Hydrodynamic Bearings [1 ed.] 9781119008064, 9781119004769

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W681-Bonneau 1.qxp_Layout 1 02/07/2014 12:07 Page 1

NUMERICAL METHODS IN ENGINEERING SERIES

This 4-volume series of books brings together the elements necessary for the realization and the validation of these tools.

Dominique Bonneau Aurelian Fatu Dominique Souchet

Hydrodynamic bearings allow the various parts of mechanical devices to be driven easily while ensuring a reliability which eliminates any risk of rupture or premature wear. When the operating conditions are severe (high or quickly varying loads, great rotational frequency) it becomes difficult to achieve this double goal without the assistance of powerful prediction digital models.

This first volume presents the rheological laws of the lubricant, the equations of hydrodynamic and elastohydrodynamic lubrication as well as the models of solution of these equations by the finite difference, finite volume and finite element methods. The algorithms are described in detail and each part is abundantly illustrated.

Dominique Bonneau is Professor Emeritus specializing in the numerical modelization of lubrication problems who has worked as a teacherresearcher at the IUT of Angoulême and at the Institute PPRIME (Laboratory of Mechanics of Solids) of the University of Poitiers-CNRSENSMA in France. Aurelian Fatu is Professor and a researcher at the Institute PPRIME specializing in the modeling of problems of lubrication for engine bearings and for systems’ sealing devices. Dominique Souchet specializes in the modeling of thermohydrodynamic lubrication of journal and thrust bearings with Newtonian or non-Newtonian fluids. He is a university lecturer and researcher at the Institute PPRIME.

www.iste.co.uk

Z(7ib8e8-CBGIBF(

Hydrodynamic Bearings

These books are of great interest both for researchers wishing to extend their knowledge of the behavior of connecting rod and crankshaft bearings of internal combustion engines and for engineers aiming to develop equipment that reduces engine energy losses while increasing their reliability. The analysis, modeling and solution methods presented here are sufficiently general to be applied to all reciprocating systems, such as, for example, piston compressors.

Hydrodynamic Bearings Dominique Bonneau Aurelian Fatu Dominique Souchet

Hydrodynamic Bearings

Series Editor Piotr Breitkopf

Hydrodynamic Bearings

Dominique Bonneau Aurelian Fatu Dominique Souchet

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

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© ISTE Ltd 2014 The rights of Dominique Bonneau, Aurelian Fatu and Dominique Souchet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2014942899 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-681-5

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Contents

FOREWORD BY J.F. BOOKER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

FOREWORD BY JEAN FRÊNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

CHAPTER 1. THE LUBRICANT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.1. Description of lubricants . . . . . . . . . . . . . . . 1.2. The viscosity . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Viscosity – temperature relationship . . . . . . 1.2.2. Viscosity – pressure relationship . . . . . . . . 1.2.3. Viscosity – pressure – temperature relationship 1.2.4. Non-Newtonian behavior. . . . . . . . . . . . . 1.3. Other lubricant properties . . . . . . . . . . . . . . . 1.4. Lubricant classification and notation. . . . . . . . . 1.5. Bibliography . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2. EQUATIONS OF HYDRODYNAMIC LUBRICATION . . . . . . . . . . . . . . .

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2.1. Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Equation of generalized viscous thin films . . . . . . . . . . . . . . 2.3. Equations of hydrodynamic for journal and thrust bearings . . . . . 2.3.1. Specific case of an uncompressible fluid . . . . . . . . . . . . . 2.3.2. Standard Reynolds equation for a journal bearing: general case 2.3.3. Reynolds equation for a thrust bearing: general case . . . . . .

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2.3.4. Equation of volume flow rate . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Equations of hydrodynamic for journal and thrust bearings lubricated with an isoviscous uncompressible fluid . . . . . . . . . . . . . . . . . . . . . 2.4. Film rupture; second form of Reynolds equation . . . . . . . . . . . . . . . 2.5. Particular form of the viscous thin film equation in the case of wall slipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Boundary conditions; lubricant supply. . . . . . . . . . . . . . . . . . . . . 2.6.1. Conditions on bearing edges . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Conditions for circular continuity . . . . . . . . . . . . . . . . . . . . . 2.6.3. Conditions on non-active zone boundaries . . . . . . . . . . . . . . . . 2.6.4. Boundary conditions for supply orifices . . . . . . . . . . . . . . . . . 2.7. Flow rate computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. First assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2. Model and additional assumptions . . . . . . . . . . . . . . . . . . . . 2.7.3. Pressure expression for the full film fringes on the bearing edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4. Evolution of the width of the full film fringes on the bearing edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5. Computation of the flow rate for lubricant entering by the bearing sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Computation of efforts exerted by the pressure field and the shear stress field: journal bearing case . . . . . . . . . . . . . . . . . . . . . . . 2.9. Computation of efforts exerted by the pressure field and the shear stress field: thrust bearing case . . . . . . . . . . . . . . . . . . . . . . . . 2.10. Computation of viscous dissipation energy: journal bearing case . . . . . 2.11. Computation of viscous dissipation energy: thrust bearing case . . . . . . 2.12. Different flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3. NUMERICAL RESOLUTION OF

THE REYNOLDS EQUATION

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3.1. Definition of the problems to be solved . . . . . . . . . . . . . . . . 3.1.1. Problem 1: determining the pressure . . . . . . . . . . . . . . . 3.1.2. Problem 2: determining of the pressure and the lubricant filling 3.1.3. Other problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The finite difference method . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Computation grid . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Discretization of standard Reynolds equation (problem 1) . . . 3.2.3. Discretization of modified Reynolds equation (problem 2) . . . 3.3. The finite volume method. . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Mesh of the film domain . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Discretization of the standard Reynolds equation (problem 1) . 3.3.3. Discretization of modified Reynolds equation (problem 2) . . .

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Contents

3.4. The finite element method . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Integral formulation of standard Reynolds equation . . . . . . . 3.4.2. Integral formulation of modified Reynolds equation . . . . . . 3.4.3. Approximation of integral formulations: method of Galerkin weighted residuals. . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Approximation of problem 1* . . . . . . . . . . . . . . . . . . . 3.4.5. Approximation of problem 2* . . . . . . . . . . . . . . . . . . . 3.5. Discretizations of time derivatives . . . . . . . . . . . . . . . . . . . 3.5.1. Discretization by finite differences . . . . . . . . . . . . . . . . 3.5.2. Discretization by time finite elements. . . . . . . . . . . . . . . 3.5.3. Adaptation of discretized expressions for equations to be solved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Comparative analysis of the different methods . . . . . . . . . . . . 3.6.1. Definition of reference problems . . . . . . . . . . . . . . . . . 3.6.2. First numerical tests . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3. Comparisons between the three discretization methods for a static case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4. Comparisons between linear and quadratic discretizations for the finite element method applied to the standard Reynolds equation. 3.6.5. Comparisons between the different discretizations of time derivatives for the modified Reynolds equation . . . . . . . . . . . . . 3.6.6. Aptitude of the various discretizations of time derivatives to follow sudden load change . . . . . . . . . . . . . . . . . . . . . . . 3.6.7. Case of a bearing under a dynamic load rotating with a frequency equal to half of the shaft frequency. . . . . . . . . . . . . . 3.7. Accounting of film thickness discontinuities . . . . . . . . . . . . . 3.8. Numerical algorithm for computing bearing axial flow rate . . . . . 3.8.1. Pressure gradient computing in the case of a finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2. Computation of axial flow rate . . . . . . . . . . . . . . . . . . 3.8.3. Algorithm for computing the axial flow rate . . . . . . . . . . . 3.8.4. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4. ELASTOHYDRODYNAMIC LUBRICATION . . . . . . . . . . . . . . . . . . .

159

4.1. Bearings with elastic structure . . . . . . . . 4.1.1. Thickness of the lubricant film . . . . . 4.1.2. Film domain discretization . . . . . . . . 4.2. Elasticity accounting: compliance matrices . 4.2.1. Surface forces due to pressure . . . . . . 4.2.2 Volume forces due to inertia effects . . .

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4.3. Accounting of shaft elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Particular case of non-conformal meshes . . . . . . . . . . . . . . . . . . . . . 4.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Foreword by J.F. Booker

Hydrodynamic lubrication is a remarkably simple concept: solid surfaces separated by a thin fluid film, thus minimizing friction and wear. By the last quarter of the 19th Century the concept had been validated experimentally by such engineers as Gustave-Adolphe Hirn (France), Beauchamp Tower (Britain), and Robert Thurston (United States). Not long afterward, the British engineer and physicist, Osborne Reynolds, had derived the governing partial differential equation that bears his name. (The Reynolds equation is the 2-dimensional result of applying thin-film approximations to 3-dimensional Navier-Stokes and continuity equations.) By the very beginning of the 20th Century Reynolds himself and later the German physicist, Arnold Sommerfeld, had worked out many of the most obvious and simple steady-state solutions. Why now, early in the 21st Century, do we need a new four-volume series entitled, “Hydrodynamic Lubrication”? The design methodology of the early 20th Century is still perfectly satisfactory for many bearings operating under steady conditions of low loads and speeds. However, the design analysis of many modern bearings must address unsteady response to dynamic loads, high temperatures, thin fluid films, surface finish limitations, structural compliance, and exotic inlet configurations (together with cavitating, piezoviscous, thermoviscous, highly nonNewtonian lubricant behavior), all in the effort to reduce power loss, wear, and cost. Some of the most extreme demands have come from designs for internal combustion engines, which face continuous pressure for greater efficiency, lower weight, and higher power density.

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In the second half of the 20th Century came gradual development of advanced design analysis techniques made possible by steady advances in computer resources and numerical methods. Spatial discretization methods gradually came to include finite difference (FD), finite element (FE), and finite volume (FV) methods. Temporal integration techniques became more elaborate and more capable of dealing with such difficulties as “stiff” systems. Lubrication problem categories progressed with increasing complexity (and lengthening acronyms) through hydrodynamic (HD), elastohydrodynamic (EHD), thermohydrodynamic (THD), and thermoelastohydrodynamic (TEHD). While there is still a place in initial design studies and/or extended whole-engine system dynamic analyses for much faster (but much more approximate) methods, detailed design analysis now requires the powerful methods of numerical analysis developed by such specialists as the authors and previously reported in multiple research publications distributed widely over time and space. An important set of these is now gathered together in one place in the form of an extended monograph — a distillation of some 25 years of work at the very forefront of the subject area carried out by the distinguished authors and others at the University of Poitiers in France. Though each chapter includes a bibliography appropriate for its subject matter, the book is not an extensive review of the work of others, except as it furthers the central development of the subject. The previous two-volume series edition in French was largely inaccessible to Anglophones (such as myself). This new English edition should thus be welcomed by a much expanded audience. Division into a sequence of four smaller and more narrowly focused volumes seems a logical and convenient way to meet the needs of a diverse audience. The first volume lays out in detail most of the necessary ingredients for modern design analyses of hydrodynamic bearings: lubricant rheology, fundamental equations of hydrodynamic lubrication and elasticity, and numerical solution techniques. The second volume takes on the inter-related matters of surface finish, mixed lubrication, and wear. The third volume extends modeling and solution techniques to include thermal effects.

Foreword by J.F. Booker

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The fourth volume addresses the specific challenges of hydrodynamic bearings in internal combustion engines, beginning with the kinematics and dynamics of the block-crankshaft-connecting rod-piston linkage, proceeding to detailed hydrodynamic lubrication analyses for each of the bearings in the chain, and culminating in a study of the application of formal experimental design to the bearing design optimization process. This approach allows the possibility of considering many more design variables and operating conditions than with conventional optimization techniques without overwhelming computing resources. Professor Jean Frêne, also of the University of Poitiers, put it perfectly in his Foreword to the French edition: “The authors should be thanked and warmly congratulated for putting together in one comprehensive book their extensive knowledge on the complex topic of the mechanism of internal combustion engines and the lubrication of various highly loaded bearings operating under transient conditions.” It is an honor and a pleasure to second the motion.

J.F. BOOKER Professor Emeritus, Cornell University Fellow, American Society of Mechanical Engineers (ASME) Fellow, Institution of Mechanical Engineers (IMechE) June 2014

Foreword by Jean Frêne

The crankshaft–rod–piston system has significantly contributed to the development and success of internal combustion engines. However, this mechanism has also always been one of the weak points of these engines, owing firstly to bearing damage and secondly to significant energy losses from friction in the bearings. Throughout the first half of the 20th Century, the design of these bearings has essentially been based on empirical methods largely derived from the experience and expertise of the designers of internal combustion engines, since no-one knew how to apply and solve the equations describing the operation of these bearings. In fact, in order to solve the Reynolds equation simply, using a direct method, the form of the lubricant film needs to be known, as does the geometry and the kinetics of the bearing. However, in the case of connecting rods and crankshaft bearings elastic deformations of structures play an important role, and it is the load applied to the bearing, which is generally known. From 1950, various researchers in Europe and United States proposed semigraphical and numerical solutions to try to describe the behavior of these bearings but it was only in the early 1980s that the first numerical approaches taking the elastic deformations of the structures into account were suggested. These approaches, based on cumbersome and time-consuming iterative methods, have furthered understanding of the importance of elastic deformations and their effects on the behavior of the bearings. Since the 1980s, the authors of these books have applied the essence of their research efforts to the study of many problems of lubrication, and particularly those arising from the operation of the connecting rods and crankshafts of internal combustion engines. Thus, it was truly useful and necessary that the authors bring

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together the essence of their work in the form of these comprehensive books, which are well researched and particularly well organized. The books begin with a comprehensive presentation of the characteristics of lubricants, which are generally non-Newtonian and whose behavior depends heavily on the temperature, on pressure and shear rate existing throughout in the bearing lubricant film. Chapter 2 of Volume 1, importantly, is devoted to the equations of fluid film lubrication and in particular to the Reynolds equation and the equations to calculate pressures and flows in the film, and the frictional torque and the load on the bearing. The boundary conditions and the circulation conditions of the lubricant in the bearing are explained as they play an essential role in determining the characteristics of the mechanism. Chapter 3 presents in detail the different numerical methods that can be used to obtain the solution of the Reynolds equation and associated equations. The choice of the finite elements method is favored because, in addition to its excellent performance, this method takes into account in a very efficient way the deformation of the elastic structures. This point concerning elastohydrodynamic lubrication is very well described in Chapter 4 of Volume 1. Two important aspects are then explained in detail. The first aspect concerns mixed lubrication, whose importance is paramount when the thickness of the lubricant film is of the same order of magnitude as the height of surface roughnesses. The second aspect involves thermal effects in the lubricant film and in the materials, which form the contact. These two aspects strongly affect the operation of the mechanism. To complete the description of the phenomena, the development of the equations and the numerical methods appropriate to their resolution, the kinematic and dynamic characteristics of the complex mobile system formed by the crankshaft, bearings and connecting rods of internal combustion engines, taking into account the elastic deformations of the structures, are presented in detail. The last chapters of this work are devoted to the application of these theories and models associated with different bearings permitting transmission of motion and forces from the piston to the crankshaft. Cases of connecting rods–crankshaft, connecting rod–piston, and crankshaft–block bearings in internal combustion engines are successively examined in detail. Solution methods are explained and

Foreword by Jean Frêne

xv

specific and detailed examples make it possible to understand how these mechanisms function. These books are of very great interest both for researchers wishing to extend their knowledge of the behavior of connecting rod and crankshafts bearings of internal combustion engines and for engineers aiming to develop equipment that reduces engine energy losses while increasing their reliability. Furthermore, the analysis, modeling and solution methods presented here are sufficiently general to be applied to all reciprocating systems, such as, for example, piston compressors. The authors should be thanked and warmly congratulated for putting together in one comprehensive work, their extensive knowledge on the complex topic of the mechanism of internal combustion engines and the lubrication of various highly loaded bearings operating under transient conditions. Jean FRÊNE Professor Emeritus at the University of Poitiers, France Member of the Academy of Technologies, France June 2014

Preface

Those who construct and use vehicles and machines have always been faced with a two-fold problem: that of allowing the various elements to move with the greatest possible ease, while still ensuring a level of solidity, which is sufficient to reduce the risk of pieces breaking or wearing out too quickly. Developments over several centuries, which were usually achieved empirically, led to the virtually universal adoption of lubricated bearings to carry rotating shafts or vehicle wheels in the middle of the 19th Century. However, the operational principle behind these bearings was only truly understood at the end of that century when O. Reynolds presented a model in 1886, which was perfectly in agreement with the experiments carried out in France by G. A. Hirn (1847) and in England by Beauchamp Tower (1883). In Reynolds’ theory, a thin lubricant film separates the surfaces of solids moving one with respect to the other. Taking into account the simplifications, which can be made due to the dimensional characteristics, the description of the thin-layer flow of the lubricating film, based on the principles of continuum mechanics, leads to the equation known as the “Reynolds equation”. This equation links the film pressure to the film thickness, to the relative velocity of the bounding bodies and to the physical properties of the lubricating fluid. With the exception of a few simple cases, the Reynolds equation cannot be solved analytically. It quickly became apparent that it needs to be handled numerically. With the power rise of computation tools, the second half of the 20th Century saw the development of algorithms for solving equations numerically, which have progressively made it possible to take an increasing number of parameters and phenomena into account in calculations. Initially, due to the simplicity of their application, finite difference methods became the preferred tool for tribologists. However, starting from the beginning of the 1970s, the finite element method was demonstrated to be highly efficient for solving elliptic partial differential equations similar to the Reynolds equation.

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Consequently, much work has been undertaken over the last 40 years on the subject of solving the Reynolds equation using the method of finite elements, giving powerful tools, which have been validated by numerous theoretical studies and by comparison with experimental results. There have not been a great number of these experiments, due to the difficult nature of the environment of lubricating films. More recently, the scope of the finite volume method has been extended to lubrication. At the end of 25 years dedicated to the study of many problems related to lubrication – in particular those related to the bearings of internal combustion engines – and the resolution of these problems using the finite element method, we considered that the time had come to take stock of the numerical tools available for the calculation of hydrodynamic bearings. An earlier version of this work was published in two volumes in French, in 2011 by Éditions Hermes-Lavoisier (Paris). The current edition in four volumes has been revised, with an added chapter on optimization techniques applied to the calculation of bearings. The content of these books is based largely on our own experience. Even if the applications developed mainly concern internal combustion engine bearings, the methods presented in them may equally be applied to other types of bearings subject to severe and/or nonstationary loadings. The lubricant is the central element in the operation of these bearings. In Chapter 1 of Volume 1, we describe the physical properties that play an essential part in the hydrodynamic phenomenon; first among these is viscosity. When the lubricating fluid is an oil (organic, mineral or synthetic), it may be considered to be virtually incompressible. As a result, we have deliberately not mentioned the issues linked to aerodynamic lubrication, where the lubricant is air or any other gas, nor have we mentioned the effect of any possible turbulence within the lubricating film. However, variations in viscosity in function of pressure (piezoviscosity) and temperature (thermo-viscosity) are phenomena essential to the applications we discuss. The non-Newtonian behavior of lubricants at high shear rates is also an important matter, especially for the bearings of high-performance engines. Equations of hydrodynamic lubrication are discussed in Chapter 2. First, the main assumptions related to thin viscous films which lead to the standard form of the Reynolds equation are recalled. In depressed areas, the lubricant cannot withstand the traction forces and it breaks, leading to the phenomena of separation and cavitation. To determine where the active and non-active zones in the pressure field are, we need a “modified” form of the Reynolds equation, which is also presented in this chapter. The remainder of this chapter discusses the definition of different boundary conditions for the film fringes and the parameters essential to the operation of hydrodynamic bearings (flow, load and friction).

Preface

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The topic of Chapter 3 is the development of numerical methods for solving the Reynolds equation. The finite difference method and finite volume method are quite simple to apply. They are in particular well suited to uniform width bearings. The comparison of discretization expressions of the Reynolds equation demonstrates that the two methods are very similar. Although it is more complex to apply, the finite element method provides better results for solving the modified form of the Reynolds equation. Chapter 3 provides a detailed description of the developments which lead to the discretization of the equation. The finite element method proves to be superior in the treatment of variable width bearings, of bearings with complex grooves or with uneven bounding body surfaces. The algorithms for these different methods are described in detail. Their respective performances are compared for simple cases for which the data is completely defined, so that they can serve as trial cases. In high-pressure zones, the fluid exerts levels of pressure on the bounding body surfaces that are high enough to deform them. The reciprocal “elastohydrodyamic” (EHD) dependency between pressure and the film thickness gives the Reynolds equation a strongly nonlinear character, which increases the complexity of solving it, as it must be solved conjointly with elasticity equations. Chapter 4 describes the techniques used to take the deformability of the constitutive elements of the bearing into account, at the bearing level. The case of the deformable bearing is characteristic of the EHD contact problems for conforming surfaces and may serve as a model for solving these. Volume 2 [BON 14a] is concerned with the study of mixed lubrication. In it, we analyze the role played by surface unevenness from both a hydrodynamic point of view and from the point of view of the contact between the roughnesses on the surfaces, presenting the corresponding numerical techniques in a detailed manner. Volume 2 also handles the issue of surface wear in this context. Volume 3 [BON 14b] contains the description of several thermo-hydrodynamic (THD) models and thermoelastohydrodynamic (TEHD) problems. It is rounded off by the description of the general algorithms of calculation software for bearings under non-stationary and severe loading. In Volume 4 [BON 14c], the final volume of the series, we discuss, in detail, the problems specific to the calculation of engine and compressor bearings. This last volume also contains a chapter on optimization techniques for the calculation of bearings with an application for calculating a big end connecting rod bearing for an internal combustion engine.

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Hydrodynamic Bearings

Acknowledgements First, our thanks go to our mentor and colleague Jean Frêne, Professor Emeritus at the University of Poitiers (France), who, in the 1980s, suggested that we work on the development and improvement of numerical models for the study of connecting rod bearings. Along with Bernard Fantino, he had, a few years earlier, initiated work on the subject. Without this initial incentive and the constant encouragement that he later offered to us, these books might never have seen the light of day. Many colleagues and graduate students of the Laboratory of Mechanics of Solids at the University of Poitiers have worked, and some continue to work, on topics related to these books. We have benefited from their contributions and we would like to thank them also. Our structures and interfaces team colleagues located at IUT Angoulême (France), and in particular Bernard Villechaise and Mohamed Hajjam, the team coordinators, have consistently provided us with the best conditions for the realization of our research activity, for which we are grateful. Postgraduate students that we have directed, wholly or partially, occupy a place of privilege for the importance of their contributions. We include here in chronological order: Stephen Mutuli, Toan Nguyen, Joseph Absi, Dominique Guines, Thierry Garnier, Stéphane Piffeteau, Virgil Optasanu, Hervé Moreau, Lê Vuong Hoang, Loubna Jeddi, Philippe Michaud, Frank Gambin, Thi Tan Hai Tran, Marco Spuria, Ramona Dragomir-Fatu and Thomas Lavie. Volume 2, which deals with mixed lubrication, is based in part on the thesis Ramona Dragomir-Fatu devoted to the study and modeling of mixed lubrication in engine bearings. This thesis was co-supervised by François Robbe-Valloire, professor at the Institute of Mechanics of Paris: his contribution to this work was essential and we want to show him here all our recognition. Our thanks also go to Thami Zeghloul and Arthur Francisco, lecturers at Angoulême University Institute of Technology (IUT), and Pier Gabriele Molari, Professor at the University of Bologna (Italy), for their contributions to the theses of Thi Tan Hai Tran, Thomas Lavie and Marco Spuria. Much of the work that is the basis for this book has been conducted through close collaboration with industrial partners. Their extensive knowledge of the devices for which the codes are developed has always been of paramount importance. We would like to offer them our special thanks. This applies in particular to Renault, with whom we have worked in uninterrupted partnership over the last 20 years. Since 1965, when he first proposed the mobility method to address the problem of hydrodynamic bearings submitted to non-stationary loads [BOO 65], and until

Preface

xxi

today [BOO 14], John (Jack) Booker, Professor Emeritus at Cornell University, has contributed immensely to the development of methods for solving lubrication problems [BOO 14]. The content of this book owes much to earlier work by Jack Booker and colleagues. Jack has done us a great honor by accepting to write a foreword for this first volume. We would like to thank him warmly. P.1. Bibliography [BON 14a] BONNEAU D., FATU A., SOUCHET D., Mixed Lubrication Bearings, ISTE, London, and John Wiley & Sons, New York, 2014.

in Hydrodynamic

[BON 14b] BONNEAU D., FATU A., SOUCHET D., Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings, ISTE, London, and John Wiley & Sons, New York, 2014. [BON 14c] BONNEAU D., FATU A., SOUCHET D., Internal Contribution Engine Bearings Lubrication in Hydrodynamic Bearings, ISTE, London, and John Wiley & Sons, New York, 2014. [BOO 65] BOOKER J.F., “Dynamically loaded journal bearings: mobility method of solution”, ASME Journal of Basic Engineering, vol. 87, no. 3, pp. 537–546, 1965. [BOO 10] BOOKER J.F., BOEDO S., BONNEAU D., “Conformal elastohydrodynamic lubrication analysis for engine bearing design: a brief review”, Proceeding of IMechE: Journal of Mechanical Engineering Science, vol. 224, Part C, pp. 2648–2653, 2010. [BOO 14] BOOKER J.F., “Mobility/impedance methods: a guide for application”, ASME Journal of Tribology, vol. 136, pp. 024501.1–8, 2014.

Nomenclature

Points, basis, repairs, links and domains M M1 M2 O Oc Oa x, y, z xc, yc, zc Ω, ΩF Ω0, Ωr Ωp ΩS ∂Ω0 ∂Ω1 ∂Ω2 ∂Ωam ∂Ωav ∂ΩS

point inside the lubricant film point on the wall 1 of the lubricant film point on the wall 2 of the lubricant film origin point of lubricant film repair (developed bearing) origin point of the repair attached to the housing (bearing center) origin point of the repair attached to the shaft Cartesian basis for the film (developed bearing) Cartesian basis for the housing film domain film domain, non-active zone film domain, active zone domain occupied by a solid boundary of a non-active zone parallel to the circumferential direction part of an active zone boundary where the pressure is imposed part of an active zone boundary where the flow rate is imposed up-flow boundary for a non-active zone down-flow boundary for a non-active zone boundary of a solid

xxiv

Hydrodynamic Bearings

Non-dimensional numbers ℜ , Re

ρUh μ

ℜ*

ℜ

Scalars

C R

m m

Reynolds number modified Reynolds number

B C Cm D Ec Ei F G H1 H2 I, I2 J, J2 L L

kg m-2 kg (Pa.s)-1 m m m2 Pa-1 s-1 m Pa-1 s-1 m m

Mxc

Nm

bearing half-width bearing radial clearance lubricant / external gas mixture density universal variable representing p else r – h discretized contact equation discretized Reynolds equation relative to node i Couette flow rate factor Poiseuille mass flow rate factor level of wall 1 at point with x, z projected coordinates level of wall 2 at point with x, z projected coordinates integrals on film thickness integrals on film thickness bearing width complete film band width that delimits a non-active zone on the bearing edge xc component for the moment at Oc of ℑ pressure torsor

Myc

Nm

yc component for the moment at Oc of ℑ pressure torsor

N Qm Qv QC

kg s-1 m3 s-1 m3 s-1

QC+

m3 s-1

QC¯

m3 s-1

R T U U1 U2

m °C m s-1 m s-1 m s-1

Pa; m

interpolation function lubricant mass flow rate lubricant volume flow rate lubricant volume flow rate per cycle passing through the bearing extremities lubricant volume flow rate per cycle outing through the bearing extremities lubricant volume flow rate per cycle entering through the bearing extremities bearing radius temperature shaft peripherical velocity for a bearing velocity of wall 1 in x direction at point (x, H1, z) velocity of wall 1 in x direction at point (x, H2, z)

Nomenclature

xxv

V VF V1 V2 W W W1 W2 Wxc

m s-1 m s-1 m s-1 m s-1 m s-1 m s-1 m s-1 N xc

squeeze velocity for a bearing circumferential velocity for boundary ∂Ωam or ∂Ωav velocity of wall 1 in y direction at point (x, H1, z) velocity of wall 2 in y direction at point (x, H2, z) shaft axial velocity for a bearing weighting function velocity of wall 1 in z direction at point (x, H1, z) velocity of wall 2 in z direction at point (x, H2, z) component of ℑ pressure torsor resultant

Wyc

N yc

component of ℑ pressure torsor resultant

h ne nne npi nx, nz

m

p pa psupply pcav pref psep qm qv r u

Pa Pa Pa Pa Pa Pa kg s-1 m-1 m2 s-1 m m s-1

v w x xam xav y

m s-1 m s-1 m m m m

z γ α β ε x, ε y

m N m-1 °Pa-1 °C-1

lubricant film thickness element number node number per element number of integration points per element number of elements for the film mesh in circumferential and axial directions pressure in the lubricant film ambient pressure out of the bearing supply pressure for a bearing depression due to saturated vapor tension reference pressure for the lubricant film depression due to surface tension mass flow rate per arc length unit for a curve volume flow rate per arc length unit for a curve lubricant filling in a non-active zone s time circumferential velocity component at a point into the film velocity squeeze component at a point into the film axial velocity component at a point into the film circumferential coordinate for a point into the film circumferential coordinate for a point of ∂Ωam circumferential coordinate for a point of ∂Ωav coordinate in the thickness direction for a point into the film axial coordinate for a point into the film lubricant surface tension piezoviscosity coefficient thermoviscosity coefficient relative eccentricity components

xxvi

Hydrodynamic Bearings

ζ x, ζ y ζ η θ λρ µ µm

rad °C-1 Pa.s Pa.s

ξ ρ ρm

kg m-3 kg m-3

τxy, τzy ω Δt Φ

rad

Pa rad s-1 s

misalignment components for the shaft into the housing parametric variable parametric variable angular coordinate for a film point for a bearing thermal correction coefficient for the density lubricant dynamic viscosity dynamic viscosity of the lubricant/gas mixture into the non-active zones parametric variable lubricant density density of the lubricant/gas mixture into the non-active zones shear stress into the lubricant film shaft angular velocity with respect to the housing time step index function identifying active and non-active zones in the lubricant film

Vectors de f n x

m N

xc , yc, zc y z {D} {bi} p, {p} pc r Δs

Pa ; m

Pa Pa

elastic deformation normal to the film wall nodal forces unit vector orthogonal to a domain boundary unit vector in the direction of the shaft surface displacement (developed bearing) unit vectors for a bearing; zc parallel to the bearing axis unit vector in the direction of the film thickness (developed bearing) unit vector equal to x ∧ y vector of nodal values for the universal variable D second member of the problem i equation discretized by the finite element method vector of pressure nodal values contact pressure residual of equations (Newton-Raphson process) solution correction (Newton-Raphson process)

Nomenclature

Torsors

ℑpressure

pressure actions exerted on the housing

ℑapplied load

loading for a shaft or thrust bearing

Matrices [A] [C] [K]

N Pa-1 m Pa-1 N m-1

integration matrix compliance matrix stiffness matrix

[P] [S] [J]

m N-1

projection matrix matrix of elementary solutions Jacobian matrix

Indices 1, 2 F H S supply amb

surfaces delimiting the film film or lubricant housing shaft, solid lubricant supply ambient medium

Acronyms EHD FE, FEM MFT, MOFT MOFP SAE

Elastohydrodynamic finite element method minimum (oil) film thickness maximum oil film pressure Society of Automotive Engineers

Other notations various various

various

values a priori known (boundary conditions) values at the time step preceding the current time step values at two time steps before the current time step

xxvii

1 The Lubricant

The role of the lubricant in internal combustion engines, and more generally in any lubricated mechanism, is to reduce friction, to evacuate heat and to hold in suspension and carry away the solid or liquid impurities formed during an operation. Reduction in friction is obtained simply by maintaining a lubricant film between the moving surfaces. This prevents them from making contact with each other, and as a result, prevents wear from occurring on the mechanical pieces. It is also very important that the lubricant can effectively evacuate the heat that comes from the friction between the various pieces, which make contact with each other, or from the combustion, while still maintaining its initial properties. Additionally, it is of great importance that the lubricant should remain effective when exposed to water, combustion acids or any other contaminant particles. Obtaining a lubricant that can fulfill these three fundamental functions is a complex task requiring a compromise between the base oil properties and the influence of the different additives intended to improve the behavior of these base oils. 1.1. Description of lubricants Depending on their origin, base oils can be classified into two main categories: mineral and synthetic. Mineral oils are obtained from crude oil and may in turn be divided into several groups, depending on the origins of the oil and the refining procedures used. Synthetic base oils are products obtained artificially, and are superior to mineral base oils, but are also more expensive than them. They perform better thermally, are more stable when faced with oxidation and are less volatile, etc. Some synthetic

2

Hydrodynamic Bearings

base oils may be mixed with mineral oils in order to obtain the required properties for volatility at high temperature and/or viscosity at low temperature. The most important requirements for a good lubricant are: – reduced volatility at operating conditions; this property is inherent to the base oil and cannot be improved by additives; – good flow characteristics at operating temperatures; this property largely depends on the base oil used, but may also be improved by adding dispersant additives and additives which modify viscosity; – good stability or capability for the lubricant to maintain its characteristics for a long time; even if this property depends to a certain extent on the base oil used, it is mostly controlled by the additives. The stability of the lubricant depends on the environment in which it is used (temperature, potential oxidation, contamination with water, with unburnt combustible, combustion soot or various acids). This is where additives really make a major contribution in the improvement of performance and the lifetime of lubricants; – compatibility with the system’s other materials; this property depends partially on the base oils, but the additives can also have a major influence here. In internal combustion engines, the performance of a lubricant is judged on its capacity to reduce friction, resist oxidation, minimize the formation of deposits and prevent corrosion and wear. To fulfill all these conditions, the lubricants used for the lubrication of internal combustion engines incorporate several types of additives: antiwear and extreme-pressure agents, antirust agents and corrosion inhibitors, detergents and dispersants, additives lowering the pour point and improving the viscosity index, antifoaming and demusifying/emulsifying agents and antioxidants. 1.2. The viscosity According to the standard ISO 3104-1994 [ISO 94; NOR 96], “the viscosity of a liquid is the property of the liquid related to the level of resistance with which its molecules oppose a force within it which causes them to slide”. Since viscosity determines the loading capacity, lubricant film thickness and losses through friction, knowledge and modeling of its variation in function of different parameters are essential to the study of bearings.

The Lubricant

3

Figure 1.1. The viscosity “concept”

A good understanding of the “concept” of viscosity may be gained by examining Figure 1.1, where a plate moves at the uniform velocity U on a lubricant film. This lubricant adheres to both the stationary and the moving surface. The lubricant may be pictured as several layers, each of these being moved by the layer above it at a speed, which is a fraction of the velocity U, proportional to the distance to the moving bounding surface (Newtonian fluid hypothesis). The force F, which needs to be applied in order to move the moving surface, is proportional to the viscosity of the fluid. Therefore, viscosity may be evaluated by measuring this force. Viscosity determined in this manner depends solely on internal friction within the lubricant and is known as dynamic viscosity µ. If at a distance y from the stationary surface the velocity of the particle of fluid is u, then the tangential constraint or the shear constraint τxy is given, in unidimensional formulation, by:

τ xy = μ

du dy

[1.1]

The force F is given by the integration of the shear constraint across the film thickness. The dimensional equation of the dynamic viscosity is: [µ] = M L-1 T-1

[1.2]

In the International System of Units (SI), the unit used is the Pascal-second (1 Pa.s = 1 kg/(m.s)). In practice, it is very often easier to measure viscosity with equipment where the measurement is influenced by the density of the fluid. A new

4

Hydrodynamic Bearings

unit might, therefore, be introduced, which is the ratio of the dynamic viscosity coefficient to the density of the fluid, named kinematic viscosity, and notated ν:

ν = µ/ρ

[1.3]

The dimensional equation of the kinematic viscosity is: [ν ] = L2 T-1

[1.4]

The unit for kinematic viscosity in the centimeter-gram-second (CGS) system of units is the Stokes (St), but the unit centistokes (cSt) is often used. In the SI system, the unit is expressed in m2/s (1 cSt = 1 mm2/s). In bearings under severe running conditions, viscosity can vary with temperature, pressure and even shear rate. 1.2.1. Viscosity – temperature relationship For oils used in the lubrication of internal combustion engines, viscosity has a decreasing exponential variation with temperature. It drops very rapidly at low temperatures and more slowly at high temperatures. In the literature, there are several analytical approximations of viscosity with temperature [SEE 06]. The following can be used: – the Reynolds relation:

μ = μ 0 e − β (T −T0 )

[1.5]

where µ0 is the dynamic viscosity at T0; – the McCoull and Walther equation [MCC 21]:

log10 ⎡⎣log10 (υcSt + c1 ) ⎤⎦ = c2 − c3 log10 T° K

[1.6]

where vcSt is the kinematic viscosity in centistokes and T°K the absolute temperature. c1, c2, and c3 are constants, which depend on the lubricant; – the Roelands equation [ROE 66]:

⎛ T ⎞ log10 [ log10 μcP + 1.200] = log10 G0 − S0 log10 ⎜1 + °C ⎟ ⎝ 135 ⎠

[1.7]

The Lubricant

5

where µcP is the dynamic viscosity in centipoises (1 cP = 10-2 dyn.s/cm2 = 10-3 Pa.s) and T°C is the temperature in °C; G0, which gives the viscosity grade, and S0 are dimensionless constants, which depend on the lubricant. This relation can be rewritten in the form [HAM 04]:

μ = μ∞

⎛ T ⎞ G0 ⎜ 1+ ° C ⎟ 10 ⎝ 135 ⎠

− S0

[1.8]

which directly gives the dynamic viscosity in Pa.s in function of the temperature in °C where μ ∞ = 6.31 × 10 − 5 Pa.s ; – the Slotte equation [SLO 81]:

μ=

a

(T − Tc )m

[1.9]

where Tc is the pour temperature, and a and m are coefficients dependent on the lubricant. The algorithms presented in the chapters which follow may include any model. In the numerical applications, the choice has often been made to use a modified Reynolds relation with a corrective viscosity µc:

μ = μ 0 e − β (T −T0 ) + µc

[1.10]

where µ is the dynamic viscosity at the temperature T, β is the thermoviscosity coefficient, and (µ0 + µc) is the viscosity at the temperature T0. Figure 1.2 shows the variation of viscosity with temperature for the various laws mentioned above. Note that the modified Reynolds relation is close to the McCoull and Walther law, and also to the Roelands law for viscosities beyond the common point (this corresponds to a temperature of 100°C for this example). Even Slotte’s law, despite its simplicity, gives a good evaluation of viscosity. Reynolds’ laws, in original form or modified, diverge from the other laws for low temperatures.

6

Hydrodynamic Bearings

Figure 1.2. Viscosity variation for SAE 30 engine oil with respect to temperature: Reynolds law: β = 0.031 °C-1; Roelands law: S0 = 1.5 G0 = 4.7; Reynolds law with limit: β = 0.031 °C-; μc = 0.0007 Pa.s; McCoull and Walther law: ρ = 840 kg/m3; c1 = 1.1131 cSt; c2 = 10.685; c3 = 4.1589 ; Slotte law: Tc = - 25 °C; a = 106; m = 3.88

1.2.2. Viscosity – pressure relationship Variation in viscosity with respect to pressure can be modeled using one of the following laws: – the Barus law [BAR 93]:

μ = μ 0 eα ( p − p 0 )

[1.11]

where α is the piezoviscosity coefficient (in Pa-1) and µ0 is the dynamic viscosity at the pressure p0 and temperature T0 ; – the power law: b

μ = μ0 (1 + ap )

[1.12]

with two coefficients a (in Pa-1) and b, which are dependent on the physico-chemical properties of the lubricant;

The Lubricant

7

– the Roelands equation [ROE 66]: z

p ⎞1 ⎛ log10 μcP + 1.200 = log10 μ0 + 1.200 ⎜1 + ⎟ 2000 ⎠ ⎝

(

)

[1.13]

where µcP is the dynamic viscosity in centipoises and p the pressure in (kg)force/cm2 (= 98.100 Pa); µ0, is the viscosity in centipoises at zero pressure; z1 is the viscositypressure index, a dimensionless constant, which varies between 0.40 and 1.20 and which, for lubricants, does not vary substantially with temperature; this relation may be rewritten in the form [LAR 00]:

μ ( p) = μ0 e

( ln μ

0

⎡ ⎛ p + 9.671 ⎢ −1+ ⎜ 1+ ⎜ cp ⎢ ⎝ ⎣⎢

)

z

⎞ 1⎤ ⎟ ⎥ ⎟ ⎥ ⎠ ⎦⎥

[1.14]

which directly gives the dynamic viscosity in Pa.s in function of the pressure in Pa where c p = 1.96 × 108 Pa.s , and the reference viscosity µ0 is expressed in Pa.s. 1.2.3. Viscosity – pressure – temperature relationship The relations presented above do not take the simultaneous variations of temperature and pressure into account. For them to be applicable to real lubricants, the coefficients of these relations should also vary with pressure for temperature, and with temperature for pressure. On the basis of experiments that he conducted, Roelands suggests a combination of the laws [1.8] and [1.14]: log10 μ cP + 1, 200 = G0

(1 + p

2000 )

C 2 log10 (1+ T° C 135 ) + D2

(1 + T°C

135 )

S0

which may be rewritten [LAR 00] as:

μ ( p, T ) = μ 0 ( T ) e

[ln μ0 (T ) + 9.671]⎡⎢−1+ (1+ p ⎣

cp

C ln (1+T° C 135 )+ DZ

)Z

⎤ ⎦⎥

[1.15]

where:

μ0 (T ) = 10−4.2+G (1+T 0

°C

135)

− S0

[1.16]

8

Hydrodynamic Bearings

Here, the viscosity is in Pa.s, the temperature is in C, and the pressure is in Pa. This law gives the viscosity in function of the four constants G0, S0, CZ and DZ. Only four experimental measurements are necessary to obtain these constants. Figure 1.3 shows the viscosity variation of a naphthenic-paraffinic oil (VG32), the coefficients of which are G0 = 4.49 ; S0 = 1.68 ; CZ = 0.010 and DZ = 0.692 [LAR 00].

Figure 1.3. Viscosity variation for a naphthenic-paraffinic oil (VG32) with respect to temperature and pressure

Relations [1.15] and [1.16] are complex. As a result, Reynolds law [1.5] or modified Reynolds law [1.10] for thermoviscosity and Barus law [1.11] or power law [1.12] for piezoviscosity are more often used for calculations regarding engine bearings. These laws may be combined in several different ways. Figure 1.4 shows that a variation close to that given by the Roelands law may be found (same parameters as for Figure 1.3) if the term for corrective viscosity µc is placed outside the product of the thermoviscosity and piezoviscosity factors.

