UNSTEADY FORCES ON A PARTICLE IN COMPRESSIBLE FLOWS

By

MANOJ KUMAR PARMAR

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

c⃝ 2010 Manoj Kumar Parmar

2

idam hi pumsas tapasah srutasya v�a

svistasya s�uktasya ca buddhi-dattayoh

avicyuto ′rthah kavibhir nir�upito

yad-uttamasloka-gun�anuvarnanam

”Learned circles have positively concluded that the infallible purpose of the

advancement of knowledge, namely austerities, study of the Vedas, sacrifice, chanting

of hymns and charity, culminates in the transcendental descriptions of the Lord, who is

defined in choice poetry.”

All paraphernalia of the cosmic universe is but an emanation from the Lord out of His

different energies because the Lord has set in motion, by His inconceivable energy, the

actions and reactions of the created manifestation. They have come to be out of His

energy, they rest on His energy, and after annihilation they merge into Him. Nothing is,

therefore, different from Him, but at the same time the Lord is always different from

them. When advancement of knowledge is applied in the service of the Lord, the whole

process becomes absolute. Therefore, all the sages and devotees of the Lord have

recommended that the subject matter of art, science, philosophy, physics, chemistry,

psychology and all other branches of knowledge should be wholly and solely applied in

the service of the Lord.

-Srimad Bhagavatam, canto 1, chapter 5, text 22. Source:

http://vedabase.net/sb/1/5/22/en.

Thus, with the permission of my preceptors I dedicate this work to original scientist, the

supreme personality of Godhead Sri Krishna.

3

ACKNOWLEDGMENTS

I am very grateful to my advisors, Profs. Haselbacher and Balachandar, for their

continued support, guidance, and encouragement.

I would like to thank Profs. Mei, Klausner, and Curtis, for serving on my thesis

committee. In particular, I would like to acknowledge helpful discussions with Prof. Mei.

I would also like to extend my thanks to labmates in Computational Multiphysics

Group. Particularly, I would like to acknowledge Hyungoo Lee, Jungwoo Kim, Thomas

Bonometti, Yoshifumi Nozaki, Yue Ling, and Subramanian Annamalai for helpful

discussions and their help during my PhD work. I also acknowledge Rajeev Jaiman

for his support and inspiration during my PhD.

Thanks are also due to my friends for their constant support, encouragement, and

help in so many ways during my PhD work. I thank my brother, Ravi, for his continuous

care and support. Last but not the least, I thank my wife Sona, for her love, care, and

pleasant company, for being very patient and understanding during my PhD work. I

thank my parents for everything they have done for me.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Particle Equation of Motion in Incompressible Flows . . . . . . . . . . . . 17

1.2.1 Creeping Motion in a Quiescent Fluid . . . . . . . . . . . . . . . . . 181.2.2 Unsteady Ambient Flow . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 Non-Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.4 Non-Linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.5 Force Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Particle Equation of Motion in Compressible Flows . . . . . . . . . . . . . 231.3.1 Quasi-Steady Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.2 Inviscid-Unsteady Force . . . . . . . . . . . . . . . . . . . . . . . . 241.3.3 Viscous-Unsteady Force . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4 Goal of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5 Dissertation Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 GENERALIZED BASSET-BOUSSINESQ-OSEEN EQUATION FOR UNSTEADYFORCES ON A SPHERE IN A COMPRESSIBLE FLOW . . . . . . . . . . . . . 30

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Solution for Impulsive Motion . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Compressibility Effect on Inviscid Unsteady Force . . . . . . . . . . . . . 342.5 Asymptotic Behaviors of Compressible Viscous Unsteady Force . . . . . 362.6 Numerical Evaluation of Viscous Unsteady Force . . . . . . . . . . . . . . 392.7 Inviscid and Viscous Unsteady Force Kernels and Numerical Confirmation 422.8 Generalization of the BBO Equation to Compressible Flows . . . . . . . . 452.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 EQUATION OF MOTION FOR A SPHERE IN NON-UNIFORM COMPRESSIBLEFLOWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Governing Equations for Flow Around a Moving Particle . . . . . . . . . . 513.3 Moving Reference Frame and Separation of Disturbance Flow . . . . . . 52

5

3.4 Scaling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Density-Weighted Velocity Transformation . . . . . . . . . . . . . . . . . . 563.6 Hydrodynamic Force due to Undisturbed Flow . . . . . . . . . . . . . . . . 573.7 Reciprocal Theorem for Compressible Perturbation Flow . . . . . . . . . . 583.8 Hydrodynamic Force due to the Disturbance flow . . . . . . . . . . . . . . 61

3.8.1 Importance of the Different Terms . . . . . . . . . . . . . . . . . . . 653.8.2 Inviscid and Viscous Kernels . . . . . . . . . . . . . . . . . . . . . 66

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 ON THE UNSTEADY INVISCID FORCE ON CYLINDERS AND SPHERESIN SUBCRITICAL COMPRESSIBLE FLOWS . . . . . . . . . . . . . . . . . . . 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Effect of Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1.1 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.1.2 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.2 Mach-Number Expansion . . . . . . . . . . . . . . . . . . . . . . . 914.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 MODELING OF THE UNSTEADY FORCE FOR SHOCK-PARTICLE INTERACTION101

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.1 Force Parameterization . . . . . . . . . . . . . . . . . . . . . . . . 1055.2.2 Importance of Inviscid Unsteady Contribution . . . . . . . . . . . . 1085.2.3 Effect of Finite Mach Number . . . . . . . . . . . . . . . . . . . . . 1125.2.4 Approximation of Ambient Flow . . . . . . . . . . . . . . . . . . . . 113

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.1 Stationary Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3.1.1 Experiments of Sun et al. . . . . . . . . . . . . . . . . . . 1185.3.1.2 Experiments of Skews et al. . . . . . . . . . . . . . . . . 124

5.3.2 Moving Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6 AN IMPROVED DRAG CORRELATION FOR SPHERES AND APPLICATIONTO SHOCK-TUBE EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . 134

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Improved Drag-Coefficient Correlation . . . . . . . . . . . . . . . . . . . . 1356.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6

7 UNSTEADY FORCES ON A PARTICLE IN VISCOUS COMPRESSIBLE FLOWSAT FINITE MACH AND REYNOLDS NUMBERS . . . . . . . . . . . . . . . . . 147

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.3 Unsteady Forces Over a Sphere at Small Mach Numbers and Finite Reynolds

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.4 Unsteady Forces Over a Sphere at Sub-Critical Mach Numbers and Finite

Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8 SUMMARY, CONCLUSIONS, AND FUTURE WORK . . . . . . . . . . . . . . . 161

8.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

APPENDIX: NUMERICAL METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . 167

A.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168A.1.1 Dimensional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 168A.1.2 Non-Dimensional Form . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.2 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.2.1 Dissipative Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.2.2 Non-Dissipative Solver . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.3 Discretization of Non-Dissipative Solver . . . . . . . . . . . . . . . . . . . 172A.3.1 Notation and Variable Arrangement . . . . . . . . . . . . . . . . . . 172A.3.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 172A.3.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 174A.3.4 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

A.4 Solution Algorithm for Non-Dissipative Solver . . . . . . . . . . . . . . . . 178A.4.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 179A.4.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 180A.4.3 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184A.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

A.5 Boundary Condition Implementation for Non-Dissipative Solver . . . . . . 190A.5.1 Solid Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190A.5.2 Farfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

A.6 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 191A.6.1 Characteristic Boundary Conditions . . . . . . . . . . . . . . . . . . 192A.6.2 Sponge Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

A.7 Moving Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 193A.7.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193A.7.2 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 193A.7.3 Transformation of the Euler Equations to Moving Reference Frame 193

A.8 Axisymmetric Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 196A.8.1 Euler Equations for Axisymmetric Flows . . . . . . . . . . . . . . . 196

7

A.8.2 Volumetric and Surface Integration . . . . . . . . . . . . . . . . . . 197

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8

LIST OF TABLES

Table page

6-1 Summary of selected test cases taken from Jourdan et al. [50] used to comparewith model. The column labeled ‘gases’ lists the gases in the driver and drivensections of the shock tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A-1 Wall boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

A-2 Farfield boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9

LIST OF FIGURES

Figure page

1-1 Examples of compressible multiphase flows. In the space shuttle, multiphaseflow arises due to the solid-propellant rocket motors because the propellant isenriched with aluminum particles. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1-2 State-of-the-art for the forces on a particle. . . . . . . . . . . . . . . . . . . . . 26

2-1 The behavior of the correction function C(τ) that accounts for the compressibilityeffect on the viscous unsteady force (see Eq. (2–22)). Results are plotted forµb = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}. The open circles represent thecurve-fit given by Eq. (2–27) evaluated for Kn′ = 10−5. . . . . . . . . . . . . . . 41

2-2 The dependence of the correction function C(τ) on the bulk viscosity. Resultsare plotted for µb/µ = {0, 1, 59/12, 10} and Kn′ = {10−2, 10−8}. . . . . . . . . . 42

2-3 Time evolution of the normalized unsteady force. Theoretical predictions (lasttwo terms of Eq. (2–23)) are plotted as solid lines for Kn′ = {10−2, 10−3, 10−4}and µb = 0. Inviscid unsteady kernel (second last term in Eq. (2–23)) is shownas dashes line. Corresponding simulation results for four different cases areshown as symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-4 τlow and τupp for µb/µ = {0, 1, 59/12, 10}. . . . . . . . . . . . . . . . . . . . . . . 45

3-1 The behavior of the correction function C v(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteady force,see Eq. (3–72), for µb/µ = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}. . . . . . . . . 67

3-2 The behavior of the correction function C s(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteady force,see Eq. (3–72), for µb/µ = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}. . . . . . . . . 68

3-3 The behavior of the correction function C v(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteady force,see Eq. (3–72), for µb/µ = {0, 1, 59/12, 10} and Kn′ = {10−2, 10−5}. . . . . . . 69

3-4 The behavior of the correction function C s(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteady force,see Eq. (3–72), for µb/µ = {0, 59/12, 10} and Kn′ = {10−2}. . . . . . . . . . . . 70

4-1 Schematic depiction of variation of freestream Mach number during computations. 84

4-2 Effect of acceleration parameter α on the unsteady force coefficient on cylinderfor M∞,0 = 0.3 and γ = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4-3 Comparison of computed results for cylinder with theoretical results of Miles(1951) for γ = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10

4-4 Evolution of non-dimensional perturbation pressure (scaled to range betweenminus and plus one at each instant) for cylinder at M∞ = 0.2 and γ = 1.4 . . . 87

4-5 Computed behavior of peak and steady-state values of F for γ = 1.4 . . . . . . 90

4-6 Comparison of computed results for sphere with theoretical results of Longhorn(1952) for γ = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4-7 Effect of γ on unsteady force coefficient on cylinder . . . . . . . . . . . . . . . . 95

4-8 Behavior of ξ(t∗ →∞, �t) defined by Eq. (4–12) . . . . . . . . . . . . . . . . . 99

5-1 Response kernels of Parmar et al. [79] . . . . . . . . . . . . . . . . . . . . . . . 109

5-2 Schematic of shock position in the symmetry plane of a spherical particle anddefinition of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5-3 Comparison of model with computations for sphere with diameter 8µm of Sunet al. [102] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5-4 Comparison of model with computations for sphere with diameter 80µm ofSun et al. [102] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5-5 Comparison of model with experiment for sphere with diameter 80 mm andcomputations for 8 mm of Sun et al. [102] . . . . . . . . . . . . . . . . . . . . . 122

5-6 Comparison of model with experiments of Skews et al. [97] . . . . . . . . . . . 125

5-7 Comparison of model with experiments and computations of Britan et al. [14] . 129

5-8 Breakdown of drag force for experimental conditions considered by Britan etal. [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5-9 Results for shock interaction with cylinder at Ms = 1.22 . . . . . . . . . . . . . . 133

6-1 Comparison of drag correlations with data of Bailey and Starr [5] assumingthat γ = 1.4 and that the particle temperature is equal to the surrounding gastemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6-2 Comparison of new drag correlation with data of Bailey and Starr Bailey andStarr [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6-3 Comparison of new drag-coefficient correlation with data of Goin and Lawrence[38] and May and Witt [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6-4 Comparison of model with experimental data of Jourdan et al. [50] Note theabscissa scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11

7-1 Time evolution of normalized unsteady force for M = 0.01, Re = 1. Bassethistory force (Eq. (7–1)), modified history force due to Mei and Adrian [68](Eq. (7–6)), inviscid unsteady force in compressible flows due to Longhorn[58] (Eq. (7–4)), and inviscid and viscous unsteady force in compressible flowsdue to Parmar et al. [81] (Eq. (7–3)) are plotted. . . . . . . . . . . . . . . . . . 150

7-2 Mesh quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7-3 Schematic depiction of variation of freestream Mach and Reynolds numberduring computations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7-4 Time evolution of normalized unsteady force for M = 0.01, Re = 1. Bassethistory force (Eq. (7–1)), modified history force due to Mei and Adrian [68](Eq. (7–6)), and inviscid and viscous unsteady force in compressible flowsdue to Parmar et al. [81] (Eq. (7–3)) are plotted. Corresponding simulationresults are shown as open circle symbols. . . . . . . . . . . . . . . . . . . . . . 154

7-5 Comparison of normalized unsteady force obtained by numerical simulationand that given by the last two terms of Eq. (7–12). Simulation results are shownas open circle symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7-6 Comparison of normalized unsteady force obtained by numerical simulationand that given by new model (the last two terms of Eq. (7–13)). . . . . . . . . . 157

7-7 Comparison of normalized unsteady force obtained by numerical simulationand that given by new model (the last two terms of Eq. (7–13)). . . . . . . . . . 158

A-1 Variable arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A-2 Time discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

12

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

UNSTEADY FORCES ON A PARTICLE IN COMPRESSIBLE FLOWS

By

Manoj Kumar Parmar

December 2010

Chair: Andreas HaselbacherCochair: S. BalachandarMajor: Aerospace Engineering

Compressible multiphase flows occur in a variety of industrial and environmental

problems. The key to improved prediction and control of these flows at the macroscale

depends on our understanding of the interaction of an individual particle in a compressible

ambient flow at the microscale. For example, simulations of compressible multiphase

flows at the macroscale require an accurate model for the time-dependent force on

a particle. At present, due to lack of fundamental knowledge, no well-founded model

exits for the evaluation of forces on a particle in a compressible flow. By contrast,

the problem of determining forces on a particle in incompressible flows has been

studied for more than 150 years and produced a comprehensive body of knowledge.

Current understanding of the forces on a particle in a compressible flow is limited to the

quasi-steady drag force, expressed in terms of a drag coefficient that is dependent on

both the Reynolds and Mach numbers. In compressible flows, unsteady interactions

between compressible waves and the particle gives rise to unsteady contributions to

the force that can be very important but remain virtually unexplored. This dissertation

attempts to lay the foundation for compressible multiphase flow by obtaining a rigorous

equation of motion for an isolated particle that is applicable in complex compressible

flows. Toward this end, we first rigorously derive the compressible extension to the

celebrated Basset-Boussinesq-Oseen equation for the unsteady motion of a particle.

We then rigorously derive the compressible extension of the Maxey-Riley-Gatignol

13

equation that accounts for the inhomogeneity of the ambient compressible flow. Through

carefully constructed simulations, finite Mach- and Reynolds- number extensions for

the quasi-steady and unsteady forces on the particle are developed. The improved

formulation is tested for shock-particle interaction.

14

CHAPTER 1INTRODUCTION

1.1 Motivation and Background

Multiphase flows are among the most interesting and challenging problems in fluid

mechanics. The study of multiphase flows is important because of its applications to

geophysical and engineering flows such as volcanic eruptions and solid-propellant

rocket motors (see fig. 1-1). General information on multiphase flows can be found in

Crowe et al. [25], Brennen [12], and Balachandar and Prosperetti [7]. The present work

focuses on a relatively little explored branch of multiphase flow, namely those in which

compressible effects are important. Relevant examples are the flow of aluminum-oxide

particles through an imperfectly expanded nozzle (see fig. 1-1B and Najjar et al. [75]),

shock-particle interactions (see Thomas [111] and Tedeschi et al. [108]), particle

removal from surfaces (see Smedley et al. [98] and Lee and Watkins [56]), needle-free

drug delivery (see Menezes et al. [69]), and detonation of multiphase explosives (see

Lanovets et al. [55], Zhang et al. [118], and Ripley et al. [86]). The overarching goal of

this work, to be discussed in more detail below, is to put the simulation of compressible

multiphase flows on a more solid theoretical footing. Before reviewing relevant prior

work, some background on multiphase flows will be discussed.

We restrict our attention to dispersed two-phase flows in which one phase is

materially disconnected, the so-called dispersed phase, and the other is materially

connected, the so-called continuous phase. The behavior of dispersed two-phase flows

is determined by the interaction between the phases. In general, the interactions occur

in the form of mass, momentum, and energy exchanges at the phase boundaries.

In the present work, attention is focused on two-phase flows in which the dispersed

phase consists of spherical particles and the continuous phase is a fluid. The particles

are assumed to be rigid and inert, so the mass transfer is neglected. In addition to

the interactions between the two phases, complexities can arise due to the interactions

15

A Mount St. Helens. Source: http://logancullen.files.wordpress.com/2010/05/mt-st-helens-erupting.jpg

B Space shuttle plume. Source: http://www.nasa.gov/

Figure 1-1. Examples of compressible multiphase flows. In the space shuttle, multiphaseflow arises due to the solid-propellant rocket motors because the propellantis enriched with aluminum particles.

within the dispersed phase, i.e., intra-phase interactions, such as collision, agglomeration,

and breakup. A two-phase flow is called dilute when only inter-phase interactions are

important. Otherwise, it is called a dense flow. Throughout this dissertation, we consider

only dilute two-phase flows. The understanding of the interactions between the phases

is complicated by disparate length and time scales involved, unsteadiness, non-linearity,

inhomogeneity, and compressibility. In previous investigations of multiphase flows,

little attention has been focused on the effects of compressibility compared to other

features in multiphase flows. Compressible flows are characterized by variable

density, non-negligible energy transfer, and wave propagation, which further add to

the difficulties.

Analytical solutions for the interactions between the particulate and the fluid

phases are possible only in the linearized regime of low-speed flows and small

perturbations propagation. In the non-linear regime of high-speed flows and large

16

perturbations, analytical solutions are seldom available, and the complex interactions

must be studied experimentally and/or computationally. The present work focuses on

studying a dilute, non-reacting suspension of rigid spherical particles in a compressible

fluid using theoretical analysis in the linearized regime and computational tools in the

non-linear regime. Through the linearity assumption, energy exchanges are negligible,

so theoretical analysis considers the effects of momentum exchanges only.

In principle, a numerical simulation of an unsteady three-dimensional flow around a

particle can provide accurate information about the flow field. The forces on the particle

can then be computed by integrating the pressure and stress distribution around the

particle. In a typical multiphase flow, however, the number of particles can range from

millions to billions. With current computational capabilities, it is impossible to resolve

the flow around every particle. As an alternative approach, a particle can be treated as

a point particle in which the details of the flow field around the particle are not resolved

and modeled instead, see Balachandar and Eaton [6]. The accuracy of the point-particle

approach depends strongly on the accuracy of the force model used in the particle

equation of motion. Force models have been studied extensively for incompressible

flows, as will be described in Section 1.2. There are many applications of two-phase

flows where compressibility is not negligible as listed above. The development of the

point particle approach for compressible flows has been limited. Because of the lack of

understanding of the forces on a particle in compressible flows, an incompressible force

model is often used, see Saito [93] and Jourdan et al. [50].

1.2 Particle Equation of Motion in Incompressible Flows

In the point-particle approach, the velocity of a particle is obtained from Newton’s

second law,

mp

dv

dt= F(t) , (1–1)

where mp is the mass of the particle (here assumed to be constant), t is time, v is the

particle velocity, and F is the force on the particle. The force on a particle can be divided

17

into surface and body forces. Examples of body forces are gravity, electrostatic, and

magnetic forces. Examples of surface forces are pressure and viscous forces. In the

current work, body forces are neglected.

The problem of determining forces on rigid bodies immersed in a fluid is a

fundamental and classical problem in fluid mechanics.

1.2.1 Creeping Motion in a Quiescent Fluid

One of the earliest contributions to the determination of forces is due to Stokes

[101]. Stokes investigated the steady rectilinear motion of an isolated sphere in a

quiescent viscous fluid based on the linearized incompressible Navier-Stokes equations

and obtained an analytical expression for the drag force as

F = 6πµav , (1–2)

where µ is the dynamic viscosity of the fluid and a is the radius of the sphere. Stokes

also extended his analysis to the unsteady case. Expressed in frequency domain, the

result for the total force on a sinusoidally oscillating sphere in a quiescent fluid is

F = 6πµav0eiωt

[1 + (1− i)

(ωa2

2ν

)1/2

− i

(ωa2

9ν

)], (1–3)

where v0 and ω are the amplitude and frequency of the oscillating velocity, respectively.

ν = µ/ρ is the kinematic viscosity, and ρ is the fluid density. The first term inside the

square brackets corresponds to the instantaneous Stokes drag given by Eq. (1–2). The

second term is what is now called the Basset force. The third term is the force due to the

added mass.

Subsequently, Basset [8], Boussinesq [10], and Oseen [77] independently examined

the time-dependent force on a sphere due to rectilinear motion in a quiescent viscous

incompressible fluid. They also based their analyses on the linearized unsteady

incompressible Navier-Stokes equations valid for creeping motion, i.e., in the limit of

vanishing Reynolds number. The resulting equation of motion for a spherical particle,

18

the so-called BBO equation, can be written as

mp

dv

dt= −6πaµv − 1

2mf

dv

dt− 6a2ρ

√πν

∫ t

−∞

1√t − ξ

dv

dt

∣∣∣∣t=ξ

dξ , (1–4)

where mf is the mass of the fluid displaced by the particle. The three terms on the

right-hand side are the quasi-steady (Stokes) drag, inviscid unsteady (added mass), and

viscous unsteady (Basset history) forces, respectively.

The added-mass force appears due to the no-penetration condition on the particle

surface and is realized instantly in the presence of particle acceleration. In general, the

added mass force can be derived analytically in the limit of potential and Stokes flow,

see Landau and Lifshitz [54]. The added-mass force Fam is typically expressed as

Fam = Cammf

dv

dt, (1–5)

where Cam is the added-mass coefficient. The added-mass coefficient is fixed for a given

particle shape. For a spherical particle, Cam = 1/2.

The Basset history force arises due to the redistribution of vorticity generated at the

particle surface. For impulsive acceleration, the history force decays monotonically.

The BBO equation of particle motion is strictly valid only in the linearized incompressible

regime for a moving particle in a quiescent fluid. A natural question is how the force

components and therefore the equation of motion changes for non-quiescent fluids,

non-uniform flows, and when nonlinearities and compressibility become important.

1.2.2 Unsteady Ambient Flow

The quasi-steady contribution to the hydrodynamic force on a moving particle in a

non-stationary ambient flow is typically parameterized in terms of the relative velocity of

the fluid with respect to the particle, urel = u− v, where u is the undisturbed fluid velocity

(in the absence of the particle) at the center of the particle. The quasi-steady drag is

found to be given by

Fqs = 6πµaurel . (1–6)

19

For a particle moving in a non-stationary fluid, the inviscid unsteady force consists

of the pressure-gradient force in addition to the added-mass force. The added-mass

force now depends on the relative acceleration of the particle with respect to the flow.

The pressure-gradient force is due to the flow acceleration in the absence of the particle.

The added-mass force Fam and the pressure-gradient force Fpg are expressed as

Fiu = Fam + Fpg = Cammf

(Du

Dt− dv

dt

)+mf

Du

Dt. (1–7)

where D/Dt is the substantial time derivative following the fluid particle.

The viscous unsteady force for the case of a moving fluid depends on the relative

acceleration of the fluid with respect to the particle. The viscous unsteady force can be

expressed as

Fvu = 6a2ρ√πν

∫ t

−∞

1√t − ξ

(Du

Dt

∣∣∣∣t=ξ

− dv

dt

∣∣∣∣t=ξ

)dξ . (1–8)

1.2.3 Non-Uniform Flow

Faxen [31] first considered the effect of non-uniform ambient flows on the force on a

particle. For steady Stokes flow, Faxen obtained the remarkable result that

Fqs = 6πµa(uS − v) , (1–9)

where (·)S

denotes an average over the surface of the spherical particle, i.e.,

fS=

1

4πa2

∮S

f dS . (1–10)

Mazur and Bedeaux [65] generalized Faxen’s results to unsteady motion of a

particle in an arbitrary incompressible Stokes flow. They presented their result in the

frequency domain. Gatignol [36] derived Faxen’s formula for unsteady forces in the time

domain. Independently, Maxey and Riley [63] derived from first principles the particle

equation of motion for non-uniform creeping flows. The Maxey-Riley-Gatignol (MRG)

20

equation of motion can be written as (neglecting body forces)

mp

dv

dt= 6πaµ

(uS − v

)+

1

2mf

(DuV

Dt− dv

dt

)+mf

DuV

Dt

+ 6a2ρ√πν

∫ t

−∞

1√t − ξ

(DuS

Dt

∣∣∣∣t=ξ

− dv

dt

∣∣∣∣t=ξ

)dξ , (1–11)

where (·)V

denotes an average over the volume of the spherical particle as

fV=

3

4πa3

∫Vf dV . (1–12)

1.2.4 Non-Linear Regime

Non-linear effects are quantified through the particle Reynolds number Re, defined

as

Re = 2urela/ν . (1–13)

At finite Reynolds numbers, non-linear effects on the quasi-steady, inviscid unsteady,

and viscous unsteady forces must be taken into account.

Because of the absence of an analytical solution for the drag force, empirical

correlations have been proposed for the quasi-steady drag. The quasi-steady drag is

generally expressed in a non-dimensional form as a drag coefficient CD , defined as

CD =Fqs

12ρu2relπa

2, (1–14)

where Fqs = ∥Fqs∥. A wealth of experimental data has been gathered and compiled

in the form of the standard drag curve, see Clift and Gauvin [21] and Clift et al. [22]

for references. The standard drag curve represents the drag coefficient on a spherical

particle as

CD =24

Re

(1 + 0.15Re0.687

)+ 0.42

(1 +

42500

Re1.16

)−1

. (1–15)

The above correlation is valid for Re / 2 · 105.

21

For finite Reynolds-number flows, it has been found that the added-mass force is

identical to that in potential and creeping flows, see Mei et al. [66], Rivero et al. [87],

Chang and Maxey [19], and Wakaba and Balachandar [114].

The Basset history force has also been extended to finite Reynolds numbers.

Odar and Hamilton [76] suggested a complex nonlinear dependence of the history

force on the relative particle acceleration. This approach was later proved incorrect by

many including Mei et al. [66], Rivero et al. [87], and Chang and Maxey [18]. Mei and

Adrian [68] obtained an expression for the history force for finite but small Reynolds

numbers by considering oscillatory flows around a particle. Kim et al. [52] made further

improvements to Mei and Adrian’s expression by considering large amplitude oscillations

of the particle. In spite of significant progress in the understanding of the history force, it

is important to recognize that the history force is problem-dependent, see Lovalenti and

Brady [60], and a universal practical expression has not yet been formulated.

1.2.5 Force Superposition

Magnaudet and Eames [62] suggested that the drag force on a particle in an

incompressible flow can be parameterized as

F(t) = Fqs(t) + Fiu(t) + Fvu(t) , (1–16)

where the terms on right-hand side represent the quasi-steady, inviscid unsteady

(added-mass and pressure gradient) and viscous unsteady (Basset history) forces. Lift,

and buoyancy/gravity forces are ignored in this work. All developments described above

were concerned with incompressible flows. No systematic studies exist for compressible

flows. In principle, Eq. (1–16) can be used for compressible flows, provided suitable

modifications are incorporated into the modeling of the forces.

22

1.3 Particle Equation of Motion in Compressible Flows

The importance of compressible effects is quantified through the Mach number,

defined as

M = urel/c , (1–17)

where c is the speed of sound of the ambient flow.

To the best of our knowledge, there has been no systematic work on deriving an

equation of motion for particles in compressible flows, There are some theoretical

investigations of unsteady forces on a particle in an inviscid compressible fluid in the

acoustic limit, i.e. M → 0, and in a viscous compressible fluid in the limit of zero Mach

and Reynolds numbers. For finite Mach numbers, attention has been focused mainly on

the quasi-steady force.

1.3.1 Quasi-Steady Drag

The representation of the quasi-steady drag coefficient in terms of the particle

Reynolds number, e.g. Eq. (1–15), is valid for incompressible flows. For compressible

flows, the incompressible drag coefficient can be assumed to be approximately valid up

to the critical Mach number. The critical Mach number is the largest freestream Mach

number relative to the particle for which the flow around particle remains subsonic. For

a spherical particle, the critical Mach number is about 0.6. In supercritical but subsonic

freestream flows, due to flow acceleration around the particle, the local flowfield

becomes supersonic and shock waves appear on the particle surface. In supersonic

flow, a detached bow shock appears upstream of the particle. The presence of shock

waves on or upstream of the particle increases the quasi-steady drag significantly.

Thus, for supercritical flow, the drag coefficient now depends also on the Mach number

in addition to the Reynolds number. For experimental work on the drag coefficient in

supercritical flows, see, e.g., Bailey and Hiatt [4], Bailey and Starr [5], and Miller and

Bailey [72]. To make such data useful for particle simulations, it needs to be expressed

in the form of an empirical relation. Several correlations have been proposed to relate

23

the particle Reynolds and Mach number to the drag coefficient, see, e.g., Crowe [24],

Walsh [116], Henderson [44], and Loth [59]. Despite the availability of correlations that

include compressibility effects, researchers are still using the correlation derived for

incompressible flows to represent the force on particles in high-speed flows, see Elperin

et al. [30] and Saito [93]. A close examination of existing correlations indicates that they

do not represent experimental data faithfully. The correlations need to be improved to

represent experimental data more faithfully.

1.3.2 Inviscid-Unsteady Force

There are many unsteady multiphase flows where compressibility is significant.

The earliest work on the unsteady inviscid force in a compressible flow is due to Love

[61] and Taylor [106]. Later Miles [71] carried out an analytical investigation of the

inviscid compressible unsteady flow due to the impulsive motion of a cylinder in the

acoustic limit. Longhorn [58] obtained the corresponding expression for a sphere.

In compressible flows, the dependence of the unsteady force on the instantaneous

acceleration is broken due to the finite wave-propagation speed. Therefore, the

unsteady force depends on the acceleration history on the acoustic time scale.

Longhorn gave following expression for the inviscid unsteady force on a rectilinearly

moving sphere

Fiu = mf

∫ t

0

dv

dt

∣∣∣∣t=ξ

e−c(t−ξ)/a cos (c(t − ξ)/a) dξ . (1–18)

Longhorn [58] showed that the amount of work done to move a sphere impulsively is

doubled compared to that in gradual motion to reach the same final velocity. Tracey

[113] extended the analytical work of Miles [71] to finite but small Mach numbers.

Longhorn’s analytical work has not been extended to finite Mach numbers. Brentner

[13] carried out numerical simulations of compressible flow about an accelerating

cylinder at M = 0.4. The focus of Brentner’s work was on the propagation of acoustic

energy from the accelerating cylinder. There is a need to extend the work of Miles and

24

Longhorn to finite Mach numbers to understand the nature of the unsteady force in

inviscid compressible flows.

1.3.3 Viscous-Unsteady Force

Zwanzig and Bixon [119] obtained in the frequency domain an expression for

the force on a spherical particle undergoing oscillatory motion in a quiescent viscous

compressible flow. A minor error in the result was corrected by Metiu et al. [70]. Similar

results were obtained by Guz [39] in Laplace space. Bedeaux and Mazur [9] derived

Faxen’s correction for the force on a spherical particle undergoing oscillatory motion in

viscous compressible flows. These results have not been derived in a form useful for

particle tracking in the time domain.

Several investigations of the viscous unsteady force in a compressible flow were

carried out in the context of shock-particle interaction. The behavior and the form

of the history force needs further investigation in compressible flows. There are few

studies on the influence of the Basset history force on the particle motion in the

shock-particle interaction, see, e.g., Forney et al. [35], Tedeschi et al. [108], Thomas

[111], and Tedeschi et al. [109]. The knowledge of the influence of the unsteady

forces on the particle motion is very important for the small tracer particles used in

the particle-imaging velocimetry (PIV) studies of supersonic flows.

The studies concluded that the Basset-history force can be many times larger than

the quasi-steady drag force in the presence of large velocity gradients. However, the

influence of the Basset history force on the particle motion may still be limited because

it exceeds the quasi-steady force only for a brief period. The incompressible form of

the history force was used in the above cited investigations because no corresponding

expression exists for compressible flow. Thus, there is a need to investigate the nature

and the form of the viscous unsteady force in compressible flow.

25

Figure 1-2. State-of-the-art for the forces on a particle.

1.4 Goal of the Present Work

An equation of motion suitable for particles in compressible flows has not been

proposed so far. In particular, the above discussion shows that the unsteady forces

on particles in compressible flows have not been studied extensively. Figure 1-2

summarizes schematically the state-of-the-art for the modeling of forces and associated

equation of motion for particles in incompressible and compressible multiphase flows.

Due to the lack of well-developed theories for the inviscid unsteady and viscous

unsteady forces in compressible flows, many computational investigations uses only

the quasi-steady force, see Carrier [17], Kriebel [53], Rudinger [89], Rudinger and

Chang [92], Pelanti and LeVeque [83], Miura and Glass [73], Elperin et al. [30], Igra and

Takayama [46], Saito [93] and Jourdan et al. [50]. The overarching goal of this work is

to study the behavior of the unsteady forces in compressible flows and develop models

for use in computational studies of compressible multiphase flows. In this thesis, using

26

theory and simulations, we carry out a sequence of investigations with the following

objectives:

1. To study and model the quasi-steady, inviscid unsteady, and viscous unsteadyforces on a spherical particle in uniform compressible Stokes flows.

2. To study and model the quasi-steady, inviscid unsteady, and viscous unsteadyforces on a spherical particle in non-uniform compressible Stokes flows.

3. To model the effect of compressibility on the quasi-steady drag in supercriticalflows.

4. To study and model the behavior of inviscid unsteady force in compressible flowsat finite Mach numbers.

5. To develop a model for shock-particle interaction.

6. To study and model the viscous unsteady force in compressible flows at finiteMach and Reynolds number.

In the current work, systematic steps are taken to study unsteady forces in

particular. It is clear, however, that the topic cannot be studied in its entirety in one

dissertation. Similar to incompressible flows, multiple dissertations are required to do full

justice to this vast topic.

1.5 Dissertation Layout

Rest of the dissertation is organized into seven chapters and an appendix.

Chapter 2: Generalized Basset-Boussinesq-Oseen equation for unsteady

forces on a sphere in a compressible flow. A theoretical investigation is carried out

using the linearized compressible Navier-Stokes equations to compute the force on a

spherical particle undergoing rectilinear unsteady motion in a quiescent homogeneous

fluid. An explicit formulation is presented for quasi-steady, inviscid unsteady, and viscous

unsteady forces that incorporates compressibility effects. This work is under review for

publication in the Physical Review Letters, see Parmar et al. [81].

Chapter 3: Equation of motion for a sphere in non-uniform compressible

flows. Using the linearized compressible Navier-Stokes equations, a theoretical

27

investigation is carried out of unsteady motion of a spherical particle in an inhomogeneous

flow. Lorentz’s reciprocal theorem is employed to obtain Faxen’s correction to the

quasi-steady, inviscid unsteady, and viscous unsteady forces. The results are used to

generalize the Maxey-Riley-Gatignol equation to compressible flows. This work is about

to be submitted to the Journal of Fluid Mechanics.

Chapter 4: The inviscid unsteady force on cylinders and spheres in compress-

ible flows. The unsteady forces on a cylinder and a sphere in subcritical compressible

flow are investigated. It is shown that the unsteady inviscid force can be more than

four times larger than that predicted from incompressible theory. This work has been

published in the Proceedings of Royal Society, see Parmar et al. [79].

Chapter 5: Modeling of the unsteady force for shock-particle interaction.

Based on the work described in the Chapter 4, a simple model for the unsteady forces

for shock-particle interaction is presented. The results are compared with experimental

and computational data for both stationary and non-stationary spheres. This work has

been published in the Shock Waves, see Parmar et al. [80].

Chapter 6: An improved drag correlation for spheres and application to

shock-tube experiments. A new correlation is presented for the drag force on spherical

particles in compressible continuum flows that represents experimental data more

faithfully than prior correlations. With this correlation, recent experimental data obtained

in shock-tubes for particle trajectory are reproduced accurately. This work has been

published in the AIAA Journal, see Parmar et al. [82].

