NUMERICAL STUDY OF VARIABLE DENSITY FLOW INTERACTION WITH A SHOCK

WAVE

By

Yifeng Tian

A DISSERTATION

Michigan State University

in partial fulﬁllment of the requirements

Submitted to

for the degree of

Mechanical Engineering – Doctor of Philosophy

2019

ABSTRACT

NUMERICAL STUDY OF VARIABLE DENSITY FLOW INTERACTION WITH A SHOCK

WAVE

By

Yifeng Tian

Fundamental understanding and modeling of multi-ﬂuid miscible Shock-Turbulence Interaction
(STI) and the corresponding post-shock turbulence are critically important to many diﬀerent ap-
plications, such as supersonic combustion, nuclear fusion, and astrophysics. This thesis presents a
comprehensive study of the multi-ﬂuid Shock-Turbulence ﬂow using accurate ﬂow-resolving, shock-
capturing, and shock-resolving simulations. The objective is to develop a better understanding of
underlying mechanisms of the variable density ﬂuid eﬀects on shock-turbulence interactions, post-
shock turbulence evolution and mixing in high speed ﬂows. Theoretical and numerical analyses of
data conﬁrm that all turbulence scales as well as the STI are well captured by the computational
method, which is based on a high order hybrid monotonicity preserving-compact ﬁnite diﬀerence
scheme. Linear Interaction Approximation (LIA) convergence tests are conducted to show that
shock-capturing numerical simulations exhibit similar converging trend to LIA predictions as more
demanding shock-resolving Direct Numerical Simulations (DNS) method.

The eﬀects of density variations on STI are studied ﬁrst by comparing the “Eulerian” results
obtained from both Eulerian (grid) and Lagrangian (particle) data for a multi-ﬂuid mixture with
the corresponding single-ﬂuid results. The comparison shows that the turbulence ampliﬁcation
by the normal shock wave is much higher and the reduction in turbulence length scales is more
signiﬁcant when strong density variations are present. Turbulent mixing enhancement by the shock
is also increased and stronger mixing asymmetry in the post-shock region is observed when there
are signiﬁcant density variations. The dominating mechanisms behind STI inﬂuence on post-shock
turbulence and mixing are identiﬁed by analyzing the transport equations for the Reynolds stresses,
vorticity, normalized mass ﬂux, and density speciﬁc volume covariance.

Statistical analyses of the velocity gradient tensor (VGT) show that the density variations also
signiﬁcantly change the turbulence structure and ﬂow topology. Compared to the single-ﬂuid case,
the correlation between rotation and strain is found to be weaker in the multi-ﬂuid case, which
is shown to be the result of complex role density plays when the ﬂow passes through the shock.
Furthermore, a stronger symmetrization of the joint PDF of second and third invariants of the
anisotropic velocity gradient tensor, as well as the PDF of the vortex stretching contribution to the
enstrophy equation, are observed in the multi-ﬂuid case. Lagrangian dynamics of the VGT and
its invariants are studied by considering particle residence times in diﬀerent ﬂow regions and the
conditional mean rate of change vectors. The pressure Hessian contributions to the VGT invariants
transport equations are shown to be strongly aﬀected by the shock wave and local density, making
them critically important to the ﬂow dynamics and turbulence structure. Lagrangian statistics of
non-inertial particles in post-shock turbulence show that the single-particle dispersion rate and
anisotropy can be correlated to the development of post-shock Reynolds stress. Lagrangian integral
time scales of ﬂuid particles with diﬀerent densities are obtained from velocity autocorrelation
functions. Particle pair dispersion generally follows the temporal scaling for isotropic turbulence.
In this thesis, the propagation of shock waves in non-uniform density media is also studied.
The 1D numerical results are shown to agree well with theoretical solutions obtained from classical
Chisnell-Whitham theory for weak shocks and linearly varying density ﬁelds. For more signiﬁcantly
varying density proﬁles, the numerical results deviate from the theoretical solutions and exhibit
additional long-wavelength oscillations, which are shown to be related to the re-reﬂected waves.
A simpliﬁed model (named CWRW) that includes the eﬀects of re-reﬂected waves is proposed to
compensate for the CW model deﬁnitions. The results obtained by the proposed CWRW model
show signiﬁcant improvement over the classical CW model. The three-dimensional (3D) eﬀects
concerning the shock propagation in 3D non-uniform density media are studied by considering
the wavenumber ratio of the streamwise and spanwise length scales in the density ﬁeld. The
asymptotic limits for wavelength ratio approaching zero and inﬁnity are investigated and used to
provide a bound for shock propagation in 3D non-uniform media.

Copyright by
YIFENG TIAN
2019

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest gratitude to my advisor Dr. Farhad Jaberi
and my co-advisor Dr. Daniel Livescu for their invaluable guidance and support through my Ph.D.
study. This work would not be possible without their patience and consistent encouragement. It
has been a truly wonderful experience to work with Dr. Jaberi and Dr. Livescu.

I would also like to thank my Ph.D. committee members Dr. Ahmed Naguib and Dr. Mohsen
Zayernouri for their constructive suggestions and comments on my research. The courses they
provided have been very helpful and prepared me with fundamental knowledge of compressible
ﬂow and numerical methods. Additionally, I would like to acknowledge Dr. Zhaorui Li for his
guidance and suggestions on the code development.

Financial support for this work was provided by Los Alamos National Laboratory, under Grant
No. 319838. Computational resources were provided by the High Performance Computing Center
at Michigan State University and Texas Advanced Computing Center (TACC) at The University of
Texas at Austin.

I would also like to thank my former colleague Dr. AbdoulAhad Validi and Dr. Husam
Abdulrahman who helped me start my research at the CFDLab. They have given me many helpful
pieces of advice on research over the past years. I would like to thank all of my beloved friends
for their emotional support and involvement in various academic and extracurricular activities with
me. Last but not least, I would like to thank my express my deepest appreciation and gratitude to
my parents and my girlfriend. This dissertation would not have been possible without their warm
love, continued patience, and endless support.

v

TABLE OF CONTENTS

LIST OF TABLES .
LIST OF FIGURES .
x
KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . xix

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

ix

.
.

.
.

.
.

.
.

.
.

.
.

.
.

.
.

CHAPTER 1 GENERAL BEHAVIOR OF VARIABLE DENSITY TURBULENCE IN-

.

.

.

.

.

.

. .

Introduction .

TERACTION WITH A NORMAL SHOCK WAVE . . . . . . . . . . . . .
1.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Governing Equations and Numerical Methodology . . . . . . . . . . . . . . . . . .
1.3 Simulations Setup and Parameters
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Results and Discussions .

1
1
4
7
9
1.4.1 Accuracy of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Convergence to LIA for Single-ﬂuid Case . . . . . . . . . . . . . . . . . . 14
1.4.3 Eﬀects of Density Variations on STI . . . . . . . . . . . . . . . . . . . . . 19
1.4.4 Reynolds Stresses .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.5 Vorticity Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.6 Mixing .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.4.7 Normalized Mass Flux and Density Speciﬁc Volume Covariance . . . . . . 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.5 Conclusions .

. .

.

.

.

.

.

.

.

.

.

.

CHAPTER 2 DENSITY EFFECTS ON THE POST-SHOCK TURBULENCE STRUC-

.

.

.

. .

Introduction .

2.1
.
2.2 Numerics and Accuracy .

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

TURE AND LAGRANGIAN FLOW DYNAMICS . . . . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. 55
2.2.1 Governing Equations and Numerical approach . . . . . . . . . . . . . . . . 55
2.2.2 Numerical Setup of Eulerian Simulation . . . . . . . . . . . . . . . . . . . 56
2.2.3 Numerical Detials of the Lagrangian Analyses . . . . . . . . . . . . . . . . 56
2.2.4 Grid and Statistical Convergence . . . . . . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3 Results and Discussions .

.

2.3.1 Density Eﬀects on the Post-shock Turbulence and Its Evolution Away

.

From the Shock .
2.3.1.1
2.3.1.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Turbulence State Immediately After the Shock . . . . . . . . . . 62
Evolution of Turbulence State Away from the Shock Wave . . . . 67
2.3.2 Topological Analysis of the Post-shock Turbulence . . . . . . . . . . . . . 74
2.3.3 Lagrangian Dynamics of VGT . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3.4 Lagrangian Particle Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 94
Single Particle Statistics . . . . . . . . . . . . . . . . . . . . . . 94
Particle Pair Statistics
. . . . . . . . . . . . . . . . . . . . . . . 98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
.
.
CHAPTER 3 EFFECTS OF DENSITY VARIATIONS ON A NORMAL SHOCK WAVE . 106

2.3.4.1
2.3.4.2
2.4 Conclusions .

. .

.

vi

.

.

Introduction . .

3.1
3.2 Numerical Methodology .

. .

.

.

.

3.4.1
3.4.2

3.2.1
3.2.2
3.2.3

3.3 Theoretical Analysis .

3.4 Results and Discussion .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
. 109
Shock-capturing Simulation . . . . . . . . . . . . . . . . . . . . . . . .
. 109
Shock-resolving Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Setup .
. 110
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
.
. 111
. 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Shock Wave Propagation in Media with Linearly Varying Density . . . .
. 120
Shock Wave Propagation in Media with Fluctuating Density . . . . . . . . 122
3.4.2.1

3.3.1 Linear Interaction Approximation . . . . . . . . . . . . . . . . . . . . .
3.3.2 Chisnell-Whitham Theory . . . . . . . . . . . . . . . . . . . . . . . . .

The Similarity of Density Fluctuations from Entropy and Com-
position Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4.2.2
Shock Propagation in Sinusoidal Density Media
. . . . . . . . . 124
3.4.2.3 CW Solution with Correction for Re-reﬂected Waves . . . . . . . 128
3.4.2.4
Shock Propagation in Random Density Media . . . . . . . . . . . 132
Shock Wave Propagation in Three-dimensional Non-uniform Density Field 133
3.4.3.1
. 135
. 139
3.4.3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.5 Conclusions .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3D Sinusoidal Density Field . . . . . . . . . . . . . . . . . . .
3D Random Density Field . . . . . . . . . . . . . . . . . . . .
.
.

3.4.3

. .

.

.

.

.

.

.

.

.

.

.

BIBLIOGRAPHY .

.

.

vii

LIST OF TABLES

Table 1.1: Details of the simulations used in the grid convergence tests.

. . . . . . . . . . . 10

Table 1.2: Cases considered for LIA convergence tests. . . . . . . . . . . . . . . . . . . . . 17

viii

LIST OF FIGURES

Figure 1.1: Scalar structure in multi-ﬂuid turbulence interacting with a Mach 2 shock
identiﬁed by the iso-surface of heavy ﬂuid mole fraction and colored with
instantaneous density ﬂuctuations. The black plane located in the middle
of the domain represents the instantaneous shock surface. The iso-surfaces
correspond to mole fraction values of (a) 0.05 (b) 0.95 (c) 0.25 and (d) 0.75.

. .

6

Figure 1.2:

Instantaneous contours of 3D pressure ﬂuctuations and shock surface resulting
from isotropic turbulence interacting with a Mach 2 shock. (a) Iso-surface
of heavy ﬂuid mole fraction (φ = 0.25) colored by instantaneous pressure
ﬂuctuations. Instantaneous shock surface colored by the ratio of local pressure
jump across the shock for (b) multi-ﬂuid A (see deﬁnition in section 1.4.3)
and (c) single-ﬂuid cases.

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

7

Figure 1.3: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09.
(a) Turbulent kinetic energy, (b) Kolmogorov length scale η, (c) streamwise
Taylor micro scale λ1 and (d) transverse vorticity variance ω(cid:48)
2ω(cid:48)
2. The region
. . . . . . . . . . .
of unsteady wrinkled shock movement is marked in grey.

. 11

Figure 1.4: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09. (a)

mass fraction variance φ(cid:48)φ(cid:48) and (b) Batchelor scale λB.

. . . . . . . . . . . . . 11

Figure 1.5: Relation between numerical shock width, δn, and grid size in the shock region.

12

Figure 1.6: Comparison of Reynolds stress obtained from single-ﬂuid simulations with
LIA results. Mt ≈ 0.24 except for Reλ = 45. (a) Streamwise Reynolds stress
and (b) transverse Reynolds stress.

. . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 1.7: Comparison of Reynolds stress obtained from multi-ﬂuid, single-ﬂuid simu-
lation with LIA results. Reλ ≈ 30. (a) Streamwise Reynolds stress and (b)
transverse Reynolds stress.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 1.8: 2D contours of instantaneous scalar and density ﬁelds for (a,b) multi-ﬂuid A
simulation and (c,d) single-ﬂuid simulation. The 2D contours are taken at the
same time step, in planes x-z.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 1.9: Plots of (a) turbulent kinetic energy, (b) Kolmogorov length scale, (c) Taylor
micro scale and (d) transverse vorticity variance for multi-ﬂuid A (red) and
single-ﬂuid (blue) simulations.

. . . . . . . . . . . . . . . . . . . . . . . . . . 22

ix

Figure 1.10: Plots of normalized (a) scalar variance, (b) Batchelor scale for multi-ﬂuid A

(red) and single-ﬂuid (blue) simulations.

. . . . . . . . . . . . . . . . . . . . . 22

Figure 1.11: Conditional expectation of scalar as a function of density at various stream-
wise location for : (a) multi-ﬂuid A and (b) single-ﬂuid cases. The conditional
mean is calculated using variables in a spanwise plane at certain streamwise
locations and then averaged over time. Peak TKE positions are diﬀerent
between multi-ﬂuid(k0x ≈ 2.0) and single-ﬂuid(k0x ≈ π) cases . . . . . . . . . 23

Figure 1.12: Conditional expectation of pressure as a function of density at various stream-
wise locations for : (a) multi-ﬂuid A close to shock, (b) multi-ﬂuid A at far
ﬁeld and (c) single-ﬂuid cases.

. . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 1.13: Conditional expectation of temperature as a function of density at various

streamwise location for : (a) multi-ﬂuid A case and (b) single-ﬂuid case.

. . . . 25

Figure 1.14: Conditional expectation of TKE as a function of density at various streamwise

location for : (a) multi-ﬂuid A and (b) single-ﬂuid case.

. . . . . . . . . . . . . 26

Figure 1.15: Conditional expectation of enstrophy as a function of density at various

streamwise location for : (a) multi-ﬂuid A and (b) single-ﬂuid case. . . . . . . . 27

Figure 1.16: Contributions of diﬀerent terms in the R11 transport equations for multi-ﬂuid
A simulation. (a) Plots of all the contributing terms in the transport equation
and (b) balance of LHS and RHS of the transport equation.

. . . . . . . . . . . 30

Figure 1.17: Contributions of diﬀerent terms in the R11 transport equations for multi-
ﬂuid A simulation averaged over 5 diﬀerent realizations (a) Plots of all the
contributing terms in the transport equation and (b) balance of LHS and RHS
of the transport equation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 1.18: Contributions of diﬀerent terms in the R22 transport equations for multi-
ﬂuid A simulation averaged over 5 diﬀerent realizations. (a) Plots of all the
contributing terms in the transport equation and (b) balance of LHS and RHS
of the transport equation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 1.19: Normalized contributions of diﬀerent terms in the R11 transport equations
averaged over 5 diﬀerent realizations for (a) multi-ﬂuid A and (b) single-ﬂuid
simulations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

. .

. .

Figure 1.20: Normalized contributions of diﬀerent terms in the R22 transport equations
averaged over 5 diﬀerent realizations for (a) multi-ﬂuid A and (b) single-ﬂuid
simulations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

. .

. .

x

Figure 1.21: Two dimensional spectra of transverse turbulent kinetic energy. The compar-

ison are made at right before shock wave and k0x ≈ 4.0.

. . . . . . . . . . . . 34

Figure 1.22: Streamwise and transverse vorticity variance for multi-ﬂuid and single-ﬂuid case 35
Figure 1.23: Contributions of diﬀerent terms in the transport equation for (a) ω(cid:48)
1ω(cid:48)
1 and

. . . . . . . . . . . . . . . . . . . . . . . 36

(b) ω(cid:48)
2ω(cid:48)
2 for multi-ﬂuid simulation.

Figure 1.24: Contributions of diﬀerent terms in the transport equation for (a) ω(cid:48)
1ω(cid:48)
1 and
. . . . . . . . . . . . . . . . . . . . . .

(b) ω(cid:48)
2ω(cid:48)
2 for single-ﬂuid simulation.

. 36

Figure 1.25: Vortex-stretching term in the enstrophy equation, normalized by ω2ω2 and

turbulence time scale TKE/

. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 1.27: Contributions of diﬀerent terms in the scalar variance(cid:103)φ
Figure 1.28: Contributions of diﬀerent terms in the(cid:103)φ

Figure 1.26: Plots of scalar Taylor micro scale for multi-ﬂuid (red) and single-ﬂuid (blue).
(cid:48)(cid:48)2 transport equations
(a) Plots of all the contributing terms in the
for multi-ﬂuid simulation.
transport equation and (b) balance of LHS and RHS of the transport equation.
(cid:48)(cid:48)2 transport equations for (a) multi-
. . . . . . . . . . . . . . . . . . . . . . . 42

ﬂuid and (b) single-ﬂuid simulations.

. 40

. 41

Figure 1.29: Comparison of scalar dissipation and advection terms obtained from multi-

ﬂuid and single-ﬂuid simulations.

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

. 43

Figure 1.30: 2D spectra of scalar. The comparison are made at various locations: (a) right
before shock wave and k0x ≈ 4.0 and (b) right before shock wave and peak
TKE position.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

. .

. .

Figure 1.31: PDFs of multi-ﬂuid A case: (a) scalar (mole fraction) and (b) density ﬂuc-
tuation at various locations. The PDFs are calculated at speciﬁed y-z planes
and then averaged over time.

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

. 45

Figure 1.32: PDFs of multi-ﬂuid B case: (a) scalar (mass fraction) and (b) density ﬂuc-
tuation at various locations. The PDFs are calculated at speciﬁed y-z planes
and then averaged over time.

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

. 45

Figure 1.33: PDFs of single-ﬂuid case: (a) passive scalar and (b) density ﬂuctuation at
various locations. The PDFs are calculated at speciﬁed y-z planes and then
averaged over time.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

.

xi

Figure 1.34: Plots of (a) normalized mass ﬂux a1 and (b) density speciﬁc volume covari-

ance − (cid:104)v ρ(cid:105) for multi-ﬂuid and single-ﬂuid simulations.

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

Figure 1.35: Contributions of diﬀerent terms in the normalized mass ﬂux a1 transport

equations for multi-ﬂuid simulation.

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

. 48

. 48

Figure 1.36: Contributions of diﬀerent terms in the density speciﬁc volume covariance b

transport equations for multi-ﬂuid simulation.

. . . . . . . . . . . . . . . . . . 49

Figure 2.1:

Instantaneous contours of vorticity and shock surface in isotropic turbulence
interacting with a Mach 2 shock. (a) Vortex structure, identiﬁed by the Q
criteria (Q = 2 (cid:104)Qw(cid:105)) and colored by the mole fraction of the heavy ﬂuid.
Fluid particles are initialized as a sheet that spans over the homogeneous
directions at a given post-shock streamwise position and allowed to develop
with the ﬂow. (b) Visualized particle sheet, convected and distorted by the
post-shock turbulence. The instantaneous shock surface is colored by the
shock intensity across the shock for (c) single-ﬂuid and (d) multi-ﬂuid cases.

. . 56

Figure 2.2: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09.
Streamwise development of (a) velocity dissipation rate ε and (b) mass frac-
tion dissipation rate εφ is shown. The region of unsteady shock move-
ment is marked in grey. Grid numbers for Grid 1 to 5 are 256×256×1024,
384×384×1024, 384×384×1536, 512×512×1536, 512×512×2048.

. . . . . . . 60

Figure 2.3: The statistical convergence for (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (DR/Dt)/(cid:104)Qw(cid:105)2
and (b) their standard deviations conditioned at point (3.0,3.0) in the (Q, R)
phase plane for multi-ﬂuid case. The number of bins are 30 × 30 (solid),
40 × 40 (dashed) and 60 × 60 (dotted).

. . . . . . . . . . . . . . . . . . . . . . 61

Figure 2.4: The statistical convergence for (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (DR/Dt)/(cid:104)Qw(cid:105)2
and (b) their standard deviations conditioned at point (3.0,3.0) in the (Q, R)
phase plane for single-ﬂuid case. The number of bins are 30 × 30 (solid),
40 × 40 (dashed) and 60 × 60 (dotted).

. . . . . . . . . . . . . . . . . . . . . . 61

Figure 2.5: Comparison of the PDFs of the normalized post-shock velocity derivatives

with a Gaussian distribution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 2.6: PDF of the strain-enstrophy angle Ψ in radian for post-shock turbulence.

. . . . 64

Figure 2.7: Conditional expectation of the magnitude of (a) rotation tensor and (b) strain

rate tensor as a function of density before and after the shock.

. . . . . . . . .

. 64

Figure 2.8:

(a) Demonstration of the baroclinic torque generation across the shock wave
and (b) PDF of the orientation between the vorticity vector and density gra-
dient in y-z direction immediately after the shock wave.

. . . . . . . . . . . . . 66

xii

Figure 2.9: Vortex structures captured using the Q-criterion, colored by density, for multi-

ﬂuid (a) pre-shock turbulence and (b) post-shock turbulence.

. . . . . . . . . . 67

Figure 2.10: Development of (a) TKE and dissipation, (b) pressure variance, (c) vortex

stretching, and (d) anisotropy (b11) of Reynolds stress and vorticity. . . . . . . . 69

Figure 2.11: Development of (a) skewness, and (b) ﬂatness, of the streamwise and trans-

verse components of velocity derivatives. . . . . . . . . . . . . . . . . . . . . . 71

Figure 2.12: PDF of the density gradient at diﬀerent streamwise locations for: (a) single-

ﬂuid case and (b) multi-ﬂuid case.

. . . . . . . . . . . . . . . . . . . . . . . . 71

Figure 2.13: PDFs of the cosine angle between eigenvectors of the strain rate tensor and
streamwise (x) axis for regions with (a) dρ/dx > 0 and (b) dρ/dx < 0 and
for multi-ﬂuid (solid lines) and single-ﬂuid (dashed lines) cases. . . . . . . . .

. 73

Figure 2.14: Iso-contour lines of joint PDFs of normalized second and third invariants
of the anisotropic part of the velocity gradient tensor, (Q∗,R∗), for (a) pre-
shock, (b) single-ﬂuid post-shock turbulence, and (c) multi-ﬂuid post-shock
turbulence. The lateral lines denote the locus of zero discriminant.

. . . . . . . 76

Figure 2.15: Iso-contour lines of post-shock (k0x ≈ 0.44) joint PDF of second and third
invariants of the anisotropic part of the velocity gradient tensor, (Q∗,R∗),
(a) regions with high density values,
in regions with diﬀerent densities.
ρ > (ρ + 90%ρ(cid:48)
rms), (b) regions with density around the post-shock mean
value, and (c) regions with low density values, ρ < (ρ − 90%ρ(cid:48)

rms).

. . . . . . 77

Figure 2.16: Color illustration of the ﬂow topology for the multi-ﬂuid STI. The ﬂow
topology is represented by the quadrants (denoted as Qi) of the joint PDF
of (Q∗,R∗).
(a) 2D color contours in the x-z plane at y = 3.14 (half
y-domain). The shock wave is located in the middle of the domain at
k0x ≈ 0.0. The ratio of the ﬂuid volume in diﬀerent quadrants in the pre-
shock region is Q1 : Q2 : Q3 : Q4=26.7%:38.7%:7.8%:26.8%. The 2D
color contours in the homogeneous y-z plane at streamwise locations of: (b)
k0x ≈ 0.2 (28.7%:34.4%:14.3%:22.6%), (c) k0x ≈ 2.0, peak TKE location
(26.7%:36.9%:11.2%:25.2%) (d) k0x ≈ 4.0 (26.3%:37.9%:9.3%:26.2%). . . . . 78

Figure 2.17: Iso-contour lines of joint PDF of (-Q∗

s , Σ∗) for (a) isotropic box turbulence
and (b,c) single-ﬂuid and multi-ﬂuid turbulence at post-shock position of
k0x ≈ 0.44.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

.

.

.

.

.

Figure 2.18: Iso-contour lines of joint PDF of (-Q∗

s , Σ∗) for diﬀerent quadrants right after

the shock wave. (a) Q2, (b) Q1 ,(c) Q3 and (d) Q4. . . . . . . . . . . . . . . . . 81

xiii

Figure 2.19: Percentage of ﬂuid particles that stay in each quadrant following particles
initialized uniformly in (a) isotropic turbulence and (b,c) single-ﬂuid and
multi-ﬂuid turbulence at post-shock position of k0x = 0.44.

. . . . . . . . . . . 83

Figure 2.20: Visualization of turbulence structure using iso-surfaces of Q∗ colored by R∗ in
the multi-ﬂuid post-shock turbulence. (a) Q1 and Q2 (vorticity-dominated re-
gion), and (b) Q3 and Q4 (strain-dominated region). The black lines represent
the instantaneous streamlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 2.21: Visualization of the temporal development (left to right) of the turbulence
structure using iso-surfaces of Q∗ colored by density for the multi-ﬂuid post-
shock turbulence. These structures are captured immediately after the shock
wave. (a) vorticity-dominated structure, and (b) strain-dominated structure.

. . 84

Figure 2.22: Contributions to the vortex stretching rate from particles starting in each
quadrant. The particles are initialized uniformly at the post-shock position
k0x ≈ 0.44 and traced downstream till the vorticity returns to an isotropic
state. (a) single-ﬂuid and (b) multi-ﬂuid cases.

. . . . . . . . . . . . . . . . . . 85

Figure 2.23: PDFs of (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (b) (DR/Dt)/(cid:104)Qw(cid:105)2 for ﬂuid particles

with diﬀerent densities at streamwise location of k0x ≈ 0.5.

. . . . . . . . . . . 87

Figure 2.24: Conditional mean rate of change vectors of (DQ/Dt/(cid:104)Qw(cid:105)3/2,DR/Dt/(cid:104)Qw(cid:105)2)
in the (Q, R) plane for (a) isotropic turbulence, (b) single-ﬂuid post-shock tur-
bulence, and (c) multi-ﬂuid post-shock turbulence at streamwise location of
k0x ≈ 0.5.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

.

.

.

.

.

.

.

Figure 2.25: Conditional mean vectors in the (Q, R) invariants plane for (a) light ﬂuid, (b)
medium density ﬂuid and (c) heavy ﬂuid at streamwise location of k0x ≈ 0.5.
Figure 2.26: Contributions to the transport equations of the VGT invariants by diﬀerent
terms for isotropic turbulence. (a) Mutual interaction among invariants, (b)
pressure Hessian term, (c) baroclinic term, and (d) viscous term.

. . . . . . . . 90

. 89

Figure 2.27: Contributions to the transport equations of the VGT invariants by diﬀerent
terms for single-ﬂuid post-shock turbulence. (a) Mutual interaction among
invariants, (b) pressure Hessian term, (c) baroclinic term, and (d) viscous term. . 92

Figure 2.28: Contributions to the dynamics of the VGT invariants by diﬀerent terms for
multi-ﬂuid post-shock turbulence. (a) Mutual interaction among invariants,
(b) pressure Hessian term, (c) baroclinic term, and (d) viscous term.

. . . . . . 93

Figure 2.29: Contributions from pressure Hessian to the dynamics of the VGT invariants
in (a) light ﬂuid region, (a) medium density ﬂuid region and (c) heavy ﬂuid
region.

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

.

.

.

.

.

.

.

.

.

. 94

xiv

Figure 2.30: Single particle dispersion rate normalized using t2 for both single and multi-

ﬂuid post-shock turbulence.

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

. 95

Figure 2.31: The temporal development of the skewness µ3 of the streamwise particle

displacement PDF. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 2.32: Lagrangian velocity autocorrelation function for particles staring immediately

after the shock wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Figure 2.33: Lagrangian velocity autocorrelation function for particles staring immediately
after the shock wave in the multi-ﬂuid STI simulations. The heavy ﬂuid
particles (dashed line) and light ﬂuid particles (dotted line) are shown for
comparison.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

.

.

.

.

.

Figure 2.34: Evolution of the pair particle dispersion rate normalized by t2 for both single-
ﬂuid (black) and multi-ﬂuid simulations (red). The direction of the initial
separation is in the spanwise direction with lyz = l0. The relative particle
dispersion in three direction are separated into streamwise direction (dotted
lines), longitudinal direction (solid line) and transverse direction (dashed
lines). The corresponding initial separation distance is (a) l0 = 0.25η, (b)
l0 = 4η and (c) l0 = 64η.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 2.35: Evolution of the pair particle dispersion rate normalized by t2 for both single-
ﬂuid (black) and multi-ﬂuid simulations (red). The relative particle dispersion
in three direction are separated into streamwise direction (dotted lines), lon-
gitudinal direction (solid line) and transverse direction (dashed lines). The
direction of the initial separation is in the spanwise direction with lx = l0.
The corresponding initial separation distance is (a) l0 = 0.25η, (b) l0 = 4η
and (c) l0 = 64η. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

.

.

.

Figure 2.36: Evolution of the pair particle velocity correlation for both single-ﬂuid (black)
and multi-ﬂuid simulations (red). Velocities in the streamwise direction (solid
lines), longitudinal direction (dashed line) and transverse direction (dotted
line) are compared separately. The direction of the initial separation is in
the spanwise direction with lyz = l0. The corresponding initial separation
distance is (a) l0 = 0.25η, (b) l0 = 4η and (c) l0 = 64η. . . . . . . . . . . . . . . 104

Figure 2.37: Evolution of the pair particle velocity correlation for both single-ﬂuid (black)
and multi-ﬂuid simulations (red). The direction of the initial separation is in
the spanwise direction with lx = l0. Velocities in the streamwise direction
(solid lines) and transverse directions (dotted and dashed lines) are compared
separately. The corresponding initial separation distance is (a) l0 = 0.25η,
(b) l0 = 4η and (c) l0 = 64η. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xv

Figure 3.1: Shock front in wrinkled shock regime (a,b) and broken shock regime (c).
Shock intensity is represented by density jump ∆ρ. Shock intensity ratio is
calculated as ∆ρ/∆ρ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

.

Figure 3.2:

(a) Demonstration of numerical setup of the 1D simulation for both shock-
(b) Space-time diagram of the
capturing and shock-resolving simulations.
1D shock propgation (thick solid line) in non-uniform density media. In the
meantime, a reﬂected wave/forward-going wave will be generated (thin solid
line), which will then interact with the post-shock non-uniform density media
at the particle path (dashed line). As a result, a re-reﬂected wave/backward-
going wave will be generated (dotted line), which will catch up with the shock
wave. .

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 3.3: Shock wave propagation in the medium with linearly varying density for

density ratios of 8 and 0.125. (a) M0 = 1.1 and (b) M0 = 4.0.

. . . . . . . . . . 122

Figure 3.4: Evolution of shock Mach number in linearly varying density ﬁeld for ρ1/ρ0 =

8 and 0.125. Initial shock Mach number M0 is 4.0. . . . . . . . . . . . . . . . . 123

Figure 3.5: Plots for ﬂow variables in streamwise directions for a Mach 3 shock propa-
gating in a sinusoidal shaped density ﬁeld. (a) Pressure, (b) density and (c)
temperature.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

. .

. .

Figure 3.6: Shock propagation in a ﬂuctuating density ﬁeld. (a,b) M0 = 2.0, density ratio

2.0, and ks = 1 and ks = 4 and (c) M0 = 1.1, density ratio 2.0, and ks = 4.

. . . 126

Figure 3.7: Comparison of the wavelength ratios between numerical results and theoret-

ical results. .

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 3.8: Comparison of the shock position ﬂuctuations between numerical results and
theoretical results. Results from shock-resolving simulations are maked using
symbols for Ms = 1.1 (black circle), Ms = 1.3 (black cross), Ms = 1.5 (black
star), Ms = 2.0 (red circle), Ms = 2.5 (red corss), Ms = 3.0 (red star).

. . . . . 128

Figure 3.9: Comparison of the relative magnitude zb/z f of the re-reﬂected waves.Results
from shock-resolving simulations are maked using symbols for At = 0.1
(black circle), At = 0.2 (black cross), At = 0.333 (black star), At = 0.4 (red
circle), At = 0.5 (red corss), At = 0.6 (red star), At = 0.9 (blue circle).

. . . . . 130

Figure 3.10: Demonstration of the catchup of the re-reﬂected waves.

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

. 132

Figure 3.11: Shock propagation in sinusoidally varying density ﬁeld. The temporal de-
velopment of shock position is compared for the shock-resolving simulation,
CW theory and the proposed CWRW theory. . . . . . . . . . . . . . . . . . . . 133

xvi

Figure 3.12: Shock propagation in random density media. Diﬀerent realizations of the
random density ﬁeld are represented using diﬀerent colors.(a) ks = 4.0 and
(b) ks = 8.0.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

.

.

.

.

.

Figure 3.13: Shock propagation in 3D sinusoidal shaped density media. The temporal
development of shock position on certain point on the shock surface is plotted.
kx = 4 for all the cases and ky = kz unless the values are indicated by the legend.136

Figure 3.14: The dependence of mean shock speed on wavenumber ratio for shock propa-

gation in 3D non-unform density media with kx = 4. . . . . . . . . . . . . . . . 137

Figure 3.15: The dependence of mean shock speed on wavenumber ratio for shock propa-

gation in 3D non-uniform density media with kx = 4.

. . . . . . . . . . . . . . 139

Figure 3.16: The dependence of L2 norm of the shock ﬂuctuations on wavenumber ratio

for shock propagation in 3D non-unform density media with kx = 4.

. . . . . . 140

Figure 3.17: The mean propagation of shock wave in 3D random density media. Red lines
represent points on the shock surface. The blue and black curves represent
the bounds at the two asymptotic limit.

. . . . . . . . . . . . . . . . . . . . . . 141

xvii

KEY TO SYMBOLS AND ABBREVIATIONS

Reλ Taylor Reynolds number
Ms Flow Mach number
Mt Turbulent Mach number
M0 reference Mach number
Pr Prandtl number

Sc Schmidt number
At Atwood number
MWi molecular weight of ith specie
µi chemical potential of ith specie
δ shock thickness
δn numerical shock thickness
η Kolmogorov length scale

ρ density
ui ﬂow velocity
p pressure

s entropy

E total energy

R speciﬁc gas constant

T temperatures

Y mass fraction of heavy ﬂuid
τi j, σi j viscous stress tensor
thermal diﬀusion vector
qi
qdi enthalpy diﬀusion vector
Ji mass fraction ﬂux vector
µ dynamic viscosity

xviii

Cp speciﬁc heat at constant pressure
Cv speciﬁc heat at constant volume
γ speciﬁc heat ratio
k0 peak wavenumber of the turbulence energy spectrum
λ Taylor micro scale
λφ scalar Taylor micro scale
λB Batchelor scale
ε dissipation
tp pass-over time
ωi vorticity
Rαα Normal Reynolds stress
ai normalized mass ﬂux
b density speciﬁc volume covariance
i particle position
x+
i particle displacement relative to the mean ﬂow
y+
i particle velocity
u+
Ai j velocity gradient tensor
Si j strain rate tensor
Wi j rotation tensor
θ dilatation

P ﬁrst invariant of the velocity gradient tensor

Q second invariant of the velocity gradient tensor

R third invariant of the velocity gradient tensor
Qw second invariant of the rotation tensor
Qs second invariant of the strain rate tensor
Ψ strain-enstrophy angle

Σ vortex stretching rate

xix

Hp
i j pressure hessian tensor
Hb
i j baroclinic tensor
Ti j viscous contribution tensor to VGT dynamics
l0 initial particle pair separation distance
lx initial particle pair separation in streamwise direction
lyz initial particle pair separation in spanwise directions
z wave intensity calculated by the pressure ratio across the wave
xs shock position
DNS Direct Numerical Simulation
RANS Reynolds Averaged Navier-Stokes
LIA Linear Interaction Approximation
STI Shock-Turbulence Interaction
TKE turbulent kinetic energy
RMI Richtmyer-Meshkov Instability
MP Monotonicity Preserving
VD variable density
RHS right hand side
LHS left hand side
IT isotropic turbulence
R-H Rankin-Hugoniot
ICF Inertial Conﬁnement Fusion
VGT velocity gradient tensor
CMT conditional mean trajectory
SFS stable-focus/stretching
UFC unstable-focus/contracting
SN/S/S stable-node/saddle/saddle
UN/S/S unstable-node/saddle/saddle
CW Chisnell-Whitham

xx

CHAPTER 1

GENERAL BEHAVIOR OF VARIABLE DENSITY TURBULENCE INTERACTION

WITH A NORMAL SHOCK WAVE

1.1 Introduction

The interaction between a normal shock wave and isotropic turbulence is an important funda-
mental problem that has been extensively studied. Understanding the physics behind this problem
is beneﬁcial to many applications such as hypersonic combustion, inertial conﬁnement fusion and
astrophysics. However, the existence of a wide range of length/time scales and other complicating
eﬀects in ﬂows involving both turbulence and shock waves have posed serious challenges on the
study of these ﬂows.