The Lubricant

9

Figure 1.4. Viscosity with respect to temperature and pressure:a) Roelands law; −α T − T + β p − p −α T − T β p− p b) μ ( p, T ) = ⎡ μ0e ( 0 ) + μc ⎤ e ( 0 ) c) μ ( p, T ) = μ0e ( 0 ) ( 0 ) + μc ; ⎣ ⎦ b −α T −T + β p − p0 ) −α (T − T0 ) + μc ⎤ ⎡⎣1 + a ( p − p0 ) ⎤⎦ μ ( p, T ) = μ0e ( 0 ) ( + μc d) μ ( p, T ) = ⎡ μ0e ⎣ −α T − T ⎦ b ( 0) ⎡⎣1 + a ( p − p0 ) ⎤⎦ + μ c µ0 = 0.51 Pa.s ; p0 = 0 ; T0 = 80 °C ; e) μ ( p, T ) = μ 0e

µc = 0.0007 Pa.s. α = 0.0375 °C-1; β = 1.43.10-8 Pa-1 ; a = 4.2.10-9 Pa-1 ; b = 5

If the pressure in the lubricant becomes significant, the laws presented above give viscosities, which are notably different at low temperatures. Figure 1.5 shows the evolution of the viscosity over a range of temperatures from 0 to 200°C and a range of pressures from 0 to 320 MPa, with the same parameters as for Figure 1.4. At a temperature of 20°C and a pressure of 320 MPa, the Roelands law gives a viscosity of 180 Pa.s, whereas the combination of the Reynolds law corrected for thermoviscosity with the power law for piezoviscosity only gives 4.06 Pa.s. However, in areas of high temperature, the results correspond well.

10

Hydrodynamic Bearings

Figure 1.5. Viscosity with respect to temperature and pressure: b −α (T −T0 ) ⎡⎣1 + a ( p − p0 ) ⎤⎦ + μc a) Roelands law ; b) μ ( p, T ) = μ0e

1.2.4. Non-Newtonian behavior The Newtonian fluid hypothesis mentioned at the beginning of section 1.2 presumes that there is a linear variation of the shear constraint τ in relation to the shear rate γ . However, experimental studies have shown that for high shear rates, the assumption of a linear variation of τ with respect to γ is no longer valid. This is known as non-Newtonian behavior, and it may also be represented by a variation in the dynamic viscosity with respect to the shear rate. In the bearings of internal combustion engines, the non-Newtonian effect diminishes viscosity when shear rate increases. While this effect is generally reversible, it may also be permanent. If this is the case, the degradation of the lubricant is due to the rupture of certain molecular chains, and especially those of the polymer additives. Pseudoplastic and viscoplastic fluids (Figure 1.6) feature among the types of fluids often used for lubrication. Threshold fluids (Bingham and Herschel–Bulkey fluids) are rarely considered for the lubrication of engine bearings. In fact, the threshold constraint of these fluids and of fats in particular is very low – at approximately 300 Pa [MUT 86] – when faced with levels of shear constraints within the lubricant film which reach 105 Pa. K and n are rheological characteristics, experimentally determined. The coefficient K is the fluid consistency index, and n is the flow index. Depending on the value of n, fluids may be classified into two groups: – shear thinning fluids for n < 1; – shear thickening fluids for n > 1.

The Lubricant

11

Figure 1.6. Non-Newtonian rheological behavior laws

Several rheological models have been developed to show these behaviors. The simplest and perhaps the most widely used of these is the Oswald model (also known as the power law) which may be written in the following form:

μ = K γ n −1 or τ = K γ n 2

[1.17]

2

⎛ ∂u ⎞ ⎛ ∂w ⎞ where γ = ⎜ ⎟ +⎜ ⎟ . ⎝ ∂y ⎠ ⎝ ∂y ⎠ The consistency K and the index n are rheological characteristics, which are determined experimentally. For a Newtonian fluid, K = µ and n = 1. This law does not provide a good representation of the variation in viscosity for low or high shear rates. The Cross equation [CRO 65] describes the non-Newtonian behavior of fluids:

μ = μ∞ +

μ0 − μ∞ γ 1+ γc

⎛ ⎜ μ − μ∞ or τ = ⎜ μ∞ + 0 γ ⎜ 1+ ⎜ γc ⎝

⎞ ⎟ ⎟γ ⎟ ⎟ ⎠

[1.18]

where µ0 is the dynamic viscosity at zero shear rate and μ∞ is the dynamic viscosity for an infinite shear rate. γ c is the shear rate at which the viscosity is medium

12

Hydrodynamic Bearings

between µ1 and µ2. To achieve a realistic representation, γ c should vary with temperature. A similar equation has been suggested by Gecim [GEC 90]:

μ = μ1

⎛ k + μ 2γ ⎞ k + μ2 γ or τ = ⎜ μ1 ⎟γ k + μ1γ ⎝ k + μ1γ ⎠

[1.19]

where k is a stability coefficient, which increases with temperature, and µ1 and µ2 are respectively the dynamic viscosities at low and high shear rates (Figure 1.7).

Figure 1.7. Viscosity variation with respect to shear rate for a 10W40 oil with different additives [GEC 90]: oil A at 75°C: μ1 = 0.0245 Pa.s ; μ2 = 0.0139 Pa.s ; k = 550 Pa ; K = 4.38 Pa.s ; n = - 0.18 ; oil B at 150°C: μ1 = 0.0043 Pa.s ; μ2 = 0.024Pa.s ; k = 80,000 Pa ; K = 34.98 Pa.s ; n = - 0.92

1.3. Other lubricant properties In the study of lubricated systems, other lubricant properties should be considered, especially if thermal phenomena are taken into account. In order to solve the equation for energy in the lubricant film (see Chapter 3 of [BON 14]), the density ρ, the specific heat Cp and the thermal conductivity coefficient k all need to be known. In much the same way as with viscosity, these parameters vary with temperature.

The Lubricant

13

The variation in density with temperature may be modeled by [LAR 00]:

ρ (T ) = ρT0 ⎡⎣1 − λρ (T − T0 ) ⎤⎦

[1.20]

where ρT0 is the density at T0 (in kg/m3) and λρ is a correction coefficient for the density with respect to temperature (which varies depending on the oil used). This coefficient varies with pressure, following the law:

λρ ( p ) = λρ 0 e−c ( p − p )

[1.21]

0

where c is a constant, which is approximately equal to 1.5 10-9 Pa-1, and λρ 0 is the correction coefficient of density with respect to the reference pressure. The variation in specific heat with respect to temperature and pressure may be written in the following form [LAR 00]: C p ( p, T ) =

⎡ a ( p − p0 ) ⎤ ⎡⎣1 + β ( p ) (T − T0 ) ⎤⎦ ⎢1 + 1 ⎥ ρ ( p, T ) ⎢⎣ 1 + a2 ( p − p0 ) ⎥⎦

ρ0C p 0

[1.22]

where Cp is the thermal capacity in J/(Kg.°C) and ρ0 and Cp 0 are the density and the thermal capacity at the reference temperature and pressure. The parameter β depends on pressure and may be expressed in the form: 2 β ( p ) = β 0 ⎡⎢1 + b1 ( p − p0 ) + b2 ( p − p0 ) ⎤⎥

⎣

⎦

[1.23]

The constants a1, a2, b1 and b2 depend on the lubricant. The variation in the thermal conductivity coefficient with temperature may be modeled by:

k (T ) = k0 ⎡⎣1 − b3 (T − T0 ) ⎤⎦

[1.24]

where k is the thermal conductivity in W/(m °C), k0 is the conductivity of the oil at the reference temperature T0, and b is a coefficient which is dependent on the lubricant. The variation in the thermal conductivity coefficient with pressure may be modeled in the same way [LAR 00]:

14

Hydrodynamic Bearings

⎡ c1 ( p − p0 ) ⎤ k ( p ) = k0 ⎢1 + ⎥ ⎣⎢ 1 + c2 ( p − p0 ) ⎦⎥

[1.25]

the constants c1 and c2 depend on the lubricant. Table 1.1 gives the coefficients of the relations above for two basic lubricants. Values for other lubricants are given by Larsson et al. [LAR 00]. Naphthenic mineral oil

S0

G0

CZ

DZ

temperature (°C) pressure (MPa) α (GPa-1) ρ at 15°C (kg/m3)

λρ0 at 40°C (°C-1)

k0 (W m-1 °C-1) c1 c2 T0 (°C) a1

b3

1.60

5.13

-1.01

0.881

1.31

4.76

20 0 40

20 400 35

80 0 19

80 400 16

20 0 26

20 400 19

6.5 10-4

917

0.118 1.54 0.33 0.00054

ρ0 Cp0 (J m-3 °C-1)

β0

22

a2

b2

0.56

b1

Paraffinic mineral oil

1.64 10-6 0.80

0.58

9.9 10-4 -0.46

0.229 0.541 80 0 18

80 400 14

884

6.6 10-4

0.137 1.72

0.54 0.00054

22 0.47

1.71 10-6 9.3 10-4 0.81

1.4

-0.51

Table 1.1. Coefficients for relations [1.21] to [1.25] for two basic oils [LAR 00]

1.4. Lubricant classification and notation There is no single way of classifying and notating lubricants. Each company tends to have its own system for doing this. However, the international classification for industrial oils is the ISO-VG classification, which contains 18 classes. The numerical system attached to a lubricant corresponds to its kinematic viscosity in centistokes (mm2/s) at 40°C (for example, the oil ISO-VG 32 has a kinematic viscosity of 32 cSt at 40°C). The classification adopted for automobile engine lubricants is that of the Society of Automotive Engineers (SAE). This classification divides the lubricants into two categories: engine oils and transmission oils. In each category, the oils are classified into grades according to dynamic viscosity at two reference temperatures: winter temperature (0°F = - 17.8°C) and summer temperature (210°F = 100°C). In addition, SAE oils may be divided into two categories: – monograde oils: the variation in viscosity corresponds to an SAE grade;

The Lubricant

15

– multigrade oils: the viscosity only varies slightly with temperature (these oils contain additives which improve the viscosity index); as a result, these oils are classified with two SAE grades, one of which must contain the letter W (winter). When an oil has two SAE grades, which have an indication of W, the lowest of these should be used. Table 1.2 shows the SAE classification of engine oils. SAE grade

Kinematic viscosity at 100°C [cSt] Minimum

Maximum

Dynamic viscosity [Pa.s] at minimum running temperature

0W

3.8

–

32.5 at – 30°C

5W

3.8

–

35 at – 25°C

10W

4.1

–

35 at – 20°C

15W

5.6

–

35 at – 15°C

20W

5.6

–

45 at – 10°C

25W

9.3

–

60 at – 5°C

20

5.6

9.3

–

30

9.3

12.5

–

40

12.5

16.3

–

50

16.3

21.9

–

60

21.9

26.1

–

Table 1.2. SAE classification for engine oils

1.5. Bibliography [BAR 93] BARUS C., “Isothermals, isopiestics and isometrics relative to viscosity”, American Journal of Science, vol. 45, pp. 87–96, 1893. [BON 14] BONNEAU D., FATU A., SOUCHET D., Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [CRO 65] CROSS M.M., “Rheology of non-Newtonian fluids: a new flow equation for pseudo plastic systems”, Journal of Colloid Science, vol. 20, pp. 417–437, 1965. [GEC 90] GECIM B.A., “Non-Newtonian effects of multigrade oils on journal bearing performance”, STLE Tribology Transaction, vol. 33, pp. 384–394, 1990. [HAM 04] HAMROCK B.J., SCHMID S.R., JACOBSON B.O., Fundamentals of Fluid Film Lubrication – Second edition, Marcel Dekker Inc., New York – Basel, 2004.

16

Hydrodynamic Bearings

[ISO 94] ISO 3104-1994, Petroleum products – Transparent and opaque liquids – Determination of kinematic viscosity and calculation of dynamic viscosity, International Organization for Standardization, Geneva, Swiss, 1994 [LAR 00] LARSSON P.O., ERIKSSON E., SJOBERG M., et al., “Lubricant properties for input to hydrodynamic & elastohydrodynamic lubrication analyses”, Proceedings of the Institution of Mechanical Engineers, vol. 214, part J, pp. 17–27, 2000. [MCC 21] MC COULL N., WALTHER C., “Viscosity-temperature chart”, Lubrication, The Texas Company, New York, vol. 7, no. 6, 1921. [MUT 86] MUTULI S., Détermination du champ de vitesse dans les films lubrifiants newtoniens et non-newtoniens, Doctorate thesis, University of Poitiers, France, 1986. [NOR 96] NORME NF EN ISO 3104, AFNOR, Produits pétroliers – Liquides opaques et transparents – Détermination de la viscosité cinématique et calcul de la viscosité dynamique, La Plaine Saint-Denis, France, 1996. [ROE 66] ROELANDS C.J.A., Correlation aspects of the viscosity-temperature-pressure relationship of lubricating oil, Doctorate thesis, University of Technology, Delft, Netherlands, 1966. [SEE 06] SEETON C.J., “Viscosity-temperature correlation for liquids”, Tribology Letters, vol. 22, no. 1, pp. 67–78, 2006. [SLO 1881] SLOTTE K.F., Wiedemanns Annalen der Physik und Chemie, Leipzig, Germany, vol. 14, pp. 13–23, 1881.

2 Equations of Hydrodynamic Lubrication

2.1. Hypothesis The generalized equation of thin viscous films is deduced from the Navier– Stokes and continuity equations. The specific form of the domain occupied by the thin film of the lubricant fluid makes it possible to obtain an equation using partial derivatives relative to pressure alone, and to eliminate the three functions which are components of the velocity vector. The developments which have led to the choices of the assumptions given below are detailed in Frêne et al. [FRE 90, FRE 97]: – the fluid medium is continuous; – the fluid is in general Newtonian. This means that the shear constraint is proportional to the shear rate; the proportionality constant – the viscosity µ – is not necessarily constant and may vary with pressure (piezoviscous fluid) and/or temperature (thermoviscous fluid). When the fluid has a non-Newtonian behavior, there is assumed to be a scalar-type relation between its viscosity and the invariants of the shear rate tensor (see Chapter 1); – the flow is laminar; – the exterior mass forces and the inertia forces within the lubricant film are negligible; – there is no slipping between the lubricant film and the bounding walls. This assumption is not unavoidable: section 2.5 presents the case where the shear constraint of the fluid in contact with the bounding walls is limited by a threshold value, which implies slippage at the walls; – the curvature radii of the bounding walls are large in relation to the mean thickness of the film;

18

Hydrodynamic Bearings

– the thickness of the film is very small in relation to the other dimensions of the domain occupied by the lubricant fluid, for example with width and diameter in the case of a shaft bearing. In the following sections the equations are presented using an extended notation in order to facilitate the reading and their discretization and transformation into numerical algorithms. In 1989, Booker published the same developments and equations using a powerful compact Cartesian tensor notation (index notation with Einstein’s summation convention) [BOO 89]. 2.2. Equation of generalized viscous thin films Figure 2.1 shows the thin film domain occupied by the lubricant. It is delimited by two bounding walls 1 and 2. The basis for projection x, y, z is chosen such that the direction y corresponds to the direction of the thickness of the film. A point M within the film has coordinates x, y and z. A point M1 on the bounding wall 1 has a coordinate H1 following y and has velocity components U1, V1 and W1 following the directions x, y and z. In the same way, a point M2 on the bounding wall 2 has a coordinate H2 following y and has velocity components U2, V2 and W2. In function of the pressure, the temperature and the shear rates, the viscosity of the lubricant may vary between one point and another, and over time, and will be denoted as µ(x, y, z, t).

Figure 2.1. Axis system and notations

Equations of Hydrodynamic Lubrication

19

In what follows, it will be assumed that:

I n ( x, y , z , t ) =

J n ( x, z , t ) =

ξn ∫H1 μ ( x, ξ , z, t ) d ξ

[2.1]

ξn dξ μ ( x, ξ , z , t )

[2.2]

y

H2

∫H

1

The pressure and the components of the velocity within the film are a function of the point M and time and are denoted as, respectively, p(x, y, z, t), u(x, y, z, t), v(x, y, z, t) and w(x, y, z, t). When the simplifying assumptions given at the beginning of the chapter are taken into account, the Navier equations are reduced to the following form:

⎧ ∂p ∂ ⎛ ∂u ⎞ ⎪ = ⎜μ ⎟ ⎪ ∂x ∂y ⎝ ∂y ⎠ ⎪⎪ ∂p ⎨ =0 ⎪ ∂y ⎪ ∂p ∂ ⎛ ∂w ⎞ ⎪ = ⎜μ ⎟ ⎩⎪ ∂z ∂y ⎝ ∂y ⎠

[2.3]

It is then possible to write the expressions of the components following x and y of the velocity at the point M as a function of the components of the pressure gradient and the integrals I0, I1, J0 and J1 at this point:

⎧ I 0 J1 ⎞ U 2 − U1 ∂p ⎛ I 0 + U1 ⎪u ( x, y , z , t ) = ⎜ I1 − ⎟+ J0 ⎠ J0 ∂x ⎝ ⎪ ⎨ I 0 J1 ⎞ W2 − W1 ∂p ⎛ ⎪ I 0 + W1 ⎪ w( x, y , z , t ) = ∂z ⎜ I1 − J ⎟ + J 0 ⎠ 0 ⎝ ⎩

[2.4]

The equation of mass conservation within the fluid is written as:

∂ρ ∂ ( ρ u ) ∂ ( ρ v) ∂ ( ρ w) + + + =0 ∂t ∂x ∂y ∂z where the density is a function ρ (x, y, z, t) of the point and time.

[2.5]

20

Hydrodynamic Bearings

By integrating this equation across film thickness, using expression [2.4] of u and w, the generalized equation of the mechanics of the thin viscous films is obtained: ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ∂ ∂H 2 ∂H + ρ1U1 1 ⎜ G ⎟ + ⎜ G ⎟ = [U 2 ( R2 − F ) + U1F ] − ρ2U 2 ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂x ∂x ∂ ∂H 2 ∂H1 + [W2 ( R2 − F ) + W1F ] − ρ2W2 + ρ1W1 ∂z ∂z ∂z ∂R2 ∂H 2 ∂H1 + − ρ2 + ρ1 + ρ2V2 − ρ1V1 ∂t ∂t ∂t

[2.6]

In this equation, ρ1 and ρ2 represent, respectively, the values of the density at the bounding walls 1 and 2. The functions R, F and G are defined by:

R ( x, y , z , t ) =

y

∫H ρ ( x, ξ , z, t )d ξ

[2.7]

1

F ( x, z , t ) =

G ( x, z , t ) =

1 J 0 ( x, z , t ) H2

∫H

1

H2

∫H

1

R ( x, ξ , z , t ) dξ μ ( x, ξ , z , t )

R( x, ξ , z, t )ξ d ξ − J1 ( x, z , t ) F ( x, z , t ) μ ( x, ξ , z , t )

[2.8]

[2.9]

R2 is the value of R at the bounding wall 2. 2.3. Equations of hydrodynamic for journal and thrust bearings As the curvature of the bounding walls is negligible, one of these – the lower surface in Figure 2.2 – can be chosen as the reference for thickness and can be assimilated to a planar surface. The component V1 of the velocity of a point on this surface will then be nil. The velocity V2 following y of a point of the bounding wall 2 may be notated V without any ambiguity. Since the bounding walls are made of solid materials which may not easily be deformed, the variation in velocity parallel to the walls is negligible with respect to the velocities themselves. This makes it possible to take bounding wall 2 as the reference for velocities in the directions x and z. The components U2 and W2 of the velocity of a point on this surface will be nil and the components following x and z of the velocity of a point on the bounding wall 1 may then be simply notated as U and W.

Equations of Hydrodynamic Lubrication

21

Figure 2.2. Axis system and notations for a developed journal bearing

Taking these new references into account, equation [2.6] is simplified, and takes the form:

∂R ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ∂ (UF ) ∂ (WF ) ∂h + + ρ 2V + 2 − ρ 2 ⎜G ⎟ + ⎜G ⎟ = ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂z ∂t ∂t where h is a function of x, z and t, which represent the thickness of the film. If it is assumed that the velocities U and W are independent of the space variables, this equation becomes:

∂R ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ∂F ∂F ∂h +W + ρ2V + 2 − ρ2 ⎜G ⎟ + ⎜G ⎟ =U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂z ∂t ∂t

[2.10]

2.3.1. Specific case of an uncompressible fluid When the fluid is assumed to be incompressible – which is the assumption generally made for internal combustion engine lubricants – expression [2.10] may be written more explicitly. According to [2.7], R, F and G become:

R ( x, y , z , t ) =

y

∫0 ρ d ξ = ρ y

22

Hydrodynamic Bearings

F ( x, z , t ) =

1 J 0 ( x, z , t )

⎡ G ( x, z , t ) = ρ ⎢ ⎢⎣

h

∫0

h

ρξ

∫0 μ ( x, ξ , z, t ) d ξ =

ρ J1 ( x, z, t ) J 0 ( x, z , t )

J 2 ( x, z , t ) ⎤ ξ2 dξ − 1 ⎥ μ ( x, ξ , z , t ) J 0 ( x, z, t ) ⎥⎦

which gives:

⎡ ∂ ( J1 / J 0 ) ⎤ ∂ ( J1 / J 0 ) ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ +W +V⎥ ⎜ G ⎟ + ⎜ G ⎟ = ρ ⎢U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂z ⎣⎢ ⎦⎥

[2.11]

2.3.2. Standard Reynolds equation for a journal bearing: general case In a cylindrical journal bearing (Figure 2.3), the domain occupied by the lubricant fluid is delimited by the bounding wall of the housing on one side and by the bounding wall of the shaft on the other. Upon initial examination, these bounding walls are cylindrical. In order to apply the formulation presented above, one of the surfaces must be taken as a reference in order to define the thickness of the film. For engine bearings, the surface of the shaft is chosen for this purpose. Thus, the parameter x represents the curvilinear abscissa in the circumferential direction of the shaft surface. The velocity U is the peripheral velocity of this same surface with respect to the housing. As the thickness h of the lubricant film is small with respect to the bearing radius, the circumference of the housing differs only slightly from that of the shaft, and x also represents the curvilinear abscissa of the housing surface. The shaft is assumed not to have any axial movement: thus, the component W for velocity is nil. Therefore, relation [2.10] takes the following form:

⎡ ∂ ( J1 / J 0 ) ⎤ ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ +V⎥ ⎜ G ⎟ + ⎜ G ⎟ = ρ ⎢U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ⎣⎢ ⎦⎥

[2.12]

By substituting the product of the bearing radius R and the angular coordinate

θ in place of the curvilinear abscissa x, this relation may also be written as:

1 ∂ R 2 ∂θ

⎡ ∂ ( J1 / J 0 ) ⎤ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ +V ⎥ ⎜G ⎟ + ⎜ G ⎟ = ρ ⎢ω ∂θ ⎝ ∂θ ⎠ ∂z ⎝ ∂z ⎠ ⎢⎣ ⎥⎦

[2.13]

Equations of Hydrodynamic Lubrication

23

where ω represents the angular velocity of the shaft with respect to the bearing housing.

Figure 2.3. Cylindrical journal bearing: notations

2.3.3. Reynolds equation for a thrust bearing: general case To guide the rotating parts of internal combustion engines, smooth-surfaced circular thrust bearings are often used in association with cylindrical bearings. This is the case, for example, for holding the crankshaft in the axial direction. The domain occupied by the film in a circular thrust bearing and the main notations are presented in Figure 2.4. The form of this domains leads to the rewriting of the Reynolds equation in cylindrical coordinates (r, θ):

⎡ ∂ ( J1 / J 0 ) ⎤ ∂p ⎞ ∂ ⎛ 1 ∂p ⎞ ∂ ⎛ +W⎥ ⎜ Gr ⎟ + ⎜G ⎟ = ρ r ⎢ω ∂r ⎝ ∂r ⎠ ∂θ ⎝ r ∂θ ⎠ ∂θ ⎢⎣ ⎥⎦

[2.14]

where ω is the angular velocity of the bounding wall 1 with respect to the bounding wall 2.

24

Hydrodynamic Bearings

Figure 2.4. Circular thrust bearing: notations

2.3.4. Equation of volume flow rate When the simplifications introduced in the preceding sections are applied, the expressions of the components u and w of velocity within the film take the following form:

⎧ ⎛ I 0 J1 ⎞ I0 ⎞ ∂p ⎛ ⎪u ( x, y , z , t ) = ⎜ I1 − ⎟ + U ⎜1 − ⎟ ∂x ⎝ J0 ⎠ J0 ⎠ ⎪ ⎝ ⎨ I 0 J1 ⎞ ∂p ⎛ ⎪ ⎪ w( x, y , z , t ) = ∂z ⎜ I1 − J ⎟ 0 ⎠ ⎝ ⎩

[2.15]

Consider a curve arc AB (Figure 2.5). The volume flow rate through this arc will be given by:

where x and z are, respectively, the unit vectors of the circumferential and axial directions and n is the normal vector of the arc AB in M.

Equations of Hydrodynamic Lubrication

25

Figure 2.5. Flow rate across a curve arc

2.3.5. Equations of hydrodynamic for journal and thrust bearings lubricated with an isoviscous uncompressible fluid In many cases, the operating conditions are sufficiently moderate in terms of velocity and loading for behavior of the lubricant to be considered to be isoviscous. With this assumption, equations [2.12] and [2.13] will take a more explicit form. The integrals J0 and J1 become:

⎧ ⎪ J 0 ( x, z , t ) = ⎪ ⎨ ⎪ J ( x, z , t ) = ⎪⎩ 1

h

1

h

∫0 μ d ξ = μ hξ

∫0 μ

dξ =

h2 2μ

and G may be written as:

⎡ hξ2 J2⎤ h3 G ( x, z , t ) = ρ ⎢ dξ − 1 ⎥ = ρ 0 J 0 ⎥⎦ 12μ ⎣⎢ μ

∫

In this way, the simplified form of the Reynolds equation is obtained for a bearing lubricated with an isoviscous uncompressible fluid:

∂ ⎛ h3 ∂p ⎞ ∂ ⎛ h3 ∂p ⎞ ∂h + 2V ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ = U ∂x ⎝ 6 μ ∂x ⎠ ∂z ⎝ 6 μ ∂z ⎠ ∂x

26

Hydrodynamic Bearings

or again by replacing the squeeze velocity V with the temporal derivative of the film thickness:

∂ ⎛ h3 ∂p ⎞ ∂ ⎛ h3 ∂p ⎞ ∂h ∂h +2 ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ = U ∂x ⎝ 6 μ ∂x ⎠ ∂z ⎝ 6 μ ∂z ⎠ ∂x ∂t

[2.16]

If the integrals I 0 and I1 are replaced with their explicit expressions:

⎧ ⎪ I 0 ( x, y , z , t ) = ⎪ ⎨ ⎪ I ( x, y , z , t ) = ⎪⎩ 1

y

y

1

y

y2 ξ dξ = 2μ μ

∫0 μ d ξ = μ ∫0

the flow rate through an arc AB may be written as:

Qv =

∫

B ⎛ h3 ∂p ⎞ h3 ∂p Uh ⎞ + ⎜⎜ − ⎟⎟ dz + ⎜⎜ − ⎟⎟ dx A A 2 ⎠ ⎝ 12 μ ∂x ⎝ 12 μ ∂z ⎠ B⎛

∫

[2.17]

For a circular thrust bearing, the Reynolds equation and the volume flow rate in cylindrical coordinates are written, respectively:

∂ ⎛ h3 ∂p ⎞ ∂ ⎛ h3 1 ∂p ⎞ ∂h ⎞ ⎛ ∂h +2 ⎟ r ⎟+ ⎜⎜ ⎜⎜ ⎟⎟ = r ⎜ ω ⎟ ∂r ⎝ 6 μ ∂r ⎠ ∂θ ⎝ 6 μ r ∂θ ⎠ ∂t ⎠ ⎝ ∂θ Qv =

∫

B ⎛ h3 ∂p ⎞ h3 1 ∂p rω h ⎞ dr − + + ⎜ ⎟ ⎜− ⎟ rdθ A ⎜ 12 μ r ∂θ A ⎜ 12 μ ∂r ⎟ 2 ⎟⎠ ⎝ ⎝ ⎠ B⎛

∫

[2.18]

[2.19]

2.4. Film rupture; second form of Reynolds equation Experimental observations in the case of bearings with a transparent exterior bounding wall, operating in stationary conditions, show that the lubricant fluid does not occupy all of the space between the bounding walls.

Equations of Hydrodynamic Lubrication

27

Dowson and Taylor [DOW 75] have presented photographs (Figure 2.6) of very structured non-active zones obtained for a bearing with a slightly off-center shaft in stationary operation. Zones filled with gas – most likely the gas from outside the bearing – are separated by zones where fluid occupies the space between the two surfaces. The pitch separating these two filled fringes is virtually constant. Dalmaz [DAL 79] has shown that it is linked to the dimensionless number µU/γ which represents the relationship between the forces due to viscosity and those due to the surface tension γ of the lubricant. If the distance between the two bounding walls increases too much – as, for example, in the case of non-conformal surfaces (gear teeth, roller bearings, cams, etc.) – the width of these filled fringes becomes too small, relative to their thickness, for them to be maintained and a spread of the lubricant is obtained which resembles that shown in the photograph on the right of Figure 2.6.

Figure 2.6. Examples of non-active zones in a lubricant film [DOW 75]

In the case of a bearing with dynamic operation, such structured distribution is unlikely. The “non-active zones” – zones filled with a fluid/gas mixture – occupy positions and quantities of space which change constantly. The schemes in Figure 2.7 show the possibilities for distribution. Since experimental observation of this phenomenon is virtually unattainable, it is not possible to be sure whether one of these structures is more often encountered than the others. In fact, it is quite likely that all of these structures may be simultaneously present in the bearing, constantly evolving from one to the other type. On the basis of the schemes in Figure 2.7, we will attempt to describe the way in which this evolution occurs for non-active zones.

28

Hydrodynamic Bearings

Figure 2.7. Possible structures of non-active zones

If in an initially “active” zone – i.e. a zone where the fluid fills the space between the bounding walls – the pressure p becomes less than the saturating vapor pressure of the fluid, the fluid cannot withstand the tensions to which it is subjected, and loses its cohesiveness. This phenomenon is known as “cavitation”: bubbles of gas appear within the fluid. This gas is a mixture of vapor from the fluid (which implies phase change) and gas that was initially dissolved within the fluid. The appearance of these bubbles progresses rapidly from the starting point of a cavitation

Equations of Hydrodynamic Lubrication

29

“seed” (impurities, microbubbles, etc.). Their growth is described by the Rayleigh– Plesset law [PLE 77, FEN 97]:

RB

d 2 RB dt 2

2

+

p −p 3 ⎛ dRB ⎞ 2γ 4 μ dRB = B − − ⎜ ⎟ ρ ρ RB ρ RB dt 2 ⎝ dt ⎠

[2.20]

where RB is the radius of the bubbles, pB is the pressure in the bubbles, assumed to be equal to the tension of the saturating vapor at operating temperature, γ is the surface tension, ρ is the density and µ is the dynamic viscosity of the lubricant. When these bubbles become sufficiently large, they coalesce to form larger bubbles and very soon there remain only thin lines of lubricant linking the bounding walls (Figure 2.7, scheme 2). Increases in the space between the bounding walls may lead to breakage of these lines, and the fluid gathering into droplets under the influence of the surface tension (Figure 2.7, scheme 3). As a result of the gravity effect or dynamic effects, these drops may be deposited on the bounding walls or, again under the influence of surface tension, layers of more or less uniform thickness may be formed (Figure 2.7, scheme 4). Of course, this is a schematic description of the phenomena. The real structure of the fluid/gas distribution is, very likely, more complex than this. If these cavitation zones which have begun to form within the fluid do not expand too much, their boundaries will not reach the bearing edges and the pressure in the gas phase will remain equal to the saturating vapor pressure. When the pressure p in the surrounding fluid increases under the effect of loading or kinematic conditions applied to the bearing, the phase change conditions for the lubricant from the liquid to the gas phase are reversed. Return to liquid state is accomplished extremely quickly. However, the reabsorption of the initially dissolved gas takes longer than it would for new cavitation conditions to appear once more. Thus, it may be considered that this gas will no longer dissolve in the liquid phase of the lubricant. If the volume occupied by the lubricant vapor is of a sufficient proportion of the overall, the disappearance of this volume leads to very rapid movement of the surrounding liquid, giving it high accelerations and enough energy to create damage when the fluid hits the bounding walls of the shaft or the bearing shell. The resulting damage leads to the appearance of small cavities on the bounding walls, known as “pitting”. However, if the cavitation zones become sufficiently large, they may reach the bearing edges. When the gaseous zones come into contact with the surrounding

30

Hydrodynamic Bearings

atmosphere, gas from outside can enter these zones and cause the pressure within them to rise. This is called “separated” lubricant film. Essentially, the surface tension controls the behavior of the liquid/gas interfaces. Thus, pressure within the fluid is linked to the value of the surface tension and the curvature of the separation surfaces, in accordance with the Laplace law (Rayleigh–Plesset law without the temporal terms). When the pressure and flow conditions lead to the disappearance of the separating zones, the expulsion of the gas toward the outside of the bearing occurs, without any resultant damage. In non-active zones, it may first be considered that the fluid filling the available space between the bounding walls is a homogeneous mixture of lubricant and gas and has a density ρm. Since stresses from shearing remain low in these zones, it may be assumed that the mixture maintains a constant viscosity µm throughout the thickness of the film. The conservation equation for the mass flow rate within the non-active zones is written1:

U

∂ ( ρ m h) ∂ ( ρ m h) +2 =0 ∂x ∂t

1 This equation may be obtained on the basis of equation [3.10], which is valid for a compressible fluid: ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ∂F ∂F ∂R ∂h +W + ρ 2V + 2 − ρ 2 ⎜G ⎟ + ⎜G ⎟ = U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂z ∂t ∂t This equation may be reduced to its second member because the pressure in non-active zones is constant (equal to the saturating vapor pressure pcav or the separation pressure psep): ∂F ∂F ∂R ∂h +W + ρ 2V + 2 − ρ 2 =0 U ∂x ∂z ∂t ∂t and by eliminating the term for axial velocity: ∂F ∂R ∂h + ρ 2V + 2 − ρ 2 =0 U ∂x ∂t ∂t If it is assumed that there is a constant viscosity µm across the film thickness, the expressions of R and F may be simplified: h h ρξ 1 1 ρ h2 ρ h R2 = R ( x, h, z , t ) = dξ = ρ d ξ = ρ h ; F ( x, z , t ) = = J 0 ( x, z , t ) 0 μ m h / μm 2 μm 2 0 which gives: U ∂ ( ρ h ) ∂ ( ρ h) + =0 ∂t 2 ∂x

∫

∫

Equations of Hydrodynamic Lubrication

31

If the filling r is defined by:

r=

ρm h ρ

[2.21]

where ρ is the density of the lubricant, the following is obtained:

U

∂r ∂r +2 =0 ∂x ∂t

[2.22]

In the active zone of the lubricant film, the pressure verifies, depending on assumptions used, one of equations [2.6], [2.10], [2.11], [2.12] or [2.16]. For bearings, equation [2.12] is important:

⎡ ∂ ( J1 / J 0 ) ∂h ⎤ ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ + ⎥ ⎜ G ⎟ + ⎜ G ⎟ = ρ ⎢U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂t ⎥⎦ ⎢⎣ In order to handle this equation and equation [2.22] at the same time, they may be gathered into a single equation, using a universal variable named D which will represent either the pressure p relative to pref, or a parameter linked to filling r:

⎡ ∂ ⎛ ∂D ⎞ ∂ ⎛ ∂D ⎞ ⎤ ⎜G ⎟ + ⎜G ⎟⎥ ⎣ ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ⎦

Φ⎢

⎡ ∂ ( J1 / J 0 ) ∂h ⎛ U ∂D ∂D ⎞ ⎤ = ρ ⎢U + + (1 − Φ ) ⎜ + ⎟⎥ ∂x ∂t ∂t ⎠ ⎥⎦ ⎝ 2 ∂x ⎢⎣

[2.23]

where Φ is an index function. Where Φ is equal to 1, equation [2.12] results. Where Φ is nil, the first member disappears and equation [2.23] becomes:

U

∂ ( J1 / J 0 + D / 2 ) ∂x

+

∂ (h + D) =0 ∂t

As is the case for an isoviscous fluid, the J1/J0 relationship is reduced to h/2 and the following is obtained:

U ∂ ( D + h ) ∂ ( D + h) + =0 ∂x ∂t 2 If D is chosen to equal r–h, then equation [2.21] results.

32

Hydrodynamic Bearings

Thus, a single equation [2.23] has been established, which is valid both for the non-active and the active zones of the film. This equation involves the universal function D and the index function Φ such that:

⎧⎪ D = p − pref ⎨ ⎪⎩Φ = 1

for active zones

[2.24]

where pref is a reference pressure equal to the cavitation or separation pressure, depending on the case involved, and:

⎧D = r − h ⎨ ⎩Φ = 0

for non-active zones

[2.25]

Because the pressure p is always greater than the cavitation or separation pressure and also because the filling r cannot be less than the space h available between the bounding walls, the following is obtained: ⎧ D ≥ 0 in active zones ⎨ ⎩ D < 0 in non active zones

[2.26]

For any point in the film, it will be necessary to verify compatibility between the sign of D and the supposed state, active or non-active, of the film at that point. Equations [2.23], [2.24] and [2.25] and inequation [2.26] together form a set which resembles that obtained for a non-lubricated contact governed by laws for frictionless contact, the condition of partial filling of non-active zones plays the role of the non-interpenetration for contact problems. 2.5. Particular form of the viscous thin film equation in the case of wall slipping The classical theory of lubrication, as has been presented in the sections above, presumes that there is no slipping between the lubricant fluid and the bounding walls. However, recent studies have experimentally demonstrated that for some types of surfaces the non-slippage condition has not been verified. Slipping may be due to the specific properties of the materials used (surface energy or tension) or simply due to the lubricant being unable to tolerate it if the shear constraint exceeds a given level. Since the shear stress always reaches its maximum at one or the other of the bounding walls (due to the parabolic profile of the velocity distribution across the film thickness), it is along the bounding walls that the threshold is first reached. In several recently published studies, the phenomenon of slippage has been used to

Equations of Hydrodynamic Lubrication

33

improve performance in terms of bearing capacity and friction for rigid bearings [FOR 05, MAG 07]. The approach taken to obtain the new form of the Reynolds equation is similar to that followed for the standard equation. Taking into consideration the common assumptions for lubrication theory (see section 2.1), the Navier–Stokes equations may be reduced to:

∂p ∂ ⎛ ∂u ⎞ ⎫ = ⎜ µ ⎟⎪ ∂x ∂z ⎝ ∂z ⎠ ⎪ ∂p ∂ ⎛ ∂v ⎞ ⎪ = ⎜ µ ⎟⎬ ∂y ∂z ⎝ ∂z ⎠ ⎪ ⎪ ∂p =0 ⎪ ∂z ⎭

[2.27]

For it to be possible to integrate the first two equations throughout the thickness of the film, the boundary conditions along the bounding walls need to be defined. To achieve this, it may be considered that each component of the slipping velocity of the bounding wall is proportional to the corresponding shear constraint2 [FOR 05], which gives:

z = 0 : u = U + αsµ ∂u z = h : u = −α h µ ∂z

∂u ∂z

; z =0

; z =h

v = αsµ

∂v ∂z

z =0

∂v v = −α h µ ∂z

[2.28]

z =h

where α a is a slippage coefficient with respect to the shaft and α c is a slippage coefficient with respect to the other surface. The circumferential component u of velocity in the film is then expressed as:

2 This assumption makes the developments which lead to the new form of the Reynolds equation easier. A definition of wall slipping based on the assumed existence of a boundary shear constraint substantially complicates matters. In this case, developments must take into account the inequations allowing definition of the zones between which slipping occurs on one or on the other or on both bounding walls and the zones without slipping.

34

Hydrodynamic Bearings

u=

h + 2α h µ ⎞ 1 ∂p 2 ⎛ U h ∂p + z −⎜ z ⎜ h + µ (α + α ) 2 µ ∂x h + µ (α + α ) ⎟⎟ 2 µ ∂x h s h s ⎠ ⎝ h + αh µ h ∂p α s µ ( h + 2α h µ ) +U − h + µ (α h + α s ) 2 µ ∂x h + µ (α h + α s )

[2.29]

A similar expression is obtained for the axial component v of velocity. If the density of the fluid is considered to be constant, the conservation of the mass implies: h

∫ 0

h

h

∂u ∂v ∂w dz + dz + dz = 0 ∂x ∂y ∂z

∫ 0

∫

[2.30]

0

The development of the first term of this equation gives: h

∫ 0

h

∂u ∂ ∂h dz = u ( z )dz − u (h) ∂x ∂x ∂x

∫ 0

2 2 ∂ ⎛ h3 h + 4hµ (α h + α s ) + 12µ α hα s ∂p ⎞ U ∂ ⎛ h 2 + 2hµα h ⎞ ⎟+ =− ⎜ ⎜ ⎟ ∂x ⎜ 12µ ∂x ⎟ 2 ∂x ⎜⎝ h + µ (α h + α s ) ⎟⎠ h ( h + µ (α a + α c ) ) ⎝ ⎠

+

[2.31]

αhµ h ∂p ∂h hµα h + 2α hα s µ2 ∂h −U 2µ ∂x ∂x h + µ (α h + α s ) h + µ (α h + α s ) ∂x

Through conducting a similar development of the second term of equation [2.30], a new form of the Reynolds equation is obtained: 2 2 ∂ ⎛ h3 h + 4hµ (α h + α s ) + 12µ α hα s ∂p ⎞ ⎜ ⎟ ∂x ⎜⎝ 12µ ∂x ⎟⎠ h ( h + µ (α h + α s ) ) 2 2 ∂ ⎛ h3 h + 4hµ (α h + α s ) + 12µ α hα s ∂p ⎞ + ⎜ ⎟ ∂y ⎜⎝ 12µ ∂y ⎟⎠ h ( h + µ (α h + α s ) )

U ∂ ⎛ h 2 + 2hµα h ⎞ = ⎜ ⎟ 2 ∂x ⎜⎝ h + µ (α h + α s ) ⎟⎠ αhµ ∂h h hµα h + 2α hα s µ 2 −U + h + µ (α h + α s ) ∂x 2µ h + µ (α h + α s )

[2.32]

⎛ ∂p ∂h ∂p ∂h ⎞ ∂h + ⎜ ⎟+ ⎝ ∂x ∂x ∂y ∂y ⎠ ∂t

Equations of Hydrodynamic Lubrication

35

Equation [2.32] is only applicable to zones where the film is continuous. For ruptured film zones, the pressure is constant and the equation is reduced to:

ρα h µ h 2 + 2hµα h ⎞ ∂h U ∂ ⎛ =0 ⎜ρ ⎟ −U 2 ∂x ⎜⎝ h + µ (α h + α s ) ⎟⎠ h + µ (α h + α s ) ∂x

[2.33]

By introducing the universal variable D and the index function Φ, which were defined in section 2.4, equations [2.32] and [2.33] are gathered together into a single equation: ⎧⎪ ∂ ⎛ h3 h2 + 4hµ (α + α ) + 12µ2α α ∂D ⎞ h s h s ⎜ ⎟ ⎜ ∂ ∂x ⎟ 12 x µ + + α α h h µ ( ) ( ) h s ⎪⎩ ⎝ ⎠

Φ⎨

2 2 ∂ ⎛ h3 h + 4hµ (α h + α s ) + 12µ α hα s ∂D ⎞ ⎫⎪ ⎜ ⎟⎬ ∂y ⎜ 12µ ∂y ⎟ ⎪ h ( h + µ (α h + α s ) ) ⎝ ⎠⎭ 2 ∂h αhµ U ∂ ⎛ h + 2hµα h ⎞ = ⎜ ⎟ −U ⎜ ⎟ 2 ∂x ⎝ h + µ (α h + α s ) ⎠ h + µ (α h + α s ) ∂x ⎧⎪ h hµα h + 2α hα s µ2 ⎛ ∂p ∂h ∂p ∂h ⎞ ⎫⎪ ∂h +Φ ⎨ + ⎜ ⎟⎬ + ⎪⎩ 2µ h + µ (α h + α s ) ⎝ ∂x ∂x ∂y ∂y ⎠ ⎪⎭ ∂t ⎧⎪U ∂ ⎛ D h2 + 2hµα ⎞ D αhµ ∂h ∂D ⎫⎪ h ⎟− U + (1 − Φ ) ⎨ + ⎜ ⎜ h h + µ (α h + α s ) ⎟ h h + µ (α h + α s ) ∂x ∂t ⎪⎬ 2 x ∂ ⎪⎩ ⎭ ⎝ ⎠

+

[2.34]

2.6. Boundary conditions; lubricant supply For a smooth circular bearing of radius R and uniform width 2B, in the configuration known as “developed bearing”, the domain over which the pressure p and filling r fields will be sought to verify equations [2.23] to [2.26] may be represented by a rectangle of length 2πR and width 2B (Figure 2.8). When Φ is equal to 1, equation [2.23] is an elliptic partial differential equation. When Φ is equal to 0, the equation becomes hyperbolic. The boundary conditions for the domain need to be defined in order to solve it, either conditions relative to the unknown function D or to its gradient. In the following sections, the boundary conditions will be defined in detail for the bearing. These may easily be transposed for the thrust bearing case.

