed stall (t/T 5 0.3); (4): Rapid pitch (t/T 5 0.46).

a

(2)

460

Computational Fluid-Structure Interaction

(A)

(B)

1 m0 = 0.05 m0 = 0.10 m0 = 0.15 m0 = 0.20

Cd difference

Cl difference

0.5

0

–0.5

–1

1 m0 = 0.05 m0 = 0.10 m0 = 0.15 m0 = 0.20

0.5

0

–0.5

0

0.2

0.4

t/T

0.6

0.8

1

–1

0

0.2

0.4

t/T

0.6

0.8

1

FIGURE 13.46 The time histories of Cl and Cd differences between the full deformation and rigid wing models for various camber amplitudes. (A) The time histories of Cl difference and (B) the time histories of Cd difference.

opposite. As for camber amplitude, 0.2 3 chord, the difference in Cl reaches about 20.38 for wake capture and about 0.57 for delayed stall mechanism, respectively. The absolute Cl values could reach to 0.1 and 1.3 for wake capture and delayed stall mechanism, respectively. The results indicate that with deformation the foil would experience positive Cl during the whole stroke and even much larger during delayed stall than that of rigid condition. Therefore the effect of deformation is believed to improve Cl values. However, one has to consider Cd at the same time. Table 13.10 shows that Cd becomes small during wake capture and large during delayed stall with the effect of deformation, which means that the foil has to consume more energy during delayed stall in order to increase lift force. In the last part of this section, we try to explain how the deformation affects wake capture and delayed stall mechanisms one by one. As observed, the first crest of the lift force appears at about t/T 5 0.1, and wake capture mechanism plays a more important role at this moment. In this mechanism, the new LEV and TEV produced in a new stroke interact with the old LEV and TEV, and the foil receives its lift force from the vortex interaction. This also means that the moving foil obtains energy from its previous period. Figs. 13.45 and 13.46 show that the wake capture mechanism becomes weak under the influence of wing deformation. The reasons for the weakening could be explained according to the generation and development of vortex around foil due to deformation. An arc shape is formed in the deformable foil, which will decrease the local angle of attack (AOA) at the leading edge, which is beneficial to the transportation of vortices. And the higher the camber amplitude is, the smaller the local AOA (see Fig. 13.47) and the easier the fluid flows around the tip of the foil, which will eventually weaken the vortex interaction. Moreover, with the condition of higher camber amplitude,

ALE FSI Model Validations and Applications Chapter | 13

461

FIGURE 13.47 Vortex contours for the explanations of wake capture with deformation effect (t/T 5 0.1) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Camber amplitude m 5 0; (B) camber amplitude m 5 0.1; (C) camber amplitude m 5 0.2; (D) a close-up of vortex contours with camber amplitude m 5 0; (E) a close-up of vortex contours with camber amplitude m 5 0.1; (F) a close-up of vortex contours with camber amplitude m 5 0.2.

462

Computational Fluid-Structure Interaction

the new LEV and TEV become weaker and directly impair vortex interaction during the wake capture mechanism. The second crest of lift force appears at about t/T 5 0.3, the lift force generated is due to the delayed stall of a new LEV. When the deformation models are adopted, the delayed stall mechanism of a new LEV will play more important role in the lift generation. There are two possible reasons for that: one is that the larger the deformation, the more easily the vortex will be attached to the foil. A larger negative pressure domain is formed on the upper surface of foil (see Fig. 13.48), which results in a larger suction force exerting on the foil upper surface. The other reason is that the arc shape of the foil at the trailing-edge adds additional lift force to the foil. The larger deformation will produce greater additional force. Fig. 13.48 shows that the duration of delayed stall for higher camber amplitude case would be longer than that for rigid case. The time duration of delayed stall is about 0.29 period for the case of camber amplitude, 0.2 3 chord, which is 1.12 times that of the rigid case. As we know, the mechanism of delayed stall contributes a larger part of the lift force in rigid wings during the hovering stroke cycle. The longer delayed stall duration due to higher camber amplitude deformation will be more advantageous for the enhancement of lift force generation. Therefore foil deformation has a beneficial effect on the lift force at this moment.

13.3.3.2 Results of Partial Deformation Foil Model The partial deformation foil model, which was proposed by Miao [98] to explore high-lift force generation during insect flight, is investigated in this section by numerical simulations. The kinematic parameters of the hovering motion are set as:A0 =c 5 3:185, f0 5 0:1, ϕ 5 0, Re 5 100, and m0 varies from 0 to 0.2 3 chord with an interval of 0.05. The study is still focused on the effects of the deformation on lift generation mechanisms by comparing the differences between the results of the deformable foil and the rigid one. The trajectory of hovering motion for partial deformation foil model is plotted in Fig. 13.49. Fig. 13.50 shows the profiles of lift force coefficient (Cl) and drag force coefficient (Cd) of the partial deformation foil model in one hovering motion cycle for different camber amplitudes. By using the same method as in the full deformation foil model, the differences between the results with various camber amplitudes and those of the rigid wing are summarized in Table 13.11 and plotted in Fig. 13.51. Again one can observe clearly from Fig. 13.50 that there are four crests and troughs for Cl and Cd curves, no matter what camber amplitude is used. However, the differences of Cl and Cd indicate significantly different profiles from those of the full deformation model. Fig. 13.51 shows that deformation leads to negative Cl difference

ALE FSI Model Validations and Applications Chapter | 13

463

FIGURE 13.48 Vortex contours for the explanations of delayed stall with deformation effect (t/ T 5 0.3) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Camber amplitude m 5 0; (B) Camber amplitude m 5 0.1 (C) Camber amplitude m 5 0.2 (D) A closeup of vortex contours with camber amplitude m 5 0 (E) A close up of vortex contours with camber amplitude m 5 0.1 (F) A close up of vortex contours with camber amplitude m 5 0.2.

values during the hovering motion cycle. This means that the deformation of the wing results in smaller lift force in the partial deformation foil model. Some explanations are given below on why the negative Cl difference value is obtained due to deformation. The first crest of the lift force (due to

464

Computational Fluid-Structure Interaction

FIGURE 13.49 The trajectory of hovering motion for partial deformation foil model. (A) Upstroke and (B) downstroke.

(A)

(B) m0 = 0.00 m0 = 0.05 m0 = 0.10 m0 = 0.15 m0 = 0.20

1

m0 = 0.00 m0 = 0.05 m0 = 0.10 m0 = 0.15 m0 = 0.20

0.5

0.5

Cd

Cl

1

0 –0.5

0

–1 –0.5

0

0.2

0.4

0.6

t/T

0.8

1

–1.5

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.50 The time histories of lift and drag coefficients for various camber amplitudes in partial deformation foil model test. (A) The time histories of lift coefficients (Cl) and (B) the time histories of drag coefficients (Cd).

wake capture mechanism) appears at t/T 5 0.1. A diversion angle is formed at the trailing edge as the deformation is considered, and it enables air flowing over the foil surface smoothly (see Fig. 13.52). The old LEV produced in previous stroke is also induced to slant down along with foil surface. Because of the change of vortex position, the interaction of the old vortex and the new LEV will become weak. It weakens wake capture mechanism, then the lift force will decrease due to deformation at this moment. The second crest of lift force (due to delayed stall mechanism) appears at t/T 5 0.3, and the effect of deformation can be explained as follows: the reverse arch camber (compared with arch camber in full deformation model)

TABLE 13.11 The Summary of Differences Between the Results of the Deformable Wings and the Rigid Wing Upstrokea