Chapter 7: Viscous unsteady forces on a sphere in compressible flow. A

computational investigation is carried out for viscous unsteady forces on a spherical

particle in compressible flows at vanishing Mach numbers but finite Reynolds numbers.

A new formulation of the viscous unsteady force is presented based on theoretical work

of Chapter 2 and the formulation of Mei and Adrian [68].

28

Chapter 8: Summary and conclusions and directions for future work.

Conclusions of the present work are presented and the directions for future work

are discussed.

Appendix A describes numerical methods used in this work.

29

CHAPTER 2GENERALIZED BASSET-BOUSSINESQ-OSEEN EQUATION FOR UNSTEADY

FORCES ON A SPHERE IN A COMPRESSIBLE FLOW

Viscous compressible flow around a sphere is considered in the limit of vanishing

Reynolds and Mach numbers. Using the analytical solution derived in earlier works, an

exact expression for the transient force on a sphere undergoing arbitrary motion with

the inclusion of compressibility effects is presented. The transient force is decomposed

into quasi-steady, inviscid unsteady, and viscous unsteady components. The influence

of compressibility on each of these components is examined. Numerical results for

the transient force are in excellent agreement with theory. The present formulation

thus offers an explicit expression for the unsteady force in the time domain and can

be considered as a generalization of the Basset-Boussinesq-Oseen equation to the

compressible flow regime that can be used in numerical simulations of compressible

multiphase flows.

2.1 Introduction

The unsteady force on a particle in accelerated motion was first analyzed by Stokes

[101], who presented an expression for the frequency-dependent force on an oscillating

spherical particle. Later Basset [8], Boussinesq [10], and Oseen [77] independently

examined the time-dependent force on a sphere due to rectilinear motion in a quiescent

viscous incompressible fluid. They based their analyses on the linearized unsteady

incompressible Navier-Stokes equations valid for creeping motion, i.e., in the limit of

vanishing Reynolds number. The resulting equation of motion for a spherical particle,

the so-called BBO equation, can be written as

mp

dv

dt= −6πaµv − 1

2mf

dv

dt− 6a2ρ

√πν

∫ t

−∞

1√t − ξ

dv

dξdξ , (2–1)

where mp is the particle mass, v(t) is particle velocity, a is the particle radius, µ is

the dynamic viscosity, mf is the mass of fluid displaced by the particle, ρ is the fluid

density, and ν = µ/ρ is the kinematic viscosity. The three terms on the right-hand

30

side are the quasi-steady (Stokes) drag, inviscid unsteady (added-mass), and viscous

unsteady (Basset history) forces, respectively. The BBO equation has been extended to

non-uniform creeping flows by Tchen [107], Maxey and Riley [63], and Gatignol [36] and

to finite Reynolds numbers by Mei and Adrian [68], Kim et al. [52], and Magnaudet and

Eames [62].

Our primary goal is to extend the BBO equation to compressible flows. The

first work relevant to our goal appears to be that of Zwanzig and Bixon [119], who

investigated the velocity-correlation function of an atom immersed in a compressible

visco-elastic liquid. A minor error in their work was corrected by Metiu et al. [70]. Temkin

and Leung [110] solved the linearized compressible Navier-Stokes equations in the

frequency domain to study the interaction between a plane monochromatic acoustic

wave and a spherical particle. A more general approach was adopted by Guz [39], who

solved the linearized compressible Navier-Stokes equations in the Laplace domain

and presented a solution valid for arbitrary particle motion. The results of these studies

are essentially identical except for differences due to simplifying assumptions and

some typographical mistakes. Felderhof [32, 33] used the frequency-space solution of

Zwanzig and Bixon [119] to investigate the motion of a particle in response to a force

impulse.

The purpose of this work is to present an explicit expression for the time-dependent

force on a spherical particle undergoing arbitrary unsteady motion on the acoustic time

scale such that compressibility effects are important. Attention is restricted to the zero

Reynolds- and Mach-number limits so that non-linear effects can be ignored. We use

previously derived solutions of the linearized compressible Navier-Stokes equations

in the Fourier/Laplace domains to determine the force on a particle in response to

a delta-function acceleration in the time domain. This force response is then used

to construct an expression for the time-dependent force on a particle undergoing

arbitrary motion. The resulting expression can be interpreted as the generalization of

31

the BBO equation to compressible flows. The generalized BBO equation allows clear

interpretation of the effect of compressibility on the quasi-steady, inviscid unsteady,

and viscous unsteady drag forces. We show that compressibility causes the inviscid

unsteady force to assume an integral representation first derived by Longhorn [58].

We obtain the effect of compressibility on the viscous unsteady force. The theoretical

results are compared with direct numerical simulations of the compressible flow around

an accelerating particle. Finally, we present the generalized BBO equation that can be

used to track particles in compressible flow and that reduces to Eq. (3–1) in the limit of

incompressible flow.

2.2 Problem Formulation

We consider the unsteady motion of a particle in a quiescent compressible

Newtonian fluid. We consider the limit of Re → 0 and M → 0 such that the perturbation

field generated by the particle motion is governed by the linearized compressible

Navier-Stokes equations. Here, M and Re are suitably defined Mach and Reynolds

numbers. Furthermore, if the Knudsen number (which is proportional to M/Re) is

small, viscous heating and adiabatic cooling effects in the energy equation can

be neglected. As a result, the temperature field decouples and the continuity and

momentum equations reduce to the form given by Zwanzig and Bixon [119],

∂ρ′

∂t+ ρ0∇ · u′ = 0 , (2–2)

ρ0∂u′

∂t+∇p′ − µ∇2u′ − (µb +

1

3µ)∇∇ · u′ = 0 . (2–3)

In Eqs. (2–2) and (2–3), properties associated with the quiescent fluid are denoted by

the subscript 0, perturbation quantities are denoted by the superscript ′, u is the velocity,

p is the pressure, and µb is the bulk viscosity. All other symbols are defined as in Eq.

(3–1). Because temperature fluctuations are neglected, the viscosities are constant and

the speed of sound

c0 =√(∂p/∂ρ)s =

√p′/ρ′ (2–4)

32

can be used as a closure relation. These linearized equations have been solved

analytically by Zwanzig and Bixon [119] and Metiu et al. [70], who obtained an explicit

expression for the force on the particle in the frequency domain. Given a general particle

motion with velocity v(t) the solution of Eqs. (2–2)–(2–4) in Laplace space can be

written as

F(s) = −mf s G(r1, r2)L(v) (2–5)

where F(s) = L(F (t)) and L(v) are the Laplace transforms of the time dependent force

F (t) and rectilinear particle velocity v(t), respectively, and mf = 4πρ0a3/3 is the mass of

fluid displaced by the particle. The transfer function G(r1, r2) is given by

G(r1, r2) =(9 + 9r1 + 2r 21 )(1 + r2) + (1 + r1)r

22

r 21 (1 + r2) + (2 + 2r1 + r 21 )r22

, (2–6)

where

r1(s) =as/c0√

1 + (µb/µ+ 4/3) νs/c20and r2(s) = a

√s

ν. (2–7)

A similar analysis was independently performed by Temkin and Leung [110] in

the context of acoustic scattering by a spherical particle. Their results are slightly

different from those by Zwanzig and Bixon [119]. The differences can be traced to an

approximation made by Temkin and Leung [110] in the definition of stress distribution

around the particle. The above results have also been derived in the recent book by Guz

[39]. However, due to typographical errors the results might appear different.

2.3 Solution for Impulsive Motion

Since the problem is linear, the force on a particle undergoing arbitrary rectilinear

motion v(t) can be expressed as a convolution integral

F (t) =

∫ t

−∞

dv

dξFδ(t − ξ) dξ , (2–8)

33

where Fδ(t) is the force response to a delta-function acceleration (i.e., corresponding to

a unit step change in particle velocity). Using Eq. (2–5), Fδ(t) can be expressed as

Fδ(s) = −mfG(r1, r2) . (2–9)

An explicit Laplace inverse transform of Eq. (2–9) and, therefore, a closed-form

expression for Fδ(t) is not readily available. Before constructing the time-domain

solution, we first analyze the limiting case of incompressible flow.

The incompressible limit is obtained by letting c0 →∞ in Eq. (2–9) to obtain

Fδ,inc(s) = −mf

9 + 9r2 + r 222r 22

, (2–10)

where r 22 can be interpreted as the Laplace variable corresponding to time non-dimensionalized

by the viscous time scale a2/ν. The corresponding expression in the time domain is

Fδ,inc(t) = −6πaµH(t)− 1

2mf δ(t)− 6a2ρ0

√πν

tH(t) , (2–11)

where H(t) is the Heaviside step function. The above expression is identical to the

BBO equation for a delta-function acceleration, cf. Eq. (3–1). Therefore, the general

compressible force given in Eq. (2–8) reduces to the correct limit for an incompressible

flow.

2.4 Compressibility Effect on Inviscid Unsteady Force

We isolate the three terms (quasi-steady, inviscid unsteady, and viscous unsteady

forces) on the right-hand side of Eq. (3–1) and investigate the effect of compressibility.

First, we consider the compressibility effect on the inviscid unsteady force. The inviscid

limit is obtained by substituting ν = 0 in Eq. (2–9) to get

Fδ,iu = −mf

1 + r1

2 + 2r1 + r 21, (2–12)

34

where r1 can be interpreted as the Laplace variable corresponding to time non-dimensionalized

by the acoustic time scale a/c0. The corresponding expression in the time domain is

Fδ,iu(τ) = −mf

c0

ae−τ cos τ H(τ) , (2–13)

where τ = c0t/a. This result for the inviscid unsteady force on a sphere impulsively set

in motion in a compressible quiescent fluid was first presented by Longhorn [58]. The

effect of compressibility on the inviscid unsteady force can be established by comparing

Eq. (2–13) with the second term on the right-hand side of Eq. (2–11). The finite speed

of sound destroys the instantaneous relationship between acceleration and force

and thus in a compressible flow the inviscid unsteady force cannot, strictly speaking,

be considered as an added-mass force. Furthermore, compressibility regularizes

the singular delta-function kernel to a smooth oscillatory exponential decay. From a

physical perspective, this can be explained by the compression and rarefaction waves

that emanate from the accelerated particle which propagate outward at finite speed.

Therefore, in a compressible flow the inviscid unsteady force is dependent not only on

the instantaneous acceleration, but on the past acceleration history. However, due to

the exponential-decay term in Eq. (2–13), the compressibility effect is significant only for

τ . 10.

The inviscid unsteady force was obtained by Longhorn [58] using the acoustic

approximation of the velocity potential equation and is thus valid only in the zero-Mach

number limit. The right-hand side of Eq. (2–13) can be considered to be the response

kernel for a delta-function acceleration for M → 0. Note that∫∞0

e−τ cos τ dτ = 1/2, and

thus over times much longer than the acoustic time scale, the net impulse on the particle

reduces to the correct limit as that given by the incompressible added-mass force. The

corresponding kernels for finite Mach numbers can be obtained through numerical

simulations, see Parmar et al. [79].

35

2.5 Asymptotic Behaviors of Compressible Viscous Unsteady Force

We now examine the effect of compressibility on the viscous unsteady force. To

study the force at arbitrary times, we resort to numerical inversion of Eq. (2–9) because

we have not found an explicit analytical Laplace inverse of the complete transfer function

G(r1, r2). With V denoting the scale of the velocity variation, we define the Reynolds and

Mach numbers as Re = ρ0Va/µ and M = V /c0. Thus we can write

Fδ(τ)

mf c0/a= −L−1 (G(R1,R2)) , (2–14)

where L−1 denotes the Laplace inverse with respect to the non-dimensional time

τ = c0t/a and

R1 =S√

1 + (µb/µ+ 4/3)Kn′Sand R2 =

√S

Kn′, (2–15)

where S = as/c0 is the non-dimensional Laplace variable and Kn′ = M/Re = µ/ρ0c0a

denotes a modified Knudsen number. It is interesting to note that the force response

depends only on the ratio of the Mach number to the Reynolds number and not on

their individual values. The expression “modified Knudsen number” is motivated by

the standard definition of the Knudsen number as Kn =√γπ/2M/Re, where γ is the

ratio of specific heats of the fluid. For air at standard conditions, γ = 1.4 and hence

Kn′ ≈ 0.67Kn. The continuum assumption is commonly taken to imply Kn < 0.01 and

thus we are interested in the force response for Kn′ . O(10−2).

Although an explicit expression for Fδ(τ) valid for arbitrary τ is not available, four

different asymptotic regimes can be identified:

Regime I: Very short time, defined as τ ≪ Kn′ ≪ 1,

Regime II: Intermediate short time, defined as Kn′ ≪ τ ≪ 1,

Regime III: Intermediate long time, defined as 1≪ τ ≪ 1/(ReM),

Regime IV: Very long time, defined as 1≪ 1/(ReM)≪ τ .

36

Explicit expressions for the force in the time domain can be obtained for the first

three regimes. As we will seen below, nonlinearity becomes important for 1/(ReM)≪ τ .

Thus the linearized Eqs. (2–2) and (2–3) and hence the solution presented here do

not describe accurately the force evolution in Regime IV. A similar situation arises in

the incompressible formulation also. It is well established that no matter how small the

Reynolds number, the decay of the viscous unsteady force at sufficiently long time will

be faster than the t−1/2 decay given by the Basset history kernel (see Mei and Adrian

[68]).

In what follows, we examine the behavior of Fδ(τ) in the first three asymptotic

regimes. The very short time behavior of Fδ(τ) can be obtained by considering the

following limit in the Laplace space: 1 ≪ Kn′|S | ≪ |S | ≪ |S |/Kn′. Correspondingly, the

transfer function can be simplified as

G(R1,R2) ∼

(2 +

√µbµ

+4

3

)√Kn′

S. (2–16)

The Laplace inverse gives the time domain force response in Regime I as

Regime I : Fδ(τ) ∼ −

(4

9+

2

9

√µbµ

+4

3

)6a2ρ0c0

√πKn′

τH(τ) for τ ≪ Kn′ ≪ 1 .

(2–17)

Comparing with the third term on the right-hand side of Eq. (2–11), which can be

written as −6a2ρ0c0√

πKn′/τ , it can been seen that compressibility modifies the

viscous unsteady force at very short times by a factor that depends on µb/µ. For

µb = 0, compressibility reduces the unsteady force by 4(1 + 1/√3)/9 ≈ 0.70. The

correction factor to the viscous unsteady force increases with increasing bulk viscosity.

Interestingly, for µb/µ = 59/12 ≈ 4.92 the correction factor is equal to unity and Eq.

(2–17) becomes identical to that in the incompressible case.

From the definition of the intermediate short time (Kn′ ≪ τ ≪ 1), the following

condition can be placed on the Laplace variable: Kn′|S | ≪ 1 ≪ |S | ≪ 1/Kn′ ≪ |S |/Kn′.

37

Then G(R1,R2) can be simplified as

G(R1,R2) ≈1 + R1

1 + (1 + R1)2+

2

R2

{1 +

2(1 + R1)

1 + (1 + R1)2+

1

1 + (1 + R1)2− 1

[1 + (1 + R1)2]2

}.

(2–18)

The dominant contribution for intermediate short time comes from the Laplace inverse of

the first two terms in the transfer function G(R1,R2), resulting in

Regime II : Fδ(τ) ∼ −mf

c0

ae−τ cos τ − 8

3a2ρ0c0

√πKn′

τH(τ) for Kn′ ≪ τ ≪ 1 .

(2–19)

The first term is same as Fδ,iu(τ) given by Eq. (2–13). Comparing the second term

with the third term on the right-hand side of Eq. (2–11), it can been seen that the

viscous unsteady force at intermediate short times is reduced by a factor of 4/9 ≈ 0.44

because of compressibility effects. Note that this reduction is independent of µb/µ. As

will be seen below in Fig. 2-1, where the results of the numerical Laplace inversion

are shown, Regime II can be observed only if Kn′ ≪ 1. With increasing Kn′, the

duration of the Regime II reduces and vanishes entirely for Kn′ ≈ 10−2. Thus the overall

effect of compressibility on the short-time behavior of the viscous unsteady force is

not as pronounced as for the inviscid unsteady force. The τ−1/2 decay observed in the

incompressible case persists and only the magnitude of the viscous unsteady force is

modified.

The intermediate long-time behavior can be obtained in a similar manner by

considering an asymptotic expansion for |S | → 0 and carrying out the Laplace inverse

we obtain

Regime III : Fδ(τ) ∼ −6πaµH(τ)− 6a2ρ0c0

√πKn′

τfor 1≪ τ ≪ 1

ReM. (2–20)

Comparing with Eq. (2–11), both the quasi-steady and the viscous unsteady forces are

recovered and found to be unaffected by compressibility effects. Strictly speaking, the

above solution for the linearized perturbation Navier-Stokes equations is valid for τ ≫ 1

38

and the additional limit of τ ≪ 1/(ReM) arises only from the neglect of the nonlinear

terms. The time scale on which nonlinear effects become significant can be estimated

as follows. In deriving the linearized form of the compressible Navier-Stokes equations,

the assumption that the inertial terms are negligible compared to the viscous terms

implies that the length scale L ≪ ν/V . If we take the length scale to grow by diffusion

as√νt, the assumption of linearized compressible Navier-Stokes equations can be

justified only for t ≪ ν/V 2. Expressed in terms of the acoustic time scale, this restriction

becomes τ ≪ 1/(ReM). Note that the above argument applies in an incompressible flow

also, and the nonlinear effects can be shown to become important for τc ≫ 1/Re, where

τc is time non-dimensionalized by the convective time scale a/V . This is consistent

with past results for incompressible flow that the Basset-history kernel is valid only for

τc ≪ 1/Re even at low Reynolds numbers (see Mei and Adrian [68]). Thus, owing to

nonlinearity, the very long time force behavior in Regime IV will be both Reynolds- and

Mach-number dependent in a complex manner and will not be addressed here.

2.6 Numerical Evaluation of Viscous Unsteady Force

In the following, we extract the compressible form of the viscous unsteady force at

arbitrary times using numerical Laplace inversion and compare it with its incompressible

form. We isolate the viscous unsteady force from the overall force expression given in

Eq. (2–14) by subtracting the quasi-steady contribution and the inviscid unsteady force,

given in Eqs. (2–12) and (2–13) in the Laplace and time domains, respectively. The

resulting viscous unsteady force in the time domain is then

Fδ,vu(τ)

mf c0/a= −

(L−1 (G(R1,R2))−

9

2Kn′ − e−τ cos τ

). (2–21)

We recast this viscous unsteady response to delta function acceleration in the following

formFδ,vu(τ)

mf c0/a= −9

2

√Kn′

πτC(τ) , (2–22)

39

where C(τ) is a compressible correction function, defined as the ratio of Fδ,vu(τ) relative

to the incompressible form of the viscous unsteady force.

Figure 2-1 shows C(τ) plotted as a function of non-dimensional time τ for various

values of Kn′ and µb = 0. In Regime I (τ ≪ Kn′ ≪ 1) the correction function approaches

0.70 at very short times. Also, we observe that C(τ) → 1 as τ → ∞ as expected.

At intermediate short times (which exist only for Kn′ ≪ 10−2) the correction function

takes a constant value of 0.44. A more complex compressibility effect can be observed

at transitional times between the different asymptotic regimes. The decrease from the

constant value at very short time to the new constant value at intermediate short time

occurs in a monotonic fashion. At about τ = O(10−2Kn′), C(τ) starts to deviate from

its limiting value of 0.70 and decreases to 0.44 at about τ = O(Kn′). The transition

from Regime II to Regime III that occurs at τ ≈ O(1) is more complex. At τ =

O(10−2), C(τ) increases rapidly irrespective of Kn′ toward a peak value of about 1.45

before decreasing in a strongly damped oscillatory manner toward unity. Thus, the

compressibility correction to the viscous unsteady force is bounded between 0.44 and

1.45.

The sensitivity of C(τ) to the bulk viscosity is assessed in Fig. 2-2. As discussed

above, the asymptotic behavior is dependent on µb/µ at very short times. Provided Kn′

is small enough, at intermediate small times C(τ) is independent of µb/µ. For all values

of Kn′, the behavior of C(τ) for τ & O(10−2) is only weakly dependent on µb/µ, as

expected.

From the analysis of Longhorn it can be inferred that when a spherical particle is

impulsively set into motion, an inviscid disturbance front of radius a + c0t = a(1 + τ)

propagates away from the particle. Initially the disturbance field is compressional/expansional

upstream/downstream of the particle, and thus contributing to a strong inviscid unsteady

force directed opposite to particle motion. As the inviscid disturbance front propagates

out an alternating sequence of compressional and expansion waves radiate out on

40

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 1020.4

0.6

0.8

1.0

1.2

1.4

1.6

τ

C(τ

)

4

9

4

9

(

1 + 1√3

)

Kn′ = 10−2

Kn′ = 10−5

Kn′ = 10−8

Kn′ = 10−11

Figure 2-1. The behavior of the correction function C(τ) that accounts for thecompressibility effect on the viscous unsteady force (see Eq. (2–22)).Results are plotted for µb = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}. The opencircles represent the curve-fit given by Eq. (2–27) evaluated for Kn′ = 10−5.

acoustic time scale, whose strength rapidly decays, thus contributing to both the

oscillatory behavior (cos τ term) and the exponential decay of the inviscid kernel (Kiu,

see Eq. (2–24)). In comparison, the viscous boundary layer thickness can be estimated

to grow as√νt = a

√τKn′. For τ & O(1) the viscous boundary layer is submerged within

the inviscid disturbance field and as a result the effect of compressibility on the viscous

unsteady force is oscillatory and strongly decays. Only for very small time ( Regime I),

when τ ≪ Kn′, the viscous boundary layer is thicker than the inviscid disturbance front.

It is only in this regime the bulk viscosity has a strong influence on the boundary layer

growth and consequently on the viscous unsteady kernel.

41

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 1020.4

0.6

0.8

1.0

1.2

1.4

1.6

τ

C(τ

)

µb/µ = 0

µb/µ = 1

µb/µ = 10

Kn′ = 10−2

Kn′ = 10−8

µb/µ = 59/12

Figure 2-2. The dependence of the correction function C(τ) on the bulk viscosity.Results are plotted for µb/µ = {0, 1, 59/12, 10} and Kn′ = {10−2, 10−8}.

2.7 Inviscid and Viscous Unsteady Force Kernels and Numerical Confirmation

Based on results presented in the previous sections, we write

Fδ

mf c0/a=

1

mf c0/a

(Fδ,qs + Fδ,iu + Fδ,vu

)= −9

2

ν

ac0− e−τ cos τ − 9

2

√Kn′

π

C(τ)√τ

. (2–23)

While the normalized inviscid unsteady force depends only on τ , the normalized viscous

unsteady force depends on both Kn′ and µb/µ through C(τ). The above force response

to a delta function acceleration can be used to define inviscid and viscous unsteady

force kernels as

Kiu(τ) = e−τ cos τ and Kvu(τ) =C(τ)√

τ. (2–24)

In the definition of Kvu we have followed the conventional notation for the Basset history

kernel KB(τ) = 1/√τ and C(τ ) is the correction function defined in Eq. (2–22).

42

To check the theoretical results and establish limits of validity, we have carried

out numerical simulations of the axisymmetric compressible flow about a spherical

particle for µb = 0. In the simulations, the spherical particle is initially stationary in

a quiescent fluid and impulsively accelerated to a final steady state. To extract the

unsteady force in the simulations, we subtract the quasi-steady force based on the

correlation CD,qs = (24/Re)(1 + 0.15Re0.687), due to Schiller and Naumann [94], from

the computed instantaneous force. The results of four simulations will be reported here.

The first three simulations are characterized by M = 10−3 and Re = {0.1, 1.0, 10.0}. The

corresponding modified Knudsen numbers are Kn′ = {10−2, 10−3, 10−4}. The results

of the simulations are shown in Fig. 2-3, where the non-dimensional force is plotted

as a function of the acoustic time τ = c0t/a. The agreement between the theory and

the simulations is excellent. The simulations capture accurately the τ−1/2 decay for

both τ ≪ 1 and τ & O(10). At intermediate times, the influence of inviscid unsteady

force becomes significant and the simulations capture this well. In fact, for Kn′ = 10−4

and 10−3, over a short time interval around τ ≈ 3.5, the unsteady force on the particle

becomes negative and is oriented along the direction of particle motion (as opposed

to the force being oriented against the direction of particle motion). This interesting

counterintuitive behavior has been observed and commented upon by Parmar et al. [79].

In the first three simulations, we have 1/(ReM) = {104, 103, 102}, respectively, and

thus the agreement between the nonlinear simulations and the linear theory is good over

the entire range of the computed time interval, as expected. We have also simulated a

case of M = 10−1 and Re = 10, corresponding to Kn′ = 10−2. The results for this case

are also plotted in Fig. 2-3. Again the agreement is excellent at small times. However,

since 1/(ReM) = 1 for this case, the effect of nonlinearity becomes important for

τ ≈ O(1) and the results of the simulation show a faster decay than the τ−1/2 behavior

predicted by the linear theory.

43

10-6 10-5 10-4 10-3 10-2 10-1 100 101 10210-3

10-2

10-1

100

101

102

103

τ

Norm

alize

dU

nst

eady

Forc

e

Kiu

Kn ′

=10 −

2Kn ′

=10 −

3Kn ′

=10 −

4

M = 10−3, Re = 0.1M = 10−3, Re = 1M = 10−3, Re = 10M = 10−1, Re = 10

Figure 2-3. Time evolution of the normalized unsteady force. Theoretical predictions(last two terms of Eq. (2–23)) are plotted as solid lines forKn′ = {10−2, 10−3, 10−4} and µb = 0. Inviscid unsteady kernel (second lastterm in Eq. (2–23)) is shown as dashes line. Corresponding simulationresults for four different cases are shown as symbols.

As τ → 0, while the inviscid kernel is equal to unity, the viscous kernel diverges as

1/√τ . At large times, the inviscid kernel decays exponentially, while the viscous kernel

decays algebraically. Thus, both at short and long times, the viscous unsteady force

dominates the inviscid unsteady force. At intermediate times, the inviscid unsteady force

becomes important and it can be shown that only for Kn′ > 5.96 × 10−2 will the viscous

unsteady force dominate the inviscid unsteady force at all times. This limiting value of

Kn′ must be viewed with caution, however, because the continuum assumption breaks

down for Kn′ > 10−2. At smaller values of Kn′, there exists an intermediate range of time,

τlow ≤ τ ≤ τupp, where the inviscid unsteady force will exceed the viscous unsteady force.

44

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-110-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Kn′

τ

µb/µ = 0

µb/µ = 1

µb/µ = 59/12

µb/µ = 10

τupp

τlow

Figure 2-4. τlow and τupp for µb/µ = {0, 1, 59/12, 10}.

Figure 2-4 presents τlow and τupp for a range of Kn′ and µb/µ. For small values of Kn′,

τupp reaches a constant value of π/2 independent of µb/µ.

2.8 Generalization of the BBO Equation to Compressible Flows

The above-presented results can be used to write a general expression for the force

on a particle undergoing arbitrary time-dependent motion v(t) in a viscous compressible

fluid. The generalization of the BBO equation to compressible flow can be expressed as

mp

dv

dt= −6πaµv−mf

∫ t

−∞Kiu

((t − ξ)

c0

a

) dvdξ

d(ξc0

a

)− 6a2ρ0

√πν

∫ t

−∞Kvu(t − ξ)

dv

dξdξ ,

(2–25)

where the inviscid and viscous unsteady force kernels are given in Eq. (2–24).

The above equation is valid in the limit of zero Reynolds and Mach numbers. The

significance of this extension is two-fold. First, it includes explicit expressions for

45

the inviscid unsteady and viscous unsteady components of the force that reduce

to their well-known counterparts in the incompressible limit. Second, the extension

can be combined with other forces such as buoyancy, gravity, lift and electrostatic

forces to give a complete equation of motion for the particle. The compressible BBO

equation can be used to track accurately large numbers of particles in Eulerian-Eulerian

or Eulerian-Lagrangian simulations of compressible multiphase flows. The above

expression also allows for back-coupling of the force onto the carrier fluid phase.

Both the inviscid and viscous compressibility corrections decay rapidly and can be

ignored for τ & 10. Therefore, if the time scale of unsteadiness is much larger than

t & 10 a/c0, the incompressible formulation is sufficient. Note that it was mentioned

earlier that the long-time integral of the inviscid unsteady kernel Kiu(τ) reduces to

the added-mass coefficient of 1/2. A similar result can be established for the viscous

unsteady kernel Kvu(τ) given by Eq. (2–24). The long-time integration of Kvu(τ) can be

shown to approach the incompressible limit,∫ t

0

Kvu(ξ) dξ −∫ t

0

KB(ξ) dξ → 0 for t ≫ 1 . (2–26)

When the proposed equation of motion (2–25) is used in practice, an expression

for C(τ) is required. The following curve-fit can be employed for this purpose, assuming

that µb = 0:

C(τ) =4

9+

4

9√3

1

1 + 2.38(

τKn′

)0.57e1.02τ/Kn

′+ 2τC1e−C2τ {cos [C3(τ − C4)] + sin [C3(τ − C4)]}

+5

9

τC5

τC5 + C6

, (2–27)

46

where

C1 = 0.96 + 1.71 exp[0.51(logKn′ + 1.25)

], C2 = 1.14 + 0.22 exp

[0.57(logKn′ + 1.45)

],

(2–28)

C3 = 0.87− 0.26 exp[0.50(logKn′ + 3.55)

], C4 = 0.25− 1.38 exp

[0.60(logKn′ + 1.98)

],

(2–29)

C5 = 3.38 + 7.82 exp[0.72(logKn′ − 0.16)

], C6 = 5.09 + 5.71 exp

[0.89(logKn′ + 2.42)

].

(2–30)

The limiting behavior of this curve fit is consistent with Eq. (2–26). The accuracy of the

curve fit can be judged from Fig. 2-1 for Kn′ = 10−5. The agreement is equally good for

other values of Kn′. The maximum error of the curve-fit is less than 1 percent.

Finally, it should also be pointed out the kernels presented in Eq. (2–24) combined

with the above correction function are appropriate in the limit of zero Reynolds and Mach

numbers. The finite Mach-number influence on the inviscid kernel has been addressed

by Parmar et al. [79]. Similarly, the correction function C(τ) can be expected to depend

on both Reynolds and Mach numbers. Mei and Adrian [68] and Kim et al. [52] have

established that non-linearities due to finite Reynolds number have a complex effect on

the viscous unsteady kernel, which also becomes dependent on the precise nature of

the acceleration and deceleration. Dependencies on the Reynolds and Mach numbers

of similar complexity can be expected to arise in the compressible viscous kernel also.

2.9 Conclusions

We have obtained an explicit equation for the time-dependent force on a spherical

particle undergoing arbitrary unsteady motion in a compressible flow. The resulting

equation of motion is the generalization of the Basset-Boussinesq-Oseen equation

to the compressible regime. The significance of this extension is that it includes

explicit expressions for the quasi-steady, the inviscid unsteady, and the viscous

unsteady components of the force, which reduce to their well-known counterparts in

47

the incompressible limit. The effect of compressibility on the inviscid unsteady force is

significant. The finite speed of sound in a compressible flow destroys the instantaneous

relationship between acceleration and the inviscid unsteady force and regularizes the

singular delta-function force to a smooth oscillatory exponential decay. The effect of

compressibility on the viscous unsteady force is modest. The overall 1/√t decay rate of

the viscous unsteady force is maintained the same as that of the Basset history force.

The modification due to compressibility appears as a multiplicative correction factor

C(c0t/a) to the Basset history force, whose value is bounded, 4/9 ≤ C(c0t/a) < 1.5

(for zero bulk viscosity). The effect of bulk viscosity is not strong and is limited to very

short times, c0t/a ≪ Kn′. The effect of compressibility on the inviscid unsteady and the

viscous unsteady is significant only up to few acoustic times, say c0t/a < 10.

48

CHAPTER 3EQUATION OF MOTION FOR A SPHERE IN NON-UNIFORM COMPRESSIBLE

FLOWS

Viscous compressible Navier-Stokes equations are considered for transient force

on a spherical particle undergoing unsteady motion in a non-homogeneous flow.

Compressible extension to Maxey-Riley-Gatignol equation is proposed for particle

dynamics in a viscous compressible flow at low Mach and Reynolds

3.1 Introduction

Stokes [101] studied the steady and oscillatory motion of a spherical particle in an

incompressible fluid and obtained expressions for the steady and frequency-dependent

drag in the limit of creeping flow. The corresponding expression in the time domain

for the drag on a particle undergoing arbitrary motion was obtained independently by

Basset [8], Boussinesq [10], and Oseen [77]. The resulting so-called BBO equation

describes the time-dependent motion of a particle in an incompressible quiescent fluid

and can be written as,

mp

dup

dt= −6πaµup −

1

2mf

dup

dt− 6a2ρ

√πν

∫ t

−∞

1√t − ξ

dup

dt

∣∣∣∣t=ξ

dξ , (3–1)

where mp, up(t), and a are the particle mass, velocity and radius, respectively, mf is

the mass of fluid displaced by the particle and ρ, µ, and ν = µ/ρ are the density and

dynamic and kinematic viscosities of the fluid, respectively. The three terms on the

right-hand side are the quasi-steady (Stokes) drag, inviscid unsteady (added-mass), and

viscous unsteady (Basset-history) forces, respectively.

The effect of ambient-flow inhomogeneity was first studied by Faxen [31], who

obtained the result that the steady drag on a spherical particle is given by 6πaµuS,

where uS is the undisturbed ambient fluid velocity averaged over the particle surface.

The influence of inhomogeneity for unsteady particle motion was first considered in

the frequency-domain solution of Mazur and Bedeaux [65]. Maxey and Riley [63] and

Gatignol [36] independently obtained an explicit expression for the Faxen correction

49

to the time-dependent force on a particle in the creeping-flow limit. The resulting

Maxey-Riley-Gatignol (MRG) equation describes the time-dependent motion of a particle

in an unsteady inhomogeneous incompressible flow.

The MRG equation and its extensions have been used extensively in the tracking

of particles, bubbles, and droplets through complex ambient flows. The MRG equation

is commonly held to be applicable to incompressible flows. However, it is valid for

compressible ambient flows also, provided that the compressibility of the perturbation

flow caused by the particle can be ignored. There are situations, such as shock-particle

interaction, where the compressibility of the perturbation flow becomes important.

Furthermore, even for Mach numbers approaching the incompressible limit, a compressible

formulation of equation of particle motion is required if the force evolution on the

acoustic time scale, i.e., t ∼ a/c , where c is the speed of sound, is of interest. In this

work, we present a generalized MRG equation that fully accounts for the compressible

nature of the perturbation flow.

The earliest effort to include the effect of compressibility on the viscous motion

of a particle appears to be that of Zwanzig and Bixon [119]. This and subsequent

works by Temkin and Leung [110], Metiu et al. [70], Felderhof [32, 33], and Guz [39]

presented exact solutions of the linearized compressible Navier-Stokes equations for

the unsteady homogeneous flow around a particle in the frequency or Laplace domains.

Recent work by Parmar et al. [81] exploited the frequency-domain solution to obtain an

explicit expression for the time-dependent force on a particle that is valid on the acoustic

time scale. The results of Parmar et al. [81] can be considered as a generalized BBO

equation that accounts for the compressibility of the perturbation flow.

In the same manner in which the MRG equation extends the BBO equation to

inhomogeneous incompressible flows, here we extend the compressible formulation

of Parmar et al. [81] to include the effects of the inhomogeneity of the ambient flow.

The simultaneous inclusion of inhomogeneity, unsteadiness, and compressibility poses

50

significant challenges even in the limit of zero Reynolds and Mach numbers. A rigorous

derivation has been made possible by the following steps: (i) Following Maxey and Riley

[63], we formulate the governing equation in the moving reference frame attached to the

particle and separate the flow into undisturbed ambient and perturbation components.

(ii) The governing equations are linearized and simplified through rigorous scaling

analysis. (iii) A density-weighted velocity transformation is the key step that transforms

the problem to an equivalent constant-density problem. (iv) The resulting linearized

compressible Navier-Stokes equations are solved by making use of the generalized

Lorentz reciprocal theorem to obtain an expression for the force on the particle in the

frequency domain. (Similar results can also be obtained by following the induced-force

approach of Bedeaux and Mazur [9].) (v) Following Parmar et al. [81], we transform the

solution to the time domain and (vi) finally by invoking Galilean invariance we obtain an

explicit expression for the time-dependent force on a particle.

The above steps are described in detail in Sections 2 to 7. The resulting compressible

equation of motion involves inviscid and viscous force kernels, which are obtained in

Section 8. The final equation of motion is presented and its limitations are discussed

in Section 9. Here we also show that the present compressible equation of motion

correctly reduces to BBO and MRG equations in the appropriate limits.