36

Hydrodynamic Bearings

Figure 2.8. Study domain: Ω0 non-active zone and Ω active zone

2.6.1. Conditions on bearing edges On the bearing edges, the pressure is equal to the ambient pressure pa outside the bearing. This pressure is generally assumed to be constant and usually equal to atmospheric pressure. However, for engine bearings, this pressure may vary slightly during the cycle. When the bearing under study is immersed in lubricant, it follows that its edges must be full. Filling r is thus equal to the thickness h of the film. This is not a situation encountered by internal combustion engine bearings, but it may nonetheless be supposed that, due to the presence of the residual lubricant maintained on the bearing edges by phenomena of capillarity and surface wetting, the film is not ruptured on the bearing edges3. Thus, the following is obtained:

⎧ p = pa ⎨ ⎩r = h

for z = ± B

[2.35]

2.6.2. Conditions for circular continuity The edges of the domain defined by x = 0 and x = 2πR form the fictional boundaries linked to the configuration of a developed bearing. For the real domain shown in Figure 2.3 these boundaries do not exist. Therefore, what is required is to simply translate the continuity between these two boundaries for all of the

3 A reducing coefficient may if necessary be applied to the filling at the bearing edges to account for the situation where wetting ability is insufficient to ensure an unbroken film.

Equations of Hydrodynamic Lubrication

37

parameters and if necessary their derivatives, except the curvilinear abscissa x. The result of this is that the following is obtained for pressure and filling:

⎧⎪ p ⎨ ⎪⎩ r

x =0 =

p

x = 2π R

x =0 =

r

x = 2π R

[2.36]

2.6.3. Conditions on non-active zone boundaries Assume that, for an initial approach, the domain of the developed bearing contains in its central part a stationary non-active zone Ω0 where the pressure is equal to the cavitation pressure pcav and the rest of the domain is an active zone Ω under a pressure greater than cavitation pressure (Figure 2.8). These two zones are separated by a boundary ∂Ω. This boundary ∂Ω may be broken down into several parts, depending on the orientation of the normal n outside Ω with respect to the velocity U of the mobile bounding wall. When the projection of n onto U is positive, this is an up-flow boundary, notated ∂Ωup. A point of ∂Ωup will have an abscissa xup. If the projection of n onto U is negative, this is a down-flow boundary, denoted as ∂Ωd. A point of ∂Ωdown will have an abscissa xdown. When the projection of n onto U is nil, the boundary is neither an up-flow nor a down-flow boundary. This type of boundary, denoted as ∂Ω0, may be encountered near the bearing edges and will be studied in detail in section 2.6, which discusses calculation of the flow rate. In a bearing subject to variable loading, the non-active zones appear, vary in size and position, and then disappear. In contrast to what was assumed in the previous section, their boundaries are, thus, not stationary, but rather are subject to a velocity of displacement VB relative to surface 2, which has been chosen as the reference surface for velocity components parallel to surface 1 (see section 2.3). Figure 2.9 shows the transition between a full film zone and a zone where the film has been ruptured by cavitation, which is assumed, for simplicity’s sake, to be separated by a boundary B, which is orthogonal in the direction x.

Figure 2.9. The non-active zone feeds the full film zone

38

Hydrodynamic Bearings

By choosing the boundary B as reference, i.e. by subtracting the velocity VB of the boundary from all of the velocities, it is easy to write the conservation of the flow rate at the transition from one zone to another. In the ruptured film zone, if it is assumed that the liquid/gas mixture behaves like a homogeneous fluid adhering to the bounding walls, the flow rate is given by the product of the filling r by the means of the relative velocities of the two walls, U – VB for bounding wall 1 and – VB for bounding wall 2. In fact, pressure in the ruptured film zone must be constant and equal to cavitation pressure, which gives a Couette-type flow with a rectilinear velocity profile. In the full film zone, the flow rate is the sum of the Couette flow rate, which is the product of the thickness h by the means of the bounding wall velocities (– VB + U – VB)/2 and the Poiseuille flow rate (parabolic velocity profile). The Poiseuille flow rate results from the pressure gradient, and, according to relation [2.17], is expressed as −

h3 dp . 12μ dx

The equality of the flow rates by length unit implies:

−

U − 2VB U − 2VB h3 dp +h =r 12μ dx 2 2

or even:

(U − 2VB )(h − r ) =

h3 dp 6 μ dx

For a rupture boundary ∂Ω, due to cavitation, the pressure from the full film side has reached the boundary pcav. This is what has caused the phase change of the fluid to occur. If it is supposed that the pressure gradient

dp is positive near the dx

boundary, there will then exist a zone in which the pressure is less than the cavitation pressure, which is contradictory. Thus, the pressure gradient is necessarily negative or nil, as is the term (U − 2VB )( h − r ) . Since the filling r is necessarily less than or equal to the available space h, it may be deduced that the velocity of the boundary verifies:

VB ≥

U 2

Equations of Hydrodynamic Lubrication

39

This condition is incompatible with stationary situations for which the boundary velocity VB is nil. Therefore, it is necessary to assume that in this case, at the same time:

r = h;

p x = B = pcav ;

∂p =0 ∂x x = B

[2.37]

This set of conditions is known as “Reynolds conditions” and is the expression of a rupture in the film. For dynamic situations, the following possibilities are available: ∂p U ⎧ =0 rupture boundary ⎪VB < 2 ; r = h ; p x= B = pcav ; ∂x x= B ⎪ ⎨ ∂p 6μ ⎪V ≥ U ; r ≤ h ; p = pcav ; = (U − 2VB )(h − r ) formation boundary B = x B ⎪ ∂x x= B h3 2 ⎩

For the condition known as “formation” of the film, the Poiseuille flow rate resulting from the non-nil pressure gradient represents a feeding of the full film zone by the ruptured film zone. These conditions may easily be transposed to define the boundary conditions at the up-flow and down-flow abscissae xup and xdown which are defined in Figure 2.8: – up-flow boundary conditions:

U ⎧ ⎪Vup < 2 ; r = h ; p x = xup = pcav ⎪ ⎨ ⎪V ≥ U ; r ≤ h ; p = pcav x = xup ⎪ up 2 ⎩

;

∂p ∂x

;

∂p ∂x

x = xup

x = xup

=0 =

6μ h3

(rupture) [2.38] (U − 2Vup )(h − r ) (formation)

– down-flow boundary conditions4: ∂p U ⎧ (rupture) =0 ⎪ Vdown > 2 ; r = h ; p x = xdown = pcav ; ∂x x = xdown ⎪ [2.39] ⎨ U 6μ ∂p ⎪V r h p p U V h r ; ; ; ( 2 )( ) (formation) ≤ ≤ = = − − cav down x = xdown ⎪ down 2 ∂x x = xdown h3 ⎩

4 The direction changes in the inequations are due to the change in orientation of the normal n at the boundary.

40

Hydrodynamic Bearings

The Reynolds boundary conditions are only valid at boundaries where rupture is due to cavitation and for a model which relates to a homogeneous mix in the ruptured film zone (schemes 1, 2 and 3 of Figure 2.7). When the fluid is deposited on the surfaces (scheme 4) the transport conditions are no longer the same, and the relations obtained are different. However, it may be assumed that cavitation initially appears in the form shown in scheme 1 and that the Reynolds conditions remain valid for determining the position of the boundaries of the rupturing by cavitation. In the case of rupture in the lubricant film through separation, the depression due to the surface tension is generally greater than the depression due to the saturating vapor tension. As a result, it is possible to have a pressure in the full film zone close to the boundary which is less than the pressure at the boundary. In this case, it is then no longer necessary to have a nil pressure gradient as a boundary condition. This situation will be encountered again in the analysis of the full film fringes at the bearing edges (see section 2.7.4.2). 2.6.4. Boundary conditions for supply orifices

Due to significant leakage flow rates at the extremities, the bearing generally needs to be supplied with lubricant. To achieve this, the lubricant under pressure is carried through pipes to orifices or grooves either on the surface of the housing or on the shaft surface. In the zone corresponding to the surface of these orifices – including any chamfers that may be present – the conditions are no longer those of thin films and the equations of hydrodynamic lubrication are no longer applicable. Only the Navier–Stokes equations enable the prediction of the velocity and pressure fields in these zones [SAN 92, BRA 93, HIL 95, HEL 03]. Figure 2.10 shows a comparison between the flow fields calculated and those observed through photographing tracers for various connection shapes between the supply pipe and the lubricant film. These results, obtained by Jeddi [JED 04], establish both the tridimensional character of the flow in the supply zone and the great influence exerted by geometrical parameters. The main aim of these studies is to determine the pressure, velocity and temperature conditions that should be applied to the transition boundary between the domain where the thin film equations can be applied and the domain where it is necessary to use complete hydrodynamic equations. As shown in Figure 2.11, the pressure distribution across the orifice width is not completely constant: on the down-flow side there is a slight overpressure due to dynamic effects, and this increases with the velocity of the sliding wall facing the orifice. As soon as the orifice boundaries are reached, pressure varies very rapidly. Within a few tenths of a millimeter, the conditions described by the thin films equation are established, as is

Equations of Hydrodynamic Lubrication

41

shown by the linear evolutions of the pressure for abscissa smaller than 1.5 mm and greater than 20.5 mm in Figure 2.11.

Figure 2.10. Examples of flow fields in the supply zone of a lubricant film [JED 04]. Influence of the connection shape: a) mounting of a non-shaped tube, b) mounting of a shaped tube and c) direct connection

Figure 2.11. Examples of pressure distributions inside a supply orifice [JED 04]

42

Hydrodynamic Bearings

For internal combustion engine bearings, these variations in pressure in transition zones between the two domains are sufficiently low in relation to mean pressure in the orifice or groove that they do not need to be taken into consideration. Due to the overpressure within the orifice or groove, the zone surrounding it is necessarily active. The boundary conditions usually used at the boundary B are, thus:

r = h;

p x = B = psupply

[2.40]

where psupply is the mean pressure in the orifice or the groove. Particular case of lubricant supply with imposed flow rate When the supply pipe is linked to a volumetric pump such as a gear pump, the flow rate is imposed, and not the pressure. There are two possible approaches: 1) The pressure gradient is considered to be constant in the area surrounding the orifice: the boundary conditions which are needed are those related to the pressure gradient, which gives, through integration, the required flow rate (equation [2.17]), h3 ∂p = C te such that 12 μ ∂n

where

⎛ h3 ∂p U i nh ⎞ − ⎜ ⎟ ds = Qsupply ⎜ 12 μ ∂n 2 ⎟⎠ orifice ⎝

∫

[2.41]

∂p is the pressure gradient grad p·n in the direction of the normal n directed ∂n

toward the inside of the orifice.

2) The pressure is considered to be constant within the orifice; the condition required is related to this pressure. Since this condition is unknown because it depends on the loss in loading in the film close to the orifice, it is necessary to use an iterative approach. On the basis of a pre-estimated supply pressure, the initial calculation of the pressure field is made. Flow rate is then calculated using relation [2.16]. Depending on the result obtained, the pressure applied is increased or decreased, and a second calculation is made. The process is repeated until the imposed flow rate is obtained with the desired level of precision. The first approach, although direct, has the disadvantage of assuming a constant pressure gradient all around the orifice. This is why the second method is preferable.

Equations of Hydrodynamic Lubrication

43

2.7. Flow rate computation

It is important to know the flow rate of a bearing in order to organize the dimensions of the supply devices, pumps, filters and pipes. It is also an important parameter for estimating effective operation of bearing lubrication, and in particular for determining the operating temperature of the bearing (see Chapter 1 of [BON 14]). The supply flow rate may be calculated using relation [2.17] applied to the contour of the orifice or the groove. This causes no problem for supplies through a groove when the discretization of the calculation domain exactly follows the contours of the groove. For very small orifices, it is difficult to make the grid for placement of the nodes used by finite difference methods to coincide with the contour of the orifice, which is generally circular or elliptical. This question is also involved in methods using finite element meshes: it is easy to build a mesh which follows the contours of an orifice. However, when this orifice is on the mobile surface, the difficulties mentioned above may be encountered when an attempt is made to adapt the mesh to each surface movement. Then the solution for this is to determine the supply flow rate using a record of the flow rates entering and exiting at either end of the bearing. It should be noted that for bearings such as internal combustion engine bearings which operate in cyclical conditions, this record does not have instantaneous significance, but rather it needs to be interpreted over the entire cycle. The calculation of the flow rate entering or exiting at the ends of the bearing is obtained on the basis of a pressure gradient at the bearing edges. In the case of a numerical solution to the Reynolds equation, the pressure field is known discretely and this applies whatever method is chosen for solving it: finite differences, finite elements, finite volumes, etc. The calculation of the pressure gradient employs the values at the discretization points and numerical schemes of the finite differences type for finite difference or finite volume methods, and the precision of the values obtained depends on the order of the schemes used. For the finite element method, interpolation functions and their derivatives are used, and precision depends on their degree, i.e. the number of nodes that the elements have. If the function that needs to be derived is continually derivable over an interval which contains enough points or elements, the values obtained for the derivatives will have the required level of precision. If the method of finite elements is used with quadratic elements which have eight nodes, a continuity of the function over the totality of the element suffices. However, it has been noted earlier that the presence of cavitation or separation zones in the lubricant film breaks the continuity of the pressure field. Because there is lubricant present on the bearing edges, these are always covered completely in

44

Hydrodynamic Bearings

film. If an element at the edges of the bearing is reached by a cavitation or separation zone which progresses from the center of the bearing toward its edges, the continuity of the pressure field within the element, which is required for the calculation of the derivative, is no longer present. The same issue occurs with finite difference and finite volume methods if fewer than two meshes next to the bearing edges are in a full film zone. One solution to this is to choose a refined discretization on the bearing edges. It will be shown in what follows that the full film zones at bearing edges are often very narrow. This leads to refinements that are incompatible with the reasonable calculation times. For this reason, a semi-analytical algorithm will be developed for the calculation of the flow rate entering at the bearing edges. 2.7.1. First assumptions

It is assumed that ambient pressure is greater than the saturating vapor tension of the lubricant. As a result, the lubricant on the bearing edges will always be in liquid phase. However, the outside of the bearing is not necessarily occupied by lubricant. Conditions from a totally submerged to a totally dry bearing are possible, through all the associated intermediary conditions, as is the presence of more or less dense lubricant fog. This variation in external conditions will be accounted for through the introduction of a mixture coefficient Cm. When Cm is not nil, there will always be a continuous zone of fluid on the bearing edges. In fact, either under the effect of the depression due to the saturating vapor tension in the case of a cavitation zone which has not joined the bearing edges, or under the effect of surface tension acting on the menisci which separate the fluid from the gaseous zones in the case of a separation zone, pressure within the fluid is less than ambient pressure. This leads to an effect of pumping exterior fluid (or a mix of fluid and gas in the case of an oil fog) toward the inside of the bearing. The width of this lubricated fringe where the pressure is less than ambient pressure varies depending on the kinematic conditions. It may widen toward the inside of the bearing, progressively filling the non-active zone. Or else, and particularly when the two bounding walls are moving away from each other, it may contract and result in a nil width. In this case, the phenomenon of pumping is not sufficient to fill all the space created by the kinematic evolution. It is of the utmost importance for the calculation of the flow rate entering and exiting at the ends of the bearing that these phenomena should be understood and taken into account. The distribution of the active and non-active zones within the film, and as a result, the calculation of the dissipated power, the operating temperature and then the bearing capacity and the thickness of the film all result from these phenomena, either directly or indirectly. The operation of a bearing without a supply orifice, as is the case with most connecting rod small end bearings, can only be clearly explained if these phenomena are taken into consideration.

Equations of Hydrodynamic Lubrication

45

2.7.2. Model and additional assumptions

Although the Cartesian coordinates correspond to the case of the bearing, the developments which follow are also applicable to the thrust bearing case, because they involve an infinitesimal stretch of the domain boundary.

Figure 2.12. Elementary domain of the full film fringe on the bearing edge

Figure 2.12 shows an elementary volume of the unruptured fluid fringe formed from the bearing edges, which is taken as the origin of the coordinate z. The coordinate x is the curvilinear abscissa of the developed bearing. This fringe is separated from the cavitation or separation zone by a boundary located at the ordinate L(x). The separation boundary may be in the form of a meniscus, as shown in the model (the case of a separation) or of a virtual surface beyond which bubbles appear (the case of cavitation). In the non-full zone, the fluid present is a proportion k of the available volume. The product k h represents the height r, i.e. the filling that the fluid would occupy if it was completely deposited on the mobile bounding wall. In the direction y, the pressure varies from the ambient pressure p0 to the pressure p(L) equal to the saturating vapor pressure or pressure due to surface tension, depending on whether cavitation or separation is involved. The saturating vapor pressure is in the order of –0.1 MPa, and the ambient pressure is taken as reference value. Pressure due to surface tension is dependent on the radius of the meniscus. This radius is of the order (h – r)/2. For a radial clearance of the order 20 µm, h – r

46

Hydrodynamic Bearings

may vary from 40 µm5 to less than 1 µm. For a surface tension of the order 0.05 N/m the pressure behind the meniscus may take values between –0.0025 MPa and –0.1 MPa. In what follows, this pressure at the boundary L(x) of the fringe will be denoted as pcav in the case of cavitation or psep in the case of separation. The values of pcav or psep will always be negative. 2.7.3. Pressure expression for the full film fringes on the bearing edges

The pressure in the full film fringe defined above verifies the Reynolds equation for thin films in its simplest form (equation [2.16])6:

∂ ⎛ h3 ∂p ⎞ ∂ ⎛ h3 ∂p ⎞ ∂h ∂h +2 ⎜ ⎟+ ⎜ ⎟ =U ∂x ⎜⎝ 6μ ∂x ⎟⎠ ∂z ⎜⎝ 6μ ∂z ⎟⎠ ∂x ∂t

[2.42]

The width of an internal combustion engine bearing is small compared to its circumference and the fringe width L(x) is of the order 1 mm. Over the same distance in the direction x, the variation in pressure is thus quite small. This observation leads to the supposition that the pressure gradient following x is insignificant in relation to the pressure gradient following z and, as a result, the Reynolds equation may be simplified as follows:

∂ ⎛ h3 ∂p ⎞ ∂h ∂h +2 ⎜⎜ ⎟⎟ = U ∂z ⎝ 6μ ∂z ⎠ ∂x ∂t

[2.43]

If it is assumed that the thickness h is independent of z, the expression of the pressure in the full film fringe may be written as:

p( z ) =

∂h ⎞ 2 3μ ⎛ ∂h U + 2 ⎜ ⎟ z + Cz + D ∂t ⎠ h3 ⎝ ∂x

The integration constants C and D are obtained on the basis of the boundary conditions:

5 40 µm corresponds to the maximum gap between the surfaces when the shaft eccentricity is maximum. 6 The developments of this chapter are intended for application in zones where the film is quite thick, which is not the case in zones under mixed lubrication. This is why flow factors have not been introduced.

Equations of Hydrodynamic Lubrication

47

– p = 0 at z = 0 (ambient pressure outside the bearing is taken as reference); – p = psep at z = L(x) in the case of separation; which gives:

p( z ) =

∂h ⎞ 3μ ⎛ ∂h z + 2 ⎟ ( z 2 − L( x) z ) + psep U 3 ⎜ ∂t ⎠ L( x) h ⎝ ∂x

[2.44]

If the fringe rupture is due to cavitation, it suffices to replace psep with pcav. The pressure gradients at z = 0 and at z = L(x) may be deduced from this expression. The first of these will be used to calculate the flow rate entering the bearing. The second will be used to determine the evolution of the width L(x) of the full film fringe as a function of time. 2.7.4. Evolution of the width of the full film fringes on the bearing edges

The pressure gradient in z = L(x) is expressed as:

psep ∂p 3μ ⎛ ∂h ∂h ⎞ = 3 ⎜U + 2 ⎟ L( x) + ∂z h ⎝ ∂x ∂t ⎠ L( x)

[2.45]

The unit volume flow rate7 in the direction z at the abscissa z = L(x) does not contain a Couette term because the bounding walls have no velocity in the direction z:

qv = −

h3 ∂p 12μ ∂z

[2.46]

By replacing the pressure gradient with its expression, the following is obtained:

∂h ⎞ 1 ⎛ ∂h h3 psep + 2 ⎟ L( x) − qv = − ⎜ U ∂t ⎠ 4 ⎝ ∂x 12 μ L( x )

[2.47]

If the geometrical and kinematical conditions at the time t0 are known, it is possible to deduce from this the value of the flow rate at the abscissa z = L(x). By writing the conservation of the mass flow rate at the passage of the interface 7 Unit flow rate: flow rate by unit of length in the direction x.

48

Hydrodynamic Bearings

between the full film fringe and the ruptured film zone, it will be possible to determine the evolution of the fringe width as a function of time. 2.7.4.1. The pressure in the full film fringe remains greater than the cavitation pressure If at the time t0 the flow rate qv given by relation [2.39] is positive, the full film fringe is moving toward the inside of the bearing. As it extends, it absorbs the fluid which partially fills the non-active zone. In the opposite case, it contracts toward the bearing edges, leaving some of the lubricant behind in the non-active zone.

Figure 2.13. The non-active zone feeds the full film fringe

Figure 2.13 shows the passage of the fluid which occupies the partially full zone toward the full film zone, taking the abscissa z = L(x) as reference. The conservation of the mass flow rate through the interface leads to the relation between the flow rate •

qm and the velocity L of the interface: •

•

− ρ h L + qm = − ρ r L If the mass flow rate is replaced by ρ qv, the volume flow rate qv by the expression [2.46], and the pressure gradient by the expression taken from equation [2.47], the following is obtained: •

L=−

⎡⎛ ∂h 1 ∂h ⎞ h3 psep ⎤ + 2 ⎟L − ⎢⎜ U ⎥ 4( h − r ) ⎣⎢⎝ ∂x 3μ L ⎦⎥ ∂t ⎠

[2.48]

The integration of this first-order differential equation will make it possible to obtain the expression of L(x) as a function of time.

Equations of Hydrodynamic Lubrication

49

By setting Λ = L2 , equation [2.40] may be written as: •

Λ+

∂h ⎞ 1 h3 psep ⎛ ∂h + 2 ⎟Λ = − ⎜U ∂t ⎠ 2(h − r ) ⎝ ∂x 6μ h − r

Taking t0 as the reference time, the solution for this equation is in the form:

⎡ t h3 psep ⎤ exp( f (ξ )) d ξ + C ⎥ ⎢⎣ t0 6 μ h − r ⎥⎦

Λ (t ) = − exp ( − f (t ) ) ⎢ ∫

where the function f ( ξ ) is defined by:

f (ξ ) =

ξ

∫t

0

⎛ ∂h ∂h ⎞ 1 +2 ⎜U ⎟d ζ ∂ζ ⎠ 2(h − r ) ⎝ ∂x

[2.49]

The integration constant C is given by the width of the fringe at the instant t0. The width of the fringe at the instant t is thus given by: 1

⎛ ⎡ t h3 psep ⎤ ⎞2 exp( f (ξ ))d ξ − L2 (t0 ) ⎥ ⎟ L(t ) = ⎜ − exp ( − f (t ) ) ⎢ ⎜ ⎟ ⎣⎢ t0 6μ h − r ⎦⎥ ⎠ ⎝

∫

[2.50]

2.7.4.2. The pressure in the full film fringe becomes lower than the cavitation pressure As the variation in pressure through the fringe is parabolic (equation [2.44]) if the pressure gradient at z = L(x) is positive, the pressure in the fringe will in part be less than the pressure at z = L(x), i.e. the separation pressure. If the lowest pressure is less than the cavitation pressure pcav, a rupture in the film by cavitation will occur. In this case, the calculation for the width of the fringe should be modified. Since the pressure for z < L(x) cannot descend lower than the value taken at z = L(x), i.e. pcav, the pressure gradient at z = L(x) can only be nil. This condition of nil gradient at the rupture abscissa is known as the “Reynolds rupture condition”, a condition which only applies to rupture by cavitation, and not by separation of the film. The boundary which separates the full film fringe from the non-active zone is thus located at the abscissa which verifies the Reynolds boundary conditions:

50

Hydrodynamic Bearings

z = L( x);

p ( L( x)) = pcav ;

∂p =0 ∂z z = L ( x )

[2.51]

In this case, the volume flow rate qv can only be nil. From relation [2.47], the expression of the fringe width in function of the other parameters can be extracted8: h3 ∂h ⎞ ⎛ ∂h 3μ ⎜ U +2 ⎟ ∂t ⎠ ⎝ ∂x

L = − pcav

[2.52]

The conservation of the flow rate through the interface •

•

ρr L = ρh L makes it possible to express the quantity of fluid abandoned by the retracting fringe, written in terms of filling r:

r=h

[2.53]

The cavitation or separation pressure pcav is negative, and therefore expression [2.39] only makes sense if the kinematic conditions are such that:

U

∂h ∂h +2 >0 ∂x ∂t

[2.54]

For example, in the case of parallel bounding walls ∂h ∂h = 0 ), these must move away from each other to create this situation ( > 0 ). ( ∂x ∂t ∂h < 0 ), this condition However, if the bounding walls are coming closer together ( ∂t may only be produced where there is sufficient divergence between them. If condition [2.54] is not satisfied, then no solution is possible and the full film fringe completely disappears, as a result of which the non-active zone extends to the bearing edges.

8 Relation [2.52] is also valid in the case of a bearing in stationary conditions. The term 2∂h/∂t simply needs to be eliminated. In this case, the inequality ∂h/∂x > 0 should be verified.

Equations of Hydrodynamic Lubrication

51

2.7.5. Computation of the flow rate for lubricant entering by the bearing sides

Relations [2.44] and [2.46] make it possible to obtain the unit mass flow rate entering at the bearing sides:

∂p ∂z qm

z =0

z =0

=

psep L

−

∂h ⎞ 3μ ⎛ ∂h + 2 ⎟L U 3 ⎜ ∂t ⎠ h ⎝ ∂x

⎛ 1 ⎛ ∂h h3 psep ∂h ⎞ = −Cm ρ ⎜ − ⎜ U + 2 ⎟L + ⎜ 4 ⎝ ∂x 12μ L ∂t ⎠ ⎝

⎞ ⎟⎟ ⎠

where L will take the value given by expression [2.50] or [2.52], depending on what is appropriate. The coefficient Cm has been introduced to account for the fact that the bearing sides have not necessarily been completely wetted by the exterior fluid. The entering volume flow rate for the lubricant alone is thus given by:

qv

z =0

=

Cm 4

⎡⎛ ∂h h3 psep ⎤ ∂h ⎞ + 2 ⎟L − ⎢⎜ U ⎥ 3μ L ⎥⎦ ∂t ⎠ ⎢⎣⎝ ∂x

[2.55]

If the fringe width L is given by expression [2.52], psep should be replaced with pcav in relation [2.55]. 2.8. Computation of efforts exerted by the pressure field and the shear stress field: journal bearing case

The equations governing the lubrication of engine bearings involve the components of the torsors of the action of the lubricant film on the bearing surfaces. Since the mass of the lubricant in the film is insignificant, the torsor acting on surface 2 is the opposite of that acting on surface 1. The components of this torsor are obtained through integration of the pressure efforts on the domain of calculation:

⎧ Wxc = p( x, z ) nixc dxdz Ω ⎪ ⎪ W yc = p( x, z )niy c dxdz ⎪ Ω ℑ pressure ⎨ ⎪ M Ox = ( OM ∧ p( x, z )n )ixc dxdz c Ω ⎪ ⎪M = ( OM ∧ p( x, z )n )iy c dxdz ⎩ Oyc Ω

∫∫ ∫∫

∫∫ ∫∫

52

Hydrodynamic Bearings

where M is the current point with the developed coordinates x and z, n is the normal unit vector at the surfaces in M and xc and yc are the basis vectors of the Cartesian coordinates (Figure 2.3). These relations may also be written as:

⎧ Wx = p ( x, z ) cos θ ( x, z ) dxdz c Ω ⎪ ⎪ ⎪ W yc = Ω p( x, z ) sin θ ( x, z ) dxdz ℑ pressure ⎨ z p( x, z ) sin θ ( x, z ) dxdz ⎪ M Ox = − c Ω ⎪ ⎪M = z p( x, z ) cos θ ( x, z ) dxdz ⎩ Oyc Ω

∫∫ ∫∫ ∫∫ ∫∫

[2.56]

Taking into account the simplifying assumptions given at the beginning of this chapter, the shear constraints within the lubricant film are expressed as:

∂u ⎧ ⎪τ xy = μ ∂y ⎪ ⎨ ⎪τ = μ ∂w ⎪⎩ zy ∂y

[2.57]

The components u and w of the velocity (Figure 2.3) are given by expression [2.15] which has taken the following simplified form in the case of the bearing:

⎧ ⎛ I 0 J1 ⎞ I0 ⎞ ∂p ⎛ ⎪u ( x, y , z , t ) = ⎜ I1 − ⎟ + U ⎜1 − ⎟ J0 ⎠ J0 ⎠ ∂x ⎝ ⎪ ⎝ ⎨ I 0 J1 ⎞ ∂p ⎛ ⎪ ⎪ w( x, y , z , t ) = ∂z ⎜ I1 − J ⎟ 0 ⎠ ⎝ ⎩ By carrying forward the expressions of u and w in [2.49], the following is obtained:

⎧ ⎛ ∂p ⎛ ∂I1 ∂I 0 J1 ⎞ ∂I 0 U ⎞ − ⎪τ xy = μ ⎜⎜ ⎜ ⎟ ⎟− ∂y J 0 ⎠ ∂y J 0 ⎠⎟ ⎪ ⎝ ∂x ⎝ ∂y ⎨ ∂p ⎛ ∂I1 ∂I 0 J1 ⎞ ⎪ ⎪ τ zy = μ ∂z ⎜ ∂y − ∂y J ⎟ 0 ⎠ ⎝ ⎩

Equations of Hydrodynamic Lubrication

53

and by replacing the integrals I0 and I1 with their expressions:

⎧ ⎪ I 0 ( x, y , z , t ) = ⎪ ⎨ ⎪ I ( x, y , z , t ) = ⎪⎩ 1

y

1

y

ξ

∫0 μ ( x, ξ , z, t ) d ξ ∫0 μ ( x, ξ , z, t ) d ξ

the following final result is obtained:

⎧ J1 ⎞ U ∂p ⎛ ⎪τ xy = ⎜y− ⎟− ∂x ⎝ J0 ⎠ J0 ⎪ ⎨ J1 ⎞ ∂p ⎛ ⎪ ⎪ τ zy = ∂z ⎜ y − J ⎟ 0 ⎠ ⎝ ⎩

[2.58]

where J0 and J1 are given by:

⎧ ⎪ J 0 ( x, z , t ) = ⎪ ⎨ ⎪ J ( x, z , t ) = ⎪⎩ 1

h

1

h

ξ

∫0 μ ( x, ξ , z, t ) d ξ ∫0 μ ( x, ξ , z, t ) d ξ

By integrating these constraints calculated at the bounding walls, it is possible to obtain the efforts exerted tangentially to those whose moment in relation to the bearing axis is expressed: – for bounding wall 1:

M 1Oz = R c

∫∫Ω τ xy

y =0

dxdz = − R

⎡ ∂p ⎛ J1 ⎞

U ⎤

∫∫Ω ⎢⎣⎢ ∂x ⎝⎜ J 0 ⎠⎟ + J 0 ⎥⎦⎥ dxdz

[2.59]

– for bounding wall 2:

M 2Oz = R c

∫∫Ω τ xy

y =h

dxdz = R

⎡ ∂p ⎛

J1 ⎞

U ⎤

∫∫Ω ⎣⎢⎢ ∂x ⎝⎜ h − J 0 ⎠⎟ − J 0 ⎦⎥⎥ dxdz

[2.60]

The integrals from [2.59] and [2.60] cover both non-active and active zones in the film. In non-active zones, the pressure gradient is nil and only the Couette term

54

Hydrodynamic Bearings

remains. The integral J0, which itself is dependent on the viscosity µ, is involved in the Couette term. Because the lubricant is mixed with gas in a proportion given by the filling coefficient r/h (see section 2.4), it may be assumed that the mean viscosity is attenuated in the same proportions, which gives the following relations9: – for bounding wall 1:

M 1Oz = − R c

⎡ ∂p ⎛ J1 ⎢ ⎜ Ω ⎢ ∂x J ⎣ ⎝ 0

⎞ r U ⎤ ⎥ dxdz ⎟+ ⎠ h J 0 ⎦⎥

∫∫

[2.61]

– for bounding wall 2:

M 2Oz = R c

⎡ ∂p ⎛

J1 ⎞

r U ⎤

∫∫Ω ⎢⎣⎢ ∂x ⎝⎜ h − J 0 ⎠⎟ − h J 0 ⎥⎦⎥ dxdz

[2.62]

Particular case of isoviscous fluids If viscosity is constant throughout the film, these expressions may be simplified and written as: – for bounding wall 1:

M 1Oz = − R c

r μU h

⎛ h ∂p

∫∫Ω ⎜⎝ 2 ∂x + h

⎞ ⎟ dxdz ⎠

[2.63]

– for bounding wall 2:

M 2Oz = R c

⎡ h ∂p

r μU ⎤ dxdz h ⎥⎦

∫∫Ω ⎢⎣ 2 ∂x − h

[2.64]

2.9. Computation of efforts exerted by the pressure field and the shear stress field: thrust bearing case

For a thrust bearing, the torsor components which act on surface 2, obtained by the integration on the calculation domain of the pressure efforts, are expressed:

9 In active zones, the filling r is equal to the thickness of the film h and the coefficient r/h has no effect.

Equations of Hydrodynamic Lubrication

55

⎧ Wz = p (r ,θ ) rdrdθ 2 ⎪ Ω ⎪ ℑ pressure ⎨ M Ox = ( OM ∧ p(r ,θ )z1 )ix 2 rdrdθ Ω 2 ⎪ ⎪M = ( OM ∧ p(r ,θ )z1 )iy 2 rdrdθ Ω ⎩ Oy2

∫∫

∫∫ ∫∫

where M is the current point with the coordinates r and θ, z1 is the normal unit vector at the surfaces in M, and xc and yc are the basis vectors of the Cartesian coordinates (Figure 2.4). These relations may also be written as:

ℑ pressure

⎧ Wz = p (r , θ ) rdrdθ 2 ⎪ Ω ⎪ 2 ⎨ M Ox2 = Ω p (r , θ ) r sin θ drdθ ⎪ ⎪M p (r , θ ) r 2 cos θ drdθ =− Ω ⎩ Oy2

∫∫ ∫∫ ∫∫

[2.65]

The shear constraints within the lubricant film are expressed as:

∂u ⎧ ⎪⎪τ rz = μ ∂z ⎨ ⎪τ = μ ∂v θz ∂z ⎩⎪

[2.66]

The components u and v of velocity (Figure 2.4) are given by the following expressions:

⎧ I 0 J1 ⎞ ∂p ⎛ ⎪u ( x, y , z , t ) = ⎜ I1 − ⎟ ∂r ⎝ J0 ⎠ ⎪ ⎨ ⎛ I 0 J1 ⎞ I0 ⎞ 1 ∂p ⎛ ⎪ ω = − + − v x y z t I r ( , , , ) 1 ⎜ ⎟ ⎜ ⎟ 1 ⎪ r ∂θ ⎝ J0 ⎠ J0 ⎠ ⎝ ⎩ By carrying forward the expressions of u and v in [2.58] the following is obtained:

⎧ ∂p ⎛ ∂I1 ∂I 0 J1 ⎞ − ⎪ τ rz = μ ⎜ ⎟ ∂r ⎝ ∂y ∂y J 0 ⎠ ⎪ ⎨ ⎡ ∂p ⎛ ∂I1 ∂I 0 J1 ⎞ ∂I 0 ω ⎤ ⎪ ⎪τ θ z = μ ⎢ ∂r ⎜ ∂y − ∂y J ⎟ − ∂θ J ⎥ ⎢⎣ ⎝ 0 ⎠ 0⎥ ⎦ ⎩

56

Hydrodynamic Bearings

and by replacing the integrals I0 and I1 with their expressions:

⎧ ⎪ I 0 ( x, y , z , t ) = ⎪ ⎨ ⎪ I ( x, y , z , t ) = ⎪⎩ 1

z

1

z

ξ

∫0 μ ( x, y, ξ , t ) d ξ ∫0 μ ( x, y, ξ , t ) d ξ

the following final result is obtained:

⎧ J1 ⎞ ∂p ⎛ ⎪τ rz = ⎜ z − ⎟ J0 ⎠ ∂r ⎝ ⎪ ⎨ J1 ⎞ rω 1 ∂p ⎛ ⎪ ⎪τ θ z = r ∂θ ⎜ z − J ⎟ − J 0 ⎠ 0 ⎝ ⎩

[2.67]

where J0 and J1 are given by: h 1 ⎧ ⎪ J 0 ( x, y, t ) == 0 μ ( x, y, ξ , t ) d ξ ⎪ ⎨ h ξ ⎪ J ( x, y , t ) = dξ 1 0 μ ( x, y , ξ , t ) ⎪⎩

∫

∫

By integrating these constraints calculated at the bounding walls, it is possible to obtain the efforts exerted tangentially on those, whose moment in relation to the rotation axis of the thrust bearing case is expressed10: – for bounding wall 1:

M 1Oz = 1

∫∫Ω

rτ θ z

z =0

rdrdθ = −

⎡ ∂p ⎛ J1 ⎞ r r 2 ω ⎤ ⎢ ⎜ ⎟+ ⎥ rdrdθ Ω ⎢ ∂θ J ⎣ ⎝ 0 ⎠ h J 0 ⎥⎦

∫∫

[2.68]

– for bounding wall 2:

M 2Oz = c

∫∫Ω

rτ θ z

z =h

rdrdθ =

⎡ ∂p ⎛ J1 ⎢ ⎜ Ω ⎢ ∂θ J ⎣ ⎝ 0

∫∫

⎞ r r 2ω ⎤ ⎥ rdrdθ ⎟− ⎠ h J 0 ⎥⎦

[2.69]

10 The coefficient r/h has been introduced to account for the filling in non-active zones.

Equations of Hydrodynamic Lubrication

57

Particular case of isoviscous fluids If viscosity is constant across the film thickness, these expressions may be simplified and written as: – for bounding wall 1:

M 1Oz = − 1

⎛ h ∂p r μ r 2ω ⎞ + ⎜⎜ ⎟ rdrdθ Ω 2 ∂θ h h ⎟⎠ ⎝

∫∫

[2.70]

– for bounding wall 2:

M 2Oz = c

⎛ h ∂p r μ r 2ω ⎞ − ⎜ ⎟ rdrd θ Ω ⎜ 2 ∂θ h h ⎟⎠ ⎝

∫∫

[2.71]

2.10. Computation of viscous dissipation energy: journal bearing case

Internal friction within the lubricant, which is proportional to its dynamic viscosity, leads to a transformation of mechanical energy into heat. The rise in lubricant temperature, which results from this, leads to a reduction in the lubricant’s viscosity. These nonlinear phenomena are examined in detail in Chapter 1 of Volume 3 [BON 14]. The calculation of the dissipation energy is obtained by the integration of the dissipated energy over an elementary volume located within the film: E=

∫∫∫Ω

i

σ ⋅ ε dx dy dz

[2.72] i

where σ is the tensor of the constraints and ε the tensor of the deformation rates. The shear rates within the lubricant film are linked to the shear constraints by the following expressions:

⎧ ∂u τ xy ⎪ = μ ⎪ ∂y ⎨ ⎪ ∂w = τ zy ⎪ ∂y μ ⎩

[2.73]

58

Hydrodynamic Bearings

It has been observed in section 2.8 that, taking into account the simplifying assumptions given at the beginning of the chapter, the shear constraints within the lubricant film may be written11 as:

⎧ J1 ⎞ r U ∂p ⎛ ⎪τ xy = ⎜ y − ⎟− ∂x ⎝ J0 ⎠ h J0 ⎪ ⎨ J1 ⎞ ∂p ⎛ ⎪ ⎪ τ zy = ∂z ⎜ y − J ⎟ 0 ⎠ ⎝ ⎩

[2.74]

By carrying forward expressions [2.73] and [2.74] in [2.72], the following is obtained:

E=

2 2 ⎡ ⎛ ∂p ⎛ J1 ⎞ r U ⎞ J1 ⎞ ⎞ ⎤⎥ 1 ⎢⎛ ∂p ⎛ ⎜ ⎜ ⎜y− ⎟⎟ + ⎟− ⎜y− ⎟ ⎟ dy dx dz 0 μ ⎢⎜ ∂x ⎝ ⎜ ∂ J h J z J 0 0 0 ⎠ ⎠ ⎟⎠ ⎥⎦ ⎠ ⎝ ⎝ ⎣⎝

∫∫Ω ∫

h

After an integration across the thickness of the film, the dissipation energy is written:

E=

2 2 2 ⎡⎛ J 2 ⎞ ⎛ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎞ r U ⎤ ⎢⎜ J 2 − 1 ⎟ ⎜ ⎜ ⎟ + ⎜ ⎟ ⎟ + ⎥ dx dz Ω ⎢⎜ J 0 ⎟⎠ ⎜⎝ ⎝ ∂x ⎠ ∂z ⎠ ⎟⎠ h J 0 ⎥ ⎝ ⎝ ⎣ ⎦

∫∫

[2.75]

where:

J n ( x, y , t ) =

h

ξn

∫0 μ ( x, y, ξ , t ) d ξ

11 The coefficient r/h has been introduced before the Couette term to account for the filling in non-active zones (see section 2.7).

