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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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2.2 Shock wave traveling in a 2D single mode vorticity field
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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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2. SHOCK VORTICITY INTERACTION
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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 Shock wave traveling in a 2D single mode vorticity field
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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2. SHOCK VORTICITY INTERACTION
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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. SHOCK VORTICITY INTERACTION
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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2. SHOCK VORTICITY INTERACTION
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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2.2 Shock wave traveling in a 2D single mode vorticity field
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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2. SHOCK VORTICITY INTERACTION
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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2. SHOCK VORTICITY INTERACTION
|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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3. SHOCK ENTROPY INTERACTION
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