3.2 Governing Equations for Flow Around a Moving Particle

The equations governing the undisturbed ambient flow in the absence of the particle

are the compressible Navier-Stokes equations,

∂ρ0∂~t

+ ~∇ · ρ0u0 = 0 , (3–2)

∂ρ0u0∂~t

+ ~∇ · ρ0u0u0 = −ρ0g+ ~∇ · σ0 , (3–3)

σ0 = −p0 I+ µ(~∇u0 + (~∇u0)T

)+(µb − 2

3µ)~∇ · u0 I , (3–4)

where the symbols are defined as above, the subscript (·)0 indicates the undisturbed

ambient flow, g is the gravitational acceleration, I is the unit tensor, and µb is the bulk

51

viscosity. The time derivative ∂/∂~t and the gradient operator ~∇ are expressed in the

fixed laboratory coordinate ~x. The Knudsen number is assumed to be small, so that the

energy equation decouples from the continuity and momentum equations. Closure is

provided by the definition of the sound speed as c20 = ∂p0/∂ρ0.

Consider a spherical particle of radius a with center at ~xp(t), translating and rotating

with velocities up(t) and p(t). The ambient flow is modified by the particle and is

governed by the following set of equations,

∂ρ

∂~t+ ~∇ · ρu = 0 , (3–5)

∂ρu

∂~t+ ~∇ · ρuu = −ρg+ ~∇ · σ , (3–6)

σ = −p I+ µ(~∇u+ (~∇u)T

)+(µb − 2

3µ)~∇ · u I , (3–7)

where c2 = ∂p/∂ρ provides closure. The boundary conditions are

u(~x, ~t) = up(~t) +p × (~x− ~xp) for |~x− ~xp(~t)| = a , (3–8)

and

ρ(~x, ~t) = ρ0(~x, ~t), p(~x, ~t) = p0(~x, ~t), u(~x, ~t) = u0(~x, ~t) for |~x− ~xp(~t)| → ∞ . (3–9)

The equation of motion of the particle is

mp

dup

d~t= mpg+

∮S

σ · n dS , (3–10)

where the integration is over the surface of the particle, n is outward unit normal vector,

and g has been assumed constant.

3.3 Moving Reference Frame and Separation of Disturbance Flow

For further analysis, it is convenient to attach the reference frame to the center of

the moving particle, henceforth termed the “moving reference frame” and denoted by

x = ~x − ~xp. The fluid velocity in the moving reference frame is v(x, t) = u(x, t) − up(t)

52

and the governing equations become

∂ρ

∂t+∇ · ρv = 0 , (3–11)

∂ρv

∂t+∇ · ρvv = −ρdup

dt− ρg+∇ · τ , (3–12)

τ = −p I+ µ(∇v + (∇v)T

)+(µb − 2

3µ)∇ · v I , (3–13)

where the time derivative ∂/∂t and the gradient operator ∇ are expressed in the moving

reference frame. When the governing equations are linearized as outlined in Section

3.4, the influence of particle translation and rotation decouples. Thus, ignoring particle

rotation, the boundary conditions become

v(x, t) = 0 for |x| = a , (3–14a)

and

v(x, t) = u0(x, t)− up(t) for |x| → ∞ . (3–14b)

The flow field (ρ, v, p) can be separated into the undisturbed flow field (ρ0, v0, p0)

and the disturbance flow field (ρ′, v′, p′). The undisturbed velocity in the moving

reference frame is v0 = u0 − up, where u0 and up are given by Eqs. (3–3) and (3–10),

respectively.

The governing equations for the disturbance flow are

∂ρ′

∂t+∇ · (ρ0v′ + ρ′v0 + ρ′v′) = 0 , (3–15)

∂

∂t(ρ0v

′ + ρ′v0 + ρ′v′) +∇ · (2ρ0v0v′ + ρ′v0v0 + ρ0v′v′ + 2ρ′v′v0 + ρ′v′v′)

= −ρ′dupdt− ρ′g+∇ · τ ′ , (3–16)

τ ′ = −p′ I+ µ(∇v′ + (∇v′)T

)+(µb − 2

3µ)∇ · v′ I , (3–17)

with boundary conditions

v′(x, t) = up(t)− u0(x, t) for |x| = a , (3–18)

53

and

v′(x, t) = 0 for |x| → ∞ . (3–19)

3.4 Scaling Analysis

We now simplify the governing equations by considering the relative importance

of the different terms. In the process, we derive a set of conditions for the simplified

governing equations to be valid. In the following, [f ] indicates the appropriate scale

of f . The undisturbed density, pressure, and relative velocity at the initial location of

the particle will be chosen as the reference density, pressure, and velocity scales, i.e.,

ρr = ρ0(0, 0), pr = p0(0, 0), and vr = |u0(0, 0)− up(0)|. Acoustic, viscous, and convective

time scales can be defined as

ta =a

c0, tv =

a2

ν, tc =

a

vr. (3–20)

Taking the radius of the particle to be the length scale, we define Mach and Reynolds

numbers as

M =vr

c0=

ta

tcand Re =

vra

ν=

tv

tc. (3–21)

The present analysis is limited to very small Reynolds and Mach numbers. In addition,

we restrict attention to the continuum regime. Therefore, the Knudsen number, which is

proportional to M/Re, is required to be small also. As a result, we express the first of our

conditions as

Condition 1: M≪ Re≪ 1 or ta ≪ tv ≪ tc . (3–22)

Next, we establish the scaling of the perturbation density from a balance of the first

two terms in Eq. (3–15). We infer that

[ρ′] ∼

ρr for t ∼ O(tc) ,

Re ρr for t ∼ O(tv) ,

M ρr for t ∼ O(ta) ,

(3–23)

54

and thus in order to restrict the density perturbations to remain small we limit attention to

only acoustic and viscous time scales. In other words, we impose the restriction that

Condition 2: t ≪ tc . (3–24)

The scale for the perturbation velocity is equal to the ambient velocity scale in the

moving reference frame, i.e., [v′] ∼ vr .

We now wish to contain the scales of spatial and temporal variations of undisturbed

ambient density and velocity fields. Let Lv and Lρ be the length scales of the undisturbed

ambient velocity and density variations, approximated as Lv ∼ vr/[∇v0] and Lρ ∼

ρr/[∇ρ0], respectively. Let the corresponding time scales be Tv ∼ vr/[∂v0/∂t] and

Tρ ∼ ρr/[∂ρ0/∂t], respectively. We require the spatial variations of the ambient velocity

to be contained over the particle radius. Similarly, we require temporal variations of the

ambient velocity to be contained on the convective time scale. These requirements lead

to

Conditions 3:

a . Lv ⇒ a

vr[∇v0] . 1 ,

a

vr. Tv ⇒ a

v 2r

[∂v0∂t

]. 1 .

(3–25)

Along the same lines, we require variations of the ambient density over the particle

radius and on the convective time scale to be contained, leading to

Conditions 4′:

a . Lρ ⇒ a

ρr[∇ρ0] . 1 ,

a

vr. Tρ ⇒ a

ρrvr

[∂ρ0∂t

]. 1 .

(3–26)

Conditions 3 and 4′ are consistent with the compressible Navier-Stokes equations

(Eqs. (3–2) to (3–4)) for the undisturbed ambient flow. It should be noted that the length

and velocity scales of the undisturbed ambient flow at the system level may be such

that the macro scale Reynolds number is large. As a result, the nonlinear effects are

important on that scale. Here we will linearize the equations by ignoring nonlinearity on

the scale of the particle. Given an ambient flow, conditions 3 and 4′ can be interpreted

55

as imposing restrictions on the allowable particle size for the linearization to be valid.

Although conditions 4′ are sufficient for linearization, as we will see below, they will be

replaced by stronger conditions to further contain the perturbation density. Linearization

further requires that

Conditions 5:

a2

vrν

[dup

dt

]. 1 ,

a2

vrν[g] . 1 .

(3–27)

The above five conditions allow Eqs. (3–15) and (3–16) to be simplified to

∂ρ′

∂t+∇ · (ρ0v′) = 0 , (3–28)

∂ρ0v′

∂t= ∇ ·

[−p′ I+ µ

(∇v′ + (∇v′)T

)+(µb − 2

3µ)∇ · v′ I

], (3–29)

with c20 = p′/ρ′.

3.5 Density-Weighted Velocity Transformation

The key to solving the linearized perturbation equations is to define a density-weighted

velocity as

V =ρ0v

′

ρr, (3–30)

and reduce the problem to one of constant uniform ambient density. To complete this

reduction we need to further constrain the ambient-density variation. The ambient-density

variation near the particle is approximated with a truncated Taylor series as

ρ0(x, t) = ρr + t

(∂ρ0∂t

)(0,0)

+ x · (∇ρ0)(0,0) , (3–31)

where (·)(0,0) indicates evaluation at x = 0 and t = 0. We require the approximation to

be valid for t ∼ O(tv). Over this time scale, the distances traveled by the convective,

shear, and acoustic waves are O(Re a), O(a), and O(Re a/M) respectively. Since we

want to simplify the density in the viscous terms, we require the truncated Taylor series

to be adequate for |x| ∼ O(a). Conditions 4′ for [∂ρ0/∂t] is sufficient for the second

term on the right-hand side of Eq. (3–31) to be of smaller order. However, we require a

56

stronger restriction on [∇ρ0] than that expressed by conditions 4′ for the Taylor series

to be convergent. Thus, we arrive at the following conditions for the ambient-density

variation that replace conditions 4′,

Conditions 4:

a≪ Lρ ⇒ a

ρr[∇ρ0]≪ 1 ,

a

vr. Tρ ⇒ a

ρrvr

[∂ρ0∂t

]. 1 .

(3–32)

With the density-weighted velocity transformation and conditions 4, Eqs. (3–28) and

(3–29) can be simplified to

∂ρ′

∂t+ ρr∇ · V = 0 , (3–33)

ρr∂V

∂t= ∇ ·

[−p′ I+ µ

((∇V) + (∇V)T

)+(µb − 2

3µ)∇ · V I

]. (3–34)

The corresponding boundary conditions for the density-weighted velocity on the particle

surface and farfield are,

V(x, t) =ρ0(x, t)

ρr(up(t)− u0(x, t)) = Vp(x, t) for |x| = a , (3–35)

V(x, t) = 0 for |x| → ∞ . (3–36)

The advantage of the transformation is that the variable density is now absent from

the equations and that the undisturbed ambient density appears only in the boundary

condition on the particle surface.

3.6 Hydrodynamic Force due to Undisturbed Flow

The total hydrodynamic force is taken to be the sum of the force F0 due to

undisturbed ambient flow and the force F′ due to the disturbance flow. The former is

defined as

F0 =

∮S

τ0 · n dS , (3–37)

where τ0 is the stress due to the undisturbed flow.

57

We simplify the hydrodynamic force due to the undisturbed flow through the

divergence theorem and employing the momentum equation of the ambient flow in the

moving reference frame to obtain

F0 =

∫Vρ0

(Dv0

Dt+

dup

dt− g)dV . (3–38)

Here, V is the volume of the particle and we define the mass of undisturbed fluid

displaced by the particle to be mf =∫V ρ0dV. Assuming the body-force density g

and particle acceleration dup/dt to be constant, the hydrodynamic force due to the

undisturbed flow can be written in the original laboratory frame of reference as

F0 =

∫Vρ0Du0

D~tdV−mf g . (3–39)

3.7 Reciprocal Theorem for Compressible Perturbation Flow

Equations (3–33) and (3–34) are identical to those solved by Zwanzig and Bixon

[119]. However, the boundary conditions given by Eqs. (3–35) and (3–36) are more

complex due to the inhomogeneous ambient flow. If ρ0 and u0 were to be homogeneous

then the solution of Zwanzig and Bixon [119] readily applies in Laplace space and

the corresponding time-domain solution is given by Parmar et al. [81]. However, the

inhomogeneous nature of the boundary condition given by Eq. (3–35) complicates the

solution.

We employ the Lorentz reciprocal theorem, see Happel and Brenner [40], which

allows determination of force on the particle in a complex Stokes flow without actually

solving for the details of the perturbation. In the present application, one just needs

the detailed solution of the homogeneous uniform ambient flow problem, and it can

be used elegantly in the reciprocal theorem to develop an expression for the desired

integral quantities in a more complex inhomogeneous flow. Peres [84] used the

reciprocal theorem to obtain a simple and elegant proof of Faxen’s theorem for steady

58

incompressible flows. Gatignol [36] and Maxey and Riley [63] also made use of the

reciprocal theorem to extend Faxen’s theorem to unsteady incompressible flows.

Here, a generalization of the reciprocal theorem to compressible unsteady flows is

required. The reciprocal theorem for a linearized compressible flow has been obtained

and exploited by several researchers, see Chow and Hermans [20], Kaneda [51], and

Cunha et al. [26]. Consider two zero Reynolds number compressible flow fields around

a spherical particle, with (ρi , vi , pi ), satisfying Eqs. (3–33) and (3–34) and the closure

relation c20 = p′i/ρ′i . Here, the subscript i = 1, 2 indicate the two different flow fields.

Let the ambient velocity fields be such that they vanish far away from the sphere and

be zero for t ≤ 0. Then the reciprocal relation between the two flows can be cast as an

integral relation, ∮S

L(v2) · L(τ1) · n dS =

∮S

L(v1) · L(τ2) · n dS , (3–40)

where L(·) is the Laplace transform, the integration is over the surface of the sphere,

and τi are the stress fields created by the respective flows that satisfy Eq. (3–13).

We choose the unknown complex flow field arising from inhomogeneous boundary

condition on the surface of the sphere to be flow field 1 (ρ1, v1, p1). For now the

inhomogeneous boundary condition is chosen to be

v1(x, t) = u1,p(x, t) for |x| = a , (3–41)

which will later be set to Eq. (3–35), corresponding to the problem to be solved. The

desired quantity of interest is the net time-dependent hydrodynamic force on the particle,

which can be expressed in the Laplace space as

L(F1) =∮S

L(τ1) · n dS . (3–42)

We let the flow field 2 (ρ2, v2, p2) to be that due to the unsteady motion of a sphere in an

otherwise quiescent fluid, whose solution is known, see Zwanzig and Bixon [119]. The

59

boundary condition for v2 is

v2(x, t) = u2,p(t) for |x| = a . (3–43)

Using Eq. (3–40) and the boundary conditions given by Eqs. (3–41) and (3–43)

leads to

L(u2,p) · L(F1) =∮S

L(u1,p(x)) · L(τ2) · n dS . (3–44)

To simplify this relation, we recognize that in the present case of axisymmetric flow

the traction vector L(τ2) · n on the surface of the sphere can be written as a linear

combination of components in the radial direction and along the direction of particle

motion as

L(τ2) · n = H1(s)(L(u2,p) · n)n+ H2(s)L(u2,p) , (3–45)

where s is the Laplace variable. The important property that we will exploit is that H1(s)

and H2(s) are independent of the details of the motion and as will be shown later they

are invariant on the surface of the sphere (i.e., independent of the spherical coordinates

θ and ϕ). Substituting Eq. (3–45) in Eq. (3–44) and rearranging,

L(F1) · L(u2,p)

= H1

∮S

(L(u2,p) · n)(L(u1,p(x)) · n) dS + H2

∮S

L(u1,p(x)) · L(u2,p) dS ,

=H1

a

∮S

[{r · L(u2,p)}L(u1,p(x))] · n dS + H2

[∮S

L(u1,p(x)) dS]· L(u2,p) ,

=H1

a

∫V∇ · [{r · L(u2,p)}L(u1,p(x))] dV+ H2

[∮S

L(u1,p(x)) dS]· L(u2,p) ,

=

[H1

a

∫V[L(u1,p(x)) + r {∇ · L(u1,p(x))}] dV+ H2

∮S

L(u1,p(x)) dS]· L(u2,p) . (3–46)

Defining volume and surface averages as

fV(x, t) =

3

4πa3

∫Vf (x− x′, t) dx′ and f

S(x, t) =

1

4πa2

∮S

f (x− x′, t) dx′ , (3–47)

60

we rewrite the above as

L(F1) =4

3πa2H1

{L(u1,p(x, t))

V+ r [∇ · L(u1,p(x, t))]

V}+ 4πa2H2

[L(u1,p(x, t))

S].

(3–48)

Thus we obtain an expression for the force on a spherical particle due to the inhomogeneous

velocity boundary condition on its surface in terms of surface and volume averages of

the inhomogeneous boundary condition.

3.8 Hydrodynamic Force due to the Disturbance flow

The detailed solution of the flow field (ρ2, v2, p2) due to a sphere undergoing

arbitrary motion u2,p(t) in a quiescent fluid has been obtained by many beginning with

Zwanzig and Bixon [119]. In Laplace space, the surface traction vector can be cast as

L(τ2) · n =− aρrs(1 + r1)r

22 − (1 + r2)r

21

r 21 (1 + r2) + (2 + 2r1 + r 21 )r22

(L(u2,p) · n)n

− aρrs(3 + 3r1 + r 21 )(1 + r2)

r 21 (1 + r2) + (2 + 2r1 + r 21 )r22

L(u2,p) , (3–49)

where

r1(s) =as/c0√

1 + (µb/µ+ 4/3) νs/c20and r2(s) = a

√s

ν. (3–50)

Defining transfer functions G v and G s ,

G s(r1, r2) =3(3 + 3r1 + r 21 )(1 + r2)

r 21 (1 + r2) + (2 + 2r1 + r 21 )r22

,

G v(r1, r2) =(1 + r1)r

22 − (1 + r2)r

21

r 21 (1 + r2) + (2 + 2r1 + r 21 )r22

, (3–51)

the traction on the sphere can be written as

L(τ2) · n = −aρrs[G v(L(u2,p) · n)n+

G s

3L(u2,p)

]. (3–52)

It is interesting to note that the normal component of the traction vector on the particle

surface is proportional to the particle velocity resolved along the normal direction. By

contrast, the traction component along the direction of particle motion is invariant along

61

the entire surface of the sphere. Comparing the above expression with Eq. (3–45), it can

be seen that H1 = −aρrsG v and H2 = −aρrsG s/3 and thereby it can be verified that H1

and H2 are dependent only on the Laplace variable as asserted in the application of the

reciprocal theorem in Eq. (3–46).

Defining the hydrodynamic force F′ due to the perturbation flow as

F′ =

∮S

τ ′ · n dS , (3–53)

where ∇ · τ ′ is given by the right-hand side of Eq. (3–34). From Eq. (3–48) we obtain an

expression for the force due to the perturbation flow in the Laplace space as

L(F′) =4

3πa3ρrs

{G sL

[Vp(x, t)

S]+ G vL

[Vp(x, t)

V+ r∇ · Vp(x, t)

V]}. (3–54)

where Vp(x, t) is the inhomogeneous velocity distribution on the particle surface,

given in Eq. (3–35). The transfer functions G s and G v apply separately to surface and

volume averages of the undisturbed ambient flow quantities and can be interpreted

as follows. The transfer function G s corresponds to the force response in Laplace

space due to a step change in the surface average of the density-weighted velocity

Vp. The transfer function G v corresponds to the force response due to a step change

in the volume-averaged quantity presented within the second square brackets on the

right-hand side of Eq. (3–54).

The above expression simplifies for a homogeneous undisturbed ambient flow

(ρ0 = constant, u0 = constant). Then, ∇ · Vp = 0, and since Vp is a constant, its surface

and volume averages are the same as the constant value. Thus, we recover the result of

Zwanzig and Bixon [119] for a uniform compressible flow,

L(F′) = −43πa3ρrs [G

v + G s ]L(Vp) = −4

3πa3ρrs GL(Vp) , (3–55)

where G is the total response function. If the undisturbed ambient flow were to be of

constant density and incompressible, then ∇ · Vp = 0. However, if the ambient flow is

62

inhomogeneous, in general VVp = V

Sp and as a result the force due to the perturbation

flow reduces to

L(F′) = −43πa3ρrs

[G sL(VS

p) + G vL(VVp )]. (3–56)

If we further assume the compressibility effects of the perturbation field to be unimportant,

G s and G v can be simplified and the above expression reduces to that given in

the frequency-domain solution of Mazur and Bedeaux [65] for an inhomogeneous

incompressible flow. The corresponding Laplace transform to the time domain will result

in the MRG equation.

Following Parmar et al. [81], Eq. (3–54) can be transformed into the time domain

and separated into quasi-steady and various unsteady contributions. The expression for

the force due to the disturbance flow can now be written as

F′(t) = 6πaµ(u0S − up)

+ 6a2√πν

∫ t

−∞K svu(t − ξ)

(dρ0u0

S

dt

∣∣∣∣t=ξ

− dρS0updt

∣∣∣∣t=ξ

)dξ

+4

3πa3

∫ t

−∞Kiu

((t − ξ)

c0

a

)( dρ0u0V

dt

∣∣∣∣t=ξ

− dρV0updt

∣∣∣∣t=ξ

)c0

adξ

+ 6a2√πν

∫ t

−∞K vvu(t − ξ)

(dρ0u0

V

dt

∣∣∣∣t=ξ

− dρV0updt

∣∣∣∣t=ξ

)dξ

+4

3πa3

∫ t

−∞Kiu

((t − ξ)

c0

a

) d2

dt2rρ0

V∣∣∣∣t=ξ

c0

adξ

+ 6a2√πν

∫ t

−∞K vvu(t − ξ)

d2

dt2rρ0

V∣∣∣∣t=ξ

dξ , (3–57)

where Kiu(ξ) is the kernel of the inviscid unsteady force and K vvu(ξ) and K s

vu(ξ) are the

viscous kernels that operate on volume and surface integrals respectively. In obtaining

the above expressions, we have used the result that the transfer functions G s and G v

can be split,

G s = G sqs + G s

vu and G v = G viu + G v

vu , (3–58)

63

where the quasi-steady part of the transfer function associated with the surface average

and the inviscid unsteady part of the transfer function associated with the volume

average, defined below, have been separated:

G sqs =

9

2r 22and G v

iu =1 + r1,inv

1 + (1 + r1,inv)2, (3–59)

where

r1,inv = limν→0

r1 = as/c0 . (3–60)

The Laplace inverse of the term involving G sqs gives rise to the quasi-steady contribution

defined in terms of the surface-averaged velocity. The Laplace inverse of the other three

transfer functions can be used to define inviscid and viscous kernels as shown below:

c0

aKiu = L−1 [G v

iu] , (3–61)

9

2a

√ν

πK svu = L−1 [G s

vu] , (3–62)

9

2a

√ν

πK vvu = L−1 [G v

vu] . (3–63)

The first two terms in Eq. (3–57) arise from the first term on the right-hand side of

Eq. (3–54). The third and fourth terms in Eq. (3–57) arise from the second term on

the right-hand side of Eq. (3–54). The last two terms in Eq. (3–57) arise from the last

term on the right-hand side of Eq. (3–54). Thus the last four terms of Eq. (3–57) are

related to volume averages. As will be shown below, the last two additional terms are

much smaller than the other contributions and can thus be ignored under the conditions

considered in this work.

We can consider different limiting behavior of the transfer functions G svu and G v

vu and

the corresponding kernels K svu and K v

vu. Consider first the inviscid limit, where

limν→0

G svu → 0 ⇒ K s

vu → 0 , (3–64)

limν→0

G vvu → 0 ⇒ K v

vu → 0 . (3–65)

64

In this limit the quasi-steady and the viscous unsteady forces vanish as expected

one recovers the inviscid unsteady component as the only force on the particle. Next

consider the incompressible limit, where

limc0→∞

G svu →

9

2r2⇒ K s

vu →1√t, (3–66)

limc0→∞

G vvu → 0 ⇒ K v

vu → 0 . (3–67)

The kernel K svu reduces to the Basset-history kernel in the incompressible limit. The

kernel K vvu is zero in the known limits of inviscid and incompressible flows and thus it

appears as a new unsteady viscous kernel only in viscous compressible flows.

In Eq. (3–57), d/dt must be carefully interpreted as the partial time derivative in

the frame attached to the particle or as the time derivative following the particle. In a

compressible flow, both the inviscid and viscous unsteady forces depend on the history

of relative acceleration weighted by appropriate kernels.

3.8.1 Importance of the Different Terms

In this section, estimates of the magnitude of the different terms in Eq. (3–57) are

presented using a simple scaling argument. We expect the quasi-steady contribution

(the first term on the right-hand side of Eq. (3–57)) to be the dominant term and it

scales as aµvr . The magnitude of the inviscid unsteady contribution (third term on the

right-hand side of Eq. (3–57)) can be obtained as follows. The inviscid kernel (to be

discussed below in Section 3.8.2) is O(1) only over an acoustic time scale of a/c0. The

time derivative of the density weighted velocity (dρ0u0V/dt) can be estimated to scale

as ρrv2r /a. Thus the inviscid unsteady term scales as ρrv

2r a

2, and when compared to the

quasi-steady force is O(Re).

In the viscous unsteady terms (second and fourth terms on the right-hand side of

Eq. (3–57)), the viscous kernels go as 1/√t and this decay is relevant only over the

viscous time scale tv , beyond which the viscous kernels decay faster due to non-linear

effect. Thus,∫K svudξ,

∫K vvudξ ∼

√tv = a/

√ν. With dρ0u0

V/dt scaling as ρrv2r /a the

65

second and fourth terms on the right-hand side of Eq. (3–57) can be estimated to scale

as ρrv2r a

2, and thus they are O(Re) compared to the quasi-steady force.

Now we consider the last two terms of Eq. (3–57). Using truncated Taylor series for

ρ0 as a function of r, we have

rρ0V ∼ rVρr + rrV ∇ρ0|r=0 = rrV ∇ρ0|r=0 . (3–68)

Using Eq. (3–32) in the above we obtain

rρ0V ≪ aρr , (3–69)

and using Eq. (3–32) again gives

O

[∂2

∂t2rρ0

V]≪ ρrv

2r

a. (3–70)

Thus, the fifth and sixth terms on the right-hand side of Eq. (3–57) are an order of

magnitude smaller than O[Re] compared to first term. So terms involving rρ0V can be

neglected compared to the other terms. It must be stressed that these additional terms

involving rρ0V were also obtained by Bedeaux and Mazur [9], but here we have argued

that these terms are asymptotically small and can be ignored.

3.8.2 Inviscid and Viscous Kernels

The kernel for the inviscid unsteady force can be explicitly expressed as

Kiu(τ) = L−1 [G viu] = e−τ cos τ , (3–71)

where τ = c0 t/a = t/ta is the dimensionless time nondimensionalized by the acoustic

time scale. The above expression for the inviscid kernel is applicable for a spherical

particle in the limit of zero Mach number and was originally derived by Longhorn

[58]. In an incompressible flow, the effect of the density-weighted fluid acceleration

dρ0u0V/dt or the particle acceleration dρV0up/dt will result in an instantaneous inviscid

force, which is termed the added-mass force. In a compressible flow, due to the finite

66

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

τ

Cv(τ)

Kn′ = 10−2

Kn′ = 10−5

Kn′ = 10−8

Kn′ = 10−11

Figure 3-1. The behavior of the correction function C v(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteadyforce, see Eq. (3–72), for µb/µ = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}.

propagation speed of the acoustic waves, the instantaneous inviscid force on a particle

is due of part history of relative acceleration (∂ρ0u0V/∂t − ∂ρV0up/∂t), weighted by

the inviscid unsteady kernel. Due to the exponential decay of Kiu, the inviscid force

due to instantaneous relative acceleration persists only for a few acoustic time scales

and for τ > 10 the inviscid effect of acceleration is negligible. Furthermore, due to

the cos τ term, the inviscid force can be negative at intermediate times and lead to the

counterintuitive behavior of the inviscid force pointing in the direction opposite to the

relative acceleration.

In the limit of zero Reynolds and Mach numbers, the kernels for the viscous

unsteady force that operate on volume and surface averaged quantities can be written

67

10-6 10-5 10-4 10-3 10-2 10-1 100 101 1020.6

0.8

1.0

1.2

1.4

τ

Cs(τ)

Kn′ = 10−2

Kn′ = 10−5

Kn′ = 10−8

Kn′ = 10−11

Figure 3-2. The behavior of the correction function C s(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteadyforce, see Eq. (3–72), for µb/µ = 0 and Kn′ = {10−2, 10−5, 10−8, 10−11}.

as

K vvu(t) = C v

(tc0

a

) 1√t

and K svu(t) = C s

(tc0

a

) 1√t. (3–72)

where 1/√t is the standard incompressible Basset history kernel. Thus, C v(τ)

and C s(τ) are correction functions to the incompressible viscous unsteady kernel.

Expressions for these correction functions can be written as

C v(τ) = L−1(G vvu)

(9

2

√Kn′

πτ

)−1

, (3–73)

C s(τ) = L−1(G svu)

(9

2

√Kn′

πτ

)−1

. (3–74)

For the case of homogeneous undisturbed ambient flow, the volume and surface

averages of the ambient flow quantities are identical and as a result the above two

68

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102-0.4

-0.2

0.0

0.2

0.4

0.6

τ

Cv(τ)

Kn′ = 10−2

Kn′ = 10−5

µb/µ = 0

µb/µ = 1

µb/µ = 59/12

µb/µ = 10

Figure 3-3. The behavior of the correction function C v(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteadyforce, see Eq. (3–72), for µb/µ = {0, 1, 59/12, 10} and Kn′ = {10−2, 10−5}.

corrections that apply to volume and surface averages can be combined into a single

correction function, C(τ) = C v(τ) + C s(τ). This compressibility correction function C(τ)

to the Basset kernel was recently obtained by Parmar et al. [81] in their derivation of the

compressible extension of the BBO equation.

In the present case of an inhomogeneous ambient flow the correction function

needs to be separated into C v(τ) and C s(τ), which are shown in Figs. 3-1 and 3-2,

respectively. The Laplace transforms of the response functions G vvu and G s

vu to the time

domain have not been obtained analytically. As a result, the Laplace inverse-transforms

were computed numerically and the numerically-evaluated correction functions are

shown in Figs. 3-1 and 3-2. Note that the corrections to the viscous kernel are also

functions of Kn′ = M/Re = ν/(ac0) and µb/µ.

69

10-6 10-5 10-4 10-3 10-2 10-1 100 101 1020.6

0.8

1.0

1.2

1.4

τ

Cs(τ)

µb/µ = 0µb/µ = 59/12µb/µ = 10

Figure 3-4. The behavior of the correction function C s(τ) that accounts for the volumeintegral contribution of the compressibility effect on the viscous unsteadyforce, see Eq. (3–72), for µb/µ = {0, 59/12, 10} and Kn′ = {10−2}.

As discussed by Parmar et al. [81], four asymptotic regimes can be identified for

C v(τ) and C s(τ):

Regime I: Very short time, τ ≪ Kn′ ≪ 1,

Regime II: Intermediate short time, Kn′ ≪ τ ≪ 1,

Regime III: Intermediate long time, 1≪ τ ≪ 1/(ReM),

Regime IV: Very long time, 1≪ 1/(ReM)≪ τ .

Explicit expressions for the correction functions in the time domain can be obtained

for the first three regimes. Nonlinearity becomes important for 1/(ReM) ≪ τ . Thus the

solution of the linearized governing equations presented here do not describe accurately

the force evolution in Regime IV.

Next we examine the asymptotic behavior of C v(τ) and C s(τ) in the first three

regimes. The very short-time behavior of the correction functions can be obtained by

70

considering the following limit in the Laplace space: 1 ≪ Kn′|S | ≪ |S | ≪ |S |/Kn′, where

S = as/c0. This results in the following simplified expressions for the transfer functions

G vvu and G s

vu,

G vvu ∼

(−1 +

√µbµ

+4

3

)√Kn′

S, (3–75)

G svu ∼ 3

√Kn′

S. (3–76)

Their Laplace inverse gives the time-domain behavior of the correction functions in

Regime I as

Regime I (τ ≪ Kn′ ≪ 1) :

C v(τ) ∼ 2

9

(−1 +

√µbµ

+4

3

),

C s(τ) ∼ 2

3.

(3–77)

It can be seen that very short-time behavior of C v(τ) depends on µb/µ. For µb = 0,

C v(τ ≪ Kn′ ≪ 1) = 2(2−√3)/(9

√3) ≈ 0.034 (see Fig. 3-1).

From the definition of the intermediate short time, the following condition can be

placed on the Laplace variable: Kn′|S | ≪ 1≪ |S | ≪ 1/Kn′ ≪ |S |/Kn′. Then G vvu and G s

vu

can be simplified as

G vvu ≈

1

r2

{−1 + (1 + r1)

1 + (1 + r1)2+

2

1 + (1 + r1)2− 2

[1 + (1 + r1)2]2

}, (3–78)

G svu ≈

1

r2

{3 +

3(1 + r1)

1 + (1 + r1)2

}. (3–79)

The dominant contribution for intermediate short time for C v(τ) and C s(τ) are

Regime II (Kn′ ≪ τ ≪ 1) :

C v(τ) ∼ −2

9,

C s(τ) ∼ 2

3.

(3–80)

It can been seen that C v(τ) at intermediate short times reduces to −2/9 ≈ 0.22

because of compressibility effects. Note that this reduction is independent of µb/µ. As

will be seen in Fig. 3-1, where the results of the numerical Laplace inversion are shown,

71

Regime II can be observed only if Kn′ ≪ 1. With increasing Kn′, the duration of Regime

II reduces and vanishes entirely for Kn′ ≈ 10−2. For C s there is no distinction between

Regime I and Regime II.

The intermediate long-time behavior can be obtained in a similar manner by

considering an asymptotic expansion for |S | → 0 and carrying out the Laplace inverse

we obtain

Regime III (1≪ τ ≪ 1/(ReM)) :

C v(τ)→ 0 ,

C s(τ)→ 1 .

(3–81)

Strictly speaking, the above solution for the linearized perturbation Navier-Stokes

equations is valid as long as τ ≫ 1. The additional limit of τ ≪ 1/(ReM) arises only

from neglecting the nonlinear terms.

In Fig. 3-2, the correction to the viscous kernel that applies to the surface average

becomes a constant equal to 2/3 for τ < 10−3 and for τ > 10 we recover C s = 1. In

the intermediate time we observe a damped oscillatory behavior. It can be observed

that for Kn′ < 10−5, the correction function C s becomes nearly independent of the

modified Knudsen number as well as bulk viscosity (also see Fig. 3-4). For Kn′ > 10−5,

we observe a weak quantitative dependence on Kn′. Here we limit attention to Knudsen

numbers smaller than 10−2, since the continuum assumption start to break down for

larger values. The dependence of C v and C s on the bulk viscosity is shown in Figs. 3-3

and 3-4.

3.9 Discussion

By combining the forces due to the undisturbed (Eq. (3–39)) and disturbance flows

(Eq. (3–57), without the smaller last two terms), we can derive an explicit equation of

motion for a sphere undergoing unsteady motion in a compressible time-dependent

inhomogeneous ambient flow. The resulting compressible extension of the MRG

72

equation can be expressed as

mp

dup

d~t= Fqs + Fiu + Fvu + Fbg , (3–82)

where the terms on the right-hand side are the quasi-steady, inviscid unsteady, viscous

unsteady and buoyancy/gravity forces,

Fqs = 6πaµ(uS0 − up) , (3–83)

Fiu =

∫Vρ0Du0

D~tdV+

4

3πa3

∫ ~t

−∞Kiu

((~t − ξ)

c0

a

)(Dρ0u0V

D~t

∣∣∣∣~t=ξ

− DρV0upD~t

∣∣∣∣~t=ξ

)c0

adξ ,

(3–84)

Fvu = 6a2√πν

∫ ~t

−∞K vvu(~t − ξ)

(Dρ0u0

V

D~t

∣∣∣∣~t=ξ

− DρV0upD~t

∣∣∣∣~t=ξ

)dξ

+ 6a2√πν

∫ ~t

−∞K svu(~t − ξ)

(Dρ0u0

S

D~t

∣∣∣∣~t=ξ

− DρS0upD~t

∣∣∣∣~t=ξ

)dξ , (3–85)

Fbg = (mp −mf )g . (3–86)

In interpreting and using the above equation of motion the following should be noted.

First, Eq. (3–82) is cast in terms of the fixed laboratory coordinate and thus t has

been replaced with ~t. Second, terms such as dρ0u0V/d~t in Eq. (3–57) do not preserve

Galilean invariance. The lack of invariance is caused by neglecting the advection terms

during linearization. Accordingly, in Eqs. (3–84) and (3–85), terms involving d/d~t have

been replaced by total derivative following the fluid element D/D~t. As pointed out by

Maxey and Riley [63] and discussed by Auton et al. [1], the difference between d/d~t and

D/D~t is asymptotically small and for strict Galilean invariance we suggest using D/D~t.

Note that the surface- and volume-averages ((·)S, (·)

V) are used inside the total

derivative following Gatignol [36]. In the solution of the linearized equations terms such

dρ0u0V/d~t can be replaced by dρ0u0/d~t

V, since time and space operations commute.

As a result in the above equation we could write the unsteady force contributions

instead involving terms like Dρ0u0/D~tV, where the average is of the total derivative.

