Jet in Supersonic Crossflow [1st ed.] 978-981-13-6024-4, 978-981-13-6025-1

Table of contents :
Front Matter ....Pages i-xiv
Introduction (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 1-25
Spatial Distribution of Gaseous Jet in Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 27-53
Flow Structures of Gaseous Jet in Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 55-101
Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 103-171
Reaction Characteristics of a Gaseous Jet in a Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 173-200
Primary Breakup of Liquid Jet in Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 201-241
Spray Characteristics of a Liquid Jet in a Supersonic Crossflow (Mingbo Sun, Hongbo Wang, Feng Xiao)....Pages 243-284

Citation preview

Mingbo Sun · Hongbo Wang · Feng Xiao

Jet in Supersonic Crossflow

Jet in Supersonic Crossflow

Mingbo Sun Hongbo Wang Feng Xiao •

•

Jet in Supersonic Crossflow

123

Mingbo Sun Science and Technology on Scramjet Laboratory National University of Defense Technology Changsha, Hunan, China

Hongbo Wang Science and Technology on Scramjet Laboratory National University of Defense Technology Changsha, Hunan, China

Feng Xiao Science and Technology on Scramjet Laboratory National University of Defense Technology Changsha, Hunan, China

ISBN 978-981-13-6024-4 ISBN 978-981-13-6025-1 https://doi.org/10.1007/978-981-13-6025-1

(eBook)

Library of Congress Control Number: 2018968093 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Hypersonic aircrafts that can fly at more than five times the speed of sound have revolutionary applications in national security and space exploration, and thus the development of hypersonic aircrafts has been described as “highest technical priority” in many countries. Supersonic combustion ramjet (Scramjet) engines are the optimal propulsion system for hypersonic aircrafts within the atmosphere. In the combustor of a scramjet engine, the fuel and the supersonic airflow must mix well in an extremely short period of time for efficient combustion. Since the gas/liquid fuel is typically injected from the wall in the scramjet combustor, gas/liquid jets in supersonic crossflow have been widely studied. Qualitative studies of jets in supersonic crossflow had been carried out using conventional experimental and numerical techniques such as high-speed photographs and RANS before 2000. As the mixing process of gas jet and atomization process of liquid jet in supersonic crossflow is very complicated, the physical mechanism and determining factors had not been well understood, requiring further research on this subject. After 2000, many advanced experimental and numerical techniques have been developed and applied to investigate jets in supersonic crossflow. For example, nanoparticle-based planar laser scattering (NPLS) technique has been developed in our research group to obtain high-fidelity resolution of eddy structures in mixing process of jet in supersonic crossflow. LES and DNS have been used in this field in the past two decades, which significantly improves our understanding of gas jet mixing and liquid jet atomization in supersonic crossflow. With the advancement of experimental and numerical techniques, jets in supersonic crossflow will be better understood, which will promote superior design of fuel injection scheme in scramjet engines. Science and Technology on Scramjet Laboratory at National University of Defense Technology has carried out huge amount of works on fuel injections, mixing, and combustion in scramjet combustors, which significantly promotes the development of scramjet engines in China. This book summarizes the research on

v

vi

Preface

jet in supersonic crossflow carried out by our group in the past 15 years and presents many state-of-the-art results and analysis in this subject. The book is aimed at graduate students majoring in aeronautical and aerospace engineering, and researcher and engineers who are working in design of scramjet engines. The prerequisite knowledge includes fluid mechanics, combustion principles, and computational fluid dynamics. The book consists of two parts. Part I includes Chaps. 2–5, presenting studies of mixing and combustion of gaseous jets in supersonic crossflow. Part II consists of Chaps. 6 and 7, detailing the research on atomization and spray of liquid jets in supersonic crossflow. Changsha, China

Mingbo Sun Hongbo Wang Feng Xiao

Acknowledgements

We would like to thank all the colleagues and students in our research group who have contributed to the understanding of the fundamentals of jets in supersonic crossflow. First and foremost we express our deep gratitude to Prof. ZhenGuo Wang, Prof. WeiDong Liu, Prof. JianHan Liang, Prof. QingLian Li, and Prof. YuXin Zhao for the significant contributions they have made to the work presented here. Particularly, we are grateful to LiYin Wu (Chap. 6), MingGang Wan (Chap. 1), DaPeng Xiong (Chap. 1), Yuan Liu (Chap. 2), ChangHai Liang (Chap. 3), YongChao Sun (Chap. 3), XiLiang Song (Chap. 4), Fan Li (Chap. 4), JinCheng Zhang (Chap. 5), ChenYang Li (Chap. 7), GuangXin Li (Chap. 7), and ChaoYang Liu (Chap. 5) for their support in the writing up of the book. We would like to thank the staff at Springer for their help and support.

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Gaseous Jets in a Supersonic Crossflow . . . . . . . . . . . . . . . . 1.2.1 Flow Structures and Jet Mixing . . . . . . . . . . . . . . . . 1.2.2 Spatial Distribution of a Gaseous Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mixing Enhancement . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Reactive Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Liquid Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . 1.3.1 Atomization Process of a Liquid Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Approaches to Liquid Jet Atomization and Spray Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Spatial Distribution of a Liquid Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Atomization Characteristics of a Liquid Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow 2.1 Experimental Measurement Technologies . . . . . . . . . . . . . 2.2 Jet Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spanwise Distribution of Gaseous Jet . . . . . . . . . . . . . . . 2.4 Near-Wall Information of Gaseous Jet . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . .

. . . .

. . . .

. . . .

1 1 2 4

. . . .

. . . .

. . . .

. . . .

8 9 11 12

....

12

....

15

....

17

.... .... ....

18 19 21

. . . . . .

27 27 29 40 45 52

. . . . . .

. . . . . .

. . . . . .

ix

x

Contents

... ...

55 58

... ... ...

58 62 65

... ...

74 74

3 Flow Structures of Gaseous Jet in Supersonic Crossflow . . . . . . 3.1 Shock Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Experimental Visualization of the Transverse Jet Flow Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Wave Structures Study by Numerical Simulation . . . . . 3.2 Upper Trailing Counter-Rotating Vortices (CRVs) . . . . . . . . . 3.3 Formation of Surface Trailing Counter-Rotating Vortex Pairs Downstream of a Sonic Jet in a Supersonic Crossflow . . . . . . 3.3.1 Instantaneous Flow Structures in Jet Near-wall Wakes . 3.3.2 Mean Flow Properties in the Jet Near-wall Wake Flowfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Flow Topology Analysis of Surface TCVPs . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

... 78 ... 89 . . . 100

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mixing Characteristics of Single Injection . . . . . . . . . 4.1.1 Mixing in the Near Field . . . . . . . . . . . . . . . . 4.1.2 Mixing in the Near-Wall Region . . . . . . . . . . 4.1.3 Mixing in the Expansion Flowpath . . . . . . . . . 4.2 Mixing Characteristics of Multiple Injections . . . . . . . 4.2.1 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Tandem Configuration . . . . . . . . . . . . . . . . . . 4.2.3 Parallel Configuration . . . . . . . . . . . . . . . . . . 4.3 Mixing Enhancement Technology . . . . . . . . . . . . . . . 4.3.1 Active Mixing Enhancement Technology . . . . 4.3.2 Passive Mixing Enhancement Technology . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

103 103 104 109 132 142 144 147 153 159 159 162 168

5 Reaction Characteristics of a Gaseous Jet in a Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow . 5.1.1 Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Unsteady Characteristics of Large-Scale Vortices . 5.1.3 Combustion Regime . . . . . . . . . . . . . . . . . . . . . 5.2 Flow and Flame Structures in the Reacting Flow . . . . . . 5.2.1 Experimental Observation . . . . . . . . . . . . . . . . . 5.2.2 Flow Structures in the Reacting Flow . . . . . . . . . 5.2.3 Analysis of Streamlines . . . . . . . . . . . . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

173 173 173 175 176 184 185 189 195 199 199

. . . . . . . . . . . . .

Contents

6 Primary Breakup of Liquid Jet in Supersonic Crossflow . . . . . 6.1 Experimental Setup of Pulsed Laser Background Imaging . . . 6.2 Experimental Analysis of Liquid Jet in Supersonic Crossflow 6.3 Interface Tracking Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Volume of Fluid Method . . . . . . . . . . . . . . . . . . . . . 6.3.2 Level Set Method . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Coupled LS and VOF Algorithm (CLSVOF) . . . . . . . 6.3.4 Comparison of VOF, LS, CLSVOF . . . . . . . . . . . . . 6.3.5 CLSVOF with Schemes of Different Order . . . . . . . . 6.3.6 Nonuniform Versus Uniform Cartesian Mesh . . . . . . 6.4 Two-Phase Flow LES Methodology for Atomization in Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Governing Equations in Liquid Phase . . . . . . . . . . . . 6.4.2 Governing Equations in Gas Phase . . . . . . . . . . . . . . 6.4.3 LES Formulation of Interface Advection Equations . . 6.4.4 Grid and Dependent Variable Arrangement . . . . . . . . 6.4.5 Numerical Methods for the Gas Flow Solver . . . . . . . 6.4.6 Numerical Methods for the Liquid Flow Solver . . . . . 6.4.7 Algorithm for LES of Atomization in Supersonic Gas Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

201 202 205 205 205 211 213 216 219 220

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

221 222 223 224 224 225 227

. . . . 229 . . . . 229 . . . . 239

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow 7.1 Experimental Study of Liquid Jets . . . . . . . . . . . . . . . . . . . . . 7.1.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Penetration Height and Cross-Sectional Distribution of a Liquid Jet . . . . . . . . . . . . . . . . . . . . 7.1.3 Spray Droplet Size and Velocity Distribution . . . . . . . 7.1.4 The Liquid-Trailing Phenomenon of a Jet Spray . . . . . 7.2 Simulation and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Computational Conditions . . . . . . . . . . . . . . . . . . . . . 7.2.3 Characteristics of a Gas Phase Flowfield . . . . . . . . . . . 7.2.4 Characteristics of the Spray Field . . . . . . . . . . . . . . . . 7.2.5 The Liquid-Trailing Phenomenon of a Jet Spray . . . . . 7.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 243 . . . 243 . . . 243 . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

245 255 263 265 265 272 274 276 278 281 282

About the Authors

Prof. Mingbo Sun is the director of Science and Technology on Scramjet Laboratory at National University of Defense Technology (NUDT) in China (Email: [email protected]). He was awarded a Doctorate in Aerospace Science and Technology from NUDT (2008), and a bachelor’s degree in Aerodynamic Engineering from NUDT (2002). His Ph.D. thesis entitled “Studies on Flow Patterns and Flameholding Mechanisms of Cavity Flameholders in Supersonic Flows” was rated as outstanding doctoral dissertation. He started his research career as a Lecturer at NUDT from 2008 and was promoted to a professor in Science and Technology on Scramjet Laboratory in 2014. His research focuses on the fluid and combustion dynamics of Scramjet engines. His expertise is in high resolution optical observation and numerical simulation of supersonic flow and combustion, with applications to scramjet combustor design. His research work covers all the procedures of the supersonic combustion, includes flow, atomization, mixing, combustion, unsteady combustion etc. He was awarded the Excellent Youth Fund of National Natural Science Foundation of China for his outstanding research in supersonic combustion. He has published 4 books and more than 160 papers, and has been granted 16 Chinese patents. Dr. Hongbo Wang is Associate Professor at National University of Defense Technology (NUDT) in China. He was awarded a Doctorate in Aerospace Science and Technology from NUDT (2012), Master of Science degree in Aerospace Science and Technology from NUDT (2007), and a bachelor’s degree in Aerodynamic Engineering from NUDT (2005). He used to be a visiting Ph.D. student in Aerospace Engineering at the University of Sheffield (UK) from 2009 to 2010. His Ph.D. thesis entitled “Combustion Modes and Oscillation Mechanisms of Cavity-Stabilized Jet Combustion in Supersonic Flows” was rated as outstanding doctoral dissertation. He started his Hypersonic Propulsion Technology research career working as a Lecturer at NUDT from 2012. He conducted research in the

xiii

xiv

About the Authors

areas of scramjet combustor design, supersonic combustion, and computational fluid/combustion dynamics. He authored over 50 publications in journals and several patents. Dr. Feng Xiao is Associate Professor at National University of Defense Technology in China. He received his B.Eng. in Flight Vehicle Design and Engineering from Tsinghua University in China (2007) and Ph.D. in Aerospace Engineering from Loughborough University in UK (2012). Over the past 10 years, Dr. Feng Xiao has been working on numerical and experimental studies of atomization process in engines. He developed an incompressible two-phase flow code for simulations of atomization in gas turbines in his Ph.D. period. After joining National University of Defense Technology, he directed his attention to the atomization in scramjet engines and developed numerical methods for large eddy simulation of atomization in supersonic flows. He also carried out experimental measurements of atomization in supersonic flows using modern optical instruments such as high-speed photography and PIV.

Chapter 1

Introduction

1.1 Background Hypersonic aircraft are flight vehicles which operate at a Mach number higher than 5, including hypersonic cruise aircraft, hypersonic cruise missiles, and reusable orbital aerospace planes. Since these hypersonic vehicles have revolutionary applications in commercial aviation, national security, and space exploration, great effort has been dedicated to their development since the 1950s (Curran 2001). Turbojets and turbofans, typically used in airliners and fighter aircraft that travel at moderate speeds, cannot propel an aircraft to hypersonic speeds. Rocket engines have been the sole practical propulsion system for hypersonic aircraft over the last century. However, rockets must carry an oxidizer onboard, imposing severe constraints on flight range and payload capacity. Scramjet engines, illustrated in Fig. 1.1, have a significantly higher specific impulse than rocket engines as the air in the atmosphere is fed into scramjet engines and used as an oxidizer. Therefore, scramjet engines represent the optimal hypersonic propulsion system for use within the atmosphere and have attracted a huge amount of research. A detailed review of the development of scramjet engine technologies in the last century in the United States, Russia, France, Germany, Japan, Australia, and other countries is given by Curran (2001). Significant progress has been made in the development and testing of scramjet engines since 2000. NASA X-43A achieved a maximum speed of close to Mach 10 with a hydrogen-powered scramjet engine on November 16, 2004, making it the fastest free-flying air-breathing vehicle. On May 1, 2013, the Boeing X-51 Waverider, powered by Pratt & Whitney Rocketdyne’s hydrocarbon-fueled (JP-7) scramjet engine (Fig. 1.1), accelerated from Mach 4.8 to Mach 5.1 and flew for 210 s, making it the longest air-breathing hypersonic flight. A review of recent scramjet propulsion research programs in Western countries is presented in Urzay (2018). Since the flow speed is very high in the combustion chamber of scramjet engines, the fuel jet and the supersonic airflow must mix well in a very short time (the order of milliseconds) in order to achieve superior combustion performance. In a scramjet engine the fuel is typically injected into a supersonic airflow from nozzles in the © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_1

1

2

1 Introduction

Fig. 1.1 Schematic of a scramjet engine (https://www.nasa.gov/centers/langley/news/factsheets/ X43A_2006_5.html)

combustor wall, as shown in Fig. 1.2, resulting in jets in supersonic crossflows. A lot of studies have been carried out to elucidate the mixing mechanism of jets in a supersonic crossflow. Liquid hydrocarbon fuel (e.g., kerosene, JP-7) and gaseous hydrogen are commonly used in scramjet engines, the choice of which depends on the practical application of the engines. Because of the inherent ease of handling, long storage life, low toxicity, low cost, and high density, liquid hydrocarbon fuel is the optimal fuel for scramjet engines of hypersonic cruising aircraft and missiles, which fly at speeds of Mach 5–8. Since hydrogen-fueled scramjet engines can operate at a significantly higher Mach number (up to 12), hydrogen is the preferred fuel for scramjet engines powering hypersonic aerospace planes into space. Therefore, both gaseous and liquid jets in a supersonic crossflow are investigated in this book and are briefly described in Sects. 1.2 and 1.3, respectively.

1.2 Gaseous Jets in a Supersonic Crossflow Jets in a subsonic crossflow are common in both nature (e.g., plumes generated by volcanoes) and human-made devices (e.g., steering jets for missiles). They have been widely studied and typical flow structures are plotted in Fig. 1.3. For gaseous jets in a supersonic crossflow, however, the interaction between jets and crossflow is much more complex, as shown in Fig. 1.4. Significant aspects of gaseous jets in a supersonic crossflow are briefly described in the following subsections, involving flow structures, spatial distribution of jet fluids, mixing characteristics, mixing enhancement, and reaction.

1.2 Gaseous Jets in a Supersonic Crossflow

(a) HyShot II scramjet (Constantine et al. 2015)

(b) Large eddy simulations of the fuel jet combustion (Larsson et al. 2015) Fig. 1.2 Schematic diagram of typical fuel injection in a scramjet engine

3

4

1 Introduction

Fig. 1.3 a Schematic flow structures of jets in a subsonic crossflow (Fric and Roshko 1994). b Mixed-fluid concentration of subsonic jets in a crossflow (Shan and Dimotakis 2006)

1.2.1 Flow Structures and Jet Mixing The flowfield formed by jets in supersonic crossflow (JISC) is very complex. With the help of flow visualization techniques, various shocks and vortical structures have been identified (Ben-Yakar et al. 2006; Gruber et al. 1997, 2000; Mcdaniel and Raves 1988). More flowfield details, such as the evolution and interaction of different vortical structures, have been obtained using numerical simulations (Chai and Mahesh 2011; Chai et al. 2015; Génin and Menon 2010). The mixing process between transverse and supersonic crossflows is closely related to flow structures. (a) Flow structures of a gaseous jet in a supersonic crossflow A typical configuration of an under-expanded sonic jet in a supersonic crossflow from a wall is illustrated in Fig. 1.5a and b. As the sonic jet is injected into the supersonic flow from the wall, a strong bow shock forms upstream of the transverse jet. The boundary layer of a supersonic crossflow at the wall

1.2 Gaseous Jets in a Supersonic Crossflow

(a) Side view of flow structures at symmetry plane. (Ben-Yakar et al. 2006)

(b) Nanoparticle-based Planar Laser Scattering (NPLS)

(c) Direct numerical simulation from Chai et al. (2015) Fig. 1.4 Flow structures of gaseous jets in a supersonic crossflow

5

6

1 Introduction

experiences an adverse pressure gradient and separates ahead of the jet, forming horseshoe vortices which persist downstream. The under-expanded jet first expands through a Prandtl–Meyer fan at the top of the jet orifice and is then compressed by the barrel shock and the Mach disk. The jet plume continuously develops into a counter-rotating vortex (CRV) pair which dominates the flow structures in the far field. Lower trailing CRVs are generated on the wall surface due to the low-pressure recirculation zone in the jet leeside and the suction of the major CRV. Viti et al. (2009) demonstrated different pairs of trailing CRVs at a transverse section aft of the barrel shock, Fig. 1.5c. The kidney-shaped CRVs formed downstream of the jet plume are the major contributors to the mixing of the injectant with the crossflow. The lower trailing vortex remains attached to the solid surface as it entrains fluid from the surrounding boundary layer. The trailing upper vortex is weaker than the other vortices and is hence more difficult to identify. A recirculation zone and a separation bubble covered by compression waves are observed ahead of the jet. The sonic jet expands into the crossflow, forming a barrel shock at the jet periphery and a Mach disk normal to the jet. A large portion of jet fluid passes through the windward and lateral barrel shocks and over the Mach disk. However, the amount of fluid passing through the leeward barrel shock is much less. Kelvin–Helmholtz instabilities are generated around the jet plume due to the high level of shear in two typical shear layers (Chai et al. 2015): one is caused by the velocity difference for the jet fluid passing through the Mach disk and the windward barrel shock; the other is formed between the jet fluid and the crossflow. The vorticity on the windward, lateral, and leeward sides sequentially decreases. Génin and Menon (2010) applied large eddy simulation (LES) to investigate the dynamics of JISC and observed the unsteadiness of the barrel shock and Mach disk. The vortical structures and pressure fluctuations of the incoming boundary layer trigger unsteady compression waves that may penetrate the jet and then deform the windward barrel shock. The deformation causes intermittent injection of jet fluid, producing additional vortical structures by Kelvin–Helmholtz instabilities. Such vortical structures are further convected along jet boundaries and enhance the shear-layer vortices. Moreover, the unsteadiness of the barrel shock alters the Mach disk and then yields velocity fluctuations in the jet fluid through the Mach disk. This also contributes to the unsteadiness of the shear layer vortices. (b) Mixing of a gaseous jet in a supersonic crossflow The mixing between a gaseous jet and a supersonic crossflow progressively occurs after the jet leaves the region surrounded by the barrel shock and the Mach disk. The near-field and far-field mixing is dominated by different physical processes. In the near field, the crossflow is entrained by generated large-scale vortices, which further break down into much smaller vortical structures due to vortical stretching and tilting. During this process mixing happens as a result of turbulent convection and diffusion. In the far field, however, mixing is mainly caused by diffusion. Pivotal coherent vortices responsible for jet mixing are summarized below:

1.2 Gaseous Jets in a Supersonic Crossflow

(a) Schematic of three-dimensional flow structures proposed by Gruber et al. (1997)

(b) Schematic of three-dimensional flow structures proposed by Dickmann and Lu (2009)

(c) Two dimensional flow structures in a slice downstream of the barrel shock (Viti et al. 2009) Fig. 1.5 Schematics of flow structures of an under-expanded sonic jet in a supersonic crossflow

7

8

1 Introduction

(1) Shear layer vortices engulf the crossflow around the jet plume. The vorticity on the windward side is much stronger than on the leeward side, which is partially attributed to the large-scale dynamics of the barrel shock and the Mach disk. (2) Jet fluid is entrained into the upstream boundary-layer separation bubble intermittently, as a result of the dynamic barrel shock. However, jet entrainment in the downstream separation bubble happens continuously. (3) Mixing under the jet plume is caused by the leeward shear layer vortices, the CRVs, and the separated boundary layer. According to planar laser-induced fluorescence (PLIF) measurements, the best instantaneous mixing in the near-field region is observed in the center of the wake region, slightly below the jet center line (VanLerberghe et al. 2000). Although a large amount of crossflow fluid is entrained into the windward shear layer vortices, large regions of unmixed fluid exist there. (4) The turbulent boundary layer enhances jet mixing. The upstream vortices in the turbulent boundary layer can interact with the bow shock and the windward jet boundary, enhancing shear layer instability and vortex breakdown (Kawai and Lele 2010).

1.2.2 Spatial Distribution of a Gaseous Jet in a Supersonic Crossflow Since fuel–air mixing can significantly affect combustion stability and efficiency in scramjets, the spatial distribution of a gaseous jet in a supersonic crossflow must be well elucidated. In the Cartesian coordinate system for describing the spatial distribution of a jet, the origin is typically located at the center of the jet exit, with the x-axis in the crossflow direction and the y-axis in the direction of the transverse jet. Then the jet distribution in the y and z directions are termed as the transverse distribution and spanwise distribution, respectively. Previous work has mainly focused on jet penetration which refers to the upper boundary of transverse distribution. Various flow visualization techniques, such as planar Rayleigh scattering, schlieren imaging, and PLIF, have been used to investigate the controlling parameters of jet penetration (Gruber et al. 2000; Ben-Yakar et al. 2006; Mcdaniel and Raves 1988). Using different techniques yields different metrics for assessing jet penetration. For example, Ben-Yakar et al. (2006) selected the visible upper edge of the jet in schlieren images to represent jet penetration while Mcdaniel and Raves (1988) defined jet penetration to be at locations that have 1% jet concentration in the crossflow direction. The momentum flux ratio J is the main controlling parameter of jet penetration. The expression for J is: J≡

ρ j U 2j ρ∞ U 2j



γ j p j M 2j 2 γ∞ p ∞ M ∞

,

(1.1)

1.2 Gaseous Jets in a Supersonic Crossflow

9

where ρ j and ρ∞ , respectively, represent the jet and crossflow densities; U j and U∞ are the corresponding velocities; γ , p, and M are the specific heat ratio, pressure, and Mach number, respectively; and p j denotes the near-field pressure after the bow shock, obtained by the Rankine–Hugoniot equations. Empirical correlations for jet penetration have been proposed (Mcdaniel and Raves 1988; Rothstein and Wantuck 1992; Gruber et al. 1997). These correlations usually include a power-law or a logarithmic fit. Even correlations with close functional forms can have constants with huge differences in value. This may be due to the diversity of the experimental setups and the differences in the definitions of jet penetration. Other parameters that can affect jet penetration have also been examined. For example, a jet with a higher molecular weight was found to have higher penetration (Gruber et al. 1997; Ben-Yakar et al. 2006). Ben-Yakar et al. (2006) attributed higher penetration to jet exit velocity. For two jets with the same momentum flux ratio, the jet with the higher molecular weight has the lower exit velocity. The eddies along the jet shear layers are then exposed to steeper velocity gradients, which speed up the breakdown of vortical structures and then increase jet penetration. Portz and Segal (2006) proposed a correlation which considered the momentum flux ratio, the molecular weight, the boundary layer thickness, and the crossflow Mach number. In this correlation, the impact of molecular weight was consistent with the observations of Ben-Yakar et al. (2006). The aforementioned correlations are all based on experiments, whereas Billig and Schetz (1966) proposed a differential equation to predict jet penetration. The equation was established by treating the jet plume as a solid body and then utilizing force balance analysis on jet segments. Provided with the density behind the barrel shock, Chai et al. (2015) found that the equation could accurately yield jet penetration in the near field.

1.2.3 Mixing Enhancement Transverse injection and axial injection are two common injection schemes in scramjets. In comparison with axial injection, transverse injection demonstrates deeper penetration and better mixing performance, but at the cost of a higher pressure loss. In order to increase combustion efficiency and reduce combustion chamber size, mixing between the fuel jet and the air crossflow should be maximized, which motivates studies of passive and active mixing enhancement techniques. (a) Passive mixing enhancement The penetration and mixing characteristics of a gaseous jet exiting from circular and elliptical injectors were compared by Gruber et al. (2000). The gaseous jet from an elliptical injector penetrates less but spreads more widely in the spanwise direction than from a circular injector, as shown in Fig. 1.6. This conclusion was validated in the numerical investigation of Wang et al. (2013).

10

1 Introduction

Fig. 1.6 Instantaneous density distribution of jet fluid injected from circular (top) and elliptical (bottom) injectors (Gruber et al. 2000)

Fig. 1.7 Flow structures of a dual-injection scheme (Lee 2006)

The simulation further showed that jet shear layer vortices are relatively small in cases with elliptical injectors. Lee (2006) studied numerically a dual injection scheme with two injectors arranged in a crossflow direction. As illustrated in Fig. 1.7, the forward jet plume blocks the crossflow, forming a low-pressure region downstream which increases the local jet-to-crossflow momentum flux ratio of the rear jet. When the distance between the two injectors is optimized, this scheme could significantly improve mixing efficiency, with an acceptable pressure loss. Takahashi et al. (2010) also investigated the mixing characteristics of a dual-injection scheme via PLIF. However, in their experiment the scheme failed to increase mixing efficiency significantly, which contradicted the conclusion of Lee (2006). This contradiction may have resulted from differences in their configurations. The distance between the two injectors may not have been optimized in Takahashi et al. (2010).

1.2 Gaseous Jets in a Supersonic Crossflow

11

Influences of various vortex generators on mixing have been explored. Wang et al. (2012a, b) visualized the vortical structures produced when a supersonic crossflow passes over a hemisphere and a finite cylinder, via NPLS. Kim et al. (2012) utilized schlieren and stereoscopic particle image velocimetry to investigate the interaction between a hypermixer and a transverse jet. Vinogradov et al. (2007) presented a comprehensive review of the applications of thin pylons in mixing enhancement. A recirculation zone formed downstream of a pylon, which can prevent the jet in the recirculation zone being directly impacted by the high-momentum crossflow. Furthermore, the large-scale turbulence in the recirculation zone may enhance jet mixing. Shock waves are found to affect the mixing process. When mixing layers encounter shock waves, a nonalignment of the density and the pressure gradients creates vorticity through baroclinic torque, and the vorticity stretches the mixing layers and improves mixing (Schetz et al. 2010). Mixing enhancement due to shock waves was also observed in other studies (Erdem et al. 2012; Huang et al. 2013, 2015). (b) Active mixing enhancement In order to achieve a superior mixing performance for jets in a supersonic crossflow, active mixing enhancement techniques have been investigated. Mechanical devices have been used to generate pulsed injection at different frequencies in order to examine the effects of jet pulsation on jet penetration and mixing characteristics (Randolph et al. 1994; Murugappan et al. 2005; Dziuba and Rossmann 2006; Cutler et al. 2013). The Reynolds-averaged Navier–Stokes (RANS) simulation by Kouchi et al. (2010) showed that there is an optimal pulsation frequency that maximizes jet penetration. Williams (2016) applied a three-dimensional wall-modeled large-eddy simulation to investigate a sinusoidally pulsed jet (16 kHz) in a supersonic crossflow.

1.2.4 Reactive Jets After the compression in the inlet section of a practical scramjet engine, the airflow at the combustor entrance has a high temperature. When the fuel is injected into the high-enthalpy airflow a reaction occurs, as shown in Fig. 1.8. Therefore, further understanding of the reactive jets in a supersonic crossflow is required. Gamba and Mungal (2015) investigated the flame structures of reactive JISC with J values of 0.3–5.0, where the hydrogen flame structures were visualized by OH planar laser-induced fluorescence (OH–PLIF). Through acquiring and analyzing OH–PLIF images at different horizontal and vertical planes (Fig. 1.9), they identified five pathways, shown in Fig. 1.10, that correspond to different fuel–air mixing patterns and combustion regimes. These pathways are consistent with the flow structures and mixing characteristics discussed in the previous sections. Micka and Driscoll (2012) reported simultaneous CH2 O/OH–PLIF imaging of stratified jet flames in a 1390-K air crossflow where J ~ 2 and investigated combustion regimes in different regions of the flowfield.

12

1 Introduction

Fig. 1.8 Simultaneous schlieren and OH–PLIF imaging of a reactive hydrogen jet in a supersonic crossflow (Ben-Yakar and Hanson 1999)

1.3 A Liquid Jet in a Supersonic Crossflow The atomization of a liquid fuel jet in an engine can increase the liquid’s surface-tovolume ratio which facilitates fast evaporation and mixing between the fuel and gas. In the combustion chamber of a scramjet, the atomization and spray characteristics of a liquid jet in a supersonic crossflow (LJSC) have significant impacts on the combustion efficiency and thrust performance of the scramjet (Nakaya et al. 2015; Tahsini and Mousavi 2015). Numerous studies of liquid jets in supersonic crossflows have been carried out since the 1960s (Kolpin et al. 1968).

1.3.1 Atomization Process of a Liquid Jet in a Supersonic Crossflow As the liquid fuel is injected into a supersonic crossflow from the wall, the liquid jet bends in the crossflow direction and disintegrates to form tiny droplets under the strong aerodynamic forces of the high-speed airflow, as shown in Fig. 1.11a. The

1.3 A Liquid Jet in a Supersonic Crossflow

13

Fig. 1.9 OH–PLIF imaging when J  5 at a y/d  0.25; b y/d  0.5; c y/d  1.0; and d y/d  3.0 (Gamba and Mungal 2015)

Fig. 1.10 Main fuel entrainment pathways in reactive JISC (Gamba and Mungal 2015)

14

1 Introduction

Fig. 1.11 a Schematic of liquid jet injection into a supersonic crossflow; b interaction of the shock and liquid jet, captured by high-speed schlieren method

strong interaction between the shocks and liquid jet during the atomization process are observed in the schlieren image shown in Fig. 1.11b.

1.3 A Liquid Jet in a Supersonic Crossflow

15

The atomization process of an LJSC is typically divided into two stages: the primary breakup and the secondary breakup. In the primary breakup process, the liquid jet exiting from the nozzle first deforms and then the liquid ligaments and droplets peel off from the liquid column from the periphery side, which is referred to as surface breakup. In the meantime, surface waves develop on the upstream side of the liquid column due to Rayleigh–Taylor instability. As the magnitude of the surface waves increases, the liquid column breaks up into large liquid clusters as a whole, referred to as column breakup. In the secondary breakup process, the large liquid structures (i.e., liquid sheets, liquid blocks, and liquid drops), resulting from the primary breakup, disintegrate into ever smaller droplets under the action of aerodynamic forces. In the farther downstream spray region the droplets disperse in the supersonic airflow.

1.3.2 Approaches to Liquid Jet Atomization and Spray Research Optical measurement technologies have been widely used in the studies of liquid jet atomization and spray as they can obtain characteristic parameters without affecting the flowfield. Based on the principle of scattering and interference, several measurement methods have been developed, such as phase doppler anemometer (PDA), laserscattering technology/laser holography, planar laser-induced fluorescence (PLIF), and particle image velocimetry (PIV). The PDA method has been extensively used in the atomization field. Koh et al. (2006) compared differences between optical imaging methods and the PDA method in liquid mass flow rate distribution, and found that the mass flow rate distribution obtained by PDA was much smaller than that obtained by optical imaging. Kim and Kim (2009) used a dual-mode PDA to measure droplet information including velocity, diameter, number density, and turbulence. Xie et al. (2013) used PDA to experimentally study the atomization characteristics of the pressure injector, including flow structure and Sauter mean diameter (SMD). It was found that within the experimental observation range, the diameter of the droplets decreased first along the direction of the injection axis and then suddenly increased. To study the liquid jet structure, Lee et al. (2010a) used PDA to measure the thin boundary of the transverse jet and obtained liquid droplet dynamic information including diameter, velocity, and number density; the penetration height and the cross-sectional distribution has been discussed. Lin et al. (2000, 2002) used PDA to study jet penetration height, and several correlations for liquid jet penetration height were developed using the method of least squares. Lin et al. (2002) also compared the effects of using different optical methods for penetration height measurements, pointing out that the penetration height obtained by PDA was generally larger than other optical measurement methods, such as high-speed photography and the shadow method, because PDA was more sensitive to a thin spray.

16

1 Introduction

Wu and Kirkendall (1998) conducted experimental studies of liquid jets from a 0.5-mm round-hole injector at subsonic conditions. PDA was used to measure droplet diameter, velocity, and volume flux. They found that large droplets existed at the center of the plume flow at a low liquid–gas momentum ratio and large droplets were mainly distributed around the plume flow at a high liquid–gas momentum ratio. The article also measured the height information of the maximum volume flux, that indicated the location of the concentrated distribution of droplets. Rachner et al. (2002) experimentally studied the penetration height and atomization characteristics of a kerosene jet in a subsonic crossflow. They attempted to establish a comprehensive model of jet penetration height and atomization. In the experiment, PDA was used to obtain droplet velocity and diameter information, and it was found that the SMD decreased in the vertical direction under most working conditions, but in a few cases the SMD decreased first and then increased in the vertical direction. The PIV method is also effective at measuring droplet velocity. It is a technique for adding tracer particles to a flowfield to capture flowfield velocity by capturing time-dependent two-frame flow images. Early PIV technology used tracer particles, several micrometers in size, and was widely used in the display and velocity measurement of low-speed flowfields (Elshamy et al. 2007). Moraitis and Riethmuller (1988) successfully applied PIV technology to the measurement of compressible flow in 1988. After several decades of development, PIV technology has made great progress both in the use of tracer particles and in the optimization and improvement of algorithms. At present, among the many speed field measurement techniques, PIV is widely used because it offers non-contact, transient, and full-field measurements (Westerweel et al. 2013). Based on the PIV technique, droplets of atomized jet were used as tracer particles to study the jet/spray transient structure and spatiotemporal development characteristics of a liquid transverse jet in a supersonic flow; a crosscorrelation algorithm was used to calculate spray-field velocity. Digital holographic microscopy was used by Lee et al. (2015) and Olinger et al. (2014) to study the rapidly developing region of several sprays near an injector in a subsonic crossflow. Digital holographic microscopy is capable of measuring drop size distribution and three-dimensional velocities in the near-injector region where diagnostics, like Phase Doppler Particle Analyzer (PDPA), are not successful in probing non-spherical droplets. Planar laser-induced fluorescence was applied by Fei et al. (2008) to visualize the atomization phenomena of a kerosene jet in a supersonic cold flow. It was demonstrated that PLIF can accurately detect the distribution of kerosene droplets. Experimental studies of LJSC are very challenging. Since the atomization process is extremely rapid and the droplets arising from liquid jet atomization are very tiny, high-spatial and high-temporal resolution is required for optical techniques. A pulsed laser background imaging method has been developed in the Science and Technology on Scramjet Laboratory (Changsha, China) to obtain a high temporal resolution of the liquid jet disintegration process in a supersonic crossflow. This method accurately visualized and extracted unstable fluctuation surface waves and the phenomenon of stripping droplets, detailed in Chap. 6.

1.3 A Liquid Jet in a Supersonic Crossflow

17

The mist resulting from atomization blocks the liquid core, making it difficult to investigate the breakup mechanism of the liquid jet core using optical approaches. As interface-tracking methods and two-phase flow modeling has achieved a lot of progress, computational fluid dynamics (CFD) can be used to further the understanding of liquid jet atomization in a supersonic crossflow. Xiao et al. (2016) have developed a two-phase flow solver to simulate liquid jet primary breakup in a supersonic crossflow using a coupled level-set and volume-of-fluid (VOF) method. Since the simulation of droplets in the spray region using interface-tracking methods is prohibitively expensive and time consuming, Lagrangian particle-tracking methods have been widely used in order to reduce computation cost. To describe the secondary breakup progress, many breakup models have been proposed in recent decades. The most widely and successfully used models for spray atomization are the TAB model (O’Rourke and Amsden 1987), the wave breakup model (Deneys and Diwakar 1987), and various modified or hybrid models based on the abovementioned two (Patterson and Reitz 1998; Im et al. 2011). Liu et al. (1993) modified the droplet drag coefficient, that uses the distortion of droplets estimated by the TAB model, and calculated the breakup of droplets using a wave model. Numerical results confirmed that spray-tip penetration is relatively insensitive to droplet breakup and drag models. Patterson and Reitz (1998) firstly suggested that droplet breakup is the result of competition between Kelvin-Helmholtz (KH) instability and Rayleigh-Taylor (RT) instability, and successfully used the KH–RT hybrid model to simulate a liquid jet in diesel. Im et al. (2011) modified the KH–RT hybrid breakup model by considering compressible effects and simulated a liquid jet in a supersonic crossflow. Reasonable jet penetration height and droplet size distribution were given, but droplet velocity distribution was not. Despite a certain gap between the results of numerical simulations and experiments, these works suggest that atomization models, stemmed from low-speed conditions, can be applied to supersonic conditions after reasonable modification.

1.3.3 Spatial Distribution of a Liquid Jet in a Supersonic Crossflow The spatial distribution of a spray characterizes the dispersion and mixing of the liquid mist in a supersonic airflow and thus significantly affects combustion performance. The spatial distribution characteristics of liquid jets include penetration height and spanwise distribution. Penetration height is defined as the farthest distance the liquid jet/spray is away from the wall in the transverse direction. Liquid jet penetration height in a supersonic crossflow has been widely studied by theoretical analyses, experimental measurements, and numerical simulations. It has been shown that the parameters affecting penetration height include liquid–gas momentum flux ratio (q), distance from jet orifice (x), jet orifice diameter (d), Mach number (Ma), Weber number (We), and

18

1 Introduction

jet angle (θ ), among which the key influencing factors were q, x, and d (Almeida et al. 2014; Sun et al. 2013; Kourmatzis and Masri 2014; Yoon et al. 2011; Mashayek et al. 2011; Im et al. 2011). Power-law function, exponential function, and logarithm function (Becker and Hassa 2002; Lin et al. 2002; Yang et al. 2012) are three typical forms of empirical correlations for jet penetration. However, different optical measurement techniques and image-processing methods lead to significant differences in the proposed empirical formulas. Therefore, a thorough study is required to further the understanding of liquid jet penetration in a supersonic crossflow. Spanwise distribution of a liquid jet is another important parameter in describing spatial distribution in an LJSC. The main factors that influence spanwise distribution include the geometric size of the nozzle, the angle of injection, and the pressure drop resulting from injection. A detailed analysis of the liquid jet spanwise distribution is presented in Chap. 7.

1.3.4 Atomization Characteristics of a Liquid Jet in a Supersonic Crossflow The size and velocity of the droplets and the gas–liquid mixture ratio are important atomization characteristics of an LJSC, which can be used to evaluate fuel/air mixing performance. The droplet size distribution of the liquid jet spray in a supersonic crossflow has been extensively investigated by experiments, simulations, and theoretical analysis. Droplet size distribution and variation have been measured in experiments and have been used to develop and validate theoretical analyses and numerical models. The simulations of an LJSC show that droplet size distribution can be approximately predicted by the K-H&R-T hybrid secondary breakup model under certain conditions. However, it needs further improvement to match experimental results well. On the theoretical aspect, there is no single distribution function to fit most of the experimental results at present, and more work needs to be done to develop a superior theoretical model that can describe the spray more comprehensively. The velocity of droplets in a spray is closely related to droplet size. Studying variations in droplet velocity is beneficial to the study of secondary atomization mechanisms. Droplet velocity can be obtained effectively using PDA measurements. Lin et al. (2004) studied characteristics of liquid water and alcohol in a supersonic crossflow. The air was mixed into the injector. The penetration height of the jet was imaged by shadowgraph method, and the droplet velocity and diameter were measured by PDA. The results indicated that in the homogeneous mixing zone, the SMD of a pure liquid jet presented an “S” shape along the vertical direction, as shown in Fig. 1.12. The spray penetration height obtained by PDA was higher than that obtained by shadowgraph, as shown in Fig. 1.13. When the gas content reached 4%, the SMD monotonously increased along the vertical direction. As the gas content increased, SMD decreased significantly. Lee et al. (2010b) conducted a liquid pulse injection test with an injection

1.3 A Liquid Jet in a Supersonic Crossflow

19

Fig. 1.12 Normalized center line distribution profiles for the SMD of pure liquid jets in an M  1.94 crossflow (Lin et al. 2004)

frequency of 35.7–166.2 Hz. A PDA was used to obtain droplet velocity and diameter at different cross sections. It was found that the boundary oscillation frequency stayed consistent with injection frequency. With an increasing frequency the spray distribution area increased. One point to mention here is that the accuracy of droplet velocity measured by PDA under supersonic conditions needs further verification.

1.4 Outline of the Book The book focuses on experimental and numerical investigations of jets in a supersonic crossflow and consists of two parts. Part 1 is comprised of Chaps. 2, 3, 4 and 5, presenting studies of mixing and combustion of gaseous jets in a supersonic crossflow. Part 2 is comprised of Chaps. 6 and 7, detailing research on the atomization and spray of liquid jets in a supersonic crossflow. Chapter 2 presents research on the penetration and diffusion of a gaseous jet in a supersonic crossflow carried out by our research group. Empirical correlations of jet penetration for engineering applications are proposed. An empirical mixing model for the analysis of a gaseous jet in a supersonic crossflow is investigated by modeling the jet center line, jet penetration boundary, and jet spanwise distribution

20

1 Introduction

Fig. 1.13 Comparison of spray penetration heights with cross-sectional distributions of liquid volume flux, droplet number density, and droplet SMD, from PDA measurements for a pure liquid jet in an M  1.94 crossflow environment (Lin et al. 2004)

characteristics. Several experimental cases are used to verify the performance of developed models. Chapter 3 presents studies of gaseous jet structures in a supersonic crossflow using advanced experimental and numerical techniques. This chapter consists of three sections, including shock structures, upper trailing counter-rotating vortices (CRVs), and lower trailing CRVs. The technique of NPLS, which has recently been developed in our laboratory, is used to obtain clear visualization of the flow structures of a gaseous jet in a supersonic crossflow. The flow structures, including a bow shock, a barrel shock, horseshoe vortex, and separation zones, are clearly observed using the NPLS technique. Direct numerical simulations of gaseous jets in a supersonic crossflow are run to elucidate the physical mechanism of upper trailing CRVs and lower trailing CRVs. Upper trailing CRVs form above the major CRVs downstream of the jet barrel shock. Streamline analysis indicates that upper trailing CRVs are related to the Mach disk. The recirculation flow in the jet leeward separation bubble forms a primary TCVP (trailing counter-rotating vortex) close to the wall. Chapter 4 presents the mixing characteristics of a gaseous jet in a supersonic crossflow. The mixing characteristics are investigated by advanced experimental techniques including NPLS and PLIF and numerical techniques, like the hybrid RANS/LES (Reynolds-averaged Navier–Stokes/large-eddy simulation) method. Then the flow patterns and mixing characteristics of multiple injections are investigated by a nanoparticle-based laser scattering imaging technique, schielien system, and surface oil-flow visualization technique. The injection schemes of tandem multi-jets and parallel multi-jets with various parameters, including injection stagnation pressure and distance between orifices, were tested and analyzed. More-

1.4 Outline of the Book

21

over, the effect of a micro-ramp vortex generator on mixing enhancement is examined by LES. Chapter 5 presents the turbulent combustion characteristics of a gaseous jet in a supersonic crossflow. This chapter focuses on the auto-ignition mechanism and turbulence/chemistry interactions. Optical observations, such as schlieren and OH–PLIF, give a macroscopic insight into the typical flow structures and reaction zone. On this basis, a hybrid RANS/LES method coupled with assumed PDF model is adopted to reproduce the three-dimensional unsteady reacting flow in a model scramjet combustor. In a high-flight Mach condition, the total enthalpy of the freestream is so high that auto-ignition may occur during the process of jet mixing with crossflow. Essentially, combustion is a mixing-limited process and thus the flame in the jet shear layer needs to be analyzed. More attention is paid to the ignition delay, the timescale of flow, and chemistry. However, the jet flame cannot be stabilized in a moderate enthalpy flow without a cavity. Therefore, a thorough understanding of the turbulent combustion mechanism is revealed. Chapter 6 presents our work on the primary breakup of a liquid jet in a supersonic crossflow. We first describe the pulsed laser background imaging method used to capture the instantaneous primary breakup morphology of a liquid jet in a supersonic crossflow. Then numerical techniques for the primary breakup in a supersonic crossflow are detailed. The interface tracking methods are evaluated. An interface tracking–based LES method for the simulation of a liquid jet primary breakup in a supersonic flow is described. The supersonic gas flow is solved using a compressible flow solver, and the liquid phase is solved using an incompressible flow solver. Appropriate boundary conditions are specified on the interface for both solvers to correctly capture the interaction between gas and liquid. The primary breakup of a liquid jet in a supersonic crossflow is simulated, and the physical mechanism of liquid jet primary atomization is examined. Chapter 7 presents studies of the spray characteristics of a liquid jet in a supersonic crossflow. The droplet size distribution and velocity distribution in the spray field after atomization are measured using the PDA and PIV techniques. The liquid jet penetration, spray velocity distribution, and cavity effects on spray characteristics are experimentally investigated. The gaseous flow structures and spray structures are obtained by high-speed schlieren and PIV methods. The liquid jet atomization and spray process of a liquid jet in a supersonic crossflow is numerically investigated using the Eulerian–Lagrangian method.

References Almeida H, Sousa JMM, Costa M (2014) Effect of the liquid injection angle on the atomization of liquid jets in subsonic crossflows. At Sprays 24:81–96. https://doi.org/10.1615/AtomizSpr. 2013008310 Becker J, Hassa C (2002) Breakup and atomization of a kerosene jet in crossflow at elevated pressure. At Sprays 12:49–68. https://doi.org/10.1615/AtomizSpr.v12.i123.30

22

1 Introduction

Ben-Yakar A, Hanson R (1999) Hypervelocity combustion studies using simultaneous OH-PLIF and Schlieren imaging in an expansion tube. In: 35th joint propulsion conference and exhibit. American Institute of Aeronautics and Astronautics, Reston, Virginia, p 47 Ben-Yakar A, Mungal MG, Hanson RK (2006) Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Phys Fluids 18:26101. https://doi.org/ 10.1063/1.2139684 Billig FS, Schetz JA (1966) Penetration of gaseous jets injected into a supersonic stream. J Spacecr Rockets 3:1658–1665. https://doi.org/10.2514/3.28721 Chai X, Iyer PS, Mahesh K (2015) Numerical study of high speed jets in crossflow. J Fluid Mech 785:152–188. https://doi.org/10.1017/jfm.2015.612 Chai X, Mahesh K (2011) Simulations of high speed turbulent jets in crossflows. In: 49th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition 2011 Constantine PG, Emory M, Larsson J, Iaccarino G (2015) Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet. J Comput Phys 302:1–20 Curran ET (2001) Scramjet engines: the first forty years. J Propul Power 17:1138–1148. https:// doi.org/10.2514/2.5875 Cutler AD, Harding GC, Diskin GS (2013) High frequency pulsed injection into a supersonic duct flow. AIAA J 51:809–818. https://doi.org/10.2514/1.J051620 Deneys R, Diwakar R (1987) Structure of high-pressure fuel sprays. SAE Paper 870598 Dickmann DA, Lu FK (2009) Shock/boundary-layer interaction effects on transverse jets in crossflow over a flat plate. J Spacecr Rockets 46:1132–1141. https://doi.org/10.2514/1.39297 Dziuba M, Rossmann T (2006) Super sonic mixing enhancement by means of high powered jet modulation. In: 44th AIAA aerospace sciences meeting and exhibit Elshamy OM, Tambe SB, Cai J, Jeng SM (2007) PIV and LDV measurements for liquid jets in crossflow. Paper presented at the AIAA 2007-1338, Nevada Erdem E, Saravanan S, Lin J, Kontis K (2012) Experimental investigation of transverse injection flowfield at March 5 and the influence of impinging shock wave. In: 18th AIAA/3AF international space planes, p 205 Fei LS, Xu SL, Wang CJ, Qiang L, Huang SH (2008) Experimental study on atomization phenomena of kerosene in supersonic cold flow. Sci China 51(2):145–152 Fric TF, Roshko A (1994) Vortical structure in the wake of a transverse jet. J Fluid Mech 279:1. https://doi.org/10.1017/S0022112094003800 Gamba M, Mungal MG (2015) Ignition, flame structure and near-wall burning in transverse hydrogen jets in supersonic crossflow. J Fluid Mech 780:226–273. https://doi.org/10.1017/jfm.2015. 454 Génin F, Menon S (2010) Dynamics of sonic jet injection into supersonic crossflow. J Turbul 11:N4. https://doi.org/10.1080/14685240903217813 Gruber MR, Nejad AS, Chen TH, Dutton JC (1997) Compressibility effects in supersonic transverse injection flowfields. Phys Fluids 9:1448–1461. https://doi.org/10.1063/1.869257 Gruber MR, Nejad AS, Chen TH, Dutton JC (2000) Transverse injection from circular and elliptic nozzles into a supersonic crossflow. J Propul Power 16:449–457. https://doi.org/10.2514/2.5609 Huang W, J-g Tan, Liu J, Yan L (2015) Mixing augmentation induced by the interaction between the oblique shock wave and a sonic hydrogen jet in supersonic flows. Acta Astronaut 117:142–152. https://doi.org/10.1016/j.actaastro.2015.08.004 Huang W, Wang Z-g Wu, J-p Li S-b (2013) Numerical prediction on the interaction between the incident shock wave and the transverse slot injection in supersonic flows. Aerosp Sci Technol 28:91–99. https://doi.org/10.1016/j.ast.2012.10.007 Im K-S, Lin K-C, Lai M-C, Chon MS (2011) Breakup modeling of a liquid jet in cross flow. Int J Automot Technol 12:489–496. https://doi.org/10.1007/s12239-011-0057-1 Kawai S, Lele SK (2010) Large-eddy simulation of jet mixing in supersonic crossflows. AIAA J 48:2063–2083. https://doi.org/10.2514/1.J050282

References

23

Kim C-H, Jeung I-S, Choi B, Kouchi T, Masuya G (2012) Flowfield characteristics of a hypermixer interacting with transverse injection in supersonic flow. AIAA J 50:1742–1753. https://doi.org/ 10.2514/1.J051588 Kim JS, Kim JS (2009) A characterization of the spray evolution by dual-mode phase doppler anemometry in an injector of liquid-propellant thruster. J Mech Sci Technol 23(6):1637–1649. Koh H, Jung K, Yoon Y, Lee K, Jeong KS (2006) Development of quantitative measurement of fuel mass distribution using planar imaging technique. J Vis 9(2):161–170 Kolpin MA, Horn KP, Reichenbach RE (1968) Study of penetration of a liquid injectant into a supersonic flow. AIAA J 6(5):853–858 Kouchi T, Sasaya K, Watanabe J, Shibayama H, Masuya G (2010) Penetration characteristics of pulsed injection into supersonic crossflow. In: 46th AIAA/ASME/SAE/ASEE joint propulsion conference & exhibit. American Institute of Aeronautics and Astronautics, Reston, VA, p 697 Kourmatzis A, Masri AR (2014) The influence of gas phase velocity fluctuations on primary atomization and droplet deformation. Exp Fluids 55:245. https://doi.org/10.1007/s00348-013-16593 Larsson J, Laurence S, Bermejo-Moreno I, Bodart J, Karl S, Vicquelin R (2015) Incipient thermal choking and stable shock-train formation in the heat-release region of a scramjet combustor. Part II: Large eddy simulations. Combust Flame 162(4):907–920 Lee I, Kang Y, Koo J (2010a) Mixing characteristics of pulsed air-assist liquid jet into an internal subsonic crossflow. J Therm Sci 19(2):136–140 Lee IC, Kang YS, Moon HJ, Jang SP, Kim JK, Koo J (2010b) Spray jet penetration and distribution of modulated liquid jets in subsonic crossflows. J Mech Sci Technol 24(7):1425–1431. https:// doi.org/10.1007/s12206-010-0418-0 Lee J, Sallam KA, Kin KC (2015) Spray structure in near-injector region of aerated jet in subsonic crossflow. J Propuls Power 25(2):11 Lee S-H (2006) Characteristics of dual transverse injection in scramjet combustor, Part 1: Mixing. J Propul Power 22:1012–1019. https://doi.org/10.2514/1.14180 Lin KC, Kennedy PJ, Jackson TA (2000) Spray penetration heights of angled-injected aerated-liquid jets in supersonic crossflows. In: 38th aerospace sciences meeting & exhibit, 1–13 Lin KC, Kennedy P, Jackson T (2004) Structures of water jets in a Mach 1.94 supersonic crossflow. In: AIAA Aerospace sciences meeting and exhibit, 2004 Lin K-C, Kennedy P, Jackson T (2002) Penetration heights of liquid jets in high-speed crossflows. In: 40th AIAA aerospace sciences meeting & exhibit. American Institute of Aeronautics and Astronautics, Reston, VA, p 2002 Liu AB, Mather D, Reitz RD (1993) Modeling the effects of drop drag and breakup on fuel sprays. Sae Paper 93 Mashayek A, Behzad M, Ashgriz N (2011) Multiple injector model for primary breakup of a liquid jet in crossflow. AIAA J 49:2407–2420. https://doi.org/10.2514/1.J050623 Mcdaniel JC, Raves J (1988) Laser-induced-fluorescence visualization of transverse gaseous injection in a nonreacting supersonic combustor. J Propul Power 4:591–597. https://doi.org/10.2514/ 3.23105 Micka DJ, Driscoll JF (2012) Stratified jet flames in a heated (1390 K) air cross-flow with autoignition. Combust Flame 159:1205–1214. https://doi.org/10.1016/j.combustflame.2011.10.013 Moraitis C, Riethmuller M (1988) Particle image displacement velocimetry applied in high speed flows. Paper presented at the 4th international symposium on applications of laser anemometry to fluid dynamic Murugappan S, Gutmark E, Carter C (2005) Control of penetration and mixing of an excited supersonic jet into a supersonic cross stream. Phys Fluids 17:106101. https://doi.org/10.1063/1. 2099027 Nakaya S, Hikichi Y, Nakazawa Y, Sakaki K, Choi M, Tsue M, Kono M, Tomioka S (2015) Ignition and supersonic combustion behavior of liquid ethanol in a scramjet model combustor with cavity flame holder. Proc Combust Inst 35(2):2091–2099

24

1 Introduction

Olinger DS, Sallam KA, Lin K-C (2014) Digital holographic analysis of the near field of aeratedliquid jets in crossflow. J Propul Power 30(6):1636–1645 O’Rourke PJ, Amsden AA (1987) The tab method for numerical calculation of spray droplet breakup. In: SAE Technical Paper Series. SAE international 400 commonwealth drive, Warrendale, PA, United States Patterson MA, Reitz RD (1998) Modeling the effects of fuel spray characteristics on diesel engine combustion and emission. In: SAE Technical Paper Series. SAE international 400 commonwealth drive, Warrendale, PA, United States Portz R, Segal C (2006) Penetration of gaseous jets in supersonic flows. AIAA J 44:2426–2429. https://doi.org/10.2514/1.23541 Rachner M, Becker J, Hassa C (2002) Modelling of the atomization of a plain liquid fuel jet in crossflow at gas turbine conditions. Aerosp Sci Technol 6(7):495–506 Randolph H, Chew L, Johari H (1994) Pulsed jets in supersonic crossflow. J Propul Power 10:746–748. https://doi.org/10.2514/3.23790 Rothstein A, Wantuck P (1992) A study of the normal injection of hydrogen into a heated supersonicflow using planar laser-induced fluorescence. In: 28th joint propulsion conference 1992, p 591 Shan JW, Dimotakis PE (2006) Reynolds-number effects and anisotropy in transverse-jet mixing. J Fluid Mech 566:47–96 Schetz JA, Maddalena L, Burger SK (2010) Molecular weight and shock-wave effects on transverse injection in supersonic flow. J Propul Power 26:1102–1113. https://doi.org/10.2514/1.49355 Sun M-B, Zhang S, Zhao Y, Zhao Y, Liang J (2013) Experimental investigation on transverse jet penetration into a supersonic turbulent crossflow. Sci China Technol Sci 56:1989–1998. https:// doi.org/10.1007/s11431-013-5265-7 Tahsini AM, Mousavi ST (2015) Investigating the supersonic combustion efficiency for the jet-incross-flow. Int J Hydrog Energ 40:3091–3097. https://doi.org/10.1016/j.ijhydene.2014.12.124 Takahashi H, Ikegami S, Masuya G, Hirota M (2010) Extended quantitative fluorescence imaging for multicomponent and staged injection into supersonic crossflows. J Propul Power 26:798–807. https://doi.org/10.2514/1.47318 Urzay J (2018) Supersonic combustion in air-breathing propulsion systems for hypersonic flight. Annu Rev Fluid Mech 50:593–627. https://doi.org/10.1146/annurev-fluid-122316-045217 VanLerberghe WM, Santiago JG, Dutton JC, Lucht RP (2000) Mixing of a sonic transverse jet injected into a supersonic flow. AIAA J 38:470–479. https://doi.org/10.2514/2.984 Vinogradov VA, Shikhman YM, Segal C (2007) A review of fuel pre-injection in supersonic, chemically reacting flows. Appl Mech Rev 60(4). https://doi.org/10.1115/1.2750346 Viti V, Neel R, Schetz JA (2009) Detailed flow physics of the supersonic jet interaction flow field. Phys Fluids 21:46101. https://doi.org/10.1063/1.3112736 Wang D-P, Zhao Y, Xia Z-X, Wang Q-H, Huang L (2012a) Experimental investigation of supersonic flow over a hemisphere. Chin Sci Bull 57:1765–1771. https://doi.org/10.1007/s11434-012-51240 Wang D-P, Zhao Y-X, Xia Z-X, Wang Q-H, Luo Z-B (2012b) Flow visualization of supersonic flow over a finite cylinder. Chin Phys Lett 29:84702. https://doi.org/10.1088/0256-307X/29/8/084702 Wang G-L, Chen L, Lu X-Y (2013) Effects of the injector geometry on a sonic jet into a supersonic crossflow. Sci China Phys Mech Astron 56:366–377. https://doi.org/10.1007/s11433-012-49842 Westerweel J, Elsinga GE, Adrian RJ (2013) Particle image velocimetry for complex and turbulent flows. Annu Rev Fluid Mech 45(1):409–436 Williams NJ (2016) Numerical investigations of a high frequency pulsed gaseous fuel jet injection into a supersonic crossflow. Doctoral dissertation, University of Tennessee Wu PK, Kirkendall KA (1998) Spray structures of liquid jets atomized in subsonic cross-flows. J Propul Power 14(2):10–17 Xiao F, Wang ZG, Sun MB, Liang JH, Liu N (2016) Large eddy simulation of liquid jet primary breakup in supersonic air crossflow. Int J Multiph Flow 87:229–240

References

25

Xie J, Gan ZW, Duan F, Wong TN, Yu MC, Zhao R (2013) Characterization of spray atomization and heat transfer of pressure swirl nozzles. Int J Therm Sci 68:94–102 Yang H, Li F, Sun B (2012) Trajectory Analysis of Fuel injection into supersonic cross flow based on Schlieren method. Chin J Aeronaut 25:42–50. https://doi.org/10.1016/S1000-9361(11)603609 Yang H, Li F, Sun B (2012) Trajectory analysis of fuel injection into supersonic cross flow based on Schlieren method. Chin J Aeronaut 25:42–50. https://doi.org/10.1016/S1000-9361(11)60360-9 Yoon HJ, Hong JG, Lee C-W (2011) Correlations for penetration height of single and double liquid jets in cross flow under high-temperature conditions. Atomiz Spr 21:673–686. https://doi.org/10. 1615/AtomizSpr.2012004212

Chapter 2

Spatial Distribution of Gaseous Jet in Supersonic Crossflow

The efficient delivery of gaseous fuel into supersonic airflow is very important in high-speed air-breathing engines, which require sufficient fuel–air mixing in the extremely short residence time within the engine combustor. The characteristics of the injected fuel plume, such as the penetration height, spanwise distribution, and concentration profiles, play an important role in ignition, flame holding, flame spreading, and combustion efficiency for a given flow path. In recent years, more and more experimental measurement technologies have been used to investigate the sonic jet in supersonic crossflow, which are given in Sect. 2.1. Jet penetration of sonic gaseous jet is discussed in Sect. 2.2. Spanwise distribution of gaseous jet and trajectory are summarized in Sect. 2.3. Moreover, near-wall information of sonic gaseous jet in supersonic crossflow is described in Sect. 2.4. Because numerical simulations of sonic jet in supersonic crossflow about penetration height, transverse distribution, and concentration profiles are usually compared with experimental results to prove the reliability of numerical simulations. (Or though numerical simulations can reproduce the detailed jet flowfield, the calculated results of penetration height, transverse distribution, and concentration profiles, however, lack the enough reliability.) Thus, we only discuss experimental research in this chapter.