Equations of Hydrodynamic Lubrication

59

Particular case of isoviscous fluids If viscosity is constant across the film thickness, the expression of the viscous dissipation energy may be simplified:

E=

⎡ h3 ⎛ ⎛ ∂p ⎞2 ⎛ ∂p ⎞2 ⎞ r μU 2 ⎤ ⎢ ⎥ dx dz ⎜ +⎜ ⎟ ⎟+ ⎟ h h ⎥ Ω ⎢12 μ ⎜ ⎜⎝ ∂x ⎟⎠ ∂ z ⎝ ⎠ ⎝ ⎠ ⎣ ⎦

∫∫

[2.76]

2.11. Computation of viscous dissipation energy: thrust bearing case

The calculation of the viscous dissipation energy for a thrust bearing is obtained by a similar method, on the basis of the expressions in cylindrical coordinates that have been established previously:

E=

h

∫∫Ω ∫0

2 2 ⎡ ⎛ 1 ∂p ⎛ J1 ⎞ ⎞ J1 ⎞ r rω ⎞ ⎤⎥ 1 ⎢⎛ ∂p ⎛ − + − − z z ⎜ ⎜ ⎜⎜ ⎟ dz rdr d θ ⎟⎟ ⎜ ⎟ J 0 ⎠ ⎟⎠ r ∂θ ⎝ J 0 ⎠ h J 0 ⎟⎠ ⎥ μ ⎢⎜⎝ ∂r ⎝ ⎝ ⎣ ⎦

After the integration across the thickness of the film, the dissipation energy is written as:

E=

2 2 ⎡⎛ J 2 ⎞ ⎛ ⎛ ∂p ⎞ 1 ⎛ ∂p ⎞ ⎞ r r 2ω 2 ⎤ ⎢⎜ J 2 − 1 ⎟ ⎜ ⎜ ⎟ + ⎥ rdr dθ ⎟+ 2 ⎜ ∂θ ⎟ ⎟ h J ⎟ ⎜ ⎝ ∂r ⎠ Ω ⎢⎜ J ⎝ ⎠ ⎥⎦ r 0 0 ⎠⎝ ⎠ ⎣⎝

∫∫

[2.77]

Particular case of isoviscous fluids If viscosity is constant across the film thickness, the expression of the viscous dissipation energy for a thrust bearing may be reduced to:

E=

⎡ h3 ⎛ ⎛ ∂p ⎞2 1 ⎢ ⎜ + 2 Ω ⎢12 μ ⎜ ⎜⎝ ∂r ⎟⎠ r ⎝ ⎣

∫∫

⎛ ∂p ⎞ ⎜ ⎟ ⎝ ∂θ ⎠

2

⎞ r μ r 2ω 2 ⎤ ⎥ rdr dθ ⎟+ ⎟ h h ⎥ ⎠ ⎦

[2.78]

2.12. Different flow regimes

Flows in thin film are generally assumed to be laminar. However, it is still possible that they might be subject to a regime change, if conditions favor this. As in all the domains of fluid dynamics, the Reynolds number of the flow makes it possible to evaluate the scale of the forces due to viscosity in relation to those that

60

Hydrodynamic Bearings

result from inertia effects. For a flow between two bounding walls separated by a mean distance h, one moving parallel to the velocity U, of a fluid of a dynamic viscosity µ and density ρ, the Reynolds number is expressed as:

ℜ=

ρUh μ

The specific form of the flow domain in thin film in the bearings has led to the definition of a modified Reynolds number which best represents the ratio of viscosity and inertia effects:

ℜ* =

ρUh C μ R

[2.79]

In this number, the relative clearance of the bearing is involved. This is the ratio between the radial clearance C and the bearing radius R. An evaluation of this number for two extreme configurations of internal combustion engines, that is, a Formula 1 engine at 20,000 revolutions per minute and a mass production petrol engine at 7,500 revolutions per minute, gives values between 0.0036 for thinner film zones and 0.75 for thicker film zones (Table 2.1). In all cases, the values are far below those that would lead to significant differences for the calculated pressures, whether the inertia effects in the film are taken into account or not [CON 79]. F1 engine

Mass production engine

20,000

7,500

Viscosity (Pa.s)

0.01

0.004

Density (kg/m3)

800

760

Radial clearance (µm)

30

24

Bearing radius (mm)

20

24

Rotational frequency (rpm)

Minimum film thickness (µm) | ℜ |

ℜ*

1 | 3.35 | 0.005

1 | 3.58 | 0.0036

Maximum film thickness (µm) | ℜ |

ℜ*

150 | 503 | 0.75

110 | 394 | 0.39

Table 2.1. Examples of Reynolds numbers

Equations of Hydrodynamic Lubrication

61

2.13. Bibliography [BON 14] BONNEAU D., FATU A., SOUCHET D., Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [BOO 89] BOOKER J.F., “Basic equations for fluid films with variable properties”, Journal of Tribology, vol. 111, pp. 475–479, 1989. [BRA 93] BRAUN M.J., CHOY F.K., ZHOU Y.M., “The effects of a hydrostatic pocket aspect ratio and its supply orifice position and attack angle on steady state flow patterns, pressure and shear characteristics”, Journal of Tribology, vol. 115, pp. 678–685, 1993. [CON 79] CONSTANTINESCU V.N., GALETUSE S., “On the contribution of inertia forces to fluid film lubrication”, Euromech Coll. 124, Hydrodynamic Lubrication in Bearings, Orbassano, Italy, 1979. [DAL 79] DALMAZ G., Le film mince visqueux dans les contacts hertziens en régimes hydrodynamique et élastohydrodynamique, Thesis, University Claude Bernard, Lyon, France, 1979. [DOW 75] DOWSON D., TAYLOR C.M., “Fundamental aspects of cavitation in bearings”, Proceedings of the 1st Leeds Lyon Symposium on Tribology, Mechanical Engineering Publications Limited, New York, NY, pp. 15–28, 1975. [FEN 97] FENG Z.C., LEAL L.G., “Nonlinear bubble dynamics”, Annual Review of Fluid Mechanics, vol. 29, pp. 201–243, 1997. [FRE 90] FRÊNE J., NICOLAS D., DEGUEURCE B., et al., Lubrification hydrodynamique – Paliers et butées, Eyrolles, Paris, 1990. [FRE 90] FRÊNE J., NICOLAS D., DEGUEURCE B., et al., Hydrodynamic Lubrication – Bearings and Thrust Bearings, Elsevier Science, Amsterdam, 1997. [FOR 05] FORTIER A.E., SALANT R.F., “Numerical analysis of a journal bearing with a heterogeneous slip/no-slip surface”, Journal of Tribology, vol. 127, pp. 820–825, 2005. [HEL 03] HÉLÈNE M., Comportement statique et dynamique de paliers hybrides: étude du champ de pression dans l’alvéole, Doctorate Thesis, University of Poitiers, France, 2003. [HIL 95] HILL D.E., BASKHARONE E.A., SAN ANDRES L.A., “Inertia effect in a hybrid bearing with a 45 degree entrance region”, Journal of Tribology, vol. 117, pp. 498–505, 1995. [JED 04] JEDDI L., Visualisation et modélisation thermohydrodynamique par éléments finis de l’alimentation des contacts lubrifiés, Doctorate Thesis, University of Poitiers, France, 2004. [MAG 07] MA G.J., WU C.W., ZHOU P., “Wall slip and hydrodynamics of two-dimensional journal bearing”, Tribology International, vol. 40, pp. 1056–1066, 2007. [PLE 77] PLESSET M., PROSPERETTI A., “Bubble dynamics and cavitation”, Annual Review of Fluid Mechanics, vol. 9, pp. 145–185, 1977. [SAN 92] SAN ANDRES L.A., VELTHUIS J.F.M., “Laminar flow in a recess of a hydrostatic bearing”, STLE Tribology Transactions, vol. 35, no. 4, pp. 738–744, 1992.

3 Numerical Resolution of the Reynolds Equation

The equations that the pressure p and the filling r in the lubricant film of a bearing should verify can be solved analytically only for some particularly simple cases. However, the supplementary assumptions are very restrictive: infinite length bearing or short bearing, is always considered to be undeformable and to be without any irregularities of shape, and the boundary conditions on the pressure field do not guarantee the conservation of the lubricant mass flow rate in the ruptured film zones. These academic solutions can be found in general works on lubrication [FRE 90, FRE 97, HAM 04, LIG 97]. For example, the bearings of contemporary internal combustion engines operate in accumulated severe conditions with reliability requirements which are such that the models for the dimensioning and prediction of behavior must take into account as many parameters as possible. In order to obtain computing codes which provide the required performance level, it is thus necessary to use the numerical methods usually employed for handling partial differential equations in physics. The finite difference method has largely been used to solve the Reynolds equation numerically, and it is still often chosen for this purpose, due to the simplicity of its application. With the development of increasingly powerful computers and the growth in demand in the domain of structure computing, the finite element method has been in continuous development for over 40 years. It has proven to be very effective, and thus was quickly adopted for handling the Reynolds equation when the finite difference method was proving difficult to use, such as, for example, in the case of variable width bearings. More recently, the finite volume method has been developed for handling problems where the equations which need to be solved express a flux conservation: conservation of heat in thermodynamics or conservation of fluid flux in fluid dynamics. It has been shown in Chapter 2 that the Reynolds equation expresses the mass flow rate conservation within the lubricant

64

Hydrodynamic Bearings

film: therefore the equations for thin film flows can be reformulated such that they can be solved using the finite volume method. Other methods have been developed for solving the problems governed by partial differential equations. There have been attempts to use these methods for solving the Reynolds equation. One of these has been the boundary finite element method (also known as the boundary integral equation method) [KHA 85]. This method requires knowledge of the Green kernel of the differential operator, which defines the equation which needs to be solved: unfortunately, apart from a few specific cases where the Reynolds equation takes a simplified form1, this kernel is unknown, and renders the use of this method ineffective. Although most of the developments presented in this work are based on the finite element method, before developing this in detail, an outline will be provided of the two other methods mentioned in this introduction: the finite difference method and the finite volume method. 3.1. Definition of the problems to be solved Study of the lubrication of hydrodynamic bearings is conducted through the successive resolution of the thin film equations presented in Chapter 2. In order to obtain a clear definition of the equations involved, three problems which will be found in the general algorithm defined in Chapter 5 of [BON 14b] will be defined. These problems are defined for a shaft bearing and can be easily transposed for a thrust bearing. 3.1.1. Problem 1: determining the pressure It is assumed that the film completely occupies a domain Ω. The surfaces covered by the supply orifices or grooves, in which the pressure is assumed to be known, are not part of this domain Ω. Film rupture is not expected to occur, even if the pressure becomes less than the cavitation pressure. The equation that needs to be solved was established in Chapter 2:

E1 ( p ) =

⎡ ∂ ( J1 / J 0 ) ∂h ⎤ ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ + ⎥=0 ⎜ G ⎟ + ⎜ G ⎟ − ρ ⎢U ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ∂x ∂t ⎦ ⎣

[3.1]

1 When the film thickness and viscosity are constant, the Reynolds equation operator [2.15] is 1 Ln(r ) reduced to the Laplacian, which has the Green kernel: 2π

Numerical Resolution of the Reynolds Equation

65

In what follows, it will be called “standard Reynolds equation”. The function p is presumed to be twice differentiable in relation to x and z in Ω, i.e. of the class C 2(Ω) [DON 03]. At any point of the domain Ω it is dependent on time and is differentiable in relation to this variable. The functions G, I2, J2 and h known at all points of the domain, are dependent on time, and are differentiable. The density ρ and the velocity U are assumed to be constant. It will be assumed that the value of the pressure p is equal to a known pressure

p on a part ∂Ω1 of the boundary ∂ Ω . This boundary condition is the one which is most often encountered in applications involving bearings, whether this is at the bearing ends or in the area surrounding the supply orifices or grooves. On the rest ∂Ω 2 of the boundary, the boundary conditions will concern the entering mass flow rate Qm : G

∂p = Qm ∂n

∂p is the pressure gradient grad p • n in the direction of the normal n outside ∂n the domain.

where

The “hydrodynamics problem” for solving the standard Reynolds equation may be expressed as follows:

⎧determine p ∈ C 2 in Ω and on ∂Ω 2 such that: ⎪ E ( p ) = 0 in Ω 1 ⎪ Problem 1⎪⎨ p = p on ∂Ω1 ⎪ ⎪ G ∂p = Q on ∂Ω m 2 ⎪⎩ ∂n

[3.2]

3.1.2. Problem 2: determining of the pressure and the lubricant filling

It is assumed that the full film zone partially occupies the domain Ω. A partition of the domain into two zones Ωp and Ωr is defined, which are not necessarily associated. The equation that needs to be solved, established in Chapter 2, is the following:

66

Hydrodynamic Bearings

⎡ ∂ ⎛ ∂D ⎞ ∂ ⎛ ∂D ⎞ ⎤ E2 ( D) = Φ ⎢ ⎜ G ⎟ + ⎜G ⎟⎥ ⎣ ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ⎦ ⎡ ∂ ( J1 / J 0 ) ∂h ⎛ U ∂D ∂D ⎞ ⎤ − ρ ⎢U + + (1 − Φ ) ⎜ + ⎟⎥ = 0 ∂x ∂t ⎝ 2 ∂x ∂t ⎠ ⎦ ⎣

[3.3]

In this equation, known as “modified Reynolds equation”, D is a universal function which represents either p − pref in Ωp, or r − h in Ωr. Φ is an index function which is equal to 1 in Ωp and 0 in Ωr. The function D is presumed to be twice differentiable in relation to x and z in Ωp and Ωr. It is dependent on time and is differentiable in relation to this variable. The functions G, I2, J2 and h known at all points of the domain, are dependent on time, and are differentiable. The density ρ and the velocity U are constant. The zones Ωp and Ωr are separated by up-flow and down-flow boundaries, the abscissa of which are respectively xup and xdown (Figure 2.8). Passing these, the functions p and r must satisfy the conditions: – boundary conditions up-flow from a non-active zone: ∂p U ⎧ (rupture) ⎪Vup < 2 ; r = h ; p x = xup = pref ; ∂x x = x = 0 up ⎪ ⎨ 1 U ∂p ⎪ V ≥U ;r ≤h; p = pref ; = ( − Vup )(h − r ) (formation) x = xup ⎪⎩ up 2 ∂x x = xup G 2

– boundary conditions down-flow from a non-active zone: U ∂p ⎧ (rupture) ⎪Vdown > 2 ; r = h ; p x = xdown = pref ; ∂x x = x = 0 ⎪ down ⎨ ∂p U 1 U ⎪V = ( − Vdown )(h − r ) (formation) ; r ≤ h ; p x = xdown = pref ; down ≤ ∂x x = xdown G 2 2 ⎩⎪

These boundary conditions may be rewritten, making the function D appear: – up-flow boundary conditions: U ∂D ⎧ (rupture) ⎪Vup < 2 ; D = 0 ; ∂x x = x− = 0 up ⎪ ⎨ 1 U ⎪ V ≥ U ; r ≤ h ; D − = 0 ; ∂D = − ( − Vup ) D x = x+ (formation) x = xup − up ⎪ up 2 2 ∂ x G x = xup ⎩

[3.4]

Numerical Resolution of the Reynolds Equation

67

– down-flow boundary conditions: ∂D U ⎧ (rupture) ⎪Vdown > 2 ; D = 0 ; ∂x x = x+ = 0 down ⎪ [3.5] ⎨ 1 U ∂D U ⎪ V ≤ ≤ = = − ; ; 0 ; ( ) (formation) r h D V D + − down x = xdown x = xdown + ⎪ down 2 ∂x x = xdown G 2 ⎩

It should be noted that in the case of film reformation boundaries, whether these are up-flow or down-flow from the ruptured film zone, D is not necessarily continuous. For rupture boundaries, D is certain to be continuous, but this does not apply to its derivative. The second “hydrodynamics problem” for solving the modified Reynolds equation may be expressed as follows:

⎧determine D ∈ C 2 in Ω and on ∂Ω 2 such that: ⎪ E ( D) = 0 in Ω 2 ⎪ Problem 2 ⎪⎨ ∂D ⎪ D = p on ∂Ω1 ; G ∂n = Qm in ∂Ω 2 ⎪ ⎪⎩ G verify [3.4] and [3.5] on boundary between Ω p and Ω r

[3.6]

As with other physics problems, this is a free boundary problem. The position of this boundary is a priori unknown, but the equations and inequations that need to be satisfied at the level of this boundary will make it possible to develop iterative search algorithms for its position. Cavitation algorithm Problem 2 expressed in this way makes it possible to determine D on the domain Ω for a partition of this domain into active zones identified by a value 1 of the index function Φ and into non-active zones in which Φ equals 0. Once D has been calculated, it is necessary to verify that the values obtained are compatible with the inequations contained in the boundary conditions [3.4] and [3.5]: ⎧D ≥ 0 ⎨ ⎩D < 0

in active zones in non active zones

Starting from an initial partition, the cavitation algorithm comprises, based on a principle of complementarity, changing the status of a point at which there is a violation of the inequation which corresponds to its prior status, and reiterating the

68

Hydrodynamic Bearings

computation of D for a new partition defined in this way. The process is continued until stability of the partition is obtained.

Figure 3.1. Cavitation algorithm

3.1.3. Other problems

During the study of the hydrodynamic lubrication of a bearing or a thrust bearing, the thickness of the lubricant film is not generally a priori known, for the following two reasons. First, under the effect of the loading (usually a function of time) applied to the shaft, the shaft moves within the housing. In order to find the position (εx, εy)2 of the center of the shaft, it is necessary to solve the equilibrium equations simultaneously with the equations from problem 1 or problem 2. With a view to establish a general algorithm which is as effective as possible, the choice has been made to handle the search for the partition into active and non-active zones independently to the search for the position of the shaft. Thus, the “problem of hydrodynamic equilibrium” will be defined as an extension of problem 1 applied to the active subdomain Ωp alone: ⎧determine ε x , ε y , p ∈ C 2 in Ω p and on ∂Ω p 2 such that ⎪ ⎪ E1 ( p ) = 0 in Ω p Problem 3 ⎪⎨ ∂p = Qm on ∂Ω p 2 ⎪ p = p on ∂Ω p1 ; G ∂n ⎪ ⎪⎩ ℑpressure = ℑloading

[3.7]

where ℑpressure and ℑloading are respectively the torsors of the pressure actions (see sections 2.7 and 2.8) and the loading applied to the bearing (see Chapter 1 of Volume 4 [BON 14a] for the load determination of internal combustion engine bearings). 2 Here x and y refers to the two directions of the bearing orthogonal to its rotation axis O z and not to the coordinates x and y of the developed bearing (in this cases x is the curvilinear abscissa in the circumferential direction and y the ordinate across the film thickness).

Numerical Resolution of the Reynolds Equation

69

In addition, under the effect of the hydrodynamic pressure generated in the lubricant film, the bounding walls may become deformed if their elasticity is taken into account. The viscous friction on the bounding walls also contributes to the deformation of the bounding walls, but this is to a lesser amplitude, and this effect may generally be disregarded. This deformability of the bounding walls makes it necessary to solve the elasticity equations at the same time as the other equations, due to the film pressure–thickness coupling. The following “problem of elastohydrodynamic equilibrium” will be defined in this way: ⎧determine ε x , ε y , ⎪ ⎪ h in Ω and on ∂Ω ⎪ p in Ω p and on ∂Ω p 2 ⎪ ⎪such that: Problem 4 ⎪⎨ E1 ( p) = 0 in Ω p ⎪ ⎪ p = p on ∂Ω ; G ∂p = Q on ∂Ω 1 m p2 ⎪ ∂n ⎪ ℑ pressure = ℑloading ⎪ ⎪ Eelasticity ( p, h) = 0 ⎩

[3.8]

where Eelasticity(p, h) = 0 represents the elasticity equations and the associated boundary conditions. This problem is studied in detail in Chapter 4. For the problems previously defined, the temperature within the film and the surrounding solids is assumed to be known a priori and usually to be constant throughout the bearing. This assumption is difficult to accept for engine bearings because of the level of power dissipated by these bearings and the phenomena of heat conduction and convection. A computation of the temperature T in the film and the solids and of the viscosity μ which results from this must be carried out at the same time as the other computations described previously. The “thermoelastohydrodynamic problem” is thus defined as an extension of the elastohydrodynamic problem:

70

Hydrodynamic Bearings

⎧determine ε x , ε y , ⎪ ⎪ h in Ω and on ∂Ω ⎪ p in Ω p and on ∂Ω p 2 ⎪ ⎪ T in Ω film , Ω solid 1 et Ω solid 2 ⎪ ⎪such that: Problem 5 ⎪⎨ E1 ( p) = 0 in Ω p ⎪ ⎪ p = p on ∂Ω ; G ∂p = Q on ∂Ω p1 m p2 ⎪ ∂n ⎪ ℑ pressure = ℑloading ⎪ ⎪ Ethermoelasticity ( p, h, T ) = 0 ⎪ ⎪⎩ Eenergy (T , h, p, μ ) = 0 ; Eheat (T ) = 0

[3.9]

Ωfilm, Ωsolid 1 and Ωsolid 2 are respectively the three-dimensional (3D) domain occupied by the lubricant and the domains occupied by the two solids surrounding the film. Ethermoelasticity(p, h, T) = 0 represents the thermoelasticity equations and the associated boundary conditions. Eenergy(T, h, p, μ) = 0 and Eheat(T) = 0 represent respectively the equation for energy in the film, and the equation of the heat in the solids and their associated boundary conditions. This problem is studied in detail in Chapters 1–3 of Volume 3 [BON 14b]. 3.2. The finite difference method

Discretization by the finite difference method is obtained by replacing the derivatives from the equation that needs to be solved with the approached expressions obtained on the basis of developments in Taylor series, generally limited to the first-order. The presentation of this below will be limited to the case of the constant width bearing. For variable width bearings, there are two possibilities for discretization: either the meshes are of constant width but the domain boundary does not follow the alignment of the computing nodes, which means that adaptations of the discretized expressions are required at this level, or the meshes are of variable width to follow the shape of the domain, but the discretization of the partial derivatives must take into account the fact that the nodes are not aligned with the axes of the orthogonal

Numerical Resolution of the Reynolds Equation

71

coordinates, which may be either Cartesian or cylindrical. If the width variation is continuous a convenient variable change may solve these difficulties. The study domain Ω is thus a rectangle if the option of developed bearing is selected. 3.2.1. Computation grid

The domain Ω occupied by the film is covered by a grid within which the distances between the nodes dx following x and dz following z have been assumed to be constant (Figure 3.2). Variable steps between nodes may be used without any special difficulty. The computing points are placed where the grid lines cross. The values on the bearing edges, thus on the upper and lower edges of the grid, are known. In order to be able to take the boundary conditions of periodicity into account at x = 0 and x = 2πR, the grid is extended by one mesh at either side horizontally.

Figure 3.2. Grid for computation: ● nodes corresponding to computational points

3.2.2. Discretization of standard Reynolds equation (problem 1)

Consider a computational point P situated toward the middle of the domain (Figure 3.3). E, N, W and S are used to designate the points of the grid surrounding the point P. Using the designated mid-step points, e, n, o and s, the following may be notated:

72

Hydrodynamic Bearings

∂ ⎛ ∂p ⎞ ⎡ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎤ ⎜ G ⎟ ⎢Ge ⎜ ⎟ − Gw ⎜ ⎟ ⎥ / Δ xP ; ∂x ⎝ ∂x ⎠ ⎣ ⎝ ∂x ⎠e ⎝ ∂x ⎠ w ⎦ p − pW p E − pP ⎛ ∂p ⎞ ⎛ ∂p ⎞ ; ⎜ ⎟ = P ⎜ ⎟ = x dx x dxw ∂ ∂ ⎝ ⎠e ⎝ ⎠w e

or:

p − pW ∂ ⎛ ∂p ⎞ ⎡ pE − pP − Gw P ⎜ G ⎟ ⎢Ge dxe dxw ∂x ⎝ ∂x ⎠ ⎣

⎤ ⎥ / Δ xP ⎦

[3.10]

Figure 3.3. Computational points

Also:

p − pS ⎤ ∂ ⎛ ∂p ⎞ ⎡ pN − pP − Gs P ⎥ / Δ zP ⎜ G ⎟ ⎢Gn dzn dzs ⎦ ∂z ⎝ ∂z ⎠ ⎣ and

U

∂ ( J1 / J 0 ) ∂x

+

( J1 / J 0 )e − ( J1 / J 0 )w ⎛ ∂h ⎞ ∂h +⎜ ⎟ UP ∂t Δ xP ⎝ ∂t ⎠ P

[3.11]

Numerical Resolution of the Reynolds Equation

73

By carrying over the expressions approached above in equation [3.1], the following is obtained: ⎡ p E − pP p − pW − Gw P ⎢Ge dx dxw e ⎣

⎤ ⎡ p N − pP p − pS ⎤ − Gs P ⎥ / Δ xP + ⎢Gn ⎥ / Δ zP dz dzs ⎦ n ⎦ ⎣ ( J1 / J 0 )e − ( J1 / J 0 )w ⎛ ∂h ⎞ − ρU P −ρ⎜ ⎟ =0 Δ xP ⎝ ∂t ⎠ P

The discretized equation of the point P is thus notated:

AP pP + AE pE + AN pN + AW pW + AS pS + Sc = 0

[3.12]

where the coefficients A E, N, W, S, P are given by: AE =

Ge Gn Gw Gs ; AN = ; AW = ; AS = dxe Δ xP dzn Δ z P dxw Δ xP dzs Δ z P AP = −

Ge Gw Gs Gn − − − dxe Δ xP dxw Δ xP dzs Δ z P dzn Δ z P

[3.13]

Sc represents the source term, which is expressed: ⎛ ∂h ⎞ Sc = − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ / Δ xP − ρ ⎜ ⎟ ⎣ ⎦ ⎝ ∂t ⎠ P

[3.14]

When the steps are constant and equal at Δx and Δz these expressions may be reduced to: AE =

Ge G G G ; AN = n2 ; AW = w2 ; AS = s 2 2 Δx Δz Δx Δz G +G G +G AP = − e 2 w − s 2 n Δx Δz

⎛ ∂h ⎞ Sc = − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ / Δ xP − ρ ⎜ ⎟ ⎣ ⎦ ⎝ ∂t ⎠ P

[3.15]

[3.16]

By writing equation [3.14] at any point where the pressure is unknown, and by taking into account the boundary conditions on the bearing edges and the supply

74

Hydrodynamic Bearings

orifices on the one hand and near to the periodicity boundaries on the other, and then by assembling the equations, the global linear system to solve is obtained. 3.2.3. Discretization of modified Reynolds equation (problem 2)

Consider the following discretized expressions:

Φ D − ΦW DW ⎤ ∂ ⎛ ∂D ⎞ ⎡ Φ E DE − Φ P DP − Go P P ⎥ / Δ xP ⎜G ⎟ ⎢Ge dxe dxw ∂x ⎝ ∂x ⎠ ⎣ ⎦ Φ D − Φ S DS ⎤ ∂ ⎛ ∂D ⎞ ⎡ Φ N DN − Φ P DP − Go P P ⎥ / Δ zP ⎜G ⎟ ⎢Gn dzn dzs ∂z ⎝ ∂z ⎠ ⎣ ⎦

[3.17]

When the point P and its neighbors E, N, W and S are situated in an active zone, all the Φ coefficients are equal to 1 and the expressions [3.10] and [3.11] from problem 1 appear. If the point P and its neighbors E, N, W and S are situated in a non-active zone, all the Φ coefficients are nil and these terms disappear. If the point P is situated in an active zone (ΦP = 1) but not one or several of its neighbors, it is then situated in an active zone but in immediate proximity to a transition boundary. For the neighbor(s) where Φ is nil, the relative pressure p – pref is nil. The discretized derivatives of the relative pressure at P thus only involve the values of the relative pressure in P and at the neighboring nodes where Φ is not nil. All these conditions are satisfied by the relations [3.17]. If the point P is situated in a non-active zone (ΦP = 0) but not one of its neighbors, it is then also situated in immediate proximity to a transition boundary. In this case, the relative pressure p – pref at P is nil and the discretized derivatives of the relative pressure at P only involve the neighboring values where Φ is not nil. In what follows, it will be seen how important the residual terms of the discretization relations [3.17] are for conserving the mass flow rate as the transition boundaries between zones are passed. In this way, a valid expression is obtained, whatever the status of the point P:

Φ D − ΦW DW Φ E DE − Φ P DP ⎡ ∂ ⎛ ∂D ⎞ ∂ ⎛ ∂D ⎞ ⎤ − Gw P P ⎜G ⎟ + ⎜G ⎟ ⎥ Ge x x z z dx x dxw Δ xP ∂ ∂ ∂ ∂ Δ ⎠ ⎝ ⎠⎦ ⎣ ⎝ e P

Φ⎢

Φ D − Φ P DP Φ D − Φ S DS + Gn N N − Gs P P dzn Δ z P dz s Δ z P

[3.18]

Numerical Resolution of the Reynolds Equation

75

The term which does not depend on D is once more approached by:

U

∂ ( J1 / J 0 ) ∂x

+

( J1 / J 0 )e − ( J1 / J 0 )w ⎛ ∂h ⎞ ∂h +⎜ ⎟ UP ∂t Δ xP ⎝ ∂t ⎠ P

[3.19]

The final term of equation [3.3] is discretized as follows:

⎛U ⎡ DP − DW D − DE ⎤ ∂DP ⎛ U ∂D ∂D ⎞ −ρ ⎜ + + (sgn U − 1) P ⎥+ ⎟ − ρ ⎜⎜ ⎢(sgn U + 1) dxw dxe ⎦ ∂t ⎝ 2 ∂x ∂t ⎠ ⎝4⎣

⎞ ⎟⎟ ⎠

This term only needs to be calculated if D represents r – h, i.e. if Φ equals 0 for the point P and its neighbors W and E. This condition is guaranteed by multiplying the corresponding terms by 1 – ΦP, 1 – ΦW and 1 – ΦE, which gives: ⎛U ⎡ (1 − Φ P ) DP − (1 − ΦW ) DW ⎡ ⎛ U ∂D ∂D ⎞ ⎤ − ρ ⎢(1 − Φ ) ⎜ + ⎟ ⎥ − ρ ⎜⎜ ⎢ (sgn U + 1) dxw ⎝ 2 ∂x ∂t ⎠ ⎦ ⎣ ⎝4⎣ (1 − Φ P ) DP − (1 − Φ E ) DE ⎤ ∂DP ⎞ + (sgn U − 1) ⎥+ ⎟ dxe ⎦ ∂t ⎠

[3.20]

∂D is always decentered ∂x toward the up-flow with respect to the velocity U due to the hyperbolic character of the equation when Φ equals 0.

In this expression, the computing of the derivative

The expression [3.20] involves the temporal derivative of DP, but only when ΦP equals 0, that is, when D represents r – h. This derivative can be evaluated using finite temporal difference: ∂DP (1 − Φ P = ∂t

The factors (1 − Φ P

t

)D

− (1 − Φ P

t − Δt

)(r

and (1 − Φ P

t − Δt

)

P t

t

)

Δt

P

− hP )

t − Δt

[3.21]

make it possible to consider a nil value

of r – h when Φ equals 1, in accordance with the cavitation algorithm. More precise forms of the discretization of the temporal derivatives are given in section 3.5.

76

Hydrodynamic Bearings

In what follows, it will be seen how important the residual terms of relations [3.20] and [3.21] are when point P is in immediate proximity to a transition boundary. By carrying over the approached expressions [3.18] to [3.21] in the equation [3.3], the following is obtained: Ge

Φ D − Φ W DW Φ D − Φ P DP Φ D − Φ S DS Φ E DE − Φ P DP − Gw P P + Gn N N − Gs P P dxe Δ xP dxw Δ xP dzn Δ z P dz s Δ z P

⎡ ( J1 / J 0 )e − ( J1 / J 0 ) w ⎛ ∂h ⎞ ⎤ − ρ ⎢U P +⎜ ⎟ ⎥ Δ xP ⎝ ∂t ⎠ P ⎦⎥ ⎣⎢

(1 − Φ P ) DP − (1 − ΦW ) DW (1 − Φ P ) DP − (1 − Φ E ) DE ⎤ U⎡ + (sgn U − 1) ⎢ (sgn U + 1) ⎥ dxw dxe 4⎣ ⎦ (1 − Φ P ) DP − (1 − Φ P t − Δt ) ( rP − hP ) t − Δt −ρ =0 Δt [3.22]

−ρ

The discretized equation for the point P is thus notated:

AP DP + AE DE + AN DN + AW DW + AS DS + Sc = 0

[3.23]

where the coefficients A E, N, W, S, P are notated: ⎡Φ G ⎤ 1 Φ G U AE = ⎢ E e + ρ (1 − Φ E ) ( sgn U − 1) ⎥ ; AN = N n 4 Δ x dx dz P n Δ zP ⎣ ⎦ e ⎡Φ G ⎤ 1 Φ G U AW = ⎢ W w + ρ (1 − ΦW ) ( sgn U + 1) ⎥ ; AS = S s Δ 4 x dx dz P s Δ zP ⎣ ⎦ w

[3.24]

⎡⎛ G ⎛G G ⎞ 1 G ⎞ 1 ⎤ AP = −Φ P ⎢⎜ e + w ⎟ +⎜ s + n ⎟ ⎥ ⎢⎣⎝ dxe dxw ⎠ Δ xP ⎝ dzs dzn ⎠ Δ z P ⎥⎦ ⎡ U ⎛ sgn U − 1 sgn U + 1 ⎞ 1 ⎤ − ρ (1 − Φ P ) ⎢ ⎜ + ⎟+ ⎥ dxw ⎠ Δt ⎦⎥ ⎣⎢ 4 ⎝ dxe

Sc represents the source term, which is expressed: ⎡ ( J1 / J 0 ) − ( J1 / J 0 ) ⎛ ∂h ⎞ e w Sc = − ρ ⎢U + ⎜ ⎟ + (1 − Φ P Δ xP ⎝ ∂t ⎠ P ⎢⎣

t − Δt

)

( rP − hP )

Δt

t − Δt

⎤ ⎥ ⎥⎦

[3.25]

Numerical Resolution of the Reynolds Equation

77

When the meshes of the grid are uniform and equal to Δx and Δz these expressions may be simplified: ⎡Φ G ⎤ 1 Φ G U ; AN = N 2 n AE = ⎢ E e + ρ (1 − Φ E ) ( sgn U − 1) ⎥ 4 Δ Δ x x Δz ⎣ ⎦ ⎡Φ G ⎤ 1 Φ G U ; AS = S 2s AW = ⎢ W w + ρ (1 − ΦW ) ( sgn U + 1) ⎥ 4 Δz ⎣ Δx ⎦ Δx

[3.26]

⎡ U ⎡G + G G +G ⎤ 1⎤ + ⎥ AP = −Φ P ⎢ e 2 w + s 2 n ⎥ − ρ (1 − Φ P ) ⎢ Δz ⎦ ⎣ Δx ⎣ 2Δ x Δt ⎦ ⎡ ( J1 / J 0 ) − ( J1 / J 0 ) ⎛ ∂h ⎞ e w Sc = − ρ ⎢U + ⎜ ⎟ − (1 − Φ P Δx ⎝ ∂t ⎠ P ⎢⎣

t − Δt

)

( rP − hP )

Δt

t − Δt

⎤ ⎥ ⎥⎦

[3.27]

Boundary conditions accounting By writing equation [3.23] for each point where D is unknown and bringing the equations together, the global system is obtained, which requires the equations from the boundary conditions in order to become complete. On the bearing edges and the periodicity boundaries, the conditions that need to be satisfied are relative to D and are easy to notate. Accounting for the boundary conditions [3.4] and [3.5] at the transition boundaries between the active and non-active zones is linked to the performance of the cavitation algorithm (Figure 3.1). The condition D = 0 is present in all situations, whether the boundary marks film formation or rupture. It is implicitly accounted for by the discretized expressions [3.24] when the resolution of the system of the equation [3.23] is integrated into the cavitation algorithm from Figure 3.1. The conditions translate the conservation of the mass flow rate on the derivative of D. In what follows, the point P will be considered to be the computing point situated in the active zone nearest to the transition boundary. Thus, at this point, D represents the pressure p – pref. For simplicity’s sake it will be assumed that: – the pitch of the mesh is uniform in both directions, equal to Δx and Δz; – the transition boundary is orthogonally oriented in the direction of movement of the sliding bounding wall; – the velocity U is positive.

78

Hydrodynamic Bearings

With this assumption, the Poiseuille flow rate orthogonal to the boundary only depends on the pressure gradient following the direction of the velocity U, i.e. the direction x. Thus, when the boundary under consideration is up-flow from the ruptured zone, the conservation condition of the mass flow rate at point P is notated: G

∂D ⎛U ⎞ + ρ ⎜ − VB ⎟ D x = x+ = 0 P − ∂x x = xP ⎝2 ⎠

The derivative of D up-flow from P can be evaluated by (DP – DW)/Δx which yields the discretized form:

Gw

DP − DW ⎛U ⎞ + ρ ⎜ − VF ⎟ DE = 0 Δx ⎝2 ⎠

This condition for the conservation of the flow rate only applies if the up-flow boundary is a formation boundary, that is, if its velocity VB verifies the inequation: VB >

U 2

which may be obtained by notating: D − DW 1⎡ U ⎞ ⎤⎡ ⎛ ⎛U ⎞ ⎤ + ρ ⎜ − VB ⎟ DE ⎥ = 0 sgn ⎜ VB − ⎟ + 1⎥ ⎢Gw P Δ 2 ⎢⎣ 2 2 x ⎝ ⎠ ⎦⎣ ⎝ ⎠ ⎦

[3.28]

If this is not the case, the boundary up-flow from the ruptured zone is a rupture boundary and the boundary conditions that need to be verified are the Reynolds ∂p conditions p – pref = 0 and = 0 r − h = 0 needs to be present on both sides of the ∂x boundary, i.e.DE = 0. Only the condition on the pressure gradient needs to be imposed: U ⎞ ⎤ ⎡ D − DW ⎤ 1⎡ ⎛ =0 sgn ⎜ VB − ⎟ − 1⎥ ⎢Gw P 2 ⎢⎣ 2 ⎠ ⎦⎣ Δ x ⎥⎦ ⎝

[3.29]

The two other boundary conditions are satisfied if the boundary is located as close as possible to the sign change of D. This condition is iteratively ensured through the cavitation algorithm (Figure 3.1). Since at the convergence of the cavitation algorithm DE = 0, condition [3.29] may also be notated:

Numerical Resolution of the Reynolds Equation

U ⎞ ⎤ ⎡ D − DW 1⎡ ⎛ ⎛U ⎞ ⎤ − ⎢sgn ⎜ VB − ⎟ − 1⎥ ⎢Gw P + ρ ⎜ − VB ⎟ DE ⎥ = 0 x 2⎣ 2 2 Δ ⎝ ⎠ ⎦⎣ ⎝ ⎠ ⎦

79

[3.30]

By adding conditions [3.28] and [3.30], all the possibilities are considered:

Gw

DW − DP ⎛U ⎞ − ρ ⎜ − VB ⎟ DE = 0 Δx ⎝2 ⎠

[3.31]

For the rupture or formation boundary situated down-flow, the derivative of D down-flow from P is evaluated by (DE – DP)/Δx and, if the same process is followed, the following is obtained: Ge

DE − DP ⎛U ⎞ + ρ ⎜ − VB ⎟ DW = 0 Δx ⎝2 ⎠

[3.32]

In a stationary operation situation, the velocity of the boundary VB is nil, the up-flow boundary at the ruptured zone is a rupture boundary and the down-flow boundary is a formation boundary. Relations [3.31] and [3.32] are thus reduced to: Gw

DW − DP D − DP U U − ρ DE = 0 ; Ge E + ρ DW = 0 Δx Δx 2 2

[3.33]

It will be shown that, in the stationary case, conditions [3.33] notated at point P situated in the active zone immediately beside the boundary are in fact ensured by the discretization relations [3.26] and [3.27]. Whether the boundary is up-flow or down-flow from the ruptured zone, ΦP = 1. Up-flow, ΦW = 1 and ΦE = 0. Down-flow, ΦW = 0 and ΦE = 1. This gives, for coefficients [3.26] and [3.27] of the equation notated at P in the stationary case: AE =

⎡Φ G Φ E Ge Φ G Φ G U⎤ 1 ; AN = N 2 n ; AW = ⎢ W w + ρ (1 − ΦW ) ⎥ ; AS = S 2 s 2 2 ⎦ Δx Δx Δz Δz ⎣ Δx

AP = −

⎡ ( J 1 / J 0 ) − ( J1 / J 0 ) ⎤ Ge + Gw Gs + Gn e w ⎥ − ; Sc = − ρ ⎢U 2 2 x Δ Δx Δz ⎢⎣ ⎥⎦

80

Hydrodynamic Bearings

Taking the values of ΦW and ΦE into account, the equation for the up-flow boundary is thus: ⎛G +G G +G −⎜ e 2 w + s 2 n x Δ Δz ⎝

⎞ G Φ N Gn Φ G DN + w2 DW + S 2s DS ⎟ DP + 2 z x Δ Δ Δz ⎠

⎡ ( J1 / J 0 ) − ( J1 / J 0 ) ⎤ e w ⎥=0 − ρ ⎢U x Δ ⎢⎣ ⎥⎦ and for the down-flow boundary: ⎛G +G Φ G G +G ⎞ G U DW − ⎜ e 2 w + s 2 n ⎟ DP + e 2 DE + N 2 n DN + ρ 2Δ x Δz ⎠ Δx Δz ⎝ Δx +

⎡ ( J 1 / J 0 ) − ( J1 / J 0 ) ⎤ Φ S Gs e w ⎥=0 DS − ρ ⎢U 2 x Δ Δz ⎢⎣ ⎥⎦

In these expressions, the terms in DN and DS represent the contributions of the Poiseuille flow rates in the transversal direction in the direction of the velocity of the bounding wall U. If the hypotheses are kept for the boundary orientation, i.e. orthogonal to the x direction, which leads to the simplified relations [3.33], these gradients are nil. Thus the coefficients ΦN and ΦS are equal to ΦP, therefore to 1, and the values of DN, DP and DS are equal. The terms in z thus disappear. After having gathered together the terms which depend on U and multiplied by Δx the following is obtained for the up-flow boundary: −

Ge + Gw G DP + w DW − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ = 0 ⎣ ⎦ Δx Δx

and for the down-flow boundary: −

Ge + Gw G ⎡ −D ⎤ DP + e DE − ρU ⎢ W + ( J1 / J 0 )e − ( J1 / J 0 ) w ⎥ = 0 2 Δx Δx ⎣ ⎦

The integrals J0 and J1 only depend on the viscosity µ and on the film thickness h. These parameters vary continuously; the same will thus be the case for the J1/J0 relationship. However D, i.e. r – h, is not continuous at a formation boundary. In the expressions above, the term which depends on the variation of J1/J0 is thus insignificant in relation to the other term of the bracket, which gives:

Numerical Resolution of the Reynolds Equation

Ge + Gw G DP + w DW = 0; Δx Δx G + Gw G U DP + e DE + ρ DW = 0 − e 2 Δx Δx

81

−

[3.34]

These conditions differ from the conditions [3.33]: Gw

DW − DP D − DP U U − ρ DE = 0 ; Ge E + ρ DW = 0 Δx Δx 2 2

However, the iterative resolution given by the cavitation algorithm (Figure 3.1) renders the values of DE nil for the up-flow boundary (rupture boundary) and the values of DP nil, whether the boundary is up or down-flow from the ruptured film zone, which gives for equation [3.33] as well as for equation [3.34]: Gw G U DW = 0; e DE + ρ DW = 0 Δx Δx 2 For a non stationary case, equations [3.31] and [3.32] involve the boundary velocity VB. The temporal derivatives discretized in relations [3.42], [3.25], [3.26] and [3.27] presume that the functions, including the index function Φ, are derivable with respect to time. Even though the index function is a binary function and cannot be derived with respect to time, it may be considered that the passage from the value 0 to the value 1 or the reverse takes place continually during the time interval Δt and thus the derivative of Φ may be expressed by: ∂Φ Φ ∂t

t

−Φ

t − Δt

Δt

A solution to this problem which is more rigorous from a mathematical point of view is obtained by expressing the temporal derivatives of Φ in terms of distributions or generalized functions (see section 3.5). In order to circumvent the problem of calculating the velocity of the transition boundary for non-stationary situations, a formulation of the problem based not on the Jacobson and Floberg [JAC 57] and Olsson [OLS 74] model (JFO model), but rather on a continuous evolution from one zone to the other has been developed by Elrod [ELR 81]. This model assumes that at any point in the film domain, the fluid is homogeneous and compressible; its density, viscosity and compressibility vary

82

Hydrodynamic Bearings

widely depending on the zones. In active zones, compressibility is very low, in fact it is that of a degassed lubricant. In contrast, in non-active zones, it is very high, due to the significant proportion of gas present in the mixture. In terms of application, this problem therefore poses a problem of the numerical order, due to the large variability of this parameter. The matrices of the linear systems that need to be solved are poorly conditioned: some terms hide others. In fact, the “compressibility” factor plays the role of a penalizing parameter in the system of equations that need to be solved, with all the difficulties that are usually encountered with these methods in relation to the choice of a value that is “neither too high nor too low”. This may lead, with the aim of efficiency, to choosing values that are far removed from the physical values. During the numerical resolution of the Elrod algorithm, even if the chosen time is very small, the continuous passage to the liquid state at the diphasic state for a computing node takes place within one computing step. In this way, the binary function Φ with its two possible values 0 and 1 remerges. As the Elrod algorithm does not require the introduction of the boundary velocity VB, it might be assumed that this is implicitly linked to the discretized “derivative” of Φ as it has been defined above, and that the discretization relations [3.24] and [3.25] are valid for all situations, stationary or not. The examples presented in section 3.6 confirm this. 3.3. The finite volume method3

The finite volume method consists of integrating, for all the adjacent subdomains, labeled “finite elementary volumes”, the system residue, often represented by partial differential equations. In contrast to the finite element method, the integration is performed directly on the strong formulation of the equations. As with the finite difference method, a presentation will be made of this method which will be limited to a rectangular domain. This is the case for a bearing of uniform width, in a developed bearing configuration, or for a thrust bearing in cylindrical coordinates. For a detailed presentation of the method, specialist works may be consulted such as, for example, the work by Versteeg and Malalasekera [VER 95]. Although the elements are surface elements, they will be labeled “volumes” as is usual when this method is used.

