Thesis Cesar

8/2/2019 Thesis Cesar

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UNIVERSIDAD NACIONAL DE EDUCACION A DISTANCIA

UNED

FACULTAD DE CIENCIAS FISICAS

DEPARTAMENTO DE FISICA FUNDAMENTAL

ADSCRIPTO A

UNIVERSIDAD DE CASTILLA-LA MANCHA

UCLM

INTERACCION DE ONDAS DE CHOQUE PLANAS

CON FLUJOS INHOMOGENEOS

TESIS DOCTORAL

AUTOR: CESAR HUETE RUIZ DE LIRA

DIRECTOR: J. GUSTAVO WOUCHUK SCHMIDT (UCLM)

TUTORA: EMILIA CRESPO DEL ARCO (UNED)

En Madrid, a 10 de Febrero de 2012


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UNIVERSIDAD NACIONAL DE EDUCACION A DISTANCIA

UNED

FACULTY OF PHYSICS SCIENCES

DEPARTMENT OF FUNDAMENTAL PHYSICS

ATTACHED TO

UNIVERSIDAD DE CASTILLA-LA MANCHA

UCLM

INTERACTION OF PLANAR SHOCK WAVES WITH

NON UNIFORM FLOWS

PH.D. THESIS

AUTHOR: CESAR HUETE RUIZ DE LIRA

DIRECTOR: J. GUSTAVO WOUCHUK SCHMIDT (UCLM)

TUTOR: EMILIA CRESPO DEL ARCO (UNED)

In Madrid, on 10th February 2012


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All of physics is either impossible or trivial. It is impossible until you

understand it, and then it becomes trivial.

Ernest Rutherford


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friends, those in the distance and those I met here, for making this journeyso special.

Of course, I am really grateful to my girlfriend Irene de Llano, whose support

was absolutely unconditional, and thanks to her, I could enjoy an amazing

and peaceful life, which are mandatory requirements for addressing this

task. She encouraged me in the most difficult situations, and thanks to

that, I am writing this right now.

I do thank my parents, Jose and Ignacia, and my brother Jose. They always

encouraged me, and provided me self-confidence for facing this work. They

also helped me in considering the realization of this thesis over the job in

the engineering company that I used to do. I also thank Prof. Juanjo

Lopez-Cela for his advice in that difficult moment.

Finally, I would like to acknowledge the Ministry of Education for the FPU

grant: AP2007-02745, and for some grants: MEC (ENE2009-09276,) and

Junta de CLM (PEI11-0056-1890), internal UCLM contract (2007-COB-

1744), and INEI.


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Contents

List of Figures xiii

1 Introduction 1

1.1 Review of past works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Shock vorticity interaction 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Shock wave traveling in a 2D single mode vorticity field . . . . . . . . . 102.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Wave equation inside the compressed fluid . . . . . . . . . . . . . 14

2.2.3 Shock pressure and shock corrugation as a function of time . . . 16

2.2.3.1 Solution of the wave equation . . . . . . . . . . . . . . . 16

2.2.3.2 Boundary and initial conditions . . . . . . . . . . . . . 17

2.2.4 Inverse Laplace transforms of Ps and s . . . . . . . . . . . . . . 19

2.2.5 Pressure perturbations radiated by the shock into the compressed

fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.6 Rotational and irrotational velocity p erturbations in the com-pressed fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.6.1 RMI growth at x = 0 . . . . . . . . . . . . . . . . . . . 25

2.2.6.2 Downstream vorticity perturbations . . . . . . . . . . . 28

2.2.6.3 Rotational velocity perturbations in the compressed fluid 30

2.2.6.4 Irrotational velocity profiles in the short wavelength

regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.7 Density perturbations generated downstream . . . . . . . . . . . 35

2.3 Shock compression of turbulent 2D/3D random vorticity fields . . . . . 38

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CONTENTS

2.3.1 General 3D upstream/downstream perturbation flows: geomet-rical representation . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 Reduction to an equivalent 2D problem . . . . . . . . . . . . . . 42

2.3.3 Interaction of a shock wave with a 2D turbulent velocity field . . 44

2.3.3.1 Average kinetic energy of the downstream perturbations 45

2.3.3.2 Acoustic energy flux . . . . . . . . . . . . . . . . . . . . 51

2.3.3.3 Density generation downstream . . . . . . . . . . . . . 57

2.3.3.4 Vorticity amplification . . . . . . . . . . . . . . . . . . . 63

2.3.4 Interaction of a shock with a 3D turbulent velocity field . . . . . 68

2.3.4.1 Average kinetic energy of the downstream perturbations 68

2.3.4.2 Acoustic energy flux . . . . . . . . . . . . . . . . . . . . 78

2.3.4.3 Density generation downstream . . . . . . . . . . . . . 84

2.3.4.4 Vorticity amplification . . . . . . . . . . . . . . . . . . . 87

2.3.5 Anisotropy of the downstream perturbations . . . . . . . . . . . 92

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3 Shock entropy interaction 97

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.2 Interaction of a planar shock with a single mode 2D density field . . . . 983.2.1 Wave Equation and Boundary Conditions . . . . . . . . . . . . . 98

3.2.2 Perturbation dynamics in the compressed fluid . . . . . . . . . . 101

3.2.3 Rotational and irrotational perturbations downstream . . . . . . 103

3.2.3.1 Richtmyer-Meshkov instability growth at the weak dis-

continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.2.3.2 Downstream vorticity perturbations . . . . . . . . . . . 105

3.2.3.3 Downstream irrotational velocity perturbations . . . . . 107

3.2.4 Asymptotic entropy field downstream . . . . . . . . . . . . . . . 108

3.3 Interaction of a planar shock with a 2D random density field . . . . . . 1093.3.1 Turbulent kinetic energy generation . . . . . . . . . . . . . . . . 110

3.3.2 Acoustic energy flux . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.3.2.1 Sound energy flux in the compressed fluid reference frame117

3.3.2.2 Sound energy flux in the shock reference frame . . . . . 118

3.3.3 Density amplification downstream . . . . . . . . . . . . . . . . . 123

3.3.4 Vorticity generation downstream . . . . . . . . . . . . . . . . . . 128

3.4 Interaction of a planar shock with a 3D random density field . . . . . . 133

3.4.1 Turbulent kinetic energy generation . . . . . . . . . . . . . . . . 133

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CONTENTS

3.4.2 Acoustic energy flux . . . . . . . . . . . . . . . . . . . . . . . . . 1393.4.3 Density amplification downstream . . . . . . . . . . . . . . . . . 141

3.4.4 Vorticity generation downstream . . . . . . . . . . . . . . . . . . 145

3.4.5 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4 Shock acoustic field interaction 153

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.2 Interaction of a planar shock with a single mode 2D acoustic field . . . . 154

4.2.1 Wave Equation and Boundary Conditions . . . . . . . . . . . . . 1544.2.1.1 Initial and boundary conditions . . . . . . . . . . . . . 157

4.2.1.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . 159

4.2.1.3 Inverse Laplace Transform . . . . . . . . . . . . . . . . 159

4.2.2 Sound waves emitted by the shock front downstream . . . . . . . 161

4.2.2.1 Pressure perturbations of the running acoustic waves

emitted downstream, when |ac| > 1/

1 M22 . . . . . 1614.2.2.2 Critical angles . . . . . . . . . . . . . . . . . . . . . . . 162

4.2.3 Velocity perturbations generated downstream . . . . . . . . . . . 164

4.2.3.1 RMI growth at x = 0 . . . . . . . . . . . . . . . . . . . 1644.2.3.2 Vorticity generation downstream . . . . . . . . . . . . . 166

4.2.3.3 Rotational velocity perturbations in the compressed fluid167

4.2.3.4 Irrotational velocity perturbations downstream . . . . . 169

4.2.4 Density perturbations generated downstream . . . . . . . . . . . 169

4.2.5 Acoustic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.3 Shock wave traveling into a random isotropic acoustic field . . . . . . . 173

4.3.1 Downstream kinetic energy . . . . . . . . . . . . . . . . . . . . . 174

4.3.2 Acoustic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3.3 Density Amplification . . . . . . . . . . . . . . . . . . . . . . . . 1864.3.4 Downstream vorticity generation . . . . . . . . . . . . . . . . . . 190