2.1 Experimental Measurement Technologies With the development of modern flow visualization methods, tracers are used to follow the flow motion in the current planar laser flow imaging technique, where the light scattering or emitting characteristics of tracers are used to reveal the structures of the flow field. Accordingly, quantitative whole field flow parameters such as density (Bergmann et al. 1998), temperature (Forkey et al. 1998), pressure (Boguszko and Elliott 2005), species concentration (Watson et al. 2000), and velocity (Stanislas et al. 2005; Stier and Koochesfahani 1999) can be measured with these methods. The advantages of these methods are that they can record the tracer information at specified spatial location, and observe local or whole characteristics of flowfield. © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_2

27

28

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

Therefore, they can realize multi-resolution three-dimensional (3D) measurement of the flowfield (Bo et al. 2012). The flow imaging methods utilizing light scattering characteristics include planar laser Mie scattering method (PLMS) (Herring and Hillard 2000), filtered Rayleigh scattering method (FRS) (Elliott et al. 2015), particle image velocimetry (PIV) (Haertig et al. 2002; Khalitov and Longmire 2002), and Doppler global velocimetry (DGV) (Meyers 1992). The light emitting methods include laser-induced fluorescence (LIF) (Meyer et al. 2002), molecular tagged velocimetry (MTV) (Falco and Chu 1988), and coherent anti-Stokes Raman spectra (CARS) (Teets 1984; Lin et al. 2014). Although these techniques have already been widely applied to experimental fluid dynamics, supersonic flow imaging still faces a lot of challenges due to influence of compressibility, shock wave, instabilities and turbulence. When measuring supersonic flowfield with Mie scattering-based particles imaging technique, the tracer particles cannot follow the large gradient variation of the flowfield, especially with the presence of shock wave and vortex. When measuring supersonic flowfield with molecular tracer-based flow imaging techniques, the output signal is rather weak and the signal-to-noise ratio (SNR) is very low, and thus ICCD (intensified CCD) is often used to improve the output light intensity, which is more expensive than the general interline transfer CCD. Moreover, the calibration method is complex, so it is difficult to realize high-resolution measurement. Development of modern laser technique, imaging technique, image process technique, and nanotechnique offers opportunities to improve supersonic flow measurement and imaging. A nanoparticle-based planar laser scattering method (NPLS) has been developed by Zhao et al. (2009) to resolve the problem of poor flow-following ability and low SNR of the current flow imaging technique encountered in supersonic flow imaging. The NPLS system is mainly composed of computer, synchronizer, CCD camera, pulse laser, and nanoparticle generator, the structure of which is schematically shown in Fig. 2.1 (Zhao et al. 2009). In NPLS, the computer controls the collaboration of the components and receives the experimental images. The input and output parameters of the synchronizer are controlled by software and the collaboration of other components is controlled by signals of the synchronizer. The timing diagrams of exposure of CCD and laser output of pulse laser can be adjusted according to the purpose of measurement. The laser beam is transformed to a sheet with cylindrical lens. The nanoparticle generator is driven by high-pressure gas, and the output particles concentration can be adjusted precisely by the driving pressure. When measuring the flow field with NPLS, the nanoparticles are injected into and mixed with the inflow of the flowfield; when the flow is established in observation window, the synchronizer controls the laser pulse and CCD to ensure synchronization of the emission of scattering laser by nanoparticles and the exposure of CCD. According to the theory of light scattering by particles, it is known that if the distance between particles is evidently larger than particle diameter, the scattering of different particles is independent. Therefore, the scattering light intensity of nanoparticle received by CCD is in proportion to locale concentration of nanoparticles on the plane of laser sheet. If the scattering laser intensity falls into linear scope of CCD, the gray scale of flow image is also in proportion to the concentration of nanoparticles. Due to the good

2.1 Experimental Measurement Technologies

29

Fig. 2.1 The schematic of NPLS (Zhao et al. 2009)

flow-following ability of nanoparticles, the variation of particle concentration can reflect the density variation and mixing structure of the flowfield if the concentration distribution of particles on the inlet of flowfield is uniform. Thus, the NPLS method is appropriate for the study of compressible flowfield and mixing flows such as mixing layer, jet, and transverse injection where the density or concentration gradient are rather high.

2.2 Jet Penetration As described by Mahesh (2013), the penetration of incompressible transverse jets is most commonly scaled with rd. For fluids of different densities, r is related to the 2 . Compressibility becomes important at high momentum flux ratio, J  ρ j V j2 /ρ∞ U∞ speeds, and jet penetration and trajectory can also depend on p j / p∞ , ρ j /ρ∞ , T j /T∞ , molecular weights M wj and M w∞ , injection Mach numbers M j , and crossflow Mach numbers M ∞ . In addition, the crossflow boundary layer thickness δ, jet velocity profile, and orifice shape can also be important. Given the fairly large parameter space, a variety of correlations for jet penetration and trajectory have been proposed. Most empirical correlations are based on the visualization of the flow. In this chapter, correlations are only discussed for cases in which the crossflow is supersonic. Early studies employing schlieren imaging used the height of the Mach disk as a measure (Spaid and Zukoski 1964; Hawkins et al. 2012), whereas subsequent studies have used measures such as the visually observable upper edge of the jet from schlieren images (Benyakar et al. 2006), the edge of the fluorescing plume at which the injectant mole fraction is approximately 1% (Mcdaniel and Raves 1986), and the intensity of planar Rayleigh scattering images corresponding to 90% of the average intensity behind the bow shock.

30

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

2 The momentum ratio J is equal to γ j p j M 2j /γ∞ p∞ M∞ . The near-field crossflow pressure ( p∞2 ) is that after the bow shock. The Rankine–Hugoniot equations yield

2γ∞ γ∞ − 1 p∞2 2 M∞ ,  − p∞ γ∞ + 1 γ∞ + 1

(2.1)

which allows J to be expressed as J

γj pj γ∞ − 1 2γ∞ − 2 . M 2j γ∞ p∞2 γ∞ + 1 M∞ (γ∞ + 1)

(2.2)

The crossflow Mach number therefore influences J by determining the pressure behind the bow shock. As noted by Papamoschou and Hubbard (1993), if p j / p∞2 is held fixed, J depends most strongly on M j , while M ∞ has a much smaller effect. In the absence of crossflow, the jet exit pressure determines the extent to which it is either underexpanded or overexpanded. Conversely, for a transverse jet, varying the exit pressure also varies the momentum flux ratio. If J is held constant and the exit pressure is varied, the jet Mach number has to be correspondingly varied. This makes assessing the effect of the exit pressure solely in terms of its effect on jet expansion difficult. Billig and Schetz (2015) considered the forces acting on a slug of jet fluid, away from shock waves when estimated the jet trajectory. Assuming vertical injection, an approximate force balance along and normal to the jet axis yielded an equation for the angle α(s) made by the jet trajectories with the horizontal direction: 2 CD b(α)ρ∞ U∞ sin2α dα − , ds 2ρ j U 2j [A2j /A(s)][ρ j /ρ(s)]

(2.3)

where the subscript j denotes quantities at the jet exit, h(s) denotes the jet width, A(s) denotes the local cross-sectional area, C D denotes a drag coefficient, and ρ(s) stands for jet density. Schetz and Billig used empirical expressions for h and C D , an assumption of an elliptical cross section of prescribed ellipticity to obtain A, and a constant value for ρ to obtain the jet trajectory. Lots of experiments have been conducted over the past few decades, and correlations for jet trajectory have been proposed. As can be seen in Table 2.1, either a power law or a logarithmic fit was used to establish the correlations. Even correlations with similar functional form, the correlations have significantly different constants. Papamoschou and Hubbard (1993) provided experimental evidence that parameters other than J can also be important. They considered helium jets in crossflowing air at Mach 2–3 and performed several experiments in which they varied J, M j , and the jet pressure and density ratios. Gruber et al. (1997) conducted experiments using planar Rayleigh scattering to study sonic helium and air jets with the same momentum flux ratio injection into the same crossflow. They found that molecular weight does not make any difference on the penetration; however, the dynamics of the large-scale structures is quite different. The large-scale motions were observed to break down in

2.2 Jet Penetration

31

Table 2.1 A compilation of correlations for the trajectories of high-speed jets Authors and references

Trajectory

Gruber et al. (1997)

y dJ

Mcdaniel and Raves (1986)

y d



Rothstein and Wantuck (1992)

y d

 2.173J 0.276 (x/d)0.281

y d

 J 0.312 ln[4.704(x/d + 0.637)]

Rogers (1971)

y d

 3.87J 0.3 (x/d)0.143

Portz and Segal (2006)

y/d j  1.362J 0.568 (x/d j − 1.5)0.276 (δ/d j )0.221

Lin et al. (2014)

y dj

 1.20



x+d/2 dJ

0.344 ρ j 2 ρ∞ M∞

0.344

ln[2.077(x/d + 2.059)]

×(Re j /Re∞ )−0.0084 (M j /M∞ )−0.025  0.32  1.16J 0.72 dx θ 0.11 j

the air jet, whereas the helium jet stayed more coherent over longer distances. They explained this behavior by noting that an appropriate convective Mach number was three times larger for helium than for air. Another study by Benyakar et al. (2006) used schlieren and OH planar laser-induced fluorescence to contrast sonic hydrogen and ethylene jets with the same momentum ratio. Similar to Gruber et al., the lighter, but faster, hydrogen jet stayed coherent for longer distances. In contrast to Gruber et al., Benyakar et al. (2006) found that the ethylene jet penetrated more. However, Benyakar et al. (2006) did not find significant differences in the dynamics of the large-scale structures between jets of different molecular weights. They attributed this behavior to the ethylene jet having lower velocities for the same J, which induces the coherent motions on the windward side to tilt in the crossflow direction, mix more, and lose their coherence. Portz and Segal (2006) summarized the predicted correlations of gaseous jet penetration in supersonic flows, and indicated that the upstream boundary layer properties, i.e., laminar/turbulent and the boundary layer thickness play important roles in the penetration of the jet. And, a lens-based schlieren system was used for penetration visualization. They gave the definition of penetration height as a function of a correlation of several independent variables, including J, the downstream distance ratio at which penetration was measured, the incoming boundary layer thickness (δ) ratio, the jet-to-air Reynolds numbers ratio, and the jet-to-air molecular weight ratio. y/d j  1.362J 0.568 (x/d j − 1.5)0.276 (δ/d j )0.221 × (Re j /Re∞ )−0.0084 (M j /M∞ )−0.025 .

(2.4)

Portz and Segal (2006) presumed that Re number ratio and molecular weight effects were negligible and removed them from the final correlation. They finally

32

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

came to those conclusions: (1) Transverse jet penetration is primarily dependent on J and, to a lesser degree, on the boundary layer thickness and air Mach number. (2) Increased air Mach number results in increased penetration due to greater effective dynamic pressure ratio at the injection location, a result of stronger bow shock generation. (3) The effect of the boundary layer thickness on near-field jet penetration is significant at low supersonic Mach numbers but decreases with increased Mach number. In a recent research, Lin et al. (2014) found their experimental data of ethylene penetration height exhibited a noticeable deviation from Gruber et al.’s (1997) correlation. They developed a new correlation to treat cases with injection angles of both 30° and 90°.  0.32 x y 0.72  1.16J θ 0.11 , dj dj

(2.5)

where θ represents the angle of the jet with the airstream. Sun et al. (2013) used NPLS to investigate the penetration of gas jet in a supersonic turbulent crossflow. Experimental injection conditions are given in Table 2.2. Figure 2.2 gives the detailed flow structures in the x-y plane of sonic jets with injection condition No. 1, No. 2, and No. 3 from Ref. Sun et al. (2013). The images are picked from one experimental test at different time points. As the injection stagnation pressure is increased, the penetration height is clearly seen to increase, while the flow patterns are kept similar. A particular character could be distinguished that the maximum eddy scale in the interface of the jet with the mainstream tends to decrease as the injection stagnation pressure decreases. As shown in Fig. 2.2, the upper-side structures in the near-field jet plume are mostly tilted upstream (counterclockwise), opposite to the expected orientation. Since the crossflow velocity is higher than the jet exit velocity, it is expected that these structures would be tilted in the clockwise direction owing to the shear, just like the far-field vortex structures in case No. 3. However, these vortices are generated at the exit of the jet where the vertical jet velocity is higher than the freestream velocity, and

Table 2.2 Experimental injection conditions

Jet exit conditions

No. 1

No. 2

No. 3

M j , Mach number

1.0

1.0

1.0

d j , diameter of the jet orifice, mm

2.0

2.0

2.0

T 0 , total temperature, K

300

300

300

m, ˙ Mass flow rate, kg/s

0.6105e−3

1.221e−3

1.832e−3

J, jet to freestream momentum flux ratio

1.7724

3.5448

5.3171

2.2 Jet Penetration

33

Fig. 2.2 NPLS images of nitrogen jet penetration into the supersonic flow with injection condition No. 1, No. 2, and No. 3, at the plane z  0 (Sun et al. 2013)

therefore resulting in a counterclockwise rotation. As the jet convects downstream, the vortex shapes are kept and transported. However, it is still expected that the large-scale structures will eventually exhibit a clockwise rotation due to the shear induced by the higher velocity crossflow. Another feature of the images is about the eddy shedding. Benyakar et al. (2006) demonstrated that for the hydrogen injection in a supersonic flow, large eddies are generated in the near field periodically and the formed eddies persist long distances downstream, while for ethylene injection, these eddies lose their coherence as the jet bends downstream. From the images shown in Fig. 2.2, the eddy shedding phenomena are more like random behavior, instead of periodic, consequent process. Although molecular weight of the injection nitrogen is close to air, eddies formed in the near field could persist long distance. These characteristics demonstrate the complexity of the transverse jet in supersonic crossflows. The eddy information could be extracted from the NPLS images, which is described schematically in Fig. 2.3. The jet eddy core, scale η, and the position (x, y) could be picked manually using an approximation method. Since there are no particles in the jet, dark color region extended from the jet exit reflects the jet struc-

34 Fig. 2.3 a Schematic description of eddy parameters, η is the eddy size, (x, y) is the spatial position, and b is an example to determine the eddy scale and position in the NPLS image (Sun et al. 2013)

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

y η3

y3 y2 y1

η2 η1

x1

o

x3

x2

x

(a) Schematic description of eddy parameters

y

o

η

x

(b) Spatial points picked in the image to determine the eddy scale and position

tures. The large eddies in the jet could be easily distinguished along the interface of the jet with the main airstream. Define the left limit point and right limit point of an eddy, the position and scale information could be extracted. Figure 2.3b gives an example to obtain them. After selecting the points from the NPLS images manually, the η-x-y diagram of eddies with 40 instantaneous images are summarized in Fig. 2.4. Figure 2.4 gives the three-dimensional distribution of η-x-y extracted data. It is useful to give further analysis for η-x and η-y correlation from the images, which are shown in Fig. 2.4b and c, respectively. A power law fit using the least squares method to give the optimal fit. For η-x, the power law fit is η  1.7292x 0.281 and for η-y, the power law fit is η  0.0026x 2.98 . Here, the tangent curves of the fits, which represent the vortex generating rates, are also given in Fig. 2.4b, c. It is seen that the vortex generating curves exist in a large number range in the near field while a low number in the far field. The average size of eddies originated near to the jet exit (jet near field) is small, the far-field eddies in the jet are about three times than the near field. The small eddies near to the jet orifice rise quickly, after the jet bends to the freestream direction, the eddy size keeps almost unchanged along the stream direction. The average size of large-scale structures becomes elongated as they march downstream by the development of compressible shear layers with the main flow. The results

2.2 Jet Penetration

35

(a) η-x-y diagram of the eddies in the No.2 transverse jet

(b) η-x diagram with the power law fit and the tangent rate

(c) η-y diagram with the power law fit and the tangent rate

Fig. 2.4 η-x-y diagram of eddies of the transverse jet in supersonic flow and the correlation of the experimental data (Sun et al. 2013)

demonstrate an important phenomenon, that is, the near field plays an important role in the eddy generation. The jet eddies are mainly formed in the near field and eddies developed in the near field or the dimension of the near-field region will determine the size of eddies in the far field. That means, if mixing enhancement device is applied, the device should be considered to install in the near field to enhance the near field region and further activate the near-field eddy scale. It is concluded that mixing process in the far field is finally effected by the near-field eddies since eddies in the far field are basically translating with little rolling. So, for jet mixing, the near field is necessary to gain enough attention, which is potentially used to enhance and control the fuel distribution and mixing. In Sun et al.’s (2013) NPLS experiments, there are no particles in the jet injectant, thus the dark region in the jet could be used to differentiate the interface between the jet and the main flow. This is better to distinguish the penetration height than Gruber et al.’s (1997), which defined jet penetration as the trajectory where the jet concentration is about 10%. This is also better than Ben-Yakar and Hanson’s (2002) which gave the “visible” jet’s penetration as measured in schlieren images

36

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

Fig. 2.5 Schematic description of jet penetration trajectory, a is the edge approximation definition and b is x-y diagram corresponding to (a) (Sun et al. 2013)

(a) schematic extraction of jet penetration

(b) the spatial points picked in the image to determine the penetration height

corresponding to 1% of the jet concentration. It is reasonable that the precision of penetration measure based on NPLS is higher than the ones based on 10% or 1% concentration identification. The jet trajectory could be extracted manually using an edge approximation definition, which is described schematically in Fig. 2.5a. After picking the points from the NPLS images, the y-x diagram of the jet interface is summarized in Fig. 2.5b. There is no interpolation or threshold from the measurements and the definition of penetration height is direct. Using this definition, the discrete data extracted from 40 instantaneous images to demonstrate the penetration height of various injection conditions are given in Fig. 2.6. In Fig. 2.6, the measured depth of jet penetration is presented, and several correlations are given to exhibit the prediction performance. It is seen that better agreement is achieved between the measured penetration heights and Rothstein–Wantuch correlation. If careful observation is given to the higher J, it is seen that Rothstein–Wantuch correlation also deviates from the measured data, and Gruber et al.’s or Lin et al.’s correlation shows a heavy deviation especially in the far field. Although Rothstein–Wantuch correlation approaches the measured jet height in a high precision, there is still possibility to promote the prediction. A modified correlation from Rothstein–Wantuch correlation gives the penetration height formula as  b a x y  c , dj J dj

(2.6)

2.2 Jet Penetration

37

(a) injection condition No.1

(b) injection condition No.2

(c) injection condition No.3 Fig. 2.6 Spanwise penetration data of the transverse jet in supersonic flow and several correlations (Sun et al. 2013)

where a, b, and c are constants and determined by using the least squares method. Considering the No. 1–3 injection conditions, to approach the least error, they obtain that a  2.933, b  0.161, and c  0.2560. Then, a modified penetration height correlation is shown as  0.161 x 2.933  0.5830 . dj J J dj J y

(2.7)

Figure 2.6 demonstrates that there is a reasonably good agreement between the measured penetration heights and the modified predicted correlation for the test conditions in the present study.

38

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

In Sun et al.’s (2013) experiments, to test the incoming boundary layer effect, the transition bans imported on the fore part of the experimental plane are removed, and then the fully developed turbulent boundary layer is returned to a laminar state. Using NPLS analysis similar to Bo et al.’s (2012), the laminar boundary layer at x  −30 mm has a boundary layer thickness δ  2.3 mm. Portz and Segal (2006) indicated that when the boundary layer thickness is small, typically less than the jet diameter, the gas penetrates similarly at the similar J; when the boundary layer is thicker than the jet diameter, the jet penetrates straight and higher. However, J is no larger than 3.0 in their research, and thus the boundary layer thickness effect might not be accurate for very high J. Figure 2.7 and 2.8 give the NPLS images of injection condition No. 2 and No. 1, respectively, with incoming laminar boundary layer. The results demonstrate that the jet penetrates the boundary layer and forms a strong bow shock. The boundary layer is lifted due to the high adverse pressure in the transverse jet. The patterns of eddies shedding in the near field and eddies transported in the far field are similar to the pattern of the incoming turbulent boundary layer. The flow character analysis is omitted here. In Fig. 2.9, the measured depth of jet penetration with incoming laminar boundary layer is presented, and better agreement is achieved using the modified correlation than original Rothstein–Wantuch correlation, Gruber correlation, and Lin correla-

No.2

Fig. 2.7 NPLS images of the transverse jet in supersonic crossflow at the plane z  0 with No. 2 injection and incoming laminar boundary layer (Sun et al. 2013)

No.1

Fig. 2.8 NPLS images of the transverse jet in supersonic crossflow at the plane z  0 with No. 1 injection and incoming laminar boundary layer (Sun et al. 2013)

2.2 Jet Penetration

39

Fig. 2.9 Transverse penetration data of the transverse jet in supersonic flow and several correlations with incoming laminar boundary layer (Sun et al. 2013)

(a) injection condition No.1

(b) injection condition No.2

tion. These results indicate that jet penetration is mainly determined by the jet flux momentum for high J, and the incoming boundary layer thickness or state plays a secondary role. In Sun et al.’s (2013) work, they evaluated the evolution of coherent structures and penetration height of gaseous transverse jet penetration into a Ma  2.7, T 0  300 K, P0  1 atm supersonic turbulent flow. The high spatiotemporal resolution coherent structures of the jet plume were obtained by utilizing the nanoparticle-based

40

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

planar laser scattering technique (NPLS). Evolving pattern of the coherent structures generated on the upwind surface of the transverse jet was analyzed based on the NPLS images. Several conclusions could be made from the analysis. (1) A modified penetration correlation is proposed based on an edge approximation definition and least squares method with various injection pressures. (2) Jet penetration is mainly determined by the jet flux momentum for high J, and the incoming boundary layer thickness or state plays a secondary role. This is similar to Portz and Segal’s (2006) conclusions.

2.3 Spanwise Distribution of Gaseous Jet In order to obtain empirical correlations of jet penetration for engineering application, a series of theoretical (Billig et al. 1971; Heister and Karagozian 1990) and empirical mixing model Yang et al. (2015) for the analysis of transverse jet in supersonic flow was presented. To solve the interaction problem, the integral forms of the two sets of the conservation equations are written by Billig et al. (1971), one for the primary stream and one for the secondary. Each set of equations includes the scalar energy and continuity equations and the vector momentum equation. If stations in the flowfield are chosen wherein the flow properties can be considered to be represented by some suitable average value, then considerable simplicity is introduced in the solution. It is the judicious choice of these particular points in the flowfield that enables the success of the subsequent unified analysis. Assuming that variables with subscripts a and b refer to primary flow conditions on faces Aa and Ab in Fig. 2.10 and subscripts j and c refer to jet flow conditions on faces Aj and Ac , the simplified forms of the equations are given as follows: Mass conservation equations: ρa u a Aa  ρb u b Ab ,

(2.8)

ρ j u j A j  ρc u c Ac ,

(2.9)

Axial momentum conservation equations:  pa Aa − pdA − pb Ab cos α  ρb u 2b Ab cos α − ρa u a2 Aa ,  p j A j cos δ j +

Ax

pdA − pc Ac cos δ ρc u 2c Ac cos δ − ρ j u 2j A j cos δ j . Ax

(2.10) (2.11)

2.3 Spanwise Distribution of Gaseous Jet

41

Fig. 2.10 Generalized model configuration for gaseous secondary injection in supersonic freestream (Billig et al. 1971)

Normal momentum conservation equations:   pdA − pb Ab sin α  ρb u 2b Ab sin α, − p∞ + pdA +  p j A j sin δ j −

N



WP

pdA − pc Ac sin δ  ρc u 2c Ac sin δ − ρ j u 2j A j sin δ j .

pdA + N

(2.12)

WS

(2.13) Energy conservation equations:     ρa u a Aa h a + u a2 /2  ρb u b Ab h b + u 2b /2 ,

(2.14)

    ρ j u j A j h j + u 2j /2  ρc u c Ac h c + u 2c /2 .

(2.15)

To solve this problem, a model was introduced by Spaid and Zukoski (1964) as shown in Fig. 2.11, where the effective shape of the injectant is represented as a quarter sphere of radius h followed by an axisymmetric half body. And several assumptions were made: (1) the secondary jet had expanded to satisfy the condition pc  pb  pa  p∞ ; (2) the pressure on the quarter sphere can be calculated by use of modified Newtonian flow; and (3) the injectant expands to achieve the ambient pressure. In the end, those equations were solved, and the correlation between jet penetration high h and the jet diameter d j can be expressed as

42

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

Fig. 2.11 Model for jet center line trajectory calculation (Billig et al. 1971)

√  1 2h Ac 2  dj Aj 

1 Ma



⎛ ⎛ ⎞⎞ 41 +1

21

γ γj −1



γγ j −1 j j p0 j γ j 4 2 2 ⎝ ⎝1 − p∞ ⎠⎠ , p∞ γa C p ∗ γj − 1 γj + 1 p0 j (2.16)

where p02 − p∞  p∞ C ∗p  1 2 ρu ∞ 2

p02 p01 − p01 p∞ 1 2 ρu ∞ 2



 1

 p∞

γ +1 Ma2 2

 γ γ−1 

γ +1 2γ Ma2 −γ +1

1  γ −1

−1

1 ρu 2∞ 2

 1

γ

γ −1 γ + 1 2 γ −1 γ +1 2 p∞ Ma  −1 2 2γ Ma2 − γ + 1 ρu 2∞   1

γ

γ −1 γ + 1 2 γ −1 γ +1 2 Ma  −1 . γ Ma2 2 2γ Ma2 − γ + 1 (2.17)

According to Eq. (2.16), jet penetration and trajectory are determined by p j / p∞ , ratio of specific heats γ j , injection Mach numbers M j, and crossflow Mach number M ∞.

2.3 Spanwise Distribution of Gaseous Jet

43

Figure 2.11 was given by Billig et al. (1971) for jet centerline trajectory calculation. And, jet trajectories were given by 

 p¯ / pa



peb / pa  p¯ / pa

(x − x2 ) D j  (y − y2 ) D j 

   cos δ¯ f d p¯ pa ,    sin δ¯ f d p¯ pa ,

(2.18)

peb / pa

where                d s Dj d s D j d D¯ D j d M¯   ,   f s D j , p¯ pa         d p¯ pa d p¯ pa d D¯ D j d M¯          d s Dj d s Dj dδ . g δ, p¯ pa         dδ d p¯ pa d p¯ pa

(2.19) (2.20)

Those equations were used to calculate the jet centerline trajectory downstream of the Mach disk. In the region downstream of the slice 3 ( p¯  pa , shown in Fig. 2.11), the flow properties within the jet remain constant while the stream continues to turn toward δ  0. Equation (2.18) can be integrated into closed form to yield

¯  tan δ/2 1 D¯ 3  y − y3 cos δ¯ − cos δ¯3 ,  K ln − ¯ ¯ Dj 2 Dj tan δ3 /2 ¯   x − x3 K K 1 D3 sin δ¯ − sin δ¯3 − +  , Dj 2 D¯ j sin δ¯ sin δ¯3

(2.21)

where

K 

− 43 π

  u¯ 3 uj

Dj D¯ 3

   γj γa

C p∗

pj pa

Mj Ma

2 .

(2.22)

In Billig et al.’s (1971) work, the jet centerline, penetration boundary, and injections spanwise distributional characteristics were given eventually. The performance of the jet penetration analysis code (JETPEN) was later improved by Billig and Schetz (2013), so that cases with injection angles other than 90° could be treated and the turbulent entrainment of freestream air into the fuel plume could be incorporated. The revised JETPEN code was developed over a wide range of conditions: freestream air, 1.4 ≤ M∞ ≤ 6.0; jet Mach number, 1.0 ≤ M j ≤ 1.8; injection angle, 15◦ ≤ θ ≤ 90◦ ; jet-to-air momentum flux ratio, 1.0 ≤ q ≤ 10; and helium and hydrogen injected into air. The applicability of the JETPEN code to higher molecular weight gases, such as hydrocarbon fuels, should be validated. The Raman scattering technique was adopted by Lin et al. (2014) to obtain timeaveraged cross-sectional concentration profiles at various freestream locations of the

44

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

ethylene plumes. The objective of his study is to obtain quantitative fuel plume data for ethylene jets delivered from injectors of a wide range of sizes, with the largest injector orifice corresponding to a large-scale combustor. The data was compared with the existing correlations for fuel plume penetration height and with the fuel plume properties predicted by the JETPEN code. The scalability of the ethylene fuel plumes was also discussed by Lin et al. (2014). Lin et al. (2014) found that the JETPEN code underpredicts the fuel plume size for ethylene jets injected at 90°. The underprediction is worse at large x  d and especially at conditions of high jet-to-air momentum flux ratio. The JETPEN code slightly overpredicts the fuel plume size for the 30° jets. But the observed discrepancies in fuel plume structure, the fuel plume penetration heights predicted by the JETPEN code agree reasonably well with the measured penetration heights. A new penetration height correlation was developed by Lin et al. (2014) to treat the cases with injection angles at 30° and 90°, which was presented in Sect. 1, Part 2. Based on the works of Hasselbrink and Mungal (2001), Torrez (2012) modeled the central line and the spread distribution of the jet in a MASIVE (Michigan air force scramjet in vehicle) code. But the MASIVE model only suits for high jet-tocrossflow momentum flux ratio (J > 1). Combining the empirical collocations and the Raman scattering images given by Lin et al. (2014), a new gas jet–crossflow mixing model was proposed by Yang et al. (2015). In his model, the jet centerline and the penetration boundary were given by  x c1x yc c  c1 J c1J , d d  x c2x b c  c2 J c2J , d d where c1  1.1 − η1 J , c1x  1/4, c1J  2/3, η1  0.07, c2  1.1 − η2 J , c2x  7/24, c1J  2/3, and η1  0.08. Besides, spatial and temporal distributions of jet components were also given by Yang et al. (2015) referenced from Hasselbrink and Mungal (2001). Schematic of the spanwise distribution of injection mixture mass fraction is shown in Fig. 2.12. The molar fraction of the injectant along the jet centreline subject to the following formula:



ρ j c3ρ u j c3u  xc c3x xc  c3 , xj ρ∞ u∞ d where x j was the mole fraction of the injectant at the outlet of the orifice, c3  3.1 − η3 J , c3ρ  1/3, c3u  −1/3, c3x  −0.9, and η3  0.02. And the injection mixture mass fraction around the jet centreline was given by wc 

x c rm . x j + (rm − 1)xc

In the formula, rm was the mass fraction ratio of injectant and the crossflow. The injection mixture mass fraction around the jet centreline could be calculated by

2.3 Spanwise Distribution of Gaseous Jet

45

Fig. 2.12 Schematic of the spanwise distribution of injection mixture mass fraction (Yang et al. 2015)

r2 , w(r )  wc exp − η4 b 2 where r 2  (x − xc )2 + (y − yc )2 + (z − z c )2 , η4  1.5. Several experimental cases were used to verify the performance of the model presented by Yang et al. (2015), as shown in Fig. 2.13, for example. Results indicate that, within the momentum flux ration of 0.5–5.3, Yang et al.’s (2015) model agreed well with the experiment image both in the jet centerline and penetration boundary, and in the area near jet centerline his model shows a higher precision in contrast with the existing models when predicting the injection mole fraction.

2.4 Near-Wall Information of Gaseous Jet The use of an oil-flow pattern allows the global acquisition of near-wall velocity directional information, which would provide a way to analysis the interactions between the jet wakes and the crossflow. In addition, the effect of the oil flow on the motion of the boundary is very small in most practical cases (Squire 2006). Liu et al. (2018) used oil-flow pattern to study the near-wall information of gas jet in a supersonic turbulent crossflow. Experimental injection conditions are given in Table 2.3. All cases lead to a sonic jet with a stagnation temperature T 0j  300 K. Jet-to-crossflow momentum flux ratio (J) is chosen as 2.3, 5.5, 7.7, 11.2, 16.0, 20.6, and 28.9, which was controlled by the injection stagnation pressure. In Liu et al.’s (2018) work, pigments are mixed with a type of special oil, painted onto the surface of the flat plate. Due to the viscosity of the flow near the wall, the oil moves slowly and eventually leaves behind skin-friction lines. The regions of reattachment or separation could be easily distinguished from diverging or converging oil-flow lines as shown in Fig. 2.14a for J  20.6. It is clearly seen in the enlarged

46

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

(a) Raman scattering images

(b) model of Yang et al. (2015)

(c) MASIVE

Fig. 2.13 Raman scattering images, model of Yang et al. (2015) and MASIVE predicted mole fraction distribution of ethylene at different x/d sections under condition of J  0.5 Table 2.3 Air jet conditions for the experiment Jet-to-crossflow momentum J

Mach number Mj

Stagnation temperature T 0j (K)

Stagnation pressure P0j (kPa)

Reynolds number Rej

2.3

1.0

300

110

6.6 × 104

5.5

1.0

300

259

1.6 × 105

7.7

1.0

300

375

2.3 × 105

11.2

1.0

300

525

3.2 × 105

16.0

1.0

300

750

4.5 × 105

20.6

1.0

300

967

5.9 × 105

28.9

1.0

300

1354

8.3 × 105

2.4 Near-Wall Information of Gaseous Jet

47

Fig. 2.14 Experimental oil-flow pattern with a circular orifice, and a schematic of the oblique recirculation zones: a skin-friction lines from the experimental oil-flow pattern with a circular orifice (J  20.6), b schematic of the oblique recirculation zones induced by the collision shock waves behind the transverse jet injected in supersonic crossflow (Liu et al. 2018)

view that V-shaped trailing recirculation occurs in the jet wakes, which consist of two oblique separation zones. In Figs. 2.14a and 2.16a, these oil flow lines clearly show the inner flow configuration of the recirculation zones. The domain shown in the dashed ellipse is the reattachment region. Downstream of the jet separation, it is seen that the separation lines are terminated on the V-shaped line. The V-shaped line downstream of the separation ending point (shown in Fig. 2.14a) represents a slip line due to the interaction of the collision shock with the boundary layer, which originates from the large velocity gradient from the lateral flow with the reattachment flow from the jet leeward. According to Liang et al.’s (2018) work on shock wave structures in the wake of sonic transverse jet in supersonic crossflow, the collision shock (marked with the pink line) intersects with the reflected shock (marked with the yellow line) which is produced when the barrel shock is reflected by the Mach disk. The shock system impinging on the wall induces the near-wall velocity slipping, as the near-wall slip lines shown in Fig. 2.14a with an enlarged view. Combining Ref. (Liang et al. 2018) and Fig. 2.14a, a schematic of recirculation zone structures in the jet wake flow is illustrated in Fig. 2.14b, which demonstrates the complex configuration of the collision shock interaction with the lateral supersonic boundary layer. The shock wave surface originates from the collision of the

48

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

supersonic flow running around the jet barrel. Since a separation bubble is induced in the jet leeward, the flow behind the collision is also blocked by the separation bubble, and a half-cone shock surface is formed. The induced recirculation extends in the spanwise direction like a foot of the incident collision shock into the incoming lateral supersonic flow. Behind the oblique separation zone, a reattachment domain occurs downstream. The separation lines of the separation zone are given in Fig. 2.14, and the slip lines between the wake and the lateral flow are shown. From the oil-flow images, it is appropriate to divide the jet wake flow field into three regions, as shown in Fig. 2.14a. Region I is the V-shaped separation zone induced by the collision shocks. Region II is the zone that is mainly characterized by the flow reattachment and the collision shock. Region III is the mixing and recovery zone. In Region II, reattachment occurs, and the velocity slips between crossflow and wake flow. There are two types of slip lines. The first type is the slip flow occurring along the collision shock, which forms the velocity difference. They denote this type of slip line as an outer slip line (outer slip line I, marked with the green line). The second type is the slip flow occurring along the slip interface of the reattached wake flow with the lateral flow passing the collision shock waves. They denote this type as an inner slip line (marked with the green line). In region III, the reflected shock takes over the work of the collision shock, and continues forming the velocity difference between crossflow and wake flow. A new outer slip line is generated along the reflected shock, and they denote this type of slip line as outer slip line II (marked with the green line). These slipping flows exchange momentum and produce sufficient turbulence that leads to a full recovery in region III. Figure 2.15 shows the surface oil-flow images of the jet cases with J  28.9 and J  7.7. It is seen that the general structures of J  28.9 and J  7.7 are similar to that of J  20.6 (Fig. 2.14a). For the J  7.7 case, the intensity of the V-shaped collision shock (marked with pink lines) and the reflected shock (marked with yellow lines) have a lower strength, and the slip line induced by the shock system declines quickly. After a short distance, the outer slip line is merged into the inner slip line between the jet wake and the lateral flow. Besides, J  7.7 case has a shorter distance from the orifice to the V-shaped collision shock and the reflected shock interaction position than J  20.6 and 28.9 cases. The separation size of J  28.9 is larger than both J  7.7 and J  20.6, which is expected since the maximum width of the separation is related to J. To obtain a quantitative analysis of the separation, the outer separation line’s boundary (as shown in Fig. 2.14a with the blue dotted line) is extracted by the points with 95% brightness of the brightest points near the outer separation line. In Fig. 2.16, they use the separation center line (marked with the red line) to present the outer separation boundary. A straight line (marked with the green line) is used to fit the above separation center line. Then, half angle θ/2 between the separation center line and the V-shaped separation zone’s center line (marked with the dash-dotted pink line) could be identified by measuring the angle between the green line and the dash-dotted pink line. In Fig. 2.14a, the starting point is defined as the crossover point of the outer separation lines, whereas the ending points are extracted from the interface where the oil lines no longer recirculate. The measurement uncertainties of

2.4 Near-Wall Information of Gaseous Jet

49

Fig. 2.15 Oil-flow images for different J values in the supersonic crossflow: a J  28.9 and b J  7.7 cases (Liu et al. 2018)

the above half angle and the length of the separation zone are mainly caused by the extraction accuracy of the outer separation lines’ boundary and the fitness accuracy of the separation center line. Therefore, in the present study, angle measurement has a system error of 3% and the length measurement has a system error of 5%. Figure 2.17a shows the outer separation lines extracted from the oil-flow images. Half angle θ/2 between the separation line and the center line and the length of the separation lines can be informed from Fig. 2.17a. Calculation results and experimental data points of skin-friction lines of separation zone with J  7.7 are compared in Ref. (Liang et al. 2018), and it is found that they are consistent. The interesting part is about the angle θ between the two separation lines (drawn by solid lines in Fig. 2.14a). A comparison is provided between the J  7.7, 20.6 and J  28.9 cases and the results indicate that θ of J  28.9 is 34.50°, which is basically equal to θ

50

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

Fig. 2.16 Schematic of the separation zone extraction process (Liu et al. 2018)

shown in Figs. 2.14a (θ  33.20°) and 2.15b (θ  34.20°). The feature demonstrates that angle θ is independent of J. This result is expected since the separation line is mainly induced by a leeward collision shock, and collision shock angle is only dependent upon the freestream Mach number. An increase in J will not change the collision shock angle and the angle between the separation lines. The jet leeward separation length and width can be identified in Fig. 2.17a. Since the separation lines are asymmetric due to the jet unsteadiness, they are treated by an averaged method. The location of the starting point of the separation region is different for the different J cases, and two parameters, x1 and x2 , are defined to describe the leeward separation length. As seen in Fig. 2.14a, x1 is the streamwise coordinate of the separation starting point; x2 is the distance between the separation starting point and the ending point. Correlations to predict the separation length and width with J can be provided according to the experiments. Since the angle θ is basically equal for all J cases, it is only necessary to provide a prediction for the separation length. A modified correlation that is referenced from the jet penetration correlation (for example, Rothstein–Wantuch correlation, Gruber correlation, etc.) gives the separation starting point and length formula as x1  A1 J B1 , d x2  A2 J B2 , d

(2.23) (2.24)

where A1 , B1 , A2 , and B2 are constants and are determined by using a least squares method. Considering eight injection conditions (J  0.0, 2.3, 5.5, 7.7, 11.2, 16.0, 20.6, and 28.9) to approach the least error, the results demonstrate that A1  0.3167, A2  4.582, B1  0.7697, and B2  0.4135. Then, the correlation for predicting the separation length is given as x2  4.582J 0.4135 d

(2.25)

2.4 Near-Wall Information of Gaseous Jet

51

Fig. 2.17 Separation length on the wall of the transverse jet in a supersonic flow and its correlation: a jet leeward separation lines for different J, b measured separation length with error bar and its exponential correlation (Liu et al. 2018)

and it is shown in Fig. 2.17b. Because the crossflow and the fuel have not mixed completely in the separation zone, correlations of the separation length and width would help us to avoid the separation zone for combustion organization.

52

2 Spatial Distribution of Gaseous Jet in Supersonic Crossflow

References Ben-Yakar A, Hanson R (2002) Ultra-fast-framing schlieren system for studies of the time evolution of jets in supersonic crossflows. Exp Fluids 32(6):652–666 Benyakar A, Mungal MG, Hanson RK (2006) Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Phys Fluids 18(2):1105–1154 Bergmann V, Meier W, Wolff D, Stricker W (1998) Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame. Appl Phys B 66(4):489–502 Billig F, Schetz J (2013) Analysis of penetration and mixing of gas jets in supersonic cross flow. In: Aiaa Materials Specialist Conference-coating Technology for Aerospace Systems, 2013 Billig FS, Schetz JA (2015) Penetration of gaseous jets injected in supersonic stream. Journal of Spacecraft & Rockets 3(11):1658–1665 Billig FS, Orth RC, Lasky M (1971) A Unified Analysis of Gaseous Jet Penetration. Aiaa Journal 9(6):1048–1058 Bo W, Liu W, Zhao Y, Fan X, Chao W (2012) Experimental investigation of the micro-ramp based shock wave and turbulent boundary layer interaction control. Phys Fluids 24(5):1166–1175 Boguszko M, Elliott GS (2005) On the use of filtered Rayleigh scattering for measurements in compressible flows and thermal fields. Exp Fluids 38(1):33–49 Elliott G, Glumac N, Carter C (2015) Molecular Filtered Rayleigh Scattering applied to combustion and turbulence. In: Aerospace Sciences Meeting and Exhibit, 2015 Falco RE, Chu CC (1988) Measurement of two-dimensional fluid dynamic quantities using a photochromic grid tracing technique. Proceedings of SPIE - The International Society for Optical Engineering 814:706–710 Forkey JN, Lempert WR, Miles RB (1998) Accuracy limits for planar measurements of flow field velocity, temperature and pressure using Filtered Rayleigh Scattering. Exp Fluids 24(2):151–162 Gruber MR, Nejad AS, Chen TH, Dutton JC (1997) Compressibility effects in supersonic transverse injection flowfields. Phys Fluids 9(5):1448–1461 Haertig J, Havermann M, Rey C, George A (2002) Particle image velocimetry in mach 3.5 and 4.5 shock-tunnel flows. AIAA J 40(6):1056–1060 Hasselbrink EF, Mungal MG (2001) Transverse jets and jet flames. Part 2. Velocity and OH field imaging. Journal of Fluid Mechanics 443(443):27–68 Hawkins PF, Lehman H, Schetz JA (2012) Structure of highly underexpanded transverse jets in a supersonic stream. Aiaa Journal 5(5):882–884 Heister SD, Karagozian AR (1990) Vortex modeling of gaseous jets in a compressible crossflow. 6(1):85–92 Herring GC, Hillard JME (2000) Flow Visualization by Elastic Light Scattering in the Boundary Layer of a Supersonic Flow Khalitov DA, Longmire EK (2002) Simultaneous two-phase PIV by two-parameter phase discrimination. Exp Fluids 32(2):252–268 Liang CH, Sun MB, Liu Y, Yang YX (2018) Shock wave structures in the wake of sonic transverse jet in supersonic crossflow. Acta Astronaut 148:12–21 Lin KC, Ryan M, Carter C, Gruber M, Raffoul C (2014) Raman Scattering Measurements of Gaseous Ethylene Jets in a Mach 2 Supersonic Crossflow. 26(3):503–513 Liu Y, Sun MB, Liang CH, Cai Z, Wang YN (2018) Structures of near-wall wakes subjected to a sonic jet in a supersonic crossflow. Acta Astronaut Mahesh K (2013) The Interaction of Jets with Crossflow. Annu Rev Fluid Mech 45(3):379–407 Mcdaniel JC, Raves J (1986) Laser-induced-fluorescence visualization of transverse gaseous injection in a nonreacting supersonic combustor. J Propul Power 4(6):591–597 Meyer TR, King GF, Martin GC, Lucht RP, Schauer FR, Dutton JC (2002) Accuracy and resolution issues in NO/acetone PLIF measurements of gas-phase molecular mixing. Exp Fluids 32(6):603–611

References

53

Meyers JF (1992) Doppler Global Velocimetry - The Next Generation? AIAA 17 th Aerospace Ground Testing Conference, paper AIAA-92-3897. In: Aiaa 17 Th Aerospace Ground Testing Conference, 1992 Papamoschou D, Hubbard DG (1993) Visual observations of supersonic transverse jets. Exp Fluids 14(6):468–476 Portz R, Segal C (2006) Penetration of Gaseous Jets In Supersonic Flows. Aiaa Journal 44(10):2426–2429 Rogers RC (1971) A study of the mixing of hydrogen injected normal to a supersonic airstream, NASA technical note D–6114 Rothstein A , Wantuck P (1992, April) A study of the normal injection of hydrogen into a heated supersonicflow using planar laser-induced fluorescence. In 28th Joint Propulsion Conference and Exhibit (p. 3423) Spaid FW, Zukoski E (1964) Secondary injection of gases in supersonic flow. In, 1964. pp 1689–1696 Squire LC (2006) The motion of a thin oil sheet under the steady boundary layer on a body. J Fluid Mech 11(2):161–179. https://doi.org/10.1017/S0022112061000445 Stanislas M, Okamoto K, Kähler CJ, Westerweel J (2005) Main results of the Second International PIV Challenge. Exp Fluids 39(2):170–191 Stier B, Koochesfahani MM (1999) Molecular Tagging Velocimetry (MTV) measurements in gas phase flows. Exp Fluids 26(4):297–304 Sun M, Zhang S, Zhao Y, Zhao Y, Liang J (2013) Experimental investigation on transverse jet penetration in supersonic turbulent crossflow. Science China Technological Sciences 56(8):1989–1998. https://doi.org/10.1007/s11431-013-5265-7 Teets RE (1984) Accurate convolutions of coherent anti-Stokes Raman spectra. Opt Lett 9(6):226–228 Torrez SM (2012) Design Refinement and Modeling Methods for Highly-Integrated Hypersonic Vehicles. Dissertations & Theses - Gradworks Watson KA, Lyons KM, Donbar JM, Carter CD (2000) Simultaneous Rayleigh imaging and CHPLIF measurements in a lifted jet diffusion flame. Combust Flame 123(1):252–265 Yang YX, Wang ZG, Sun MB, Wang HB (2015) Empirical mixing model of transverse gaseous jet in supersonic jet in supersonic flow. journal of Aerospace Power 30(1392–1399 Zhao YX, Yi SH, Tian LF, Cheng ZY (2009) Supersonic flow imaging via nanoparticles. Sci China Ser E-Tech Sci 52(12):3640–3648

Chapter 3

Flow Structures of Gaseous Jet in Supersonic Crossflow

Transverse injection into supersonic flow is one of the most fundamental canonical flows for supersonic propulsion community, which has been studied to enhance the understanding of supersonic turbulent mixing of jet fuel and combustion in scramjet engine combustors. It includes many flow features of interest, such as the threedimensionality, the shock structures, the flow separation and recirculation, the wallbounded free shear layer phenomenon, and the jet wakes. A typical topology of a sonic jet in supersonic crossflow from a circular orifice in the plate is shown in Fig. 3.1 (Dickmann and Lu 2008). Many interesting structures can be observed, such as shock waves, vortices, contact surfaces, shear layers, and their interactions. In the near-field region upstream of the jet orifice, the bow shock is generated by the impeding effect of the jet, inducing horseshoe vortex in the windward region. The jet forms a major counter-rotating vortex pair (CVP) that dominates the mixing of far-field regions. A barrel shock ends at the Mach disk. The most interesting parts are a local leeward separation bubble and the secondary shock. This chapter introduces the shock waves and vortex structures of sonic injection into supersonic crossflow and consists of three sections, including shock structures, upper trailing CRVs, and lower trailing CRVs. Large-eddy simulation (LES) by Chai et al. (2015a) successfully captured shock systems and vortex structures of an underexpanded sonic jet. Figure 3.2 shows density gradient contours on the symmetrical and horizontal slices, in which the turbulent boundary layer has been fully developed. And the bow shock, barrel shock, Mach disk and detailed vortex structures can be observed. The recirculation region upstream of the jet induces a series of compression waves that merge into a ‘separation shock’. Aλ shape shock consists of a separation shock and a bow shock. A barrel shock and Mach disk are formatted due to the penetration and expansion of the sonic jet. On the windward side of the jet, the shear layer rolls up into vortices which detach from the jet boundary and are shed downstream. Coherent flow structures are observed downstream of the jet after the jet/crossflow interaction. These coherent vortices appear to be the Kelvin–Helmholtz vortices that originate from two shear layers. One is the shear layer between the jet fluid that passes through the Mach disk and the windward barrel shock, and the other is due to the velocity © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_3

55

56

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Fig. 3.1 Schematic diagrams of a transverse jet in supersonic crossflow showing some of the dominant flow feature (Dickmann and Lu 2008)

Fig. 3.2 Three-dimensional view of instantaneous density gradient magnitude contours on the symmetry plane (showing shocks and compression waves) and horizontal plane close to the wall (showing the effect of the jet on the incoming turbulent features) for the sonic jet in supersonic crossflow (Chai et al. 2015a)

difference between the jet fluid that passes through the Mach disk and the ambient crossflow. Figure 3.3 shows the iso-surface of the second invariant of the velocity gradient tensor (Q-criterion) colored by the streamwise velocity (u), which visualizes the instantaneous vortical flow features. The flow is highly unsteady and composed of turbulent eddies of different sizes. Thin longitudinal vortices shed from the jet upstream shear layer are clearly visible. Also observed are hairpin-shaped vortices close to the wall downstream of the jet, which is characteristic of turbulent boundary layers (Chai et al. 2015a). Although the flowfield in the boundary layer was demonstrated, no further details were discussed on the jet wake flow. Won et al. (2010) ENREF_1 used detachededdy simulation (DES) to reveal the vortex evolvement under the jet conditions

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

57

Fig. 3.3 Isocontours of Q-criterion colored by streamwise velocity (u) depicting the instantaneous vortical flow features for the sonic jet (Chai et al. 2015a)

of Ben-Yakar’s experiment (Ben-Yakar et al. 2006). Chai et al. (2015b) conducted LES on flowfield of a sonic jet in a supersonic crossflow. In Wang et al. (2013) and Zhang et al. (2016), hybrid RANS/LES methods were validated and used to study three-dimensional jet mixing and the effects of micro-ramp on jet mixing in supersonic crossflow. Direct numerical simulations are conducted to study transverse jet in supersonic flows and the upper trailing CVP (TCVP) which is located above the major CVP is detailed (Sun and Hu 2018a). The main conclusion was that the upper trailing CVPs are related to baroclinic effects caused by the deflecting Mach disk. Combining experimental results with simulations, the shock structures and the jet vortices resulting from a sonic transverse jet injected into supersonic crossflow have been well-identified. In addition, formation mechanisms of the major CVP and the horseshoe vortices have also been well explained. In the near-wall jet wake flow, a TCVP is the generally agreed flow feature. Rana et al. (2011) discussed this TCVP and indicated that the TCVP is due to the low-pressure recirculation zone in the jet leeward and the suction action of the major CVP. The TCVP rotates in the opposite direction to major CVP and is dependent upon the freestream Mach number. Kawai and Lele (2010) inferred that TCVP is the pair of boundary layer separation vortices along the symmetric plane induced by the suction of major CVP. Chai et al. (2015b) presented in-plane streamlines which reflected TCVPs but they did not give further discussion. Viti et al. (2009) described the flow features in details and deduced that

58

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

TCVP originates from the second counter-rotating vortex of the upstream separation region and is energized by the shear layer of the barrel shock. From the above analysis, it is found that there is no widely accepted conclusion on formation mechanism of surface trailing CVP currently. Though many researchers referred to suction of major CVP and separation, no works were done to mark separation domain and identify TCVP located in favorable or adverse flow. And paradox largely exists in the description of different literatures. For example, in Fig. 3.1 secondary shock is located downstream of the leeward separation and the possible separation bubble induced by the secondary shock is neglected. And it is still argued whether TCVP originates from upstream separation (Viti et al. 2009) or leeward separation (Rana et al. 2011). Furthermore, the detailed evolution of TCVP in the far-field have not been investigated. As known, it is important to know the near-wall jet wake to establish a clear physical understanding of the jet in supersonic crossflow. Experiments have not been able to demonstrate this, while RANS or LES cannot capture detailed turbulence behavior in the near-wall region. In the present work, DNS is conducted to investigate a sonic jet in a supersonic crossflow at Ma  2.7 and to obtain the actual turbulence behind the jet near to the boundary layer. The injected air jet is modeled at two momentum flux ratios J  2.3 and J  5.5. Both cases lead to a sonic jet with a stagnation temperature T 0i  300 K. More details can be found in Sun and Hu (2018b). Besides, some interesting structures have been obtained, including shock wave structures and some vortices (such as upper trailing counter-rotating vortices, lower trailing counter-rotating vortices), which are detailed in the following three sections.

