SHOCK-TURBULENCE INTERACTION AND …fr407rm7881/...SHOCK-TURBULENCE INTERACTION AND...

SHOCK-TURBULENCE INTERACTION AND

RICHTMYER-MESHKOV INSTABILITY IN

SPHERICAL GEOMETRY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND

ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Ankit Bhagatwala

August 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/fr407rm7881

© 2011 by Ankit Vijay Bhagatwala. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/fr407rm7881

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Sanjiva Lele, Primary Adviser


Brian Cantwell


Parviz Moin

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

The canonical problems of shock-turbulence interaction and Richtmyer-Meshkov in-

stability (RMI) are central to understanding the hydrodynamic processes involved in

Inertial Confinement Fusion (ICF). Over the last few decades, there has been con-

siderable analytical, computational and experimental work on the planar versions of

these problems. In spite of the problem of interest being spherical in nature, there

have been few studies in any of the three areas for these problems. It is not clear a

priori, that the conclusions drawn from planar versions of these problems carry over

to the spherical domain. The research presented here represents a first attempt to

understand the hydrodynamic processes involved in an Inertial Fusion Engine (IFE)

from capsule implosion to interaction of the resulting shock waves with the chamber

gases.

To abstract the key hydrodynamic components from the complex physics involved

in an IFE, three canonical problems are identified and simulated: Interaction of a

blast wave with isotropic turbulence, interaction of a converging shock with isotropic

turbulence and RMI in spherical geometry. The last problem is a hydrodynamic

abstraction of the capsule implosion itself, while the first two problems attempt to

model the late stage interaction of fusion induced shock waves with chamber gases.

Simulations of the Taylor blast wave and converging spherical shock waves propa-

gating through a region of compressible isotropic turbulence show that the turbulent

fluctuations are either significantly attenuated or unchanged depending on the initial

strength of the shock wave. It is shown through Eulerian simulations and Lagrangian

v

tracking of particles that both these effects are primarily related to the vorticity-

dilatation term in the vorticity transport equation. The turbulence length scales

associated with this problem are defined and the effect on them quantified. Turbu-

lence also distorts the shock, which can lead to substantial local variations in shock

strength and asphericity. Transverse vorticity amplification is compared with linear

planar shock-turbulence theory. Aspects that distinguish spherical shock-turbulence

interaction from the planar case are stressed.

For the converging shock problem, both converging and reflected phases of the

shock are studied. Effect of the reflected phase of the shock is found to be quite

different from the expanding shock in the Taylor blast wave-turbulence interaction

problem. Vorticity and turbulent kinetic energy are amplified due to passage of

the shock. Similar to the latter problem, the vorticity-dilatation term is primarily

responsible for the observed behavior. This is confirmed through Eulerian and La-

grangian statistics. Transverse vorticity amplification is compared with linear planar

shock-turbulence theory. The smallest eddies, represented by the Kolmogorov scale,

decrease in size after passing through the converging shock and this is shown to be

related to a decrease in kinematic viscosity and increase in dissipation behind the

converging shock. Distortion of the shock due to turbulence is also investigated and

quantified. Turbulence also affects maximum compression achieved at the point of

shock reflection, when the shock radius is at a minimum. This decrease in compres-

sion is quantified by comparing with pure shock simulations.

Simulations of the Richtmyer-Meshkov instability for an Air-SF6 interface in

spherical geometry have been carried out with three-dimensional ‘egg-carton’ type

interface perturbation. Parametric variation with respect to initial shock Mach num-

ber has been studied. Effect of shock Mach number on growth rates of the mixing

layer, vorticity, turbulent kinetic energy (TKE) and maximum compression achieved

have been studied. Different regimes for spherical RMI have been identified: con-

verging shock, reflected shock and turbulent mixing. Mixing is found to be more

efficient at higher initial shock Mach numbers. Baroclinic generation was found to

vi

be the driving force behind vorticity dynamics, with higher magnitudes observed at

higher Mach numbers. Evolution of TKE is seen to keep pace with vorticity and has

similar dynamics. Maximum compression was almost unchanged with and without

interface perturbation at weak to moderate initial shocks. For strong shocks, interface

perturbation decreases maximum compression achieved.

vii

Acknowledgements

This work was supported by the Department of Energy under the SciDAC project

with grant number DE-FC02-06-ER25787. Additional support from the Department

of Aeronautics and Astronautics at Stanford is also acknowledged. Computational

resources were provided by the Argonne Leadership Computational Facility under the

INCITE award. We thank Dr. Johan Larsson for providing the routines for gener-

ating isotropic turbulence, used extensively in this work and also for many helpful

discussions. We would also like to thank Prof. Tim Colonius at Caltech for sugges-

tions on the windowing technique used in chapter 5. Thanks are also due to members

of the reading committee, Prof. Brian Cantwell and Prof. Parviz Moin. Lastly, I

would like to acknowledge my advisor Prof. Sanjiva Lele for his guidance and help

over the years.

viii

Nomenclature

Abbreviations

DNS Direct Numerical Simulation

ICF Inertial Confinement Fusion

LDDRK Low Dissipation and Dispersion Runge-Kutta

LES Large Eddy Simulation

NASA National Aeronautics and Space Administration

PDF Probability Density Function

R-H Rankine-Hugoniot

RMI Richtmyer-Meshkov Instability

RTI Rayleigh-Taylor Instability

SGS Subgrid Scale (model)

TKE Turbulent Kinetic Energy

WENO Weighted Essentially Non-Oscillatory Scheme

UCLA University of California Los Angeles

Greek Symbols

α Guderley exponent

βf Physical bulk viscosity coefficient

βh Artificial bulk viscosity coefficient

γ Ratio of specific heats for ideal gases for single fluid

ix

γeff Effective gamma for a mixture of two species

∆ Grid spacing

ǫ Turbulent kinetic energy dissipation rate

η Kolmogorov (dissipation) length scale

κf Physical thermal conductivity coefficient

κh Artificial thermal conductivity coefficient

λ0 Representative energy containing length scale for isotropic turbulence

µf Physical shear viscosity coefficient

µh Artificial shear viscosity coefficient

ωr Radial component of vorticity

ωt Tangential component of vorticity

Ω Enstrophy (Domain averaged vorticity)

ρ Density

σ Sponge strength parameter

τ0 Representative time scale for isotropic turbulence

τij Viscous stress tensor

τs Representative shock time scale

θ Dilatation

χ Shock asphericity parameter

Roman Symbols

c0 Speed of sound in still medium

et Total energy per unit mass

e Internal energy per unit mass

h Enthalpy per unit mass

l0 Turbulence integral length scale

ls Length scale associated with shock capturing

P Pressure

t0 Time of turbulence launch

x

ts Time taken for shock to reach origin after launch

Re Reynolds number

Pr Prandtl number

Sc Schmidt number

qi Conductive heat flux in the i th Cartesian direction

r Radius

Rs Radius of spherical shock wave

S Contraction of the strain rate tensor

Sij Symmetric part of velocity strain-rate tensor

s Entropy per unit mass

Ur Mean radial velocity

ur Radial component of fluctuating velocity field

ut Tangential component of fluctuating velocity field

u0 RMS value of fluctuating turbulent velocity

ui Velocity component in the i the Cartesian direction

Ym Mass fraction of mth species

Subscripts, Superscripts and Accents

〈·〉 Tangential average, plotted as a function of radius

(·) Gaussian or compact filter

(·)0 Characteristic value of a field associated with turbulence alone

(·)r Radial component of a given vector field

(·)t Tangential component of a given vector field

(·)s Post-shock value of a field associated with a spherical shock

(·)ref Reference value used for non-dimensionalization

xi

Contents

Abstract v

Acknowledgements viii

Nomenclature ix

1 Introduction 1

1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Overview and Accomplishments . . . . . . . . . . . . . . . . . . . . . 8

2 Methodology 12

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Artificial Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Convective terms and control of aliasing errors . . . . . . . . . 21

2.2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Code performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Numerical tests 29

3.1 Method of Manufactured Solutions . . . . . . . . . . . . . . . . . . . 29

xii

3.1.1 Verifying order of spatial discretization . . . . . . . . . . . . . 31

3.1.2 Verifying accuracy of time stepping . . . . . . . . . . . . . . . 32

3.2 Taylor-Green vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Shu-Osher problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Shock-Vorticity/Entropy wave interaction . . . . . . . . . . . . . . . . 38

3.5 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Strong Convecting vortex . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 Blast Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Taylor Blast wave - Turbulence Interaction 54

4.1 Base flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.2 Taylor Blast Wave . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Spherical shock-turbulence interaction . . . . . . . . . . . . . . . . . 63

4.2.1 Tangentially averaged radial profiles . . . . . . . . . . . . . . . 65

4.2.2 Interaction of turbulence with a Taylor blast wave . . . . . . . 66

4.2.3 Mean flow fields . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.4 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . 68

4.2.5 Kolmogorov scales . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.6 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.7 Particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.8 Velocity variances . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.9 Thermodynamic fluctuations . . . . . . . . . . . . . . . . . . . 75

4.2.10 Instantaneous profiles . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.11 Time evolution of the shock . . . . . . . . . . . . . . . . . . . 79

4.2.12 Effect of shock strength: Weak Taylor blast wave . . . . . . . 80

4.3 Comparison with linear theory . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Converging Shock - Turbulence Interaction 86

5.1 Base flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xiii

5.3 Spherical shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Converging shock . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Spherical shock-turbulence interaction . . . . . . . . . . . . . . . . . 94

5.5 Tangentially averaged radial profiles . . . . . . . . . . . . . . . . . . . 95

5.6 Converging shock-turbulence interaction . . . . . . . . . . . . . . . . 96

5.6.1 Mean flow fields . . . . . . . . . . . . . . . . . . . . . . . . . . 97


5.6.3 Kolmogorov scales . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6.4 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6.5 Particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6.6 Velocity variances . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.6.7 Thermodynamic fluctuations . . . . . . . . . . . . . . . . . . . 108

5.6.8 Instantaneous profiles . . . . . . . . . . . . . . . . . . . . . . . 109

5.6.9 Time evolution of the shock . . . . . . . . . . . . . . . . . . . 111

5.7 Comparison with linear theory . . . . . . . . . . . . . . . . . . . . . . 112

5.8 Maximum compression . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Spherical RMI 120

6.1 Spherical shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Richtmyer-Meshkov Instability in spherical geometry . . . . . . . . . 124

6.2.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2.2 Mean flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


6.2.4 Mixing zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2.5 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.6 Turbulent Kinetic Energy . . . . . . . . . . . . . . . . . . . . 144

6.2.7 Maximum compression . . . . . . . . . . . . . . . . . . . . . . 148

6.2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Conclusions 153

7.1 Recommendations for future work . . . . . . . . . . . . . . . . . . . . 156

xiv

A Symmetry boundary conditions 158

A.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.1.1 First derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.1.2 Second derivative . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.2 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.2.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.2.2 Antisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

A.2.3 Gaussian filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B Outflow boundary conditions 165

B.1 Characteristic-based boundary conditions . . . . . . . . . . . . . . . . 165

B.2 Sponge treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

References 169

xv

List of Tables

2.1 Coefficients for the first derivative for pentadiagonal and tridiagonal

based compact schemes. . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Coefficients for the second derivative for pentadiagonal and tridiagonal

based compact schemes. . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Coefficients for the LDDRK4 and Kennedy et. al time stepping schemes. 22

3.1 Color and line legend for the plots. . . . . . . . . . . . . . . . . . . . 33

4.1 Comparison of turbulence parameters at time of shock launch with

those at time of turbulence launch. . . . . . . . . . . . . . . . . . . . 58

4.2 Comparison of theoretical vs. best fit value of the exponent for pure

Taylor blast wave flow. The theoretical values are Taylor’s exponent

for the Taylor blast wave. . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Comparison of initial conditions for Taylor blast wave shock-turbulence

interaction cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Evolution of flow parameters shock with time for the strong (case

M60G512) and weak (case M7G512) Taylor blast wave-turbulence in-

teraction. Subscript ‘ ps’ stands for post-shock values and are com-

puted at r = Rs−0.3l0 behind the shock. Turbulent Reynolds number

is defined as Ret = 〈urms〉psl0/〈ν〉ps. Ret before the shock is 140. . . 66

4.5 Evolution of vorticity variance components and anistropy with time

for blast wave-turbulence interaction (Cases M60G256 and M7G256).