73

Although these terms also preserve Galilean invariance, the total derivative and the

surface or volume averages do not commute. On the basis of the present linearized

derviation we cannot rationally choose terms such as Dρ0u0V/D~t over Dρ0u0/D~t

V, since

asymptotically they both are of the same order.

The above equation is valid for unsteady inhomogeneous ambient flows where ρ0

and u0 are functions of both space and time. The form of the above equation of motion

could have been anticipated, since it appears to combine the features of MRG equation

of motion and the compressible BBO equation of Parmar et al. [81]. For example,

the volume and surface averages are similar to those arising in the incompressible

MRG equation of motion for an inhomogeneous ambient flow, while the inviscid and

viscous compressible corrections are the same as those obtained in the compressible

BBO equation (although the viscous correction separates into two components

that apply independently for the volume and surface averages). Nevertheless, the

above equation is new and for the first time incorporates the combined effects of

unsteadiness and inhomogeneity in the ambient velocity and density on the inviscid and

viscous unsteady forces on a spherical particle in a compressible flow in a consistent

manner. The rigorous derivation is crucially dependent on the density-weighted velocity

transformation.

Most importantly equation (3–82) is cast in the time domain and thus can be readily

used to track particle. It is particularly useful when the compressibility of the perturbation

flow and the force evolution on the acoustic time scale are important.

The equation of motion reduces to the correct simplified forms when appropriate

approximations are made. (i) If the spatial variation of the ambient flow as seen by the

particle is negligibly small, the volume and surface averages can be replaced by the

corresponding undisturbed ambient values at the center of the particle and Eq. (3–82)

reduces to the equation for a homogeneous ambient flow. (ii) If we further assume

the ambient flow to be steady, the equation of motion reduces to the generalized BBO

74

equation for the compressible flow given by Parmar et al. [81]. (iii) Instead, if ambient

flow variations and particle acceleration occur only on time scales much larger than the

acoustic time scale, then the expression within the parentheses in the second term on

the right-hand side of Eq. (3–84) can be moved out of the integral and for t/ta & 10

the inviscid unsteady kernel Kiu(ξ) can be integrated to obtain the incompressible

added-mass force2

3πR3

(Dρ0u0

V

D~t− DρV0up

D~t

). (3–87)

The resulting inviscid unsteady force is in agreement with the expression derived by

Eames and Hunt [29] for the force on a sphere moving unsteadily in a compressed flow.

In a similar fashion, for t/ta & 10 the viscous unsteady kernel K svu(~t) in Eq. (3–85)

reduces to the incompressible Basset-history kernel, while K vvu(~t) becomes zero. (iv)

If we further assume ρ0 to be constant, as would be appropriate in an uniform density

incompressible flow, the resulting simplified equation of motion can be shown to be

identical to the MRG equation. (v) If, in addition, we assume the ambient flow u0 to be

constant we obtain the BBO equation.

The following limitations of Eq. (3–82) must be highlighted. First, this equation

of motion is strictly valid only in the limit of Re → 0 and M → 0. At finite Reynolds

and Mach numbers linearization of the perturbation equations will be invalid, since

Condition 1 (Eq. (3–22)) will be violated. As a result rigorous analytic solutions are

unavailable at finite Reynolds and Mach numbers. However, empirical relations

have been developed in the context of incompressible perturbation flows, and their

development has been functionally based on the corresponding zero Reynolds number

MRG equation. In particular, the superposition of the total force on the particle into

quasi-steady, inviscid unsteady and viscous unsteady contributions has been adopted

under nonlinear conditions as well. For example, at finite Reynolds numbers the Stokes

drag expression is replaced by the standard drag correlation, which is an empirical

function of Re. For a compressible flow finite Reynolds and Mach number empirical

75

corrections to the quasi-steady drag have been discussed by Parmar et al. [82]. The

inviscid unsteady force, by definition, is not influenced by the Reynolds number and

the Mach number dependence of Kiu(t) has been discussed by Parmar et al. [79]. The

nonlinear dependence of the viscous kernels can be expected to be strong and complex.

For example, in the incompressible limit it is know that the viscous kernel decays faster

than 1/√t and the actual behavior of the kernel depends not only on Re, but also on

the nature of acceleration or deceleration. When compressibility effects are taken into

account it is expected that the viscous kernels, K svu(t) and K v

vu(t) will depend on both Re

and M in a complex manner, whose behavior needs further investigation.

Limitations posed by the other conditions presented in Sections 3.4 and 3.5 can

similarly be analyzed. From the inviscid kernel and the compressibility correction to

the viscous kernels it is clear that the compressibility effects are significant only over

about 10 acoustic time scales. Since in the continuum regime we require ta ≪ tv , the

restriction C2 (Eq. 4.5) to times of the order of the viscous time scale is not a serious

limitation in the investigation of compressibility effects.

Restrictions C3 and C4 on the spatio-temporal variation of the ambient density

and velocity will limit the strict validity of the equation of motion to sufficiently small

particles. It is clear that in applications such as shock-particle interaction, the scale of

the ambient flow variation is likely to be smaller than the particle. Yet, it is encouraging

that a simplified version of the above compressible inviscid unsteady force has been

quite successful in reproducing the force evolution during the passage of a shock wave

over stationary particles, see Parmar et al. [80].

It thus appears that the above defined volume and surface averages provide a

good approximation for the effective ambient flow quantities seen by the particle even

when they vary strongly on the scale of a particle, as it can happen frequently when

compressible flow features interact with the particle.

76

The above linear formulation does not include contributions from vorticity-induced

lift forces. In a compressible flow, contributions from both ambient shear and density

gradients can exist. For example, as shown by Eames and Hunt [28], additional “lift-like”

inviscid forces arise due to baroclinic vorticity production, even under steady motion

of a particle through a field of non-uniform density. Such forces can make additional

contributions in case of shock-particle interaction and have not been included in Eq.

(3–82).

Finally, we want to place the present results in the context of the prior works of

Bedeaux and Mazur [9] and Maxey and Riley [63], which have greatly motivated our

efforts. First, the governing linearized perturbation equations (3–33) and (3–34) that

we solve are somewhat different from those solved in Bedeaux and Mazur [9]. The

differences, although asymptotically small and negligible under the present scaling

analysis, have allowed us to apply the density-weighted velocity transformation and

obtain the above results. Here, following Maxey and Riley [63], we solve the governing

equations in the moving reference frame attached to the particle and separate the force

contributions for the undisturbed and the disturbance flows. This allows exact treatment

of particle boundary condition and precise extraction of buoyancy/gravity forces as well

as pressure-gradient forces. Finally, through the Laplace inverse we present our results

in the time domain and hence readily usable for particle tracking.

Note that Maxey and Riley [63] approximated the surface and volume averages

in terms of the Laplacian of ambient velocity at the particle center. This approximation

requires a restriction on the spatial variation of ambient velocity that is more stringent

than conditions 3. This more stringent restriction is necessary to employ the following

truncated Taylor-series expansion of the ambient velocity in approximating the volume

and surface integrals,

v0(x, t) = v0(x, t)(0,0) + x · (∇v0)(0,0) +1

2xx : (∇∇v0)(0,0) . (3–88)

77

In this conditions 3 must be replaced by the stronger condition

a≪ Lv ⇒ a

vr[∇v0]≪ 1 . (3–89)

This stronger restriction stated in Maxey and Riley [63] can be relaxed if we leave the

volume and surface averages in Eq. (3–57) unmodified.

78

CHAPTER 4ON THE UNSTEADY INVISCID FORCE ON CYLINDERS AND SPHERES IN

SUBCRITICAL COMPRESSIBLE FLOWS

The unsteady inviscid force on cylinders and spheres in subcritical compressible

flow is investigated. In the limit of incompressible flow, the unsteady inviscid force on a

cylinder or sphere is the so-called added-mass force which is proportional to the product

of the mass displaced by the body and the instantaneous acceleration. In compressible

flow, the finite acoustic propagation speed means that the unsteady inviscid force

arising from an instantaneously applied constant acceleration develops gradually and

reaches steady values only for non-dimensional times c∞t/R & 10, where c∞ is the

freestream speed of sound and R is the radius of the cylinder or sphere. In this limit, an

effective added-mass coefficient may be defined. The main conclusion of our study is

that the freestream Mach number has a pronounced effect on both the peak value of the

unsteady force and the effective added-mass coefficient. At a freestream Mach number

of 0.5, the effective added-mass coefficient is about twice as large as the incompressible

value for the sphere. Coupled with an impulsive acceleration, the unsteady inviscid

force in compressible flow can be more than four times larger than that predicted from

incompressible theory. Furthermore, the effect of the ratio of specific heats on the

unsteady force becomes more pronounced as the Mach number increases.

4.1 Introduction

In an incompressible flow, well-established analytical expressions exist for the

steady and unsteady (added-mass and history) forces on cylinders and spheres in the

Stokes and inviscid limits (see, e.g., Landau & Lifshitz [54]). Analytical and empirical

extensions of the quasi-steady, added-mass, and history forces to finite Reynolds

numbers have been studied extensively (see, e.g., Crowe et al. [25] and Magnaudet

& Eames [62]). In the compressible regime, attention has generally been focused on

the steady drag force. Detailed parameterizations of the steady drag force in terms of

Mach and Reynolds numbers have been considered (see, e.g., Bailey & Hiatt [4] and

79

the references cited therein). However, our understanding of unsteady forces in the

compressible regime, arising either from the acceleration of the cylinder or sphere or

from the acceleration of the surrounding fluid, is limited.

The earliest fundamental contributions to the study of unsteady forces in the

compressible regime appear to be due to Love [61] and Taylor [106]. Miles [71]

investigated the motion of a cylinder impulsively started from rest based on the acoustic

approximation of the velocity potential equation. He treated the cases of transient motion

generated by a constant force applied over a finite time interval as well as an impulsively

applied velocity. Independently, Longhorn [58] considered the unsteady motion of a

sphere based on the same approximation. The work of Longhorn was considered by

Ffowcs Williams & Lovely [34], who determined analytically the acoustic field produced

by a sphere accelerated impulsively from rest. Both Miles and Longhorn pointed out the

limitations of the conventional added-mass concept in describing the inviscid force in

compressible flows.

These limitations are rooted in the relationship between the added-mass force

and the instantaneous acceleration. In an incompressible flow, the added-mass force

depends only on the instantaneous acceleration. In a compressible flow, on the other

hand, the inviscid force develops on an acoustic time scale R/c∞, where R is the

radius of the cylinder or sphere and c∞ is the speed of sound in the ambient fluid in the

farfield. The results of Miles and Longhorn show that under constant acceleration,

the inviscid force reaches a constant value for c∞t/R & 10. If an added-mass

coefficient is computed based on this constant long-time force, values of 1.0 and 0.5

are recovered for the cylinder and sphere, respectively. These values are consistent

with the low-Mach-number limit implicit in the acoustic approximation employed by Miles

and Longhorn. It should also be noted that in a compressible flow the force evolution in

response to constant acceleration is non-monotonic. As a result, at intermediate times

80

the instantaneous force on the cylinder or sphere can be substantially larger than the

constant final force.

Other relevant work was performed by Tracey [113], who extended Miles’s analytical

work on the motion of an impulsively started cylinder to finite Mach number. Numerical

simulations of compressible flow about an accelerating cylinder at finite Mach numbers

were performed by Brentner [13]. However, the focus of Brentner’s study was on the

propagation of acoustic energy as the cylinder accelerated impulsively from rest to a

Mach number of 0.4.

The objective of this work is to extend the results of Miles and Longhorn to finite

freestream Mach numbers. We will investigate the effect of Mach number on the

non-monotonic evolution and asymptotic long-time constant value of the unsteady force

in response to a sudden constant acceleration. To this end, we solve numerically the

Euler equations in a frame of reference attached to the cylinder and sphere, prescribe

their motion, and compute the drag coefficient. We thus take an approach similar to

Brentner, but our goal is the determination of forces and not the propagation of the

acoustic field. We restrict our attention to the subcritical Mach number regime in this

article.

4.2 Numerical Method

The numerical method solves the Euler equations in integral form cast in a frame

of reference attached to the cylinder or the sphere. The spatial discretization is based

on the flux-difference splitting method of Roe (1981) (see [41] for reference) and the

weighted essentially non-oscillatory reconstruction described by Haselbacher [41]. The

discrete equations are integrated in time using the four-stage Runge-Kutta method.

The basic methodology employed in this work has been applied to several unsteady

compressible flows and demonstrated good agreement with theory and experimental

data, see, e.g., Haselbacher et al. [43].

81

For the cylinder, a two-dimensional hexahedral grid of O-type topology is used

with 386 cells around the circumference. Relative to the cylinder radius R, the radial

grid spacing adjacent to the cylinder surface is �r/R = 1.624 × 10−2, thus producing

cells with aspect ratios of nearly unity. The radial stretching of grid cells is adjusted

such that each layer of cells consists of approximately square cells to minimize internal

wave reflections. We have employed grids consisting of up to 1,760,160 cells to assess

grid-independence of our solutions. The results shown below were obtained on a grid of

110,010 cells.

For the sphere, a hexahedral grid consisting of six blocks is used. Each block

contains 100 × 100 cells on the sphere surface and 320 cells in the radial direction.

Relative to the sphere radius R, the radial grid spacing on the surface is �r/R ≈ 2.9 ×

10−2. As for the cylinder grid, the radial stretching of cells is adjusted such that each

layer consists of approximately cuboid cells. The results shown below were obtained

with a very fine grid of 19,200,000 cells. For both cylinder and sphere computations,

the characteristic boundary conditions of Poinsot & Lele [85] are applied at the outer

boundary, located at 200R.

4.3 Results

Our objective is to extract the time-dependent inviscid force on a cylinder or sphere

in response to suddenly imposed acceleration at finite Mach numbers. To accomplish

this, the cylinder or sphere is first held fixed and a steady-state solution is obtained at

the chosen farfield Mach number M∞,0. At some time t0, we impose on the cylinder

or the sphere a constant acceleration a in the direction opposite to the ambient flow.

We maintain the constant acceleration for a finite time interval tf − t0 and remove it

thereafter. The duration of acceleration, non-dimensionalized in terms of the acoustic

time scale, is chosen to be c∞(tf − t0)/R = 20, which is sufficient for the inviscid force

to reach a constant value. The instantaneous relative Mach number increases linearly

82

during the period of acceleration and reaches a value M∞,0 + δ by the end of the interval,

where δ = a(tf − t0)/c∞.

In all cases considered, the non-dimensional acceleration α = aR/c2∞ is chosen

carefully to satisfy the following two competing requirements. First, we limit the value

of α such that δ, the change in Mach number, is kept small. This allows interpretation

of the resulting time-dependent force on the cylinder or sphere to be at a fixed or

frozen Mach number of M∞,0. Accordingly, we drop the subscript 0 and simply write

M∞ in the following. Figure 4-2 shows the time evolution of the non-dimensional force

on a cylinder for several values of the non-dimensional acceleration α starting from

M∞ = 0.3. Here, the force (per unit width) is non-dimensionalized as F = F/(mf a),

where mf is the mass of the fluid displaced per unit width of the cylinder. It can be

seen that provided α is sufficiently small, the non-dimensional force is observed to

be independent of the actual value of α. With increasing magnitude of acceleration,

e.g., for α = 1.2 × 10−3, the increase in relative Mach number over the duration of

acceleration has a significant influence on the net force and therefore the result can no

longer be considered to correspond to a frozen Mach number of 0.3. At even higher

accelerations, the instantaneous Mach number exceeds the critical value of about

0.398 and the effect of locally supersonic flow around the cylinder results in a steady

increase in the force. Clearly, α should be maintained sufficiently small to extract the

time-dependent force at a frozen Mach number. The second requirement is that the rate

of acceleration must not be too small, for otherwise the resulting force will be very weak

with low signal-to-noise ratio. In all the cases considered here, a range of acceleration

satisfies the two competing requirements. Provided α is chosen to lie within this range,

the resulting appropriately non-dimensionalized force is independent of α.

Note that the farfield Mach-number range investigated here is limited to values

below the critical Mach numbers of about 0.398 and 0.6 for the cylinder and sphere,

respectively. Below the critical farfield Mach number, the steady flows around the

83

tti tf

M∞

M∞

,0M∞,0(1 + δ)

Figure 4-1. Schematic depiction of variation of freestream Mach number duringcomputations.

cylinder and the sphere remain subsonic everywhere. Therefore, the steady-state

inviscid drag force is identically zero before the application of the acceleration as well as

after the removal of the acceleration following the decay of transients.

4.3.1 Effect of Mach Number

In an incompressible flow, a cylinder or sphere with a constant acceleration

a experiences an inviscid force of magnitude CMmf a opposite to the direction of

acceleration, where CM is the added-mass coefficient. For a cylinder CM = 1 and

for a sphere CM = 0.5. The added-mass force is realized instantaneously upon the

application of acceleration due to the infinite acoustic propagation speed implicit in the

incompressibility assumption. The force is proportional to the applied instantaneous

84

0 5 10 15 20 25 30 35-0.5

0.0

0.5

1.0

1.5

2.0 α=4.8E-3α=2.4E-3α=1.2E-3α=6.0E-4α=3.0E-4α=6.0E-5α=3.0E-5

˜ F

τ

Figure 4-2. Effect of acceleration parameter α on the unsteady force coefficient oncylinder for M∞,0 = 0.3 and γ = 1.4

acceleration and ceases to exist once the acceleration is removed. As described in §1,

once compressibility effects become important, the force is no longer dependent on only

the instantaneous acceleration.

4.3.1.1 Cylinder

In figure 4-3 we plot the time-dependent force on the cylinder as a function of

non-dimensional time τ = c∞(t − t0)/R for farfield Mach numbers ranging from

zero to 0.39. Guided by incompressible results, the dimensional force F per unit

cylinder width has been non-dimensionalized as Fcy = F/(mf a). Also plotted in the

figure is the theoretical result of Miles [71] corresponding to the limit M∞ → 0. The

non-monotonic time evolution of the inertial force in response to constant acceleration is

clearly visible. After about 12 acoustic time units the non-dimensional force Fcy reaches

a constant value of 1.0, which corresponds to the added-mass coefficient of a cylinder

85

0 5 10 15 20 25 30 35-0.5

0.0

0.5

1.0

1.5

2.0

2.5

˜ F

τ

A Force coefficient (for legend see (B))

0 5 10 15-0.25

0.00

0.25

0.50

0.75

1.00

M∞=0.39M∞=0.38M∞=0.37M∞=0.34M∞=0.30M∞=0.25M∞=0.20M∞=0.00 (Miles)

d˜ F/d

τ

τ

B Derivative of force coefficient

Figure 4-3. Comparison of computed results for cylinder with theoretical results of Miles(1951) for γ = 1.4

86

-2 -1 0 1 20

1

2

3

1.00.80.60.40.20.0

-0.2-0.4-0.6-0.8-1.0

x/R

y/R

A τ = 0.56

-2 -1 0 1 20

1

2

3

x/R

y/R

B τ = 1.67

-2 -1 0 1 20

1

2

3

x/R

y/R

C τ = 2.78

-2 -1 0 1 20

1

2

3

x/R

y/R

D τ = 11.12

Figure 4-4. Evolution of non-dimensional perturbation pressure (scaled to rangebetween minus and plus one at each instant) for cylinder at M∞ = 0.2 andγ = 1.4

in incompressible flow. In other words, for τ & 12 the acoustic disturbance arising from

the sudden onset of acceleration has radiated sufficiently far away that the near field is

approximated well by an incompressible potential flow. At intermediate times, however,

the inviscid force is observed to be substantially larger. For instance, Miles’ solution

yields a peak value of Fcy ≈ 1.17 at τ ≈ 3.1.

87

The normalized perturbation pressure (difference between the instantaneous

pressure and the initial steady pressure distributions, normalized to range between

minus and plus one) is plotted in figure 4-4 for M∞ = 0.2 at the four different times

marked by black circles in figure 4-3(A). From the compressible form of Bernoulli’s

equation, the perturbation pressure can be shown to have two contributions. The first

contribution arises from the time-dependence of the velocity potential. The second

contribution is due to changes in the square of the velocity. In figure 4-4, the influence

of the first contribution dominates at early times and a fore-aft asymmetry can be seen

clearly. At later times, with the second contribution becoming increasingly important, the

perturbation pressure increases and appears to be more fore-aft symmetric.

As indicated by figure 4-3(A), the qualitative behavior of the unsteady force remains

the same at all subcritical Mach numbers. The asymptotic long-time constant value

and the peak value of the non-dimensional force increases with M∞. For both the peak

and asymptotic values of F the effect of Mach number is substantial, as can be seen

from figure 4-5(A). The peak instantaneous force at M∞ = 0.39 is about 2.4 times as

large as that predicted from incompressible theory. The asymptotic long-time constant

value of the non-dimensional force can be seen to increase steadily from a value of unity

at M∞ → 0 to about 2.04 at M∞ = 0.39. This steady value can be considered as an

effective added-mass coefficient for compressible flow.

The ratio of peak to long-time steady force for the different Mach numbers is also

shown in figure 4-5(A). It can be seen that this ratio is nearly constant at about 1.17-1.19

for the range of Mach numbers considered. With increasing Mach number, the time at

which F reaches a peak increases. For example, for M∞ = 0.39, the peak occurs at

τ ≈ 5.26. Correspondingly, the approach to a constant value is also slightly delayed.

Interestingly, once the acceleration is turned off, the return to zero force occurs in a

manner similar to when the acceleration is first applied.

88

It is convenient to differentiate the non-dimensional force presented in figure 4-3(A)

and define a kernel as Kcy(τ) = dFcy/dτ . The resulting response kernels for the

different Mach numbers are presented in figure 4-3(B). The kernel can be interpreted

readily in terms of the non-dimensional inviscid force per unit width on a cylinder

subjected to an impulsive jump in relative velocity of u0 given as Kcy(τ = tc∞/R) =

F/(mf (c∞u0/R)). In fact, this is the form in which Miles had presented his result for

M∞ → 0. As shown by Miles, in this limit the kernel satisfies the integral equation∫ τ

0

(τ − ξ + 1)2√(τ − ξ)(τ − ξ + 2)

Kcy(ξ) dξ =√τ(τ + 2). (4–1)

The asymptotic solutions to the integral equation for small and large τ were also

obtained by Miles as

Kcy(τ ,M∞ → 0) =

1− 1

2τ − 1

16τ 2 +

5

48τ 3 if τ ≪ 1,

− 2

τ 3− 62− 24 ln 4τ

τ 5+O

(ln2τ

τ 7

)if τ ≫ 1.

(4–2)

The non-monotonicity seen in figure 4-3(A) translates to the kernel being positive for

a short duration, then becoming negative, and slowly approaching zero. This behavior

has interesting implications. The response to an impulsive jump in cylinder velocity is an

initial non-dimensional force of unit magnitude (Kcy(τ → 0)→ 1), which decays rapidly

with time. Initially the force is opposite to the direction of cylinder acceleration. But

after some time as Kcy changes sign, the inviscid hydrodynamic force on the cylinder is

along the direction of impulsive acceleration. The time at which F peaks in figure 4-3(A)

corresponds to the zero-crossing time in figure 4-3(B). The slower approach to steady

state with increasing M∞ is clearly visible.

4.3.1.2 Sphere

The time evolution of the non-dimensional force on the sphere for varying M∞ is

plotted in figure 4-6(A). The non-dimensional force is defined to be F = F/(mf a), where

mf is the mass of the fluid displaced by the sphere. The analytical result of Longhorn

89

0.0 0.1 0.2 0.3 0.41.0

1.2

1.4

1.6

1.8

2.0

2.2

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Peak, computationsSteady-state, computationsRatio peak to steady-stateJanzen-Rayleigh expansion to O(M∞

2)˜ F

max/

˜ Fm

ax,M

∞=

0or

CM

,eff

˜ Fm

ax/C

M,e

ff

M∞

A Cylinder

0.0 0.1 0.2 0.3 0.4 0.51.0

1.2

1.4

1.6

1.8

2.0

1.0

1.2

1.4

1.6

1.8

2.0Peak, computationSteady-state, computationRatio peak to steady-state

˜ Fm

ax/

˜ Fm

ax,M

∞=

0or

CM

,eff/C

M∞

=0

˜ Fm

ax/C

M,e

ff

M∞

B Sphere

Figure 4-5. Computed behavior of peak and steady-state values of F for γ = 1.4

90

[58] corresponding to the limit M∞ → 0 is also shown in the figure. In this limit, after the

initial transient, the long-time value of non-dimensional force settles at 0.5 consistent

with the added-mass coefficient of a sphere in incompressible flow. As for the cylinder,

the approach to the constant force is non-monotonic. The peak value of about 0.6 is

reached at a non-dimensional time of τ ≈ 1.6.

The effect of Mach number on the peak value of the unsteady force coefficient and

the effective added-mass coefficient are shown in figure 4-5(B). The effect of Mach

number is again substantial. The long-time asymptotic value for M∞ = 0.5 is about 0.97

and a peak value of about 1.2 is reached at τ ≈ 3.1. As before, the approach to steady

state is delayed with increasing Mach number. However, compared to the cylinder, the

steady state is approached more rapidly.

We differentiate the non-dimensional force presented in figure 4-6(A) and define

a kernel as Ksp(τ) = dFsp/dτ . The resulting response kernels are presented in figure

4-6(B) as a function of Mach number. The inviscid force on a sphere subjected to an

instantaneous jump in relative velocity is then given by mf (c∞u0/R)Ksp, where u0 is the

jump in relative velocity. In the limit M∞ → 0, Longhorn obtained

Ksp(τ ,M∞ → 0) = exp τ cos τ . (4–3)

The approach to steady state is oscillatory, but the rapid exponential decay masks the

oscillatory behavior. The oscillatory nature of the kernel can be discerned from the

computational results in figure 4-6(B), at least for the higher values of M∞. It may be

conjectured that Kcy is also oscillatory. (Such behavior can be seen in figure 4-3 for

M∞ = 0.39.)

4.3.2 Mach-Number Expansion

Provided the flow around the cylinder or sphere is irrotational, the velocity field

can be expressed in terms of a velocity potential ϕ. In the compressible regime, the

91

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

1.2

˜ F

τ

A Force coefficient (for legend see (B))

0 2 4 6 8 10-0.25

0.00

0.25

0.50

0.75

1.00

M∞=0.50M∞=0.45M∞=0.40M∞=0.35M∞=0.30M∞=0.25M∞=0.20M∞=0.00 (Longhorn)

d˜ F/d

τ

τ

B Derivative of force coefficient

Figure 4-6. Comparison of computed results for sphere with theoretical results ofLonghorn (1952) for γ = 1.4

92

governing equation for the velocity potential can be expressed as

c2∇2ϕ =∂2ϕ

∂t2+

[∂

∂t+

1

2(∇ϕ · ∇)

](∇ϕ)2, (4–4)

where the speed of sound c is given by

c2 = c20 − (γ − 1)

[∂ϕ

∂t+

1

2(∇ϕ)2

], (4–5)

with c0 denoting the stagnation speed of sound. It is understood that (∇ϕ)2 = (∇ϕ) ·

(∇ϕ). We non-dimensionalize Eq. (4–4) with R as the length scale, M∞c0 as the velocity

scale, and T as an as of yet unspecified time scale. Denoting the non-dimensional

quantities by a tilde, the resulting equation can be expressed as

∇2ϕ− R2

c20T2

∂2ϕ

∂t2=

M∞R

c0T

∂(∇ϕ)2

∂t+ (γ − 1)

M∞R

c0T

∂ϕ

∂t∇2ϕ

+γ − 1

2M2

∞(∇ϕ · ∇ϕ) ∇2ϕ+M2

∞2∇ϕ · ∇(∇ϕ)2.

(4–6)

Immediately after the sudden onset of a constant acceleration a, irrespective of its

magnitude, the appropriate time scale will be such that the propagation of acoustics and

the associated time derivative terms are both important. However, after the acoustics

have propagated sufficiently far away from the cylinder or sphere, the appropriate time

scale for the variation of the ambient flow is T = M∞c0/a. Then it can be deduced that

compared to ∇2ϕ the next three terms in Eq. (4–6), all of which involve time derivatives,

scale as α2/M2∞, α, and α, respectively. Provided that the non-dimensional acceleration

α = aR/c20 satisfies the condition α ≪ M∞, the three terms can be ignored and the

resulting equation is

∇2ϕ =γ − 1

2M2

∞(∇ϕ · ∇ϕ) ∇2ϕ+M2

∞2∇ϕ · ∇(∇ϕ)2. (4–7)

Thus, after an initial transient, the compressible potential flow around the cylinder

can be considered quasi-steady and expressed in terms of the Janzen-Rayleigh

93

expansion (see Oswatitsch [78]) to O(M4∞) as

ϕ

U∞R=

(r +

1

r

)cos θ +M2

∞

[(13

12r− 1

2r 3+

1

12r 5

)cos θ −

(1

4r− 1

12r 3

)cos 3θ

],

where the time dependence enters only through the instantaneous freestream Mach

number M∞. The above expansion can be substituted into the compressible form of

the Bernoulli equation to obtain a Mach-number expansion for pressure, which can be

integrated around the cylinder to obtain the force. An analytic expansion for the effective

added-mass coefficient can then be obtained as

CM,e� = 1 +M2∞ +O(M4

∞), (4–8)

which is also plotted in figure 4-5(A). Unfortunately, with the present numerical

simulation it is not possible to go to much smaller Mach numbers and the difference

between the above expansion and the numerical results increases with increasing M∞.

Nevertheless, the above simple theory supports the qualitative behavior of increasing

added-mass effect at finite Mach number.

The role of specific-heat ratio γ can now be examined. In the limit M∞ → 0, Eq.

(4–6) reduces to

∇2ϕ− R2

c20T2

∂2ϕ

∂t2= 0 . (4–9)

This is the fundamental equation underlying the works of Miles and Longhorn. It

indicates that in the limit M∞ → 0 the flow and force evolution in response to infinitesimal

acceleration will be independent of γ. On the other hand, if we assume M∞ to be

finite, then we find that at short times the velocity potential is given by the complete

Eq. (4–6) and the leading-order term including the specific-heat ratio is O(M∞). At

long times, however, Eq. (4–7) applies and the leading-order term depending on the

specific-heat ratio scales as O(M2∞). Thus, the effect of the specific-heat ratio on the

effective added-mass coefficient can be expected to be weak at small values of M∞.

94

0 5 10 15 20 25 30 35-0.5

0.0

0.5

1.0

1.5

2.0

2.5 γ=1.67γ=1.4γ=1.2

M∞=0.2

M∞=0.34

M∞=0.39˜ F

τ

Figure 4-7. Effect of γ on unsteady force coefficient on cylinder

These theoretical results are corroborated by our computations. In figure 4-7,

we present the effect of varying γ for three different values of M∞ on the unsteady

force for a suddenly accelerated cylinder. As with the previously presented results, the

acceleration is small enough that quasi-steady conditions are maintained. It can be seen

clearly that for M∞ = 0.2, the effect of γ is indiscernible. For M∞ = 0.39, however, the

effect of γ is noticeable for both the peak and steady-state values of the force coefficient.

4.4 Discussion

In the following, we discuss some specific questions related to the results presented

in §3. We focus in particular on the relevance of these results to predicting particle

motion using force laws.

The first question that arises is: Under what conditions will compressibility effects

on unsteady forces be important? The relative importance of inviscid (added-mass

and pressure gradient) and viscous (history) unsteady forces arising from particle

95

acceleration compared to the dominant quasi-steady drag scales as 1/(ρ + CM) and

1/√ρ+ CM , respectively, where ρ is the ratio of particle density to surrounding fluid

density and CM is the added-mass coefficient (Bagchi & Balachandar [2]). Thus, in the

context of a particle moving in air, owing to the large density ratio encountered in most

practical situations, the unsteady forces arising from particle acceleration are typically

ignored.

It is important to note that the unsteady forces arising from the acceleration of

the surrounding fluid do not follow the above scalings. For example, in the case of a

particle injected into an accelerating flow, the ratio of inviscid and viscous unsteady

forces to quasi-steady drag can be shown to scale as Re d/L and√Re d/L, where

Re is Reynolds number based on particle diameter d and relative velocity and L is the

typical length scale of ambient flow variation (Bagchi & Balachandar [2]). Note that

the density ratio does not appear in these estimates. The above scaling, although

developed for an incompressible flow, will be appropriate even if compressibility effects

are important. Thus, situations can exist where the motion of a finite-size particle

in a rapidly accelerating compressible flow can be influenced by unsteady forces,

irrespective of the particle to fluid density ratio. For example, Tedeschi et al. [108] and

Thomas [111] assessed the influence of the history force on the motion of a particle

through a shock wave theoretically and numerically using the Basset-Boussinesq-Oseen

equation (Crowe et al. [25]) and observed the instantaneous history force to be many

times larger than the viscous drag force.

The second question is: How would the above results be used in practice? A simple

model of the compressible inviscid force arising from the unsteady motion of the particle

or the surrounding fluid can be written as follows

Fiu(t) = mf

∫ t

−∞K

(c∞(t − χ)

R;M∞

)(Du

Dt− dv

dt

)d(c∞χ

R

)+mf

Du

Dt, (4–10)

96

where Du/Dt is the ambient fluid acceleration evaluated at the particle position and

K and mf are chosen appropriately for the cylinder and the sphere. As cautioned by

Miles [71], Longhorn [58], Yih [117], and several others, we refrain from calling the first

term on the right-hand side of the above equation the ‘added-mass force,’ since the

time-dependent nature of the force does not always reduce to the form of a constant

mass multiplying the instantaneous acceleration. In the case of a constant acceleration,

for c∞t/R ≫ 1 the above integral reduces to a constant times acceleration, thus

permitting interpretation in terms of an effective added-mass coefficient, which reduces

to the incompressible value as M∞ → 0.

We next address the question of how much the incompressible results for the

added-mass force are modified by compressibility. As already seen in figures 4-5, the

effective added-mass coefficient more than doubles as the Mach number increases in

the subcritical range. Furthermore, as pointed out by Miles and Longhorn, the effective

influence of the unsteady inviscid force will be stronger if the acceleration is large over a

short period of time than if it is small over a long period of time. To explore this behavior

at finite Mach number, consider a problem similar to that described at the beginning of

§3: A cylinder or sphere is held fixed in a steady ambient flow of farfield Mach number

M∞. At time t = 0, the cylinder or sphere is given a constant acceleration a in the

direction opposite to the ambient flow until t = �t, when the acceleration is removed.

The resulting force on the cylinder or sphere is given by Eq. (4–10). From this force, the

work done by the cylinder or sphere at t = t∗ can be expressed as

W (t∗, �t) =

∫ �t

0

mf a t

[∫ t

0

a K

(c∞(t − χ)

R;M∞

)d(c∞χ

R

)]dt

+

∫ t∗

�t

mf a�t

[∫ �t

0

aK

(c∞(t − χ)

R;M∞

)d(c∞χ

R

)]dt.

(4–11)

97

Rearranging the integrals we obtain W (t∗, �t) = mf ξ(t∗, �t)(a�t)2/2, where

ξ(t∗, �t) =2

(�t)2

∫ �t

0

t

∫ c∞t/R

0

K(ζ;M∞) dζdt

+2

�t

∫ t∗

�t

∫ c∞t/R

c∞(t−�t)/R

K(ζ;M∞) dζdt.

(4–12)

The work done is equal to the kinetic energy imparted to the fluid due to the acceleration.

Thus, for t∗ → ∞, mf ξ can be interpreted as the added-mass due to the fluid whose

velocity changed by a�t because of the acceleration of the cylinder or sphere.

The variation of ξ(t∗ →∞, �t) for varying Mach numbers is presented in figure 4-8.

For the sphere, Longhorn (1952) showed that in the limit of M∞ → 0, the result reduces

to

ξ(t∗ →∞, �t) = 1 +1

(�t)2[1− exp(−�t) cos(�t)− exp(−�t) sin(�t)] . (4–13)

At finite Mach numbers, ξ was obtained through numerical integration of Eq. (4–12)

using the kernels presented in figures 4-3(B) and 4-6(B). In the limit of sustained slow

acceleration (i.e., �t → ∞, a → 0), the force on the cylinder or sphere remains constant

except for the initial and final transients, and ξ approaches the effective added-mass

coefficient CM,e� presented in figures 4-5(A) and (B). However, ξ increases as the

duration of acceleration is reduced, and in the limit of �t → 0 we observe a doubling of

the added mass. In fact, in the limit of �t → 0, the contribution from the first term in Eq.

(4–12) becomes zero and it can be shown that

ξ(t∗ →∞, �t → 0) = 2ξ(t∗ →∞, �t →∞) = 2CM,e� . (4–14)

Thus, the combined effect of finite Mach number and impulsive acceleration can

intensify the added-mass effect to more than four times of what would be predicted

based on incompressible theory.