3.1 Shock Structures In this section, NPLS (Nanoparticle-based Planar Laser Scattering) (Zhao et al. 2009, 2016) and oil-flow techniques were employed to achieve experimental visualization. The typical structures, including a bow shock, a barrel shock, horseshoe vortex, and separation zones, were clearly observed by the NPLS techniques. Characters of gaseous jet in near-wall region were identified based on the oil-flow results. RANS (Reynolds-averaged Navier–Stokes) simulation was used to identify the intersection among the collision shock, the Mach disk, the reflected shock, and the barrel shock.

3.1.1 Experimental Visualization of the Transverse Jet Flow Structures This study combines the NPLS technology (developed by Zhao et al. 2009) and the oil-flow technology to conduct the experiments in the Mach 2.95 suction-type supersonic wind tunnel of the National University of Defense Technology. In the oil-

3.1 Shock Structures

59

Table 3.1 Airflow conditions for the experiment, including the dimensional boundary layer (BL) thicknesses and Reynolds number at the air inflow Mach number

Stagnation temperature

Stagnation pressure

BL 99% thickness

Reynolds number

Ma

T0

P0

δi

Reδ

2.95

300 K

101 kPa

6 mm

3.5 × 104

Fig. 3.4 Flow structures of a flat plate turbulent boundary layer measured by NPLS (Liang et al. 2018)

Table 3.2 Air jet conditions for the experiment Jet-to-crossflow momentum J

Orifice diameter D

Mach number M j

Stagnation temperature T 0j

Stagnation pressure P0j

Reynolds number Rej

7.7

2 mm

1.0

300 K

375 kPa

2.3 × 105

flow experiment, a special kind of oil is made into a pigment and spread evenly over the surface of the flat plate. The oil-flow slowly and eventually forms skin-friction lines due to the viscous nature of the fluid near the wall in the experiment. As shown in Table 3.1, the air inflow parameters are set in accordance with the Ma  2.95 with stagnation pressure P0  101325 Pa, stagnation temperature T 0  300 K. The freestream velocity U ∞ is estimated as 605 m/s, and the Reynolds number Reδ based on the inflow boundary layer thickness, is 3.5 × 104 . Figure 3.4 shows a NPLS image of the turbulent boundary layer. The boundary layer thickness is estimated as δ  6.0 mm, which is analyzed by Wang et al. (Zhao et al. 2009; Wang and Wang 2016; Wang et al. 2016) where the same experimental condition was used. In the experiment, nitrogen is injected at a sonic velocity from a jet orifice with a diameter of 2 mm. The jet properties are set to correspond to the injection parameter as shown in Table 3.2. All cases lead to a sonic jet with stagnation temperature T 0j  300 K. The momentum ratio J is regarded as an important parameter that has a dominant effect on the jet penetration. 2 J  ρ j V j2 /ρ∞ U∞

In the experiment, J is set to 7.7. The jet Reynolds number Rej  U j d/ν j is 2.3×105 .

60

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Figure 3.5 gives NPLS images on the symmetry slice and two vertical slices (y  2 mm and y  4 mm), showing the instantaneous flow structures of the sonic jet

Fig. 3.5 Instantaneous streamwise flow structures of sonic jet into supersonic crossflow at a z  0 mm, b y  2 mm and c y  4 mm slice (Liang et al. 2018)

3.1 Shock Structures

61

Fig. 3.6 Skin-friction lines from experimental oil-flow pattern (Liang et al. 2018)

into supersonic crossflow. From the symmetry slice, the bow shock and the inflow turbulent boundary layer upstream of the jet can be clearly observed. While the large-scale structures including K-H vortex and the highly turbulent wake structures dominate the downstream region. Nanoparticles shown in the wake structures come from the crossflow since nitrogen jet do not carry any nanoparticle. It is found that, from the vertical slices, the jet interacts intensely with the upcoming boundary layer. Around the two flanks of the jet, flow wrinkles are quite dense, and this is due to the stirring function of the horseshoe vortex, which originates from the upstream separation zone. Downstream of the jet, the crossflow that crosses the jet impinges in trailing wakes. Figure 3.6 is an instantaneous image of the oil flow. In the images of the oil-flow experiments, the separation zone, the reattachment zone, and the skin-friction lines are clearly observed. It is indicated that the V-shaped separation bubble is induced by the V-shaped collision shock. It is clearly seen from the enlarged view that the herringbone trailing recirculation occurs in the jet wakes regions, which consists of two oblique separation regions. Besides, the skin-friction lines could be seen, which

62

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

reflects the interaction between the boundary layer and the wall. Crossflow hits the bow shock and the flow lines deflect outward along the bow shock.

3.1.2 Wave Structures Study by Numerical Simulation Liang et al. (2018) used RANS simulation to identify the intersection among the collision shock, the Mach disk, the reflected shock, and the barrel shock. At the symmetrical interface, the reflected shocks can be divided into upper and lower parts and the lower part can interact with the collision shock and have influence on the flowfield as shown in Fig. 3.7 Pressure contours at (a) y/D  3 slice (b) z/D  0 slice (b). Figure 3.7 shows pressure contours at slices y/D  3 and z/D  0. It is clear that the barrel shock wrapping around the jet passes through the Mach disk and induces the reflected shock. The reflected shock is also barrel-like and spreads to the periphery. In the streamwise direction, the reflected shock can be divided into upper and lower parts, and the lower part intersects with the collision shock.

Fig. 3.7 Pressure contours at a y/D  3 slice, b z/D  0 slice (Liang et al. 2018)

(a) y/D=3 slice

(b) z/D=0 slice

3.1 Shock Structures

63

The location of the shocks is changing with the increase of the wall-normal distance rising so that the impact on the flowfield is different, depending on the intensity of the collision shock and the position of the reflected shock. In order to better analyze the three-dimensional structure of the collision shock, density gradient contour images are built, as shown in Figs. 3.8 and 3.9. Figure 3.8 is a density gradient perpendicular to the streamwise, including the x/D  4.5, 8 and

Fig. 3.8 Steady density gradient contours at a x/D  4.5, b x/D  8, c x/D  10 slices and steady Mach number contours on d x/D  10 slice (Liang et al. 2018)

64

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Fig. 3.9 Steady density gradient contours on different slices, a y/D  0.08, b y/D  0.5, c y/D  1 and d y/D  1.5 (Liang et al. 2018)

10 slices. As the transverse slice moves downstream, the barrel shock disappears and the collision shock gradually forms. On the x/D  10 slice (Fig. 3.8c), there is a curve with a higher density gradient below the collision shock, which converges from the wall to the central slice and reaches the highest point in the middle. Comparing Fig. 3.8c and d, it can be concluded that the curve is the shear layer of the jet and the crossflow in the wake region, which is beyond the trailing CVP. At the x/D  8 slice, the reflected shock, formed by the reflection of the barrel shock, and the collision shock begins to intersect. When the reflected shock intersects with the collision shock at x/D  10 slice, the reflected shock is deflected downward, and the shear layer becomes relatively flat after intersecting with the reflected shock. Figure 3.9 shows steady density gradient contours of the spanwise slice as a comparison with Fig. 3.8, starting from y/D  0.08 slice to y/D  1.5 slice. The width of collision shock decreases gradually in the spanwise direction and the collision shock is outside the shear layer. As the distance from the wall increases, the interval between collision shock and shear layer decreases gradually. As shown in Fig. 3.9c, d, both the collision shocks deflect to the opposite direction of the center line due to the interaction with the reflected shock. And the shear layer, influenced by the collision shock, is deflected in the same direction. It is interesting to note that the front of the collision shocks gradually intersects with the barrel shock with the increasing height. After the intersection, the barrel shock is deflected by the collision shock, creating a concave shape in the leeward zone, as indicated by the red circle.

3.1 Shock Structures

65

Fig. 3.10 Shock wave structures behind the transverse jet (Liang et al. 2018)

Figure 3.10 demonstrates the complex configuration of collision shock interaction with reflected shock. The collision shock wave surface originates from the collision of the supersonic flow running around the jet barrel. With the wall-distance rising, the collision shock intersects with the reflected shock which is produced when the barrel shock is reflected by the Mach disk. Besides, the collision shock intersects with the barrel shock slightly, which influents the shape of the Mach disk.

3.2 Upper Trailing Counter-Rotating Vortices (CRVs) In the research of Sun. et al. (2018a, b), direct numerical simulations were conducted to investigate physical structures of a transverse sonic air jet injected into a supersonic air crossflow at a Mach number of 2.7. Simulations were run for two different jetto-crossflow momentum flux ratios (J) of 1.85 and 5.5. The work aims at producing a detailed physical analysis of trailing CRVs in the jet interaction with supersonic crossflow (JISC) field. Such an analysis can improve the understanding of the relevant flow structures responsible for the mixing of the injectant with the crossflow. The air inflow parameters are set in accordance with the Ma  2.7 experiments of Sun et al. (2013) with stagnation pressure P0  101,325 Pa, stagnation temperature T 0  300 K. The bottom wall 99% boundary layer thickness, which is the same for all simulations, is estimated to be δ i  5.12 mm, with the compressible (including density variations) boundary layer displacement and momentum thicknesses δ *i  1.75 mm, θ  0.38 mm and Reynolds numbers Re*δ  15,367, Reθ  3337. Time-averaged analysis is presented for flow visualization in Fig. 3.11 showing the contours of streamwise vorticity at different x-locations for J  1.85 and J  5.5. It is seen that in the vicinity downstream of the jet, there are several pairs of TCRVs around the major CRVs in Fig. 3.11a and d. The upper and lower TCRVs

66

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

rotate in an opposite direction against major CRVs. The lower surface TCRVs mainly are attached on the wall surface and some induced counter-rotating vortices exist in the region between the major CRVs and the lower TCRVs. Those induced TCRVs develop and merge into the pair of the major CRVs in the downstream, as shown in Fig. 3.11b, c, and d, e). The major CRVs grow in size but decrease in magnitude. The definition of upper TCRVs here is not the same as that denoted by Viti et al. [5]. The upper TCRVs in this paper are similar to Trailing vortex 1 mentioned by Viti et al. [5]. The pair of upper TCRVs defined by Viti et al. could be indicated as top TCRVs in this study. The magnitude of top TCRVs is very weak and could not be well recognized in Fig. 3.11a and d. Viti et al. gave a slice of vorticity magnitude of trailing vortex 2 and did not show the rotating direction. The generation of the upper TCRVs and the detailed flow topology in three-dimensionality was not unveiled. It is not clear how the opposite rotation is energized. The lower TCRVs have been discussed a lot in previous studies and usually thought to be due to the low-pressure recirculation zone and the effects of the major CRVs. The upper TCRVs will be investigated in detail. To clearly demonstrate the CRVs, three-dimensional iso-surfaces of streamwise vorticity is shown in Fig. 3.12 for J  1.85 and J  5.5, respectively. In these figures, arrows with solid line point to major CRVs and arrows with dashed-line point to upper TCRVs. It is identified that the upper TCRVs are located above the major CRVs. For J  1.85, the upper TCRVs extend only in a very short streamwise distance, which means that the pair of upper TCRVs are a common feature but depend on J. Higher J leads to a more apparent upper TCRV phenomenon. It is concluded that the upper TCRVs have an opposite rotating direction against the major CRVs. As the jet plume evolves, the upper TCRVs disappear and only major CRVs exist in the far-field, which reflects a merging process in the downstream region. Time-averaged contours of Mach number on the z/D  0 slice with streamlines are shown in Fig. 3.13a, b for J  1.85 and J  5.5. The bow shock and barrel shock are clear in these figures. The solid line arrow points to the Mach disk. The three-dimensional barrel shock and Mach disk are shown in Fig. 3.13c for J  5.5. Transparent slice of z/D  −1.0 contoured with Mach number is also shown in Fig. 3.13c. Slice of z/D  −1.0 corresponds to central location of the upper TCRV shown in Fig. 3.11d. As shown in Fig. 3.13c, behind the Mach disk, flow is recompressed to a lower Mach number and the slow-moving injectant fluid comes in contact with the high-speed crossflow fluid aft of the Mach disk. Stream ribbons colored by local streamwise vorticity are given in Fig. 3.14 to demonstrate the trailing CRVs in the jet plume. Stream ribbons roll along different directions, which reflect the local vorticity. In Figure 3.14a, b, the ribbons exhibit that upper TCRVs are related to the Mach disk formed in the expanded jet plume. Black arrows shown in Fig. 3.14a indicate the lateral rolling direction. As the stream ribbons penetrate the lateral side of the Mach disk, the stream gets energized and acquires enough torque from the local shear field to twist. This twisting corresponds to the generation of the upper trailing vortex. In the region near to the center line plane, the ribbons are not obviously twisted, which demonstrates that there is no strong torque generation in the center line plane of the Mach disk. The major CRVs

3.2 Upper Trailing Counter-Rotating Vortices (CRVs)

67

Upper TCRVs Major CRVs

Lower TCRVs

(a) J=1.85, x/D=3

(b) J=1.85, x/D=6

(c) J=1.85, x/D=9

Induced TCRVs

(d) J=5.5, x/D=6

(e) J=5.5, x/D =12

(f) J=5.5, x/D =18

Fig. 3.11 Time-averaged streamwise vorticity contours at streamwise slices at different axial locations for J  1.85 and 5.5, respectively (Sun and Hu 2018a)

originate from the lateral shear layer between the barrel shock and the freestream, as shown in Fig. 3.14c). The stream ribbons come from the lateral side of the injector and twist when going around the jet barrel shock. Part of the ribbons come from the crossflow and are entrained to the edge of the major CRV region. It is inferred that as the barrel shock detaches from the wall surface, it creates the low-pressure region which absorbs the lateral flow. As shown in Fig. 3.14a–c, the stream ribbons of the upper TCRVs and major CRVs bind together into a rope-like structure in the far-field, which represents that they merge into a single vortex as they trail downstream. These ribbons twist and roll around a vortex core in a common longitudinal axis, which corresponds to the major merged vorticity shown in Figs. 3.11f and 3.12b, 3.14c

68

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

(a) J=1.85

(b) J=5.5

Fig. 3.12 Iso-surfaces of time-averaged streamwise vorticity (x) for J  1.85 and 5.5 respectively, dark iso-surface corresponds to x  −2.0, grey iso-surface corresponds to x  2.0, non-dimensionalized by freestream velocity and characteristic length (D). Solid line arrows point to major CRVs and dashed-line arrows point to upper TCRVs (Sun and Hu 2018a)

demonstrates the lower TCRVs and the induced TCRVs located between the major CRVs and the lower TCRVs. Ribbons of the lower surface TCRVs move across the separation bubble and run into the low-pressure region behind the jet. The magnitude of rotation of the ribbons is very large since a highly twisted structure exists in the ribbons. The ribbons of the induced TCRVs originate from the crossflow near to the symmetric plane and they gain strength as they are convected downstream of the lateral side of the barrel shock plume. Figure 3.15 present front views of stream ribbons which penetrate the upper TCRVs, major TCRVs, induced TCRVs, and lower surface TCRVs. The representative stream ribbons also correspond to the ribbons shown in Fig. 3.14a–c. Stream ribbons penetrating the major CRVs in the x/D  6 slice follow the rotating motion and swirl into the CRV core in the far-field. Stream ribbons penetrating the upper TCRV center are rolled by the major CRVs, merge into major CRVs and arrive at a higher height in the downstream. Stream ribbons penetrating the induced TCRVs are rolled from a position under the major CRVs to a top position due to the effects of major CRVs. An interesting phenomenon is that stream ribbons penetrating the lower surface TCRVs are not rolled into the major CRVs and only get a slight lift near the outlet of the flowfield, which indicates that the lower TCRVs are not affected significantly by major CRVs and only dissipate in the far-field. Generation of major CRVs, induced TCRVs and lower surface TCRVs are not the main point of this study and further analysis of them is omitted here. Figure 3.16a, b give the Mach number iso-surface superimposed with streamlines 1–4 for J  5.5. Streamlines 1 and 3 are colored by local streamwise vorticity and indicate that jet fluid penetrates the upper trailing vortex center. Streamlines 2 and 4 originate from the lateral side of the jet orifice and penetrate the major CRVs. Stream-

3.2 Upper Trailing Counter-Rotating Vortices (CRVs)

69

(a) J=1.85, slice of z/D=0.0 and arrow points to Mach disk

(b) J=5.5, slice of z/D=0.0 and arrow points to Mach disk, same legend with (a)

finlike structure

(c) J=5.5, iso-surface of the barrel shock (Ma=3.1) and z/D=-1.0 slice, same legend with (a) Fig. 3.13 Time-averaged contours and iso-surface of Mach number of J  1.85 and 5.5 (Sun and Hu 2018a)

70

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

(a) stream ribbons penetrating the Mach disk, shown with slice x/D=6 (left) and transparent iso-surface of barrel shock (right), colored by streamwisevorticity. Black arrows point out the rotating direction.

(b) stream ribbons running around the jet barrel shock, shown with slice x/D=6 (left) and transparent iso-surface of barrel shock (right) Fig. 3.14 Stream ribbons in the jet plume to demonstrate the TCRVs of J  5.5, colored by local streamwise vorticity (Sun and Hu 2018a)

3.2 Upper Trailing Counter-Rotating Vortices (CRVs)

71

(c) lower TCRVs near to the wall surface and the induced TCRVs,shown with slice x/D=6 (left) and transparent iso-surface of barrel shock (right) Fig. 3.14 (continued)

(a)

(b)

Upper TCRVs

Lower TCRVs

Major CRVs

Induced TCRVs

Fig. 3.15 Stream ribbons of J  5.5 to demonstrate the merging process of the upper TCRVs and the induced TCRVs with major CRVs (a) and the motion of lower surface TCRVs (b), colored by local streamwise vorticity, shown with slice x/D  6. Same legend as Fig. 3.11 is used (Sun and Hu 2018a)

72

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

lines 2 and 4 are rolled by the merged CRVs. The voriticities shown on streamlines 1 and 3 are opposite, similar on streamlines 2 and 4, which demonstrates the counterrotating characteristics. The voriticities on streamlines 1 and 2 are opposite in the same lateral side, which means that the generation mechanism of streamlines 1 and 2 is different. The barrel shock is clearly shown in Fig. 3.16b by the iso-surface of Mach number 2.8. Figure 3.16c gives the barrel shock and streamlines of J  1.85 and the flow topology is similar to J  5.5. The contrary part is that streamlines from the Mach disk of J  1.85 quickly have a sign change in vorticity since the upper trailing vorticity is weak and the upper TCRVs get merged into the major CRVs in a short distance away from the barrel shock. A summary of the vortical structures in the present study is shown in Fig. 3.17. This figure shows a schematic of the crossflow section at a location aft of the barrel shock. A system of four pairs of counter-rotating vortices forms ahead of the injector, along the barrel shock wave and immediately downstream of the Mach disk. The pair of upper TCRVs is formed by the interaction of the jet fluid passing through the Mach disk with the crossflow running around the barrel shock. The upper and induced vortex systems trail downstream and finally merge into the major CRVs in the downstream far-field. The lower surface TCRVs are attached to wall and dissipate in the far-field without apparent interaction with major TCRVs. Figure 3.18 demonstrates the local pressure and density contour on the x/D  4.5 slice of the J  5.5 case, which reflects the circumstance streamlines 1–4 running through. As the barrel shock from the wall surface blocks the mainstream, it creates the low-pressure and low-density region on the jet leeward side. It is seen that streamlines 1 and 3 are located near the bounds of the low-pressure and density region which means high pressure and density gradient at this location. Figure 3.18c gives the contour of the local streamwise vorticity production via baroclinic term on x/D  4.5 slice, which reflects the circumstance near to the region aft of the Mach disk. It is seen that streamlines 1 and 3 experience a definitely different region of baroclinic torque from streamlines 2 and 4. Streamlines 1 and 3 penetrate the region with high magnitude of the baroclinic term, while streamlines 2 and 4 penetrate the region with a nearly zero baroclinic term. To clearly show the baroclinic term on the streamlines, Fig. 3.19 is given with iso-surface of Mach number for identifying barrel shock and Mach disk. It is seen that streamlines 2 and 4 gain vorticity production at the windward side of the barrel shock and the vorticity production drops significantly at the lateral of the barrel shock. Streamlines 1 and 3 gain the vorticity production at aft region in the lateral of the Mach disk and the voriticity production is opposite to each other. Recalling Fig. 3.14a, the sign of the vorticity production just corresponds to the rotation direction of the stream ribbons. Streamline 5 shown in Fig. 3.19 is located near to the center line plane of the jet plume. The vorticity production on streamline 5 approaches zero in the jet barrel shock and after the Mach disk. This indicates that the vorticity on streamlines 1 and 3 is due to the shear effects between the flow penetrating the Mach disk and the high-speed flow from the jet lateral side. As streamlines 1–4 convect downstream, the vorticity production decreases to zero, which means there is no further baroclinic torque pumped into the vorticity equation. In the downstream region, vorticity stretches

3.2 Upper Trailing Counter-Rotating Vortices (CRVs)

73

(a)

2 1

3 4

(b)

(c) 2 1 3 4

Fig. 3.16 Iso-surface of Mach number (Ma  2.8) demonstrating barrel shock and Mach disk, combining with streamlines colored by local streamwise vorticity, a oblique view of the streamlines originating from jet orifice of J  5.5 with iso-surface of Mach number (Ma  2.8), b Iso-surface of Mach number (Ma  2.8) of J  5.5, c Iso-surface of Mach number 2.8 is set transparent to show the streamlines of J  1.85 (Sun and Hu 2018a)

or bends and the streamlines only inherit the rotation momentum acquired in the near-field region. Streamline 1 is energized by positive vorticity production aft of the Mach disk. But as the upper TCRVs merges into the major CRVs by their suction, the vorticity changes into a negative value, as shown in Fig. 3.16b. It is inferred that the opposite rotating direction of the upper trailing vorticity against the major CRVs is induced by baroclinic production aft of the Mach disk and disappears downstream by the suction of the major CRVs. Based on the numerical results and the analysis, it is concluded that the threedimensional jet flow structures lead to different vorticity production and local stream-

74

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Mach disk Upper trailing CRVs

Major CRVs Merged CRVs

Induced vortex Lower trailing CRVs

Fig. 3.17 Schematic of the flowfield at a transverse section aft of the barrel shock (Sun and Hu 2018a)

wise rotating directions in the spanwise cross-section. The pair of upper trailing CRVs has a different generation mechanism than the major CRVs that originate from lateral plume. The lateral flow of the Mach disk rotates in an opposite direction against the major CRVs due to the local baroclinic torque. All the trailing CRVs (lower TCRVs excluded) and the major CRVs merge into a pair of major CRVs in the downstream far-field. The definition of upper TCRVs in the paper of Sun et al. is not the same as that denoted by Viti et al. [5]. The upper TCRVs in the paper of Sun et al. are similar to Trailing vortex 1 mentioned by Viti et al. [5]. The pair of upper TCRVs defined by Viti et al. could be indicated as top TCRVs the paper of Sun et al. In the study of Viti, one of these trailing vortices stemming from the separation region isthe upper trailing vortex. This vortex is formed by the recirculating fluid close to the plane of symmetry, and it follows the leading edge of the barrel shock away from the solid surface.

3.3 Formation of Surface Trailing Counter-Rotating Vortex Pairs Downstream of a Sonic Jet in a Supersonic Crossflow 3.3.1 Instantaneous Flow Structures in Jet Near-wall Wakes Typical instantaneous results of λ2 iso-surface in the jet flowfield of J  5.5 and J  2.3 are shown in Fig. 3.20, colored by the local density. It shows clearly the mixing process of the low-density jet plume with the mainstream and the incoming boundary layer. Jet plume density is low in the near-field and increases as the jet plume is convected downstream. The sonic jet with higher J (=5.5) has a more intense interaction with the incoming flow, which results in a larger separation region

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

Streamline 1

Streamline 2

Streamline 3

Streamline 4

(a) pressure contour on slice at x/D=4.5, J=5.5

Streamline 1

Streamline 2

Streamline 1

75

Streamline 3

Streamline 2

Streamline 4

(b) density contour on slice at x/D=4.5, J=5.5

Streamline 3

Streamline 4

(c) contour of baroclinicstreamwisevorticity production on slice at x/D=4.5, J=5.5 Fig. 3.18 Density and pressure contours on slices with different locations aft of the barrel shock, lines of streamwise vorticity distribution are superimposed on Fig. 3.15a–b) (Sun and Hu 2018a)

76

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Fig. 3.19 Iso-surface of Mach number (Ma  3.1) demonstrating barrel shock and Mach disk, combining with streamlines colored by local baroclinic vorticity production (Sun and Hu 2018a)

2 1 3 4

5

2 1 5 3 4

and a higher penetration height. Typical instantaneous simulated results of density contours at z/D  0.0 in the flowfield of J  5.5 and J  2.3 are shown in Fig. 3.21. In particular, the shock and turbulent structures are well-identified. In Fig. 3.21 it is seen that twisted envelope shock appears behind the jet in both cases and it is clearer in J  2.3 case. As described in many literatures (Mahesh 2013; Morkovin et al. 1952), the envelope shock originates from the pressure gradients around the expanding jet. A strong mixing occurs in the jet wake flow and turbulent structures are identical both in near-field and far-field. Near-wall density fields are shown in Fig. 3.22 on horizontal planes. Turbulent streaks are exhibited in Fig. 3.22a, b at y/D  0.08, which reflect the local turbulence, and their response to the jet injection will be analyzed in another paper. There appear herringbone separation zones (wrapped by solid lines representing zero streamwise velocity) which extends both in streamwise and spanwise direction, as shown in Fig. 3.22 at y/D  0.08 slices. In the lateral downstream of the jet orifice, local density magnitude decreases significantly, which implies that expansion occurs near the jet leeward. High-velocity fluid moves downstream around the jet and interacts with the shock generated by the collision of the flow around the jet barrel. The collision shock waves are very

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

77

(a) J=5.5

(b) J=2.3 Fig. 3.20 Instantaneous iso-surfaces of vortex structures (λ2  −0.3) at J  5.5 and J  2.3 colored by local density (Sun and Hu 2018b)

78

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

(a)

(b)

Envelope shock

Fig. 3.21 Density contour at z/D  0.0 slice of jet interaction with supersonic crossflow, a J  5.5, b J  2.3, isoline of u/U ∞  0.0 is superposed in red color (Sun and Hu 2018b)

clear in density contours on y/D  1.0 slice of J  5.5 and y/D  0.75 slice of J  2.3 and they are twisted due to local turbulence. It is sure that these collision shocks induce the herringbone separation in the near-wall region and they have a complex cross interaction. The collision shock exhibits a three-dimensional configuration since the separation zone is herringbone and there are two separation oblique wings. In Fig. 3.22a, b, it is seen that the herringbone separation region is confined and flow reattaches downstream of the cross point of the wings. In the vicinity of the symmetric plane, a recovery region is embayed by the herringbone separation region near the center line. The herringbone separation region for J  5.5 is larger than the J  2.3 case and their flow patterns are similar.

3.3.2 Mean Flow Properties in the Jet Near-wall Wake Flowfield Time-averaged density contours at z/D  0.0 slice are shown in Fig. 3.23. It is seen that the averaged flow patterns are similar for J  5.5 and 2.3. The penetration height, barrel shock size, and jet leeward separation size of J  5.5 is larger than J  2.3. The envelope shock related to expanding jet barrel is clear on time-averaged contour. Skin-friction coefficient contours are shown at y/D  0.0 in Fig. 3.24a, b. The solid isolines superposed on contour correspond to u/U ∞  −0.0001 and the region surrounded by isolines represents a separated zone. It clearly shows that the mean herringbone separation region is limited in the cramp of the collision shock waves. In the vicinity of the symmetric plane, flow reattaches more quickly than outside, resulting in a V-shaped reattachment line (as shown in Fig. 3.24a) behind

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

79

(a) density contour at y/D =0.08 slice of J=5.5

(b) density contour at y/D =0.08 slice of J=2.3

Collision shock

(c) density contour at y/D =1.0 slice of J=5.5

Collision shock

(d) density contour at y/D=0.75 slice of J=2.3 Fig. 3.22 Instantaneous streamwise velocity streaks at different horizontal planes of jet interaction with the incoming boundary layer for J  5.5 and J  2.3 cases, isolines of u/U ∞  0 are superposed (Sun and Hu 2018b)

80

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

(a)

(b)

Envelope shock

Fig. 3.23 Time-averaged density contours on z/D  0.0 plane for a J  5.5 and b J  2.3 case, isolines of u/U ∞  0.0 are superposed

the separation zone. The herringbone separation size of J  5.5 is larger than J  2.3 and it is expected that maximum width of the separation is related to J. The interesting part is the half-span declining angle of the mean separation line (drawn by a solid line in Fig. 3.24a, b) with jet center line. A comparison is given between J  2.3 case and J  5.5 case and it is found that θ 1 shown in Fig. 3.24a of J  5.5 is basically equal to θ 2 shown in Fig. 3.24b of J  2.3, that is, θ 1 ≈ θ 2 ≈ 15.6°. The feature demonstrates that separation line is independent on J. This could be expected since separation line is mainly induced by leeward collision shock and collision shock angle is only dependent upon the freestream Mach number. The increase of J or orifice diameter would not change the collision shock angle and half-span declining angle of separation line. The deflection angle θ 3 (≈28.0°) of the V-shaped reattachment line (shown in Fig. 3.24a) with jet center line is larger than θ 1 and θ 2 . In Fig. 3.24c, d, time-averaged collision shock waves and reattached shock waves are identified. Reattached shock locates behind the collision shock and between them exists a low-density field, which reflects local expansion and reattachment in separation/shock interaction. As discussed above, reattached shock waves reveal the three-dimensionality of the herringbone separation zone. Figure 3.25 represents the streamlines on the crossflow planes downstream of the jet plume for J  2.3 and J  5.5, respectively. The major CVP appears due to the interaction of the freestream flow with the jet fluid and grows in size further downstream. As the major CVP starts to grow in size as in Fig. 3.25 another small pair of vortices, named lower trailing CVP, occurs near to surface. The lower trailing CVP is denoted as surface trailing CVP in this paper, and for brevity it is referred to as trailing CVP (TCVP). This TCVP is usually thought to be due to the low-pressure recirculation zone and the suction effect of the major CVP. Figure 3.25 shows that

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

81

θ1 θ3 V-shape reattached zone

(a) skin friction coefficient contour at y/D=0.0 slice of J=5.5

θ2

(b) skin friction coefficient contour at y/D=0.0 slice of J=2.3

Reattached shock Collision shock

(c) density contour at y/D=1.0 slice of J=5.5

Reattached shock

Collision shock

(d) density contour at y/D=0.75 slice of J=2.3 Fig. 3.24 Time-averaged skin-friction coefficient and density contours on different horizontal planes for J  5.5 and J  2.3 case, isolines of u/U ∞  0.0 are superposed (Sun and Hu 2018b)

TCVP rotates in the opposite direction to the major CVP and disappears as the jet plume lifts up downstream, which is an apparent feature for the far-field. The interesting part lies in the correlation of separation bubble with TCVP. Separation lines of u/U ∞  0.0 are shown in Fig. 3.25a, b and there is no separation zone in Fig. 3.25c, d. The enlarged view of the window in Fig. 3.25a clearly demonstrates that TCVP mainly locates in the separation bubble, or in the recirculation zone based on the separation bubble. This phenomenon is definitely in existence for both J  5.5 and J  2.3 case. In Fig. 3.25b, the separation bubble is pushed aside from the center line and its size decreases to be very small.

82

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

J=5.5

u=0

J=5.5

u=0

J=2.3

u=0

J=2.3

u=0

(a) x/D=3.5 for J=5.5 and x/D=3.5 for J=2.3

J=5.5

J=2.3

(c) x/D=12 for J=5.5 and x/D=8 for J=2.3

(b) x/D=8 for J=5.5 and x/D=5.5 for J=2.3

J=5.5

J=2.3

(d) x/D=18 for J=5.5 and x/D=12 for J=2.3

Fig. 3.25 Contours of the mean flow streamwise velocity and streamlines on cross-planes of J  5.5 and J  2.3, isolines of u/U ∞  0.0 are superposed (Sun and Hu 2018b)

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

83

Combing with Fig. 3.24, it is found that the separation bubble has a threedimensional configuration. In the jet leeward, the separation bubble has a large size especially in height. In the separation wing of the herringbone bubble, though the separation covers a large area, the height of the separation bubble is small and the whole separation region is attached to wall surface. The TCVP in Fig. 3.25b is very clear to occur in domain with u/U ∞ > 0.0, which is obviously kept outside of the separation bubble and it is quite different from TCVP shown in Fig. 3.25a. This is very interesting since no researchers mentioned this before. Literatures about jet in supersonic crossflow usually gave fuzzy description about TCVP in the jet leeward and led to an impression that TCVP is a single character both in the near-field and far-field due to the suction of the major CVP. A careful observation exhibits that in the separation bubble of J  2.3 case in Fig. 3.25b, there is still a recirculation zone around small separation bubble. And it is clear that TCVP near to center line is not in the recirculation zone but kept some distance from it. In Fig. 3.25c separation bubble vanishes and TCVR is totally located in favorable flow region. In Fig. 3.25d TCVP disappears completely in the far-field. Figure 3.26 gives the streamwise vorticity contours at different x-locations. It is seen that the magnitude of vorticity of TCVP is comparable to major CVP in the jet vicinity and even larger than major CVP at x/D  8. The vorticity of surface TCVP in Fig. 3.26a and b has the same sign while their configuration is varied. Thus very interesting questions are presented here, such as, whether the TCVP shown in Figs. 3.25a and 3.26a is occurring in the same vortex tubes with the TCVP shown in Figs. 3.25b and 3.26b, since the two TCVPs shown in Fig. 3.25a and b or Fig. 3.26a and b are located in the separated and reattached regions, respectively. Since Figs. 3.25 and 3.26 only demonstrate the 2D flow character on different cross-section slices in the flowfield, it is difficult to answer this question before detailed three-dimensional flow topology is unveiled. From Figs. 3.23, 3.24, and 3.25 it is seen that flowfields of J  2.3 and J  5.5 are similar. Since J  5.5 case provides stronger jet interaction, the J  5.5 flowfieldis carefully examined. First, velocity profiles at the same wall-normal distance are shown in Fig. 3.27 for different streamwise locations. The velocity profiles have a W-shape (the arrow points) corresponding to the herringbone separation. It is seen that in Fig. 3.27a, b at x/D  3 and 6, the velocity downstream of the jet is negative (dottedline in Fig. 3.27 represents u/U ∞  0.0), which corresponds to a recirculation. Size of recirculation region approaches nearly zero at y/D  0.4, as shown in Fig. 3.27c. Along the streamwise direction on y/D  0.08 plane, the velocity increases and at x/D  9 a positive velocity field is observed. At x/D  14, it is seen that the velocity profile has basically recovered and an earlier recovery occurs in the reattachment region near to symmetric plane. At other wall-normal locations (y/D  0.4 and y/D  0.6), this phenomenon is also apparent. The W-shaped width in velocity profile along wall-normal direction converges significantly from y/D  0.08 to y/D  0.6, which means the region affected by the jet is decreased in y-axis. On y/D  0.6 plane, the velocity profile at x/D  6 is completely positive, which indicates the recirculation is located under y/D  0.6 plane. Separation width from the velocity

84

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

J=5.5

J=2.3

(a) x/D =3.5 for J =5.5 and x/D =3.5 for J =2.3

J=5.5

J=2.3

( b) x/D =8 for J =5.5 and x/D =5.5 for J =2.3

Fig. 3.26 Contours of the time-averaged streamwise vorticity on cross-planes of J  5.5 and J  2.3 (Sun and Hu 2018b)

profiles also converges from y/D  0.08 to y  0.2 plane, which reflects that the separation is likely to be a bump on the surface. Several conclusions could be made from above analysis. The first is that TCVP shown in Fig. 3.25a and b, c is located in totally different velocity regime. TCVP in Fig. 3.25a is in a separation region while the TCVP in Fig. 3.25b, c is in a zone with positive velocity. From this point, these TCVPs are inevitably different in formation. It is appropriate to denote TCVP shown in Fig. 3.25a as primary TCVP and that in Fig. 3.25b as secondary TCVP. The second is that separation region has a complex three-dimensional configuration, the edge of the separation region is located below y/D  0.6 and separation bubble has a decreased width along y-axis. Time-averaged pressure and Mach number contours at various locations along with streamlines are shown in closer view in Fig. 3.28, which are consistent with the analysis in Fig. 3.27. Separation line of u/U ∞  −0.0001 is shown in Fig. 3.28a with solid line. It is seen that the herringbone recirculation has two wings, which corresponds to two oblique symmetric separation regions. Between the wings, there is a zone with positive velocity, which means that a reattachment occurs and it is embayed by the separated zone, as shown in the dotted-line window. A reattached point with u/U ∞  −0.0001 in the center line is also marked in Fig. 3.28a. The streamlines clearly show the inner flow structure of recirculation zones. The pressure

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

85

(a) y/D=0.08

(b) y/D=0.2

(c) y/D=0.4

(d) y/D=0.6

Fig. 3.27 Time-averaged velocity profile at different wall-normal location of J  5.5 case, dottedline represents u/U ∞  0.0 (Sun and Hu 2018b)

contour shown in Fig. 3.28a demonstrates that a low pressure exists in jet leeward due to the blockage and expansion effects from the jet. Crossflow runs around the jet and impinges together in the jet leeward, which leads to a local pressure increment due to the collision effects. It is identified that pressure after the separation line drawn in Fig. 3.28a has an increment which implies a shock/boundary layer interaction occurring. It is observed in Fig. 3.28b that the crossflow near the wall in the jet lateral is supersonic. This could be imagined since separation region ahead of the jet pushes crossflow high away from surface and a sudden expansion after the separation bubble occurs in the negative y-axis. The crossflow running into the jet leeward experiences further expansion in the spanwise direction. These two types of expansion cause the flow speed to become supersonic, and it is sure that the collision shock would have a foot standing in the recirculation. As shown in Fig. 3.28a–c, when the wall-normal

86

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

distance increases from y/D  0.0 to 0.5, the separation region ends further upstream and this is due to the three-dimensionality of the separation bubble. At y/D  0.5, there is no obvious recirculating flow on this slice. The separation region in the jet wake and the reattachment region can also be identified in Fig. 3.28d, which shows Mach number contours and a superposed separation line. In Fig. 3.28, it is confirmed again that the TCVP in Fig. 3.25a, b are different vortices since the velocity field is completely different. The counter-rotating character of primary TCVP is not only related to the recirculation in the spanwise direction but also connected to the recirculating flow in an adverse streamwise direction. From Fig. 3.28a, b, it is seen that the herringbone region fades some distance behind the separation wing. It is expected since streamlines running around the jet impinge on the wingtip of the herringbone separation and result in a wing interception, which indicates that the interaction intensity of major CVP reduces to be very little at the wingtip location since major CVP has been lifted to a high position. Due to the impingement of the lateral supersonic flow around the jet, the trailing wingtip is converged. At the same time the reattachment flow extends in the reattachment zone (shown in the dashed-line window of Fig. 3.28a), impinges the separation bubble and blows off the near-wall separated flow. It is identified since in Fig. 3.28a streamlines in the reattachment region run into separation wing region and the downstream zone in the far-field. The cross interaction in this wingtip region leads to the termination of the oblique trailing separation wings. Downstream of the vicinity of reattachment region, there is an expansion zone in which diverging streamline impinges with crossflow and forms a slip line, as shown in Fig. 3.28a. This expansion zone is attached to wall since no such phenomenon is found in Fig. 3.28c. The reattachment supersonic flow impinges on the wall and leads to a pressure increment in the downstream region of the herringbone separation bubble. As shown in Fig. 3.24c, weak reattachment shock is generated after the flow expansion over the separation wing. Iso-surfaces of the streamwise velocity of J  5.5 and J  2.3 cases are shown in Fig. 3.29 to demonstrate the three-dimensional near-wall wake structure, colored by the local wall-normal distance. The three-dimensional separation bubble looks like a herringbone hill. The jet leeward separation bubble is a united region and further downstream there are two separate separation wings. The peak of the oblique wing corresponds to the edge of the separation zone. The V-shaped valley between the separation wings corresponds to the reattachment region, which has an obvious threedimensional structure. It is expected that the flow experiences an expansion when going over the top of the separation hills and falls into the V-shaped reattachment valley. Combining with the analysis of two-dimensional streamlines on y/D  0.2 plane, as shown in Fig. 3.28b, it is indicated that the reattached flow directly supports the secondary TCVP downstream of the reattachment valley, which is different from primary TCVP. J  2.3 case is shown in Fig. 3.29b and it is seen that lower J case has a similar character of flow configuration, though the size is smaller than the J  5.5 case. A careful observation shows that the size of jet leeward separation of J  2.3 case is almost kept unchanged comparing to J  5.5 case, while the separation wing is much shorter. This phenomenon further confirms that collision shock is independent upon J, which leads to a leeward separation in similar size.

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

Separation line

87

Reattachment point Slip line

Jet leeward region

Expansion

(a) pressure contour with streamlines at y/D=0.0

Slip line

(b) Mach number contour with streamlines at y/D=0.2

(c) Mach number contour with streamlines at y/D=0.5

(d) Mach number contour with streamlines at mid-span plane z/D=0.0 Fig. 3.28 Contours of time-averaged pressure and Mach number contours and streamlines on different planes of J  5.5. Separation line u/U ∞  0.0 is superposed (Sun and Hu 2018b)

88

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

(a)

Separation wing

Reattachment Valley

Leeward separation bubble

(b)

Leeward separation bubble

Fig. 3.29 Iso-surface of streamwise velocity (u/U ∞  0.0) colored by local wall-normal distance (y/D) of a J  5.5 and b J  2.3 cases (Sun and Hu 2018b)

The schematic of recirculation zone structures in jet wake flow is illustrated in Fig. 3.30. The bow shock ahead of the jet and the horseshoe vortex that wraps around the base of the jet and the major CVP are well known, thus they are omitted and will not be discussed in detail in this paper. Figure 3.30a shows a classic flow topology of an oblique shock wave interaction with a supersonic boundary layer, which demonstrates the recirculating flow has an opposite direction to the incoming flow. After the separation bubble peak, expansion occurs and is followed by a reattachment shock. Reattachment shock in Fig. 3.24c, d and pressure increment in Fig. 3.28a can be explained by this schematic. Figure 3.30b demonstrates the complex configuration

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

89

of collision shock interaction with the lateral supersonic boundary layer. It should be mentioned that since collision shocks have been twisted by the induced separation, shock structures could not be observed as regularly as described in Fig. 3.30b. The collision shock surface is located under major CVP and oblique shock surfaces are generated along the streamwise direction vertically to the wall. Since the separation bubble is induced in the jet leeward, the flow behind collision is also blocked by the separation bubble and a shock surface is formed in the wall-normal direction. These shock surfaces merge and form a three-dimensional unified half-cone-like shock surface, as shown in red color in Fig. 3.30b. As shown in Fig. 3.30b, the induced herringbone recirculation extends in the spanwise direction like a foot of the incident collision shock into the incoming lateral supersonic flow. Behind the oblique separation wing, the reattachment domain colored in grey in the sketch could be deduced to exist in the V-shaped valley. It is predicted that at least three streams come into the reattachment valley. Two streams are from the flow running around the jet lateral and across the oblique separation wing, and the other one is from flow going through jet leeward and across the reattachment point. In the next section, three-dimensional streamlines will be given to demonstrate these flow patterns. Figure 3.30b shows the separation and demonstrates the TCVP on slice cuts in the recirculating flowfield, which is consistent with primary TCVP assumption. This means that primary TCVP occurs in the separation bubble to rotate and recirculate. Secondary TCVP cannot be explained by using Fig. 3.30b. In the next section, three-dimensional streamlines are used to analyze the flow topology, especially to discuss secondary TCVP.

3.3.3 Flow Topology Analysis of Surface TCVPs In this section, three-dimensional time-averaged streamlines running into jet wake region are illustrated and their flow patterns are analyzed. The key purpose is to unveil the mechanism of formation of surface TCVPs. Figure 3.31 gives representative streamlines which originate from an oblique line parallel to separation wing on the iso-surface of u/U ∞  0.05 in the jet lateral. These streamlines reflect flow motions in the jet wakes. It is found that different types of flow patterns form in the wake. Some streamlines run around the jet barrel and concentrate in the jet leeward and rise up into far-field. Some streamlines run across the leeward separation bubble, recirculate and converge at the jet leeward. Some others run across the separation wing, rotate and move into downstream region. It is necessary to understand clearly flow motion in the jet wakes, especially the flow topologies corresponding to surface TCVPs. Figure 3.32 demonstrates different types of detailed flow patterns separately for clarity. Streamlines are generated in the same way with Fig. 3.31. Figure 3.32a shows the streamlines originating from the supersonic crossflow converging in the jet leeward and going directly downstream, which is denoted as Type I pattern. These streamlines rise up from the jet leeward following the suction of the major CVP and

90

3 Flow Structures of Gaseous Jet in Supersonic Crossflow Reflected shock Incident shock Expansion wave Boundary layer Compression wave

Reattachment shock Separation bubble Sonic line

(a) classic schematic of an incident oblique shock interacting with a supersonic boundary layer, modified from Ref(Gaitonde 2015) Collision Shock

en

tz

on

e

Reattachment point

Re

att

ac

hm

Horseshoe vortex

Separation wing Recirculation zone Streamlines Leeward separation bubble Jet orifice

(b) separation induced by collision shocks and primary TCVP on a slice cut from the herringbone recirculation zone Fig. 3.30 Schematic of herringbone oblique recirculation zones induced by collision shock waves behind the transverse jet injected into supersonic crossflow (Sun and Hu 2018b)

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

91

Streamlines originating from this line on isosurface of u/U∞=0.05

Fig. 3.31 Flow topology demonstrated by three-dimensional streamlines in the jet wake of J  5.5, with iso-surface of streamwise velocity (u/U ∞  0.05) colored by local wall-normal distance (Sun and Hu 2018b)

run into the core of the jet plume in the far-field. Analysis shows that these streamlines mainly originate from the supersonic crossflow in the vicinity of the symmetric plane. Combining with Fig. 3.28a, b, it is concluded that the streamlines in the jet lateral region pass the expanded supersonic zone and collide with another supersonic stream from the opposite side, which converges the flow and generates V-shaped collision shock waves. This collision also leads to generation of the jet leeward separation and the further induce the herringbone trailing separation in the boundary layer, as described in Fig. 3.30b. Figure 3.32b demonstrates the Type II flow pattern with transparent iso-surface of u/U ∞  0.05 to reveal the flow in the interior of the separation bubble. For this pattern, the streamlines go across the jet leeward separation bubble and cluster in the reattachment valley, and then recirculate following the leeward separation bubble which acts as a tunnel to let streamlines penetrate. The penetration procedure is very clear with streamlines colored by local wall-normal distance. Combining with Fig. 3.28a, it is inferred that streamlines of Type II in the vicinity of the reattachment point come into the recirculation and move against the freestream direction. This recirculating flow along the oblique separation tunnel is just what has been described as the primary TCVP in above. The streamlines concentrate in the jet leeward and rise

92

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Type I: Streamlines concentrate in the jet leeward and rise up

(a) Type I pattern, streamlines concentrate in the jet leeward and rise up following the suction of the CVP

Type II: Streamlines recirculate in leeward separation bubble and rise up

(b) Type II pattern, streamlines recirculate through leeward separation bubble, concentrate in the jet leeward and rise up Fig. 3.32 Flow patterns (Type I, II, III, IV) demonstrated by streamlines in jet (J  5.5) wake to demonstrate different types of trailing vortices, with transparent iso-surface of u/U ∞  0.05, colored by local wall-normal distance (Sun and Hu 2018b)

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

Type III: Streamlines go across separation bubble and rotate in reattachment valley

(c) Type III pattern, streamlines run across separation wing and roll to form a vortex rope

Type IV: a combined pattern; Streamlines recirculate in separation wing and rotate in reattachment valley

(d) Type IV pattern, streamlines run across separation wing and go through the separation zone and rotate into the vortex rope shown in type III Fig. 3.32 (continued)

93

94

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

up due to the major CVP. Type II streamlines rotate when penetrating the recirculation zone. This is due to the rotation character of the recirculating flow in the spanwise direction, which has been well described in Fig. 3.30a. Figure 3.32c demonstrates the Type III flow pattern. This pattern is believed to well explain secondary surface TCVP shown in Fig. 3.25b, c. For type III, the jet lateral flow is supersonic ahead of separation line and the herringbone separation wing acts as a micro-ramp vortex generator. Those streamlines go over the separation wing bubble or separation hill and rotate to form a swirling flow and roll into a concentrated vortex rope, to form the secondary TCVP. This procedure only can be displayed in a three-dimensional wake configuration with three-dimensional streamlines. From the analysis given in the above, it is known that secondary TCVP is located in the reattachment valley with a positive velocity. This flow pattern is definitely different from Type II, which mainly locates in separation zone with a negative velocity. In the separation wing, there is a combined flow pattern which has the partial character of Type II and III, denoted as Type IV. For this pattern, streamlines go over the wing, run into the reattachment valley and recirculate through the separation wing interior downstream of the reattachment point. Further, these streamlines penetrate out of the separation wing and follows secondary surface TCVP to move downstream. Type IV has the recirculating character of Type II and rotating character of Type III. Figure 3.24 indicates that the primary TCVP shown in Fig. 3.25a mainly corresponds to Type II flow pattern. For Fig. 3.25b, existence of the separation wing on slices indicates a combined pattern of Type III and IV. For Fig. 3.25c, there is no separation and the secondary TCVP comes from the combined effects of Type III and IV. The main difference between Type III and Type II is that Type III streamlines go over the separation bubble and directly run into the reattachment valley, which means the streamlines mainly move outside of the separation region. Streamlines of Type II penetrate in leeward separation bubble and rotate following the swirling vortex in the recirculation region. Streamlines in Type III and IV rotate in the reattachments valley outside of the separation and further lose the swirling strength downstream of the reattachment valley due to the disappearance of herringbone separation and the local dissipation, which corresponds to the terminating of secondary surface TCVP. Decay of TCVP strength in the far-field can also be deduced from vortex dynamics since the two counter-rotating vortex in TCVP induces each other and they lift off the surface as a result of upwash. The main feature of micro-ramp wake region is presented in Fig. 3.33 to demonstrate the formation of secondary TCVPs. Babinsky et al. (2009) presented a schematic (Fig. 3.34a) to show that a primary pair of counter-rotating vortices rolls up gradually and gets strengthened on the two sides of the micro-ramp, which finally converge near the trailing edge of the ramp. Wang et al. (2015) traced the streamlines and found that streamlines in the inner boundary layer were engulfed into the primary vortex core (shown in Fig. 3.34b) and streamlines in outer layer simply passed over the ramp. The jet lateral flow shown in Fig. 3.34c near to wall passes separation wing bubble, where an upwind ramp plays the same role with micro-ramp in the boundary layer. The formation of the primary vortex in micro-ramp flow is just same

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

(a)

95

(b)

Fig. 3.33 Micro-ramp flow features in a supersonic boundary layer, a sketch of vortices generated around micro-ramp (one side only for clarity) (Babinsky et al. 2009), b streamlines passing around the micro-ramp, originating from supersonic stream at wall parallel plane with 2% of boundary layer thickness (Wang et al. 2015; Sun and Hu 2018b)

Type III and IV, forming secondary surface TCVP Type II , forming primary surface TCVP

Fig. 3.34 Flow topology (Type II, III and IV) demonstrated by streamlines in jet wake with transparent iso-surface of u/U ∞  0.0 colored by local pressure of J  5.5 case (Sun and Hu 2018b)

with secondary surface TCVP in jet wake flow. Many literatures have discussed the mechanism of micro-ramp flow, detailed analysis is omitted here for brevity. It is inferred that local pressure difference plays a dominant role in the flow recirculation or rotation. Figure 3.34 demonstrates the pressure contour on the isosurface of u/U ∞  0.0 with representative streamlines of Type II, III, and IV. In the jet leeward, a bow low-pressure region exists due to the blockage and expansion

96

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

effects from the jet barrel. The pressure recovers at the reattachment valley. Together with Fig. 3.28a, it is indicated that streamlines from the jet lateral region run into the reattachment valley would increase the local pressure to a higher level over the surrounding area. The high pressure in the reattachment region would push the streamlines into the herringbone recirculation region. Type II streamlines go across the separation bubble and are pushed into the jet leeward recirculation zone. The streamlines recirculate and rotate into the jet leeward. Type IV streamlines are also pushed by the high pressure of the reattachment region into the oblique separation wing. Streamlines of Type III go over the separation wing and directly acquire the rotation momentum by the blockage of high pressure in reattachment valley. These streamlines swirl downstream of the herringbone zone and would not roll into any recirculation region. Figure 3.35 gives the front view of these three streamlines. If compared to Fig. 3.25, Type II streamlines represent the rotation of the primary TCVP, while Type III streamlines represent the rotation of the secondary TCVP. The streamlines on slices in Fig. 3.25 are the projection of the three-dimensional streamlines in field. Type II streamline rotates as well as recirculates in jet leeward separation bubble. Type IV streamline recirculates in the separation wing first and then follows secondary TCVP to move. The analysis further validates the conclusion that primary TCVP and secondary TCVP are different patterns in the jet wake flow. Figure 3.36 shows a cluster of streamlines which originate from an oblique line parallel to separation wing on the iso-surface of u/U ∞  0.05 and 0.5 from jet lateral side. It is found that streamlines with different patterns are driven by flow and move into different regions. Streamlines in Fig. 3.36a demonstrate secondary TCVP structure and Fig. 3.36b demonstrate primary TCVP. It is interesting to see that on the wingtip of the separation bubble, a new shedding vortex is induced and streamlines rotate from the wingtip and the swirling strength fades some distance downstream. In Fig. 3.36c, d with streamlines originating from iso-surface of u/U ∞  0.5, both Type II and III flow pattern are found. The main difference of Fig. 3.36c from Fig. 3.36a is that the acquired rotation magnitude decays earlier, which indicates that the separation ramp mainly collects the low-momentum streamlines from the inner boundary layer. The function is the same as the micro-ramp described by Wang et al. (2015). For Fig. 3.36d, the streamlines cover a larger area in separation wing than Fig. 3.37b, which means that crossflow in the outer boundary layer enters into the bottom of the separation bubble in Type II pattern. In Fig. 3.36d, it is found that some streamlines go over the separation bubble, run into the reattachment valley and move into far-field directly, which corresponds to the prediction discussed in the above. This also reflects that the flow running into reattachment valley has different origins at different wall-normal location. To clearly demonstrate the surface TCVPs, iso-surface of streamwise vorticity is shown in Fig. 3.37a for J  5.5 case. Arrows with solid line points to primary and secondary surface TCVPs, respectively. It is identified that the surface TCVPs are located near to wall. It is seen that the surface TCVPs have an opposite rotating direction to major CVP. The primary TCVP in leeward separation zone has a same rotation direction with secondary TCVPs in reattachment zone. That is why usually it is not easy to distinguish them if three-dimensional streamlines are not given.

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

(a) Type II showing primary TCVP

97

(b) Type III showing secondary TCVP

(c) Type IV showing a combined character of primary and secondary TCVP Fig. 3.35 Front view of typical streamlines of Type II, III, and IV in jet wake with a Mach number slice at x/D  40 to demonstrate TCVPs of J  5.5 (Sun and Hu 2018b)

Separation line is imposed at y/D  0.0 slice to identify separation zone. It is shown that secondary TCVP is separate from primary TCVP in the jet leeward, especially in the separation wing. It is concluded that primary TCVP is a single CVP which occurs in the herringbone recirculation and has no direct relation with secondary TCVP. Figure 3.37b shows a schematic of vortex tubes in the near-wall wake to demonstrate that primary TCVP and secondary TCVP. It is seen that primary TCVP is formed in the herringbone recirculation including jet leeward separation bubble and the oblique separation wing. Secondary TCVP is formed on the ramp leeward of the

98

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Secondary TCVP

Induced vortex on the wingtip

(a) iso-surface of u/U∞=0.05, Type III flow

(b) iso-surface ofu/U∞=0.05, Type II flow

(c) iso-surface ofu/U∞=0.5, Type III flow

(d) iso-surface ofu/U∞=0.5, Type II flow

Fig. 3.36 Streamlines passing the separation bubble (from an oblique back view), originating from lateral side on iso-surfaces of u/U ∞  0.05 and u/U ∞  0.5 (Sun and Hu 2018b)

separation bubble adjacent to the reattachment valley. The separation bubble serves as a micro-ramp vortex generator, which is well analyzed in a previous study (Wang et al. 2015). The inner boundary layer flow would roll and form the vortex tubes in the downstream field which corresponds to secondary TCVP.

3.3 Formation of Surface Trailing Counter-Rotating Vortex …

99

Separation streamwise vorticity

Trailing streamwise vorticity

Separation line

(a) iso-surface of streamwisevorticityΩx=-5 and 5, non-dimensionalized by freestream velocity and jet orifice diameter

Secondary TCVP Induced vortex Collision shock surface

3D Recirculation region

Jet orifice Primary TCVP

(b) schematic of vortex tubes (primary and secondary TCVPs) in jet wake Fig. 3.37 Iso-surface of streamwise vorticity and schematic of vortex tubes (primary and secondary surface TCVPs) in jet wakes (Sun and Hu 2018b)

100

3 Flow Structures of Gaseous Jet in Supersonic Crossflow

Based on the numerical results and the analysis, it is concluded that the flow in the jet wake has very complex three-dimensional structures including the collision shock, herringbone separation, reattachment valley, and two different TCVPs which are generated by different mechanism in the near-wall jet wake flow.