The subscript ‘ ps’ stands for quantities evaluated downstream of the

shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xvi

4.6 Evolution of velocity variance components and anistropy with time for

blast wave-turbulence interaction (Cases M60G256 and M7G256). The

subscript ‘ ps’ stands for quantities evaluated downstream of the shock. 75

5.1 Comparison of turbulence parameters at time of shock launch with

those at time of turbulence launch. . . . . . . . . . . . . . . . . . . . 88

5.2 Comparison of theoretical vs. best fit value of exponents for converging

pure shock. The theoretical values are the Guderley exponents for a

converging shock. αfit represents a two-parameter fit with respect to

Rs0 and ts. α′fit represents a one parameter fit with respect to Rs0. . 91

5.3 Comparison of initial conditions for converging shock-turbulence inter-

action cases. Grid sizes include buffer zone needed for windowing. . 95

5.4 Evolution of flow parameters with time for converging shock-turbulence

interaction (Case C2G288). The subscript ‘ s’ stands for shock values

and are computed at r = Rs(t). Turbulent Reynolds number is defined

as Ret = 〈urms〉sl0/〈ν〉s. Ret before the shock is 140. . . . . . . . . . 97

5.5 Evolution of vorticity components and anistropy with time for con-

verging shock-turbulence interaction (Case C2G288). The subscript ‘

s’ stands for shock values and are computed at r = Rs(t) and ‘ ps’

stands for quantities evaluated downstream of the shock where they

attain their maximum value. . . . . . . . . . . . . . . . . . . . . . . 104

5.6 Evolution of fluctuating velocity components and Reynolds stress anistropy

with time for converging shock-turbulence interaction (Case C2G288).

The subscript ‘ s’ stands for shock values and are computed at r =

Rs(t) and ‘ ps’ stands for quantities evaluated downstream of the shock

where they attain their maximum value. . . . . . . . . . . . . . . . . 108

5.7 Comparison of pressure and density ratios at maximum compression

for a pure shock with shock propagating through turbulence . . . . . 116

5.8 Time delay for maximum compression with and without turbulence for

different cases. Time is taken at the point when shock radius is at its

minimum and instantaneous pressure is at its maximum. . . . . . . . 117

xvii

6.1 Comparison of theoretical vs. best fit value of exponents for converging

pure shock. The theoretical values are the Guderley exponents for a

converging shock. αfit represents a two parameter fit with respect to

Rs0 and ts, while α′fit represents a fit with respect to Rs0 alone. . . . 121

6.2 Initial conditions. Initial radius of shock,Rs0, Initial radius of inter-

face, R0, Initial wavenumber of perturbation, k0, k0R0 = 33, At-

wood number, At = (ρ2 − ρ1)/(ρ2 + ρ1) = 0.67, Shock Mach num-

ber, Ms = Rs/c0, Velocity impulse to interface due to shock, ∆v/c0 =

2(M2s − 1)/(γ + 1)Ms. c0 is the speed of sound in air, ts is the time

taken for the shock to reach the origin. . . . . . . . . . . . . . . . . 126

6.3 Table of different regimes for spherical Richtmyer-Meshkov instability

and time ranges when they are applicable. . . . . . . . . . . . . . . . 126

6.4 Comparison of pressure and density ratios at maximum compression

for spherical RMI with shock passing through an unperturbed interface 149

6.5 Time delay for maximum compression with and without interface per-

turbation for different cases. . . . . . . . . . . . . . . . . . . . . . . 150

xviii

List of Figures

1.1 Four main stages of Inertial Confinememt Fusion (ICF). (a) Laser

beams impinge on the outer metal coating (ablative layer) of the cap-

sule, heating it. (b) Surface layer heats up and blows off, setting off an

imploding shock wave that compresses the fuel. (c) Point of maximum

compression is reached with density and pressure several orders of mag-

nitude above the base state. (d) Ignition is initiated at the core and

spreads throughout the capsule. Image source: Lawrence Livermore

National Laboratories (llnl.gov) . . . . . . . . . . . . . . . . . . . . . 2

1.2 Schematics of a (a) Taylor blast wave and (b) Converging shock prop-

agating through a region of isotropic turbulence. . . . . . . . . . . . 3

2.1 Modified wavenumber for the first derivative scheme (a) Pentadiago-

nal scheme (dashed) (b) Tridiagonal scheme (dash-dotted) (c) Spectral

(solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Modified wavenumber for the second derivative scheme (a) Pentadiago-

nal scheme (dashed) (b) Tridiagonal scheme (dash-dotted) (c) Spectral

(solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Plot of the window function and the buffer region. . . . . . . . . . . 26

2.4 (a) Strong and (b) Weak scaling for the code. Dashed lines indicate

ideal scaling behavior in both cases. . . . . . . . . . . . . . . . . . . 27

3.1 Order verification for Case 1 . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Order verification for time stepping scheme . . . . . . . . . . . . . . 32

xix

3.3 Mean quantities for the Taylor-Green vortex on a 643 grid. The zero

subscript denotes the initial value. The semi-analytical result of Bra-

chet et. al. [7] are the black symbols. Figure reproduced from Johnsen

et. al. [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Velocity spectra for the Taylor-Green vortex on 643 grid at t = 5. (a)

Convergence of the reference solution using the HYBRID code in stan-

dard mode (solid) and with eighth-order accurate dissipation (dashed)

on 2563 (black), 1283 (blue), and 643 (cyan). (b) Comparison between

the different schemes. The reference solution is that obtained on the

2563 grid using the HYBRID code. Figure reproduced from Johnsen

et. al. [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Density profiles for the Shu-Osher problem with ∆x = 0.05 at t = 1.8.

The reference solution is the seventh-order accurate WENO solution

with ∆x = 6.25× 10−3. Figure reproduced from Johnsen et. al. [31]. 36

3.6 Entropy profiles for the Shu-Osher problem with ∆x = 0.05 at t = 1.8

(close-up of the solution just downstream of the shock). The ref-

erence solution is the seventh-order accurate WENO solution with

∆x = 6.25 × 10−3. (a) HYBRID, STAN, STAN-I, NASA and ref-

erence solutions. (b) UCLA, WENO and reference solutions. Figure

reproduced from Johnsen [?] [31]. . . . . . . . . . . . . . . . . . . . . 37

3.7 Velocity profiles for the Shu-Osher problem using the STAN andWENO

codes, with ∆x = 0.05 at t = 1.8 (close-up of the solution just down-

stream of the shock). The reference solution is the seventh-order accu-

rate WENO solution with ∆x = 6.25× 10−3. Figure reproduced from

Johnsen et. al. [31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 (a) ψ = 450 at t = 25. (b) ψ = 750 at t = 32. The results from the

WENO and STAN-I codes are not shown, but are similar to those of

the HYBRID and STAN codes, respectively. Instantaneous vorticity

profiles for the shock-vorticity/entropy wave interaction at y = 0 (k2 =

1). Figure reproduced from Johnsen et. al. [31]. . . . . . . . . . . . . 40

xx

3.9 Mean quantities for the shock-vorticity/entropy wave interaction (ψ =

450). Shock location: xs = 3π/2. The reference is the linear analysis

solution of Mahesh et. al. [55]. (a) Mean-square kinetic energy (k2 =

1) (b) Mean-square vorticity (k2 = 1). (c) Mean-square kinetic energy

(k2 = 2). (d) Mean-square vorticity (k2 = 2). Figure reproduced from

Johnsen et. al. [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Mean quantities for the shock-vorticity/entropy wave interaction (ψ =

750). Shock location: xs = 3π/2. The reference is the linear analysis

solution of Mahesh et. al.[55].(a) Mean-square kinetic energy (k2 = 1)

(b) Mean-square vorticity (k2 = 1). (c) Mean-square kinetic energy

(k2 = 2). (d) Mean-square vorticity (k2 = 2). Figure reproduced from

Johnsen et. al. [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.11 Comparison of Dilatation and artificial bulk viscosity spectra for 643

isotropic turbulence. Plus: 2563 DNS. Square: Constant coefficient

model. Circle: Modified coefficient model . . . . . . . . . . . . . . . 47

3.12 Comparison of RMS quantities for 643 isotropic turbulence. Solid line

- 2563 DNS, Dashed line - 643 Modified coefficient model. Dotted line

- 643 Constant coefficient model . . . . . . . . . . . . . . . . . . . . 47

3.13 Comparison of slices of dilatation and density contours: Dilatation on

the left and Density to the right. Top to bottom: Constant Cβ, 643

grid, Modified Cβ, 643 grid and 2563 grid DNS. Same contour and

grayscale levels are used in both plots. . . . . . . . . . . . . . . . . . 48

3.14 Dilatation spectrum at t/τ = 4 for the isotropic turbulence problem on

a 643 grid for different artificial diffusivity methods. Black solid: 2563

DNS; Red dash: Cook [12]; black dash-dot: Mani et. al. [56]; magenta

dash (thin): Bhagatwala & Lele [4]. . . . . . . . . . . . . . . . . . . . 49

3.15 Absorption of a compressible, convecting strong vortex at boundary

treated with windowing. Plots shows contours of vorticity. Same con-

tour levels have been used in all plots. . . . . . . . . . . . . . . . . . 50

xxi

3.16 Traces of (a) Dilatation and (b) Pressure along one of the principal axes

for the convecting vortex problem. t/τv = 0.2 (black-solid), t/τv = 0.5

(red-solid), t/τv = 0.7 (blue-solid), t/τv = 1.2 (black-dashed), t/τv =

1.5 (red-dashed), t/τv = 2.2 (blue-dashed) . . . . . . . . . . . . . . . 51

3.17 Absorption of a spherical shock wave at a boundary treated with win-

dowing. Plots shows contours of dilatation. Same contour levels have

been used in all plots. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.18 Traces of (a) Dilatation and (b) Pressure along one of the principal axes

for the blast wave problem. t/τs = 0.2 (black-solid), t/τs = 0.5 (red-

solid), t/τs = 0.7 (blue-solid), t/τs = 1.2 (black-dashed), t/τs = 1.5

(red-dashed), t/τs = 2.2 (blue-dashed) . . . . . . . . . . . . . . . . . 53

3.19 Zoomed view of figure 3.18(b) around the foot of the shock at early

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 Schematic of the simulation domain for Taylor blast wave-turbulence

interaction problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 (a) Evolution of velocity derivative skewness and (b) Enstrophy with

the time at which the shock is launched, marked. (c) Converged 3D

kinetic energy spectra at time of shock launch in Kolmogorov variables.

1283 (dashed); 2563 (dash-dotted); 5123 (solid) . . . . . . . . . . . . 56

4.3 Converged spectra for (a) Energy (b) Density fluctuations (c) Vorticity

(d) Dilatation. 1283 (dashed); 2563 (dash-dotted); 5123 (solid) at t =

tshock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Energy spectrum showing decomposition into dilatational and solenoidal

modes for turbulence field at t = tshock. Dilatational TKE (dashed) and

total TKE (solid). Thatched region shows solenoidal component. . . 58

4.5 Evolution of (a) Turbulent kinetic energy (TKE) and (b) Thermody-

namic quantities (density, pressure, temperature) for pure turbulence

(without shocks) in DNS mode on a 2563 grid. Legend for plot (b):

Density (black-solid), Pressure (blue-solid), Temperature (red-solid).