We now comment on the use of the unsteady inviscid force given in equation (4.1).

Consider the motion of a cylinder or sphere in response to an external force Fext in a

98

0 5 10 15 20 25 30 35 40 45

1

2

3

4

M∞=0.38M∞=0.37M∞=0.34M∞=0.25M∞=0.20M∞=0.00 (Miles)

ξ(t

∗→

∞,∆

t)

∆t

A Cylinder

0 5 10 15 20 25 30 35 40 45

1.0

1.2

1.4

1.6

1.8

2.0

ξ(t∗

→∞

,∆t)

/ξ(t

∗→

∞,∆

t→

∞)

∆t

B Cylinder, normalized (for legend see (A))

0 5 10 15 20 25

1

2

3

4

M∞=0.50M∞=0.45M∞=0.40M∞=0.35M∞=0.30M∞=0.25M∞=0.20M∞=0.00 (Longhorn)

ξ(t

∗→

∞,∆

t)

∆t

C Sphere

0 5 10 15 20 25

1.0

1.2

1.4

1.6

1.8

2.0

ξ(t∗

→∞

,∆t)

/ξ(t

∗→

∞,∆

t→

∞)

∆t

D Sphere, normalized (for legend see (C))

Figure 4-8. Behavior of ξ(t∗ →∞, �t) defined by Eq. (4–12)

stagnant inviscid compressible fluid. The motion is governed by

mp

dv

dt+mf

∫ t

−∞K

(c∞(t − χ)

R;M∞

)dv

dtd(c∞χ

R

)= Fext, (4–15)

where mp is the mass of the cylinder (per unit width) or of the sphere and mf is the mass

of the fluid displaced by the cylinder (per unit width) or the sphere. If the external force

is small and maintained over a long period of time, the resulting acceleration of the

cylinder or sphere is given by Fext/(mp +mfCM,e�), which reduces to the incompressible

99

result in the limit M∞ → 0. On the other hand, if the external force is large and of very

brief duration with a net impulse of I , the resulting velocity change due to the impulse is

given by I/(mp + mfCM) in an incompressible flow. In a compressible flow, however, an

asymptotic constant velocity will be reached on an acoustic time scale after the impulse,

and if an added-mass coefficient were to be computed in comparison with the above

incompressible result, it will be dependent on both M∞ and the density ratio ρ. Similar

behavior can be observed in the results of Tracey (1988). Clearly, as cautioned by other

authors, the concept of added mass is fraught with difficulty in compressible flow and

it is advantageous to simply consider Eq. (4–10) as an expression for the unsteady

inviscid force.

Finally, we note that in an incompressible flow, the added-mass force has been

shown to be independent of the Reynolds number or viscous effects (see Rivero et

al. [87], Chang & Maxey [19], Mougin & Magnaudet [74], Bagchi & Balachandar [2],

Bagchi & Balachandar [3], and Wakaba & Balachandar [114]). The instantaneous

nature of the added-mass force in incompressible flow precludes any interaction with the

viscous response to the acceleration. In a compressible flow, the effect of the Reynolds

number Re on the unsteady inviscid force will depend on the time scales. The acoustic,

inertial, and viscous time scales are R/c∞, R/U and R2/ν, respectively, where U is the

characteristic relative velocity and ν is the kinematic viscosity of the fluid. The ratio of

viscous to acoustic time scales will be proportional to Re/M∞ where Re = UR/ν and

thus at sufficiently high Reynolds number the unsteady inviscid force can be expected to

be independent of Reynolds number. However, at finite Re, quasi-steady and unsteady

(history) components of the hydrodynamic force must be taken into account also.

100

CHAPTER 5MODELING OF THE UNSTEADY FORCE FOR SHOCK-PARTICLE INTERACTION

The interaction between a particle and a shock wave leads to unsteady forces that

can be an order of magnitude larger than the quasi-steady force in the flow field behind

the shock wave. Simple models for the unsteady force have so far not been proposed

because of the complicated flow field during the interaction. Here, a simple model is

presented based on the work of Parmar et al. (Phil. Trans. R. Soc. A, 366, 2161-2175,

2008). Comparisons with experimental and computational data for both stationary

spheres and spheres set in motion by shock waves show good agreement in terms of

the magnitude of the peak and the duration of the unsteady force.

5.1 Introduction

The interaction between a particle and a shock wave has been studied extensively

due to its practical importance, see, e.g., [46, 95], and many others. As the shock wave

propagates into a gas-particle mixture, the gas velocity increases instantaneously

across the shock. By contrast, the particle velocity approaches the post-shock gas

velocity only slowly due to the finite inertia of the particles. This leads to the so called

frozen, relaxation, and equilibrium regimes that have been discussed in detail by

Carrier [17], Soo [100], Kriebel [53], and Rudinger [89]. Several investigations, see,

e.g., [27, 46, 47, 50, 57, 88, 90, 103], have documented careful measurements of the

time-dependent particle motion behind a shock wave. One important observation is

that the particles generally approach the equilibrium state faster than predicted by the

standard drag relation. Thus, the above studies have inferred that the drag force on the

particle, and hence the drag coefficient, are substantially enhanced in the post-shock

flow.

Direct time-resolved measurements of the force exerted by a shock wave propagating

over a particle are very challenging. The measurements need to resolve very small

forces over a duration of milliseconds with a resolution of microseconds. Techniques

101

with the required accuracy and resolution have only been developed relatively recently.

Tanno et al. [104, 105] and Sun et al. [102] used an accelerometer installed inside the

sphere. Bredin and Skews [11] and Skews et al. [97] used a stress-wave drag balance

and reported time-dependent force measurement on a stationary particle with an

estimated error of less than 15%. These experiments have shown that the force on the

particle increases significantly as the shock wave passes over it. The peak force on the

particle can be more than an order of magnitude larger than the steady-state force in the

post-shock flow. Furthermore, it was observed that the transition from the peak force to

the steady-state force can be non-monotonic.

The propagation of a shock wave over a spherical obstacle such as a particle is

very complicated, consisting of regular and irregular shock-wave reflection, diffraction,

and focusing phenomena, see [15, 102, 104]. With recent advances in numerical

methods and computer performance, highly accurate direct numerical simulations

of shock-particle interaction have been accomplished. One example is the work of

Sun et al. [102], who obtained good agreement with their own measurements. The

simulations captured the increase of the instantaneous drag force by more than

an order of magnitude as the shock wave propagates over the particle as well as

the non-monotonic approach to the steady state. Sun et al. [102] observed that the

maximum drag coefficient occurs slightly after the shock-wave reflection changes from a

regular to a Mach reflection and that the minimum drag coefficient appears to be due to

the focusing of the diffracted shock wave at the rear of the sphere.

In many applications, shock waves interact with millions to billions of particles. In

such situations, the detailed resolution of the interaction between the shock wave and

each particle is not feasible. Instead, one needs to resort to the Eulerian-Lagrangian

point-particle approach and track the trajectory of a sufficiently large number of particles

as they interact with the shock wave. The key component of this approach is a model

that accurately predicts the instantaneous force on the particles. The commonly used

102

standard-drag model, see, e.g., [21], parameterizes the drag coefficient in terms of

the particle Reynolds number based on the particle diameter and the velocity of the

particle relative to that of the surrounding gas. Even with compressibility corrections that

account for finite-Mach-number effects, see, e.g., [59], the standard-drag correlation is

appropriate only for the quasi-steady state that is established long after the passage

of the shock wave. The large increase in instantaneous drag and the subsequent

non-monotonic approach to steady state is clearly due to the unsteady effect resulting

from the propagation of the shock wave over the particle. This strong time-dependent

component of the force cannot be accounted for with quasi-steady force models.

In incompressible flows, it has been well established that unsteadiness, either due

to acceleration of the particle or of the surrounding fluid, gives rise to additional inviscid

(pressure-gradient and added-mass) and viscous (Basset history) forces, see, e.g.,

[54]. Recently, our understanding of and models for these forces have been extended

to finite particle Reynolds numbers (see, e.g., [19, 62, 74, 114]). However, the existing

unsteady-force parameterizations are developed for the incompressible limit, and are

therefore incapable of accurately predicting the strong variation in drag force as the

shock wave propagates over the particle.

The earliest investigations of unsteady forces in the compressible regime, due to

Love [61] and Taylor [106], focused on the inviscid contribution only. Subsequently, Miles

[71] and Longhorn [58] considered the effect of compressibility on unsteady inviscid

forces on a cylinder and a sphere, respectively, in the zero-Mach-number limit using the

acoustic approximation of the velocity potential equation. More recently, Parmar et al.

[79] extended the results of Miles and Longhorn to finite Mach number.

It is important to note that the conventional concept of added mass is of limited

use in compressible flows, because compressibility destroys the straightforward

dependence of the inviscid unsteady force on the instantaneous acceleration. Instead,

in a compressible fluid, a particle accelerated impulsively as a(t) = δ(t), where δ(t)

103

is the Dirac delta function, is subjected to a time-dependent inviscid force that decays

on the acoustic time scale c∞/R, where c∞ is the speed of sound in the ambient fluid

and R is the particle radius. The inviscid unsteady force can be represented in terms

of a history integral that uses a kernel to weight the history of the particle acceleration

relative to the fluid acceleration. The decay of the kernel in terms of the non-dimensional

time τ = c∞t/R depends on both the geometry of the particle and the Mach number M

formed from the velocity of the particle relative to that of the fluid and c∞. The kernels

for a cylinder and a sphere in the limit of zero Mach number were obtained by Miles [71]

and Longhorn [58], respectively. The corresponding kernels at finite but subcritical Mach

numbers (M . 0.4 for a cylinder and M . 0.6 for a sphere) were obtained by Parmar

et al. [79]. They observed that as the Mach number increases, the kernel changes such

that the effective inviscid unsteady force can be more than double of its value in the

zero-Mach-number limit.

The objective of this paper is to present a physics-based force model for unsteady

compressible flows that combines the inviscid unsteady force expressed in terms of

a history integral with the standard quasi-steady drag force. We will demonstrate that

the resulting simple model provides the ability to predict the time-dependent force

on a spherical particle as a shock wave propagates over it. Recent time-resolved

measurements of the force exerted by a shock wave propagating over a stationary

sphere by Tanno et al. [104, 105], Sun et al. [102], Bredin and Skews [11], and Skews

et al. [97] and their companion numerical simulation results will be used to evaluate the

accuracy of the force model. In these experiments and computations, the shock-wave

Mach number Ms is sufficiently low that the Mach number of the flow behind the shock

wave is subcritical, thus permitting the use of the finite-Mach-number kernels obtained

by Parmar et al. [79]. Results from the force model will also be presented for the case

of a sphere set in motion by the impact of a shock wave and compared to experimental

and computational data of Britan et al. [14]. Despite its simplicity, the model appears to

104

capture the essential features of the unsteady force during the shock-particle interaction

remarkably well in all cases. The Eulerian-Lagrangian point-particle approach with

the proposed force model can thus be used as an efficient approach to compute

compressible multiphase flows involving shock waves propagating through suspensions

containing large numbers of particles.

5.2 Force Model

5.2.1 Force Parameterization

Magnaudet and Eames [62] suggested that the force F(t) on a particle in an

incompressible flow can be parameterized for a range of particle Reynolds numbers as

F(t) = Fqs(t) + Fiu(t) + Fvu(t) + Fl(t) + Fbg(t) , (5–1)

where the terms on the right-hand side represent quasi-steady, inviscid unsteady

(added-mass and pressure-gradient), viscous unsteady (Basset history), lift, and

buoyancy/gravity forces, respectively. Equation (5–1) is based on the assumption that

the particle is much smaller than a characteristic length scale of the surrounding flow;

this will be discussed further below. We emphasize here that the inviscid unsteady

force Fiu consists of two contributions, namely the added-mass and pressure-gradient

forces. The former is the force arising from the no-penetration condition on the particle

surface whenever the relative acceleration between the particle and the ambient fluid

is non-zero. The latter may be interpreted as the force that existed due to an ambient

pressure gradient in the absence of the particle, i.e., if the particle were replaced by the

fluid.

The above parameterization is applicable in the compressible regime also

provided appropriate modifications are made to the modeling of the different terms.

In a compressible flow, the quasi-steady drag will depend on both the Reynolds and

Mach numbers. An empirical relation for the quasi-steady drag coefficient on a spherical

105

particle in incompressible flow can be expressed as

CD,qs(Re) =∥Fqs∥

12ρf ∥ur∥2πR2

=24

Reϕ(Re) + 0.42

(1 +

42500

Re1.16

)−1

, (5–2)

where Re = 2ρf ∥ur∥R/µ is the particle Reynolds number based on the fluid density ρf ,

the relative velocity ur = u− v between the particle and the surrounding gas, the particle

diameter 2R, and the dynamic fluid viscosity µ, see [21]. In the above, ρf and µ are the

density and dynamic viscosity of the surrounding gas and ϕ(Re) = 1 + 0.15Re0.687,

see [94]. Provided that the particle Mach number M = ∥ur∥/c∞ remains below the

critical value (approximately 0.6 for a sphere), the flow around the particle is entirely

subsonic. Therefore, the effect of compressibility is relatively small and the use of an

incompressible relation for the drag coefficient is justified. Above the critical Mach

number, the effect of compressibility cannot be neglected and modified forms of Eq.

(5–2) must be considered, see, e.g., [59]. We restrict attention to subcritical Mach

numbers in this article. (Under supercritical conditions, preliminary simulations indicate

that the inviscid unsteady force is negligible compared to changes in the quasi-steady

drag as the relative velocity is varied. A possible, but perhaps only partial, explanation

for this observation is that the quasi-steady drag force Fqs , which now includes a

substantial contribution from shock waves, increases faster than the unsteady inviscid

force Fiu with the Mach number.)

As discussed in the introduction, the dependence of the inviscid unsteady force

on the instantaneous acceleration is lost with the introduction of compressibility.

Instead, the inviscid unsteady force on a particle depends on a weighted integral of

the acceleration history. Following the suggestion of Parmar et al. [79], the inviscid

unsteady force on a particle in a compressible flow can be modeled as

Fiu(t) =

∫ t

−∞K

(Dmf u

Dt− dmf v

dt

)d(c∞χ

R

)+

∫ρfDu

DtdV , (5–3)

106

where K = K(c∞(t − χ)/R;M) is the compressible inviscid unsteady force kernel,

M is the particle Mach number, mf is the mass of the fluid displaced by the particle,

D(·)/Dt is the rate of change following the undisturbed ambient fluid, and d(·)/dt is rate

of change following the particle. Note that the above equation incorporates the effect

of density changes in the ambient fluid as suggested by Eames & Hunt [29] and that

the second term on the right-hand side is the pressure-gradient force. The decay of the

kernel in terms of the non-dimensional time τ = c∞t/R depends on both the geometry

of the particle and the Mach number M. For a spherical particle, an explicit expression

for the inviscid unsteady force kernel in the zero-Mach-number limit was obtained by

Longhorn [58] as

Ksp(τ) = exp(−τ) cos τ . (5–4)

The corresponding kernels at finite but subcritical Mach numbers were presented by

Parmar et al. [79]. The kernels were obtained from computed time histories of the force

on a particle in response to suddenly imposed constant accelerations once a steady

state is established at a given Mach number. Provided the acceleration is sufficiently

small, so that the change in Mach number during the period of constant acceleration is

negligible, the extracted kernels can be interpreted to be associated with the given Mach

number. The kernels obtained by Parmar et al. [79] for cylindrical and spherical bodies

over a range of Mach numbers are shown in Fig. 5-1.

The viscous unsteady force in the compressible regime will depend on both the

particle Reynolds and Mach numbers and also takes the form of a history integral.

The history kernel for the viscous unsteady force has been extensively studied in

incompressible flow. Its form has been shown to be quite complicated and problem-dependent,

see [67, 68]. It can be expected that compressibility further complicates the viscous

history force. We are not aware of any existing models for the viscous unsteady force

that take into account compressibility effects.

107

In the present case of a particle subjected to a planar shock wave, the lift force is

zero and we ignore the viscous unsteady and buoyancy/gravity forces in the following.

The viscous unsteady force is neglected because of the apparent lack of suitable

models for compressible flows. Future studies will focus on the development of models

for the viscous unsteady force for compressible flows.

Thus the present model is based on the following simplified form of Eq. (5–1),

F(t) = Fqs(t) + Fiu(t) , (5–5)

where Fqs(t) and Fiu(t) are determined from Eqs. (5–2) and (5–3), respectively. To

make this model useful from a practical perspective, the ambient flow as experienced

by the particle must be characterized. Before we address the characterization of the

ambient flow in Section 5.2.4, we discuss the importance of the inviscid unsteady force

as well as the effect of finite Mach numbers.

5.2.2 Importance of Inviscid Unsteady Contribution

It is generally assumed that the ratio of inviscid and viscous unsteady forces to the

quasi-steady drag force to scale as the fluid-to-particle density ratio and therefore can be

ignored for solid particles in typical gas flows, see, e.g., [91, 99]. Here it is important to

separate unsteady forces due to acceleration of a particle from those due to acceleration

of the ambient flow. To see why, consider the quasi-steady force given by Eq. (5–2) to be

the dominant contribution. Then it can be shown that the resulting particle acceleration

in response to this force will be inversely proportional to the particle density. Therefore,

the unsteady force arising from particle acceleration will scale as Fqs/β, where β = ρp/ρf

is the particle-to-fluid density ratio.

However, if the ambient flow acceleration is imposed externally, such as during a

shock-particle interaction, the unsteady force arising from the ambient flow acceleration

need not follow the above scaling, see [3]. If the velocity, length, and time scales of the

ambient flow variation are denoted by U, L, and T = L/U, the relative importance of the

108

0 5 10 15-0.2

0.0

0.2

0.4

0.6

0.8

1.0

M=0.39M=0.38M=0.37M=0.34M=0.30M=0.25M=0.20M=0.00 (Miles)d

˜ F/d

τ

τ

A Cylinder

0 2 4 6 8 10-0.2

0.0

0.2

0.4

0.6

0.8

1.0

M=0.50M=0.45M=0.40M=0.35M=0.30M=0.25M=0.20M=0.00 (Longhorn, eq. (4))d

˜ F/d

τ

τ

B Sphere

Figure 5-1. Response kernels of Parmar et al. [79]

109

unsteady force arising from the ambient flow acceleration compared to the quasi-steady

drag can be expressed as2

9

R

L

Re

ϕ(Re), (5–6)

where it is assumed that ∥ur∥ ∼ O(U) and the quasi-steady drag force included

only the first term on the right-hand side of Eq. (5–2). Therefore, at finite Reynolds

numbers, unsteady forces on a particle can be important provided that the length scale

of the ambient flow variation is not much larger than the particle diameter. The above

argument is consistent with recent experiments, discussed below, where the peak force

on a particle as the shock wave propagates over it is observed to be 10-20 times larger

than the corresponding quasi-steady force. The peak drag force occurs slightly after

the shock-wave reflection changes from regular to Mach reflection and before the shock

wave reaches the equator. Subsequently, the drag decays non-monotonically to the

quasi-steady value on the acoustic time scale. Furthermore, this result is independent of

the particle-to-fluid density ratio.

Having established the relative magnitude of the unsteady inviscid and quasi-steady

forces, we now turn attention to the importance of the inviscid unsteady force on the

motion of a particle. In the incompressible limit, the equation of motion of the particle

considering only the inviscid forces can be expressed as

ρpdv

dt=

1

2ρf

(Du

Dt− dv

dt

)+ ρf

Du

Dt, (5–7)

where consistency with Eq. (5–3) is obtained if we take

Ksp(τ) = H(τ)δ(τ), (5–8)

where H(t) is the Heaviside step function and δ(t) is the Dirac delta function. In this

limit, the particle acceleration is directly proportional to the instantaneous acceleration of

the ambient flow. Thus, if we consider the ambient flow to jump from u1 = 0 to a uniform

velocity of u2 > 0, the corresponding jump in the particle velocity due to inviscid forces

110

can be estimated to bev2

u2=

3

2β + 1, (5–9)

where β is the particle-to-fluid density ratio.

With the introduction of compressibility, the simple linear dependence of the inviscid

unsteady force on instantaneous acceleration is lost and the force must be expressed

in terms of the history integral (see Eq. (5–3)). The equation of motion for the particle,

accounting for only the inviscid forces, can now be written as

βdv

dt+ K ∗ dv

dt=

Du

Dt+ K ∗ Du

Dt, (5–10)

where mf is assumed to be a constant and the following notation for the convolution

integral has been adopted:

f (t) ∗ g(t) =∫ t

−∞f (t − χ)g(χ) dχ . (5–11)

In the zero-Mach-number limit the analytic kernel obtained by Longhorn [58] given

by Eq. (5–4) applies. The equation of motion can be integrated with the above kernel

to again obtain Eq. (5–9) for the velocity jump of the particle due to inviscid forces.

The above simple analysis clearly illustrates the following two important points. First,

the inviscid unsteady contribution and the compressibility of the flow are of critical

importance in accurately accounting for the rapid increase in the drag force on the

particle that occurs on the acoustic time scale. Second, when considering the integrated

effect on particle motion, the contribution from inviscid forces is proportional to the

fluid-to-particle density ratio β−1. Of course, the particle velocity will continue to change

and approach the post-shock gas velocity due to the action of viscous forces. While the

inviscid unsteady force acts on a very short acoustic time scale, the viscous force acts

on a much longer time scale.

111

5.2.3 Effect of Finite Mach Number

As discussed in Section 5.2.1, the effect of the Mach number on quasi-steady drag

is quite weak in the subcritical regime. In the context of a shock wave propagating over

a stationary particle, the flow around the particle will remain subcritical provided that

Ms . 1.5. The effect of the Mach number on the inviscid unsteady force appears through

the dependence of the kernel on the Mach number. From Fig. 5-1 it can be seen that

with increasing Mach number in the subcritical regime the rate of decay of the kernel

becomes smaller, suggesting a more extended influence of unsteadiness. As a result,

the peak force at M = 0.5 was observed to be about twice as large as that at M → 0.

Furthermore, the time at which the peak force occurs is also nearly double (Parmar et

al. [79]). It can therefore be expected that the Mach number of the post-shock flow plays

an important role in determining the magnitude and timing of the peak drag force as the

shock wave propagates over the particle.

The impact of the Mach number on the integrated effect of the inviscid unsteady

force can now be investigated. As in Parmar et al., we first define an effective added-mass

coefficient as

CM,e�(M) =

∫ ∞

0

K

(c∞t

R;M

)d

(c∞t

R

). (5–12)

In the zero-Mach-number limit, we recover the added-mass coefficient for incompressible

flow as CM,e�(M → 0) → 0.5. Using the finite-Mach-number kernels, such as those

presented in Fig. 5-1, the effective added-mass coefficient can be calculated for M > 0.

In particular, for a spherical particle we observe CM,e�(M → 0.5) ≈ 1.0, i.e., double its

value in the zero-Mach-number limit.

To obtain the jump in particle velocity, we integrate the equation of motion in time

from −∞ to∞. Because the kernel K has compact support, equation (5–10) can be

integrated using the convolution theorem to obtain the following exact relation,

v2

u2=

1 + CM,e�(M)

β + CM,e�(M). (5–13)

112

This expression is valid for finite Mach numbers and is independent of the manner in

which the fluid velocity varies from the quiescent state ahead of the shock wave to u2

behind the shock wave. The only limitation is that the change in Mach number incurred

during the change in fluid velocity is assumed to be small enough so that mf can be

assumed to be constant and that the effective added-mass coefficient can be defined for

the given Mach number. In the incompressible limit, Eq. (5–13) reduces to Eq. (5–9). If

the particle density is much larger than that of the surrounding gas, i.e., if β → ∞, we

obtain from Eqs. (5–9) and (5–13),(v2

u2

)M

=2

3(1 + CM,e�(M))

(v2

u2

)M=0

, (5–14)

and hence the jump in particle velocity for a post-shock Mach number of 0.5 will be

approximately 4/3 times larger than the jump for incompressible flow.

5.2.4 Approximation of Ambient Flow

The force parameterization given by Eq. (5–3) assumes the particle to be

sufficiently small compared to the length scales of the ambient flow. This allows

unambiguous definitions of c∞, M, and mf as the speed of sound, particle Mach

number, and displaced mass of the ambient fluid as experienced by the particle. On the

other hand, if the ambient flow variations occur on a scale comparable to the size of the

particle, unambiguous definitions of these quantities are not possible. This is certainly

the case for a shock-particle interaction, since the shock wave can be interpreted as a

discontinuity with a thickness of the order of the mean free path, see, e.g., [96]. Although

the flow ahead of and behind the shock wave can be considered homogeneous, the

ambient flow is strongly inhomogeneous as the shock wave passes over the particle.

Therefore, unambiguous values of the quantities listed above as well as of quantities

such as the Mach number, density, and viscosity of the ambient flow as “seen” by the

particle are not easily defined. One needs to resort to modeling and identify models that

best capture the underlying physics.

113

xs(t)

xp(t)

x′

s(t)

us

v(t)

R

A(t)

V1(t)V2(t)

12 shock wave

Figure 5-2. Schematic of shock position in the symmetry plane of a spherical particleand definition of variables

The present model is based on a simplified one-dimensional description of the

interaction of the shock wave with the particle. More specifically, we assume the shock

wave to be planar, infinitely thin, and to propagate over the volume occupied by the

particle in its undisturbed form, as illustrated schematically in Fig. 5-2. The implications

of these simplifications are discussed in Section 5.4. At any time t, the particle can

therefore be thought to be divided into two parts corresponding to the homogeneous

states ahead of and behind the shock wave. In the following, these states are denoted

by the subscripts 1 and 2, respectively. The cross-sectional area of the particle cut by

the shock wave is denoted by A(t) and the volumes of the particle ahead of and behind

the shock are denoted by V1(t) and V2(t), respectively.

114

To describe the ambient flow experienced by the particle, we define the reference

time t = 0 to correspond to the instant when the shock wave makes contact with the

particle. Before this reference time, the particle is entirely immersed in the quiescent

ambient fluid upstream of the shock wave. As the shock wave moves over the particle,

the particle experiences both regions, and hence the effective ambient flow seen by the

particle is necessarily a combination of the upstream and downstream conditions. Once

the shock wave has moved past the particle the ambient flow seen by the particle is

given by the downstream state. The position of the shock wave with respect to the front

of the particle is given by

ξs(x′s(t)) =

x ′s(t)

R= 1 +

1

R(xs(t)− xp(t))

= 1 +1

R

(ust −

∫ t

−∞v(ζ) dζ

),

(5–15)

where xs(t) and xp(t) are the positions of the shock wave and (the centroid of) the

particle, respectively, us is the ambient shock velocity (assumed to be constant), and

v(t) is the velocity of the particle obtained from Newton’s second law,

v(t) =1

mp

∫ t

−∞F (ζ) dζ , (5–16)

where mp is the mass of the particle and F (t) is the unsteady drag force determined

from Eq. (5–5). (We use the scalar force F (t) in Eq. (5–16) for the x-component of the

vector force F(t) in Eq. (5–5) because we assume the interaction to be one-dimensional.)

We propose a simple model for the ambient flow velocity and fluid properties “seen”

by the particle to be an average of the upstream and downstream flow conditions,

weighted by the swept volume. Elementary geometry allows the swept volume V2(x′s(t))

115

to be expressed in terms of x ′s(t) as

V2(x′s(t))

V=

0 if ξs 6 0,

14(3− ξs)ξ

2s if 0 < ξs 6 2,

1 if 2 < ξs ,

(5–17)

where V = V1(x′s(t)) + V2(x

′s(t)) and A(x ′

s(t)) follows similarly. Therefore, the fluid

velocity u(x ′s(t)) as “seen” by the particle can be expressed as

u(x ′s(t))− u1

u2 − u1=

0 if ξs 6 0,

V2(x′s)

Vif 0 < ξs 6 2,

1 if 2 < ξs .

(5–18)

The flow acceleration can be defined as

1

u2 − u1

du(x ′s(t))

dt=

0 if ξs 6 0,

A(x ′s)

V

dx ′s

dtif 0 < ξs 6 2,

0 if 2 < ξs .

(5–19)

The density ρ(x ′s(t)) and the rate of change of density (dρ(x ′

s(t))/dt)/(ρ2 − ρ1) are

defined analogously. The above expressions are used to evaluate the instantaneous

particle Reynolds and Mach numbers in Eqs. (5–2) and (5–3),

Re(t) =2ρf (xp(t))(u(xp(t))− v(t))R

µ, (5–20)

M(t) =u(xp(t))− v(t)

c(xp(t)), (5–21)

where µ is the dynamic viscosity (assumed to be a constant for simplicity), and c(xp(t))

is the speed of sound at the particle location. The latter is computed from the constancy

of the total temperature in a frame of reference moving with the shock and the velocity

determined from Eq. (5–18). The instantaneous Mach number is used to select the

116

appropriate kernel in the integration of Eq. (5–3). The kernels are tabulated as a function

of M and τ for the values given in the legend of Fig. 5-1(B); values in between the

tabulated data points are interpolated linearly.

The use of the volume ratio in Eq. (5–18) to model the velocity and density “seen”

by the particle is somewhat arbitrary. Other models are possible, of course. For

example, the surface-area ratio S2/S may be more appropriate for the computation

of the quasi-steady drag according to the Faxen correction, see [36, 63]. We have found

the results obtained with the surface-area ratio to be similar to those obtained with the

volume ratio.

Finally, we note that the pressure-gradient force in Eq. (5–3) can be computed as∫ρfDu

DtdV = −

∫∂p

∂xdV = (p2 − p1)A(x

′s(t))

=2γ

γ + 1

(M2

s − 1)p1A(x

′s(t)) ,

(5–22)

for a finite-sized particle where γ is the ratio of specific heats and Ms = us/c1 is the

shock-wave Mach number.

5.3 Results

The model described in Section 5.2 is assessed by comparison with computational

and experimental results for the interaction of shock waves with spheres. First, we

consider the case of a stationary spherical particle subjected to a normal shock wave.

The predicted force on the particle is compared with experimental and computational

results. Second, we consider the case of a spherical particle set in motion by the

impact of a shock wave. The predicted particle position and velocity are compared

against experimental and numerical results. All cases are based on a ratio of specific

heats of γ = 1.4 and a specific heat of the gas of Cp = 1004.64 J/(kgK). The results

are presented in terms of the drag coefficient CD(t) = F (t)/(12ρ2u

22πR

2) and the

non-dimensional time τs = ust/R. With this definition of the non-dimensional time, the

undisturbed shock wave has propagated past the sphere at τs = 2.

117

5.3.1 Stationary Sphere

In the following, we compare the present model to the experiments and simulations

of Sun et al. [102] and to the experiments of Skews et al. [97]. The former allow us

to study the effect of sphere diameter on the unsteady force, the latter the effect of

shock-wave Mach number.

5.3.1.1 Experiments of Sun et al.

In the experiments of Tanno et al. [104] and Sun et al. [102], the unsteady force

exerted on a sphere with diameter 8 cm was measured for a shock wave with Ms =

1.22, p1 = 101325 Pa, and T1 = 293.15 K. Simulations based on the axisymmetric

compressible Navier-Stokes equations for spheres with diameters ranging from 8µm

to 8 mm were also carried out by Sun et al. [102]. The Reynolds number based on

particle diameter and flow conditions behind the shock is 49 for the smallest sphere

and increases to 4.9 × 105 for the largest sphere. Correspondingly, the Knudsen

number varies from 9.4 × 10−3 to 9.4 × 10−7, justifying the continuum assumption

in the simulations. Sun et al. found that the computations agreed very well with the

experiments. For the larger spheres, the experiments and computations showed a

brief period of negative drag due to shock-wave focusing at the rear of the sphere. For

smaller spheres, negative drag was not observed because the viscous contribution

becomes more pronounced.

The unsteady drag coefficients computed by Sun et al. are presented in Figs.

5-3(A)-5-5(A) for the diameters 8µm, 80µm, and 8 mm. For the largest diameter, the

experimental result for 80 mm is also shown. The rapid increase of the drag coefficient

following the impact of the shock wave is immediately apparent. The drag coefficient

is observed to exhibit a maximum at τs ≈ 1, followed by a decrease to a minimum at

τs ≈ 5 − 6, and a slow approach to the constant drag coefficient due to the flow behind

the shock wave. For the larger spheres, the peak drag coefficient is more than an order

of magnitude larger than the quasi-steady-state value. For the smallest sphere, the peak

118

drag coefficient is about five times larger than the quasi-steady-state value. This is a

clear indication that unsteady effects are very strong as the shock wave propagates

over the sphere and that the prediction of the drag force based on quasi-steady drag

formulae strongly underpredicts the actual time-dependent force on the sphere. The

maximum drag coefficient was shown by Sun et al. [102] to occur slightly after the time

when the shock-wave reflection changes from a regular to a Mach reflection and before

the shock wave reaches the equator. The minimum drag coefficient was explained by

Tanno et al. [105] and Sun et al. [102] as being due to the focusing of the shock wave at

the rear of the sphere.

Also plotted in Figs. 5-3(A)-5-5(A) are the predictions of the unsteady drag

coefficient based on Eqs. (5–5), (5–2), and (5–3) with the finite-Mach-number kernels

of Parmar et al. [79] (see Fig. 5-1(B)), the zero-Mach-number kernel of Longhorn [58]

(Eq. (5–4)), and the incompressible kernel (Eq. (5–8)). In the figure legends, these

results are labeled by “Model,” “Model M → 0,” and “Model M = 0,” respectively. The

overall agreement between the computations, experiments of Sun et al. [102], and the

present model in terms of the order of magnitude of the maximum and minimum drag

coefficients, the times at which they occur, and the approach to steady state is good.

In fact, the agreement is better than expected given that the present model has been

constructed based on data gathered from homogeneous flows. The drag-coefficient

minima predicted by the present model are due to the negative values of the kernel

shown in Fig. 5-1.

Closer investigation of the results presented in Figs. 5-3(A)-5-5(A), reveals some

discrepancies. First, we note that the model overpredicts the peak drag coefficient

for the spheres with diameters 80 mm and 80µm, but underpredicts the peak drag

coefficient for the sphere with diameter 8µm. These trends lead to slight over- and

underprediction of the drag coefficient for 1 . τs . 2. In particular, it is noteworthy that

while the agreement between the model and the computations is quite good for 2 .

119

0 2 4 6 8 10

0

2

4

6

8

10

ComputationModelModel M→0Model M=0

τs

CD

A Total drag

0 2 4 6 8 10

0

2

4

6

8

10TotalQuasi-steadyPressure-gradientHistory

τs

CD

B Breakdown of model terms

Figure 5-3. Comparison of model with computations for sphere with diameter 8µm ofSun et al. [102]

120

0 2 4 6 8 10

0

2

4

6

8

10

ComputationModelModel M→0Model M=0

τs

CD

A Total drag

0 2 4 6 8 10

0

2

4

6

8

10TotalQuasi-steadyPressure-gradientHistory

τs

CD

B Breakdown of model terms

Figure 5-4. Comparison of model with computations for sphere with diameter 80µm ofSun et al. [102]

121

0 2 4 6 8 10

0

2

4

6

8

10

ComputationModelModel M→0Model M=0Experiment

τs

CD

A Total drag

0 2 4 6 8 10

0

2

4

6

8

10TotalQuasi-steadyPressure-gradientHistory

τs

CD

B Breakdown of model terms

Figure 5-5. Comparison of model with experiment for sphere with diameter 80 mm andcomputations for 8 mm of Sun et al. [102]

122

τs . 5 for the diameters 80 mm and 80µm, the model underpredicts the computational

results for the diameter 8µm. This underprediction is related to the narrower peak and

faster decay predicted by the model compared to that observed in the computations. It

appears that there are also other sources for the discrepancy because the computation

does not exhibit a pronounced minimum in the drag coefficient.

The figures show clearly that with the present simple model of the ambient-flow

acceleration “seen” by the particle, even the results obtained with the incompressible

kernel predict reasonably well the large increase in the drag force as the shock

wave passes over the sphere. However, the inviscid unsteady force predicted by the

incompressible kernel is zero for τs > 2 because the ambient flow is then taken to be

the steady post-shock state, and hence the drag force is given by the quasi-steady

contribution only. The rapid rise of the drag force and its non-monotonic decay over

several acoustic time scales to the quasi-steady state observed in the measurements

and are captured well by the compressible kernels. In the experiments of Sun et

al. [102], the Mach number behind the shock wave is quite low at M2 = 0.31 and

therefore the difference between the results predicted by the zero-Mach-number kernel

of Longhorn [58] and the finite-Mach-number kernel of Parmar et al. [79] is small. The

peak drag coefficients computed with these kernels are basically identical and lower

than those predicted by the incompressible kernel. The time at which the peak drag

coefficient occurs is predicted to be larger by the kernels of Longhorn [58] and Parmar et

al. [79] compared to that predicted by the incompressible kernel. The primary difference

between the drag-coefficient variation obtained from the kernels of Longhorn [58] and

Parmar et al. [79] is that the latter show a slower decay of the drag coefficient and are

closer to the computations and experiments during that phase.