4.3.5 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

4.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5 Closure 199

5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

A Bessel series expansion 205

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CONTENTS

B Analytical Integrals 209

References 221

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List of Figures

2.1 Shock vorticity interaction scheme . . . . . . . . . . . . . . . . . . . . . 12

2.2 Complex path integration . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Shock pressure behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Interface x-velocity evolution . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Vorticity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Post shock x-velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Entropy field downstream . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.8 Vector velocity plot upon entropy spots . . . . . . . . . . . . . . . . . . 37

2.9 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.10 Shear wave orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.11 Shock isotropic vorticity field interaction in 2D . . . . . . . . . . . . . . 44

2.12 Comparison with Rotman for shock/turbulence in air . . . . . . . . . . 47

2.13 2D Kinetic energy amplification . . . . . . . . . . . . . . . . . . . . . . . 48

2.14 Strong shock and highly compressible limits for 2D kinetic energy . . . . 51

2.15 Polar plot for the acoustic energy flux radiated . . . . . . . . . . . . . . 55

2.16 2D Relative acoustic energy flux for different gases . . . . . . . . . . . . 56

2.17 2D Relative acoustic energy flux . . . . . . . . . . . . . . . . . . . . . . 57

2.18 Strong shock and highly compressible limits for 2D acoustic energy flux 582.19 Density generation contributions in air . . . . . . . . . . . . . . . . . . . 59

2.20 2D Density generation factor . . . . . . . . . . . . . . . . . . . . . . . . 60

2.21 Strong shock and highly compressible limits for 2D density generation . 63

2.22 Vorticity amplification contributions for air . . . . . . . . . . . . . . . . 64

2.23 Vorticity amplification factor . . . . . . . . . . . . . . . . . . . . . . . . 65

2.24 Strong shock and highly compressible limits for 2D vorticity amplification 69

2.25 Shock isotropic vorticity field interaction in 3D . . . . . . . . . . . . . . 69

2.26 Kinetic Energy amplification K3D contributions . . . . . . . . . . . . . 71

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LIST OF FIGURES

2.27 Kinetic energy amplification. Comparison with experiments and simu-lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.28 Sonic and total contribution to the kinetic energy amplification factor . 73

2.29 Strong shock asymptotic of the acoustic and total contribution to the

kinetic energy amplification factor . . . . . . . . . . . . . . . . . . . . . 75

2.30 High compressibility limit for the acoustic and total contribution to the

kinetic energy amplification factor . . . . . . . . . . . . . . . . . . . . . 78

2.31 Averaged longitudinal acoustic flux emitted by a shock moving into air . 79

2.32 Relative averaged acoustic flux emitted by a shock. Comparison with

Ribner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.33 Relative acoustic flux as a function in 3D . . . . . . . . . . . . . . . . . 81

2.34 Strong shock and highly compressible limits for 3D acoustic energy flux 83

2.35 Density generation 3D contributions in air . . . . . . . . . . . . . . . . . 85

2.36 3D Density generation factor . . . . . . . . . . . . . . . . . . . . . . . . 86

2.37 Strong shock and highly compressible limits for 3D density generation . 87

2.38 Vorticity amplification contributions for air . . . . . . . . . . . . . . . . 89

2.39 Vorticity amplification factor in 3D . . . . . . . . . . . . . . . . . . . . . 90

2.40 Strong shock and highly compressible limits for 3D vorticity amplification 93

2.41 Downstream velocity anisotropy parameter . . . . . . . . . . . . . . . . 942.42 Downstream velocity-isotropy curve . . . . . . . . . . . . . . . . . . . . 94

2.43 Downstream vorticity anisotropy parameter . . . . . . . . . . . . . . . . 95

3.1 Shock entropy interaction scheme . . . . . . . . . . . . . . . . . . . . . . 100

3.2 Asymptotic and exact shock pressure evolution . . . . . . . . . . . . . . 104

3.3 Post shock x-velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.4 Vector velocity plot upon entropy spots . . . . . . . . . . . . . . . . . . 110

3.5 Shock isotropic entropy field interaction in 2D . . . . . . . . . . . . . . . 111

3.6 Turbulent kinetic energy generated downstream separated in its different

contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.7 Acoustic and total 2D kinetic energy generation . . . . . . . . . . . . . . 113

3.8 Acoustic and total contribution for the kinetic energy in the strong shock

limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.9 Acoustic and total contribution for the kinetic energy in the high com-

pressibility limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.10 Averaged longitudinal-acoustic energy flux emitted by the shock . . . . 119

3.11 Polar plots of the dimensionless flux qx as a function of the angle of

emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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1. INTRODUCTION

The study of shock waves with monochromatic disturbances ahead was also stud-ied by Mckenzie (8). He considered the shock interaction with vorticity, entropy and

acoustic waves independently, showing how these three modes generated different sets

of vorticity, entropy and sonic flux downstream. All the works mentioned so far studied

the asymptotic evolution of the shock waves through those perturbed fields, and hence,

the asymptotic profiles for the entropy spots and vortex structure behind. They do not

consider the transient behavior until the asymptotic regime is achieved. In 1992, Kevla-

han et al (9) solved analytically the interaction of a shock front with vortical structures

in the weal shock limit. In a subsequent paper (10), he developed a new theory for the

propagation of weak shocks into 2D non uniform flows, where he considered the nonlinear second order terms, but the equations could not be solved analytically, except

for a few simple situations. In a posterior work (11), he solved the vorticity generation

for finite amplitude shock waves interacting with different kind of perturbations ahead.

It was remarked there that, thanks to the baroclinic generation when shocking density

disturbed fields, vorticity can be generated behind the shock even if the flow ahead is

irrotational.

The shock interaction with non uniform flows has been studied experimentally since

more than twenty years (12, 13, 14) by means of different configurations. In 1988, Hes-

selink and Sturtevant (15) investigated the propagation of weak shock waves through a

statistically uniform random medium in a shock tube. They used schlieren and shadow-

graph technique to visualize the density variations across the shock, and also determine

the pressure waves emitted by the corrugates shock. J. Keller and W. Merzkirch (16)

did experiments in a shock tube using the speckle photography method. They found

(as theoretically predicted) an amplification of the turbulence intensity by the normal

shock wave, and they have also shown that the density micro-scale increases across the

shock. Honkan and Andreopoulos (17) focused on interactions of a normal shock with

grid-generated turbulence in a shock tube. The spectral analysis indicated that large

eddies are amplified more than small eddies during interactions with shock waves ofthe same strength. A few years later, Barre et al (18) developed a similar experiment

and they were able to measure the kinetic energy amplification quite accurately. They

used a new method to generate quasi-homogeneous isotropic turbulent supersonic flow

for strong Mach numbers (M1 = 3) in air. It is shown there that, in such conditions,

the kinetic energy is amplified a factor 2 just behind the shock and a factor 1.5 in the

far field. In Chapter 2, we compare the analytical linear theory with the results of this

experiment.

Besides the experimental study, with the advancement of modern supercomputers,

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1.1 Review of past works

the problem has been addressed with the use of sophisticated numerical simulationcodes. In 1991, Rotman (19) studied numerically the interaction of shock waves on

a turbulent flow, by means of Large-Eddy Simulation (LES). The results shown there

agreed with the previous analytical works and the analytical results presented here,

where the most important feature is the increase of the turbulence through the shock

in a wide range of parameters. Since the work published with Kevlahan (9), Mahesh

fruitfully contributed with other numerical works (20, 21, 22). In the first work (20),

Mahesh, Lele and Moin studied the response of anisotropic turbulence to an oblique

shock passage using the Rapid Distortion Theory (RDT). They found out that, besides

shock strength, the anisotropy and the shock inclination are important in determiningthe evolution of turbulence. RDT was previously developed by Jacquin et al (23),

however, the authors themselves recognized the inadequacy of their treatment for the

problem here considered. In a subsequent paper (21), they studied the interaction of a

planar shock wave with an isotropic field of acoustic waves by Direct Numerical Simula-

tions (DNS). In a later work (22), Mahesh et al studied the role of entropy fluctuations

upstream on the shock/turbulence interaction numerically. They use Morkovin hypoth-

esis to correlate the vorticity and entropy disturbances, analyzing the combined effect

of the baroclinic and bulk compression to the shock passage.