References Babinsky H, Li Y, Pitt Ford CW (2009) Microramp control of supersonic oblique shockwave/boundary-layer interactions. AIAA J 47(3):668–675. https://doi.org/10.2514/1.38022 Ben-Yakar A, Mungal MG, Hanson RK (2006) Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Phys Fluids 18(2):026101. https://doi. org/10.1063/1.2139684 Chai X, Iyer PS, Krishnan M (2015a) Numerical study of high speed jets in crossflow. J Fluid Mech 785:152–188 Chai X, Iyer PS, Mahesh K (2015b) Numerical study of high speed jets in crossflow. J Fluid Mech 785:152–188. https://doi.org/10.1017/jfm.2015.612 Dickmann DA, Lu FK (2008) Shock/boundary-layer interaction effects on transverse jets in crossflow over a flat plate. J Spacecr Rockets 46(6):1132–1141 Gaitonde DV (2015) Progress in shock wave/boundary layer interactions. Progress Aerosp Sci 72:20 Kawai S, Lele SK (2010) Large-eddy simulation of jet mixing in supersonic crossflows. AIAA J 48(9):2063–2083 Liang CH, Sun MB, Liu Y, Yang YX (2018) Shock wave structures in the wake of sonic transverse jet into supersonic crossflow. Acta Astronaut Mahesh K (2013) The interaction of jets with crossflow. Annu Rev Fluid Mech 45(1):379–407. https://doi.org/10.1146/annurev-fluid-120710-101115 Morkovin MV, Pierce CA Jr, Craven CE (1952) Interaction of a side jet with a supersonic main stream Rana ZA, Thornber B, Drikakis D (2011) Transverse jet injection into a supersonic turbulent crossflow. Phys Fluids 23(4):046103. https://doi.org/10.1063/1.3570692 Sun M, Zhang S, Zhao Y, Zhao Y, Liang J (2013). Experimental investigation on transverse jet penetration into a supersonic turbulent crossflow. Sci China Technol Sci 56(8): 1989–1998. Sun MB, Hu ZW (2018a) Generation of upper trailing counter-rotating vortices of a sonic jet in a supersonic crossflow. AIAA J 56(3):1047–1059. https://doi.org/10.2514/1.J056442 Sun M, Hu Z (2018b) Formation of surface trailing counter-rotating vortex pairs downstream of a sonic jet in a supersonic cross-flow. J Fluid Mech 850:551–583. https://doi.org/10.1017/jfm. 2018.455 Viti V, Neel R, Schetz JA (2009) Detailed flow physics of the supersonic jet interaction flow field. Phys Fluids 21(4):046101. https://doi.org/10.1063/1.3112736 Wang QC, Wang ZG (2016) Structural characteristics of the supersonic turbulent boundary layer subjected to concave curvature. Appl Phys Lett 108(11):97 Wang H, Wang Z, Sun M, Qin N (2013) Hybrid Reynolds-averaged Navier-Stokes/large-eddy simulation of jet mixing in a supersonic crossflow. Sci China Technol Sci 56(6):1435–1448. https://doi.org/10.1007/s11431-013-5189-2 Wang B, Liu WD, Sun MB, Zhao YX (2015) Fluid redistribution in the turbulent boundary layer under the microramp control. AIAA J 53(12):3777–3787. https://doi.org/10.2514/1.J054074 Wang QC, Wang ZG, Zhao YX (2016) On the impact of adverse pressure gradient on the supersonic turbulent boundary layer. Phys Fluids (1994–present) 28(11):116101 Won S-H, Jeung I-S, Parent B, Choi J-Y (2010) Numerical investigation of transverse hydrogen jet into supersonic crossflow using detached-eddy simulation. AIAA J 48(6):1047–1058. https:// doi.org/10.2514/1.41165

References

101

Zhang Y, Liu W, Sun M (2016) Effect of microramp on transverse jet in supersonic crossflow. AIAA J 54(12):4043–4045. https://doi.org/10.2514/1.J055338 Zhao YX, Yi SH, Tian LF, Cheng ZY (2009) Supersonic flow imaging via nanoparticles. Sci China 52(12):3640–3648 Zhao Y, Liang J, Zhao Y (2016) Vortex structure and breakup mechanism of gaseous jet in supersonic crossflow with laminar boundary layer. Acta Astronaut 128:140–146

Chapter 4

Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

The limited flow residence time inside a supersonic combustor makes the mixing of jet fuel and a crossflow a serious problem in air-breathing supersonic engines. The mixing process is the primary factor that restricts combustion performance. Increased efficiency of fuel–air mixing can lead to engine size and weight reductions. Transverse injection from a wall orifice is one of the simplest and most promising configurations to enhance the mixing process between fuel and supersonic air. This process has attracted increased attention since the early 1960s. The mixing characteristics of a gaseous jet in a supersonic crossflow are detailed in this chapter.

4.1 Mixing Characteristics of Single Injection The single injection scheme is the simplest strategy for the mixing process in a scramjet engine. This has been studied both theoretically and experimentally. Computational fluid dynamics plays an important role in the design and assessment of fuel injection strategies, due to the high costs involved in flight testing and ground experimental tests, as well as the difficulty of taking measurements. Reynolds-averaged Navier–Stokes (RANS) simulations have been employed to capture many of the mean flow features (Viti et al. 2009), and the large-eddy simulation (LES) approach (Schaupp and Friedrich 2010; Kawai and Lele 2010) has been utilized to include the unsteady structure of a jet plume, as well as the generation of mixing and combustion processes (Peterson and Candler 2015). The data set obtained by Spaid and Zukoski (2012) is more suitable for turbulence model validation than the data set obtained by Aso et al. (2006), and has been employed by Huang et al. (2012), Sriram and Mathew (2006), Chenault and Beran (2012), and Sriram and Mathew (2008) to validate the influence of the turbulence model on the simulation of twodimensional transverse injection flowfield properties. Aso et al.’s configuration used three-dimensional effects in the flowfield and strong leading-edge shock waves, and Rizzetta (2013) utilized it to assess the effect of the compressibility correction. The predicted results obtained with the compressibility correction showed better agree© Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_4

103

104

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

ment with experimental data, with the peak pressure level upstream of the jet being reduced. The compressibility effect seems to weaken in the far field. Furthermore, the Schmidt number proved to have an important impact on simulation results (He et al. 1999; Boles et al. 2011; Ivanova et al. 2013) as well as the turbulence model (Rana et al. 2011a). A turbulent Schmidt number of 0.2 is recommended for simulations of a jet in a crossflow. However, several important issues still need to be clarified, including mixing in the near field, mixing in the near-wall region, and mixing in the expansion flowpath.

4.1.1 Mixing in the Near Field Jet and crossflow interactions in the near field are crucial to jet mixing and thus deserve special attention. A sonic underexpanded transverse jet injection into a Mach 1.6 supersonic crossflow was investigated numerically by Wang et al. (2013) using a hybrid RANS/LES method. The general theoretical framework has been discussed and tested in early work (Wang et al. 2012; Sun et al. 2008; Wang et al. 2011; Sun et al. 2009). In addition, calculated results for δ/D  0.775 have been compared with existing experimental results to validate present numerical models. Figure 4.1 shows the instantaneous

(a) Experiment, PLIF images

(b) Calculation

Fig. 4.1 Representative snapshots of a jet fluid in the center plane (Wang et al. 2013)

4.1 Mixing Characteristics of Single Injection

105

flowfields of numerical and experimental results. The vortical structures in the windward and leeward jet boundaries, observed in the experiments, are well captured by present calculations. Figure 4.2 provides comparisons of time-averaged velocity distributions for calculations and experiments at jet downstream locations. The calculated results agree reasonably well with experimental data except that a significant discrepancy is observed close to the wall in the region immediately downstream of the jet in streamwise velocity profiles. The same discrepancy is also observed in the LESs of Kawai and Lele (2008a) and Kawai and Lele (2009). One possible reason for this may be the uncertainty involved in the experimental data and the unmatched Reynolds numbers used in calculations. Figure 4.3 shows the instantaneous flowfield of a transverse jet with a supersonic incoming turbulent boundary layer. Typical low-speed streaks and turbulent structures in the boundary layer, as well as the bow shock, separation shock, and

Fig. 4.2 Comparison of streamwise and wall-normal velocities for calculations and experiments (Santiago and Dutton 1997) at jet downstream locations x/D  2, 3, 4, and 5 (Wang et al. 2013)

Fig. 4.3 Transverse jet with a supersonic incoming turbulent boundary layer. Density gradient magnitude in the center plane is z/D  0 and streamwise velocity contours in the wall-parallel plane close to the wall are y/D  0.155 (Wang et al. 2013)

106

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

large-scale structures in the jet boundaries are clearly observed. Also, the interaction of the turbulent structures in the incoming boundary layer and the transverse jet can be clearly seen—these are expected to be important to the quick breakdown of the jet. Two regions of vortex formation which form hairpin-like structures are identified in the windward and leeward jet boundaries, as observed by Kawai and Lele (2008) and (2010). These vortices play an important role in determining the behavior of jet fluid stirring and subsequent mixing. The vortices appearing on the windward portion of the jet boundary are of a larger scale with more hairpin-like structures than from the leeward portion of the jet boundary. Moreover, the vortex formation on the windward portion of the jet boundary has a closer coupling with the dynamics of the incoming boundary layer and shocks (bow shock, separation shock, barrel shock, and Mach disk). Therefore, the following analyses are focused on the dynamics of the vortex formation on the windward portion of the jet boundary. Representative snapshots of jet fluid overlapped with the divergence of velocity contours in the center plane are shown in Fig. 4.4, demonstrating the deformation of shock structures and associated vortex forming and shedding (snapshots taken over consistent time intervals). The divergence of the velocity contours represents the presence of shock structures. Several important characteristics accompanied by the forming and shedding of large-scale vortices on the windward portion of the jet boundary can be observed, including the large-scale dynamics of the bow shock, deformation of the barrel shock and Mach disk, generation of acoustic waves, and the formation and movement of a kink in the barrel shock due to the appearance of a local shock wave around the upstream boundary of the jet. More interestingly, associated with the deformation of the barrel shock and Mach disk, the two triple lines (the kink in the barrel shock and the intersection of the barrel shock and Mach disk) move toward each other along the barrel shock and then coalesce into a single line, as seen in snapshots 4 and 5. This also indicates that the old windward portion of the barrel shock is replaced by a new one. That is, an update of the barrel shock occurs during vortex shedding. Meanwhile, some jet fluid is entrained into the separated region upstream of the jet, which may support ignition of a reactive species at that point, as observed by Benyakar et al. (2006). Let us take a closer look at the mechanisms of vortex formation and mixing. Figure 4.5 shows corresponding snapshots of streamwise velocity contours overlapped with the divergence of velocity contours in the center plane. It is observed that a large-scale vortex forms mainly by intermittent impingement of high-speed fluid on the upstream boundary of the jet. Before vortex formation, an evident shock in the upstream boundary of the jet does not exist, and the windward boundary of the barrel shock starts far away, and at a small angle, from the wall, as shown in snapshot 1. Therefore, the upstream half of the jet can expand upstream, leading to a region of low or negative streamwise velocity around the upstream jet boundary, just above the barrel shock. When a high-speed fluid impinges on the upstream boundary of the jet, the jet shear layer starts to deflect and a local shock wave appears within the jet around its upstream boundary due to blockage of the supersonic jet. Next, the local shock grows and connects with the original barrel shock, forming a kink in the

4.1 Mixing Characteristics of Single Injection

107

Fig. 4.4 Snapshots of jet fluid overlapped with the divergence of velocity contours in the center plane, t  0.5D/u ∞ (Wang et al. 2013)

barrel shock. Once the kink appears, it moves downstream along the barrel shock with an accompanying large-scale vortex that is directly generated by the shearing between the incoming high-speed fluid and the upper low-speed jet fluid. Associated with further deformation of the barrel shock and Mach disk, the high-speed fluid continues to accelerate, as a result the vortex continues to evolve, and the incoming fluid is entrained into the jet boundary. During vortex formation and shedding, a strong acoustic wave is generated and propagates upstream toward the bow shock. This may induce additional oscillations of the bow shock. Figure 4.4 and Fig. 4.6 show the corresponding pressure contours and streamlines near the jet exit. It is observed that a high-pressure region around the upstream boundary of the jet appears and moves upward along the barrel shock during vortex formation and shedding, resulting from the interaction of the jet and the upstream separated region. This may be responsible for sustaining the acceleration of the high-speed fluid. After vortex shedding, this new high-pressure region replaces the original high-pressure region behind the bow shock. Figure 4.4 shows the distortion of the streamlines in the jet and the upstream separated region accompanied by the deformation of the barrel shock and the upstream separated region. Interestingly, what can also be seen is that a secondary recirculation zone between the upstream separated region and the jet evidently develops during vortex formation, inducing the entrainment of jet fluid into the upstream separated region.

108

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.5 Snapshots of streamwise velocity contours overlapped with the divergence of velocity contours in the center plane, t  0.5D/u ∞ (Wang et al. 2013)

Based on studying jet-mixing characteristics, the following conclusions can be derived. The large-scale vortices on the windward portion of the jet boundary are produced mainly by intermittent impingement of the incoming high-speed fluid on the relatively low-speed region of the upstream jet boundary, where the interaction between the upstream separated region and the jet supplies favorable pressure conditions for sustaining the acceleration of the high-speed fluid during vortex formation. Associated with this, incoming fluid is entrained into the jet boundary and large-scale mixing occurs. Meanwhile, a secondary recirculation zone, between the upstream separated region and the jet, is observed to develop during vortex formation, inducing entrainment of jet fluid into the upstream separated region.

4.1 Mixing Characteristics of Single Injection

109

Fig. 4.6 Snapshots of pressure contours overlapped with the divergence of velocity contours in the center plane, t  0.5D/u ∞ (Wang et al. 2013)

4.1.2 Mixing in the Near-Wall Region A dispute has existed for a long time as to whether higher values of J would increase the downstream near-wall injectant concentration. In order to assess this problem, detailed information on the injectant mass fraction distribution is required. This can only be revealed through accurate simulations. The mixing status downstream of a transverse sonic jet in a supersonic crossflow at a Mach number of 2.7 was studied by Sun and Hu (2018a). Direct numerical simulations were performed to investigate the transport of a passive scalar of a jet fluid for jet-to-crossflow momentum flux ratios (denoted as J) of 1.85 and 5.5.

4.1.2.1

Instantaneous Flow Structures

Typical contours of instantaneous density and jet fluid at the midspan plane z/D  0.0 of J  1.85 and J  5.5 are shown in Fig. 4.7. The results reveal detailed structures of unsteady jet penetration in a Mach-2.7 supersonic crossflow. In particular, the bow shock, barrel shock, and separation bubbles are clearly identified, features that have been noticed in previous simulations (Kawai and Lele 2010; Boles et al. 2011; Rana

110

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) J=1.85, density contour (top) and injectant mass fraction contour (bottom)

Leeward region with low mass fraction

Farfield nearwall mixing

(b) J=5.5, density contour (top) and injectant mass fraction contour (bottom) Fig. 4.7 Contours for J  1.85 and J  5.5 at the midspan plane (z/D  0). Density is normalized by the freestream density and the injectant mass fraction is normalized by the injectant mass fraction in the jet orifice. Isolines of u/U ∞  0.0 are superposed in red (Sun and Hu 2018a)

et al. 2011b; Won et al. 2012; You et al. 2013). It is seen that the windward structures tilt and fold as they are convected downstream. These structures engulf large amounts of freestream fluid. The mixing of the injectants with the freestream fluids is achieved at the periphery of the jet. Further downstream, these vortical structures break down, enhancing the local mixing process. It can be seen from Fig. 4.7 that there is a low-injectant mass fraction region in the jet lee as the injectant is mostly moved upward due to its wall-normal high momentum—the size of this region increases as

4.1 Mixing Characteristics of Single Injection

111

J increases. Further downstream, the jet fluid approaches the wall and the injectant mass fraction increases compared with that in the near field of the jet lee. As the jet plume is fully three-dimensional, it is not enough to conclude that the mechanism of near-wall mixing is only based on midspan plane contours. Instantaneous injectant mass fraction contours on the cross-view planes (CVP) are shown in Fig. 4.8 for both J  1.85 and J  5.5. It is observed that downstream of the injection, there exists a region in the jet lee where the injectant mass fraction has a low magnitude. The jet fluid in the jet plume twists into two branches according to the effects of the major counter-rotating vortex pair (CVP) and mixes with the freestream in the near field. Near the wall in the jet near field, jet fluid exhibits strong variation across-span. In the near-wall region on the cross-plane, like x/D  20.0 for the J  5.5 case, the mass fraction in the vicinity of the midspan plane z/D  0.0 is clearly lower than regions further away from z/D  0.0. This means that there is an isolation zone between the CVP branches that corresponds to the two counter-rotating vortices of the major CVP. Here the isolation zone is denoted as a “gap” here. In the far field, like in the cross-plane x/D  40.0 for the J  5.5 case, the jet fluid mixes with the air stream sufficiently and the “gap” is smeared in the near-wall region. At x/D  40.0, near-wall injectant mass fraction variation is not apparent along the span, which means the mixing in the far field is enhanced in the near-wall region. Traditionally, it was believed that the injectant distribution was mainly dominated by the flow in a jet’s near field. These instantaneous contours are qualitative. Further discussion will be carried out considering time-averaged results in sect. 4.1.2.2. Near-wall density and injectant mass fraction contours are shown in Fig. 4.9 on horizontal planes. As can be seen in the slices of y/D  0.1 and 1.0 in Fig. 4.9a–d, the injectant is mostly distributed in the plume CVP branches of the jet for both cases. Turbulent streaks are observed in Fig. 4.9a, b at slices y/D  0.1. There appear herringbone separation zones (wrapped by solid lines representing zero streamwise velocity) which extend both in the streamwise and spanwise directions. In the lateral downstream of the jet orifice, local density decreases significantly, which implies that expansion occurs near the jet lee. It can be seen from Fig. 4.9a, b that the herringbone separation zone is confined and flow reattaches downstream of the cross point of the herringbone wings. Near the symmetrical plane, a recovery region is embayed by the herringbone separation zone near the center line. Although similar in flow pattern, the herringbone separation region for J  5.5 is larger size than the J  1.85 case. It can be seen from the contours of the mass fraction that the injectant transport in the jet near field is affected by the near-wall separation and the recovery process directly. Little injectant is entrained into the jet leeward near-wall separation region. As the flow develops further downstream, the large pockets containing the injectant break up and significantly mix with the main air stream. In the slice y/D  0.1, shown in Fig. 4.9a, it is seen that jet fluid is entrained into the far-field zone with streaky patterns downstream of x/D ≈ 18.0 for J  1.85. The same phenomenon only occurs at x/D ≈ 30.0 for J  5.5, as demonstrated by the blue dotted ellipse on the plane y/D  0.1 of Fig. 4.9b. It is inferred that the reattachment in the near-wall region of the jet fluid occurs earlier for J  1.85 than J  5.5. From Fig. 4.9c, d, it is observed that

112

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

J=1.85

major CVP branches

J=5.5

‘gap’ smeared ‘gap’ between the plume CVP branches Fig. 4.8 Instantaneous contours of injectant mass fraction for J  1.85 and 5.5, showing cross-view planes at x/D  0.0, 6.67, 13.33, 20.0, 26.66, 33.33, and 40.0 (Sun and Hu 2018a)

the jet plume branches extend a longer streamwise distance before fully breaking down for J  5.5 than for J  1.85 at slice y/D  1.0. It is suggested that mixing of the jet plume CVP branches with the air stream occurs more quickly for J  1.85 than for J  5.5 in the near-wall region.

4.1 Mixing Characteristics of Single Injection

113

Injectant entrained into farfield nearwall streaks

a) y/D=0.1, J=1.85, density(top) and injectant mass fraction (bottom)

Injectant entrained into farfield nearwall streaks

b) y/D=0.1, J=5.5, density(top) and injectant mass fraction (bottom) Fig. 4.9 Instantaneous contours of density and injectant mass fraction for J  1.85 and J  5.5 on different wall-parallel planes. Isolines of u/U ∞  0.0 are superposed in red (Sun and Hu 2018a)

114

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Major CVP branches breakup into small structures

c) y/D=1.0, J=1.85, density (top) and injectant mass fraction (bottom)

d) y/D=1.0, J=5.5, density (top) and injectant mass fraction (bottom) Fig. 4.9 (continued)

4.1 Mixing Characteristics of Single Injection

4.1.2.2

115

Time-Averaged Mixing Characteristics

Time-averaged injectant mass fraction and Mach number distributions in the midspan plane z/D  0.0 are shown in Fig. 4.10 for both cases. The jet penetrates further into the mainstream as the injection momentum ratio is increased. In the study by Sun et al. (2013), the jet penetration was related to the momentum ratio from experimental data: 2.933  x 0.161 y  0.5830 DJ J DJ

(4.1)

where x represents the streamwise distance to the jet orifice and y is the normal distance to the wall. Gruber et al. (2011) suggested a power-law fit, based on jet

J=1.85

J=5.5

(a) Jet fluid mass fraction contours and iso-lines for J=1.85 and J=5.5 cases, z/D=0.0, ● represents data from penetration correlation (4.1), ■ represents data from penetration correlation (4.2)

J=1.85

J=5.5

(b) Mach number contours for J=1.85 and J=5.5 cases, z/D=0.0

Fig. 4.10 Time-averaged mass fraction and Mach number distribution contours on the midspan z/D  0.0 plane for J  1.85 and J  5.5 (Sun and Hu 2018a)

116

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

concentration identification; however, Benyakar et al. (2006) and Sun et al. (2013) found that their experimental data deviated from the above power-law. As discussed in the previous study, better agreement was achieved with the correlation of Rothstein and Wantuck (1992):    x y 0.6373 (4.2)  J −0.6985 ln 4.704J + DJ DJ J Figure 4.10a shows the contour of the time-averaged injectant mass fraction for J  1.85 and J  5.5. The boundary of the penetrating jet is identified based on mass fraction (Y i  0.02) and is obtained and compared using Eqs. (4.1) and (4.2). It is seen that the current penetration line agrees well with the previous experimental correlations for both J  1.85 and J  5.5. In Fig. 4.10b, in-plane streamlines are superposed on Mach number contours. These streamlines clearly show the upstream and downstream recirculation zones of the jet. It can be noted from Fig. 4.10b that most of the injectants pass through the windward side of the jet plume and the Mach disk and then are diverted toward the freestream flow where mixing occurs. A recirculation zone exists in the jet lee. It is inferred that the flowfield within the recirculation zone has a low mass exchange with the flow outside. The magnitude of the injectant mass fraction in the recirculation is smaller than outside. As seen in Fig. 4.10a, at z/D  0.0, the size of the low-speed region in the jet lee becomes larger as J increases. The low mass fraction region, wrapped by the isoline Y i  0.02 in the near-wall region, is enlarged due to the expansion of the low-speed region shown in Fig. 4.10b. The streamlines on z/D  0.0 show that air freestream concentrates in the jet lee and leads to an upwash flow. The streamlines near to the leeward separation run across the low-speed region and reattach to the wall. Combining with the instantaneous contours shown in Figs. 4.8 and 4.9, it is conjectured that the low mass fraction region above the jet lee separation under the jet plume is caused by the gap between the CVP branches. Further analysis of the three-dimensional jet configuration will be considered here. Figure 4.11 shows the time-averaged injectant mass fraction on different cross sections, consistent with the major CVP structure. Downstream of the jet, the region of well-mixed injectant increases as the major CVP increases in size for larger J. The major CVP is smeared by the turbulence originating from the jet CVP breakdown. A careful observation shows that the main body of the jet plume penetrates further into the near field (x/D < 20.0) and the gap between the plume CVP branches is enlarged for J  5.5 compared with J  1.85. The injectant in the jet far field distributes more uniformly than the jet near field. The jet fluid in the near-wall region has a lower concentration than in the plume core. The jet flow needs to be analyzed thoroughly to clarify the mixing process and far-field injectant distribution. The time-averaged injectant distributions on wall-parallel planes y/D  0.1, 1.0, and 2.0 are shown in Fig. 4.12. A herringbone separation zone is formed in the jet leeward region. As seen from the comparison of separation zones J  5.5 and J  1.85 at y/D  0.1 (shown in Fig. 4.12a), the jet leeward separation increases in size as J increases. As discussed in Sect. 3.2, in the separation zone, mass exchange with

4.1 Mixing Characteristics of Single Injection

117

J=1.85

Skeleton of Region I

J=5.5 ‘gap’ smeared ‘gap’ between the plume branches

Fig. 4.11 Time-averaged contours of injectant mass fraction for J  1.85 and J  5.5, showing cross-view planes at x/D  0.0, 6.67, 13.33, 20.0, 26.66, 33.33, and 40.0 (Sun and Hu 2018a)

the outer flow decreases, leading to a larger zone of low injectant mass fraction for J  5.5 in the jet lee than for J  1.85. It is clear at y/D  0.1 there is little injectant entrained into the jet leeward near-wall separation. In the recovery zone, the injectant mass fraction increases obviously compared with in the separation zone. At y/D  1.0, the injectant-rich zone is near to the jet orifice and the injectant is diluted in the far field. Along the centerline, downstream of the jet, there exists a low injectant mass fraction zone, which corresponds to the gap between the major CVP branches, as also shown in Fig. 4.11. For J  5.5, the injectant mass fraction in the jet lee is clearly smaller, and the main pockets of the jet fluid, indicated by the dotted ellipse in Fig. 4.12b, are located further downstream than J  1.85 on y/D  1.0. It is clear that the concentrated injectant-rich region of J  1.85 locates in an upstream position compared with J  5.5. This is mainly due to the smaller penetration height of lower J, as shown in Fig. 5. In the far field at y/D  1.0 and 2.0, as shown in Fig. 4.12b, c, the magnitude of the injectant mass fraction of J  5.5 is higher than J  1.85. This

118

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow (a) y/D=0.1 J=5.5 J=1.85

(b) y/D=1.0 J=5.5 J=1.85

(c) y/D=2.0 J=5.5 J=1.85

Fig. 4.12 Comparison of the time-averaged injectant mass fraction contours on the y/D  0.1, 1.0, and 2.0 planes for J  1.85 and J  5.5 (Sun and Hu 2018a)

is very interesting since it indicates that the magnitude of the injectant mass fraction changes in a different way in the near field and far field when J increases. Contours of the time-averaged Mach number with in-plane streamlines on the wall-parallel plane y/D  1.0 are shown in Fig. 4.13. On this plane the streamlines run around the jet body and merge downstream in the jet far field. The streamlines are similar on slices y/D  0.1 and 2.0 (not shown here for brevity). Together with the results shown in Fig. 4.12, it is concluded that the jet fluid convection corresponds to the streamlines but the local mass fraction is determined by the injectant regions penetrated by streamlines.

J=5.5 J=1.85

Fig. 4.13 Comparison of the time-averaged Mach number contour with streamlines on the y/D  1.0 planes between J  1.85 and J  5.5 (Sun and Hu 2018a)

4.1 Mixing Characteristics of Single Injection

119

Figures 4.11 and 4.12 show that there is a three-dimensional low-injectant region that exists in the near field under the jet plume, as indicated in Fig. 4.11 by the dotted lines on different cross sections. This region consists of the gap between the plume CVP branches and the leeward recirculation zone near the wall. When J increases, the jet fluid concentration in this region (denoted as Region I in this chapter) tends to decrease, which is opposite to the trend of other regions (denoted as Region II) in the flowfield. It is inferred that the jet leeward separation zone in Region I, located near the wall, increases in size as J increases. The size increase in the jet lee separation zone would push the injectant further away from the wall and form a zone with a low injectant mass fraction. The other part of Region I over the separation zone increases with J since the jet penetration is enhanced, and the gap between plume CVP branches is enlarged, as seen in Fig. 4.11 for J  5.5 compared with J  1.85. The injectant mass fraction in this “gap” is lower than in the lateral side, as shown in Figs. 4.11 and 4.12. Figure 4.14 show the contours of the local Mach number with in-plane streamlines and injectant mass fraction in the cross sections downstream of the jet plume. At x/D  6.0, the major and surface trailing CVPs near the wall appear due to the interaction of the crossflow with the jet fluid. The major CVP grows in size further downstream. The surface trailing CVP is located in Region I (shown in the dotted ellipse in Fig. 4.14c) and disappears (shown in Fig. 4.14d) as the jet plume is lifted up downstream. It can be clearly seen in Fig. 4.14 that the major CVP increases in size for J  5.5 and the jet plume has a larger width and height for J  5.5 than for J  1.85. At x/D  6.0 it is seen that the injectant mass fraction of J  5.5 in the plume is larger than J  1.85, while the injectant mass fraction in the near-wall region is lower for J  5.5. In the far field (x/D  27.5) of J  5.5 the injectant mass fraction has a higher value both in the plume and near-wall region. The above analysis shows that for lower values of J (J 1.85) Region I remains in a smaller region and the jet penetration height is smaller, resulting in an induced near-wall zone with a higher injectant mass fraction in Region I. Higher values of J (J  5.5) bring much more injectant close to the wall downstream of Region I and therefore a higher injectant mass fraction is expected. In the far field especially, flow runs over the jet leeward separation zone and recovers, where the effect of Region I vanishes and local mixing is promoted by small vortices from the breakup of the major and trailing CVPs. The injectants are well mixed and have a higher mass fraction magnitude in the far field for higher values of J (J  5.5), including the near-wall region. Next, the injectant mass distributions are shown to quantify the qualitative conclusions presented above. Profiles of the time-averaged injectant mass fraction are shown in Fig. 4.15a at different streamwise locations. In the jet near field, the injectant mass fraction of J  5.5 is smaller than J  1.85 for y/D < 2.0 (indicated by the blue dashed line in Fig. 4.15a), which strongly suggests that in the jet near field more injectant is entrained into the near-wall zone at lower J. On the contrary, in the jet far field, also shown for x/D  27.5 and 40.0 in Fig. 4.15a, the injectant mass fraction of J  5.5 is always larger than the J  1.85 case. This contradicts the suggestion given by previous research (Sun et al. 2015), which considered a local injectant lean

120

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

J=5.5

J=1. 85

J=5.5

J=1.85

Region I

(a) x/D =6.0, jet fluid distribution

J=5.5

J=1.85

(c) x/D =6.0, Mach number contour

(b) x/D =27.5, jet fluid distribution

J=5.5

J=1.85

(d) x/D =27.5, Mach number contour

Fig. 4.14 Comparison of the time-averaged injectant mass fraction and Mach number contours with local in-plane streamlines between J 1.85 and J  5.5 on cross sections of x/D  6.0 and 27.5 (Sun and Hu 2018a)

4.1 Mixing Characteristics of Single Injection

121

(a) Time averaged injectant mass fraction profiles of J=1.85 (solid line) and J=5.5 (dashdotted line), x/D =8.5, 15, 27.5, 40 (from left to right), respectively

(b) Time averaged streamwise velocity profiles of J=1.85 (solid line) and J=5.5 (dashdotted line), x/D =8.5, 15, 27.5, 40 (from left to right), respectively

Fig. 4.15 Comparison of injectant mass fractions (a) and time-averaged streamwise velocities (b) of J  1.85 and J  5.5 at x/D  8.5, 15, 27.5, and 40 (from left to right), respectively. Solid lines, J  1.85, z/D  0; dash-dotted lines, J=5.5, z/D  0; Solid line with triangle () J  1.85, z/D  3.45; dash-dotted line with circle (◯), J  5.5, z/D  3.45 (Sun and Hu 2018a)

result for the incoming flow when a high J is conducted, and attributed downstream extinction to the fuel-lean status. Time-averaged streamwise velocity distributions for J  1.85 and J  5.5 are shown in Fig. 4.15b. It is seen that at the outlet of x/D  40.0, the profiles of the streamwise velocity remain similar for the two cases with a small deformation seen for J  5.5 due to a stronger interaction. Based on Fig. 4.15, explaining the experiments of Sun et al. (2015), the current simulation reveals that a local injectant-rich region appears in the near-wall far field, which essentially results in flame extinction in the far field (where x/D > 120.0 in Sun et al. (2015)) when J varies from 1.6 to 2.5. As is well known, a jet plume has a three-dimensional configuration and the mass fraction profiles in the midspan of z/D  0.0, 3.45, are not enough to describe the injectant distribution. Figure 4.16 gives a comparison of the mean injectant mass

122

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) x/D =8.5

(c) x/D =27.5

(b) x/D =15.0

(d) x/D =40.0

Fig. 4.16 Comparison of the mean injectant mass fractions of J  1.85 and J  5.5 at y/D  1.0 and 2.0 at different streamwise locations. Solid lines, J  1.85, y/D  1.0; dash-dotted line, J  5.5, y/D  1.0; solid line with circle (◯), J  1.85, y/D  2.0; dash-dotted line with diamond (♦), J  5.5, z/D  2.0 (Sun and Hu 2018a)

fraction profiles of J  1.85 and J  5.5 over the spanwise direction at different streamwise locations. First, it is seen that close to the symmetric line z/D  0.0 in the near field (x/D  8.5 and 15.0), the injectant mass fraction is always lower than the peak of the same profile for both cases. The peak corresponds to the core of the major CVP. Therefore, a “gap” with a low injectant mass fraction between CVP branches is expected. Second, in the far field (x/D  27.5 and 40.0), the injectant mass fraction near z/D  0.0 is approximately equal to the peak value of the same profile, which indicates a mixing and smeared effect with the development of the major CVP. The injectant mass fraction of J  1.85 at y/D  1.0 and x/D  8.5 is larger than that of J  5.5 over the spanwise region of z/D ∈ [0, 2.15], which is due to the higher penetration and the larger “gap” between the major CVP branches for J  5.5. While

4.1 Mixing Characteristics of Single Injection

123

over z/D ∈ [2.15, 3.5], the injectant mass fraction of J  1.85 is smaller than for J  5.5, since higher values of J would bring more injectants into the core of the major CVP. At y/D  2.0 and x/D  8.5, over the spanwise region of z/D ∈ [0, 1.25], the injectant mass fraction of J  1.85 is higher than for J  5.5 while it is smaller at z/D ∈ [1.25,3.5], which indicates a decrease of “gap” width at a higher wall-normal position. The gap between the CVP branches of J  5.5 lasts longer and has a longer distance than J  1.85, and that is why the injectant mass fraction of J  5.5 at x/D  15.0 is smaller near to the symmetric plane, as shown in Fig. 4.16b. The low mass fraction zone near to the symmetric plane increases with J, corresponding to an extension of Region I, as analyzed in Sect. 4.1.2.2. In Fig. 4.16c and d at x/D  27.5 and 40.0, the injectant mass fraction of J  5.5 is higher than for J  1.85 across the whole span, which means that after the recovery of Region I, more injectant is entrained into the near-wall region in the far field for higher J. Figure 4.17 shows the mass fraction distribution of four representative lines of J  5.5 and J  1.85 along the x-axis. Injectant mass fraction on these lines clearly reflects the phenomenon that in the near field a low mass fraction zone exists. At y/D  1.0, the mass fraction in Region I of J  1.85 is larger than for J  5.5, something that remains true at y/D  2.0 and z/D  0.0. However, at y/D  2.0 and z/D  2.0, this phenomenon disappears since at x/D > 0.0 the injectant mass fraction of J  5.5 is always larger than J  1.85. This reveals the three-dimensional configuration of Region I. The results shown in Figs. 4.15, 4.16, and 4.17 demonstrate that a jet with lower values of J (J  1.85) leads to a higher injectant concentration in the near-wall region located in the jet near field (Region I) but produces a lower concentration in the far field. To better describe the mixing characteristics of the injectant distribution, the area excluding Region I is defined as Region II, as shown in Fig. 4.18. In Region I, there exists two zones. One zone is associated with the near-wall separation in the jet lee. The other zone is associated with the gap between the plume CVP branches. Region I expands as J becomes larger and less injectant is entrained into Region I, which explains the decrease of the injectant mass fraction with an increased J in this region. On the contrary, the injectant mass fraction in Region II increases with J since higher J brings more injectants. In the far field of Region II, the injectant is entrained into the near-wall zone and has a higher mass fraction magnitude when a larger J is set. From a direct point view, Region I is thought to be caused by the blockage of the jet body in a supersonic crossflow. It is especially important to note that when J becomes high, the jet body expands. When mixing enhancement is concerned in the near field, it is useful to change the jet blockage to alter jet penetration and the wake flow, perhaps leading to jet arrays or staged injection (Huang 2016). Figure 4.18 only provides a two-dimensional schematic and the three-dimensional configuration of Region I and Region II will be discussed in sect. 4.1.2.3.

124

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(b) y/D =1.0

(a) y/D = 0.5

(c) y/D =2.0

Fig. 4.17 Comparison of the time-averaged injectant mass fractions of J  1.85 and J  5.5 at y/D  0.5, 1.0, and 2.0, and z/D  0.0 and 2.0 along the streamwise direction (Sun and Hu 2018a)

4.1.2.3

Three-Dimensional Configuration and Streamline Analysis

Transport of the injectants toward the wall in the jet near field and far field is further analyzed in this section using three-dimensional streamlines to demonstrate the injectant transport process and the resulting distribution in the near-wall zone. Figure 4.19 presents an oblique view and the corresponding front view of the flowfield containing the streamlines originating from the inside of the jet orifice and the streamlines aiming toward the near-wall position at the exit for J  1.85. The streamlines are marked with spheres colored according to the local time-averaged injectant mass fraction. The J  5.5 case shows similar topology and therefore it is not given here. In Fig. 4.19a and c the streamlines exhibit significant curvatures, which indicate the effects of the major CVP. Contours of the time-averaged injectant

4.1 Mixing Characteristics of Single Injection

125

Bow shock

Ma>1

Barrel shock

Region I size increases with J

Boundary layer

Separated region

Fuel jet Recirculation zone

Recirculation zone

Region II: mass fraction increases in farfield as J increases

Boundary layer Region I: mass fraction decreases with increasing J in nearfield

Fig. 4.18 Schematic of the mass fraction changes in Regions I and II with increasing values of J (Sun and Hu 2018a)

mass fraction on the x/D  40.0 slice give an indication of the extent of injectant transport. Figure 4.19a and c show streamlines originating from the jet orifice and indicate the motion and mixing of the jet fluid. The streamlines originating in the jet orifice follow the major CVP into the core of the jet plume. A close observation indicates that the major CVP generally pumps injectant away from the wall. In the far field, the streamlines follow the lifting and rotating major CVP and bring injectants to a higher position away from the near-wall region. Considered overall, the streamlines inherit the rotating movement from the near field while they have little deformation or twist, and go directly downstream, in the far field. Figure 4.19b and d show the representative streamlines of J  1.85, traced back from far-field near-wall positions at y/D  0.5 and y/D  1.0 on the x/D  40.0 slice. It is seen that those streamlines originate from the crossflow in the lateral side running around the jet body. Only a small amount of the injectant in the near-wall region is entrained to the far field. Streamlines run around the jet body and gain injectants when penetrating the injectant-rich region. Streamlines in Fig. 4.19b do not exhibit strong rotating motion since the intensity of the major CVP greatly decreases at the lateral side of the jet plume. Ferrante et al. (2011) conducted an LES to study a helium sonic jet in supersonic crossflows and found almost no helium near the wall, whereas experiments showed a non-zero mole fraction of helium below the main core of the jet. They attributed this to the absence of a wall-normal mass flux model in their simulations. In the current simulation, the detailed turbulence and jet fluid mixing is well resolved and the injectant distribution in the near-wall region is clearly identified, without the necessity to conduct flux models on the wall. To more clearly illustrate the injectant entrainment to the near-wall region, Fig. 4.20 gives the streamlines passing the lines at x/D  −6.0, 6.0, 19.0, and 32.0 in

126

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) 3D streamlines originating from jet orifice,

(b) 3D streamlines aiming at y/D=0.5 and 1.0

oblique view

(nearwall region) at the exit, oblique view

(c) 3D streamlines originating from jet orifice, front

(d) 3D streamlines aiming at y/D=0.5 and 1.0

view

(nearwall region) at the exit, front view

Fig. 4.19 Three-dimensional streamlines originating inside the jet orifice and streamlines aiming toward the near-wall position on the exit of J  1.85, colored by local injectant mass fraction. The iso-surface of Y i  0.4 is also given (Sun and Hu 2018a)

the y/D  0.5 plane. It shows that the streamlines originating from the lateral side of the crossflow run around the jet body and approach the near-wall region in the near field and far field. These streamlines, colored by injectant mass fraction, show that part of the jet fluid, especially the injectant adjacent to the jet orifice, is entrained and mixed with the air stream. Figure 4.20a shows the representative streamlines originating from the crossflow at y/D  0.5 and x/D  −6.0, upstream of the jet. It can be seen that some of

4.1 Mixing Characteristics of Single Injection

127

(a) Streamlines passing the wall parallel plane y/D=0.5 at x/D=-6.0 (Red line)

(b) Streamlines passing the wall parallel plane y/D=0.5 at x/D=6.0 (Red line) Fig. 4.20 Oblique view of three-dimensional streamlines for J  1.85 passing through the lines on the wall-parallel plane y/D  0.5 at different streamwise locations, colored by injectant mass fraction (Sun and Hu 2018a)

128

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(c) Streamlines passing the wall parallel plane y/D=0.5 at x/D=19.0 (Red line)

(d) Streamlines passing the wall parallel plane y/D=0.5 at x/D=32.0 (Red line) Fig. 4.20 (continued)

4.1 Mixing Characteristics of Single Injection

129

the injectants are “trapped” in the downside region of the jet owing to the major CVP and the lower fluid velocity in the center of the jet lee. This phenomenon explains the upwash velocity under the jet plume in the two-dimensional symmetric slice z/D  0.0 in Fig. 4.10b. Streamlines in the jet lateral are found to entrain the injectant to the near-wall region in the far field. In Fig. 4.20b, representative streamlines, which pass the near-wall position at y/D  0.5 and x/D  6.0, come from the lateral side of the crossflow and wrap around the jet orifice. The streamlines near the jet orifice concentrate in the jet lee and move to a higher position due to upwash. Some of the streamlines coming from the lateral crossflow run into the nearwall region downstream. Streamlines passing the y/D  0.5 plane in the far field, shown in Fig. 4.20c and d, wholly come from the crossflow in the jet lateral and gain a downwash velocity approaching the wall in the far field. In Fig. 4.20c, the streamlines running around the jet orifice are slightly affected by the upwash flow in the far field. In Fig. 4.20d, streamlines in the far field demonstrate the injectants in the near-wall region come from the entrainment of lateral crossflow. As a summary of the analysis of Fig. 4.19 and Fig. 4.20, the streamlines originating from the jet orifice follow the major CVP and penetrate into the core of the jet plume. The jet fluid transported to the near-wall region downstream of the jet originates primarily from the crossflow in the upstream lateral of the jet, which runs around the jet orifice and gains downwash velocity approaching the wall in the far field. Next, further detailed analysis is given to explain Region I and Region II configurations. Figure 4.21 shows the iso-surface of the injectant mass fraction Y i  0.1 for J  1.85, indicating the configuration of Region II since Region I is under the jet plume near the wall. Region I is near to the wall in the jet near field. Region II includes the major CVP plume region and the near-wall zone in the far field.

Fig. 4.21 Transparent iso-surface of injectant mass fraction Y i  0.1 for J  1.85, indicating Region II, colored by wall-normal distance. Oblique view (left) and back view (right) (Sun and Hu 2018a)

130

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

To clearly show Region I, the iso-surface is shown cut through so that the jet near-field near-wall configuration can be observed. By slicing the Y i iso-surface, iso-surfaces of Y i  0.1 and 0.05 for J  5.5 and J  1.85 are shown in Fig. 4.22a, b, respectively. The dotted circle marks the skeleton of Region I in the jet lee. Figure 4.22a demonstrates that Region I has a bump with a low mass fraction extending into the gap underneath the CVP branches. From the streamline analysis in Sect. 3.4, streamlines run around the jet orifice and little injectant is entrained into Region I. In Fig. 4.22c, iso-surfaces of the streamwise velocity u/U ∞  0.0 of J  5.5 and J  1.85, colored by local wall-normal distance, are shown to demonstrate a three-dimensional near-wall separation configuration. Further downstream of the jet leeward separation bubble exist two separate separation wings. The peak of the oblique wing corresponds to the edge of the separation zone. Comparing J  5.5 with J  1.85, it can be seen that the lower value of J has similar flow configuration characteristics to J  5.5, although they are smaller in size. A detailed formation mechanism for the separation bubble can be found in Sun and Hu (2018b), which provides detailed structures of separation interactions with the collision shock and the induced surface trailing CVPs in the jet wake. From Fig. 4.22c it is observed that the separation zone is fully three-dimensional, re-defining the separation zone in Region I, shown in Fig. 4.18. Figure 4.23 gives the three-dimensional configurations of Region I and Region II based on the above analysis. Region I is located under the jet plume near the wall, containing the jet leeward separation zone and the gap between the plume CVP branches. In the far field of Region II, the jet plume expands and approaches the wall, which leads to a mass fraction increase in the near-wall region of Region II. The reattachment line on the wall also represents the end of Region I. As analyzed above, higher values of J lead to an expansion of Region I, resulting in a lower injectant concentration in Region I, but producing a higher concentration in the far field (the far field of Region II). In summary, when J increases, the change of the injectant concentration or mass fraction varies near the wall along the streamwise direction. In Region I, the change of the injectant concentration or mass fraction is opposite to the change in J, while in the near-wall zone of Region II, the change of injectant concentration or mass fraction is synchronous with the change in J. This simulation clarifies the dispute over whether higher values of J would increase the downstream near-wall injectant concentration or not. Based on the analysis above, conclusions can be drawn as follows. The simulations provide instantaneous and time-averaged flow features including coherent structures, streamlines, and mixing characteristics in the near field and far field downstream of the jet injection. The time-averaged jet penetrations are compared with experimental correlations and good agreements are observed. It is shown that the large-scale major CVP dominates near-field mixing, breaking into smaller eddies in the far field providing an enhanced mixing effect. Analysis of the streamlines originating from the jet orifice suggests that significant jet fluid entrainment can be attributed to the major CVP structures which pump injectants into the core of the jet plume. Jet fluid in the near-wall region of the far field is entrained by streamlines originating from

4.1 Mixing Characteristics of Single Injection

131

(a) Iso-surface of the injectant mass fraction Yi=0.1 near the wall with a slice cut view, from left to right (J=1.85 and J=5.5)

(b) Iso-surface of the injectant mass fraction Yi=0.05 near the wall with a slice cut view, from left to right (J=1.85 and J=5.5)

(c) Iso-surface of the streamwise velocity u/U∞=0.0 near the wall, representing the jet lee separation zone, from left to right (J=1.85 and J=5.5)

Fig. 4.22 Iso-surface of injectant mass fraction and streamwise velocity of Region I in the jet lee, colored by wall-normal distance (Sun and Hu 2018a)

132

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Crosssections of Region II

Region II: mass fraction increases as J increases Crosssections of Region I

Ma=2.7 Fuel jet

Region I: mass fraction decreases with increasing J in nearfield

Fig. 4.23 Schematic of the three-dimensional configurations of Regions I and II (Sun and Hu 2018a)

lateral crossflow, which travel around the jet and mix with the injectants downstream of the jet orifice. Profiles of injectant mass fraction are compared in detail to quantify the distribution in the near-wall region downstream of the jet orifice. Detailed analysis shows there are two regions downstream of the jet orifice, denoted as Regions I and II. A three-dimensional schematic is presented to explain the formation of Regions I and II. Region I consists of the jet leeward separation zone and the gap between plume CVP branches. Region II consists of the major CVP region and the near-wall region in the far field. An injectant zone with a low mass fraction exists in Region I. As J becomes higher, the size of Region I increases and less injectant is entrained into it, leading to a decreased mass fraction zone in the near field. Region II increases in size as J gets higher and meanwhile has a higher mass fraction in the far-field near-wall region. Current simulations reveal variations in the near-wall injectant mass fraction with increased values of J along the streamwise direction, clarifying the dispute whether higher values of J increase the near-wall injectant concentration or not.

4.1.3 Mixing in the Expansion Flowpath In a scramjet combustor, the fuel may be injected from an expansion wall, thus jet mixing in the expansion flowpath is of practical importance. Mixing characteristics of a transverse jet injection into supersonic crossflows through an expansion plate were investigated by Liu et al. (2016) using an LES, where the expansion effects on the mixing were analyzed emphatically by comparison with a flat plate counterpart. An adaptive central-upwind weighted essentially non-oscillatory (WENO) scheme along

4.1 Mixing Characteristics of Single Injection

133

with a multi-threaded and multi-process MPI (Message Passing Interface)/OpenMP parallel was adopted to improve the accuracy and efficiency of the calculations. A progressive mesh refinement study was performed to assess grid resolution and solution convergence.

4.1.3.1

Flow Characteristics

In order to explore the mixing process in the flowfield with a wall expansion, it is necessary to study the flow characteristics first. Figure 4.24 shows an instantaneous snapshot of density gradient magnitude at the midline and streamwise planes together with the streamwise velocity contours obtained at the wall-parallel plane close to the expansion wall of all the flowfields. The essential flow structures are captured well by high-accuracy LES, including turbulent vortices in the STBL, expansion waves induced by the expansion corner, lambda shock, bow shock, barrel shock, and large-scale turbulent vortices. Typical low-speed streaks in the supersonic turbulent boundary layer (STBL) and instantaneous reverse-flow regions located ahead of the bow shock and behind the jet plume Jets in supersonic crossflow (JISC) can clearly be observed in the wall-parallel plane. Based on previous studies, it was found that large-scale vortices generated by the interaction between the transverse jet and supersonic crossflows mainly contributes to the mixing process in the near field (Liu et al. 2015). So, it is very significant to recognize the three-dimensional vortical structures in the JISC section of expansion flowfields. Figure 4.25a displays the iso-surfaces of passive scalar (injectant fraction) at y  0.25, colored by vorticity magnitude. The vorticity magnitude at the windward side of the jet shear layer is distinctly higher than in other regions. Furthermore, a λ2 -criterion (Terzi et al. 2009) is adopted to vividly identify the vortical structures in the JISC section. The iso-surfaces of the second largest eigenvalue with λ2  −0.5, colored by streamwise velocity, are shown in Fig. 4.26b. Some thicker vortex tubes that represent large-scale vortices are recognized at the windward shear layer close to the injection, and thinner ones exist at the leeward side and downstream of the flowfield. It is proven that large-scale vortices in the windward shear layer induced by

Fig. 4.24 Density gradient magnitude contours at midline and streamwise planes together with streamwise velocity contours at the wall-parallel plane close to the expansion wall (Liu et al. 2016)

134

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.25 Instantaneous jet and vortex structures: a the iso-surfaces of passive scalar at y  0.25, colored by vorticity magnitude; b the iso-surfaces of the second largest eigenvalue with λ2  −0.5, colored by streamwise velocity (Liu et al. 2016)

K–H instability break up into smaller scale vortices downstream. Kelvin-Helmholtz (K–H) instability is a convection instability and the oscillations are not self-excited but require a continuous source of disturbance to persist. High levels of velocity shear result in K–H instability, and the impact of supersonic crossflows on the transverse jet may provide the disturbance source to maintain this instability. The inherent mechanism of a transverse jet mixing with supersonic crossflows is very complicated, but the shear vortices must be one of the most important factors. This stimulates us to further reveal the generation and evolvement of large-scale vortices. Figure 4.26 shows the consecutive instantaneous contours of density gradient magnitude and passive scalar at the midline plane, both with the same inter-framing time. It indicates that large-scale coherent structures are intermittently generated from the boundaries of the recirculation zone ahead of the bow shock, and then are intensively stretched and distorted by the mainstream during the process of propagating along jet boundaries. The convection of large-scale vortices facilitates the mixing of the jet and mainstream by rolling up and enlarging the contact area. In particular, the cores of vortices correspond to high levels of density gradient, which

4.1 Mixing Characteristics of Single Injection

135

Fig. 4.26 Consecutive instantaneous contours of density gradient magnitude (left) and passive scalar (right) at the midline plane with an inter-framing time of t  1U/d (Liu et al. 2016)

means scalar diffusion is occurring. However, in the downstream of the flowfield, the scale of the vortices changes very little, as the jet velocity is close to that of the mainstream. Instantaneous contours reveal some noticeable phenomenon in the flowfield with wall expansion, while real advances in developing a deep understanding of flow mechanisms will necessarily come through the analysis of some statistics. The timeaveraged statistics including passive scalar distributions and turbulent kinetic energy (TKE) are obtained by averaging instantaneous flowfields in 300 non-dimensional time units. Passive scalar distribution and isolines at different streamwise planes overlapped with streamlines are shown in Fig. 4.27. Similar to the case of the flat plate, the wallnormal momentum of a transverse jet induces a pair of large-scale counter-rotating vortices, regarded as the main phenomena for mixing, especially for the far field. Secondary wake vortices are observed close to the wall but appear to play a minor role in the mixing process. TKE contours at the middle and different streamwise planes are displayed in Fig. 4.28. There are three high-TKE regions observed around the jet plume, and these high-intensity regions correspond to regions where strong vortical structures are observed in the instantaneous flowfield. Further downstream of the jet plume, the TKE dissipates gradually. Jet fluid is progressively diluted in regions where high TKE is observed, suggesting that turbulent vortices are responsible for turbulent stirring and subsequent mixing.

4.1.3.2

Effects of Wall Expansion

The flow and mixing process in the flowfield with wall expansion is very similar to that with a flat plate, but there are some distinctions between the two when considered in detail, especially in terms of statistics. In addition to instantaneous structures and

136

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.27 Passive scalar distribution at streamwise planes overlapped with streamlines and isolines of passive scalar (Liu et al. 2016)

Fig. 4.28 Contours of turbulent kinetic energy at the middle and streamwise planes (Liu et al. 2016)

4.1 Mixing Characteristics of Single Injection

137

time-averaged results, a series of comparative studies of turbulent kinetic energy and the root mean square of passive scalar between the two cases was carried out. What is unique is that a conditional average method was used to predict the volume integral of the passive scalar root mean square with a variation in Mach number. Figure 4.29 gives comparisons of the instantaneous density gradient magnitude and passive scalar contours at the midline plane between the two cases. In the expansion flowfield, a series of expansion waves are generated from the corner, accelerating the fluid. Subsequently, the scale of instantaneous turbulent vortices in the boundary layer seems slightly different. The most surprising phenomenon is that large-scale vortices break up into smaller ones as soon as they roll up from the shear layer. This process in the case of a flat plate may need more spatial distance. The interaction of a jet and crossflows is enhanced due to the effect of wall expansion, so the fluid shear increases to accelerate the process of breaking up large-scale vortices. The entrainment behavior of large-scale vortices dominates mixing in the near field and affects the whole mixing process downstream (Sun et al. 2013). It can be assumed that the mixing process in the shear layer is suppressed to some extent by the quick breakup of large-scale vortices in the expansion flowfield. Next, some time-averaged analyses are used to explore the effects of wall expansion. First of all, Fig. 4.30 gives a comparison of the mean wall-pressure distributions at the midline. It indicates that the static pressure at the midline of the wall decreases abruptly owing to bulk expansion after the corner, then rises gradually in the separation region. A sudden drop in static pressure level occurs at the leeward side of the jet plume, and then recovers to a basic constant value downstream. The favorable pressure gradient is caused by wall expansion, so that wall pressure decreases compared to the flat plate case. For the expanded compressible turbulent boundary layer, its velocity profile and properties change subject to a favorable pressure gradient. Comparisons of mean streamwise velocity contours at the midplane between the case of a flat plate and expansion plate are shown in Fig. 4.31. The fluid is accelerated after passing through the expansion corner and the streamwise velocity in the jet wake is higher. Two recirculation zones exist in the expansion flowfield—the one upstream of the injection

Fig. 4.29 Comparison of the instantaneous density gradient magnitude (top) and passive scalar (bottom) contours at the midline plane (Liu et al. 2016)

138

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

is closer to the wall, but the scale of the other, at the leeward side, is larger. To study the properties of the expanded turbulent boundary layer, the streamwise velocity profiles along the wall-normal direction, extracted at x/d  1, are given in Fig. 4.32. Rapid distortion theory indicates that the rapid recovery of the near-wall region leads to the proposal that a new internal layer has been formed after the expansion and that the boundary layer has been relaminarized, attributed to inertia (Dussauge and Gaviglio 1987). When fluid passes through the expansion corner, the streamwise velocity along the wall-normal direction increases. It is clear that increased streamwise velocity of the expanded turbulent boundary layer is beneficial to suppressing

Fig. 4.30 Comparison of mean-wall-pressure distributions at the midline (Liu et al. 2016)

2.5 Flat plate Expansion plate

2

p/p

1.5

1

0.5

0 -5

0

5

10

15

20

x/d

Fig. 4.31 Comparison of mean streamwise velocity contours at the midplane (Liu et al. 2016)

25

4.1 Mixing Characteristics of Single Injection Fig. 4.32 Comparison of mean streamwise velocity profiles along the wall-normal direction at x/d  1(Liu et al. 2016)

139

2 Flat plate Expansion plate

yn /d

1.5

1

0.5

0 0

0.5

1

1.5

u/u

flow separation, leading to a recirculation zone closer to the wall. As the upstream turbulent boundary layer interacts with the jet plume, the instability of the windward jet shear layer will be affected, followed by a change in mixing. Jet penetration height is often used as a representative index to evaluate the flowfield characteristics of a transverse jet in a crossflow, because it has an important influence upon the mixing and combustion efficiency in a scramjet combustor. In current comparisons, 1% of the jet concentration is extracted as a jet penetration height, as shown in Fig. 4.33. As a whole, jet penetration height in the far field is slightly lower than in the flat case and empirical data. The favorable pressure gradient reduces static pressure and accelerates the fluid after the expansion corner. Then the kinetic energy of the mainstream increases, based on conservation of energy, leading to a low-momentum flux ratio of jet to freestream. In the previous study, the jet penetration height was monotonically increasing with the momentum flux ratio of jet to freestream. So, a moderate decline in the flowfield with wall expansion is shown. There is no doubt that the mixing characteristics of a jet injected into a supersonic crossflow are controlled by flow behavior. Therefore, the flow acceleration and changes in boundary-layer properties caused by favorable pressure gradients have a great effect on the mixing process. Figure 4.34 shows a comparison of passive scalar distributions and corresponding root mean square contours at the middle plane. The root mean square of passive scalar at the windward side of the jet plume in an expansion flow is lower. This indicates that the scalar fluctuation recedes, so it seems that a favorable pressure gradient suppresses the fluctuation intensity of passive scalar. Meanwhile, the shorter residence time of a flow, caused by flow acceleration, reduces the rate of molecular diffusion. In this case, wall expansion is not conducive to the mixing of a jet in a supersonic crossflow.