A breakdown of the terms in the model of Parmar et al. is shown in Figs. 5-3(B)-5-5(B).

In the legends of these figures, the label “History” denotes that part of the inviscid

unsteady force given by the first term on the right-hand side of Eq. (5–3). In the present

123

model, the quasi-steady drag increases monotonically and is fully established at τs = 2.

Because the pressure-gradient drag is zero for τs ≥ 2, the existence of a negative

drag force for a limited time is determined by the relative magnitude of the quasi-steady

drag and the minimum value of the inviscid unsteady drag force. The viscous unsteady

contribution has been ignored in the present model. As can be expected, its contribution

is likely to be higher for the smaller spheres and thus explains, at least partially, the

larger discrepancy seen for the smallest sphere of diameter 8µm and also the very slow

decay in the computed drag coefficient for τs > 10.

5.3.1.2 Experiments of Skews et al.

Skews et al. [97] measured the unsteady drag on a sphere of diameter 5 cm for

shock waves with Ms = 1.08 and 1.31 and p1 = 83000 Pa and T1 = 293.15 K. The

corresponding sphere Reynolds numbers varied from about 105 to 6× 105 and the Mach

numbers M2 of the post-shock flow varied from 0.13 to 0.40.

The behavior of the drag force predicted by the present model is compared to the

measurements in Fig. 5-6. The large oscillations in the measured force may be due to

chaotic vortex shedding behind the sphere, which is relevant at the higher Reynolds

numbers considered here. The time-averaged mean value of the quasi-steady force

on the sphere long time after the passage of the shock is reasonably well captured

at both Mach numbers. The model captures the rapid increase and decrease of the

drag coefficient reasonably well. It is interesting that the peak is underpredicted for

Ms = 1.08 and overpredicted for Ms = 1.31. The difference, thus, cannot be satisfactorily

explained by the neglected viscous unsteady contribution. The difficulty of measuring

the unsteady force, associated uncertainties, and the shortcomings of the force model

perhaps contribute to the difference.

As with the results for the experiments of Sun et al. [102], the incompressible

kernel overpredicts the peak drag coefficient compared to the kernels of Longhorn [58]

and Parmar et al. [79]. The differences between the results obtained with the latter

124

0 5 10 15-10

0

10

20

30

40ExperimentModelModel M→0Model M=0

τs

CD

A Ms = 1.08

0 5 10 15-1

0

1

2

3

4

5

6

7ExperimentModelModel M→0Model M=0

τs

CD

B Ms = 1.31

Figure 5-6. Comparison of model with experiments of Skews et al. [97]

125

two kernels are restricted to the decay of the drag coefficient and approach to the

quasi-steady state, with the predictions by the kernel of Parmar et al. [79] being slightly

better.

5.3.2 Moving Sphere

Next we consider the motion of a sphere following the impact of a shock wave. The

majority of experimental data for spheres set in motion by shock waves are not suitable

for assessing the present model. The reason is that most experiments consider large

particle-to-fluid density ratios that result in negligible motion of the particle in the early

stages of the interaction with the shock wave. Here we make use of the experimental

data of Britan et al. [14] to validate the present model. Britan et al. [14] measured the

motion of a sphere of diameter 38 mm and a density of ρp = 89.4 kg/m3 in response

to a shock wave with Ms = 1.5 in a shock tube with a cross-sectional area of 64 cm2.

The relatively large blockage ratio leads to brief choking of the flow and reduces the test

time due to reflection of the shock wave from the side walls. Further disturbances are

introduced by the support that protrudes from the shock-tube floor to hold the sphere

prior to the arrival of the shock wave. Despite these deficiencies, the experiment of

Britan et al. [14] is interesting because it shows clearly the effect of the shock wave on

the motion of the sphere.

Britan et al. [14] also computed the sphere motion using a simple one-dimensional

model that included only the quasi-steady drag force and the drag-coefficient correlation

of Gilbert et al. [37], CD,qs = 0.48 + 28Re−0.85, for particle motion in the post-shock flow.

The results of Britan et al. [14] were presented in terms of a non-dimensional time τ ′s that

was zero when the undisturbed shock wave moved past the sphere, i.e., τ ′s = τs − 2.

They noted that their computations with an initial condition of zero initial velocity at

τ ′s = 0 gave poor agreement with the experimental data, indicating that the use of the

quasi-steady drag force only is insufficient. Britan et al. observed that an initial velocity

of 11 m/s had to be applied at τ ′s = 0 to obtain good agreement with the measured

126

evolution of the sphere position with time. This suggests that the sphere gained, nearly

instantaneously, a velocity of about 11 m/s during the interval −2 ≤ τ ′s ≤ 0. The present

model can be used to predict the motion of the sphere even during the initial interaction

with the shock wave.

The motion of the sphere predicted by the present model is presented in Fig.

5-7 along with the experimental data and computations of Britan et al. [14]. If we

take the density ratio ρp/ρ2 to be 32, the sphere trajectory evaluated with Eq. (5–16)

compares well with the experimental data. Model results obtained using density ratios

of ρp/ρ2 = 26 and 38 are also presented to examine sensitivity to variations in the gas

density. (The conditions ahead of the shock were not specified by Britan et al. [14])

It is clear that the initial velocity of 11 m/s suggested by Britan et al. [14] based on

their measurements is consistent with the initial velocities predicted with the present

inviscid unsteady force model. Therefore, the present model provides a rational way of

estimating the velocity imparted to the sphere due to the action of the inviscid unsteady

force.

The breakdown of the drag force as a function of the non-dimensional time τs and

the non-dimensional position xp/(2R) are shown in Fig. 5-8. It can be seen very clearly

that the inviscid unsteady force is negligible compared to the quasi-steady force for

xp/(2R) & 0.08, indicating that quasi-steady conditions are established before the

sphere has moved about a tenth of a diameter. The rapidity with which quasi-steady

conditions are attained explains why Britan et al. [14] were able to compute accurate

results for the sphere position and velocity with the quasi-steady drag force once the

initial velocity jump was taken into account.

It is noted that the experimental data exhibit time-dependent oscillations in the

particle position and velocity. Britan et al. [14] attributed the oscillations to shock waves

reflected from the shock-tube side walls. Vortex shedding behind the sphere may also

127

contribute to the oscillatory motion. Neither of these effects are captured by the present

model and hence no oscillations are present in the model predictions.

5.4 Discussion

The results presented in Section 5.3 clearly demonstrate that including the inviscid

unsteady force is crucial to capturing the peak in the unsteady force on particles due to

the interaction with a shock wave. In fact, as already mentioned, the results obtained

with the present model demonstrate much better agreement than might be expected

given that the model does not explicitly incorporate the details of the shock-wave

diffraction, reflection, and focusing processes.

The better-than-expected agreement can be explained as follows. We consider only

the case of a stationary spherical particle, since the argument is similar for a particle

set in motion by a shock wave. Following the impact of the shock wave on the front of

the sphere, the pressure near the front stagnation point is that behind a reflected shock

wave. Assuming that the sphere does not move, the pressure at the rear stagnation

point remains unchanged until the shock wave reaches the rear end. Thus at the very

early stages of the interaction process, the force on the sphere is determined only by

an increase in pressure near the front of the sphere. The inviscid unsteady force model

of Parmar et al. [79], however, is based on the compressible potential flow resulting

from an unsteady homogeneous ambient flow and hence approximates the actual

interaction in at least two ways. First, since the model does not incorporate the details

of the shock-interaction process, it underpredicts the pressure loading on the front of

the sphere. Second, the model is based on a simultaneous decrease in pressure at the

rear of the sphere as given by the potential flow due to the homogeneous ambient flow

acceleration. The underprediction of the pressure increase at the front combined with

the premature reduction of the pressure at the rear result in a reasonably accurate net

force prediction.

128

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

1.2ExperimentComp., Britan et al., vi=11m/sComp., Britan et al., vi=0Model, ρp/ρ1=26Model, ρp/ρ1=32Model, ρp/ρ1=38

sphere trajectories (D=38mm, M=1.5)

τ ′

s

xp/(

2R)

A Sphere trajectory

0 10 20 30 40 500.00

0.05

0.10

0.15

0.20ExperimentComp., Britan et al., vi=11m/sComp., Britan et al., vi=0Model, ρp/ρ1=26Model, ρp/ρ1=32Model, ρp/ρ1=38

sphere trajectories (D=38mm, M=1.5)

τ ′

s

v/u

2

B Sphere velocity

Figure 5-7. Comparison of model with experiments and computations of Britan et al. [14]

129

0 2 4 6 8 10-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0TotalQuasi-steadyPressure-gradientHistory

τs

CD

A As a function of non-dimensional time

0.00 0.02 0.04 0.06 0.08 0.10-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0TotalQuasi-steadyPressure-gradientHistory

xp/(2R)

CD

B As a function of non-dimensional distance moved

Figure 5-8. Breakdown of drag force for experimental conditions considered by Britan etal. [14]

130

At present, it is not clear whether the same level of agreement will hold for higher

shock-wave Mach numbers. Detailed numerical simulations indicate that for shock-wave

Mach numbers that lead to supercritical but subsonic post-shock Mach numbers, vortex

shedding occurs that makes a clear identification of inviscid unsteady forces difficult.

Furthermore, simulations not reported on here show that the inviscid unsteady force

is negligible compared to the change in the quasi-steady drag force at supersonic

post-shock Mach numbers.

The local minimum of the drag coefficient before the slow approach to the

quasi-steady-state value, was attributed by Tanno et al. [104] to shock-wave focusing

at the rear of the sphere. The local force minimum occurs after the undisturbed shock

wave has left the sphere, i.e., for τs > 2. Therefore, at the time of minimum force, the

contribution of the pressure-gradient term is identically zero and the minimum is only

determined by the relative magnitude of the inviscid unsteady and quasi-steady drag

terms. It seems reasonable to assume that the quasi-steady drag model is accurate,

given that the long-time behavior of the measured drag force is accurately captured by

the present model. Therefore, it can be concluded that the present model for the inviscid

unsteady drag force is reasonably effective in predicting the local minimum of the drag

coefficient, although the shock-focusing process is not taken into account. This in turn

suggests that the kernel and the present model for the ambient-flow acceleration seen

by the particle are effective.

The present model can also be applied to predict the inviscid unsteady force

exerted by shock waves on other particle geometries provided the appropriate inviscid

force kernels and a model for ambient flow acceleration seen by the particle are

developed. For the case of a cylinder subjected to a shock wave the results obtained

with the present model are compared with inviscid computations using the numerical

method described in [41, 43] in Fig. 5-9 for Ms = 1.22. As with the previously presented

results for the sphere, the overall agreement in terms of the peak value of the drag

131

coefficient, the time at which the peak value is attained, and the long-time behavior is

good.

We close the discussion by outlining how the model presented in this article could

be applied to the numerical simulation of shock waves interacting with millions to billions

of particles. In such simulations, shock waves are usually captured with shock-capturing

schemes and hence smeared over 2-5 cells. Thus particles interact with regions in

which the solution varies rapidly, but over length scales much larger than the particle

diameter. Within the framework of the Eulerian-Lagrangian point-particle approach, the

force decomposition given by Eq. (5–1) can be used with the ambient fluid properties

evaluated at the particle center, where the quasi-steady drag is evaluated from Eq.

(5–2), and the inviscid unsteady force is evaluated from Eq. (5–3) using the kernel

presented in Fig. 5-1(B). Simplified descriptions of the unsteady force, such as those

based on the kernel by Longhorn [58], see Eq. (5–4), or the incompressible kernel are

possible. The impact of the simplified descriptions on the unsteady drag coefficient was

discussed in Sections 3.1 and 2.3, respectively. If the particle diameter is comparable to

or larger than the width of the shock wave, the approach outlined in Section 5.2.4 can be

used.

132

0 5 10 15-2

0

2

4

6

8

10

12

ComputationModel

cylinder inviscid shock loading for MS = 1.22

τs

CD

A Comparison of model and computation

0 5 10 15-2

0

2

4

6

8

10

12

TotalPressure-gradientHistory

cylinder inviscid shock loading for MS = 1.22

τs

CD

B Breakdown of model terms

Figure 5-9. Results for shock interaction with cylinder at Ms = 1.22

133

CHAPTER 6AN IMPROVED DRAG CORRELATION FOR SPHERES AND APPLICATION TO

SHOCK-TUBE EXPERIMENTS

A new correlation is presented for the drag force on spherical particles in compressible

continuum flows. The new correlation represents experimental data more faithfully than

prior correlations. With this correlation, experimental data obtained in shock tubes

by Jourdan et al. (Proc. R. Soc. A, 463:3323-3345, 2007) for the particle velocity

behind shock waves can be reproduced accurately. This suggests that the higher drag

coefficients observed in several experiments involving shock-particle interaction could

simply be a consequence of compressibility.

6.1 Introduction

The drag coefficient on a spherical particle subjected to incompressible steady

uniform flow has been studied extensively. Empirical correlations exist that can very

accurately capture the drag coefficient as a function of Reynolds number, see, e.g.,

Clift and Gauvin [21]. These correlations are accurate provided that the particle is

smaller than the smallest length scale of the ambient flow and that the relative Mach

number, based on the velocity of the particle relative to that of the ambient flow, is

quite small. As the relative Mach number increases, compressibility effects become

important and the drag coefficient depends on both the Reynolds and Mach numbers.

Several authors have presented drag-coefficient correlations for finite Mach numbers,

see, e.g., Henderson [44] and Loth [59]. In this note, we first assess the accuracy of the

correlations of Henderson and Loth using the data collected by Bailey and Starr [5] and

develop an improved drag correlation that is quite accurate over a range of Reynolds

and Mach numbers.

We then validate the improved correlation for shock-particle interaction using the

recent shock-tube experiments of Jourdan et al. [50]. Shock-tube experiments are

often used to determine the drag coefficients of a single spherical particle or a cloud

of spherical particles, see, e.g., Ingebo [47], Selberg and Nicholls [95], Rudinger [90],

134

Sommerfeld [99], Igra and Takayama [46], Rodriguez et al. [88], Devals et al. [27],

and Jourdan et al. [50]. The drag coefficients for single particles obtained from such

experiments are often observed to be considerably higher than those given by the

standard-drag curve that describes the drag coefficient in the limit of quasi-steady

incompressible conditions. Discrepancies have been ascribed to several sources:

acceleration effects [46, 47], interference between particles [90], trajectory perturbations

[90], boundary layers on shock-tube walls [88], turbulence [112], surface roughness [95],

unsteady “shear waves” [50], and support mechanisms for particles prior to the arrival of

the shock wave [50]. Here we use the recent data of Jourdan et al. [50] as an example

and demonstrate that the experimentally observed increase in the drag coefficient is

primarily due to compressibility effects.

6.2 Improved Drag-Coefficient Correlation

Several models for the quasi-steady drag coefficient that incorporate a dependence

on the relative Mach number are available in the literature. The model of Henderson [44]

includes three correlations depending on whether the relative Mach number is below

unity or above 1.75, with linear interpolation in-between. Henderson demonstrated the

superior accuracy of his correlation compared to those of Carlson and Hoglund [16]

and Crowe [24]. Recently, Loth [59] presented a model that includes two correlations

depending on whether the relative Reynolds number is above or below 45, which is

taken to be the limit between rarefaction- and compressibility-dominated regimes.

The correlations of Henderson and Loth are compared with the data of Bailey and

Starr [5] in Figs. 6-1A and 6-1B. For a given Mach number, Henderson’s correlation

decreases monotonically with increasing Re and thus fails to capture the rise in the drag

coefficient as the critical Reynolds number is approached. Loth’s correlation shows a

consistent early rise as the Reynolds number increases for a given Mach number.1

1 It should be noted that Eq. (25b) in Loth’s article is missing the term 2√π/3s.

135

Furthermore, there is an overlap in Loth’s correlation for 0.89 ≤ M ≤ 1.0 that does

not exist in the experimental data. For Henderson’s correlation, the largest deviation

from the data of Bailey and Starr is about 16% for Reynolds numbers below 104. For

Loth’s correlation, the largest error is about 55%, concentrated around a Mach number

of about 0.9. Discrepancies of this magnitude call for the development of an improved

drag-coefficient correlation.

Here we propose an improved correlation based on the following assumptions: (i)

We limit our attention to continuum flows, i.e., we assume that the Knudsen number

Kn =M

Re

√πγ

2< 0.01, (6–1)

where M is the Mach number based on relative velocity, Re is the Reynolds number

based on the relative velocity and particle diameter, and γ is the ratio of specific heats.

(ii) The particle temperature is assumed constant and equal to the surrounding gas

temperature. (iii) The drag correlation should approach the standard drag relation given

by Clift and Gauvin [21] in the limit of zero Mach number,

CD,std(Re) =24

Re

(1 + 0.15Re0.687

)+ 0.42

(1 +

42500

Re1.16

)−1

. (6–2)

(iv) Attention is restricted to subcritical Reynolds numbers, i.e., Re / 2 · 105, above

which the attached boundary layer becomes turbulent. (v) We focus on the range

0 6 M 6 1.75 and make use of the extensive data compiled by Bailey and Starr [5].

The resulting improved drag-coefficient correlation consists of three separate

correlations for subcritical (0 6 M 6 Mcr ≈ 0.6), supersonic (1 < M 6 1.75) and

intermediate (Mcr 6 M 6 1) Mach-number regimes:

CD(Re,M) =

CD,std(Re) +(CD,Mcr

(Re)− CD,std(Re)) MMcr

if M 6 Mcr,

CD,sub(Re,M) if Mcr < M 6 1.0,

CD,sup(Re,M) if 1.0 < M 6 1.75 .

(6–3)

136

Re

CD

101 102 103 104 105 106

0.5

1.0

1.5

2.0

2.5 M=0.10M=0.49M=0.72M=0.82M=0.89M=0.96M=1.05M=1.15M=1.25M=1.75Standard dragBailey & Starr data

A Correlation of Henderson [44]

Re

CD

101 102 103 104 105 106

0.5

1.0

1.5

2.0

2.5 M=0.10M=0.49M=0.72M=0.82M=0.89M=0.96M=1.05M=1.15M=1.25M=1.75Standard dragBailey & Starr data

B Correlation of Loth [59]

Figure 6-1. Comparison of drag correlations with data of Bailey and Starr [5] assumingthat γ = 1.4 and that the particle temperature is equal to the surrounding gastemperature.

137

This separation is motivated by the following observations. (i) For subcritical Mach

numbers, the flow around a spherical particle is shock-free and thus the drag coefficient

is only weakly affected by compressibility effects. (ii) For supercritical but subsonic

Mach numbers, an annular shock wave of limited radial extent exists on the sphere and

the drag coefficient becomes more strongly dependent on the Mach number. (iii) For

supersonic Mach numbers, a bow shock exists that leads to a large increase in drag

coefficient. (It should be noted that the bow shock does not appear at precisely sonic

conditions, but the above simple separation into regimes is sufficient for our modeling

purposes.) (iv) The upper limit of M = 1.75 is used because that is the maximum

Mach number for which Bailey and Starr presented data. This limit is sufficient for our

purposes because we are mainly interested in unsteady shock waves accelerating

initially stationary particles. For such interactions, the largest possible Mach number is

M ≈ 1.89 for γ = 1.4.

In the subcritical regime, the drag coefficient is expressed as a linear interpolation

between the drag coefficients at M = 0 and M = Mcr. (It should be noted that we

do not make use of the data of Bailey and Starr for M < 0.6 because it exhibits drag

coefficients that lie below the standard drag curve.) Based on the functional form of Eq.

(6–2), CD,Mcr(Re) is expressed as

CD,Mcr(Re) =

24

Re

(1 + 0.15Re0.684

)+ 0.513

(1 +

483

Re0.669

)−1

. (6–4)

Note the similarity of the coefficients compared to Eq. (6–2) due to the weak influence of

the compressibility in this regime, as stated above.

In the supersonic regime, the drag coefficient is expressed as a nonlinear

interpolation between the drag coefficients at M = 1 and M = 1.75,

CD,sup(Re,M) = CD,M=1(Re) +(CD,M=1.75(Re)− CD,M=1(Re)

)ξsup(M, Re) , (6–5)

138

where

CD,M=1(Re) =24

Re

(1 + 0.118Re0.813

)+ 0.69

(1 +

3550

Re0.793

)−1

, (6–6)

CD,M=1.75(Re) =24

Re

(1 + 0.107Re0.867

)+ 0.646

(1 +

861

Re0.634

)−1

, (6–7)

and

ξsup(M, Re) =

3∑i=1

fi ,sup(M)− fi ,sup(1)

fi ,sup(1.75)− fi ,sup(1)

3∏j =i

j=1

logRe− Cj ,sup

Ci ,sup − Cj ,sup

, (6–8)

with

f1,sup(M) = 0.126 + 1.15M − 0.306M2 − 0.007M3 − 0.061 exp

(1−M

0.011

), (6–9)

f2,sup(M) = −0.901 + 2.93M − 1.573M2 + 0.286M3 − 0.042 exp

(1−M

0.01

), (6–10)

f3,sup(M) = 0.13 + 1.42M − 0.818M2 + 0.161M3 − 0.043 exp

(1−M

0.012

), (6–11)

and

C1,sup = 6.48, C2,sup = 8.93, C3,sup = 12.21. (6–12)

In the intermediate regime (supercritical and subsonic), the drag coefficient is

expressed as a non-linear interpolation between the drag coefficients at Mcr and M = 1,

CD,sub(Re,M) = CD,Mcr(Re) +

(CD,M=1(Re)− CD,Mcr

(Re))ξsub(M, Re) , (6–13)

where

ξsub(M, Re) =

3∑i=1

fi ,sub(M)− fi ,sub(Mcr)

fi ,sub(1)− fi ,sub(Mcr)

3∏j =i

j=1

logRe− Cj ,sub

Ci ,sub − Cj ,sub

, (6–14)

with

f1,sub(M) = −0.087 + 2.92M − 4.75M2 + 2.83M3 , (6–15)

f2,sub(M) = −0.12 + 2.66M − 4.36M2 + 2.53M3 , (6–16)

f3,sub(M) = 1.84− 5.13M + 6.05M2 − 1.91M3 , (6–17)

139

Re

CD

101 102 103 104 105 106

0.5

1.0

1.5

2.0

2.5 M=0.62M=0.72M=0.82M=0.89M=0.96M=1.05M=1.15M=1.25M=1.75Standard dragBailey & Starr dataKn=0.01

Figure 6-2. Comparison of new drag correlation with data of Bailey and Starr Bailey andStarr [5].

and

C1,sub = 6.48, C2,sub = 9.28, C3,sub = 12.21. (6–18)

The range of applicability of the improved correlation is limited to Re 6 2 · 105,

M 6 1.75, and Kn < 0.01. The improved correlation is compared to the data of Bailey

and Starr in Fig. 6-2. Within the range of applicability, the largest deviation between

the improved correlation and the data of Bailey and Starr is only 2.5% for M > 0.6.

This deviation is substantially smaller than those for the correlations of Henderson and

Loth. In Fig. 6-3, a comparison with the earlier data of Goin and Lawrence [38] and May

and Witt [64] is shown. It can be seen that even at lower Reynolds numbers, the new

drag-coefficient correlation agrees quite well with experimental data.

140

Re

CD

102 103 104 1050.3

0.5

0.7

0.9

1.1

1.3

1.5 M=0.20M=0.33M=0.46M=0.60M=0.75M=0.89M=0.98M=1.00M=1.50Clift & GauvinGoin and LawrenceMay and Witt

Kn=0.01

Figure 6-3. Comparison of new drag-coefficient correlation with data of Goin andLawrence [38] and May and Witt [64].

6.3 Validation

We validate the new drag correlation using the shock-particle interaction experiments

of Jourdan et al. [50]. Compared to other shock-tube experiments, the measurements of

Jourdan et al. [50] are more suitable because (i) boundary-layer effects are eliminated

by hanging the spherical particles from a spider-web thread, (ii) a large part of the

particle trajectory is recorded using multiple shadowgraphs in a single run, and (iii)

interference between particles is minimized by testing no more than three particles

simultaneously. Table 6-1 lists the cases from the experiments of Jourdan et al. [50] that

we use to validate our model. (See Table 2 in Ref. [50] for a complete list of cases and

the report by Jourdan and Houas [49] for more details.) The cases include experimental

141

Table 6-1. Summary of selected test cases taken from Jourdan et al. [50] used tocompare with model. The column labeled ‘gases’ lists the gases in the driverand driven sections of the shock tube.

M dp ρpCase Gases

− mm kg/m3

148 air/air 0.47− 0.66 1.92 1130166a air/air 0.47− 0.61 1.96 1204181a helium/air 1.28− 1.60 0.62 1096184a helium/air 0.70− 1.53 1.60 25

conditions leading to subsonic and supersonic particle Mach numbers behind the shock

wave.

The time history of the particle velocity measured by Jourdan et al. is compared

against the velocity computed from the particle equation of motion

mp dup

dt= F (t) , (6–19)

where mp is the mass of the particle, up is the particle velocity, and F (t) is the drag

force on the particle. We assume that the drag force on the particle is given by the

quasi-steady drag that depends on the instantaneous relative velocity between the

particle and ambient fluid as

F (t) =π

8ρ2(u

g2 − up(t))|ug2 − up(t)|(dp)2CD(Re(t),M(t)) (6–20)

where ρ2 and ug2 are the density and velocity of the gas behind the shock (assumed to

be constants), dp is the diameter of the particle, CD(Re(t),M(t)) is given by Eq. (6–3),

and the relative Reynolds and Mach numbers are

Re(t) =ρ2 |ug2 − up(t)| dp

µ2and M(t) =

|ug2 − up(t)|a2

, (6–21)

where µ2 and a2 are the dynamic viscosity and speed of sound of the gas behind

the shock. Other forces, such as those arising from unsteady effects and gravity, are

142

negligible for the time scales of interest in this study. See Parmar et al. [79, 80] for more

information on why the inviscid unsteady force can be neglected.

The results are presented in Fig. 6-4, where the particle velocity up normalized by

the post-shock gas velocity u2 is plotted against the nondimensional time τs = ust/dp.

(Note that τs = 1 corresponds to the time required for the shock wave to propagate over

the particle.) As can be seen from Figs. 6-4A and 6-4B, the particle velocity is predicted

quite accurately with the new drag-coefficient correlation for cases 148 and 166a.

Because the relative Mach numbers are subsonic and mostly subcritical, the difference

between the results obtained with the standard and the new drag-coefficient correlations

are relatively small. The standard drag-coefficient correlation leads to better results than

Henderson’s correlation. The explanation for this result is that Henderson’s correlation

under-predicts the standard-drag coefficient curve for the range of Mach and Reynolds

numbers encountered in cases 148 and 166a, see Fig. 6-1A. Figures 6-4C and 6-4D

show results for cases 181a and 184a, where the relative Mach number after the

passage of the shock wave over the particle is supercritical. The results obtained with

both the new correlation and Henderson’s correlation agree well with the experimental

data.

Figures 6-4C and 6-4D show results for the cases where the relative Mach number

after the passage of the shock wave over the particle is supercritical. As the result,

the compressibility effect on the quasi-steady drag is significant. These cases show

that the differences between the results obtained with the standard and the new

drag-coefficient correlations are substantial. In both cases, the new drag-coefficient

correlation accurately captures the evolution of the normalized particle velocity.

The results for the four cases suggest that the discrepancies between the

experiments of Jourdan et al. [50] and the standard drag correlation can be traced

to compressibility effects.

143

τs (× 10-5)

up /u2g

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35Standard drag correlationNew drag correlationExperiment

A Case 148

τs (× 10-5)

up /u2g

0 2 4 6 80.00

0.05

0.10

0.15

0.20

0.25Standard drag correlationNew drag correlationExperiment

B Case 166a

τs (× 10-5)

up /u2g

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25Standard drag correlationNew drag correlationExperiment

C Case 181a

τ s (× 10-5)

up /u2g

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8Standard drag correlationNew drag correlationExperiment

D Case 184a

Figure 6-4. Comparison of model with experimental data of Jourdan et al. [50] Note theabscissa scaling.

144

6.4 Discussion

As mentioned in the introduction, it has been argued that unsteadiness could

contribute to the increased drag on the particle observed in shock-particle interaction

experiments [46, 47]. Two sources of unsteadiness can be identified in shock-particle

interactions. First, as the shock wave moves over the particle, the ambient flow

around the particle rapidly changes from the quiescent condition ahead of the shock

wave to the usually uniform post-shock flow. Recent experiments by Sun et al. [102]

and Skews et al. [97] and the theoretical model of Parmar et al. [80] have shown

that the unsteady contribution to the force on the particle immediately following the

shock-particle interaction can be an order of magnitude larger than the quasi-steady

force. Furthermore, the unsteady force decays rapidly on an acoustic time scale and

its effect can be ignored for τs ' O(5 − 10). Referring to Fig. 6-4, it can be seen that

the unsteady contribution to the force is insignificant on the the time scale on which

the particle attains a significant fraction of the post-shock gas velocity. The velocity

gained by the particle as a fraction of the past-shock gas velocity due to the unsteady

inviscid force can be shown to be proportional to the gas-to-particle density ratio [80].

The second source of unsteadiness arises from the acceleration of the particle that

takes place on the particle time scale ρ(dp)2/18νg. The ratio of the unsteady force

due to particle acceleration to the quasi-steady force, however, again scales as the

gas-to-particle density ratio. Thus it is clear that unsteady effects are insignificant for

the velocity evolution of the particle following its interaction with the shock wave if the

particle density is substantially larger than the gas density. This is indeed the case in the

experiments of Jourdan et al. [50] and in most shock-tube studies.

6.5 Conclusions

An improved correlation for the drag coefficient of spherical particles was presented.

The new correlation represents the available experimental data more accurately than

previous correlations. The model is capable of reproducing particle velocities following

145

the impact of shock waves measured in the experiments of Jourdan et al. [50] This

suggests that the increased drag coefficients observed in the shock-tube experiments

of Jourdan et al. are simply due to compressibility effects and not due to unsteadiness.

It is hypothesized that the discrepancies seen in at least some of the earlier shock-tube

experiments are also due to compressibility effects. Additional carefully designed and

executed experiments are required to verify this hypothesis.

146

CHAPTER 7UNSTEADY FORCES ON A PARTICLE IN VISCOUS COMPRESSIBLE FLOWS AT

FINITE MACH AND REYNOLDS NUMBERS

Effects of nonlinearity are explored for unsteady forces on a particle in viscous

compressible flows. Theoretical investigations are used to study linearized flows in

Chapters 2 and 3. Such theories are not possible for finite M and Re flows. Chapter

4 presented inviscid unsteady forces in finite but subcritical Mach-number flows using

carefully designed numerical simulations. This chapter presents results from high fidelity

numerical simulations used to study the inviscid unsteady and viscous unsteady forces

on a sphere in a compressible flows at finite M and Re.

7.1 Introduction

The unsteady force on a particle in accelerated motion was first analyzed by Stokes

[101], who presented an expression for the frequency-dependent force on an oscillating

spherical particle. Later Basset [8], Boussinesq [10], and Oseen [77] independently

examined the time-dependent force on a sphere due to rectilinear motion in a quiescent

viscous incompressible fluid. They based their analyses on the linearized unsteady

incompressible Navier-Stokes equations valid for creeping motion, i.e., in the limit of

vanishing Reynolds number. The resulting equation of motion for a spherical particle,

the so-called BBO equation, can be written as

mp

dv

dt= −6πaµv − 1

2mf

dv

dt− 6a2ρ

√πν

∫ t

−∞KBH(t − ξ)

dv

dξdξ , (7–1)

where

KBH(t) =1√t, (7–2)

and mp is the particle mass, v(t) is particle velocity, a is the particle radius, µ is the

dynamic viscosity, mf is the mass of fluid displaced by the particle, ρ is the fluid density,

and ν = µ/ρ is the kinematic viscosity. The three terms on the right-hand side are the

quasi-steady (Stokes) drag, inviscid unsteady (added-mass), and viscous unsteady

(Basset history) forces, respectively.

147

Extension of BBO equation to compressible creeping flows is recently proposed by

Parmar et al. [81]. The generalized particle equation of motion in compressible flows

becomes

mp

dv

dt= −6πaµv −mf

∫ t

−∞KLH

((t − ξ)

c0

a

) dvdt

dξc0

a− 6a2ρ

√πν

∫ t

−∞KPHB(t − ξ)

dv

dξdξ ,

(7–3)

where c0 is the speed of sound, KLH(t) is the inviscid unsteady kernel first found

by Longhorn [58], and KPHB(t) is the viscous unsteady kernel in compressible flow

proposed by Parmar et al. [81]. KLH(t) and KPHB(t) are given by

KLH(τ) = e−τ cos τ (7–4)

and

KPHB(t) =C(c0t/a)√

t, (7–5)

where τ = c0t/a and C(c0t/a) is the compressible correction-function to Basset-history

force whose expression can be found in Parmar et al. [81].

Non-linearity effects for unsteady forces were studied by Mei and Adrian [68],

Lovalenti and Brady [60], and Kim et al. [52] in incompressible flows. The BBO equation

has been extended to finite Reynolds numbers by Mei and Adrian [68], Kim et al. [52],

and Magnaudet and Eames [62]. First important finding was that added-mass force is

independent of Re, see Mei et al. [66], Rivero et al. [87], Chang and Maxey [19], and

Wakaba and Balachandar [114]. Second, it was found that the Basset history force is

not uniformly valid for all Reynolds numbers. The viscous-unsteady force was found

to be dependent on Re. Even for creeping flows (Re → 0), the long time decay rate

is affected by non-linearity. Mei and Adrian [68] proposed a modified expression for

viscous-unsteady force which behaves like Basset history force for short times (1/t1/2)

and for long times it decays at a faster rate (1/t2). For finite Re flow also, short time

behavior can be predicted by Basset history force. The time scale on which nonlinear

148

effects become significant can be estimated as follows. In deriving the linearized form

of the compressible Navier-Stokes equations, the assumption that the inertial terms

are negligible compared to the viscous terms implies that the length scale L ≪ ν/V ,

where ν is kinematic viscosity and V is scale of flow velocity. If we take the length scale

to grow by diffusion as√νt, where t is time, the assumption of linearized Navier-Stokes

equations can be justified only for t ≪ ν/V 2. Expressed in terms of the convective

time scale, this restriction becomes tc = tV /L ≪ 1/Re. Particle equation of motion

incorporating modified history-kernel proposed by Mei and Adrian [68] can be written as,

mp

dv

dt= −6πaµvϕ(Re)− 1

2mf

dv

dt− 6a2ρ

√πν

∫ t

−∞KMA(Re; t − ξ)

dv

dξdξ , (7–6)

where

KMA(Re; t) =(K

−1/2BH (t) + K

−1/2NL (Re; t)

)−2

, (7–7)

where

KNL(Re; t) =

(√32ν3f 6Hπv 6

)1

t2, (7–8)

where fH = 0.75 + 0.105Re. For finite Reynolds-number flows ϕ(Re) = 1 + 0.15Re0.687,

see Schiller and Naumann [94]. The short and long time limits are as follows,

limt≪a/v

KMA(t)→ KBH(t) , (7–9)

limt≫a/v

KMA(t)→ KNL(t) . (7–10)

In the limit of short time the modified history kernel of Mei & Adrian reduces to Basset

history. Equation 7–6 can be considered an extension of BBO equation to non-linear

finite Reynolds-number flows.

A force kernel can be interpreted as force due to delta-function acceleration. Figure

7-1 shows normalized forms of above mentioned inviscid- and viscous-unsteady forces

vs non-dimensional time τ for M = 0.01 and Re = 1. The forces are normalized by

mf c0/a.

149

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

BBOMei andAdrianLonghornParmar et al.

Figure 7-1. Time evolution of normalized unsteady force for M = 0.01, Re = 1. Bassethistory force (Eq. (7–1)), modified history force due to Mei and Adrian [68](Eq. (7–6)), inviscid unsteady force in compressible flows due to Longhorn[58] (Eq. (7–4)), and inviscid and viscous unsteady force in compressibleflows due to Parmar et al. [81] (Eq. (7–3)) are plotted.

Corresponding extension of particle equation of motion to non-linear compressible

flows has not been studied so far. Here we carry out carefully designed numerical

simulations to study unsteady forces on a particle in finite Mach- and Reynolds-number

flows. As discussed earlier, in a compressible flow, both inviscid- and viscous-unsteady

forces have an integral representation depending on history of relative acceleration

of the fluid with respect to the particle. A straight-forward separation of inviscid- and

viscous-unsteady forces is not possible. Numerical methodology used to study the

unsteady forces is described in Section 7.2. Next, results from numerical simulation

for M = 0.01 and Re = 1, 10 are described in Section 7.3. Motivated by the finding in

incompressible flows that added-mass force is independent of Reynolds number, we

model inviscid unsteady force in a compressible flow to be independent of Reynolds

150

number. A new kernel for viscous unsteady force is proposed based on work of Parmar

et al. [81] and Mei and Adrian [68]. Results for finite Mach- and Reynolds-numbers are

give in Section 7.4. Finally, conclusions are presented in Section 7.5.