Ribner (1954) proposed a theoretical model for the description of Shock-Turbulence Interaction
(STI), in which Eulerian Rankine-Hugoniot equations were solved. In this theory, referred to as
Linear Interaction Approximation (LIA), the pre-shock turbulence was assumed to be made of
small amplitude disturbances, so nonlinear and viscous eﬀects were excluded from the analysis.
This allowed the changes of diﬀerent modes across the shock to be separately investigated. The
ampliﬁcation of turbulent kinetic energy (TKE) and the reduction of turbulence length scales by the
shock are successfully predicted by LIA and they are treated as the upper limits for the interaction
with nonlinear and viscous eﬀects. Early Direct Numerical Simulation (DNS) of shock-isotropic
turbulence by Lee et al. (1993) (ﬂow Mach number Ms (cid:54) 1.2) showed reasonably good agreement
with LIA for the vorticity ampliﬁcation when the turbulent Mach number, Mt, was kept small.
However, due to the low resolution of the simulations, quantities which peak behind the shock
were not compared to LIA. Jamme et al. (2002) and Mahesh et al. (1997) have also used DNS
to study the eﬀects of diﬀerent upstream ﬂow parameters on STI. Lee et al. (1997), Larsson and
Lele (2009), Larsson et al. (2013) have conducted turbulence-resolving simulations for diﬀerent
ﬂow Mach numbers using a high order shock-capturing scheme and found good agreement with

1

LIA results for some of the statistics, like turbulent kinetic energy and vorticity variance, but a
poor match of individual Reynolds stress components with LIA was observed. More recently, Ryu
and Livescu (2014) have conducted shock-resolving DNS for a wide range of parameters and have
shown that DNS results converge to LIA when the ratio of shock thickness, δ, to Kolmogorov
length scale, η, becomes small, which was achieved by decreasing Mt. This work has established
the reliability of LIA as a prediction tool for low Mt STI, even at low upstream Reynolds numbers.
LIA and shock-capturing simulation data have been used for a detailed study of the turbulent
energy ﬂux in the post-shock region by Quadros et al. (2016). Using LIA to alleviate the need to
resolve the shock allows studying of post-shock turbulence at arbitrarily high shock Mach numbers
(Livescu and Ryu, 2016). However, Shock-LIA studies (following the notation from Livescu
and Ryu, 2016) do not address the decay away from the shock or situations where the nonlinear
and/or viscous eﬀects can have an eﬀect on the interaction. On the other hand, shock-resolving
DNS studies are limited in the range of Reynolds and shock Mach numbers achievable by today’s
computers. Turbulence-resolving shock-capturing simulations could extend the parameter range
and ﬂow complexity (Jammalamadaka et al., 2014) achievable by shock-resolving DNS, provided
that the reliability of the tool can be established for such problems. This is also addressed in this
paper.

The early theoretical study of small amplitude turbulent ﬂuctuations conducted by Kovasznay
(1953) showed the existence of three diﬀerent components in compressible turbulence: vorticity,
acoustic and entropic. The eﬀect of upstream entropy ﬂuctuations on the shock-turbulence inter-
action was further studied by Mahesh et al. (1997) and Jamme et al. (2002). Their results show
that a negative correlation between the upstream velocity and temperature will cause a higher am-
pliﬁcation of turbulent kinetic energy and vorticity variance. The presence of acoustic ﬂuctuations
in the upstream ﬂow is found to cause less ampliﬁcation of turbulent kinetic energy but more
ampliﬁcation of transverse vorticity variance (Jamme et al., 2002; Mahesh et al., 1995). Hannappel
and Friedrich (1995) observed in their study that the turbulent ﬂuctuations of the transverse vor-
ticity increase more and those of streamwise vorticity increase less due to the interaction with the

2

shock wave when the inﬂow compressibility increases. Shock waves also enhance mixing (Menon,
1989; Kim et al., 2003), a well-known eﬀect used for increasing the fuel-air mixing in practical
combustion/propulsion systems (e.g. Budzinski et al., 1992; Yang et al., 1993).

In the above STI studies, the pre-shock turbulence is single-ﬂuid, so the density variations
are due to either thermodynamic or acoustic ﬂuctuations. In order to understand the process of
hypersonic combustion where strong density variations exist in the fuel-air mixture, the multi-ﬂuid
variable density eﬀects (i.e. due to compositional variations) should be taken into consideration.
In a hypersonic reacting ﬂow, the species mixing enhancement is aﬀected by the density variation
and expected to be diﬀerent from that of a passive scalar. For example, strong mixing asymmetry
exists in variable density turbulence (but not observed in single-ﬂuid ﬂow), as predicted by Livescu
and Ristorcelli (2008) and Livescu et al. (2010). Such mixing asymmetry is further ampliﬁed after
passing through the shock wave as shown in ﬁgure 1.1. For a system like this, where nonlinear
eﬀects become dominant due to the strong density variations, LIA becomes ineﬀective, leaving
high-order numerical simulation as the feasible approach. Li and Jaberi (2012) have recently
developed a new shock-capturing ﬁnite diﬀerence method based on a Monotonicity-Preserving
(MP) (Suresh and Huynh, 1997) scheme and have tested the method for diﬀerent supersonic ﬂows.
This work is based on their method, extended to multi-component ﬂows. Our previous study has
shown general characteristics of multi-ﬂuid STI (Tian et al., 2017d). The goal of current study is to
develop a better understanding of the density eﬀects on the shock-turbulence interaction and scalar
mixing. The conﬁguration addressed here, where the shock is stationary and the turbulence is fed
through the inlet of an open-ended domain, is also relevant to the Richtmyer-Meshkov Instability
(RMI) as it represents a statistically-stationary version of this problem. The classical RMI problem
describes the transient linear and the subsequent nonlinear growth of interface and mixing layer
(see Zabusky, 1999; Brouillette, 2002) , while the current study resembles the re-shock problem
where variable density eﬀects are important for the shock-driven mixing (e.g. Lombardini et al.,
2011). In the case where the upstream density ﬁeld is isotropic, converged statistical data can be
obtained with signiﬁcantly reduced computational cost.

3

The rest of the paper is organized as follows. The governing equations are presented in section
1.2 and the simulations parameters in section 1.3. In order to establish the accuracy of current
simulations, the idea of scale separation between Kolmogorov length scale and shock thickness
(Ryu and Livescu, 2014) (in shock-capturing simulations it becomes numerical shock thickness
δn) is adopted to limit the viscous eﬀects during the interaction, and thus determine whether the
interaction is correctly resolved. Combining grid convergence tests with theoretical analysis, we
prove in subsection 1.4.1 of the main results section, section 1.4, that the statistics are independent
of the mesh and the simulations are reliable. LIA convergence tests are conducted in subsection
1.4.2 to show that LIA limits can be approximated using shock-capturing simulations when certain
criteria are satisﬁed. The eﬀects of density variations on STI are investigated by comparing current
results with that of a single-ﬂuid simulation in subsection 1.4.3. For detailed analysis of mechanisms
and physics behind this problem, the structure and budgets of important turbulence quantities are
examined in subsections 1.4.4 and 1.4.5. The simulation data are then used to address mixing and
modeling in variable density turbulence in subsection 1.4.6 and 1.4.7. Finally, section 1.5 presents
the conclusions of the study.

1.2 Governing Equations and Numerical Methodology

In this work, the following dimensionless compressible Navier-Stokes equations for continuity,
momentum, energy and mass fraction, in a binary ﬂuid system following the ideal gas equation of
state, are solved numerically:

∂U
∂t

+

∂(F − Fv)

∂x1

+

∂(G − Gv)

∂x2

+

∂(H − H v)

∂x3

= 0,

p =

ρRT
γM2
0

(1.1)

(1.2)

The solution vector U = {ρ, ρu1, ρu2, ρu3, E, ρY} contains the primary variables: density, ρ,
velocity components, ui, total energy, E, and mass fraction of the heavy ﬂuid, Y. t, x1, x2, x3 are time

4

and the three coordinate axes. In equation 1.2, R and T denote the speciﬁc gas constant and tem-
perature. M0 is the reference Mach number, which is generated from the non-dimensionalization.
The inviscid ﬂux vectors (F,G,H) and the viscous ﬂux vectors (Fv,Gv,H v) are deﬁned as

(cid:110)
(cid:110)
(cid:110)

F =

G =

H =

and

ρu1, ρu2
ρu2, ρu1u2, ρu2
ρu3, ρu1u3, ρu2u3, ρu2

1 + p, ρu1u2, ρu1u3,(E + p)u1, ρYu1
2 + p, ρu2u3,(E + p)u2, ρYu2
3 + p,(E + p)u3, ρYu3

(cid:111)
(cid:111)
(cid:111)

,

,

(1.3a)

(1.3b)

(1.3c)

(1.4a)

(1.4b)

(1.4c)

Fv = {0, τ11, τ21, τ31, uiτi1 − q1 − qd1,−J1} ,
Gv = {0, τ12, τ22, τ32, uiτi2 − q2 − qd2,−J2} ,
H v = {0, τ13, τ23, τ33, uiτi3 − q3 − qd3,−J3}

The viscous stress tensor τi j, thermal diﬀusion vector qi, enthalpy diﬀusion vector qdi and mass
fraction ﬂux vector Ji are calculated by the following equations:

(cid:18) ∂ui

∂xj

τi j =

µ
Re0

+

∂uj
∂xi

− 2
3

∂uk
∂xk

δi j

(cid:19)

(1.5)

(1.6)

(1.7)

(1.8)

qi =

−µCp
(γ − 1)M2

0 Re0Pr

∂T
∂xi

qdi =

−µT
(γ − 1)M2

0 Re0Sc

∂Cp
∂xi

Ji =

−µ
Re0Sc

∂Y
∂xi

5

Figure 1.1: Scalar structure in multi-ﬂuid turbulence interacting with a Mach 2 shock identiﬁed by
the iso-surface of heavy ﬂuid mole fraction and colored with instantaneous density ﬂuctuations.
The black plane located in the middle of the domain represents the instantaneous shock surface.
The iso-surfaces correspond to mole fraction values of (a) 0.05 (b) 0.95 (c) 0.25 and (d) 0.75.

where Re0 is reference Reynolds number. The ratio of speciﬁc heats is γ = Cp/Cv = 1.4. The ﬂuid
viscosity is assumed to be dependent on dimensionless temperature as µ = T0.75 for all species in
the ﬂow. The Prandtl and Schmidt numbers are Pr = Sc = 0.75. This set of non-dimensional fully
compressible equations are solved numerically in the conservative form. The inviscid ﬂuxes are
computed by the ﬁfth-order MP scheme as described in (Li and Jaberi, 2012) with the exception
that, here, the mass fraction equation is also computed in the conservative form using the MP
scheme. The ﬂuxes are divided into forward and backward parts via Lax-Friedrichs ﬂux splitting
method after projecting them into the characteristics ﬁeld. Both parts are then reconstructed using
the MP scheme and combined before projecting back to the physical space. The same tolerance
value (10−10) for the MP scheme as in Li and Jaberi (2012) is used for this study. The viscous and
diﬀusive ﬂuxes are calculated by the sixth-order compact scheme (Lele, 1992) coupled with the
ﬁfth-order one-sided compact boundary scheme of Cook and Riley (1996). Time advancement is
achieved via the classical 3rd-order Runge-Kutta scheme.

6

Figure 1.2: Instantaneous contours of 3D pressure ﬂuctuations and shock surface resulting from
isotropic turbulence interacting with a Mach 2 shock. (a) Iso-surface of heavy ﬂuid mole fraction
(φ = 0.25) colored by instantaneous pressure ﬂuctuations. Instantaneous shock surface colored by
the ratio of local pressure jump across the shock for (b) multi-ﬂuid A (see deﬁnition in section
1.4.3) and (c) single-ﬂuid cases.

1.3 Simulations Setup and Parameters

Figure 1.2(a) shows the three-dimensional (3D) iso-surface of heavy ﬂuid mole fraction from
the base multi-ﬂuid STI simulation, which is colored by local pressure ﬂuctuations. This high-
lights the setup of the current simulations:
the dimensions of the computational domain are
(L1, L2, L3)=(4π,2π,2π); the streamwise direction is denoted by x (or x1) and transverse directions
are denoted by y and z (or x2 and x3); the normal shock is nearly stationary and is initialized in the
middle of the domain consistent with the laminar Rankine-Hugoniot relations. Periodic boundary
conditions in transverse directions are implemented and a buﬀer layer is set at the end of the domain
from x = 4π to x = 6π. Turbulence is assumed to be homogeneous in transverse directions and
isotropic before the shock wave. Averages are computed over homogeneous directions to obtain
statistics of the ﬂow. Reynolds averages are denoted by an overbar, f , while Favre averages are

denoted by a tilde, (cid:101)f ; the corresponding ﬂuctuations around these averages are denoted by f (cid:48) and

f (cid:48)(cid:48).

To provide inﬂow conditions, isotropic turbulence ﬁelds are superposed on a uniform mean

7

1/2/(cid:113)

i

ﬂow with Mach number of 2.0 and then advected through the inlet using Taylor’s hypothesis. At
relatively low turbulent Mach numbers Mt = u(cid:48)
iu(cid:48)
γRT, the Taylor’s hypothesis is a good
approximation for upstream isotropic turbulence. Moreover, the turbulence is allowed to develop
before reaching the shock wave so as to achieve a realistic state. We use the same method as that
of Ristorcelli and Blaisdell (1997), and generate the turbulence by a separate temporal simulation
of decaying isotropic turbulence. The velocities for the isotropic box simulations are initialized
randomly following a 3D Gaussian spectral density function. The mean velocity is set to zero.
The peak value of the energy spectrum is k0 = 4.0. The temperature ﬁeld is uniform initially
and the initial pressure is calculated by solving the Poisson equation. The simulation is then
conducted until the skewness of velocity derivative reaches around −0.5 and the ﬂatness reaches
around 4.0. The range of Mt values of the ﬁnal fully developed turbulence is 0.03-0.38 and that
iu(cid:48)
u(cid:48)
i/3λ/µ, is 9-45. The Taylor micro
of Reynolds number based on Taylor micro scale, Reλ = ρ
scale is computed from viscosity, µ, and turbulent energy dissipation, , as λ = (15µu(cid:48)2
rms/)0.5 and
u(cid:48)
iu(cid:48)
rms = (u(cid:48)
i/3)0.5. These isotropic turbulence boxes are taken from the same temporally decaying
isotropic turbulence simulations at diﬀerent times.

(cid:113)

The variable density (multi-ﬂuid) eﬀects arise from variations in composition ﬁeld, by correlat-
ing the density to an isotropic scalar ﬁeld (heavy ﬂuid mole fraction or mass fraction). The scalar
ﬁeld was generated as a random ﬁeld following a Gaussian spectrum with a peak at ks = 8.0 and
has double-delta probability density function (PDF) distribution, so that the scalar value is either
0.0 or 1.0. The scalar ﬁeld was then smoothed out by solving the pure diﬀusion equation, until the
scalar gradients became well resolved on the mesh. The resulting scalar ﬁeld was allowed to decay
in fully-developed isotropic turbulence simulations for about one eddy turnover time as a passive
scalar (denoted as φ). The density ﬁeld is then calculated by making X = φ for case A (where X
is the mole fraction of the heavy ﬂuid) or Y = φ for case B and using the ideal gas equation of
0/RT. The ratio of the molar masses of the two ﬂuids
state in its multicomponent form, ρ = pγM2
is 1.78, resulting in an Atwood number, At = (MW2 − MW1)/(MW2 + MW1), of 0.28. This value
of the Atwood number was chosen such that the variable density eﬀects are non-negligible, yet

8

the interaction with the shock wave is still in the wrinkled-shock regime. Higher At values cause
signiﬁcant distortions of the shock front and even breaking of the shock, depending on the speed
of sound in the light ﬂuid.

A ‘buﬀer’ layer is set at the end of the domain from x = 4π to x = 6π to prevent reﬂections
of waves from the outﬂow boundary. In this region, the ﬂow variables are smoothly dissipated to
a laminar solution through a damping function, eﬀectively controlling the unphysical oscillations
around the outﬂow (Larsson and Lele, 2009). A constant back pressure calculated from Rankine-
Hugoniot equations is applied as the outﬂow boundary condition. This outﬂow boundary condition
has been shown to cause small movement of the shock wave (Larsson and Lele, 2009). However,
the Mt value is low in the ﬁnal simulations, and the shock drifting speed is calculated to be less
than 0.2% of the mean ﬂow speed, so we are conﬁdent that the shock movement does not aﬀect the
statistics presented.

1.4 Results and Discussions

In this section, the variable density (VD) eﬀects on the STI and the mechanisms behind
the interaction are discussed in details. However, before discussing these eﬀects, the accuracy
of the simulations is established by grid convergence tests and analysis of various ﬂow variables.
Convergence to LIA is then studied for shock-capturing simulations and the criteria for convergence
are identiﬁed. The comparison between multi-ﬂuid and single-ﬂuid cases is then made for various
turbulent statistics. A series of in-depth analyses of turbulence budgets for important turbulence
and mixing quantities like turbulent kinetic energy and variance of mole fraction are presented for
better understanding of VD eﬀects on STI.

To eliminate the statistical variability, most results are space averaged over homogeneous
directions and time averaged for more than two pass-over times after the ﬂow reaches a statistically
steady state, which is achieved after the acoustic wave propagate from the outlet to the shock wave
(Larsson et al., 2013). The pass-over time is calculated as tp = 2π/u1,u + 2π/u1,d, where u1,u
and u1,d are the pre-shock and post-shock mean streamwise velocities. Instantaneous results are

9

Mesh info
1024 × 2562
1024 × 3842
1536 × 3842
1536 × 5122
2048 × 5122

Grid-1
Grid-2
Grid-3
Grid-4
Grid-5

∆xd/∆y Reλ Mt
0.09
0.09
0.09
0.09
0.09

2.0
1.3
2.0
1.5
2.0

45
45
45
45
45

(kmaxη)min

0.96
1.45
1.45
1.93
1.93

η/δn
0.93
0.93
1.40
1.40
1.86

Table 1.1: Details of the simulations used in the grid convergence tests.

also presented in the form of constant property surfaces and energy spectra when needed. For all
the results presented in this section, the coordinate is shifted such that the shock is approximately
located at k0x = 0.0.

1.4.1 Accuracy of Numerical Results

In the shock-turbulence interaction problem, the grid resolutions used for both turbulence and shock
wave are critical to the accuracy of results. Here, the inﬂow turbulence is generated separately
using 2563 grid points and has a kmaxη value around 2.3, where kmax is the maximum turbulence
wavenumber and η is the Kolmogorov length scale. kmaxη is usually taken to be greater than 1.5 to
resolve small-scale turbulence. It is relatively easy to construct a mesh to fully resolve turbulence
in the pre-shock region. However, the post-shock small-scale turbulence is expected to decrease
in size, especially in the shock normal direction. Therefore, to ensure that the smallest post-shock
turbulence scales are well resolved, a detailed grid convergence test was conducted using 5 diﬀerent
meshes, as shown in table 1.1. The calculation of kmaxη is based on the largest grid size among all
three directions to give safer and more conservative estimates.

In ﬁgure 1.3, several large and small scales turbulent statistics obtained from the convergence
test are shown. The region of unsteady wrinkled shock movement is marked as a grey area in
this and following ﬁgures. Due to the unsteady shock movement and wrinkled shock surface, the
averaged shock thickness is much larger than the instantaneous numerical shock thickness. As the
mesh gets reﬁned from Grid-1, Grid-3 to Grid-5 with the same compression ratio, all turbulent

10

Figure 1.3: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09. (a)
Turbulent kinetic energy, (b) Kolmogorov length scale η, (c) streamwise Taylor micro scale λ1 and
(d) transverse vorticity variance ω(cid:48)
2ω(cid:48)
2. The region of unsteady wrinkled shock movement is
marked in grey.

Figure 1.4: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09. (a) mass
fraction variance φ(cid:48)φ(cid:48) and (b) Batchelor scale λB.

11

k0x-505101520TKE00.050.10.150.2Grid-1Grid-2Grid-3Grid-4Grid-5k0x-505101520η00.0050.010.0150.02Grid-1Grid-2Grid-3Grid-4Grid-5k0x-505101520λ100.050.10.150.20.250.3Grid-1Grid-2Grid-3Grid-4Grid-5k0x-505101520ω′2ω′20102030405060Grid-1Grid-2Grid-3Grid-4Grid-5(a)(b)(c)(d)k0x-505101520φ′φ′00.010.020.030.040.05Grid-1Grid-2Grid-3Grid-4Grid-5k0x-505101520λB00.020.040.060.080.10.12Grid-1Grid-2Grid-3Grid-4Grid-5(a)(b)Figure 1.5: Relation between numerical shock width, δn, and grid size in the shock region.

statistics converge, proving the suﬃciency of grid resolution. Results from meshes with diﬀerent
compression ratios, i.e. Grid-5 and Grid-4, are compared to make sure that the grid is ﬁne enough
to resolve the decreased length scale in x-direction. Statistics that characterize the behavior of mass
fraction are also examined. Figure 1.4 shows the mass fraction variance and the Batchelor scale,
representing large scale and small scale statistics of mixing, respectively. Again, these statistics are
grid converged and the errors are small when using Grid-5. The convergence rate of the Batchelor
scale is slower than for all the other statistics, a consequence of VD eﬀects.

Larsson et al. (2013) have investigated how the large scales are aﬀected by the ﬁnite com-
putational box size. Their results show that the eﬀects of box size are minimal for the range of
parameters analyzed. For current study, the same peak wavenumber k0 = 4 for the energy spectra
and the same box size are used, so the eﬀects from ﬁnite computational box size should also be
negligible.

Another important issue in STI simulations is whether the interaction between shock wave and
turbulence is well captured. When a shock capturing scheme is used, a scale separation between
shock width and turbulence length scales is desirable. As suggested by Ryu and Livescu (2014),

12

∆x00.0050.010.0150.020.025δn00.0050.010.0150.020.0250.030.0350.04the ratio of Kolmogorov length scale to numerical shock width, δn, is considered to be an important
parameter to assess this separation, where δn is calculated as (u1,u − u1,d)/|∂u1/∂x1|max, and
|∂u1/∂x1|max denotes the maximum magnitude of streamwise velocity gradient. In the coarsest
simulation using Grid-1, the ratio of pre-shock Kolmogorov length scale to numerical shock width
is estimated to have a value of about 0.93. However, since the Kolmogorov length scale is calculated
using average dissipation, locally, turbulence eddies can become commensurate with the numerical
shock thickness. Indeed, using instantaneous dissipation values to estimate the minimum turbulence
length scale, yields a ratio with the numerical shock thickness that is much smaller. In order to
achieve a better scale separation, the numerical shock width needs to be reduced to a much smaller
value than that corresponding to the Grid-1 simulation. The dependence of δn on the grid size is
investigated by conducting a series of tests using a range of grid sizes around the shock region.
The results are shown in ﬁgure 1.5, which indicates that the numerical shock width is linearly
correlated with the grid size, as expected. Therefore, by reﬁning the mesh in x-direction, the scale
separation can be controlled. Results from Grid-2 and Grid-3 show good resolution of post-shock
Kolmogorov length scale with a kmaxη value around 1.45, but considerably diﬀerent post-shock
prediction of ω(cid:48)
2ω(cid:48)
2. This diﬀerence can be attributed to the eﬀects of scale separation ratio (0.93
for Grid-2 and 1.40 for Grid-3) as shown in table 1.1. Further increasing the scale separation
ratio has a small eﬀect on the resolution of post-shock statistics, e.g. from Grid-4 (1.4) to Grid-5
(1.86). To conﬁrm this, we have also examined another set of simulations with lower Reλ but wider
range of scale separation ratios, and noticed that having a larger than 1.4 scale separation ratio
has minimal eﬀect on the statistics presented here. These results are not shown here for brevity.
Based on these results, we can safely say that for the ﬁnal simulation with Grid-5, which has a scale
separation ratio of 1.86, the scale separation ratio is large enough to correctly resolve STI for the
current shock-capturing method.

13

1.4.2 Convergence to LIA for Single-ﬂuid Case

In the fully resolved STI simulations of Ryu and Livescu (2014), it is shown that as the ratio of shock
thickness, δ, to Kolmogorov length scale, η, becomes small, the DNS results converge to the LIA
solutions. This suggests that the scale separation between δ and η can be a criterion for controlling
the viscous eﬀects on the interaction. In Livescu and Ryu (2016), high Reynolds number / high shock
Mach number post-shock turbulence data are generated using the LIA procedure, which resolved
the issue of high or prohibitive computational cost in fully resolved DNS. For shock-capturing
simulation, such convergence has not been investigated in previous studies.
In DNS, the scale
separation is controlled by the ratio δ/η , which can be calculated as δ/η = 7.69Mt/(Re0.5
(Ms −1))
and it was varied in Ryu and Livescu (2014) by changing Mt, which also minimized the non-
linear eﬀects. In shock-capturing simulations, δn/η, Mt, Reλ, and Ms are, in general, independent
parameters; however, the same general principle can be applied to study the convergence of the
numerical results to LIA.

λ

Unlike DNS, where an overlap between shock width and turbulence scales does represent a
physical problem, albeit diﬀerent than the LIA limit, shock-capturing simulations need to have
separation of scales to ensure that the numerical shock-capturing algorithm does not alter the
physics of the problem. For current simulations, the numerical shock thickness depends on the
mesh size in streamwise direction instead of any turbulent statistics, so it can be independently
varied without changing Mt and Reλ. The shock thickness for diﬀerent shock-capturing schemes
is aﬀected by the amount of numerical dissipation introduced around shock. It is shown in Li and
Jaberi (2012) that the MP5 scheme has less numerical dissipation than the WENO scheme, so the
numerical shock thickness is smaller for MP5. Generally, the shock jump is represented by 2 to
3 grid points and the numerical shock thickness calculated using (u1,u − u1,d)/|∂u1/∂x1|max is
around 1.6∆x. By changing ∆x, the scale separation ratio can be independently varied. Figures
1.3 and 1.4 (ﬁgures correspond to multi-ﬂuid cases, but the ideas are the same) show that for a
ratio η/δn greater than 1.40, there is no signiﬁcant error in the post-shock statistics. Compared to
shock-resolving simulations, this feature of shock-capturing simulations relaxes the requirement of

14

fully resolved shock proﬁle and makes the simulations cheaper without diminishing the accuracy of
turbulent statistics. Larsson (2010) took a diﬀerent approach towards this problem. A theoretical
model of the error introduced by the shock-capturing scheme was used to predict the post-shock
turbulence statistics. The error of post-shock Reynolds stresses was found to scale with grid
spacing to the second order. Using this model, a critical value of k0∆x can be calculated to
control the error of Reynolds stresses for a certain shock-capturing scheme. Compared to this
method, the scale separation criterion naturally takes into account the artiﬁcial dissipation added
in the shock region by using numerical shock thickness as a shock length scale, which makes it a
simpler criterion. Typically, a scale separation ratio η/δn ≥ 1.4 is recommended for current shock-
capturing simulations. We also recognize that, even though the scale separation ratio provides
guidance in conducting shock-capturing turbulence resolving simulations, the speciﬁc value for the
scale separation criterion is not universal. It depends on the shock-capturing method and the way
the numerical shock is quantiﬁed.

While δn/η is still an important parameter to assess the accuracy of the results, it is unlikely
that the results in the shock region have the same physical meaning as those from shock-resolving
DNS. Therefore, to study the convergence to LIA in shock-capturing simulations, Mt and Reλ
need to be considered separately, instead of being one single parameter as in shock-resolving DNS.
Single-ﬂuid simulations are conducted covering a wide range of Mt and Reλ parameter space to
study the convergence to LIA. Reλ immediately upstream of the shock varies between 9 and 45
and Mt right before the shock ranges from 0.03 to 0.38. More details of the LIA convergence tests
can be found in table 1.2. The minimum scale separation in these tests is δn/η = 1.4 to ensure
accuracy.

2u(cid:48)
1 and u(cid:48)
1u(cid:48)
In ﬁgure 1.6, the streamwise and transverse Reynolds stress components (u(cid:48)
2)
obtained from the single-ﬂuid simulations are compared with the LIA solution. All the plots are
normalized by the values immediately upstream of the shock wave. The eﬀects of Reλ are ﬁrst
considered and the Mt values are kept around 0.24 (except for Reλ = 45 case, for which Mt = 0.09).
The results for u(cid:48)
1u(cid:48)
1 are shown on ﬁgure 1.6(a). As Reλ increases, the post-shock peak values of

15

1 converge to LIA prediction in some region behind the shock. For u(cid:48)
1u(cid:48)
u(cid:48)
2u(cid:48)
2, the post-shock value
reaches the peak right after the shock wave, and then it monotonically decreases. The peak right
after the shock can be aﬀected by shock thickness, shock corrugation and unsteady shock movement,
so it is not a good representation of the shock ampliﬁcation of turbulence. As suggested by Ryu
2 at corresponding u(cid:48)
and Livescu (2014), the values of u(cid:48)
2u(cid:48)
1u(cid:48)
1 peak positions are examined instead
2u(cid:48)
to evaluate the convergence. The trend of converging to LIA prediction for u(cid:48)
2 is shown in ﬁgure
1.6(b), and it qualitatively matches with DNS results. But compared to u(cid:48)
1u(cid:48)
1, the convergence of
u(cid:48)
2u(cid:48)
2 is slower. This can be explained by the fact that Mt values in these simulations are not low
enough for u(cid:48)
2u(cid:48)
2 to approximate the LIA solution. As Mt increases, the shock wave becomes more
wrinkled and it strongly aﬀects the convergence of u(cid:48)
2u(cid:48)
2, since this quantity is more sensitive to
shock wrinkling (Lee et al., 1997).

To further examine the role of the shock wrinkling, the eﬀects of Mt on the convergence to
LIA predictions are examined by setting the Reλ immediately before the shock wave around 30 and
varying Mt. In ﬁgure 1.7, the normalized plots of Reynolds stresses are compared and it can be seen
that the ampliﬁcation ratios of u(cid:48)
1u(cid:48)
1 for all the cases are very close (but diﬀerent than the LIA limit)
and those of u(cid:48)
2u(cid:48)
2 converge to LIA as Mt decreases. This indicates that for current shock capturing
2u(cid:48)
simulations, the nonlinear eﬀects induced by high Mt are more important for u(cid:48)
1u(cid:48)
2 than for u(cid:48)
1.
The results are also consistent with our previous statement that u(cid:48)
2u(cid:48)
2 is more sensitive to Mt. When
2 exhibits the same converging trend, but u(cid:48)
2u(cid:48)
comparing to DNS results, u(cid:48)
1u(cid:48)
1 does not converge to
the LIA limit at Reλ = 30. It seems the LIA limit of u(cid:48)
1u(cid:48)
1 cannot be approximated by decreasing
Mt at low Reλ and Reλ needs to be large enough for convergence to be achieved. This is conﬁrmed
by another test with low Mt (0.09) and higher Reλ (45) as shown in ﬁgure 1.6. For Reλ greater than
45 and Mt lower than 0.1, the match to LIA results is fairly good, suggesting a minimum value of
Reλ around 45 needed for the streamwise Reynolds stress component to converge.

In summary, the results show that shock-capturing simulations exhibit similar converging trend
to LIA solutions as shock-resolving DNS. LIA solutions for individual Reynolds stresses can be
approached provided that there is some separation of scale between turbulence scales and shock

16

LIA convergence case Reλ Mt
0.09
0.24
0.24
0.24
0.03
0.09
0.17
0.24
0.38

Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9

Case 1(ﬁnal case)

45
30
16
9
30
30
30
30
30

Table 1.2: Cases considered for LIA convergence tests.

1), and Mt is low enough (for u(cid:48)
1u(cid:48)
width, Reλ is large enough (for u(cid:48)
2u(cid:48)
2). On the other hand,
the approach to the LIA prediction is diﬀerent for the streamwise and spanwise components of
the Reynolds stress tensor. For the transverse component, the converging trend agrees very well
with the DNS results as we vary either Mt or Reλ. However, the streamwise component does not
converge to the LIA prediction at low Reynolds numbers by varying Mt, which is diﬀerent from the
DNS study by Ryu and Livescu (2014). This is attributed to the fundamental diﬀerences between
shock-capturing turbulence-resolving simulations and shock-resolving DNS. A further test then
shows that the LIA limit of u(cid:48)
1u(cid:48)
1 can be approached at both large Reλ (45) and low Mt (0.09),
which represents a regime that viscous and nonlinear eﬀects are not important for the prediction
of u(cid:48)
1u(cid:48)
1. Based on these ﬁndings, it would be safer to extend the single-ﬂuid STI to multi-ﬂuid
STI at higher Reλ. The requirement on Reλ makes it relatively more expensive for shock-capturing
simulation to approach LIA than full DNS. However, shock-capturing simulations can still achieve a
large scale separation between Kolmogorov length scale and physical shock thickness using coarser
grids than shock-resolving DNS. In practice, using coarser grid for simulating the shock should
dominate over the requirement of increased grid resolution for the turbulence to be able to capture
a higher Reynolds number ﬂow. Therefore, shock-capturing simulations should be an eﬀective tool
for studying STI so long as the results have been carefully veriﬁed as shown above.

17

Figure 1.6: Comparison of Reynolds stress obtained from single-ﬂuid simulations with LIA
results. Mt ≈ 0.24 except for Reλ = 45. (a) Streamwise Reynolds stress and (b) transverse
Reynolds stress.

Figure 1.7: Comparison of Reynolds stress obtained from multi-ﬂuid, single-ﬂuid simulation with
LIA results. Reλ ≈ 30. (a) Streamwise Reynolds stress and (b) transverse Reynolds stress.