140

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

To quantitatively evaluate the effects of wall expansion on mixing characteristics, passive scalar fluctuations, mixing efficiency, and total pressure recovery coefficients are analysed in the following text. The volume integral of passive scalar root mean square VIYrms with variation in flowfield Mach number is displayed in Fig. 4.35. This can be used to represent the distribution of passive scalar fluctuations. The main concentration of values in the case of a flat plate is near Mach 1.6, while the peak in the expansion flowfield corresponds to about Mach 1.7. The scalar fluctuations are mainly located in regions that are close to the Mach number of the freestream, where the jet interacts with crossflows strongly. Wall expansion results in scalar fluctuations occuring within a higher Mach number range. Mixing efficiency parameter is usually used to quantificationally evaluate mixing degree, indicating the fraction of the reactant that would react if brought to chemical equilibrium with the air, defined as (Wang et al. 2013):  Yr ρud A (4.3) ηm   Yρud A and

Sun et al Flat plate Expansion plate

H0.99 /d

6

3

0

5

10

15

20

25

30

x/d Fig. 4.33 Comparison of jet penetration heights along with empirical data (Liu et al. 2016)

Fig. 4.34 Comparison of passive scalar distributions and corresponding root mean square contours at the middle plane (Liu et al. 2016)

4.1 Mixing Characteristics of Single Injection Fig. 4.35 Volume integral of the passive scalar root mean square with variation in Mach number (Liu et al. 2016)

141

Flat plate Expansion plate

4

VIYrms

3

2

1

0 1

1.2

1.4

1.6

1.8

2

Ma

Yr 

Y, Ystoic



1−Y 1−Ystoic

 Y ≤ Ystoic , Y ≥ Ystoic

(4.4)

where Y is the fuel mass fraction, Y r is the fuel fraction mixed in a proportion that can react, and Y stoic is the fuel stoichiometric mass fraction. The value Y stoic of ethylene is used here considering that its molecular weight is similar to that of inert air. The total pressure recovery is used to evaluate the total pressure loss during the mixing process, which is defined as (Wang et al. 2013):  P0 ρudA (4.5) σrec   P∞0 ρud A Figure 4.36 shows comparisons of mixing efficiency and total pressure recovery coefficient between flat plate and expansion plate cases. Figure 4.36a shows that there is no distinct difference in mixing efficiency near the jet injection between these two cases, where entrainment by large-scale coherent structures in the windward shear layer contributes to rapid mixing. Wall expansion has no significant impact on the generation and evolution of large-scale vortices close to the jet injection, so mixing efficiency changes very little. In the regions away from the jet injection, large-scale vortices break up into smaller ones and jet velocity is close to the supersonic mainstream. Entrainment of counter-rotating vortices and molecular diffusion control mixing. Owing to an increase of mainstream velocity with wall expansion, entrainment and molecular diffusion become more difficult, leading to a decrease in mixing efficiency. In Fig. 4.36b it can be seen that total pressure decreases along the streamwise direction and that total pressure loss in the expansion case represents an increase of 0.64% compared with the flat plate case. The blockade effect of a transverse jet

142

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a)

(b) 1

0.8

Flat plate Expansion plate

Total pressure recovery

Mixing efficiency

Flat plate Expansion plate

0.6

0.4

0.2

0

10

15

20

x/d

25

30

0.99

0.98

0.97

0.96

0.95

10

15

20

25

30

x/d

Fig. 4.36 Mixing efficiency and total pressure recovery coefficient (Liu et al. 2016)

on a supersonic crossflow causes a loss of total pressure. The higher the velocity in the flowfield, the more total pressure loss is induced. After investigating the effects of wall expansion on instantaneous vortical structures and statistical characteristics, the main findings were as follows. Large-scale vortices at the windward side of the shear layer in the near field gradually break up into smaller ones in the far field. The K–H instability is enhanced by the interaction of a sonic jet with accelerated crossflows caused by wall expansion, which leads to large-scale vortices breaking up more quickly. With the flow passing through the expansion corner, the favorable pressure gradient causes falling static pressure and a markedly higher streamwise velocity, resulting in the separation region ahead of jet plume being closer to the wall. Comparing this with the results of a flat plate, the jet penetration height is slightly lower, mixing efficiency in the near field is almost the same, but in the far field is lower; at the same time, in our current comparison, wall expansion brings about a 0.64% greater total pressure loss at the exit.

4.2 Mixing Characteristics of Multiple Injections The mixing and combustion properties of a multiport injection scheme are very different from those of a single injection scheme. Blockage effects, due to the momentum flux of the front injection flow, have a great impact on the rear injection flow (Lee 2015a), as do preheating effects, due to the chemical reactions of the front injection flow having an effect on the combustion process of the rear injection flow (Lee 2015b). Accurate fuel distributions in the flowpath are crucial for the design of an injection system (Jacobsen et al. 2015), and a staged-injection configuration can enhance the mixing of the primary fuel jet by allowing interactions between two adjacent jets—promoting penetration, induced by the secondary jet’s push up to the

4.2 Mixing Characteristics of Multiple Injections

143

primary (Takahashi et al. 2015). Thus, this has been combined with other flame holders in the flowpath design of scramjet combustors (Chakraborty et al. 2003). At the same time, this technique enhances wall cooling and reduces total pressure losses and the generation of injection-induced shock (Pudsey and Boyce 2015). Furthermore, a smaller secondary jet has been employed downstream of the primary jet by Viti et al. (2004) to increase the normal force on the flat plate and decrease the nose-down pitching moment. Pudsey et al. (2013a, b) numerically investigated the film-cooling drag reduction performance of a small-scale multiport injector array and the interactions between adjacent jets that are responsible for variations in downstream performance, i.e., drag reduction, heat transfer, and mixing efficiency. Large three-dimensional recirculation zones exist in this region, and the effective upstream flow for each injector influences the generation of normal CVPs. They found that global drag reduction has a strong linear dependence on injection mass flow rate, with drag decreasing with increased flow rate, as well as heat transfer rate and mixing efficiency. However, it only has a slight dependence on streamwise spacing, and it generally increases with increased streamwise spacing, as well as heat transfer and mixing efficiency. Lee (2015a, b) comprehensively investigated the mixing and combustion characteristics of dual transverse injection in a scramjet combustor. Consideration was given to the influences of the jet-to-crossflow momentum flux ratio and injector distance. He found that the dual injection system improved mixing and combustion efficiency as well as penetration depth, and that it would induce a greater stagnation of pressure loss. At the same time, there exists an optimal injector distance for the improvement of mixing and combustion efficiency—this increases with an increase of the jet-to-crossflow momentum flux ratio. Takahashi et al. (2015) extended the fluorescence ratio technique for processing planar laser-induced fluorescence data for quantitative imaging of the injectant molefraction and density in staged-injection configurations, as well as investigating the influences of injection species and secondary pulsed-injection. They found that the air–air case shows higher penetration compared with others after secondary injection, and the effect of secondary injection occurs earlier in the air–primary injection than in the He-primary injection. At the same time, a hysteresis exists in penetration height for pulsed injection—the ascending phase is beneficial for fuel penetration. This phenomenon is the same as that observed in the mode transition process (Huang et al. 2014). Davitian et al. (2015) explored the stability properties of the shear layer for a single gaseous jet in a crossflow or transverse jet in order to control jet penetration and spread—periodic jet forcing has been proposed as well. The controlled supersonic swirling injector (CSSI) has been proved to be able to achieve good mixing and entrainment (Murugappan et al. 1971, 2005). However, a staged-injection was not able to offer significant advantages in terms of mixing—a conclusion different from that obtained by Gao and Lee (2011). This may be a result of differences in the geometric configuration and boundary conditions employed. Section 4.2 reveals the flow patterns and mixing characteristics of multiple injections with tandem multi-ports and parallel multi-ports in a supersonic vitiated air flow, both numerically and experimentally. Using the nanoparticle-based laser-scattering

144

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

imaging technique, the schlieren system, and the surface oil-flow visualization technique, the injection schemes of tandem multi-ports and parallel multi-ports were tested using various parameters, including injection stagnation pressure and distance between orifices (Sun et al. 2011).

4.2.1 Baseline In order to study the flow patterns and mixing characteristics of different multiple injection schemes and to compare the effects of key parameters, several conditions were considered as baseline. The freestream conditions of the nozzle exit and the nitrogen injection conditions used are provided in Table 4.1. For the experiments reported in this section, the injection stagnation pressures were 1.2 and 1.8 MPa, denoted as No. 1 and No. 2 injection conditions, respectively—the stagnation temperature was 300 K. No. 1 and No. 2 injection conditions were considered for the basic testing of single injections. Figures 4.37, 4.38, 4.39, and 4.40 illustrate a series of experimental results for a single-injection scheme with injection conditions No. 1 and No. 2. The general characteristics of a single-injection flowfield are evident in these images. From the planar laser-scattering (PLS) images of Figs. 4.37 and 4.38, features of large-scale coherent structures can clearly be seen. Large-scale eddies are generated at the early stages of the jet–freestream interaction and bend in response to the crossflow. In the region of bending, the velocity gradient between the high-speed freestream and the low-speed jet flow leads to stretching of the large-eddy structures. This stretching process also enlarges eddy structures and causes the freestream to be engulfed by the jet shear layer. As the injection total pressure is increased, penetration height is clearly seen to increase. The width of the injection beam was found to increase, as seen in top-view PLS images. Considering the schlieren images, shown in Fig. 4.39, the bow shock upstream of the jet obviously intensified when No. 2 injection conditions were used. From Fig. 4.40, by applying an oil tracer mixture on the wall surface, two separation lines were identified, wrapped around the injector from the windward recirculation. The oil between the two lines was not observed to penetrate either line as it moved outward and around the jet. The outer line also demonstrated the shape

Table 4.1 Experimental conditions (Sun et al. 2011)

Nozzle exit

No. 1

No. 2

Stagnation temperature, K (T 0 )

1404

300

300

Ratio of specific heat (γ )

1.34

1.40

1.40

Stagnation pressure, MPa (P0 )

0.5

1.2

1.8

Mach number (M)

2.0

1.0

1.0

4.2 Mixing Characteristics of Multiple Injections

(a) No. 1 injection condition

145

(b) No. 2 injection condition

Fig. 4.37 Side-view NPLS images in the center line plane of a single-injection scheme with different injection pressures (Sun et al. 2011)

(a) No. 1 injection condition

(b) No. 2 injection condition

Fig. 4.38 Top-view NPLS images in the y  4-mm plane of a single-injection scheme with different injection pressures (Sun et al. 2011)

(a) No. 1 injection condition

(b) No. 2 injection condition

Fig. 4.39 Schlieren images of a single-injection scheme with different injection pressures (Sun et al. 2011)

of the bow shock upstream of the jet. The distinctive shape of these lines is usually referred to as the vortex system, resulting from flow separation in this region as a horseshoe vortex. Downstream of the separation lines, the region wrapped around the injector showed a clean, smooth surface, meaning a high-speed field existed around the jet. Since the bow shock was declined or significantly oblique, velocity in this field might be supersonic. At the downstream boundary of the jet port, two lines formed, along which oil accumulated. The lines were observed to diverge as they progressed downstream. These lines perhaps represented a boundary between the flow upstream of the jet that moved around the jet and the reattaching flow downstream of the injector port. Since a low-speed field existed and there was possibe separation in the jet leeward region, the two lines perhaps demonstrated shapes associated with reattaching shock, generated from the interaction of supersonic flow around the jet and the low-speed field downstream of the jet port. Figures 4.41, 4.42, 4.43, and 4.44 illustrate a series of numerical results for a single-injection scheme with No. 1 and No. 2 injection conditions. From the figures it can be seen that the flow patterns and mixing characteristics revealed by the cal-

146

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.40 Surface oil-flow images of a single injection scheme with No. 2 injection conditions (Sun et al. 2011)

culations compare well with experimental images. From Figs. 4.41 and 4.42, it can be seen that jet penetration height is increased when injection stagnation pressure is increased. The jet beam expands wider and farther into the mainstream. From Fig. 4.43, it can be seen that although a low-angled injector is used in the study, the bow shock is clear and the intensity of the bow shock upstream of the jet is increased with higher injection pressure. Compared with experimental images, the large-scale structures along the jet surface are not well resolved, due to the steady RANS calculation. The static pressure behind the jet at the initial injection port, and under the jet beam during the bending process, is lower than that of the mainstream—the pressure gradually recovers as the jet expands downstream. Obviously, as the injection pressure increases the low-pressure region will enlarge. From the velocity field and the streamline image of Fig. 4.44, it can be seen that the separation zone upstream of the jet propagates downstream and forms some separation lines. As the separation zone wraps around the jet, it converges on itself at a point aft of the jet. A separation occurs immediately downstream of this point, then the flow reattaches aft of the separation, generating the reattachment shock seen in Fig. 4.44. The secondary separation zone continues propagating downstream and acquires a velocity recovery region. The flow patterns have been well described elsewhere (Dickmann and Lu 2006) and are identical to the patterns obtained here. The simulation agrees favorably with experimental results. Figure 4.45 shows the mass-averaged mixing efficiency and total pressure recovery coefficient of a single-injection scheme along the streamwise direction for No. 1 and No. 2 injection schemes. It can be observed from the figures that the mixing efficiency of a No. 2 injection is higher than that of No. 1, while the total pressure recovery coefficient of No. 2 is lower than No. 1 downstream of the x  20-mm plane. The results correspond to analyses based on the flow patterns and mixing characteristics, indicating that the definition of the mixing efficiency and the total pressure recovery coefficient is feasible for comparing mixing performance in this case.

4.2 Mixing Characteristics of Multiple Injections

(a) No. 1 injection condition

147

(b) No. 2 injection condition

Fig. 4.41 Iso-surfaces of species mass fraction structures (z  0.05) and contours on the wall surface of a single-injection scheme with No. 1 and No. 2 injection conditions (Sun et al. 2011)

(a) No. 1 injection condition

(b) No. 2 injection condition

Fig. 4.42 Slices of species mass fraction contours on x  20-, 40-, and 60-mm planes and the wall surface of a single-injection scheme with No. 1 and No. 2 injection conditions (Sun et al. 2011)

(a) No. 1 injection condition

(b) No. 2 injection condition

Fig. 4.43 Center line slices of pressure contours on the z  0-mm plane of a single-injection scheme with No. 1 and No. 2 injection conditions (Sun et al. 2011)

4.2.2 Tandem Configuration The diameter of the injection orifice is denoted as “D” in this context. The number of injector ports is denoted as “N.” The distance between centers of injectors is denoted as “T ” when tandem injection geometry is used. For a multiple injection scheme with two ports in a tandem row, parallel with the freestream direction, the distance between the injection orifices is T /D  5 and 10. Sun et al. (2011) investigated the mixing characteristics of multiple injections in a tandem configuration. They

148

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) No. 1 injection condition

(b) No. 2 injection condition

Fig. 4.44 Mach number contours and streamlines on the y  0.05-mm plane of a single-injection scheme with No. 1 and No. 2 injection conditions (Sun et al. 2011)

Fig. 4.45 Mixing efficiencies and pressure recovery coefficients of a single-injection scheme along the wall for No. 1 and No. 2 injection conditions (Sun et al. 2011)

found that decreasing the distance between the tandem ports reduced the pressure and velocity of the stream, upstream of the second jet, resulting in an increased penetration height of the second injection and a quick mixing of the whole field. Figures 4.46, 4.47, 4.48, and 4.49 illustrate a series of experimental results for the N2T5 and N2T10 scheme with No. 2 injection conditions. From the nanoparticlebased planar laser scattering (NPLS) images of Figs. 4.46 and 4.47, it can be seen that enhanced mixing of the injection occurs just downstream of the first injector for both schemes. The majority of the injectant from the second injector appears to penetrate further from the wall and expand farther than the first jet. In addition, further observation demonstrates that with a smaller distance between the tandem injectors in the N2T5 scheme, the penetration of the second jet increases more than in the N2T10 scheme. This might be due to the effect of the wake flow of the first jet. It can be concluded that the pressure behind the first jet is relatively low, and flow in the region behind the first jet has a low velocity. If the second jet is located in

4.2 Mixing Characteristics of Multiple Injections

(a) N2T5 scheme

149

(b) N2T10 scheme

Fig. 4.46 Side-view NPLS images of the center line plane of N2T5 and N2T10 schemes with No. 2 injection conditions (Sun et al. 2011)

(a) N2T5 scheme

(b) N2T10 scheme

Fig. 4.47 Top-view NPLS images of the y  4-mm plane of N2T5 and N2T10 schemes with No. 2 injection conditions (Sun et al. 2011)

(a) N2T5 scheme

b) N2T10 scheme

Fig. 4.48 Schlieren images of N2T5 and N2T10 schemes with No. 2 injection conditions (Sun et al. 2011)

the wake region of the first jet, the pressure on the upwind surface of the second jet would reduce greatly and this would result in a higher penetration into the flowfield of the second jet. When the second jet penetrates higher than the region affected by the first jet, and encounters the mainstream, the jet will be compressed by the main flow and quickly bend. In the bending process, the jet is sheared by the main flow and large-scale eddies are generated in the interface where the jet and freestream interact (as shown in Fig. 4.46). When the distance between the tandem injectors increases, especially if the second jet is outside of the wake region of the first jet, the pressure and velocity will be recovered at the location of the second jet. At this time the dynamic pressure on the upwind surface of the second flow shows almost no difference compared with the first jet which is exposed in the main flow. Therefore, the penetration height of the second jet only increases a little. The bending process covers a long distance due to the weakened interaction of the second jet with the mainstream. Considering the schlieren images shown in Fig. 4.48, for the N2T5 scheme, the bow shock upstream of the second jet is not apparent, which means the near-wall

150

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) N2T5 scheme

(b) N2T10 scheme Fig. 4.49 Surface oil-flow images of N2T5 and N2T10 schemes with No. 2 injection conditions (Sun et al. 2011)

windward surface of the second jet is mainly located in the subsonic region formed by the first jet. When the second jet penetrates higher into the flowfield, a shock wave originates from the supersonic mainstream. While for N2T10 scheme, the bow shock of the second jet appears near to the wall, which means that the flow has recovered to a supersonic state upstream of the second jet. Looking at the surface oil-flow images shown in Fig. 4.49, the shock wave originates from the windward region of the second jet in the N2T10 scheme, where the flow might be supersonic and compress the second jet, resulting in a long penetration distance. From this point of view, a greater distance between tandem injectors would decrease the interaction of the twin jets and prevent adequate mixing effects for the second jet. Figures 4.50, 4.51, 4.52, and 4.53 illustrate a series of numerical results for N2T5 and N2T10 schemes under No. 2 injection conditions. From Figs. 4.50 and 4.51, it can be seen that the penetration height of the second jet is greater than the first jet for both schemes. However, for the N2T5 scheme, which has a shorter tandem distance than the N2T10 scheme, the second jet beam expands both wider and farther into the mainstream. From the crossflow planes shown in Fig. 4.51, it can be seen that at x  60 mm the injectant of the N2T5 scheme penetrates higher than the N2T10 scheme and the N2T5 interface of injectant with air stream covers a larger area, indicating more efficient mixing in the N2T5 scheme than in the N2T10 scheme. From Figs. 4.52 and 4.53, it can be observed that for the N2T5 scheme, the second jet is located completely in the leeward region of the first jet, resulting in a lower pressure and lower streamwise velocity upstream of the second jet than in the N2T10 scheme. Thus, the second jet of the N2T5 scheme acquires a higher penetration into

4.2 Mixing Characteristics of Multiple Injections

(a) N2T5 scheme

151

(b) N2T10 scheme

Fig. 4.50 Iso-surfaces of species mass fraction structures (z  0.05) and contours on the wall surface with No. 2 injection conditions for tandem schemes (Sun et al. 2011)

(a) N2T5 scheme

(b) N2T10 scheme

Fig. 4.51 Slices of species mass fraction contours on the x  20-, 40-, and 60-mm planes and the wall surface with No. 2 injection conditions for tandem schemes (Sun et al. 2011)

the mainstream than the N2T10 scheme, and the bow shock formed by the interaction with the mainstream has a greater declined angle to the freestream than the first jet. For the N2T10 scheme, the distance between tandem injectors increases and it can be seen from Fig. 4.52 Center line slices of pressure contours on the z  0 plane with No. 2 injection conditions for tandem schemes and Fig. 4.53 that the second jet is located in the pressure and velocity recovery region downstream of the first jet. The pressure and velocity conditions on the windward surface of the second jet are not very different to those of the first jet. Therefore, the penetration height of the second jet increases little and the bow shock upstream demonstrates a similar declined angle as the first jet. The flow patterns and mixing characteristics revealed by the numerical calculations compare favorably with experimental images. Figure 4.54 illustrates the mixing efficiencies and total pressure recovery coefficients of the N2T5 and N2T10 schemes along the mainstream direction under No. 2 injection conditions. It can be seen that the mixing efficiency of the N2T5 scheme is always higher than the N2T10 scheme over the whole region, which means a smaller distance between the tandem injectors could efficiently increase the penetration and mixing effects of the second jet, and further increase the whole mixing performance. Obviously in Fig. 4.54b it can be seen that the total pressure recovery coefficient of the N2T5 scheme is lower than that of the N2T10 at the location downstream of

152

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) N2T5 scheme

(b) N2T10 scheme

Fig. 4.52 Center line slices of pressure contours on the z  0 plane with No. 2 injection conditions for tandem schemes (Sun et al. 2011)

(a) N2T5 scheme

(b) N2T10 scheme

Fig. 4.53 Mach number contours and streamlines on the y  0.05-mm plane with No. 2 injection conditions for tandem schemes (Sun et al. 2011)

Fig. 4.54 Mixing efficiencies and pressure recovery coefficients of the N2T5 and N2T10 schemes along the wall with No. 2 injection conditions (Sun et al. 2011)

the second jet. However, the pressure recovery for the N2T5 scheme processes more quickly than the N2T10 scheme since at x  60 mm the pressure of both schemes has recovered to the same level. Based on this characteristic, the smaller the distance between tandem injectors the more efficient and rapid the mixing.

4.2 Mixing Characteristics of Multiple Injections

153

4.2.3 Parallel Configuration The distance between the centers of injectors is denoted as “S” when parallel injection geometry is used. For a multiple-injection scheme with three injection orifices in a parallel row perpendicular to the freestream direction, the distance between the injection orifices is S/D  2.5, 5, and 10. For small distances between parallel multi-orifices, the bow shock waves upstream of the injected jets connect with each other and the air stream entering the gap between the jets is insufficient, resulting in a decrease in the mixing effect. Large distances between parallel multi-orifices decrease the interaction between injection jets. For mixing enhancement, there should be an optimized distance between the parallel injection orifices. Figures 4.55, 4.56, 4.57, and 4.58 illustrate a series of experimental results for N3S2.5, N3S5, and N3S10 schemes with No. 2 injection conditions. From the NPLS images of Figs. 4.55 and 4.56, it can be seen that the center jet clearly extrudes side jets for the N3S2.5 and N3S5 schemes. For the N3S2.5 scheme, the three jets almost connect with one another and combine—a process that blocks the mainstream significantly. Additionally, since the air stream would not be able to penetrate the core of the parallel jets, the mixing effect would be weakened. For the N3S5 scheme, the center jet interacts with the side jets while still allowing the air to enter the jet—a process that is efficient in terms of enhancing the mixing of jets with air. Also, the interaction between the center jet with the side jets induces a disturbance that promotes the generation of large structures on the jet surface, taking advantage of the mixing enhancement offered by large eddies. For the N3S10 scheme, the center jet has little interaction with the side jets—the three jets develop downstream separately, having a similar mixing character to the single port injection. However, by comparing the N3S5 scheme with the N3S10 scheme, it is hard to tell which would acquire a higher mixing efficiency, since no quantitative data has been obtained. Considering the schlieren images shown in Fig. 4.57, seeing the effects of the change of distance between the parallel injectors on the bow shock location is not obvious, and all the schemes demonstrate similar flow patterns from the side view. From the oil-flow surface image of Fig. 4.58, when the parallel distance is small (the N3S2.5 scheme), the separation lines upstream of the three jets form into a joint line. The flow pattern of the three jets on the wall surface is similar to that of a single jet. From this point of view, the pressure loss of the N3S2.5 scheme would be higher. For the N3S5 scheme, the obvious streamlines appear on the surface, caused by interactions between multiple jets. In addition, for large parallel distances (the N3S10 scheme), each jet maintains the characteristics of a single jet, revealing a pattern showing no disturbance between multiple injectors. Due to limited flowfield information, the mixing characteristics cannot be clearly identified from only NPLS, schlieren, and surface oil-flow images. These images cannot give flow structures in detail. Therefore, it is useful to apply computational fluid dynamics (CFD) to the flowfields of the various injection schemes and compare them in order to discuss the effects of key parameters on flow patterns and mixing characteristics.

154

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.55 Top-view NPLS images of the y  4-mm plane of different parallel injection schemes with No. 2 injection conditions (Sun et al. 2011)

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme

x=20mm:

x=40mm:

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme

Fig. 4.56 End-view NPLS images of the x  20- and 40-mm planes for different parallel injection schemes with No. 2 injection conditions (Sun et al. 2011)

4.2 Mixing Characteristics of Multiple Injections

(a) N3S2.5 scheme

155

(b) N3S5 scheme

(c) N3S10 scheme Fig. 4.57 Schlieren images of different parallel injection schemes with No. 2 injection conditions (Sun et al. 2011)

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme Fig. 4.58 Surface oil-flow images of different parallel injection schemes with No. 2 injection conditions (Sun et al. 2011)

156

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme

Fig. 4.59 Iso-surfaces of species mass fraction structures (z  0.05) and contours on the wall surface with No. 2 injection conditions for parallel injections (Sun et al. 2011)

Figures 4.59, 4.60, 4.61, and 4.62 illustrate a series of numerical results for the N3S2.5, N3S5, and N3S10 schemes under No. 2 injection condition. From Figs. 4.59 and 4.60, it can be seen that the center jet clearly interacts with the side jets for the N3S2.5 and N3S5 schemes, but not the N3S10 scheme. From the crossflow planes shown in Fig. 4.60, it can be seen that for the N3S2.5 scheme, the side jets extrude the center jet and lift it to a higher position some distance downstream of the injection orifices. In addition, due to the extrusion, the center lines of the core of the side jets rotate outward from the center jet in the crossflow planes. However, for the N3S5 scheme, due to the ejecting effects of the center jet, the center lines of the core of side jets rotate inward to the center jet in the crossflow planes. From Figs. 4.61 and 4.62, it can be seen that the bow shock wave upstream of the multi-jets in the N3S2.5 or N3S5 schemes connect with one another, forming a joint shock. Additionally, the N3S2.5 scheme acquires a more intensive bow shock due to the smaller distance between the injectors. For the N3S10 scheme, the center jet interacts much more weakly with the side jets and each jet develops downstream, with similar properties to that of a single jet injection. From Fig. 4.61 it can be seen that the bow shock of the N3S2.5 scheme is the most intensive and the N3S10 scheme the least, indicating that a greater distance between parallel jets could reduce pressure loss. From the mass fraction distribution on the crossflow planes in Fig. 4.60, it is seen that for the N3S5 scheme more air enters the gap between the parallel jets than in the N3S2.5 scheme, indicating that the mixing effect would be increased. However, for the N3S10 scheme, excessively large distances between parallel jets decrease the interactions between the injection jets, the result being that the mixing effect might not be enhanced.

4.2 Mixing Characteristics of Multiple Injections

(a) N3S2.5 scheme

157

(b) N3S5 scheme

(c) N3S10 scheme

Fig. 4.60 Slices of species mass fraction contours on the x  20-, 40-, and 60-mm planes and the wall surface with No. 2 injection conditions for parallel injections (Sun et al. 2011)

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme

Fig. 4.61 Center line slices of pressure contours on the z  0 plane with No. 2 injection conditions for parallel injections (Sun et al. 2011)

158

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

(a) N3S2.5 scheme

(b) N3S5 scheme

(c) N3S10 scheme

Fig. 4.62 Mach number contours and streamlines on the y  0.05-mm plane with No. 2 injection conditions for parallel injections (Sun et al. 2011)

Figure 4.63 gives the mass-averaged mixing efficiency and total pressure recovery coefficient for the N3S2.5, N3S5, and N3S10 schemes along the mainstream direction for No. 2 injection conditions. For parallel injection, starting at the injection ports, the total pressure recovery coefficient of the N3S10 scheme is highest due to it having the lowest pressure loss induced by the bow shock, as shown in Fig. 4.63b. However, downstream of x  20 mm, the pressure recovery coefficients seem to be almost the same at different locations, meaning that pressure loss is not a significant problem for parallel schemes. When errors are considered the differences in losses are negligible. The most interesting thing is the mixing efficiency of the parallel injections. It is observed in Fig. 4.63a that the N3S5 scheme acquires the highest mixing efficiency over the whole region, and the mixing efficiencies of the N3S2.5 and N3S10 schemes are almost the same. Combined with the above analysis on the flow patterns and mixing characteristics of parallel injections, it is clear that less air is engulfed into the jet core and consequently the mixing effect suffers, although for the N3S2.5 scheme the center jet has a greater penetration height. For the N3S5 scheme, the parallel injectors have a suitable distance between them, ensuring interaction between jets as well as ensuring that enough air is blowing into the jet gap to provide a higher mixing performance. In a summary, too narrow or too wide a distance between the parallel injectors will provide a poor mixing effect. An intermediate distance offers a better mixing performance with an acceptable pressure

4.2 Mixing Characteristics of Multiple Injections

159

Fig. 4.63 Mixing efficiencies and pressure recovery coefficients of the N3S2.5, N3S5, and N3S10 schemes along the wall with No. 2 injection conditions (Sun et al. 2011)

loss. Experimental and numerical studies both indicate that there should be a proper optimized distance between the parallel injection orifices for mixing enhancement.

4.3 Mixing Enhancement Technology In a scramjet engine, the velocity of supersonic inflow is fast, and thus the time available for mixing and combustion is of the order of a few milliseconds. It is difficult to achieve stable and efficient combustion in such a short time in the combustion chamber. It requires solving many key technological issues, such as those associated with fuel injection and mixing enhancement technology, ignition technology, and flame stabilization technology. Since the residence time of fuel is extremely short and the stability of the supersonic mixing layer is strong (the expansion ratio is only one-third of the incompressible shear layer with the same density ratio), mixing enhancement technology must be developed to achieve rapid mixing. Mixing enhancement technology in scramjet engines can be classified into two categories: active mixing enhancement technology and passive mixing enhancement technology.

4.3.1 Active Mixing Enhancement Technology Pulsed injection has been considered a promising approach for jets in a crossflow to enhance jet penetration and the subsequent mixing process. Large-scale vortex structures due to a pulsed jet can induce deeper penetration and a larger mixing zone compared with counterparts in a steady jet case. The introduction of a pulsed jet transversely into a supersonic crossflow is shown schematically in Fig. 4.64.

160

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.64 Side-view schematic of combustion with a pulsed jet

First, the corresponding jet-to-crossflow momentum flux ratio J is defined as: J

ρ j U 2j 2 ρ∞ U∞

(4.6)

The pulsation waveform shape can be in a variety of forms. Based on high-speed (Kouchi et al. 2010) and low-speed (Davitian et al. 2010) studies, a sine wave is chosen as the typical waveform for the pulsed jet. The pulsed jet-to-crossflow momentum flux ratio can be described as: J p  Ja + Jm sin(2π f t)

(4.7)

where Jm and f represent the pulsation amplitude and frequency of the momentum flux ratio, respectively, and Ja is the mean value of the pulsed momentum flux ratio. Figure 4.65 shows the variation of the flowfield in one pulse period. A bow shock wave is generated before the jet when a pulsed jet injects into a supersonic crossflow, as shown in Fig. 4.65a. At the same time, the turbulent boundary layer of the incoming flow will be separated by an inverse pressure gradient. Because of the interaction between the pulsated injection and the separated flow in the foot-region of the bow shock, a compression wave (CW1 ) propagates upward behind the bow shock. The compression wave is connected to the large-scale jet shear vortex (SV1 ) caused by the shear instability of the upwind jet. When the jet extends to the crossflow, the barrel shock (BS) and Mach disk (MD) are generated and a local reflected shock (RS1 ) and large-scale vortices (SV1 ) are formed. With the development of the shock wave, compression wave, and large-scale jet shear vortex, other compression waves CW2 and reflection shocks RS2 , related to the jet shear vortex SV2 , are formed periodically. These similar structures also appear successively in 2T /4 and 3T /4 during the pulse period.

4.3 Mixing Enhancement Technology

161

Fig. 4.65 Variation in the flowfield in one pulse period during one pulse cycle (Shi et al. 2016)

By changing the large-eddy characteristics of the jet shear layer, the mixing characteristics can be improved. The pulsed jet promotes unsteady characteristics and interactions between the compression wave, expansion wave, and vortex. During the evolution of the interaction between the pulsed jet and supersonic cross flow, the bow shock propagates upward with the compression wave. Bow shock waves show obvious oscillations and deformations. In addition, large-scale coherent vortices, associated with pulsed jets, such as SV1 and SV2 in Fig. 4.65, are formed due to the shear roll-up of the upwind jet, which can entrain the crossflow and enhance mixing and penetration of the jet. Shi et al. (2016) investigated a pulsed jet injected into a supersonic crossflow by utilizing a large-eddy simulation (LES) technique. In the study, a mixing area ψ is employed and defined to quantify the mixing property (Maddalena et al. 2006; Pudsey et al. 2013a, b): ∫(αρuYk )dydz 1 if Yk < 0.5 α  1−Yk (4.8) ψ if Yk ≥ 0.5 ∫(αρuYk )dydz Yk

162

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

where u is the streamwise velocity and α is determined by the mass fraction Y K . Figure 4.66 shows the mixing area for the pulsed and steady jet. Identified in this figure are the mixing area increments between pulsed and steady jets, indicating the mixing enhancement of a pulsed jet.

4.3.2 Passive Mixing Enhancement Technology The earliest scramjet fuel injection method is vertical fuel injection from the wall. This type of injection can provide fast near-field mixing and high fuel penetration, and the recirculation zone generated by the jet can stabilize the flame. However, the jet generates a strong shock wave, resulting in a large total pressure loss, and the local wall heat load is also high. At the same time, the compressive effect is not conducive to the generation and shedding of vortices in the fuel/air mixture layer, which inhibits suction mixing and slows down the mixing and combustion of the far field. In order to reduce the shock intensity and total pressure loss of vertical injection, small-angle injection can be used. It has been shown that the near-field mixing effect is weakened with a decreased angle, but the far-field mixing and the overall mixing effect are basically unchanged. Parallel injection minimizes shock intensity and total pressure loss, but the mixing efficiency is so low that combustion efficiency is very low. In order to enhance the penetration and mixing efficiency of wall jets, researchers adopted some wall-assisted measures, commonly used with a vortex generator scheme. Micro-ramp Vortex Generator Recently, a kind of micro-ramp vortex generator shown in Fig. 4.67 has been used in the study of shock wave and boundary layer interactions (Wang et al. 2012; Sun et al. 2014). The tiny ramp is lower than the boundary layer thickness, thus it can

Fig. 4.66 Profiles of the mixing area (ψ) for pulsed and steady jets (Shi et al. 2016)

4.3 Mixing Enhancement Technology

163

be immersed into the boundary layer. Obviously, the total pressure loss is smaller compared with a traditional ramp. Besides this, the most remarkable discovery is that the micro-ramp decreases the momentum flux in wake flow. If the jet orifice is placed downstream of the micro-ramp, the low momentum wake flow may provide ideal conditions for increasing jet penetration. Based on the idea mentioned above, Zhang (2017) used a large-eddy simulation (LES) method to study a micro-ramp vortex generator submerged in an incoming boundary layer. The micro-ramp vortex generator was placed upstream of the jet orifice. He also studied the effect of a micro-ramp on transverse jet penetration and mixing of a transverse jet. (a) Analysis of the flowfield structure of a vortex generator The ratio of jet momentum to incoming flow is the main factor that determines the penetration and mixing efficiency of fuel injection. Therefore, the influence of a delta-wing vortex generator on incoming flow momentum is considered here. Figure 4.68 shows the momentum distribution of a time-averaged flowfield. The momentum in the boundary layer increases with an increase in normal height. When the fluid passes through the vortex generator, the momentum decreases sharply, with the momentum increasing upon development of the wake downstream. Therefore, based on the distribution characteristics of the wake momentum downstream of the vortex generator, nozzle holes are placed in the wake downstream of the vortex generator in order to improve penetration of the jet plume. In order to find the cause of the sudden decrease of fluid momentum downstream of the micro-delta-wing vortex generator, the following analysis on the flowfield under the action of the vortex generator is provided. Assuming that the fluid that flows through the two sides of the micro-delta-wing vortex generator is under adiabatic isentropic expansion, the following relationship is obtained: δ  υ(Ma2 ) − υ(Ma1 )

(a) configuration of micro - ramp

(4.9)

(b) jet orifice location

Fig. 4.67 Configuration and location of a micro-ramp (Zhang 2017)

164

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.68 The distribution of momentum (Zhang 2017)

where δ is the deflection angle, υ is the Plant Mayers function, Ma1 is the incoming Mach number, and Ma2 is the Mach number after expansion. Because: 1− T1  T2 1−

k−1 2 λ k+1 1 k−1 2 λ k+1 2

(4.10)

the momentum ratio before and after deflection is as follows: ρ1 u 21  ρ2 u 22



1− 1−

k−1 2 λ k+1 1 k−1 2 λ k+1 2

k  k−1

Ma12 Ma22

(4.11)

Here, δ  24◦ and Ma1  2.7. υ(Ma1 )  44◦ and λ1  1.891. So: υ(Ma2 )  ρ u2 Ma1  0.645, uu 21  0.868. 68◦ , ρ1 u 12  3.313, ρρ21  4.412, TT21  1.811, Ma 2 2 2 The above analysis shows that the momentum decreases to 0.301, the density decreases to 0.227, the temperature decreases to 0.552, the Mach number increases by 1.55 times, and the velocity increases by 1.152 times. Therefore, after the micro-deltawing vortex generator, although the Mach number increases, ρ and T are reduced. The reduction of T leads to a decrease in the speed of sound, and thus u increases less than ρ, resulting in the decrease of the overall momentum. The above analysis is the reason why the momentum of the airflow decreases rapidly after encountering the micro-delta-wing vortex generator.

4.3 Mixing Enhancement Technology

165

In addition to the low-momentum downstream wake of the delta-wing vortex generator, another notable feature is the existence of a pair of inverted vortices. The reversal vortex pair dominates the momentum and mass exchange in the wake. If the nozzle is placed in the wake, the fuel distribution of the injected gas will be affected by the reversal vortex pair. Therefore, it is necessary to study the reversal vortex pair in the wake. Figure 4.69a shows the time-averaged velocity distribution nephogram at different cross sections downstream of the micro-vortex generator and the density distribution nephogram at a normal cross section. A symmetrical low-density region clearly exists on both sides of the vortex generator, caused by the acceleration of expanded air on both sides of the vortex generator. There is a cylindrical vortex generator wake region on cross sections of the direction of flow. It can be seen that there is a velocity deficit in the wake. With the wake developing downstream, momentum increases gradually, and flow velocity increases accordingly. At the same time, the wake will gradually rise to the mainstream in the process of developing downstream due to the effect of inverted vortex pairs. Figure 4.69b shows a pair of inverted vortices in the wake of the micro-vortex generator. The fluid under the wake is rolled-up due to the action of the inverted vortices, and the fluid in the mainstream flows toward the wall. Under the action of the inverted vortices, a pair of secondary vortices with opposite directions is induced near the wall. Figure 4.70 shows the iso-surface of the eddy structure of an instantaneous flowfield in an LES numerical simulation, in which the color represents normal height. A series of K–H vortices can be clearly seen in the wake of the vortex generator downstream. The vortex structure is mainly induced by the strong shear between the low-speed wake flow and the high-speed peripheral mainstream. There is no obvious K–H vortex near the vortex generator. As the wake develops downstream, large-scale K–H vortices begin to appear and gradually rise to the mainstream, then K–H vortices begin to destabilize and break up. At the same time, the time-dependent calculations of fluctuations in density and temperature scalars show that large-scale K–H vortices appear periodically in the wake and keep the same periodicity downstream.

(a)

(b)

Fig. 4.69 a The velocity and density distributions. b The vorticity and streamline distributions (Zhang 2017)

166

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

As a consequence of research, it has been found that a micro-ramp vortex generator can clearly promote mixing of injection gas and mainstream gas as shown in Fig. 4.71. (b) Influence of the vortex generator layout and combination methods on injection The low momentum swirl wake, induced by the micro-vortex generator, can effectively improve the penetration and mixing effect of a wall-jet injection flow. Zhang (2017) conducted experiments and simulations on the relative position and relative distance of the vortex generator and the orifice, revealing the influence of the arrangement of the vortex generator and the orifice on flowfield characteristics, jet penetration, and mixing effect. By comparing the jet plume of the vortex generator before and after the nozzle hole, it was found that the lift effect and mixing effect of the physical oblique plane of the vortex generator for the jet plume are limited. But when the nozzle hole is

Fig. 4.70 Vortex structure in an instantaneous flowfield (Zhang 2017) Fig. 4.71 Mixing coefficients for different cases (Zhang 2017). sj-0 represents a single nozzle injection scheme without a vortex generator, rp-8 represents a single nozzle injection scheme with a vortex generator, and 8 (nozzle diameter) represents the distance between the trailing edge of the vortex generator and the nozzle hole

4.3 Mixing Enhancement Technology

167

Fig. 4.72 Mixing coefficient for different cases (Zhang 2017). Sj-0 represents an example of wall injection without a vortex generator, rj-8 represents an example of a nozzle downstream of the vortex generator, and fj-2.5 represents an example of an orifice upstream of the vortex generator, where fj represents the front jet; 2.5 (nozzle diameter) indicates the distance from the orifice to the vortex generator

located downstream of the vortex generator, the penetration of the jet plume and the mixing effect of gas injection can be significantly increased and improved as shown in Fig. 4.72. Under the scheme that the nozzle hole is located downstream of the vortex generator, Zhang (2017) used a numerical simulation method to compare variations in the jet plume that resulted from changing the distance between the vortex generator and the nozzle orifice, so as to understand the role the vortex generator plays as a control mechanism of the jet plume. (The distance between vortex generator trailing edge and nozzle hole was X/d  0.25, 4, 8, and 16, respectively recorded as rp-0.25, rp-4, rp-8, and rp-16). A schematic diagram is shown in Fig. 4.73. The penetration of the average results of each study was extracted (penetration being defined as a 10% nitrogen mass fraction contour). The results are shown in Fig. 4.74. The comparison shows that penetration was the highest in the rp-8 scheme, followed by the rp-16, rp-4, sj-0, and rp-0.25 schemes. This apparent difference indicates that the distance between the vortex generator and the orifice has a significant effect on the penetration of the jet plume. Due to the change in the distance between the vortex generator and the orifice, the “state” of the wake of the vortex generator near the orifice is different, e.g., the momentum and the thickness of the wake, resulting in differences in jet plume penetration.

168

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Fig. 4.73 Jet orifice location for different cases (Zhang 2017)

Fig. 4.74 Mixing coefficients for different cases (Zhang 2017)

References Aso S, Kawai M, Ando Y (2006) Experimental study on mixing phenomena in supersonic flows with slotinjection. Aerosp Sci Meet 2006:1323–1324 Benyakar A, Mungal MG, Hanson RK (2006) Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Phys Fluids 18(2):1105–1154 Boles JA, Edwards JR, Bauerle RA (2011) Large-Eddy/Reynolds-Averaged Navier-Stokes simulations of sonic injection into Mach 2 crossflow. In: AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, 2011, pp 1444–1456 Chakraborty D, Roychowdhury AP, Ashok V, Kumar P (2003) Numerical investigation of staged transverse sonic injection in Mach 2 stream in confined environment. Aeronaut J 107(1078):719–729 Chenault CF, Beran PS (2012) K-e and Reynolds stress turbulence model comparisons for twodimensional injection flows. AIAA J 36(8):1401–1412 Davitian J, Hendrickson C, Getsinger D, MCloskey RT, Karagozian AR (2010) Strategic control of transverse jet shear layer instabilities. AIAA J 48:2145–2156 Davitian J, Hendrickson C, Getsinger D, M’Closkey RT, Karagozian AR (2015) Strategic control of transverse jet shear layer instabilities. AIAA J 48(9):2145–2156 Dickmann DA, Lu FK (2006) Jet in supersonic crossflow on a flat plate

References

169

Dussauge JP, Gaviglio J (2006) The rapid expansion of a supersonic turbulent flow—role of bulk dilatation. J Fluid Mech 174(174):81–112 Ferrante A, Matheou G, Dimotakis P (2011) LES of an inclined sonic jet into a turbulent crossflow at Mach 3.6. J Turbul 12(12) Gao ZX, Lee CH (2011) Numerical research on mixing characteristics of different injection schemes for supersonic transverse jet. Science China Technological Sciences 54(4):883–893 Gruber MR, Nejad AS, Chen TH, Dutton JC (2011) Transverse injection from circular and elliptic nozzles into a supersonic crossflow. J Propul Power 16(3):449–457 He G, Guo Y, Hsu AT (1999) The effect of Schmidt number on turbulent scalar mixing in a jet-incrossflow. Int J Heat Mass Transf 42(20):3727–3738 Huang W (2016) Transverse jet in supersonic crossflows. Aerosp Sci Technol 50:183–195 Huang W, Liu WD, Li SB, Xia ZX, Liu J, Wang ZG (2012) Influences of the turbulence model and the slot width on the transverse slot injection flow field in supersonic flows. Acta Astronaut 73(2):1–9 Huang W, Yan L, Tan JG (2014) Survey on the mode transition technique in combined cycle propulsion systems. Aerosp Sci Technol 39:685–691 Ivanova EM, Noll BE, Aigner M (2013) A numerical study on the turbulent Schmidt numbers in a jet in crossflow. J Eng Gas Turbines Power 135(2013):011505 Jacobsen LS, Schetz JA, Ng WF (2015) Flowfield near a mulitiport injector array in a supersonic flow. J Propul Power 16(2):216–226 Kawai S, Lele SK (2008) Mechanisms of jet mixing in a supersonic crossflow: a study using largeeddy simulation. In: AIAA Paper, 2008 Kawai S, Lele SK (2009) Large-eddy simulation of jet mixing in a supersonic turbulent crossflow. J Fluid Mech Kawai S, Lele SK (2010) Large-eddy simulation of jet mixing in supersonic crossflows. AIAA J 48(9):2063–2083 Kouchi T, Sasaya K, Watanabe J, Sibayama H, Masuya G. (2010) Penetration characteristics of pulsed injection into supersonic crossflow. AIAA Paper, 2010–6645 Lee SH (2015a) Characteristics of dual transverse injection in scramjet combustor, Part 1: Mixing. J Propul Power 22(5):1012–1019 Lee SH (2015b) Characteristics of dual transverse injection in scramjet combustor, Part 2: Combustion. J Propul Power 22(5):1020–1026 Liu C, Wang Z, Wang H, Sun M (2015) Numerical investigation on mixing and combustion of transverse hydrogen jet in a high-enthalpy supersonic crossflow. Acta Astronaut 116:93–105 Liu C, Wang Z, Wang H, Sun M (2016) Mixing characteristics of a transverse jet injection into supersonic crossflows through an expansion wall. Acta Astronaut 129:161–173 Maddalena L, Campioli TL, Schetz JA (2006) Experimental and computational investigation of light-gas injectors in a Mach 4.0 crossflow. J Propul Power 22(5):1027–1038. https://doi.org/10. 2514/1.19141 Murugappan S, Gutmark E, Carter C, Donbar J, Gruber M, Hsu KY (1971) Transverse supersonic controlled swirling jet in a supersonic cross stream. AIAA J 44(44):290–300 Murugappan S, Gutmark E, Carter C (2005) Control of penetration and mixing of an excited supersonic jet into a supersonic cross stream. Phys Fluids 17(10):729 Peterson DM, Candler GV (2015) Hybrid Reynolds-averaged and large-eddy simulation of normal injection into a supersonic crossflow. J Propul Power 26(3):533–544 Pudsey AS, Boyce RR (2015) Numerical Investigation of transverse jets through multiport injector arrays in a supersonic crossflow. J Propul Power 26(6):1225–1236 Pudsey AS, Boyce RR, Wheatley V (2013a) Hypersonic viscous drag reduction via multiporthole injector arrays. J Propul Power 29(5):1087 Pudsey AS, Boyce RR, Wheatley V (2013b) Influence of common modeling choices for high-speed transverse jet-interaction simulations. J Propul Power 29(5):1076–1086. https://doi.org/10.2514/ 1.B34750

170

4 Mixing Characteristics of a Gaseous Jet in a Supersonic Crossflow

Rana ZA, Thornber B, Drikakis D (2011a) On the importance of generating accurate turbulent boundary condition for unsteady simulations. J Turbul 12(12) Rana ZA, Thornber B, Drikakis D (2011b) Transverse jet injection into a supersonic turbulent cross-flow. Phys Fluids 23(4):585 Rizzetta D (2013) Numerical simulation of slot injection into a turbulent supersonic stream. AIAA J 30(30):2434–2439 Rothstein AD, Wantuck PJ (1992) A study of the normal injection of hydrogen into a heated supersonic flow using planar laser-induced fluorescence. AIAA 92 Santiago JG, Dutton JC (1997) Velocity measurements of a jet injected into a supersonic crossflow. J Propul Power 13(2):264–273 Schaupp C, Friedrich R (2010) Large-eddy simulation of a plane reacting jet transversely injected into supersonic turbulent channel flow. Int J Comput Fluid Dyn 24(10):407–433 Shi H, Wang G, Luo X, Yang J, Lu X-Y (2016) Large-eddy simulation of a pulsed jet into a supersonic crossflow. Comput Fluids 140:320–333. https://doi.org/10.1016/j.compfluid.2016.10.009 Spaid FW, Zukoski EE (2012) A study of the interaction of gaseous jets from transverse slots with supersonic external flows. AIAA J 6(2):205–212 Sriram AT, Mathew J (2006) Improved prediction of plane transverse jets in supersonic crossflows. AIAA J 44(2):405–408 Sriram AT, Mathew J (2008) Numerical simulation of transverse injection of circular jets into turbulent supersonic streams. J Propul Power 24(1):45–54 Sun M, Hu Z (2018a) Mixing in nearwall regions downstream of a sonic jet in a supersonic crossflow at Mach 2.7. Phys Fluids 30(106102) Sun M, Hu Z (2018b) Formation of surface trailing counter-rotating vortex pairs downstream of a sonic jet in a supersonic crossflow. J Fluid Mech 850(33 Sun M, Wang Z, Liang J (2008) Flame characteristics in a supersonic combustor with hydrogen injection upstream of a cavity flameholder. J Propul Power 24(4):9 Sun M, Geng H, Liang J, Wang Z (2009) Mixing characteristics in a supersonic combustor with gaseous fuel injection upstream of a cavity flameholder. Flow Turbul Combust 82(2):271–286 Sun M, Lei J, Wu H, Liang J, Liu W, Wang Z (2011) Flow patterns and mixing characteristics of gaseous fuel multiple injections in a non-reacting supersonic combustor. Heat Mass Transf 47(11):1499–1516 Sun M, Zhang S, Zhao Y, Zhao Y, Liang J (2013) Experimental investigation on transverse jet penetration into a supersonic turbulent crossflow. Sci China Technol Sci 56(8):1989–1998 Sun Z, Schrijer FFJ, Scarano F, van Oudheusden BW (2014) Decay of the supersonic turbulent wakes from micro-ramps. Phys Fluids 26(2):25115. https://doi.org/10.1063/1.4866012 Sun M, Cui X, Wang H, Bychkov V (2015) Flame flashback in a supersonic combustor with ethylene injection upstream of cavity flameholder. J Propul Power 31(2):6 Takahashi H, Ikegami S, Masuya G, Hirota M (2015) Extended quantitative fluorescence imaging for multicomponent and staged injection into supersonic crossflows. J Propul Power 26(4):798–807 Terzi DAV, Sandberg RD, Fasel HF (2009) Identification of large coherent structures in supersonic axisymmetric wakes. Comput Fluids 38(8):1638–1650 Viti V, Bowersox R, Neel R, Schetz J, Wallis S (2004) Jet interaction with a primary jet and an array of smaller jets. AIAA J 42(7):1358–1368 Viti V, Neel R, Schetz JA (2009) Detailed flow physics of the supersonic jet interaction flow field. Phys Fluids 21(4):296 Wang H, Qin N, Sun M, Wang Z (2011) A hybrid LES (Large Eddy Simulation)/assumed sub-grid PDF (Probability Density Function) model for supersonic turbulent combustion. Sci China Tech Sci 51(10):13 Wang H, Qin N, Sun M, Wang Z (2012) A dynamic pressure-sink method for improving large eddy simulation and hybrid Reynolds-averaged Navier-Stokes/large eddy simulation of wall-bounded flows. Proc Inst Mech Eng Part G J Aerosp Eng 226(9):14 Wang H, Wang Z, Sun M, Qin N (2013) Hybrid Reynolds-averaged Navier-Stokes/large-eddy simulation of jet mixing in a supersonic crossflow. Sci China Technol Sci 56(6):1435–1448

References

171

Won SH, Jeung IS, Parent B, Choi JY (2012) Numerical investigation of transverse hydrogen jet into supersonic crossflow using detached-eddy simulation. AIAA J 48(6):1047–1058 You Y, Luedeke H, Hannemann K (2013) Injection and mixing in a scramjet combustor: DES and RANS studies. Proc Combust Inst 34(2):2083–2092 Zhang YJ (2017) Research on flow and mixing mechanism of wall gaseous jet in supersonic crossflow. PHD

Chapter 5

Reaction Characteristics of a Gaseous Jet in a Supersonic Crossflow

Most of the previous studies were performed in conventional wind tunnels by accelerating cold air into a supersonic crossflow, namely in low-velocity and low–total enthalpy flow conditions. However, a real supersonic combustor environment at flight speeds beyond Mach 8 can only be simulated using impulse facilities due to the required high total enthalpy. Due to the auto-ignition process, a stable turbulent jet flame is achieved in a supersonic crossflow without any flame holders. The resulting turbulent combustion is very complicated, involving a series of physical and chemical processes. As a result, it is necessary to reveal the combustion characteristics of a gaseous jet in a heated supersonic crossflow. Ben-Yakar et al. (2006) visualized the typical structures in supersonic reactive flows using the schlieren system, including shocks and large-scale vortices. OH radicals can show the reacting front, distributed in the windward shear layer and near-wall region. These experimental results lead to a good description of jet combustion in a heated supersonic crossflow. However, due to the lack of available experimental data, it is difficult to elucidate the inherent dynamics of how jet combustion is stabilized in the high-enthalpy crossflow. Therefore, a wall-modeled large-eddy simulation of the transverse hydrogen jet in a supersonic crossflow was carried out. The experimental results can be used to validate the reliability of numerical data, and the unsteady evolution process of large-scale vortices and turbulent combustion characteristics can be examined.

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow 5.1.1 Flow Conditions Based on schlieren imaging and OH planar laser-induced fluorescence (OH–PLIF), Ben-Yakar et al. (2006) studied the temporal and spatial evolution of a transverse jet in a supersonic flow. In particular, significant differences in the near-flowfield © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_5

173

174

5 Reaction Characteristics of a Gaseous Jet …

characteristics of hydrogen and ethylene jets at similar jet-to-freestream momentum flux ratios were observed. The experimental facility consisted of a flat plate and an injector. A sonic jet of hydrogen, with a static temperature of 246 K and a static pressure of 490 kPa, was vertically injected into a Mach-3.38 gaseous flow through an orifice with a diameter of 2 mm. The temperature and pressure of the freestream gas flow was 1290 K and 32.4 kPa, respectively. The jet exit conditions corresponded to a jet-to-freestream momentum flux ratio of J  1.4. First, flow dynamics in the near field of the transverse jet in the heated supersonic flow are examined. In particular, the emphasis of our present study is on the combustion regime. To illustrate the characteristics of a turbulent jet flame, numerical investigations were carried out based on the experiments of Ben-Yakar et al. (2006). In order to reduce the computation cost, the three-dimensional computational domain was confined to a finite local region near the injection port, which comprised of a solid surface that represented a flat plate with a circular hole as the injection port. Figure 5.1 shows the schematic diagram of the three-dimensional computational domain, which refers to the region from 5D upstream to 10D downstream of the injection port in the streamwise direction. The size of the domain is 10D and 12D in the normal and spanwise directions, respectively. The inflow variables were obtained by a two-dimensional preliminary method using Reynolds-averaged Navier-Stokes (RANS) simulation, the outputs of which at a distance of 20D from the flat plate leading edge were used for the inlet profile parameters of the three-dimensional calculation to decrease computation cost and simulation time. In order to capture the coherent structures and instability of the reacting flowfield, the mesh was mainly

Fig. 5.1 Schematic diagram of computational domains and boundary conditions (Liu et al. 2015)

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow

175

concentrated in the vicinity of the jet exit where coherent structures may initially generate.