3 Part of the text and some of the figures for this section are taken from the thesis of Abdelghani Maoui [MAO 08].

Numerical Resolution of the Reynolds Equation

83

3.3.1. Mesh of the film domain

The domain Ω is broken down into rectangular elements (Figure 3.4). In the example presented, the rectangles are of the same size (Δx, Δz), but this is not essential. The computational points are placed in the center of the elements. In order to take the boundary conditions at the bearing edges into account, points are also placed on the bearing edge.

Figure 3.4. Mesh of the domain: ● nodes at the center of typical finite volumes, × nodes at

the periodicity section

The integration of the Reynolds equation is carried out on each typical volume, P, and its adjacent volumes, following the integration directions (Figure 3.5). The periodicity condition following the direction x is ensured by a line of supplementary virtual volumes on the edges x = 0 and x = 2πR (Figure 3.4). A virtual volume on (x = 0, z) corresponds to a typical volume on (x = 2πR, z), and a virtual volume on (x = 2πR, z) corresponds to a typical volume on (x = 0, z). Definition of the mesh elements – dx and dz: distance between the centers of the control volumes; – xp and zp: coordinates of the centers of the control volumes; – xc and zc: coordinates of the interfaces of the control volumes.

84

Hydrodynamic Bearings

Figure 3.5. Typical volume and adjacent volumes

Figure 3.6. Typical control elements

3.3.2. Discretization of the standard Reynolds equation (problem 1)

On the typical control volume (Figure 3.6) the standard Reynolds equation [3.1] is integrated: n e

⎡∂ ⎛

∂p ⎞

∂⎛

∂p ⎞

∂ ⎛

∫ ∫ ⎢⎢ ∂x ⎜⎝ G ∂x ⎟⎠ + ∂z ⎜⎝ G ∂z ⎟⎠ − ρ ∂x ⎜⎝U s w

⎣

∂ ( J1 / J 0 ) ∂x

+

∂h ⎞ ⎤ ⎟ ⎥dx dz = 0 ∂t ⎠ ⎦⎥

[3.35]

Numerical Resolution of the Reynolds Equation

85

If it is assumed that the parameters vary linearly from the center of the element to the center of the neighboring elements, the following is obtained: n e

∂ ⎛

∂p ⎞

⎡

∂⎛

∂p ⎞

⎡

∫ ∫ ∂x ⎜⎝ G ∂x ⎟⎠ dxdz = ⎢⎣G

e

s w

n e

∫ ∫ ∂z ⎜⎝ G ∂z ⎟⎠ dxdz = ⎢⎣G

n

s w

p − pW pE − p P − Gw P dxe dxw

⎤ ⎥ Δ zP ⎦

p N − pP p − pS ⎤ − Gs P ⎥ Δ xP dzn dzs ⎦

These two terms represent the Poiseuille flow rates entering into the control volume from the sides and coming from adjacent volumes. n e

−∫ ∫ ρ s w

∂ ⎛ ∂ ( I 2 / J 2 ) ∂h ⎞ + ⎟ dxdz ⎜U ∂x ⎝ ∂x ∂t ⎠

∂h = − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ Δ z P − ρ P Δ xP Δ z P ⎣ ⎦ ∂t

The first term of the second member of the expression above represents the Couette flow rate entering the control volume, coming from the volumes on either side in the direction x. The second term represents the entering flow rate in the control volume, due to the vertical movement of its upper bounding wall. By carrying over the above expression in equation [3.35], the discretization equation for each typical control volume is obtained, a conservation equation for the entering and exiting mass flow rate:

Ap pP + AE pE + AN pN + AW pW + AS pS + Sc = 0 where the coefficients AI ; AE =

I ≡ P, E, N, W, S

are written:

Ge G G G Δ zP ; AN = n Δ xP ; AW = w Δ z P ; AS = s Δ xP dxe dzn dxw dzs ⎛G G ⎞ AP = − ⎜ e + w ⎟ Δ z P ⎝ dxe dxw ⎠

[3.36]

⎛G G ⎞ − ⎜ s + n ⎟ Δ xP ⎝ dz s dzn ⎠

[3.37]

Sc is the source term; its expression is given by: ∂h S c = − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ Δ z P − ρ P Δ xP Δ z P ⎣ ⎦ ∂t

[3.38]

86

Hydrodynamic Bearings

When the volumes are of uniform sizes for each direction, simply notated Δx and Δz, these expressions are reduced to: AE = Ge

Δz Δx Δz Δx ; AN = Gn ; AW = Gw ; AS = Gs Δx Δz Δx Δz Δz Δx − ( Gs + Gn ) AP = − ( Ge + Gw ) Δx Δz

∂h S c = − ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ Δ z P − ρ P Δ xP Δ z P ⎣ ⎦ ∂t

[3.39]

[3.40]

These relations are not applicable to volumes at bearing edges which are of halfsize. It should be noted that the coefficients [3.39] and [3.40] are equal to the coefficients [3.15] and [3.16] given by the finite difference method, multiplied by the product Δx Δz. Except for the nodes near to the bearing sides and the supply orifices, for which Δz will be replaced by Δz/2, and Δx by Δx/2 for supply zone boundaries orthogonal to the x direction, the equations that need to be solved will thus be alike except for this multiplier. In fact, the difference between the two methods when they are applied to problem 1 is the placement of the computing points. By notating equation [3.36] for each typical volume, and by taking into account the specific characteristics of the volumes on the bearing edges on the one hand and near to the periodicity boundaries on the other, and then by assembling the equations, the global linear system is obtained. 3.3.3. Discretization of modified Reynolds equation (problem 2)

If the modified Reynolds equation [3.3] is integrated on the typical control volume: n e

⎛

⎡∂ ⎛

∂D ⎞

∂ ⎛

∂D ⎞ ⎤

∫ ∫ ⎜⎝Φ ⎣⎢ ∂x ⎝⎜ G ∂x ⎠⎟ + ∂z ⎝⎜ G ∂z ⎠⎟⎦⎥ s w

⎡ ∂ ( J1 / J 0 ) ∂h ⎛ U ∂D ∂D ⎞ ⎤ ⎞ − ρ ⎢U + + (1 − Φ ) ⎜ + ⎟ ⎥ ⎟ dx dz = 0 ∂x ∂t ⎝ 2 ∂x ∂t ⎠ ⎦ ⎠⎟ ⎣

[3.41]

the expressions obtained for each term will depend on the status of the control volume, i.e. the value of the binary function Φ.

Numerical Resolution of the Reynolds Equation

87

For volumes where the film is “full”, the following may be written: n e

∂ ⎛

Φ E DE − DP

⎡

∂D ⎞

∫ ∫ ∂x ⎜⎝ G ∂x ⎟⎠ dxdz = ⎢⎣G

e

dxe

s w

− Gw

DP − ΦW DW dxw

⎤ ⎥ Δ zP ⎦

In fact, for the volume P, the function Φ equals 1. For neighboring volumes, if the status is “full film”, Φ equals 1 and D represents p – pref, and if the status is “ruptured film”, D represents r – h but p – pref equals 0. This is implicitly considered because Φ equals 0. For volumes where the film is “ruptured” this term should be nil. Since, in this case ΦP is nil, it is sufficient to notate: n e

∂ ⎛

⎡

∂D ⎞

∫ ∫Φ ∂x ⎝⎜ G ∂x ⎟⎠ dxdz = ⎢⎣G

e

Φ E DE − Φ P DP dxe

s w

− Gw

Φ P DP − ΦW DW ⎤ dxw

⎥ Δ zP ⎦

to have a valid notation, whatever the status of the volume P. In the same way: n e

∂⎛

⎡

∂D ⎞

∫ ∫ Φ ∂z ⎜⎝ G ∂z ⎟⎠ dxdz = ⎢⎣G

n

Φ N DN − Φ P DP dzn

s w

− Gs

Φ P DP − Φ S DS ⎤ dzs

⎥ Δ xP ⎦

As was the case for problem 1: n e

∂ ⎛

∫ ∫ ρ ∂x ⎜⎝U

∂ ( J1 / J 0 ) ∂x

s w

+

∂h ⎞ ⎟ dxdz ∂t ⎠

∂h = ρU ⎡( J1 / J 0 )e − ( J1 / J 0 ) w ⎤ Δ z P + ρ P Δ xP Δ z P ⎣ ⎦ ∂t

The final term of equation [3.41] is notated: n e

⎡

⎛ U ∂D ∂D ⎞ ⎤ + ⎟ dx dz = ∂x ∂t ⎠ ⎥⎦

∫ ∫ − ρ ⎢⎣(1 − Φ ) ⎜⎝ 2 s w

⎛U ⎡ (1 − Φ P ) DP − (1 − ΦW ) DW − ρ ⎜ ⎢ (sgn U + 1) ⎜4 dxw ⎝ ⎣ + (sgn U − 1)

(1 − Φ P ) DP − (1 − Φ E ) DE ⎤ dxe

∂DP ⎞ ⎟⎟ Δ xP Δ z P ⎥+ ⎦ ∂t ⎠

[3.42]

88

Hydrodynamic Bearings

∂D is always decentered ∂x toward the up-flow with respect to the velocity U due to the hyperbolic character of the equation when Φ equals 0.

In this expression, the computation of the derivative

The expression [3.42] involves the temporal derivative of DP, but only when ΦP equals 1, that is, when D represents r – h. This derivative can be evaluated using finite temporal difference: ∂DP ∂t

DP

t

− ( rP − hP )

t − Δt

Δt

By carrying over the discretized expressions in the equation [3.42], the following equation is obtained:

Ap pP + AE pE + AN pN + AW pW + AS pS + Sc = 0

[3.43]

where the coefficients AI ; I ≡ P, E, N, W, S are notated:

Φ G U ⎡ ⎤ Δz AE = ⎢Φ E Ge + ρ (1 − Φ E ) ( sgn U − 1) Δ xP ⎥ P ; AN = N n Δ z P 4 dzn ⎣ ⎦ dxe Φ G U ⎡ ⎤ Δz AW = ⎢ΦW Gw + ρ (1 − ΦW ) ( sgn U + 1) Δ xP ⎥ P ; AS = S s Δ z P 4 dzs ⎣ ⎦ dxw ⎡ ⎛G G G G ⎞ AP = − ⎢Φ P ⎜ e + w + s + n ⎟ ⎢⎣ ⎝ dxe dxw dzs dzn ⎠

[3.44]

⎤ ⎛ U ⎛ sgn U − 1 sgn U + 1 ⎞ 1 ⎞ − ρ (1 − Φ P ) ⎜ ⎜ + + ⎟ Δ xP ⎥ Δ z P ⎟ ⎜4 dxw ⎠ Δt ⎠⎟ ⎝ ⎝ dxe ⎦⎥

The source term Sc is expressed: Sc = − ρ ⎡U ⎣

(( J / J ) − ( J / J ) ) 1

0 e

1

⎛ ⎛ ∂h ⎞ + ⎜ ⎜ ⎟ − (1 − Φ P ⎝ ⎝ ∂t ⎠ P

0 w

Δ )

t− t

(rP − hP )

Δt

t − Δt

⎤ ⎞ ⎟ Δ xP ⎥ Δ z P ⎥⎦ ⎠

[3.45]

Writing equation [3.43] for each typical volume and bringing the equations together leads to the global system. This system is completed through the addition of equations from the boundary conditions. On the bearing and on the supply orifices

Numerical Resolution of the Reynolds Equation

89

boundaries on the one hand, and on the periodicity boundaries on the other, the conditions that need to be satisfied are relative to D and are easy to notate. As with the finite difference method, accounting for the boundary conditions [3.4] and [3.5] at the transition boundaries between the active and non-active zones is linked to the running of the cavitation algorithm. The condition D = 0, present in all situations, that the boundary marks a film formation or rupture, is accounted for by the system of discretized equations when it is associated with the cavitation algorithm. In what follows, it is assumed that the “active” or “non-active” state concerns a volume as a whole. The transition boundaries between the zones thus follow the contours of a set of volumes. The conditions which show the conservation of the mass flow rate relate to the derivative of D. When a film rupture boundary located up-flow from the ruptured ∂D zone is involved, the condition = 0 is ensured by writing for the first volume of ∂x the ruptured film zone: ∂D 2 ( Dw − DW ) =0 Δx ∂x However, as DP represents r – h in the volume P, the same must be the case for the volume W. Yet, in this adjacent volume situated in the active zone the filling r is equal to the thickness h therefore r – h is nil. The above relation may now be written as: ⎡ U ⎛ − ⎢sgn ⎜ VB − 2 ⎝ ⎣

⎞ ⎤ 2 DW =0 ⎟ − 1⎥ ⎠ ⎦ Δx

When a film formation boundary situated up-flow from the ruptured zone is involved, the boundary condition [3.4] is notated:

⎤ ∂D ⎡ 6μ U ⎞ ⎤ ⎡ 2D 6μ ⎛ + 3 (U − 2VB ) D x = x+ ⎢sgn ⎜ VB − ⎟ + 1⎥ ⎢ − W + 3 (U − 2VB ) DP ⎥ = 0 − w 2 ⎠ ⎦ ⎣ Δ x hw ∂x x = xw h ⎝ ⎣ ⎦ These two conditions may combine to form only one if they are added together:

−

DW ⎡ μ ⎛ U U + ⎢3 − VB − − VB 2 Δ x ⎣ hw3 ⎜⎝ 2

⎞ ⎤ ⎟ DP ⎥ = 0 ⎠ ⎦

[3.46]

90

Hydrodynamic Bearings

When the rupture or formation boundary is situated down-flow, the following is obtained:

DE ⎡ μ ⎛ U U − ⎢3 − VB − − VB 2 Δ x ⎣ he3 ⎜⎝ 2

⎞ ⎤ ⎟ DP ⎥ = 0 ⎠ ⎦

[3.47]

Equations [3.46] and [3.47] involve the boundary velocity VB. As with the finite difference method, it will be assumed that the velocity is implicitly accounted for by the discretization of the “derivative” of the function Φ. The examples in section 3.6 confirm this assumption. 3.4. The finite element method 3.4.1. Integral formulation of standard Reynolds equation

First of all let us consider the problem 1, defined in section 3.1.1. Let us define the integral form on the domain Ω:

⎛ ⎡ ∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ⎤ E * ( p, W ) = ∫ W ⎜ − ⎢ ⎜ G ⎟ + ⎜ G ⎟ ⎥ ⎝ ⎣ ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ⎦ Ω ⎡ ∂ ( J1 / J 0 ) ∂h ⎤ ⎞ + ρ ⎢U + ⎥ ⎟ dΩ ∂x ∂t ⎦ ⎠ ⎣

[3.48]

where W is a “weighting” function defined on Ω. Any function p which verifies the integro-differential equation

E* ( p,W ) = 0 whatever the function W, is a solution to the equation with partial derivatives [3.1] at any point of the domain (method of weighted residuals). Demonstration is straightforward if the possibility of the existence is considered of a point of the domain for which equation [3.1] is not verified. It is then easy to find a function W such that E*(p, W) ≠ 0. The method of weighted residuals is not the only possible approach for handling the Reynolds equation using the finite element method. At the beginning of the 1970s, a variational approach was developed and the equivalence of the two methods was established [BOO 72].

Numerical Resolution of the Reynolds Equation

91

For it to be possible to evaluate the integral [3.48], the function p must always be twice differentiable, but its second derivative does not necessarily need to be. It is enough for this second derivative to be of integrable square, that is, that p belongs to the Sobolev space H2(Ω)4 [DON 03]. An integration by parts of the terms containing G makes it possible to reduce the order of derivability of the function p which is sought. This is the “weak formulation”, the functional E(p, W) of which is defined by: ⎛ ∂W ∂p ∂W ∂p + + ρW E ( p, W ) = ∫ G ⎜ ⎝ ∂x ∂x ∂z ∂z Ω

⎡ ∂ ( J 1 / J 0 ) ∂h ⎤ ⎞ + ⎥ ⎟ dΩ ⎢U ∂x ∂t ⎦ ⎠ ⎣

[3.49]

For it to be possible to evaluate the integral, the weighting function W and the sought function p simply need to belong to the space H1(Ω) [DON 03]. During integration by parts, two boundary terms appeared: ⎡ ∂ ⎛ ∂p ⎞ ⎤ ⎡ ∂W − ∫ W ⎢ ⎜ G ⎟ ⎥ dxdz = ∫ ⎢ ⎣ ∂x ⎝ ∂x ⎠ ⎦ Ω Ω ⎣ ∂x

⎛ ∂p ⎞ ⎤ ⎜ G ⎟ ⎥ dxdz − ⎝ ∂x ⎠ ⎦

∫

∂Ω

⎡ ∂ ⎛ ∂p ⎞ ⎤ ⎡ ∂W − ∫ W ⎢ ⎜ G ⎟ ⎥ dxdz = ∫ ⎢ z z ∂ ∂ ⎝ ⎠ ⎣ ⎦ Ω Ω ⎣ ∂z

⎛ ∂p ⎞ ⎤ ⎜ G ⎟ ⎥ dxdz − ⎝ ∂z ⎠ ⎦

∫

∂Ω

⎛ ∂p ⎞ W ⎜ G ⎟ nx d ( ∂Ω ) ⎝ ∂x ⎠ ⎛ ∂p ⎞ W ⎜ G ⎟ n z d ( ∂Ω ) ⎝ ∂z ⎠

They can be brought together in the form: −

∫

∂Ω

⎡ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎤ W ⎢⎜ G ⎟ nx + ⎜ G ⎟ nz ⎥d ( ∂Ω ) = − x ∂ ⎠ ⎝ ∂z ⎠ ⎦ ⎣⎝

∫

∂Ω

WG

∂p d ( ∂Ω ) ∂n

By omitting this contour term in functional [3.3], it is implied that either W or ∂p is nil on ∂ Ω . The first assumption will be used for parts ∂Ω1 of ∂ Ω where ∂n the pressure is known. The second assumption will be used for parts of ∂Ω 2 where the normal pressure gradient – or the flow rate that is proportional to it – is nil. For example, in the case of a problem that needs to be solved which has a symmetry axis, the study may be conducted on a half-domain with a nil pressure gradient condition at the symmetry axis.

4 The Sobolev space H k(Ω) represents the set of functions whose derivatives up to the order k are of integrable square on Ω, that is, belong to L 2(Ω) [DON 03].

92

Hydrodynamic Bearings

If an imposed non-nil flow rate condition has to be applied to a part of ∂Ω 2 , this contour integral should not be neglected. The problem for solving the standard Reynolds equation may be expressed as follows: ⎧determine p ∈ H 1 in Ω and on ∂Ω such that: 2 ⎪ Problem 1* ⎪⎨ p ∈ C0 ; p = p on ∂Ω1 ⎪ ⎪⎩ ∀W ∈ C0 ; W = 0 on ∂Ω1 ; E ( p, W ) = ∫ ∂Ω2 W Qm d ( ∂Ω )

[3.50]

3.4.2. Integral formulation of modified Reynolds equation

Now let us consider problem 2. On the basis of a one-dimensional model, it will be established that the boundary conditions of active and non-active zones which are relevant to the derivative of D are natural, that is implicit, for an integro-differential operator which still needs to be defined.

Figure 3.7. One-dimensional film domain

3.4.2.1. Modelization of a one-dimensional lubricant film Consider a one-dimensional domain [AE] composed of active and non-active stationary zones, as shown in Figure 3.7. On this domain of dimension 1, equation [3.3] takes the following simplified form:

Φ

⎡ d ( J1 / J 0 ) ∂h d ⎛ dD ⎞ ⎛ U dD ∂D ⎞ ⎤ + + (1 − Φ ) ⎜ + ⎜G ⎟ = ρ ⎢U ⎟⎥ ∂t dx ⎝ dx ⎠ dx ⎝ 2 dx ∂t ⎠ ⎦ ⎣

[3.51]

In the active zone, Φ equals 1 and the equation that needs to be solved is the Reynolds equation:

Numerical Resolution of the Reynolds Equation

93

d ( J1 / J 0 ) ∂h d ⎛ dp ⎞ −ρ =0 ⎜ G ⎟ − ρU ∂t dx ⎝ dx ⎠ dx In the non-active zone, the function Φ equals 0 and equation [3.51] becomes (see section 2.4): U dr ∂r + =0 2 dx ∂t

Solving equation [3.51] is equivalent to solving the integro-differential equation:

∫

E A

⎡ d ⎛ dD ⎞ ⎛ d ( J1 / J 0 ) ∂h ⎞ ⎛ U dD ∂D ⎞ ⎤ + ⎟ − (1 − Φ ) ρ ⎜ + N ⎢Φ ⎜ G ⎟ − ρ ⎜U ⎟ ⎥ dx = 0 dx ∂t ⎠ ⎝ 2 dx ∂t ⎠ ⎦⎥ ⎝ ⎣⎢ dx ⎝ dx ⎠

for any function N defined on the interval AE, or in addition taking into account the specific qualities of each zone: B ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ −∫ ρ N ⎜ + ⎟ dx A ∂t dx ⎝2 ⎠

⎡ d ⎛ dD ⎞ C ⎛ d ( J1 / J 0 ) ∂h ⎞ ⎤ + ∫ N ⎢ ⎜G + ⎟ ⎥ dx ⎟ − ρ ⎜U B dx ∂t ⎠ ⎥⎦ ⎢⎣ dx ⎝ dx ⎠ ⎝ E ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ −∫ N⎜ + ⎟ dx = 0 C dx ∂t ⎝2 ⎠ In what follows, the values at B–, B+, C– and C+ will designate the values obtained at B and C by prolonging by continuity on the basis of the values in the interval situated on the side indicated by the sign. An integration by parts on the interval BC of the term dependent on G gives: B ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ −∫ ρ N ⎜ + ⎟ dx A ∂t dx ⎝2 ⎠ C⎡ ⎛ d ( J1 / J 0 ) ∂h ⎞ ⎤ dN dD dD ⎞ dD ⎞ ⎛ ⎛ − ∫ ⎢G + ρ N ⎜U + ⎟ ⎥ dx + ⎜ NG − ⎜ NG ⎟ ⎟ B dx dx dx t dx dx ⎠ x = x ( B+ ,t ) ∂ − ⎝ ⎠ x = x ( C ,t ) ⎝ ⎝ ⎠ ⎦⎥ ⎣⎢

E ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ − ∫ ρN ⎜ + ⎟ dx = 0 C dx ∂t ⎝2 ⎠

[3.52]

94

Hydrodynamic Bearings

To ensure continuity of the mass flow rate, function D must satisfy conditions [3.4] and [3.5] at the boundaries of the active and non-active zones related to the derivative of D: – condition at B: i ⎛ dD ⎞ ⎛ U ⎞ + ⎜ ρ D⎟ − ( ρ D ) x = x ( B − ,t ) B = 0 ⎜G ⎟ ⎝ dx ⎠ x = x ( B+ ,t ) ⎝ 2 ⎠ x = x ( B− ,t )

– condition at C: i ⎛ dD ⎞ ⎛ U ⎞ +⎜ ρ D⎟ − ( ρ D ) x = x ( C + ,t ) C = 0 ⎜G ⎟ ⎝ dx ⎠ x = x (C − ,t ) ⎝ 2 ⎠ x = x (C + ,t )

where B and C are the respective velocities of the boundaries B and C. These conditions may be respectively multiplied by some NB and NC weighting functions, on the condition that NB is non-nil at B and NC is non-nil at C: i dD ⎞ U ⎞ ⎛ ⎛ B=0 + ⎜ NB ρ D ⎟ − ( NB ρ D) ⎜ NBG ⎟ − x = x ( B ,t ) dx ⎠ x = x ( B+ ,t ) ⎝ 2 ⎠ x = x ( B− ,t ) ⎝

[3.53]

i dD ⎞ U ⎞ ⎛ ⎛ C=0 + ⎜ NC ρ D ⎟ − ( NC ρ D ) ⎜ NC G ⎟ + = x x ( C , t ) dx ⎠ x = x (C − ,t ) ⎝ 2 ⎠ x = x (C + ,t ) ⎝

[3.54]

By adding the equations [3.52], [3.53] and [3.54], a unique integro-differential equation is obtained which integrates the conservation conditions of the mass flow rate as it passes from one zone to the other: B ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ −∫ ρ N ⎜ + ⎟ dx A ∂t dx ⎝2 ⎠ C⎡ ⎛ d ( J1 / J 0 ) ∂h ⎞ ⎤ dN dD dD ⎞ dD ⎞ ⎛ ⎛ − ∫ ⎢G + ρ N ⎜U + ⎟ ⎥ dx + ⎜ NG − ⎜ NG ⎟ ⎟ B dx dx ⎠ x = x ( C − ,t ) ⎝ dx ⎠ x = x ( B+ ,t ) ∂t ⎠ ⎥⎦ ⎝ ⎢⎣ dx dx ⎝ E ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ − ∫ ρN ⎜ + ⎟ dx C ∂t dx ⎝2 ⎠ i U ⎞ dD ⎞ ⎛ ⎛ + ⎜ N BG + ⎜ NB ρ D ⎟ − ( NBρ D) B ⎟ − x = x ( B ,t ) 2 ⎠ x = x ( B− ,t ) dx ⎠ x = x ( B+ ,t ) ⎝ ⎝ i dD ⎞ U ⎞ ⎛ ⎛ + ⎜ NCG + ⎜ NC ρ D ⎟ − ( NC ρ D ) C =0 ⎟ + = ( , ) x x C t 2 ⎠ x = x ( C + ,t ) dx ⎠ x = x ( C − ,t ) ⎝ ⎝

Numerical Resolution of the Reynolds Equation

95

In order to make the terms dependent on G at B and C disappear, the weighting functions NB and NC are chosen such that: NB = − NC = N which yields: B C⎡ ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ ⎛ d ( J1 / J 0 ) ∂h ⎞ ⎤ dN dD −∫ ρ N ⎜ + + ρ N ⎜U + ⎟ ⎥ dx ⎟ dx − ∫B ⎢G A ∂t ∂t ⎠ ⎦⎥ dx dx ⎝2 ⎠ ⎝ ⎣⎢ dx dx i E ⎛ U d ( 2 J1 / J 0 + D ) ∂ ( h + D ) ⎞ U ⎞ ⎛ −∫ N⎜ + − ( N ρ D ) x = x ( B− ,t ) B dx + ⎜ N ρ D ⎟ ⎟ C ∂t dx 2 ⎠ x = x ( B− ,t ) ⎝ ⎝2 ⎠ i U ⎞ ⎛ − ⎜ Nρ D⎟ + ( N ρ D ) x = x ( C + ,t ) C = 0 2 ⎠ x = x ( C + ,t ) ⎝

[3.55]

The relation: ∂ ∂t

(∫

c (t )

b (t )

)

f ( x, t )dx = ∫

c (t )

b (t )

∂ ( f ( x, t ) ) dx + f (b(t ), t )b(t ) − f (c(t ), t )c(t ) ∂t

makes it possible to transform the terms in

∫

B A

∫

C

∫

E

B

C

∂D : ∂t

ρN

i i ∂D ∂ B dx = ∫ ρ NDdx + ( ρ ND ) x = x ( A+ ,t ) A− ( ρ ND ) x = x ( B− ,t ) B A ∂t ∂t

ρN

i i ∂D ∂ C dx = ∫ ρ NDdx + ( ρ ND ) x = x ( B+ ,t ) B − ( ρ ND ) x = x ( C − ,t ) C ∂t ∂t B

ρN

i i ∂D ∂ E dx = ∫ ρ NDdx + ( ρ ND ) x = x ( C + ,t ) C − ( ρ ND ) x = x ( E − ,t ) E ∂t ∂t C

Since the points A and E have a position which is independent of time, the terms i

i

in A and E are nil. Equation [3.55] becomes:

96

Hydrodynamic Bearings i ∂ B U d ( 2 J1 / J 0 + D ) dx − ∫ ρ N ( h + D ) dx + ⎡⎣ ρ N ( h + D ) ⎤⎦ B − A x = x ( B ,t ) ∂t A 2 dx i i C⎡ C d ( J1 / J 0 ) ⎤ ∂ dN dD − ∫ ⎢G + ρ NU dx − ∫ ρ Nhdx − [ ρ Nh ] x = x ( B + ,t ) B + [ ρ Nh] x = x (C − ,t ) C ⎥ B ∂t B dx ⎣ dx dx ⎦ B

−∫ ρ N

i U d ( 2 J1 / J 0 + D ) ∂ E dx − ∫ ρ N ( h + D ) dx − ⎡⎣ ρ N ( h + D ) ⎤⎦ C + C x x C t ( , ) = ∂t dx 2 i i U ⎞ U ⎞ ⎛ ⎛ + ⎜ Nρ D⎟ − ( N ρ D ) x = x ( B− ,t ) B − ⎜ N ρ D ⎟ + ( N ρ D ) x = x ( C + ,t ) C = 0 2 ⎠ x = x ( B− ,t ) 2 ⎠ x = x (C + ,t ) ⎝ ⎝ E

−∫ N C

As the thickness h of the film is continuous at B and C, as well as the function N, the equation may be simplified: B

−∫ ρ N A

C⎡ d ( J1 / J 0 ) ⎤ U d ( 2 J1 / J 0 + D ) dN dD dx − ∫ ⎢G + ρ NU ⎥ dx B dx dx 2 ⎣ dx dx ⎦

U d ( 2 J1 / J 0 + D ) ∂ B ∂ C dx − ∫ ρ N ( h + D ) dx − ∫ ρ Nhdx C dx 2 ∂t A ∂t B U ⎞ U ⎞ ∂ E ⎛ ⎛ − ∫ ρ N ( h + D ) dx + ⎜ N ρ D ⎟ −⎜ Nρ D⎟ =0 2 2 ⎠ x = x ( C + ,t ) ∂t C ⎝ ⎠ x = x ( B − ,t ) ⎝ E

−∫ N

In this equation, there remain two terms defined respectively at B– and C+. An dD on the intervals AB and CE: integration by parts of the terms in dx .

B

d ( J1 / J 0 )

A

dx

− ∫ ρ NU

B

dx + ∫ ρ A

U dN U ⎞ U ⎞ ⎛ ⎛ −⎜ Nρ D⎟ Ddx + ⎜ N ρ D ⎟ 2 dx 2 ⎠ x = x ( A, t ) ⎝ 2 ⎠ x = x ( B− ,t ) ⎝

C⎡ E E d ( J1 / J 0 ) ⎤ d ( J1 / J 0 ) dN dD U dN − ∫ ⎢G + ρ NU dx + ∫ ρ Ddx ⎥ dx − ∫C ρ NU B C dx dx dx dx 2 dx ⎣ ⎦ ∂ B U ⎞ U ⎞ ∂ C ⎛ ⎛ +⎜ Nρ D⎟ −⎜ Nρ D⎟ − ∫ ρ N ( h + D ) dx − ∫ ρ Nhdx A ∂ t ∂ 2 2 t B ⎝ ⎠ x = x ( C + ,t ) ⎝ ⎠ x = x ( E ,t )

−

∂ E U ⎞ U ⎞ ⎛ ⎛ −⎜ Nρ D⎟ =0 ρ N ( h + D ) dx + ⎜ N ρ D ⎟ ∫ C ∂t 2 ⎠ x = x ( B− ,t ) ⎝ 2 ⎠ x = x ( C + ,t ) ⎝

makes it possible to dispose of these two terms:

Numerical Resolution of the Reynolds Equation

97

B C⎡ d ( J1 / J 0 ) ⎤ ⎡ d ( J1 / J 0 ) 1 dN ⎤ dN dD D ⎥ dx − ∫ ⎢G − ∫ ρU ⎢ N − + ρ NU ⎥ dx A B dx dx 2 dx ⎦ ⎣ ⎣ dx dx ⎦ E ⎡ d ( J1 / J 0 ) 1 dN ⎤ ∂ B ∂ C D ⎥ dx − ∫ ρ N ( h + D ) dx − ∫ ρ Nhdx − ∫ ρU ⎢ N − C A dx 2 dx ⎦ ∂t ∂t B ⎣

−

U ⎞ U ⎞ ∂ E ⎛ ⎛ −⎜ Nρ D⎟ =0 ρ N ( h + D ) dx + ⎜ N ρ D ⎟ ∫ C 2 ⎠ x = x ( A, t ) ⎝ 2 ⎠ x = x ( E ,t ) ∂t ⎝

If the domain AE represents a developed bearing, the values of D at A and E are equal, and the two last terms of the above expression are eliminated. If the domain AE is not looped, it is sufficient to choose the functions N that are nil at A and E. Using the index function Φ, which identifies each zone, and after a sign change and the elimination of the last two terms, this equation may be reduced to:

∫

E A

⎡ ⎛ d ( J1 / J 0 ) ∂N D ⎞ ⎤ dN dD + ρU ⎜ N − (1 − Φ ) ⎢Φ G ⎟ ⎥ dx dx dx dx ∂x 2 ⎠ ⎥⎦ ⎢⎣ ⎝ ∂ E + ∫ ρ N [ h + (1 − Φ ) D ]dx = 0 ∂t A

[3.56]

Equation [3.56] implicitly integrates the flow rate conservation conditions over the whole one-dimensional domain of the film. 3.4.2.2. Modelization of a two-dimensional lubricant film Consider the integral form defined on the domain Ω of the lubricant film: ⎛ ⎡ ∂W ⎛ ∂D ⎞ ∂W ⎛ ∂D ⎞ ⎤ ∂W D ⎤ ⎞ ⎡ ∂ ( J1 / J 0 ) E ( D, W ) = ∫ ⎜ Φ G ⎢ − (1 − Φ ) ⎟d Ω ⎜G ⎟ ⎥ + ρU ⎢W ⎜ ⎟+ ⎜ ∂x ∂x 2 ⎥⎦ ⎟⎠ ⎣ Ω⎝ ⎣ ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ⎦ ∂ + ∫ ρW [ h + (1 − Φ ) D ] d Ω ∂t Ω [3.57]

where W is a weighting function defined on Ω. By analogy with equation [3.56] for the one-dimensional case, it may be deduced that the conservation conditions for the mass flow rate within the lubricant film will be implicitly satisfied if this functional is used to define the problem of the search for the pressure p and the filling r. Returning to the problem as it was defined in the previous section, with, in particular, the boundary conditions relevant to pressure for part ∂Ω1 of the

98

Hydrodynamic Bearings

boundary and to the flow rate on the complementary part ∂Ω 2 , the new form of problem 2 which needs to be solved in order to find p and r can be defined: ⎧determine D in Ω and on ∂Ω such that: 2 ⎪ *⎪ Problem 2 ⎨ D ∈ C0 ; D = p on ∂Ω1 ⎪ ⎪⎩ ∀W ∈ C0 ; W = 0 on ∂Ω1 ; E ( D, W ) =

[3.58]

∫

∂Ω 2

W Qm d ( ∂Ω )

where E(D, W) is the functional defined by [3.57]. The cavitation algorithm described in section 3.1.2 must of course be associated with solving problem 2*. 3.4.3. Approximation of integral formulations: method of Galerkin weighted residuals

The previously defined problems 1* and 2* involve integro-differential operators which involve the functions p or D which are being sought and the weighting functions W. Let it be assumed that the problem to be resolved is the problem 1* defined in section 3.3.1. The sought function p is firstly replaced by a linear combination of predefined functions: n

p = ∑ N i ai

[3.59]

i =1

The weighting functions W must satisfy the conditions listed in the problems that need to be solved. The Galerkin (or Boubnov–Galerkin) method consists of choosing the Ni functions as the weighting functions. In this way, for the n weighting functions Ni, the corresponding integral equation may be written. By replacing p with the combination [3.59] a system of n equations is obtained, of which the n unknowns are the coefficients ai. The requirement to have a precise approximation of the sought function should lead to the choice of sophisticated Ni functions, which in turn might lead to difficulties as regards the evaluation of the integrals. In order to avoid these difficulties the domain is split into many subdomains with simple form– the “elements” – and simple polynomial functions are defined on each of these subdomains, for example interpolation functions. In certain situations, it is necessary to choose weighting functions that are different to the interpolation functions. In this case, the method is known as the Petrov–Galerkin method. It will be seen that this choice needs to be partially made for problem 2*.

Numerical Resolution of the Reynolds Equation

99

3.4.4. Approximation of problem 1*

3.4.4.1. Domain decomposition into elements Usually, the width of hydrodynamic bearings is constant. The domain which corresponds to the lubricant film is thus a cylindrical revolution surface or, if the configuration of the developed bearing is chosen, a rectangle, the width of which corresponds to the axial direction of the bearing and the length in the circumferential direction. The decomposition of this rectangular surface may be conducted with elements of different shapes, most usually triangles and rectangles. For bearings of variable width, without discontinuity in width, the rectangles will be replaced with quadrangles, elements which are topologically identical to rectangles. When the width of the bearing is discontinuous, decomposition using only quadrangles may prove difficult and a mix of quadrangles and triangles (or only triangles) may be necessary. The decomposition of the domain into elements is known as “meshing”. The elements are interconnected at points situated on their edges known as “nodes”. There are always nodes at the corners of the elements. Depending on the required degree for the interpolation functions, nodes may also be placed at other points along the element edges, or even inside them. Figure 3.8 shows an example of a mesh for the surface of a film for a connecting rod big end bearing, in a cylindrical representation (a) or in a developed bearing representation. As with all the applications presented in this work, the mesh only uses quadrangles.

Figure 3.8. Example of mesh for the film domain

100

Hydrodynamic Bearings

3.4.4.2. Shape functions, interpolation functions and parametric representation Consider an element, for example quadrangular in shape (Figure 3.9). This element has nne nodes located on its edges; one of these may also be at its center. The coordinates x and z of a point P situated in the element are obtained on the basis of the coordinates of the nne nodes by a relation which involves nne polynomial functions Ni: nne

nne

i =1

i =1

x = ∑ N i (ξ ,η ) xi ; z = ∑ N i (ξ ,η ) zi

[3.60]

The polynomials used for the interpolation of x and z are defined on a prototypical square element in function of the parametrical coordinates ξ and η which vary between –1 and 1. The parametrical coordinates of the central node of the element are thus nil.

Figure 3.9. Parametric representation of an element

These polynomials used to define the real form of the element are known as shape functions. When only the corner nodes are used (nne = 4), the polynomials are first degree. The elements are said to be linear. If the nodes located half-way along the side5 of the elements and possibly the central node are also used, the elements are known as quadratic or parabolic. In the works of Dhatt et al. [DHA 05] or Zienkiewicz and Taylor [ZIE 00a], definitions of the shape functions Ni for triangular elements with 3 and 6 nodes and quadrangular elements with 4, 8 and 9 nodes may be found. It should be noted that: 5 The nodes along the side are in the middle of the side only in the parametrical space, in which either ξ = 0 or η = 0.