18

k0x05101520u′1u′100.511.522.53LIAsingle-ﬂuid:Reλ=45single-ﬂuid:Reλ=30single-ﬂuid:Reλ=16single-ﬂuid:Reλ=9k0x05101520u′2u′200.511.522.53LIAsingle-ﬂuid:Reλ=45single-ﬂuid:Reλ=30single-ﬂuid:Reλ=16single-ﬂuid:Reλ=9(a)(b)k0x05101520u′1u′100.511.522.5LIAsingle-ﬂuid:Mt=0.03single-ﬂuid:Mt=0.09single-ﬂuid:Mt=0.167single-ﬂuid:Mt=0.237single-ﬂuid:Mt=0.38k0x05101520u′2u′200.511.522.5LIAsingle-ﬂuid:Mt=0.03single-ﬂuid:Mt=0.09single-ﬂuid:Mt=0.167single-ﬂuid:Mt=0.237single-ﬂuid:Mt=0.38(a)(b)1.4.3 Eﬀects of Density Variations on STI

In this section, the eﬀects of density variations on STI are examined in detail by comparing the
results obtained from variable density or ‘multi-ﬂuid’ cases with those obtained from a reference
‘single-ﬂuid’ simulation. As demonstrated in the previous section, a minimum value Reλ ≈ 45 is
needed for streamwise Reynolds stress to converge to LIA prediction. At lower Reλ, the convergence
trend of u(cid:48)
1u(cid:48)
1 is diﬀerent from fully resolved DNS studies (Ryu and Livescu, 2014). Therefore, the
value of Reλ = 45 was chosen for the results presented in the rest of the paper. These results were
obtained using Grid-5 from table 1.1 and are grid converged, as shown in section 1.4.1. For the
multi-ﬂuid case, two diﬀerent ways are used to generate density variations at inﬂow: (a) the heavy
ﬂuid mole fraction is correlated with the random scalar used for initialization and (b) the heavy
ﬂuid mass fraction is correlated with the random scalar. In the following discussions, these two
cases will be referred to as the multi-ﬂuid A and multi-ﬂuid B cases. The random scalar ﬁelds used
to generate density in both multi-ﬂuid cases are the same and have symmetric PDFs. Evidently, this
initialization of density ﬁeld will make the density PDF of multi-ﬂuid A case symmetrical and that
of multi-ﬂuid B case asymmetrical. Below, the multi-ﬂuid A case will be treated as the default case,
because it is a better representation of most mixing processes, in which the initial volumes occupied
by the ﬂuids in the mixture are speciﬁed. Moreover, the reactants in a stoichiometric combustion
process are commonly introduced based on their mole fractions, so a speciﬁed mole fraction ﬁeld
at the inlet is preferred. The multi-ﬂuid B case is also of interest to us, as it features a diﬀerent
inﬂow density structure (for the same density ratio) compared to the default case. The single-ﬂuid
reference simulation was conducted using the same inﬂow conditions for turbulent variables except
density. In this reference case, the mass fraction of the heavy ﬂuid is set to 1.0. At the same time, a
passive scalar equation, which is the same as the mass fraction equation in the multi-ﬂuid case, is
solved for comparison. This case is referred to as just the single-ﬂuid case and used as a reference
to study the eﬀects of VD on STI. For all cases, the turbulence is allowed to adjust itself to the
scalar ﬁeld in the pre-shock region before interacting with the normal shock.

In ﬁgures 1.2(b) and (c), the instantaneous shock fronts colored by the ratio of local pressure

19

Figure 1.8: 2D contours of instantaneous scalar and density ﬁelds for (a,b) multi-ﬂuid A
simulation and (c,d) single-ﬂuid simulation. The 2D contours are taken at the same time step, in
planes x-z.

jump as obtained from multi-ﬂuid A and single-ﬂuid cases are compared. As expected, the shock
wave of multi-ﬂuid case becomes more wrinkled and has stronger pressure jump variation. Figure
1.8 shows the instantaneous 2D contours of density and scalar for multi-ﬂuid A (heavy ﬂuid mole
fraction) and single-ﬂuid (passive scalar) cases. After scaling the contours to the same range,
ﬁgures 1.8(a) and (c) show that the density variation is much stronger in the multi-ﬂuid case due to
the compositional change. In contrast, the variation of density in the reference (single-ﬂuid) case
results from thermodynamic ﬂuctuation, which makes it very small for the parameters considered
here. When examining the passive scalar ﬁelds and mass fraction ﬁelds (ﬁgure 1.8(b) and (d)) for
each case, we observe that the scalar ﬁelds have similar structure and magnitude before the shock
wave as they are generated similarly at inﬂow. After passing through the shock wave, the scalar
ﬁelds and mixing are diﬀerent. Evidently, when the density variation is signiﬁcant in pre-shock
turbulence, the mixing enhancement by shock is stronger.

Figures 1.9-1.15 show several important ﬂow statistics obtained from the multi-ﬂuid and single-

20

ﬂuid simulations. These statistics are gathered by averaging over homogeneous directions and time,
so that they only depend on the streamwise direction.

Averaged ﬂow statistics are compared in ﬁgure 1.9 for single-ﬂuid and multi-ﬂuid A cases.
Before the shock wave, all cases yield the same results. This observation not only conﬁrms that the
inﬂow conditions are somewhat similar in these cases, but also implies that for current simulation,
the eﬀect of density variations on turbulence is small in the pre-shock region. When comparing
the multi-ﬂuid turbulent kinetic energy and vorticity variance with the corresponding single-ﬂuid
values in ﬁgures 1.9(a) and (d), it is noted that the ampliﬁcation in these turbulent statistics is much
more signiﬁcant in the multi-ﬂuid cases. Furthermore, the multi-ﬂuid turbulent kinetic energy
reaches a peak around k0x ≈ 2.0, which is closer to the shock than k0x ≈ π for single-ﬂuid case.
Figures 1.9(b) and (c) show the comparison for streamwise turbulence Taylor micro length scale,
λ1, and Kolmogorov length scale, η. The reduction in turbulence length scales across the shock
wave is evident in these ﬁgures; the multi-ﬂuid cases show more reduction than the single-ﬂuid
case. Note that the changes in turbulence statistics in multi-ﬂuid cases are expected to depend on
the scalar structure and the Atwood number (e.g. Lombardini et al., 2011); these are not discussed
in this paper.

In ﬁgure 1.10, statistics related to the scalar ﬁeld (heavy ﬂuid mass fraction for multi-ﬂuid case
A and passive scalar for the single-ﬂuid case) and mixing are compared. Both scalar variance
φ(cid:48)φ(cid:48) and Batchelor scale λB are shown. λB is calculated based on the scalar dissipation and is the
representative of the smallest scales in the scalar ﬁeld. The scalar statistics are normalized by the
values immediately before the shock wave. After passing through the shock wave, the faster decay
of scalar variance for the multi-ﬂuid case indicates stronger shock enhancement of scalar mixing.
The Batchelor scale, however, shows a more complex behavior. Unlike the Kolmogorov length
scale, the same reduction ratio of Batchelor scale across the shock wave is observed between the
multi-ﬂuid and single-ﬂuid cases. After passing through the shock wave, the Batchelor scales of
multi-ﬂuid cases exhibit a transient process of decreasing before returning to the pre-shock value,
during which an even smaller structure of the scalar ﬁeld is generated. In the single-ﬂuid case,

21

Figure 1.9: Plots of (a) turbulent kinetic energy, (b) Kolmogorov length scale, (c) Taylor micro
scale and (d) transverse vorticity variance for multi-ﬂuid A (red) and single-ﬂuid (blue)
simulations.

Figure 1.10: Plots of normalized (a) scalar variance, (b) Batchelor scale for multi-ﬂuid A (red)
and single-ﬂuid (blue) simulations.

22

k0x051015TKE00.050.10.150.2multi-fluidsingle-fluidk0x051015η00.0050.010.0150.020.025multi-fluidsingle-fluidk0x051015λ100.050.10.150.20.250.3multi-fluidsingle-fluidk0x051015ω′2ω′20102030405060multi-fluidsingle-fluid(a)(b)(c)(d)k0x051015φ′φ′00.20.40.60.81multi-fluidsingle-fluidk0x051015λB00.20.40.60.811.2multi-fluidsingle-fluid(a)(b)Figure 1.11: Conditional expectation of scalar as a function of density at various streamwise
location for : (a) multi-ﬂuid A and (b) single-ﬂuid cases. The conditional mean is calculated
using variables in a spanwise plane at certain streamwise locations and then averaged over time.
Peak TKE positions are diﬀerent between multi-ﬂuid(k0x ≈ 2.0) and single-ﬂuid(k0x ≈ π) cases

however, the Batchelor scale increases monotonically back to its pre-shock value. We also note
that after k0x ≈ 10.0, the multi-ﬂuid λB values are larger than the single-ﬂuid values as the faster
mixing immediately after the shock smooths out the small scalar scales. The diﬀerent behaviors of
scalar variance and Batchelor scale in the post-shock region can be explained by using the scalar
transport equation. This will be discussed in Section 4.5. We also note that results from multi-ﬂuid
case B show almost the same behavior as those from multi-ﬂuid case A for the above statistics,
despite their completely diﬀerent density PDFs.

It was shown in ﬁgure 1.1 that the turbulence structure is aﬀected very diﬀerently by the shock in
regions with diﬀerent density values. To further understand this behavior, conditional expectations
of several turbulence quantities, conditioned on the density, are calculated and examined. Figures
1.11 - 1.15 show these conditional means at various locations for the multi-ﬂuid A and single-ﬂuid
cases. These statistics are calculated in selected spanwise planes (y-z planes) and then averaged in
time. The conditional mean of scalar, shown in ﬁgure 1.11(a), conﬁrms the correlation of density
ﬁeld to the scalar ﬁeld (heavy ﬂuid mole fraction in this case). The dependence of mole fraction on
density is almost linear on average, which suggests that the density ﬂuctuations are mainly caused
by compositional variations, instead of thermodynamics ﬂuctuations. After passing through the

23

ρ−ρ-0.6-0.4-0.200.20.40.6φ0.10.20.30.40.50.60.70.80.9rightbeforeshockpeakTKEk0x≈4.0k0x≈9.24ρ−ρ-0.1-0.0500.050.1φ00.20.40.60.81rightbeforeshockpeakTKEk0x≈4.0k0x≈9.24(b)(a)Figure 1.12: Conditional expectation of pressure as a function of density at various streamwise
locations for : (a) multi-ﬂuid A close to shock, (b) multi-ﬂuid A at far ﬁeld and (c) single-ﬂuid
cases.

shock wave, a decrease in the slope is observed due to enhancement of density variation across the
shock wave, which translates into an increased extent of the x axis values. This results from a larger
local variation of the shock strength, with the subsequent ampliﬁcation of the post-shock values in
the multi-ﬂuid case. For comparison, ﬁgure 1.11(b) shows that the correlation between the passive
scalar and density is very low for the single-ﬂuid case, which is expected.

Figure 1.12 shows the development of the correlation between pressure and density ﬂuctuations
as the ﬂow passes through the shock and moves away from the shock. Little correlation between
pressure and density is observed before the shock, as expected. The increase in the correlation
between pressure and density can be mostly attributed to the acoustic ﬁeld. Due to the fast
evolving nature of the acoustic ﬁeld in STI (as evidenced in the rapid decay predicted by LIA), it
is worth comparing the pressure-density correlation close to the shock, where the transient eﬀects
dominate, with the far ﬁeld behavior. Figure 1.12(a), shows the development of the conditional
mean of pressure close to the shock wave. Right after the shock, at k0x ≈ 0.64, there exists
a strong positive correlation, which is the direct eﬀect of shock wave on turbulence. During
the post-shock ﬂow/turbulence development, RMS value of pressure ﬂuctuations, p(cid:48)
rms, decreases
rapidly, indicated by the decreasing range of pressure ﬂuctuations. This process corresponds to the
conversion of potential energy to turbulent kinetic energy. This is conﬁrmed by the fact that p(cid:48)
rms
remains almost unchanged after reaching the peak TKE position. In the far ﬁeld (as shown in ﬁgure
1.12b), the pressure ﬂuctuations become less correlated with density ﬂuctuations in the mixed ﬂuid
regions. The loss of pressure-density correlation is hypothesized to be caused by the nonlinearity

24

ρ−ρ-0.500.51P−P-2024rightbeforeshockk0x≈0.64k0x≈0.825peakTKEρ−ρ-0.6-0.4-0.200.20.40.6P−P-1-0.500.511.5rightbeforeshockpeakTKEk0x≈5.0k0x≈9.24ρ−ρ-0.1-0.0500.050.1P−P-1-0.500.51rightbeforeshockpeakTKEk0x≈5.0k0x≈9.24(a)(b)(c)Figure 1.13: Conditional expectation of temperature as a function of density at various streamwise
location for : (a) multi-ﬂuid A case and (b) single-ﬂuid case.

introduced by the strong density ﬂuctuations and the coupling between diﬀerent modes.

Figure 1.12(c) shows the density-pressure correlation at various positions for the single-ﬂuid
case. In contrast to the multi-ﬂuid case, the small perturbation assumption is satisﬁed, so the mode
decomposition by Kovasznay (1953) can be applied. Based on the way the inﬂow turbulence is
generated (Ristorcelli and Blaisdell, 1997), the pre-shock turbulence is dominated by vorticity and
acoustic modes. In this case, the pressure and density ﬂuctuations should follow the isentropic
process: γ(γ − 1)ρ(cid:48)/ρ = (γ − 1)p(cid:48)/p (Kovasznay, 1953). For current single-ﬂuid case results, the
density and pressure ﬂuctuations indeed follow the isentropic process scaling after being normalized
by local mean, in agreement with previous work (Lee et al., 1993; Mahesh et al., 1997). After
passing through the shock wave, the isentropic scaling is no longer satisﬁed due to the generation
of entropy mode, which is featured by a negative correlation between density and temperature
ﬂuctuations (ﬁgure 1.13b). This should be contrasted to the study by Mahesh et al. (1997) whose
focus is on the interaction between turbulent boundary layer and shock wave. In such case, the
inﬂow ﬂuctuations satisfy the weak form of Morkovin’s hypothesis (Morkovin, 1962), instead of
following the isentropic relations. For the current case, despite the generation of entropy mode, the
pressure ﬂuctuations cannot be neglected even at k0x = 9.24 and, therefore, Morkovin’s hypothesis
is not satisﬁed throughout the domain.

25

ρ−ρ-0.8-0.6-0.4-0.200.20.40.60.8T−T-0.04-0.03-0.02-0.0100.010.020.030.040.05rightbeforeshockpeakTKEk0x=5.0k0x=9.24ρ−ρ-0.1-0.0500.050.1T−T-0.015-0.01-0.00500.0050.010.0150.02rightbeforeshockpeakTKEk0x=5.0k0x=9.24(a)(b)Figure 1.14: Conditional expectation of TKE as a function of density at various streamwise
location for : (a) multi-ﬂuid A and (b) single-ﬂuid case.

For a complete discussion on thermodynamic properties, the conditional expectation of temper-
ature for both multi-ﬂuid A case and single-ﬂuid case are shown in ﬁgure 1.13. For the multi-ﬂuid
case, the direct eﬀect of upstream Mach number ﬂuctuations on temperature is strong, resulting in
a positive correlation between density and temperature. The generated entropy mode as predicted
by LIA in the single-ﬂuid case seems to be overshadowed by this eﬀect even further downstream.
For the single-ﬂuid case, the scaling between density and temperature ﬂuctuations satisﬁes that of
an isentropic process before the shock wave. Through the interaction with the shock wave, negative
correlation is established as previously mentioned. However, the correlation is not fully linear,
which can be attributed to the presence of the acoustic mode and the coupling between this and the
entropy mode.

In ﬁgure 1.14(a), the conditional expectation of the turbulent kinetic energy, TKE, is shown.
We note that TKE has a preferential distribution in the relatively high or low density regions in
the post-shock regions. One possible explanation is that in high and low density regions, the local
sound speed has diﬀerent values from that of average sound speed, so that the local shock velocity,
u1,s becomes nonzero (in the reference frame of the laminar shock wave) and changes signiﬁcantly.
The local movement of the shock surface then further changes the post-shock velocity, u1,d and
makes it deviate from the averaged post-shock velocity u1,d. The deviation from u1,d or u(cid:48)
i in low

26

ρ−ρ-0.6-0.4-0.200.20.40.6TKE00.10.20.30.40.50.60.7rightbeforeshockpeakTKEk0x≈4.0k0x≈9.24ρ−ρ-0.1-0.0500.050.1TKE0.060.080.10.120.140.160.18rightbeforeshockpeakTKEk0x≈4.0k0x≈9.24(a)(b)Figure 1.15: Conditional expectation of enstrophy as a function of density at various streamwise
location for : (a) multi-ﬂuid A and (b) single-ﬂuid case.

and high density regions is much larger in magnitude than that in regions with moderate density,
which results in larger TKE in the high and low density regions. We have computed the conditional
average of u(cid:48)
i on density. Results agree very well with our explanation for conditional TKE. We
also note that TKE is larger in the light ﬂuid regions compared to heavy ﬂuid regions. This is due
to the low inertia of the light ﬂuid, which responds faster to the changes in the local strain and, thus,
accelerates faster (Livescu and Ristorcelli, 2009; Livescu et al., 2010) . This explanation is also
applicable to ﬁgure 1.14(b) for single-ﬂuid case, which shows a preferential distribution of TKE
in the lower density ﬂuid region before the shock wave. After passing through the shock wave, a
stronger ampliﬁcation in the high and low density regions is noted but this is relatively weak, which
agrees with the previous arguments.

The correlation between vorticity and density has a completely diﬀerent behavior than TKE.
Evidently, ﬁgure 1.15(a) shows that more vorticity is generated in the mixed ﬂuid regions. Before
reaching the shock wave, the mixing process is relatively slow, so that there are still large regions
with pure or partially mixed ﬂuids and only narrow regions with fully mixed ﬂuids. In these regions,
the density gradients remain large. Through the interaction with the shock wave, the large density
gradients lead to the generation of vorticity through the baroclinic torque (see Eq. 1.10). For
the single-ﬂuid case (ﬁgure 1.15b), the distribution of vorticity is not aﬀected much by the shock

27

ρ−ρ-0.8-0.6-0.4-0.200.20.40.60.8√ωiωi2345678910rightbeforeshockpeakTKEk0x=5.0k0x=9.24ρ−ρ-0.1-0.0500.050.1√ωiωi234567rightbeforeshockpeakTKEk0x=5.0k0x=9.24(a)(b)because of the absence of large density variations.

The discussion above using conditional expectations emphasizes the VD eﬀects on the modiﬁ-
cation of the turbulence structure. These eﬀects reveal the underlying physics of multi-ﬂuid STI.
It’s worth mentioning that the characteristics of conditional expectations for multi-ﬂuid B case (not
shown) are similar to those of the multi-ﬂuid A case.

1.4.4 Reynolds Stresses

αu(cid:48)(cid:48)

The modiﬁcation of Favre-averaged normal Reynolds stress componentsu(cid:48)(cid:48)

α or Rαα (no summa-
tion over α) across the shock wave is an important feature of STI. As mentioned in previous section,
the pre-shock behavior of Reynolds stresses is very similar for multi-ﬂuid and single-ﬂuid cases and
is dominated by viscous dissipation. The variable density eﬀects on the viscous dissipation before
interacting with the shock are small at the Atwood number considered here. After passing through
the shock wave, the main contributing mechanisms to the transient development of Reynolds stresses
are the pressure-velocity correlation and pressure-strain correlation (Lee et al., 1997; Jamme et al.,
2002; Larsson et al., 2013). This corresponds to the decay of the acoustic mode generated by STI.
However, when the variable density eﬀect is introduced, the dominating terms in the Reynolds
stress budgets are unknown, especially after passing through the shock wave. To investigate this,
the budgets of the Favre-averaged normal Reynolds stresses need to be considered. Before further
discussing the turbulence budgets, we need to diﬀerentiate the modiﬁcation of turbulence by the
shock from the evolution away from the shock. For shock-capturing simulations, when there is an
overlap in scale, turbulence and TKE budgets in the shock region cannot be correctly captured.
However, when there is a scale separation between the shock and the turbulence, as is the case here,
the numerical viscous eﬀects become, in principle, negligible during the interaction. Thus, the
interaction becomes physical; however, it also becomes uninteresting from the point of view of the
budget equations. Nevertheless, the evolution away from the shock is accurately described by the
current approach and, for the VD case, has unknown evolution mechanisms that can be elucidated
by the budget equations. Thus, the following discussions are focused on the evolution away from

28

∂(ρ(cid:101)uk Rαα)/2

∂xk

= −ρRαk

∂xk

∂(cid:101)uα
(cid:124)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
+p(cid:48) ∂u(cid:48)
α
(cid:18) ∂p
(cid:19)
∂xα
term II
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

− ∂ταk
∂xk

term I

−u(cid:48)(cid:48)

∂xα
term V

α

(p(cid:48)u(cid:48)(cid:48)
α)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

− ∂
∂xα
term III
(ρu(cid:48)(cid:48)
αu(cid:48)(cid:48)

αu(cid:48)(cid:48)
k/2)

− ∂
∂xk

−τ(cid:48)
αk

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂u(cid:48)
α
∂xk
term IV
αku(cid:48)(cid:48)
(τ(cid:48)
α)
∂
∂xk
term VII

+

term VI

the shock, with the immediate post-shock value treated as boundary value. Livescu et al. (2009)
derived the Reynolds stress budgets for compressible turbulence as following:

(1.9)

The left hand side of Eq. 1.9 contains terms describing the advection of Reynolds stresses by
mean velocity, and it is balanced by summation of all terms on the right hand side, which includes: (I)
production; (II) turbulent pressure-strain; (III) velocity-pressure transport; (IV) viscous dissipation;
(V) production due to mean pressure gradient and stress gradient; (VI) turbulent transport, and (VII)
viscous diﬀusion, respectively. Figures 1.16- 1.20 show the contributions of individual terms in the
Rαα equation and the summation of all the right hand side terms, denoted by RHS. As mentioned
before, all the statistics are time- and space-averaged to achieve converged results. The streamwise
Reynolds stress R11 budgets are shown in ﬁgure 1.16 (the results of multi-ﬂuid A and multi-ﬂuid
B cases are close, so only multi-ﬂuid A results are shown). We note that in the post-shock region,
there exists a strong transient process that is dominated by the combination of pressure-strain (II)
and pressure-velocity (III) correlations. These correlations are related to pressure and quickly
decrease, which is hypothesized to correspond to the fast decay of the acoustic ﬁeld generated by
STI. All the other terms remain small compared to these three terms in the entire post-shock region.
Figure 1.16(b) gives the balance of R11 transport terms. The good balance of advection and RHS
in both shock zone and post-shock zone conﬁrms the accuracy of the simulation.

After the initial strong transient process, the pressure-strain(II) and pressure-velocity (III) terms
exhibit long-lasting small magnitude ﬂuctuations and the dissipation (IV) becomes the leading order
term. These ﬂuctuations are partly due to the limited size of the inﬂow turbulence box and can
be smoothed by repeating the simulations using additional realizations of the isotropic turbulence
box and then averaging the results. In order to remove the eﬀects from limited turbulence box size,

29

Figure 1.16: Contributions of diﬀerent terms in the R11 transport equations for multi-ﬂuid A
simulation. (a) Plots of all the contributing terms in the transport equation and (b) balance of LHS
and RHS of the transport equation.

we have conducted 5 more new simulations using diﬀerent realizations of statistically identical
isotropic inﬂow turbulence and then averaged the budget terms for all these realizations in addition
to time. Other methods have also been used to reduce the statistical variability, such as the blending
method in Larsson and Lele (2009) or an auxiliary forced IT simulation in Ryu and Livescu (2014).
It can be seen in ﬁgure 1.17 that the secondary ﬂuctuations are greatly smoothed. All following
budget plots in this subsection are averaged over ﬁve realizations.

In Figure 1.18, the R22 budgets are presented. For R22, the pressure-velocity(III) term vanishes
due to averaging in homogeneous directions, but the pressure-strain remains dominant in the
transient region. When comparing this term for R11 and R22 budgets, we note a similar behavior,
with a quick return to zero while oscillating around x-axis, but opposite sign. After k0x ≈ 2.0,
the dissipation starts to become the leading contributing term in the R22 budget just like the R11
budget. The good balance of R22 budget is shown in ﬁgure 1.18(b).

To further illustrate the eﬀects of multi-ﬂuid density variations on Reynolds stresses, the
contributing terms are normalized by the corresponding Reynolds stress values at the same location,
as shown in ﬁgure 1.19. We note that both multi-ﬂuid and single-ﬂuid values are now of similar
magnitude, in contrast to the large diﬀerences in their absolute values. The single-ﬂuid budgets
agree qualitatively well with those from previous STI studies (Lee et al., 1997; Jamme et al., 2002;

30

k0x-202468-0.500.511.5Pressure-StrainPressure-VelocityDissipationProductionProduction (by mean pressure and stress gradient)Turbulent DiffustionViscous Diffusionk0x-202468-0.500.511.5AdvectionRHS(a)gu′′1u′′1(a)(b)gu′′1u′′1Figure 1.17: Contributions of diﬀerent terms in the R11 transport equations for multi-ﬂuid A
simulation averaged over 5 diﬀerent realizations (a) Plots of all the contributing terms in the
transport equation and (b) balance of LHS and RHS of the transport equation.

Figure 1.18: Contributions of diﬀerent terms in the R22 transport equations for multi-ﬂuid A
simulation averaged over 5 diﬀerent realizations. (a) Plots of all the contributing terms in the
transport equation and (b) balance of LHS and RHS of the transport equation.

31

k0x-202468-0.500.511.5Pressure-StrainPressure-VelocityDissipationProductionProduction (by mean pressure and stress gradient)Turbulent DiffustionViscous Diffusionk0x-202468-1-0.500.511.5AdvectionRHS(a)gu′′1u′′1(a)(b)gu′′1u′′1k0x-202468-0.6-0.5-0.4-0.3-0.2-0.10Pressure-StrainDissipationProductionProduction (by mean pressure and stress gradient)Turbulent DiffustionViscous Diffusionk0x-202468-0.6-0.5-0.4-0.3-0.2-0.10AdvectionRHS(a)gu′′2u′′2(a)(b)gu′′2u′′2Figure 1.19: Normalized contributions of diﬀerent terms in the R11 transport equations averaged
over 5 diﬀerent realizations for (a) multi-ﬂuid A and (b) single-ﬂuid simulations.

Larsson et al., 2013). One diﬀerence is the magnitude of the dissipation term, which is relatively
small compared to Larsson et al. (2013). To examine the diﬀerence, a preliminary simulation in a
similar parameter range (Mt ≈ 0.15 and Ms = 1.5) was performed (not shown here) and the results
were similar to Larsson et al. (2013). Therefore the diﬀerence can be attributed to diﬀerent shock
and turbulent Mach numbers.

We note that the pressure-strain and pressure-velocity terms return to zero faster in the multi-ﬂuid
case, proving that the transient process for the multi-ﬂuid case is shorter than that for the single-ﬂuid
case, which is in agreement with our previous observation. After the ﬁrst large transient process
of post-shock adjustment, we note that in both cases there is a secondary lower-level transient
process that lasts very long in the post-shock region, with magnitude close to the dissipation term,
especially for the multi-ﬂuid case. This long-lasting small magnitude transient process makes the
development of RANS models for these terms challenging. This observation is in agreement with
Gréa et al. (2016), even though the jumps themselves can be well predicted by some of the current
models (Schwarzkopf et al., 2016).

Other than the time-averaged and space-averaged statistics, the instantaneous planar data in
homogeneous directions could also provide valuable insights into the turbulence statistics and
structure. Figure 1.21 shows the comparison of transverse kinetic energy spectra for single and

32

k0x012345678-2-101234567Pressure-StrainPressure-VelocityDissipationRHSProductionProduction (by mean pressure and stress gradient)Turbulent DiffustionViscous Diffusionk0x012345678-2-101234567Pressure-StrainPressure-VelocityDissipationRHSProductionProduction (by mean pressure and stress gradient)Turbulent DiffustionViscous Diffusion(a)gu′′1u′′1(a)gu′′1u′′1Figure 1.20: Normalized contributions of diﬀerent terms in the R22 transport equations averaged
over 5 diﬀerent realizations for (a) multi-ﬂuid A and (b) single-ﬂuid simulations.

multi-ﬂuid cases at diﬀerent locations. The positions selected include: right before the shock
(k0x ≈ −0.72) and at k0x ≈ 4.0. Again, because the shock surface is wrinkled, the location
of the plane cannot be right before the shock everywhere, which will aﬀect the accuracy of the
calculations. In order to gather the spectrum information right before the shock, the instantaneous
shock wave position and shape are ﬁrst calculated. Then the spectra are taken at the plane closer
overall to the shock wave.

In ﬁgure 1.21, the normalized spectra are compared at positions before the shock and at
k0x ≈ 4.0 for each case. Right before the shock wave, spectra for single- and multi-ﬂuid cases
are nearly identical. After passing through the shock wave, at k0x ≈ 4.0, the small scale energy
is enhanced for both cases, agreeing well with early studies by Lee et al. (1993). However, the
ampliﬁcation of the small scales is stronger in the multi-ﬂuid case at the same spatial location.

1.4.5 Vorticity Variance

Vorticity is another important variable that is strongly aﬀected by STI. Earlier studies on STI have
2ω(cid:48)
3ω(cid:48)
2 (or ω(cid:48)
proven that the transverse vorticity variance ω(cid:48)
3) is ampliﬁed immediately after the
shock wave, while the streamwise component ω(cid:48)
1ω(cid:48)
1 stays unchanged. This is then followed by
1 and a decrease in ω(cid:48)
1ω(cid:48)
an increase in ω(cid:48)
2ω(cid:48)
2 as the turbulence returns to isotropy downstream of

33

k0x012345678-3-2-10123Pressure-StrainDissipationRHSProductionTurbulent DiffustionViscous Diffusionk0x02468-3-2-10123Pressure-StrainDissipationRHSProductionTurbulent DiffustionViscous Diffusion(a)gu′′2u′′2gu′′2u′′2(b)Figure 1.21: Two dimensional spectra of transverse turbulent kinetic energy. The comparison are
made at right before shock wave and k0x ≈ 4.0.

the shock. The ampliﬁcation of ω(cid:48)
2ω(cid:48)
2 predicted by LIA has been reasonably captured by previous
shock-resolving DNS studies (Ryu and Livescu, 2014). Current shock-capturing simulations also
show good agreement on the ampliﬁcation ratio of ω(cid:48)
2ω(cid:48)
2 with LIA (outside shock zone). When the
eﬀect of variable density (due to compositional change) is introduced, the modiﬁcation of vorticity
variance by shock can no longer be predicted by LIA. In ﬁgure 1.22, the vorticity variance for
both single-ﬂuid and multi-ﬂuid cases are plotted. As demonstrated before, the ampliﬁcation of
transverse vorticity variance by the shock is greater when density variations are large. For the
streamwise vorticity variance, both cases show values unaﬀected by the shock immediately after
the shock, while the multi-ﬂuid values increase faster downstream.

The governing equation for vorticity variance can be derived from the momentum equation and
is used here to identify the mechanisms responsible for the change in vorticity. After rearrangement,
this equation takes the following form (Lee et al., 1993):

34

k100101102E22(k)+E33(k)10-710-610-510-410-310-210-1multi-ﬂuidk0x≈−0.72multi-ﬂuidk0x≈4.0single-ﬂuidk0x≈−0.72single-ﬂuidk0x≈4.0Figure 1.22: Streamwise and transverse vorticity variance for multi-ﬂuid and single-ﬂuid case

(ω

uj

∂
∂xj

(cid:48)2
α ) = 2ω(cid:48)

αω(cid:48)

j

j

−ω

(cid:48)2

α

−ω

(cid:48)2

α

∂uα
∂xj

+2ω(cid:48)

αω(cid:48)

∂u(cid:48)
α
∂xj

term II
/ρ2

term I
+2εi jk ω(cid:48)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

term V
(cid:48)2
α u(cid:48)

+2εi jk ω(cid:48)

∂uk
∂xk
term III

∂u(cid:48)
k
∂xk
term IV
∂τkq
∂xq

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:18) 1
(cid:19)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

term VI
∂u(cid:48)
+2ωαω(cid:48)
α
∂xj

αu(cid:48)
−2ω(cid:48)

− ∂
∂xj

∂ωα
∂xj

∂p
∂xk

∂ ρ
∂xj

ω

α

∂
∂xj

α

ρ

α

term VII

term VIII

term IX

j

j

−2ωαω(cid:48)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂u(cid:48)
k
∂xk

α

term X

(1.10)

The terms on the LHS of Eq. (1.10) represents the convection or transport of vorticity variance
by the mean velocity and the RHS terms are: (I) production by mean velocity; (II) production
by turbulent stretching; (III) vorticity-mean dilatation; (IV) vorticity-turbulent dilatation; (V)
baroclinic torque; (VI) viscous diﬀusion/dissipation; (VII) turbulent transport ; (VIII) mixed
velocity-vorticity gradient; (IX) mixed vorticity-velocity gradient; and (X) vorticity-dilatation.

35

k0x-50510152005101520multi-ﬂuidω′1ω′1multi-ﬂuidω′2ω′2single-ﬂuidω′1ω′1single-ﬂuidω′2ω′2Figure 1.23: Contributions of diﬀerent terms in the transport equation for (a) ω(cid:48)
1 and (b) ω(cid:48)
1ω(cid:48)
2ω(cid:48)
for multi-ﬂuid simulation.

2

1 and (b) ω(cid:48)
1ω(cid:48)
Figure 1.24: Contributions of diﬀerent terms in the transport equation for (a) ω(cid:48)
2ω(cid:48)
for single-ﬂuid simulation.

2

The last three terms vanish due to the averaging in homogeneous directions, but they are shown in
the equations and ﬁgures for completeness.

The main contributing terms in Eq. (1.10) are shown in ﬁgures 1.23 and 1.24. Previous DNS
study by Jamme et al. (2002) considered the vorticity variance budgets inside the shock wave.
Here, the post-shock region is of interest. It can be seen that the dominating mechanisms in the
post-shock development region of the ﬂow are production by turbulent stretching (II) and viscous
dissipation (VI) terms. These two terms have competing eﬀects on vorticity variance. Turbulent
stretching production adds a positive contribution while viscous dissipation decreases the vorticity

36

k0x0246810-200-150-100-50050100150200250300AdvectionTurbulent-strechingviscous diffusionRHSk0x0246810-400-300-200-1000100AdvectionTurbulent-strechingviscous diffusionRHS(a)ω′1ω′1(b)ω′2ω′2k0x0246810-30-20-10010203040506070AdvectionTurbulent-strechingViscous diffusionRHSk0x0246810-100-80-60-40-2002040AdvectionTurbulent-strechingViscous diffusionRHS(a)ω′1ω′1(b)ω′2ω′2variance. The relative importance of these two terms describes diﬀerent behaviors of ω(cid:48)
1ω(cid:48)
1 and
1ω(cid:48)
ω(cid:48)
2. In ω(cid:48)
2ω(cid:48)
1 transport equation, term (II) is more signiﬁcant than term (VI), resulting in a net
increase in ω(cid:48)
1ω(cid:48)
1 . On the contrary, the magnitude of term (VI) is larger than term (II) for ω(cid:48)
2ω(cid:48)
2,
causing the spanwise vorticity variance to decrease. It is worth mentioning that, other than the
above dominating terms, terms like baroclinic torque (V) or mean velocity stretching (I) also make
small contributions to the development of spanwise vorticity variance immediately after the shock,
but they virtually have no eﬀect on the streamwise vorticity.