7.2 Numerical Methodology

Two different numerical methods are used in this investigation, namely the

dissipative solver and the non-dissipative solver. The numerical methods solves the

compressible Navier-Stokes equations in integral form cast in a frame of reference

attached to the sphere.

In the dissipative solver the spatial discretization is based on the flux-difference

splitting method of Roe (1981) (see [41] for reference) and the weighted essentially

non-oscillatory reconstruction described by Haselbacher [41]. The discrete equations

are integrated in time using the four-stage Runge-Kutta method. The methodology

of dissipative solver employed in this work has been applied to several unsteady

compressible flows and demonstrated good agreement with theory and experimental

data, see, e.g., Haselbacher et al. [43].

The non-dissipative solver is based on the formulation given by Hou and Mahesh

[45]. Details of this method is given in Appendix.

Axisymmetric formulation is used to simulate flow around a sphere. A two-dimensional

hexahedral grid of O-type topology is used with 200 cells around the semi-circumference.

Relative to the cylinder radius a, the radial grid spacing adjacent to the cylinder surface

is �r/a = 1.5 × 10−4. and is gradually increased to reach aspect ratio of nearly unity far

from cylinder. The radial stretching of grid cells is adjusted such that near the particle

surface high aspect ratio cells are created to capture boundary layer growth and away

from particle each layer of cells consists of approximately square cells to minimize

internal wave reflections. We have employed grids consisting of up to 124, 800 cells to

assess grid-independence of our solutions. The results shown below were obtained on

151

x

y

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

A High aspect ratio cells near cylinder surface.

x

y

0 2 4 6 8 100

2

4

6

8

B Nearly square cells away from cylinder.

Figure 7-2. Mesh quality.

152

tti

M∞,Re ∞

M∞

,0,Re ∞

,0

M∞,0(1 + δ), Re∞,0(1 + δ)

Figure 7-3. Schematic depiction of variation of freestream Mach and Reynolds numberduring computations.

grids of 62, 400 cells (coarse mesh) and 124, 800 cells (fine mesh). Fig. 7-2 shows the

near-field mesh and far-field mesh for coarse mesh.

For the computations, the characteristic boundary conditions of Poinsot & Lele [85]

are applied at the outer boundary, located at 100a.

We divide the computations into two stages. In stage one, the sphere is held fixed

and a steady-state solution is obtained for M∞ = M∞,0 and Re∞ = Re∞,0. In stage two,

we impose a delta-function negative acceleration in the x-direction on the sphere at time

ti . The sphere will thus attain a relative Mach and Reynolds numbers M∞,0(1 + δ) and

Re∞,0(1 + δ) respectively, where δ > 0, as depicted schematically in Fig. 7-3. We choose

δ ≪ 1 to ensure that the change in Mach and Reynolds number is small and that we can

therefore isolate the effect of the initial freestream Mach and Reynolds numbers on the

drag force.

7.3 Unsteady Forces Over a Sphere at Small Mach Numbers and Finite ReynoldsNumbers

In Chapter 2 results of numerical simulation for M = 10−3 and Re = 0.1, 1.0, 10

were shown. For such low Mach- and Reynolds-number range the numerical simulation

153

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

BBOMei andAdrianParmar et al.

Figure 7-4. Time evolution of normalized unsteady force for M = 0.01, Re = 1. Bassethistory force (Eq. (7–1)), modified history force due to Mei and Adrian [68](Eq. (7–6)), and inviscid and viscous unsteady force in compressible flowsdue to Parmar et al. [81] (Eq. (7–3)) are plotted. Corresponding simulationresults are shown as open circle symbols.

results were in excellent agreement with results of theoretical investigation of linearized

compressible Navier-Stokes equations for the time range shown there. For sufficiently

long time, the non-linear effects results in deviation from the theory based on linearized

flow. To get non-linear effects earlier in time, here we choose M = 10−2 and Re =

1, 10, 100.

Figure 7-4 shows numerically obtained normalized unsteady force, existing theories,

and empirical correlations for M = 0.01 and Re = 1. For short-time (τ ≪ 1), numerical

data follows compressibility-modified history-force (Parmar et al. [81]) closely. For

long-time (τ ≫ 1), numerical data follows modified history-kernel of Mei and Adrian [68].

Modeling inviscid-unsteady force as independent of Reynolds number we can separate

viscous-unsteady force. For the M = 0.01 flow considered here, inviscid-unsteady

154

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

A M = 0.01, Re = 1.

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

B M = 0.01, Re = 10.

Figure 7-5. Comparison of normalized unsteady force obtained by numerical simulationand that given by the last two terms of Eq. (7–12). Simulation results areshown as open circle symbols.

155

kernel can be taken same as Longhorn’s kernel given by Eq. (7–4). A new kernel can

be proposed for viscous-unsteady force in compressible flows at finite Reynolds-number

to mimic observed behavior. In the new kernel we replace Basset-history force in the

formula of KMA(t) (Eq. (7–7)) by recently found viscous-unsteady kernel in compressible

flows KPHB(t) (Eq. (7–5)) by Parmar et al. [81]. We can write new viscous-unsteady

kernel as

K newvu (Re; t) =

(K

−1/2PHB (t) + K

−1/2NL (Re; t)

)−2

(7–11)

A new particle equation of motion valid for very small Mach-numbers and finite but

small Reynolds-numbers can be written as,

mp

dv

dt= −6πaµvϕ(Re)−mf

∫ t

−∞KLH

((t − ξ)

c0

a

) dvdt

dξc0

a

− 6a2ρ√πν

∫ t

−∞K newvu (Re; t − ξ)

dv

dξdξ , (7–12)

Figures 7-5A and 7-5B show comparison of numerically obtained unsteady force

and that predicted by Eq. (7–12) for M = 0.01, Re = 1 and M = 0.01, Re = 10

respectively. Numerical data and model predictions are in excellent agreement.

7.4 Unsteady Forces Over a Sphere at Sub-Critical Mach Numbers and FiniteReynolds Numbers

We carry out numerical simulations for finite but subcritical Mach-number and finite

Reynolds-number flows. Following four cases are considered:

1. M = 0.1 and Re = 10

2. M = 0.2 and Re = 20

3. M = 0.3 and Re = 30

4. M = 0.4 and Re = 40

Figures 7-6A, 7-6B,7-7A, and 7-7B show numerically obtained unsteady force

due to a delta-function acceleration. As mentioned earlier, we model inviscid-unsteady

force to be independent of Reynolds number and dependent only on Mach number as

156

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

ModelSimulation

A M = 0.1, Re = 10.

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

ModelSimulation

B M = 0.2, Re = 20.

Figure 7-6. Comparison of normalized unsteady force obtained by numerical simulationand that given by new model (the last two terms of Eq. (7–13)).

157

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

ModelSimulation

A M = 0.3, Re = 30.

10-4 10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

101

102

τ

Norm

alizedUnsteadyForce

ModelSimulation

B M = 0.4, Re = 40.

Figure 7-7. Comparison of normalized unsteady force obtained by numerical simulationand that given by new model (the last two terms of Eq. (7–13)).

158

a parameter. Effect of Mach number on the inviscid-unsteady kernel was studied by

Parmar et al. [79]. Here we use the same model to represent inviscid-unsteady kernel

denoted by Kiu(M; τ) and separate viscous-unsteady force. The viscous-unsteady force

for a viscous-compressible flow at finite Mach- and Reynolds-number depends on both

Mach and Reynolds-numbers. As a first approximation, we model viscous-unsteady

force as only a function of Reynolds number using the new kernel K newvu (Re; t) used in

Section 7.3 in Eq. (7–7). An approximate particle equation valid for finite Mach- and

Reynolds-number flows can be written as

mp

dv

dt= −6πaµvϕ(Re)−mf

∫ t

−∞Kiu

(M; (t − ξ)

c0

a

) dvdt

dξc0

a

− 6a2ρ√πν

∫ t

−∞K newvu (Re; t − ξ)

dv

dξdξ , (7–13)

Figures 7-6A, 7-6B,7-7A, and 7-7B also show prediction using last two terms of Eq.

(7–13). For very small time, numerically obtained unsteady force follow compressibility-modified

history force (last term of Eq. (7–3)). Unlike the case of weak Mach number flows

considered in Section 7.3, the numerically obtained unsteady-force in finite Mach-number

flows considered here does not follow prediction by new kernel K newvu (Re; t) for long time.

This deviation is result of finite Mach-number effect on viscous-unsteady kernel which

has been neglected in first approximation considered here. For long time, however, the

decay rate of 1/t2 is maintained. The magnitude of offset increases with Mach number.

7.5 Conclusions

Unsteady forces in viscous-compressible flows at finite Mach- and Reynolds-number

are studied. As discussed in Chapter 2, solution of linearized compressible Navier-Stokes

equations is valid for early time τ < 1/(ReM) even for finite Mach- and Reynolds-number

flows. We can identify three regimes for Mach and Reynolds numbers.

1. Regime I: Very small Mach and Reynolds numbers, i.e., M ≪ 1 and Re ≪ 1 suchthat M/Re≪ 1 to ensure continuum assumption.

2. Regime II: Small Mach-numbers and finite Reynolds-numbers.

159

3. Regime III: Large Mach- and Reynolds-numbers.

For regime I, τ < 1/(ReM) is satisfied for up to very large time and thus linearized

solution are valid till long time. For regime II, τ < 1/(ReM) is violated early in time

and non-linear effects causes solution to deviate from linear solution. Since Mach

number is still small, its effect on non-linear regime at large time is weak and negligible.

In the limit of small Mach number (M ≪ 1) and finite Reynolds-number, a new

particle equation of motion is proposed (Eq. (7–12)). Standard drag correlation of

Schiller and Naumann [94] is used for quasi-steady drag. Inviscid-unsteady force is

modeled independent of Reynolds number as given by Longhorn [58]. A new kernel is

proposed for viscous-unsteady force incorporating non-linear effects depending only

on Reynolds number as parameter. The predictions for unsteady forced using Eq.

(7–12) are in excellent agreement with numerically obtained results. For finite Mach-

and Reynolds-number flow in Regime III, as a first approximation, a particle equation

of motion is proposed consisting of standard drag correlation of Schiller and Naumann

[94], Mach-number dependent inviscid-unsteady kernels of Parmar et al. [79], and

Reynolds-number dependent viscous-unsteady kernel proposed in Section 7.3 (Eq.

(7–11)). Finite Mach-number effects on non-linear behavior at intermediate and large

time for viscous-unsteady kernel is not modeled in this work. Work is in progress to

model the effects of finite Mach-number on viscous-unsteady kernel.

160

CHAPTER 8SUMMARY, CONCLUSIONS, AND FUTURE WORK

8.1 Summary and Conclusions

The main results and conclusions arrived at in this work can be summarized as

follows:

BBO equation for compressible creeping flows. A theoretical analysis is carried

out for the linearized compressible Navier-Stokes equations. An exact solution is

obtained for the force acting on a sphere undergoing unsteady rectilinear motion in a

homogeneous fluid. The total force is expressed as the sum of the quasi-steady, inviscid

unsteady, and viscous unsteady forces. The quasi-steady force is identical to the well

known Stokes drag in the incompressible limit. The inviscid unsteady force is very

different from that encountered in incompressible flows. The singularity of the inviscid

unsteady force in the incompressible limit is regularized for the case of compressible

flows because of the finite speed of sound. The expression for the inviscid unsteady

force in viscous compressible flow is identical to that obtained by Longhorn [58] for a

sphere accelerating in an inviscid compressible fluid in the acoustic limit (M → 0). The

inviscid unsteady force is represented by an integral that depends on the acceleration

history. The effect of compressibility on the viscous unsteady force is found to be

moderate. The viscous unsteady force is modified for short times only (c0 t/a ≤ 10),

after which it reverts to the incompressible behavior, with a decay rate of t−1/2. An

exact analytical expression for the viscous unsteady force in a compressible flow cannot

be obtained. A curvefit to numerically inverted data is proposed that is accurate to

within 1%. With the expression for the forces, an equation of motion is proposed as

the compressible extension of the Basset-Boussinesq-Oseen (BBO) equation. The

proposed particle equation of motion can be used in Lagrangian tracking of point

particles in compressible flows in the limit Re≪ 1 and M≪ 1.

161

MRG equation for compressible creeping flows. The theoretical analysis

underlying the derivation of the BBO equation for compressible creeping flows is

extended to investigate the force on a sphere undergoing unsteady motion in a

non-uniform compressible flow. A systematic approach is adopted to simplify the

governing equations. A density weighting technique is used to incorporate density

changes. Faxen’s theorem is derived for the force on the sphere using the reciprocal

theorem for viscous compressible flows. Faxen corrections are derived for the

quasi-steady, inviscid unsteady, and viscous unsteady forces. The Faxen corrections

appear as surface and volume averages of quantities of the flow relative to the particle.

In contrast to incompressible flows, the volume-average quantities appear in the

inviscid unsteady as well as the viscous unsteady forces. With the expression for

the forces, an equation of motion is proposed that can be considered an extension of the

Maxey-Riley-Gatignol (MRG) equation to compressible flows.

Inviscid unsteady force at finite Mach numbers. Carefully designed numerical

simulations are carried out to investigate the inviscid unsteady force for cylinders

and spheres at finite but subcritical freestream Mach numbers. This work can be

considered an extension of the work of Miles [71] and Longhorn [58] As noted by Miles

and Longhorn, the concept of added-mass is not applicable to compressible flows

because the dependence of the force on the instantaneous acceleration is broken. A

response kernel Kiu can be defined that represents the force response to an impulsive

acceleration of unit magnitude. The total force for a general acceleration case can be

obtained by a convolution integral of the response kernel and the acceleration. For finite

Mach numbers, the behavior of the inviscid unsteady response kernels is similar to that

in the acoustic limit. For constant acceleration and long times, an effective added-mass

coefficient can be derived. The effective added-mass coefficient for finite but subcritical

Mach numbers increases to a value that is nearly double the added-mass coefficient in

an incompressible flow. In a subcritical compressible flow, the inviscid unsteady force

162

can be up to four times larger than that in incompressible flows. The effect of the ratio of

specific heats is found to increase with increasing Mach numbers.

Shock-particle interaction model. Shock-particle interaction usually leads to large

unsteady forces. A simple model to account for the unsteady force in shock-particle

interaction was constructed. Results obtained with this model clearly show that including

the inviscid unsteady force is crucial to capture the peak force acting on a particle. The

simple model predicts the unsteady force with good accuracy when compared with

recent experimental data on stationary particles. The overall agreement in terms of the

peak value of the drag coefficient, the time at which the peak value is attained, and the

long-time behavior is good. For the moving particles it is observed that the primary effect

of the inviscid unsteady force is to give an initial impulsive increase in particle velocities.

Drag-coefficient correlation for finite Mach numbers. An improved correlation

for the drag coefficient of spherical particles was constructed. The new correlation

represents available experimental data more accurately than previously published

correlations. The model is capable of accurately reproducing particle velocities

measured in the experiments following the impact of a shock wave. This suggests

that the increased drag coefficient observed in many shock-tube experiments is simply

a consequence of compressibility effects and not due to unsteadiness as is sometimes

claimed.

Unsteady forces on a particle in a finite Reynolds and finite Mach flow. High

fidelity numerical simulations have been carried out to measure the forces on an

accelerating particle in finite Mach- and Reynolds-number flows. Mach numbers upto 0.4

and Reynolds numbers upto 40 are considered in this work. Solution of the linearized

compressible Navier-Stokes equations is valid for early time c0t/a < 1/(ReM) even

for finite Mach- and Reynolds-number flows. We can identify three regimes for Mach

and Reynolds numbers. (i) Regime I is characterized by very small Mach and Reynolds

numbers, i.e., M≪ 1 and Re≪ 1 such that M/Re≪ 1 to ensure continuum assumption.

163

In this regime, linear theory is valid till long time. (ii) Regime II is characterized by small

Mach-numbers and finite Reynolds-numbers. In this regime, c0t/a < 1/(ReM) is violated

early in time and non-linear effects causes solution to deviate from linear solution. Since

Mach number is still small, its effect on non-linear regime at large time is weak and

negligible. A new kernel for viscous-unsteady force is proposed which accounts for

compressibility-correction for short times and incorporates Reynolds-number effects for

large time. The predictions for unsteady forces using proposed unsteady kernels are in

excellent agreement with numerically obtained results. (iii) Regime III is characterized

by large Mach- and Reynolds-numbers. For finite Mach- and Reynolds-number flow in

this regime, as a first approximation, inviscid-unsteady force is modeled as dependent

only on Mach number and viscous-unsteady force is modeled as dependent only on

Reynolds number. A particle equation of motion is proposed consisting of standard drag

correlation of Schiller and Naumann [94], Mach-number dependent inviscid-unsteady

kernels of Parmar et al. [79], and Reynolds-number dependent new viscous-unsteady

kernel proposed here. A good agreement is found in overall behavior of numerically

obtained force and that predicted by the model. Work is in progress to model the effects

of finite Mach-number on viscous-unsteady kernel.

8.2 Future Work

A systematic study has been undertaken in this work to study particle equation

of motion in compressible flows. As described in Section 8.1, significant progress has

been made in our understanding of forces, particularly unsteady forces, on a particle in

compressible flows. However, there are still many open questions left for future studies.

Suggested directions for future work are:

Inviscid unsteady force in supercritical and supersonic flows. In a supercritical

but subsonic flow, a shock wave appears on the particle significantly increasing the

drag. Moreover, the symmetrically positioned shock waves on the surface of the particle

soon become unstable and result in highly unsteady flow with vortex shedding even

164

in inviscid flows. Supersonic flow around the particle is characterized by a detached

bow shock causing a strong wave or form drag. Investigations need to be carried out to

determine the form and magnitude of the inviscid unsteady force in super-critical and

supersonic flows compared to inviscid quasi-steady drag. Our preliminary investigations

show that the inviscid unsteady force is very small compared to the quasi-steady drag in

supercritical and supersonic flows.

Investigation of history force in compressible flows. Further work is required

to study the history force in compressible viscous flows at finite Mach and Reynolds

numbers. The integral nature of both the inviscid and the viscous unsteady forces in

compressible flow makes their separate extraction very difficult. Thus a composite

kernel can be proposed that represents the total unsteady force in a compressible flow.

Our preliminary results show that the short-time behavior of the viscous unsteady force

can be obtained from linear theory. At intermediate times, non-linear effects appear that

cause the viscous unsteady kernel for finite Mach and Reynolds numbers to deviate

from that obtained using linear theory. The long time decay rate is 1/t2, the same as

that observed in incompressible flows. But the compressible viscous unsteady kernel is

found to be larger than the incompressible kernel for long time.

Shock-particle interaction including viscous unsteady force. Using the

understanding developed in this dissertation on the viscous unsteady force, shock-particle

interaction model needs to be improved. Our preliminary investigation shows that the

inviscid-unsteady force is the dominant force, an order of magnitude larger than the

viscous-unsteady force, during initial period of shock-particle interaction.

Effect of finite accelerations. For incompressible flows, Kim et al. [52] modified

the kernels for the viscous unsteady force presented by Mei and Adrian [68] to account

for large accelerations. A similar study is needed for compressible viscous flows.

Spherical bubbles with variable slip. The present work is focused on the study

of particle motion with no-slip conditions on the particle surface in viscous flows and

165

perfect slip conditions in inviscid flows. A bubble with no contamination on the surface

(perfect slip) in a viscous compressible flow has not been studied. However, such

studies have been undertaken in incompressible flows and found to lead to different

viscous unsteady forces. Also, variable slip conditions need to be explored. Our

preliminary work shows that new viscous-unsteady forces arise in compressible flows,

similar to that in incompressible flows, due to variable slip on particle surface.

Rigid non-spherical particles. Particle shapes are rarely perfectly spherical. This

work focused on spherical particles as an idealization. Further work is needed to extend

the present study to non-spherical particle shapes.

166

NUMERICAL METHODOLOGY

The main focus of the present work is to study the unsteady force on particles

in viscous compressible flows. For flows with finite Reynolds and Mach numbers

theoretical analysis is not possible. Hence, an accurate numerical methodology is

needed. The numerical method should have the following properties:

1. Capable of solving viscous compressible flows with second order accuracy,

2. Able to simulate M→ 0 as well as finite M flows,

3. Capture shocks, and

4. Accurate boundary conditions.

Another factor affecting the accuracy of numerical results is the quality of the mesh

being used. Significant time has been invested to develop an accurate numerical

method and high quality grids.

Two solution methods have been used for the numerical investigations. The

solution methods will be called “Dissipative solver” and “Non-dissipative solver”

in this dissertation. The Dissipative solver was developed by Prof. Haselbacher

(see Haselbacher [42]) and extended in this work. The Non-dissipative solver was

developed from scratch in this work. Specific capabilities that have been added to both

these solvers include (i) solving governing equations in moving reference frame, (ii)

axisymmetric computation to reduce computational cost, and (iii) characteristic boundary

conditions and sponge layers.

This appendix describes the solution methods.

167

A.1 Governing Equations

A.1.1 Dimensional Form

The governing equations are the three-dimensional time-dependent compressible

Navier-Stokes equations, expressed in Cartesian tensor notation,

∂ρ

∂t+

∂ρui∂xi

= 0 (A–1)

∂ρui∂t

+∂ρuiuj∂xj

= − ∂p

∂xi+

∂τij∂xj− ρai (A–2)

∂ρE

∂t+

∂ρHui∂xi

=∂τijuj∂xi

− ∂qi∂xi− ρuiai (A–3)

where ρ is the density, ui is the i th component of the velocity vector, p is the pressure,

τij is the ij th component of the stress tensor, ai is the i th component of the acceleration

vector for the moving reference frame, E is the total energy, H is the total enthalpy,

and qi is the i th component of the heat-flux vector. The summation convention is used

throughout this document.

The total energy is given by

E = CvT +1

2uiui (A–4)

where Cv is the specific heat at constant volume and T is the static temperature. The

total enthalpy is given by

H = CpT +1

2uiui (A–5)

where Cp is the specific heat at constant pressure. The equation of state is

p = ρRT (A–6)

where R = Cp − Cv is the gas constant.

The stress tensor is given by

τij = µ

(∂ui∂xj

+∂uj∂xi

)− 2

3µδij

∂uk∂xk

(A–7)

168

where µ is the dynamic viscosity and δij is the Kronecker delta. The heat-flux vector is

given by

qi = −κ∂T

∂xi(A–8)

where κ is the thermal conductivity.

The speed of sound is defined as

c =√γRT =

√γp/ρ (A–9)

where γ = Cp/Cv .

A.1.2 Non-Dimensional Form

The independent and dependent variables are non-dimensionalized through

x i =xi

Lref(A–10)

t =ureft

Lref(A–11)

ρ =ρ

ρref(A–12)

u i =ui

uref(A–13)

p =p − pref

ρrefu2ref(A–14)

T =T

Tref

(A–15)

µ =µ

µref(A–16)

ai =ai

u2ref/Lref(A–17)

where the notation (·) denotes a non-dimensional variable and the subscript ref

indicates a reference value. It is assumed that

pref = ρrefRTref (A–18)

and

c2ref = γRTref = γpref/ρref (A–19)

169

Substituting these definitions into the dimensional form of governing equations gives

∂ρ

∂t+

∂ρ u i∂x i

= 0 (A–20)

∂ρ u i∂t

+∂ρ u iuj∂x j

= − ∂p

∂x i+

1

Reref

∂τ ij∂x j− ρai (A–21)

and

M2ref

{∂

∂t

(p +

γ − 1

2ρ u iu i

)+

∂

∂x j

[(γp +

γ − 1

2ρ u iu i

)uj

]}+

∂u i∂x i

=(γ − 1)M2

ref

Reref

∂τ ijuj∂x i

+1

RerefPr

∂

∂x i

(µ∂T

∂x i

)− (γ − 1)M2

refρuiai (A–22)

where

Reref =ρrefurefLref

µref(A–23)

Mref =uref

cref(A–24)

Pr =µCp

κ(A–25)

The equation of state becomes

ρT = γM2refp + 1 (A–26)

The speed of sound becomes

c2 =

(c

cref

)2

=1

ρ

(γM2

refp + 1)= T (A–27)

A.2 Solution Methods

A.2.1 Dissipative Solver

The dissipative solver solves the dimensional form of the governing equations in

integral form on arbitrary unstructured grids consisting of tetrahedra, hexahedra, prisms,

and pyramids. The spatial discretization is based on the flux-difference splitting method

of Roe (1981) and a simplified weighted essentially non-oscillatory reconstruction

170

described by Haselbacher [41]. The discrete equations are integrated in time using the

four-stage Runge-Kutta method. The basic methodology employed in this work has been

applied to several unsteady compressible flows and demonstrated good agreement with

theory and experimental data, see, e.g., Haselbacher et al. [43].

The solver is fully parallel and has been shown to give excellent scalability on

thousands of processors. Details for the Dissipative solver can be found in Haselbacher

[42].

A.2.2 Non-Dissipative Solver

A non-dissipative flow solver based on the methodology presented by Hou &

Mahesh (2005) has also been developed. It also operates on arbitrary unstructured

grids. The non-dissipative solver solves the non-dimensional form of the governing

equations in integral form. Fig. A-1 shows the storage of variables. The velocity

components, pressure, and density are colocated in space at cell centroid. The

face-normal velocity is stored at face centers. Note that Density, pressure, and

temperature are staggered in time compared to the velocity. This makes the discretization

symmetric in space and time, resulting in a non-dissipative solution method. A

pressure-correction method is used to solve the governing equations. A predictor-corrector

type iterative algorithm is implemented. The numerical scheme is implicit and second-order

accurate in space and time. The Hypre library (Lawrence Livermore National Laboratory

2003) is used to solve the linear system in parallel.

A few advantages of this solver are that the numerical scheme is (i) non-dissipative,

(ii) kinetic-energy conserving, and (iii) the solver can tackle finite but subcritical Mach as

well as M → 0 flows.

A limitation of the solver is that the supercritical flows cannot be simulated due to

lack of the shock-capturing capability. Adding such a capability is work in progress.

171

A.3 Discretization of Non-Dissipative Solver

The Non-dissipative solver is based on the method of Hou and Mahesh [45]

(hereafter referred to as HM)) with some modifications. In HM thermal flux in energy

equation was treated explicitly. It was found that this makes numerical method unstable.

We use implicit formulation for the thermal flux. A detailed derivation for discretization

and solution algorithm is presented in current section and Section A.4, respectively.

A.3.1 Notation and Variable Arrangement

The notation to be used below follows these guidelines:

• Superscripts denote time levels. Thus ϕn+1 denotes the value of ϕ at time leveln + 1. Superscripts can be combined to indicate iterates. Thus ϕn+1,q denotes theqth iterate of ϕ at time level n + 1. Similarly, the superscript ∗ denotes a predictedvalue. Thus ϕn+1,∗ denotes the predicted value of ϕ at time level n + 1.

• Subscripts denote spatial locations or components of a vector or tensor. Theindices i and j denote components of a vector or tensor. Greek subscripts indicatevariables assigned to a specific control volume. Thus ϕα denotes the variable ϕ incontrol volume α. The subscript k denotes the index of the face shared by controlvolumes α and β.

The summation convention is not applied to spatial locations, i.e., k , α, and β.

The variable arrangement is shown in Fig. A-1. Note that the velocity components

and the thermodynamic variables are stored at the same location in space, but are

staggered in time. In addition, the velocity normal to a control-volume face is stored at

the same time-level as the control-volume velocity.

In the following, the (·) notation is dropped for convenience; all variables are

understood to be non-dimensional.

A.3.2 Continuity Equation

The continuity equation is integrated over α × [tn+1/2, tn+3/2] to give∫ n+3/2

n+1/2

∫α

∂ρ

∂tdV dt +

∫ n+3/2

n+1/2

∫α

∂ρui∂xi

dV dt = 0 . (A–28)

172

uni,α, ρ

n+1/2α , pn+1/2

α , T n+1/2α

α β

vn

k

Figure A-1. Variable arrangement.

ui ρ, p, T

n

n+ 1

n+ 2

n + 1/2

n + 3/2

n + 5/2

Figure A-2. Time discretization.

173

Expressed the surface integral as a sum of integrals over the faces of the control

volume, and approximating the surface integral on each face to give

(ρn+3/2α − ρn+1/2α

)Vα +

∫ n+3/2

n+1/2

(∑k

ρkvk �sk

)dt = 0 , (A–29)

where with the face-normal velocity is defined as v = uini , The time integral is

approximated by the mid-point rule,

ρn+3/2α − ρ

n+1/2α

�t+

1

Vα

∑k

ρn+1k v n+1k �sk = 0 , (A–30)

where �sk is the area of face k and �t = tn+3/2 − tn+1/2.

At each face, ρn+1k is obtained from

ρn+1k =ρn+1α + ρn+1β

2(A–31)

and the value ρn+1α is obtained by interpolation in time as

ρn+1α =ρn+1/2α + ρ

n+3/2α

2(A–32)

and similary for ρn+1β , so we obtain(Vα

�t+

1

4

∑k

v n+1k �sk

)ρn+3/2α +

1

4

∑k

ρn+3/2β v n+1k �sk

=

(Vα

�t− 1

4

∑k

v n+1k �sk

)ρn+1/2α − 1

4

∑k

(ρn+1/2β

)v n+1k �sk (A–33)

A.3.3 Momentum Equation

The momentum equation is integrated over α × [tn, tn+1] to give

∫ n+1

n

∫α

∂ρui∂t

dV dt +

∫ n+1

n

∫α

∂ρuiuj∂xj

dV dt

= −∫ n+1

n

∫α

∂p

∂xidV dt +

1

Reref

∫ n+1

n

∫α

∂τij∂xj

dV dt −∫ n+1

n

∫α

ρai dV dt (A–34)

174

Using the approximations introduced in the discretization of the continuity equation for

the time-derivative, convective, and diffusive terms, we obtain

[(ρui)

n+1α − (ρui)

nα

]Vα + �t

∑k

(ρui)n+1/2k v

n+1/2k �sk

= −∫ n+1

n

∫α

∂p

∂xidV dt +

�t

Reref

∑k

τn+1/2ij ,k nj ,k �sk − (ρai)

n+1/2α Vα�t (A–35)

where it is assumed that �t ≡ tn+3/2 − tn+1/2 = tn+1 − tn.

Following HM, the pressure term is approximated as∫ n+1

n

∫α

∂p

∂xidV dt = Vα�t

(∂p

∂xi

)n+1/2

α

(A–36)

where it should be noted that the divergence theorem is not used. Following [115], the

pressure gradient appearing in Eq. (A–36) is approximated as(∂p

∂xi

)n+1/2

α

=1

2

[(∂p

∂xi

)n

α

+

(∂p

∂xi

)n+1

α

](A–37)

=1

4

[(∂p

∂xi

)n−1/2

α

+ 2

(∂p

∂xi

)n+1/2

α

+

(∂p

∂xi

)n+3/2

α

](A–38)

Equation (A–38) can be interpreted as a filtered pressure gradient or as the gradient of a

filtered pressure ~pn+1/2α defined by

~pn+1/2α =1

4

(pn−1/2α + 2pn+1/2α + pn+3/2α

)(A–39)

With Eqs. (A–36) and (A–39) and by dividing by �t (and not by Vα�t like HM to simplify

the implementation) we obtain the discrete momentum equation,

Vα

�t

[(ρui)

n+1α − (ρui)

nα

]+∑k

(ρui)n+1/2k v

n+1/2k �sk

= −Vα

4

[(∂p

∂xi

)n−1/2

α

+ 2

(∂p

∂xi

)n+1/2

α

+

(∂p

∂xi

)n+3/2

α

]+

1

Reref

∑k

τn+1/2ij ,k nj ,k �sk

− (ρai)n+1/2α Vα (A–40)

175

HM do not describe how the viscous terms are computed. The following outlines a

plausible way to compute the viscous terms. To find a form of the viscous stress suitable

for discretization, it is more convenient to consider the traction vector

ti = τijnj (A–41)

where the indices associated with time discretization and face location are ignored. It is

a simple matter to show from Eq. (A–7) that

t1 = µ∂u1∂n

+1

3µ∂uk∂xk

n1 + µ

(∂u2∂x1

n2 −∂u2∂x2

n1

)+ µ

(∂u3∂x1

n3 −∂u3∂x3

n1

)(A–42)

t2 = µ∂u2∂n

+1

3µ∂uk∂xk

n2 + µ

(∂u1∂x2

n1 −∂u1∂x1

n2

)+ µ

(∂u3∂x2

n3 −∂u3∂x3

n2

)(A–43)

t3 = µ∂u3∂n

+1

3µ∂uk∂xk

n3 + µ

(∂u1∂x3

n1 −∂u1∂x1

n3

)+ µ

(∂u2∂x3

n2 −∂u2∂x2

n3

)(A–44)

where∂ui∂n

=∂ui∂xj

nj (A–45)

It is convenient to write

ti = ti ,N + ti ,D + ti ,T (A–46)

where, referring to Eqs. (A–42), (A–43), and (A–44), ti ,N represents the first term,

ti ,D represents the second term, and ti ,T represents the third and fourth terms. The

significance of these terms is as follows: The first term will be large because it captures

the wall-normal variation in boundary layers (provided control-volume faces are aligned

with the wall), the second term will be small unless large density gradients exist, and the

contributions of the third and fourth terms to the flux imbalance of a control volume will

be zero if the viscosity is constant.