Pressure and Temperature offset by 0.2 and 0.4 respectively. . . . . . 60

xxii

4.6 Density schlieren wave diagram for expanding pure shock. Quantity

plotted is 10log|∇ρ| . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Comparison of shock radius for expanding pure shock with Taylor’s

2/5th power law scaling (best-fit with respect to S(γ)). (a) Linear plot

(b) Log plot. (× - symbol) - Best fit of a 2/5th power law curve. (o -

symbol) - Radius from 2563 simulation. . . . . . . . . . . . . . . . . 62

4.8 (a) Pressure (b) Density (c) Radial velocity profiles for strong Taylor

blast wave (case M60G512). t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed),

t/τ0 = 0.18 (dash-dotted). . . . . . . . . . . . . . . . . . . . . . . . . 67

4.9 Comparison of tangentially averaged radial profiles from 1283, 2563 and

5123 simulations (cases M60G128, M60G256 and M60G512) at t/τ0 =

0.04 to show convergence. (a) Radial vorticity and (b) Tangential

vorticity. 1283 (+ -symbol); 2563 (× -symbol); 5123 (∗ -symbol) . . . 68

4.10 Probability density function (PDF) for strong Taylor blast wave from

1283, 2563 and 5123 simulations (cases M60G128, M60G256 and M60G512)

at t/τ0 = 0.04 to show convergence. (a) Density (b) Vorticity magni-

tude (c) Turbulent kinetic energy. 1283 (+ -symbol); 2563 (× -symbol);

5123 (∗ -symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.11 (a) Kolmogorov scale η, (b) Kinematic viscosity ν (c) Dissipation rate

ǫ for strong Taylor blast wave (case M60G512). ν and ǫ are plotted in

log scale. t/τ0 = 0 (dotted), t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed),

t/τ0 = 0.18 (dash-dotted). . . . . . . . . . . . . . . . . . . . . . . . . 70

4.12 Evolution of (a) Radial vorticity (b) Tangential vorticity and (c) Vor-

ticity anisotropy for the strong Taylor blast wave (Case M60G512).

t/τ0 = 0 (dotted), t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18

(dash-dotted) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.13 Vorticity sources and sinks for the strong Taylor blast wave (Case

M60G512) at t/τ0 = 0.06. (a) Linear plot (b) Log plot. Absolute val-

ues are used for the log plot. Baroclinic generation: (∇p × ∇ρ)τ 20 /ρ2( - symbol), Vorticity dilatation: −ω(∇ · u)τ 20 (♦ - symbol) . . . . . 72

xxiii

4.14 Distribution functions of (a) Particle density, (b) Vorticity and (c)

Potential vorticity for the strong Taylor blast wave (Case M60G256).

t/τ0 = 0 (solid), t/τ0 = 0.02 (dashed), t/τ0 = 0.2 (dash-dotted) . . . 74

4.15 Evolution of radial and tangential velocity variances and Reynolds

stress anisotropy for strong shock case M60G512. t/τ0 = 0 (dotted),

t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted) . 76

4.16 Evolution of radial and tangential velocity variances and Reynolds

stress anisotropy for weak shock case M7G512. t/τ0 = 0 (dotted),

t/τ0 = 0.08 (solid), t/τ0 = 0.3 (dashed), t/τ0 = 0.6 (dash-dotted) . . 76

4.17 (a) Pressure and (b) Density RMS plots for strong blast wave case

M60G256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.18 Traces of (a) Pressure (b) Density and (c) Radial velocity for strong

blast wave-turbulence interaction case M60G256. t/τ0 = 0 (dotted),


4.19 Traces of (a) Pressure (b) Density and (c) Radial velocity for weak

blast wave-turbulence interaction case M7G256. t/τ0 = 0 (dotted),


4.20 Evolution of (a) Shock pressure ratio, (b) Shock radius and (c) Shock

asphericity, for the strong Taylor blast wave (Case M60G256). Shock

with Turbulence (+ symbol), Pure shock (o symbol) . . . . . . . . . 79

4.21 Evolution of (a) Shock pressure ratio, (b) Shock radius and (c) Shock

asphericity, for the weak Taylor blast wave (Case M7G256). Shock

with Turbulence (+ symbol), Pure shock (o symbol) . . . . . . . . . 80

4.22 Slices along x− y plane at z = L/2 of pressure contours for (a) Weak

(Case M7G256) and (b) Strong (Case M60G256) Taylor blast waves

with a linear colormap. Same mean shock radius in both plots, however

contour levels for both plots are not the same. . . . . . . . . . . . . 81

xxiv

4.23 Transverse vorticity profiles with horizontal axes translated to shock

radius and rescaled to shock corrugation amplitude. (a) Weak case

(M7G256), t/τ0 = 0.08 (solid), t/τ0 = 0.2 (dashed), t/τ0 = 0.3 (dash-

dotted) (b) Strong case (M25G256), t/τ0 = 0.02 (solid), t/τ0 = 0.06

(dashed), t/τ0 = 0.18 (dash-dotted) . . . . . . . . . . . . . . . . . . . 82

4.24 Comparison of transverse vorticity amplification ratios with linear the-

ory. (a) Taylor blast wave. Case M7G256 (x symbol), Case M16G256

(o symbol), Case M60G256 (+ symbol). Solid line in (a) refers to the

linear theory result from Lee et al [43] and (b) refers to the Rankine-

Hugoniot relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.25 Contours of dilatation superimposed on slices along x − y plane at

z = L/2 of Mach number contours for (a) Weak (Case M7G256) and

(b) Strong (Case M60G256) Taylor blast waves. Plots are at different

times and different colormaps, but same mean radius. . . . . . . . . 83

5.1 Schematic of the simulation domain for Converging shock-turbulence


5.2 Density schlieren wave diagram for converging pure shock. Quantity

plotted is 10log|∇ρ|. Shock initial Mach number Ms0 = 1.4. ts is time

taken for shock to reach the origin, i.e. ts = tRs=0. . . . . . . . . . . 91

5.3 Comparison of shock radius for converging pure shock with an assumed

best-fit for Guderley’s power law scaling (best-fit with respect to Rs0).

Points below the dashed line are excluded when making the fit. (a)

Linear plot (b) Log plot. (× - symbol) - Best fit of Guderley’s power

law curve. (o - symbol) - Radius from 2883 simulation. Line closer to

x− axis in the log plot corresponds to reflected shock while the line

further from x− axis represents the converging shock. . . . . . . . . 92

xxv

5.4 (a) Pressure, (b) Density and (c) Entropy and (d) Radial velocity pro-

files for converging shock (Case C2G288). Converging phase: t/τ0 =

0.04 (black-dotted), t/τ0 = 0.16 (black-solid), t/τ0 = 0.31 (black-

dashed), t/τ0 = 0.42 (black-dash/dotted). Reflected phase: t/τ0 = 0.52

(blue-dotted), t/τ0 = 0.60 (blue-solid), t/τ0 = 0.65 (blue-dashed),

t/τ0 = 0.71 (blue-dash/dotted). . . . . . . . . . . . . . . . . . . . . . 98

5.5 Plot of convective term in entropy equation u · ∇s (Case C2G288).

Converging phase: t/τ0 = 0.04 (black-dotted), t/τ0 = 0.16 (black-

solid), t/τ0 = 0.31 (black-dashed), t/τ0 = 0.42 (black-dash/dotted).

Reflected phase: t/τ0 = 0.52 (blue-dotted), t/τ0 = 0.60 (blue-solid),

t/τ0 = 0.65 (blue-dashed), t/τ0 = 0.71 (blue-dash/dotted). . . . . . . 99

5.6 Comparison of tangentially averaged radial profiles from 1443, 2883 and

5763 simulations (cases C2G144, C2G288 and C2G576) at t/τ0 = 0.24

to show convergence. (a) Radial vorticity and (b) Tangential vorticity.

1443 (+ -symbol); 2883 (× -symbol); 5763 (∗ -symbol) . . . . . . . . 100

5.7 Probability density function (PDF) for converging shock-turbulence

interaction from 1443, 2883 and 5763 simulations at t/τ0 = 0.24 to

show convergence. (a) Density (b) Vorticity magnitude (c) Turbulent

kinetic energy. 1443 (+ -symbol); 2883 (× -symbol); 5763 (∗ -symbol) 101

5.8 (a) Kolmogorov scale η, (b) Kinematic viscosity ν and (c) Dissipation

rate ǫ for converging shock (Case C2G288). ν and ǫ are plotted in log

scale. For legend please refer to figure 5.4. . . . . . . . . . . . . . . . 102

5.9 Slices of total vorticity superimposed on dilatation contours showing

shock location at mid plane for (a) Converging shock at t/τ0 = 0, (b)

Reflected shock at t/τ0 = 0.9 for case C2G288. . . . . . . . . . . . . 103

5.10 Evolution of (a) Radial vorticity, (b) Tangential vorticity and (c) Vor-

ticity anisotropy for the converging shock (Case C2G288). For legend

please refer to figure 5.4. . . . . . . . . . . . . . . . . . . . . . . . . 105

xxvi

5.11 Magitudes of vorticity sources and sinks for the converging phase (Case

C2G288) at t/τ0 = 0.16(black) and reflected phase at t/τ0 = 0.65

(blue). (a) Linear plot (b) Log scale. (∇P × ∇ρ)/ρ2 ( - symbol),

−ω(∇ · u) (♦ - symbol). . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.12 Distribution functions of (a) Particle density, f(ρ), (b) Vorticity, f(ω)

and (c) Potential vorticity, f(ω/ρ) for converging shock (Case C2G144).


5.13 Evolution of (a) Radial velocity variance, (b) Tangential velocity vari-

ance and (c) Reynolds stress anisotropy for the converging shock (Case

C2G288). For legend please refer to figure 5.4. . . . . . . . . . . . . 109

5.14 (a) Pressure and (b) Density RMS plots for moderate strength con-

verging shock case C2G288 . . . . . . . . . . . . . . . . . . . . . . . 110

5.15 Traces of (a) Pressure (b) Density and (c) Radial velocity for con-

verging shock-turbulence interaction case C2G288. Converging phase:

t/τ0 = 0.04 (black-dotted), t/τ0 = 0.16 (black-solid), t/τ0 = 0.31

(black-dashed), t/τ0 = 0.42 (black-dash/dotted). Reflected phase:

t/τ0 = 0.52 (blue-dotted), t/τ0 = 0.60 (blue-solid), t/τ0 = 0.65 (blue-

dashed), t/τ0 = 0.71 (blue-dash/dotted). . . . . . . . . . . . . . . . . 110

5.16 Evolution of (a) Shock radius and (c) Shock asphericity, for converging

(Case C2G288). Shock with Turbulence (+ symbol), Pure shock (o

symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.17 Slices of dilatation contours. (a) Ms0 = 1.4 (Case C1G288), (b) Ms0 =

2.5 (Case C2G288) and (c) Ms0 = 4.0 (Case C3G288) (d) Pure shock

at Ms0 = 2.5 at different times but same mean shock radius, Rs/Rs0 =

0.45. Contour levels are not same across plots. . . . . . . . . . . . . 113

5.18 Comparison of transverse vorticity amplification ratios with linear the-

ory and shock pressure and density ratios with the Rankine-Hugonoit

relation. Case C1G288 (x symbol), Case C2G288 (o symbol), Case

C3G288 (+ symbol). Solid line in (a) refers to the linear theory result

from Lee et al [43] and in (b) refers to Rankine-Hugoniot relation. . 114

xxvii

5.19 Pressure and density ratios at maximum compression in presence of

turbulence compared to pure shock case. Quantities plotted are (ρmax/ρ0)s−t

/ (ρmax/ρ0)p−s ( - symbol) and (Pmax/P0)s−t / (Pmax/P0)p−s (♦ -

symbol). 2883 grid (solid line); 1443 grid (dashed line) s − t: shock-

turbulence, p− s: pure-shock . . . . . . . . . . . . . . . . . . . . . . 116

6.1 Schematic of the simulation domain for Converging shock-turbulence


6.2 Comparison of shock radius for converging pure shock with an assumed

best-fit for Guderley’s power law scaling. Line closer to x− axis cor-

responds to the reflected shock, line farther from x− axis corresponds

to convverging shock. Plot shows the α′fit fit. Points below the dashed

line are excluded when making the fit. (a) Linear plot (b) Log plot

(× - symbol) - Best fit of Guderley’s power law curve. (o - symbol) -

Radius from 1283 simulation. . . . . . . . . . . . . . . . . . . . . . . 122

6.3 Evolution of spherical RMI for MI = 1.8. Contours of mass fraction,

Converging shock regime: (a) t/ts = 0, (b) t/ts = 0.8. Post-reshock

regime: t/ts = 2.4, t/ts = 3.1 . . . . . . . . . . . . . . . . . . . . . . 125

6.4 Evolution of spherical RMI forMI = 1.8. Slices of density, ρ/ρunshockedair

(a) Converging shock regime, t/ts = 0.4, (b) t/ts = 0.8, Post-reshock

regimes, (c) t/ts = 2.4, (d) t/ts = 3.1 . . . . . . . . . . . . . . . . . . 127

6.5 Evolution of spherical RMI forMI = 1.8. Slices of air mass fraction, Y

(a) Converging shock regime, t/ts = 0.4, (b) t/ts = 0.8, Post-reshock

regimes, (c) t/ts = 2.4, (d) t/ts = 3.1 . . . . . . . . . . . . . . . . . . 128

6.6 Evolution of spherical RMI for MI = 1.8. Slices of vorticity variance

magnitude, sqrt(ω′2i ) (a) Converging shock regime, t/ts = 0.4, (b)

t/ts = 0.8, Post-reshock regimes, (c) t/ts = 2.4, (d) t/ts = 3.1 . . . . 129

xxviii

6.7 Tangentially averaged radial profiles of pressure for converging and re-

flected shocks for MI = 1.8 case. (a) Converging, t/ts = 0.11 (dotted),

t/ts = 0.35 (solid), t/ts = 0.69 (dashed), t/ts = 0.92 (dash-dotted)