Camber Amplitude

Difference of Cl

Difference of Cd

Downstrokea

(1)

(2)

(3)

(4)

0.05

2 0.04

2 0.10

2 0.05

0.00

0.1

2 0.07

2 0.16

2 0.13

0.15

2 0.11

2 0.26

0.2

2 0.13

0.05

2 0.01

(1)

(2)

(3)

(4)

0.02

2 0.08

2 0.07

0.00

0.01

2 0.04

2 0.21

2 0.13

0.00

2 0.17

0.01

2 0.09

2 0.31

2 0.18

0.00

2 0.34

2 0.26

2 0.00

2 0.15

2 0.39

2 0.23

0.00

2 0.09

0.08

2 0.05

2 0.01

0.10

0.08

0.06

0.1

2 0.14

2 0.20

0.19

2 0.14

2 0.14

0.15

0.19

0.06

0.15

2 0.22

2 0.26

0.25

2 0.19

2 0.22

0.21

0.25

0.11

0.2

2 0.30

2 0.30

0.36

2 0.21

2 0.29

0.26

0.36

0.14

Note: (1): wake capture (t/T 5 0.1); (2): induce Jet (t/T 5 0.2); (3): delayed stall (t/T 5 0.3); (4): rapid pitch (t/T 5 0.46).

a

466

Computational Fluid-Structure Interaction

(A) 1

1 m0 = 0.05 m0 = 0.10 m0 = 0.05 m0 = 0.05

0.8 0.6 Cd difference

0.5 Cl difference

(B)

m0 = 0.05 m0 = 0.10 m0 = 0.15 m0 = 0.20

0

–0.5

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

–1

0

0.2

0.4

0.6 t/T

0.8

1

–1

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.51 The time histories of Cl and Cd differences between the partial deformation and rigid wing models for various camber amplitude cases. (A) The time histories of Cl difference and (B) the time histories of Cd difference.

increases the difficulty of LEV adhering to the foil surface (see Fig. 13.53). With the deformation, the delayed stall LEV could not generate much lift force like that in the rigid model. Moreover, the change of AOA at the trailing-edge makes air flowing below the foil smoothly. Thus the flow only generates small lift force for the foil. Therefore deformation plays a negative effect on lift force generation at this moment. In order to prove further, the camber direction of the partial deformation foil model leads to negative lift forces, the reverse camber partial deformation foil model (the camber direction is the same as that in full deformation foil model) is implemented. The results of Cl and Cd have been plotted in Fig. 13.54. As expected, the effect of deformation during delayed stall mechanism provides positive lift force when camber direction is the same as that adopted in full deformation foil model.

13.3.3.3 Results of Hinge Connected Two-Rigid-Plate Model In this section, numerical examples are designed to investigate the hinge connected two-rigid-plate model, especially the effect of the trailing-edge flapping (TEF). The kinematic parameters of the hovering motion are set as: A0 =c 5 3:185, f0 5 0:1, ϕ 5 0, Re 5 100, and β L 5 45 degrees and β T are set to 15, 30, and 60 degrees. The results presented in this section is only for β T 5 30 degrees. The rest are simulated for power consumption work in the next section. The rotational center is at the tip of the foil. The trajectories of hovering motion for hinge connected two-rigid-plate model is plotted in Fig. 13.55. In the section, one will mainly focus on the influence of the trailing edge on the lift force generation mechanism, and the results will be discussed as follow.

ALE FSI Model Validations and Applications Chapter | 13

467

FIGURE 13.52 Vortex contours for the explanations of wake capture with deformation effect in the partial deformation foil model (t/T 5 0.1) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Camber amplitude m 5 0, (B) camber amplitude m 5 0.1, (C) camber amplitude m 5 0.2, (D) a closed up of vortex contours with camber amplitude m 5 0, (e) a closed up of vortex contours with camber amplitude m 5 0.1, and (F) a closed up of vortex contours with camber amplitude m 5 0.2.

Fig. 13.56 shows the time histories of Cl and Cd for different deflection phases in the hinge connected two-rigid-plate model. Four crests appear in the hinge connected two-rigid-plate model. However, the occurrence of the first crest is delayed in the case of ϕ 5 2 120 degrees, while the appearance

468

Computational Fluid-Structure Interaction

FIGURE 13.53 Vortex contours for the explanations of delayed stall with deformation effect in the partial deformation foil model (t/T 5 0.3) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Camber amplitude m 5 0, (B) camber amplitude m 5 0.1, (C) camber amplitude m 5 0.2, (D) a closed up of vortex contours with camber amplitude m 5 0, (E) a closed up of vortex contours with camber amplitude m 5 0.1, and (F) a closed up of vortex contours with camber amplitude m 5 0.2.

of the first crest in the case of ϕ 5 60 degrees is advanced. The possible reason is deflection phase angle of TEF affects transportation of old vortices. The vortex contours at t/T 5 0.1 for three different deflection phases are plotted in Fig. 13.57. When the direction of TEF is toward to the convecting

ALE FSI Model Validations and Applications Chapter | 13 (A) 1.5

(B) 2 m0 = 0.00 m0 = –0.10 m0 = –0.20

469

m0 = 0.00 m0 = –0.10 m0 = –0.20

1.5 1

1

Cd

Cl

0.5 0.5

0 –0.5 –1

0

–1.5 –0.5

0

0.2

0.4

0.6

t/T

0.8

1

–2

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.54 The time histories of lift and drag coefficients for various camber amplitudes in partial deformation foil model (but reverse camber direction) test. (A) The distribution of lift coefficients and (Cl) (B) the distribution of drag coefficients (Cd).

direction of the old vortices in front of the foil, the old vortices are blocked or delayed in advancement, which enhances wake capture mechanism, thus results in higher lift force for the foil (see Fig. 13.57A). If the direction of TEF is reversed, the old vortices are pass by the foil faster in vortex shedding, which leads to wake capture happening earlier (see Fig. 13.57B). The second crest of lift force appears at t/T 5 0.3. Vortex contours at that moment are plotted in Fig. 13.58 for analyzing the lift force generation mechanism. In the case of ϕ 5 60 degrees, it is observed that the LEV is adhering to the foil better, while the TEF blocks and delays the advancement of the old vortices. Thus larger lift force is generated in that case. As for the case of ϕ 5 2 120 degrees, the situation is just the opposite, and the lift force at that moment is even lower than that obtained from the rigid foil model.