To ensure strong coupling and to damp oscillations on the grid scale, ti ,N at a face k

is approximated as (∂ui∂n

)k

=ui ,β − ui ,α

�nk(A–47)

176

where �nk = ∥rβ − rα∥. The derivatives appearing in ti ,D and ti ,T can be computed using

averaged cell gradients, i.e.,(∂ui∂xj

)k

=1

2

[(∂ui∂xj

)α

+

(∂ui∂xj

)β

](A–48)

Note that using this approach to compute the normal derivative would lead to decoupling

on some uniform grids. Cell gradients are approximated by application of the Green-Gauss

theorem as (∂ui∂xj

)α

=1

2Vα

∑k

ui ,β nj ,k�sk (A–49)

A.3.4 Energy Equation

The energy equation is integrated over α × [tn, tn+1] to give

M2ref

∫ n+1

n

∫α

∂

∂t

(p +

γ − 1

2ρuiui

)dV dt

+M2ref

∫ n+1

n

∫α

∂

∂xj

[(γp +

γ − 1

2ρuiui

)uj

]dV dt +

∫ n+1

n

∫α

∂ui∂xi

dV dt

=(γ − 1)M2

ref

Reref

∫ n+1

n

∫α

∂τijuj∂xi

dV dt

+1

RerefPr

∫ n+1

n

∫α

∂

∂xi

(µ∂T

∂xi

)dV dt

− (γ − 1)M2ref

∫ n+1

n

∫α

ρuiai dV dt (A–50)

Using the approximations introduced in the approximation of the continuity equation for

the time-derivative, convective, and diffusive terms, and Eq. (A–39) applied to the face

177

pressure, we obtain

M2ref

[(pn+1α +

γ − 1

2ρn+1α (uiui)

n+1α

)−(pnα +

γ − 1

2ρnα (uiui)

nα

)]Vα

+M2ref�t

∑k

(γ~p

n+1/2k +

γ − 1

2ρn+1/2k (uiui)

n+1/2k

)vn+1/2k �sk +�t

∑k

vn+1/2k �sk

=(γ − 1)M2

ref

Reref�t∑k

(τijuj)n+1/2k ni ,k �sk +

1

RerefPr�t∑k

(µ∂T

∂xi

)n+1/2

k

ni ,k �sk

− (γ − 1)M2ref (ρuiai)

n+1/2α Vα�t (A–51)

As with the viscous terms in the momentum equation, HM did not describe the

discretization of the viscous and conduction terms in the energy equation. The

approximation of the viscous terms in Eq. (A–51) builds on those developed for the

momentum equation. Because the stress tensor τij is symmetric, we can write

τijujni = τjiujni = tjuj (A–52)

The conduction term is discretized as(∂T

∂xi

)n+1/2

k

ni ,k =

(∂T

∂n

)n+1/2

k

=T

n+1/2β − T

n+1/2α

�nk(A–53)

Hence we obtain

M2ref

[(pn+1α +

γ − 1

2ρn+1α (uiui)

n+1α

)−(pnα +

γ − 1

2ρnα (uiui)

nα

)]Vα

+M2ref�t

∑k

(γ~p

n+1/2k +

γ − 1

2ρn+1/2k (uiui)

n+1/2k

)vn+1/2k �sk +�t

∑k

vn+1/2k �sk

=(γ − 1)M2

ref

Reref�t∑k

(tjuj)n+1/2k �sk +

1

RerefPr�t∑k

(µ∂T

∂n

)n+1/2

k

�sk

− (γ − 1)M2ref (ρ ui ai)

n+1/2α Vα�t (A–54)

A.4 Solution Algorithm for Non-Dissipative Solver

Referring to Fig. A-2, we seek to advance the velocities from n to n + 1 and the

density, pressure, and temperature from n + 1/2 to n + 3/2 by solving the discrete

178

equations in an iterative fashion. Accordingly, the superscript n + m, q denotes the

qth iterate of a given variable at the time n + m. The equations are solved using a

predictor-corrector approach based on a pressure correction. We define

pn+3/2,q+1 = pn+3/2,q + δpn+3/2 (A–55)

A.4.1 Continuity Equation

With this notation, Eq. (A–33) becomes(Vα

�t+

1

4

∑k

v n+1,qk �sk

)ρn+3/2,q+1α +

1

4

∑k

ρn+3/2,q+1β v n+1,qk �sk

=

(Vα

�t− 1

4

∑k

v n+1,qk �sk

)ρn+1/2α − 1

4

∑k

ρn+1/2β v n+1,qk �sk (A–56)

which can be written as

aραρn+3/2,q+1α +

∑k

aρβρ

n+3/2,q+1β = bρα (A–57)

where

aρα =Vα

�t+

1

4

∑k

v n+1,qk �sk (A–58)

aρβ =

1

4v n+1,qk �sk (A–59)

bρα =

(Vα

�t− 1

4

∑k

v n+1,qk �sk

)ρn+1/2α − 1

4

∑k

ρn+1/2β v n+1,qk �sk (A–60)

179

A.4.2 Momentum Equation

Predictor Step. Using Eqs. (A–41) and (A–46), Eq. (A–40) can be written as

Vα

�t

[(ρui)

n+1,q+1α − (ρui)

nα

]+∑k

(ρui)n+1/2,q+1k v

n+1/2,q+1k �sk

= −Vα

4

[(∂p

∂xi

)n−1/2

α

+ 2

(∂p

∂xi

)n+1/2

α

+

(∂p

∂xi

)n+3/2,q+1

α

]

+1

Reref

∑k

(tn+1/2,q+1i ,N,k + t

n+1/2,q+1i ,D,k + t

n+1/2,q+1i ,T,k

)�sk

− (ρai)n+1/2α Vα (A–61)

where

tn+1/2,q+1i ,N,k = µk

un+1/2,q+1i ,β − u

n+1/2,q+1i ,α

�nk(A–62)

The velocity un+1,q+1i is predicted from

Vα

�t

[(ρui)

n+1,∗α − (ρui)

nα

]+∑k

(ρui)n+1/2,∗k v

n+1/2,qk �sk

= −Vα

4

[(∂p

∂xi

)n−1/2

α

+ 2

(∂p

∂xi

)n+1/2

α

+

(∂p

∂xi

)n+3/2,q

α

]

+1

Reref

∑k

(tn+1/2,∗i ,N,k + t

n+1/2,qi ,D,k + t

n+1/2,qi ,T,k

)�sk

− (ρai)n+1/2α Vα (A–63)

where the superscript ∗ denotes the predicted value at q + 1 and only the first traction

term is treated implicitly. This is because the other terms would lead to a larger

bandwidth of the linear system below. In Eq. (A–63), we define

(ρui)n+1/2,∗k =

1

2

[(ρui)

nk + (ρui)

n+1,∗k

](A–64)

and

vn+1/2,qk =

1

2

(v nk + v n+1,qk

)(A–65)

180

HM do not specify how (ρui)nk and (ρui)

n+1,∗k appearing in Eq. (A–64) are computed. A

plausible approach is

(ρui)nk =

1

2

[(ρui)

nα + (ρui)

nβ

](A–66)

(ρui)n+1,∗k =

1

2

[(ρui)

n+1,∗α + (ρui)

n+1,∗β

](A–67)

where ρnα is interpolated from

ρnα =1

2

(ρn−1/2α + ρn+1/2α

)(A–68)

In the algorithm of HM, it appears to be implicitly assumed that

ρn+1,q+1α = ρn+1,∗α (A–69)

so that

ρn+1,∗α =1

2

(ρn+1/2α + ρn+3/2,q+1α

)(A–70)

With the obvious changes in subscripts, ρn+1,q+1β and ρn+1,∗β are computed similarly.

The implicitly treated traction term is expressed as

tn+1/2,∗i ,N,k = µk

un+1/2,∗i ,β − u

n+1/2,∗i ,α

�nk(A–71)

and with the definition

un+1/2,∗i ,α =

uni ,α + un+1,∗i ,α

2(A–72)

we get

tn+1/2,∗i ,N,k =

µk2�nk

(un+1,∗i ,β − un+1,∗i ,α + uni ,β − uni ,α

)(A–73)

With these definitions, Eq. (A–63) can be written as

aρui

α (ρui)n+1,∗α +

∑k

aρui

β (ρui)n+1,∗β = bρui

α (A–74)

181

where

aρui

α =Vα

�t+

1

4

∑k

vn+1/2,qk �sk +

1

Reref

∑k

µk2�nk

1

ρn+1,∗α

�sk (A–75)

aρui

β =1

4vn+1/2,qk �sk −

1

Reref

∑k

µk2�nk

1

ρn+1,∗β

�sk (A–76)

and

bρui

α = bρui

α,1 + bρui

α,2 + bρui

α,3 + bρui

α,4 (A–77)

where

bρui

α,1 = −Vα

4

[(∂p

∂xi

)n−1/2

α

+ 2

(∂p

∂xi

)n+1/2

α

+

(∂p

∂xi

)n+3/2,q

α

](A–78)

bρui

α,2 =1

Reref

∑k

[tn+1/2,qi ,D,k + t

n+1/2,qi ,T,k +

µk2�nk

(uni ,β − uni ,α

)]�sk (A–79)

bρui

α,3 =

(Vα

�t− 1

4

∑k

vn+1/2,qk �sk

)(ρui)

nα −

1

4

∑k

(ρui)nβ v

n+1/2,qk �sk (A–80)

bρui

α,4 = −Vα (ρai)n+1/2α (A–81)

From (ρui)n+1,∗α , we obtain

un+1,∗i ,α =(ρui)

n+1,∗α

ρn+1,∗α

=(ρui)

n+1,∗α

ρn+1,q+1α

(A–82)

by Eq. (A–69) and ρn+1,q+1α is given by Eq. (A–70).

Corrector Step. Subtracting Eq. (A–63) from Eq. (A–61) gives

(ρui)n+1,q+1α = (ρui)

n+1,∗α − �t

4

(∂δp

∂xi

)n+3/2

α

(A–83)

where the contributions from the inviscid and viscous terms were neglected. HM present

the equivalent of Eq. (A–83) as being exact and do not discuss why the contributions

from the convective and diffusive terms can be neglected. From Eq. (A–83) one can

182

obtain

un+1,q+1i ,α = un+1,∗i ,α − �t

4ρn+1,q+1α

(∂δp

∂xi

)n+3/2

α

(A–84)

where, on account of Eq. (A–69), the appearance of a density ratio multiplying un+1,∗i ,α

is avoided. ρn+1,q+1α is computed from Eqs. (A–69) and (A–70). The kinetic energy

becomes, to O(δp2),

(uiui)n+1,q+1α = (uiui)

n+1,∗α − un+1,∗i ,α

�t

2ρn+1,q+1α

(∂δp

∂xi

)n+3/2

α

(A–85)

Now HM claim that the face-normal velocity can be computed from Eq. (A–84) by

dotting it with the face normal ni ,k . This approach is not rigorous because each face has

two adjacent cells. Instead, one may define

v n+1,q+1k =1

2

[un+1,q+1i ,α + un+1,q+1i ,β

]ni ,k

= v n+1,∗k − �t

8

[1

ρn+1,q+1α

(∂δp

∂xi

)n+3/2

α

+1

ρn+1,q+1β

(∂δp

∂xi

)n+3/2

β

]ni ,k

(A–86)

where

v n+1,∗k =1

2

(un+1,∗i ,α + un+1,∗i ,β

)ni ,k (A–87)

By assuming that ρn+1,q+1α = ρn+1,q+1β = ρn+1,q+1k , one obtains

v n+1,q+1k = v n+1,∗k − �t

8ρn+1,q+1k

[(∂δp

∂xi

)n+3/2

α

+

(∂δp

∂xi

)n+3/2

β

]ni ,k (A–88)

but this is not desirable because it will very likely lead to decoupling of the face-normal

velocity and pressure gradient. So the only way to derive HM’s expression for v n+1,q+1k

is to forego the rigorous derivation and substitute a pressure gradient normal to the face

which is not computed from a simple average,

v n+1,q+1k = v n+1,∗k − �t

4ρn+1,q+1k

(∂δp

∂n

)n+3/2

k

(A–89)

where

ρn+1,q+1k =1

2

(ρn+1,q+1α + ρn+1,q+1β

)(A–90)

183

where Eq. (A–70) is used to compute ρn+1,q+1α . The face-normal pressure gradient is

computed as (∂δp

∂n

)n+3/2

k

=δp

n+3/2β − δp

n+3/2α

�nk(A–91)

where �nk = ∥rβ − rα∥ and rα is the position vector of the centroid of control volume α.

This guarantees strong coupling but is in general highly inaccurate (even inconsistent)

on distorted grids.

A.4.3 Energy Equation

The approximations in Eq. (A–54) will be described term-by-term.

Time-Derivative Term. The time-derivative term becomes (excluding the common

factor M2refVα), by using the appropriate averages and Eq. (A–55),[

1

2

(pn+3/2,qα + δpn+3/2α + pn+1/2α

)+

γ − 1

2ρn+1,q+1α (uiui)

n+1,q+1α

]−[1

2

(pn+1/2α + pn−1/2

α

)+

γ − 1

2ρnα (uiui)

nα

](A–92)

where ρnα and ρn+1,q+1α are computed from Eqs. (A–68) and (A–70). Now using Eq.

(A–85), we obtain (cf. Eq. (23) in HM)

1

2

(pn+3/2,qα + δpn+3/2α + pn+1/2α

)+

γ − 1

2

(ρn+1,q+1α (uiui)

n+1,∗α − ρnα (uiui)

nα

)− γ − 1

4un+1,∗i ,α �t

(∂δp

∂xi

)n+3/2

α

− 1

2

(pn+1/2α + pn−1/2

α

)(A–93)

and with Eq. (A–49) we arrive at

δpn+3/2α

2− γ − 1

8Vα

un+1,∗i ,α �t∑k

δpn+3/2β ni ,k�sk

+γ − 1

2

(ρn+1,q+1α (uiui)

n+1,∗α − ρnα (uiui)

nα

)+

1

2

(pn+3/2,qα − pn−1/2

α

)(A–94)

184

Convective Term. The convective term becomes (excluding the common factor M2ref�t),

using Eq. (A–39),

∑k

[γ

4

(pn−1/2k + 2p

n+1/2k + p

n+3/2,q+1k

)+

γ − 1

8ρn+1/2k

(un+1,q+1i ,k + uni ,k

) (un+1,q+1i ,k + uni ,k

)]v n+1,q+1k + v nk

2�sk (A–95)

and using Eqs. (A–55) and (A–89),

∑k

[γ

(~pn+1/2k +

δpn+3/2k

4

)+

γ − 1

8ρn+1/2k

(un+1,q+1i ,k + uni ,k

) (un+1,q+1i ,k + uni ,k

)][1

2

(v n+1,∗k + v nk

)− �t

8ρn+1,q+1k

(∂δp

∂n

)n+3/2

k

]�sk (A–96)

Now it is tacitly assumed that Eq. (A–84) applies not only to velocities at cell centers, but

also to velocities at faces. Hence we write

un+1,q+1i ,k = un+1,∗i ,k − �t

4ρn+1,q+1k

(∂δp

∂xi

)n+3/2

k

(A–97)

where ρn+1,q+1k is given by Eq. (A–90) and therefore

un+1,q+1i ,k + uni ,k = un+1,∗i ,k + uni ,k −�t

4ρn+1,q+1k

(∂δp

∂xi

)n+3/2

k

(A–98)

With the definition

un+1/2,∗i ,k =

uni ,k + un+1,∗i ,k

2(A–99)

we obtain, to O(δp2),

(un+1,q+1i ,k + uni ,k

) (un+1,q+1i ,k + uni ,k

)= 4 (uiui)

n+1/2,∗k − �t

ρn+1,q+1k

un+1/2,∗i ,k

(∂δp

∂xi

)n+3/2

k

(A–100)

With the definitions

vn+1/2,∗k =

1

2

(v n+1,∗k + v nk

)(A–101)

�k = γ~pn+1/2k +

γ − 1

2ρn+1/2k (uiui)

n+1/2,∗k (A–102)

185

the convective term can be expressed as

∑k

[�k +

γ

4δp

n+3/2k − γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

un+1/2,∗i ,k

(∂δp

∂xi

)n+3/2

k

][vn+1/2,∗k − �t

8ρn+1,q+1k

(∂δp

∂n

)n+3/2

k

]�sk (A–103)

Equation (A–49) cannot be used to approximate(

∂δp∂xi

)n+3/2k

because the latter is a face

gradient. HM appear to employ the relation(∂δp

∂xi

)n+3/2

k

=

(∂δp

∂n

)n+3/2

k

ni ,k (A–104)

which neglects contributions of tangential components. Multiplying out and using Eq.

(A–91) to approximate the face-normal derivative, we obtain to O(δp2),

∑k

[�kv

n+1/2,∗k �sk − �k

�t

8ρn+1,q+1k

(δp

n+3/2β − δpn+3/2α

) �sk�nk

+γ

4vn+1/2,∗k δp

n+3/2k �sk

− γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

(vn+1/2,∗k

)2 (δp

n+3/2β − δpn+3/2α

) �sk�nk

](A–105)

Approximating δpn+3/2k as

δpn+3/2k =

1

2

(δpn+3/2α + δp

n+3/2β

)(A–106)

and collecting terms gives

∑k

�kvn+1/2,∗k �sk

+ δpn+3/2α

∑k

(γ

8vn+1/2,∗k +�k

�t

8ρn+1,q+1k

1

�nk+

γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

(vn+1/2,∗k

)2 1

�nk

)�sk

+∑k

(γ

8vn+1/2,∗k − �k

�t

8ρn+1,q+1k

1

�nk− γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

(vn+1/2,∗k

)2 1

�nk

)δp

n+3/2β �sk

(A–107)

186

Divergence Term. The divergence term can be expressed as (excluding the common

factor �t) ∑k

vn+1/2k �sk =

∑k

v n+1,q+1k + v nk2

�sk (A–108)

and using Eq. (A–101) gives

∑k

vn+1/2k �sk =

∑k

[vn+1/2,∗k − �t

8ρn+1,q+1k

(∂δp

∂n

)n+3/2

k

]�sk (A–109)

Using Eq. (A–91) and expanding leads to

∑k

vn+1/2k �sk =

∑k

vn+1/2,∗k �sk + δpn+3/2α

∑k

�t

8ρn+1,q+1k

�sk�nk−∑k

�t

8ρn+1,q+1k

δpn+3/2β

�sk�nk

(A–110)

Viscous and Heat-Flux Terms. The viscous terms do not require further simplification.

Thermal terms are treated differently than in HM. An implicit formulation is developed for

thermal flux. Thermal flux can be expressed as (excluding the common factor �tRerefPr

)

∑k

µ∂T

∂n�sk =

∑k

µT

n+1/2β − T

n+1/2α

�n�sk (A–111)

Like pressure, temperature can be expressed as

T n+1/2 =1

4

(T n−1/2 + 2T n+1/2 + T n+3/2,q+1

)T n+1/2 =

1

4

(T n−1/2 + 2T n+1/2 + T n+3/2,q + δT

), (A–112)

where δT is related to δp through equation of state as follows

δT =γM2

ref

ρδp . (A–113)

187

Moving Reference Frame Source Terms. Source terms can be expressed as

(excluding the common factor −(γ − 1)M2refVα�t))

(ρuiai)n+1/2α = (ρai)

n+1/2α u

n+1/2i ,α = (ρai)

n+1/2α

(un+1,q+1i ,α + uni ,α

)2

(A–114)

Using Eq. (A–84)

(ρuiai)n+1/2α = (ρai)

n+1/2α u

n+1/2,∗i ,α − �t

8

(ρai)n+1/2α

ρn+1,q+1α

(∂δp

∂xi

)n+3/2

α

(A–115)

Using Eq. (A–49)

(ρuiai)n+1/2α = (ρai)

n+1/2α u

n+1/2,∗i ,α − �t

16Vα

(ρai)n+1/2α

ρn+1,q+1α

∑k

δpn+3/2β ni ,k�sk (A–116)

Final Form. Collecting terms and dividing through by �t (again we do not divide by Vα

to simplify the implementation) gives, finally,

aδpα δpn+3/2α +∑k

aδpβ δpn+3/2β = bδpα (A–117)

where

aδpα = aδpα,1 + aδpα,2 + aδpα,3 (A–118)

and

aδpα,1 =M2

ref

2

Vα

�t(A–119)

aδpα,2 = M2ref

∑k

(γ

8vn+1/2,∗k +�k

�t

8ρn+1,q+1k

1

�nk+

γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

(vn+1/2,∗k

)2 1

�nk

)�sk

(A–120)

aδpα,3 =∑k

�t

8ρn+1,q+1k

�sk�nk

(A–121)

where vn+1/2,∗k is computed from Eq. (A–102) and ρn+1,q+1k is given by Eq. (A–90).

Similarly,

aδpβ = aδpβ,1 + aδpβ,2 + aδpα,3 + aδpα,4 (A–122)

188

where

aδpβ,1 = −γ − 1

8M2

refun+1,∗i ,α ni ,k�sk (A–123)

aδpβ,2 = M2ref

(γ

8vn+1/2,∗k − �k

�t

8ρn+1,q+1k

1

�nk− γ − 1

8�t

ρn+1/2k

ρn+1,q+1k

(vn+1/2,∗k

)2 1

�nk

)�sk

(A–124)

aδpβ,3 = −�t

8ρn+1,q+1k

�sk�nk

(A–125)

aδpβ,4 = −(γ − 1)

16M2

ref�t(ρai)

n+1/2α

ρn+1,q+1α

ni ,k�sk (A–126)

and

bδpα = bδpα,1 + bδpα,2 + bδpα,3 + bδpα,4 + bδpα,5 (A–127)

where

bδpα,1 = −M2ref

Vα

�t

[γ − 1

2

(ρn+1,q+1α (uiui)

n+1,∗α − ρnα (uiui)

nα

)+

1

2

(pn+3/2,qα − pn−1/2

α

)](A–128)

bδpα,2 = −M2ref

∑k

�kvn+1/2,∗k �sk (A–129)

bδpα,3 = −∑k

vn+1/2,∗k �sk (A–130)

bδpα,4 =(γ − 1)M2

ref

Reref

∑k

(tjuj)n+1/2k �sk +

1

RerefPr

∑k

(µ∂T

∂n

)n+1/2

k

�sk (A–131)

bδpα,5 = −(γ − 1)M2refVα (ρai)

n+1/2α u

n+1/2,∗i ,α (A–132)

A.4.4 Summary

The following is a summary of the necessary steps in the solution approach for each

time step n:

1. Initialize iterates: un+1,0i ,α = uni ,α, ρn+3/2,0α = ρn+1/2α , pn+3/2,0α = p

n+1/2α , T n+3/2,0

α =

Tn+1/2α , v n+1,0k = v nk .

2. Solve Eq. (A–57) for ρn+3/2,q+1α .

189

3. Solve Eq. (A–74) to get the predicted momenta (ρui)n+1,∗α .

4. Compute predicted cell-centered velocities un+1,∗i ,α from Eq. (A–82).

5. Compute predicted face-normal velocities v n+1,∗k from Eq. (A–87).

6. Solve Eq. (A–117) for δpn+3/2α .

7. Update pn+3/2,q+1α from Eq. (A–55).

8. Compute gradient of pressure correction from Eq. (A–49) with ϕ = δpn+3/2.

9. Update (ρui)n+1,q+1α from Eq. (A–83).

10. Update un+1,q+1α from Eq. (A–84).

11. Check for convergence of density, pressure correction, and velocities. If converged,proceed to next iteration (n ← n + 1, and go to Step 1), otherwise carry outadditional predictor-corrector step (q ← q + 1, and go to Step 2).

A.5 Boundary Condition Implementation for Non-Dissipative Solver

A.5.1 Solid Walls

In this work, wall boundary condition arise at the particle surface. For inviscid flow,

particle surface acts as a slip wall and for viscous flow it acts as a no-slip wall. On a

slip wall, the fluid must have a zero velocity normal to the wall, i.e., vb = 0. On a no-slip

wall, (ui)b = 0, and thus both the normal and tangential components of fluid velocity are

zero. The pressure and density at the wall are extrapolated from the interior cell. Velocity

gradients at the boundary face can be computed from(∂ui∂xj

)n∣∣∣∣b

=(ui)

nα

�nnj , (A–133)

where �n be distance of boundary face from interior cell centroid. Stresses at the

boundary face can be computed using Eqs. (A–41), (A–42), (A–43), (A–44) and velocity

gradients from Eq. (A–133). The contributions from a boundary face to the global

system matrix for continuity and momentum equation are shown in table A-1, where ti ,D

and ti ,T are given by Eq. (A–46) and �sb is area of boundary face. A slip wall can be

distinguished from a no-slip wall by making µ = 0.

190

At a solid wall boundary, a ghost cell is assumed in which the pressure and density

are assigned the same values as that in interior cell and the velocity is assigned as the

mirror image of velocity in interior cell. Thus in Eq. (A–117) δpβ = δpα can be assumed,

and so aβ terms go to aα. Not considering source terms due to moving reference

frame, the contributions from wall boundary face to the global system matrix for energy

equation are shown in table A-1.

Table A-1. Wall boundary condition.

aα bαContinuity eq. 0 0

Momentum eqs. 1Reref

µ

�nρn+1,∗α

�sb1

Reref(µun

i ,α

�n+ ti ,D + ti ,T )�sb

Energy eq. −γ−18M2

refun+1,∗i ,α ni ,k�sk

1RerefPr

(µ∂T

∂n

)n+1/2k

�sk

For an isothermal boundary ∂T/∂n is computed as (Tα − Tb)/�n. Boundary with

an imposed heat flux are handled easily by replacing temperature derivative at the wall.

A.5.2 Farfield

Farfield boundary condition arise at the outer boundary where uniform flow

conditions are assumed to apply. This assumption is valid if the outer boundary is

placed at a large distance from the particle. Let ρb, pb, (ui)b denote the uniform flow

conditions imposed at the farfield boundary. The face-normal flow velocity at the

boundary is vb = (ui)b(ni)b, The contributions from a farfield boundary faces to global

system matrix are shown in table A-2, where �sb is area of boundary face.

Table A-2. Farfield boundary condition.

aα bαContinuity eq. 0 −ρbvb�sb

Momentum eqs. 0 −ρb(ui)bvb�sbEnergy eq. 0 −

(γpb +

γ−12ρb(uiui)b + 1

)vb�sb

A.6 Absorbing Boundary Conditions

The present work studies the flow around particle in a fluid of infinite expanse.

Ideally, the outer boundaries should be place at infinity. To reduce computational cost,

191

artificial boundaries are used to truncate the infinite domain. Invariably, there are

spurious reflections from the boundaries. Such reflections can corrupt the solution near

the particle. Absorbing boundary conditions are needed to avoid or at least reduce

spurious wave reflections from artificial boundaries.

A.6.1 Characteristic Boundary Conditions

Characteristic boundary conditions for the Euler and the Navier-Stokes equations

based on the NSCBC method of Poinsot and Lele [85] are used. Traditionally NSCBC

boundary conditions were implemented in finite-difference formulation. In the current

work, NSCBC is implemented in the Dissipative solver in a finite-volume formulation.

Gradients of solution variables at the boundary are required to estimate the wave

strengths in the NSCBC method. These gradients are computed using a higher-order

finite-difference approximation.

A.6.2 Sponge Layer

Sponge layers are specific type of absorbing layers used to damp the disturbances

prior to interaction with an boundary. One simple way to achieve this is by adding a

linear friction term to the governing equations to damp the difference of solution with

respect to some reference solution,

∂q

∂t+ L(q) = σ(q− qref) . (A–134)

where q is solution vector, L is a non-linear operator, σ is friction coefficient, and qref is

reference solution state.

The friction coefficient can be a function of space. Typically, σ varies smoothly from

zero to a large value near the boundary to avoid spurious reflection from the interface

between the sponge layer and the interior domain. See Israeli and Orszag [48] and

Colonius [23] for more details on sponge layers.

192

A.7 Moving Reference Frame

This section contains derivation of source terms in governing equations in moving

reference frame. Euler equations are taken as example to show derivation, though

source terms due to moving reference frame remains same in Navier-Stokes equations.

A.7.1 Euler Equations

Euler equations can be written as

∂ρ

∂t+∇ · (ρu) = 0 (A–135)

∂ρu

∂t+∇ · (ρuu) = −∇(p) (A–136)

∂ρE

∂t+∇ · (ρEu) = −∇ · (pu) (A–137)

where u is fluid velocity, ∇ is divergence operator and other symbols have usual

meaning as explained in Section A.1.1.

A.7.2 Coordinate Transformation

Consider a stationary reference frame (x , y , z , t) and a moving reference frame

(x ′, y ′, z ′, t ′) moving with velocity up = (up,x , up,y , up,z). Variables in moving and stationary

reference frames are related as follows,

t ′ = t r′ = r − rp (A–138)

where rp = (xp, yp, zp). Derivatives are related as follows,

∇ = ∇′ ∂

∂t=

∂

∂t ′− up · ∇′ (A–139)

A.7.3 Transformation of the Euler Equations to Moving Reference Frame

Take continuity equations∂ρ

∂t+∇ · (ρu) = 0 (A–140)

193

Transforming to moving reference frame

∂ρ

∂t ′− up · ∇′(ρ) +∇′ · (ρ(u′ + up)) = 0 ,

∂ρ

∂t ′− up · ∇′(ρ) +∇′ · (ρu′) +∇′ · (ρup) = 0 ,

∂ρ

∂t ′+∇′ · (ρu′) = 0 . (A–141)

Re-writing momentum equation

∂ρu

∂t+∇ · (ρuu) = −∇(p) . (A–142)

Transforming momentum equation to moving reference frame

∂ρ(u′ + up)

∂t ′− up · ∇′ (ρ(u′ + up)) +∇′ · (ρ(u′ + up)(u

′ + up)) = −∇′ (p) ,

∂ρu′

∂t ′+

∂ρup∂t ′− up · ∇′ (ρ(u′ + up)) +∇′ · (ρ(u′u′ + u′up)) + up · ∇′ (ρ(u′ + up)) = −∇′ (p) .

(A–143)

Further rearranging

∂ρu′

∂t ′+∇′ · (ρu′u′) + ρ

∂up∂t ′

+ up

[∂ρ

∂t ′+∇′ · (ρu′u′)

]= −∇′ (p) ,

∂ρu′

∂t ′+∇′ · (ρu′u′) = −∇′ (p)− ρ

∂up∂t ′

(A–144)

where the expression with in square bracket is identically zero being continuity equation.

Consider energy equation

∂ρE

∂t+∇ · (ρEu) = −∇ · (pu) (A–145)

Transforming to moving reference frame

∂ρE

∂t ′− up · ∇′ (ρE) +∇′ · (ρEu′) +∇′ · (ρEup) = −∇′ · (pu′)−∇′ · (pup) ,

∂ρE

∂t ′+∇′. (ρEu′) = −∇′ · (pu′)− up · ∇′(p) (A–146)

194

Energy ρE can be expanded as

ρE =1

2ρu2 +

p

γ − 1=

1

2ρu′2 +

1

2ρup

2 + ρu′.up +p

γ − 1= ρE ′ +

1

2ρup

2 + ρu′.up (A–147)

where

ρE ′ =1

2ρu′2 +

p

γ − 1. (A–148)

Substituting above expression in energy equation

∂ρE ′

∂t ′+

1

2

∂ρup2

∂t ′+

∂ρu′ · up∂t ′

+∇′ · (ρE ′u′) +∇′ ·(1

2ρup

2u′)+∇′ · (ρ(u′.up)u′)

= −∇′ · (pu′)− up · ∇′(p) (A–149)

Rearranging

∂ρE ′

∂t ′+∇′ · (ρE ′u′) +

1

2up

2

[∂ρ

∂t ′+∇′ · (ρu′)

]+ up ·

[∂ρu′

∂t ′+∇′ · (ρu′u′) +∇′(p) + ρ

∂up∂t ′

]= −∇′ · (pu′)− ρu′ · ∂up

∂t ′(A–150)

The expressions inside first and second square brackets are identically being continuity

and momentum equations respectively.

∂ρE ′

∂t ′+∇′ · (ρE ′u′) = −∇′ · (pu′)− ρu′ · ∂up

∂t ′(A–151)

In summary, the Euler equations in moving reference frame are

∂ρ

∂t ′+∇′ · (ρu′) = 0 ,

∂ρu′

∂t ′+∇′ · (ρu′u′) = −∇′ (p)− ρ

∂up∂t ′

,

∂ρE ′

∂t ′+∇′ · (ρE ′u′) = −∇′ · (pu′)− ρu′ · ∂up

∂t ′(A–152)

Corresponding Navier-Stokes equations in moving reference frame are given in Eq.

(A–1).

195

A.8 Axisymmetric Computations

This Section contains derivation of axisymmetric flow equations and details of their

implementation in 2D codes. Euler equations are used as an example in the following

description, though same arguments apply for Navier-Stokes equations also.

A.8.1 Euler Equations for Axisymmetric Flows

Euler equations in the cylindrical coordinates can be written as,

∂ρ

∂t+

1

r

∂

∂r(rρur) +

1

r

∂

∂θ(ρuθ) +

∂

∂z(ρuz) = 0 (A–153)

∂ρur∂t

+1

r

∂

∂r

(rρur

2)+

1

r

∂

∂θ(ρuruθ) +

∂

∂z(ρuruz) = −

∂p

∂r(A–154)

∂ρuθ∂t

+1

r

∂

∂r(rρuruθ) +

1

r

∂

∂θ

(ρuθ

2)+

∂

∂z(ρuθuz) = −

1

r

∂p

∂θ(A–155)

∂ρuz∂t

+1

r

∂

∂r(rρuruz) +

1

r

∂

∂θ(ρuθuz) +

∂

∂z

(ρuz

2)= −∂p

∂z(A–156)

∂ρe

∂t+

1

r

∂

∂r(rρeur) +

1

r

∂

∂θ(ρeuθ) +

∂

∂z(ρeuz) = −

(1

r

∂rpur∂r

+1

r

∂puθ∂θ

+∂puz∂z

)(A–157)

Simplifying above equations for the axisymmetric case, i.e. uθ = 0 and ∂∂θ

= 0,

∂ρ

∂t+

1

r

∂

∂r(rρur) +

∂

∂z(ρuz) = 0 (A–158)

∂ρur∂t

+1

r

∂

∂r

(rρur

2)+

∂

∂z(ρuruz) = −

∂p

∂r(A–159)

∂ρuz∂t

+1

r

∂

∂r(rρuruz) +

∂

∂z

(ρuz

2)= −∂p

∂z(A–160)

∂ρe

∂t+

1

r

∂

∂r(rρeur) +

∂

∂z(ρeuz) = −

(1

r

∂rpur∂r

+∂puz∂z

)(A–161)

Rearranging terms,

∂ρ

∂t+

1

r

∂

∂r(r(ρur)) +

∂

∂z(ρuz) = 0 (A–162)

∂ρur∂t

+1

r

∂

∂r

(r(ρur

2 + p))+

∂

∂z(ρuruz) =

p

r(A–163)

∂ρuz∂t

+1

r

∂

∂r(r(ρuruz)) +

∂

∂z

((ρuz

2 + p))= 0 (A–164)

∂ρe

∂t+

1

r

∂

∂r(r(ρe + p)ur) +

∂

∂z((ρe + p)uz) = 0 (A–165)

196

Mapping z-axis and r-axis of the cylindrical coordinates to x-axis and y-axis of the

Cartesian coordinates respectively, we can express Euler equations as

∂ρ

∂t+

∂

∂x(ρux) +

1

y

∂

∂y(y(ρuy)) = 0 (A–166)

∂ρux∂t

+∂

∂x

((ρux

2 + p))+

1

y

∂

∂y(y(ρuxuy)) = 0 (A–167)

∂ρuy∂t

+∂

∂x(ρuxuy) +

1

y

∂

∂y

(y(ρuy

2 + p))=

p

y(A–168)

∂ρe

∂t+

∂

∂x((ρe + p)ux) +

1

y

∂

∂y(y(ρe + p)uy) = 0 (A–169)

Defining Q,Fx,Fy and H as

Q =

ρ

ρux

ρuy

ρe

; Fx =

ρux

ρux2 + p

ρuxuy

(ρe + p)ux

; Fy =

ρuy

ρuxuy

ρuy2 + p

(ρe + p)uy

; H =

0

0

p

y

0

,

(A–170)

where Fx represents the flux in the x-direction and Fy represents the flux in the

y-direction. Flux in the θ direction is zero. Euler equations can now be expressed

as,∂

∂t(Q) +

∂

∂x(Fx) +

1

y

∂

∂y(yFy) = (H) (A–171)

A.8.2 Volumetric and Surface Integration

A typical finite volume cell in the cylindrical coordinates can be chosen in such a

way that faces of control volume are along curvilinear coordinate directions. Face areas

are Ax = ydydθ,Ay = ydxdθ and Aθ = dxdy and cell volume is V = ydxdydθ. Integrating

the Euler equations over the control volume would yield,

∂

∂t(Q)V + (Fx)Ax + (Fy)Ay = (H)V (A–172)

197

Substituting AX ,Ay and V in the above equation yields,[∂

∂t(Q) ydxdy + (Fx) ydy + (Fy) ydx = (H)ydxdy

]dθ (A–173)

∂

∂t(Q) ydxdy + (Fx) ydy + (Fy) ydx = (H)ydxdy (A–174)

∂

∂t(yQ) dxdy + (yFx) dy + (yFy) dx = (yH)dxdy (A–175)

Solving axi-symmetric Euler equation Eq. (A–171) in the cylindrical coordinates is

equivalent to solving following equation in 2D cartesian coordinate system.

∂

∂t(yQ) +

∂

∂x(yFx) +

∂

∂y(yFy) = (yH) (A–176)

There are three possible ways to solve axisymmetric Euler equations.

1. First is to modify Q,Fx,Fy and H quantities by multiplying each by y and integratingover 2D cartesian mesh. Integration over 2D mesh can be equivalently done byusing single layered 3D mesh with single layer of unit length in z-direction providedno flux computation is done on z-faces of each cell.

2. Second is to modify the mesh in such a way to get a factor of y in volume andsurface areas. This can be done by scaling z-coordinates of top layer of 3D meshdescribed above by value of y -coordinate. In the original mesh a control volumemade by faces along curvilinear coordinate lines would have V = dx .dy .1 (as widthalong z-direction is 1), Ax = dy .1, Ay = dx .1 and Az = dxdy . Az is not needed asz-faces are virtual faces. Now consider modified mesh. These geometric quantitiesbecome V = ydxdy , Ax = ydy and Ay = ydx . So while using modified mesh flowquantities Q,Fx,Fy and H need not be modified. Integration can be done as in Eq.(A–172). Substituting for geometric quantities we get.

∂

∂t(Q) ydxdy + (Fx) ydy + (Fy) ydx = (H)ydxdy (A–177)

which is same as integrating Eq. (A–172) in the Cartesian coordinate system onthe original mesh.

3. Third is to simply scale volumes and surface areas w/o actually distorting themesh.

198

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206

BIOGRAPHICAL SKETCH

Manoj Parmar was born in 1981 in Baseri, Rajasthan, India. His early schooling till

7th grade was done in Baseri. Thereafter his family moved to Sri Ganganagar and later

part of schooling was continued at Central school at Sri Gangananagar. He received

Bachelor of Technology and Master of Technology degrees from the Indian Institute of

Technology (IIT), Mumbai in 2003.

He joined the University of Illinois at Urbana-Champaign (UIUC) in 2004 for further

studies. He earned M.S. degree in Aerospace Engineering in 2006 and continued to

study towards Ph.D degree. Later he moved to University of Florida, Gainesville in 2007

to continue his Ph.D. study.

Manoj Parmar received his Ph.D. degree from the University of Florida in the fall

of 2010. He continued as postdoc in the University of Florida. Manoj Parmar has been

married to Sona Parmar since December 2008.

207