Numerical Resolution of the Reynolds Equation

101

– each function Ni equals 1 at the node i and 0 at the other nodes; – the sum of the nne shape functions is 1 at any point in the element. The various parameters involved in the equations that need to be solved, both those that are known, such as, for example, the thickness of the film h and those that are not known such as the function sought, for example the pressure p, must also be defined on the basis of the values taken at the nodes of the mesh using interpolation functions. It is convenient to take the same polynomials as for the shape functions. In this case, the elements are known as isoparametric. This choice will always be made for the applications presented in this work. Thus, the following is obtained: nne

nne

nne

i =1

i =1

i =1

h = ∑ N i (ξ ,η ) hi ; μ = ∑ N i (ξ ,η ) μi ; ... ; p = ∑ N i (ξ ,η ) pi

[3.61]

3.4.4.3. Discretization of integral equations The Boubnov–Galerkin method consists of weighting the equation that needs to be solved by as many weighting functions as there are unknowns resulting from the approximation of the function being sought. For problem 1, these unknowns are thus nodal values of the pressure p, except for nodes located on the domain boundaries, where the pressure is known and the weighting functions are the corresponding interpolation functions. Let i be one of the mesh nodes. As the interpolation functions have been defined for the interpolation in an element, at the node i, a priori common for several elements, the functions relative to this same node and defined in each of the contiguous elements are assembled. The definition of each function is such that the assembled function is continuous and equal to 0 on the contour of the assembled elements. Beyond these elements, the function Ni obtained in this way is prolonged by continuity at the value 0 as shown in Figure 3.10.

Figure 3.10. Representation of the weight function Ni

102

Hydrodynamic Bearings

According to relation [3.48], the corresponding weighted equation is thus notated: ⎛ ∂N ∂p ∂N i ∂p ⎡ ∂ ( J1 / J 0 ) ∂h ⎤ ⎞ Ei = ∫ G ⎜ i + + ρ N i ⎢U + ⎥ ⎟ dΩ = 0 ∂x ∂t ⎦ ⎠ ⎝ ∂x ∂x ∂z ∂z ⎣ Ω

The integration over the entire domain is obtained by adding together the integrations on each element: ne ne ⎛ ∂N ∂p ∂N i ∂p ⎡ ∂( J1 / J 0 ) ∂h ⎤ ⎞ + + ρ Ni ⎢U + ⎥ ⎟ dΩ = 0 Ei = ∑ Intk = ∑ ∫ G ⎜ i ∂ ∂ ∂ ∂ ∂x ∂t ⎦ ⎠ x x z z ⎝ ⎣ k =1 k =1 Ω k

[3.62]

where ne is the number of mesh elements. The computation of each elementary integral is carried out in the space of the parametric coordinates, i.e. on the square [-1, 1] × [-1, 1]:

Intk i = ∫∫ f ( x, z ) dxdz = ∫

1

1

∫ f (ξ ,η ) det Jdξ dη

−1 −1

Ωk

where det J is the determinant of the Jacobian matrix defined by: ∂x ∂ξ = J≡ ∂ (ξ ,η ) ∂z ∂ξ ∂ ( x, z )

∂x ∂η ∂x ∂z ∂z ∂x = − ∂z ∂ξ ∂η ∂ξ ∂η ∂η

[3.63]

The transition from the real variables x and z to the parametric variables ξ and η requires formulae of the transformation of the derivatives: ∂z ⎧ ∂N i ⎫ ⎪⎪ ∂x ⎪⎪ ∂η 1 ⎨ ∂N ⎬ = ∂x det J ⎪ i⎪ − ⎪⎩ ∂y ⎪⎭ ∂η

∂z ∂ξ ∂x ∂ξ

−

⎧ ∂N i ⎫ ⎪ ⎪ ⎪ ∂ξ ⎪ ⎨ ⎬ ⎪ ∂N i ⎪ ⎪⎩ ∂η ⎭⎪

[3.64]

Computing of the elementary integrals is necessarily numerical. To achieve this, the functions that need to be integrated are evaluated at several points located in the element. The choice of the Gauss points for the quadrangular or Hammer points for triangular elements makes it possible to optimize the precision of the result (see

Numerical Resolution of the Reynolds Equation

103

[DHA 05] or [ZIE 00a] for the position and the weighting coefficients wm of these points). Thus, an elementary integral is written: npi ⎡ ⎛ ∂N ∂p ∂N i ∂p ⎞ ⎛ ∂ ( J1 / J 0 ) ∂h ⎞ ⎤ Intk = ∑ wm ⎢G ⎜ i + + ρ Ni ⎜U + ⎟ ⎥ det J ⎟ ∂x ∂t ⎠ ⎦ ⎝ m =1 ⎣ ⎝ ∂x ∂x ∂z ∂z ⎠

where all the functions are evaluated at npi integration points of coordinates ξm and ηm on the basis of nodal values. Starting with the evaluation of the derivatives of x and z: ∂x (ξ ,η ) ∂ξ

nne

=∑ j =1

∂z (ξ ,η ) ∂ξ

nne

=∑ j =1

∂N j (ξ ,η ) ∂ξ ∂N j (ξ ,η ) ∂ξ

xj ; zj ;

∂x (ξ ,η ) ∂η

∂z (ξ ,η ) ∂η

nne

=∑

∂N j (ξ ,η ) ∂η

j =1

nne

=∑

∂N j (ξ ,η ) ∂η

j =1

xj zj

where the index m of the integration point has been omitted to simplify notation, the Jacobian is then calculated, and following this, the derivatives ∂N/∂x and ∂N/∂z using relations [3.63] and [3.64]. The factor G at the integration point m is calculated on the basis of the nodal values: nne

Gm = ∑ N j (ξ m ,η m ) G j j =1

In the case of an isothermal computing with a Newtonian fluid, this term is notated: G=

h3 12μ

It is assumed that the film thickness h is part of the data of problem 1. There are thus two possible ways of evaluating G: either h is calculated by interpolation at the integration point, then G is calculated, or G is calculated at the element nodes and then interpolated: 3

e hj h 3 ; Gm = ∑ N j (ξm ,ηm ) G j hm = ∑ N j (ξ m ,η m ) h j ; Gm = m or G j = 12μ 12 μ j =1 j =1

nne

The first method is more accurate.

nn

104

Hydrodynamic Bearings

The term ∂(J1/J0)/∂x is written:

∂ ( J1 / J 0 ) ∂x

m

nne

∂N j (ξm ,ηm )

j =1

∂x

=∑

(J

1j

/ J0 j )

It may also be calculated as follows: ∂ ( J1 / J0 ) ∂x

m

=

J0m

∂J ∂J1 − J1m 0 ∂x m ∂x m

( J0m )2

⎛ nne ⎞⎛ nne ∂N (ξ ,η ) ⎞ ⎛ nne ⎞⎛ nne ∂N (ξ ,η ) ⎞ j j ⎜ N j (ξ ,η ) J0 j ⎟⎜ ⎟ ⎜ J1 j − N j (ξ ,η ) J1 j ⎟⎜ J0 j ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂x ∂x j =1 ⎠⎝ j=1 ⎠ ⎝ j=1 ⎠⎝ j=1 ⎠ =⎝

∑

∑

∑

∑

2

⎛ nne ⎞ ⎜ N j (ξ ,η ) J0 j ⎟ ⎜ ⎟ ⎝ j =1 ⎠

∑

Even if this second form is more precise, the complexity of the computation means that the first form is preferable. The factors ∂p/∂x and ∂p/∂z are written:

∂p (ξ ,η ) ∂x

nne

∂N j (ξ ,η )

j =1

∂x

=∑

pj ;

∂p (ξ ,η ) ∂z

nne

∂N j (ξ ,η )

j =1

∂z

=∑

pj

i.e. in function of the nodal values of the pressure. However, for the most part, these are unknown. By isolating these nodal values, the coefficients are obtained for the system of discretized equations. The terms in which known pressure values (boundary conditions) are involved will be carried over in the second member of the equations. By introducing all of the expressions in the integral [3.62] the discretized form of the equation Ei is obtained: ne npi

∑∑ w k =1 m =1

m

⎡ ⎛ ∂N i nne ∂N j ⎞ ∂N nne ∂N j pj + i ∑ pj ⎟ ⎢Gm ⎜ ∑ ∂z m j =1 ∂z m ⎠ ⎣⎢ ⎝ ∂x m j =1 ∂x m ⎛ ∂ ( J1 / J 0 ) ∂h ⎞ ⎤ + ρ N im ⎜ U + m ⎟ ⎥ det J m = 0 ∂x ∂t ⎠ ⎦⎥ m ⎝

The set of the discretized equations constitutes a linear system whose unknown vector {p} represents the n unknown nodal pressures:

Numerical Resolution of the Reynolds Equation

[ A1]{ p} = {b1}

105

[3.65]

A term of the matrix [A1] is defined by: nne ne npi ⎛ ∂N ∂N j ∂N ∂N j ⎞ A1ij = ∑∑ wm Gm ∑ ⎜ i + i ⎟ det J m ∂z m ∂z m ⎠ k =1 m =1 j =1 ⎝ ∂x m ∂x m

[3.66]

A component of the vector {b1} is defined by: ne npi ⎡ nne ⎛ ∂N ∂N j ⎛ ∂ ( J1 / J 0 ) ∂N ∂N j ⎞ ∂h ⎞ ⎤ b1i = −∑∑ wm ⎢Gm ∑ ⎜ i + i + m ⎟ ⎥ det J m ⎟ p j + ρ Nim ⎜ U ∂x ∂t ⎠ ⎥⎦ m k =1 m =1 ⎢⎣ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ ⎝ [3.67]

where p j represents the non-nil nodal pressures imposed. 3.4.4.4. Discretization of time derivatives A term dependent on the temporal derivative of the film thickness at the integration points is involved in the definition of bi. If the film thickness is analytically known with respect to time, this derivative is replaced by its exact analytic form. If this is not the case, several numerical techniques may be used to evaluate the time derivatives. The simplest of these involves an approximation of the derivative by finite differences.

∂hm hm t − hm ∂t Δt

t − Δt

where Δt represents the time step. The values of hm

t

and hm

t − Δt

are obtained, like

all the other parameters, by interpolation on the basis of the nodal values. More precise expressions of the discretization of the temporal derivatives are given in section 3.5. 3.4.5. Approximation of problem 2*

3.4.5.1. Discretization of the integral equation of problem 2 The discretization of problem 2* by the finite element method follows the same global approach as was used for problem 1*. In the two subdomains the equation to be solved takes totally different form and mathematical identity:

106

Hydrodynamic Bearings

– in Ωp, Φ is equal to 1 and the partial differential equation [3.3] is elliptical, of the type of non-stationary diffusion equations [DHA 05]; – in Ωr, Φ is equal to 0 and the equation with partial derivatives [3.3] is hyperbolic, of the type of non-stationary convection equations [DHA 05]. While the finite element method based on the Boubnov–Galerkin method of weighted residuals is very well suited to solve partial differential equations of elliptical type, hyperbolic convection-type equations require up-flow decentering techniques [ZIE 00a; DHA 05]. The Petrov–Galerkin method mentioned above is one of the effective solutions for this problem. The discretized form of the equation of problem 2* for the node i is notated: ne npi

∑∑ w k =1 m =1

m

⎡ nne ⎛ ∂Wi ∂N j ∂Wi ∂N j ⎞ + ⎢Gm ∑ ⎜ ⎟Φ j D j ⎣⎢ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ ∂ ( J1 / J 0 ) ⎛ ⎞⎤ 1 ∂Wi nne N jm (1 − Φ j ) D j ⎟ ⎥ det J m + ρU ⎜ Wim − ∑ ∂x 2 ∂x m j =1 m ⎝ ⎠ ⎥⎦

+

[3.68]

nne ⎛ ⎞ ∂ ne npi wm ρWim ⎜ hm + ∑ N j m (1 − Φ j ) D j ⎟ det J m = 0 ∑∑ ∂t k =1 m =1 j =1 ⎝ ⎠

It should be noted that the index function Φ is evaluated at the nodes in relation with the cavitation algorithm presented in section 3.1.2. The weighting functions Wi will be equal to the interpolation functions Ni for equations relevant to the nodes i for which Φ is equal to 1. If this is not the case (Φ = 0) the weighting functions will be polynomial functions which are decentered toward up stream, the side from which the fluid is coming, driven by the sliding bounding wall. The functions used are given in the appendix. 3.4.5.2. Linear system In relation [3.68] terms are involved which are dependent on the temporal derivative at the nodes of the elements of the function D when it represents filling. To make the components of the system of equations that need to be solved clear, it will be assumed that this derivative is approximated by first order finite differences6: ∂D j ∂t

Dj

t

− Dj

t − Δt

Δt

6 Section 3.5 gives more developed forms of the temporal derivatives.

[3.69]

Numerical Resolution of the Reynolds Equation

107

where Δt represents the time step. The derivative of the thickness h is obtained analytically where this is possible (section 3.4.4.4) or numerically on the basis of the values of hm t and hm t −Δt , which are obtained, like all the other parameters, by interpolation on the basis of the nodal values. Using the discretization expression [3.69] of the temporal derivative for a time step, the ith discretized equation is notated: ne npi

⎡ nne ⎛ ∂Wi ∂N j ∂Wi ∂N j ⎞ + ⎟GmΦ j D j ⎣ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠

∑∑ w ⎢⎢∑ ⎜ k =1 m =1

m

⎛ ∂ ( J1 / J 0 ) ⎞ 1 ∂Wi nne + ρU ⎜ Wim − N jm (1 − Φ j ) D j ⎟ ∑ ∂x 2 ∂x m j =1 m ⎝ ⎠ nne ⎤ ⎛ ⎞ 1 + ρWim ⎜ hm − hm + ∑ N j m (1 − Φ j ) D j − D j ⎟ ⎥ det J m = 0 Δt j =1 ⎝ ⎠ ⎦⎥

(

[3.70]

)

where the underlined functions are evaluated at the preceding time step. As with problem 1*, the set of discretized equations forms a linear system of unknown vector {D}:

[ A2]{D} = {b2}

[3.71]

A term of the matrix [A2] is defined by: ne npi ⎡ nne ⎛ ∂W ∂N j ∂Wi ∂N j ⎞ + A2ij = ∑∑ wm ⎢ ∑ ⎜ i ⎟GmΦ j k =1 m =1 ⎣⎢ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ nn ⎤ 1 ⎛ U ∂Wi ⎞ e −ρ ⎜ − Wim ⎟ ∑ N j m (1 − Φ j ) ⎥ det J m x t Δ 2 ∂ m ⎝ ⎠ j =1 ⎦

[3.72]

A component of the vector {b2} is defined by: ne npi hm − hm ⎛ ∂ ( J1 / J 0 ) ⎡ nne ⎛ ∂W ∂N j ∂Wi ∂N j ⎞ + + b 2i = − ∑∑ wm ⎢ ∑ ⎜ i ⎟Gm p j + ρWim ⎜⎜ U ∂x Δt m k =1 m =1 ⎣⎢ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ ⎝ nne 1 ⎛ U ∂Wi ⎞⎤ − ρ ∑ N jm ⎜ rj − h j − Wim (1 − Φ j ) D j ⎟ ⎥ det J m Δt ⎝ 2 ∂x m ⎠⎦ j =1

(

)

[3.73]

where p j represents the nodal value of the pressure when it is imposed and non-nil at the node j, Φ j , which is equal to 1 and rj , represents the nodal value of the

⎞ ⎟⎟ ⎠

108

Hydrodynamic Bearings

filling when it is imposed and less than the film thickness at the node j, where Φ j is equal to 0. 3.5. Discretizations of time derivatives

Several numerical techniques can be used to evaluate the temporal derivatives involved in the equations that need to be solved in the problems presented in section 3.1. These techniques may be classified into two categories: those based on an approximation of the derivatives by finite differences, and those based on a formulation by finite temporal elements. 3.5.1. Discretization by finite differences

3.5.1.1. First-order discretization The derivative of the film thickness h will be taken as an example. If it is assumed that this is a continually derivable function of time, a Taylor series development would make it possible to notate the value of h at the time t – Δt in function of the values of h and its derivatives at the time t. By limiting the development to the first-order, the approximation expression of the derivative is obtained: ∂h h(t ) − h(t − Δt ) ∂t Δt

[3.74]

where Δt represents the time step. The disadvantage of this first-order approximation is that it centers the approached value of the derivative on the time half-step before that of the computing. 3.5.1.2. Second-order discretization Consider the times t – Δt1 – Δt2, t – Δt1 and t. Δt1 and Δt2 respectively represent the latest and the next-to-latest time step. By notating the Taylor development limited to the second-order at the times t – Δt1 and t – Δt1 – Δt2, it is possible to eliminate the term which depends on the second derivative in t and to obtain a more precise approximation of the first derivative:

Δt + Δt2 Δt1 ∂h (2Δt1 + Δt2 ) h(t ) − 1 h(t − Δt1 ) + h(t − Δt1 − Δt2 ) ∂t Δt1 (Δt1 + Δt2 ) Δt1Δt2 Δt2 (Δt1 + Δt2 )

[3.75]

Numerical Resolution of the Reynolds Equation

109

This expression becomes simpler when the time steps are identical: ∂h 3h(t ) − 4h(t − Δt ) + h(t − 2Δt ) ∂t 2Δt

[3.76]

It should be noted that these approximations make it possible to evaluate the derivative at the time which corresponds to that of the computation of {p}, in contrast to the first-order approximation. When the time steps are big and/or the variation of the function that needs to be derived is rapid, this second-order approximation is to be recommended. 3.5.2. Discretization by time finite elements

Whatever the method of discretization chosen – finite difference, finite volume, or finite element – the equation obtained ([3.23]–[3.24] or [3.43]–[3.45] or [3.70]) can be put in the form: [E]D +

∂F =0 ∂t

[3.77]

where [E] is a matrix function of time and D the vector of the discretization parameters which depend on the time of the universal function D, the product [E] D represents the non-derived terms with respect to time. F is a time function which represents the terms that are derived with respect to time. D can be a part of F and no product of terms dependent on time is involved in F. 3.5.2.1. First order discretization The functions D, [E] and F are assumed to be known at the preceding time step, which will be chosen to be the time 0. The resolution to the equation [3.77] must lead us to obtain the vectorial function D and F at the time step in progress, separated by the time interval Δt from the preceding time step. If it is assumed that: ED +

∂F =0 ∂t

110

Hydrodynamic Bearings

represents one of the system’s equations [3.77]7, solving the vectorial equation [3.77] is equivalent to solving a set of integral equations of the form:

∫

Δt

0

∂F ⎞ ⎛ W ⎜ ED + ⎟dt = 0 ∂t ⎠ ⎝

for any weighting function W. An integration by parts gives:

∫

Δt

0

WEDdt + [WF ]0 − ∫ Δt

Δt

0

∂W Fdt = 0 ∂t

[3.78]

It is assumed that the functions D, E and F evolve linearly in the interval Δt: D ( t ) = D1 + F ( t ) = F1 +

t

Δt t

Δt

( D2 − D1 ) ; E ( t ) = E1 +

t

Δt

( E2 − E1 ) ;

( F2 − F1 )

where D1 = D(0), D2 = D(Δt), E1 = E(0), E2 = E(Δt), F1 = F(0) and F2 = F(Δt). D(t), E(t) and F(t) may also be written: t ⎞ t t ⎞ t ⎛ ⎛ D ( t ) = ⎜1 − ⎟ D1 + D2 ; E ( t ) = ⎜1 − ⎟ E1 + E2 ; Δt Δt ⎝ Δt ⎠ ⎝ Δt ⎠ t ⎞ t ⎛ F ( t ) = ⎜ 1 − ⎟ F1 + F2 Δ Δ t t ⎝ ⎠

which thus clearly shows the two interpolation functions on the temporal element [0, Δt]: N1 ( t ) = 1 −

t

Δt

; N 2 (t ) =

t

Δt

Equation [3.78] is thus notated:

7 In this notation, D represents the vector of discretized values of D and E represents the corresponding coefficients for the chosen equation. The developments following this are presented as if scalar quantities were involved. Vectorial notation will lead to the same results.

Numerical Resolution of the Reynolds Equation

∫

Δt

0

t t ⎛ ⎞⎛ ⎞ W ⎜ E1 + ( E2 − E1 ) ⎟ ⎜ D1 + ( D2 − D1 ) ⎟dt + W (Δt ) F2 − W (0) F1 Δt Δt ⎝ ⎠⎝ ⎠ Δt ∂W ⎛ t ⎞ −∫ ⎜ F1 + ( F2 − F1 ) ⎟dt = 0 0 Δt ∂t ⎝ ⎠

111

[3.79]

In order to give a clear sense to the equation obtained, the weighting function must be chosen with a derivative that is not identically nil on the interval [0, Δt], which excludes constant functions. The following function will be taken: W (t ) =

t

Δt

which leads to the Galerkin scheme [ZIE 00a]. By introducing this function in [3.79]:

∫

Δt

0

Δt 1 ⎛ t ⎛ t t t ⎞⎛ ⎞ ⎞ ⎜ E1 + ( E2 − E1 ) ⎟ ⎜ D1 + ( D2 − D1 ) ⎟dt + F2 − ∫0 ⎜ F1 + ( F2 − F1 ) ⎟dt = 0 Δt ⎝ Δt Δt Δt ⎝ Δt ⎠⎝ ⎠ ⎠

and by integrating, the discretized form of the equation [3.77] is obtained:

( 3E2 + E1 ) D2 + ( E2 + E1 ) D1 +

6 (F − F ) = 0 Δt 2 1

[3.80]

It should be noted that the process followed gives a weak formulation of the problem of the temporal derivative. In fact, the derivability of F is no longer required. Terms in the form of temporal derivative in which the nodal value of Φ appear are involved in equation [3.68]. However, from one time step to the next, this index function may change from the value 0 to the value 1 and vice versa. The value that should be taken into account is at the time step in progress. 3.5.2.2. Second-order discretization A relation which uses the values of the time step in progress and of the two preceding time steps may also be established. For this, a temporal element with three nodes is considered, which corresponds to the latest time interval Δt1 and the one preceding this Δt2. To simplify the developments, it is assumed that the time 0 corresponds to the time step which precedes the most recent one. The following is then notated:

112

Hydrodynamic Bearings

D1 = D(0); D2 = D(Δt1); D3 = D(Δt1+ Δt2) E1 = E(0); E2 = E(Δt1); E3 = E(Δt1+ Δt2) F1 = F(0); F2 = F(Δt1); F3 = F(Δt1+ Δt2) It is assumed that the functions D, E and F follow a quadratic development in the interval Δt1 + Δt2:

D ( t ) = D1 + ad t + bd t 2 ; E ( t ) = E1 + ae t + be t 2 ; F ( t ) = F1 + a f t + a f t 2 The coefficients ad and bd, ae and be, and af and bf are expressed respectively on the basis of the nodal values of D, E and F. For example, for ad and bd the following is obtained: ad =

⎤ 1 ⎡ 2k + 1 k +1 k D1 + D2 − D3 ⎥ ⎢− k k + 1 ⎦⎥

Δt2 ⎣⎢ k ( k + 1)

1 bd = Δt2 2

⎡ 1 ⎤ 1 1 D1 − D2 + D3 ⎥ ⎢ k k + 1 ⎥⎦ ⎢⎣ k ( k + 1)

[3.81]

where k represents the relationship between the two time steps: k=

Δt1 Δt2

Equation [3.78] is thus notated:

∫

Δt1 + Δt2

0

W ( E1 + ae t + be t 2 )( D1 + ad t + bd t 2 )dt + W (Δt1 + Δt2 ) F3 − W (0) F1 −∫

Δt1 + Δt2

0

∂W ( F1 + a f t + b f t 2 )dt = 0 ∂t

[3.82]

The solution is sought in t = Δt1 + Δt2, i.e. at the third node of the temporal element; the values are known at the two other nodes. The weighting function chosen is thus a quadratic function equal to 1 at the third node and nil at the two others:

W (t ) =

⎞ 1 t ⎛ t −k⎟ ⎜ k + 1 Δt2 ⎝ Δt2 ⎠

[3.83]

Numerical Resolution of the Reynolds Equation

113

By carrying over this function in [3.82] and integrating, the second-order temporal discretization relation is obtained: 2−k 3− k 4−k 2 E1 D1 + ( E1ad + ae D1 )( k + 1) Δt2 + ( E1bd + ae ad + be D1 )( k + 1) Δt2 2 6 12 20 − k 5−k 6 3 4 [3.84] + be bd ( k + 1) Δt2 4 ( ae bd + be ad )( k + 1) Δt23 + 30 42 Δt 10 ⎡ k (−4k + 7) F3 − (k + 1) 2 F2 + ( k 2 − k + 1) F1 ⎤ = 0 ; k = 1 + ⎣ ⎦ k t t + Δ Δ 1 ( ) 2

2

This expression links the values of the parameters in the time step in process (D3, E3 and F3 are involved in the coefficients a and b) with the values at the latest time step calculated (D2, E2 and F2) and also with those at the time step before the latest one (D1, E1 and F1). When the time steps are identical (k = 1), by replacing the coefficients a and b with their simplified expressions, this relation may be reduced to the following form: −3E1 D1 − 8 E1 D2 − 3E1 D3 − 8 E2 D1 + 16 E2 D2 + 20 E2 D3 − 3E3 D1 + 20 E3 D2 + 39 E3 D3 +

35 [3F3 − 4 F2 + F1 ] = 0 Δt

[3.85]

3.5.3. Adaptation of discretized expressions for equations to be solved

Depending on the method for temporal discretization which is used, the expressions discretized by one of the three methods of spatial discretization – finite differences, finite volumes or finite elements – need to be modified. For example, in the case of a spatial discretization by finite elements and a temporal discretization by first-order finite element, expressions [3.72] and 3.73] will be replaced by the following: ne npi ⎡ nne ⎛ ∂W ∂N j ∂Wi ∂N j ⎞ A2ij = ∑∑ wm ⎢ ∑ 3 ⎜ i + ⎟ 3Gm + Gm Φ j k =1 m =1 ⎢⎣ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ nn ⎤ 6 ⎛ U ∂Wi ⎞ e −ρ ⎜ 4 − Wim ⎟ ∑ N j m (1 − Φ j ) ⎥ det J m ⎝ 2 ∂x m Δt ⎠ j =1 ⎦

(

)

114

Hydrodynamic Bearings

ne npi ⎡ nne ⎛ ∂W ∂N j ∂Wi ∂N j ⎞ + b 2i = −∑∑ wm ⎢ ∑ ⎜ i ⎟ 3Gm + Gm p j + Gm + Gm p j k =1 m =1 ⎢⎣ j =1 ⎝ ∂x m ∂x m ∂z m ∂z m ⎠ ⎛ 4 hm − hm + 2 hm − hm ⎞ ⎜ ∂ 4I2 / J 2 + 2I2 / J 2 ⎟ + ρWim ⎜ U + ⎟ x t Δ ∂ m ⎜ ⎟ ⎝ ⎠

((

(

−ρ

U 2

nne

∑N j =1

jm

)

(

)

(

)

(

)

) )

⎤ ∂Wi 6 nne 4rj + 2rj − 4h j − 2h j + ρ ∑ N jmWim (1 − Φ j ) D j ⎥ det J m Δt j =1 ∂x m ⎦

(

)

In these expressions, the parameters with single underline have been evaluated at the latest calculated time step prior to the time step in progress, and those with double underline have been calculated at the time step before the latest one. When the film thickness is known analytically, the term

(

)

(

4 hm − hm + 2 hm − hm

) can be

Δt dh dh advantageously replaced by 4 + 2 , which uses the derivatives of the time step dt dt in progress and from the time step preceding this. Housing radius

20

mm

Bearing width

20

mm

Radial clearance

25

µm

3,000

rpm

0.5

MPa

Ambient pressure

0

MPa

Dynamic viscosity

0.01

Pa.s

Rotational frequency Supply pressure

Table 3.1. Reference data

3.6. Comparative analysis of the different methods 3.6.1. Definition of reference problems

The numerical tests to be carried out must make it possible to validate the discretization algorithms of the various methods and also to compare the performances in terms of precision and computing time. To achieve this, the model used must be simple enough to allow any necessary analytical derivations.

Numerical Resolution of the Reynolds Equation

115

The bearing used is circular, without any defect in shape, and of a constant width. Its dimensions are close to those of a connecting rod big end bearing for a small cylinder capacity engine. The lubricant is Newtonian and isoviscous. The various parameters used for all of the tests are given in Table 3.1. Two problems are considered: – stationary operation of the bearing at imposed eccentricity; – non-stationary operation: the center of the shaft circles at constant rotation velocity. The first problem corresponds to the case of a bearing supporting a shaft which is turning at constant velocity and which supports a constant load. The second problem corresponds to the case of a rotative load of constant amplitude such as that, for example, produced by an imbalance. If the reference data (Table 3.1) and the eccentricity remain the same, this second problem presents the interest of having the same solution for maximum pressure, load and flow rate as the first problem. In fact, if the shaft surface is taken as the reference surface for rotation and the bearing shell surface is taken as the reference surface for the film thickness, then a stationary problem identical to the first problem may be obtained. This makes it possible to test the effectiveness of the algorithms for non-stationary problems.

Figure 3.11. Pressure field Sommerfeld solution: a) “developed bearing” representation; b) cylindrical representation

3.6.2. First numerical tests

The first computations are carried out with a discretization by finite differences of the stationary problem, without seeking active and non-active zones (Problem 1 defined in section 3.1.1). The first aim is to verify the validity of the program carried out, and in particular that the domain looping has been taken into account.

116

Hydrodynamic Bearings

Figure 3.12. Pressure field. Gümbel solution

Figure 3.11 presents the pressure field obtained for a relative eccentricity8 (εx|εy) = (0.6|0.6). The correct continuity of the pressure field at the bearing cut should be noted. This is necessary for discretization. A representation in cylindrical coordinates ensures the continuity of the pressure field, but visibility is not complete (Figure 3.11(b)). This solution, known as “Sommerfeld solution” presents an anti-symmetry with respect to the direction of minimum film thickness. It makes negative pressures appear, which are unacceptable from the physics point of view, – 20.3 MPa in the present case. If these negative pressures are eliminated, the “Gümbel solution” is obtained. This is shown in Figure 3.12. This pressure field situates the film rupture at the angular position which corresponds to the minimum film thickness. A new improvement is contributed by the Christopherson algorithm: during an iterative resolution of the equations by the Gauss–Seidel method [GOL 96], the negative pressures are eliminated as they appear. The pressure field shown in Figure 3.13 is obtained. The maximum pressure is 22.2 MPa.

Figure 3.13. Pressure field. Christopherson solution

8 See note 2 about the notation (εx|εy).

Numerical Resolution of the Reynolds Equation

117

The solution to problem 1 (standard Reynolds equation) by the Christopherson algorithm does not ensure conservation of the mass flow rate. Whatever the lubricant supply conditions, the film forms at the point of maximum thickness, as is shown by the pressure fields presented in Figure 3.14, obtained for a relative eccentricity (εx|εy) = (0.4|0.4) and various supply conditions.

Figure 3.14. Change of feeding parameters Christopherson solution. a) film thickness; b) without lubricant supply; c) supply orifice centered at 85°; d) supply orifice centered at 265°

The same cases handled by resolution of the modified Reynolds equation (problem 2 presented in section 3.1.2) give the pressure fields shown in Figure 3.15. It should be noted that for the case without supply orifice, the re-supply of lubricant is solely by the sides of the bearing, using the difference between the ambient pressure and the cavitation pressure (0.1 MPa).

Figure 3.15. Change of the lubricant supply parameters. Solution of “problem 2” (section 3.1.2)

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Hydrodynamic Bearings

The resolution of problem 2 also gives the “filling” of the bearing with lubricant, as shown in Figure 3.16 for the three cases.

Figure 3.16. Change of the lubricant supply parameters. Lubricant filling

Tables 3.2 and 3.3 give a few results which show the dispersion of values depending on the method used to solve the problem. The Sommerfeld dimensionless number is directly linked to the loading by the following relation:

( )

μ LDN R C S=

2

W

[3.86]

The dimensionless friction torque given in the table is linked to the friction torque on the dimensioned shaft by:

Cf =

Cf

( )

μ LDN R C

[3.87]

Numerical Resolution of the Reynolds Equation

Lubricant supply Without

80–90

260–270

Method

Pressure (MPa)

Load (N)

Sommerfeld number

Torque

Attitude angle (°)

Gümbel

2.164

688.8

0.3717

14.55 24.58

58.3

Christoph.

2.225

719.4

0.3559

18.14 24.59

54.9

Problem 2

2.210

699.5

0.3660

21.12

46.3

Gümbel

2.210

713.0

0.3591

17.11 24.56

52.8

Christoph.

2.225

717.0

0.3570

20.72 24.55

51.1

Problem 2

2.221

725.4

0.3529

22.89

46.3

Gümbel

2.169

706.3

0.3624

14.68 24.64

64.0

Christoph.

2.229

732.3

0.3496

18.22 24.64

60.4

Problem 2

2.229

753.1

0.3399

21.57

57.0

119

Table 3.2. Lubricant supply effects with respect to the solving method

Lubricant supply Without

80–90

260–270

Method

Outing flow (dm3/min)

Gümbel

0.0946

Entering flow (dm3/min) 0.0702

Supply flow (dm3/min)

Error (%)

0

25.8

Christoph.

0.0962

0.0374

0

61.1

Problem 2

0.0598

0.0598

0

0

Gümbel

0.0953

0.0681

0.0201

7.4

Christoph.

0.0962

0.0369

0.0163

44.7

Problem 2

0.0727

0.0549

0.0178

0

Gümbel

0.1351

0.0677

0.0431

18.0

Christoph.

0.1369

0.0355

0.0429

42.7

Problem 2

0.1192

0.0544

0.0649

0

Table 3.3. Lubricant supply effects with respect to the solving method

Dispersion at maximum pressure is low: a disparity of less than 2.7% in relation to the value given by the resolution of the modified Reynolds equation which ensures the conservation of the flow rate (labeled “problem 2”) and this may thus be taken as a reference method. As regards the load that the pressure field can support, the disparity reaches 6.2 % for the Gümbel solution. With the Christopherson solution, which better localizes the film rupture boundary, the disparity is reduced to 2.8 %.

120

Hydrodynamic Bearings

The friction torque on the shaft is obtained by the integration of the shear constraint at this level. For the active part of the film, the computations do not present any more dispersion than those of the load, because the pressure fields used are approximately the same. In the non-active zones, the computation should involve the filling with lubricant (see section 3.7). The Gümbel and Christopherson methods do not allow this to be calculated. In order to be able to compare this to the solution to “problem 2”, which accounts for filling, two values are calculated for the Gümbel and Christopherson methods: in the first column, the non-active zone is assumed to be empty, and in the second, it is assumed to be full. These two extreme conditions make it possible to range the value obtained with a partial filling. The values obtained for the attitude angle – the angle between the direction of the load and the direction of the line of the centers of the shaft and the housing – have a maximum disparity of 12° with respect to the solution given by problem 2. Only the method based on the resolution of “problem 2” makes it possible to obtain an exact balance between the flow rate entering the bearing through the supply orifice if there is one, and by the bearing edges, and the flow rate exiting by the bearing edges under the effect of hydrodynamic pressure. The flow rate coming from the supply orifice is labeled “supply flow” in Table 3.3 and is calculated using relation [3.33] applied to the perimeter of the orifice. The Gümbel and Christopherson methods do not give an order of magnitude for the flow rate: for a bearing without supply the error on the balance of flow rates in relation to the exiting flow rate is more than 60 %. These first computations have been carried out through a discretization by finite differences, but the results would be the same using the other methods. They show that the calculated value of the maximum pressure is not very sensitive to the method used for determining the active and non-active zones of the lubricant film. The values calculated for the load and the attitude angle are more affected but still remain acceptable. However, the friction torque and the flow rate cannot be obtained with an acceptable level of precision using the Gümbel or Christopherson algorithms. Only an algorithm based on the resolution of the modified Reynolds equation that makes it possible to calculate filling with lubricant enables the precise computing of these parameters. These are very important for the study of hydrodynamic bearings and in particular for the computation of the energy dissipated by them (see Chapters 1–3 of [BON 14b] about thermal aspects). To conclude, the numerical study of the lubrication of hydrodynamic bearings must imperatively be based on a resolution of the modified Reynolds equation (problem 2).

Numerical Resolution of the Reynolds Equation

121

3.6.3. Comparisons between the three discretization methods for a static case

The bearing studied remains that defined by Table 3.1 with a relative eccentricity of 0.5657 ((εx|εy) = (0.4|0.4)). The pressure and the filling of the bearing are sought (problem 3.2). A rectangular supply orifice is located on the bearing shell between the angle 80° and the angle 90° in the circumferential direction and between – L/4 and L/4 in the axial direction. This form has been used in order to have perfect adjustment of the mesh or the elements at the edges of the orifice and in order to make it possible to obtain a precise computation of the flow rate entering the bearing through the orifice (relation [3.33]). A precise computation of the entering and exiting flow rates for the bearing may be obtained in this way. The density of the mesh is progressively increased, from 4 to 40 meshes, volumes or linear elements in the axial direction and 36 to 360 in the circumferential direction.

Figure 3.17. Evolution of the computing time

The increase in computing time in function of the number of equations is identical for the three methods, as can be seen in Figure 3.17. In fact, the structure of the matrix of the linear systems to be solved is virtually the same for the three methods: a computing point is only connected to the four neighboring nodes for finite difference methods and finite volume methods, and to the eight neighboring nodes for the finite element method. This leads to a matrix of a band-width structure

122

Hydrodynamic Bearings

of 2 nz + 1 for the first two methods, and 2 nz + 3 for the finite element method, where nz is the number of unknown parameters in the axial direction of the bearing. Resolution using the Crout method [GOL 96, DHA 05] uses this structure and makes it possible to have low calculation times despite the large number of equations reached. When there are a large number of equations, the number of iterations necessary to the search for the partition of the domain into active and non-active zones becomes stable at around 10 and the computing time increases proportionally to the square of the number of equations. Figure 3.18 shows the evolution of the load capacity calculated by the three methods in function of the number of computing points. The progressions are very similar and the three methods tend toward the same value.

Figure 3.18. Evolution of the load with respect to the mesh density

When the density of the mesh becomes sufficient, the finite difference and finite element methods show the same convergence of the friction torque (Figure 3.19). For the finite volume method, a numerical deviation appears.

Numerical Resolution of the Reynolds Equation

123

Figure 3.19. Evolution of the friction torque with respect to the mesh density

Figure 3.20 shows the different flow rates calculated: exiting the bearing at its ends, entering the bearing at its ends and through the supply orifice (feeding). The convergence toward the same values, when the mesh density increases, is obtained with all three methods. The balance between the entering flow rate and the exiting flow rate is perfectly obtained from the fourth level of refinement (2,115 computing points) for the finite difference methods and from the sixth level (5,112 computing points) for the finite volume method. For the finite element method, the perfect record is not obtained: after having decreased in the same proportion as for the other methods, the error becomes stable at around 0.4 %. For a higher eccentricity (0.9192), the relative error on the flow rate balance becomes stable at around 4% for the finite difference and finite element methods and at 6% for the finite volume method. It may be surprising to encounter such an error for the finite volume method, which is based on a discretization, which in theory ensures conservation of the flow rate. It should simply be recalled here that the computation of the flow rate for these tests uses the pressure gradients calculated on the basis of the pressure values at the bearing edges and on the lines adjacent to the computing points. When a non-active zone encroaches on the narrow bands defined in this way, the computation is erroneous (see sections 3.6 and 3.7).

124

Hydrodynamic Bearings

Figure 3.20. Evolution of the flow rate with respect to the mesh density

3.6.4. Comparisons between linear and quadratic discretizations for the finite element method applied to the standard Reynolds equation

It has been shown in section 3.6.3 that precisely obtaining the flow rate and the friction torque requires, whatever the discretization method, a significant number of computing points, in the order of several thousand points. With such a number of points the computation time for solving the Reynolds equation is in the order of several tenths of a second on an Intel Q9450 processor at 2.6 GHz. It may be acceptable if this solving needs to be made only quite rarely, for example, once for each time step. The general resolution algorithm for an EHD problem for an engine bearing (see Chapter 5 of [BON 14b]) leads to solving the standard Reynolds equation for a known partition of the domain several hundred times per time step, i.e. several dozen thousand times per engine cycle. In addition, the nonlinearity introduced by the thickness dependency on the hydrodynamic film pressure leads to the use of a Newton–Raphson type process [ZIE 00b] with a full Jacobian matrix. The band structure of the matrices which made the computing times mentioned above possible is lost. The imperatives for efficiency in terms of precision and computing time thus require that the rank of linear systems that need to be solved should be reduced as much as possible. To this end, a comparison between the linear and quadratic discretizations by finite elements is presented below. The bearing studied is that defined by Table 3.1 with a relative eccentricity of 0.8485 ((εx|εy) = (0.6|0.6)). The active domain, i.e. the computing domain is assumed to cover the angular sector between 0° and 60° and then between 180° and 360°. For the linear elements, the density of the mesh varies from 4 to 40 elements in the axial

Numerical Resolution of the Reynolds Equation

125

direction and 24 to 240 elements in the circumferential direction, i.e. from 96 to 9,600 elements. For the quadratic elements, the density of the mesh varies from 2 to 20 elements in the axial direction and from 12 to 120 elements in the circumferential direction, i.e. from 24 to 2,400 elements. For quadratic elements with eight nodes, a corner node of an element is connected to the 20 nodes of the four elements to which it belongs, while for linear elements with four nodes, a corner node is connected to only four nodes. As a result of the greater connectivity of the quadratic elements, the band width of the system matrix to be solved is bigger and the computing time in function of the number of nodes increases more rapidly than with linear elements (Figure 3.21). The asymptotic value of the maximum pressure (22.16 MPa) approached less than 0.5 % by the second refinement level for both discretizations, i.e. for CPU times less than 0.002 s (Figure 3.22). The effectiveness of quadratic elements is more apparent if the evolution of the load carried by the bearing is observed (Figure 3.23). With quadratic elements, the asymptotic value of the load is approached to a distance of less than 0.4% by the first discretization (45 equations) and in a CPU time of 0.00031 s. At the second refinement level in quadratic elements (233 equations and 0.0017 s CPU time), the obtained value approaches the asymptotic value to a distance of less than 0.001%. With linear elements, such a level of precision cannot be attained: at the highest level of refinement (9,321 equations and 0.172 s CPU time) the value obtained is still at a distance of 0.6% from the asymptotic value. The discretization methods by finite difference and finite volume presented in this chapter are first order discretization methods and give similar results in terms of precision and computing times to the finite linear element method (Figures 3.17 and 3.18).