13.3.4 Energy Consumptions for Different Models Energy consumption is one of the most important considerations in the aerodynamic design of MAVs. Good MAV design has to meet the demands of not only lift force required for cruising but also minimized energy consumption. In this section, energy consumption is studied for the three deformable foil models described above and the effect of deformation on energy consuming is investigated. The method for calculating energy consumption in the rigid model is to obtain the work, which supports the foil to complete the hovering action [102] by using the following formula: ð 1 T Cd 3 UðtÞ 3 dt ð13:16Þ P5 T 0

470

Computational Fluid-Structure Interaction

FIGURE 13.55 The trajectories of hovering motion for hinge connected two-rigid-plate model when β T 5 30 degrees. (A) Upstroke for hinge connected two-rigid-plate model when φ 5 60 degrees, (B) downstroke for hinge connected two-rigid-plate model when φ 5 60 degrees, (C) upstroke for hinge connected two-rigid-plate model when φ 5 2 120 degrees, (D) downstroke for hinge connected two-rigid-plate model when φ 5 2 120 degrees.

where U is the velocity component of the rigid foil in the moving direction, Cd is drag force coefficient, and T is the period of hovering motion. However, the above methodology needs to be modified as the effect of deformation is included in studying hovering motion. In this chapter, we follow the below formula to investigate the work required within T: ð I 1 T P5 ðUðTÞUnðtÞÞpðtÞdlðtÞdt ð13:17Þ T 0 l

ALE FSI Model Validations and Applications Chapter | 13 (A)

(B) 1.5

471

2.5 2

RIGID FAI = 60 FAI = –120

1

1.5

RIGID FAI = 60 FAI = –120

1

Cd

Cl

0.5 0.5

0 –0.5 –1

0

–1.5 –2

–0.5 0

0.2

0.4

0.6

t/T

0.8

1

–2.5

0

0.2

0.4

0.6

0.8

1

t/T

FIGURE 13.56 The time histories of lift and drag coefficients for various camber amplitudes in hinge connected two-rigid-plate model. (A) The distribution of lift coefficients (Cl) and (B) the distribution of drag coefficients (Cd).

where U is velocity vector at a moving boundary node, n is the normal vector of a moving boundary element surrounding the node, p is pressure at the node and l the boundary element’s length. By applying the formula, energy consumption for three kinds of deformation foil models is calculated and summarized in Tables 13.12, 13.13, and 13.14. In order to explore energy consumption in hovering motion for different deformation foil models, we plot all the results obtained by numerical simulations, which includes Cl, Cd, Cl/Cd, work, and ratio of work/Cl (work needed per unit Cl) in Figs. 13.59 and 13.60. For the purpose of comparison, Fig. 13.59 includes the results of the full and partial deformation foil models, while Fig. 13.60 includes TEF research results. Fig. 13.59 shows that Cl and Cd increase with camber amplitude in the full deformation foil model, while the tendency is totally reverse in the partial deformation one. Although work increases with camber amplitude, there exists an optimized value for the ratio of work/Cl in the full deformation foil model. The results indicate that larger camber amplitude will not necessarily lead to better performance for the full deformation foil model. Among the cases, we simulated here, it seems the optimized camber amplitude is 0.1 3 chord. Fig. 13.60 shows that TEF deflection phase strongly influences unsteady aerodynamic performances of the foil studied, especially for Cd. The simulation results of the hinge connected two-rigid-plate model confirm that TEF deflection directly affects the strengths of LEV and TEV, thus the entire vortex shedding process. Lift enhancement can reach up to 33.5% just by the TEF deflection in the hinge connected two-rigid-plate model. However, one has to note that drag force becomes higher with the increase of lift force. Although the TEF deflection, such as the case of ϕ 5 2 120 degrees, could reduce drag force to lower levels, lift force seems too small and cannot be improved easily.

472

Computational Fluid-Structure Interaction

FIGURE 13.57 Vortex contours for different TEF for hinge connected two-rigid-plate model (t/T 5 0.1) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Rigid, (B) φ 5 60 degrees, (C) φ 5 2 120 degrees, (D) a close up of vortex contours in (A), (E) a close up of vortex contours in (B), and (F) a close up of vortex contours in (C).

13.3.5 Remarks In this section, the effects of deformation on lift force generation mechanisms of wings in hovering flight are investigated. Three foil deformation models: full deformation model, partial deformation model, and hinged

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FIGURE 13.58 Vortex contours for different TEF for hinge connected two-rigid-plate model (t/T 5 0.3) (the blue and red colors indicate clockwise and counter-clockwise vortices, respectively). (A) Rigid, (B) φ 5 60 degrees, (C) φ 5 2 120 degrees, and (D) a close up of vortex contours in (A), (E) a close up of vortex contours in (B), and (F) a closed up of vortex contours in (C).

connected two-rigid-plate model are adopted in the research work. The prescribed deformation changes foil morphing during hovering motion in camber and angle of incidence. The effects of camber amplitude and rotation location on their aerodynamic performances and flow structures are studied in details. The numerical results obtained show that the deformation of foils

474

Computational Fluid-Structure Interaction

TABLE 13.12 Energy Consumptions for Full Deformation Foil Model Camber Amplitude

Cl

Cd

Cl/Cd

p

p/Cl

0.00

0.3528

0.5373

0.6566

0.4267

1.2095

0.05

0.3909

0.5526

0.7074

0.4367

1.1172

0.10

0.4225

0.5818

0.7262

0.4534

1.0731

0.15

0.4332

0.5993

0.7228

0.4780

1.1034

0.20

0.4535

0.6468

0.7011

0.5194

1.1453

TABLE 13.13 Energy Consumptions for Partial Deformation Foil Model Camber Amplitude

Cl

Cd

Cl/Cd

p

p/Cl

0.00

0.3577

0.5908

0.6055

0.4267

1.1929

0.05

0.3245

0.5484

0.5917

0.3902

1.2025

0.10

0.2806

0.4973

0.5642

0.3498

1.2466

0.15

0.2425

0.4597

0.5275

0.3290

1.3567

0.20

0.1967

0.4172

0.4715

0.2900

1.4743

TABLE 13.14 Energy Consumptions for Hinge Connected Two-Rigid-Plate Model φ (degree)

βT (degree)

Cl

Cd

Cl/Cd

p

p/Cl

60

15

0.3561

0.6827

0.5216

0.5049

1.4179

30

0.3634

0.7336

0.4954

0.5581

1.5358

60

0.3573

0.8053

0.4437

0.6320

1.7688

15

0.3020

0.5330

0.5666

0.3561

1.1791

30

0.3067

0.5526

0.5550

0.3616

1.1790

60

0.2373

0.4928

0.4815

0.3051

1.2857

2 120

indeed affects their unsteady aerodynamic performances during their hovering motion. Foil morphing due to deformation makes LEV and TEV generation and development processes, as well as the lift force generation mechanisms different from those of the rigid foil model. For the full deformation foil model, the effect of deformation enhances its lift force during the

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FIGURE 13.59 Energy consumption profiles for full and partial deformation foil models. (A) Energy consumption profiles for full deformation foil model and (B) energy consumption profiles for partial deformation model.

FIGURE 13.60 Energy consumption profiles for hinge connected two-rigid-plate models. (A) φ 5 60 degrees and (B) φ 5 2 120 degrees.

wake capture and delayed stall. There is an optimized camber amplitude, which is 0.1 3 chord among those cases simulated. The results obtained from the partial deformation foil model show that Cl and Cd decrease with the increase of camber amplitude. The effect of deformation does not enhance lift force due to unfavorable camber, as we discussed in this chapter. But TEV is significantly changed by the local AOA due to the deformation of the foil. The results of the hinge connected two-rigid-plate model indicate that the TEF could have an important control of the LEV and TEV generation and development. TEF deflection directly affects the strength of the LEV and TEV, thus altering the entire vortex shedding process. As a result, its lift enhancement can reach up to 33.5% just by the TEF deflection alone.

476

Computational Fluid-Structure Interaction

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FURTHER READING B. Ghozlani, Z. Hafsia, K. Maalel, Numerical study of flow around an oscillating diamond prism and circular cylinder at low Keulegan-Carpenter number, J.Hydrodyn., Ser. B 24 (5) (2012) 767
775.

Index Note: Page numbers followed by ‘f’ and ‘t’ refer to figures and tables, respectively.