5.1.2 Unsteady Characteristics of Large-Scale Vortices Schlieren images provide a visual observation of both the instantaneous and timeaveraged characteristics of the flowfield. Instantaneous images reveal some unsteady properties of vortices and shock structures in non-reacting conditions. Figure 5.2 shows two instantaneous schlieren images of hydrogen and ethylene jets injected into a supersonic crossflow. Freestream fluid flows from left to right and the fuel jets enter from the bottom wall. Large-scale eddies are periodically generated in the early stage of the jet/freestream interaction. Although such eddies exist in both cases, significant differences are observed as they develop in the downstream region. In the hydrogen case, these structures preserve their coherence in the far field while in the ethylene case they disappear beyond ~12 jet diameters downstream. Since this observation is consistent with all the obtained visualizations, it is not a schlieren contrast issue, but is very likely related to the enhanced mixing characteristics of the flowfield. The ethylene structures are larger and penetrate deeper into the crossflow. Besides the bow shock, additional weak shock waves are formed around the ethylene

Fig. 5.2 Examples of hydrogen (a) and ethylene (b) injections into a supersonic crossflow (nitrogen) (Ben-Yakar et al. 2006)

176

5 Reaction Characteristics of a Gaseous Jet …

eddies indicating their subsonic motion relative to the freestream. Comparison also shows that the bow shock is almost merged with the jet, close to the injection location with a very small standoff distance, and curves sharply downstream. Its local shape appears to depend strongly on large-scale shear layer structures, especially close to the jet exit where the freestream behind the steep bow shock is subsonic. As a result, the bow shock demonstrates position fluctuations, which are moderate in the hydrogen case but significant in the ethylene case. The most impressive observations are the coherent structures that are easily identified in instantaneous schlieren images. The large-scale jet shear-layer vortices are important due to their role in near-field mixing. These intermittently form eddies that appear to enlarge and engulf freestream fluid as they travel downstream with the flow. Taking the hydrogen jet combustion as an example, Ben-Yakar et al. (2006) studied the temporal evolution of large eddies utilizing a high-speed camera (Fig. 5.3). Hydrogen large-scale coherent structures survive over a long distance. Coherence of these shear-layer eddies can be seen in Fig. 5.3, constituting consecutive schlieren images from a single experiment. Close to the jet exit, the circumferential rollers rise periodically, creating gaps between the eddies. The evolution of these eddies occurs primarily through engulfment of the crossflow fluid into the jet but also through merging/pairing of smaller eddies in the beginning of the shear layer (eddy number 3 in Fig. 5.3). Beyond 3–4 jet diameters downstream, the separations between eddies become constant and no further merging is visible. The energetic structures elongate in the transverse direction while the crossflow fluid fills the braid regions between eddies. More attention is given to the unsteady characteristics of the reacting flows, especially for the near-field region. Figure 5.4 provides consecutive instantaneous contours of the H2 mass fraction overlapped with pressure isolines and temperature along with the stoichiometric isolines in the middle plane. Large-scale coherent structures are intermittently generated by the interaction of a transverse jet and a supersonic crossflow. It seems that Kelvin–Helmholtz instabilities should be responsible for the generation and evolution of large-scale vortices in the windward shear layer. With the development of vortices, the bow shock ahead of the jet plume fluctuates. During the transport process to the downstream region, the windward shear vortices are distorted and stretched, eventually breaking into smaller scale vortices. Through the bow shock, the compression effect leads to an increase in temperature. Once the local temperature is high enough, the reactant ignites, with the flame being sustained without any flame holders.

5.1.3 Combustion Regime To gain further insight into the coherence and mixing properties of the injection flowfield we examine the ignition characteristics of a hydrogen jet using OH–PLIF. Figure 5.5 shows schlieren and lateral OH–PLIF images superimposed into a single image (Fig. 5.5c). The region containing the OH molecules in the image is clearly

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow

177

Fig. 5.3 Consecutive schlieren images of an under-expanded hydrogen injection into a supersonic crossflow (nitrogen) (Ben-Yakar et al. 2006)

178

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.4 Consecutive instantaneous contours of the H2 mass fraction overlapped with pressure isolines and temperature along with the stoichiometric isolines in the middle plane (t  1 µs) (Ben-Yakar et al. 2006)

shown, indicating the location of the reaction area. The images show an isolated thin flamelet along the outer edge of the plume. The center of the plume itself has no OH signals, indicating poor mixing of the air with the core of the hydrogen jet. The OH radicals are primarily produced in the hot separation region upstream of the jet exit, behind the bow shock, and are convected downstream with the shear-layer vortices. Figure 5.6 shows a set of instantaneous elevation images collected at four different heights above the jet exit (white points in the figure indicate the center of the jet exit). The results show that OH existed around the jet flow, while no OH formed in the center of the plume. The bottom image, at 1 jet diameter above the plate, shows two main features: (1) assuming the flame is near the jet–air interface, the jet propagates very rapidly in the lateral direction (up to 8 jet diameters); and (2) the OH concentration in the downstream jet remained almost unchanged. In contrast, the other three images obtained at 2, 2.5, and 3 jet diameter heights show the same trend in the side view, that is, the OH signal level decreases as the jet moves down.

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow

179

Fig. 5.5 Simultaneous images visualizing hydrogen injection into a supersonic crossflow. a Schlieren image; b OH–PLIF image; c overlaid OH–PLIF and schlieren image (Ben-Yakar and Hanson 1999)

Since the total enthalpy of the freestream in the experiment approaches 4 MJ/kg, namely the total temperature is about 4000 K, auto-ignition of the transverse H2 jet is achieved. The structures of the jet flame are visualized using OH–PLIF, and combustion is maintained without a flame holder. The time-averaged and instantaneous flame structures are discussed basing on numerical results, and then the auto-ignition effect and flame-holding mechanism are further revealed. Figure 5.7 shows the instantaneous OH distribution superimposed with stoichiometric isolines and an OH–PLIF image in the center plane. The corresponding contours of OH in axial and normal planes are shown in Fig. 5.8. The OH radical, as an intermediate production, appears during the reaction of H2 with air and its sharp gradient is generally assumed to correspond to the flame front. In both experimental and numerical results, the OH radical appears in the boundary layer and shear layer on the windward side of the jet plume, but is rarely visualized on the leeward side. The instantaneous contours of the non-dimensionalized heat release rate (HRR) superimposed with stoichiometric isolines are shown in Fig. 5.9. The HRR in a thick layer referring to the upstream boundary layer and windward shear layer facing toward the freestream is clearly higher than in other regions. Heat is intensively released in regions confined within the stoichiometric isolines or along the lower

180

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.6 Instantaneous top-view OH–PLIF images obtained at different heights above the injection plate: a y/d  3; b y/d  2.5; c y/d  2; and d y/d  1 (Ben-Yakar and Hanson 1999)

side of the isolines, especially in the upstream region. It is further believed that heatreleasing chain reactions consuming OH radicals mainly occur in rich-fuel regions below the stoichiometric isolines, whereas the chain-branching reactions creating OH radicals absorb heat. The other significant discovery is that the stoichiometric isolines coincide with the periphery of the jet shear layer. The convection of large-scale vortices facilitates combustion by enhancing the mixing of the fuel and airstream and enlarging the reacting area. Therefore, it is very important to understand the flow mechanism before flame holding is researched. The instantaneous flame structures are shown based on OH and HRR contours, and the results indicate that the flame mode in the upstream boundary and windward shear layer may be different from downstream of the jet plume. In order to reveal the flame-holding mechanism, it is necessary to investigate flame modes. In Figs. 5.10 and 5.11 the mass fraction profiles of H2 , O2 , and OH, in an averaged flowfield at different midline locations corresponding to upstream and downstream of the jet, respectively, are extracted in the normal direction. It is clearly seen from Fig. 5.10 that the reacting region in the upstream boundary and windward shear layer is dominated by a diffusion flame mode, where the OH has peak values, while O2 and H2 approach zero from opposite directions. The leakage of OH distribution at x/D  1, with more than one peak, is induced by an oscillation of the flame’s location,

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow

181

Fig. 5.7 Numerical OH radical distribution superimposed with stoichiometric isolines (top) and experimental OH–PLIF image (bottom) in the center plane (Liu et al. 2015)

due to the instability of the windward shear layer. Profiles of the OH mass fraction in Fig. 5.11, with two distinct peaks, show that two main reacting regions exist downstream of the jet plume. One is on the windward side of the jet plume, where O2 and H2 decrease from opposite directions with a diffusion flame existing upstream of the flowfield. The other is in the near-wall region, where O2 and H2 extensively coexist, indicating that the flame basically occurs in the fuel-rich, partially premixed, mode. In summary, the upstream boundary layer and shear layer are dominated by a diffusion flame and at the near-wall region on the leeward side of the jet plume exists a fuel-rich, partially premixed, flame. The consecutive instantaneous temperature contours in the center plane, overlapped with stoichiometric isolines, are shown in Fig. 5.12, with a time interval of 1 µs. Thick bands with high temperatures exist in the periphery of the boundary layer and windward shear layer above the stoichiometric isolines. The O2 and H2 immediately mix by diffusion in the boundary layer, where the temperature is high and residence time of flow is long. Conditions of auto-ignition are achieved and combustion occurs. However, the combustion process in the shear layer is more complex and is powerfully coupled with turbulence. Two typical packets in the windward

182

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.8 Instantaneous contours of the OH radical overlapped with stoichiometric isolines in axial and normal planes (Liu et al. 2015)

shear layer near the injection were tracked to analyze the auto-ignition process. The air stream behind the bow shock, with a high temperature and pressure, is entrained into the boundary of the jet by large-scale coherent structures and mixes with the fuel rapidly. At the same time, as a high-enthalpy air stream impacts the transverse jet, the static temperature and pressure increases further with ignition delay times. Auto-ignition promptly occurs as soon as stoichiometric conditions are satisfied. Packet 1, having a high temperature, is formed in the shear layer near injection. It reacts with the transformation of jet shear vortices. The stoichiometric conditions are easily achieved and the area of reaction is largely increased with the effect of large-scale vortices, so combustion occurs intensely. Figure 5.13 shows the instantaneous and average contours of the H2 O mass fraction overlapped with stoichiometric isolines in the center and axial planes. H2 O is the final combustion product of H2 and O2 . In both instantaneous and averaged contours, H2 O that is mainly created by a heat-releasing chain reaction in the rich-fuel region distributes within the stoichiometric isolines and in regions covered by stoichiometric isolines. Combustion products then travel to the interior of the jet plume due to the effect of counter-rotating vortices (CRVs).

5.1 Evolution of a Hydrogen Jet in a Supersonic Crossflow

183

Fig. 5.9 Instantaneous HRR contours superimposed with stoichiometric isolines in the center plane (top) and axial and normal planes (bottom) (Liu et al. 2015)

Fig. 5.10 Profiles of mass fractions at x/D  −2 and 1, corresponding to the upstream boundary layer and windward shear layer (Liu et al. 2015)

184

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.11 Profiles of mass fractions at x/D  5 and 8, corresponding to downstream of the jet plume (Liu et al. 2015)

Fig. 5.12 Instantaneous temperature contours in the center plane overlapped with stoichiometric isolines (Liu et al. 2015)

5.2 Flow and Flame Structures in the Reacting Flow Although the unsteady characteristics of the supersonic turbulent jet flame have been revealed in the above studies, the dynamics that drive the reacting flow are still unclear. In this section, flow and combustion characteristics in the far field of a sonic jet, issuing transversely into a high-enthalpy flow, is investigated, corresponding to the experimental work of Gamba and Mungal (2015). Based on experimental

5.2 Flow and Flame Structures in the Reacting Flow

185

Fig. 5.13 Instantaneous and averaged contours of H2 O overlapped with stoichiometric isolines in the center and axial planes (Liu et al. 2015)

conditions, the stagnation temperature of heated Mach-2.4 air approaches 3000 K. In this test case, its static temperature and pressure are 40 kPa and 1400 K, respectively. A hydrogen jet, whose static temperature and pressure are 250 K and 1074.5 kPa, respectively, is injected into a crossflow through a 2-mm injector.

5.2.1 Experimental Observation The structure of the shock system for our current experiment is shown in Fig. 5.14, where an instantaneous schlieren image is presented. It shows an instantaneous snapshot for the J  5 case. The image captures the incoming flow as it develops over the flat plate (indicated by the black outline) and the near field of the transverse jet. Well upstream of the injector, two shock wave systems are observed. First, a lip shock is formed by the interaction of the incoming supersonic flow with the boundary layer developing on the flat plate; second, a set of oblique waves is observed to originate from the flat plate itself. The second shock wave system does not result from injection, nor does it originate by the mating of the sharp leading edge and the plate. Its

186

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.14 Schlieren image of the flow and shock structure around transverse under-expanded jets in a supersonic crossflow for J  5 (Gamba and Mungal 2015)

Fig. 5.15 Side-view OH–PLIF images on the center plane for a J  0.3, b J  2.7, and c J  5.0 (Gamba and Mungal 2015)

nature and origin could not be identified. However, it does not appear to interact with the transverse jet; thus, it is not expected to affect the jet. OH–PLIF imaging is used to investigate the instantaneous structure of the reaction regions on several side- and plan-view planes. Figure 5.15 shows an example of the instantaneous reaction zone marked by OH–PLIF for the symmetrical center plane of the transverse jet for three cases of J. Under current flow conditions and for sufficiently high values of J, three major flow features are observed: (1) an intermittently reacting recirculation region upstream of the jet, (2) a reactive shear layer on the windward side of the transverse jet, and (3) a highly reactive boundary layer that extends laterally for several diameters off the center plane of the jet.

5.2 Flow and Flame Structures in the Reacting Flow

187

Fig. 5.16 Averaged side-view OH–PLIF image on the center plane for a J  0.3 and b J  5.0 (Gamba and Mungal 2015)

Figure 5.16 shows time-averaged OH–PLIF images for J  0.3 and J  5.0, computed over a small number of repetitions. In spite of the fact that the average is clearly not statistically converged, it demonstrates the repeatability of the flow features shown by the set of instantaneous images provided previously. Furthermore, it demonstrates the intermittent nature and corrugation of the reacting shear layer that results in a relatively broad region where the reacting shear layer typically exists. For the case of a low J value, what we refer to here as the reacting shear layer can exist anywhere between the wall and 3D from the wall. For cases of high J values, the reacting shear layer exists in a well-defined band. The thickness of this band is seen to grow from the injection point in the region of maximum deflection, and then remain nearly unchanged as the jet develops in the wake—where this band is approximately 3D wide. To better quantify the properties of the reacting shear layer, some of its morphological properties have been investigated statistically. This was done by extracting OH layer thickness, location, and orientation from the set of instantaneous images used to construct the average field shown in Fig. 5.16. In particular, the OH layers in the shear layer were extracted and are shown in Fig. 5.17, where in the inset an OH— PLIF image is shown along with the positions of some profiles (labelled A–J), while the main figure shows the (normalized) layer profiles (A–J) plotted as a function of s/λ, where s is the coordinate across the profile and λ is the width of the profile, defined as the full-width at half-maximum of a Gaussian curve fitted to the profile. The solid line shown in Fig. 5.17 indicates a normalized Gaussian, demonstrating the similarity between the central portion of the profiles and a Gaussian curve. It should be noted, however, that the resemblance of the profiles to a Gaussian is primarily attributed to spatial resolution limitations (blurring) introduced by the imaging system used. To approximately account for some of these limitations, image-blurring estimation was used to correct the measured layer thickness.

188

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.17 Normalized profiles of selected OH layers on the center plane of J  5.0 (Gamba and Mungal 2015)

To investigate this flow feature in more detail, plan-view OH–PLIF imaging at different wall-normal locations was carried out. Examples at selected wall-normal locations (y/d  0.25, 0.5, 1, and 3) are shown in Fig. 5.18 for the J  5.0 case. Based on the estimate of the boundary layer thickness at the injection point given previously, and from side-view OH images, the measurement plane at y/d  0.25 is well within the (undisturbed) boundary layer, while the plane at y/d  0.5 is approximately at the edge of it. Thus, for each figure, panels (a–c) identify the structure of the OH regions within the boundary layer, while panel (d) identifies the structure somewhere in the shear layer. It should be noted that the flow features in these figures have been labelled with an alphabetical letter followed by a number (e.g., C1), such that the letter indicates the corresponding flow features across different J cases while the number identifies the three cases reported in the set of figures. The morphological characteristics of the near-wall OH layer begin to significantly change for the J  5.0 case where some drastic modification takes place. The main difference in the J  5.0 case is that stabilization appears upstream of the injection point (see C3 in Fig. 5.18a), along with the existence of a significantly large OH footprint near the wall. The set of plan-view OH images indicates that ignition in the recirculation region in front of the transverse jet occurs only at large J (in this case for J  5.0) with a characteristic bow shape, while broadly distributed OH is visible downstream of the injector for most of the J cases. As shown in Fig. 5.18a, b, this reacting recirculation region wraps around the base of the jet with a characteristic “hole” just in front of the injector. Very close to the wall (y/d  0.25, see Fig. 5.18a) the OH is distributed uniformly and no structure is observed; on the contrary, at higher locations (y/d  0.5, see Fig. 5.18b) the OH is organized in stretched loops (see B3) connected to the OH layers of the windward shear layer (F3), mixed with regions characterized by OH streaks originating at a virtual point (e.g., C3 in Fig. 5.18b). Analysis of the ignition

5.2 Flow and Flame Structures in the Reacting Flow

189

Fig. 5.18 OH–PLIF imaging on selected plan-view planes for J  5.0: a y/d  0.25, b y/d  0.5, c y/d  1.0, and d y/d  3.0 (Gamba and Mungal 2015)

and flame-holding capabilities of different wall–injection configurations identified the upstream recirculation as a suitable location for flame stabilization due to the long residence time and near-stagnation conditions. Therefore, the length of the upstream recirculation region (which increases with J), the low velocities associated with it, and the blockage effect of the jet (which increases with J) would play an important role in this process. Another mechanism supporting ignition in the separation region is the favorable mixing that occurs as a result of the interaction of the issuing jet and the system of recirculation regions, along with the shorter ignition times expected at lean conditions in a supersonic non-premixed combustion of a cold fuel mixed with a hot oxidizer, that could be observed in the upstream recirculation region.

5.2.2 Flow Structures in the Reacting Flow To further describe the flow characteristics of a jet injected into a heated crossflow, we conducted a wall-modeled large-eddy simulation. The computational domain in this simulation is confined to a local region, a schematic of which is provided in Fig. 5.19. It is divided into two parts, namely the jet region and buffer zone. Fine mesh

190

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.19 Side-view of the experimental configuration and corresponding boundary conditions (Liu et al. 2018)

is distributed in the jet zone, so that large-scale vortices and unsteady process can be captured. A buffer zone is set in the downstream region, to prevent the numerical disturbance from propagating upstream along the supersonic boundary layer. As an important intermediate product in air–hydrogen reactions, the hydroxyl (OH) radical is often used as a marker of the reacting front. Figure 5.20 compares the instantaneous OH distribution from PLIF and a large-eddy simulation (LES) in the middle plane. Except for the near-wall region, the OH radicals mainly concentrate in a thin windward shear-layer, which is corrugated by the turbulence. A large number of OH radicals are observed at the leeward side of the jet in the LES result, which is the most significant difference noticed when making a comparison with the PLIF image. To further illustrate this conclusion, a comparison is made of the instantaneous OH radials at the y/d  0.25 plane between LES and PLIF (Fig. 5.21). It seems that the fuel entrained into the recirculation zone is so deficient that the chemical reactions are very weak in the experiment. Therefore, almost no OH signal appears in this region. According to the above analysis, the auto-ignition process should be responsible for the chemical reactions, stabilizing the non-premixed combustion in the windward shear layer and near-wall region. Figure 5.22 displays the instantaneous contours of the OH radicals in different streamwise planes overlapped with stoichiometric isolines. The OH radicals are mainly distributed at the fuel-lean side of stoichiometric isolines. As illustrated in Fig. 5.19, the under-expanded jet expands at the lip of the orifice before it is compressed by the barrel shock and the Mach disk. Due to the blocking effect of the transverse jet, a bow shock is formed in front of the jet plume. Thus, the resultant flowfield involves complex three-dimensional unsteady shocks, turbulence, and subsequent interactions. Figure 5.23 presents the iso-surface of the second invariant of the velocity-gradient tensor, colored by instantaneous temperature. Large-scale vortices are generated at the windward side of jet wakes, giving rise to a rapid mixing between the jet and supersonic crossflow. Subjected to the adverse pressure gradient induced by the bow shock, the upstream supersonic boundary layer is separated. As a result, horseshoe vortices are formed within the separated region, where a small amount of the fuel is entrained. It is known that high temperatures and low-flow residence times prevail, providing a perfect reacting environment. However,

5.2 Flow and Flame Structures in the Reacting Flow

191

Fig. 5.20 Comparison of the instantaneous OH distribution using PLIF (top) and LES (bottom) in the middle plane (Liu et al. 2018)

Fig. 5.21 Comparison of the instantaneous OH distribution using PLIF (right) and LES (left) at the y/d  0.25 plane (Liu et al. 2018)

the local temperature in the jet wake is lower than in the near-wall region, leading to a longer delay time for the ignition of the reactant. Figure 5.24 shows time-averaged Mach number distributions with streamlines at the midline, wall-normal, and cross-view planes (CVP). The Mach number contours clearly show shock structures, such as the front bow shock, barrel shock, Mach disk, reflected shock from the triple line (the intersection of the barrel shock and Mach disk), and separation shock at the front of the bow shock. The upstream separation shock is not as strong as the other shocks, the contours slightly change from dark gray to light gray (from preshock to postshock) and interact with the bow shock. The streamlines show that most of the jet fluid passes through the barrel shock and Mach disk and then turns downstream. Upstream of the jet, a recirculation region is observed. The recirculation forms a horseshoe separation vortex ahead of the foot of the bow shock. The horseshoe vortex curves sideways from the midline plane. The top view shows that the streamlines diverge laterally after the crossflow deflects through

192

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.22 Instantaneous contours of OH radicals in different streamwise planes overlapped with stoichiometric isolines (x/d  − 2.5, 11.5, 25.5, and 39.5) (Liu et al. 2018)

Fig. 5.23 Iso-surface of the second invariant of the velocity-gradient tensor (Q  0.1) colored by instantaneous temperature (Liu et al. 2018)

5.2 Flow and Flame Structures in the Reacting Flow

193

Fig. 5.24 Time-averaged flow structures and streamlines: a contour of the Mach number in the middle plane overlapped with isolines of u  0; b contour of the Mach number in the normal plane (y/d  1) overlapped with isolines of u  0 (Liu et al. 2018)

the bow shock and then converge downstream due to the low pressure downstream of the jet. Two pairs of counter-rotating vortices are clearly visible downstream of the jet injection in the cross view of the time-averaged flowfield; one pair are the counter-rotating jet vortices, the other are the boundary-layer separation vortices along the symmetric plane, induced by the suction of the counter-rotating jet vortices. Figure 5.25 shows the contours of the local Mach number with in-plane streamlines and injectant mass fraction in cross sections downstream of the jet plume. At x/d  18, the major and surface trailing counter-rotating vortex pairs (CVP) near the wall appear due to the interaction of the crossflow with the jet fluid. The major CVP grows in size further downstream. The surface trailing CVP is located in the nearwall region and disappears as the jet plume is lifted up downstream at x/d  45. In the far field, the injectant mass fraction has a high value both in the plume and the near-wall region. The time-averaged contours of the injectant mass fraction on wall-parallel planes y/d  0.1, 1.0, and 2.0 are shown in Fig. 5.26. A herringbone separation zone is formed in the jet leeward region. A large zone of low injectant mass fraction is distributed in the recirculation zone, leading to a rapid mass exchange with the outer flow. It is clear that at y/d  0.1, there is little injectant entrained into the jet leeward nearwall separation. In the recovery zone, the injectant mass fraction increases obviously compared with that in the separation zone. At y/d  1.0, the rich injectant zone is near to the jet orifice and the injectant is diluted in the far field. Along the center line, downstream of the jet, there exists a low injectant mass fraction zone, which corresponds to the gap between the major CVP branches. The injectant mass fraction in the jet lee is significantly smaller, and the main pockets of the jet fluid. In Fig. 5.27, iso-surfaces of the time-averaged streamwise velocity u/u∞  0, colored by local wall-normal distance overlapped with three-dimensional streamlines, are shown to demonstrate the three-dimensional near-wall separation configuration.

194

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.25 Time-averaged contours of the injectant mass fraction and Mach number on cross sections of x/d  18 and 45. a x/d  18, Mach number overlapped with streamlines; b x/d  45, Mach number overlapped with streamlines; c x/d  18, injectant mass fraction with sonic lines; and d x/d  45, injectant mass fraction with sonic lines (Liu et al. 2018)

There are two recirculation zones existing in the flowfield. One is ahead of the injector, induced by the separation of the boundary layer. The other is located at the leeward side of the jet plume, which is λ shaped. The reflux can prolong the residence time of the fluid and plays an important role in flame stabilization. Further downstream of the jet leeward separation bubble exists two separate separation wings. The peak of the oblique wing corresponds to the edge of the separation zone. A detailed formation mechanism for the separation bubble can be found in our work above, which identified detailed structures in the separation interaction with the collision shock and the induced surface trailing CVPs in the jet wake. After the bow shock, the supersonic crossflow deflects upward accompanied by a decrease in Mach number. Notably, the velocity gradually increases owing to a favorable pressure gradient. The jet fluid expands and is compressed by the barrel shock and Mach disk. Then, the fuel rapidly mixes with the crossflow, assisted by turbulent fluctuations and counter-rotating vortices (Sun and Hu 2018).

5.2 Flow and Flame Structures in the Reacting Flow

195

Fig. 5.26 Comparison of the time-averaged injectant mass fraction contours on the y/d  0.1, 1.0, and 2.0 planes for J  5 (Liu et al. 2018)

5.2.3 Analysis of Streamlines Transport of the injectants toward the wall in the jet near field and far field is further analyzed in this section using three-dimensional streamlines to demonstrate the injectant transport process and the resulting distribution in the near-wall zone. Figure 5.28 presents a front view of the flowfield containing the streamlines originating from the inside of the jet orifice and those aiming toward the near-wall position at the exit. The streamlines are marked with spheres colored by local time-averaged injectant mass fraction. In Fig. 5.28, the streamlines exhibit significant curvatures, which indicate the effects of the major CVP. Contours of the time-averaged injectant mass fraction on the x/d  40.0 slice give an indication of the extent of the injectant transport. Figure 5.28 shows streamlines originating from the jet orifice and indicates the motion and mixing of the jet fluid. The streamlines originating in the jet orifice follow the major CVP into the core of the jet plume. Close observation indicates

196

5 Reaction Characteristics of a Gaseous Jet …

Fig. 5.27 Iso-surface of time-averaged streamwise velocity (u/u∞  0) colored by normal distance to the bottom wall overlapped with three-dimensional streamlines (Liu et al. 2018)

Fig. 5.28 Iso-surface front view of the H2 mass fraction Y i  0.8 and three-dimensional streamlines originating from a jet orifice (Liu et al. 2018)

5.2 Flow and Flame Structures in the Reacting Flow

197

that the major CVP generally pumps the injectant away from the wall. In the far field, streamlines follow the lifting and rotating major CVP and bring the injectants to a higher position away from the near-wall region. Taking an overall view, the streamlines inherit the rotating movement from the near field while they have little deformation or twist and go directly downstream in the far field. To more clearly illustrate injectant entrainment to the near-wall region, Fig. 5.29 shows streamlines passing the lines at x/d  0.6 and 30 on the y/d  0.5 plane. It shows that streamlines originating from the lateral side of the crossflow run around the jet body and approach the near-wall region in the near field and far field. These

Fig. 5.29 Oblique view of three-dimensional streamlines passing through lines on the wall-parallel plane y/d  0.5 at different streamwise locations, colored by injectant mass fraction. a Streamlines passing the wall-parallel plane y/d  0.5 at x/d  0.6; and b streamlines passing the wall-parallel plane y/d  0.5 at x/d  30 (Liu et al. 2018)

198

5 Reaction Characteristics of a Gaseous Jet …

streamlines, colored by injectant mass fraction, show that part of the jet fluid, especially the injectant adjacent to the jet orifice, is entrained and mixed with the air stream. Figure 5.29a shows the representative streamlines which originate from the crossflow at y/d  0.5 and x/d  0.6, upstream of the jet. It is seen that some of the injectants are “trapped” in the downside region of the jet owing to the major CVP and the lower fluid velocity in the center of the jet lee. Streamlines in the jet lateral are found to entrain the injectant to the near-wall region in the far field. In Fig. 5.29b, streamlines in the far field demonstrate that the injectants in the near-wall region come from the entrainment of the lateral crossflow. As a summary of the above analysis, the streamlines originating from the jet orifice follow the major CVP and penetrate into the core of the jet plume. The jet fluid transported to the near-wall region downstream of the jet originates primarily from the crossflow in the upstream lateral of the jet, which runs around the jet orifice and gains a downwash velocity to approach the wall in the far field.

Fig. 5.30 Comparison of injectant mass fraction and time-averaged streamwise velocity at x/d  13, 24, and 46 (from left to right) (Liu et al. 2018)

5.2 Flow and Flame Structures in the Reacting Flow

199

Profiles of the time-averaged injectant mass fraction are shown in Fig. 5.30a–c at different streamwise locations. In the jet near field, the injectant mass fraction is small, which strongly suggests that in the jet near field more injectant is entrained into the near-wall zone. On the contrary, in the jet far field, also shown for x/d  24 and 46, the injectant mass fraction is always larger. The local injectant lean in the incoming flow leads to downstream extinction due to the fuel-lean status. Timeaveraged streamwise velocity distributions are shown in Fig. 5.30d–f. It can be seen that at the outlet of x/d  46, the profiles of the streamwise velocity remain similar, with a small deformation due to a stronger interaction. Based on Fig. 5.30, the current simulation reveals that a local injectant-rich region appears in the near-wall far field, which essentially results in flame extinction in the far field.

5.3 Conclusions Numerical and experimental investigations are performed to reveal the mixing process and flame-holding mechanisms for an unsteady reacting flowfield with a transverse jet into a high-enthalpy crossflow. A diffusion flame with auto-ignition dominates the reacting region of the boundary layer and windward shear layer where a supersonic crossflow stagnates. It then propagates downstream of the jet plume, holding the flame stable within the total flowfield. Additionally, it is found that J indirectly controls many of the combustion processes. At low values of J, the flame is lifted and stabilizes in the wake close to the wall, possibly by auto-ignition after some partial premixing occurs; most of the heat release occurs at the wall in regions where OH occurs over broad regions. At high values of J, the flame is anchored at the upstream recirculation region and remains attached to the wall within the boundary layer where OH remains distributed over broad regions; a strong reacting shear layer exists where the flame is organized in thin layers. Stabilization occurs in the upstream recirculation region that forms as a consequence of the strong interaction between the bow shock, the jet, and the boundary layer. In general, this interaction controls jet penetration, dominating the fluid dynamic processes and thus stabilization.

References Ben-Yakar A, Mungal MG, Hanson RK (2006) Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Phys Fluids 18(2):026101–026116. https://doi.org/10.1063/1.2139684 Ben-Yakar A, Hanson R (1999) Hypervelocity combustion studies using simultaneous OH-PLIF and schlieren imaging in an expansion tube. In: 35th joint propulsion conference and exhibit, p 2453 Gamba M, Mungal MG (2015) Ignition, flame structure and near-wall burning in transverse hydrogen jets in supersonic crossflow. J Fluid Mech 780:226–273. https://doi.org/10.1017/jfm.2015. 454

200

5 Reaction Characteristics of a Gaseous Jet …

Liu C, Wang Z, Wang H, Sun M (2015) Numerical investigation on mixing and combustion of transverse hydrogen jet in a high-enthalpy supersonic crossflow. Acta Astronaut 116:93–105. https://doi.org/10.1016/j.actaastro.2015.06.023 Liu C, Yu J, Wang Z, Sun M, Wang H, Grosshans H (2018) Characteristics of hydrogen jet combustion in a high-enthalpy supersonic crossflow. Phys Fluids (submitted) Sun M-b, Hu Z-w (2018) Mixing in nearwall regions downstream of a sonic jet in a supersonic crossflow at Mach 2.7. Phys Fluids 30:1–18

Chapter 6

Primary Breakup of Liquid Jet in Supersonic Crossflow

In a liquid hydrocarbon-fuelled supersonic combustion ramjet (scramjet) engine, the atomization and mixing of liquid fuel in the supersonic airflow determine the engine ignition reliability and combustion performance. Since the gas/liquid fuel is typically injected from the wall in the scramjet combustor, gas/liquid jets in supersonic crossflow have been widely studied. As the liquid jet atomization process of transverse liquid jet in supersonic crossflow is very complicated, the physical mechanism and determining factors have not been well understood, requiring further research on this subject. A large number of experimental studies have been carried out on the liquid jet atomization in supersonic airflows. Since the liquid jet disintegrates very quickly into droplets of several microns, it requires very high temporal and spatial resolution for experimental techniques to well capture the breakup process. The primary breakup structures of liquid jet in supersonic crossflow captured by conventional experimental technique (Ghenai et al. 2009) can be blurred due to the long exposure time when the liquid interface can move several or tens of pixels. Pulsed laser background imaging (PLBI) method is developed to obtain high temporal resolution of liquid jet disintegration process in supersonic crossflow. In this method, the nanosecond pulsed light is used to clearly capture the instantaneous atomization structures in supersonic crossflows, eliminating the blurring effects due to the long exposure time. The mist resulting from the atomization blocks the liquid core, making it difficult to investigate the breakup mechanism of the liquid jet core using optical instruments. The numerical methods for two-phase flows have made significant progress and contributed a lot to the understanding of the atomization mechanism (Gorokhovski and Hermann 2008; Desjardins et al. 2013). In order to accurately predict the atomization process, the interface deformation and breakup must be properly resolved. Three popular interface tracking methods including volume of fluid (VOF) method, level set (LS) method, coupled level set, and volume of fluid (CLSVOF) method are examined in this chapter. Two approaches can be used to simulate the liquid jet atomization in supersonic gas flow: one is to treat both the liquid and gas as compressible fluid (Chang et al. 2013); the other is to treat the liquid as incompressible fluid and the gas as compressible © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_6

201

202

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

fluid (Caiden et al. 2001). Since the sound speed is much higher in the liquid phase than in the gas phase, the time step must be very small to satisfy the CFL condition when the liquid is treated as compressible. Therefore, the second approach is applied here to achieve high calculation efficiency. In this chapter, the experimental setup of PLBI method is first described, which is followed by experimental analysis of liquid jet primary breakup in supersonic crossflow. Then, the interface tracking methods and two-phase LES formulation for simulation of atomization in compressible gas flows are described. In the end, liquid jet primary atomization in supersonic air crossflow is simulated, and the physical mechanism of atomization mechanism is analyzed.

6.1 Experimental Setup of Pulsed Laser Background Imaging The experimental facility for the study of liquid jet primary breakup in supersonic crossflow using PLBI method is shown in Fig. 6.1. Experiments are run in a twodimensional supersonic wind tunnel with a designed Mach number of 2.1 as shown in Fig. 6.1a. High-pressure air enters into test section after accelerated through 2D Laval nozzle. The test section is a rectangular channel with a length of 200 mm and a cross-sectional dimension of 60 mm × 40 mm. Windows are opened on each four walls. The bottom one is installed with the test plate while others inlaid with K9 optical glasses for observation and imaging. The nozzle is vertically mounted on the test plate to provide a low turbulent liquid jet. The nozzle diameter is 0.5 mm. The Mach number (Ma), stagnation temperature (T 0 ) and stagnation pressure (P0 ) of supersonic flow in the test section are 2.1, 300 K and 891 kPa, respectively. The layout of optical system used in PLBI method is illustrated in Fig. 6.1b. A highenergy pulse laser beam is first transformed to light sheet through convex lens and cylindrical lens, and then, the produced light sheet is injected into the scattering system with sol medium. The incident laser sheet will get scattered and refracted before going out through the transparent glass window in the system, under the influence of Tyndall effect caused by the sol medium. The scattering causes the disordering of laser propagation direction and phase, making the planar light source uniform and eliminating the interference effect of laser. Then, a uniform light source is formed and can be set as the uniform background. The dimension of generated planar light source is 250 × 100 mm, with a pulse width of 7 ns. CCD camera is located directly in front of planar light source. The dimension of CCD pixel space is 4000 × 2672 pixel, and the gray scale is 256. CCD camera coordinates with Sigma105 prime lens to carry out imaging of physical space of 96 mm × 64 mm at the shooting distance of object distance of 100 mm. The depth of field of imaging system is 30 mm, which can fully cover the spray area. The movement distance of the object within pulsed illumination time of 7 ns is no more than 0.3 pixel, which can ensure the instantaneous image of liquid jet primary breakup morphology without smearing.

6.1 Experimental Setup of Pulsed Laser Background Imaging

(a) Schematic of the supersonic wind tunnel

(b) Scheme of Pulsed Laser Background Imaging Method Fig. 6.1 Schematic of experimental facility

203

204

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Figure 6.2a shows the raw image of background with supersonic crossflow but without liquid jet. Figure 6.2b–d shows the gray histogram of successive five rows as shown by the yellow line in Fig. 6.2a, successive five columns as shown by the blue line in Fig. 6.2a and the whole background image, respectively. It is shown that the gray distribution of background image obtained by this method generally complies with the normal distribution, with consistent characteristics in the whole field. This confirms that the scattering system can eliminate the interference effect of laser and produce a uniform planar light source for observation of the instantaneous atomization structures.

400

300

Meanrow=176.8

200

Varrow=65.1

100

(a) Raw Image of Background 0 100

50

0

150

200

250

(b) Histogram of Row x 10

4

200 8 150

100

Meancol=177.3

6

Mean=176.4

Varcol=66.3

4

Var=67.3

50

2 0

0 0

50

100

150

200

(c) Histogram of Column

250

0

50

100

150

200

(d) Histogram of Background

Fig. 6.2 Gray scale distribution characteristics of background image

250

6.2 Experimental Analysis of Liquid Jet in Supersonic Crossflow

205

6.2 Experimental Analysis of Liquid Jet in Supersonic Crossflow Instantaneous primary breakup morphology of liquid jet in supersonic crossflow captured by PLBI technique is presented in Fig. 6.3. As the liquid jet exits the nozzle, the liquid column undergoes deformation under the actions of strong aerodynamic forces from the supersonic gas crossflow. The liquid ligaments are observed to strip off the liquid column as shown in Fig. 6.3a, which is surface breakup. Surface waves develop on the upwind side of the liquid column. As the magnitude of the surface waves increases, the liquid column disintegrates into large liquid blocks as shown in Fig. 6.3b, which is column breakup. Figure 6.4a, b shows the instantaneous liquid jet primary breakup morphology in supersonic crossflow at t and t + 2 μs, observing corresponding atomization structures at the two time instant. The liquid structures A, B, C, D at t as shown in Fig. 6.4a evolve into liquid structures A , B , C , D at t + 2 μs as shown in Fig. 6.4b. Figure 6.4a shows the evolution of interface topology in a time period of 2 μs. By measuring the movement of these interface topology, the velocity of these protruding liquid structures on the upwind side of the liquid jet can be measured and is shown in Fig. 6.5. The velocity component U in the crossflow direction is observed to increase along the liquid jet due to the acceleration of the liquid structures under the strong drag forces from the supersonic crossflow. The velocity component V in the vertical direction (perpendicular to the wall) increases first and then decreases along the liquid jet. After the liquid jet exits the nozzle, the protruding liquid structures on the upwind side of the liquid jet is accelerated by the supersonic crossflow, and thus, the velocity magnitude of these liquid structures always increases along the jet as shown by velocity vectors in Fig. 6.5. As the liquid jet bends more, the vertical component of the velocity decreases. The highest vertical velocity of these protruding liquid structures on the liquid jet upwind side is observed when the liquid jet bends by 45°.

6.3 Interface Tracking Methods 6.3.1 Volume of Fluid Method The VOF function F is defined as the volume fraction occupied by the liquid. The fully (temporally and spatially) resolved evolution of F is governed by ∂F ∂F + Ui  0. ∂t ∂ xi

(6.1)

Due to the discontinuity of the volume fraction function across the interface, excessive numerical diffusion can be introduced when applying conventional numerical schemes to solution of the VOF advection equation. A special procedure is required

206

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.3 Instantaneous primary breakup morphology captured by PLBI technique

(a) surface breakup

(b) column breakup

to minimize this error. In this procedure, the interface is first reconstructed in each cell where 0 < F < 1, and then, the fluid flux through each face of these cells is calculated geometrically using knowledge of the interface location to provide sharp interface advection. The method is described below in 2D for convenience; it is easily generalized to 3D.

6.3 Interface Tracking Methods

207

Fig. 6.4 a Atomization structures at t, b atomization structures at t + 2 μs, and c evolution of interface topology in 2 μs

Fig. 6.5 Velocities of liquid structures on the upwind side of the liquid jet: a velocity component U in the crossflow direction, and b velocity component V in the vertical direction

208

6.3.1.1

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Interface Reconstruction

In the present VOF method, the interface is reconstructed using a piecewise linear interface calculation (PLIC) method (Rider and Kothe 1998). In 2D, for example, the linear interface within a cell has the standard representation: n x x + n y y  α,

(6.2)

where (n x , n y ) are components of the unit normal vector (pointing from liquid into gas), and α is the shortest distance from the chosen coordinate origin (cell corner) to the linear interface (Fig. 6.6). Interface reconstruction therefore has two steps: (1) calculation of the unit normal vector and (2) calculation of the shortest distance α. The interface normal vector is calculated using a central difference method (Pilliod and Puckett 2004) in the present implementation of pure VOF method. Although more advanced approaches have been proposed (Pilliod and Puckett 2004; Scardovelli and Zaleski 2003; Aulisa et al. 2007), they are not implemented here due to their relatively high complexity and computational cost (especially in 3D). Given the unit normal vector, the position of the linear interface (α) is constrained by the volume fraction. When nx > 0 and ny > 0 as in Fig. 6.6, the compatibility relation between F and φ may be written as F x y 

      α − n y y 2 α2 α − n x x 2 −H (α − n y y) 1 − H (α − n x x) , 2n x n y α α

(6.3)

Fig. 6.6 Typical shape of liquid region when the interface truncates a single cell

6.3 Interface Tracking Methods

209

where x and y are cell dimensions, and H(x) is the Heaviside function [a 3D version of this equation is given in Gueyffier et al. (1999)]. Calculating the volume fraction F from α and a specified normal vector is explicit and straightforward. However, this is not the case when solving the inverse problem of determining the α which corresponds to specified values of the volume fraction (from the current VOF solution) and normal vector. Newton’s method converges fast, but it is complex to calculate the derivatives of the function. The bisection method is simple, but it converges slowly. Brent’s algorithm is an appropriate method to find a bracketed root of a general one-dimensional function, when the function’s derivative is difficult to compute. Brent’s method combines the sureness of bisection with the speed of a higher order method (Press 2000) and thus has been used here. In practice, there are many cases when nx < 0 or ny < 0. This can be tackled by a coordinate transform, with the linear fit changed to n ∗x x ∗ + n ∗y y ∗  α ∗ n ∗x  |n x |, n ∗y  |n y |, α ∗  α − min(0, n x x) − min(0, n y y) x ∗  sgn(n x )x − min{0, sgn(n x )x} y ∗  sgn(n y )y − min{0, sgn(n y )y}.

(6.4)

Given the volume fraction F, n*x and n*y , α ∗ is calculated from Eq. (6.3) and α is then obtained via: α  α ∗ + min(0, n x x) + min(0, n y y).

6.3.1.2

VOF Advection Scheme

Advection schemes for solution of volume tracking equations can be classified into two categories: operator split and unsplit methods. Relative to an unsplit advection algorithm, the operator split method is easier to implement and is well documented, and has thus been implemented in the current code. The philosophy of the operator split method is to advect the volume fraction F sequentially in each coordinate direction. Following the operator split method from Sussman and Puckett (2000), the 2D VOF advection equation is discretized as follows:   Fi,n j + (t/x) G i− 1 , j − G i+ 1 , j 2 2   F˜ i, j  1 − (t/x) u i+ 1 , j − u i− 1 , j 2 2     t ˜ t ˜ ˜ ˜ + v , (6.5) Fi,n+1  F + − G − v G F 1 1 1 1 i, j i, j j i, j− i, j+ i, j+ i, j− y y 2 2 2 2 where G i+ 1 , j  u i+ 1 , j Fi+ 1 , j and G˜ i, j+ 1  vi, j+ 1 F˜ i, j+ 1 denote liquid volume 2

2

2

2

fluxes through the right and top faces of cell (i, j).

2

2

210

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.7 Liquid volume flux calculation through right face (ui+1/2,j > 0, udt  ui+1/2,j t)

In order to advect the volume fraction for cell (i, j) in the x-direction, the liquid volume fluxes through right and left faces Gi+1/2,j and Gi−1/2,j must be calculated. As shown in Fig. 6.7, the shaded region is the liquid volume (defined as V i+1/2,j ) that crosses the right face during time t. The linear interface in the cell (with point A as the coordinate origin) is represented by Eq. (6.2), with α being the shortest distance from point A to the interface. By a shift of coordinate system, we obtain the shortest distance from point B to the interface as α B  α − n x (x − udt). Given α B , nx and ny , the volume of the shaded region is calculated as V

1 i+ 2 , j



      α B − n y y 2 αB 2 α B − n x udt 2 −H (α B − n y y) 1 − H (α B − n x udt) . 2n x n y αB αB

(6.6) Then, the liquid volume flux through the right face Gi+1/2,j becomes G i+ 1 , j  2

Vi+ 1 , j 2

t y

.

(6.7)

The liquid volume flux through all faces is calculated in a similar manner. The ˜ the interface is then volume fraction is first advected in the x-direction, obtaining F; reconstructed based on F˜ and then advected in the y-direction, obtaining the final volume fraction F n+1 . By alternating the starting sweep direction at each time step, the above operator split algorithm can achieve second-order accuracy. For the 3D version of the method see Sussman and Puckett (2000).

6.3 Interface Tracking Methods

211

6.3.2 Level Set Method The LS function φ is interpreted as the signed distance from the interface satisfying ∇φ  1. The interface is defined by φ  0, with φ > 0 representing liquid and φ < 0 representing air. φ is evolved (in a fully resolved sense) by solving the simple advection equation: ∂φ ∂φ + Ui  0. ∂t ∂ xi

(6.8)

To maintain the signed distance property, the reinitialization equation is also solved    ∂ϕ ∂ϕ ∂ϕ  S(ϕ0 ) 1 − ∂τ ∂ xk ∂ xk

S(ϕ0 ) 

ϕ0 ϕ0 2 +d 2

,

(6.9)

where ϕ0  ϕ(xi , τ  0)  φ(xi , t),d  max(x, y, z), τ represents pseudotime, and S(ϕ0 ) is a modified sign function. After solving this equation to steady state in the interface vicinity, φ is replaced by ϕ.

6.3.2.1

LS Advection Scheme

Since the fluid is incompressible, the LS equation can also be written as ∂ φ ∂ (Ui φ) +  0. ∂t ∂ xi

(6.10)

For temporal discretization of this equation, the second-order Adams–Bashforth scheme has been implemented in pure LS method:     3 ∂ u n φn ∂ vn φn 1 ∂ u n−1 φ n−1 ∂ v n−1 φ n−1 φ n+1 − φ n − + + + . (6.11) t 2 ∂x ∂y 2 ∂x ∂y By applying a finite volume approach, the following formulation is obtained:  n  n n n 3t G i+1/2, j − G i−1/2, j G i, j+1/2 − G i, j−1/2 n φi,n+1 +  φ − j i, j 2 x y  n−1  n−1 n−1 n−1 t G i+1/2, j − G i−1/2, j G i, j+1/2 − G i, j−1/2 + + , 2 x y

(6.12)

where G i+1/2, j  u i+1/2, j φi+1/2, j is the flux of φ through the (i, j) cell right face. Since φ is unknown at cell faces, φi+1/2, j needs to be approximated. Due to the hyperbolic characteristic of the LS equation, an upwind-biased scheme is an appro-

212

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

priate choice. Schemes of different orders have been implemented and compared in the present study; these are briefly outlined here: First-order The first-order approximation to φi+1/2, j is the simple upwind scheme:

φi+1/2, j 

φi, j if u i+1/2, j ≥ 0 φi+1, j if u i+1/2, j < 0.

(6.13)

Second-order A suitable second-order method is the total variation diminishing (TVD) scheme (Versteeg and Malalasekera 2007): ⎧   ⎨ φi, j + 1  φi, j −φi−1, j / φi+1, j −φi, j (φi+1, j − φi, j ) if u i+1/2, j ≥ 0 2 x −x x −x  i, j i−1, j i+1, j i, j  φi+1/2, j  ⎩ φi+1, j + 1  φi+2, j −φi+1, j / φi+1, j −φi, j (φi, j − φi+1, j ) if u i+1/2, j < 0. 2 xi+2, j −xi+1, j xi+1, j −xi, j (6.14) A Van Leer Flux limiter is used (r ) 

r + |r | . 1 + |r |

(6.15)

Fifth-order For a high-order method, the fifth-order upwind-biased WENO scheme of Ren et al. (2003) was selected; for details of this scheme see Ren et al. (2003).

6.3.2.2

Discretization of the Reinitialization Equation

A second-order Adams–Bashforth scheme was used for temporal discretization of the reinitialization equation:   1   3 ϕ n+1 − ϕ n  S(ϕ0 ) 1 − |∇ϕ n | − S(ϕ0 ) 1 − |∇ϕ n−1 | . τ 2 2

(6.16)

Here, the pseudo-time step was set to be τ  30% of minimum cell dimension. Since the reinitialization equation is hyperbolic, a Godunov scheme is recommended by Herrmann (2008) and Javierre et al. (2007) for solving this PDE to maintain the signed distance property away from the interface. The term |∇φ| is approximated with (Javierre et al. 2007):

6.3 Interface Tracking Methods

213

⎧ 2 2 2 2 ⎪ ⎪ ⎨ max(a+ , b− ) + max(c+ , d− ) if ϕi, j > 0 2 2 |∇ϕ| max(a− , b+2 ) + max(c− , d+2 ) if ϕi, j < 0 ⎪ ⎪ ⎩ 0 otherwise,

(6.17)

where (e.g.,): a-  min(a, 0), a+  max(a, 0), and: a  dx− , b  dx+ , c  d y− , d  d y+ . In the following, only the discretization of dx− and dx+ is described, and other terms can be discretized in a similar way. First-order The first-order approximation to dx− and dx+ at node (i, j) is the upwind scheme: dx− 

ϕi, j − ϕi−1, j xi, j − xi−1, j

dx+ 

ϕi+1, j − ϕi, j . xi+1, j − xi, j

(6.18)

Second-order The second-order approximation to dx− and dx+ is the second-order ENO scheme of Yue et al. (2003):   ϕi, j − ϕi−1, j + (xi, j − xi−1, j ) min mod h i−1, j , h i, j xi, j − xi−1, j   ϕi+1, j − ϕi, j dx+  + (xi, j − xi+1, j ) min mod h i, j , h i+1, j xi+1, j − xi, j

sign( p) min(| p|, |q|) if pq > 0 min mod ( p, q)  0 otherwise.

dx− 

The divided difference h i, j is defined as   ϕi+1, j − ϕi, j ϕi, j − ϕi−1, j /(xi+1, j − xi−1, j ). h i, j  − xi+1, j − xi, j xi, j − xi−1, j

(6.19) (6.20) (6.21)

(6.22)

Fifth-order For a high-order method, the fifth-order HJ WENO scheme by Jiang and Peng (2000) was selected; for details of this scheme see Jiang and Peng (2000), Osher and Fedkiw (2003).

6.3.3 Coupled LS and VOF Algorithm (CLSVOF) 6.3.3.1

Algorithm for the Present CLSVOF Method

The detailed algorithm of the CLSVOF method used here is as follows (this is also illustrated pictorially in Fig. 6.8):

214

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.8 Flowchart for the CLSVOF method (Xiao 2012)

• Initialize the LS and VOF functions at the time step n  0: φ n and F n • Reconstruct the interface in cut cells where 0 < F n < 1. The interface normal vector is calculated from the LS function, and the position of the interface within the cell is constrained by the VOF function. • Advect the VOF function F n in the first coordinate direction (selected by cycling ˜ based on the reconstructed interthrough x, y, z after each time step) to obtain F, n ˜ face. Advect the LS function φ in the same direction to calculate φ. ˜ ˜ • Reconstruct and constrain the interface in the cut cells, based on F and φ. ˜˜ based • Advect the VOF function F˜ in the second coordinate direction to obtain F, on the reconstructed interface. Advect the LS function φ˜ in the same direction to ˜˜ obtain φ. ˜˜ • Reconstruct and constrain the interface in the cut cells based on F˜˜ and φ. ˜ • Advect the VOF function F˜ in the third coordinate direction to obtain F n+1 , based on the reconstructed interface. Advect the LS function φ˜˜ in the same direction to obtain (φ n+1 )∗ . • Reconstruct and constrain the interface in the cut cells based on F n+1 and (φ n+1 )∗ . • Reset the LS value in the cut cells to be the signed distance from the reconstructed ∗ interface by (e.g., in 2D): (φi,n+1 j )  α − n x x/2 − n y y/2. Perform a reinitialization step to obtain the final LS function φ n+1 with a recovered signed distance property.

6.3 Interface Tracking Methods

215

In the CLSVOF method, an operator split approach is preferred for LS advection in order to be consistent with the scheme adopted for the VOF equation. The (2D) algorithm for the operator split method for the LS equation is as follows:   φi,n j + (t/x) G i−1/2, j − G i+1/2, j ˜   φ i, j  (6.23) 1 − (t/x) u i+1/2, j − u i−1/2, j  t   t  ˜ ˜ ˜ G φ˜ i, j vi, j+1/2 − vi, j−1/2 , + (6.24)  φ + − G φi,n+1 i, j−1/2 i, j+1/2 i, j j y y where G i+1/2, j  u i+1/2, j φi+1/2, j , and φi+1/2, j is approximated as in Sect. 6.3.2.1.

6.3.3.2

Calculation of the Normal Vector in CLSVOF

In CLSVOF, the normal vector is calculated from the LS function. A least squares technique was recommended by Sussman and Puckett (2000) in their CLSVOF method. Ménard et al. (2007) also adopted this method, but modified it to resolve thinner ligaments by appropriate choice of the stencil. It was observed in Ménard et al. (2007) that a nine-point stencil failed to locate the interface correctly when two interfaces crossed the stencil domain, and a six- or four-point stencil was more appropriate. The practice suggested in Ménard et al. (2007) is followed in the present method; the unit normal vector is calculated by discretizing the LS gradient via a finite difference scheme, with corrections made to improve the ability to resolve thin ligaments as explained below. The straightforward way to discretize m i  ∂∂φxi is via a central difference scheme, e.g., in 2D the normal vector components at node (i, j) are approximated by m x  m cx 

φi+1, j − φi−1, j 2x

m y  m cy 

φi, j+1 − φi, j−1 . 2y

(6.25)

The unit normal vector ni can then be obtained via (in 2D): nx  −

mx mx 2 + m y2

ny  −

my mx 2 + m y2

.

(6.26)

However, the above scheme results in a large error in the computed normal vector when two interfaces approach each other as shown in Fig. 6.9. The level set values at nodes (i, j), (i + 1, j), (i, j + 1), (i, j − 1) are determined by the interface which crosses cell (i, j), while the level set value at node (i − 1, j) is determined by the interface which crosses cell (i − 1, j). Thus, when approximating the normal vector of the interface for cell (i, j), only nodes (i, j), (i + 1, j), (i, j + 1), (i, j − 1) should be used while node (i − 1, j) should be excluded. In this case, mx should be approximated by a forward difference, and my by a central difference:

216

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.9 Normal vector calculation for two closely located interfaces

mx 

φi+1, j − φi, j x

my 

φi, j+1 − φi, j−1 . 2 y

(6.27)

Thus, the calculation of the normal vector can be improved by appropriate choice from one of three finite difference schemes: central difference, forward difference, and backward difference. The forward and backward difference expressions for m i  ∂φ at node (i, j) are (2D): ∂ xi φi+1, j − φi, j φi, j+1 − φi, j m +y  x y − φ − φi, j−1 φ φ i, j i−1, j i, j m− m− . x  y  x y m +x 

(6.28) (6.29)

If the value of m +x is similar to that of m − x , the LS values at nodes (i − 1, j), (i, j) and (i + 1, j) should be determined by a single interface and a central difference scheme can be used (m x  m cx ). Otherwise, if the value of m +x deviates considerably + − from that of m − x (e.g., |m x − m x | ≥ 0.01), the LS values at (i − 1, j) or (i + 1, j) should be determined by a different interface to the one that crosses cell (i, j). The node which produces the larger value of the derivative  the one whose LS value  is +  is determined by the interface crossing cell (i, j). If m +x  > m − x , then m x  m x ; . otherwise m x  m − x

6.3.4 Comparison of VOF, LS, CLSVOF Stretching of a 2D liquid disk in a prescribed single vortex flowfield as described in Ménard et al. (2007) has become a standard benchmark test case for performance assessment of interface tracking methods and is used here to evaluate and compare “pure” LS, “pure” VOF, and CLSVOF methods. A liquid disk of radius r  0.15 is

6.3 Interface Tracking Methods

217

Fig. 6.10 2D liquid disk in a single vortex flow

placed in a single vortex velocity field in a unit size box, with the center of the disk located at (0.5, 0.75) as shown in Fig. 6.10. In the ideal (exact) solution, the velocity field stretches the disk into an ever thinner ligament shape. In this section, fifth-order WENO schemes are used in evolving and re-initializing LS for both pure LS and CLSVOF results; the performance of numerical schemes of different orders in the CLSVOF scheme will be reviewed in Sect. 6.3.5. The WENO scheme of Ren et al. (2003) was used for LS advection, and the HJ WENO scheme of Jiang and Peng (2000) was used for LS reinitialization. Figure 6.11 shows the predicted interface shape for the deformed liquid disk at t  3 obtained from the three interface tracking methods on a uniform mesh of 128 × 128. Table 6.1 provides the errors in the liquid mass enclosed by the interface predicted by the three methods at t  3. It is obvious that the pure VOF and CLSVOF methods can conserve mass accurately while the pure LS method induces considerable mass error (31%). Figure 6.11b shows that liquid blobs are generated numerically in the tail of the liquid ligament in the pure VOF method. A similar generation of numerical blobs is also observed in CLSVOF predictions when the normal vector is calculated from a simple central difference, as shown in Fig. 6.11c. The generation of this numerical breakup in the tail of the liquid mass is mainly caused by the inaccurate approximation of the normal vector when two interfaces approach each other. Figure 6.11d shows the interface shape captured by CLSVOF with the alternative normal vector calculation method described above; this improves the solution significantly, reducing considerably the size of the numerical breakup zone when the ligament is under-resolved. Although the pure VOF method can be improved to enable as good resolution of the thin ligament on the same mesh as the present CLSVOF method by using a second-order interface reconstruction method (see López et al. 2004), this method is very complex and difficult to extend to 3D. The

218

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.11 Interface at t  3, vortex test a LS, b VOF, c CLSVOF normal vector from central difference, and d CLSVOF improved normal vector calculation Table 6.1 Mass error at t  3 in single vortex test, 128 × 128 mesh

Method Mass error (%)

LS 31

VOF 6×

10−4

CLSVOF 5 × 10−4

CLSVOF method is thus preferred for use in the current two-phase flow simulations because of its combined superiority in (i) straightforward implementation in 3D, (ii) capability of resolving thin ligaments, and (iii) accuracy of conserving mass. Due to the use of an operator split method, the extension of the CLSVOF method to 3D is straightforward, and thus not detailed here.