(b) Reflected, t/ts = 1.33 (dotted), t/ts = 1.52 (solid), t/ts = 1.71

(dashed), t/ts = 1.92 (dash-dotted). Arrow indicates location and

propagation direction of shock. . . . . . . . . . . . . . . . . . . . . . 130

6.8 Tangentially averaged radial profiles of density for converging and re-

flected shocks for MI = 1.8 case. (a) Converging, t/ts = 0.11 (dotted),

t/ts = 0.35 (solid), t/ts = 0.69 (dashed), t/ts = 0.92 (dash-dotted)

(b) Reflected, t/ts = 1.33 (dotted), t/ts = 1.52 (solid), t/ts = 1.71

(dashed), t/ts = 1.92 (dash-dotted). Arrow indicates location and


6.9 Tangentially averaged radial profiles of radial velocity for converging

and reflected shocks for MI = 1.8 case. (a) Converging, t/ts = 0.11

(dotted), t/ts = 0.35 (solid), t/ts = 0.69 (dashed), t/ts = 0.92 (dash-

dotted) (b) Reflected, t/ts = 1.33 (dotted), t/ts = 1.52 (solid), t/ts =

1.71 (dashed), t/ts = 1.92 (dash-dotted). Arrow indicates location and


6.10 Traces of flow variables along x-axis showing convergence for single

mode perturbation during converging phase on different grids at t/ts =

0.52. (a) Mass fraction (b) Density (c) Pressure and (d) Radial velocity.

1283 (black), 2563 (blue), 5123 (red). . . . . . . . . . . . . . . . . . . 134

6.11 Artificial properties on different grids at t/ts = 0.52. (a) Artificial

shear viscosity (b) Artificial bulk viscosity (c) Artificial conductivity

(d) Artificial diffusivity. 1283 (black-solid); 2563 (blue-solid); 5123 (red-

solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.12 Zoomed in view of log of artificial diffusivity. . . . . . . . . . . . . . 136

xxix

6.13 (a) Comparison of mixing zone width. MI = 1.2 (red), MI = 1.8

(black), MI = 3.0 (blue). Width normalized by initial perturbation

amplitude. (b) Profiles of 〈Y 〉(1− 〈Y 〉) for MI = 1.8 case, Converging

(black) and post-reshock (blue) phases. Y is the mass fraction of Air.

(a) Converging, t/ts = 0.2 (dotted), t/ts = 0.36 (solid), t/ts = 0.52

(dashed), t/ts = 0.68 (dash-dotted). (b) Post-reshock regimes, t/ts =

2.23 (dotted), t/ts = 2.88 (solid), t/ts = 3.1 (dashed), t/ts = 3.74

(dash-dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.14 Probability density functions (PDF) of species mass fraction in the

mixing zone during turbulent mixing regime. A value of 0 represents

pure SF6 and a value of 1 represents pure air. (a) MI = 1.2, (b)

MI = 1.8, (c) MI = 3.0. t/ts = 2.8 (solid), t/ts = 3.1 (dashed),

t/ts = 3.4 (dash-dotted) for all three cases. . . . . . . . . . . . . . . 138

6.15 Evolution of PDF based mean and variance of the mass fraction Y , with

time for different initial shock Mach numbers. (a) Mean (b) Variance.

MI = 1.2 (red), MI = 1.8 (black), MI = 3.0 (blue). . . . . . . . . . . 139

6.16 Comparison of Enstrophy and TKE.MI = 1.2 (red),MI = 1.8 (black),

MI = 3.0 (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.17 Radial profiles of vorticity magnitude for shock withMI = 1.8 (a) Con-

verging, t/ts = 0.2 (dotted), t/ts = 0.36 (solid), t/ts = 0.52 (dashed),

t/ts = 0.68 (dash-dotted). (b) Post-reshock regimes, t/ts = 2.23 (dot-

ted), t/ts = 2.88 (solid), t/ts = 3.1 (dashed), t/ts = 3.74 (dash-dotted). 141

6.18 Radial magnitude profiles of vorticity budget for shock with MI = 1.8

during converging phase at t/ts = 0.36 (black) and turbulent mixing

phase at t/ts = 3.1 (blue). (a) Linear (b) Log scale. (∇P ×∇ρ)/ρ2 (- symbol), −ω(∇ · u) (♦ - symbol). . . . . . . . . . . . . . . . . . . . 142

6.19 PDF profiles of vorticity magnitude during the turbulent mixing phase

after reshock during turbulent mixing regime. MI = 1.2 (red), MI =

1.8 (black), MI = 3.0 (blue). t/ts = 2.8 (solid), t/ts = 3.1 (dashed),

t/ts = 3.4 (dash-dotted) for all three cases. . . . . . . . . . . . . . . . 143

xxx

6.20 Temporal evolution of turbulent kinetic energy (TKE) and dissipation.

MI = 1.2 (red), MI = 1.8 (black), MI = 3.0 (blue). . . . . . . . . . . 144

6.21 Traces of pressure from t/ts = 1.1 to t/ts = 3.1 for MI = 1.8 case.

Lines in blue show phase during which reflected shock is still within

simulation domain. Lines in black are traces at later time showing

formation of secondary shocks from coalescing waves moving inwards 145

6.22 Radial profiles of TKE for shock with MI = 1.8 (a) Converging (b)

Post-reshock regimes. For legend, please refer to figure 6.17 . . . . . 146

6.23 PDF profiles of TKE during the turbulent mixing phase after reshock

during turbulent mixing regime. MI = 1.2 (red), MI = 1.8 (black),

MI = 3.0 (blue). t/ts = 2.8 (solid), t/ts = 3.1 (dashed), t/ts = 3.4

(dash-dotted) for all three cases. . . . . . . . . . . . . . . . . . . . . 147

6.24 Temporal profiles of turbulent Reynolds number, Ret = q4/ǫν where q

denotes turbulent kinetic energy. MI = 1.2 (red), MI = 1.8 (black),

MI = 3.0 (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.25 Pressure and density ratios at maximum compression in presence of

turbulence compared to pure shock case. Quantities plotted are (ρmax/ρ0)RMI

/ (ρmax/ρ0)unperturbed ( - symbol) and (Pmax/P0)RMI / (Pmax/P0)unperturbed

(♦ - symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xxxi

Chapter 1

Introduction

The problem of a shock wave traveling through a region of turbulence and other in-

homogeneities is one of interest in several practical applications, most importantly

Inertial Confinement Fusion (ICF), supernova explosion and shockwave lithotripsy

among others. Several studies in the past have focused on planar shock-turbulence

interaction and planar Richtmyer-Meshkov instability as the hydrodynamic abtrac-

tion of these problems. However, such canonical problems do not address the more

practical cases of interest, wherein the shock is spherical. Literature on such problems

in a turbulent regime is quite scarce, although there is a rich body of theoretical and

numerical work on converging and expanding spherical shocks through a quiescent

medium.

The application that motivates this study is ICF (Lindl [46], Lindl et. al. [47],

Keefe [33]) , where nuclear fusion is initiated in a small fuel capsule by compressing it

through high energy laser beams. The term “Inertial Confinement” refers to the fact

that the target capsule is held in place by the lasers that initiate the fusion process,

i.e there is no independent suspension mechanism. This is to be contrasted with

Magnetic confinement, whereby the fuel is restrained by magnetic fields.

Figure 1.1 shows the different stages of the ICF process. During the first state,

the material on the surface is uniformly heated by the lasers. During stage two, it

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Four main stages of Inertial Confinememt Fusion (ICF). (a) Laser beamsimpinge on the outer metal coating (ablative layer) of the capsule, heating it. (b)Surface layer heats up and blows off, setting off an imploding shock wave that com-presses the fuel. (c) Point of maximum compression is reached with density andpressure several orders of magnitude above the base state. (d) Ignition is initiatedat the core and spreads throughout the capsule. Image source: Lawrence LivermoreNational Laboratories (llnl.gov)

blows off outwards, sending an imploding shock wave towards the center of the cap-

sule. This converging shock compresses the fuel as it travels inward. The third, stage

which is the most crucial, occurs when the shock wave reaches the center, compress-

ing the fuel to its maximum density and temperature, initiating fusion. During stage

four, the fusion reaction spreads throughout the capsule yielding, in theory, several

times more energy than was needed to heat up the target.

The primary issues with ICF are energy delivery to the target, symmetry of the

imploding fuel, premature heating of the fuel before point of maximum compression

is reached and mixing between fuel layers due to hydrodynamic instabilities such as

Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM). To maximize yield from the

ICF process, the converging shock wave needs to focus symmetrically so as to gener-

ate the maximum possible density. There has been considerable research into the first

three problems, solving them to a large extent by using beam smoothing techniques

3

(a) (b)

Figure 1.2: Schematics of a (a) Taylor blast wave and (b) Converging shock propa-gating through a region of isotropic turbulence.

and beam diagnostics. The hydrodynamic instability problem however, remains a

crucial roadblock in achieving sustainable fusion. This is the main motivation behind

the study of the Richtmyer-Meshkov instability in this work.

The second aspect of this problem involves the design of a practical power plant

using ICF as its core energy generating mechanism. One such idea is the so called

Inertial Fusion Engine (IFE) plants. A continuous stream of targets would be deliv-

ered to the fusion chamber, about 10 − 15 shots per second and the resulting heat

and radiation would be captured to drive a conventional steam turbine.

A detailed hydrodynamic simulation of the IFE chamber is a key step towards

achieving sustained energy output using the fusion based power plant described above.

The shock wave that implodes the fuel expands out into the chamber gas. It hits the

walls of the chamber, heating up the gas surrounding it. This potentially drives


another imploding shock, this time into the chamber gas, resulting in a series of ex-

panding and converging shocks. One of the key issues is mitigating the effect of the

post-reaction outward propagating shock wave. There is a danger that this shock, if

strong enough, could damage the sensitive laser beam tubes. At the end of a shot, the

chamber gas is at a high temperature and pressure. In addition, it is turbulent and

has significant density gradients. This could be deleterious to the sensitive lasers that

have to be able to reach the next target with pinpoint accuracy. Driver beams for

the next target should be able to propagate and be focused in this preshot chamber

environment.

As can be imagined the implosion of the capsule, the ensuing fusion, and series

of expanding and convering shock waves involve a wide range of complex physical

phenomena. To incorporate all of these into a numerical simulation would be very

difficult. We therefore seek to abstract the hydrodynamic components and define a

set of key canonical problems that attempt to simuate the hydrodynamic aspect of

this extremely complex process. To that end, for problems presented in this thesis,

we neglect radiation, ionization, and walls.

The question then remains, as to what hydrodynamic processes can be abstracted

and simulated in a simplified form. We define three such problems, that form the

basis of this study. The first is a Taylor blast wave propagating through a region of

isotropic turbulence, mimicking the post-reaction outward propagating shock wave

from the capsule into the fusion chamber gas. The second is a converging shock prop-

agating through a region of isotropic turbulence, mimicking the imploding shock wave

initiated at the wall of the fusion chamber. Figure 1.2 shows schematics of the first

two problems. Lastly, we attempt to simulate the hydrodynamic abstraction of the

capsule implosion itself, the Richtmyer-Meshkov instability in spherical geometry. It

bears clarification that we do not attempt to match parameters from the actual ICF

problem, for which the parameter regime (density ratios, shock strength, time scales,

laser drive parameters etc.) is too extreme for the numerical setup we use. Instead,

we use the setup similar to that used to study planar RMI, but adapted to spherical

1.1. PREVIOUS WORK 5

geometry.