A Accuracy order, 355 357 Advanced rotation style, 446 Aerodynamic forces of deformable foils in hovering motion, 453 475 deformation effects on lift force generation mechanisms, 472 475 deformation wing models in hovering motion, 454 456 energy consumptions for different models, 469 471 lift force generation mechanisms for deformable wings, 457 469 of rigid foils in hovering motion, 442 453 grid dependency study, 443 446 lift force generation mechanisms for rigid wings, 446 453 Aeroelastic flutter, large-eddy simulation of, 395 406 Agglomeration-based MG algorithm, 96 ALE method. See Arbitrary Lagrangian Eulerian method (ALE method) ALE FSI model validations and applications aerodynamic forces of deformable foils in hovering motion, 453 475 of rigid foils in hovering motion, 442 453 validation of incompressible ALE solvers, 409 441 Angle of attack (AOA), 460 462 Aortic valve, 316 317, 316f Arbitrary Lagrangian Eulerian method (ALE method), 4, 6 7, 127, 409 discretization methods, 128 131 governing equations, 127 128 parallelization of ALE code with moving boundary, 131 134, 132f

Area-weighted contribution calculation, 79, 81 82 Artificial compressibility method, 36 37 parameter, 409 410, 421 Artificial timestep (Δτ), 36 Artificial waves mechanism, 36 Axial velocity component, 392, 393f

B Background mesh, 105 107, 172f, 194f Benchmark lid-driven cavity flow problem, 147 BHV. See Bioprosthetic heart valve (BHV) Bi-leaflet mechanical heart valve, serial computation of, 282 298 model properties of St. Jude bi-leaflet mechanical heart valve, 283 286 pressure contours, 288f, 294f and fluid shear-stress distributions, 291f, 298f results for closing phase, 292 296 for opening phase, 286 292 time-dependent inlet velocity profile for opening and closing phases, 285f velocity vectors, 290f, 296f Bioprosthetic heart valve (BHV), 311 314. See also St. Jude medical aortic bi-leaflet mechanical heart valve (SJM aortic bi-leaflet MHV) closing phase, 339 351 computational model, 339 340, 339f fluid pressure and stress distribution, 345f positions for velocity extraction of plane, 350f pressure contours, 343f results, 340 351

481

482

Index

Bioprosthetic heart valve (BHV) (Continued) valve orifice area, 352f velocity vector plots, 350f opening phase, 314 339 computational model, 314 315, 315f extracted positions of velocity profiles, 335f initial and boundary conditions, 317 319 material and geometrical properties of valve leaflet, 316 317, 316t mesh for bioprosthetic aortic valve leaflet, 317f pressure contours, 328f results, 319 339 valve orifice area, 335f velocity profiles extracted for, 337f, 351f velocity vector plots, 334f parallel computation, 311 353 Body force, 102 104 components, 53 54 Boundary conditions, 25 29, 28f, 50 far-field conditions, 26 28 for preconditioned systems, 28 29 solid wall/slip conditions, 28 Boundary node projection algorithm, 86, 86f Boundary simulations, 421 Bounding box, 110 111

C Camber amplitude, 460 462, 463f, 464f, 469f Camber deformation, 455 CarboMedics bi-leaflet heart valve, 262 263 Cardiac cycle, 311 314 Cartesian coordinate system, 40 41 Cartesian mesh, 67 68 Cat-eye flow state, 418f Cauchy’s equations, 4, 53 54 Cavitation, 231 235 CB scheme. See Characteristic-based scheme (CB scheme) CCWV. See Counter-clockwise vortex (CCWV) Cell-based method, 17 18, 56 57 Cell-centered approach, 37 Cell-centered approximation, 3 4 Cell-centered scheme, 26 Cell-vertex approach, 37 CFD. See Computational fluid dynamics (CFD) CFL. See Courant Friedrichs Lewy (CFL) Channel flow, 423 3D, 365 368 with collapsible walls, 420

Characteristic-based scheme (CB scheme), 44 Chimera mesh method, 7 Chorin’s method, 40 41 Circular cylinder, viscous unsteady flow past, 160 171, 162f Circular disk, rotating and translating, 178 181 streamlines and velocities contour plots, 182f Clap motion, 451 453, 453t Clap stroke, 449 450 Clockwise vortex (CWV), 446 448 Coarse cell, 71 72 Coarser grids, 63 64, 66, 305f Collapsible channel, 441 Collapsible walls, 420 Communication time, 98, 415, 423 427 Computational structural dynamics solver, validation of, 370 382 deformation of point-loaded fixed-free cantilever structures, 370 376 3D fixed-free cantilever immersed in fluid flow, 376 382, 377f Computation time, 98 Computational domain, 262 263, 306 307, 355 357, 360, 361f Computational fluid dynamics (CFD), 1 3, 63, 101, 146, 273, 409 Computational solid mechanics (CSM), 3 Computational structural dynamics (CSD), 3 4 mathematical formulation constitutive relationship for stress and strain, 54 convergence acceleration techniques, 60 62 discretization of displacement equations, 56 60 displacement formulation, 55 governing equations, 53 54 Computational times (CPU times), 69, 97 98, 381, 382f Conservation equations, 16 17 Convection fluxes, 119 Convergence acceleration techniques, 24 25, 48 50 implicit residual smoothing, 49 50, 61 62 local time-stepping in pseudo time, 48 49, 60 61 multigrid method, 50, 62 Core mesh components, 89 90 Core node, 88 89 Correction transfer operators, 82 84 from coarse mesh to fine mesh, 83f from coarse nodes to fine node, 83f

Index Counter-clockwise vortex (CCWV), 446 448 Courant Friedrichs Lewy (CFL), 61, 319 condition, 48 49 CPU times. See Computational times (CPU times) CSD. See Computational structural dynamics (CSD) CSM. See Computational solid mechanics (CSM) CWV. See Clockwise vortex (CWV) Cyclic clap, 449 450 Cylinder oscillation, 434 435

D Damping, 53 54 exponential damping functions, 58 viscous, 58 Data structure for different multigrid level, 66 67 for exchange of ghost nodes variables and gradients, 93 95 communication between core and ghost nodes, 94t DD-MG approach. See Domain decomposition combined with multigrid approach (DD-MG approach) Deflection phase, 456 Deformable foil aerodynamic forces of deformable foils in hovering motion, 453 475 models, 469 Deformable wings, lift force generation mechanisms for, 457 469, 465t Deformation effects on lift force generation mechanisms, 472 475 of membrane, 423, 424f of point-loaded fixed-free cantilever structures, 370 376 wing models in hovering motion, 454 456 full deformation foil model, 454 455 hinge connected two-rigid-plate model, 456 partial deformation foil model, 456 Delayed rotation style, 446 Delayed stall mechanism, 457 458, 464 466 Direct forcing method, 101 102 Discrete solution, 377 381 Discretization methods, 16 24, 128 131 of displacement equations, 56 60 control volume within tetrahedron, 56f edge-based method

483

and cell-based method, 17 18 for term FCG, 129 matrix-free, implicit, dual-time-stepping algorithm, 22 24 moving mesh algorithm in ALE, 129 130, 130f Roe’s TVD scheme for inviscid flux in edge-based method, 18 20 smoothing of node displacement, 131f spring analogy method, 130 131 time integration, 24 upwind-biased interpolation, 20 22 Discretized dynamic equation, 84 Disk disk-enclosure gap effects, 394 storage system, 384 thickness effects, 394 Displacement equations, discretization of, 56 60 control volume within tetrahedron, 56f Displacement formulation, 55 Distance function, 129 130 Domain decomposition, 87 95, 88f data structure for exchange of ghost nodes variables and gradients, 93 95 grid files with local numbering, 92 93 identification of ghost nodes and overlapping elements, 89 92 METIS, 88 89 Domain decomposition combined with multigrid approach (DD-MG approach), 96 Dot product, 70 71, 71f, 110 111 Drag coefficients, 450 451 Drag force coefficient, 457 458, 469 471 Driven-cavity problem, 150 152 Dual-time-stepping, 145 146 algorithm, 22 24, 44 47 scheme, 58 Dynamic equilibrium equation, 53 54