Comparison of vorticity variance budgets between multi- and single-ﬂuid cases can be made to
understand the eﬀects of density variations. In both cases, production by turbulent stretching (II)
and viscous dissipation (VI) terms are the main mechanisms for modeling vorticity in the post-shock
region. Generally, the dominating terms in the multi-ﬂuid case have much larger magnitudes than
those in the single-ﬂuid case. As for speciﬁc vorticity components, both multi-ﬂuid and single ﬂuid
cases have increasing turbulent stretching (II) and decreasing viscous diﬀusion term (VI) close
to the shock (ﬁgures 1.23(a) and 1.24(a)) for ω(cid:48)
1ω(cid:48)
1. The increase in the magnitude of viscous
diﬀusion (VI) seems to be not as fast as that of turbulent stretching (II). Due to this, the multi-
ﬂuid A case shows greater contribution from the turbulent stretching (II) than from the viscous
diﬀusion term (VI) in the vicinity of shock. The increase in the magnitude of the viscous term
(VI) is closely connected to the increase of streamwise vorticity variance. For transverse vorticity
variance budgets, the turbulent-stretching term (II) has a diﬀerent behavior; in the multi-ﬂuid case
it increases slowly after the shock and in the single-ﬂuid case it decreases almost immediately
after shock. The ω(cid:48)
2ω(cid:48)
2 viscous term (VI) for the single-ﬂuid case decreases in magnitude after the
shock, which results from the decrease of the vorticity variance. In contrast, the same term for the
multi-ﬂuid case remains almost unchanged for a short distance before the decrease in magnitude,
indicating a diﬀerent transient process compared to the single-ﬂuid case.

Ribner (1954); Lee et al. (1993); Larsson and Lele (2009); Livescu and Ryu (2016) showed
that STI leads to a two-dimensionalization of the ﬂow immediately after the shock, i.e. the ﬂow
becomes locally axi-symmetric. The vortex stretching term is therefore the smallest immediately

37

Figure 1.25: Vortex-stretching term in the enstrophy equation, normalized by ω2ω2 and
turbulence time scale TKE/

after the shock. This term then increases as the ﬂow evolves away from the shock and returns
to a 3-D structure. Following the expression of the vortex-stretching term in Livescu and Ryu
(2016), the turbulent stretching term can be decomposed into contributions from eigenvectors and
eigenvalues of strain rate. After normalizing the turbulent stretching term in the enstrophy equation
using ω(cid:48)
2ω(cid:48)
2 and the turbulence time scale TKE/ (where  is the rate of dissipation), the eﬀects
of the eigenvectors and eigenvalues of the strain rate can be isolated. As shown in ﬁgure 1.25,
the magnitude of turbulent stretching decreases to around zero across the shock wave. During the
transient post-shock process, the multi-ﬂuid case exhibits a faster return to 3-D isotropic turbulence
structure, indicated by the faster increase in the normalized turbulent stretching term. Further
downstream, the multi-ﬂuid A case reaches its peak value much sooner than the single-ﬂuid case.
The behavior of the normalized turbulent stretching term indicates that the contributions to the
return to 3-D turbulence not only come from increased enstrophy ampliﬁcation, but also from
the change of alignment between vorticity and strain rate tensor eigenvectors. This diﬀerence is
hypothesized to be the result of the nonlinear eﬀects introduced by large density variations.

38

k0x05101500.511.5multi-fluid A casesingle-fluid case1.4.6 Mixing

When turbulence passes through a shock wave, the enhancement of mixing by STI is usually
expected (see ﬁgure 1.10).
In this section, the mixing characteristics of multi-ﬂuid turbulence
interacting with the normal shock wave are studied in more details. When the variable density
eﬀect was introduced, the modiﬁcation of turbulence by the shock was shown to be very diﬀerent
from that of single-ﬂuid case. This leads to changes in basic features of the characteristics of mixing
after the shock.

Figure 1.26 compares the scalar Taylor micro scale of heavy ﬂuid mole fraction or passive scalar
(λφ = (φ(cid:48)φ(cid:48)/(∂φ/∂x1)2)0.5) between multi-ﬂuid and single-ﬂuid cases. Both cases are normalized
using the values right before the shock. Immediately after the shock, the same decrease in λφ is
observed. These results and those shown in ﬁgure 1.10(b) for the Batchelor scale indicate that the
modiﬁcation of scalar length scales by the shock wave is not dependent on the density variations
in the pre-shock turbulence. Away from the shock, the multi-ﬂuid ﬂow exhibits a transient region
with slowly decreasing λφ until k0x ≈ 3.0. After the transient region, λφ for the mole fraction starts
to increase. On the contrary, λφ for the single-ﬂuid case continuously increases after the shock. It
was shown in ﬁgure 1.10 that the Batchelor scale λB for the multi-ﬂuid case becomes even larger
than that of the single-ﬂuid case close to the end of the domain. This is not observed for λφ and the
multi-ﬂuid case λφ values remain small throughout the post-shock region. This suggests that the
characteristics of the scalar ﬁeld as aﬀected by the shock and density variations are very diﬀerent
at small and intermediate length scales.

The transport equation for Favre-averaged scalar (passive scalar or heavy ﬂuid mass fraction)
variance is considered here to study the characteristics of scalar mixing for multi-ﬂuid turbulence
interaction with the shock wave. This equation is rearranged in the following form (Livescu et al.,
2000):

39

Figure 1.26: Plots of scalar Taylor micro scale for multi-ﬂuid (red) and single-ﬂuid (blue).

∂ ρ(cid:101)uj(cid:103)φ(cid:48)(cid:48)2

∂xj

∂xj

j φ(cid:48)(cid:48) ∂(cid:101)φ
= −2ρu(cid:48)(cid:48)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

term I
+2φ(cid:48)(cid:48) ∂Mj
∂xj

term V

−2M(cid:48)
j

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

∂φ(cid:48)
∂xj

term II

(ρu(cid:48)(cid:48)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
j φ(cid:48)(cid:48)2)

− ∂
∂xj

term III

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

+2 ∂
∂xj

j φ(cid:48))

(M(cid:48)

term IV

(1.11)

where Mj = (µ/Re0Sc)∂φ/∂xj. The LHS term in Eq. (4.3) represents the convection of the scalar
variance by the mean velocity and the terms on the RHS are: (I) production; (II) scalar dissipation;
(III) turbulent diﬀusion; (IV) viscous diﬀusion; and (V) compressibility.

Figure 1.27 shows that for the multi-ﬂuid case A the most important term is scalar dissipation (II).
In the vicinity of the shock wave, there is a transient process where the production (I) and turbulent
transport terms ((III)+(IV)) make relatively small contributions to the overall balance. These two
terms reduce to zero around k0x ≈ 5.0 and after that the advection is only balanced by scalar
dissipation. In contrast, the only dominating term for the single-ﬂuid case is the scalar dissipation

40

k0x051015λφ0.20.40.60.81multi-ﬂuidsingle-ﬂuidFigure 1.27: Contributions of diﬀerent terms in the scalar variance(cid:103)φ

(cid:48)(cid:48)2 transport equations for
multi-ﬂuid simulation. (a) Plots of all the contributing terms in the transport equation and (b)
balance of LHS and RHS of the transport equation.

throughout the post-shock region as shown in ﬁgure 1.28(b). To make a better comparison between
the two cases, the magnitude of scalar dissipation and advection terms are shown together in ﬁgure
1.29. Right after the shock, the same value for scalar dissipation is observed. As the ﬂow moves
away from the shock, the scalar dissipation for the multi-ﬂuid case starts to decrease, while that
of the single-ﬂuid case keeps increasing. However, after k0x ≈ 3.0, the scalar dissipation in the
multi-ﬂuid case starts to increase with a faster rate than that in the single-ﬂuid case. In the region
before k0x ≈ 10.0, the scalar dissipation of the multi-ﬂuid ﬂow is higher in magnitude, which
indicates enhanced mixing.

Another important feature of scalar mixing in the STI conﬁguration is that, unlike velocity ﬁeld
statistics, the direct eﬀects of the shock on the scalar statistics are almost the same in the single-
and multi-ﬂuid cases. This can be explained by the fact that the Rankine-Hugoniot relation for the
scalar is linear: φu=φd . This feature of scalar transport prevents direct changes in the scalar ﬁeld
by the shock. This trend exists so long as the pre-shock turbulent Mach number Mt and density
ratio in the multi-ﬂuid case is low. These criteria are satisﬁed in our reported simulations, so the
modiﬁcations of scalar statistics across the shock remain the same regardless of the VD eﬀects.
Furthermore, if certain scalar statistics only uses the data in y-z plane, that statistics is continuous
across the shock (e.g. scalar variance). For completeness, we need to mention that the evolution of

41

k0x051015-0.12-0.1-0.08-0.06-0.04-0.0200.020.04ProductionScalar DissipationCompressibility related termTransport termk0x051015-0.12-0.1-0.08-0.06-0.04-0.0200.020.04AdvectionRHS(a)gφ′′φ′′(b)gφ′′φ′′Figure 1.28: Contributions of diﬀerent terms in the(cid:103)φ

(b) single-ﬂuid simulations.

(cid:48)(cid:48)2 transport equations for (a) multi-ﬂuid and

the scalar away from shock is indeed aﬀected by the VD eﬀects.

The structure of the scalar ﬁeld can be studied by examining the scalar spectra. Figure 1.30
shows the comparison of the scalar (passive scalar or heavy ﬂuid mole fraction) variance spectra
at three locations: right before the shock, at peak TKE, and at k0x ≈ 4.0. The spectra right before
the shock wave match well with each other. However, downstream of the shock at k0x ≈ 4.0,
the multi-ﬂuid case shows more ampliﬁcation of small-scale ﬂuctuations (ﬁgure 1.30a). The more
signiﬁcant ampliﬁcation of these ﬂuctuations is accompanied by smaller scalar length scales λφ
and λB and higher rate of scalar dissipation. Comparison of the scalar variance spectra at the
peak TKE position shows almost the same proﬁles for the single- and multi-ﬂuid cases (ﬁgure
1.21b). We already argued that the modiﬁcation of scalar statistics (obtained by averaging over
planes perpendicular to the mean ﬂow direction) by the shock is not very sensitive to the pre-shock
turbulence. The collapse of scalar variance spectra immediately after the shock at peak TKE
position implies the scalar evolution away from the shock is mainly controlled by the post-shock
turbulence.

A mixing asymmetry in variable density turbulence was observed for Rayleigh-Taylor turbulence
and its mechanisms are studied by Livescu and Ristorcelli (2008, 2009); Livescu et al. (2010). Due
to the diﬀerential accelerations experienced by the ﬂuids with diﬀerent molecular weights, pure

42

k0x051015-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.06ProductionScalar DissipationCompressibility related termTransport termRHSk0x051015-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.06ProductionScalar DissipationCompressibility related termTransport termRHS(a)gφ′′φ′′(b)gφ′′φ′′Figure 1.29: Comparison of scalar dissipation and advection terms obtained from multi-ﬂuid and
single-ﬂuid simulations.

Figure 1.30: 2D spectra of scalar. The comparison are made at various locations: (a) right before
shock wave and k0x ≈ 4.0 and (b) right before shock wave and peak TKE position.

43

k0x051015-6-5-4-3-2-1012multi-fluid advectionmulti-fluid dissipationsingle-fluid advectionsingle-fluid dissipationgφ′′φ′′k100101102Sc(k)10-610-410-2multi-ﬂuidk0x≈−0.72multi-ﬂuidk0x≈4.0single-ﬂuidk0x≈−0.72single-ﬂuidk0x≈4.0k100101102Sc(k)10-610-410-2multi-ﬂuidk0x≈−0.72multi-ﬂuidpeakTKEsingle-ﬂuidk0x≈−0.72single-ﬂuidpeakTKE(a)(b)heavy ﬂuid mixes slower than the pure light ﬂuid in contrast to the Boussinesq approximation.
Such non-Boussinesq eﬀects in turbulence statistics are already observed for current multi-ﬂuid
STI results and a similar mixing asymmetry can be observed by comparing the PDFs of scalars
(heavy ﬂuid mole fraction, heavy ﬂuid mass fraction and passive scalar) and density. Figures 1.31,
1.32, 1.33 show the PDFs of scalar and density as obtained from the spanwise 2D plane data at
various streamwise positions for multi-ﬂuid A (mole fraction), B (mass fraction), and single-ﬂuid
(passive scalar) cases. For the multi-ﬂuid case A, ﬁgure 1.31 shows that the PDF of the inﬂow
heavy ﬂuid mole fraction is symmetrical. Before reaching the shock, the eﬀect of multi-ﬂuid
density variations on the mixing asymmetry is negligible. After passing through the shock wave,
however, the initially symmetric mole fraction PDF starts to become skewed and the skewness
keeps increasing during the post-shock development. The skewed PDFs suggest that light ﬂuid
vanish faster than the heavy ﬂuid. This is also shown in the density PDFs as the low density tails
of the PDFs vanish faster. In the multi-ﬂuid case B (ﬁgure 1.32), a similar behavior is observed
for density PDF evolution. Initially, a symmetrical mass fraction PDF results in a skewed density
PDF. The interaction with the shock itself has only a small eﬀect on the asymmetry. Away from the
shock, the skewness in the density PDF increases. This asymmetry in the multi-ﬂuid cases is related
to the larger kinetic energy in the light ﬂuid regions (see ﬁgure 1.14a above). This preferential
distribution of TKE suggests that the mixing is stronger in the low density regions. In contrast,
ﬁgure 1.33 shows that the single-ﬂuid scalar and density PDFs remain symmetrical throughout the
domain.

i/ρ = −u(cid:48)(cid:48)

1.4.7 Normalized Mass Flux and Density Speciﬁc Volume Covariance
The normalized mass ﬂux ai = ρu(cid:48)
i and density speciﬁc volume covariance b = −v(cid:48)ρ(cid:48),
where v = 1/ρ, are important quantities for the modeling of mixing in compressible turbulent
ﬂows with signiﬁcant inhomogeneity and density variations (Schwarzkopf et al., 2016). Both of
these two terms appear as a closure in the Favre-averaged Reynolds stress equations. Even though
in the current study, the terms involving ai and b may not be important to Reynolds transport

44

Figure 1.31: PDFs of multi-ﬂuid A case: (a) scalar (mole fraction) and (b) density ﬂuctuation at
various locations. The PDFs are calculated at speciﬁed y-z planes and then averaged over time.

Figure 1.32: PDFs of multi-ﬂuid B case: (a) scalar (mass fraction) and (b) density ﬂuctuation at
various locations. The PDFs are calculated at speciﬁed y-z planes and then averaged over time.

on average, the contributions from these terms might be signiﬁcant locally. More importantly,
as the density ratio of two ﬂuids increase, the interaction with the shock tends to generate strong
inhomogeneity, particularly when the shock becomes unstable and breaks. In these conditions, the
behavior and modeling of ai and b will be of great signiﬁcance. Considering these factors, the
behavior of normalized mass ﬂux and density speciﬁc volume covariance are further investigated
in this section.

Figure 1.34 compares a1 and b for both single- and multi-ﬂuid cases. The magnitude of these
two terms in multi-ﬂuid turbulence is comparably larger than those in the single-ﬂuid case. In ﬁgure

45

xheavy00.20.40.60.8100.511.522.533.5inletrightbeforeshockpeakTKEk0x=5.0k0x=9.24ρ−ρ-0.6-0.4-0.200.20.40.600.511.522.533.5inletrightbeforeshockpeakTKEk0x=5.0k0x=9.24(a)(b)φ00.20.40.60.8100.511.522.5inletrightbeforeshockpeakTKEk0x=4.0k0x=7.1ρ−ρ-0.6-0.4-0.200.20.400.511.522.533.5inletrightbeforeshockpeakTKEk0x=4.0k0x=7.1(a)(b)Figure 1.33: PDFs of single-ﬂuid case: (a) passive scalar and (b) density ﬂuctuation at various
locations. The PDFs are calculated at speciﬁed y-z planes and then averaged over time.

1.34(a), the pre-shock values of normalized mass ﬂux are shown to be small. After passing through
the shock wave, the streamwise normalized mass ﬂux is greatly ampliﬁed, indicating a strong
inhomogeneity in the streamwise direction, generated by the VD eﬀects. This can be explained
by the mismatch between the mean density jump in the simulated STI and the predicted value
by R-H relation as observed in Larsson et al. (2013) for single-ﬂuid ﬂows. Immediately after the
shock, the mean density jump is lower than the laminar R-H prediction. The mean density then
slowly returns to the laminar R-H mean density as the turbulence decays downstream of the shock,
introducing a small mean density inhomogeneity in the streamwise direction. This is strongly
ampliﬁed in the multi-ﬂuid simulations, as the wrinkling of the shock surface and variations in
shock strength increase the deviation from the R-H values. Further downstream, a1 attains a very
high magnitude for a short distance before decreasing, in agreement with previous observation that
the mean density returns to R-H value and the ﬂow becomes isotropic. In ﬁgure 1.34(b), the density
speciﬁc volume covariance for both cases are compared. As b is a measure of the mixing state, its
behavior is expected to be similar to that of scalar variance (ﬁgure 1.10a) outside the shock region,
at least when the non-Boussinesq eﬀects are not large. This is conﬁrmed by ﬁgure 1.34(b), which
shows that b slowly decreases before the shock and much faster after passing through the shock.
For the single-ﬂuid case, the density speciﬁc volume correlation remains statistically unimportant

46

φ00.20.40.60.8100.511.522.53inletrightbeforeshockpeakTKEk0x=5.0k0x=9.24ρ−ρ-0.1-0.0500.050.10510152025inletrightbeforeshockpeakTKEk0x=5.0k0x=9.24(a)(b)throughout the whole domain.

The transport equations for the normalized mass ﬂux components ai, which appear in unclosed

form in the transport equations for Reynolds stresses, are (Livescu et al., 2009):

(ρ(cid:101)ujai) =

(ρai) +

∂
∂t

∂
∂xj

term I

bPi(cid:124)(cid:123)(cid:122)(cid:125)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
+ρv(cid:48) ∂p(cid:48)
∂xi
term II
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

(aiaj)

∂
∂xj

+ρ

∂
∂xj

−ρaj

(cid:102)(u(cid:48)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
i − ai)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

term III
(ρ(cid:48)u(cid:48)
j) + ρu(cid:48)
iu(cid:48)

−( ∂
∂xj

∂u(cid:48)
j
∂xj

)

i

term V

term VI

+

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

iu(cid:48)
(ρ(cid:48)u(cid:48)
j − Ri j)

∂ ρ
∂xj

1
ρ

term IV

(1.12)

where Pi = ∂p/∂xi−∂τi j/∂xj. Figure 1.35 shows the terms in streamwise normalized mass ﬂux
equation. The contributing terms are normalized by its local value so that their relative magnitude
can be shown. Right after the shock, terms (II) and (VI) are the leading order terms and have
opposite sign. In the vicinity of the shock wave, positive net contributions from these two terms are
observed, which correspond to the sharp increase of a1 after the shock wave. Further downstream,
term (II) decreases and term (VI) increases. After k0x ≈ 2.0, term (II) becomes negative and keeps
decreasing. Term (VI) approaches 0 after k0x ≈ 5.0, making term (II) the only dominating term.
The overall behavior of a1 is dominated by term (II), while term (VI) contributes to the transient
process after the shock interaction and aﬀects the peak position of a1. The good balance of the
advection term with all the RHS terms once again conﬁrms the accuracy of the simulations.

The density speciﬁc volume covariance b, is a measure of the mixing state. This is an unclosed
term in the normalized mass ﬂux equations. Its governing equation has the form (Livescu et al.,
2009):

b +(cid:101)uj

∂
∂t

∂b
∂xj

= 2aj

(cid:124)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:125)

∂b
∂xj

term I

(cid:169)(cid:173)(cid:171)u(cid:48)
(cid:170)(cid:174)(cid:172)
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

j ρ(cid:48)v(cid:48)

∂
∂xj

+ρ

ρ

term III

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
+2ρv(cid:48) ∂u(cid:48)

j
∂xj

term IV

(1.13)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

−2aj(1 + b) ∂ ρ
∂xj

1
ρ

term II

47

Figure 1.34: Plots of (a) normalized mass ﬂux a1 and (b) density speciﬁc volume covariance
− (cid:104)v ρ(cid:105) for multi-ﬂuid and single-ﬂuid simulations.

Figure 1.35: Contributions of diﬀerent terms in the normalized mass ﬂux a1 transport equations
for multi-ﬂuid simulation.

48

k0x-505101520-0.02-0.0100.010.020.03multi-ﬂuida1single-ﬂuida1k0x-505101520-0.0100.010.020.030.040.05multi-ﬂuidsingle-ﬂuid(a)(b)k0x123456-20-10010203040AdvectionRHSIIIIIIIVVVIFigure 1.36: Contributions of diﬀerent terms in the density speciﬁc volume covariance b transport
equations for multi-ﬂuid simulation.

Figure 1.36 shows all terms in the transport equation of b (normalized by its local value). The
only important term in this equation is term (IV). Terms (I), (II) and (III) have nonzero contributions
close to the shock in the post-shock region, but their total contribution to b is negligible. The
‘transient’ region ends around k0 ≈ 3.0; after that term (IV) becomes the only nonzero term and
remains nearly constant. It can be concluded that, the modeling of term (IV) is critical to the RANS
simulation of variable density mixing in STI. Figure 1.36(b) shows the good balance between the
advection and RHS terms.

1.5 Conclusions

A hybrid numerical method, which combines a monotonicity-preserving scheme with a compact
scheme, is used for high-order numerical simulations of isotropic multi-ﬂuid turbulence interacting
with a planar normal shock wave. A reference simulation of single-ﬂuid turbulence is also conducted
with similar conditions. The main objective is to study the eﬀects of density variations on the shock-
turbulence interaction and mixing. Convergence tests are conducted to establish the accuracy of
numerical results using diﬀerent meshes with a wide range of grid sizes. The computed statistics

49

k0x123456-10-8-6-4-202AdvectionRHSIIIIIIIVare found to be independent of the grid when the turbulence after the shock is well resolved and the
scale separation between numerical shock thickness and turbulent scales is adequate.

To further establish shock-capturing simulation as an eﬀective tool in studying STI, LIA con-
vergence tests are conducted to show that shock-capturing simulations exhibit similar converging
trend to LIA predictions. As Reλ increases, both the streamwise and spanwise components of
Reynolds stress tensor show converging trends to LIA predictions. Stronger Mt eﬀects on the
LIA convergence of the spanwise component are observed. The streamwise component, however,
shows no converging trend toward LIA predictions at low Reλ. Further results show LIA limits for
streamwise component can be approached using turbulence-resolving shock-capturing simulations
with a minimum Reλ around 45 and a coarser grid compared to shock-resolving DNS. This advan-
tage over DNS makes the shock-capturing simulations cheaper and at the same time able to achieve
accurate predictions.

The comparison between two multi-ﬂuid cases and a single-ﬂuid case is made to identify and
explain the eﬀects of major density variations on the ﬂow structure and turbulence statistics in
the STI conﬁguration. Due to the strong nonlinear eﬀects introduced by density variations, the
modiﬁcation of turbulence statistics by the normal shock is shown to be diﬀerent from that of
the single-ﬂuid case and LIA prediction, with an increased ampliﬁcation of turbulence variables
(like TKE and vorticity variance) and a more signiﬁcant reduction in turbulence length scales.
Turbulent mixing enhancement by the shock is also more signiﬁcant in the multi-ﬂuid cases. The
redistribution of turbulence statistics across the shock wave is another important feature of multi-
ﬂuid STI, as reﬂected in the changes of conditional expectations. For thermodynamic variables,
the isentropic relations of ﬂuctuations are satisﬁed for the single-ﬂuid case before the shock. After
passing through the shock wave, both acoustic and entropy modes are important for the correlations
of thermodynamic properties. For the multi-ﬂuid cases, the nonlinear eﬀects and coupling between
diﬀerent modes complicates the analysis and further investigation is needed. It is also found that
TKE has a preferential distribution on the pure ﬂuid regions after the shock wave, while vorticity
ampliﬁcation is strongest in the mixed ﬂuid areas. This diﬀerence is attributed to the diﬀerent

50

roles density plays for these quantities. The eﬀects from diﬀerent density PDFs in the pre-shock
turbulence are examined by comparing two multi-ﬂuid cases. Both multi-ﬂuid cases exhibit very
similar behavior across the shock even though their density ﬁelds are diﬀerent. Other parameters
such as At and ks may also aﬀect the interaction, but they are not the focus of current study.

A detailed study of turbulent budgets for Reynolds stresses, vorticity variance, and scalar
variance is conducted and the dominating mechanisms in the post-shock region are identiﬁed. The
rapid increase in the streamwise Reynolds stress component after the shock wave is mainly caused
by the pressure-strain and velocity-pressure correlation, whereas the decrease in the transverse
Reynolds stress is mostly due to pressure-strain term, which are consistent with the observations
made in the single-ﬂuid simulation. Away from the shock wave, as transient processes vanish,
the viscous dissipation becomes dominating and causes both Reynolds stresses to decrease. After
being normalized using the corresponding local Reynolds stresses values, the contributing terms
become similar in magnitude. The energy spectra of transverse kinetic energy are diﬀerent at
the same absolute position k0x ≈ 4.0, conﬁrming multi-ﬂuid and single-ﬂuid cases have diﬀerent
post-shock length scales. For the vorticity variance, the dominating terms are turbulence stretching
(positive contribution) and viscous diﬀusion (negative contribution). Turbulence stretching plays
an important role in understanding the post-shock behavior: it attains small values right after the
shock due to the two-dimensionalization of turbulence; during the return to a three-dimensional
structure, the normalized turbulence stretching for the multi-ﬂuid case increases faster and reaches
the peak earlier compared to the single-ﬂuid case. The mechanism that is responsible for enhanced
mixing has been identiﬁed to be the increased scalar dissipation. The modiﬁcation of scalar
statistics, as measured by changes in variables like Batchelor scale, scalar Taylor micro scale, scalar
dissipation across the shock wave is shown to be similar in single- and multi-ﬂuid simulations,
which is attributed to the linear Rankine-Hugoniot relation for the scalar (φu=φd). The mixing
asymmetry is more pronounced after the interaction with the shock for the multi-ﬂuid cases. This
can be explained by the preferential distribution of TKE in the low density regions. Finally, the
mechanisms that are responsible for the post-shock development of turbulence are identiﬁed and

51

used to assess the main characteristics of variable density compressible ﬂows.

52

CHAPTER 2

DENSITY EFFECTS ON THE POST-SHOCK TURBULENCE STRUCTURE AND

LAGRANGIAN FLOW DYNAMICS

2.1 Introduction

In the ﬁrst chapter, we have examined the turbulence statistics, turbulence budgets, conditional
statistics, and energy spectrum in the multi-ﬂuid STI and found that the nonlinear eﬀects from
the density variations signiﬁcantly change the turbulence properties in both physical and spectral
spaces. There have been few other detained study of multi-ﬂuid or variable density STI in the
literature. For example, Jin et al. (2015); Huete et al. (2017) considered a reactive shock wave in a
premixed mixture and used LIA and shock-capturing simulations to study the detonation-turbulence
interaction. However, there still exist many gaps in our knowledge of the variable density eﬀects
on the post-shock turbulence structure and ﬂow topology.

In this chapter, we will present a comprehensive study of the density eﬀects on the post-shock
turbulence structure from two perspectives: 1) the velocity gradient tensor and its dynamics and
2) Lagrangian statistics of non-inertial ﬂuid particles. The properties of the velocity gradient
tensor (VGT) determine a wide variety of turbulence characteristics, such as the ﬂow topology,
deformation of material volume, energy cascade, and intermittency. Understanding both the VGT
ﬁeld immediately after the shock-wave and its dynamics as the ﬂow evolves away from the shock
wave are also crucial to the development of subgrid-scale models that can accurately describe the
shock interaction and return-to-isotropy eﬀects. Chong et al. (1990) has proposed an approach to
classify the local ﬂow topology and structure using the invariants of VGT. The dynamical behavior
of the VGT has been studied for incompressible ﬂows using the Lagrangian evolution of the
invariants along conditional mean trajectories (CMT) (Meneveau, 2011). The statistics regarding
the invariants of VGT and their Lagrangian dynamics have been used to understand the structure
of turbulence in many canonical ﬂows, such as isotropic turbulence, turbulent boundary layer and

53

mixing layers (e.g. Chong et al., 1998; Ooi et al., 1999; Wang and Lu, 2012). Previous studies on
single-ﬂuid STI have examined some of the statistics of the PDF of VGT. Ryu and Livescu (2014);
Livescu and Ryu (2016) took a step further to investigate the turbulence structure and vorticity
dynamics based on the examination of VGT invariants. By taking advantage of the Shock-LIA
procedure, they extracted the statistics of VGT and its invariants for a wide range of shock Mach
numbers. On the other hand, the dynamics of VGT as the turbulence evolves away from the shock
wave cannot be examined using the Shock-LIA procedure. Our preliminary numerical studies on
variable density STI have revealed some important features of VGT (Tian et al., 2017c, 2018).
The relation between velocity and scalar ﬁeld has also been studied by Boukharfane et al. (2018)
to better understand mixing in this ﬂow conﬁguration. However, these studies have not yet fully
revealed the variable density eﬀects on the post-shock turbulence/scalar structure.

It is well-known that Lagrangian statistics used to understand the transport of a localized
source of passive scalars. Essentially, if a group of marked ﬂuid particles is released from a
localized source of passive contaminants, it is well-known that the mean concentration ﬁeld can
be expressed in terms of "on-particle" statistics, and the concentration variance is similarly related
to "two-particle" statistics of the relative motion between ﬂuid particle pairs. These Lagrangian
statistics have been extensively studied using DNS in the past few years due to the diﬃculty in
experimental measurement of ﬂuid particles. Yeung and Pope (1989) has used DNS database to
extract single-particle Lagrangian statistics for isotropic turbulence. Squires and Eaton (1991)
has also presesnted one-particle Lagrangian statistics for homogeneous shear ﬂow. Particle-pair
dispersion statistics have been reported by Yeung (1994); Ishihara and Kaneda (2002); Yeung and
Borgas (2004); Biferale et al. (2005). These Lagrangian statistics have been used for the validation
of Richardson’s theory on particle dispersion (Richardson, 1926) and the development of stochastic
models (Durbin, 1980; Thomson, 1990; Komori et al., 1991; Pope, 2002), PDF methods (Pope,
1994) and so on. To extend the applicability of these models, researchers have been studying the
particle dispersion in other anisotropic and inhomogeneous type of ﬂows. Shen and Yeung (1997);
Yeung (1997) studied the particle dispersion in homogeneous turbulent shear. Choi et al. (2004)

54

then reported the Lagrangian statistics in turbulent channel ﬂow. The progress in theoretical and
numerical progress in Lagrangian particle study has been reported in the review papers Sawford
(2001); Yeung (2002); Salazar and Collins (2009); Toschi and Bodenschatz (2009).

This study aims at using the database of the turbulence-resolving shock-capturing simulations
of multi-/single- ﬂuid STI generated in the ﬁrst chapter to: 1) further study the density eﬀects on the
turbulence structure immediately after the shock wave, 2) perform the ﬁrst Lagrangian simulations
of this conﬁguration to understand the dynamical behavior of VGT away from the shock, and 3)
perform an Lagrangian analyses of the particle dispersion in the post-shock turbulence. The paper
is organized as follows. Details of the simulations and the testing conducted to assess the accuracy
of results are discussed in section 2.2. Results are presented in section 2.3 and concluding remarks
are made in section 2.4.

2.2 Numerics and Accuracy

In this section, we ﬁrst brieﬂy discuss the numerical approach used for shock-capturing
turbulence-resolving simulations in our previous study (Tian et al., 2017a), from which we have
extracted the VGT statistics addressed in this paper. The conﬁguration of the variable-density STI
extension is described next, followed by a discussion of the new Lagrangian simulations used to
examine the VGT dynamics away from the shock.

2.2.1 Governing Equations and Numerical approach

The conservative form of the dimensionless compressible Navier-Stokes equations for ﬂows with
two miscible species (i.e. continuity, momentum, energy, and species mass fraction transport
equations) were solved numerically together with the perfect gas law using a high-order hybrid
numerical method (Tian et al., 2017a). The inviscid ﬂuxes for the transport equations were
computed using the ﬁfth-order Monotonicity Preserving (MP) scheme, as described in Suresh and
Huynh (1997); Li and Jaberi (2012); Tian et al. (2017a). The molecular transport terms were
calculated using the sixth-order compact scheme (Lele, 1992). The 3rd-order Runge-Kutta scheme

55

Figure 2.1: Instantaneous contours of vorticity and shock surface in isotropic turbulence
interacting with a Mach 2 shock. (a) Vortex structure, identiﬁed by the Q criteria (Q = 2 (cid:104)Qw(cid:105))
and colored by the mole fraction of the heavy ﬂuid. Fluid particles are initialized as a sheet that
spans over the homogeneous directions at a given post-shock streamwise position and allowed to
develop with the ﬂow. (b) Visualized particle sheet, convected and distorted by the post-shock
turbulence. The instantaneous shock surface is colored by the shock intensity across the shock for
(c) single-ﬂuid and (d) multi-ﬂuid cases.

was used for time advancement.

2.2.2 Numerical Setup of Eulerian Simulation

The numerical setup of the Eulerian simulation has been explained in detail in the ﬁrst chapter.

2.2.3 Numerical Detials of the Lagrangian Analyses

For the current study, we have tracked more than 4.5 million particles that are initialized uniformly
at various streamwise positions (cid:174)x0, and calculated various statistics concerning turbulence structure
following their trajectories. The aim is to understand the evolution of ﬂow structures following
ﬂuid particles. Figure 2.1 (a) marks with red lines a typical streamwise plane where particles are

56

initialized. The particles are then convected by the instantaneous turbulent velocity obtained by
turbulence-resolving shock-capturing simulations (Tian et al., 2017a) and moved to another plane
marked by the blue lines. At this stage, the initially ﬂat particle sheet is distorted by the turbulence
as shown in ﬁgure 2.1 (b).

The ﬂuid particles are non-inertial and follow the local ﬂow velocity. The corresponding

transport equations for particle positions x+

i are:

i (t| (cid:174)x0, t0)
dx+
dt
i (t| (cid:174)x0, t0) = ui(x+
i , t)
u+

= u+

i (t| (cid:174)x0, t0),

(2.1a)

i (t| (cid:174)x0, t0) can be obtained from the Eulerian velocity ﬁeld ui(x+

(2.1b)
i (t| (cid:174)x0, t0) represents the positions of the particles at time t that are initialized at (cid:174)x0 and time
where x+
i , t) by
t0. The particle velocity u+
interpolation. The interpolation is based on the cubic spline scheme, whose accuracy in predicting
particle positions has been studied in Yeung and Pope (1988). In the STI conﬁguration, there is a
sharp change of the ﬂow velocity at the shock, which deteriorates the interpolation accuracy. To
achieve accurate interpolation of the particle velocity, the domain is partitioned into three diﬀerent
regions as shown in ﬁgure 2.1 (a): pre-shock, shock, and post-shock regions. The instantaneous
shock surface is identiﬁed using the sensor: s = −θ/(|θ| + (cid:104)ωiωi(cid:105)0.5
yz ) > 0.5, where θ = ∂ui/∂xi
is the dilatation, ωi = i jk ∂uk/∂xj is the vorticity, and (cid:104)(cid:105)yz represents the instantaneous average
over the homogeneous directions. After the instantaneous shock region is identiﬁed, the pre- and
post-shock turbulence ﬁelds can be separated for interpolation. Lagrangian dynamics of particles
across the shock wave is not considered in this study.