Figure 3.21. Evolution of computing time with respect to the discretization order

126

Hydrodynamic Bearings

Figure 3.22. Evolution of the maximum pressure with respect to the discretization order

Figure 3.23. Evolution of the load with respect to the discretization order

The maximum pressure in the lubricant film is one of the main parameters for exploiting the results. A precision of several per cent is in general sufficient for correct interpretation. However, the load is a parameter, which is involved in the process of resolution: as opposed to the computings, which were carried for these tests, it is not the eccentricity that is imposed during the computing of the hydrodynamic lubrication of a bearing (see Chapter 5 of [BON 14b]), but rather the

Numerical Resolution of the Reynolds Equation

127

load carried by the shaft. The eccentricity is the result of the computing. Experience has shown that a high level of precision needs to be obtained for this eccentricity, because the minimum thickness of the film – which is an essential parameter among the results sought – depends on it to a large extent. Obtaining eccentricity with precision requires precise equilibrium between the load applied and that which results from the calculated pressure field. These precision requirements, together with the computing time requirements mentioned at the beginning of this section, lead to the use of the finite element method with quadratic elements for the discretization of the standard Reynolds equation. As regards the modified Reynolds equation used to find the partitioning of the domain into active and non-active zones, finite difference and finite volume methods may be used. If the finite element method is chosen, linear elements remain preferable. In fact, quadratic elements are not well suited to the handling of the equation reduced to its hyperbolic part for non-active zones.

3.6.5. Comparisons between the different discretizations of time derivatives for the modified Reynolds equation

The first test problem used involves imposing a circular movement at the shaft center at a frequency equal to that of its rotation. The bearing remains that defined in Table 3.1. The eccentricity is 0.5657. The supply of the bearing is through a rectangular orifice located between 80° and 90° in the circumferential direction and between 5 mm and 15 mm in the axial direction. The mesh used has 3,675 computing points for the finite difference and finite element methods, and 3,850 for the finite volume method. The first results obtained are calculated using second order temporal discretization by finite difference (section 3.5.1.2). Figure 3.24 shows the evolution of maximum pressure during the second cycle calculated. Due to the non-stationarity of the solution, the first computing steps of the first cycle are erroneous. The three methods – finite difference, finite volume and finite element – give the same evolution. The drop in maximum pressure for the first third of the cycle is due to passage through the alimentation orifice in the pressure field, as shown in Figure 3.25, which corresponds to the loading angle 60°.

128

Hydrodynamic Bearings

Figure 3.24. Evolution of the maximum pressure along the cycle

Figure 3.25. Supply orifice located in the hydrodynamic pressure field

The evolution of the load (Figure 3.26) is also identical for the three methods of spatial discretization. The load is influenced by the position of the supply orifice throughout the whole cycle. When the supply orifice is located in the a priori nonactive zone, there develops around it an active zone which contributes, along with the pressure in the orifice itself, to reducing the resultant load. This phenomenon is very pronounced in this example due to the small difference between the maximum pressure (2.26 MPa) and the supply pressure (0.5 MPa). Load is a parameter, which requires precise computing (see section 3.6.4). Since this parameter is more sensitive to the computing techniques used, it is the only one that is considered for the tests presented below. To have a reliable reference value, the orifice is removed and the eccentricity is fixed at 0.8. In this case, the nonstationary solution with a turning eccentricity and constant amplitude is identical to

Numerical Resolution of the Reynolds Equation

129

the stationary solution to the nearest turning, with a dephasing, which is dependent on the time step, as is shown in Figure 3.27, which depicts the corresponding pressure fields and filling. The loads calculated in both cases are equal, and so are the maximum pressures, the friction torques and the flow rates. Only the attitude angle follows the rotation of the shaft center in the non-stationary case.

Figure 3.26. Evolution of the load along the cycle

Figure 3.27. Lubricant filling and pressure fields. a) stationary, b) non-stationary, shaft center at 0°, c) shaft center at 100° and d) shaft center at 200°

130

Hydrodynamic Bearings

To be able to compare the results for the non-stationary case with a stationary reference solution, it is necessary to consider a case without a supply orifice. The parameters common to the two cases, stationary and non-stationary, are given in Table 3.4. Three cycles are calculated. Figure 3.28 shows the evolution of the load over the three cycles. After a transitory phase corresponding to approximately one half-turn of the shaft, which is a phase required to establish the filling of the bearing with lubricant, the load becomes stable at the values given in Table 3.5. This table also gives the CPU time necessary for the stabilization of the load. The methods used are those that are described in section 3.5: – FD1: first order temporal finite differences. – FD2: second order temporal finite differences. – FE1: first order temporal finite elements. – FE2: second order temporal finite elements. Housing radius Bearing width Radial clearance Rotational frequency Ambient pressure Dynamic viscosity Number of nodes

20 20 25 3,000 0 0.01 4,877

mm mm µm rpm MPa Pa.s

Table 3.4. Reference data

Figure 3.28. Evolution of the load for an angular step of 3°

Numerical Resolution of the Reynolds Equation

FD1 step (°) 24 12 6 3 2 1

FD2

FE1

131

FE2

load (N)

CPU (s)

load (N)

CPU (s)

load (N)

CPU (s)

load (N)

CPU (s)

A

2,172

2.76

2,678

2.84

1,473

3.21

4,338

3.21

N

7,301

3.09

3,319

2.81

4,628

2.81

5,337

3.20

A

2,593

4.12

2,746

3.14

2,130

3.95

2,905

3.87

N

4,791

3.95

2,861

3.32

3,874

3.87

3,025

3.89

A

2,735

7.62

2,782

6.10

2,470

5.90

2,801

7.43

N

3,726

6.25

2,807

5.92

3,368

6.64

2,826

6.96

A

2,778

9.14

2,796

10.5

2,641

10.6

2,800

11.5

N

3,246

10.3

2,801

10.4

3,087

12.3

2,804

11.8

A

2,788

12.8

2,799

14.3

2,696

15.7

2,801

16.6

N

3,094

14.1

2,801

14.4

2,993

15.2

2,803

16.2

A

2,797

20.9

2,801

21.9

2,750

22.4

2,802

27.4

N

2,946

22.1

2,802

21.7

2,896

24.6

2,802

28.2

Table 3.5. Stabilized load and CPU time with respect to the computing step and the time discretization method

The film thickness at the point of angular coordinate θ has an analytic expression:

h(θ , t ) = C ⎡⎣1 − ε cos (θ − ωc t ) ⎤⎦ where C represents the radial clearance, ε the relative eccentricity and ωc the angular velocity of the shaft center. The term dh/dt can thus be evaluated either analytically (A): dh(θ , t ) = −Cεωc sin (θ − ωc t ) dt

or numerically (N) using the numerical derivation schemes for each method. The chosen angular steps for computing correspond to steps of the shaft rotation of 24°, 12°, 6°, 3°, 2° and finally 1°. The reference load (2808 N) is given by the stationary problem. All the values given in the table are stabilized values: the first-order methods converge toward values far removed from those of the reference solution when the computing step is too large.

132

Hydrodynamic Bearings

Figure 3.29. Evolution of the relative load difference for an angular step of 3°

Figure 3.29 shows the evolution of the difference in load obtained relative to the reference load for the angular step of 3°. Only second-order methods give an error level of less than 1%. The third computing cycle needs to be reached before a load evolution is obtained, which no longer shows the history of the approach to the stationary solution. Then the relative difference fluctuates but does not exceed 0.2%. If smaller computing steps are considered, there is noticeable reduction in the relative disparity in the load (Figure 3.30) Fluctuation is still present and remains of the same order. This fluctuation is more or less pronounced depending on the temporal discretization methods (Figure 3.31). The periodicity of this fluctuation is due to an interference between the spatial discretization and the temporal discretization: the localization of the lubricant film formation and rupture boundaries may only be obtained using the spatial discretization mesh. The probability of correspondence between this mesh and theoretical position of the boundaries is extremely small: depending on whether the active zone in the film is a little too spread out or covering a smaller area, the load calculated is greater or lesser than the theoretical load. The periodicity of the spatial mesh and the time steps can thus lead to repetitive combinations, as can be seen in Figures 3.29–3.31. By changing the step of the spatial mesh – the number of nodes increases from 3675 to 5250 – the periodicity disappears and the disparity with respect to the reference load is on average five times smaller (Figure 3.32).

Numerical Resolution of the Reynolds Equation

133

None of the solutions obtained can be considered to be a theoretical reference: even the load of the stationary case, which was taken as a reference is dependent on the mesh used to calculate it and on the better or worse level of correspondence between the mesh and the film formation and rupture boundaries.

Figure 3.30. Evolution of the relative load difference for an angular step of 1 degree

Figure 3.31. Evolution of the relative load difference for an angular step of 1°

134

Hydrodynamic Bearings

Figure 3.32. Evolution of the relative load difference for an angular step of 1°. Effect of the node number (3675–5250)

Among the various discretization and spatial methods analyzed, only the secondorder methods give a value for the load, which is close to the value expected. The first-order discretization method by finite differences gives a satisfactory result (disparity of less than 2 % for angular steps of less than 6°, but this on condition that the analytical derivative of the film thickness is used (Figure 3.33).

Figure 3.33. Load evolution with respect to the discretization method for the time derivative and to the time step

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135

Figure 3.34. Load evolution with respect to the CPU time

If it is considered that the aim is to obtain the load to a distance of 2%, it is important to know which method will produce better results in terms of computing time. Figure 3.34 shows the evolution of the load in function of the CPU time for each of the methods, and for each of the angular steps. Note that the second-order temporal finite difference discretization methods enter the fork in less than 4 CPU seconds, followed by the temporal finite element discretization methods (7 s). The first order finite difference method with analytical derivation of the film thickness has a similar performance (9 s). This is an interesting point to note, because this method is easy to program. In Figure 3.35, the CPU time for obtaining a load at 2 % from the reference value is shown. For the four second-order methods, this goal is reached with an angular step of 6°. For the first-order discretization by temporal finite difference method and analytical derivation of the thickness, a computing time of 5° is necessary. For the five methods used, the decrease in the angular step leads to an increase in the computing to obtain the same goal at 2%. Therefore, it may be disadvantageous from the point of view of the CPU time to take angular steps that are too small, even if the computing is halted on a predefined convergence test, which it might be thought would be reached more quickly by having better temporal discretization, i.e. smaller steps.

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Hydrodynamic Bearings

Figure 3.35. Evolution of the CPU time required to obtain load values that differ less than 2% from reference values

3.6.6. Aptitude of the various discretizations of time derivatives to follow sudden load change

Internal combustion engine bearings, and in particular those for diesel engines, undergo variations in load, which can be very rapid. The necessity to reduce computing time leads to avoiding angular steps, which are too small. The quality of the results obtained in zones with rapid variation might then be questionable. With the aim of analyzing this point, the bearing under study in this section is placed in a non-stationary configuration defined as follows: the shaft turns at constant velocity, but its center follows a trajectory defined by Figures 3.36 and 3.37. Between the shaft rotation angles 0 and 30° the center is at the same rotation trajectory with a constant relative eccentricity equal to 0.8. From this point onward, the center of the shaft ceases its rotation movement, but rather diametrally crosses the circle of possible positions with three different velocities, first medium during the separation from the bounding wall, then more rapid in the central zone, and finally slow for the phase of approach to the opposite wall. At this stage, which corresponds to the shaft rotation angle 180°, the center is immobilized at a relative eccentricity 0.8 until the shaft rotation reaches the angle 240°. The following phase, at constant eccentricity, involves catching up on the shaft center’s delay by giving it a rotation velocity that is greater than that of the shaft. This is caught up at the angle

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137

330°, and from this angle onwards the movement in the beginning phase of the cycle is retrieved.

Figure 3.36. Trajectory of the shaft center along the cycle

Figure 3.37. Position of the shaft center along the cycle

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Hydrodynamic Bearings

In contrast to the case of an engine bearing, in this case, it is the trajectory of the shaft center that is imposed, and not the load. This makes it possible to have kinematical changes that are instantaneous and thus more dramatic than those encountered in a real situation. It also makes it possible to provide conditions, which are easy to reproduce. In addition, perfectly defined kinematics makes it possible to provide an argued interpretation of the phenomena of loading variation. The bearing and operation parameters are displayed in Table 3.6. Lubricant supply is only at the bearing edges. Three cycles are calculated. Figure 3.38 shows the evolution in the load during the third cycle for a computing step of 3°. At the angle 750° (or 30° if the 720° which correspond to the first two cycles are not taken into account) the load suddenly becomes nil. In fact, at this moment, the rotation movement of the shaft center is interrupted and replaced by a normal movement at the bounding wall. The sudden distancing of the bounding walls in the zone under pressure leads to film rupture of the whole active zone. At the same time, in the diametrically opposite zone, the bounding walls come closer together. However, because in this zone the film was ruptured with a low residual quantity of lubricant, no pressure field can be created while the film thickness has not sufficiently decreased. This situation, created artificially by the kinematics chosen for the present tests, is also found for small end bearings during the load inversions which occur at the two reverses in acceleration of the piston in the middle of the exhaust and admission phases (see section 3.3, Volume 4 [BON 14a]). Housing radius

20

mm

Bearing width

20

mm

Radial clearance

25

µm

3,000

rpm

Ambient pressure

0

MPa

Dynamic viscosity

0.01

Pa.s

Number of nodes

4,877

Rotational frequency

Table 3.6. Reference data

When the surface of the shaft is sufficiently close to that of the housing, the film is recreated and the load increases rapidly between the angles 780 and 810 (or 60 to 90°). At the angle 810 (90°) the velocity of movement toward the bounding wall is suddenly divided by six, which leads to a dramatic reduction in the load, from 6,000 N to less than 2,000 N: in this phase, the dominant effect comes from the squeezing of the film. Then the load increases more and more rapidly when the shaft

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139

comes closer to the bounding wall. At the angle 900 (180°) the approaching velocity toward the bounding wall is cancelled out: the “oil wedge” effect alone remains, and the load drops significantly. A quasi-stationary situation is maintained up until the angle 960 (240°). The low variation in load that is noticed during this phase is due to the change in extension of the lubricant film. Indeed, due to the history of filling with lubricant, obtaining a stabilized solution may take more than one shaft turn (see section 3.6.5).

Figure 3.38. Load evolution for an angular step of 3°

When the rotation movement of the shaft center begins again at the angle 960 (240°), the load drops again. This is due to a rapid movement of the zone potentially under pressure: because the rotation velocity of the shaft center becomes greater than that of the shaft, the “convergent” changes to the other side with respect to the point of lubricant film minimum thickness. The pressure field disappears from the up-flow side where the “oil wedge” was previously and needs to build itself up again on the other side in a zone, which was on the down-flow side and thus where the film was ruptured with very little lubricant present. To recreate a film, it is necessary to wait until this potential zone moves to a region, which contains enough lubricant: this happens at the angle 990 (270°), which is 30° later. The film builds itself up again at this point, and a new quasi-stationary situation follows. This situation, where the pressure field moves from one side to the other side of the zone of minimum film thickness, occurs with connecting rod big end bearings at low regime and in particular with diesel engine bearings toward the end of the combustion phase (see section 2.6.1 of [BON 14a]). During the entire compression phase and the first part of the combustion phase, the load has remained turned toward the same

140

Hydrodynamic Bearings

direction, in this way building up a delay on the shaft rotation. While this delay is being caught up toward the end of the combustion phase, the loading vector turns more rapidly than the shaft, which should thus lead to a reversing of the pressure field profile. The incapacity to establish a hydrodynamic pressure field which is sufficiently large due to the lack of lubricant, while the load that needs to be carried remains high, is thus the cause for the extremely low film thickness, which may be observed at this part of the cycle. The return of the shaft center to the initial rotation velocity, which is slightly less than the preceding velocity (factor 0.75), results in a new drop in the load. Numerical instabilities appear during the changes in kinematical conditions at the angles 810, 960, 990 and 1,050° (Figure 3.39). These changes are more or less pronounced depending on the discretization methods for the temporal derivatives. First order methods with numerical derivation of the film thickness, and the firstorder method by temporal finite elements with an analytical derivative of the thickness give erroneous quasi-stationary solutions. The rising front toward the angle 990 differs by 6° i.e. two computing steps, depending on whether the film thickness derivatives are calculated analytically or numerically. The second-order approximation methods for temporal derivation show more oscillations, with, however, quicker stabilization for the finite difference method with analytical derivative of the film thickness.

Figure 3.39. Numerical instabilities due to sudden changes of kinematical conditions. Angular step of 1°

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141

Changing from an angular step of 3° to an angular step of 1° makes it possible to greatly attenuate the numerical oscillations of the load, which occur when there are sudden changes of kinematical conditions (Figures 3.40 and 3.41). The disparity in localization of the rising front (toward the angle 990) between the solutions obtained by methods where numerical derivative of the thickness with respect to numerical derivative methods remains equal to two computing steps, i.e. 2°. The relative behavior of the different approximation methods for the temporal derivatives, as regards numerical instabilities, remains the same.

Figure 3.40. Numerical instabilities due to sudden changes of kinematical conditions. Angular step of 1

All the results presented above, were obtained on the basis of a spatial discretization by finite differences. As shown in Figures 3.41 and 3.42, the results show a same numerical behavior, which is approximately the same if the spatial discretization uses finite elements. However, in this case, there are less pronounced numerical instabilities for second-order methods of discretization of the temporal derivatives by finite elements. The analytical expression of the film thickness in function of time is not always available. In this case, the film thickness needs to be derived numerically and the second order method of approximation for the temporal derivatives by finite differences provides the best results. When the derivative of the film thickness is calculated analytically, the secondorder approximation method for the temporal derivatives by finite elements is the most stable.

142

Hydrodynamic Bearings

Figure 3.41. Details of numerical instabilities. Angular step of 1°. Spatial discretization by finite differences

Figure 3.42. Details of numerical instabilities. Angular step of 1°. Spatial discretization by finite elements

It should be remembered that the case studied in this section presents velocity discontinuities that could not be reproduced in any real situation. For an engine bearing, even in the most severe situations from this point of view (diesel engine connecting rod), passages will be more “gentle” because the load is the given element and the position of the shaft center results from this, with a damping effect

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143

due to the viscous film itself. The numerical oscillations will be imperceptible if the chosen angular step is sufficiently small and if the best differentiation method is used. 3.6.7. Case of a bearing under a dynamic load rotating with a frequency equal to half of the shaft frequency

Figure 3.43 shows the carrying load capacity of a bearing of which the dimensioning and operation parameters are those given in Table 3.6. The shaft center is located at a constant eccentricity equal to 0.8. At angle 0 of the computing, the rotation velocity of the center of the shaft is nil, and the load is that of a stationary operation (2,808 N). During the first shaft turn following the time 0 of the computing, the center of the shaft takes a rotation movement, which increases linearly until a frequency equal to half that of the shaft is reached. Following this, the rotation frequency of the shaft center is maintained constant (1,500 turns/min). Note that the load decreases linearly with the rise in frequency of the position of the shaft center, and then oscillates around a very low value (between 50 and 60 N). This slight residual load is due to the fact that the film is partially ruptured in the minimum thickness zone and down-flow from it, as can be seen in Figure 3.44, which shows the filling field in lubricant. Although the pressure is not greater at any point than the reference ambient pressure, which is the pressure found on the bearing edges (Figure 3.45), the zone where the film is ruptured and where the pressure is equal to cavitation pressure creates a light-lifting force oriented toward the zone diametrically opposite.

Figure 3.43. Load when the rotational frequency of the shaft center becomes equal to half of the shaft frequency

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Hydrodynamic Bearings

Figure 3.44. Lubricant filling field when the rotational frequency of the shaft center is equal to half of the shaft frequency

Figure 3.45. Pressure field when the rotational frequency of the shaft center is equal to half of the shaft frequency

The incapacity of the lubricant film to create pressures above ambient pressure when the kinematical conditions are those described above may be easily explained if we consider a reference frame, which is linked neither to the housing nor the shaft, but is at the point of minimum thickness (Figure 3.46). Relatively to this frame, the two bounding walls move with equal velocities in opposite directions. In all film sections, the mean flow rate is nil, with a linear velocity profile (Couette flow) and thus without generation of pressure.

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145

Figure 3.46. Velocity field in the film when the rotational frequency of the shaft center is equal to half of the shaft frequency

To illustrate this particular case the flow field is reproduced in Figure 3.47, visualized by tracers between two cylinders that turn in opposite directions at frequencies, which give the bounding walls velocities of the same amplitude. On the basis of stroboscoped views of the same flow (Figure 3.48), the velocity may be measured at numerous points in the lubricant film. In Figure 3.49, the linear distribution of the velocity can be found, which characterizes a Couette flow with a quasi-nil flow rate in each section.

Figure 3.47. Flow field in the lubricant film between two cylinders rotating in opposite directions [BON 86]

Figure 3.48. Stroboscoped view of the flow field [BON 86]

Hydrodynamic Bearings

Film thickness h in mm Velocity Uin cm/s Flow rate Q in mm3/s/mm

Film thickness

146

Velocity

Figure 3.49. Velocity field measured from the stroboscoped views [BON 86]

It should be emphasized that this phenomenon of absence of pressure may occur even for a film, which is amply supplied with lubricant, in contrast to the other rupture phenomena for carrying load described in section 3.6.5. At some points in the cycle, internal combustion engine bearings may take this kinematical configuration as a result of changes in orientation of the load. However, these are short-lived situations and will be less problematic than those that result from a lack of lubricant. 3.7. Accounting of film thickness discontinuities

The second member of the Reynolds equation contains the term

∂ ( J1 / J 0 )

∂x 1 ∂h which, when the fluid is isoviscous, is reduced to . When one or the other of 2 ∂x the bounding walls has discontinuities in thickness as a result of the presence of grooves, pockets or texturization, the thickness h is no longer continuous and cannot be derived for the discontinuities. The finite difference method assumes that the functions that need to be discretized are sufficiently differentiable, for it would be possible to replace all the derivatives with an approximation by finite difference, through application of the theorem of finite increments (Rolle’s theorem). The only way out of the problem

Numerical Resolution of the Reynolds Equation

147

posed by the discontinuities in h is to assume that the passage from one thickness to another takes places continuously, as is shown in Figure 3.50.

Figure 3.50. Film thickness discontinuity

The evaluation of the derivative will thus depend on the size of the mesh. If, in addition, the discontinuity is located precisely at the level of a node, the evaluation of the Poiseuille terms of the Reynolds equation will lead to the problem of the choice of the value that should be given to h: (h1 + h2)/2 would seem to be the most logical choice. The finite volume method also assumes that all the parameters vary continuously from one center of the control surface to the next. The consequences and solutions are therefore the same as for the finite difference method. For the finite element method, the evaluation of the parameters during the numerical integration process is at the integration points located within the elements (in general Gauss points). The flexibility of the method in terms of the choice of element shape makes it possible to make the boundaries of these coincide with the thickness discontinuity lines even if they are oblique, as is the case for bearings with herringbone grooves. The problem of the choice of the value that should be given to h is no longer relevant: it is the value that corresponds to the element. However, the ∂h evaluation of the term at requires special handling. In fact, in the case of the pad ∂x ∂h bearing with a middle step (Rayleigh pad bearing), the derivative within the ∂x elements is nil at all points. The numerical procedure described in section 3.4 leads to a solution, which is identically nil. The solution to this problem is once more 1 ∂h given by the weak formulation: it is enough to integrate the term dependent on 2 ∂x by parts to carry over the derivation onto the weighting function which has all the necessary derivability properties. The new form of the integral equation is obtained in this way:

148

Hydrodynamic Bearings

⎛ ⎡ ∂W ⎛ ∂D ⎞ ∂W ⎛ ∂D ⎞ ⎤ ∂W E ( D, W ) = ∫ ⎜ Φ G ⎢ ⎜G ⎟ ⎥ − ρU ⎜ ⎟+ ⎜ ∂x Ω⎝ ⎣ ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ⎦ ∂ + ∫ ρW [ h + (1 − Φ ) D ]d Ω ∂t Ω

D⎤⎞ ⎡ ⎢ J1 / J 0 + (1 − Φ ) 2 ⎥ ⎟⎟d Ω ⎣ ⎦⎠

[3.88] which leads to a new notation of the component i of the vector {b1}: ⎡ nne ⎛ ∂N i ∂N j ∂N ∂N j ⎞ ⎤ + i ⎢Gm ∑ ⎜ ⎟ pj ⎥ ∂z m ∂z m ⎠ ⎥ j =1 ⎝ ∂x m ∂x m ⎢ b1i = −∑∑ wm det J m ⎢ ⎛ ∂N k =1 m =1 ∂hm ⎞ ⎥ im ⎢− ρ ⎜U ⎥ ( J1 / J 0 )m − Nim ∂x ∂t ⎟⎠ ⎥⎦ ⎢⎣ ⎝ ne npi

[3.89]

Figure 3.51 gives an example of a thickness field for a bearing with herringbone grooves and a pressure field obtained by a discretization by finite elements. The shaft is in a centered position. The effect of the herringbone grooves should be noted, which generate pressure fields that contribute to the stability of this type of bearing, even operating without a load [ABS 94, BON 94].

Figure 3.51. Thickness and pressure fields for a herringbone bearing L/D = 1, C/R = 0.001, hr/C = 0.4 and pmax/psupply = 4.74

Numerical Resolution of the Reynolds Equation

149

3.8. Numerical algorithm for computing bearing axial flow rate 3.8.1. Pressure gradient computing in the case of a finite element discretization

It was established in Chapter 2 that, for a bearing, the flow rate through an arc

AB is given by: B⎛ B⎛ ∂p Uh ⎞ ∂p ⎞ Qv = ∫ ⎜ −G + ⎟ dz + ∫A ⎜ −G ⎟ dx A ∂x 2 ⎠ ∂z ⎠ ⎝ ⎝

If it is an axial flow rate exiting at one of the bearing ends, this relation becomes: Qv = ∫

2π R

0

∂p ⎞ ⎛ ⎜ −G ⎟ dx ∂z ⎠ ⎝

In what follows, the unitary flow rate will be notated: qv = −G

∂p ∂z

[3.90]

In a resolution process by finite elements, this relation is discretized in a summation on the neb curvilinear elements at the bearing edges obtained on the basis of the mesh elements of the domain Ω, as shown in Figure 3.52. For this illustration, the domain has been broken down into quadrangular elements with eight nodes.

Figure 3.52. Flow rate evaluation: integration points inside the surface element, integration points on the bearing edge

150

Hydrodynamic Bearings

Thus, the following equation is obtained: neb npi

Qv = − ∑∑ − wm Gm k =1 m =1

∂p Δ xm ∂z m

[3.91]

where npi is the number of integration points and wm the weight of these points. The computing requires the evaluation of the pressure gradient at the integration points on the basis of the pressure in the surface element in question. To obtain this, the interpolation functions are used: nne ∂p ∂ (ξ ,η ) = ∑ Ni (ξ ,η ) pi ∂z i =1 ∂z

[3.92]

However, this computing only gives a precise result for the points, which were used for the computing of the surface integrals during the constitution of the matrix [A1] and the vector {b1}. For other points, and in particular for those that are involved in relation [3.92], the values obtained may be completely erroneous [ZIE 00a]. First, the computing of the pressure gradients will be performed at all the integration points situated in the original surface element. Then, on the basis of the values obtained, an extrapolation will follow, to obtain the value of the gradients at the integration points located on the element edges. For 2 × 2 integration points, the extrapolation function is bilinear. For a point M of parametric coordinates (ξ, η), the following can be written as:

f (ξ ,η ) = a0 + a1ξ + a2η + a3ξη

[3.93]

If the integration points are symmetrically located in the parametric space with respect to the center of the element, and numbered as in Figure 3.11, the coefficients ai are given on the basis of the values of f at the four points by: a0 = ( f1 + f 2 + f3 + f 4 ) / 4

a1 = ( − f1 + f 2 − f 3 + f 4 ) / ( 2d12 )

a2 = ( − f1 − f 2 + f 3 + f 4 ) / ( 2d13 )

[3.94]

a3 = ( f1 − f 2 − f 3 + f 4 ) / ( d12 d13 )

where d12 is the distance of the points 1 and 2 and d13 the distance of the points 1 and 3. Similar relations can be obtained for other choices of integration points.

Numerical Resolution of the Reynolds Equation

151

The algorithm for computing the pressure gradients at the integration points on the element edges is shown in Figure 3.53. Computing of pressure gradients at point M Determine element k which contains point M From pressure at nodes of element k For the npi integration points in element k Compute pressure gradients at this point (relation 3.92) End Compute the interpolation polynomial coefficients (relation 3.94) Compute the parametric coordinates (ξ, η) of M in element k Extrapolate the gradients (relation 3.93) Figure 3.53. Algorithm for computing the pressure gradients on an element edge

3.8.2. Computation of axial flow rate

In the parts of the bearing where there are cavitation or separation zones, the computation of the entering flow rate will, at first, require the computation of the band width using the relation established in Chapter 2: 1

⎛ ⎡ t h3 psep ⎤ ⎞2 L(t ) = ⎜⎜ − exp ( − f (t ) ) ⎢ ∫ exp( f (ξ ))d ξ − L2 (t0 ) ⎥ ⎟⎟ t ⎣ 0 6μ h − r ⎦⎠ ⎝

[3.95]

The computation of the function f(ξ) which is involved in this relation can only be performed numerically or approached if it is assumed that the integration interval [t0, t] is small. The second method is described in what follows. The filling r is the fluid present outside the full film fringe and is thus independent of time in the integration interval [t0, t] under consideration. If it is ∂h assumed that in the same interval the thickness gradient has a negligible ∂x variation, the integral: f (ξ ) = ∫

ξ

t0

∂h ⎞ 1 ⎛ ∂h +2 ⎜U ⎟d ζ ∂ζ ⎠ 2( h − r ) ⎝ ∂x

152

Hydrodynamic Bearings

may be written as: ξ ∂h ∂ζ 1 ⎡ ∂h ξ d ζ ⎤ U dζ ⎥ +2 ⎢ t0 h − r 2 ⎣ ∂x t 0 h − r ⎦

∫

f (ξ )

∫

As the variation of h is small in the interval [t0, t] the first integral of the above expression is evaluated by:

∫

ξ

t0

d ζ ξ − t0 h − r h0 − r

which yields:

f (ξ )

⎛ hξ − r ⎞ 1 ∂h ξ − t0 + ln ⎜ U ⎟ 2 ∂x h0 − r ⎝ h0 − r ⎠

[3.96]

where h0 and hξ are, respectively, the film thicknesses at the times t0 and ξ. By introducing the function f(ξ) simplified in this way in the integral of [3.95]: h3 psep exp( f (ξ ))d ξ t0 6 μ h − r

Int (t ) = ∫

t

the following is obtained:

Int (t ) =

⎡ t hξ 3 ⎤ U ∂h ξ − t0 exp( )d ξ ⎥ ⎢ ∫t 0 6μ (h0 − r ) ⎢⎣ hξ − r 2 ∂x h0 − r ⎥⎦ psep

Again using the hypothesis that t is close to t0, the integral Int(t) is evaluated by linearization around t0: •

•

Int (t ) Int (t0 ) + (t − t0 ) Int (t0 ) = (t − t0 ) Int (t0 ) which yields:

psep h03 t − t0 Int (t ) = 6μ h0 − r Finally, by carrying over the expressions of f(t) and of Int(t) in [3.95], the following is obtained: 1 ⎛ h0 − r ⎞ 2

1

psep h03 t − t0 ⎞ 2 ⎛ U ∂h t − t0 ⎞ ⎛ 2 ⎜ ⎟ L(t ) = ⎜ ⎟ exp ⎜ − ⎟ L (t0 ) − 6μ h0 − r ⎟ ⎝ ht − r ⎠ ⎝ 4 ∂x h0 − r ⎠ ⎜⎝ ⎠

[3.97]

Numerical Resolution of the Reynolds Equation

153

3.8.3. Algorithm for computing the axial flow rate

It is assumed that the domain is broken down into quadrangular elements with eight nodes. The flow rate of the bearing is obtained by an integration of the unitary flow rate at y = 0, which is an ordinate assumed to define the bearing edges (Figure 3.52). The bearing edges are thus broken down into curvilinear elements, which correspond to the domain elements. The function “unitary flow rate” is evaluated at the integration points of these curvilinear elements, in general Gauss points. The algorithm below is defined for one of the bearing edges. If the bearing is not symmetrical9 it is necessary to apply it for computing the flow rate for each side of the bearing. The computing is carried out at each time step t1. The preceding time step, t0, must be sufficiently close to t1 for the simplifying hypotheses introduced in the developments exposed previously to be acceptable. The following parameters are presumed to be known at time t0 at any computing point of the abscissa x: – the width L(x, t0) of the fringe if there is a ruptured film zone at the abscissa x; this value is eventually nil (see section 3.2.4.2); – the thickness h(x, t0) between the bounding walls; – the filling r(x, t0); if there is no ruptured film zone at the abscissa x the filling is equal to the thickness. The algorithm for determining the partition into active and non-active zones (see section 3.1.2) has made it possible to calculate the following parameters at the time t1 at any computing point of the abscissa x: ∂h ; – the thickness h(x, t1) between the bounding walls, and the slope ∂x – the velocity of the sliding bounding wall U, and the squeezing velocity

∂h ; ∂t

– the filling r(x, t1); if there is no ruptured film zone at the abscissa x the filling is equal to the thickness. For the points for which the status is unchanged between t0 and t1, r(x, t1) is equal to r(x, t0)10; 9 A bearing is not symmetrical if it does not have a symmetry plane orthogonal to the Oz axis of the bearing or if, despite having a symmetry plane, the result of the load applied to the shaft is not in this symmetry plane, which leads to misalignment of the plane. 10 As the times t0 and t1 are close, the variation in filling due to transportation of the fluid in the non active zone between these two times is disregarded.

154

Hydrodynamic Bearings

– the separation pressure psep or the cavitation pressure pcav; for a point which, at the time t0, has an “active11” status and is not located in the full film fringe at the bearing edges, the reference pressure for the computing of the rupture conditions of the film is equal to pcav; for a point located in the full film fringe, the reference pressure is equal to psep; for a point that has the status “non-active” at the time t0, the reference pressure is equal to psep if it belongs to a separation zone or equal to pcav if it does not. The algorithm is shown in Figure 3.54.

Figure 3.54. Algorithm for computing the axial flow rate

11 A point has the status “active” if it belongs to a full film zone.

Numerical Resolution of the Reynolds Equation

155

3.8.4. Example

Figure 3.55 shows an example of evolutions of the full film fringe for a connecting rod big end bearing of the half-width 9.7 mm. The graphs are traced at every 4° of motor angle, from 602° to 625°. A full film zone occupies the central region of the bearing and another, which is not very large is located further up-flow at the supply orifice. These are characterized by a fringe width, which is equal to the bearing width. The rest of the bearing is partially full. These regions are separated by film rupture or formation boundaries, depending on the kinematical conditions. For example, the boundary located on the right of the supply zone is a film rupture boundary because its advancing velocity is of the same order as that of the surface of the crank, which carries the supply orifice. For this boundary to transform into a formation boundary it needs to advance at a velocity that is twice that of the crank pin surface (see section 3.4.3). For the opposite reasons, the boundary down-flow from the main ruptured film zone is a film formation boundary.

Figure 3.55. Example of evolutions for full film fringe width and axial flow

3.9. Bibliography [ABS 94] ABSI J., Etude des paliers à rainures: approche expérimentale et simulation numérique, Doctoral Thesis, University of Poitiers, France, 1994. [BON 86] BONNEAU D., Formation du film lubrifiant dans les contacts à alimentation non surabondante. Aspects expérimentaux et théoriques, Doctoral Thesis of Physics, University of Poitiers, France, 1986.

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[BON 94] BONNEAU D., ABSI J., “Analysis of aerodynamic journal bearings with small number of herringbone grooves by finite element method”, Journal of Tribology, vol. 116, pp. 698–704, 1994. [BON 14a] BONNEAU D., FATU A., SOUCHET D., Internal Combustion Engine Bearings Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [BON 14b] BONNEAU D., FATU A., SOUCHET D., Thermo-hydrodynamic Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [BOO 72] BOOKER J.F., HUEBNER K.H., “Application of finite-element methods to lubrication: an engineering approach”, ASME Journal of Lubrication Technology, vol. 14, pp. 313–323, (errata: vol. 98, p. 39, 1976) 1972. [DHA 05] DHATT G., TOUZOT G., LEFRANÇOIS E., Méthode des éléments finis, Hermes, Lavoisier, Paris, 2005. [DON 03] DONEA J., HUERTA A., Finite Element Method for Flow Problems, John Wiley & Sons Ltd., Chichester, 2003. [ELR 81] ELROD H.G., “A cavitation algorithm”, ASME Journal of Lubrication Technology, vol. 103, pp. 350–354, 1981. [JAC 57] JACOBSON B., FLOBERG L., The Finite Journal Bearing Considering Vaporisation, Chalmers Tekniska Hoegskolas Hnndlingar, Goteborg, Sweden, vol. 190, pp. 1–116, 1957. [OLS 74] OLSSON K., On Hydrodynamic Lubrication with Special Reference to NonStationary Cavitation, Chalmers University of Technology, Goteborg, Sweden, 1974. [FRE 90] FRÊNE J., NICOLAS D., DEGUEURCE B. et al., Lubrification hydrodynamique, Paliers et butées, Eyrolles, Paris, 1990. [FRE 97] FRÊNE J., NICOLAS D., DEGUEURCE B., et al., Hydrodynamic Lubrication – Bearings and Thrust Bearings, Elsevier Science, Amsterdam, 1997. [GOL 96] GOLUB G., VAN LOAN C., Matrix Computation, 3rd ed., The Johns Hopkins Press Ltd, London, 1996. [HAM 04] HAMROCK B.J., SCHMID S.R., JACOBSON B.O., Fundamentals of Fluid Film Lubrication, 2nd ed., Marcel Dekker Inc., New York – Basel, 2004. [KHA 85] KHADER M.S., “A generalized integral numerical solution method for lubrication problems”, Journal of Tribology, vol. 107, pp. 92–96, 1985. [LIG 97] LIGIER J.-L., Lubrification des paliers de moteur, Editions Technip, Paris, 1997. [MAO 08] MAOUI A., Etude numérique et expérimentale du comportement thermoélastohydrodynamique des joints à lèvre en élastomère, Doctoral Thesis, University of Poitiers, Paris, 2008.

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157

[VER 95] VERSTEEG H.K., MALALASEKERA W., An Introduction to Computational Fluid Dynamics. The Finite Volume Method, Longman Scientific & Technical, Harlow, England, 1995. [ZIE 00a] ZIENKIEWICZ O.C., TAYLOR R.L., The Finite Element Method: The Basis, 5th ed., vol. 1, Butterworth-Heinemann Ed., Oxford, 2000. [ZIE 00b] ZIENKIEWICZ O.C., TAYLOR R.L., The Finite Element Method: Solid and Structural Mechanics, 5th ed., vol. 2, Butterworth-Heinemann Ed., Oxford, 2000.

4 Elastohydrodynamic Lubrication

The mechanical pieces, which make up the mobile coupling of an internal combustion engine, are dimensioned so as to be able to withstand, without damage, the efforts that they have to transmit. Until around the beginning of the 1980s, they were generously dimensioned, and it was not considered necessary to take modification in the thickness of the lubricant film around bearings due to their elasticity into account. In a bid to reduce energy loss through friction, compact engines were developed. At the same time, power has increased, either through higher compression levels (especially for diesel engines) or through higher regimes. The increase in the load carried out by the bearings leads to a strong reduction in the minimum thicknesses of the film. The increase in rotation regimes has as its direct consequence the increase in accelerations and inertia effects that result from it. It is then obvious that the assumption of rigid bearings for which the lubricant film thickness is completely defined by geometrical and kinematic parameters is no longer satisfactory. Elasticity of the structures must also be taken into account. The reciprocal dependency of the thickness of the lubricant film and the field of the wall constraints leads to the theory of elastohydrodynamic lubrication (EHD lubrication). This theory was initially developed for Hertzian contacts. Due to the non-conformity of the surfaces, which are involved in these contacts (sphere-plane, cylinder-plane, sphere-cylinder, etc.), the contact zone where the lubricant film becomes established is very small and the pressure reached is very high, greatly exceeding 1 GPa. The fact that the lubricated contact zone is of small dimension in relation to the size of the elastic bodies makes it possible to calculate their deformation by combination of solutions obtained on semi-infinite bodies. For smooth bearings, the surfaces of the solids are conformal, and the totality of the space between the solids defines the lubricant film zone. The pressure, which dominates, is less than that for Hertzian contacts, in the order of 100 MPa for

160

Hydrodynamic Bearings

internal combustion engine bearings, but it acts over large surfaces. The significant deformations of solids, which result from this, can no longer be obtained by the combination of analytical solutions. The complexity of forms means that modelings by finite elements of the elastic behavior of the solids around the bearing are required. If linear elastic behavior is assumed, compliance can be represented by a matrix directly linking the wall constraints to the deformation of the film surface. This chapter describes the procedures followed for obtaining compliance matrices for the two solids which delimit the film on a bearing. The outer solid – which will be simply called “housing” in what follows – is usually made up of fretted bearing shells in a housing reamed into an engine block-cap assemblage or a connecting rodcap body. In the case of connecting rod small end, or piston humps, the bored hole, in which a bushing may possibly be fretted, is manufactured out of a single piece. The inner solid is more simply constructed. It is always made up of a single body, crankpin, crankshaft, wrist pin, etc. In what follows, it will be labeled “shaft”. The assumption of elastic linearity considers that the materials have linear behavior, and also that the contacts between the various elements making up the housing take place without unsticking or slippage. In the case of contact, where unsticking or slippage occurs, compliance matrices need to be calculated for each part of the assemblage (see section 2.4 of Volume 4 [BON 14a]). For materials with nonlinear behavior (for example, visco-elastic materials), the use of precalculated compliance matrices is no longer possible. 4.1. Bearings with elastic structure 4.1.1. Thickness of the lubricant film Figure 4.1 shows the domain occupied by the lubricant film for a cylindrical bearing. For an aligned bearing with perfectly cylindrical rigid surfaces and a radial clearance C, the nominal thickness hn of the film at each point of cylindrical coordinates (θ, z) may be notated:

(

hn (θ ) = C 1 − ε x cos θ − ε y sin θ

)

[4.1]

where εx and εy are the coordinates of the center of the shaft in the reference frame OC, xC, yC linked to the housing (or bearing shell).

Elastohydrodynamic Lubrication

161

Yc

R

πR

εy

Oc

h Oa

εx

0 / 2πR

θ

Xc

z

Zc

Figure 4.1. Bearing film thickness

When the bearing is misaligned, the nominal thickness also varies, following the axial direction, and is expressed as:

(

) (

)

hn (θ , z ) = C 1 − ε x cos θ − ε y sin θ − ζ y cos θ − ζ x sin θ z

[4.2]

where ζx and ζy are the (small) angles between the projections of the housing and shaft axis, respectively, in the planes (yC, zC) and (zC, xC). When the bearing operates in a non-stationary regime, the parameters εx, εy, ζx and ζy are in function of time. Generally, the contributions due to the defects of shape hd1 and hd2 of the two surfaces are added to the nominal thickness. These are initially present for design reasons (for example, bore reliefs1 or “lemon-shaped” bearing shells) or appear during operation (wear and tear). The modifications in thickness dt1 and dt2, which result from thermic dilations, which may or may not be under the effect of a uniform temperature field (global thermal – local thermal, see Chapters 1–3 of [BON 14b]) and the modifications in thickness de1 and de2, which result from elastic deformations, are also added to the nominal thickness.