E Eberle approach, 42 Edge-based method, 17 18 Roe’s TVD scheme for inviscid flux, 18 20 for term FCG, 129 Ekman layer, 386, 388f Elastic forces, 5 strain, 54

484

Index

Elasticity equation theory, 53 54 Elliptic equation, 36 37 Elliptical chord, 444 Energy consumption for different models, 469 471 for full deformation foil model, 474t for hinge connected two-rigid-plate model, 474t in hovering motion, 471 for partial deformation foil model, 474t profiles for full and partial deformation foil models, 475f for hinge connected two-rigid-plate models, 475f Energy equation, 11, 33 34 Error analysis, 304 305 Euler equation, 40 41, 63 Eulerian background mesh, 7 Eulerian mesh, 105 106 nodes, 101 Eulerian method, 4 6 IMM, 5 6 IOM, 5 Eulerian solution, 101 Exponential damping functions, 130 Eye-catching phenomenon, 427 428

F Far-field conditions, 26 28 for preconditioned systems, 28 29 FDM. See Fictitious domain method (FDM); Finite difference method (FDM) FEM. See Finite element method (FEM) Fictitious domain method (FDM), 4, 102 “Field extension” task, 125 126 Fine-grid partitions, 96, 97f residuals, 65 Finite difference method (FDM), 3 Finite element method (FEM), 3 Finite volume method (FVM), 3 Finite-volume discretization, 56 Finite-volume formulation, 37 39, 38f Finite-volume numerical discretization, 37 38 Five-stage Runge Kutta scheme, 190 Five-stage Runge Kutta time-integration algorithm, 102 103 Fixed cylindrical enclosure steady and unsteady air flows between two rigid disks corotating in, 383 395, 384f

three-dimensional steady flow, 386 388 three-dimensional unsteady flow, 389 395 Flapping frequency, 443 kinematics, 443 Flexible membrane, 280 Flight speed, 443 Fling motion, 449 453, 451f, 453t stroke, 449 450 Flow computations, 435 436 domain, 421 equations, 26 field variables for OG, 106 over immersed fixed membrane, 365 369, 368f induced by oscillating sphere in closed cavity, 357 359 without moving grid, 421, 422f, 423f switching, 438 441 phenomenons, 438 441 in two-dimensional collapsible channel, 420 430 Flow field domain zoning, 67 76, 68f area of zones smaller than largest element, 69f correlation between number of zones and CPU time, 70f interconnectiveity relationship between meshes, 70 76, 78f negative mapping produced by algorithm, 68f Flow over two-dimensional oscillating circular cylinder, 430 441 flow regimes, 431f model validations, 431 441 comparison of velocity x-component, 434f isolines of pressure and vorticity, 433f mesh grids at different phase positions, 432f velocity vector fields and streamline, 435f viscous flow around oscillating cylinder, 432f vorticity isolines, 436f, 440f Flow-field variables transfer operators, 78 80 transfer of flow-field values from fine mesh to coarse mesh, 79f

Index transfer of variables from fine nodes to coarse node, 79f Fluid domain, 306 307 flows, 420 421 governing equations for, 11 14 3D fixed-free cantilever, 376 382, 377f forces, 122 123 interactions between fluid and structure meshes, 113 118 nondimensional equations of fluid motion, 13 14 pressure, 320, 324f stress, 320, 324f tensor, 123 124 Fluid structure interaction (FSI), 1, 5 6, 113, 187 191, 286 290, 420 421 algorithm, 183 fluid domain, 187 structure domain, 187 191 discretization of structure equation, 189 191 hemodynamics forces and moments, 188 189 Fluid structure interface. See Fluid structure interaction (FSI) Force calculation on wall boundary, 131 132 Forcing function on coarse grid, 65 FORGEN grid-generation program, 299 300 Fortran 90 Source Codes communication data creation, 138 example communication module, 139 ghost nodes creation, 136 local numbering, 137 overlapping element creation, 135 parallel boundary movement smoothing, 140 143 Freshly cleared cells, 124 125 FSI. See Fluid structure interaction (FSI) Full deformation foil model, 454 455, 455f, 457 462, 458f, 459t, 460f FVM. See Finite volume method (FVM)

G Gauge pressure, 13 Ghost cell method (GCM), 4 Ghost fluid method (GFM), 4 Ghost fluid nodes, 109 113 Ghost nodes, 88 89 data structure for exchange of ghost nodes variables and gradients, 93 95, 94t

485

identification, 89 92 cavity mesh decomposing into partitions, 92f mesh decomposing into four partitions, 91f Governing equations, 33 36, 53 54, 127 128 Gradients, 39, 57 data structure for exchange of ghost nodes variables and, 93 95 Green’s theorem, 39, 57 Grid convergence study, 273 275, 357 for parallel solver, 298 305 3D lid-driven cavity steady flow, 146 147 viscous unsteady flow past circular cylinder, 161, 162f Grid dependency study, 410 411, 443 446, 445f grid information for grid dependence analysis, 445t history of normalized residual vs. number of time-steps, 411f trajectory of wing hovering motion, 444f velocity v along x, 412f Grid files with local numbering, 92 93 Grid-generation programs, 299 300, 300f

H Harmonic oscillation, 431 circular cylinders motion, 430 Higher-order MUSCL like interpolation, 120 121, 121f Hinge connected two-rigid-plate model, 456, 457f, 466 469, 470f, 471f Hooke’s law, 54 Hovering flight, 442 Hovering motion aerodynamic forces of deformable foils in, 453 475 deformation wing models in, 454 456 lift force generation mechanisms for deformable wings in, 457 469 aerodynamic forces of rigid foils in, 442 453 grid dependency study, 443 446 lift force generation mechanisms for rigid wings in, 446 453 Hybrid method, 7 Hyperbolic equations, 25, 34 35, 48 49

I IBM. See Immersed boundary method (IBM) icore function, 93

486

Index

ighst function, 93 IMM. See Immersed membrane method (IMM) IMM FSI model validations and applications for compressible flows large-eddy simulation of aeroelastic flutter, 395 406 rate of convergence γ calculated for error norms, 357t steady and unsteady air flows between two rigid disks corotating, 383 395, 384f validation of baseline compressible flow solver with IMM, 355 369 of computational structural dynamics solver, 370 382 Immersed boundary method (IBM), 4 Immersed membrane method (IMM), 4 6, 109, 273, 360, 369f based fluid-structure interaction, 121 126 FSI model validations and applications for compressible flows large-eddy simulation of aeroelastic flutter, 395 406 rate of convergence γ calculated for error norms, 357t steady and unsteady air flows between two rigid disks corotating in fixed cylindrical enclosure, 383 395, 384f validation of baseline compressible flow solver, 355 369 validation of computational structural dynamics solver, 370 382 FSI model validations and applications for incompressible flows grid convergence study for parallel solver, 298 305 parallel computation of bioprosthetic heart valve, 311 353 serial computation of bi-leaflet mechanical heart valve, 282 298 two-dimensional immersed membrane computations, 273 282 validation of parallel solver, 306 311 FSI solution, 122f ghost fluid nodes, 109 113 ghost nodes and ghost values, 110f interactions between fluid and structure meshes, 113 118 one-sided MUSCL like extrapolation, 118 121 properties of vortices of flow field with, 370t