To extract single-particle dispersion statistics in the post-shock turbulence, we have sampled
particles that are uniformly distributed at the y − z plane with k0x ≈ 0.64, and followed them to
the end of the computational domain. This streamwise location is chosen such that the particles are
away from the wrinkled shock wave so that the interpolation scheme is not aﬀected by the shock
jump. At the same time, we would like to ensure the particles are close enough to the shock to
capture the eﬀects of the post-shock transient process on particle dispersion. On top of that, the ﬂow

57

has reached a statistically steady-state before the Lagrangian particle statistics are being computed.
This allows us to initialize the particles in three homogeneous dimensions y, z and t with k0x being
ﬁxed. The total lifespan of the each of the particles, tend − tstart, can be approximately calculated
as 2π/ud, where ud is post-shock mean velocity.

For particle pair statistics, the separation vectors need to be clearly deﬁned due to the anisotropy
of the post-shock turbulence, in contrast to that in isotropic turbulence. Due to the homogeneity
in the spanwise direction, the separation vectors can be divided into streamwise and spanwise
directions. Here we denote the separation vector as (lx, ly, lz), where lx, ly, and lz are the relative
distance between the particles. Again, the homogeneity in the spanwise direction allows us to
rewrite the separation vector as (lx, lyzsinα, lyzcosα), where lyz is the relative distance in y − z
plane and α is the angle between separation vector and z-axis. The only dominating parameters are
lyz and lx. The angle α should be irrelevant in particle pair statistics and can be any random value.
2rad. The particle separation vectors can then be
For convenience, we have only use α = 0 and π
deﬁned as (lx,0,0) and (0, lyz,0) or (0,0, lyz), where the latter two are equivalent.

2.2.4 Grid and Statistical Convergence

The accuracy of the numerical results is addressed in this subsection through a series of convergence
tests. To ensure that all the turbulence length scales are well resolved, a grid convergence test was
conducted in Tian et al. (2017a). Here, we summarize these results for completeness, together with
additional convergence results for small-scale quantities. Figure 2.2 shows the velocity dissipation
rate ε and mass fraction dissipation rate εφ as a function of the normalized streamwise direction k0x
for a series of meshes. The grey regions in the following ﬁgures indicate the unsteady shock region,
inside which the results are aﬀected by the shock wrinkling and unsteady shock movement. As the
grid is reﬁned in all three directions, both quantities display convergence, proving the accuracy of
the turbulence database. Another aspect that needs to be considered is the scale separation between
the numerical shock thickness δn and the Kolmogorov length scale η as suggested in our previous
study (Tian et al., 2017a). With the ﬁnest mesh (512×512×2048), the scale separation ratio η/δn is

58

around 1.9, which is suﬃcient for resolving the interaction between the numerical shock wave and
small-scale turbulent motions. Therefore, in the current study, we have obtained all the statistics
from the turbulence ﬁeld based on the ﬁnest grid to ensure accuracy. Secondly, LIA convergence
tests were conducted following Ryu and Livescu (2014) to show that the shock-capturing simulations
can capture the correct limits. Turbulent Mach number Mt and Taylor Reynolds number Reλ were
varied for the canonical single-ﬂuid simulations, covering a wide range of parameter space. The
shock-capturing simulation results do converge to LIA predictions for individual Reynolds stress
components as long as certain conditions are satisﬁed (Tian et al., 2017a). This was the ﬁrst time that
the asymptotic values for individual Reynolds stresses were approximated using shock-capturing
simulations.

Statistical convergence is another important factor that needs to be addressed before any further
analysis. To reduce the statistical variability, all the results that are based on the Eulerian data are
space-averaged over homogeneous directions and time-averaged for around two pass-over times.
The averaging is performed after the ﬂow has reached a statistically steady state to eliminate the
eﬀects of transient processes (Larsson et al., 2013). For the Lagrangian statistics, the number
of ﬂuid particles needs to be large enough for statistical convergence, especially for conditional
averaged statistics. We deﬁne the conditional averaged quantity X(A, B), which depends on the
random variables A and B using the following deﬁnition:

(cid:104)X|(A = A0, B = B0)(cid:105) = (cid:104)X|(A0 − 1

(B0 − 1

1
2 ∆A),
2 ∆A) ≤ A < (A0 +
1
2 ∆B)(cid:105)

2 ∆B) ≤ B < (B0 +

(2.2)

where ∆A and ∆B are the bin sizes. The conditional statistics are obtained by ensemble averaging
(denoted by (cid:104)(cid:105)) over all the ﬂuid particles that fall into the bins. The number of samples needed to
achieve statistical convergence will be examined for diﬀerent bin sizes. Figure 2.3 and 2.4 show the
Dt (cid:105)/(cid:104)Qw(cid:105)2 and
convergence of two important conditional Lagrangian statistics (cid:104) DQ
their standard deviation (see section 2.3.3 for deﬁnitions), depending on the number of particles

Dt (cid:105)/(cid:104)Qw(cid:105)3/2, (cid:104) DR

59

Figure 2.2: Results of multi-ﬂuid grid convergence tests at Reλ = 45 and Mt = 0.09. Streamwise
development of (a) velocity dissipation rate ε and (b) mass fraction dissipation rate εφ is shown.
The region of unsteady shock movement is marked in grey. Grid numbers for Grid 1 to 5 are
256×256×1024, 384×384×1024, 384×384×1536, 512×512×1536, 512×512×2048.

in each bin. For the multi-ﬂuid case, we note that the convergence of both conditional means and
standard deviations can be achieved when using around 10,000 particles, larger than that needed
for the canonical single-ﬂuid case as shown in ﬁgure 2.4. This suggests that the variable density
eﬀects make the simulations more computationally demanding. The eﬀects of the bin sizes are also
examined by comparing three diﬀerent set of bin numbers 30× 30, 40× 40 and 60× 60 in the (Q, R)
phase plane at the same point (3.0,3.0). These bin numbers correspond to the following bin sizes:
(1.3,1.3), (1.0,1.0) and (0.67,0.67). Our analysis indicate that the statistics converge to almost the
same value when the sample size is large enough. In the present study, we uniformly sampled more
than 4.5 million particles and made sure that there are at least 10,000 particles in each sample bin
with the number of bins being 40 × 40 ((∆Q, ∆R) = (1.0,1.0)).

2.3 Results and Discussions

The variable density eﬀects on the post-shock turbulence structure and dynamics are examined
in this section. The results obtained from the multi-ﬂuid STI simulation are compared with those of
a reference single-ﬂuid case and standard isotropic turbulence. First, the post-shock turbulence state
and its evolution away from the shock wave are examined to identify the variable density eﬀects.

60

k0x01020ε050100150k0x01020εφ00.020.040.060.080.10.12Grid-1Grid-2Grid-3Grid-4Grid-5(a)(b)Figure 2.3: The statistical convergence for (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (DR/Dt)/(cid:104)Qw(cid:105)2 and (b)
their standard deviations conditioned at point (3.0,3.0) in the (Q, R) phase plane for multi-ﬂuid
case. The number of bins are 30 × 30 (solid), 40 × 40 (dashed) and 60 × 60 (dotted).

Figure 2.4: The statistical convergence for (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (DR/Dt)/(cid:104)Qw(cid:105)2 and (b)
their standard deviations conditioned at point (3.0,3.0) in the (Q, R) phase plane for single-ﬂuid
case. The number of bins are 30 × 30 (solid), 40 × 40 (dashed) and 60 × 60 (dotted).

61

number of samples in each bin102103104-0.4-0.3-0.2-0.100.10.20.3number of samples in each bin1021031040.511.522.533.5(a)(b)σ(DRDt)/hQwi2σ(DQDt)/hQwi3/2DRDt/hQwi2DQDt/hQwi3/2number of samples in each bin102103104-1.5-1-0.500.5number of samples in each bin1021031040.811.21.41.61.82(a)σ(DQDt)/hQwi3/2σ(DRDt)/hQwi2DRDt/hQwi2DQDt/hQwi3/2(b)The results are based on temporal- and spatial-averaged statistics obtained from the Eulerian data.
Then, the ﬂow topology is analyzed to further understand the post-shock turbulence evolution. The
dynamics that dominate the transient evolution of post-shock turbulence structure are examined
using the Lagrangian equation of VGT and Lagrangian data collected for sample ﬂuid particles.

2.3.1 Density Eﬀects on the Post-shock Turbulence and Its Evolution Away From the Shock

2.3.1.1 Turbulence State Immediately After the Shock

In our earlier study on multi-ﬂuid STI, we have focused on turbulent length scales, Reynolds
stresses, vorticity variance, scalar variance, and their budgets (Tian et al., 2017d,a) as the ﬁrst
quantities to examine and compare with the single-ﬂuid STI predictions. It was found that, in the
variable-density case, the small-scale turbulent motions are further enhanced after the interaction
with the shock. In addition, the vortex stretching mechanism is more suppressed, but it experiences
a faster increase during the development away from the shock. In this section, we take a step further
to examine the post-shock turbulence structure in details.

The PDFs of streamwise and spanwise (averaged over the two periodic directions right after
the shock at k0x = 0.5) longitudinal velocity derivatives for both single-ﬂuid and multi-ﬂuid cases
are shown in ﬁgure 2.5 alongside with the Gaussian distribution as a reference. The non-Gaussian
nature of the velocity gradient PDFs and their connection to the energy cascade and intermittency are
well documented in the turbulence literature. Here, we note that immediately after the shock wave,
the PDF of the spanwise velocity gradient for both cases remains negatively skewed, as in isotropic
turbulence. The streamwise component, however, becomes more symmetric and Gaussian-like due
to the interaction with the shock wave. This indicates that the energy transfer to small scales is
suppressed in the streamwise direction. We also note that the density has a relatively weak eﬀect
on the velocity derivatives PDFs since the single-ﬂuid and multi-ﬂuid cases have similar PDFs.

The preferential ampliﬁcation of the transverse components of the rotation and strain rate tensors
is a well-known eﬀect in STI and has been extensively studied for the canonical single-ﬂuid ﬂows

62

Figure 2.5: Comparison of the PDFs of the normalized post-shock velocity derivatives with a
Gaussian distribution.

(Mahesh et al., 1997; Ryu and Livescu, 2014; Livescu and Ryu, 2016). This ampliﬁcation can lead
to an increase in the correlation between the two quantities. When density variations are introduced,
Tian et al. (2017a) showed that the transverse components of vorticity variance are further enhanced
across the shock wave while the streamwise component retains almost the same value. To better
understand the variable density eﬀects on post-shock turbulence, the PDF of the strain-enstrophy
angle, Ψ, is considered in ﬁgure 2.6. Ψ is calculated using Ψ =tan−1(Si jSi j/(Wi jWi j)), where
Si j = 1/2(Ai j + Aji) and Wi j = 1/2(Ai j − Aji) are the strain and rotation tensors and Ai j = ∂ui/∂xj
is the velocity gradient tensor. In isotropic turbulence, the PDF of Ψ peaks at 90◦ (Jaberi et al.,
2000), indicating a strain dominated ﬂow.
In single-ﬂuid post-shock turbulence, the PDF of Ψ
exhibits a shift of the peak from 90◦ to around 45◦, as the shock Mach number increases. This has
been observed by Livescu and Ryu (2016) and is interpreted as the increase in correlation of strain
and rotation. However, in the multi-ﬂuid case, the peak still occurs at relatively large angles and
the increase in correlation is not as pronounced as that in the single-ﬂuid case, at the same shock
Mach number. Figure 2.6 implies that the rotation and strain are ampliﬁed diﬀerently by the shock
when large density variations are present, which compromises the correlation between the above
two quantities.

63

∂u/∂x-8-6-4-20246810-410-310-210-1100Gaussian∂u/∂xpost-shock(m)∂v/∂ypost-shock(m)∂u/∂xpost-shock(s)∂v/∂ypost-shock(s)Figure 2.6: PDF of the strain-enstrophy angle Ψ in radian for post-shock turbulence.

Figure 2.7: Conditional expectation of the magnitude of (a) rotation tensor and (b) strain rate
tensor as a function of density before and after the shock.

64

Ψ(rad)00.511.5PDF00.20.40.60.81multi-ﬂuidpost-shocksingle-ﬂuidpost-shockρ−ρ-1-0.500.5111.522.533.544.55pre-shockWijWijpost-shockWijWijρ−ρ-1-0.500.51123456789pre-shockSijSijpost-shockSijSij(a)(b)To further understand the eﬀects of the density variations on the strain and rotation tensors as the
ﬂow interacts with the shock wave, the pre- and post-shock values of the conditional expectations
of the magnitudes of the two tensors are shown in ﬁgure 2.7 for the multi-ﬂuid case. As observed,
the correlation between vorticity (and strain rate) with density is completely diﬀerent before and
after the shock wave. Through the shock wave, the ampliﬁcation of vorticity is stronger in the
mixed ﬂuid regions with near average density, but weaker in the pure ﬂuid regions. This is not
observed in the single-ﬂuid simulation. One mechanism that might be responsible for this behavior
is the baroclinic torque: (∇ρ × ∇p)/ρ2 in the vorticity transport equation. Consider a wrinkled
shock surface extracted from the multi-ﬂuid simulation as shown in ﬁgure 2.8 (a); a strong pressure
gradient ∇p exists through the shock wave as marked in the ﬁgure; large density gradients ∇ρ also
exist, especially in the mixed ﬂuid regions. Since the pre-shock density ﬁeld is isotropic, ∇ρ and
∇p can be locally misaligned, becoming a source of vorticity generation through the baroclinic
torque. In the pure ﬂuid regions or single-ﬂuid simulation, however, the density gradients are much
smaller, so that the cross product of ∇p and ∇ρ is also small. To conﬁrm this, the PDF of the angle
between the density gradient in the transverse direction and the vorticity vector is examined in
ﬁgure 2.8 (b). Before the shock wave, it is noted that the PDF of the cosine of the angle between the
density gradient and vorticity vectors is slightly tilted towards the 0-degree angle. After the shock
wave, the multi-ﬂuid case exhibits a stronger tendency of vorticity vector being perpendicular to
the density gradient. In contrast, the alignment is completely diﬀerent after the interaction with
the shock wave for the single ﬂuid case, as the two vectors are more likely to be parallel to each
other. This provides evidence that density gradients and baroclinic torque play important roles in
establishing the preferential deposition of vorticity across the shock wave.

Figure 2.9 can help visualize the changes in the vortex structures across the shock wave. The
vortex tubes are captured using the Q-criterion and are colored by their local density. Figure 2.9
(a) shows the vortex structures for pre-shock multi-ﬂuid isotropic turbulence. For the visualized
vortex tubes, there are no identiﬁable eﬀects from the density variations; the vortex tubes are not
preferentially distributed due to the density eﬀects. However, the interaction with the shock has a

65

Figure 2.8: (a) Demonstration of the baroclinic torque generation across the shock wave and (b)
PDF of the orientation between the vorticity vector and density gradient in y-z direction
immediately after the shock wave.

clear eﬀect on the post-shock vortical structures (ﬁgure 2.9 b). Immediately behind the shock wave,
the vortex tubes are aligned in the spanwise directions, which has been observed in previous STI
studies. More importantly, the vortex tubes also get aligned with the density iso-surfaces, meaning
that the vorticity becomes perpendicular to the density gradient. This is consistent with the earlier
analysis of the baroclinic torque. As a consequence, the post-shock vorticity ﬁeld will enhance the
mixing between adjacent density regions which, in return, will weaken the transverse vortical ﬁeld
and accelerate the return to an isotropic state. This coupling is further explored in the next section.
On the other hand, ﬁgure 2.7 (b) shows that the magnitude of the strain rate tensor tends to be
stronger in the heavy ﬂuid regions and weaker in the light ﬂuid region. This trend is hypothesized
to be related to the dependence of shock strength on the pre-shock density. Tian et al. (2017a)
showed that shock strength is positively correlated with density. In the highest density regions, the
shock compression is also stronger, while it is weaker in the smallest density regions, leading to
the observed trend in the ampliﬁcation of the strain rate tensor magnitude. This trend is diﬀerent
from that observed for the vorticity, which is explained above. As a result, the trend of the strain-
enstrophy angle PDF peaking around 45◦, observed in the single ﬂuid case at higher shock Mach
numbers is weakened in the multi-ﬂuid case.

66

Figure 2.9: Vortex structures captured using the Q-criterion, colored by density, for multi-ﬂuid (a)
pre-shock turbulence and (b) post-shock turbulence.

2.3.1.2 Evolution of Turbulence State Away from the Shock Wave

The evolution of variable density turbulence away from the shock wave involves many coupled
nonlinear processes, some of which have been discussed through the turbulence budgets analysis
in Tian et al. (2017a). In this section, the evolution of turbulence structure will be examined in
details.

Figure 2.10 shows the development of some of the fundamental turbulence statistics. The
evolution of these statistics helps in the understanding of the general characteristics of single- and
multi-ﬂuid STI. The enhancement of turbulent kinetic energy (TKE) and dissipation due to the
interaction with the shock is a well-known eﬀect in STI. With the introduction of strong density
variations, the eﬀect becomes larger, as shown in ﬁgure 2.10 (a). Away from the shock wave,
TKE of the multi-ﬂuid case exhibits a similar transient behavior compared to the single-ﬂuid case,
but it reaches its peak earlier (k0x ≈ 2.0 versus k0x ≈ 3.1). The mechanism that dominates this
transient process has been shown to be related to the fast decay of the acoustic waves (Tian et al.,
2017a). To highlight the decay of the acoustic waves, ﬁgure 2.10 (b) shows the pressure ﬂuctuations
variance as a function of the streamwise position. The ampliﬁcation of pressure ﬂuctuation across
the shock wave is noted, agreeing with Sethuraman et al. (2018). The acoustic wave is stronger

67

in the multi-ﬂuid case immediately after the shock wave. This is related to the shock intensity
ﬂuctuations induced by the strong density variations (Tian et al., 2017a). As a result, the decay of
the acoustic ﬁeld is also faster for the multi-ﬂuid case, causing a faster increase of TKE. After the
post-shock transient pressure adjustment, the multi-ﬂuid case still exhibits larger absolute pressure
ﬂuctuations. However after normalizing with ρu(cid:48)u(cid:48), the pressure ﬂuctuations become somewhat
similar in magnitude in these two cases.

In ﬁgure 2.10 (c), the vortex stretching term Σ is decomposed into streamwise and spanwise
directions to explore the axisymmetric state and return-to-isotropy of post-shock turbulence. Pre-
vious studies (Livescu and Ryu, 2016; Tian et al., 2017a) have shown that the normalized vortex
stretching term reaches a low value after passing through the shock wave, indicating a tendency
towards an axisymmetric state. Figure 2.10 (c) shows that, without normalization, the absolute
values of the vortex stretching terms are magniﬁed in both single- and multi-ﬂuid cases, more so
for the spanwise component. The two components then undergo a transient process, where they
ﬁrst increase and cross each other, before the ﬂow returns to an isotropic state.

k ω(cid:48)

k − δi j

In order to quantitatively study the evolution of turbulence anisotropy, we consider here the
Reynolds stress anisotropy tensor deﬁned as bi j = u(cid:48)
ku(cid:48)
j/u(cid:48)
iu(cid:48)
k − δi j
3 (Livescu et al., 2002). A similar
iω(cid:48)
anisotropy tensor, di j, can also be deﬁned for the vorticity ﬁeld, as di j = ω(cid:48)
j/ω(cid:48)
3 . Due
to the homogeneity in spanwise directions, the diagonal components of the anisotropy tensor are
related by b22 = b33 = −0.5b11, so only b11 is discussed. The near zero value of b11 ≈ 0.0 is
an indication that ﬂow has reached an isotropic state, while b11 ≈ −1/3 means that the turbulent
ﬁeld has a tendency towards a 2D axisymmetric state. Figure 2.10 (d) shows that d11, a small-scale
turbulent variable, reaches around -0.3 in the multi-ﬂuid case, which is lower than that observed in
the single-ﬂuid case. This indicates that variable density intensiﬁes the trend towards axisymmetric
state for small-scale turbulence. On the other hand, the stronger turbulent stretching mechanism
as observed in ﬁgure 2.10 (c), makes the return to isotropy much faster in the multi-ﬂuid case as
compared to that in the single-ﬂuid case. For Reynolds stresses, which are large scale turbulence
statistics, the multi-ﬂuid ﬂow reaches a quasi-isotropic state immediately after the shock wave

68

Figure 2.10: Development of (a) TKE and dissipation, (b) pressure variance, (c) vortex stretching,
and (d) anisotropy (b11) of Reynolds stress and vorticity.

(b11 ≈ 0.0), while single-ﬂuid turbulence exhibits a tendency towards an axisymmetric state.
Evidently, the variable density eﬀects on the post-shock turbulence ﬁelds are diﬀerent at small and
large scales. Additionally, the quasi-isotropic state of the multi-ﬂuid turbulence is not stable and
is modiﬁed in the post-shock transition. Due to the energy transfer between the acoustic ﬁeld and
solenoidal turbulence ﬁeld, R11 quickly increases, causing b11 to become larger than zero. The
anisotropy reaches its maximum value around the peak TKE position (k0x ≈ 2.0) and then slowly
decreases. For the single-ﬂuid case, b11 keeps increasing till k0x ≈ 13.0, even though the acoustic
eﬀects almost vanish after peak TKE location (k0x ≈ π).

In ﬁgure 2.11, the developments of skewness and the ﬂatness of the longitudinal velocity
gradients are examined before and after the interaction with the shock wave. They show how the
non-Gaussian behavior of the velocity ﬁeld and speciﬁcally VGT are aﬀected by the combined shock
and density eﬀects. For isotropic turbulence, the skewness of the longitudinal velocity gradient

69

0510150246810ǫ(m)ǫ(s)TKE(m)TKE(s)05101501234p′p′(m)p′p′(s)k0x051015050100150200Σx(m)Σyz(m)Σx(s)Σyz(s)k0x051015-0.4-0.3-0.2-0.100.10.2Reynoldsstress(m)Reynoldsstress(s)vorticity(m)vorticity(s)0510151234(a)(b)(c)(d)prms/(ρu′u′)should be around -0.5, which is observed to be true in the pre-shock region for both single and
multi-ﬂuid cases for both streamwise as well as spanwise components (ﬁgure 2.11 a). Immediately
after the interaction with shock, diﬀerent components of the derivative skewness tensor are shown
to be modiﬁed in diﬀerent ways. The streamwise component for both single-ﬂuid and multi-ﬂuid
cases approaches to values very close to 0.0, which is consistent with the tendency towards a
two-dimensional axisymmetric state observed above. As the turbulence evolves away from the
shock wave, the streamwise velocity derivative skewness decreases rapidly. Due to the strong
density variations, the multi-ﬂuid case exhibits a faster decrease before k0x = 5.0, after which it
slowly increases towards the −0.54 value. The shock modiﬁcation of the skewness of the transverse
derivative is relatively small for the single-ﬂuid case. For the multi-ﬂuid case, the longitudinal
transverse velocity derivative becomes less negatively skewed, with a value of around -0.25. This
diﬀerence can be attributed to stronger shock intensity variations and shock wrinkling in the multi-
ﬂuid case. Away from the shock wave, for both cases, the skewness of ∂v/∂ y increases ﬁrst until it
reaches a peak and then slowly decreases. Comparably, the multi-ﬂuid case exhibits a shorter but
more intensiﬁed transition. At the end of the domain, however, the spanwise derivative skewness
is still larger than −0.5, as the ﬂow is still anisotropic. Figure 2.11 (b) shows the development
of longitudinal velocity derivative ﬂatness across and after the shock wave. Immediately after the
shock, the ﬂatness of the streamwise component decreases in value while that of the spanwise
component increases. Similar to the skewness, the eﬀect of density variations is relatively small on
the ﬂatness of streamwise component at the Atwood number considered in this study. On the other
hand, the density variations in the multi-ﬂuid case make the increase in transverse component less
signiﬁcant, with the pre- and post-shock values being almost the same. Away from the shock wave,
the ﬂatness of the longitudinal streamwise velocity derivative increases, returning to its pre-shock
value, while the increase is much faster for the multi-ﬂuid case. For the transverse longitudinal
derivative component, the ﬂatness slowly decreases after a small change.

From the results above, it can be stated that the variable density eﬀects are not strongly
manifested immediately after the shock wave for some of the statistics, but they play an important

70

Figure 2.11: Development of (a) skewness, and (b) ﬂatness, of the streamwise and transverse
components of velocity derivatives.

Figure 2.12: PDF of the density gradient at diﬀerent streamwise locations for: (a) single-ﬂuid
case and (b) multi-ﬂuid case.

71

k0x051015skewness-1-0.500.51∂u/∂x(m)∂v/∂y(m)∂u/∂x(s)∂v/∂y(s)k0x051015ﬂatness34567(a)(b)∂ρ/∂xi-1-0.500.51012345∂ρ/∂xi-15-10-505101500.10.20.30.40.5∂ρ/∂x1post-shock∂ρ/∂x2post-shock∂ρ/∂x1peakTKE∂ρ/∂x2peakTKE(a)(b)role in the post-shock adjustment. To get an insight into this behavior, density gradient PDFs are
examined at various streamwise positions in ﬁgure 2.12. Before the shock wave, the PDFs of the
density gradients are symmetric in all three directions for both single- and multi-ﬂuid cases (not
shown). For the single-ﬂuid case, after passing through the shock wave, the density gradients’
PDFs remain symmetrical, but the streamwise component PDF becomes wider due to the shock
compression. As the turbulence develops away from the shock wave to the peak TKE position,
the density gradients’ PDFs still remain symmetrical and become narrower, which is related to
the fast decay of the acoustic waves. For the multi-ﬂuid case, the density gradients’ PDFs are
strongly ampliﬁed through the shock wave, but the changes away from the shock are relatively
small, because the density variations are controlled by the compositional changes instead of the
acoustic waves. More importantly, the streamwise component becomes negatively skewed. We
have observed in our previous study (Tian et al., 2017a) that the post-shock streamwise velocity is
correlated with the density. Thus, as the ﬂow develops away from the shock wave, both the density
and velocity gradients become negatively skewed. This will be further discussed and explained in
the next subsection.

The nonlinear eﬀects of the shock wave, magniﬁed by large density variations have been shown
to cause the preferential deposition of turbulent quantities in regions with diﬀerent densities (Tian
et al., 2017a). This non-uniform turbulence distribution can further induce asymmetry in the
development of post-shock turbulence, such as the skewed density distribution and asymmetric
mixing. To understand the mechanism responsible for the skewness of the streamwise density
gradient, we further examine here the orientation of the eigenvectors of strain rate tensor Si j. The
PDFs of the cosines of the angles between the three eigenvectors with the streamwise direction,
conditioned on regions with positive or negative density gradients, are plotted in ﬁgure 2.13. The
eigenvalues of the strain rate tensor are γ1, γ2 and γ3 with the convention that γ1 < γ2 < γ3.
The angles between these eigenvectors and streamwise direction are denoted by χ1, χ2 and χ3.
For the multi-ﬂuid case, in the positive density gradient regions, the extensive (γ3-) eigenvector
is more likely to be aligned with the streamwise direction (ﬁgure 2.13 a). The intermediate (γ2-)

72

Figure 2.13: PDFs of the cosine angle between eigenvectors of the strain rate tensor and
streamwise (x) axis for regions with (a) dρ/dx > 0 and (b) dρ/dx < 0 and for multi-ﬂuid (solid
lines) and single-ﬂuid (dashed lines) cases.

eigenvector is misaligned with the streamwise direction and the compressive (γ1-) eigenvectors
have no preferential alignment. This implies that the density ﬁeld is generally being stretched in
the streamwise direction, making the magnitude of the density gradient smaller. On the other hand,
the alignment of the γ1− and γ3− eigenvectors with the streamwise directions is reversed in the
negative density gradient regions as shown in ﬁgure 2.13 (b). The density ﬁeld is then compressed
so that the magnitude of the density gradient is increased. This asymmetry in the alignment is
induced by the VD nonlinear eﬀects through the shock wave. This analysis serves as an explanation
for the negatively skewed PDF of density gradient in the multi-ﬂuid case. For the single-ﬂuid case,
the asymmetry in the eigenvector alignment is weak and vanishes quickly away from the shock
wave. This implies that even though density variations do not aﬀect some turbulence statistics in
a straightforward manner, they modify the topology and structure of turbulence immediately after
the shock and continue to manifest their eﬀects in the post-shock turbulence evolution.

73

cosχi00.20.40.60.81PDF012345cosχ1cosχ2cosχ3cosχi00.20.40.60.81012345(a)(b)2.3.2 Topological Analysis of the Post-shock Turbulence

To further characterize the turbulence structure behind the shock wave, we have analyzed the
invariant space of the VGT. The second and third invariants (denoted as Q∗ and R∗) of the
anisotropic/deviatoric part of the VGT can reveal important features of the ﬂow topology. The
anisotropic part of the VGT is calculated using the formula A∗
= Ai j − θ/3I, where Ai j = ∂ui/∂xj
i j
is the VGT and θ = ∂ui/∂xi is the trace of the VGT of dilatation. Correspondingly, the second and
third invariants can be calculated using the following formulas:

Q∗ = −1
2 A∗
i j A∗
R∗ = −1
3 A∗
i j A∗
jk A∗

ki

ji

(2.3a)

(2.3b)

Similarly, the invariants of the symmetric and skew-symmetric parts of the anisotropic velocity
i j, can also be calculated using the corresponding form of equation 2.3.
s) and (Q∗
w) here. The following equations relate the above variables

gradient tensor, S∗
They are denoted as (Q∗
for the anisotropic part of the velocity gradient tensor (Ooi et al., 1999):

i j and W∗
s, R∗

w, R∗

Q∗ = Q∗
s + Q∗
w
∗
∗
i S∗
s − ω
i j ω
j

R∗ = R∗

(2.4a)

(2.4b)

w

= ωi

where ω∗
is the vorticity vector. The scalar variables Q∗
s and Q∗
w denote the local
i
dissipation rate (−Q∗
i jS∗
s = 1/2S∗
= 1/2W∗
i j) and enstrophy (Q∗
i jW∗
i j), respectively. Conse-
quently, Q∗ represents the diﬀerence between enstrophy and dissipation (Chu and Lu, 2013).
Similarly, R∗
ki represents the production of dissipation due to strain ﬁeld and
ω∗
i S∗
i j ω∗
j is the vortex stretching contribution to the enstrophy (Chu and Lu, 2013). Therefore,
R∗ represents the diﬀerence between enstrophy production and dissipation production. Based
on the local values of Q∗ and R∗, four types of local ﬂow topologies can be identiﬁed: stable-
focus/stretching (SFS), unstable-focus/contracting (UFC), stable-node/saddle/saddle (SN/S/S) and

s = −1/3S∗
i jS∗

jk S∗

74

unstable-node/saddle/saddle (UN/S/S). For isotropic turbulence, the joint PDF of (Q∗, R∗) has the
well-known tear-drop shape. This has been further observed in other fully developed turbulent
ﬂows, such as boundary layers, mixing layers, and channel ﬂows (Pirozzoli and Grasso, 2004;
Wang et al., 2012). This type of distribution of Q∗ and R∗ is an indicator that the turbulence is
more likely having a local topology of stable-focus/stretching or an unstable-node/saddle/saddle. In
ﬁgure 2.14 (a), it is shown that the joint PDF of normalized second and third invariants, Q∗/(cid:104)Qw(cid:105)
and R∗/(cid:104)Qw(cid:105)3/2, has the same tear-drop shape in the pre-shock ﬂow. Using shock-LIA and DNS
data, Ryu and Livescu (2014) showed that for single-ﬂuid STI, the (Q∗, R∗) distribution is signiﬁ-
cantly modiﬁed by the shock wave, with a tendency towards symmetrization of the joint PDF. This
indicates that the regions with stable-focus/compression and stable-node/saddle/saddle (ﬁrst and
third quadrant) are more likely to occur as turbulence develops a 2-D axisymmetric ﬂow structure.
To understand the variable density eﬀects on this shock-induced symmetrization, the joint PDFs
of (Q∗,R∗) for both single-ﬂuid and multi-ﬂuid post-shock turbulence are compared in ﬁgure 2.14
(b,c).

Figure 2.14 (b) shows the joint distribution for the single-ﬂuid post-shock turbulence. The
dashed lines denote the locus of zero discriminant of A∗, where Q∗ and R∗ satisfy 27R∗2/4+Q∗3 = 0.
Compared to the pre-shock joint PDF, there is a tendency towards symmetrization, with more points
located in the ﬁrst and third quadrants. This agrees very well with previous shock-LIA results (Ryu
and Livescu, 2014). Similar to single-ﬂuid STI, multi-ﬂuid STI demonstrates a tendency towards
symmetrization of the (Q∗, R∗) distribution. However, the multi-ﬂuid distribution is slightly more
symmetric and has a larger variance, with more points away from the axes. This implies that more
extreme "events" exist in the post-shock multi-ﬂuid turbulence. It also agrees with our previous
study on multi-ﬂuid STI that shows ampliﬁcation of TKE to be stronger when there are signiﬁcant
density variations (Tian et al., 2017d,a).

The density eﬀects on the post-shock joint PDF of second and third invariants are further
explored by comparing the conditional distribution, conditioned on regions with diﬀerent densities,
in ﬁgure 2.15 (a)-(c). Figure 2.15 (a) corresponds to regions with relatively high density (ρ >

75

Figure 2.14: Iso-contour lines of joint PDFs of normalized second and third invariants of the
anisotropic part of the velocity gradient tensor, (Q∗,R∗), for (a) pre-shock, (b) single-ﬂuid
post-shock turbulence, and (c) multi-ﬂuid post-shock turbulence. The lateral lines denote the
locus of zero discriminant.

(ρ + 90%ρ(cid:48)
rms)), 2.15 (b) to regions with density around the post-shock mean value, and 2.15 (c)
to low density regions (ρ < (ρ − 90%ρ(cid:48)
rms)). For consistency check, the joint PDFs corresponding
to these regions are also computed for the pre-shock ﬂow (not shown) and found to be close to
the single-ﬂuid PDFs. After the shock wave, the joint PDFs demonstrate signiﬁcant diﬀerences
between regions with diﬀerent densities. In regions with density closer to that of post-shock mean
density, the distribution of invariants appears to be very similar to that shown in ﬁgure 2.14 (c). But
for regions with higher density (ﬁgure 2.15 (a)), the joint PDF becomes more symmetric compared
to the overall ﬂow or single-ﬂuid case. There is a much larger portion of data points having a local
topology of stable-node/saddle/saddle, and fewer data points fall into the ﬁrst and second quadrants,
indicating larger strain-dominated regions. On the other hand, the post-shock regions with low-
density values (ﬁgure 2.15 (c)) exhibit features similar to that of isotropic turbulence, with almost
the same tear-drop shape, only with a larger variance or a wider distribution. The quantitative
diﬀerence is hypothesized to be related to the higher shock strength variation in the multi-ﬂuid
case. It was observed in our previous studies (Tian et al., 2017b), that the local shock strength is

76

R∗-10010Q∗-10-50510R∗-4-2024-8-6-4-202468R∗-4-2024-8-6-4-202468(a)(b)(c)Figure 2.15: Iso-contour lines of post-shock (k0x ≈ 0.44) joint PDF of second and third invariants
of the anisotropic part of the velocity gradient tensor, (Q∗,R∗), in regions with diﬀerent densities.
(a) regions with high density values, ρ > (ρ + 90%ρ(cid:48)
post-shock mean value, and (c) regions with low density values, ρ < (ρ − 90%ρ(cid:48)

rms), (b) regions with density around the

rms).

positively correlated with the pre-shock density. With a stronger shock, the two-dimensionalization
eﬀect on the post-shock turbulence should also appear stronger in the high-density regions (Livescu
and Ryu, 2016). For low-density regions, the smaller two-dimensionalization eﬀect reduces the
symmetrization trend. Moreover, the relatively lower inertia in these regions leads to a faster
response to the local strain ﬁeld (Livescu et al., 2010), which could make a faster return to isotropic
turbulence. The diﬀerent characteristics of (Q∗, R∗) joint PDF in regions with diﬀerent densities
provide additional evidence for the previous argument made about the density role on the preferential
ampliﬁcation of the strain and rotation tensors.