Lee et al (24), extended the study of weak shock/turbulence interaction by DNS.

They studied the shock distortion and thickness as well as the turbulent spectra modifi-

cation. In a subsequent work (25), they completed the analysis on such interactions by

choosing different kind spectra ahead, and studying the modification of these spectra

by the shock wave passage. It is clear how DNS has complemented the study of shock

traveling into non uniform flows (26, 27). Recently, Larsson et al (28) and Bermejo-

Moreno (29) studied a more realistic shock turbulence interaction by means of DNS and

LES. They did not consider a purely isotropic vorticity field, but a canonical spectrum

where vorticity ahead is related with the density fluctuations upstream by the Reynolds

stress. Lately, Sub-Grid Scale (SGS) codes have been developed in order to complementLES, because SGS includes gas compressibility together with more accurate micro-scale

effects. Besides, this kind of interactions has also been studied by means of Implicit

Large-Eddy Simulation (ILES) by Thornber et al (30), and Reynolds-Averaged Navier-

Stokes (RANS) models (31).

Another important problem that relates shock dynamics occurs when a planar shock

wave hits a localized perturbation at a material interface separating two homogeneous

fluids, triggering the Richtmyer-Meshkov Instability (RMI) (32, 33, 34, 35, 36, 37, 38,

39, 40, 41, 42, 43, 44, 45, 46, 47, 48). The RMI is essentially driven by the vorticity

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2

Shock vorticity interaction

2.1 Introduction

In this chapter, we focus on the interaction of a planar shock wave with a pure vortic-

ity field. The chapter is structured as follows. In section 2.2, the basic blocks needed

to construct the 2D single-mode model are presented and developed. Exact temporal

evolution of the different perturbed quantities is shown since t = 0+ , and the explicit

asymptotic expressions are derived. The generation of vorticity, entropy and sound

waves downstream is discussed in detail. The RMI growth that ensues at the weak dis-

continuity formed at the boundary that separates the turbulent field with the initially

quiescent fluid is also described. In section 2.3, these results are used to calculate the

quantities averaged over an either 2D or 3D isotropic pre-shock velocity distribution.

Euler angles are used to represent the most general divergence-free velocity/vorticity

field in front of the shock, in terms of elementary shear waves. This representation is

greatly simplified by an adequate rotation on the plane of the shock to deal with the

isotropic spectrum upstream. The ratio of the averages of the kinetic energy (down-

stream/upstream) are obtained either for a 2D or a 3D pre-shock turbulent vorticity

field and are studied as explicit functions of the shock strength and the compressibility

of the gas. The contribution of the sonic waves is also discussed in detail. The sound

energy flux radiated into the fluid downstream is analyzed in both the compressed fluid

reference and the shock front reference frames. An analysis of the amplification of

the vorticity perturbations is presented. An anisotropy parameter is defined both for

the downstream velocity and vorticity fields. It is found that the compressed velocity

field may remain isotropic for specific choices of the two parameters that govern the

dynamics of the interaction, and M1, whereas the post-shock vorticity field is always

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2. SHOCK VORTICITY INTERACTION

anisotropic. In section 2.4 we conclude with a summary. It is emphasized that allthe line and surface plots shown in this work are generated using exact closed-form

analytical formulas. An example of the analytical integrals is given in the Appendix B.

2.2 Shock wave traveling in a 2D single mode vorticity

field

2.2.1 Boundary conditions

We refer to Fig.2.1(a) where we consider a shock wave that comes from the left withvelocity Dx, as measured in the laboratory frame of reference (coordinates variablesx and y). The fluid is an ideal gas, which is homogeneous for x < 0 and is onlyperturbed in the half-space x > 0 [see Fig.2.1(a)]. Therefore, the surface locatedat x = 0 is a weak contact discontinuity that separates two regions inside the samefluid. The perturbations consist of steady-state, rotational,and divergence-free velocity

fluctuations. In the space x < 0, the velocity of the compressed fluid measured in thelaboratory is Ux, the density and pressure ahead of the shock are 1, p1, and theircompressed values are 2, p2. The shock Mach number with respect to the fluid ahead

of the shock is denoted by M1, and that with respect to the compressed fluid by M2.The sound speed ahead of the shock is c1, and the compressed fluid sound speed is

c2. Before the shock arrives to the interface x = 0, the following relationships hold

between the quantities at both sides of the shock front:

R =21

=D

D U =(+ 1) M21

( 1) M21 + 2, (2.1)

p2p1

=2M21 + 1

+ 1, (2.2)

c2c1

=

2M21 + 1(+ 1) R

=

2M21 + 1 ( 1) M21 + 2

(+ 1) M1, (2.3)

M2 =D U

c2=

( 1) M21 + 22M21 + 1

. (2.4)

To the right of the surface x = 0, the perturbed velocity field is described by:

v1x = v1x(x, y) = c2u1 cos

kxx

cos(kyy) ,v1y = v1y(x

, y) = c2u1kxky

sin kxx

sin(kyy) , (2.5)

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2.2 Shock wave traveling in a 2D single mode vorticity field

where x in the above two equations is the normal coordinate measured in the laboratoryframe and u1 = u1 (kx, ky) is a dimensionless parameter that quantifies the magnitude

of the velocity perturbations. To remain within the limits of validity of the linear

theory, we assume that u1 1. It can be easily verified that the above velocity field issolenoidal (.v1 = 0). The quantities kx and ky are the longitudinal and transversewave-numbers, defined as:

kx =2

x, ky =

2

y, (2.6)

where x and y are the characteristic longitudinal and transverse lengths of the vor-

ticity profile upstream. The vorticity associated to the velocity components given byEq.(2.7) is:

1(x, y) = kyc2u1

1 +

kxky

2cos

kxx

sin(kyy) . (2.7)Once the shock enters the space x 0, the velocity perturbations in front of it willinduce post-shock velocity fluctuations downstream, and its shape will be distorted,

generating at the same time pressure and density perturbations in the compressed

fluid. We assume that these fluctuations are always much smaller than the background

values corresponding to a non distorted shock.

We introduce the following definitions, where the subscript 1 always refers to quan-tities in front of the shock front and the subscript 2 refers to the quantities behind it.

We factor out the linear dependence with the dimensionless parameter u1:

v1xc2

= u1 cos

kxx cos(kyy) ,

v1yc2

= u1kxky

sin

kxx sin(kyy) ,

22

= u1(x, t)cos kyy , (2.8)

p22c22 = u

1p(x, t)cos kyy ,v2x

c2= u1vx(x

, t)cos kyy ,

v2yc2

= u1vy(x, t)sin kyy ,

where we have defined: v1x = cos(kxx), and v1y = sin(kxx). In Fig.2.1(a), we show the

perturbation field before the incident shock refraction at the weak contact discontinuity

located at x = 0. The position of the unperturbed shock in the laboratory frame isxs(t) = Dt for t < 0. We assume that the shock enters the half space filled with the

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we write:

= vx

(ky x) vy ,

vx

= p(ky x)

,

vy

= p , (2.9)

representing, respectively, the mass, x-momentum, and y-momentum conservation equa-

tions. Besides, we assume adiabatic flow in the fluid behind the shock, and therefore,

the following relationship holds between the density and pressure perturbations:p

=

. (2.10)

At t = 0+ a shock is transmitted into the space to the right and a neutrally stable

sound wave is reflected back into the space x < 0. The transmitted shock is deformed as

it propagates into the disturbed fluid. As a consequence, pressure fluctuations will be

generated behind its surface which propagate downstream in the form of sound waves.

If the shock/turbulence interaction is started by a planar shock wave entering the weak

perturbation field from a quiescent fluid, as shown in Fig.2.1, then none of these waves

gets reflected to the right, when they arrive at the interface x = 0. This constraintis known as the isolated shock boundary condition. Examples of different boundary

conditions, applicable when the incident shock is driven by a piston, are found in ( 44).