While ICF is the primary motivation for this study, spherical shocks are encoun-

tered in a variety of other scientific applications, most prominent among them being

implosion of type II supernovae and shock-wave lithotripsy used for treatment of kid-

ney stones, where a converging shock is focused on the stone to break it. Recently,

this technique has also found application in opthalmology, gene therapy and food

preservation.

1.1 Previous Work

There have been several studies on the interaction of a planar shock wave with a

turbulent inflow. Small amplitude fluctuations were decomposed into entropy, vor-

tical and acoustic modes by Kovasznay [37]. There has been considerable work on

acoustic wave generation behind the shock wave (Ribner [66, 67, 68], Moore [59], Ker-

rebrock [35], Chang [10], McKenzie & Westphal [57]). Ribner [66] analytically solved

the linearised Navier-Stokes equations with only incoming vortical disturbances. Lee

et al [42, 43] performed a series of DNS calculations confirming Ribner’s linearised

analysis. Mahesh et al [53, 54] later included the effect of entropy and acoustic fluc-

tuations as well in the incoming turbulent field. Recently, Larsson and Lele [40] have

carried out a series of DNS calculations with much higher turbulent Mach numbers

and Reynolds numbers. Hesselink and Sturtevant [20] have conducted experiments

where a normal shock passes through a turbulent mixture of Helium and R12. Barre

et. al. performed experiments where a Mach 3 supersonic flow interacted with a

normal shock. They observed an increase in fluctuating velocities in the longitudinal

direction in agreement with Ribner’s linear theory.

The classical point source explosion model was introduced by Taylor [84] and Von

Neumann [88], wherein the deposition of energy can be assumed to take place at

a point in space, so that the shock wave propagates spherically outwards into an


undisturbed medium. This widely studied problem was first proposed independently

by Sedov [78] and Taylor [84]. Taylor [84] first numerically solved the system of

nonlinear ODEs governing the self-similar solution of the blast wave. Later Sakurai

[75] generalized his result to shock waves of moderate strength. Whitham [89] gave

a general theory that governs shock behavior at very large distances when it has

become very weak. Brode [8] was among the first to generate numerical results for

spherical shock waves. The problem is also extensively discussed in Zel’dovich [95]

and Barenblatt [2]. Barenblatt [2] describes a extremely fast thermal wave, resulting

from the temperature rise due the sudden energy deposition that propagates ahead

of the hydrodynamic shock. This wave is not resolved in our simulations as the time

step restrictions it would impose are quite severe. Kurdyumov et al. [38] also study

heat propagation from a time-dependent concentrated energy source in a gas. They

distinguish between two time scales, the energy deposition time scale and the char-

acteristic acoustic time scale and investigate the effect of varying the ratio between

those two.

Recently Ghosh and Mahesh [18] have studied the interaction of a laser-induced

plasma with isotropic turbulence. In their study, laser energy is deposited asymmet-

rically over a finite region which heats up the air and causes a shock wave to form and

propagate through it. They consider the effects of ionization and dissociation, which

leads to the formation of a plasma core. This plasma core is unstable at late times be-

cause of the highly asymmetric nature of energy deposition and it eventually rolls up

on itself. They do a parametric variation on initially deposited energy and Reynolds

number and conclude that the initial energy does not alter the flow qualitatively, but

the plasma core does not roll up for very low Reynolds number. Experiments on ex-

ploding and imploding cylindrical and spherical shock waves have been performed by

Hosseini & Takayama [25, 26]. Cylindrical and spherical shock waves were produced

at the center of a shell by means of an exploding charge. These then hit the walls of

a containment vessel and reflect back inward, generating a converging shock.

The converging shock was first studied by Guderley [19] who assumed self-similar

1.1. PREVIOUS WORK 7

behavior as the shock propagated inwards and derived an exponent which describes

the shock radius as a function of time. Chisnell [14, 15] has also extensively studied

converging shocks and in his recent work has derived an approximate analytic solution

for the Guderley exponent as well for the flow behind the shock wave. More recently

Ponchaut et al [63] have improved upon Guderley’s solution by considering a series

expansion, allowing them to relax the strong shock approximation used by Guderley

in that work. Van Dyke and Guttmann [86] studied a converging shock driven by

an inward moving spherical piston. Payne [61] was one of the first to numerically

simulate converging shocks in a cylindrical domain. An extensive list of self-similar

converging shock solutions has been compiled by Sachdev [74]. Sachdev [74] discusses

both numerical and analytical approaches that have been made over the decades in

fine detail, for both expanding and converging shocks.

Most existing literature on the Richtmyer-Meshkov instability (RMI) deals with

the planar version of this instability. Starting with the landmark paper my Richtmyer

[69], who studied the linearized compressible Euler equations and proposed an ana-

lytical expression for the growth rate. Meshkov [58] performed the first experiments

and showed that the observed growth rates were much lower than predicted by the

impulsive model of Richtmyer [69]. Samtaney & Zabusky [76] showed that misaligned

pressure and density gradients are responsible for baroclinic deposition of vorticity at

the interface which then grows with time. The RMI differs from the Rayleigh-Taylor

instability (RTI) in that the interface is unstable irrespective of the direction of prop-

agation of the shock, whereas RTI is only unstable when acceleration is directed from

heavy to the light fluid.

There have been seminal experiments by Vetter & Sturtevant [87] and Jacobs [29].

Experiments using the NOVA laser at Lawrence Livermore have been reported by

Holmes et. al. [22]. That study also presents comparison with theory and numerical

simulations. RMI arising from radiatively driven shocks has been studied by Dimonte

et. al. [16]. Holmes et. al. [23] have also performed a numerical study using front-

tracking. Landmark LES simulations have been performed by Hill et. al. [21].


The review article by Brouillette [9] discusses the fundamental physical processes

occuring in RMI, as well as an overview of analytical, numerical and experimental

approaches to the problem. Recent interest has focused on different forms of density

inhomogeneity, such as a cylinder or a sphere being impulsively accelerated by a shock.

Such interaction are generally classified under “shock-bubble” interactions. A recent

review by Ranjan et. al describes progress in that field. However, for the problem

of interest here, the geometry is spherical. This study attempts for the first time, an

investigation of 3D spherical RMI. Lombardini et. al. [49] have recently carried out

RMI simulations in cylindrical geometry. RMI for a spherical axisymmetric flow was

investigated by Dutta et. al. [17]. Lombardini [50] has done an extensive study of 3D

cylindrical RMI in the linear and nonlinear regimes. Experiments in cylindrical RMI

have recently been carried out by Hosseini & Takayama [27] who study evolution of

the instability at different Atwood numbers for same initial shock strength. In this

study we focus on the nonlinear regime, as it is more relevant to mixing and turbulence

and also requires a smaller physical domain, transition to turbulence occuring sooner.

1.2 Overview and Accomplishments

The principal objective of this study is to define and simulate a set of canonical prob-

lems to understand shock-turbulence interaction and Richtmyer-Meshkov instability

in spherical geometry, with a view to understand the hydrodynamic aspects of ICF.

These problems have been extensively studied in planar regime, though it is not clear

that the results carry over into the spherical regime, which is where the application

problems are. The main contributions are sumarized below.

• An modified version of artificial viscosity that further localizes dissipation around

shocks is proposed and shown to work well.

• First of their kind simulations of a blast wave interacting with isotropic tur-

bulence and a converging shock interacting with turbulence have been carried

out and compared and contrasted with the planar shock-turbulence interaction

problem.

1.2. OVERVIEW AND ACCOMPLISHMENTS 9

• First of their kind simulations of the Richtmyer-Meshkov instability in spherical

geometry have been carried out. This study expands on previous studies by

carrying out full 3D RMI simulations.

The points above are explained in short detail below.

The traditional artificial bulk viscosity formulation of Cook [12] is shown to be

excessively dissipative for compressible turbulence problems. A modified formulation

for artificial bulk viscosity is proposed which retains the shock capturing capability of

the original formulation while significantly improving performance in terms of acous-

tic motions and thermodynamic fluctuations for high Mach number turbulence.

The blast-wave turbulence interaction problem was a first attempt at studying

shock-turbulence interaction in spherical geometry. It was found to be different from

planar shock-turbulence in several aspects. Vortical fluctuations were found to be

highly attenuated in the wake of the shock. Kolmogorov scales increased in the

post-shock region. Both these behaviors are contrary to planar shock-turbulence in-

teraction and were investigated in detail through Eulerian and Lagrangian statistics.

Effects of turbulence on shock structure were quantified. Shock Mach number was

found to have a significant impact on both turbulence and shock distortion. Com-

parisons were made with linear theory from planar shock-turbulence interaction.

The analysis carried out for the blast wave-turbulence interaction was extended

to the problem of a converging shock interacting with turbulence. This problem pre-

sented several challenges from a simulation standpoint. Isotropic turbulence is very

difficult to sustain with nonperiodic boundaries. None of the widely used boundary

condition approaches were found satisfactory for this purpose. On the other hand,

simulating a converging shock with periodic boundaries corrupts the solution in the

simulation domain due to waves traveling from neighboring periodic images. A new

approach called windowing, developed and refined by Colonius & Ran [11] was used

to overcome these problems. This approach allows one to retain periodic boundaries

while drastically reducing errors due to reflection. Another challenge was to eliminate


the contact surface arising from the shock-tube like initial conditions used to launch

the converging shock, which creates additional complexity in an already complex

problem. The shock tube equations were re-derived to compute diaphragm pressure

and density ratios so as to eliminate the contact surface.

The problem was studied well past reflection of the converging shock from the

origin. Vorticity was found to be amplified and Kolmogorov scales decreased in the

post-shock region of the converging shock. This behavior is similar to planar shock-

turbulence interaction, but the degree of amplification was different, and this is shown

by again comparing the amplification rates with linear theory. Lastly, shock distor-

tions were found to be significant, irrespective of initial shock strength, and were

retained till late times.

The RMI study represents a first attempt at simulating Richtmyer-Meshkov in-

stability in spherical geometry. Similar to planar Richtmyer-Meshkov simulations,

an egg-carton shaped initial perturbation with a constant length scale is devised for

spherical geometry. It is to be noted that using spherical harmonics to generate initial

conditions analogous to the product of harmonics used for planar ICs will not give

a constant lengthscale throughout the perturbation field. The focus is on the post-

reshock turbulent phase, where turbulent mixing, vorticity generation and evolution,

evolution of turbulent kinetic energy are studied. Maximum compression achieved is

an important metric for ICF type applications. This is quantified as a function of

initial shock Mach number.

The remaining chapters are organized as follows. Chapter 2 describes the govern-

ing equations and details of numerical methods used for the various problems. Im-

provements to artificial viscosity based shock capturing are also described. Chapter

3 describes the various numerical tests performed to validate the code. Comparison

of the numerical algorithms used in this work is compared with other widely used

algorithms for the same problems. Chapter 4 deals with the first canonical problem

of interaction of a Taylor blast wave with isotropic turbulence. Chapter 5 continues

1.2. OVERVIEW AND ACCOMPLISHMENTS 11

with interaction of a converging shock with isotropic turbulence. Chapter 6 describes

simulations of Richtmyer-Meshkov instability in spherical geometry. Lastly, chapter

7 concludes the study with remarks for future work.

Chapter 2

Methodology

This chapter describes the governing equations, boundary conditions and numerical

methods employed to solve the different problems described in this thesis. A Carte-

sian grid is used for all problems in this work. Since the problems of interest are all

spherical in nature, a legitimate question to ask therefore is, why a spherical grid was

not used. There are several numerical issues associated with spherical grids. First

and foremost are the singularities, both polar and radial, which necessitate special

treatment at those points. This would require either interpolation or skipping singu-

lar points, which would introduce errors. The non-uniform placement of grid points

on conventional latitude-longitude grids puts severe restrictions on the time step for

explicit time stepping schemes. Often, filtering is required at the poles, where the

meridians converge, which considerably degrades the quality of the numerical solution.

Domain decomposition and consequently, parallel scalability is difficult to achieve on

a spherical grid.

Each problem has its unique characteristics that necessitate a different approach.