In ﬁgure 2.16, the planar distribution of the ﬂow topologies are shown. Here, Qi refers to
the quadrants on the joint PDF of (Q∗,R∗), which amounts to a representation of the local ﬂow
topology. Figure 2.16 (a) presents the 2D visualization of the ﬂow topology in a typical x − z
plane. The regions occupied by diﬀerent quadrants are marked using diﬀerent colors. Evidently,
the vorticity-dominated regions (Q1 and Q2) cover a large portion of the ﬂow and have more
compact shapes. These regions are connected by UN/S/S areas (Q4), which are more elongated.

77

R∗-202Q∗-4-3-2-101234R∗-202-4-3-2-101234R∗-202-4-3-2-101234(a)(b)(c)Figure 2.16: Color illustration of the ﬂow topology for the multi-ﬂuid STI. The ﬂow topology is
represented by the quadrants (denoted as Qi) of the joint PDF of (Q∗,R∗). (a) 2D color contours in
the x-z plane at y = 3.14 (half y-domain). The shock wave is located in the middle of the domain
at k0x ≈ 0.0. The ratio of the ﬂuid volume in diﬀerent quadrants in the pre-shock region is
Q1 : Q2 : Q3 : Q4=26.7%:38.7%:7.8%:26.8%. The 2D color contours in the homogeneous y-z
plane at streamwise locations of: (b) k0x ≈ 0.2 (28.7%:34.4%:14.3%:22.6%), (c) k0x ≈ 2.0, peak
TKE location (26.7%:36.9%:11.2%:25.2%) (d) k0x ≈ 4.0 (26.3%:37.9%:9.3%:26.2%).

The SN/S/S (Q3) areas can be located either at the edge of Q4 regions or in-between diﬀerent Q4
regions. These strain-dominated regions seem to be more fragmented than the compact vorticity-
dominated regions. Figure 2.16 (b)-(d) show the 2D contours of Qi in the homogeneous (y − z)
planes at diﬀerent streamwise locations after the shock. These locations are also marked on ﬁgure
2.16 (a).
Immediately after passing through the shock wave, the volume fractions of diﬀerent
quadrants are calculated to be Q1 : Q2 : Q3 : Q4 = 28.7% : 34.4% : 14.3% : 22.6%, indicating
a trend towards symmetrization in the joint PDF. This can also be observed in the 2D contours in
ﬁgure 2.16 (b). Moreover, the characteristic length scales associated with the regions occupied by

78

diﬀerent quadrants are decreased across the shock wave. As the ﬂow evolves away from the shock
wave, the distribution slowly changes back to the pre-shock shape but still with smaller turbulence
length scales. Most of ﬂuid in diﬀerent quadrants return to the pre-shock values at k0x = 4.0. The
re-orientation of the ﬂow structures into the streamwise direction is also noted in ﬁgure 2.16 (a),
consistent with the return to isotropy of the ﬂow. However, the rates at which diﬀerent ﬂow features
return to an isotropic state are slightly diﬀerent. The dynamics of ﬂow and the return-to-isotropic
turbulence process will be examined in detail in the next section using the Lagrangian statistics.

In our previous study (Tian et al., 2017a), the (2-D) axisymmetric state of multi-ﬂuid post-shock
turbulence is explored by considering the normalized turbulent stretching term in the enstrophy
budget equation.
It was shown that this term is close to zero immediately after the shock and
increases after the shock, more so in the multi-ﬂuid case. In the current paper, we have shown that
the multi-ﬂuid joint PDF of (Q∗, R∗) becomes more symmetric. Below, we continue to explore
how the vortex stretching rate is aﬀected by the change in turbulence to an axisymmetric state and
diﬀerent ﬂow topologies.

w (Ooi et al., 1999). In ﬁgure 2.17, the joint PDF of (−Q∗

The rate at which the vorticity is stretched or contracted, i.e. the normalized vortex stretching
rate, can be calculated based on the VGT invariants using the formula: Σ∗ = wiSi j wj = (R∗
s −
R∗)/Q∗
s,Σ∗) is plotted to investigate the
eﬀects of the strain ﬁeld on the vortex stretching rate. Positive and negative Σ∗ values correspond
to the vortex being stretched or contracted. Figure 2.17 (a) shows the joint PDF of (−Q∗
s,Σ∗) for the
isotropic turbulence. The results agree very well with those of Ooi et al. (1999), which indicates
that the ﬂow favors positive Σ∗ values or an overall vortex stretching, especially in the strong strain
dominated regions. Here, we compare the results from isotropic turbulence to those from single-
ﬂuid and multi-ﬂuid post-shock turbulence to understand the shock and variable density eﬀects.
We note that in ﬁgure 2.17 (b), the joint PDF becomes more symmetric around Σ∗ = 0.0 after
passing through the shock wave. For the multi-ﬂuid case, as shown in ﬁgure 2.17 (c), the joint PDF
becomes almost fully symmetric, especially at lower −Q∗
s values. This symmetry has a strong eﬀect
on the overall vortex stretching rate for the multi-ﬂuid post-shock turbulence because the positive

79

Figure 2.17: Iso-contour lines of joint PDF of (-Q∗
single-ﬂuid and multi-ﬂuid turbulence at post-shock position of k0x ≈ 0.44.

s , Σ∗) for (a) isotropic box turbulence and (b,c)

and negative Σ∗ values tend to cancel each other through averaging. Moreover, the variances of
the stretching term are almost the same for single and multi-ﬂuid cases, meaning that the lower
stretching rate is mainly due to changes in the turbulence structure (especially more negative Σ∗
regions), and not simply the decrease in the magnitude of Σ∗.

To understand the contribution from diﬀerent topological states to the vortex stretching, the
joint PDF of (−Q∗
s,Σ∗) is conditioned on diﬀerent quadrants for the multi-ﬂuid case. Figure 2.18
(a,b) shows the joint PDF for Q1 and Q2 regions, or areas with a local topology of SFS and UFC.
It can be observed that in these rotation-dominated regions, the magnitude of the vortex stretching
rate Σ∗ is relatively small. Moreover, Q1 is dominated by the negative Σ∗ and Q2 is dominated by
positive Σ∗ areas, which can be inferred from their corresponding topologies. However, for Q3 and
Q4 (ﬁgure 2.18 c,d), the magnitude of the vortex stretching rate is larger than that in the rotation-
dominated regions (ﬁgure 2.18 a,b). In Q3, the joint PDF is relatively symmetric and seems to be
slightly biased towards negative vortex stretching rates. Q4, on the other hand, is dominated by
positive vortex stretching. Overall, the results explain the lower averaged vortex stretching rate in
the multi-ﬂuid case through enhancement of Q1 and Q3 regions.

80

Σ∗-202−Q∗s05101520Σ∗-20205101520Σ∗-20205101520(a)(b)(c)Figure 2.18: Iso-contour lines of joint PDF of (-Q∗
shock wave. (a) Q2, (b) Q1 ,(c) Q3 and (d) Q4.

s , Σ∗) for diﬀerent quadrants right after the

2.3.3 Lagrangian Dynamics of VGT

The Lagrangian particle method has been widely used for studying many important aspects of
turbulence, such as ﬂuid particle dispersion, clustering, and mixing. Lagrangian analysis of the
VGT is also proven beneﬁcial in the study of turbulence (e.g. Chong et al., 1998; Ooi et al., 1999;
Meneveau, 2011; Wang and Lu, 2012). Even though the turbulence structure at diﬀerent post-shock
locations can be extracted from the Eulerian data, the dynamics that dominate the development
of VGT are still missing.
In this section, we use non-inertial Lagrangian particles/tracers that
move with the local ﬂow velocity because their statistics can reﬂect important transient turbulent
dynamics, which are diﬃcult to study using Eulerian data (Yeung, 2002). More importantly, in the
context of variable density ﬂows, the Lagrangian statistics enable us to diﬀerentiate among particles

81

-3-2-10123−Q∗s051015-3-2-10123051015Σ∗-3-2-10123−Q∗s051015Σ∗-3-2-10123051015(a)(b)(c)(d)with diﬀerent densities and investigate their dynamics separately. Lagrangian data are used here to
perform a detailed analysis of the post-shock turbulence structure and its dynamics in the STI with
and without signiﬁcant density ﬂuctuations.

The ﬁrst result considered is the timescale of particle motions related to diﬀerent ﬂow topologies.
In ﬁgure 2.19, the percentage of ﬂuid particles that remain in their starting quadrants are plotted
over time so that we can identify the residence time of particles for diﬀerent turbulence structures.
In ﬁgure 2.19 (a), the percentages of ﬂuid particles are plotted for decaying isotropic turbulence as
a reference. It is noted that Q3 and Q4, which are the strain-dominated regions, have the smallest
associated residence times among all the four quadrants, with Q3 time being the smaller of the two.
For the rotation-dominated regions, the residence times are expectedly longer, especially for Q2.
The residence times for single-ﬂuid and multi-ﬂuid simulations can be inferred from ﬁgure 2.19
(b,c). For both cases, Q3 always has the least residence time and Q2 has the largest one. Comparing
all three cases, the particles in the multi-ﬂuid and single-ﬂuid post-shock turbulence are shown to
evolve faster away from the original quadrant than the particles in isotropic turbulence. This is
hypothesized to be due to the fact that post-shock turbulence still experiences a return-to-isotropy,
while the isotropic turbulence has already reached an equilibrium among all the quadrants. Between
the two post-shock turbulence ﬁelds, the multi-ﬂuid case presents a relatively shorter residence time.
This agrees with the fact that the multi-ﬂuid turbulence exhibits a faster return-to-isotropy than the
single-ﬂuid turbulence (Tian et al., 2017a).

The rotation-dominated regions are sometimes identiﬁed as coherent structures like stretched
vortex tubes, as shown in ﬁgure 2.20 (a). When the particles travel through these structures,
they tend to stay longer due to the coherency of the surrounding ﬂow. This is conﬁrmed by the
circulating streamlines in such structures (black lines in ﬁgure 2.20 (a)). On the other hand, the
strain-dominated regions are usually lacking such coherency, making the residence times smaller.
The local streamlines pass through such structures instead of circulating around them. Figure 2.21
presents an example of the temporal development of the above-mentioned structures. The evolution
of a vortex tube in the post-shock turbulence is tracked and visualized as it moves away from the

82

Figure 2.19: Percentage of ﬂuid particles that stay in each quadrant following particles initialized
uniformly in (a) isotropic turbulence and (b,c) single-ﬂuid and multi-ﬂuid turbulence at
post-shock position of k0x = 0.44.

shock wave in ﬁgure 2.21 (a). As expected, the vortical structure maintains its shape, except that
it is being stretched and reoriented by the local ﬂow ﬁeld. Moreover, the vortex tube surface is
almost parallel with the iso-surface of the density ﬁeld, i.e. perpendicular to the density gradient.
This is consistent with the discussion in section 2.3.1.1 regarding the bulk of vorticity generation
across the shock wave. As the vortex tube evolves away from the shock wave, the reorientation of
the density gradient by the vortex is also observed. In ﬁgure 2.21 (b), a strain-dominated structure
is visualized using the iso-surface of negative Q∗. It can be clearly seen that such structures lack
temporal coherency since they tend to be become fragmented as they evolve.

In ﬁgure 2.22, the contributions to the normalized vortex stretching rate from particles that
are initialized in each of the four quadrants are plotted following these particles. As expected, at
t = 0.0, particles from Q2 and Q4 add positively to the vortex stretching rate, while those from
Q1 have a negative vortex stretching rate contribution on average. This is in good agreement with
the joint PDFs of (−Q∗
s,Σ∗), shown in ﬁgure 2.18. For Q3, the initial contribution is close to zero.
As the ﬂuid particles move with the turbulent ﬂow, their contributions to the vortex stretching also
change. It can be seen that the fast increase in vortex stretching can be mainly attributed to the

83

t00.5100.20.40.60.81Q1Q2Q3Q4t00.5100.20.40.60.81t00.5100.20.40.60.81(a)(b)(c)Figure 2.20: Visualization of turbulence structure using iso-surfaces of Q∗ colored by R∗ in the
multi-ﬂuid post-shock turbulence. (a) Q1 and Q2 (vorticity-dominated region), and (b) Q3 and Q4
(strain-dominated region). The black lines represent the instantaneous streamlines.

Figure 2.21: Visualization of the temporal development (left to right) of the turbulence structure
using iso-surfaces of Q∗ colored by density for the multi-ﬂuid post-shock turbulence. These
structures are captured immediately after the shock wave. (a) vorticity-dominated structure, and
(b) strain-dominated structure.

84

Figure 2.22: Contributions to the vortex stretching rate from particles starting in each quadrant.
The particles are initialized uniformly at the post-shock position k0x ≈ 0.44 and traced
downstream till the vorticity returns to an isotropic state. (a) single-ﬂuid and (b) multi-ﬂuid cases.

particles originating in Q1 and Q3. Particles starting in Q4 have an increasing vortex stretching
contribution for a short period before their combined/average contributed value starts to decrease.
The behavior is qualitatively similar for the single-ﬂuid case, but the changes are smaller in this
case. For both cases, the vortex stretching contribution from the initial Q2 particles decreases in
time.

To further understand this behavior, the Lagrangian equations of the VGT and its invariants
are considered. The time evolution of Ai j for ﬂuid particles can be obtained by taking the spatial
derivatives of the Navier-Stokes equations. In dimensionless form, it can be written as (Chu and
Lu, 2013):

∂ Ai j
∂t

+ uk

∂ Ai j
∂xk
DAi j
Dt

+ Aik Ak j = −Hi j + Ti j
= −Aik Ak j − Hi j + Ti j

(2.5a)

(2.5b)

with

85

) = − 1
ρ2
)

∂p
∂xi
∂σik
∂xk

Hi j =
Ti j =

σi j =

∂
∂xj
∂
∂xj
µ
Re0

(1
ρ
(1
ρ

(cid:18) ∂ui

∂xj

+

∂uj
∂xi

− 2
3

∂uk
∂xk

δi j

∂ ρ
∂xj

∂p
∂xi

1
ρ

∂p2
∂xi∂xj

= Hb

i j + Hp
i j

+

(cid:19)

(2.6a)

(2.6b)

(2.6c)

where Re0 is the reference Reynolds number. From here, the dynamic equations for the three

invariants of the VGT, P, Q, and R can be derived in the following form (Chu and Lu, 2013):

DP
Dt
DQ
Dt
DR
Dt

+ Hb

ii − Tii
+ Ai j Hp

= (P2 − 2Q) + Hp
ii
= (PQ − 3R) + (PHp
ii
= PR + (QHp
ii
+Ai j Ajk Hb

ji) + (PHb
+ Ai j Ajk Hp

ii + Ai j Hb
ki) + (QHb
+ PAi j Hp
ji
ki) + (−QTii − PAi jTji − Ai j AjkTki)

ji) + (−PTii − Ai jTji)
ii + PAi j Hb
ji

where the three invariants of VGT are deﬁned as:

P = −tr(Ai j)
Q = −1
2
R = −det(Ai j)

(tr(Ai j)2 − tr(Ai j Ajk))

(2.7a)

(2.7b)

(2.7c)

(2.8a)

(2.8b)

(2.8c)

Here, tr(Ai j) and det(Ai j) denote the trace and determinant of a tensor. Note that instead of the
deviatoric part of the VGT, the dynamic equations for the full VGT are considered. The reason is
that due to the variable density eﬀects and shock compression, the incompressibility condition is
not satisﬁed especially when Mt and At become large. More importantly, the contributions from
the isotropic part of the VGT to the ﬂow dynamics are still unknown and need to be explored for
STI.

86

Figure 2.23: PDFs of (a) (DQ/Dt)/(cid:104)Qw(cid:105)3/2 and (b) (DR/Dt)/(cid:104)Qw(cid:105)2 for ﬂuid particles with
diﬀerent densities at streamwise location of k0x ≈ 0.5.

The dynamical equations can be divided into contributions by four diﬀerent parts: I) mutual-
i j and IV) viscous term

interaction among invariants, II) pressure Hessian, Hp
Ti j. The statistics regarding these terms can be extracted from the Lagrangian data.

i j, III) baroclinic, Hb

Some general features of the Lagrangian dynamics of the VGT invariants are examined through
the PDFs of their material derivatives. The variable density eﬀects can be identiﬁed by comparing
the PDFs corresponding to regions with diﬀerent densities (ﬁgure 2.23). In the light ﬂuid regions,
the PDFs of DQ/Dt and DR/Dt have narrower tails, while the tails are larger in the heavy ﬂuid
regions. Another important observation is that the skewness of DQ/Dt is diﬀerent in the light
and heavy ﬂuid regions. Heavy ﬂuid particles have a positively-skewed PDF, similar to the overall
ﬂow. On the other hand, the DQ/Dt skewness resulting from light ﬂuid particles is negative. This
implies that heavy ﬂuid particles are more likely to move towards rotation-dominated regions and
vice versa. These diﬀerences can be attributed to diﬀerent return-to-isotropy, experienced by ﬂuid
particles with diﬀerent densities.

The Lagrangian dynamics of the turbulence and the evolution of ﬂow topology are further
examined here by considering the conditional mean rate of change of Q and R in the invariants

87

(DQ/Dt)/hQwi3/2-20-100102010-410-310-210-1100(DR/Dt)/hQwi2-20-100102010-410-310-210-1100heavyﬂuidlightﬂuidmean(a)(b)Figure 2.24: Conditional mean rate of change vectors of (DQ/Dt/(cid:104)Qw(cid:105)3/2,DR/Dt/(cid:104)Qw(cid:105)2) in the
(Q, R) plane for (a) isotropic turbulence, (b) single-ﬂuid post-shock turbulence, and (c) multi-ﬂuid
post-shock turbulence at streamwise location of k0x ≈ 0.5.

plane (Ooi et al., 1999). The rates of change are used to form a vector at each point in the invariants
plane. The trajectories implied by these vectors can be followed to understand the return-to-isotropy.
The procedure used to obtain the conditional mean vectors (CMVs) in this study is similar to that in
Ooi et al. (1999). Based on the conditional averages introduced in equation 2.2, X(Q, R) represents
a statistical quantity that is conditioned on Q and R. The statistical convergence concerning the bin
sizes and the number of samples in each bin has been discussed in section 2.2.4.

Figure 2.24 shows the normalized conditional mean vectors (DQ/Dt/(cid:104)Qw(cid:105)3/2,DR/Dt/(cid:104)Qw(cid:105)2)
for diﬀerent ﬂows. The vectors obtained from isotropic turbulence data are shown in ﬁgure 2.24 (a)
for reference. It can be seen that the CMVs exhibit a circulating behavior in the (Q, R) plot around
the origin in the clockwise direction, indicating that the ﬂow evolves from SFS to UFC, UN/S/S,
SN/S/S then back to SFS on average. This circulating behavior represents the Lagrangian dynamics
in fully developed turbulence that maintains the tear-drop shape of the (Q, R) distribution. This has
been observed in many incompressible/compressible canonical turbulent ﬂows (Ooi et al., 1999;
Chu and Lu, 2013). The CMVs for single-ﬂuid and multi-ﬂuid post-shock turbulence are shown in
ﬁgure 2.24 (b) and (c). Evidently, the joint PDF of (Q, R) becomes more symmetric due to shock
compression. From the Lagrangian point of view, the circulating behavior as seen in ﬁgure 2.24 (a)
isotropic turbulence disappears. The particles in Q2 tend to have an increasing Q and decreasing
R, resulting in an overall trend of getting away from the original point, instead of circulating and

88

R/hQwi3/2-1-0.500.51Q/hQwi-2-1.5-1-0.500.511.52R/hQwi3/2-1-0.500.51-2-1012R/hQwi3/2-0.500.5-1-0.500.51(a)(b)(c)Figure 2.25: Conditional mean vectors in the (Q, R) invariants plane for (a) light ﬂuid, (b) medium
density ﬂuid and (c) heavy ﬂuid at streamwise location of k0x ≈ 0.5.

then moving toward Q1. This trend in the second quadrant represents an increase of enstrophy.
The particles in Q1 have similar dynamics as in isotropic turbulence and tend to move downward
in the (Q, R) plane toward the zero discriminant curve. The particles in Q3 are more likely to
move straight up towards Q2, while those in Q4 are likely to move away from the original point
following the direction of the zero discriminant line. The overall behavior formed by these particles
demonstrates the return-to-isotropy process, with an enlarging head in the second quadrant and
elongating tail in the fourth quadrant, anticipating the formation of the classic tear-drop shape.

The density eﬀects can be further examined by conditioning the (DQ/Dt, DR/Dt) vector ﬁeld
on the local density. Figure 2.25 (a) shows the CMVs for the light ﬂuid regions. The light ﬂuid
particles retain the circulating motion, except that the particles in Q3 and Q4 are likely to go straight
left instead of following the zero discriminant line. In general, the ﬂow dynamics in the light ﬂuid
regions are less aﬀected by the shock wave. For the medium density ﬂuid regions (ﬁgure 2.25
b), the circulating motion disappears. On the right side of the (Q, R) plane (R > 0), which is
the strong dissipation-production region based on equation 2.4, the ﬂuid particles tend to move
downward, resulting in lower Q values. On the left side of the (Q, R) plane (R < 0), which is the
enstrophy-production dominated region, the ﬂuid particles tend to move to the left, indicating an
increased enstrophy-production. The overall downward-moving behavior of the medium density

89

R/hQwi3/2-0.200.2Q/hQwi-1-0.500.51R/hQwi3/2-0.200.2-1-0.500.51R/hQwi3/2-0.200.2-1-0.500.51(a)(b)(c)Figure 2.26: Contributions to the transport equations of the VGT invariants by diﬀerent terms for
isotropic turbulence. (a) Mutual interaction among invariants, (b) pressure Hessian term, (c)
baroclinic term, and (d) viscous term.

ﬂuid particles is indicative of decreasing vorticity. This is possibly due to the fact that vorticity
is preferentially ampliﬁed in the medium density region across the shock wave. After passing the
shock wave, the vorticity will decrease as the correlation between density and vorticity vanishes.
Figure 2.25 (c) shows the CMVs for the heavy ﬂuid regions. Interestingly, the heavy ﬂuid particles
exhibit counterclockwise motion. The heavy particles start from Q3 and move to Q4, Q1, and
ﬁnally to Q2. This implies that they become vorticity dominated due to the fast depletion of strain.
Evidently, density plays an important role in the development of the ﬂow topology in the post-shock
region, so special attention should be paid to the modeling of variable density STI.

To better understand the underlying mechanisms that cause the behavior highlighted above,

90

-1-0.500.51Q/hQwi-2-1012-1-0.500.51-2-1012R/hQwi3/2-1-0.500.51Q/hQwi-2-1012R/hQwi3/2-1-0.500.51-2-1012(a)(b)(c)(d)the dynamic equations (2.7) governing the vector (DQ/Dt, DR/Dt) are examined. The right hand
sides of these equations can be divided into four types of contributing terms: (a) mutual interaction
among invariants (term I), (b) pressure Hessian (term II), (c) baroclinic (term III), and (d) viscous
term (term IV). As a reference, these terms are shown in ﬁgure 2.26 for the isotropic turbulence.
The variable Q tends to be ampliﬁed in the enstrophy-production dominated region due to the
eﬀects of vortex stretching mechanism and is decreased in the dissipation-production dominated
region due to self-ampliﬁcation of the strain rate tensor. On the other hand, the mutual eﬀects on
R are small because the ﬁrst invariant P is usually small and the positive and negative values are
likely to cancel each other. The contributions from the pressure Hessian (ﬁgure 2.26 b) tend to
move the particles away from an asymptotic line, ending up amplifying the magnitude of R. For the
current simulation, the asymptotic line starts from Q2 and ends in Q4 with a slope of around -2.5.
This result agrees well with that observed in turbulent boundary layers (Chu and Lu, 2013). The
baroclinic contributions are very small in the post-shock turbulence as shown in ﬁgure 2.26 (c). The
viscous eﬀects as shown in ﬁgure 2.26 (d) and as expected are reducing the magnitudes of Q and R
and pushing the particles towards the origin. This has been observed in various types of turbulence
(Ooi et al., 1999; Chu and Lu, 2013). The combined eﬀects from the four above mechanisms
determine the circulating behavior of the conditional mean of (DQ/Dt, DR/Dt) vectors.

After interaction with the shock wave, the conditional mean vectors in the (Q, R) plane are
diﬀerent from those in the pre-shock isotropic turbulence. Figure 2.27 shows the results for the
single-ﬂuid case. By comparing it with ﬁgure 2.26, we note that even though the conditional
means of (DQ/Dt, DR/Dt) vectors are diﬀerent, the contributions from mutual interaction (ﬁgure
2.27 a), baroclinic term (ﬁgure 2.27 c) and viscous term (ﬁgure 2.27 d) are very similar. The
only term that qualitatively distinguishes the post-shock turbulence from isotropic turbulence is
the pressure Hessian term as shown in ﬁgure 2.27 (b). In the post-shock single-ﬂuid turbulence,
the asymptotic line that separates the vectors into two regions with diﬀerent behaviors disappears.
Instead, the pressure Hessian term tends to move the particles away from the origin in Q1 and Q2,
thus increasing Q and |R| values of the particles. In Q3 and Q4, the vectors are parallel to the left

91

Figure 2.27: Contributions to the transport equations of the VGT invariants by diﬀerent terms for
single-ﬂuid post-shock turbulence. (a) Mutual interaction among invariants, (b) pressure Hessian
term, (c) baroclinic term, and (d) viscous term.

zero discriminant line, making the particles move from Q3 to Q4, and then to Q1.

For multi-ﬂuid post-shock turbulence, the pressure Hessian term is also the only term that is
qualitatively diﬀerent than that in isotropic turbulence (ﬁgure 2.28). Despite the increased density
and pressure gradient in the multi-ﬂuid case, the baroclinic term is still considerably smaller than
all the other terms. In Q1 and Q2, an asymptotic line similar to that in isotropic turbulence seems
to exist, which repels the vectors away from it, causing an increase in |R| values. In Q3 and Q4, this
term tends to move the ﬂuid particles along the right zero discriminant line from Q4 to Q3. The
further conditioned pressure Hessian term based on the local densities in ﬁgure 2.29 indicates that
ﬂuid particles with diﬀerent densities exhibit very diﬀerent behaviors in respond to the pressure
Hessian dynamics. Speciﬁcally, the pressure Hessian generally moves the heavy particles toward

92

-1-0.500.51Q/hQwi-2-1012-1-0.500.51-2-1012R/hQwi3/2-1-0.500.51Q/hQwi-2-1012R/hQwi3/2-1-0.500.51-2-1012(a)(b)(c)(d)Figure 2.28: Contributions to the dynamics of the VGT invariants by diﬀerent terms for
multi-ﬂuid post-shock turbulence. (a) Mutual interaction among invariants, (b) pressure Hessian
term, (c) baroclinic term, and (d) viscous term.

the regions with larger Q values. In Q3 and Q4, it also moves the heavy ﬂuid particles towards
the R > 0 plane. For the light ﬂuid particles, the pressure Hessian term tends to make them move
towards regions with larger |R| values in the ﬁrst and second quadrant. In Q3 and Q4, the ﬂuid
particles move from Q4 to Q3. Last but not the least, the ﬂuid particles with medium density seem
to exhibit similar behavior to light ﬂuid particles, except that in Q1, where the pressure Hessian
contribution is moving the ﬂuid particles towards the regions with large Q values. Examining ﬁgure
2.25 and ﬁgure 2.29 together, we observe that the diﬀerent particle dynamics in the (Q, R) plane in
regions with diﬀerent densities are mainly due to diﬀerences in the pressure Hessian contribution.

93

-0.4-0.200.20.4Q/hQwi-1-0.500.51-0.4-0.200.20.4-1-0.500.51R/hQwi3/2-0.4-0.200.20.4Q/hQwi-1-0.500.51R/hQwi3/2-0.4-0.200.20.4-1-0.500.51(a)(b)(c)(d)Figure 2.29: Contributions from pressure Hessian to the dynamics of the VGT invariants in (a)
light ﬂuid region, (a) medium density ﬂuid region and (c) heavy ﬂuid region.

2.3.4 Lagrangian Particle Dispersion

2.3.4.1 Single Particle Statistics

In this section, single particle statistics such as particle dispersion rate, autocorrelation of velocity
will be presented. The Lagrangian statistics of ﬂuid particles reveal important features of turbulent
ﬂow and are considered in this study for better understanding of post-shock ﬂow dynamics. We
will ﬁrst consider single particle statistics, which are closely related to the mean concentration of
a passive scalar. Stochastic models for particle dispersion has been proposed to predict various
statistics.

Due to the existence of mean ﬂow, it would be reasonable to examine the particle displacement
i (t) =
relative to the mean ﬂow. Thus, the relative particle dispersion can be calculated using y+
i (t) − ui(t − t0). Note that the only non-zero component of the mean velocity is the streamwise
x+

component. The particle dispersion rate can then be calculated using Di j(t) =(cid:10)y+

diagonal components of the dispersion rate tensor Di j(t) are plotted in ﬁgure 2.30. Due to the
homogeneity in the spanwise direction, D22 is equal to D33, so only D22 is shown in the ﬁgure.
On top of that, the displacement variance is normalized using t2 to better visualize the temporal
development. Figure 2.30 compares the single particle dispersion rate between single- and multi-

α(t)(cid:11). The

α(t)y+

94

R/hQwi3/2-0.4-0.200.20.4Q/hQwi-1-0.500.51R/hQwi3/2-0.4-0.200.20.4-1-0.500.51R/hQwi3/2-0.4-0.200.20.4-1-0.500.51(a)(b)(c)ﬂuid post-shock turbulence. For the multi-ﬂuid case, both the streamwise and spanwise particle
dispersion follows a t2 development. Later downstream, the streamwise particle dispersion rate
increases and reaches a peak around t − t0 ≈ 0.5 and then decreases to a t scaling at larger time.
On the other hand, the spanwise dispersion decreases from t2 to t scaling at larger time. Till the
end of the domain, the streamwise particle dispersion is always larger than the that in the spanwise
direction. For the single-ﬂuid case, similar temporal scaling for each component of the particle
dispersion is observed. However, the relative dispersion rate between the streamwise and spanwise
direction are diﬀerent.
It is noted that in the single ﬂuid simulations, the particle dispersion is
initially faster in the spanwise direction. As the particles move downstream, D11 starts to grow and
eventually becomes larger than D22. This shows that the dispersion anisotropy becomes diﬀerent
due to the variable density eﬀects. The diﬀerence in the early time anisotropy can be related to the
anisotropy in the Reynolds stress. The early time particle dispersion can be approximated using
iu(cid:48)
j(t − t0)2. Therefore, the early time anisotropy in the particle dispersion
is strongly correlated with the Reynolds stress and this is also supported by the Eulerian results
shown in ﬁgure 2.10

Di j =(cid:10)y+

α(t)(cid:11) ≈ u(cid:48)

α(t)y+

Most of previous studies on Lagrangian particle dispersion are conducted for fully homogeneous

Figure 2.30: Single particle dispersion rate normalized using t2 for both single and multi-ﬂuid
post-shock turbulence.

95

10-2100Dαα/t20.020.040.060.080.10.120.14D11/t2(m)D22/t2(m)D11/t2(s)D22/t2(s)turbulence, so the third-order moment of the particle displacement PDF is zero. For post-shock
turbulence, the ﬂow is only homogeneous in the spanwise directions, so the third-order moment or
skewness of the PDF is expected to be zero in those directions. In the streamwise direction, the
turbulence experiences complicated transient process as it evolves downstream, so it is not known
what the skewness of the particle displacement PDF is. In ﬁgure 2.31, the skewness of particle
displacement in the streamwise direction is shown for both cases. We note that the skewness of the
dispersion µ3 is non-zero. Immediately after the shock wave, the multi-ﬂuid case has a negative
skewness of around -0.15. Then it increases to around -0.12 and remains at the value till the end
of the domain. For the single-ﬂuid case, however, the skewness is positive immediately after the
shock wave. It then increases for a short period before decreases to a negative value. Due to the fact
that many stochastic models assume a Gaussian process for particle dispersion, the non-Gaussian
behavior of particle dispersion in streamwise direction needs to be addressed.

It is well known that Lagrangian velocity autocorrelation plays an important role in the de-
velopment of the particle dispersion (Yeung, 1994), which can be calculated using the formula:

Figure 2.31: The temporal development of the skewness µ3 of the streamwise particle
displacement PDF.

96

t00.511.52µ3-0.2-0.100.1µ3,x′1(m)µ3,x′1(s)(cid:68)
(t0)(cid:69)1/2(cid:68)

i (t0)u+
u+
u+2
j

(cid:68)

u+2
i

j (t)(cid:69)
(t)(cid:69)1/2 . For any homogeneous turbulence, the autocorrelation function

ri j(t0, t) =
will only depend on the time delay τ = t − t0. However, the ﬂow is not homogeneous in the
streamwise direction for post-shock turbulence, so the autocorrelation will depend on both t and t0.
Here, we will focus on how the particle velocity autocorrelation from immediately after the shock
wave. Figure 2.32 shows the diagonal components of the Lagrangian particle velocity autocorre-
lation. Again, due to the homogeneity in the spanwise direction, only r11 and r22(t) are shown.
Comparing single-ﬂuid with multi-ﬂuid case, we can clearly see that both velocity components in
the single-ﬂuid case have a stronger correlation than the multi-ﬂuid case. For the multi-ﬂuid case,
the velocity autocorrelation functions for both velocity components are very similar to each other.
For the single-ﬂuid case, the spanwise velocity has a much stronger correlation than the streamwise
component.

Results from the ﬁrst chapter have shown that the density variations in the multi-ﬂuid mixture
cause a diﬀerential distribution of turbulence statistics in regions with diﬀerent densities. For the
Lagrangian study, it would be reasonable to diﬀerentiate heavy and light ﬂuid particles to understand
the density eﬀects. In ﬁgure 2.33, the Lagrangian velocity autocorrelation of heavy and light ﬂuid

Figure 2.32: Lagrangian velocity autocorrelation function for particles staring immediately after
the shock wave.

97

t00.511.52rαα00.20.40.60.81r11(t0,t)(m)r22(t0,t)(m)r11(t0,t)(s)r22(t0,t)(s)particles are compared with the overall average. We note that for streamwise velocity, both heavy
and light ﬂuid particles have weaker correlations than the average. For the spanwise velocity,
however, the autocorrelation functions for particles with diﬀerent densities are very similar.

2.3.4.2 Particle Pair Statistics

In this section, we will present the particle pair statistics such as the particle dispersion rate
and particle pair velocity correlation to address the eﬀects of density variation on particle pair
dispersion. As described in section 3.2, the particle pair statistics depend strongly on the initial
separation direction. For simpliﬁcation, we will only address the initial separation direction as
streamwise (lx = l0) or spanwise (

z = lyz = l0).