The sound perturbations radiated by the corrugated front go into the fluid downstream,

eventually reaching the surface x = 0, and are transmitted into the space x < 0 without

being reflected towards the shock. Those perturbations will travel at the local sound

velocity behind the neutral sound wave mentioned above. Pressure and normal velocity

perturbations are continuous on both sides of it. However, the distorted front generates

vorticity and entropy perturbations and the neutral sound wave to the left does not.

Hence, the x-derivative of vy and the density perturbation are discontinuous at x = 0.The perturbed conservation equations across the neutral sound wave that travels to

the left are the simplest to write:

vx + p = 0 ,

vy = 0 , (2.11)

and both are valid just to the right of the left-facing sound wave, at the position

x = c2t, in the reference frame of the compressed fluid. There remain the boundaryconditions at the corrugated shock front moving to the right. The unperturbed shock

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moves along the x-axis with a velocity c2 tanh , which for the shocked gas is lowerthan the speed of the shock front with respect to the compressed fluid, M2c2. It is

noted that the new shock front coordinate (s) can be actually written in terms of the

downstream shock Mach number as: tanh s = M2. Further, from Eq.(2.16), we see

that:

= rs cosh s =rs

1 M22. (2.17)

We define the auxiliary function h by :

h =

1

r

p

, (2.18)

and the equations of motion [Eqs.(2.9), (2.10)] can be rewritten in the new coordinates

r, . The mass conservation is recast as:

sinh h + cosh pr

+cosh

r

vx

sinh vxr

+ vy = 0 , (2.19)

while the xmomentum equation is given by:

sinh

r

vx

+ cosh vx

r

+ cosh h

sinh

p

r

= 0 , (2.20)

and the ymomentum equation is rewritten as:

sinh r

vy

+ cosh vyr

= p . (2.21)

The shock/vorticity interaction problem here is analyzed within the limits of validity of

linear theory. Hence, up to a first order expansion in the small parameter u1, either p,

vx and downstream are proportional to cos kyy, which appears as a common factor

in Eqs.(2.19), and (2.20) and can be therefore dropped out from both sides of those

equations. Similarly, the terms that compose the tangential momentum conservationequation are proportional to sin kyy, which can also be canceled out. Therefore, we

have omitted the y dependence in all the equations, for simplicity, thus only relating

quantities that depend exclusively on r and (or equivalently, on x and t). The last

equations can be combined into a system of coupled differential equations involving p

and h:

r2p

r2+

p

r+ r p =

h

, (2.22)

rh =p

. (2.23)

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It is clear that the system of equations above is equivalent to the wave equation down-stream, which we also write here in Cartesian coordinates:

2p

2=

2p

(kyx)2 p . (2.24)

To get the solutions of this problem, we use the Laplace transform. The advantage is

that we get exact analytic closed form expressions for the Laplace transforms of the

perturbation fields. After some algebra in the complex plane, we get the complete

temporal evolution with an inverse transformation, and the exact asymptotic behavior

by looking at the singularities of the Laplace functions.

2.2.3 Shock pressure and shock corrugation as a function of time

2.2.3.1 Solution of the wave equation

For any quantity (, r), we define its Laplace transform (indicated with capital letters)

by:

(, s) =

0

(, r)exp(sr)dr . (2.25)

It is convenient to make the variable change s = sinh q and transform Eqs.(2.22)-(2.23),

following:

q

cosh q P

+

H

= 0 , (2.26)

cosh q P

+

H

q= 0 . (2.27)

As discussed in (40), a first integral is obtained for the above system of equations, and

written as:

cosh (q + ) P (, q + ) + H(, q + ) = 2F1(q) , (2.28)

cosh (q ) P (, q ) H(, q ) = 2F2(q) . (2.29)

The functions F1 and F2 must be determined with the help of the boundary and initialconditions. As discussed in (40, 41), it can be seen that F1 represents the sound

perturbations radiated by the shock into the compressed fluid and F2 corresponds to

the sound waves that arrive to the shock from behind. These waves might be evanescent

or traveling fronts, depending on the ratio kx/ky, the shock Mach number and the gas

compressibility. In this work, we consider an isolated shock, to which no so. After some

additional algebra, the following decomposition for the pressure and pressure gradient

fluctuations is obtained:

P (, q) =F1 (q ) + F2

cosh q, (2.30)

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H(, q) = F1 (q ) F2 . (2.31)The expressions shown in Eqs.(2.30) and (2.31) are valid in the whole compressed fluid

in the space 0 x xs(t) = (D U)t. To get the values of F1 and F2 (equivalently,the values of P and H), we have to work out the boundary and initial conditions either

at the shock front and at the surface x = 0.

2.2.3.2 Boundary and initial conditions

We transform the boundary conditions [Eqs.(2.12)-(2.15)] into a system of coupled par-

tial differential equations for the shock pressure perturbation and the shock corrugation.

From now on, the variables that are evaluated at the shock front coordinate are de-

noted with the subscript s. We follow Richtmyer (32) and take the time derivative of

Eq.(2.13) along the shock front trajectory. After using Eqs.(2.9) together with the rest

of the boundary conditions at the shock [Eqs.(2.12)-(2.15)], we arrive to the system:

hs() = M21 + 1

2M21 M2

1 M22dpsd

2

1 M22+ 1

s() +

+2

1 M22(+ 1) M2

kxky

sin

R

kxky

M2

, (2.32)

dsd

=(+ 1)

4M2ps() + cos

R

kxky

M2

. (2.33)

We use the coordinates defined in Eqs.(2.16), noting that = rs/

1 M22 accordingto Eq.(2.17), and recast the system above into:

hs(rs) =

M21 + 1

2M21 M2

dpsdrs

2

1 M22+ 1

s(rs) +(R 1) 0

Rsin(0rs) , (2.34)

dsdrs

=+ 1

4M21 M22

ps(rs) +cos(0rs)

1 M22

, (2.35)

where 0 is a dimensionless frequency that takes into account the periodicity of the

pre-shock velocity field, and is given by:

0 =RM21 M22

kxky

. (2.36)

We take the Laplace transform of Eqs.(2.34) and (2.35), use s = sinh q, and arrive to

the following system:

Hs(q) = M21 + 1

2M21 M2 sinh q Ps(q) ps0

2

1 M22+ 1

s(q) +(R 1)20

R sinh2 q + 20, (2.37)

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2.2.4 Inverse Laplace transforms ofPs and s

To solve the pressure and velocity profiles in the compressed fluid (that is, the sound

pressure waves, the vorticity generated at each position, the spatial structure of the

longitudinal and transverse velocities, etc.), we must obtain the dynamical evolution of

the shock pressure perturbation, as the shock moves to the right. This information may

be obtained after inverting the expressions for Ps and s in the complex plane. From

the Laplace transform theory, we know that the function ps(rs) is formally calculated

from the integral in the complex plane:

ps(rs) = 12i

c+ici

Ps(s) exp(srs)ds , (2.44)

where c is a real number to the right of the singularities ofPs(s), and i is the imaginary

unit (i2 = 1). To get an algebraic expression from (2.41)-(2.44), we close the integra-tion contour to the left and use the Residue Theorem, taking care of the singularities

enclosed by the integration path (see Fig.2.2). For a shock moving into an ideal gas, the

only singularities of Ps(s), as can be seen from (2.41), are the branch points at s = i,and the poles at s = i0. The branch point singularities represent the generation

Figure 2.2: Integration path on the complex plane.

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of evanescent sound wave perturbations, that decay asymptotically in time like t3/2,much in the same way as Bessel functions. On the other hand, the imaginary poles give

rise to asymptotic constant amplitude oscillations. These permanent oscillations of the

shock ripple are due to the perturbations in velocity distributed periodically upstream.

Besides, the denominator 2M21 M2 s

s2 + 1 +

M21 + 1

s2 + M21 does never contribute

with additional singularities that could result in permanent oscillations for an ideal gas

equation of state. We obtain:

ps(rs) = 2

1

0fp(z)cos(z rs) dz +

2v

1

0fp(z)

cos(z rs) cos(0 rs)20

z2 dz ,

(2.45)where, the auxiliary function fp is given by:

fp(z) =4M41 M

22 z

2

1 z24M21 M

22 z

2(1 z2) + M21 + 1 z2 M21 2 . (2.46)From the expressions above, we can also get the formulas for the asymptotic functions.