relation between structure and fluid domain, 123f treatment of fluid nodes, 111f two-dimensional example, 125f validation of baseline compressible flow solver with, 355 369 flow induced by oscillating sphere in closed cavity, 357 359 flow over immersed fixed membrane, 365 369, 368f steady and unsteady flows past circular cylinder, 360 365 steady flow induced by rotating sphere in quiescent air, 355 357 Immersed object method (IOM), 4 5, 172 173 computations with OGs, 102 108, 172 181 rotating and translating circular disk, 178 181 rotating square cylinder, 176 178 viscous unsteady flow past circular cylinder, 172 176 Implicit algorithm, 22 24, 44 47 Implicit dual-time-stepping scheme, 183 Implicit residual smoothing, 49 50, 61 62 Implicit schemes, 21 22, 44, 57 58 Incompressible ALE solver validation, 409 441 flow in two-dimensional collapsible channel, 420 430 flow over two-dimensional oscillating circular cylinder, 430 441 overlapping grid elements technique, 441 steady flow in two-sided, lid-driven cavity, 409 420 Incompressible flow solver, mathematical formulation for artificial compressibility method, 36 37 boundary conditions, 50 convergence acceleration techniques, 48 50 finite-volume formulation, 37 39 governing equations, 33 36 matrix-free, implicit, dual-time-stepping algorithm, 44 47 time integration, 47 48 upwind characteristics-based method, 40 44 upwind-biased interpolation, 39 40 Incompressible unsteady Navier Stokes governing equations, 127 128

Index Indexing array pointer, 66 67 Induced jet phenomenon, 450 451 Inflow angle, 310 311 Intercluster communication or computer overload, 420 Interconnectiveity relationship between meshes, 70 76, 78f Internal boundary, 368 369, 369f, 370t Inverse distance weighted method, 125 126 Inviscid flux, 38 Roe’s TVD scheme for, 18 20 IOM. See Immersed object method (IOM) IOM FSI model validations and applications. See also ALE FSI model validations and applications computation of St. Jude medical aortic bileaflet mechanical heart valve, 181 269 computations with IOM with OGs, 172 181 serial/parallel single grid and multigrid computations, 145 171 3D lid-driven cavity steady flow, 145 159 viscous unsteady flow past circular cylinder, 160 171

J Jacobi iterative method, 24 25, 49 50, 61 62

K KC number. See Keulegan Carpenter number (KC number) Keulegan Carpenter number (KC number), 430 Kinematic parameters, 457 Kinematic viscosity, 444 Kinetic viscosity of fluid, 430 Krafczyk’s simulation geometry of valve used in, 268f time-dependent inlet velocity in, 268f

L Lagrangian formulation, 6 7 Lagrangian mesh, 7, 105 106 Lagrangian phase, 6 7 Large eddy simulation turbulence model (LES turbulence model), 186 187, 318 319

487

of ONERA M6 wing flutter, 402 406 of steady aerodynamic computation of M6 wing, 397 401 Lattice-Boltzmann method, 262 263 Leading-edge vortex (LEV), 442 443 Left ventricle (LV), 292 296 LES turbulence model. See Large eddy simulation turbulence model (LES turbulence model) LEV. See Leading-edge vortex (LEV) Lid-driven cavity, 409 420 flow problem, 145, 153 154 Lift coefficients, 450 451, 451f Lift force coefficient, 457, 462 463 generation mechanisms for deformable wings, 457 469 results for full deformation foil model, 457 462 results of hinge connected two-rigid-plate model, 466 469 results of partial deformation foil model, 462 466 generation mechanisms for rigid wings in hovering motion, 446 453 single rigid wing in hovering motion, 446 449 two rigid wings in hovering motion, 449 453 Lift generation mechanism, 442 443, 446 448, 453, 462 Linear strain displacement formulation, 55 Linear viscous damping, 53 54 Local time-stepping in pseudo time, 48 49, 60 61 Loose-coupled methods, 191 Low Reynolds numbers, 412 Low-dispersion numerical algorithms, 25 Low-speed flows, 13 preconditioning formulation, 14 16 LV. See Left ventricle (LV)

M M6 wing flutter computation, 402 LES of steady aerodynamic computation, 397 401 Mach number distributions, 405, 406f Master processor, 133 Mathematical formulation(s), 65 66

488

Index

Mathematical formulation(s) (Continued) for computational structural dynamics, 53 54 constitutive relationship for stress and strain, 54 convergence acceleration techniques, 60 62 discretization of displacement equations, 56 60 displacement formulation, 55 governing equations, 53 54 for incompressible flow solver, 33 36 artificial compressibility method, 36 37 boundary conditions, 50 convergence acceleration techniques, 48 50 finite-volume formulation, 37 39 governing equations, 33 36 matrix-free, implicit, dual-time-stepping algorithm, 44 47 time integration, 47 48 upwind characteristics-based method, 40 44 upwind-biased interpolation, 39 40 Matrix-free algorithm, 22 24, 44 47 computation, 59 60 implicit dual-time-stepping scheme, 191 implicit scheme, 44 MAVs. See Micro air vehicles (MAVs) Mechanical boundary conditions, 3 4 Mechanical heart valve (MHV), 5, 282 283 Mesh(es) convergence study, 445 446 density, 371 372 interconnectiveity relationship between, 70 76 different sign computed, 74f fine node within coarse boundary edge, 73f node belonging to neighboring edge, 75f node does not fall within 2D triangle cell, 72f node falls within 2D triangle cell using dot product, 71f projected lengths for transfer operator algorithms, 76f mesh-partitioning technique, 87 mesh-to-mesh transfer operators, 77 84 correction transfer operators, 82 84 flow-field variables transfer operators, 78 80 residual transfer operators, 80 82 movement, 128 129 MeshFile, 88 89

MeshFile.epart.Nparts, 88 89 MeshFile.npart.Nparts, 88 89 Message passing, 94 Message-passing interface (MPI), 87, 88f METIS, 88 89 MG. See Multigrid (MG) MG-DD approach. See Multigrid domain decomposition approach (MG-DD approach) MHV. See Mechanical heart valve (MHV) Micro air vehicles (MAVs), 442 Momentum equations, 13 Moving boundary, 131 134 discontinuity of geometry during smoothing process, 133f force calculation on wall boundary, 131 132 information stored in master process, 134t partition boundaries movement, 132 134 problems, 4 7 ALE method, 6 7 Eulerian method, 4 6 hybrid method, 7 Moving flexible membrane, 280 282 Moving grid, 134, 422 423 Moving mesh algorithm, 431 433 in ALE, 129 130, 130f Moving object velocity, 5 Moving rigid membrane, 279 280 MPI. See Message-passing interface (MPI) mpi_bsend() routine, 94 95 mpi_comm_world, 95 mpi_recv() routine, 94 95 Multigrid (MG), 145, 150 152, 360 basic concepts, 62 64, 64f codes, 87 data structure for different multigrid level, 66 67 mathematical formulations, 65 66 mesh-to-mesh transfer operators, 77 84 method, 25, 50, 63 parallel single grid and MG computations, 153 159 parallelization, 96 97, 97f serial single grid and MG computations, 147 153 for structural dynamic solver, 84 86 zoning of flow field domain, 67 76, 68f Multigrid domain decomposition approach (MG-DD approach), 96 Navier Stokes equations, 1, 11 12, 15, 21 22, 33 34, 36, 102 103, 129, 145, 183, 189, 191, 355