6.3 Interface Tracking Methods

219

6.3.5 CLSVOF with Schemes of Different Order While the second-order PLIC scheme is well established as adequate for VOF advection, numerical schemes of different order have been proposed and used for LS advection and reinitialization in both pure LS and CLSVOF methods. In general, fifth-order schemes have been preferred. Ménard et al. (2007) suggested that a fifthorder WENO scheme should be used to evolve LS in their CLSVOF method for accuracy. In the CLSVOF method proposed by Park et al. (2006), the fifth-order WENO was again used in discretizing the LS reinitialization equation, based on the observation of Croce et al. (2004) that WENO showed superior performance than other ENO schemes in the pure Level Set method. However, WENO schemes for LS advection (Ren et al. 2003) and reinitialization (Jiang and Peng 2000) were developed and have generally been used only on uniform Cartesian meshes. Thorough assessment/validation of WENO schemes on the typically highly nonuniform meshes that would be inevitable in complex applications has not been reported. A further drawback of WENO is that it is computationally expensive. It was therefore decided to survey the performance of different numerical schemes in the present CLSVOF approach to establish the optimum scheme for LS evolution in the sense of low computational cost and sufficient accuracy. The first test case considered to illustrate the problem was the transport of a liquid disk in a uniform velocity field using a pure LS method. A liquid disk of radius 0.25 with center at point (−0.5,0) was placed in the domain [− 1,1] × [−1,1]. The fixed velocity field was u  0.05, v  0. After 20 s, the circle should have been transported unchanged to a position with its center at point (0.5,0). All simulations of this test case were run on a uniform mesh of 100 × 100 cells with a time step of 0.08 s. Figure 6.12 shows the liquid disk captured by the pure LS method with three schemes of different order used for both advection and reinitialization steps. The fifth-order scheme can capture the liquid disk well with small mass loss (see Table 6.2) in this simple test case. The second-order scheme loses considerable mass (7.5%) while the first-order scheme loses nearly all the mass.

Fig. 6.12 Tansported liquid disk in uniform flow pure LS method with various discretization schemes (black circle—exact solution): a fifth-order WENO; b second-order; and c first-order

220 Table 6.2 Mass error when transporting a liquid disk

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Method

Fifth-order

Second-order

First-order

Pure LS (%)

0.3

7.5

99.7

CLSVOF (%)

3.2 × 10−4

3.0 × 10−4

3.1 × 10−4

Fig. 6.13 Tansported liquid disk in uniform flow with CLSVOF method with various discretization schemes (black circle—exact solution): a fifth-order WENO; b second-order; and first-order

Figure 6.13 and Table 6.2 show the results from the CLSVOF method for the same test problem. The CLSVOF method captures the liquid disk transport very well, and the three different-order schemes used for LS advection and reinitialization show no effect. Table 6.2 shows that the CLSVOF method conserves mass accurately, even with a first-order scheme for the evolution of the LS function. In conclusion, a fifth-order scheme for LS advection and reinitialization is clearly necessary in a pure LS method to reduce the mass error. However, low-order schemes can be used for LS evolution in the CLSVOF method where mass conservation is determined by VOF evolution. It is not immediately clear whether this result is only relevant to interface capture with a prespecified velocity field as in the simple test case chosen in this section, or also holds when the two-phase velocity field is part of the calculation. It is demonstrated in Xiao (2012) that when curvature is calculated from the LS function as in the current methodology a second-order method for LS evolution in a CLSVOF scheme was indeed found to be the optimum choice, and has been used in all simulation results shown below.

6.3.6 Nonuniform Versus Uniform Cartesian Mesh Due to the choice of a second-order LS evolution scheme, it is very straightforward to implement the CLSVOF method on a nonuniform Cartesian mesh. This can be used to provide fine resolution in regions where it is needed and avoid the thin ligament numerical breakup observed in Fig. 6.11. The benefits of using a nonuniform mesh are illustrated here via a 3D simulation of sphere deformation. A sphere of radius

6.3 Interface Tracking Methods

221

Fig. 6.14 Sphere deformation in a single vortex velocity field at t  T /2; second-order CLSVOF: a uniform mesh; b nonuniform mesh

0.15 was placed in the domain [1,1,1] with its center at point (0.35,0.35,0.35). The velocity field was prescribed by u(x; y; z; t)  2 sin2 (π x) sin(2π y) sin(2π z) cos(π t/T ) v(x; y; z; t)  2 sin(2π x) sin2 (π y) sin(2π z) cos(π t/T ) w(x; y; z; t)  2 sin(2π x) sin(2π y) sin2 (π z) cos(π t/T ),

(6.30)

where T  3 s. In this single vortex velocity field, the liquid sphere is stretched into a thin distorted membrane, reaching its maximum deformation at t  T /2, and then reverses back to the original spherical shape. Figure 6.14 shows that on a uniform 160 × 160 × 160 mesh the thin liquid film displays evidence of numerical breakup, with spurious ligaments formed in the middle section of the deformed shape at its maximum deformation (t  T /2); on a nonuniform mesh with the same number of cells no such breakup is observed due to the improved resolution.

6.4 Two-Phase Flow LES Methodology for Atomization in Supersonic Flow In the LES formulation of atomization in supersonic gaseous flow, the liquid and gas are, respectively, treated as incompressible and compressible flows (Xiao et al. 2016). The interface is resolved directly (probably under-resolved for certain tiny liquid structures) using the coupled Level Set and VOF method, and the usual spatially filtered LES formulation is employed in the single-phase flow regions. Appropriate discretization schemes are used at the phase interface to properly capture the interaction between the gas and the liquid.

222

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

6.4.1 Governing Equations in Liquid Phase In the current study of two-phase flow modeling, the liquid phase is assumed to be incompressible and immiscible with gas. The filtered continuity equation in the liquid phase is ∂ u¯ i  0, ∂ xi

(6.31)

where the variable with an overbar represents the resolved (filtered) component, and a box filter is used here. The filtered momentum equation in the liquid phase is SGS ∂(u¯ i ) ∂(u¯ i u¯ j ) 1 ∂ p¯ 1 ∂(τ¯ i j −τi j ) 1 ST + − + + gi + F i . ∂t ∂x j ρ ∂ xi ρ ∂x j ρ

(6.32)

Here, gi is gravitational acceleration, and the filtered viscous stress is calculated from   1 ∂ u¯ i ∂ u¯ j .. (6.33) τ i j  2μ S i j S i j  + 2 ∂ x j ∂ xi sgs

The sub-grid-scale (SGS) stress tensor τi j (τiSGS  ρu i u j − ρ u¯ i u¯ j ) is modeled j by a simple Smagorinsky eddy viscosity approach 1 SGS − τkk δi j  −2μr S i j τiSGS j 3

μr  ρ (CS )2 S S 



2 Si j Si j ,

(6.34)

where  represents the filter width and is equal to the cube root of the local cell volume; the value of the Smagorinsky constant C S is set to 0.1 in all the calculations. The resolved surface tension force is computed by ∂H F¯iST  σ κ¯ ∂ xi  1 if ¯ H (φ)  0 if

κ¯ 

∂ n¯ i ∂ xi

φ¯ > 0 φ¯ ≤ 0.

n¯ i  − 

1 ∂ φ¯ ∂ φ¯ ∂ xk ∂ xk

∂ φ¯ ∂ xi (6.35)

Here, σ is the surface tension coefficient, κ¯ is the interface curvature, and n¯ i is the interface normal vector pointing from the liquid phase into the gas phase. φ is LS function which is the signed distance from the interface, with φ  0 representing the interface, φ > 0 in liquid, and φ < 0 in gas. Proper modeling of SGS interface movement and SGS surface tension has not been developed yet in the current formulation and is the subject for future research.

6.4 Two-Phase Flow LES Methodology …

223

6.4.2 Governing Equations in Gas Phase In the LES of atomization in supersonic flows, the gas phase is compressible, and the filtered governing equations are as follows:   sgs ¯ i j − τ¯i j + τi j ∂ ρ¯ ∂(ρ¯ u˜ i ) ∂(ρ¯ u˜ i ) ∂ ρ¯ u˜ i u˜ j + pδ + + 0 0 (6.36) ∂t ∂ xi ∂t ∂x j    sgs sgs ˜ ∂ ρ¯ E˜ ∂ ρ¯ E + p¯ u˜ i + q¯i − u˜ j τ¯ ji + Hi + σi 0 (6.37) + ∂t ∂ xi p¯  ρ¯ R T˜ , where an overbar represents spatial filtering as in the liquid phase, and a tilde repre ¯ is the total energy per unit mass which is the sum sents Favre average ( f˜  ρ f ρ).E of the specific internal energy and kinetic energy, i.e., E  e + 21 u i u i  cv T + 21 u i u i where cv is the specific heat at constant volume. The stress tensor τ¯i j and the heat flux vector q¯i are computed by     1˜ 1 ∂ u˜ i ∂ u˜ j ˜ ˜ ˜ τ¯i j  2μ(T ) Si j − Skk δi j + Si j  3 2 ∂ x j ∂ xi   c p μ T˜ ∂ T˜ q¯i  −k(T˜ ) , (6.38) k(T˜ )  ∂ xi Pr where k is heat conductivity and c p is the specific heat at constant pressure. Pr is Prandtl number and is set to 0.72 for air in the simulations. The viscosity coefficient μ is given by Sutherland’s law: μ(T˜ )  μ0



T˜ T0

1.5

T0 + 110 , T˜ + 110

(6.39)

where μ0 is the reference viscosity at the reference temperature μ0 . μ0  1.716 × 10−5 Pa s and T0  273.15 K are used for air here. sgs SGS stress tensor τi j (τiSGS  ρu i u j − ρ¯ u˜ i u˜ j ) in the gas phase is also modeled j by Smagorinsky eddy viscosity approach.    1 SGS 1˜ 2 ˜ ˜ ˜ μ τ τiSGS − δ  −2μ − δ  ρ ) 2 S˜ i j S˜ i j , S S  S S (C ij r ij kk i j r S j 3 kk 3 (6.40) sgs

sgs

where Hi and σi viscous work.

are, respectively, sub-grid-scale enthalpy flux and sub-grid-scale

224

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

⎧   ⎨ H sgs  ρ¯  u i E − u˜ i E˜ + ( pu i − p¯ u˜ i ) i , ⎩ σ sgs  − τ u − τ¯ u˜  i

sgs

where Hi

ij

j

ij

(6.41)

j

sgs

and σi j are set to zero currently for simplicity.

6.4.3 LES Formulation of Interface Advection Equations The VOF function F is defined as the volume fraction occupied by the liquid. The evolution of the resolved VOF function is governed by ∂F ∂F + u iL  0. ∂t ∂ xi

(6.42)

The liquid velocity field u iL is constructed from the resolved velocity u¯ i using an extrapolation and divergence-free algorithm. It is demonstrated in Xiao (2012) that the use of u iL for interface advection can produce more accurate results than u¯ i in simulations of droplet deformation and breakup. The resolved LS function φ is evolved by a similar advection equation: ∂φ ∂φ + u iL  0. ∂t ∂ xi

(6.43)

To maintain the signed distance property (∇φ  1), a reinitialization equation is solved:    ∂ϕ ∂ϕ ∂ϕ ϕ0  S(ϕ0 ) 1 − , S(ϕ0 )  ∂τ ∂ xk ∂ xk ϕ0 2 +d 2

(6.44)

¯ i , t), d  max(x, y, z). τ represents pseudowhere ϕ0  ϕ(xi , τ  0)  φ(x time, and S(ϕ0 ) is a modified sign function. After solving this equation to steady state in the interface vicinity, φ is replaced by ϕ.

6.4.4 Grid and Dependent Variable Arrangement In the present two-phase modeling, the gas flow is resolved by a finite difference method, and the liquid flow is solved using a finite volume method on a Cartesian grid. In the following, the methods on a 2D grid are described for illustration; extension to 3D is straightforward. The grid and dependent variable arrangement are shown in Fig. 6.15. The variables used by the liquid flow solver are arranged in a staggered

6.4 Two-Phase Flow LES Methodology …

225

Fig. 6.15 Grid and variables arrangement. Green-shaded region is pressure CV; gray-shaded region is x-momentum CV; and yellow-shaded region is y-momentum CV (Xiao et al. 2016)

manner: the pressure p, LS φ, and VOF F are located at cell centers; the velocity components u (uL ) and v (vL ) are located at corresponding faces. The variables used by the gas flow solver (pressure P, velocity components U and V , temperature T , and density ρ) are all located at cell corners.

6.4.5 Numerical Methods for the Gas Flow Solver The supersonic gas flow is solved by the LES code developed by Sun et al. (2008, 2009). A second-order total variation diminishing (TVD) Runge–Kutta method proposed by Shu (1999) is used for temporal discretization of the compressible flow governing equation. The time step is constrained by setting the Courant–Friedrichs–Lewy (CFL) number to 0.4. A fifth-order WENO scheme developed by Jiang and Shu (1996) is used here for spatial discretization of inviscid fluxes. A second-order central difference scheme is applied to discretize the viscous terms. The LES code for compressible flows using the finite difference method has been validated in Sun et al. (2008, 2009). In order to solve the gas flow, boundary conditions should be specified at the interface. However, the variables used by the compressible flow solver are specified

226

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

y

∆xi-1

∆xi

∆xi+1 x

∆yj+1

Gas

∆yj

Vi+1/2, j-1/2 Pi+1/2, j-1/2 Ui+1/2, j-1/2

∆yj-1

Fig. 6.16 Illustration of boundary conditions specification at the interface for the gas flow solver. Shaded region is gas phase (Xiao et al. 2016)

instead at the cell corners in the liquid phase for convenience in discretization of the governing equations as shown in Fig. 6.16. The velocity in the liquid region (φi+1/2, j−1/2 > 0) given by the liquid velocity field constructed in the liquid flow solver as follows: Ui+1/2, j−1/2  Vi+1/2, j−1/2 

L L u i+1/2, j−1 y j + u i+1/2, j y j−1

y j + y j−1 L L vi+1, j−1/2 x i + vi, j−1/2 x i+1

xi + xi+1

.

(6.45) (6.46)

Since the energy equation is not solved in the liquid phase in the current formulation, the temperature in the liquid phase is set to a constant Tliquid : Ti+1/2, j−1/2  Tliquid .

(6.47)

The pressure Pi+1/2, j−1/2 is obtained by extrapolation from gas phase to liquid phase. And the density   is calculated from the equation of state for the gas ρi+1/2, j−1/2  Pi+1/2, j−1/2 RTi+1/2, j−1/2 . The velocity components (Ui+1/2, j−1/2 , Vi+1/2, j−1/2 ) and fluid properties (ρi+1/2, j−1/2 , Pi+1/2, j−1/2 , Ti+1/2, j−1/2 ) can be assumed to belong to the ghost gas at that node.

6.4 Two-Phase Flow LES Methodology …

227

After the velocity components and fluid properties are specified in the liquid phase as above, the gas flow can be solved using the compressible flow solver in the gas region.

6.4.6 Numerical Methods for the Liquid Flow Solver A first-order forward-Euler projection method was used for temporal discretization of the liquid flow governing equations. First, an intermediate velocity is computed from convection, diffusion, and gravitational terms: n

n SGS ∂(u in u nj ) u i∗ − u in 1 ∂(τi j + τi j ) − + n + gi . δt ∂x j ρ ∂x j

(6.48)

Then, the Poisson equation is solved to calculate pressure field P n+1 : ∂ ∂ xi



1 ∂ p n+1 ρ n ∂ xi

 

1 ∂u i∗ . δt ∂ xi

(6.49)

In the end, the intermediate velocity field is updated by the pressure gradient to obtain the velocity at time step n + 1 which satisfies the continuity equation: u in+1 − u i∗ 1 ∂ p n+1 − n . δt ρ ∂ xi

(6.50)

Since the gas phase has a much smaller density and viscosity than the liquid phase, the velocity gradient in the gas phase is typically much larger than liquid phase. In order to tackle the discontinuity of the velocity gradient at the interface, a spatial discretization scheme using the extrapolated liquid velocity is developed in Xiao (2012, 2016). In this numerical scheme, the interface velocity is approximated by the velocity in the neighboring liquid cell rather than the linear interpolation of velocities in the neighboring liquid and gas cells. An artificial liquid velocity field is constructed from the resolved physical velocity in the liquid phase by an extrapolation technique, and the extrapolated liquid velocity is used in the calculation of the convection term and eddy viscosity for cells in the vicinity of the interface to reduce numerical errors. Since the velocity u i in control volumes which change from gas to liquid is set to the extrapolated liquid velocity u iL , u iL must satisfy the continuity equation, making the divergence-free step necessary for the extrapolated liquid velocity. The surface tension is incorporated in the discretization of the pressure gradient using Ghost Fluid Method (Fedkiw et al. 1999; Kang et al. 2000), which can reproduce the physical pressure discontinuity across the interface. Readers are referred to Xiao (2012, 2016) for more details on the numerical methods used in the incompressible flow solver. In order to solve the liquid flow, the pressure and shear stress forces should be known at the interface in mathematical sense. However, for convenience in numerical

228

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

∆xi-1

∆xi

∆xi+1

x ∆yj+1 vi, j+1/2 ui-1/2, j

pi, j Liquid

∆yj

∆yj-1

Fig. 6.17 Illustration of boundary conditions specification at the interface for the liquid flow solver. Shaded region is liquid phase (Xiao et al. 2016)

calculation, the boundary conditions for the liquid flow solver are specified in the neighboring gas cells as shown in Fig. 6.17. In order to solve the pressure Poisson equation in the liquid region, the pressure in the neighboring gas cell (i, j) (φi, j < 0) must be given. pi, j is computed by averaging the gas pressures at the cell corners which are resolved by the gas flow solver: Pi−1/2, j−1/2 (φi−1/2, j−1/2 ) + Pi+1/2, j−1/2 (φi+1/2, j−1/2 ) + Pi−1/2, j+1/2 (φi−1/2, j+1/2 ) pi, j 

+ Pi+1/2, j+1/2 (φi+1/2, j+1/2 ) (φi−1/2, j−1/2 ) + (φi+1/2, j−1/2 ) + (φi−1/2, j+1/2 ) + (φi+1/2, j+1/2 )

 Θ(φ)  φi−1/2, j−1/2 

(6.51) 1 if φ ≤ 0 0 if φ > 0

(6.52)

φi−1, j−1 xi y j + φi, j−1 xi−1 y j + φi−1, j xi y j−1 + φi, j xi−1 y j−1 . (xi−1 + xi )(y j−1 + y j )

(6.53) In order to calculate the shear stress acting on the interface by the gas, the velocity in the adjacent gas momentum control volumes (φi−1/2, j < 0, φi, j+1/2 < 0) should be specified for the liquid flow solver. u i−1/2, j and vi, j+1/2 are calculated by averaging the velocity at the cell corners which are obtained from the gas flow solver: u i−1/2, j 

Ui−1/2, j−1/2 + Ui−1/2, j+1/2 2

(6.54)

6.4 Two-Phase Flow LES Methodology …

vi, j+1/2 

Vi−1/2, j+1/2 + Vi+1/2, j+1/2 . 2

229

(6.55)

After the pressure and velocity components are specified in the gas cells adjacent to the interface, the incompressible flow solver developed in Xiao (2012) can be used to calculate the liquid flow. The successive over-relaxation (SOR) Gauss–Seidel method is used here to solve the pressure Poisson equation in the liquid phase.

6.4.7 Algorithm for LES of Atomization in Supersonic Gas Flows The complete algorithm for LES of atomization in supersonic airflow may be detailed for a single time step as follows: • Based on the interface represented by the LS function, specify the boundary conditions for the gas flow at the interface using the resolved liquid variables. • Resolve the gas flow to the next time step by a finite difference method. • Based on the interface represented by the LS function, specify the boundary conditions for the liquid flow at the interface using the resolved gas variables. • Resolve the liquid flow to the next time step by a finite volume method. technique. Ensure • Construct the liquid velocity field u iL using an extrapolation  continuity for the extrapolated liquid velocity (∂u iL ∂ xi  0) by a divergencefree step. • Based on u iL , advect the LS and VOF functions to the next time step using the CLSVOF algorithm developed by Xiao (2012). • Repeat for further time steps.

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow The primary atomization of a water jet in a supersonic air crossflow is simulated and analyzed in this section. Four test cases are simulated with flow conditions and characteristic parameters listed in Table 6.3. D is the diameter of the water jet. ρG and T G are, respectively, the density and static temperature of the freestream airflow. The static pressure and viscosity of the freestream airflow can be calculated, respectively, from Equation of State and Sutherland’s law. U G is the freestream velocity of the air inflow. V L is the injection speed of the water jet which has a uniform and laminar profile at the inlet in all the simulations. The fluid properties of water are ρ L  1000 kg/m3 and μ L  1 × 10−3 Pa s. The temperature of the water is set by TL  300 K. The surface tension coefficient is σ  0.072 N/m. Three nondimensional characteristic parameters are defined as follows: liquid/gas

230

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Table 6.3 Flow conditions and characteristic parameters of simulated test cases D (mm)

ρG (kg/m3 )

TG (K)

UG (m/s)

VL (m/s)

q

We

Ma

Weeff

Case 1

0.1

1.516

159

531

Case 2

0.2

0.2

300

500

42

4

594

2.1

211

17.32

6

139

1.44

Case 3

0.2

0.29

300

79

600

17.32

3

290

1.728

129

Case 4

0.2

0.29

300

600

25

6

290

1.728

129

momentum flux ration (q  ρ L V 2L /(ρ G U 2G )), Weber number (We  ρ G U 2G D/σ ), and Mach number (Ma = U G /a, a is the sound speed in the gas). Definition of Weeff will be given below. For test case 1, the simulation domain is [0, 30D] × [0, 15D] × [−8D, 8D] in the x, y, and z directions, and has 480 blocks as shown in Fig. 6.18. Each block has 39 × 27 × 42 cells, resulting in a total of 21 million cells in the whole domain. In order to provide good resolution of the initial stages of primary breakup, a uniform fine meshwas used in the region [10D, 20D] × [0, 10D] × [−2D, 2D] with a cell size of D 30; in the outer region of the domain, an expanding mesh was used to reduce computational cost. The water jet is injected from the domain bottom (y  0), with the jet center located at (10.7D, 0, 0). In order to model the effect of the boundary layer the gas flow, a mean velocity profile of turbulent boundary layer   in   1 7  2y D / if y < D 2 (U UG  ) is currently specified at the inlet for the 1 else gas. A recycling method will be used in the future research to reproduce a realistic turbulent boundary layer. The simulation is run on 480 processors for 2 days. The water jet atomization in a supersonic crossflow with the same airflow conditions as test case 1 has been experimentally studied by our research group (Wang et al. 2014). Icing was not observed in the primary breakup region and was only observed far downstream. This implies that the water temperature just undergoes a moderate decrease in the primary breakup process though the temperature of the supersonic airflow is very low. Therefore, it is valid to neglect the solution of the energy equation in the liquid phase in the current simulation. Figure 6.19 shows the morphology of liquid jet primary breakup in the supersonic airflow of Mach 2.1. The pressure contours in the gas flow demonstrate that a strong shock wave forms in front of the liquid column. Moderate shock waves are also observed ahead of the liquid drops/ligaments produced in the atomization process. Two breakup modes (column breakup and surface breakup) are shown in the simulated liquid jet primary breakup. In the column breakup, the liquid column breaks up into clusters as a whole as the surface waves develop on the windward side of the liquid jet. It is demonstrated in Fig. 6.19 (also the movie 1) that the strong shock wave oscillates moderately as liquid clusters disintegrate from the liquid jet. The surface breakup, i.e., liquid droplets shedding from the liquid column, is also well reproduced in the simulation.

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow

231

Fig. 6.18 Simulation domain with 480 blocks (Ma  2.1, We  594) (Xiao et al. 2016)

Figure 6.20 presents the velocity vectors and pressure contours at slice z  0. Figure 6.20a shows that a circulation region forms near the wall upstream of the liquid jet due to the wall boundary layer in the gas flow, agreeing well with the experimental observation (Beloki Perurena et al. 2009). As the gas flows around the liquid column, a high-pressure region is formed in front of the liquid jet, and a low-pressure region and strong turbulence are observed behind the liquid jet. Due to the pressure difference between the windward and leeward sides, the liquid jet is bent downward in the crossflow direction. A zoomed-in view of the vector field in the liquid column breakup region is presented in Fig. 6.20b. As the magnitude of the surface wave grows on the upstream surface of the liquid jet, gaseous vortices develop in the troughs of the surface waves, introducing aerodynamic flow features which may enhance the primary breakup process. The large eddies and their interaction with the interface captured in the simulations clearly demonstrate the advantage of LES over RANS for primary breakup modeling. Figure 6.21 shows the instantaneous vortical structures using the invari  second 2 2  1 , see ance of the velocity gradient tensor (Q  2 ∇ • U − tr ∇ U Chakraborty et al. (2005) for more on vortex identification schemes). The vortices in the turbulent wall boundary layer and wake region behind the liquid jet are reasonably reproduced by the current LES. Furthermore, two kinds of vortices are obviously observed on the liquid jet. The first kind of vortices denoted by “1” in Fig. 6.21 is aligned with the liquid column on the lateral side, which corresponds to the surface break. The second kind of vortices denoted by “2” in Fig. 6.21 is aligned in the span-

232

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.19 Morphology of liquid jet primary breakup and the pressure contours (Ma  2.1, We  594). The time interval between two adjacent subfigures is 2 × 10−6 s (Xiao et al. 2016)

Fig. 6.20 Velocity vectors and pressure contours at z  0 (Ma  2.1, We  594) (Xiao et al. 2016)

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow

233

2

1

Fig. 6.21 Isosurface of the second invariance of the velocity gradient tensor (colored by green) together with the liquid jet structure (colored by red) (Ma  2.1, We  594) (Xiao et al. 2016) (the picture in the upper-left corner is simulation from Mahesh 2013)

wise (Z) direction on the windward side of the liquid column, which corresponds to the column breakup. It is interesting to note that these two kinds of vortices are analogous to those obtained by Mahesh (2013) in simulations of single-phase jet in crossflow shown in the upper-left corner of Fig. 6.21. It is also shown in Fig. 6.21 that vortical structures are generated around the liquid drops/ligaments. Since the Weber numbers in test cases 2–4 are smaller than that in test case 1, a slightly coarse mesh is used to simulate test cases 2–4. The simulation domain is [0, 26D] × [0, 18D] × [−8D, 8D] in the x, y, and z directions, and has 240 blocks. Each block has 26 × 20 × 62 cells, resulting in a total of 7.7 million cells in the whole domain. A uniform fine mesh was used in the region [7.9D, 17.5D] × [0, 10D] × [−2.1D, 2.1D] with a cell size of 0.045D; an expanding mesh was used in the outer region of the domain. The water jet is injected from the domain bottom (y  0), with the jet center located at (8.6D, 0, 0). A mean velocity profile of turbulent boundary layer as in test case 1 is specified for the gas inflow. Each simulation was run on 240 processors for 2 days. Figures 6.22, 6.23, and 6.24 show the instantaneous liquid jet structures and pressure contours predicted in simulations of test cases 2–4. The breakup process of the liquid jet in these three test cases is less aggressive than that

234

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.22 Morphology of liquid jet primary breakup and the pressure contours in plane z  0 (Ma  1.44, We  139) (Xiao et al. 2016)

in test case 1 due to the weaker disintegrating aerodynamic forces in comparison with restoring surface tension as characterized by the smaller Weber number. The distance between the shock and the liquid column (δ as shown in Fig. 6.22) in test case 2 is obviously larger than that in test case 1. δ is measured from the four LES simulations and is compared with the experimental data for standing shock waves ahead of cylinders and spheres from Liepmann and Roshko (1957) in Fig. 6.25. Besides flowing peripherally around the liquid column, the gas behind the shock also flows upward along the upstream side of the liquid jet. This flow relaxation results in a smaller shock wave detachment distance than that for a cylinder. Figures 6.23 and 6.24 demonstrate that as liquid velocity (q) decreases, the liquid jet bends more aggressively in the crossflow direction, resulting in stronger flow relaxation and a smaller shock wave detachment distance. Therefore, δ in test case 3 (q  3) is smaller than test case 4 (q  6) as shown in Fig. 6.25 though both test cases have the same Ma and We. The predicted liquid jet outer boundaries in Fig. 6.26 illustrate that the simulated liquid jets of test cases 2 and 4 (which have the same q) have similar bending angles, indicating that the strength of flow relaxation is similar. Data for test cases 2 and 4 in Fig. 6.25 show that δ decreases as Ma grows when the liquid jet bending angle has more or less the same magnitude. Though the liquid jet of test case 1 (q  4) has higher penetration than test case 3 (q  3) as shown in Fig. 6.26, δ in test case 1 is smaller than test case 3 as shown in Fig. 6.25, also implying that

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow

235

Fig. 6.23 Morphology of liquid jet primary breakup and the pressure contours in plane z  0 (Ma  1.728, We  290, q  3) (Xiao et al. 2016)

δ decreases as Ma grows. This is consistent with the experimental observations for cylinders and droplets. It is shown in Figs. 6.19 and 6.22, 6.23, 6.24 that the pressure upstream of the liquid column is significantly higher than that downstream of the liquid column. Due to this high-pressure difference, the lighter gas strongly accelerated the high-density liquid phase, making the liquid column subject to the Rayleigh–Taylor instability. The surface waves arising from the Rayleigh–Taylor instability are clearly observed on the upstream side of the liquid jet from a top view as shown in Fig. 6.27. It is shown in experiments (Gopala 2012; Sallam et al. 2004) and current LES that the surface wave wavelength increases as the liquid jet bends downstream in the crossflow direction. Therefore, the wavelength λ S is measured as soon as the surface wave first appears as shown in Fig. 6.27, where the liquid jet is still nearly straight (vertical) and only undergoes slight deformation. Figure 6.27 only illustrates where λ S is measured, and λ S is actually defined and measured along the liquid jet as in Sallam et al. (2004) and Xiao et al. (2013). As the gas passes through the normal shock ahead of the liquid column, the kinetic energy (momentum flux) of the gas flow reduces. Since it is the gas flow behind the shock that directly interacts with the liquid column, the aerodynamic forces acting on the interface should be characterized by the gas flow behind the shock. Therefore, an effective Weber number is defined here by

236

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Fig. 6.24 Morphology of liquid jet primary breakup and the pressure contours in plane z  0 (Ma  1.728, We  290, q  6) (Xiao et al. 2016) Fig. 6.25 Shock wave detachment distance at different Mach numbers predicted by LES (experimental data from Liepmann and Roshko 1957; Xiao et al. 2016)

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow

237

Fig. 6.26 Outer boundaries of the calculated liquid jets (x 0 is the nozzle center coordinate) (Xiao et al. 2016)

Fig. 6.27 Top view of surface waves on upstream side of liquid jet (Ma  2.1, We  594) (Xiao et al. 2016)

238

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

W eeff 

2 ρG,2 UG,2 D

σ

,

(6.56)

where ρG,2 and UG,2 are the density and velocity behind the normal shock. The change in density and velocity across the normal shock satisfies the following correlation (White 2003): UG ρG,2 (γ + 1)Ma 2   , ρG UG,2 2 + (γ − 1)Ma 2

(6.57)

where γ is the specific heat ratio of the gas and is set to be 1.4 for air. Thus, the following correlation between Weeff and We exists for liquid jet primary breakup in supersonic airflow (Ma > 1): W eeff 

2 ρG,2 UG,2 D

σ



2 + (γ − 1)Ma 2 ρG UG2 D 2 + (γ − 1)Ma 2  W e. (6.58) σ (γ + 1)Ma 2 (γ + 1)Ma 2

The corresponding Weeff for the simulated test cases computed from the above equation is given in Table 6.3. The predicted surface wavelength versus effective Weber number is plotted in Fig. 6.28 and is compared with experimental measurements (Sallam et al. 2004) and LES predictions (Xiao et al. 2013) for liquid jet in subsonic gas crossflow where W eeff  W e. The wavelength predicted by LES of liquid jet in supersonic gas flows fits very well into one line with that obtained by LES of liquid jet in subsonic crossflow. The wavelength of the simulated surface as a power-law function of effective Weber number (i.e.,  waves decreases −0.45 ), with the power-law component accurately predicted by LES λ S D ∝ W eeff in comparison with the experimental data from Sallam et al. (2004). Since the liquid column is significantly accelerated by the gas phase due to the strong pressure imbalance between the upward and leeward sides of the liquid column as shown in Fig. 6.20, the Rayleigh–Taylor instability can be induced. And the wavelength of the Rayleigh–Taylor instability can be computed by  λR-T  2π

3σ , ρL a

(6.59)

where a is the acceleration of the liquid column due to the drag of the gaseous flow: a

2 C D,e 21 ρG,2 UG,2 D

ρ L 41 π D 2



2 2C D,e ρG,2 UG,2

ρL π D

,

(6.60)

where C D,e is the effective drag coefficient for the deformed liquid column. C D,e is correlated to the Weber number by C D,e ∝ W e−0.1 for a liquid jet in subsonic flows −0.1 is assumed to be also valid here in supersonic Xiao et al. (2013), and C D,e ∝ W eeff

6.5 Simulations of Liquid Jet Primary Breakup in Supersonic Crossflow

239

Fig. 6.28 Variation of surface wavelength versus effective Weber number (Xiao et al. 2016). Experimental data from Sallam et al. (2004) and LES for subsonic crossflow from Xiao et al. (2013)

flows. Thus, the following correlation between the Rayleigh–Taylor wavelength and effective Weber number can be derived λR-T  D



6π 3 −0.5 −0.45 W eeff ∝ W eeff . C D,e

(6.61)

The correlation between λR - T and W eeff is the same as that between λ S and W eeff , implying that the surface waves on the upstream side of the liquid jet in supersonic airflow arise from the Rayleigh–Taylor instability. And as the surface waves develop further downstream, the liquid jet disintegrates in a column breakup mode.

References Aulisa E, Manservisi S, Scardovelli R, Zaleski S (2007) Interface reconstruction with least-squares fit and split advection in three-dimensional cartesian geometry. J Comp Phys 225:2301–2319 Beloki Perurena J, Asma CO, Theunissen R, Chazot O (2009) Experimental investigation of liquid jet injection into Mach 6 hypersonic crossflow. Exp Fluids 46:403–417 Caiden R, Fedkiw RP, Anderson C (2001) A numerical method for two-phase flow consisting of separate compressible and incompressible regions. J Comp Phys 166:1–27 Chakraborty P, Balachandar S, Adrian RJ (2005) On the relationships between local vortex identification schemes. J Fluid Mech 535:189–214 Chang CH, Deng XL, Theofanous TG (2013) Direct numerical simulation of interfacial instabilities: a consistent, conservative, all-speed, sharp-interface method. J Comp Phys 242:946–990 Croce R, Griebel M, Schweitzer MA (2004) A parallel level-set approach for two-phase flow problems surface tension in three space dimensions. Universitaet Bonn

240

6 Primary Breakup of Liquid Jet in Supersonic Crossflow

Desjardins O, McCaslin JO, Owkes M, Brady P (2013) Direct numerical and Large-Eddy simulation of primary atomization in complex geometries. At Sprays 23:1001–1048 Fedkiw R, Aslam T, Merriman B, Osher S (1999) A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J Comp Phys 152:457–492 Ghenai C, Sapmaz H, Lin CX (2009) Penetration height correlations for non-aerated and aerated transverse liquid jets in supersonic cross flow. Exp Fluids 46(1):121–129 Gopala Y (2012) Breakup characteristics of a liquid jet in subsonic crossflow. Ph.D. thesis, Georgia Institute of Technology Gorokhovski M, Hermann M (2008) Modeling primary atomization. Ann Rev Fluid Mech 40:343–366 Gueyffier D, Li J, Nadim A, Scardovelli R, Zaleski S (1999) Volume-of fluid interface tracking with smoothed surface stress methods for three dimensional flows. J Comp Phys 152:423–456 Herrmann M (2008) A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids. J Comp Phys 227:2674–2706 Javierre E, Vuik C, Vermolen FJ, Segal A (2007) A level set method for three dimensional vector stefan problems: dissolution of stoichiometric particles in multi-component alloys. J Comp Phys 224:222–240 Jiang GS, Peng D (2000) Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J Sci Comput 21(6):2126–2143 Jiang GS, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comp Phys 126:202–228 Kang M, Fedkiw R, Liu XD (2000) A boundary condition capturing method for multiphase incompressible flow. J Sci Comput 15:323–360 Liepmann HW, Roshko A (1957) Elements of gasdynamics. Wiley, New York López J, Hernández J, Góez P, Faura F (2004) A volume of fluid method based on multidimensional advection and spline interface reconstruction. J Comp Phys 195:718–742 Mahesh K (2013) The interaction of jets with crossflow. Ann Rev Fluid Mech 45:379–407 Ménard T, Tanguy S, Berlemont A (2007) Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary breakup of a liquid jet. Int J Multiph Flow 33:510–524 Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New York Park SW, Kim S, Lee CS (2006) Breakup and atomization characteristics of mono-dispersed diesel droplets in a cross-flow air stream. Int J Multiph Flow 32:807–822 Pilliod JE, Puckett EG (2004) Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J Comp Phys 199:465–502 Press WH (2000) Numerical recipes in Fortran 77 Ren YX, Liu ME, Zhang H-X (2003) A characteristic-wise hybrid compact-weno scheme for solving hyperbolic conservation laws. J Comp Phys 192:365–386 Rider WJ, Kothe DB (1998) Reconstructing volume tracking. J Comp Phys 141:112–152 Scardovelli R, Zaleski S (2003) Interface reconstruction with least-squares fit & split Eulerian Lagrangian advection. Int J Numer Method Fluids 41:251–274 Sallam KA, Aalburg C, Faeth GM (2004) Breakup of round non-turbulent liquid jets in gaseous crossflow. AIAA J 42:2529–2540 Shu CW (1999) High order ENO and WENO schemes for computational fluid dynamics. In: Barth TJ, Deconinck H (eds) High-order methods for computational physics. Springer, Berlin, 439–582 Sun MB, Geng H, Liang JH, Wang ZG (2009) Mixing characteristics in a supersonic combustor with gaseous fuel injection upstream of a cavity flameholder. Flow Turbul Combust 82:271–286 Sun MB, Wang ZG, Liang JH, Geng H (2008) Flame characteristics in a supersonic combustor with hydrogen injection upstream of a cavity flameholder. J Propul Power 24:688–696 Sussman M, Puckett EG (2000) A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J Comp Phys 162:301–337 Versteeg HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite volume method, 2nd edn. Pearson Ed. Ltd

References

241

Wang ZG, Wu L, Li Q, Li C (2014) Experimental investigation on structures and velocity of liquid jets in a supersonic crossflow. Appl Phys Lett 105:134102 White FM (2003) Fluid mechanics. McGraw-Hill, Boston Xiao F (2012) Large Eddy Simulation of liquid jet primary breakup. Ph.D. thesis, Loughborough University Xiao F, Dianat M, McGuirk JJ (2013) Large Eddy Simulation of liquid-jet primary breakup in air crossflow. AIAA J 51:2878–2893 Xiao F, Wang ZG, Sun MB, Liang JH, Liu N (2016) Large eddy simulation of liquid jet primary breakup in supersonic air crossflow. Int J Multiph Flow 87:229–240 Yue W, Lin CL, Patel VC (2003) Numerical simulation of unsteady multidimensional free surface motions by level set method. Int J Numer Methods Fluids 42:853–884

Chapter 7

Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

7.1 Experimental Study of Liquid Jets 7.1.1 Experimental Methods In this chapter, phase doppler anemometer (PDA) and particle image velocimetry (PIV) systems were used to experimentally study the characteristics of liquid jets. The two-dimensional PDA system (Dantec Dynamics, Skovlunde, Denmark) with its transmitting optics and receiving optics was used to measure the droplet diameter and velocity of a transverse liquid jet. Focal lengths were 500 mm for the transmitting optics and 1000 mm for the receiving optics, respectively, and the secondary scattering angle was 35°, as shown schematically in Fig. 7.1. A BSA Flow and Particle Processor, operating with BSA Flow V.5 Software, controlled the PDA system and data acquisition. The droplet velocities varied significantly with the distance from the injector, so the velocity parameters were set individually. The transverse and vertical velocity range was set from [−150, 400] m/s to [100, 600] m/s and from [−80, 80] m/s to [0, 150] m/s, respectively, considering the effect of the distance from the injector. Figure 7.2 shows a histogram of the single-point measurement result. It recorded the dynamic information of 62,844 droplets in total during the 10 recording period at measurement position x/d  120, y/d  20. The result shows that most droplet diameters range from 0 to 30 µm. This means that the liquid jet has broken up into small droplets and secondary atomization has been completed. With increased recording times, the counts on the histogram present a similar distribution, suggesting that PDA results have good repeatability and reliability. To keep the data reliable and save testing time, the PDA system in this experiment was set to acquire 20,000 particles or to measure for 5 s. A PIV device, capable of inclined imaging, was used in this test. As shown in Fig. 7.3, the cross-sectional spray is instantaneously illuminated by a pulsed laser through the side glass, meanwhile the spray field is imaged by CCD camera. The image of the spray on the cross section can be cleared by adjusting the angle between © Springer Nature Singapore Pte Ltd. 2019 M. Sun et al., Jet in Supersonic Crossflow, https://doi.org/10.1007/978-981-13-6025-1_7

243

244

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.1 Schematic of a PDA system and experimental setup

the central axis of the lens and the photosensitive element in the camera. The image deformation caused by the inclination of the camera can be modified on the basis of the Scheimpflug principle. The pulsed laser instrument used in the test was a Nd:YAG double-pulse laser, whose single-pulse width was 7 ns, having a maximum pulse energy of 500 mJ. The CCD camera used was a high-resolution double-exposure frame-straddling camera, with a resolution of 4000 × 2672 pixels and minimum frame-straddling time of 200 ns. The time control precision of the synchronous controller was 250 ps, to ensure that only one pulsed laser was sent within the exposure time of the CCD camera. The optical lens used in the test was a Nikon 35-mm fixed-focus lens that

7.1 Experimental Study of Liquid Jets

245

Fig. 7.2 Measurement result at x/d  120, y/d  20

could realize full-frame imaging of a physical space with dimensions 85 mm × 57 mm, with a spatial resolution of 21.4 µm/pixel.

7.1.2 The Penetration Height and Cross-Sectional Distribution of a Liquid Jet 7.1.2.1

The Penetration Height of a Liquid Jet

The liquid jet penetration height is an important parameter to evaluate the mixing characteristics of the spray and airflow. Many previous studies have been completed to obtain liquid jet penetration height and predict spay trajectory. The empirical formulas used in such studies are listed in Table 7.1. In Tong’s study, the effects of three key factors, i.e., the pressure drop of the liquid jet, the diameter of the injector, and the angle of the transverse liquid jet, on the injection characteristics of a transverse liquid jet in a cold supersonic crossflow were analyzed in detail. They found that with an increasing pressure drop the penetration height and the span expansion area of the jet increased. The penetration height was proportional to the angle between the jet and the crossflow. As for the shock wave caused by the injection of the liquid fuel, its angle went down proportionally with a decrease in the three key factors mentioned above. The total pressure drop of the

246

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Laser(532 nm and 7 ns) Gas flow Ma=2.85

Computer

Injector Test section

CCD 4000×2672pixel

Fig. 7.3 Schematic of the PIV system and experimental setup (Wu et al. 2016)

7.1 Experimental Study of Liquid Jets

247

Table 7.1 The empirical formula of jet penetration height in a supersonic crossflow Year

Authors

Empirical formula

Mach number

Method

1973

Kush and Schetz (1973)

h/d 

2.0, 4.0

Schlieren

1978

Baranovsky and Schetz (1980)

h/d  1.32q 0.5 ln(1 + 6(x/d)) sin(2θ/3)

3

Shadowgraph

2002

Lin et al. (2002a)

h/d  3.94q 0.47 (x/d)0.21

1.94

Shadowgraph

2004

Lin et al. (2004)

h/d  4.73q 0.3 (x/d)0.3

1.94

PDPA

2009

Perurena et al. (2009)

h/d  3.5q 0.3 (x/d)0.38

6

High-speed photography

2009

Ghenai et al. (2009)

h/d  3.88q 0.4 (x/d)0.2

1.5

High-speed photography

2012

Tong (2014)

2014

Li (2013)

2016

Wu (2016)

2 h/d  1.48q 0.577 (x/d)0.198   y 0.29 x 0.26 2.1 d  4.14q d  2.1 h d   2.95 − 0.85 · (γ − 0.5) ·   0.26 q 0.44 · x d ·

6q 0.49

Shadowgraph PIV Pulse background light

PDPA: Phase Doppler Particle Analyze; PIV : particle image velocimetry

crossflow decreased simultaneously with a decrease in the jet pressure, a reduction in the injector diameter, and a reduction in the injection angle. They employed a jet penetration height definition and jet flow boundary extraction method, outlined in their work, and combined multiple groups of experimental results. The empirical formulae for the penetration curve for vertical injection and the penetration height curve, considering the effect of the injection angle, are summarized below:  0.769 1 2 2 x 0.198 ρ u sin θ y l l 2  1.053   2 1 d d ρg u g − u l cos θ

(7.1)

2

To describe the oscillation of a liquid jet boundary, Wu (2016) proposed a new method to investigate the spray boundary. Figure 7.4 shows the overlay curves of the transient boundary of the jet/spray at different moments in time. It indicates that the traditional jet penetration curve changes with time, which is why it is hard to describe the process of boundary oscillation accurately.

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.4 Overlay of penetration curves at various moments (Wu et al. 2017)

y/d

248

10

0 0

10

20

30

40

50

x/d

To describe the oscillation distribution of the jet/spray, a non-dimensional parameter, the spray proportion, is defined as:  tliquid (7.2) γ  lim t→∞ t where t liquid is the sum of the time of liquid phase existence at any one position in the test section and t is the total time represented by sampling images. Spray proportion γ , being similar to intermittency factor, describes the characteristics of the spray distribution and boundary oscillation intuitively. The spray field can be divided into three regions according to γ , as shown in Fig. 7.5. γ  0 represents a non-liquid phase area, i.e., a pure gaseous region, whereas γ  1 indicates a pure liquid region in liquid phase all the time. The remainder, where γ ∈ (0, 1), represents the boundary oscillation region. Therefore, the boundary band, as a new concept for describing the liquid spray boundary, was proposed. The boundary band, composed of lines of γ , is a region

Samples：120 frames Contour

Isoline map

Pure air field

Ma=2.1

Boundary band

Pure spray field

Liquid Fig. 7.5 Spray regions separated according to spray proportion (Wu et al. 2017)

7.1 Experimental Study of Liquid Jets

249

between the gaseous and liquid regions. The traditional jet penetration boundary is considered as γ  0.

7.1.2.2

Cross Section of a Liquid Jet

According to Tong’s research (Tong 2012), span assembled injectors are favorable to increased span expansion. Compared with the single injector case, the penetration height of a liquid jet using a span assembled injector went up slightly, but the total pressure drop of the crossflow clearly increased. The streamwise assembled double injectors contributed to the increase in penetration height while cutting down the total pressure of the crossflow. Comparing these two kinds of assembled injectors, the streamwise assembled case obtained larger penetration heights while the span assembled case made the jet boundary grow wider in the tunnel. To deepen our understanding of a liquid jet, Wu adopted Stereoscopic Particle Image Velocimetry (SPIV) to image the cross section of the spray. In supersonic crossflows, factors such as turbulence, gas flow pulsation, and the strong shear action of a liquid jet caused irregular variation in the jet distribution over time and space. The spray’s cross-sectional distribution was found to have unsteady characteristics. Figure 7.6 shows the cross-sectional distribution of the spray at four random moments. It can be seen that these distributions are very different at different moments in time. Note that only the distribution of one side of the central symmetrical section (z  0 plane) is shown in the figure. In order to describe the oscillation distribution phenomenon of the jet, the spray proportion of non-dimensional parameter γ is introduced. The spray proportion γ is defined as the ratio of the time period when the space is  occupied by liquid spray to total time, and the mathematical expression is γ  tspray t. In the above formula tspray indicates the time when the spatial point is surrounded by liquid spray, and “t” the total time. It can be seen from the definition that the spray proportion at the spatial point that is always surrounded by spray is 1, and the spray proportion of gas in the mainstream region is constantly 0. To acquire the γ distribution of the cross-sectional spray field, features of n images of random cross-section distributions of the spray are extracted and processed based

Fig. 7.6 Cross-sectional distribution of the spray at four random moments in time (Wu et al. 2017)

250

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

on the Otsu method. As defined, the gray value of the area surrounded by the spray is 255 (gmax ), and the gray value of the major gas stream area is 0 (gmin ). Statistics can be applied to n image samples. The γ value of each spatial pixel can be calculated using Eq. (7.3). When n (the number of samples) is large enough, the calculated probability is exactly the value of γ . In the current analysis, the number of samples is n  150.  1 

gi lim γ  gmax n→∞ n n 1

(7.3)

Figure 7.7 shows the process and the result of calculating the γ value corresponding to a cross section. In the figure, the symmetrical distribution result is obtained using the mirror method, where the left part is the cloud of the γ distribution and the right part is the contour of the same—the corresponding γ values are 0.01, 0.1, 0.3, 0.5, 0.7, 0.9, and 0.99, respectively. The colorized cloud map of the γ distribution is composed of three features, a pure red part of γ  1, a pure blue part of γ  0, and a color gradient part of the spray proportion ranging between 0 and 1, in which the region of spray proportion γ > 0 is the region that the jet/spray might reach. The spray proportion intuitively reflects the distribution and oscillation characteristics of a liquid jet/spray. A spray proportion constant with a value of 1 shows that the pure liquid jet, or the mixing process of liquid and gas, always occurs in this region. This region is considered a constant region surrounded by spray, called the “spray core region.” A spray proportion that has a constant value of 0 shows that the liquid can never reach this region. This region is in a gaseous phase, without any mixture of gas and liquid, called the “constant gas region.” Spray proportions ranging from 0 to 1 show that the jet/spray might be within this region. The jet/spray boundary oscillates within the range of this region and is called the “spray boundary zone.” The value and spatial variation gradient of the spray proportion reflects the intensity of the jet oscillation distribution.

Fig. 7.7 Process and results of calculating the γ value corresponding to a cross section (Wu et al. 2017)

7.1 Experimental Study of Liquid Jets

251

It is worth noting that the cross-sectional γ contour plot has a distribution shaped like an omega, “.” We consider that the cross-sectional distribution consists of two parts, spray body and spray foot, where the spray foot refers to the spray part that is close to the jet wall and extends outward in the z+ direction, as shown in Fig. 7.6. According to previous studies, the jet liquid column, under the effect of the gas stream, will bend downward and form a spray (Wu et al. 2016). Meanwhile, plenty of small droplets will separate from the jet column as well as from larger droplets and liquid filaments generated by the breakup (Wu et al. 2016). These small droplets are likely to turn around and, under the effect of pressure disturbance, partly separate from the major spray, being blown to, and gathering at, the bottom wall. The shape of the spray body in the cross-sectional distribution is decided by the diffusion of the major spray as well as by its mixture with gases, while the small droplets, which come from the jet column and have been blown to the bottom wall, are the principle component of the formation of the spray foot.

Regularity of Cross-Sectional Distributions Figure 7.8 shows how the spray’s cross-sectional distribution changes with streamwise distance x, with d and q remaining unchanged. Obviously, as x increases, the spray throughout the cross section gradually radiates outward, which means that the spray’s cross-sectional area increases continuously. Furthermore, when x < 50 the variation in amplitude is larger than that when x > 50. This is because the atomization and mixture of the liquid are completed at position x  50. At that point, the spray develops downward and is further mixed with gases under shear action, slowly spreading outward (Kolpin et al. 1968). This mixing and spreading process is much milder than the gas–liquid interaction in the near field. In addition, when x is larger,

Fig. 7.8 Impact on spray cross-sectional distribution when x changes (d and q remain constant) (Wu et al. 2017)

252

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

the spray foot will expand spanwise (in the z direction) and its distance to the wall will increase. In Fig. 7.9, the effects of different q values on spray cross-sectional distribution are compared, with x and d remaining constant. The area of the cross section increases considerably as q increases, which is because both the liquid flow flux and the initial momentum of the jet along both the axial direction and the circumferential direction increase after leaving the orifice. As q increases, the spray foot tends to shrink inward along the span direction. The reason for this is that the diffusion of the jet along both the span and vertical directions, after leaving the orifice, is a continuous process. When the distance to the orifice exit is smaller, the span width of the spray will be smaller as well. At this moment, the droplets from the jet will reach the bottom wall under the effect of gas disturbance, consequently, the width of the span distribution becomes smaller. The span width of the spray foot formed by the small droplets, which have traveled farther from the orifice, is larger. Under the aerodynamic force of the crossflow, however, if the small droplets are far from the orifice they are more unlikely to be blown to the bottom wall.

Fig. 7.9 Impact on spray cross-sectional distribution when q changes (d and x remain constant) (Wu et al. 2017)

7.1 Experimental Study of Liquid Jets

253

Model for the Spray Body in the Cross Section A mathematic model of the cross-sectional distribution was built based on parameter γ . The γ contour on the cross section can be described in the form of a piecewise function. As shown in Fig. 7.10, the mathematical meaning of the segment point “S” corresponds to a peak in the γ contour in the y direction, where the coordinate is (ys , zs ). Taking the segment point as the boundary, the upper part is the spray body and the lower part the spray foot. On the same cross section, the ys values, corresponding to different γ values, change very little. To simplify the model, the mean of the ys values corresponding to different γ values is regarded as the vertical coordinate of the segment point. By making a statistical analysis of ys in different cases (using least squares fitting), the ys value can then be predicted and obtained. As can be seen from Eq. (7.4), the ys value decreases to some degree as q increases, but increases with increasing x/d. The mean squared error of Eq. (7.4) is 0.11.   0.5  ys d  0.32 · q −0.2 · x d (z/d)2 (y/d − m)2 + 1 a2 [c · (y/d − b) + b − m]2     2  z d  k y d − ys d + z s d

(7.4) (7.5) (7.6)

By taking the segment point as the boundary, the spray body of the contour is described by an egg-shaped curve, while the spray foot is described by a parabola. The coordinates of the segment point meet two functions, as indicated in Eqs. (7.5)

Fig. 7.10 Model of the spatial distribution of the spray in the cross section (Wu et al. 2017)

254

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

and (7.6). In this work, the constant coefficients of the model considering the spray body are determined by the following analysis. The geometrical properties of the egg-shaped curve, represented by Eq. (7.5), are that it is bilaterally symmetrical about the y-axis—the coordinate of the center of the egg-shaped curve is (0, m). Coefficient “a” represents the longest distance (width) that can be reached. Coefficient “b” represents the highest distance (length) that can be reached. The deformation coefficient of the curve is represented by “c.” For the special case where γ  0, “a” is the lateral extension at the streamwise position of x and “b” is the penetration height at the same position. For any γ contour line, there is an optimum set of values for a, b, c, and m that makes Eq. (7.5) a best-fit result to the real spatial distribution—see the five curves plotted in Fig. 7.10. During optimization of the fitting coefficient, Eq. (7.7) is taken as the objective function. The minimum value of which is then taken as the optimization objective, where (yi , zi ) represents the points of the γ contour line obtained according to test results, and N is the total number of points. ⎞2 ⎛ 1 ⎝ (z i /d)2 (yi /d − m)2 + − 1⎠ t(x, d, q, γ )  N a2 (c · (yi /d − b) + b − m)2

(7.7)

The constant coefficients corresponding to different γ values are obviously different. This suggests that the constant coefficients a, b, c, and m are functions of γ . As can be discovered from the above distribution regularity, a, b, c, and m are, at least, functions of x/d and q. In conclusion, when the egg-shaped curve is used to describe the distribution of γ contour lines on the cross section, there are five characteristic parameters: the deformation coefficient c, the width a, the height b, the vertical coordinate of the center m, and the vertical coordinate of the segment point ys . Among these, a, b, c, and m are all related to r, x/d, and q, while ys is only related to x/d and q.