We have to distinguish between the long (0 < 1) and short (0 > 1) wavelength regimes.

From (2.41), the asymptotic behavior of ps is found from the residues at the poles

s = i0:

ps(rs) =

el1 cos(0 rs) + el2 sin(0 rs) , 0 1

es cos (0 rs) , 0 1 ,(2.47)

where rs =

1 M22 , according to (2.17). The coefficients el1, el2, and es are formallyequivalent to those obtained in (44, 91, 92, 94):

el1 =2M21 M2

M21

M21 + 1

20

v

4M41 M22

20

1 20

+

M21

M21 + 1

202 ,

el2 =4M41 M

22 0

1 20 v4M

41 M

22

20

1 20

+

M21 M21 + 1 202

,

es = 2M21 M2 v

2M21 M20

20 1 +

M21 + 1

20 M21. (2.48)

It is easy to see that the asymptotic expressions above are actually functions of the

argument 0rs Dkxt. Besides, it is worth to note here that the only difference betweenthe coefficients in the above equations and the corresponding ones in is the dependence

of v on the shock strength, gas compressibility, and on the dimensionless frequency

0. We show the time evolution of the shock front pressure perturbations for different

regimes, comparing the exact solution with the asymptotic expressions. The parameter

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that distinguishes both cases (long and short wavelength) is the dimensionless frequency0. We show the shock front pressure fluctuations in Fig.2.3(a) for a shock moving into

an ideal gas with = 7/5, the shock Mach number is M1 = 3, and kx/ky = 1/5, which

gives 0 = 0.41662 (long wavelength). The red curve is the exact solution given by

Eq.(2.45), and the black curve is the asymptotic solution (2.47). In Fig.2.3(b) we show

the shock pressure perturbations for the same gas and shock, but for a short wavelength

case (kx/ky = 1, 0 = 2.08310), distinguishing between the exact and asymptotic

functions of time. In Appendix A, it is shown an alternative way of calculating the

0 20 40 60 80 1001

0.5

0

0.5

1

ps( 1)ps

kyDt

(a)

0 20 40 60

0.4

0.2

0

0.2

0.4

0.6

ps( 1)ps

(b)

kyDt

Figure 2.3: (a): Shock front pressure perturbations originating when a planar strong

shock (M1 = 3) moves into an incompressible vorticity field in an ideal gas with = 7/5.

The ratio of longitudinal to transverse wave-numbers is kx/ky = 1/5, which gives 0 =

0.41662. (b): The wave-number ratio is kx/ky = 1, which gives 0 = 2.08310.

shock pressure evolution by means of Bessel series expansion. Now we study the shock

corrugation amplitude as the shock moves into the half space filled with the vorticity

perturbations. We can easily get the complete time evolution of the shock front ripple

for the single mode 2D case. This can be obtained from the direct inverse Laplace

transform of Eq.(2.42), or substituting Eq.(2.45) into Eq.(2.35) and integrate on the

variable rs. We get:

s(rs) = (+ 1)2M2

1 M22

10

fp(z)

zsin(z rs) dz +

sin(0 rs)

0

1 M22+

+(+ 1) v

2M2

1 M22

10

fp(z)

z 0

0 sin(z rs) z sin(0 rs)

20 z2

dz . (2.49)

The asymptotic expression for the shock ripple can also be obtained by analyzing the

poles of the Laplace transform Eq.(2.42) or substituting Eq.(2.47) into Eq.(2.43) and

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when (2.55) [or (2.54)] is satisfied, the asymptotic pressure fluctuations are describedin space and time by:

p (, r) = es cos (r) =

= es cos [cosh (0 s) sinh(s 0) kyx] . (2.57)

Here, es is the asymptotic amplitude of the pressure perturbations at the shock front

defined in (2.48). Looking inside the argument of the last equation, we realize that the

function given in (2.57) represents propagating planar fronts in the (x, ) plane. If 0 s), the

sound waves are emitted to the left. In the absence of a reflecting surface at x = 0,

those waves escape towards x = . Furthermore, the frequency of oscillation of thecompressed fluid particles downstream (when 0 > 1) can be seen to be equal to:

1 = cosh (0 s) = 0 M2

20 11 M22

, (2.58)

and the longitudinal wave-number of the sonic fronts (kacx ) is equal to:

kacxky

= sinh(s 0) = M20

20 11 M22

, (2.59)

which is positive (and hence representing waves that travel to the right) if 0 < s,

and negative (representing waves to the left) if 0 > s. We remind that the frequency

of oscillation of the compressed fluid particles is 1 < 0, due to the Doppler shift.

The boundary condition at x = 0 is that no sound waves arriving there get reflected.

If the reflection was possible, because of the presence of a piston, for example, the

reflected waves could have reached the shock from behind. For this to happen, 0

should be greater than a certain value that would depend on the shock strength andthe fluid compressibility. This possibility (as well as the possibility of having multiple

reverberations between the shock and the reflecting surface) has been discussed in and

will be left for future study.

2.2.6 Rotational and irrotational velocity perturbations in the com-

pressed fluid

As the shock moves into the perturbed fluid upstream, additional velocity fluctuations

are generated downstream. In this subsection we study the velocity field generated

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behind the shock front. At first, we discuss the velocity perturbations as a function oftime at the surface x = 0. In fact, the initial discontinuity in tangential velocity at that

position will evolve like the contact surface of a RMI problem. In the next subsection,

we analyze the vorticity deposition in the compressed gas and determine the rotational

velocity profiles in both regimes of short and long wavelength. Finally, the velocity

profiles associated to the sound waves emitted by the shock front are obtained.

2.2.6.1 RMI growth at x = 0

To study the velocity perturbations that develop at the surface x = 0, we consider

the x-momentum equation in (2.13) and evaluate it at = 0. We take its Laplace

transform, and with the help of (2.28)-(2.31), and (2.41) we get the Laplace transform

of the velocity fluctuations at the interface:

Vxi = Vx (x = 0, s = sinh q) = cosh (q + s) Ps(q + s)sinh q

. (2.60)

After some algebra, we arrive to:

Vxi =

M22 + 1

s2 + 1s + 2M2s

2 + M2

s M22 s2 20 + 1 + 2M2s

s2 + 1 + s2 + 20

00

M22

s2 20 + 1

+ 2M2s

s2 + 1 + s2 + 20

+ 20

10s2 +

s2 + 1s 11, (2.61)

where the coefficients 00, 10, 11 and 20 are respectively:

00 =2M21 M2 + M

21 + 1

M2(2M21 M22 + 4M

21 + 2)

ps0 ,

10 = M21

5M22 + 1

+ M22 + 1

2M2

M21

M22 + 2

+ 1 ,

11 = M21 2M22 + 1 + M22

2M2

M21

M22 + 2

+ 1

,

20 = M21

M22 1

M21

M22 + 2

+ 1

v . (2.62)

The exact temporal evolution since = 0+ is formally given by:

vxi() =1

2i

c+ici

Vx(x = 0, s) exp(s) ds , (2.63)

where c is any real number to the right of all the singularities of the Laplace function

Vx(x = 0, s). The procedure to calculate the integral in the complex plane above is the

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in formal agreement with Eqs.(91) and (92) of (44). In Fig.2.4 we plot the longitudinalvelocity at the interface. In the first case, the shock wave does not emit stable acoustic

waves downstream, and the velocity at the interface reaches a constant value, which

means that the interface ripple grows linearly as a RMI. For the second figure, the shock

ripple oscillates getting a constant sonic radiation downstream [0 1/

1 M22 ], andhence, an oscillatory pulse is superposed to the growth rate vxi .

0 5 10 15 20 25 300.4

0.5

0.6

0.7

0.8

0.9

vxivxi

(a)

0 5 10 15 200.2

0

0.2

0.4

0.6

vxi( 1)vxi

(b)

Figure 2.4: (a): Longitudinal interface velocity when M1 = 3, = 7/5 and the wave-

number ratio is kx/ky = 1/5, which gives 0 = 0.41662. (b): The ratio of longitudinal to

transverse wave-numbers is kx/ky = 1, which gives 0 = 2.08310.