Index

N Newton’s second law of motion, 187 188 Newtonian fluid, 33 34, 185 186 Nodal graph partitioning, 88 89 Nondimensional form, 34 parameters, 421 vector form, 35 Young’s modulus of elasticity, 421 Nonlinear Euler equations, 25 Nonnested mesh method, 63 64 npart function, 93 nprtn function, 93 nsenrev function, 93

O Object domain, 109 110 Octree search algorithm, 67 68 OGs. See Overlapping grids (OGs) One-sided MUSCL like extrapolation, 118 121, 118f ONERA M6 wing, 395 396, 397f computational grids, 398f flutter responses, 405f large-eddy simulation of aeroelastic flutter for, 395 406 LES of ONERA M6 wing flutter, 402 406 schematic of computational domain, 398f time history of drag coefficient, 399f of lift coefficient, 399f of moment coefficient, 399f time-averaged Mach number distribution, 400f time-averaged pressure coefficients distributions, 402f Optimal speedup, 420 Oscillating circular cylinder, 431, 436 438 Oscillating sphere in closed cavity, flow induced by, 357 359 Overlapping element identification, 89 92, 91f, 92f Overlapping grids (OGs), 172 173 computations with IOM with, 172 181 method, 5, 7

P Parallel computation of bioprosthetic heart valve, 311 353 domain decomposition, 87 95, 88f

489

measuring performance, 97 99 parallel efficiency, 99 speedup, 98 multigrid parallelization, 96 97, 97f parallelization strategy, 87 Parallel efficiency, 99 Parallel performance, 415 420, 418t, 419t, 423 430 collapsible channel, 427t, 428t efficiency and speedup normalized by 4 processors, 429t measured parallel efficiency, 429f measured parallel speedup, 429f share of computing time and speedup gradient, 419f speedup and efficiency diagram, 419f velocity contour plot, 426f Parallel single grid and multigrid computations, 153 159 comparison between percentage computation and communication time for Re, 156f total simulation wall-clock time for different number of processors, 155f velocity profiles, 159f Parallel solver, 298 305 grid convergence study for, 298 303 computational domain for oscillating sphere, 299f computational domain for rotating sphere, 306f error analysis, 304 305 flow fields, 304f mesh partition for grid convergence study, 302f validation, 306 311 evolution history for flow field, 309f mesh partition for steady flows inducing by rotating sphere, 307f numerical results, 313f polar inflow angles, 312t steady flows due to rotating sphere, 310 311, 312f unsteady flows at Reynolds number of 100, 307 309 unsteady results, 310f Parallel unsteady flow computation, 161 171 coarse level tetrahedral grids for three-level MG computations, 162f instantaneous streamlines plots for all planes in z-direction, 166f lift and drag coefficients vs. time for flow

490

Index

Parallel unsteady flow computation (Continued) over circular cylinder, 163f over 2D circular cylinder, 168f over 3D circular cylinder, 167f lift coefficient, drag coefficient, and Strouhal number for unsteady flow, 164t performance for unsteady flow past circular cylinder, 171f streamlines patterns, 165f streamlines plots for different planes in z-direction, 169f vorticity iso-surfaces for Re, 167f, 170f Parallel-MG method, 105, 266 267 Parallel-SG computation, 105 Parallelization, 411 412 of ALE code with moving boundary, 131 134, 132f moving boundary, 131 134 moving grid, 134 strategy, 87 Partdmesh programs, 88 89 Partial deformation foil model, 456, 456f, 462 466, 464f, 466f Partition, 89, 90f boundaries movement, 132 134 partitioned procedures, 191 Partnmesh programs, 88 89 Point-loaded fixed-free cantilever structures deformation, 370 376 schematic of flexural deformation test of fixed-free cantilever, 371f three-dimensional cantilever, 373 376, 374t two-dimensional cantilever, 370 373, 371t Point-tracing technologies, 453 454 Poisson’s ratio, 371 372 Postprocessor, 2 3 Preconditioned compressible flow solver, mathematical formulation for boundary conditions, 25 29 far-field conditions, 26 28 far-field conditions for preconditioned systems, 28 29 solid wall/slip conditions, 28 convergence acceleration techniques, 24 25 discretization methods, 16 24 governing equations for fluid flows, 11 14 low-speed preconditioning formulation, 14 16 preconditioning matrix, 30 31 Preconditioned system, 22 far-field conditions for, 28 29

Preconditioning matrix, 30 31 prepgrid3D, 88 89 prepmesh3D, 88 89 Preprocessing, 2 3 Preprocessor, 2 3 Pressure contours, 423, 425f, 446 449 gradient, 307 309 Prolongation operators, 80, 82 Prosthetic heart valve, 181 183 Pseudo time, local time-stepping in, 48 49, 60 61 Pseudo-unsteady pressure, 45

Q Quadtree search method, 104 105 Quiescent air, steady flow inducing by rotating sphere in, 355 357 recevert function, 93

R Reimann invariants, 28 29 Residual transfer operators, 80 82, 80f, 81f, 82f Restriction transfer operator, 78 80 Reynolds number (Re), 145 146, 160, 284 286, 298 299, 306, 310 311, 357 359, 395 396, 409 410, 412, 430, 437f, 438f, 439f, 443 444 Riemann problem, 18 Rigid foil aerodynamic forces of rigid foils in hovering motion, 442 453 model, 472 475 Rigid wings, 453 lift force generation mechanisms for, 446 453 rms. See Root-mean-square (rms) Roe-averaged Jacobian matrix, 38 Roe’s approximate Riemann solver, 38 Roe’s averaging, 18 Roe’s TVD scheme for inviscid flux, 18 20 Root-mean-square (rms), 391 392 Rotating circular disk, 178 181 Rotating square cylinder, 176 178, 179f, 180f Rotation function, 444 motion, 454 455 Runge Kutta scheme, 45 46, 58, 65 Runge Kutta time-integration algorithm, 24, 47 48, 60

Index

S Search algorithm, 104 105, 235 Self-induced vibration, 160 Semi-rigid wing models, 453 454 sendvert function, 93 Separation bubble, 173 174 Serial single grid and multigrid computations, 147 153, 150f, 151f, 153f SG. See Single grid (SG) Single grid (SG), 87, 145, 148 150, 364 Single program multiple data (SPMD), 87, 92 93 Single Program Multiple Data programming paradigm, 153 154 Single rigid wing in hovering motion, 446 449, 446f, 447f, 449f SJM aortic bi-leaflet MHV. See St. Jude medical aortic bi-leaflet mechanical heart valve (SJM aortic bi-leaflet MHV) Slip boundary condition, 319 Solid wall/slip conditions, 28 Solver, 2 3 Speedup, 98 and efficiency, 428 430 gradient, 419 420, 419f SPMD. See Single program multiple data (SPMD) Spring analogy method, 130 131 Squeeze flow, 231 235 St. Jude medical aortic bi-leaflet mechanical heart valve (SJM aortic bi-leaflet MHV), 5, 181 183, 184f, 260 269. See also Bioprosthetic heart valve (BHV) closing phase, 227 260 background mesh, immersed leaflet and valve housing for, 233f change of leaflet angular velocity and angular position, 237f combination of background mesh and OG with leaflet, 234f OG with aortic valve leaflet, 233f plan view for vorticity iso-surfaces, 232f plots for closing phase at different degrees, 244f, 246f, 248f, 252f, 256f, 260f, 264f, 266f pressure contour plots for, 240f pressure on outer and inner leaflet surfaces, 241f rotating moment acting on leaflet, 236f velocity profile, 242f computation, 181 183 coupling of fluid and structure domains, 191, 192f