(cid:113)

+ l2

l2
y

+(2)
j

+(1)
y
i

The ﬁrst thing to examine is the diagonal components of the relative particle dispersion tensor
i j(t) =
Dr
with diﬀerent magnitudes and directions of the initial separation vectors.
Figure 2.34 shows that the pair particle dispersion rate for particles that are initialized in the
spanwise direction with initial separation distance of l0 = 0.25η, 4η and 64η. The results in

y

(cid:68)

(cid:69)

Figure 2.33: Lagrangian velocity autocorrelation function for particles staring immediately after
the shock wave in the multi-ﬂuid STI simulations. The heavy ﬂuid particles (dashed line) and
light ﬂuid particles (dotted line) are shown for comparison.

98

00.511.52rαα(t0,t)00.20.40.60.81r11(t0,t)(m)r22(t0,t)(m)αα ≈(cid:68)(u+(1)

α

α

− u+(2)

)2(cid:69) (t − t0)2. On top of that, multi-ﬂuid case (red lines) shows an

ﬁgure 2.34 are normalized with t2 to better visualize the temporal scaling. At early times with
t − t0 < tη, the particle dispersion for all the initial separation distances follows the t2 scaling.
This is expected due to the fact that the early time dispersion is determined by the relative velocity
variance, Dr
overall larger dispersion rate than the single-ﬂuid cases with the same initial separation distance. At
intermediate times, interesting transient processes are observed for both single-ﬂuid and multi-ﬂuid
cases. The anisotropy in the particle dispersion is mainly manifested in this transient process.
For both single- and multi-ﬂuid cases, the particle dispersion in three directions starts to have
diﬀerent behaviors. After this transient adjustment, the particle dispersion in three direction starts
to converge to similar temporal scaling. In ﬁgure 2.34 (b), the initial particle separation distance is
4η. At intermediate times, diﬀerent transient processes are also shown for the particle dispersion
in three directions. Some similarity in the transient process between ﬁgure 2.34 (a) and (b) are
observed. This is possibly due to the fact that the Reynolds number considered in this study is not

Figure 2.34: Evolution of the pair particle dispersion rate normalized by t2 for both single-ﬂuid
(black) and multi-ﬂuid simulations (red). The direction of the initial separation is in the spanwise
direction with lyz = l0. The relative particle dispersion in three direction are separated into
streamwise direction (dotted lines), longitudinal direction (solid line) and transverse direction
(dashed lines). The corresponding initial separation distance is (a) l0 = 0.25η, (b) l0 = 4η and (c)
l0 = 64η.

99

large (Reλ ≈ 40). For large initial separation distance (ﬁgure 2.34 (c)), we note that the temporal
scaling of relative particle dispersion is very similar to the single-particle dispersion results (ﬁgure
2.30). This indicates that with large initial separation, the correlation between the two particles is
too small, so the two-particle dispersion is very similar to that of single-particle dispersion.

22 ≈ Dr

Figure 2.35 shows the same relative particle dispersion statistics for initial particle separation
in the streamwise direction with lx = l0. Due to the homogeneity in the spanwise directions, we
note that for all the cases in ﬁgure 2.35, Dr
33. At both early and large times, the temporal
scaling agrees with previous observations. At the intermediate time, the anisotropy due to the
diﬀerent initialization direction starts to play a role in the relative particle dispersion. For both
lx = l0 = 0.25η and 4η cases, we observe that the Dr
33) goes through a diﬀerent transient
process than Dr
11 during intermediate time. Through this transient process, the streamwise particle
dispersion becomes larger than in the spanwise direction before they reach large time temporal
scaling. For large initial separation distance, however, the transient process is less pronounced and
the temporal scaling goes from t2 to t. For all the cases, it seems that the density variations only
aﬀect the magnitude of the particle dispersion rate and anisotropy, but no eﬀects on the temporal
scaling.

22 (Dr

The Lagrangian particle relative velocity correlation is an important statistics that plays a central
role in determining the relative particle dispersion (Salazar and Collins, 2009). Figure 2.36 shows
relative velocity correlation for initial separation in the spanwise direction (lyz = l0) with initial
separation distance l0 equals to 0.25η, 4.0η and 64η. For small initial separation, the velocities
of the particle pair remain highly correlated till the end of the domain. Comparing each of the
velocity components, the correlation is stronger for the streamwise velocity. For initial separation
of l0 = 4η, the particle pair velocity correlation for all components are strong. As the particle evolve
downstream, the particle pair velocities slowly de-correlate, but this process is even slower for the
streamwise component. For large initial separation distance, the particle pair velocity correlation
is very low. The streamwise velocity undergoes a short-lived transient process before slowing
growing back. For the other velocity components, the correlation keeps decreasing after the very

100

short transient processes. Comparing single-ﬂuid with the multi-ﬂuid case, it seems that the density
variations only aﬀects the magnitude of velocity correlation.

For initial separation in the spanwise direction (lx = l0), the particle pair velocity correlation is
shown in ﬁgure 2.37. With small initial separation distance, the behaviors of the velocity correlation
coeﬃcient are very similar to those observed in ﬁgure 2.36, except the faster decorrelation. At
intermediate initial separation, all velocity components experience transient processes of increasing
and then decreasing correlation. On top of that, the length scales of the transient processes are
larger for the streamwise velocity component. For large initial separation distance, all the correlation
coeﬃcients increase as the particles evolve away from the shock wave. The implies that at this scale,
the separation distance is larger than the transient process length scale, so the transient process
won’t aﬀect the particle velocity correlation.

2.4 Conclusions

Accurate shock-capturing turbulence-resolving simulations are conducted together with Eu-
lerian and Lagrangian particle tracking post-processing methods to investigate the interaction of
isotropic turbulence with a normal shock wave for both single-ﬂuid and a mixture of diﬀerent
density ﬂuids. The main objective is to develop a better understanding of the variable density and
shock eﬀects on the post-shock turbulence structure. Grid and LIA convergence tests are conducted
to establish the numerical accuracy of the simulated data. The results show that the turbulence
statistics are grid-converged and the LIA prediction can be approached, indicating good accuracy
of the current computational method. Statistical convergence is also tested for both Eulerian and
Lagrangian statistics.

The density eﬀects on the post-shock turbulence structure are ﬁrst studied using Eulerian data.
The non-Gaussianity of the velocity gradient tensor (VGT) is studied by examining the PDFs of
velocity gradient components. The preferential ampliﬁcation of rotation and strain rate tensors is
found to be aﬀected by the density variations, leading to a weaker correlation between the two
tensors in the multi-ﬂuid case. This is shown to be caused by diﬀerent roles that density plays

101

on the modiﬁcation of rotation and strain rate tensors across the shock wave. The skewness and
ﬂatness of VGT components before and after the shock wave are then examined to understand the
evolution of VGT. It is shown that density eﬀects are weak across the shock, but are stronger in the
post-shock development. The density variations are also shown to cause the preferential alignment
between eigenvectors of the strain rate tensor and density gradient vector, which then modiﬁes the
skewness of the velocity gradient and density gradient PDFs.

The density eﬀects on the turbulence topology are then examined by ﬁrst comparing the joint
PDF of the second and third invariants of the velocity gradient tensor. The pre-shock joint PDF
has the classic tear-drop shape. However, after the shock wave, a tendency towards symmetrization
of the joint PDF, as in single-ﬂuid STI, is observed for the multi-ﬂuid case, with more data points
falling into the ﬁrst and third quadrants. After conditioning the joint PDFs based on ﬂuid density,
large diﬀerences among heavy, medium, and light ﬂuid regions are observed. In the heavy ﬂuid
regions, the joint PDF becomes almost completely symmetrical with an increasing portion of data in
the third quadrant. In contrast, the majority of the light ﬂuid data points have a similar distribution
to that of isotropic turbulence. A connection between low vortex stretching and the joint Q∗,R∗
statistics is established for the post-shock turbulence, by considering the contribution to vortex
stretching rate from each quadrant.

Furthermore, Lagrangian ﬂuid particles are used to track the development of the turbulence
and VGT after the interaction with the shock. The Lagrangian dynamics of the VGT are also
examined by using the conditional mean rate of change of the invariants of VGT. The results show
that particles in Q3 have the least residence time, while those in Q2 have the longest residence
times. Due to the return-to-isotropy experienced in the post-shock turbulence, the residence times
are smaller than those in isotropic turbulence, especially in the multi-ﬂuid case. It is also shown
that particles starting in quadrants Q1 and Q3 play an important role in recovering of the vortex
stretching rate. After interacting with the shock wave, the clockwise circulating behavior as
observed in the isotropic turbulence disappears in both single- and multi-ﬂuid cases. Our analysis
highlights the mechanism through which post-shock turbulence recovers the classical tear-drop

102

shape, with an enlarging head in the second quadrant and elongating tail in the fourth quadrant.
The contributions from diﬀerent terms in the dynamic equations of VGT invariants compared with
isotropic turbulence show that the pressure Hessian term is critical to the topological evolution of
turbulence. The pressure Hessian term is also shown to be strongly dependent on the local density
in the multi-ﬂuid case, resulting in completely diﬀerent dynamics in regions with diﬀerent densities.
Lagrangian statistics of non-inertial particles in post-shock turbulence show that the single-particle
dispersion rate and anisotropy can be correlated to the development of post-shock Reynolds stress.
The particle dispersion in the streamwise direction is found to be non-Gaussian, with the skewness
of the dispersion PDF dependent on the density variations. Lagrangian integral time scales of ﬂuid
particles with diﬀerent densities are obtained from velocity autocorrelation functions. Particle pair
dispersion generally follows the temporal scaling for isotropic turbulence.

103

Figure 2.35: Evolution of the pair particle dispersion rate normalized by t2 for both single-ﬂuid
(black) and multi-ﬂuid simulations (red). The relative particle dispersion in three direction are
separated into streamwise direction (dotted lines), longitudinal direction (solid line) and
transverse direction (dashed lines). The direction of the initial separation is in the spanwise
direction with lx = l0. The corresponding initial separation distance is (a) l0 = 0.25η, (b) l0 = 4η
and (c) l0 = 64η.

Figure 2.36: Evolution of the pair particle velocity correlation for both single-ﬂuid (black) and
multi-ﬂuid simulations (red). Velocities in the streamwise direction (solid lines), longitudinal
direction (dashed line) and transverse direction (dotted line) are compared separately. The
direction of the initial separation is in the spanwise direction with lyz = l0. The corresponding
initial separation distance is (a) l0 = 0.25η, (b) l0 = 4η and (c) l0 = 64η.

104

Figure 2.37: Evolution of the pair particle velocity correlation for both single-ﬂuid (black) and
multi-ﬂuid simulations (red). The direction of the initial separation is in the spanwise direction
with lx = l0. Velocities in the streamwise direction (solid lines) and transverse directions (dotted
and dashed lines) are compared separately. The corresponding initial separation distance is (a)
l0 = 0.25η, (b) l0 = 4η and (c) l0 = 64η.

105

CHAPTER 3

EFFECTS OF DENSITY VARIATIONS ON A NORMAL SHOCK WAVE

3.1 Introduction

In previous chapters, we studied the eﬀects of shock wave on variable density ﬂow. In this
chapter, we will focus on the shock movement in a variable density environment. Extreme casses
involving complex shock surface topology and shock breakup will not be considered. When a
normal shock wave propagates through a non-uniform media, the strength and geometry of the
shock wave, as well as the media itself, will be modiﬁed through their interactions. This is an
important fundamental problem in the study of Inertial Conﬁnement Fusion (ICF), supersonic and
hypersonic combustion, astrophysics, and other applications (Ranjan et al., 2011; Hesselink and
Sturtevant, 1988; Bisnovatyi-Kogan and Silich, 1995). In an early theoretical study, Chisnell (1955)
investigated the one-dimensional propagation of shock waves through a non-uniform (variable
density) media and derived an expression for the shock Mach number as a function of time. Later,
Whitham (1958) developed a more general analytical solution for a medium with both varying
density and pressure. These earlier formulations of shock propagation, even though useful, suﬀer
from a number of diﬃculties, one of which originates from the fact that the waves generated from
the post-shock region, i.e.
re-reﬂected waves are neglected. The re-reﬂected waves eﬀects are
accounted for in the study conducted by Bird (1961), who performed numerical simulations with
the method of characteristics and compared them with the Chisnell-Whitham (as referred to in Bird
(1961)) results and asymptotic solutions. Soukhomlinov et al. (2002) further examined the eﬀects
of temperature gradients on shock propagation. Their results were in good agreement with Bird’s
results. The above approaches can be applied to non-uniform ﬂows with large density ﬂuctuations.
In the limit of weak ﬂuctuations, this problem can be linearized to obtain the LIA prediction. To
address the eﬀects of pre-shock non-uniform density ﬁeld, McKenzie and Westphal (1968) studied
the interaction of isobaric entropy waves with an oblique shock. Mahesh et al. (1997) then treated

106

the 3D density ﬂuctuations as a combination of vorticity and entropy waves and conducted a linear
analysis. Griﬀond (2005); Huete et al. (2011) has considered the pre-shock non-uniform density
ﬁeld as a mixture of gases with diﬀerent molecular weights and extended the linear analysis to multi-
component ﬂows. These LIA results are capable of predicting both unsteady shock propagation
and post-shock ﬂuctuations, but most of the discussions have been focused on the later.

Numerical examination on this fundamental problem has been focused on the shock interaction
with turbulence or Shock-Turbulence Interaction (STI). The canonical STI is an example of shock
propagation in non-uniform media. Here, the non-uniformity in the pre-shock region is mainly
due to turbulent motions. In this type of problem, the shock propagation can reach a quasi-steady-
state for a range of parameters, so that the statistical behavior of turbulence through the shock
or in the post-shock region is of interest. Such a ﬂow has been studied using shock-capturing
and shock-resolving simulations for limiting cases (e.g. Lee et al., 1993, 1997; Larsson and Lele,
2009; Larsson et al., 2013; Ryu and Livescu, 2014; Livescu and Ryu, 2016). On the other hand,
there have been only a few numerical studies on shock propagation in non-uniform density ﬁeld.
Mahesh et al. (1997); Jamme et al. (2002) have studied the pre-shock vorticity-entropy ﬂuctuations
interaction with a normal shock wave, which is strongly related to shock/turbulent boundary layer
interaction. Recently, our group has investigated the shock interaction with multi-ﬂuid variable
density turbulence, and has shown that shock front, shock intensity, and post-shock turbulence are
strongly aﬀected (ﬁgure 3.1 a,b) by the non-uniformities in density, even at low Atwood number At
(Tian et al., 2017a,d). By further increasing the density ratio, the interaction exhibits a broken shock
regime similar to the eﬀect seen at high turbulent Mach number Mt values (Larsson et al., 2013)
(ﬁgure 3.1 c). In previous studies, the interaction was studied in regimes where the shock motion
was negligible (Ryu and Livescu, 2014) or the constant backpressure outﬂow condition was used to
stabilize the shock motion even at high Mt values or density ratios (Larsson et al., 2013) so that the
statistical analysis could be carried out. In this study, the transient shock propagation in variable
density ﬂow without boundary eﬀects is of interest, as it resembles the conditions considered in the
early theoretical work and practical applications of interest (e.g. ICF). We aim to test the available

107

theories on predicting the shock wave and post-shock ﬂow under various conditions using accurate
ﬂow resolving shock resolving/capturing simulations. We will then extend the existing theories
to account for the eﬀects of the re-reﬂected waves for 1D problems, with the hope of laying the
theoretical foundation for more complicating 3D problems.

The rest of the paper is organized as follows. The numerical details will be presented in section
3.2, including the governing equations, numerical methods, and simulation setup. In this study,
we will conduct both shock-capturing and shock-resolving simulations and assess their accuracy
in predicting shock motions. The theoretical approaches will then be described in section 3.3.
Both LIA and Chisnell-Whitham (CW) theory will be presented and discussed. In section 3.4, the
numerical results will be compared with the theoretical results for various types of non-uniform
density ﬂows. The predictability of the CW theory will be assessed based on its agreement
with numerical results. An improved model will then be proposed and the 3D eﬀects on shock
propagation will be addressed. The main ﬁndings and conclusions are summarized in the last
section.

Figure 3.1: Shock front in wrinkled shock regime (a,b) and broken shock regime (c). Shock
intensity is represented by density jump ∆ρ. Shock intensity ratio is calculated as ∆ρ/∆ρ.

108

3.2 Numerical Methodology

3.2.1 Shock-capturing Simulation

Numerical detials of the shock-capturing simulations have been described in detial in the ﬁrst
Chapter.

3.2.2 Shock-resolving Simulation

Shock-resolving simulations are conducted by computing both the viscous and inviscid ﬂuxes using
the sixth-order compact scheme. To ensure numerical stability, the viscous terms are solved in
the nonconservative form and the inviscid ﬂuxes are solved in the skew-symmetric form (Johnsen
et al., 2010). A newly developed six-order compact scheme with a fourth order boundary scheme
is implemented for the calculation of ﬁrst-order derivative (Brady and Livescu, 2018). On top
of that, the nonconservative form of the viscous ﬂuxes contains the second-order derivative of
primitive variables, which will be calculated using the six-order compact scheme with a fourth

Figure 3.2: (a) Demonstration of numerical setup of the 1D simulation for both shock-capturing
and shock-resolving simulations. (b) Space-time diagram of the 1D shock propgation (thick solid
line) in non-uniform density media. In the meantime, a reﬂected wave/forward-going wave will be
generated (thin solid line), which will then interact with the post-shock non-uniform density
media at the particle path (dashed line). As a result, a re-reﬂected wave/backward-going wave will
be generated (dotted line), which will catch up with the shock wave.

109

order boundary scheme as described in Lele (1992). To fully resolve all ﬂow scales, including
shock width (δ), the mesh is further reﬁned around the shock region. At least 12 grid points are
used in the streamwise direction to resolve the shock proﬁle, in contrast to the shock-capturing
simulations, where the shock wave is numerical and is represented by only 2-3 grid points.

3.2.3 Simulation Setup

The numerical methods discussed above were developed and tested for 3D simulations. In this
study, we will only consider the 1D cases. As shown in ﬁgure 3.2 (a), the dimension of the
computational domain is 4π in the streamwise direction. The normal shock wave is initialized in
the middle of the domain at x = 2π. For the shock-resolving simulation, the shock is initialized
using the laminar shock proﬁle as derived in Thompson (1971). A moving reference of frame at the
same speed as the laminar shock wave is used so that the shock stays relatively stationary throughout
the simulations. A buﬀer layer in the streamwise direction is set at the end of the open end of the
domain. Since in this study, we are only considering shock propagation with no boundary eﬀects,
the simulations will end before the outﬂow acoustic waves propagate to the shock wave.

At the inﬂow, a non-uniform density ﬁeld is superposed on a uniform mean ﬂow that matches
the laminar pre-shock condition for a shock wave with shock Mach number Ms. Here, we will use
the overbar f to denote the mean and f (cid:48) to denote the ﬂuctuations. The density non-uniformity ρ(cid:48)
is introduced by correlating the mass fraction (or mole fraction) of the binary ﬂuid to a scalar ﬁeld.
Diﬀerent types of density non-uniformity will be considered and discussed in the results section. To
quantify the non-uniform density ﬁeld, we will use the peak wave number of the density spectrum,
ks and Atwood number At = (MW2 − MW1)/(MW2 + MW1) to describe the length scales and
magnitude of the non-uniform medium, where MW1 and MW2 are the molecular weights of binary
mixture.

110

3.3 Theoretical Analysis

In this section, the theoretical approach developed in previous studies and its extension to
the non-uniform medium will be summarized. Here we will categorize the non-uniform density
interaction with shock wave into two diﬀerent regimes based on whether the linear interaction
assumption holds true in the analysis. Here, we will present and discuss the theories for both
interaction regimes, i.e. Linear Interaction Approximation for the linear regime and Chisnell-
Whitham theory for the nonlinear regime.

3.3.1 Linear Interaction Approximation

The interaction between non-uniform density medium and a shock wave in the linear regime has been
studied by considering the pre-shock ﬂuctuations as entropy ﬂuctuations or entropy waves (Mahesh
et al., 1997). The foundation of this analysis is the Kovasznay decomposition of compressible
turbulence (Kovasznay, 1953): small-amplitude turbulence ﬂuctuations can be decomposed into
vorticity, acoustic and entropy waves. Both acoustic and entropy waves contribute to the density
non-uniformity, but in this section, we will only focus on the shock interaction with entropy
wave induced non-uniform density ﬁeld due to its resemblance to the multi-component variable
density eﬀects. The interaction of these inﬁnitesimally small waves with a shock wave is assumed
to be linear, so that the linearized Rankin-Hogoniot jump condition can be used to derive the
magnitude of post-shock waves. Here, we will re-examine the theoretical analysis of shock/entropy
wave interaction by Mahesh et al. (1997) and extend it to include the multi-component density
ﬂuctuations.

111

The compressible multi-component Euler equations can be obtained from equation 1.1:

+

∂ ρui
∂t
∂pui
∂xi

+

∂p
∂t

+

∂ ρ
∂t
∂ ρuiuj

∂ ρui
= 0
∂xi
= − ∂p
∂xj
∂xi
= −(γ − 1)p ∂ui
∂xi
= 0

∂ ρYiuj
∂xj

+

∂ ρYi
∂t

(3.1a)

(3.1b)

(3.1c)

(3.1d)

In the Euler equations, the energy equation is reorganized to form an equation for pressure. The
mass fraction equations can be grouped to derive an equation for speciﬁc gas constant R by using
the relation: R = RunivΣ

:

Yi
MWi

∂ ρR
∂t

+

∂ ρRuj
∂xj

= 0

(3.2)

The linearized form of equation 3.1 can be written for small amplitude ﬂuctuations by ignoring
, the linearized

nonlinear terms. By introducing the entropy ﬂuctuation (γ−1) s(cid:48)
Euler equations for ﬂuctuations can be expressed as:

= p(cid:48)
p − γ ρ(cid:48)

µiY(cid:48)

− Σ

R

T

ρ

i

∂ ρ(cid:48)
∂t
∂u(cid:48)
i
∂t

+ ui

+ uj
∂s(cid:48)
∂t
∂R(cid:48)
∂t

∂ ρ(cid:48)
∂xi
∂u(cid:48)
i
∂xj

= −ρ
= −1
ρ
∂s(cid:48)
∂xj
∂R(cid:48)
∂xj

+ uj

+ uj

∂u(cid:48)
i
∂xi
∂p(cid:48)
∂xi
= 0

= 0

(3.3a)

(3.3b)

(3.3c)

(3.3d)

Note that for multi-component media, the entropy ﬂuctuations also include the chemical poten-
. It can be easily seen that the linearized Euler equation for s(cid:48) is a linear
ρ and

tial of the species −Σ
combination of the temperature contributions to the entropy ﬂuctuations (γ − 1) s(cid:48)

p − γ ρ(cid:48)
= p(cid:48)

µiY(cid:48)

T

i

t
R

112

µiY(cid:48)

i

T

c
R

= −Σ

the composition contributions (γ − 1) s(cid:48)
. The composition part of the entropy wave has
the same behavior as the speciﬁc gas constant R(cid:48), due to the fact they are both linear combinations
of the mass fraction equations. For the convenience of further discussion on shock interaction,
we will separate the two parts of the entropy ﬂuctuations and only consider the temperature part
of entropy wave: s(cid:48)
t. Equations 3.3 can then be further reorganized to obtain the Kovasznay’s
decomposition into diﬀerent waves and its extension to multi-component ﬂows.

+ uj

+ uj

+ uj

∂ω(cid:48)
i
∂t
∂s(cid:48)
t
∂t
∂R(cid:48)
∂t
∂2p(cid:48)
∂xi∂t

∂2p(cid:48)
∂t2

+ 2ui

= 0

= 0

∂ω(cid:48)
i
∂xj
∂s(cid:48)
t
∂xj
∂R(cid:48)
= 0
∂xj
∂2p(cid:48)
∂xi∂xj

= 0

+ uiuj
−a2 ∂2p(cid:48)
(cid:114)
∂xi∂xi

(3.4a)

(3.4b)

(3.4c)

(3.4d)

where ωi = i jk ∂uk/∂xj is vorticity and a =
γp
ρ is the sound speed. Four types of independent
waves can be clearly observed from equations 3.4 with their corresponding wave speed. The
vorticity wave (equation 3.4a) is convected by the mean ﬁeld and is only related to the solenoidal
part of the velocity ﬁeld, which does not contribute to the density ﬂuctuations. The density non-
uniformity comes from the other three types of waves: entropy wave (equation 3.4b), ’composition
wave’ (equation 3.4c) and acoustic wave (equation 3.4d). The acoustic waves consist of pressure
ﬂuctuations p(cid:48), density ﬂuctuations ρ(cid:48)
i,d and they
propagate at local sound speed with respect to the ﬂow. The entropy wave is isobaric and can
be expressed as the combination of density and temperature ﬂuctuations and they satisfy ρ(cid:48)
et/ρ =
−T(cid:48)/T. Another source of the density ﬂuctuations is the composition ﬂuctuations or ’composition
waves’, which satisfy ρ(cid:48)
c/ρ = −R(cid:48)/R. Equation 3.4 shows that in the linear and inviscid regime,
the composition wave in multi-component ﬂows has the same behavior as the entropy wave. Both

a and the dilatational velocity ﬂuctuations u(cid:48)

113

types of waves aﬀect the density ﬂuctuations and are convected by the mean ﬂow. Combining all
the waves, the linearized equation of state can be expressed as:

p(cid:48)
p
p(cid:48)
p

=

=

ρ(cid:48)
ρ
ρ(cid:48)
a
ρ

+

+

T(cid:48)
T
ρ(cid:48)
et
ρ

+

R(cid:48)
R
ρ(cid:48)
c
ρ

+

T(cid:48)
T

+

+

R(cid:48)
R

(3.5a)

(3.5b)

Following the procedure in Mahesh et al. (1997), we will express the pre-shock variable density
ﬁeld as a combination of diﬀerent plane waves.
In this study, we are interested in the shock
interaction with non-acoustic non-uniform density variations, so we neglect pre-shock vorticty
wave and acoustic wave and only considered entorpy wave and composition wave. The extensions
of Mahesh’s analysis are marked by {}.

u(cid:48)
1,u
u1,u
u(cid:48)
2,u
u1,u
ρ(cid:48)
u
ρu

T(cid:48)
u
Tu
p(cid:48)
u
pu
R(cid:48)
u
Ru

= 0

= 0
= Aeeik(mx+l y−u1,ut)

Aceik(mx+l y−u1,ut)(cid:111)

(cid:110)

+

= −Aeeik(mx+l y−u1,ut)

(cid:110)−Aceik(mx+l y−u1,ut)(cid:111)

= 0

=

(3.6a)

(3.6b)

(3.6c)

(3.6d)

(3.6e)

(3.6f)

(3.6g)

Here the subscript u represents the pre-shock variables. Ae and Ac are the magnitude of the
entropy wave and composition wave. The variables m = cosφ and l = sinφ are related to the
incident angle of the plane wave to the shock. The normal shock will be deformed by the waves and
the displacement is denoted using xs = ξ(y, t). Correspondingly, the linearized Rankine-Hugoniot

114

(RH) jump condition can be extended to include the composition wave:
u + RuT(cid:48)
u)
( R(cid:48)
u
Ru

M(cid:48)
1 =

)

=

3

(cid:41)

u(cid:48)
1,d − ξt
u1,u

=

=

=

u(cid:48)
2,d
u1,u
ρ(cid:48)
d
ρd

p(cid:48)
d
pd

=

R(cid:48)
d
Rd

=

−

−

R(cid:48)
u
Ru

)

)

+

)

)

2

+

2

)

ξy



u(cid:48)
2,u
u1,u

+

(γ + 1)M2
1

u(cid:48)
(cid:40)
2(TuR(cid:48)
M1 − γM1
1,u
u1,u
2u1,u
u(cid:48)
(T(cid:48)
M1 − M1
− M1
)
1,u
u
2
2
u1,u
Tu
(u(cid:48)
(γ − 1)M2
1 − 2
1 − ξt
(γ + 1)M2
u1,u
1
(T(cid:48)
u
Tu
( R(cid:48)
u
(γ + 1)M2
Ru
1
2(M2
1 − 1)
(γ + 1)M2
1
(u(cid:48)
1 − ξt
4
(γ − 1)M2
u1,u
1 + 2
−(γ − 1)M2
(T(cid:48)
1 + 4
u
(γ − 1)M2
Tu
1 + 2
( R(cid:48)
1 + 4
u
Ru
1 + 2
(u(cid:48)
1 − ξt
u1,u
(T(cid:48)
u
Tu
( R(cid:48)
u
Ru

(γ − 1)M2
4γM2
1
1 − (γ − 1)
2γM2
1
1 − (γ − 1)
2γM2
1
1 − (γ − 1)

−(γ − 1)M2



2γM2

2γM2

2γM2

)

)

)



)

(3.7a)

(3.7b)

(3.7c)

(3.7d)

(3.7e)

(3.7f)

Here, we assume that the speciﬁc heat ratio γ remains constant in the mixture. As we can see,
the pre-shock entropy wave and composition wave will generate other types of waves, including

115

vorticity wave, acoustic wave and entropy wave. The post-shock variable ﬂuctuations can therefore
be written as:

u(cid:48)
1,d
u1,u
u(cid:48)
2,d
u1,u

p(cid:48)
d
pd
ρ(cid:48)
d
ρd

T(cid:48)
d
Td

R(cid:48)
u
Ru

= Fei ˜k xeik(l y−mu1,ut)
Geik(mr x+l y−mu1,ut)
= Hei ˜k xeik(l y−mu1,ut)
Ieik(mr x+l y−mu1,ut)
ei ˜k xeik(l y−mu1,ut)
K
γ
ei ˜k xeik(l y−mu1,ut)
K
γ
+Qeik(mr x+l y−mu1,ut)

=

=

Seik(mr x+l y−mu1,ut)(cid:111)
(cid:110)−Seik(mr x+l y−mu1,ut)(cid:111)

−Qeik(mr x+l y−mu1,ut)

Kei ˜k xeik(l y−mu1,ut)

γ − 1
γ

(cid:110)

+

=

=

with r being the compression ratio of velocity:

r =

u1,u
u1,d

(3.8a)

(3.8b)

(3.8c)

(3.8d)

(3.8e)

(3.8f)

(3.9)

The boundary conditions across the shock wave will yield the equations for shock velocity:

= Leik(l y−mu1,ut)

ξt
u1,u
ξy = − l
m

Leik(l y−mu1,ut)

(3.10a)

(3.10b)

(3.10c)

The magnitude of the post-shock waves can then be related to the pre-shock waves through the

116

linearized RH equations:

F + G − L = −B1L − B2(Ae + Ac)
+ (Q + S) = −C1L − C2(Ae + Ac)
K = −D1L − D2(Ae + Ac)

K
γ

H + I = −E1
S = Ac

l
m

L

F = αK

(3.11a)

(3.11b)

(3.11c)

(3.11d)

(3.11e)

(3.11f)

α, B1, B2, C1, C2, D1 and D2 are considered as constants that depend on γ and M1. Based on
these extensions of Mahesh et al. (1997), we are able to theoretically calculate the magnitude of
the post-shock waves. From equations 3.11, we notice some physical interpretations regarding the
post-shock ﬂuctuations. Firstly, due to the linearity of the R-H jump conditions for mass fraction,
the magnitude of the post-shock composition wave S is the same as the pre-shock magnitude Ac.
Composition waves are conserved through the shock and cannot be generated by other types of
waves. Secondly, both entropy wave and composition wave can generate acoustic and vorticity
ﬁeld through interaction with the shock wave. On top of that, there exist similarities in the variable
density eﬀects from the two parts. The pre-shock variable density eﬀects from both parts with a
magnitude of Ae + Ac (equation 3.6 c,d) can be grouped together on the right hand side (RHS)
of equation 3.11 (a-c). This implies that if the total pre-shock density ﬂuctuations are ﬁxed, i.e.
Ac + Ae =constant, the post-shock pressure, density and velocity ﬂuctuations will be the same,
regardless of the individual magnitude of Ae and Ac. This has also been observed using transfer
matrix by Griﬀond (2005). On the other hand, the eﬀects both entropy wave and composition wave
on post-shock entropy wave will be diﬀerent depending on the values Ac and Ae. The magnitude
of the composition wave is the same through the shock wave, while that of entropy wave is aﬀected
by both Ae and Ac. Note that the above analyses are based on the linearized Euler equations and
the assumptions that speciﬁc heat ratio γ is constant. If scale separation between the shock wave
and pre-shock disturbances is not satisﬁed, the viscous eﬀects can play a role in the interaction,

117

which will make the post-shock ﬁeld deviate from the LIA limit (Ryu and Livescu, 2014). Or if γ
varies in the binary mixture, the analysis will also be further complicated (Griﬀond, 2005).

3.3.2 Chisnell-Whitham Theory

In many of the engineering applications, At or the magnitude of the density non-uniformity is too
large, which will invalidate the assumptions of LIA. For example, the density ﬂuctuations ρ(cid:48)/ρ
are O(1) in the hydrogen/air mixture in the scramjet engine. In the nonlinear regime, a diﬀerent
approach is needed to predict shock propagation in non-uniform density media. Early theoretical
studies by Chisnell (1955); Whitham (1958) has considered the large density ﬂuctuations as a
sequence of weak contact discontinuities and derived a relation between the density change of dρ
with the shock strength change dz, where z is the ratio of pressure across the shock wave. The
underlying assumption is that the shock is quasi-equilibrium and the density change across the
contact discontinuity is weak.

Figure 3.2 (b) shows a wave diagram of the shock propagation in a non-uniform density media.
As the shock wave encounters a density change dρ, its Mach number and propagation speed will be
modiﬁed. In the meantime, a reﬂected wave/forward-going wave will be generated (marked as thin
solid line), which will then interact with the post-shock non-uniform density media at the particle
path (dashed line). As a result, a re-reﬂected wave/backward-going wave will be generated (dotted
line), which will propagate in the same direction as the shock wave and eventually catch up the
shock wave.

The relation between upstream density jump dρu and dz can be written as (Chisnell, 1955):

where λ =(cid:112)(γ − 1)/(γ + 1) is a constant. In equation 3.12, the re-reﬂected waves are neglected.

( 1 + λ2z
z(1 + λ2))1/2

2
z − 1

1
ρu

dρu
dz

λ2 + z

=

2
z − 1

−

1

+

(3.12)

The diﬀerential equations can be integrated to obtain the explicit relation between shock strength

118

and density.