On the other hand, the Laplace Transform for the interface ripple holds si(s) = Vxi.

Therefore, we can obtain the asymptotic interface ripple evolution by integration the

longitudinal velocity field at x = 0. It is:

i( 1) =

i0 + vxi 2 , 0 M2/(1 M22 )

i0 + vxi 2 es

211

21

sin(12) , 0 M2/(1 M22 ) ,(2.67)

where the initial value for the asymptotic ripple evolution is i0 = lims0d

ds

s2

i(s)

,which can be expressed as:

i0 = ps0

M22 1

M21 M22

M21 (2M2 + 1) + 1

M21

2M22 + 1

+ M222 +

+ v2M41 M2

M22 1

M42

20 5

M22 220 + 1 + 20M21

2M22 + 1

+ M22

2 20 M22

20 1

2 v

2M41 M32

M22 1

M42

20 1

M22 220 3 + 20M

21 2M

22 + 1 + M

22

2

20 M22

20 1

2. (2.68)

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2.2.6.2 Downstream vorticity perturbations

Thanks to the conservation of tangential momentum across the shock front, it is not

difficult to see that vorticity will be always generated behind a corrugated shock wave.

This vorticity will remain frozen to the fluid elements in the absence of viscosity, which

we assume to be the case. The total vorticity in the compressed fluid will be composed

of two terms, one due to an amplification of the upstream vorticity and the other

one will be the result of the shock ripple oscillations. The first term, as will be seen

in this subsection, is the result of the shock compression of the upstream vortices,

in the direction of shock motion. The shock compression reduces the characteristic

xlength of the pre-shock eddies by a factor 1/R = 1/2. On the other hand, thesecond contribution comes from the shock oscillations, induced by the perturbations

upstream. In fact, as the shock ripple oscillates, lateral pressure gradients are generated

behind the shock, which induce an additional lateral velocity on the compressed fluid

particles. Once the shock arrives at the point x, at the time t0(x) = x/(D U), thevorticity that corresponds to the fluid element at that point will remain frozen to that

fluid particle. We define the dimensionless vorticity for our 2D problem as:

(x, y) =2kyc2

= g (kyx)sin(kyy) , (2.69)

where the function g is given by:

g (kyx) =

vy

kyx

t=t0(x)

+ (vx)t=t0(x) . (2.70)

Both terms in the previous equation are evaluated just behind the shock front position,

at the time t0 when the shock arrives to that position. The space derivative of the

tangential velocity can be obtained by taking the time derivative of Eq.(2.15) following

the shock trajectory. We get:

vy

kyx

x=xs

=

R

1 + k2xk2y

1

cos(Rkxx) +

+

M21 1

R 2M21

2M21 M2

ps (x, t0(x)) , (2.71)

where use has been made of the linearized Rankine-Hugoniot conditions [Eqs.(2.12)-

(2.15)]. After some additional algebra, we obtain:

g (kyx) = 1 cos(Rkxx) + 2 ps

rs =

kyx

1 M22M2

, (2.72)

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and Eqs.(2.9) and (2.10) can be rewritten in the form:

2v

2= 2D [g(x)sinh(kyy) z] + 22Dv , (2.92)

where the product symbol indicates vector product and the operator 22D is definedby:

22D =2

(kyx)2 +

2

(kyy)2 . (2.93)

Evaluating the xcomponent of (2.92), we get:2vx2 = 22Dvx + g(x)cos(kyy) . (2.94)

The solution to this last equation can always be put in the form:

vx(x,y ,t) vacx (, x)cos(kyy) + u(kyx)cos(kyy) , (2.95)

where the function vacx satisfies the homogeneous wave equation in 2D:

2vacx2

=2vacx

(kyx)2 vacx . (2.96)

The function u(kyx) satisfies (2.79) and is also formally given by (2.84) in the short

wavelength regime. Summing up, Eq.(2.95) expresses the fact that the velocity field isdecomposed in two parts: a steady rotational contribution, which is also divergence-

free and has been calculated in the last subsection, plus a second term which does not

add vorticity but travels in the compressed fluid with the local sound speed. When

0 < 1, this potential contribution is actually an evanescent sound wave that vanishes

asymptotically in time like t3/2 at any position x, and it is only the rotational partu(kyx)cos(kyy) that remains as a permanent velocity perturbation when the shock is

far away. On the other hand, when 0 > 1, the solutions to (2.96) add a non zero

contribution to vx consisting of constant amplitude oscillations, for t t0(x). Theseoscillations travel as planar fronts with a definite orientation in space, that is a functionof0. In fact, in accordance to the previous discussion, the function v

acx can be written,

asymptotically in time, as:

vacx (x, t t0) = Qac cos

kyx

21 1 + 1

, (2.97)

where Qac has been defined in (2.64). Eq.(2.97) can be verified substituting it into

Eq.(2.96). Further, we recognize the wave-number vector of the traveling sound wave,

defined bykac = (k

acx , ky) , (2.98)

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where |kacx | = ky21 1. The sonic front described by (2.97) actually represents thesuperposition of two planar waves running in opposite directions along the yaxis,each of which has a ray inclined with respect to the xdirection through an angleac. According to (2.59), the value of ac is given by:

tan ac =kykacx

=

1 M22

M20

20 1. (2.99)

To conclude this subsection, we write both components of the total velocity field when

0 1, for x y and t t0(x):

vx(x, t) = Qsrot cos (Rkxx) + Qac cos(kacx x 1) ,vy(x, t) = Qsrot

Rkxky

sin(Rkxx) Qac

1 M2220 1 M20

sin(kacx x 1) , (2.100)

where Qac is given by (2.64) and Qsrot is given by (2.90). When 1 0 < 1/

1 M22 ,

kacx > 0 and the waves travel to the right. For 0 > 1/

1 M22 , kacx < 0 and theymove to the left. If 0 = 1/

1 M22 , the waves that escape from the shock actually

travel along the ydirection in the frame co-moving with the compressed fluid. Toillustrate Eq.(2.100), we show the normal velocity (vx) for a short wavelength case in

Fig.2.6(b) for a case in which the sound waves emitted by the front fill the whole space

downstream. The velocity curves are shown quite far from the weak discontinuity at

x = 0, and enough time has passed such that the shock entered its own asymptotic

regime, oscillating with the dimensionless frequency 0

1 M22 . We show the normalvelocity profiles given by vx in Eq.(2.100), at different values of the dimensionless time

. The dashed vertical lines indicate the instantaneous shock front positions at the

corresponding values of . We can see that the velocity field is a superposition of an

average steady profile [corresponding to up in Eq.(2.89)] plus an oscillation driven by

the sound waves in Eq.(2.97).

2.2.7 Density perturbations generated downstream

As can see from the Rankine-Hugoniot Eqs.(2.12)-(2.15), density perturbations are

generated behind the shock. In linear theory we decompose the density field as the

sum of entropic and acoustic contributions:

(kyx, ) = en(kyx) + ac(kyx, ) . (2.101)

The dimensionless acoustic fluctuation is directly given by Eq.(2.57):

ac(kyx, ) = p(kyx, ) , (2.102)

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Each choice of the incidence angle defines a set of shear waves in front of the shock(the elements within this set have different values of the angle ). It is also easy to

see that the intervals 0 /2 and /2 are physically indistinguishable.Besides, the value 0 = 1 defines a critical value for the incidence angle of the shear

wave upstream, which we call cr. For angles of incidence greater than cr, the pressure

perturbations behind the shock front are evanescent, as they would be associated to

values 0 < 1. On the other hand, for < cr, the shock oscillates and radiates traveling

sound waves downstream, in agreement with previous works. The value of cr is given

by:

sin cr = M21

+ 12M41 + (3 ) M21 2

. (2.126)

Furthermore, we note that the dimensionless velocity factor of the upstream field u1

defined for the single mode 2D problem before, is changed into v1k sin sin for the 3D

problem. The normalization v1k = v1k/c2 is understood, and the sub-index k indicates

the dependence of the quantity v1 on k = |k|.