491

FSI, 187 191 geometry of valve used in Krafczyk’s simulation, 268f model properties, 183 187, 283 286, 285f, 286f opening phase, 192 227, 194f, 195f change of leaflet angular velocity and angular position, 198f plots for opening phase at different degrees, 205f, 209f, 213f, 217f, 221f, 225f, 229f, 231f pressure contour plots for, 201f pressure on outer and inner leaflet surfaces, 202f rotating moment acting on leaflet, 197f velocity profile, 203f time-dependent inlet velocity in Krafczyk’s simulation, 268f velocity vector plot, 269f, 270f ST2UNST grid-generation program, 299 300 Staggered procedures, 191 Stain, constitutive relationship for stress and, 54 Steady flows, 25, 362 363, 362f. See also Transonic flow cat-eye flow state, 418f contour plot for vertical velocity, 413f, 414f domain geometry of two-sided, lid-driven cavity, 410f grid dependency study, 410 411 parallel performance, 415 420 past circular cylinder, 360 365 due to rotating sphere, 310 311 streamlines, 416f, 417f between two rigid disks, 383 395, 384f in two-sided, lid-driven cavity, 409 420 validation, 275 278, 277f velocity v over x, 415f Steady viscous flow, 173 174 Stokes number, 430 Strains, 55 Stress, constitutive relationship for stain and, 54 Strouhal number (St), 160 Structural dynamic solver, multigrid method for, 84 86 Structural dynamics, 53 54 “Suspected array” nodes, 110 111 Symmetric vortex shedding, 434 435 Symmetrical rotation hovering motion, 446 448 style, 446

492

Index

T Tag argument, 95 Taylor series expansion, 46, 58 Taylor-Go¨rtler-like vortices (TGL vortices), 145 TECPLOT tool, 3 TEF. See Trailing-edge flapping (TEF) TETRAKE fluid solver, 121 122 TEV. See Trailing-edge vortex (TEV) TGL vortices. See Taylor-Go¨rtler-like vortices (TGL vortices) Third-order characteristics upwind scheme, 145 146, 160 Third-order characteristics-based scheme, 176 179 Third-order MUSCL interpolation, 20 21 Three-dimension (3D), 16 calculations, 392, 395 cantilever, 373 376, 374t circular cylinder mesh, 172 173, 172f equations, 16 fixed-free cantilever immersed in fluid flow, 376 382, 377f iso-surface of axial velocity component, 392, 393f lid-driven cavity steady flow, 145 159 geometry for lid-driven cavity flow, 146f grid convergence study, 146 147 parallel single grid and multigrid computations, 153 159 serial single grid and multigrid computations, 147 153 Navier Stokes solver, 89 steady flow, 386 388 unsteady flow, 389 395 disk-enclosure gap effects and disk thickness, 394 Three-grid system, 64 Three-level MG method, 375, 375f, 376f Time integration, 24, 47 48 Time marching, 129 Time-accurate time-marching scheme, 34 35 Time-critical simulations, 410 411 Time-dependent computations, 430 Time-dependent solution, 36 Tip vortex (TiV), 443 Total simulation wall-clock time, 98 99 Traditional FE methods, 4 Traditional Quasi-steady aerodynamic theory, 442 443 Trailing-edge flapping (TEF), 456, 466 Trailing-edge vortex (TEV), 446 448 Translating circular disk, 178 181

Translation motion, 454 455 Translational amplitude, 444 Transonic flow. See also Steady flows large-eddy simulation of aeroelastic flutter for ONERA M6 wing in, 395 406 geometry and calculation parameters, 396 397 LES of ONERA M6 wing flutter, 402 406 LES of steady aerodynamic computation of M6 wing, 397 401 Transvalular pressure difference, 352 353 Two rigid wings in hovering motion, 449 453 Two-dimension (2D), 273 cantilever, 370 373, 371t ellipse wing geometry, 443 444 elliptical wing simulation, 443 flow in two-dimensional collapsible channel, 420 430 channel flow with collapsible walls, 420 developing flow without moving grid, 421 geometry of collapsible channel, 421f moving grid, 422 423 parallel performance, 423 430 flows, 273 foil models, 454 immersed membrane computations, 273 282 comparison of flow field with immersed membrane and flow field, 280t comparison of X-component velocity profiles, 279f convergence history of immersed membrane and internal boundary, 278f flow field of grid, 274f geometry of 2D channel model, 274f grid convergence study, 273 275 motion of rigid membrane, 281f moving flexible membrane, 280 282 moving rigid membrane, 279 280 parameters of flexible membrane, 282t properties of vortices of three grids, 275t steady flow validation, 275 278, 277f steady flow with immersed flexible membrane, 282f Navier Stokes solver DELTINKE, 277 278 rigid wing, 454 456 Two-level MG, 65 66

Index

U Unit normal vectors, 70 71, 110 111 Unsteady flows, 25, 34 35, 363 365, 367f. See also Viscous unsteady flow past circular cylinder air flows between two rigid disks, 383 395, 384f lift and drag coefficients and Strouhal number for, 367t past circular cylinder, 360 365 at Reynolds number, 307 309 Unsteady viscous flow, 174 176 Unstructured grids, 61 62, 70, 101 Upwind characteristics-based method, 40 44, 41f Upwind-biased discretization scheme, 66 Upwind-biased interpolation, 20 22, 39 40

V V-cycle, 64, 64f Validation of incompressible ALE solvers, 409 441 flow in two-dimensional collapsible channel, 420 430 flow over two-dimensional oscillating circular cylinder, 430 441 overlapping grid elements technique, 441 steady flow in two-sided, lid-driven cavity, 409 420 of parallel solver, 306 311 Vertical contours, 448 449 Viscous term, 38 39 Viscous unsteady flow past circular cylinder IOM with OGs, 172 176 mesh, OG, and 3D immersed circular cylinder mesh, 172f steady viscous flow, 173 174 unsteady viscous flow, 174 176 serial/parallel single grid and multigrid computations, 160 171 grid convergence study, 161, 162f

493

parallel unsteady flow computation, 161 171 VOF method. See Volume of fluid method (VOF method) Volume of fluid method (VOF method), 4 Von Ka´rma´n vortex street phenomenon, 160, 164 Vortex contours, 446 448, 467f, 468f, 472f, 473f motion, 434 435 shedding and motion, 438 441 phenomenon, 160, 235, 364, 365f process, 471 vortex vortex interaction, 446 448 Vorticity contours, 448 449, 452f surfaces, 255 258

W W-cycle, 64, 64f Wake capture mechanism, 442 443, 457 458, 461f Wall-clock time, 97 98, 154 157 Wash-out effect, 327 333 Wave speed (λk), 41 Weather Prediction by Numerical Process, 1 Wing deformation, 457 458 Wing wake interaction, 442 443

Z Zoning of flow field domain, 67 76, 68f area of zones smaller than largest element, 69f correlation between number of zones and CPU time, 70f interconnectiveity relationship between meshes, 70 76, 78f negative mapping produced by algorithm, 68f