The Taylor blast wave-turbulence interaction problem is solved with periodic bound-

ary conditions. The converging shock problem is also solved with periodic boundary

conditions, but with an additional windowing procedure to damp outgoing waves from

the neighboring images so that they do not affect the solution in the domain of inter-

est. The spherical RMI problem is a multi-fluid problem and requires an additional

12

2.1. GOVERNING EQUATIONS 13

species evolution equation. In order to save on computational costs, it is solved on

an octant instead of the entire sphere. This necessitates exact symmetry boundary

conditions on three inner planes that pass through the origin and outflow boundary

conditions on the three outer planes of the simulation domain. Note that derivative

schemes used for the spherical RMI problem are different from those used for the blast

wave and converging shock-turbulence interaction problems. The specific details and

rationale behind them are given in later chapters.

2.1 Governing equations

2.1.1 Navier-Stokes equations

The Navier-Stokes equations for a perfect gas in Cartesian coordinates are solved.

Due to the strong nature of the shocks, shock-capturing is needed without which the

code would blow up. This is provided through artificial viscosity following Cook [12],

but with a modified coefficient for bulk viscosity as described in Bhagatwala & Lele

[4].

The dependent variables are normalized as

ui =u∗ia∗ref

(2.1)

ρ =ρ∗

ρ∗ref(2.2)

p =p∗

ρ∗refa∗2ref

(2.3)

and the space and time variables are non-dimensionalized as

xi =x∗iL∗ref

(2.4)

t =t∗a∗refL∗ref

(2.5)

14 CHAPTER 2. METHODOLOGY

The non-dimensional Navier-Stokes equations are as follows

∂ρ

∂t+

∂

∂xj(ρuj) = 0 (2.6)

∂

∂t(ρui) +

∂

∂xj(ρuiuj + pδij) =

∂τij∂xj

(2.7)

∂

∂t(ρet) +

∂

∂xj((ρet + p)uj) = − ∂qj

∂xj+

∂

∂xj(uiτij) (2.8)

∂ρY

∂t+

∂

∂xj(ρY uj + Jj) = 0 (2.9)

The viscous stress τij and the heat flux qi are

τij =µ

Reref

(∂ui∂xj

+∂uj∂xi

)

+µ

Reref

(

β − 2

3

)∂uk∂xk

δij (2.10)

qi = − µ

RerefPr

∂T

∂xi(2.11)

where the mass fluxes J are given by

Jj = −ρ(

Dj∂Y

∂xj− Y

(

D1∂Y

∂xj+D2

∂(1 − Y )

∂xj

))

(2.12)

(2.13)

For the multi-fluid spherical RMI problem, the energy equation has an additional

flux term, so that qi becomes

qi = − µ

RerefPr

∂T

∂xi+

k=N∑

k=1

hkJk (2.14)

where the last term on the right hand side is the interdiffusional enthalpy flux as

described in Cook [13]. The enthalpy of an individual species is given by

hk = ek + Pk/ρk (2.15)

2.1. GOVERNING EQUATIONS 15

Reynolds and Prandtl numbers are Reref = ρ∗refL∗refa

∗ref/µ

∗ref and Pr = c∗pµ

∗/κ∗.

The physical shear viscosity is assumed to follow the power law µ = T n where n = 0.76

and the physical bulk viscosity is set to zero in accordance with Stokes’ assumption.

Bulk viscosity has a significant effect on physical shock thickness and hence this

assumption is not always valid. For instance H2 has βf/µf ∼ 30, CO2 has βf/µf ∼1000. For the problems of interest in this work, air is the primary fluid for which

βf/µf = 0.4. Artificial bulk viscosity βh is usually two to three orders of magnitude

higher than physical shear viscosity. Both would primarily act around shocks. In our

case, βh >> βf and βf can therefore be neglected following Stokes’ assumption. But

in general, care must be exercised in making this assumption. Following Cook [12],

artificial terms are now added to the physical viscosity coefficients. µ = µf + µh,

β = βf + βh, κ = κf + κh and D = Df +Dh.

The mass flux equations above are an approximation for the exact term Jm =

ρm(um−u), where the subscript m denotes the mth species. The second term on the

right hand side is to ensure the Σm=Nm=1 Jm = 0 as needed for continuity. However, for

a binary system, this term goes to zero. Also for a binary system, D12 = D1 = D2,

which implies J1 = −J2, i.e flux of one species into a control volume is exactly

balanced by the outflux of the other species to ensure continuity.

2.1.2 Artificial Viscosity

In this study, an explicit subgrid scale (SGS) model is not used. For problems in-

volving isotropic turbulence, all turbulence scales are resolved. The only unresolved

features are the eddy-shocklets which are captured by the shock-capturing artificial

viscosity described below. Since the purpose of an SGS model is to provide dissipa-

tion in unresolved regions of the flow, artificial viscosity can be said to act as an SGS

model in this case. Large driving shocks for the blast wave and converging shocks are

also captured using the same scheme.

The artificial viscosity terms, as given in Cook [12], are as follows.

µh = Cµρ|∇4S|∆6 βh = Cβρ|∇4S|∆6 κh = Cκρc

T|∇4e|∆5 (2.16)


The artificial diffusivity terms are as follows

Dm = CDcs|∇4Ym|∆4 + CY cs[Ym − 1]H(Ym − 1)− Ym[1−H(Ym)]∆ (2.17)

where H is the Heaviside function, ∆ is the grid spacing, S =√S : S is the

magnitude of the symmetric strain rate tensor, Cµ = 0.002, Cκ = 0.01, CD = 1.0 and

CY = 100. The coefficient for artificial bulk viscosity is given by

This formulation satisfies D1 = D2 required for binary diffusion.

The overbar indicates a Gaussian filter defined in Cook [12] as

f(x) =

∫ L

−L

G(|x− ξ|;L)f(ξ)d3ξ, (2.18)

where

G(ζ ;L) =e−6ζ2/L2

∫ L

−Le−6ζ2/L2dζ

, L = 4∆ (2.19)

The purpose of this filter is to ensure smooth behavior of the artificial quantities,

which are likely to have very high degree of oscillations due the high-order laplacian

and the absolute value operator. On a discrete grid, the Gaussian filter described

above can be formulated as

fi =3565

10368fi +

3091

12960(fi−1 + fi+1) +

1997

25920(fi−2 + fi+2)

+149

12960(fi−3 + fi+3) +

107

103680(fi−4 + fi+4)

The effective gamma formulation is used for computing the ratio of specific heats

and it is calculated as follows

γeff =Cp,eff

Cv,eff

2.2. NUMERICAL METHOD 17

=Y1Cp1 + Y2Cp2

Y1Cv1 + Y2Cv2

=

Y1γ1γ1−1

+ ǫ Y2γ2γ2−1

Y1

γ1−1+ ǫ Y2

γ2−1

where ǫ = M1/M2 is the ratio of gas constants or molecular weights of the two

gases.

Cβ =1

2

(

1− tanh

(

2.5 + 10∆

c∇.u

))

× (∇.u)2(∇.u)2 + Ω2 + ǫ

(2.20)

where Ω =√

(ωiωi) is the vorticity magnitude and ǫ = 10−7. As described in

Bhagatwala & Lele [4], this choice of coefficient considerably decreases the excess

damping of dilatational and thermodynamic fluctuations caused by the traditional

formulation in Cook [12]. This coefficient is discussed in further detail in the next

chapter.

2.2 Numerical method

Having described the governing equations solved for single-fluid and multi-fluid prob-

lems, we now describe the numerical methods used to solve the discretized equations.

For the blast wave problem, periodic boundary conditions are used. Since the shock

front is the fastest propagating wave in the domain, the periodicity does not affect

the shock and as long is it is confined within the domain, there is no interference

with neighboring boxes. This is true only the blast wave problem. The converging

shock-turbulence and spherical RMI problems necessitate more sophisticated bound-

ary conditions described later. All results shown in this paper are over the time

horizon when the shock is still within the original box. The reason this configuration

was chosen, was that isotropic turbulence is difficult to maintain in a non-periodic

domain. Precise boundary conditions would need to be specified and enforced. For

the converging shock, this is not the case and hence, a technique for absorbing out-

going waves called windowing is used which still allows us the use periodic boundary


conditions. This technique is described in detail in the next section. For the multi-

fluid spherical Richtmyer-Meshkov Instability problem, outflow boundary conditions

with a damping sponge layer are used at the three faces forming the exit boundary

while symmetry boudnary conditions are used at the three other faces. These are

described in the appendices.

2.2.1 Spatial discretization

For the blast wave-turbulence and converging shock-turbulence interaction problems,

a sixth order compact finite difference scheme optimized for high wavenumber resolu-

tion is used to compute spatial derivatives in all three directions as described in detail

in Lui [51] and formulated in Lele [44]. The optimized scheme uses a pentadiagonal

stencil on the LHS. The spherical RMI problem is solved in Euler mode and to save

on computational costs a sixth order tridiagonal-based scheme is used. Both these

schemes are described below.

First derivative

The general form of the first derivative for the compact scheme is

β1f′

i−2+α1f′

i−1+ f′

i +α1f′

i+1+β1f′

i+2 = c1fi+3 − fi−3

6∆x+ b1

fi+2 − fi−2

4∆x+ a1

fi+1 − fi−1

2∆x(2.21)

The coefficients for the pentadiagonal scheme have been optimized to get a scheme

that is formally sixth order with very good wavenumber resolution properties. The

coefficients for the tridiagonal scheme are constrained to obtain a formally sixth order

scheme, but with no optimization. Table 2.1 below gives these coefficients for both

these schemes. The modified wavenumber for both schemes is plotted in figure 2.1

Second derivative

The general form of the compact scheme for the second derivative is


Coefficient Pentadiagonal Tridiagonal

a1 1.367577724399269 1.555555555555556b1 0.8234281701082790 0.1111111111111111c1 0.01852078348668660 0α1 0.5381301488732363 0.3333333333333333β1 0.06663319012388112 0

Table 2.1: Coefficients for the first derivative for pentadiagonal and tridiagonal basedcompact schemes.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

k∆x

k′

∆x

Figure 2.1: Modified wavenumber for the first derivative scheme (a) Pentadiagonalscheme (dashed) (b) Tridiagonal scheme (dash-dotted) (c) Spectral (solid)


Coefficient Pentadiagonal Tridiagonal

a2 0.3855624784762861 0.0383252421459301b2 1.486194872274852 0.272727272727273c2 0.09330361806154306 0α2 0.4442052422604100 0.181818181818182β2 0.0383252421459301 0

Table 2.2: Coefficients for the second derivative for pentadiagonal and tridiagonalbased compact schemes.

β2f′′

i−2 + α2f′′

i−1 + f′′

i + α2f′′

i+1 + β2f′′

i+2 = c2fi+3 − 2fi + fi−3

9∆x2+ b2

fi+2 − 2fi + fi−2

4∆x2

+a2fi+1 − 2fi + fi−1

∆x2

The optimized coefficients for the sixth order pentadiagonal and unoptimized sixth

order tridiagonal schemes are given in the table 2.2 below. The modified wavenumber

for both schemes is shown in figure 2.2

2.2.2 Temporal discretization

The smallest time scales in the flows of interest here are dictated by the acoustic waves

associated with the dilatational modes of turbulence. The use of high order schemes

has advantages due to lower dispersion and dissipation errors. For the single fluid

problems involving isotropic turbulence, we use the fourth order, two-step, eleven-

stage Low Dissipation and Dispersion RK4 (LDDRK4) scheme of Stanescu & Habashi

[83]. For the spherical RMI problem, we use a 5-stage RK4 scheme of Kennedy et.

al [34], chosen for its broader stability properties and computational economy. Both

these schemes solve the ODE

du

dt= F (u, t) (2.22)

The discretized version of this equation is


0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

k2∆x

k′′

∆x

Figure 2.2: Modified wavenumber for the second derivative scheme (a) Pentadiagonalscheme (dashed) (b) Tridiagonal scheme (dash-dotted) (c) Spectral (solid)

ti−1 = tn−1 + ci∆t

∆ui = αi∆ui−1 +∆tF (ui−1, ti−1)

ui = ui−1 + βi∆ui

where i refers to the substep number for the particular scheme and n refers to

the current global time step. Superscript refers to the global overall time index and

subscript refers to a substep which is a part of the current time step. Coefficients for

both schemes are given in table 2.3

2.2.3 Convective terms and control of aliasing errors

The skew-symmetric form of the convective term is employed for the momentum

equations.