2.3.3 Interaction of a shock wave with a 2D turbulent velocity field

Before facing the interaction of a shock wave with a 3D isotropic random density field,

it is convenient to study the simpler 2D case (see Fig.2.11). With this idea, in this

section we compute the averages for the kinetic energy, sonic energy flux, density and

vorticity behind the shock. Although this case may be considered non realistic, it is

Figure 2.11: Shock isotropic vorticity field interaction in 2D.

a convenient reference problem that might help with the understanding of numerical

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2.3 Shock compression of turbulent 2D/3D random vorticity fields

simulations. Rotman (19) studied this situation with the help of large scale eddysimulations (LES), in which he considered the interaction of a planar shock with two

types of upstream inhomogeneities: vorticity and density. We concentrate here on the

velocity fluctuations as the density case will be the subject of next chapter. In our

model, this 2D problem may be thought of as if the wave number vector k had only

components kx and ky and simultaneously were vz = 0. This case is a particular subset

corresponding to the choice: = /2 and 0 . This gives us a velocity fieldcontained in the {x, y} plane and a vorticity vector only directed along the zaxis.Assuming that the wavenumber k = (kx, ky) is uniformly distributed along the unit

semicircle, it is easy to see that the probability of a particular orientation is just d/.

2.3.3.1 Average kinetic energy of the downstream perturbations

Before proceeding, it is necessary to define the non dimensional velocities. For the

downstream rotational perturbations, we define:

vrot2x = c2u1vrot2x ,

vrot2y = c2u1vrot2y = c2u1R

kxky

vrot2x , (2.127)

and for the sound waves:

vac2x = c2u1vac2x ,

vac2y = c2u1vac2y . (2.128)

In general, the amplitudes of the transformed velocities and of the sound wave pressure

perturbations downstream will be complex numbers with real and imaginary parts,

because the input to the shock equations is chosen as a complex exponential. However,

it is not difficult to realize that the absolute value of the complex amplitudes are the

same as the absolute values of the 2D downstream velocities obtained in the previous

section 2.2. We note that the small dimensionless amplitude is u1 = v1k sin , afterusing (2.121) with = /2. We can therefore write, for the rotational part:

|vrot2x | = |Qrot (, M1, 0) |v1k sin ,|vrot2y | = R

kxky

|Qrot (, M1, 0) |v1k sin . (2.129)

The function Qrot is defined with the aid of (2.88) and (2.90). Similarly, we have for

the sound waves emitted downstream:

|vac2x

|=

|Qirrot (, M1, 0)

|v1k sin ,

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|vac2y | = |Qirrot (, M1, 0) |21 1

v1k sin ,

vac2z = 0 , (2.130)

where 1 and Qac have been defined in (2.58) and (2.66), respectively. The rotational

kinetic energy of the compressed fluid particles (per unit mass) is:

Trot2D =1

2

vrot2x

2+

vrot2y2

=

=v21k

2 1 + Rkxky

2

|Qrot|2 sin2 , (2.131)

and the sound waves contribution is:

Tac2D =1

2

(vac2x)

2 +

vac2y2

=v21k

2

2121 1

Q2ac sin2 . (2.132)

It is noted that the functions defined above can be thought either as functions of or

0, through the relationship that couples 0 and given by (2.125). The amplification

coefficient is defined as the ratio between the averaged downstream kinetic energy and

the pre-shock average kinetic energy per fluid particle:

K2D =4

v21k

/2

0

Trot2D + T

ac2D

d , (2.133)

where we have used the fact that the intervals 0 /2 and /2 arephysically equivalent for the calculation of the kinetic energy. The last integral is

better decomposed as:

K2D = Kl2D + K

s2D + K

ac2D , (2.134)

where, expressing each term as integrals over 0, we can write:

Kl

2D =

4

v21k10

Trot2D M31 R

3/2M21 1RM21 + M21 1 202 d0 ,Ks2D =

4

v21k

1

Trot2D M31 R

3/2

M21 1RM21 +

M21 1

202 d0 ,

Kac2D =4

v21k

1

Tac2DM31 R

3/2

M21 1RM21 +

M21 1

202 d0 , (2.135)

in which Kl2D stands for the integral of the rotational velocities over the long wavelength

interval 0 0 1, Ks2D stands for the rotational integral over the short wavelengthinterval 1

0 1, the amplification saturates at the value given by Eq.(2.241), but

for = 1 it grows without limit like:

W3D ( = 1, M1 1) = 2M41

5+

2M315

+ O

M21

. (2.248)

In Fig.2.39, we show a 2D map of W3D as function of and M1. We see that the

vorticity grows unbounded in the limits of very strong shock and highly compressible

gases in agreement with the previous equations and the graphs shown in Fig.2.24(a)

and Fig.2.24(b). The black curve corresponds to a shock moving into air ( = 7/5).The weak shock limit for arbitrary values of is given by:

W3D(, M1 1 1) = 1 + 8+ 1

(M1 1) + O

(M1 1)2

. (2.249)

2.3.5 Anisotropy of the downstream perturbations

One of the questions that we may try to answer using the model presented here regards

the isotropy of the compressed turbulent field. Being that there is a privileged direction

in space, given by the direction of the shock motion, it seems natural to think that this

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2.3 Shock compression of turbulent 2D/3D random vorticity fields

104

103

102

101

100

101

102

100

101

102

103

104

105

106

107

108

W3D(M1 1)W3D(M1 = 20)W3D(M1 = 10)

1 (a)

102

101

100

101

102

100

101

102

103

104

105

106

107

108

W3D( = 1)W3D(( = 1.001)W3D( = 1.01)

M1 1 (b)

Figure 2.40: (a): Vorticity amplification a function of for M1 1 and different strongshock strengths. (b): Relative acoustic flux as a function of M1 for different values of in

the highly compressible limit.

fact may modify the isotropy of the upstream perturbations to some degree, which could

be worth quantifying. Different choices of the anisotropy parameter have been used

in the existing literature, which are actually ad hoc definitions aimed to characterize

the level of anisotropy of the perturbations downstream. We define, with the same

liberty, the following parameter:

v =v2 2v2

v2 + 2v2= 1

4

v2xv2 + v2x

. (2.250)

Here, v refers to the velocity component parallel to the shock surface, and v is actuallyequal to vx. The anisotropy parameter varies between -1 and 1, and a zero value implies

isotropy downstream. The upper limiting value of v = 1 is actually reached in the

limit of strong shock and high compressibility, 1, M1 , where we know thatthe longitudinal motion of the compressed particles is suppressed. In this limit, the

velocity vectors of the compressed fluid elements would be inclined towards the shock

surface. In Fig.2.41 we show the parameter v as a function of and M1. We see that

there are no situations for which this parameter might reach the limit v =

1, which

would correspond to a completely longitudinal velocity field. This is quite natural,

as we know that 1/2 of the kinetic energy (with components parallel to the shock

surface) passes unchanged for any value of and any shock strength. In between both

limits, we have the possibility of making v = 0, corresponding to isotropic velocity

perturbations downstream. This condition defines an isotropy curve (besides the trivial

curve that is drawn at M1 = 1), which is shown in Fig.2.42. Interestingly, we note the

existence of an asymptote of this curve for very strong shocks, which defines a critical

value = 2.3646. For fluids with > , there is no shock wave that leaves thevelocity perturbations isotropic during compression. If the gas under study is air, we

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Figure 2.41: Downstream velocity anisotropy parameter [Eq.(2.250)] as a function of

and M1.

would instead find regions of lateral and longitudinal anisotropy. As can be seen, for

a shock moving into air with M1 = 1.9332, the isotropy of the velocity perturbationswill not be altered. Having defined an anisotropy parameter for the velocity field, it is

1 1.9332 5 10 20 50 100 200 5001

1.5

2

2.5

3

= 7/5

= 2.3646

Lateral anisotropyv < 0

M1

Longitudinal anisotropyv > 0

v = 0

Figure 2.42: Downstream velocity-isotropy curve, defined by the condition: v = 0 we

also show the regions of longitudinal and transverse anisotropy.

straightforward to do similarly with the vorticity perturbations. We have already seen

in Eqs.(2.123) and (2.124) that isotropy of the upstream velocities implies isotropy of

the vorticity perturbations. As the shock may substantially change the isotropy of the

velocity field, it is natural to ask whether the vorticity field isotropy is modified to a

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2.4 Summary

similar extent. Similarly as in Eq.(2.250), we define a downstream vorticity anisotropyparameter:

=2 222 + 22

=W3D 1W3D + 1

. (2.251)

We know that W3D in the limit of very strong shocks and highly compressiblefluids (M1 1, 1), which makes = 1 in that limit. In fact, in that case,the vorticity is essentially parallel to the shock front, as discussed previously. Besides,

Figure 2.43: Downstream vorticity anisotropy parameter [see Eq.(2.251)] for different

values of and M1.

we know that the transverse vorticity amplification factor is always greater than 1

for any values of and M1. Hence, it will be always > 0, which means that the

vorticity downstream is always essentially laterally anisotropic. In Fig.2.43 we can see

the behavior of on the (, M1) plane.