Coefficient Models and Three-Dimensional Reconstruction In order to reconstruct the three-dimensional distribution of the spray in a supersonic crossflow, it is necessary to correctly determine how the constant coefficients a, b, c, and m of the cross-sectional distribution model change with operating parameters. Coefficient models are also required. It is known from the physical meaning of γ , as well as from the analysis above, that the optimal fitting parameters a and b, corresponding to the γ  0 contour line, traditionally present lateral extension and penetration height. A great number of studies have shown that the penetration height and the lateral extension depend mainly on x/d and q, and that they satisfy the power-law function. It can be seen from the γ contour that the variation amplitudes of a and b become larger as the absolute value of γ – 0.5 increases, with the center at γ  0.5. It is beneficial that coefficient models containing γ are established based on the penetration height

7.1 Experimental Study of Liquid Jets

255

model, to determine the quantitative relations between the constant coefficients a, b, c, and m and the operating parameters. a(x/xd, q, γ )  1.33 · q 0.21 · (x/xd)0.33 − 0.76 · q 0.33 · (x/xd)0.23 ·  1.33 · q 0.21 · (x/xd)0.33

|γ −0.5| γ −0.5

· |γ − 0.5|1.1 γ  0.5 γ  0.5

(7.8) b(x/xd, q, γ )  −0.5| · |γ − 0.5|1.1 γ  0.5 3.34 · q 0.36 · (x/xd)0.25 − 0.83 · q 0.2 · (x/xd)0.32 · |γγ −0.5 (7.9)  0.25 0.36 3.34 · q · (x/xd) γ  0.5 m(x/xd, q, γ )  −0.5| 1.93 · q 0.48 · (x/xd)0.15 − 0.24 · q 0.25 · (x/xd)0.35 · |γγ −0.5 · |γ − 0.5|1.1 γ  0.5  0.15 0.48 · (x/xd) γ  0.5 1.93 · q

     c  0.25 · x d − 1.7 · q − 5 × 10−2

(7.10) (7.11)

The coefficient models are expressed in Eqs. (7.8–7.11), where, a, b, and m are functions of q, x/d, and γ , and the deformation coefficient c is only related to q and x/d. All the coefficients in the equations are calculated using the least squares fitting method, based on the test data listed in Table 7.1. The mean squared errors of Eqs. (7.8–7.11) are 0.35, 0.44, 0.53, and 0.08, respectively. In conclusion, Eqs. (7.4), (7.6), and (7.8–7.11) constitute the model of the threedimensional spatial distribution of the spray. When γ  0, the model describes the outer three-dimensional boundary of the spray, with no liquid outside the boundary. When γ  1, the model describes the inner three-dimensional boundary, and the area inside the boundary always covered by the spray. When 0 < γ < 1, the model describes the unsteady oscillation of the spray, namely, the real-time transient spray boundary is oscillating between the inner boundary and the outer boundary. The value of γ can be used to describe, quantitatively, such an unsteady oscillation. The model presented in this chapter was used to predict the oscillating distribution of a cylindrical water jet, vertically injected into a Mach 2.1 crossflow. The results are shown in Fig. 7.8. In a verification test, the nozzle diameter “d” was 1 and “q” was 2.147. Figure 7.8 shows that the spray distribution calculated by the model fits well with test results. The mean squared errors of jet penetration and the boundary of the spray core predicted by the model are 0.29 and 0.18, respectively. The maximum relative error of the two models did not exceed 10%.

7.1.3 Spray Droplet Size and Velocity Distribution Liquid jet injection and breakup in a supersonic crossflow, determines combustion performance and many other processes, playing a fundamental role in scramjets. This topic was studied quite extensively in the 1970s and 1980s in literature regarding

256

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

the development of ramjet and scramjet combustors for aerospace plane applications (Maddalena et al. 2006; Grossman et al. 2008; Schetz et al. 2010; Hewitt and Schetz 2015). Relative studies mainly focused on the breakup process and the mechanisms of jets, the spatial distribution of the spray plume, droplets with different sizes, and the velocity distribution of droplets in a flow. Due to the extreme environment, optical imaging methods were regarded as the most efficient methods to visualize macrostructures in a spray field. Direct camera techniques and schlieren/shadowgraph methods were primarily applied (Sallam et al. 2004, 2006) to monitor liquid jet injection and breakup in a supersonic crossflow. These studies improved various empirical correlations for spray penetration using power-law (Eslamian et al. 2014; Bolszo et al. 2014; Lin et al. 2010; Wu et al. 1997), exponential (Chen et al. 1993), and logarithmic forms (Becker and Hassa 2002). As scramjet technology developed, there became an increased demand to acquire detailed information in test chambers, for example, information on droplet distributions. However, this was quite difficult for investigators to achieve in the 1970s and 1980s as technological limits existed at the time. As a result of the huge development of optical measurement technology, Nejad and Schetz (1983) and Schetz et al. (2010) applied a diffractively scattered light method to measure droplet size within a liquid plume in a supersonic crossflow. Their work represented the beginning of subsequent droplet size measurement research. PDAs were regarded the most efficient method to access both the size and velocity information of droplets in flowfields. They have been widely used to study the structure of sprays to improve atomizer design, and explore the structure of liquid jets in subsonic crossflows (Eslamian et al. 2014; Wu et al. 1997; Mashayek et al. 2008, 2011; Fuller et al. 2000). However, PDAs have not been used to explore liquid jets in supersonic crossflows other than in the work of Lin et al. (2002a) and Masutti et al. (2009). Experiments were carried out in a two-dimensional supersonic wind tunnel with a designed Mach number of 1.86. The facility and Cartesian coordinate system were the same as specified by Wang et al. (2014). The experimental parameters of a supersonic stream and liquid jet are listed in Table 7.2. The Dantec three-component PDA, shown in Fig. 7.11 (left), along with a 5-W argon iron laser and a P80 processor, were used in experiments. The clear aperture and the focal length of the transmitter were 112 and 500 mm, respectively. The initial beam diameter was 4.0 mm and the beam separation was 70 mm. The waist diameters of the probe volume for wavelengths of 488 and 515 nm were 64.4 and 67.9 µm respectively. The receiver had a clear aperture diameter of 112 mm and a collecting lens with a focal length of 500 mm. The collected light passed through a 50-µm slit to further reduce probe volume. Light was collected at 45° from the transmitter (secondorder refractive scattering). Droplet sizes in the symmetry plane were measured. In a spray plume zone with high number density, droplet properties were extracted from at least 2000 droplets at each location to reduce experimental uncertainty. Measurements were taken from 30 injector diameters downstream to avoid nonspherical particles. In practice, all the spherical validations at each measuring point remained above 95% in these experiments.

7.1 Experimental Study of Liquid Jets

257

Table 7.2 Experimental parameters of a supersonic stream and liquid jet Parameter number

Supersonic crossflow

1.

Mach number

1.86

Water jet Nozzle diameter (mm)

1.0

2.

Mass flow rate (kg/s)

2.65

Ratio of length to diameter

2

3.

Stagnation temperature (K)

300

Injecting pressure (MPa)

2

4.

Stagnation pressure (MPa)

1.128

Mass flow rate (g/s)

32

5.

Velocity (m/s)

496

Momentum flux (kg/(m s2 ))

2.61 × 106

6.

Momentum flux (kg/(m s2 ))

5.45 × 105

–

Fig. 7.11 Experimental setup: wind tunnel and PDA system (left); distribution of measuring points (right) (Wu et al. 2015)

Measurements were performed at several discrete points in the symmetry plane, shown in Fig. 7.11 (right). Measurement points were located in four streamwise lines (x/d  30, 60, 90, and 120) starting from y  9 mm and increasing at a y of 2 mm in the y+ direction until the liquid volume flux fell below 0.01 cc/s per cm2 for each streamwise line. PDAs kept a complete record of the arrival time and physical diameters of all droplets passing through the measured volume. Figure 7.12 (left) shows the results at point x/d  30, y/d  9. In this figure, the letter “D” represents the physical diameters of droplets. The word “Count” indicates the number of droplets corresponding to various diameters. Figures with variform shapes show different droplet size distributions at different locations. The arithmetic average diameter (D10 ) and Sauter mean diameter (SMD) were calculated from the experimental data. The mean diameter distribution of all measuring points is shown in Fig. 7.12 (right). Figure 7.13 shows the distribution of D10 and SMD in the flowfield, where the y-axis is transformed non-dimensionally by dividing by the local penetration height for every streamwise line. As shown in Fig. 7.13, D10 is concentrated between 7 and

258

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.12 The raw results from a PDA at location x/d  30, y/d  19 (left); diameter distribution of all measuring points (right) (Wu et al. 2015) Fig. 7.13 D10 and the SMD distribution in the transverse direction (Wu et al. 2015)

17 µm and peaks at locations near to the wall. D10 indicates a large → small → slightly large pattern in the y direction. The distribution of SMD shows a similar characteristic to D10, giving maximum values close to the wall. At x/d  30, the SMD indicates an evident large → small → large pattern and shows a “C” shape in the y+ direction. In the leeward area, near the wall, there is a relatively weak gas–liquid interaction, which results in the location of some droplets with larger diameters. With an increase in y/h, as a result of gas–liquid interaction enhancement, droplet diameters present a correspondingly decreasing trend. How-

7.1 Experimental Study of Liquid Jets

259

ever, compared with small droplets, a few large ones can penetrate deeper into a supersonic crossflow because of their greater momentum. With a further increase of y/h to the boundary (the maximum is 1, corresponding to the location of the edge of the spray), the SMD increases rather than decreases. As a result, the SMD distribution takes the form of a “C” shape at x/d  30 location. With downstream movement, SMD decreases slightly with an increase in y/h, and its distribution gradually changes from a “C” shape to an “I” shape. This phenomenon can be explained through further atomization of large droplets close to the edge of the spray. The droplets at the edge of the spray are adjacent to the supersonic mainstream, where the gas–liquid interaction is more intense compared with that at other y/h locations. These droplets break into smaller droplets with downstream movement. Therefore, at the x/d  60 location, the distribution of SMD shows a slight “C” shape, decreasing along the y+ axis gradually until reaching the valley, and then increasing suddenly at the boundary. However, the increasing amplitude is smaller than at the x/d  30 location. On the x/d  90 and x/d  120 lines, the SMD generally remains unchanged in the transverse direction, taking the form of a curve with an “I” shape, always containing droplets with uniform diameters. The SMD decreases significantly with the development of atomization in the 30 < x/d < 60 region. In the x/d > 60 region, the SMD becomes constant. These results indicate that the primary atomization of droplets is complete before x/d  60. To further explain the profile of the mean diameters, all physical droplet diameters are statistically analyzed at each measuring point. Each measuring point has the same sample size of 2000. As illustrated in Fig. 7.12 (right), the droplet distribution along the streamwise direction, at various positions (y  9, y  11, y  13, y  15, y  17, y  19, and spray boundary), was considered. Since the profiles were very similar, only the statistic results of four positions on the boundary are illustrated in detail in Fig. 7.14. At point (x/d, y/d)  (30, 19), there are more small droplets with diameters smaller than 10 µm and more large droplets with diameters larger than 80 µm compared with the other three points. The percentage of the defined large and small droplets both decreased along the streamwise direction.

Fig. 7.14 Statistical results of the four measuring points at the edge of the spray (“vol%” means the cumulative percentage of droplet volume; “Num%” means the cumulative percentage of droplet number) (Wu et al. 2015)

260

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

An assumption—the existence of breakup and coalescence—is proposed to explain the variation in the droplet distributions. As the liquid jet is injected to a supersonic crossflow from a circular hole, a great number of small droplets with D < 10 µm are peeled off due to the stronger gas–liquid interaction in the near-nozzle region compared with other regions, which is reflected in the greater proportion of small droplets with D < 10 at x/d  30. At the same time, the jet column fractures and breaks up into a number of large droplets. Because the second breakup has not been completed in the x/d < 30 field, there are more large droplets at x/d  30. The SMD decreases rapidly because these large droplets break into smaller droplets with downstream movement. Accompanying the breakup of large droplets, small droplets are continually accelerated, driven by supersonic air. The relative velocity between the gas and small droplets decreases, and the Weber number (We) becomes lower. These small droplets coalesce with one another because of collisions and surface tension. This coalescence leads to a decrease in the proportion of small droplets with D < 10, seen at measuring points x/d  60, 90, and 120. Nevertheless, another possible explanation is that small droplets move from the edge of the spray into inner regions. However, this can be discarded because of the following analysis. Droplet size data for every x/d line were combined and plotted in Fig. 7.15. The quantity percentages of droplets shows that along the x+ axis, the number of small droplets decreases. The cumulative percentage curves of droplet numbers also show this phenomenon. In addition, the cumulative percentage curves of volume indicate that the mass of the large droplets with D > 80 µm decreased with downstream movement as well. This all highlights that the decrease of small and large droplets along the streamwise direction is a global, not a local, phenomenon. As a result, the assumption that the movement of droplets influences the measured results can be excluded. So, it can be concluded that the breakup of large droplets and the coalescence of small droplets continuously occurs with downstream movement in a supersonic crossflow. It should be noted that as x/d increases from 90 to 120,

Fig. 7.15 Statistical results of droplets at the same x/d (Q_P denotes the percentage of the droplets in a certain diameter range) (Wu et al. 2015)

7.1 Experimental Study of Liquid Jets

261

the characteristics discussed above are not obvious. This may be because secondary breakup has been completed at x/d  90, where the breakup of large droplets and the coalescence of small droplets achieved a balance. Based on the former discussion, two conclusions can be reached, as follows. First, with downstream movement, the SMD distribution in the y direction gradually shows an “I”-shaped curve, converted from a “C”-shaped one. For “C” type, the value near the periphery of the spray is greater due to a small number of large droplets. Fewer small droplets lead to larger droplet sizes near the wall. For an “I”-type curve, the atomization results in more uniform diameters. Second, the breakup of large droplets and the coalescence of small droplets continuously occurs throughout the whole atomization process in a supersonic crossflow. The variation of droplet distributions can be explained perfectly by the phenomenon of breakup and coalescence. Figure 7.16 shows the average velocity contour based on instantaneous images but excluding the red part, which is artificially added to represent the mainstream with a velocity of 530 m/s. The average velocity represents the arithmetic average of 100 results of transient velocities. A water jet is vertically injected into a supersonic crossflow at x/d  0. Due to a strong momentum exchange between the gas and water, the spray is accelerated to about 350 m/s over a short distance (x/d < 15), equal to 66% of the mainstream velocity. The velocity maintains growth along the x direction (x/d > 15), though at a relatively slow rate. The contour presents a stratification, that is, the upper velocity close to the mainstream is significantly higher than the lower near the wall. The stratification is gradually weakened with the downstream movement of liquid. The spray is divided into three regions. A high-speed region directly contacts the mainstream, where the gas–liquid interaction is the strongest, and the water can be easily and significantly accelerated. Numerous small droplets are produced because of vortex shedding and breakdown in this region. Due to the quick response behavior of small droplets, the spray velocity in the high-speed region is maximal. Conversely, the spray in the low-speed region near the wall cannot contact with the mainstream due to the fact it is surrounded by spray. Thus, the growth rate of velocity is relatively small, and the value is minimal. A transition region exists between the high-speed region and the low-speed region. This region is characterized by a large velocity gradient.

Fig. 7.16 Average velocity contours of the spray (Wang et al. 2014)

262

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.17 Velocity contours of U and V . a U is the velocity component of the average velocity in the x direction. b V is the velocity component of the average velocity in the y direction (Wang et al. 2014)

Figure 7.17 presents U and V velocity contours. U is the velocity component of the average velocity in the x direction and V is the component in the y direction. It is shown that U is much larger than V . The distribution of U is similar to that of the average velocity. The peripheral spray velocity U is maximal, corresponding to the high-speed region, while the innermost spray velocity U is minimal, corresponding to the low-speed region. It is observed from the velocity vectors of U that the streamwise velocity increases in the y+ direction. The velocity U grows relatively slowly in the high-/low-speed region while in the transition region it does not—the velocity gradient in the transition region is much larger than in other regions. Due to the strong gas–liquid interaction, U increases rapidly in the short distance x/d < 15, immediately after water leaves the orifice. However, in the far downstream (x/d > 15), the growth of U tends to be slow. This is because the supersonic flow continues to accelerate the spray from the periphery to inner areas. The contour of V indicates that a high-velocity spray concentrates in the nearperiphery area, since the jet itself has a large momentum along the y direction, just after injecting from the nozzle. Despite the strong impact and compression applied by the gas, high-momentum droplets can still penetrate deeper into the mainstream. The darker area of Fig. 7.17b displays larger values of V . The spray with maximum V appears in the near-periphery area. However, the periphery of the spray, contacting with the mainstream, has small V values. This is due to an intense gas–liquid momentum exchange resulting in the depletion of momentum in the y direction. As the spray moves downstream, the maximum velocity V region maintains its close

7.1 Experimental Study of Liquid Jets

263

proximity to the upper position. In addition, V decreases while the distribution tends to be uniform. Basically, all the V velocities are positive—considered to be the main factor influencing penetration. To conclude, the instantaneous structures and velocity distribution of the spray in a supersonic crossflow was studied using the PIV technique. It was found that surface waves saliently exist in the near-injector region. With the growth of surface waves, the liquid jet finally breaks up at troughs, establishing spray formations with particular shapes. The far-field spray showed lots of span vortex structures, completely different from structures in the near-injector region. The spray velocity increased rapidly in the vicinity of the orifice, mainly because of intense gas–liquid interaction, whereas, the velocity increased slowly in the far field where the mainstream gradually accelerated the spray.

7.1.4 The Liquid-Trailing Phenomenon of a Jet Spray Although existing experimental work has made explicit progress, an in-depth understanding of the physics of the two-phase flow in the spray plume is still challenging due to the complexity of its flowfield. To our knowledge, little work has been carried out to explain the gas–liquid interaction process of a liquid jet in a supersonic crossflow. In our supersonic low-noise wind tunnel (Zhao et al. 2016), we obtained a series of experimental images with a pulse laser background light, as shown in Fig. 7.18. The detailed near-filed structure of the liquid jets was clearly observed. Subsequently, using a planar laser scattering (PLS) system, we observed very detailed flowfield characteristics of a jet spray in a supersonic crossflow, as shown in Fig. 7.19. As shown in this figure, under the strong influence of a supersonic airflow, the liquid jet rapidly breaks up into small droplets, and there appears some ribbon-shaped spray structure on the leeward side of the jet plume. This is known as a liquid-trailing phenomenon.

Fig. 7.18 The near-filed structure of liquid jets with a pulse laser background light (Wu 2016)

264

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.19 The liquid-trailing phenomenon of jet spray observed using PLS (Li et al. 2017)

The liquid-trailing phenomenon is consistent with the experimental results of Perurena et al. (2009). They investigated a liquid jet in a Mach 6 hypersonic crossflow and analyzed the impact of the injection shape and injection momentum ratio on the penetration height and lateral extension. As shown in Fig. 7.20, deflection of the jet, due to the high-pressure distribution on its surface above the injection point, is noticeable. The liquid jet was clearly diverted from a vertical direction to a horizontal direction. At about 10 injector diameters downstream of the injection, the vertical component decreased to negligible values. The fundamental driving force of the liquid-trailing phenomenon and the mixing process between droplets and freestream air comes from the gas–liquid interaction. Therefore, it is important to study the gas–liquid interaction process to explain experimental observations and to understand the essence of the atomization and mixing process in a two-phase flowfield. Since the experimental data is very limited, the following section will reveal the essential mechanism behind the liquid-trailing phenomenon through numerical simulation.

Fig. 7.20 Stream traces of a liquid jet, after ensemble-PIV analysis (Perurena et al. 2009)

7.2 Simulation and Analysis

265

7.2 Simulation and Analysis 7.2.1 Mathematical Models 7.2.1.1

Gas Phase Governing Equations

The Favre-filtered compressible Navier–Stokes conservation equations of continuity, momentum, and total energy (with a contribution of the dispersed phase included) for the gas phase, can be written as: ∂ ρ¯ ∂(ρ¯ u˜ i ) + 0 ∂t ∂ xi

  sgs ¯ i j − τ¯i j + τi j ∂ ρ¯ u˜ i u˜ j + pδ

∂(ρ¯ u˜ i )  F˜˙s,i + ∂t ∂x j 

 ˜ + p¯ u˜ i + q¯i − u˜ j τ¯ ji + Hisgs + σisgs ∂ ρ ¯ E ˜ j ∂ ρ¯ E +  Q˜˙ s ∂t ∂ xi

(7.12)

(7.13)

(7.14)

where ρ¯ is the filtered density, u˜ i is the filtered velocity, E˜ is the filtered total energy, p¯ is the filtered pressure, τ¯i j is the filtered laminar viscous stress tensor, and q¯i is the energy flux caused by heat conduction and mass diffusion. Here, “–” denotes spatial filtering and “~” denotes filtering. Favre filtering of flow parameter f can be  Favre sgs sgs ¯ τi j is the unclosed (sub-grid scale) stress tensor, Hi is defined by f˜  ρ f ρ. sgs the unclosed enthalpy flux vector, and σi j is the unclosed heat flux vector. F˜˙s,i and Q˜˙ are respectively the momentum and energy source terms caused by the effect of s

droplet motion. The sub-grid scale terms in the Favre-filtered equations require closure by establishing a turbulence model. Due to the high mesh resolution required, it is difficult for a large-eddy simulation (LES) to accurately simulate the flows in the nearwall region at high Reynolds numbers. So, a hybrid Reynolds-averaged Navier–Stokes (RANS)/LES method, blending the Spalart–Allmaras (S–A) RANS model and Yoshizawa sub-grid scale (SGS) model, is adopted—the former used in near-wall regions, the latter in regions away from the solid wall. The fifth-order WENO (weighed essentially non-oscillation) scheme is adopted for the inviscid fluxes, and viscous fluxes are calculated using the fourth-order central difference scheme. The time integration is performed by means of a third-order accuracy total-variation-diminishing (TVD) Runge–Kutta method. The CFL (Courant–Friedrichs–Lewy) number is fixed at 0.5. Notably, the numerical methods and models of a gas phase integrated into the present two-phase code have been validated through early simulation studies of gas phase jet mixing in a supersonic crossflow, supersonic cavity flow, supersonic jet combustion, and other supersonic flows or combustion problems.

266

7.2.1.2

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Liquid Phase Equations

Governing Equations of Droplet Motion The liquid phase is represented by Lagrangian droplets, carrying diameter and velocity information, tracked by the trajectory method. Since real spray contains a very large number of droplets, tracking and solving for every droplet evolution in space and time within the gas field can be computationally expensive. Therefore, a sampling technique is employed, whereby a characteristic group of droplets is gathered into a single unit termed as a “computational droplet.” Each computational droplet represents a group of real droplets having the same diameter and velocity. For simplicity, the void fraction occupied by spray droplets is not taken into account. The interactions between droplets (collision and coalescence processes) are neglected due to the assumption of a dilute flow in the present study. The droplet motion is simulated using Basset–Boussinesq–Oseen (BBO) equations. Gravity, Coriolis force, centrifugal force, Basset force, and virtual mass effect are all neglected, only the drag force is considered in the present simulation. Under these assumptions, the Lagrangian droplet equations for position and velocity are: dXl  Ul dt   dUl ml  Dl Ug − Ul dt

(7.15) (7.16)

where Xl is the position vector of the droplet centroid, Ul is the droplet velocity vector, Ug is the gas phase velocity interpolated to droplet location, subscript l denotes the parameters of   liquid phase, and subscript g denotes the parameters of gas phase. Dl Ug − Ul is the drag force on a droplet, modeled by the drag coefficient Cd :    Dl  π dl2 Cd ρg Ug − Ul  8

(7.17)

where dl is the diameter of the droplets and ρg is the gas phase density. The drag coefficient Cd plays a significant role in the prediction of droplet trajectory. However, in most spray research, the drag coefficient is given for subsonic conditions (Jangi et al. 2015; Jones et al. 2014) or is based on the hypothesis of rigid spheres for droplets (Im et al. 2011). When droplets move in a supersonic flow, their deformation and the effect of compressibility may affect the drag coefficient significantly, so an empirical model developed by Henderson is utilized, where the drag coefficient of a rigid sphere Cds is calculated based on the local relative Mach number Md and droplet Reynolds number Red : Cds  f (Md , Red )

(7.18)

7.2 Simulation and Analysis

267

where Md is evaluated based on the relative velocity between droplets and surround  ing gas Ur  Ug − Ul  and local sonic speed of the gas phase and Red is estimated based on Ur , droplet diameter, free stream density, and viscosity: Md 

Ur , ag

Red 

ρg dl Ur μg

(7.19)

On the other hand, the droplets undergo significant flattening and are no longer spherical as soon as they enter a supersonic airflow. To quantify the effect of droplet deformation on drag coefficient, the distortion parameter Q in the TAB model (O’Rourke  and Amsden 1987) is adopted. The normalized radial droplet deformation Q  x (Cb r ) indicates the level of droplet deviation from spherical forms, and its time evolution can be described by the forced, damped linear harmonic oscillator:  2 2ρg Ug − Ul d2 Q 8σl 5μl dQ  − Q− dt 2 3ρl r 2 ρlr 3 ρlr 2 dt

(7.20)

where x denotes the radius increase of the equator from its equilibrium position, r is the droplet radius, and Cb  0.5 is determined from theoretical considerations and experimental results. The external forcing term is given by the aerodynamic droplet–gas interaction, the damping is due to liquid viscosity μl , and the restoring force is supplied by the surface tension σl . When there is a distortion Q, the effect of droplet deformation can be modeled using an effective droplet cross-sectional area  S f  πr 2 (1 − Cb Q), and a simple correction of the drag coefficient was proposed as:  Cd  Cds (1 − Cb Q)

(7.21)

In the present study, the droplet temperature is assumed to be constant without evaporation effects, so the droplet energy equation is not considered, which is a realistic assumption to make based on the experimental conditions described by Lin et al. (2004). The new position and velocity of droplets are integrated from Eqs. (7.15) and (7.16) using a fourth-order Runge–Kutta time-stepping algorithm. After obtaining the new position, an efficient droplet-locating algorithm (Chorda et al. 2002) was adopted to relocate droplets on corresponding control volume cells. A node-based linked list of droplets (Shams et al. 2010) was created to reduce the computational time consumption for searching and judgment between the two phases. The interprocessor strategy of droplet sharing and transfer (Capecelatro and Desjardins 2013) was implemented to perform parallel computing of droplets. Spray Breakup Model The process of a fuel jet breaking up is indispensable in a practical spray combustion system in order to create a large liquid surface-tovolume ratio, facilitating fast evaporation. The TAB model (O’Rourke and Amsden 1987; Trinh et al. 2007; Tanner and Weisser 1998) based on the oscillation and

268

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

deformation of a droplet and the KH–RT hybrid breakup model (Im et al. 2011) based on the linearized stability analysis of a jet column are widely used in spray research. The linearized stability process and deformation and oscillation all exist in the liquid jet when it is injected vertically into a supersonic crossflow. Therefore, in the present study, the KH model was used to calculate the droplet-stripping process, the TAB model was adopted to simulate oscillation and deformation, and after a critical breakup time, the RT breakup model and TAB model competed to simulate the secondary droplet breakup process. For the KH breakup model, from the numerical solution of a general dispersion equation, the maximum growth rate ΩKH , and its corresponding wavelength ΛKH , are related to the pertinent properties of liquid and gas by:

 0.5 0.34 + 0.38W eg1.5 ρlrp3    ΩKH (7.22) σl (1 + Oh) 1 + 1.4TP0.6    1 + 0.45Oh 0.5 1 + 0.4TP0.7 ΛKH  9.02 (7.23)

0.6 rp 1 + 0.87W eg1.67 The parameters above are defined as: ρg Ur2 rp ρl Ur2 rp ρl dl Ur W eg  , W el  , Rel  , σl σl μl √  W el , TP  Oh W eg Oh  Rel where rp is the parent droplet radius, W eg and W el are the gas and liquid Weber numbers, respectively, Rel is the liquid Reynolds number, Oh is the Ohnesorge number, and TP is the Taylor number. This model postulates that if the wavelength of the fastest growing wave meets the condition B0 ΛKH ≤ rp , there will be mass stripping from the parent droplet. The rate of change of droplet radius is given by:   rp − rc drp − (7.24) dt τKH  where rc is the child droplet radius, τKH  3.726B1 rp (ΛKH ΩKH ) is the droplet breakup time, and B1  1.73 is a constant. The liquid mass shed from the parent droplet is tracked during the computation. New droplets are produced when the shed mass reaches or exceeds 3% of the parent droplet mass, and the radius of the new droplets is assumed to be proportional to the wavelength of the fastest growing wave rc  0.61ΛKH . Existing research rarely mentions how to determine the velocity and position of child droplets after breakup. Our numerical tests found that the location and velocity of child droplets have obvious effects on the final results. Here, the deformation of the

7.2 Simulation and Analysis

269

droplet is concerned with evaluation of the location of child droplets, and a velocity redistribution strategy is proposed to estimate droplet velocity after KH breakup. In KH breakup mode, child droplets are stripped from the edge of the parent droplet, and always viewed on the leeward side (Theofanous et al. 2007; Li et al. 2004; Kim and Hermanson 2012), so it is not reasonable to locate child droplets in the center of a parent droplet. A simple expression proposed here is: Xc  X p + an f ⊥ + bn f

(7.25)

rp , b  rp (1 − Cb Q) 1 − Cb Q

(7.26)

a√

As shown in the schematic diagram in Fig. 7.21, the parent droplet deforms into a disk shape under the impact of the gas phase. Xc and Xp denote the position vectors of both the child droplet and parent droplet, respectively. a and b denote the semi-major axis and the semi-minor axis of the elliptical parent droplet. n f ⊥ is the unit vector randomly distributed in a plane normal to the relative velocity vector Ur and n f is the unit vector in the direction of Ur . From the results of high-speed photography (Theofanous et al. 2007; Li et al. 2004), it has been found that child droplets are always at downstream locations of the parent droplet, indicating that child droplets move faster than parent droplets. The resultant velocity Ul of a droplet group after breakup can be calculated by Eq. (7.16). It can be considered as a combination of two components Ul  Ujet + Udg , where Ujet is the jet exit velocity vector and Udg is the sub-component velocity vector obtained from airflow acceleration. The droplet-stripping process of a liquid jet mainly occurs in the near-injector region, where the relative velocity between the two phases is very high. Compared with the aerodynamic drag force, surface tension and viscous force can almost be negligible, so that the convex bulge generated on the surface of a parent droplet acquires a larger acceleration than that of parent droplet itself. According to Eqs. (7.16) and (7.17), droplet acceleration is inversely proportional

Fig. 7.21 Schematic diagram of child droplet location (Li et al. 2017)

270

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

to droplet diameter, so at the instant of breakup  the velocityrelationship between child and parent droplets is estimated as Uc,dg Up,dg  dp dc . Using a velocity redistribution strategy and according to the law of momentum conservation, at the moment of breakup, the sub-component velocity Udg is redistributed to child and parent droplets thus: Uc,dg  Up,dg 

N dc3 + dp3 K N dc3 + dp3

Udg , Uc  Ujet + Uc,dg

1 N dc3 + dp3 Udg , Up  Ujet + Up,dg K K N dc3 + dp3

(7.27) (7.28)

 where K  dp dc , N is the number of child droplets stripped from the parent droplet, subscript c denotes child droplet parameters, and subscript p parent droplet parameters. A critical breakup time (Pilch and Erdman 1987), empirically validated by experimental data in a liquid column breakup process, was adopted in the present model as the active boundary for the RT model:  −0.25   1 + 2.2Oh 1.6 tbRT  1.9 W eg − 12

(7.29)

If the dimensionless time after start-of-injection is greater than the critical breakup time tbRT , the RT mode is activated for each droplet. The RT breakup model also uses the fastest growing disturbances to determine when and how droplets will break up. The growth rate ΩRT and wavelength ΛRT of the most unstable surface waves are: 

  1.5 0..5 −ap ρl − ρg 2 ΩRT  √ ρl + ρg 3 3σs    −a p ρl − ρg Λ RT  2πC RT 3σl

(7.30)

(7.31)

where ap is the droplet acceleration and CRT is an adjustable constant equal to 0.3. The wavelength ΛRT is compared with the distorted droplet diameter. If the wavelength is smaller than the droplet diameter, RT waves are assumed to be growing on the surface of the droplet. The amount of time inwhich waves grow is tracked and compared with droplet breakup time τRT  Cτ ΩRT , where Cτ is a constant equal to 1. If the RT waves have been growing for a time greater than the droplet breakup time, the parent droplet breaks up into a collection of smaller droplets that have radii rc  0.5ΛRT . The TAB model was adopted to model the deformation and oscillation of the droplets and simulate the secondary breakup process competing with the RT model. The governing equation for an oscillating, distorting droplet has been given in Eq. (7.20). This model assumes that a droplet will breakup if the normalized defor-

7.2 Simulation and Analysis

271

mation of the droplet grows to Q  1. The SMD of child droplets can be determined from the conservation of droplet energy during breakup: rp 7 ρlr 3 ˙ 2  + Q rc 3 8σl

(7.32)

The Rosin–Rammler distribution is adopted for droplet sizes after breakup. The initial deformation condition for new droplets is Q n+1  Q˙ n+1  0, and its velocity is equal to the combination of the velocity of the parent and a velocity component Cb r Q˙ that is randomly distributed in a plane normal to the path of the parent droplet. Unlike the stripping process in KH mode, both RT mode and TAB mode are explosion-type breakup modes in which the droplets formed are all significantly smaller than the original droplet. So, the position vector of child droplets can be given by considering droplet deformation as Xc  Xp + φan f ⊥ , where φ ∈ [0, 1] is a random parameter. Interphase Exchange Terms The presence of interphase exchange terms accounts for two-way coupling between the continuous and dispersed phases. The gas phase governing equation is calculated using the Eulerian coordinate system, while the droplet motion equation is calculated using the Lagrange coordinate system. Droplets are not restricted to lie on Eulerian grid points where the gas phase properties are known. So, a tri-linear interpolation method is adopted to estimate the gas phase properties at an individual droplet location. First, the control volume where the droplet is located should be known, and the flowfield parameters F(x g ) of its eight grid points should be acquired. Then the interpolation function S(xl , x g ), determined by the relative positions the of droplets and the control volume is used in the following general form: F(xl ) 

8

S(xl , x g )F(x g )

(7.33)

g1

where x g is the Eulerian grid point and F is the gas phase property known at grid point x g . For example, the schematic diagram of two-dimensional data exchange between coordinates is given in Fig. 7.22. Since reaction and evaporation are not considered in the present study, the interphase source terms that appear on the right-hand side of the gas phase governing equations are the momentum source term F˜˙s,i and the energy source term Q˜˙ s . Let D f denote the number of real droplets contained in the computational droplet, then the volume-averaged source terms caused by droplet motion are computed by summing the contribution from all droplets as:

272

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.22 Schematic diagram of two-dimensional data exchange between coordinates (Li et al. 2017)

⎧ Nd

⎪   ⎪ ˜˙  − 1 ⎪ F Dl u g,i − u l,i,k D f ⎪ s,i ⎪ ⎨ V k

Nd ⎪

⎪   ⎪ ˜˙  − 1 ⎪ Dl Ug − Ul,k Ul,k D f Q ⎪ s ⎩ V k

(7.34)

where i  x, y, z, V is the volume of the control cell, the summation index k is over all droplets located in the control volume, and Nd is the total number of computational droplets in the control volume.

7.2.2 Computational Conditions The present simulation was performed to investigate the gas–liquid interaction and mixing mechanism of a liquid jet in a supersonic crossflow. To validate the code, all test conditions were based on the experimental work of Lin et al. (2004). Figure 7.23 shows a schematic diagram of the computational domain and experimental conditions. Water was used as the test liquid, with a density of 988 kg/m3 , viscosity of 2.67 × 10−3 kg/(m s), and surface tension of 0.072 N/m. The water was injected into a crossflow of Mach number M∞  1.94, the static temperature and pressure of which are 304.1 K and 29 kPa, respectively, and the freestream velocity U∞ was  2  7.0, and the 678.13 m/s. The gas–liquid momentum ratio was q0  ρl vl2 ρ∞ v∞ injection velocity v j was derived as 32.73 m/s. The detailed parameters of jet-exit and crossflow are shown in Table 7.3. Considering the computational cost, the computational domain was confined to a limited local region near the injector nozzle, comprised of a solid surface representing the flat plate with a circular hole as the injector. The computational domain was designed (L x × L y × L z  200 mm × 40 mm × 40 mm) in three dimensions. The injector nozzle, with a diameter of 0.5 mm, was located in the center of the bottom

7.2 Simulation and Analysis

273

Fig. 7.23 Schematic diagram of the computational domains and experimental conditions (Li et al. 2017) Table 7.3 Physical parameters in the experimental work of Lin et al. (2004)

Supersonic crossflow

Liquid jet-exit flow

Mach number, M∞

1.94

Gas–liquid momentum ratio, q0

7.0

Velocity, U∞

678.13 m/s

Injection velocity, v j

32.73 m/s

Static pressure, P∞

29 kPa

Water density, ρl

998 kg/m3

Static temperature, T∞

304.1 K

Injector nozzle diameter, d

0.5 mm

floor, 50 mm downstream of the leading edge. The mesh was refined near the injector nozzle and the near-wall region. The number of grid points was 481 × 201 × 201 in the x, y, and z directions, providing a grid resolution of x + ≈ 10–50, y + ≈ 1–50, and z + ≈ 10–50 basing on the wall stress τw at the inlet. This resolution may be coarse for a strict wall-resolved LES but is suitable for a hybrid RANS/LES approach. The total number of computational droplets tracked in the computational domain was approximately 1 million. For the gas phase, a no-slip, no-penetration adiabatic condition was imposed at the wall, a supersonic inflow condition was used at the inlet section, and the other boundaries were treated by the extrapolation method. For the liquid phase, the Tsuji empirical formula (Tsuji et al. 1987) was used to calculate droplet–wall interaction.

274

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

7.2.3 Characteristics of a Gas Phase Flowfield The characteristics of a gas phase flowfield and liquid phase flowfield may be the two key factors that lead to the liquid-trailing phenomenon of a jet spray. They will be analyzed in the following two sections. In a supersonic two-phase flow, the aerodynamic force provides the driving force for the breakup and the movement of droplets, so the gas phase flowfield has a critical influence on the evolution of the liquid jet. The gas phase flow pattern determines the droplet movement pattern, at the same time it is also significantly affected by the liquid jet. To study the interaction between the two phases, it is first necessary to identify the essential characteristics of the gas phase flowfield. An instantaneous snapshot of the velocity contours in the central plane is given in Fig. 7.24. Apparently, a bow shock appears in front of the liquid jet. It is observed that the airflow velocity decreases significantly after passing the liquid jet. There may be two reasons for this. One is the shock, which leads to a sharp decline in gas phase velocity. The other is the kinetic energy transfer from gas phase field to liquid phase field. Figure 7.25 shows the averaged results of the gas phase velocity normalized by the freestream velocity. Figure 7.25a gives the averaged distribution of gas velocity in the central plane at different y positions. It can be seen that gas velocity decreases rapidly after the shock. In addition, the velocity has a stronger slowdown when the position gets closer to the wall region. After that, the gas velocity rebounds a little, and then decreases rapidly due to the airstream flowing into the liquid spray region where gas–liquid momentum exchange is very strong. After the droplets obtain a large enough momentum, the gas–liquid momentum exchange becomes weaker, and then the gas velocity gradually recovers to a higher value. The second drop in gas velocity starts at the front region of the spray, and the gas velocity at the leading edge of the spray increases quickly with increasing y. Figure 7.25b gives the averaged distribution of gas phase velocity in the central plane at different x positions. The continuous decrease of gas velocity from the freestream to the liquid spray zone can be seen clearly. However, for positions x  25 mm and x  50 mm, the gas velocity acquires a local increase in the near-wall region, mainly caused by the effect of a

Fig. 7.24 Instantaneous snapshot of velocity contours in the central plane (Li et al. 2017)

7.2 Simulation and Analysis

275

Fig. 7.25 Averaged distribution of gas velocity: a along the x-axis at different y positions and b along the y-axis at different x positions (Li et al. 2017)

three-dimensional flow around the liquid jet. In the middle region of the spray, it can be clearly seen that the gas velocity increases with increasing x. To further explore the gas–liquid interaction process and understand the mixing and dispersion of the liquid spray, Fig. 7.26 shows the streamlines of the gas phase and the velocity vectors of droplets in the central plane. Because of the bow shock, the streamlines are deflected upward when they encounter the spray zone. But before long, the air stream flows obliquely downward and is deflected toward the nearwall region by the effects of droplets. The velocity vectors of the droplets show that they move obliquely upward. This result has been confirmed in experimental work (Ragucci et al. 2009; Gopala et al. 2009). In the central plane, due to the discrete

Fig. 7.26 Streamlines of the gas phase and the droplet vectors in the central plane (Li et al. 2017)

276

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

characteristics of the liquid jet after breakup, the flow streamlines may intersect with the trajectory of the droplets, which is different from the gas jet in a supersonic crossflow (Chai and Mahesh 2011; Kawai and Lele 2009). Thus, when there is higher pressure at the front region of the spray and a relatively low pressure inside the liquid spray, there will be a deflection tendency of the airflow toward the spray interior. The deflection of the airflow will in turn apply an aerodynamic force to the droplets, letting them acquire an inclined downward acceleration. This may be the first basic condition required for the liquid-trailing phenomenon.

7.2.4 Characteristics of the Spray Field The aerodynamic force acting on the droplets determines the evolution process and breakup process of the droplets, so it can be used to deepen the understanding of the mixing and distributing processes ofdroplets. The aerodynamic force acting on  2 Ug − Ul  /8, where the greatest d C ρ the droplets can be expressed as Dl  π d g  l  influencing factor is the relative velocity Ug − Ul . Therefore, this relative velocity can be used to analyze the strength of the gas–liquid interaction and reflects the potential position of breakup. Figure 7.27 shows the instantaneous distribution of the gas–liquid relative velocity in the central plane, where all droplets are colored and sized by relative velocity magnitude and the relative velocity is normalized by the freestream velocity. Results show that the droplets with a normalized relative velocity larger than 0.5 are mainly distributed in the x  5–20 mm region, mainly at the leading edge of the spray field. While in the inner region or far downstream region, the relative velocity is relatively low. Figure 7.28 shows the averaged distribution of the gas–liquid relative velocity at different locations. For a fixed position of y, the maximum relative velocity always appears at the leading edge, and these maximum values initially increase with increasing y, reaching their peak values around y  15 mm, and then decrease. From the longitudinal distribution, it can be seen that relative velocity profiles tend to be

Fig. 7.27 Instantaneous distribution of the gas–liquid relative velocity in the central plane (Li et al. 2017)

7.2 Simulation and Analysis

277

Fig. 7.28 Averaged distribution of gas–liquid relative velocity: a along the x-axis at different y positions and b along the y-axis at different x positions (Li et al. 2017)

Fig. 7.29 The instantaneous distribution of normalized droplet deformation in the central plane (Li et al. 2017)

more uniform with increasing x, indicating that the gas–liquid interaction has become weaker. A larger deformation of droplets indicates a greater possibility of breakup. Figure 7.29 shows the instantaneous distribution of normalized droplet deformation in the central plane, where all droplets are colored and sized by droplet deformation. Additionally, Fig. 7.30 shows the averaged distribution of normalized droplet deformation at different locations. It indicates that the droplets with normalized deformation Q ≥ 0.8 are mainly distributed in the x < 10-mm region. For a fixed position of y, droplet deformation declines rapidly with increasing x, and becomes stable at x > 10 mm. In the x > 10-mm region, due to the reduction of gas–liquid relative velocity, the droplet deformation decreases gradually and becomes steady under the action of viscous force and tension. This is consistent with the conclusion that the atomization process mainly occurs within the x < 50-mm region noted by Lin et al. in their experimental study (Lin et al. 2004). In addition, the droplets with larger deformation are mainly distributed at the leading edge region of the spray field,

278

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.30 Averaged distribution of normalized droplet deformation: a along the x-axis at different y positions and b along the y-axis at different x positions (Li et al. 2017)

while droplets in the inner region of the spray field have smaller deformation due to their lower relative velocity and the weaker interaction between the two phases. The results of gas–liquid relative velocity and droplet deformation show that the strongest gas–liquid interaction takes place at the front region of the spray field, as does the breakup of droplets. This forms the second basic condition required for the liquid-trailing phenomenon.

7.2.5 The Liquid-Trailing Phenomenon of a Jet Spray Based on the above analysis, we try to explain the formation process of the liquidtrailing phenomenon using numerical results. Figure 7.31b shows the streamlines of airflow and the distribution of droplets in the central plane, where all droplets are colored and sized by diameter. The experimentally observed liquid-trailing phenomenon of the jet spray in Fig. 7.31a is well captured. As shown in this figure, under the strong influence of supersonic airflow, the liquid jet rapidly breaks up into small droplets of different sizes. The small droplet groups mainly concentrate at the front region of the jet. On the other hand, the air stream flows obliquely downward when it encounters the liquid spray because of the pressure difference. The inclined downward movement of the air stream lets the droplets acquire an inclined downward acceleration. Since the droplets after breakup are different sizes, and thus have different accelerations, droplet groups shape into a series of liquid-trailing structures in the local airflow direction. To show this trailing phenomenon of droplets clearly, its mechanism is provided in Fig. 7.32. The solid arrows in the schematic diagram represent the direction of the gas phase while the dotted arrows represent the direction of droplet movement.

7.2 Simulation and Analysis Fig. 7.31 The liquid-trailing phenomenon of a jet spray revealed a experimentally and b via simulation (Li et al. 2017)

Fig. 7.32 The mechanism driving the liquid-trailing phenomenon (Li et al. 2017)

279

280

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Fig. 7.33 Evolutionary process of the liquid-trailing phenomenon (t  24 µs) (Li et al. 2017)

To explain the liquid-trailing phenomenon in detail, the movements of all child droplets from the same parent droplet are tracked and labeled with the same notation, namely as a “family number.” The distribution of child droplets with the same family number at different moments in time is shown in Fig. 7.33. In this figure there are three characteristic droplet families which are colored and sized by diameter—the background is the corresponding instantaneous spray plume in the central plane. This figure shows that many child droplets, of different size, are formed after breakup of the parent droplet. Due to the inertia effect, the child droplets still move obliquely upward. Based on the above analysis, these child droplets will acquire an extra acceleration under the influence of the local airflow—its direction is inclined downward. Since the sizes of child droplets are different, the smaller droplets with better response characteristics move faster in the local airflow direction than larger droplets. As a result, all the child droplets from the same parent droplet are arranged according to size in the direction of the local airflow, as shown in Fig. 7.33. By comparing

7.2 Simulation and Analysis

281

the droplet distributions at different moments, it can be found that the trailing angle of child droplets gradually increases, due to the fact that the larger droplets have a greater inclined upward momentum, while the inertia of the smaller droplets is relatively small, so that the smaller droplets are easier to move obliquely downward with the local airflow. Consequently, the larger droplets are mainly distributed in the outer region of the spray field, and the smaller droplets mainly locate in the inner and bottom of the spray field. However, as the group of droplets moves downstream, the larger droplets are mainly located in the outer region of the liquid spray where the gas–liquid interaction is strong. Under the influence of air stream, the larger droplets will further break up into small droplets. Whereas, the smaller droplets are mainly located in the center of the spray field where the gas–liquid interaction is weaker, so they rarely experience further breakup.

7.2.6 Conclusions This research mainly focuses on the interaction process between a liquid jet and a supersonic flow. The Eulerian–Lagrangian approach and modified breakup models were adopted to numerically study droplet deformation, breakup, and mixing in a supersonic crossflow. Significant conclusions can be summarized as follows: 1. There are strong interactions between the airflow and liquid spray. Under the influence of spray, a bow shock wave forms ahead of the jet. Airflow passes through the shock and the spray zone, so there are two decreases in the velocity of airflow. The streamlines of a supersonic flow can intersect with the trajectory of droplets and deflect to the wall after entering the spray zone in the center plane. 2. The most intense gas–liquid interaction process occurs near the front edge of a jet. Were the gas–liquid relative velocity and the droplet non-dimensional deformation both reach maximum values, represents the greatest possibility of fragmentation. The gas–liquid relative velocity decreases gradually after x > 30 mm, and the interaction between the airflow and liquid droplets weakens, resulting in a gradual decrease in liquid droplet non-dimensional deformation. 3. The simulation results successfully revealed the drawing phenomena of liquid jet breakup, discovered in previous experiments. Under the influence of airflow, the child droplets form groups after fragmentation. They are arranged by size and the direction of arrangement is similar to the local airflow in the spray area. In the process of downstream movement, the drawing angle of sub-droplet groups increases, and larger droplets located externally further break up due to severe gas–liquid interaction.

282

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

References Baranovsky SI, Schetz JA (1980) Effect of injection angle on liquid injection in supersonic flow. AIAA J 18(6):625–629 Becker J, Hassa C (2002) Breakup and atomization of kerosene jet in crossflow at elevated pressure. At Spray 12(1/2/3):49–67 Bolszo CD, McDonell VG, Gomez GA, Samuelsen GS (2014) Injection of water-in-oil emulsion jets into a subsonic crossflow: an experimental study. At Sprays 24(4):303–348 Capecelatro J, Desjardins O (2013) An Euler-Lagrange strategy for simulating particle-laden flows. J Comput Phys 238(2013):1–31 Chai XC, Mahesh K (2011) Simulations of high speed turbulent jets in crossflow. Paper presented at the 49th AIAA aerospace sciences meeting including the New Horizons Forum and Aerospace Exposition Chen T, Smith C, Schommer D, Nejad A (1993) Multi-zone behavior of transverse liquid jet in high-speed flow. Paper presented at the 31st aerospace sciences meeting & exhibit, Reno, NV Chorda R, Blasco JA, Fueyo N (2002) An efficient particle-locating algorithm for application in arbitrary 2D and 3D grids. Int J Multiph Flows 28(6):1565–1580 Eslamian M, Amighi A, Ashgriz N (2014) Atomization of liquid jet in high-pressure and hightemperature subsonic crossflow. AIAA J 52(7):1374–1385 Fuller RP, Wu PK, Kirkendall KA, Nejad AS (2000) Effects of injection angle on atomization of liquid jets in transverse airflow. AIAA J 38(1):64–72 Ghenai C, Sapmaz H, Lin CX (2009) Penetration heigh correlations for non-aerated and aerated transverse liquid jets in supersonic crossflow. Exp Fluids 46(1):121–129 Gopala Y, Oleksandr B, Lubarsky E, Zinn BT (2009) Liquid jet in crossflow-measurement of the velocity field of the near field continuous medium. Paper presented at the 47th AIAA aerospace sciences meeting Grossman PM, Schetz JA, Maddalena L (2008) Flush-wall, diamond-shaped fuel injector for high Mach number scramjets. J Propul Power 24(2):259–266 Hewitt P, Schetz JA (2015) Atomization of impinging liquid jets in a supersonic crossflow. AIAA J 21(2):178–179 Im KS, Lin KC, Lai MC, Chon MS (2011) Breakup modeling of a liquid jet in cross flow. Int J Automot Technol 12(4):489–496 Jangi M, Solsjo R, Johansson B, Bai X-S (2015) On large eddy simulation of diesel spray for internal combustion engines. Int J Heat Fluid Flow 53(2015):68–80 Jones WP, Marquis AJ, Vogiatzaki K (2014) Large-eddy simulation of spray combustion in a gas turbine combustor. Combust Flame 161(2014):222–239 Kawai S, Lele SK (2009) Dynamics and mixing of a sonic jet in a supersonic turbulent crossflow. Center for Turbulence Research Kim Y, Hermanson JC (2012) Breakup and vaporization of droplets under locally supersonic conditions. Phys Fluids 24(7):1–24 Kolpin MA, Horn KP, Reichenbach RE (1968) Study of penetration of a liquid injectant into a supersonic flow. AIAA J 6(5):853–858 Kush EA, Schetz JA (1973) Liquid jet injection into a supersonic flow. AIAA J 11(9):1223–1224 Li C (2013) Structure characteristic of transverse liquid jet in supersonic crossflow. National University of Defense Technology (Chinese) Li GJ, Dinh TN, Theofanous TG (2004) An experimental study of droplet breakup in supersonic flow: the effect of long-range interactions. Paper presented at the 42nd AIAA aerospace sciences meeting and exhibit Li P, Wang Z, Sun M, Wang H (2017) Numerical simulation of the gas-liquid interaction of a liquid jet in supersonic crossflow. Acta Astronaut 134(2017):333–344 Lin KC, Kennedy P, Jackson T (2002a) Penetration heights of liquid jets in high-speed crossflows. Paper presented at the 40th AIAA aerospace sciences meeting & exhibit, AIAA 2002-0873

References

283

Lin KC, Kennedy P, Jackson T (2004) Structures of water jets in a Mach 1.94 supersonic crossflow. Paper presented at the 42nd AIAA aerospace sciences meeting and exhibit, Reno, Nevada Lin KC, Ryan M, Carter C, Gruber M, Raffoul C (2010) Raman scattering measurements of gaseous ethylene jets in Mach 2 supersonic crossflow. J Propul Power 26(3):503–513 Maddalena L, Campioli TL, Schetz JA (2006) Experimental and computational investigation of light-gas injectors in Mach 4.0 crossflow. J Propul Power 22(5):1027–1038 Mashayek A, Jafari A, Ashgriz N (2008) Improved model for the penetration of liquid jets in subsonic crossflows. AIAA J 46(11):2674–2686 Mashayek A, Behzad M, Ashgriz N (2011) Multiple injector model for primary breakup of a liquid jet in crossflow. AIAA J 49(11):2407–2420 Masutti D, Bernhardt S, Asma CO, Vetrano MR (2009) Experimental characterization of liquid jet atomization in Mach 6 crossflow. Paper presented at the 39th AIAA fluid dynamics conference, San Antonio, Texas Nejad AS, Schetz JA (1983) Effects of viscosity and surface tension on a jet plume in supersonic crossflow. AIAA J 22(4):458–459 O’Rourke PJ, Amsden AA (1987) The Tab method for numerical calculation of spray droplet breakup. Paper presented at the international fuels and lubricants meeting and exposition Perurena JB, Asma CO, Theunissen R, Chazot O (2009) Experimental investigation of liquid jet injection into Mach 6 hypersonic crossflow. Exp Fluids 46(3):403–417 Pilch M, Erdman CA (1987) Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop. Int J Multiph Flow 13:741–757 Ragucci R, Picarelli A, Sorrentino G, Martino Pd (2009) Validation of droplets behavior model by means of PIV measurements in a cross-flow atomizing system. Paper presented at the 32nd meeting on combustion in Italian Section of the Combustion Institute Sallam KA, Aalburg C, Faeth GM (2004) Breakup of round nonturbulent liquid jets in gaseous crossflow. AIAA J 42(12):2529–2540 Sallam KA, Ng C L, Sankarakrishnan R (2006) Breakup of turbulent and non-turbulent liquid jets in gaseous crossflows. Paper presented at the 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada Schetz JA, Maddalena L, Burger SK (2010) Molecular weight and shock-wave effects on transverse injection in supersonic flow. J Propul Power 26(5):1102–1113 Shams E, Finn J, Apte SV (2010) A numerical scheme for Euler-Lagrange simulation of bubbly flows in complex systems. Int J Numer Meth Fluids 1:1–41 Tanner FX, Weisser G (1998) Simulation of liquid jet atomization for fuel sprays by means of a cascade drop breakup model. Paper presented at the international congress and exposition Theofanous TG, Li GJ, Dinh TN, Chang C-H (2007) Aerobreakup in disturbed subsonic and supersonic flow fields. J Fluid Mech 593(2007):131–170 Tong Y (2012) Breakup process and injection characteristic of transverse liquid jet in crossflow. National University of Defense Technology (Chinese) Tong Y (2014) Experimental investigation on injection characteristics of assembled transverse injectors in supersonic crossflow. J Natl Univ Def Technol 36(2):73–80 Trinh HP, Chen CP, Balasubramanyam MS (2007) Numerical simulation of liquid jet atomization including turbulence effects. J Eng Gas Turbines Power 129:920–928 Tsuji Y, Morikawa Y, Tanaka T (1987) Numerical simulation of gas-liquid two-phase flow in a two-dimensional horizontal channel. Int J Multiph Flow 13(5):671–684 Wang Z, Wu L, Li Q, Li C (2014) Experimental investigation on structures and velocity of liquid jets in a supersonic crossflow. Appl Phys Lett 105(13):4 Wu L (2016) Breakup and atomization mechanism of liquid jet in supersonic crossflows. National University of Defense Technology (Chinese) Wu PK, Kirkendall KA, Fuller RP (1997) Breakup processes of liquid jets in subsonic crossflows. J Propul Power 13(1):64–73

284

7 Spray Characteristics of a Liquid Jet in a Supersonic Crossflow

Wu L, Wang Z, Li Q, Zhang J (2015) Investigations on the droplet distributions in the atomization of kerosene jets in supersonic crossflows. Appl Phys Lett 107(10):104103 Wu L, Wang Z, Li Q, Li C (2016) Study on transient structure characteristics of round liquid jet in supersonic crossflows. J Vis, 1–5 Wu L, Chang Y, Zhang K, Li Q, Li C (2017) Model for three-dimensional distribution of liquid fuel in supersonic crossflows. Paper presented at the 21st AIAA international space planes and hypersonic systems and technologies conference, Xiamen, China Zhao YH, Liang JH, Zhao YX (2016) Non-reacting flow visualization of supersonic combustor based on cavity and cavity-strut flameholder. Acta Astronaut 121(2016):282–291