LDDRK4 Step 1

i α β c

1 0.0 0.26874543888713438496 0.02 −0.60512264332862261228 0.8014706973220802933 0.268745438887134384963 −2.04375640234761394333 0.50515704269422722538 0.585228069295243034694 −0.74069990637544192841 0.56235680379000296407 0.682706644784246788215 −4.42317651302968168941 0.05900655127758823335 1.1646854837729261436

LDDRK4 Step 2

i α β c

1 0.0 0.11584888181285561688 0.02 −0.44127377153877382565 0.37287699051652864918 0.115848881812855616883 −1.073982008079781868 0.73795368921435295698 0.324185036404128068534 −1.7063570791256758809 0.57981109366311039538 0.619320820351777923685 −2.7979293162682443056 1.031284991300145194 0.803447266633590790596 −4.0913537120919160454 0.15 0.91841664452065965078

RK4 Kennedy et. al [34]

i α β c

1 0.0 0.1028639988104959 0.02 −0.4801594388477952 0.7408540575766301 0.10286399881049593 −1.404247195200064 0.7426530946683799 0.48798998783301924 −2.016477077503357 0.4694937902357905 0.68851772315620365 −1.056444269767207 0.1881733382887932 0.9023816453077341

Table 2.3: Coefficients for the LDDRK4 and Kennedy et. al time stepping schemes.


1

2

∂

∂xj(uiuj) +

1

2uj∂ui∂xj

(2.23)

This form achieves energy conservation in the inviscid limit (Horiuti [24], Zang

[94]). Theoretical analysis by Blaisdell et al. [6] shows that aliasing errors are reduced

by using a skew-symmetric formulation. In addition, after every RK4 substep, an

eigth order filter as described in Cook [12] is applied to all the conserved variables to

control aliasing errors stemming from the high-order laplacian based shock-capturing

scheme. This filter, based on a pentadiagonal compact stencil like the derivative

scheme, is designed to remove the top 10% of wavenumbers in a sharp, spectral-like

manner, so that the frequency of filter application does not affect the result. The

filter is given by

βfi−2 + αfi−1 + fi + αfi+1 + βfi+2

= afi +b

2(fi−1 + fi+1) +

c

2(fi−2 + fi+2) +

d

2(fi−3 + fi+3)

+e

2(fi−4 + fi+4)

where fi is the filtered variable and the coefficients are

α = 0.66624

β = 0.16688

a = 0.99965

b = 1.33304

c = 0.0.33348

d = 8× 10−5

e = −10−5

The absolute value operator and the fourth order Laplacian needed for localization,


lead to strong oscillations in the artificial viscosity field. The filter helps damp these

oscillations, which can potentially make the simulation unstable.

2.2.4 Boundary Conditions

For the blast wave-turbulence interaction problem, use of periodic boundary condi-

tions can be justified on the grounds that since the outward propagating shock wave

is the fastest wave in the domain, there is no corruption from the neighboring peri-

odic images until that wave reaches the boundaries of the simulation domain. Hence,

for simulations of isotropic turbulence and the Taylor blast wave, periodic boundary

conditions are used.

Such an argument cannot be made for the converging shock-turbulence interaction

problem. This flow generates, in addition to an inward propagating shock wave, an

outward propagating expansion wave which will reach the boundaries of the simula-

tion domain before the converging shock reaches the origin. This would corrupt the

solution at the boundaries during the reflected shock phase of the problem. Therfore

special treatment is needed at the boundaries to absorb this expansion wave without

reflections or corruption from periodic images. For this purpose, periodic boundary

conditions with windowing are used as described below.

For the spherical RMI problem, outflow boundary conditions with sponge treat-

ment are used. These are described in detail in Appendix B.

Windowing

The domain is periodic in all three coordinate directions. The grid is uniform and

Cartesian. To ensure that outgoing expansion waves from neighboring periodic images

do not interfere with the simulation, we use a technique called windowing as described

in Colonius & Ran [11]. The governing equations are multiplied by a windowing func-

tion and then closed by modeling the original dependent variables as functions of the

windowed version. This approach can be shown to be conceptually similar to grid


stretching and damping disturbances in a buffer region near the boundary.

The windowing function is of a form widely used in signal processing literature.

Similar to Colonius & Ran [11], the function we use is obtained by multiplying three

separate windowing functions in three coordinate directions.

H(x) = h1(x1/L1)h2(x2/L2)h3(x3/L3) (2.24)

The vector of dependent variables transforms as Q = H(x)Q. If Fj(Q) are the

flux vectors, starting from the usual Navier-Stokes written as

∂Q

∂t+∂Fj(Q)

∂xj= 0 (2.25)

one can obtain the following equation

∂Q

∂t+

3∑

j=1

(hj + (∆x)2h′′

j )∂Fj(Q)

∂xj= 2atyp(Mtyp + 1)

3∑

j=1

(∆x)2|h′′

j |∂2Q

∂xj∂xj+H.O.T

(2.26)

where atyp and Mtyp are the characteristic sound speed and Mach number for the

flow of interest. For the window function, we use the tanh based function prescribed

in Colonius & Ran [11]. We use the same grid spacing in all three directions, denoted

here by ∆x.

h(x) =1

2

(

tanh

(

2s

[x/L− ǫ

2ǫ

])

− tanh

(

2s

[x/L− 1 + ǫ

2ǫ

]))

(2.27)

The function takes the value unity in the interior of the domain and smoothly

falls to zero at the edges. Therefore, in the interior, equation (2.26) reduces to the

familiar equation (2.25) as desired.

Figure 2.3 shows the windowing function with the buffer zone marked. ǫ is the

buffer zone width, and also a measure of the computational cost of adding a buffer,

so that ǫ = Nbf/ntot. For infinite buffer size, we have a theoretically infinite domain

with perfect resolution, but also at infinite expense. We therefore seek the smallest


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

w(x

)

ε ε

L

Figure 2.3: Plot of the window function and the buffer region.

buffer size that yields sufficiently accurate results. For the simulations reported in

the paper, we use ǫ = 0.1 and the slope parameter s = 2 which corresponds to a very

smoothly decaying window function.

2.2.5 Particles

To understand how individual fluid particles behave, Lagrangian tracking particles

are introduced in the flow. These fluid particles “see” the local fluid velocity and

advect according to it. If rp(t) is the position vector of a particle at any time t, its

motion is governed bydrp(t)

dt= up(t) (2.28)

where rp = (xp, yp, zp) is the location of the particle and up = (up, vp, wp) is the local

fluid velocity interpolated to the particle location. The particle trajectories are inte-

grated using the same LDDRK4 scheme as described in the previous section. Other

flow properties are calculated at particle locations using a simple trilinear interpola-

tion scheme.

2.3. CODE PERFORMANCE 27

101

102

103

104

100

101

102

# processors

Spe

edup

(a)

101

102

103

104

105

0

5

10

15

20

25

30

35

# processors

Tim

e/st

ep (s

ec)

(b)

Figure 2.4: (a) Strong and (b) Weak scaling for the code. Dashed lines indicate idealscaling behavior in both cases.

2.3 Code performance

The code developed as part of this work is fully parallel. The domain for all three

problems has equal length in all three directions. The domain is decomposed in all

three coordinate directions. A typical run is on a 2563 grid with 4096 processors.

The code has excellent strong and weak scaling properties. Weak scaling is measured

by keeping the problem size per processor constant, increasing the overall number of

grid points and number of processors used. Strong scaling is measured by keeping

the overall problem size same and increasing the number of processors.

Figure 2.4 shows strong and weak scaling for the code. The strong scaling study

is a limited one because the full problem was too large to be fit on a single processor

as required by the definition. Hence, 64 processors were used as a benchmark to com-

pare the speed up. As can be seen, there is some loss of parallel efficiency (∼ 20%)

at the maximum number of processors due to higher communication overhead, but

considering the small problem size per processor (4096 grid points/processor), this

can be considered extremely good strong scaling behavior. This allows us to use a

large number of processors for a given problem, drastically decreasing the total wall


clock time requirement, while ensuring that the number of processor-hours consumed

is almost the same if we had used fewer number of processors.

Weak scaling behavior is also very good, the decrease in efficiency going from 64

processors to 32, 768 processors is only ∼ 15%, which can be considered excellent

scaling behavior. Good performance on weak scaling allows us to use large grid sizes,

typical of DNS studies (upto 5123 in the present case) while ensuring that the cost per

time step remains the same, hence ensuring computational efficiency even for large

problem sizes. Without it, we would be severely limited in the largest problem size

we can simulate, hence making a true DNS very difficult.

Chapter 3

Numerical tests

The code (STAN-I) has been validated by evaluating the numerical method used

with a suite of problems involving shocks and turbulence and compared with other

existing numerical techniques for solving such problems. The methods against which

the code has been compared are the traditional artificial viscosity formulation of Cook

[12] (STAN), Hybrid WENO/Central difference scheme of Larsson [41] (HYBRID), a

pure WENO scheme with entropy fix for upwinding by Jiang & Shu [30] and Sanders

et. al. [77] (WENO), an adaptive characteristic-based filter method of Yee and

Sjogreen [91, 92, 93, 81] (NASA) and the shock fitting method of Zhong (UCLA).

Most figures in this chapter have been reproduced from Johnsen et. al. [31]. The

author would like to thank Eric Johnsen and Johan Larsson for providing the image

files.

3.1 Method of Manufactured Solutions

The method of manufactured solutions (MMS) is a widely used technique for checking

the accuracy of implementation of the Euler equations or the convective and pressure

terms in the Navier-Stokes equations. It was formally given that name in a series of

papers (Roache [71, 72, 73], Knupp & Salari [36], Oberkampf et al. [60]). Recently,

it was used for verification by Shunn & Ham [80] for the variable density flow solver

CDP. The method also helps in ensuring the accuracy of the spatial discretization

29

30 CHAPTER 3. NUMERICAL TESTS

scheme. In this code an optimized 6th order compact finite difference scheme is

used. If the implementation is correct, the order of accuracy of the scheme should be

retrieved as the grid is made finer.

The basic idea consists of assuming arbitrary solutions for one or more of the

different variables on the left hand side of the equations, i.e ρ, ρu, ρv, ρw and ρet.

When plugged into the equations, they are obviously not satisfied. They give rise to a

right hand side consisting of derivatives of these assumed quantities. These are then

cancelled by adding ”source” terms to the RHS making it equal to zero. This means

that the time rate of change of these variables is zero. If all the convective terms are

implemented correctly in the code, adding these so called source terms and marching

the solution forward in time should keep the variables unchanged from their values

at t = 0. To illustrate the idea consider a very simple equation.

∂u

∂t=∂u

∂x(3.1)

Assume the domain is periodic. Plugging in an arbitrary solution u = sin(x),

gives us a cos(x) on the right hand side. So if we subtract cos(x) from the right hand

side we can make it zero. i.e

∂u

∂t=∂u

∂x− cos(x) = 0 (3.2)

for u = sin(x). If there was any error in the implementation of the RHS, such as

wrong partial derivative or additional spurious terms or omitted terms, addition of

the source terms would not make it zero and the solution would grow if it was marched

in time. This indicates an error. This also serves as a check for the order of accuracy

of the method. Due to roundoff, the right hand side will not exactly be zero, but it

will be zero to within the order of the derivative scheme. Then as the mesh is made

finer, the error should decrease according to the order of the method.

Tests were made with domain sizes ranging from 243 to 1283 to test the correct-

ness and accuracy of the implementation. It must be noted that the simplest possible

manufactured solution is used here. For the technique to be effective, a series of such

solutions are needed to activate every term on the RHS of every equation. This was,

3.1. METHOD OF MANUFACTURED SOLUTIONS 31

101

102

10−10

10−9

10−8

10−7

10−6

10−5

Grid points, N

L inf e

rror

NumericalTheoretical

6th order

Figure 3.1: Order verification for Case 1

in fact carried out for this code, though only results from one of the tests are shown in

this chapter. Choosing other manufactured solutions yield similar results. It must be

pointed out that the results are somewhat sensitive to the particular manufactured

solution(s) used. In general, more complicated the right hand side, greater the nu-

merical round-off errors. However, the order of accuracy of the discretization scheme

should still be maintained.

3.1.1 Verifying order of spatial discretization

The method of manufactured solutions can also be used to verify the order of accuracy

of the spatial discretization. If time stepping errors are sufficiently small, the error

after subtracting the source terms from the right hand side should decrease according

to the order of the spatial discretization scheme when grid resolution is increased.