2.4 Summary

In the shock/vorticity interaction, each shear wave defines a plane on which rota-

tional/incompressible velocity perturbations are prescribed, with a given wave-number

vector k. The shear wave plane can be described in space in terms of the angles that

specify the orientation of the vector k in front of the shock. By properly rotating the

coordinate axis, to exploit the isotropy of the upstream perturbations, all the important

statistical averages can be easily calculated integrating over the solid angle defined byk. In this way, explicit analytic expressions in terms of elementary functions of and

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3

Shock entropy interaction

3.1 Introduction

In this chapter, we solve the interaction of a planar shock wave with a linear field

of isotropic density perturbations. The mathematical formalism is the same as the

one used in Chapter 2 for the shock vorticity interactions. The pre-shock density

fluctuations are assumed to be much smaller than the background density, which is

a reasonable hypothesis in ICF targets. We can decompose the pre-shock density

spectrum in its Fourier modes and study the interaction of the shock with a single-

mode density field. Later, a superposition is carried out to study the effect of the

shock compression over the whole spectrum. Assuming adiabatic conditions, there is

no energy exchange among upstream modes and the different modes can be considered

uncorrelated. Statistical averages are obtained by integrating over the angles that

define the orientation of the pre-shock perturbation wavenumber vector in space, as

done in Chapter 2. We will restrict our results to isotropic spectra in front of the

shock. In section 3.2, the 2D single-mode shock interaction theory is developed, and

we show the exact shock dynamics with its corresponding asymptotic expressions. The

RMI ensuing at the weak discontinuity (x = 0) is also considered. In section 3.3, the2D isotropic random density field is discussed. We obtain exact expressions for the

kinetic energy and the vorticity downstream fluctuations. The density and internal

energy amplification behind the shock wave as well as the acoustic energy flux emitted

by the shock front are calculated exactly. These quantities are simplified to closed

analytical expressions for the interesting limits: very strong shocks (M1 1) andhighly compressible gases (1 1). In section 3.4, we particularize all the expressionsobtained in the previous section 3.3 for the 3D pre-shock density field. Besides, for the

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3. SHOCK ENTROPY INTERACTION

3D problem we characterize the anisotropy of the downstream velocity perturbations.In section 3.5, we summarize the main conclusions. All the quantities calculated in

this work can be explicitly worked out as closed form analytical expressions in terms

of elementary functions, for any values of and M1. These equations can be found in

the EPAPS file of (92).

3.2 Interaction of a planar shock with a single mode 2D

density field

3.2.1 Wave Equation and Boundary Conditions

We consider the interaction of an isolated shock wave with a single mode 2D den-

sity perturbed field. In Fig.3.1, a schematic plot of the problem is presented, where

Fig.3.1(a) refers to a traveling shock wave before it hits the interface (x = 0). Wechoose the time origin such that the shock is incident at t = 0 at the surface x = 0in the laboratory frame. The fluid is an ideal gas with adiabatic exponent . The

shock comes from the left (x = ) and travels with velocity D x, as measured inthe laboratory frame. In the uniform half-space x < 0, the density and pressure aheadthe shock are 1, p1 respectively, and 2, p2 are the values behind it. The velocity

of the compressed fluid is Ux, also measured in the laboratory. The upstream soundspeed is c1, and its downstream value is c2. The shock Mach number with respect to

the upstream gas is M1 = D/c1 1 and the shock Mach number with respect to thecompressed fluid is M2 = (D U)/c2 1. We denote the upstream values with thesubscript 1 and the downstream values with the subscript 2. Before the shock arrives

at the interface x = 0, the quantities at both sides of the shock front are related byEqs.(2.1)-(2.4). The relationships that mach the quantities across the shock are the

same at the zeroth order, independently on the kind of perturbation. In the right half-

space, the perturbed density field is described by 1(x, y) = 1k cos(kxx)cos(kyy)where x and y are the longitudinal and transverse coordinates as measured in the lab-oratory system of reference. To remain within the limits of validity of the linear theory,

we assume k 1, where k is a function of k =

k2x + k2y, as isotropy is assumed

for the pre-shock perturbations. The longitudinal and transverse wave numbers are

defined, respectively by: kx = 2/x and ky = 2/y, with x and y characteristic

wavelengths of the upstream perturbations. Once the shock is in the half-space x 0,the density profile in front of it will induce density, pressure and velocity fluctuations

downstream, and its shape will be distorted [see Fig.3.1(b)]. We define the followings

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3.2 Interaction of a planar shock with a single mode 2D density field

dimensionless perturbation functions, factoring out the small parameter k:

22

= k(x, t) cos(kyy) ,

p22c

22

= k p(x, t)cos(kyy) ,

v2xc2

= kvx(x, t)cos(kyy) ,

v2yc2

= kvy(x, t)sin(kyy) ,

2zkyc2 =

kz sin(kyy) . (3.1)

In Eq.(3.1) t is the time, and x is the longitudinal coordinate as measured in the com-

pressed fluid frame. From now on, the linearized equations of motion will be written

and solved in a system of reference that co-moves with the compressed fluid particles.

The quantities vx and vy correspond to the longitudinal and transverse velocities re-

spectively, and and p represent the dimensionless density and pressure perturbations.

We also define the dimensionless time = kyc2t. At t = 0+, a shock is transmitted to

the right into the perturbed half-space and a neutrally stable sound wave is reflected

back inside the region x < 0, as shown in Fig.3.1(b). As the shock wave travels in anon-uniform fluid, the shock will be corrugated. We define the shock ripple amplitude

s(y, t) as its deviation from planarity. The shock ripple oscillates in time, generating

pressure fluctuations that propagate with the local sound speed into the compressed

fluid. The sound waves generated by the shock oscillation can be stable or evanescent

waves, depending on the ratio kx/ky , the shock Mach number M1 and the gas com-

pressibility (44). As a boundary condition, we assume that no sound wave hits the

shock surface from behind, that is, the shock travels isolated (91, 92, 94). At the sur-

face x = 0, pressure and normal velocity are continuous on both sides of it. However,

the distorted front generates vorticity and entropy perturbations and the neutral soundwave to the left does not. Hence, the x-derivative of vy and are generally discontin-

uous at x = 0. It is not difficult to get the relationships just to the right of the left

traveling sound wave [See. Eq.(2.11)]. The boundary conditions at the shock are ob-

tained after linearizing the Rankine-Hugoniot (RH) conditions and using the continuity

of the tangential velocity. We write them here for the particular case of density/entropy

pre-shock modulation at constant pressure and zero velocity perturbation ahead of the

shock:dsd

=+ 1

4M2p M2R

2k , (3.2)

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Figure 3.1: (a) The incident planar shock travels with velocity Dx in the laboratory

reference frame for t < 0. (b) The transmitted corrugated shock moves into the disturbed

fluid at the compressed fluid frame for t > 0 with velocity (D U)x. Density, vorticityand acoustic fluctuations are generated behind it. A stable sound wave is reflected to the

left and it travels with velocity c2 in this frame.

vx =M21 + 1

2M21 M2p M2(R 1)

2k , (3.3)

vy = M2(R 1)s , (3.4)

=1

M21 M22

p + k , (3.5)

where Eq.(3.2) re

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