1 The bore reliefs of bearing shells designate very shallow chamfers (in the order of 10 µ for a bearing of diameter 50 mm) created at the approach to the joint plane in order to prevent the problems that could be caused by a fault in the alignment of the bearing shells.

162

Hydrodynamic Bearings

h (θ , z ) = hn (θ , z ) + hd 1 (θ , z ) + hd 2 (θ , z ) + dt1 (θ , z ) + dt 2 (θ , z ) + de1 (θ , z ) + d e 2 (θ , z )

[4.3]

The elastic deformations of the surfaces 1 (housing) and 2 (shaft) have two causes: the effect of the constraints the lubricant film exerts on the bounding walls and the inertia effects of the acceleration fields applied to the structures. For bounding wall constraints, the calculations show that the tangential shear constraints have a negligible effect in the case of internal combustion engine bearings2. Only the normal constraints at the bounding wall (equal and opposite to pressure) are thus taken into consideration for the calculation of the deformation of the housing or shaft surfaces. Deformations due to inertia effects resulting from the acceleration fields may in turn be divided into two categories: those due to displacements of a rigid body, such as, for example, the displacement of the connecting rod as described in Chapter 1 of [BON 14a], and those due to structure vibrations. Movements of rigid bodies play a significant role in connecting rod big end bearings in engines at high regime (see Chapter 2 of [BON 14a]). However, they may be disregarded for shafts (crank pins, crank pivots and piston axis) due to the short, fat shape of these. Deformations due to vibrations play a significant role in bearing shafts, mainly because of the misalignments that they bring about. 4.1.2. Film domain discretization Whatever the mode of discretization used for the discretization of the Reynolds equation – finite difference, finite volume or finite element – the calculation domain is shown (for a developed bearing) by a quadrangle of the length 2π R and a width which is usually constant. The meshing of this domain gives a network of nF computing points, with a step that may be variable. Whatever method is used, this network of points defines a meshing into elements which cover the film surface; the points are the nodes of these elements. Depending on what is required, these elements may be either linear triangles with three nodes or quadratic triangles with six nodes, or they can be linear quadrangles with four nodes or quadratic quadrangles with eight nodes.

2 The same is not the case for elastomeric structures, such as lip seals, for which the deformations due to tangential forces need to be taken into account.

Elastohydrodynamic Lubrication

163

For the writing of discretized equations, knowledge is required of the film thickness (relation [4.3]) at every node i or the mesh, and thus, in particular, of the deformation due to pressure for each bounding wall, respectively, de1(i) and de2(i). As the procedure followed for calculating the deformations is the same for both surfaces, in what follows, it will not be deemed necessary to indicate which surface is in question. 4.2. Elasticity accounting: compliance matrices 4.2.1. Surface forces due to pressure In order to calculate with ease the deformation of the film bounding walls, a matrix relation will be established between the vector p of the pressures at the nodes and the vector de of the elastic deformations at these same nodes: de = [C] p

[4.4]

where [C] is the compliance matrix sought. The vectors p and d are defined for all the nF nodes of the Reynolds equation discretization mesh. The calculation of the matrix [C] is made through discretization into finite volume elements of the solid, housing or shaft under consideration. Usually, these tridimensional meshes are obtained with structure calculation software involving automatic meshers. The elements most often used are linear tetrahedra with four nodes and quadratic tetrahedra with 10 nodes. In this case, the meshing of the surface, which delimits the film, comprises triangular elements with three or six nodes, without a particular relationship to the meshing of the film itself. In addition, when the solids corresponding to the housing and the shaft are considered together, the meshes of the respective surfaces are generally different from one surface to another. The calculation of the compliance matrix [C] is conducted through matrix operations, which transform the vector p defined on the film mesh into a vector pS defined on the bounding wall surface mesh of the solid under consideration, and inversely a vector dS defined on the surface mesh is transformed into the vector de. These operations are described below. Consider an element ΩF of the film mesh and an element ΩS of the solid surface mesh, for example, a quadratic quadrilateral for the film and a linear triangle for the bounding wall of the solid, as is shown in Figure 4.2. The pressure is known at the nodes i of the quadrilateral. Using the interpolation functions N iF of the discretization process by finite elements, the pressure can be evaluated at the nS nodes m of the surface of the solid contiguous to the film:

164

Hydrodynamic Bearings

pS m =

nnef

∑ NiF (ξm ,ηm ) pi i =1

;

m = 1, nF

[4.5]

where nnef is the number of nodes per element of the film mesh (eight in the example from Figure 4.2), and ξm and ηm represent the parametric coordinates of the point m in the element ΩF. The set of relations [4.5] can be notated in the matrix format: pS = [P]

[4.6]

The projection matrix [P] is of the dimension nS × nF.

Figure 4.2. Location of a surface mesh node inside the film mesh

4.2.1.1. Computing the nodal deformation at the solid surface nodes by combination of elementary solutions The most direct method for obtaining the deformation dS of the solid surface involves determining a basis of nS elementary solutions s of the dimension nS. The grouping together of these solutions will form a matrix [S] of the rank nS and the following will be obtained: dS = [S] f

[4.7]

where f is the vector of the nodal forces applied to the nodes. These are obtained by numerical integration of the pressures, element by element. The force fm at the node m is given by:

fm =

nnes

⎞

e =1 ⎝ pg l =1

⎠

neS

⎛

∑ ⎜⎜ ∑ ∑ wpg NmS NlS pS l det J pg ⎟⎟

[4.8]

where pg is the number of integration points per element (these, in general, are Gauss points for the quadrangular elements or Hammer points for the triangular elements [DHA 05], [ZIE 00]), wpg and det Jpg are the corresponding weights and Jacobians, and nnes is the number of nodes per element of the bounding wall mesh

Elastohydrodynamic Lubrication

165

for the solid under consideration (three in the example from Figure 4.2). By defining the integration matrix [A] as:

[ Aml ] =

⎛

⎞

e =1 ⎝ pg

⎠

neS

∑ ⎜⎜ ∑ wpg NmS NlS det J pg ⎟⎟

[4.9]

the set of relations [4.8] is written as: f = [A] pS

[4.10]

The matrix [A] is of the dimension nS. Bringing together the relations [4.6], [4.7] and [4.10], the displacement dS takes the form: dS = [S] [A] [P] p

[4.11]

Each node of the film mesh is located in a surface mesh element (Figure 4.3).

Figure 4.3. Location of a film mesh node inside the solid wall mesh

The deformation at the node i is given by interpolation of the deformations at the nodes of the element of the wall mesh for the bounding solid – which may be housing or shaft – that contains the node i:

dF i =

nnes

∑ N kS (ξi ,ηi ) d S k k =1

This relation defines a projection matrix [P*] which links the displacement vectors dF and dS defined respectively on the mesh of the film and the bounding wall: dF = [P*] dS

[4.12]

166

Hydrodynamic Bearings

with a term of [P*] given by:

P *ij =

neS

nnes

e =1

k =1

(

∑ δ ei ∑ N kS ξi ,ηi

)

[4.13]

where δei equals 1 if the node i belongs to the element e and 0, if not; j is the number of the kth node of the mesh element for the bounding wall which contains the node i of the film mesh. The matrix [P*] is rectangular of the dimension nF × nS. The force fm is normal to the bounding wall: thus, the corresponding elementary solution sm should be calculated with a unitary force that is also normal to the bounding wall (Figure 4.4). After having defined the necessary boundary conditions3, and in particular the conditions for maintenance of the solid, a calculation by finite elements gives the three components of the displacement of all the mesh nodes, from which will only be used the components um and vm following x and y (the axial displacement is not involved in the calculation of the film thickness) for only those nodes which are contiguous to the film. On the basis of these components, a projection on the radial direction gives the sought solution sm (Figure 4.4):

s j m = u j m cosθ j + v j m sin θ j ;

j = 1, nS

[4.14]

Figure 4.4. Calculus of the radial displacement

3 If the bearing is geometrically symmetrical and the load applied does not induce misalignment, a half-model of the structure is sufficient. In this case, the boundary conditions must take this symmetry into account (zero normal displacement on the symmetry plane).

Elastohydrodynamic Lubrication

167

In addition to the deformation of the cylinder, which represents the housing or the shaft, the displacements u and v include a rigid body displacement of this cylinder. In the case of a connecting rod big end bearing, as this cylinder is maintained by a clamping placed approximately half-length along the connecting rod, the displacement due to flexion of the rod is around 10 times the maximum displacement due to the deformation of the cylinder alone. Figure 4.5 shows the elementary solution obtained for a transversal load: at (1) the housing appears circular and simply displaced upward due to flexion of the rod. In developed bearing configuration, (2) the solution appears to be a sinusoidal surface. A reprocessing of the solution consists of removing the mean displacement in the direction x on the one hand and in the direction y on the other, before carrying out the projection in the radial direction. This time the graphic representation (3) clearly shows the deformation of the cylindrical surface with strong ovalization, and the punching due to the normal unitary force applied to the bearing surface.

(b)

(c)

(d)

Figure 4.5. Elementary solution for a transversal loading, for a connecting rod big end bearing: a, b) with mean displacement c, d) without mean displacement

Let um and vm be the mean of the components um and vm of the solution sm. The averaged unitary solution is notated:

(

)

(

)

s j m = u j m − um cos θ j + v j m − vm sin θ j ;

j = 1, nS

or even:

s j m = s j m − um cosθ j − vm sin θ j ;

j = 1, nS

[4.15]

168

Hydrodynamic Bearings

All of the recentered solutions s m constitute a compliance matrix [ S ] of rank nS which makes it possible to calculate the elastic deformation d s of the considered surface on the basis of the nodal forces:

d S = ⎡⎣ S ⎤⎦ f By introducing the integration matrix, the following is obtained:

d S = ⎡⎣ S ⎤⎦ [ A ] p S

[4.16]

The difference between d S (relation [4.10]) and d S is given by:

(

)

d S − d S = [S] − ⎡⎣ S ⎤⎦ [ A ] p S which, projected onto the film mesh, gives:

(

)

d F − d F = [ P*] [S] − ⎡⎣ S ⎤⎦ [ A ] p S Relations [4.13], [4.15] and [4.9] give the detailed expression of a term of this vector: neS

nnes

e =1

k =1

( dF − dF )i = ∑ δ ei ∑ N kS (ξi ,ηi ) ( um cosθ j + vm sin θ j ) ⎛

⎞

e =1 ⎝ pg

⎠

neS

∑ ⎜⎜ ∑ wpg N mS NlS det J pg ⎟⎟ pS l Since the mean displacements do not depend on the node i, this relation may also be notated: nS

neS

⎛

⎞

( dF − dF )i = cosθi ∑ um ∑ ⎜⎜ ∑ wpg NmS NlS det J pg ⎟⎟ pS l m =1

e =1 ⎝ pg

⎛ + sin θi vm ⎜ ⎜ m =1 e =1 ⎝ nS

neS

∑ ∑∑ pg

⎠ ⎞ w pg N mS NlS det J pg ⎟ pS l ⎟ ⎠

[4.17]

Elastohydrodynamic Lubrication

169

If the relation [4.17] is compared to that of the nominal film thickness (relation [4.1]), it may be observed that the use of a basis of non-recentered elementary solutions (matrix [S]) for expressing the deformation of the housing has the same effect as modifying the definition of the components εx and εy of the shaft center. In this case, the calculated values for these parameters must compensate for the displacement of the housing center, because this displacement is not taken as a reference. Even if all the other results remain unchanged, this choice leads to plotted shaft orbits which do not have the appearance which would be expected.

Figure 4.6. Load applied to a connecting rod big end bearing and shaft center trajectory

Figures 4.6 and 4.7 give an illustration of the records obtained for the case of a connecting rod big end bearing, using the matrices [S] and [ S ]. The calculations have been made for a relatively rigid connecting rod (the connecting rod with conventional geometry shown in Figure 4.4, whose rigidity has been multiplied by 10).

170

Hydrodynamic Bearings

This results in a relative orbit – i.e. after subtraction of the mean displacement – of the shaft center that rarely exits the circle for radius 1, which corresponds to the limit of the displacement domain of an undeformable bearing (Figure 4.6). This orbit, shown with a dotted line, clearly demonstrates the greater compliance on the cap side for the connecting rod. When the mean displacement of the elementary solutions is conserved, the orbit greatly exceeds the circle of radius 1, and this is particularly in the transversal direction, due to the bending of the rod. For this very rigid connecting rod, the displacement of the shaft center reaches over three times the radial clearance. By restoring the initial compliance to the connecting rod, the trajectory in the transversal sense exceeds 30 times the radial clearance! Figure 4.7 shows that the evolutions of maximum pressure and minimum thickness are the same, whether the elementary solutions are recentered (in dotted lines) or not (in an unbroken line).

Figure 4.7. Minimum film thickness and maximum pressure: ––––– non-centered elementary solutions; – – – – centered elementary solutions

Elastohydrodynamic Lubrication

171

Thus, the compliance matrix [C] is calculated by:

[C] = [ P*] ⎡⎣S⎤⎦ [ A][ P]

[4.18]

where the terms of the matrices [P*], [ S ], [A] and [P] are given respectively by relations [4.13], [4.15], [4.9] and [4.5]. 4.2.1.2. Calculus of nodal deformations at solid surface nodes by combination of modal solutions In 1989, Kumar, Booker and Goenka [KUM 89] introduced an evaluation method for the deformation of the housing of a bearing, which begins with a decomposition on a basis of modal solutions. This method was subsequently applied by J. F. Booker and his colleagues to many problems involving bearings with elastic structure, and especially connecting rod big end bearings. In 2002, Booker and Boedo [BOO 01] published an article in which there is a global overview of this method, and in which all the references to relevant articles may be found. Only the main points of these will be presented here. Based on an assumed quasi-static linear elastic behavior, the relation between the efforts applied and the displacements which result from them may be notated: [K] dS = f

[4.19]

where [K] represents the opposite of the compliance matrix [S] (relation [4.7]), i.e. the rigidity matrix of the structure under consideration, reduced by condensation to the nodes alone, which may be located on either the housing or the shaft surface. The essential approximation introduced by the choice of a modal basis involves replacing the displacement vector dS defined on the nS nodes of the housing or shaft surface, with a vector dS’ of the significantly smaller dimension: dS = [T] dS’

[4.20]

where [T] is a transformation matrix of the dimension nS × mS. The mS columns of the matrix [T] can be chosen arbitrarily, but must be able to represent in the best possible way the displacements of the bearing surface, whether these result from a rigid body displacement or from the elastic deformation of the structure. By analogy with relation [4.19], the vector dS’ must verify a relation of the form: [K’] dS’ = f’

[4.21]

172

Hydrodynamic Bearings

If we write: [K’] = [T]t [K] [T]

;

f’ = [T]t f

[4.22]

the vector f is obtained by resolution of the equation [4.21], a resolution which is quick because the matrix [K’] is small. If [T] satisfies the general problem of finding specific modes [BOE 95]: [T]t [K] [T] = [T]t [A] [T] [Λ] where [A] is the matrix of masses of the structure and [Λ] the diagonal matrix made up of the nS kernel values of the system, then the matrices [K’] and [A’] are also diagonal and the transformation [T] obtained is independent of the meshing (in the same way as the form of the specific modes is independent from the meshing). An example of transformation [T] of which the columns are made up of the 11 first “useful” modes – the modes which represent the bearing displacements in the axial direction are eliminated – is given by Boedo and Booker [BOE 01]. Figure 4.8 shows the 10 first modes, and then modes 20, 30, 40 and 50, listed in increasing order of the kernel values for the connecting rod big end bearing, shown in Figure 4.9. Each mode has been subjected to a post-treatment similar to that described previously for the elementary solutions (see section 4.2.1.1): recentering of the components u and v and then projection onto the radial direction (Figure 4.4). For reasons of the symmetry of the model and of the load applied, only modes which are symmetrical with respect to the median plane are used. The modes corresponding to rigid body displacements have been eliminated. One of the difficulties of the method is in the choice of the modes that should be used. The selection process can employ an iterative approach, by progressively adding modes until a stabilized numerical solution is obtained [BOE 01]. This process, which should be followed for all new forms of the structure, may eventually be quite heavy to manage. In the case of structures with complex forms, the number of modes necessary to represent local deformations with a high gradient may be quite large. For the example shown in Figures 4.8 and 4.9, modes up to the order 50 are required.

Elastohydrodynamic Lubrication

1

2

3

4

5

6

7

8

9

10

20

30

40

50

Figure 4.8. Modal basis for the calculus of the deformation of a connecting rod big end bearing. Model with a symmetry plane

173

174

Hyydrodynamic Bearings

Figu ure 4.9. Conneccting rod big en nd bearing for which w modes are shoown in Figure 4.8; 4 half-model

4.2.2 Vo olume forces due d to inertia effects Intern nal combustiion engine bearings b may sometimes be subjectedd to high accelerattion fields. Thhis is especiallly the case forr connecting rod r big end beearings of high speeed engines. In I this case, and a independeently of any vibratory v phennomenon, the strucctures are subbject to deform mations which h are a resultt of the volum me forces brought about by inerrtia effects. Inn order to mak ke the EHD caalculation phaase of the bearing less l cumbersoome, the calcuulation of thesse inertia effeects may be annticipated by follow wing the proceess described below for con nnecting rod big b end bearinng. Let us u consider thee reference fraames for the engine e block, the crank shaft and the connectiing rod defined in Chapter 4 (Figure 4.1) of [BON 14aa]: – ℜ0 ≡ {O, x0, y0, z0}: engine bllock (solid 0);; – ℜ1 ≡ {O, x1, y1, z0}: crank shaaft (solid 1); – ℜ2 ≡ {A, x2, y2, z0}: connecting rod (solid 2), A is thee center of thee big end bearing. The velocity and acceleration of a point P of the connnecting rod, with the coordinaates (x, y) in thhe reference frrame ℜ2 may be written as: i

i

V02 (P) = R θ y1 + ϕ [ − y x 2 + x y 2 ]

Elastohydrodynamic Lubrication

175

i2 i2 ⎛ ii ⎞ ii Γ02 (P) = R ⎜ θ y1 −θ x1 ⎟ + ϕ [ − y x 2 + x y 2 ] − ϕ [ x x 2 + y y 2 ] ⎜ ⎟ ⎝ ⎠

where R is the radius of the crank shaft,, θ the crank shaft angle (x0, x1) relative to the engine block, ψ the connecting rod angle (x1, x2) with respect to the crank shaft and ϕ the connecting rod angle (x0, x2) with respect to the engine block. If it is assumed that the engine is functioning at constant velocity (absence of acyclism) the acceleration of the point P may thus be written in the basis B2: i2 ii ⎞ ii i2⎞ ⎛ i2 ⎛ i2 Γ02 (P) = − ⎜ R θ cosψ + x ϕ + y ϕ ⎟ x 2 + ⎜ R θ sinψ + x ϕ − y ϕ ⎟ y 2 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ i

ii

and by introducing the expression of ϕ and ϕ : i2 ⎡ ⎛ R cos θ Γ02 (P) = − θ ⎢ R cosψ + x ⎜⎜ ⎢ ⎝ cos ϕ ⎣

2 ⎤ ⎞ R ⎛ R cos 2 θ sin ϕ ⎞ ⎥ x + + y sin θ ⎜ ⎟ ⎟⎟ ⎟⎥ 2 cos ϕ ⎜⎝ cos 2 ϕ ⎠ ⎠⎦

⎡ ⎛ R cos θ R ⎛ R cos 2 θ sin ϕ ⎞ + θ ⎢ R sinψ + x ⎜ sin θ + ⎟ − y ⎜⎜ 2 ⎜ ⎟ ⎢ cos ϕ ⎝ cos ϕ ⎝ cos ϕ ⎠ ⎣ i2

⎞ ⎟⎟ ⎠

2⎤

⎥ y2 ⎥ ⎦ [4.23]

where R =

R in which L is the distance between the big end and small end centers L

of the connecting rod.

i

For a unitary angular velocity ( θ = 1), the acceleration at any point of the structure of the connecting rod may be calculated independently of the motor rotation velocity. If the domain occupied by the connecting rod is broken down into neV finite elements, this acceleration field leads to a vector of nodal forces f defined on all the mesh nodes, given by the integration of the volume forces:

fi =

⎛ ⎜ ⎜ e =1 ⎝

neV

∑∑ pg

⎞ w pg NiV Γ02 ( P ) ρe det J pg ⎟ ⎟ ⎠

[4.24]

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Hydrodynamic Bearings

where P corresponds to the integration points (Gauss points), NiV is the shape function of the node i in the element e, ρe is the density of the material of the element e, and dΩe is the volume element. The shape of the “elementary” solution obtained by applying the load f is i

independent of the rotation velocity θ (which is chosen equal to unit) but not of the chosen θ angle, an angle on which φ and ψ are also dependent, by the relations:

sin ϕ = d − R sin θ ; cos ϕ = 1 − (d − R sin θ ) 2 ; ψ = θ − ϕ where d =

d in which d is the alignment defect between the cylinder axis and the L

center of the crank shaft.

By varying the angle θ with a constant step, for example, of 1°, a set of solutions is obtained which, after recentering and projection (see section 4.2.1.1), forms the columns of the “matrix” for inertial deformations. During the EHD calculation, the inertial deformation at the angle θ and for a i

rotation velocity θ will be obtained by: 2

i ⎡θ ⎤ −θ θ − θi d(θ )= θ ⎢ i +1 di + di +1 ⎥ θi +1 − θi ⎣ θi +1 − θi ⎦

[4.25]

where di and di+1 are the elementary solutions which correspond to the angles θi and θi+1 which enclose the angle θ. i

Figure 4.10 represents the elementary inertial solutions ( θ =1) obtained for the connecting rod shown in Figure 4.4, in a set-up where the piston pin and the crankshaft pin are concurrent (d = 0), for the engine angles 0, 90, 180 and 270 . At the angle θ = 0, which corresponds to the top dead center, the angles φ and ψ are also nil, and the acceleration at a point P of the connecting rod has the expression (relation [4.23]):

(

Γ 02 (P) = − R + x R

2

)x

2

2

− y R y2

Elastohydrodynamic Lubrication

177

The ratio R between the radius of the crankshaft and the length of the connecting rod is in the order of 0.32 (Table 1.1 of [BON 14a]). At the big end bearing, x is in the order of R. Thus, the acceleration has a negative component following x2 which is clearly dominant with respect to the component following y2. As the inertia effect is in the opposite direction to acceleration, the connecting rod cap is pushed back in the direction of the acceleration. This results in a compression of the housing in the axial direction. At the engine angle 180° (bottom dead center), the acceleration of a point P of the connecting rod becomes:

(

Γ02 (P) = R − x R

2

)x

2

2

− y R y2

This time the acceleration for the points, which surround the big end bearing, is globally positive in the direction of x2. The center of the cap moves away from the body of the connecting rod, which results in an ovalization of the housing in the axial direction.

Figure 4.10. Deformation due to inertia for a connecting rod big end bearing at engine angles 0, 90, 180 and 270°

For the engine angles 90 and 270°, symmetrical deformations of the housing are obtained, for which distortion is more accentuated. The diameter at its greatest

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Hydrodynamic Bearings

dimension (at around 45° in the direction x2) is increased by 1.074 × 10-4 µm for an angular velocity equal to 1 rad s-1. This gives an increase in the diameter of 42.4 µm, of the order of the radial clearance of the bearing, for a rotation frequency of 6,000 rpm and of 169.6 µm at 12,000 rpm. 4.3. Accounting of shaft elasticity The calculation of the compliance matrix of the shaft follows the same procedure as for that of the housing. A part of the shaft – the crankshaft, crank pivots, piston axis, etc. – is delimited and meshed. Maintenance constraints (clamping) are defined at a distance, which is sufficiently far from the active surface of the bearing, in order not to disturb the elementary solutions that have to be calculated. If the bearing is geometrically symmetrical and the load applied does not bring about misalignment, a half-model of the shaft is sufficient. In this case, the boundary conditions applied to the shaft should include accounting for this symmetry (normal displacement in the zero symmetry plane). Once calculated, the elementary solutions are recentered and projected on the radial direction. After interpolation, (matrix [P], relation [4.5]), integration (matrix [A], relation [4.9]) and projection (matrix [P*], relation [4.13]), the compliance matrix for the shaft is obtained; it is defined with respect to the reference point of the shaft.

Figure 4.11. Location of a node of the film mesh at time t inside the film mesh at time 0

The EHD calculation of the bearing is carried out in the reference frame of the housing. Thus, it is necessary to express the lubricant film thickness in this same reference frame. The deformation of the shaft surface is involved in this calculation. If the shaft is a revolution body, the calculation of its deformation may be carried out directly with the compliance matrix expressed in the shaft reference frame. This calculation is satisfactory for most applications. However, if the shaft is asymmetrical with respect to its rotation axis, a direct calculation is no longer possible. The procedure that should be adopted may then be the following. On the basis of the pressure field obtained at the nodes of the film mesh at the time t, the pressure at the nodes of the same mesh is firstly calculated, but at the time 0.

Elastohydrodynamic Lubrication

179

For a connecting rod big end bearing, the angle – ψ measures the shaft rotation with respect to the housing (Figure 1.1 of [BON 14a]). The rotation to which the pressure field needs to be subjected in order to be correctly placed in the film mesh at the time 0 is thus the angle ψ. The coordinates (xi, zi) of a node i of the film mesh of the developed bearing become, after notation, (xi + R ψ, zi), where R is the shaft radius. This point is situated in an element e of the film mesh at the time t by its parametric coordinates ξi and ηi. The pressure pi at this point is obtained by interpolation:

pi =

nnef

∑ N kF (ξi ,ηi ) pk

[4.26]

k =1

where nnef is the number of nodes per element of the film mesh (eight in the example of Figure 4.11) and N kF is the interpolation function of the film mesh relative to the node k of the element e. The deformation of the shaft can then be calculated (relation [4.18]) in the mesh at the time 0:

d = [ P*] ⎡⎣ S ⎤⎦ [ A ][ P ] p

[4.27]

All that remains is to deduce by an opposite process the deformation expressed in the mesh at the time t by applying a rotation of an angle – ψ or a translation – R ψ in the developed bearing configuration. For this, each node i of the mesh at the time t is situated in an element of the film mesh at the time 0 by its parametric coordinates ξi and ηi, and the deformation di at this point is obtained through interpolation:

di =

nnef

∑ NkF (ξi ,ηi )dk

[4.28]

k =1

If the mesh is regular – all the elements are rectangular and have the same dimensions – only the parametric coordinates ξi need to be calculated. If, in addition, the time step leads to displacements of the mesh, which correspond to a multiple of the element size in the circumferential direction, the position of the nodes at the time t and the time 0 coincides. In this case, the operations described above do not require interpolation. In a more general case of an irregular mesh and/or a variable time step, the relative positions ξi and ηi of the mesh nodes at the time t in the mesh at the time 0

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Hydrodynamic Bearings

and ξ*i and η*i of the mesh nodes at the time 0 in the mesh at the time t need to be recalculated at the start of each new time step, giving the vectors ξ, η, ξ* and η*. These vectors will be used during the entire time step for the many updates of deformations which require the relations [4.26], [4.27] and [4.28]. 4.4. Particular case of non-conformal meshes The meshes of the housing and shaft surface have nodes that do not necessarily coincide with the nodes of the film mesh. Section 4.1 describes the standard procedure for putting in concordance these meshes. All the operations together lead to the expression of the elastic deformation d of the surface under the effect of the hydrodynamic pressure p:

d = [ P*] ⎡⎣ S ⎤⎦ [ A ][ P ] p

[4.29]

Figure 4.12. Film mesh (ΩF) and solid surface mesh (ΩS) with excessive non-conformity

Figure 4.13. Meshes for film (ΩF) and for solid surface (ΩS) with coincident nodes

This relation may also be used to calculate the pressures which result from the contact between the surfaces involved when the load is severe. In this case, an

Elastohydrodynamic Lubrication

181

increase in the pressure at the node i must result in an increase in the deformation at this same node. Let us consider the node i of Figure 4.12. It is not involved in the calculation of the pressure at the mesh nodes on the solid surface. Any change in pressure at this node will have no effect on the calculation of the deformation of the surface. Even with mesh with a higher level of conformity, it can occur that the pressure at certain nodes is not involved in the calculation of the deformation. Figure 4.13 illustrates the case of a film mesh in quadratic elements with eight nodes, whereas the mesh of the solid surface is in linear elements with three nodes (left-hand image) or with triangular quadratic elements with six nodes (right-hand image). In the case of linear triangles, the pressure at the nodes located at the middle of the sides of the film elements will not be used. In the case of quadratic triangles, all the pressures at the film mesh nodes are used, either directly, or to interpolate the value of the node which is in central position. On the basis of the deformations calculated at the nodes of the triangular elements, the values at the nodes located in the middle of the sides of the film elements are then obtained by linear interpolation. The fact that these values are in the alignment of the values obtained at the corners of the film elements makes facets appear on the deformation and then on the film thickness. This problem can only be avoided by choosing either a structure surface mesh for which the number of triangular linear elements is eight per film mesh element, as shown in Figure 4.14, or by choosing meshes that exhibit total conformity and are made up of the same elements.

Figure 4.14. Conformal meshes, quadratic for the film (ΩF) and linear for the solid surface (ΩS)

If the use of non-conformal meshes cannot be avoided, a modification of the calculation of the vector of the forces f makes it possible to use all the values of the pressures of all the nodes of the film mesh. The calculation of f, given by relations [4.6] and [4.10], may be notated:

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Hydrodynamic Bearings

f = [A] [P] p in which the calculation of the pressure at the nodes of the mesh of the solid surface which comes before the calculation of the nodal forces by integration, is replaced by a calculation where the integration of the nodal forces is carried out firstly by integration on the film elements:

fF i =

⎛ ⎜ ⎜ e =1 ⎝

neF

nnef

∑ ∑∑ pg l =1

⎞ w pg N mF NlF pl det J pg ⎟ ⎟ ⎠

[4.30]

where neF is the number of elements of the film mesh, pg the number of integration points by element and nnef is the number of nodes per element of the film mesh (eight in the example of Figure 4.12). By defining the integration matrix [A’] with: neF ⎛

⎞

e =1 ⎝ pg

⎠

[ A 'il ] = ∑ ⎜⎜ ∑ wpg NiF NlF det J pg ⎟⎟

[4.31]

the vector of the nodal forces on the film mesh is: fF = [A’] p

[4.32]

The calculation of the nodal forces at the mesh nodes of the structure surface is obtained through decomposition of the nodal forces at the nodes of the film mesh. For this, let us consider the node i of Figure 4.12. This point is located in the element of the surface mesh (triangular element with three nodes in the case of Figure 4.12) by its parametric coordinates ξi and ηi. The contribution of the nodal force fF i to the nodal forces at the nodes of the element e (in gray in Figure 4.12) of the surface mesh is given by:

(

f ki = N kS ξi ,ηi

)f

Fi

[4.33]

The sum of the forces fki for the nodes of the element e which contains the node i is equal to the force fF i. The summing of all the elements of relations [4.33] gives:

f ji =

neS

∑ δ ei N kS (ξi ,ηi ) f F i

[4.34]

e =1

where δei equals 1 if the element e contains the node i and 0 if not, k is the local number in the element e of the node j. The set of relations [4.34] makes it possible to

Elastohydrodynamic Lubrication

183

define a matrix [P’] of passage of the nodal forces on the film mesh nodes to the nodal forces on the nodes of the solid surface mesh.

P ' ji =

neS

∑ δ ei N kS (ξi ,ηi )

[4.35]

e =1

Finally, the vector f of the nodal forces on the mesh of the solid surface is obtained by the relation: f = [P’] [A’] p

[4.36]

If the film mesh is denser than that of the bounding wall – which is the case in Figure 4.12 – any modification in the pressure at a node, whatever the film mesh, will lead to a modification of the vector f. On the other hand, if the bounding wall mesh is denser than the film mesh, part of the nodal forces, which make up the vector f, will remain nil, and this applies to whatever the pressure field is. Figure 4.15 gives an example of this: the nodes of the surface mesh shown in gray are not associated with a nodal force. The deformation, which results from this, then has local undulations due to the more pronounced pushing at the loaded nodes. Since the process of projection of the deformations onto the film mesh (relation [4.28]), for the same reason, only uses the values of the deformations at the nodes where the forces are nil, the values obtained are close to the more pronounced deformations. Thus, the average value of the deformation is slightly altered.

Figure 4.15. Mesh of the solid surface (ΩS) with a higher density than the film one (ΩF)

184

Hydrodynamic Bearings

4.5. Bibliography [BOE 95] BOEDO S., BOOKER J.F., WILKIE M.J., “A mass conserving modal analysis for elastohydrodynamic lubrication”, Proceedings of the 21st Leeds-Lyon Symposium on Tribology. Lubricants and Lubrication, Elsevier, pp. 513–523, 1995. [BOE 01] BOEDO S., BOOKER J.F., “Finite element analysis of elastic engine bearing lubrication: application”, Revue Européenne des Eléments Finis, vol. 10, pp. 725–739, 2001. [BON 14a] BONNEAU D., FATU A., SOUCHET D., Internal Combustion Engine Bearings Lubrication in Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [BON 14b] BONNEAU D., FATU A., SOUCHET D., Hydrodynamic Bearings, ISTE, London and John Wiley & Sons, New York, 2014. [BOO 01] BOOKER J.F., BOEDO S., “Finite element analysis of elastic engine bearing lubrication: theory”, Revue Européenne des Eléments, vol. 10, pp. 705–724, 2001. [DHA 05] DHATT G., TOUZOT G., LEFRANÇOIS E., Méthode des éléments finis, Hermes, Lavoisier, Paris, 2005. [KUM 89] KUMAR A., BOOKER J.F., GOENKA P.K., “Dynamically loaded journal bearings: a modal approach to EHL design analysis”, Proceedings of the 15th Leeds-Lyon Symposium on Tribology: Tribological Design of Machine Elements, Elsevier, Amsterdam, pp. 305– 315, 508–510, 1989. [ZIE 00] ZIENKIEWICZ O.C., TAYLOR R.L., The Finite Element Method. Volume 1, The Basis, Butterworth-Heinemann, Oxford, 2000.

Appendix

A.1. Weighting functions for the modified Reynolds equation The modified Reynolds equation makes it possible to ensure the conservation of the mass flow rate in non-active zones of the lubricant film. The specific form that the equation takes in these zones requires, during the resolution process through the finite element method, use of weighting functions, which are decentered in the up-flow direction. In a developed bearing configuration, the domain occupied by the lubricant film has the form of a band of length 2π R and of a width that is generally constant. For the resolution of the modified Reynolds equation (equation [3.3]), the domain is broken down into quadrangular isoparametric elements with eight nodes, the sides of which are parallel in the directions x and z. A parametric transformation transforms each element into a square centered at the origin, the side of which is 2 (Figure A.1). In active zones (full film zones) quadratic elements with eight nodes are used. In the works devoted to the finite element method, the expressions of the shape (and interpolation) functions relative to the quadratic quadrangular elements may be found [DHA 05, ZIE 00]. These functions are used as weighting functions for these elements.

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Hydrodynamic Bearings

Figure A.1. Isoparametric quadratic element and linear subelements

In non-active zones, the equation that needs to be solved (equation [3.3]) only involves the unknown function D in the form of a primary derivative relative to x. The necessary decentering of weighting functions in the up-flow direction relative to x is facilitated if the original quadratic element is broken down into four linear subelements, as indicated in Figure A.1. For a square element occupying the square [0, 1] × [0, 1] such as the subelement 3 in Figure A.1 the shape and linear interpolation functions are simply expressed: N1 (ξ ,η ) = (1 − ξ )(1 − η ) ; N 2 (ξ ,η ) = ξ (1 − η ) N3 (ξ ,η ) = ξη ; N 4 (ξ ,η ) = (1 − ξ )η

where the nodes 1, 2, 3 and 4 occupy the positions indicated in the circles in Figure A.1. In this case, for each subelement, the weighting functions are decentered in the up-flow direction and have the following expressions: W1 (ξ ,η ) = ⎣⎡1 − ξ + 3sgn (U ) ξ (ξ − 1) ⎦⎤ (1 − η ) W2 (ξ ,η ) = ⎡⎣ξ − 3sgn (U ) ξ (ξ − 1) ⎤⎦ (1 − η ) W3 (ξ ,η ) = ⎡⎣ξ − 3sgn (U ) ξ (ξ − 1) ⎤⎦ η W4 (ξ ,η ) = ⎡⎣1 − ξ + 3sgn (U ) ξ (ξ − 1) ⎤⎦ η

where U is the velocity of the sliding bounding wall

Appendix

187

A.2. Bibliography [DHA 05] DHATT G., TOUZOT G., LEFRANÇOIS E., Méthode des éléments finis, Hermes, Lavoisier, Paris, 2005. [ZIE 00] ZIENKIEWICZ O.C., TAYLOR R.L., The Finite Element Method: The Basis, 5th ed., vol. 1, Butterworth-Heinemann Education, Oxford, 2000.

Index

A, B, C active zone, 27, 28, 30–32, 36, 37, 44, 48–50, 53, 54, 56, 58, 66–68, 74, 77, 79, 82, 89, 92–94, 115, 120, 122, 123, 127, 128, 132, 138, 153 ambient pressure, 36, 44, 45, 47, 117, 143, 144 Barus law, 6, 8 bearing edges, 29, 36, 37, 40, 43–50, 71, 73, 77, 83, 86, 120, 123, 138, 143, 149, 153, 154 boundary conditions, 33, 35, 39, 40, 42, 46, 49, 63, 65–67, 69–73, 77, 78, 83, 88, 89, 92, 97, 104, 166, 178 cavitation algorithm, 67, 75, 77, 78, 81, 89, 98, 106, 163, 168, 171, 178 computation time, 124 connecting rod, 44, 99, 115, 139, 142, 155, 160, 162, 167, 169, 171, 172, 174–177, 179 contact zone, 159 Couette flow, 38, 85, 144, 145 crank pin, 155, 162 Crout method, 122 D, E damage, 29, 30, 159 density, 3, 12, 13, 19, 20, 29–31, 34, 60, 65, 66, 81, 121–124, 176, 183

developed bearing, 35–37, 45, 68, 71, 82, 97, 99, 162, 167, 179 diesel engine, 136, 139, 142, 159 discretized equation, 73, 76, 89, 104, 107, 163 dissipated energy, 57 power, 44 down flow boundary, 79, 80 dynamic viscosity, 3–7, 10, 11, 14, 29, 57, 60 EHD problem, 124 elastic deformation, 161, 162, 163, 168, 171, 180 elastohydrodynamic lubrication, 159 elementary solution, 164, 166, 167, 169, 170, 172, 176, 178 engine, 6, 8, 10, 14, 15, 21, 22, 36, 42, 43, 46, 51, 60, 68, 69, 115, 124, 136, 138, 142, 146, 159, 160, 162, 174–177 block, 160, 174, 175 cycle, 124 entering flow, 85, 123, 151 F, G, H film rupture, 89, 116, 119, 138, 155 thickness, 2, 3, 20, 26, 30, 32, 57, 59, 60, 64, 68, 80, 103, 105, 108, 114, 115, 117, 131, 134, 135, 138, 140, 141, 146, 152, 159, 161, 163, 166, 169, 170, 178, 181

190

Hydrodynamic Bearings

finite difference method, 43, 63, 64, 70, 82, 86, 89, 90, 121, 123, 135, 140, 146, 147 element method, 43, 63, 64, 82, 90, 105, 106, 121–124, 127, 147 volume method, 43, 44, 63, 64, 82, 121–123, 127, 147 flow factor, 46 regime, 59 formation boundary, 78, 79, 80, 89, 90, 155 friction torque, 118, 120, 122–124, 129 frictionless contact, 32 Galerkin method, 98, 101, 106 Gauss point, 102, 147, 153, 164, 176 hydrodynamic lubrication, 17, 40, 68, 126 I, J, L inertia effects, 60, 159, 162, 174 integration points, 103, 105, 147, 149, 150, 151, 153, 164, 176, 182 interpolation function, 43, 98, 99, 100, 101, 106, 110, 150, 163, 179 isothermal, 103 isoviscous fluid, 31, 54, 57, 59 Jacobian matrix, 102, 124 joint plane, 161 leakage flow, 40 load, 68, 115, 119, 120, 122, 125, 126, 128–136, 138–143, 146, 148, 153, 159, 166, 167, 169, 172, 176, 178, 180 lubricant filling, 65 supply, 35, 42, 117, 118 M, N, O mass flow, 30, 47, 48, 51, 63, 65, 74, 77, 78, 85, 89, 94, 97, 117 maximum pressure, 115, 116, 119, 120, 125–129, 170 mean pressure, 42 minimum film thickness, 116, 139 misalignment, 153, 166, 178

mixed lubrication, 46 modal basis, 171 modified Reynolds equation, 66, 67, 74, 86, 92, 117, 119, 120, 127 Navier, 17, 19, 33, 40 Newtonian fluid, 3, 10, 11, 103 nodal force, 164, 168, 175, 182, 183 non active zone, 153 orbit, 170 P, R phase change, 28, 29, 38 piezoviscosity, 6, 8, 9 Poiseuille flow, 38, 39, 78, 80, 85 power law, 6, 8, 9, 11 pressure field, 40, 42, 43, 51, 54, 63, 116, 117, 119, 120, 127–129, 138, 139, 148, 178, 183 gradient, 19, 38–43, 46–49, 53, 65, 78, 91, 123, 150, 151 projection matrix, 164, 165 radial clearance, 45, 60, 131, 160, 170, 178 rupture boundary, 38, 78, 79, 81, 155 S, T SAE classification, 15 separation, 30, 32, 40, 43–45, 47, 49, 50, 136, 151, 154 shaft center trajectory, 169 elasticity, 178 shape function, 100, 101, 176 shear rate, 4, 10–12, 17, 18, 57 stress, 32, 51, 54 thickening fluid, 10 thinning fluid, 10 standard Reynolds equation, 65, 71, 84, 90, 92, 117, 124, 127 supply conditions, 117 flow,43, 120 zone, 40, 41, 86, 155

Index

surface tension, 27, 29, 30, 40, 44, 45 thermal conductivity, 12, 13 thermoviscosity, 5, 8, 9 thickness discontinuity, 147 thrust bearing, 20, 23–26, 35, 45, 54, 56, 59, 64, 68, 82 time derivative, 105, 108, 127, 134, 136 U, V, W uncompressible fluid, 21, 25 up flow boundary, 66, 78, 79, 80, 81 variable load, 37

191

viscosity, 2–12, 14–18, 27, 30, 54, 57, 59, 60, 64, 69, 80, 81, 114, 130, 138 index, 2, 15 viscous dissipation, 57, 59 volume flow, 24, 26, 47, 48, 50, 51 force, 174, 175 wall slipping, 32, 33 weighting function, 91, 94–98, 101, 106, 110–112, 147