ρ = A

(z − 1)2
(λ2 + z)

(cid:40)(cid:112)
(cid:112)

λ2 + 1/z + λ
λ2 + 1/z − λ

(cid:41)2λ/(cid:113)(1+λ2)(cid:40)√

√

λ2 + 1 −(cid:112)
λ2 + 1 +(cid:112)

λ2 + 1/z
λ2 + 1/z

(cid:41)2

(3.13)

where constant A can be calculated from the initial condition. In this formulation, the pre-shock
density non-uniformity is assumed to be either entropy ﬂuctuations or composition ﬂuctuations.
This implies that in the nonlinear interaction regime, the density non-uniformity resulted from both
temperature ﬂuctuations and multi-component eﬀects are also equivalent. On top of that, the shock
strength z is a direct function of ρ, meaning that shock strength should be independent of the shape
of density proﬁle. This equation can then be numerically integrated over any given density proﬁles
to obtain the temporal evolution of the post-shock medium, which can then be used as a boundary
condition for the solution of the post-shock turbulent ﬁeld. Due to the fact that the dependence of
shock strength on density is implicit, z needs to be numerically solved using an iterative method. In
this study, the numerical solution of z is considered to be converged when the error is smaller than
10−8. Here for simplicity, the dependence of z on ρ will be represented using implicit function
z = g(ρ). Therefore, the shock position can be obtained using the following integration:

Þ
Þ (cid:112)
Þ (cid:112)

xs =

=

=

dxs =

usdt =

γP/ρ(xs)

γP/ρ(xs)

Þ

a(xs)M(xs)dt

Þ
((z − 1)(γ + 1)
+ 1)dt
((h(xs) − 1)(γ + 1)

2γ

(cid:115)
(cid:115)

+ 1)dt

2γ

(3.14)

By integrating the above equations numerically, the shock position xs as a function of t can be

obtained.

The interaction between weak contact discontinuity with a shock wave will also generate
Its magnitude can be expressed as

acoustic waves that propagate away from the shock wave.
(Chisnell, 1955):

119

z f = 1 +

dz
z

(3.15)

where dz can be related to the density jump through equation 3.12. When this elementary
wave interacts with the post-shock non-uniform ﬁeld, the re-reﬂect waves with be generated, which
will propagate towards the shock wave and change the shock strength. Chisnell (1955) derived the
magnitude of the re-reﬂected waves as:

here ρ(z, z0) can be calculated using the following equation (Chisnell, 1955):

zb = 1 − 1

1

4 α(z, z0)
α(z, z0) = ( ρ(z,z0)

dz
z dz0

ρ(z,z0) ∂ ρ(z,z0)
∂z0
ρ(z,z) )1/4
Þ z

(3.16)

(3.17)

(3.18)

−1/4(z, z0) = ρ

ρ

−1/4(z0, z0) − 1
4γ

−1/4(x, x) dx
x

ρ

x=z0

where ρ(z, z) is the density immediately after the shock wave for a given shock intensity z. Based
on these equations, the magnitude of the fast and slow acoustic waves can be calculated through
numerical integration for a given density proﬁle.
In this study, we aim to test the applicability
of these theoretical results to various types of problems with the hope of ﬁnding a universal and
simple model for the shock propagation in non-uniform density ﬁeld.

3.4 Results and Discussion

In this section, the numerical results obtained from both shock-capturing and shock-resolving

simulations will be examined and compared with theoretical results.

3.4.1 Shock Wave Propagation in Media with Linearly Varying Density

In this section, the simple case of shock propagation in a linearly varying density ﬁeld is considered.
The aim is to verify the accuracy of current shock-capturing simulations in predicting shock

120

propagation in variable density media. Pseudo-1D simulations for linearly varying density proﬁle
in the weak shock limit of M0 = 1.1 are performed ﬁrst. The ratio of the density at the beginning the
ﬁeld to that at the end of the ﬁeld is 8 (for linearly decreasing density ﬁeld) and 0.125 (for linearly
increasing density ﬁeld). Figure 3.3 (a) compares the time development of shock position obtained
from numerical results and CW solution. The x-axis denotes time and the upper curve and lower
curve represent cases with increasing and decreasing density ﬁeld, respectively. As observed, in the
weak shock limit, the results from current shock-capturing simulations are in excellent agreement
with the CW solution for both decreasing and increasing density proﬁles. On the other hand, the
match with the CW solution is less than satisfactory for a strong shock (M0 = 4). In this case, the
numerical results for the case with increasing density gradually deviate from the CW solution in
time (ﬁgure 3.3), resulting in a ﬁnal diﬀerence of around 5%. The case with decreasing density
shows a relatively better agreement at M0 = 4.

To further understand the discrepancy, we compare the results from current simulations with
those obtained by Bird (1961), who used the method of characteristics. The exact same conditions
are used as in Bird (1961), for Mach number 4.0 and density ratios of 8 and 0.125. The comparison
of the shock Mach number as a function of distance is shown in ﬁgure 3.4. Evidently, the current
results agree better with those obtained by the method of characteristics for the increasing density
case. On the other hand, for the linearly decreasing density case, all three methods give comparable
predictions. It was argued in Bird (1961) that the inﬂuence of re-reﬂected waves, which is not
considered in the CW solution, is diﬀerent for diﬀerent density proﬁles. This explains the diﬀerent
performance of the CW model in diﬀerent situations. Our numerical results agree well with those
obtained by the method of characteristics and also with CW theory when the re-reﬂected waves are
not important. Note that the solutions are obtained by solving the Navier-Stokes instead of Euler.
This implies that the diﬀusion eﬀects are indeed week through the interaction for the parameter
range considered in these cases.

121

3.4.2 Shock Wave Propagation in Media with Fluctuating Density

The fundamental understanding of shock propagation in 1D ﬂuctuating density is critical to the
study of shock-turbulence/shock-boundary layer interactions. This problem provides a simpliﬁed
conﬁguration for the more complex 3D case with turbulence.

The propagation of shock wave into non-uniform density with a sinusoidal shaped proﬁle
is called the "Shu-Osher" problem, which can be theoretical solved using the formulas in the
linear interaction regime. This has been a classical problem for the validation of shock-capturing
numerical schemes (Shu and Osher, 1989). In this section, we will use it as a simpliﬁed case for
the study of shock propagation in randomly ﬂuctuating density ﬁeld, especially in the nonlinear
regime.

3.4.2.1 The Similarity of Density Fluctuations from Entropy and Composition Fluctuations

In section 3.3, the similarities and diﬀerences entropy wave and composition wave through shock
wave have been discussed under certain conditions using theoretical derivation. In this section, we
will use the numerical results to compare the density eﬀects from both temperature ﬂuctuations
and composition ﬂuctuations.

Here, we will brieﬂy describe the numerical setup for the "Shu-Osher" problem. The shock

Figure 3.3: Shock wave propagation in the medium with linearly varying density for density ratios
of 8 and 0.125. (a) M0 = 1.1 and (b) M0 = 4.0.

122

is initialized in a ﬂuid with ρ = 1.0 at x ≈ π; the shock Mach number is 3.0; then a ﬂuid with
density proﬁle given by ρ = 1.0 + Atsin(ksx) is superposed on the mean ﬂow at the inlet, where
At = 0.25 is the Atwood number. This density proﬁle is obtained using two diﬀerent methods: a)
the composition of the ﬂow is varied with ﬁxed T and p and b) the temperature is varied with ﬁxed
Yi and p. The shock-resolving method is used for the simulations.

Figure 3.5 shows the pressure, density and temperature in the streamwise direction. As shown
in the ﬁgure, the post-shock pressure and density from both cases are exactly the same. This
is in good agreement with the previous theoretical analyses. We also note that the post-shock
density ﬁeld has both short and long wavelength ﬂuctuations. This is caused by the superposition
of post-shock entropy/composition waves and acoustic waves and the short and long wavelength
ﬂuctuations are due to their diﬀerent characteristic speed. The ratio of the wavelength is a function
of the shock Mach number.
Its dependence on Ms is plotted in ﬁgure 3.7 and conﬁrmed with
numerical results. On the other hand, the post-shock temperature ﬂuctuations are diﬀerent and the

Figure 3.4: Evolution of shock Mach number in linearly varying density ﬁeld for ρ1/ρ0 = 8 and
0.125. Initial shock Mach number M0 is 4.0.

123

x00.20.40.60.81Ms234567present workChisnell-WhithamBird (1961)ρ1=8M0=4ρ0=1ρ1=0.125phase of the temperature ﬂuctuations are reversed. This shows the diﬀerence between composition
waves and entropy waves in the interaction with a normal shock wave.

3.4.2.2 Shock Propagation in Sinusoidal Density Media

It was shown in the last section that the non-uniform density eﬀects from both temperature or
composition ﬂuctuations through the shock wave are similar. Therefore in this section, we will only
consider the variable composition cases.

In ﬁgures 3.6 (a,b), the temporal development of shock position is shown for a Mach 2 shock
and small density ratio of 2:1 (At = 0.33) at diﬀerent wavenumbers of ks = 1 and 4. The
CW solution is obtained by integrating equation 3.14. As shown in the ﬁgure, the CW solution
predicts a shock motion that resembles the sine wave variation in the density proﬁle, but with a
constant mean velocity relative to the unperturbed shock that moves the shock wave upstream.
The match between the numerical and theoretical solution is fairly good during the ﬁrst (sine)
wave cycle. However, after the ﬁrst cycle, the shock wave acquires additional oscillatory motions

Figure 3.5: Plots for ﬂow variables in streamwise directions for a Mach 3 shock propagating in a
sinusoidal shaped density ﬁeld. (a) Pressure, (b) density and (c) temperature.

124

instead of moving upstream following the original sine wave only, as predicted by the theory.
Thus, the current shock-capturing simulations are predicting long-wavelength oscillatory motions
superposed on the original sinusoidal motion. Shock-resolving simulations are also conducted to
verify this observation. As seen in ﬁgure 3.6 (b), shock-resolving results match very well with
the shock-capturing results, proving that the deviation from CW solution and the long-wavelength
oscillatory motions are both physical. The ratio of the wavenumber, kl, of the ﬁrst harmonic of
these motions to the sine wave wavenumber, ks, is denoted as rl = kl/ks ≈ 1/4. This ratio appears
the to be the same for ks = 1 and 4. To explain these long-wavelength motions, we consider two
velocities that are related to the behavior of shock: 1) the velocity at which the pre-shock medium
approaches the shock wave vp, and 2) the velocity of the re-reﬂected waves approaching the shock
wave, vr, relative to the shock speed. vp carries the density ﬂuctuations responsible for moving
the shock wave. vr can be found from the local sound speed behind the shock. The re-reﬂected
waves can eventually overtake the initial shock and aﬀect its strength and speed (Bird, 1961). The
ratio of vr/vp is denoted as rv, and we ﬁnd that rv ≈ rl. For a weaker shock, as shown in ﬁgure
5(c), rv and rl have small magnitudes: rv ≈ rl ≈ 1/12. Thus, the early time agreement with the
CW model is improved and the long-wavelength variation in shock position is also clearer. It can
be inferred that the re-reﬂected waves causing the long-wavelength motions of the shock (Bird,
1961) are responsible for deviation from the theoretical solution. For a ﬂuctuating density proﬁle,
it seems that the re-reﬂected waves play a more important role in changing the shock strength and
the shock motion, making the shock wave further deviate from the CW model.

This hypothesis can be veriﬁed by theoretically deriving the wavelength ratio of the two eﬀects.
This is done by matching the frequency of the waves and then relate the wavelength to the wave
speed:

(cid:115)

(

1
s −(γ−1)/2
γM2
1+(γ−1)M2
s /2

− 1)

rms =

2+(γ−1)M2
s
(γ+1)M2
s

(3.19)

From equation 3.19, we can see that the wavelength ratio is only a function of shock Mach

125

number Ms and speciﬁc heat ratio γ. In this study, we have ﬁxed γ to be 1.4, so the wavelength
ratio will only be a function of Ms. To verify the applicability of this equation, we have conducted
numerical simulations covering a wide of range of Ms. The numerical results (black circle) are
compared with equation 3.19 (black solid line) in ﬁgure 3.7. As we can see, the theoretically

derived wavelength ratio decreases as the Ms increases, reaching an asymptotic value ((cid:112)2γ +
(cid:112)γ − 1)/(cid:112)γ − 1. The numerical results from shock-resolving simulations are marked in the ﬁgure

using black circles. Results show that the agreement with the theoretical derivation is very good,
which shows that the long-wavelength movement of the shock is caused by the re-reﬂected slow
acoustic waves.

It has been pointed out that in ﬁgure 3.6, the CW prediction of the ﬁrst sinusoidal waves agrees
very well with the numerical results. We have shown that the wavelength ratio decreases to an
asymptotic limit as the shock Mach number Ms increases. For γ = 1.4, the asymptotic value is
around 3.65, which still provides large enough separation between the two types of waves. This

Figure 3.6: Shock propagation in a ﬂuctuating density ﬁeld. (a,b) M0 = 2.0, density ratio 2.0, and
ks = 1 and ks = 4 and (c) M0 = 1.1, density ratio 2.0, and ks = 4.

126

means that the CW solution that neglects the re-reﬂected waves could provide a good estimation for
the short wave-length shock oscillation. The magnitude of the short wavelength shock oscillation
can be obtained by integrating equation 3.14 over the very ﬁrst sin wave. The wavelength of the
pre-shock density proﬁle is ﬁxed, so the results will be a function of shock Mach number Ms and
Atwood number At.

The shock position oscillation from both numerical simulations and numerical integration of
CW solutions are compared in ﬁgure 3.8. The CW theoretical results show that as the At number
increases, the shock oscillation becomes larger. For cases with diﬀerent Ms, lower Ms case will
have a relatively larger shock oscillation. The numerical results are also marked in the ﬁgures. It
can be seen that the numerical results agree with the theoretical results. This conﬁrms that the
CW solutions provide as a reasonably good method for predicting the 1D shock oscillation in the
nonlinear regime.

Figure 3.7: Comparison of the wavelength ratios between numerical results and theoretical results.

127

3.4.2.3 CW Solution with Correction for Re-reﬂected Waves

In the above analyses, the eﬀects of the re-reﬂected waves are neglected in the CW solutions.
Without taking the re-reﬂected waves into consideration, the theory still gives acceptable predictions
for short wave-length shock oscillations. To further improve the applicability of the theory, the
eﬀects from the re-reﬂected waves need to be considered. Chisnell (1955) has considered the
catching up of the re-reﬂected waves for the linearly increasing density proﬁles. However, the
formulas are not closed, making it impossible to theoretically calculate the shock strength. In this
section, we will extend some general formulas in Chisnell (1955), i.e. equations 3.15 and 3.16 to
propose a theoretical model for the estimation of the re-reﬂected wave eﬀects on shock propagation
in ﬂuctuating density proﬁles.

The magnitude of the generated acoustic waves immediately after the shock wave can be
estimated by numerically integrating equation 3.15. The magnitude of the wave can be expressed
as:

Figure 3.8: Comparison of the shock position ﬂuctuations between numerical results and
theoretical results. Results from shock-resolving simulations are maked using symbols for
Ms = 1.1 (black circle), Ms = 1.3 (black cross), Ms = 1.5 (black star), Ms = 2.0 (red circle),
Ms = 2.5 (red corss), Ms = 3.0 (red star).

128

Þ x

0

dz
z(xs)

z f (xs) − 1 =

(3.20)

Similarly, the magnitude of the re-reﬂected waves can be estimated by integrating 3.16 over dz0

Þ z

0

and then dz:

zb(xs) − 1 = −

1
4 α(z, z0)

1

ρ(z, z0)

∂ ρ(z, z0)
∂z0

dz
z

dz0

(3.21)

(λ2+z)
1+λ2z

and ρ(z, z) is the post-shock density when the shock strength is z and can be calculated using the
formula ρ(z, z) = ρ
. From these equations, the proﬁles of the numerically integrated waves
can be obtained. To estimate the applicability of these estimations, the results from shock-resolving
simulations will be compared with the theoretical results. In order to decompose the post-shock
waves from numerical simulations, a wave decomposition method as proposed by Payri et al.
(1995) is used. The forward propagating (or downstream the shock wave) waves (p f )or backward
propagating waves (pb) can be calculated using:

p f = p0[1
2

(1 + ( p
p0

) γ−1
2γ (1 +

γ − 1
2

u1
a

)] 2γ
γ−1

pb = p0[1
2

(1 + ( p
p0

) γ−1
2γ (1 − γ − 1
2

u1
a

)] 2γ
γ−1

(3.22)

(3.23)

Therefore, based on the deﬁnition of wave strength z, z f can be calculated to using p f ,max/p f ,min
and zb can be calculated as pb,max/pb,min. The relative importance of the re-reﬂected waves can
be estimated using the ratio of the magnitude of the two waves zb/z f .

The ratio zb/z f from both theoretical derivation and numerical results are plotted in ﬁgure
3.9. The theoretical results predict that as the initial shock Mach number Ms increases, the ratio
of zb/z f becomes smaller. And as the magnitude of density non-uniformity At increases, the
relative importance of re-reﬂected waves zb also becomes weaker. Numerical results, as marked in
the ﬁgure using symbols, matches very well with the theoretical results for both shock-capturing

129

simulations and shock-resolving simulations. This proves that the prediction of the magnitude of
the post-shock re-reﬂected waves provided by Chisnell (1955) is accurate. On the other hand, the
relative magnitude of the re-reﬂected wave decreases as the shock Mach number Ms increases and
At increases.

The correction to the shock propagation was proposed by (Chisnell, 1955) to compensate for
the eﬀects of the re-reﬂected waves. The strength of the re-reﬂected waves zb can be incorporated
into the original equation. We have shown that the magnitude of the re-reﬂected waves can be
predicted using the above formulation. Here, we will propose several simpliﬁcations of Chinsnell’s
correction to the shock propagation.

As shown in ﬁgure 3.10, the elementary re-reﬂected wave generated by a forward-going wave
dz1 interacting with the non-uniform density ﬁeld generated by dz0 can be expressed using equation
3.21. This elementary wave will propagate upstream and eventually changing the shock intensity at
dz. Therefore, the modiﬁcation of shock strength dz is not only aﬀected by the pre-shock density
change dρ, but also aﬀected by the re-reﬂected waves along the negative characteristic lines, which

Figure 3.9: Comparison of the relative magnitude zb/z f of the re-reﬂected waves.Results from
shock-resolving simulations are maked using symbols for At = 0.1 (black circle), At = 0.2 (black
cross), At = 0.333 (black star), At = 0.4 (red circle), At = 0.5 (red corss), At = 0.6 (red star),
At = 0.9 (blue circle).

130

1 − dzÞ z0=z

z0=0

1 − dzrere =

is marked as the dash-dotted line in ﬁgure 3.10. From here, we will use dzre to represent the
modiﬁcation of the shock wave strength after considering the re-reﬂected waves. After getting
the magnitude of the re-reﬂected wave dzrere that catches up with the shock wave, dzre can be
estimated using:

= ( 2

dρu
ρu

z−1( 1+λ2z
z−1 − 1
+ 2
λ2+z
z(1+λ2))1/2dzrere
z−1( 1+λ2z
+ 4

z(1+λ2))1/2)dzre

(3.24)

The only remaining question is the estimation of dzrere along the negative characteristic line.
In Chisnell’s theory (Chisnell, 1955), the magnitude of the re-reﬂected wave which catches up the
incident shock is:

1
z1

dz0

1

ρ(z1,z0) ∂ ρ(z1,z0)

∂z0

dz1
dz

(3.25)

4 α(z, z1)
1
The only unknown in the integration is dz1
dz , which will make the Chisnell’s formulation not closed.
This ratio dz1
dz represents the portion of the elementary re-reﬂected waves that will catch up and
then modify the strength of the initial shock wave. This is determined by the relative speed of
the shock wave and the re-reﬂected wave. Here we assume that the post-shock acoustic speed is
constant so that the characteristic lines are straight, which implies dz1
= constant. The
dz
above assumption is reasonable when the shock wave is propagating in a ﬂuctuating density ﬁeld
with a ﬁxed mean value. By making the above assumption, one can easily integrate the re-reﬂected
waves along the characteristic line without the need of solving for the full post-shock ﬁeld.

= ub−us
us

After obtaining the magnitude of the re-reﬂected waves, it can be substituted into equation 3.24
to obtain dzre, and therefore the modiﬁed shock strength zre(t). The modiﬁed shock position can
then be calculated by replacing z(t) with zre(t) in the numerical integration of equation 3.14. The
modiﬁed CW solution will be referred to as CW with re-reﬂected waves (CWRW).

Figure 3.11 shows the results for shock-resolving simulation, CW, and CWRW model. It can be
seen that by including the eﬀects of re-reﬂected waves, the CWRW model can signiﬁcantly improve

131

the prediction of shock propagation in a sinusoidally shaped density ﬁeld. Evidently, both the mean
deviation from the CW solution and the long-wavelength oscillation are predicted by the model.

3.4.2.4 Shock Propagation in Random Density Media

The above analysis based on simpliﬁed density proﬁles can provide some basic assessment of the
CW and CWRW models. However, due to the nonlinear nature of the problem, one should not
expect that in a random realistic density media, the CWRW model solutions would still be valid.
In this section, we will conduct an analysis on a randomly generated density proﬁles that satisfy a
given spectrum and compare both CW and CWRW models with the numerical results.

Here, we will use the same spectrum as used in Tian et al. (2017d) for the 1D density ﬁeld.
The characteristic length scale of the density ﬁeld is modiﬁed by changing the peak wave number
of the spectrum. The generated density ﬁeld will then be used as the inﬂow condition for both
numerical simulation and numerical integration of CW and CWRW models. Figure 3.12 shows

Figure 3.10: Demonstration of the catchup of the re-reﬂected waves.

132

the shock propagation in a realization of the random density ﬁeld with ks = 4.0 and At = 0.5. The
numerical solutions are plotted using solid black lines and CW solutions are represented as dashed
lines. As we can observe, the CW solutions predicted an overall similar proﬁle as the numerical
simulations. However, they still gradually deviate from the numerical solution, the same as we have
observed in ﬁgure 3.6 (b). We also note that the long wavelength oscillation are not visible. On
the other hand, the CWRW model shows improvement over the CW solution and agrees better with
the shock-resolving simulation predictions. The results conﬁrm that the CWRW model provides a
signiﬁcant improvement over the CW theory by including the re-reﬂected wave eﬀects in 1D shock
propagation.

3.4.3 Shock Wave Propagation in Three-dimensional Non-uniform Density Field

In most real applications, the non-uniform density distribution/media is three-dimensional. The
interaction between small amplitude 3D density ﬂuctuations with a normal shock wave has been

Figure 3.11: Shock propagation in sinusoidally varying density ﬁeld. The temporal development
of shock position is compared for the shock-resolving simulation, CW theory and the proposed
CWRW theory.

133

theoretically studied by Griﬀond (2005); Huete et al. (2011) using linear theory. These studies
have been focusing on the prediction of the post-shock ﬁeld rather than the shock propagation and
wrinkling. The amplitude of density ﬂuctuations or At is usually too large in real-life applications,
making the linear theory inappropriate for predicting shock motions.

In the above discussions, we have considered the application of CW theory and CWRW model
in predicting shock propagation in 1D non-uniform density media. The advantage of CWRW model
is that it can be extended to the nonlinear regime. When the 3D spatial eﬀects in the spanwise
directions play a role, one may not simply treat each point on the 3D shock as individual 1D normal
shock. In this section, we will present results and discussions to describe the 3D eﬀects of shock
propagation in non-uniform media.

Figure 3.12: Shock propagation in random density media. Diﬀerent realizations of the random
density ﬁeld are represented using diﬀerent colors.(a) ks = 4.0 and (b) ks = 8.0.

134

3.4.3.1

3D Sinusoidal Density Field

k2
y

+ k2

(cid:113)

kyz , where kyz =

The shock propagation in a realistic 3D random non-uniform media is understandably complicated.
To simplify the problem, here we consider a 3D sinusoidal shaped density ﬁeld that satisﬁes
ρ(x, y, z) = 1.0 + Atsin(kx x + ky y + kzz). By using this 3D density ﬁeld, one can prescribe At and
kx, ky, kz to modify the intensity and length scale of the non-uniform density ﬁeld. The anisotropy
of the non-uniform density can also be changed by modifying the ratio of the streamwise to spanwise
length scales rk = kx
z . Using coordinate transformation, one may show
that the 3D sinusoidal shaped density ﬁeld is essentially a 2D plan wave with At, kx, and kyz
being the important parameters. When the wavenumber ratio rk is equal to one, the density ﬁeld
becomes isotropic. When rk increases and approaches inﬁnity, the density ﬁeld becomes quasi-1D.
On the other hand, when rk approaches zero, the spanwise shock surface wrinkling will dominate
the shock propagation. Fundamental understandings of these two limits and how the shock behaves
when the ratio rk is varied are critically important to the modeling of shock propagation in 3D
non-uniform media. The following results and discussions on the 3D eﬀects are divided into two
parts. The limit when scale separation ratio rk approaches inﬁnity will be studied ﬁrst, followed by
an examination of the limit when rk goes to zero.

In ﬁgure 3.13, the instantaneous shock positions for several selected points on the shock surface
are plotted as a function of time. We have veriﬁed that the selected points on the shock surface
can represent the overall behavior of the shock wave, so the following discussions and conclusions
may apply to the whole 3D shock surface. For all the cases in ﬁgure 3.13, At is very close to 0.5
and the length scale in the streamwise direction is ﬁxed with kx = 4. The 3D eﬀects are studied
by varying ky and kz. Unless otherwise stated, ky is equal to kz. The results form CW theory
and CWRW models are also plotted. Note that both CW theory and CWRW models are for 1D
shock propagation, so in this 3D case, we have made the assumption that the selected point on the
shock wave is only aﬀected by the non-uniform density ﬁeld right before that point. This essentially
implies that the corresponding point on the shock is moving independently like a 1D normal shock.
We note that when ky, kz is reduced to zero, the 3D simulation becomes quasi-1D. As expected,

135

the predicted values fall onto the CWRW model curve, which deviate from CW theory on average.
As ky, kz are increased from 0 to 1 and 2, the results do not show signiﬁcant deviations from the
quasi-1D (ky = kz = 0) case and CWRW model. This indicates that with a wavelength ratio greater
√
2, the shock motion can be approximated by the 1D theory and the 3D eﬀects are
than or equal to
generally negligible. Further increasing k y, kz to 3, however, signiﬁcantly changes the behavior of
the shock wave. In addition to the sinusoidal-shaped ﬂuctuations, the shock wave moves forward
with a mean velocity similar to that predicted by the CW theory. The sudden change in the shock
2) is hypothesized to be caused
behavior when the wavenumber ratio is decreased from
by the nonlinear eﬀects induced by the 3D non-uniform density ﬁeld. When the density ﬁeld
becomes more isotropic, the eﬀects from the post-shock re-reﬂected waves seems to be weakened.
Additionally, the 3D eﬀects somehow further increase the mean shock propagation speed when
ky, kz are further increased. We have conducted similar simulations for kx = 1 and 2 with same
wavenumber ratio. The results (not shown) are found to be in good agreement with the results

√
√
2 to 4/(3

Figure 3.13: Shock propagation in 3D sinusoidal shaped density media. The temporal
development of shock position on certain point on the shock surface is plotted. kx = 4 for all the
cases and ky = kz unless the values are indicated by the legend.

136

t00.511.52Xs-1.2-1-0.8-0.6-0.4-0.20ky=0(1D)ky=1ky=2ky=3ky=3,kz=4ky=5,kz=0ky=4ky=5ky=6ky=10,kz=0ky=8ky=16Chisnell-WhithamCWRWabove, implying that the wavenumber ratio is the only important parameter.

To study the other limit (when rk approaches zero), the length scales in the spanwise directions
are further decreased by increasing ky, kz. By doing so, we observe a smooth increase in the mean
shock speed until ky = kz = 8 or a wavenumber ratio of 0.35. Results from a smaller wavenumber
ratio have almost the same mean propagation speed. This suggests that when the wavenumber ratio
is less than 0.35, the mean shock propagation speed reaches another limit and in this limit, the 3D
eﬀects in the streamwise direction do not play a role in the mean shock propagation.

To better show the dependence of the shock propagation speed to the wavelength ratio, all the
results for kx = 4 with varying ky and kz are plotted on ﬁgure 3.14. The shock speed is normalized
using the mean sound speed in the variable density media. A hyperbolic tangent function is ﬁt to
the data points using least square ﬁtting. The two limits on mean shock propagation for wavelength
ratios rk = 0 and ∞ are not well captured by the ﬁtted function but can be inferred from ﬁgure 3.14.
For the limit where rk approaches inﬁnity, the mean shock propagation speed reaches a small non-
zero value, much smaller than that predicted by the CW theory. This limit is essentially a pseudo-1D
case, which can be numerically approached by setting ky = kz = 0. For this limit, the improved

Figure 3.14: The dependence of mean shock speed on wavenumber ratio for shock propagation in
3D non-unform density media with kx = 4.

137

rk01234Us/a00.020.040.060.080.10.120.140.16CWRW model proposed earlier gives a better prediction of the mean shock propagation speed than
the CW theory. For the other limit, results show that when rk approaches 0, the shock propagation
speed becomes independent of the wavelength ratio and reaches a maximum speed around Mach
0.175. Dimensional analysis show that the mean shock speed will be a function of At and Ms, i.e.
Us(At, Ms). For this limit, the smallest wavenumber ratio achieved in the numerical simulation is
0.044. Unlike the other limit, one can not simply set the streamwise wavenumber kx = 0, because
this will result in a broken shock wave. It is also problematic to continuously increase the spanwise
wavenumber ky, kz in simulations, because the wavelength of the ﬂuctuation will keep decreasing
and eventually reaching a scale where the diﬀusion eﬀects become very important. In that case, the
shock wave propagation speed will depend on Reynolds number Re. Previous simulations satisfy
the scale separation criteria between the characteristic shock length scale and non-uniform density
length scale, so the viscous eﬀects can be neglected in the shock interaction. This essentially means
that the above observations are for Reynolds number being inﬁnitely large across the shock. When
the small Re starts to aﬀect the results, the predicted Us will not be the correct inviscid rk = 0 limit.
In this study, the limit rk = 0 can only be approximated through simulations with rk = 0.044 with
reasonably large length scale separation.

The dependence of asymptotic value of mean shock propagation at the limit of rk ≈ 0 on Atwood
number is also studied with ﬁxed initial shock Mach number. The results are plotted in ﬁgure 3.15.
We note that the mean propagation speed increases when the Atwood number is increased, which
is mainly due to nonlinear eﬀects. When At = 0, the mean shock propagation speed is zero, so the
shock would remain stationary. As the Atwood number increases, the nonlinear eﬀects introduced
through the non-uniform density ﬁeld will increase, causing the increase in mean shock propagation
speed. In 1D cases, the re-reﬂected waves cancel out part of the nonlinear eﬀects, making the shock
propagation speed smaller than the 3D cases.

By removing the mean propagation from the instantaneous shock position, one can obtain the
magnitude of ﬂuctuations in shock position at diﬀerent spanwise locations, which can be used
to describe the shock surface wrinkling. Figure 3.16 shows the magnitude of the ﬂuctuations in

138

shock position (L2(∆xs)) as a function of the wavelength ratio. A least-square-ﬁt to the function
(c(1)rk + c(2)) × ec(3)r2
k − c(2) is also shown. Here we note that L2(∆xs) peaks around rk ≈ 0.57
and continuously decreases as the wavelength ratio increases. The asymptotic limit at rk = ∞,
representing the 1D case, is not accurately captured by the ﬁtted function but can be reasonably
well predicted using the CWRW model. On the other hand, when the wavelength ratio decreases
and approaches zero, the spanwise length scale of the non-uniform density ﬁeld becomes smaller.
Numerical results show that this will cause the shock oscillations to decrease. As the wavelength
ratio approaches this limit, the ﬂuctuations in the spanwise directions become more important to
the shock wrinkling. Therefore, the decrease in length scale of the spanwise density ﬂuctuations
causes the decrease in the shock surface wrinkling.

3.4.3.2

3D Random Density Field

For a 3D random density ﬁeld, the temporal shock propagation becomes extremely diﬃcult to
predict, because there exists a wide range of spanwise and streamwise length scales. Despite the
diﬃculty in the accurate prediction of shock position, previous results show that for a ﬁxed At and

Figure 3.15: The dependence of mean shock speed on wavenumber ratio for shock propagation in
3D non-uniform density media with kx = 4.

139

At0.20.30.40.50.60.7Xs00.10.20.30.40.50.60.7Ms, the shock propagation speed should be bounded by the two limits obtained in the previous
section. In this study, a shock-capturing simulation was conducted to study shock propagation in a
randomly generated density ﬁeld. The generation of the 3D density ﬁeld has been described in the
ﬁrst chapter. Here, we have kept the peak wavenumber of the density spectrum to be ks = 4.0. The
Atwood number is set to be 0.5.

Figure 3.17 shows the mean shock propagation for many individual points on the shock surface.
The two bounds for the Atwood number of 0.5 and shock Mach number of 2 are also shown.
Interestingly, the results show that for each individual point on the shock, the mean propagation is
bounded by the two limits, even though the density ﬁeld is generated randomly. This conﬁrms that
even though predicting the instantaneous shock surface response to the random density ﬂuctuations
is very challenging, one can still provide bounds for the shock position at every point on the shock
surface.

3.5 Conclusions

In this study, accurate ﬂow-resolving, shock-resolving/shock-capturing simulations are con-
ducted together with theoretical analyses to study the shock propagation in non-uniform density

Figure 3.16: The dependence of L2 norm of the shock ﬂuctuations on wavenumber ratio for shock
propagation in 3D non-unform density media with kx = 4.

140

rk012345L2(∆Xs)0.010.020.030.040.050.060.07media. Theoretical analyses show that in both linear and nonlinear regime, there exist similarities
between shock propagation in non-uniform density ﬁelds induced by entropy and composition ﬂuc-
tuations. In the nonlinear regime, the CW theory gives explicit solutions for the shock propagation
in 1D non-uniform density media by neglecting the post-shock ﬁeld eﬀects, i.e.
the re-reﬂected
waves. Pseudo-1D numerical simulations for shock propagation in linearly varying density proﬁles
show reasonably good agreement with the CW solutions as well as the results in the literature,
which conﬁrms the accuracy of our current numerical results. For oscillatory density proﬁles,
the numerical results further deviate from the theoretical solutions and exhibit additional long-
wavelength oscillations, which are shown to be related to the re-reﬂected waves. The magnitude of
the re-reﬂected waves is numerically integrated by extending the original Chisnell’s formulation,
which shows excellent agreement with the numerical results. A simpliﬁed model that includes
the eﬀects of re-reﬂected waves is then proposed to compensate for this deviation. The results
obtained by the proposed model show signiﬁcant improvement over the shock wave propagation in
ﬂuctuating density ﬁeld predicted by the classical CW theory.

The shock propagation in 3D non-uniform density media is also studied using a simpliﬁed 3D

Figure 3.17: The mean propagation of shock wave in 3D random density media. Red lines
represent points on the shock surface. The blue and black curves represent the bounds at the two
asymptotic limit.

141

t00.511.52Xs-0.8-0.6-0.4-0.200.2sinusoidal shaped density proﬁle. The 3D eﬀects are investigated by varying the wavenumber ratio
of the streamwise and spanwise wavenumbers of the non-uniform density ﬁeld. The asymptotic
limit for wavelength ratio approaching zero and inﬁnity are investigated and used to understand
the 3D eﬀects on the shock propagation.
It is shown that when the wavenumber ratio is larger
than 1.4, the results are very close to the proposed CWRW model, which implies one can treat
each point on the shock wave as an independent 1D shock. Decreasing the wavenumber ratio will
result in an increasing shock propagation speed. When the wavenumber ratio approaches zero, the
shock propagation speed reaches another asymptotic limit. The shock propagation speed is shown
to be bounded by the two limits for rk approaches 0 and inﬁnity at any ﬁxed At and Ms. The
magnitude of the shock wrinkling is also found to be dependent on rk, reaching its maximum value
at rk ≈ 0.57 for At = 0.5 and Ms = 2. Numerical study of shock propagation in 3D random density
ﬁeld indicates that the shock propagation speed on any point at shock surface is bounded by the
two limits observed in our 3D simulations with sinusoidal density proﬁle.

142

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