This was done for both sets of manufactured solutions discussed above.


100

10−6

10−5

10−4

10−3

10−2

10−1

Time step

L inf e

rror

Numerical

Theoretical

4th order

Figure 3.2: Order verification for time stepping scheme

3.1.2 Verifying accuracy of time stepping

The method of manufactured solutions can also be used for time stepping schemes in

the same way as for spatial schemes. An arbitrary right hand side (as a function of

time) is assumed and when the equation is integrated using whatever time stepping

scheme is used in the code, it gives an approximation to the exact (definite) integral.

If the right hand side is chosen suitably, the exact value of this definite integral can

be calculated. When subtracted from the calculated value, the error should behave as

the order of the method, i.e it should go down with finer time steps as the order of the

scheme dictates. In this code a 4th order low dissipation and dispersion Runge-Kutta

time stepping scheme is used.

3.2 Taylor-Green vortex

The inviscid Taylor-Green problem is a simple means of studying turbulence ampli-

fication by vortex stretching and consqeuent production of small scales. The initial

3.2. TAYLOR-GREEN VORTEX 33

Code Color Line style

Reference black varyingSTAN red dashedSTAN-I magenta dashed (thin)HYBRID blue solidWENO cyan solid (thin)NASA green dashed-dottedUCLA black dotted

Table 3.1: Color and line legend for the plots.

condition is a highly anisotropic, well-resolved, incompressible velocity field with uni-

form pressure and density. No dissipation apart from that intrinsic in the numerical

scheme is provided. The flow evolves into increasingly small scale structures with no

lower bound. From a numerical perspective, this problem is a very good test for the

stability of a method to highly under-resolved motions. Two metrics compared here

are kinetic energy decay and enstrophy growth.

The three-dimensional Euler equations are solved with gas constant γ = 5/3. The

domain xi ∈ [0, 2π) is periodic and the grid spacing is ∆xi = 2π/64. The initial

conditions are

ρ = 1 ,

u1 = sinx1cosx2cosx3 ,

u2 = −cosx1sinx2cosx3 ,

u3 = 0 ,

p = 100 +[cos(2x3) + 2][cos(2x1) + cos(2x2)]− 2

16

The numerical solutions are compared with the semi-analytical solutions of Bra-

chet et. al. [7]. For the ideal inviscid, incompressible problem, the kinetic energy

remains constant and enstrophy grows rapidly.

The STAN code lies somewhere in between the HYBRID/NASA and WENO

results. The improved STAN-I results are identical to those of STAN for this problem,


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t

〈ρuiu

i〉/〈ρuiu

i〉 0

(a)

0 2 4 6 8 100

5

10

15

20

25

30

t〈ω

iωi〉/

〈ωiω

i〉 0(b)

Figure 3.3: Mean quantities for the Taylor-Green vortex on a 643 grid. The zerosubscript denotes the initial value. The semi-analytical result of Brachet et. al. [7]are the black symbols. Figure reproduced from Johnsen et. al. [31]

since the solenoidal velocity field is insensitive to the bulk viscosity. On the present

grid, both methods agree with the semi-analytical results for the enstrophy growth.

Next, the velocity spectra at t = 5 are shown in figure 3.4. No analytical spec-

trum is known, so the results are compared with a converged spectrum. The range

of interest here is k ≤ 32, given that 643 grids are employed. To generate a refer-

ence solution, the velocity spectrum was computed on a sequence of grids using the

HYBRID code in both non-dissipative form and with the addition of an eighth-order

accurate dissipation term, which effectively removes numerical noise. The results of

this exercise are shown in Figure 3.4, where it is clear that the lower wavenumbers

converge.

Both the STAN and STAN-I codes agree with the reference for k . 16 and then

underpredicts the spectrum, while the WENO code agrees only for k . 8 despite

being formally high-order accurate. The dealiasing filter in the STAN and STAN-

I codes directly affects only the very highest wavenumbers. The damping for k &

16 therefore implies either that the cumulative effect of the many filter operations

decreases the effective bandwidth, or that nonlinear processes distribute the damping

3.3. SHU-OSHER PROBLEM 35

0 0.2 0.4 0.6 0.8 10.6

0.7

0.8

0.9

1

1.1

1.2

1.3

k

Eu

(a)

101

10−6

10−4

10−2

k

Eu

(b)

Figure 3.4: Velocity spectra for the Taylor-Green vortex on 643 grid at t = 5. (a)Convergence of the reference solution using the HYBRID code in standard mode(solid) and with eighth-order accurate dissipation (dashed) on 2563 (black), 1283

(blue), and 643 (cyan). (b) Comparison between the different schemes. The referencesolution is that obtained on the 2563 grid using the HYBRID code. Figure reproducedfrom Johnsen et. al. [31]

to lower wavenumbers.

3.3 Shu-Osher problem

The Shu-Osher problem is a one dimensional canonical shock-turbulence interaction

in which a Mach 3 shock wave interacts with a sinusoidal density field (Shu & Osher

[79]). Pressure is constant on either side of the shock, which gives an entropy wave

downstream of the shock. The Euler equations are solved on a domain x ∈ [−5, 5]

with initial conditions

(ρ, u, p) =

(3.857143, 2.629369, 10.333333) x < −4

(1 + 0.2sin(5x)), 0, 1) x ≥ −4

This problem corresponds to a M = 3 shock moving into a field with a small

density (or entropy) disturbance. The solution is compared to a reference solution


−4 −2 0 2 40.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

ρ

Figure 3.5: Density profiles for the Shu-Osher problem with ∆x = 0.05 at t = 1.8.The reference solution is the seventh-order accurate WENO solution with ∆x =6.25× 10−3. Figure reproduced from Johnsen et. al. [31].

with ∆x = 6.25× 10−3.

The goal of this problem is to test the capability to accurately capture a shock

wave, its interaction with an unsteady density field, and the waves propagating down-

stream of the shock. The domain is discretized withN = 200 grid points. The solution

is integrated to t = 1.8 and compared with the constant coefficient bulk viscosity and

a converged reference solution as shown in figures. The amplitude of the post shock

entropy waves, position of the shock and position of the contact discontinuity sepa-

rating the entropy and acoustic waves are all well captured just as with the constant

coefficient bulk viscosity.

Figure 3.5 shows the density, figure 3.6 shows the entropy, and figure 3.7 shows

the velocity, all at t = 1.8. The entropy is given by ∆s/cv = ln (p/ργ), where cv is the

specific heat at constant volume. The interaction between the shock and the entropy

disturbance generates both acoustic and entropy waves downstream of the shock.

The acoustic waves are strong enough to steepen into weak shock waves. At the

3.3. SHU-OSHER PROBLEM 37

0.5 1 1.5 2 2.5

0.2

0.3

0.4

0.5

0.6

0.7

x

∆s/c v

(a)

0.5 1 1.5 2 2.5

0.2

0.3

0.4

0.5

0.6

0.7

x

∆s/c v

(b)

Figure 3.6: Entropy profiles for the Shu-Osher problem with ∆x = 0.05 at t = 1.8(close-up of the solution just downstream of the shock). The reference solution is theseventh-order accurate WENO solution with ∆x = 6.25×10−3. (a) HYBRID, STAN,STAN-I, NASA and reference solutions. (b) UCLA, WENO and reference solutions.Figure reproduced from Johnsen [?] [31].

−3 −2 −1 0 1 22.45

2.5

2.55

2.6

2.65

2.7

2.75

x

u

Figure 3.7: Velocity profiles for the Shu-Osher problem using the STAN and WENOcodes, with ∆x = 0.05 at t = 1.8 (close-up of the solution just downstream of theshock). The reference solution is the seventh-order accurate WENO solution with∆x = 6.25× 10−3. Figure reproduced from Johnsen et. al. [31].


given time, the shock location is xs ≈ 2.39, the location of the contact discontinuity

at the leading entropy wave is xc ≈ 0.69, and the location of the leading acoustic

wave is xa ≈ −2.75. The initial entropy waves have wavelength λ1 = 2π/5, which by

conservation of mass in a frame moving with the main shock is approximately 0.33

in the post-shock region. For the present grid, the entropy waves behind the shock

have 6.5 points per wavelength.

There is little appreciable difference between the original and improved artificial

diffusivity methods. The methods relying on a bulk viscosity (STAN and STAN-I)

achieve higher amplitudes of the entropy waves than the other methods. This is an

effect of the bulk viscosity only affecting the dilatational velocity field, whereas the

upwinding-based methods add dissipation to all modes.

3.4 Shock-Vorticity/Entropy wave interaction

The general version of the Shu-Osher problem is the interaction of a 2D vortic-

ity/entropy wave with a normal shock. The two-dimensional Euler equations are

solved with γ = 1.4 on the domain x1 ∈ [0, 4π], x2 ∈ [−π, π), with ∆x1 = π/50 and

∆x2 = π/16. Periodic boundaries are used in the x2-direction; x1 = 0 is a supersonic

inflow and x1 = 4π is a subsonic outflow. Different techniques are employed to avoid

acoustic reflections from the outflow, including an extension of the domain with a

sponge region. First a one-dimensional base solution corresponding to a M = 1.5

shock is defined as

(ρ, u1, p) =

(ρL, uL, pL) = (1, 1.5, 0.714286), x1 < 3π/2,

(ρR, uR, pR) = (1.862069, 0.8055556, 1.755952), x1 ≥ 3π/2.

A combined vorticity/entropy wave is superposed onto the base flow. The initial

data then becomes

ρ = ρ+ ρLAe cos(k1x1 + k2x2),

3.4. SHOCK-VORTICITY/ENTROPY WAVE INTERACTION 39

u1 = u1 + uLAv sinψ cos(k1x1 + k2x2),

u2 = −uLAv cosψ cos(k1x1 + k2x2),

p = p,

and the conditions at the inflow boundary x1 = 0 are

ρ = ρL + ρLAe cos(k2x2 − k1uLt),

u1 = uL + uLAv sinψ cos(k2x2 − k1uLt),

u2 = −uLAv cosψ cos(k2x2 − k1uLt),

p = pL.

For the present study,

k1 =k2

tanψ, Ae = Av = 0.025, (3.3)

and k2 = 1, 2. This inviscid problem has no length scale other than k2; hence an

increase of k2 corresponds to an effectively coarser grid.

All codes do well in the ψ = 450 case, but in the ψ = 750 case all codes show

evidence of post-shock oscillations. The oscillations persist downstream for the codes

such as HYBRID and APS3DID that do not have any filtering, whereas the filter in

the STAN codes removes these oscillations as they travel downstream.

Figures 3.9 and 3.10 show the kinetic energy and mean-square vorticity down-

stream of the shock averaged in span and over one time period for ψ = 450, 750 and

k2 = 1, 2. The values are normalized with respect to the conditions upstream of the

shock. In the ψ = 450 case, the vorticity seems to converge to the linearized solution

as the grid is refined (k2 = 1 as discussed earlier). In the ψ = 750 case, the codes

yield larger vorticity amplification than the linear analysis and seem to converge to a

value different that of the linearized solution; all shock-capturing codes yield higher

amplifications on the effectively coarser grid; this behavior is most likely related to

the post-shock oscillations.


1 1.5 2 2.5 3

−0.1

−0.05

0

0.05

0.1

x1/π

ω3

(a)

1 1.5 2 2.5 3−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

x1/πω3

(b)

Figure 3.8: (a) ψ = 450 at t = 25. (b) ψ = 750 at t = 32. The results from the WENOand STAN-I codes are not shown, but are similar to those of the HYBRID and STANcodes, respectively. Instantaneous vorticity profiles for the shock-vorticity/entropywave interaction at y = 0 (k2 = 1). Figure reproduced from Johnsen et. al. [31].

3.5 Isotropic turbulence

High turbulence Mach number and high Reynolds number isotropic turbulence sim-

ulations were carried out. The Mach number was Mt = 0.6 and the Taylor scale

Reynolds number was Reλ = 100. This Mach number is high enough for the flow

to spontaneously form shocklets and the Reynolds number is high enough to have a

broad range of scales for vortical motions. The properties of this field are discussed in

further detail in the following chapter. For this chapter, we will focus on the artificial

viscosity modification and improvements in representing the small dilatational scales

of turbulence. It has been noted in other studies (Johnsen et. al [31]), that artificial

bulk viscosity causes excessive dissipation of dilatational motions and gives erroneous

RMS profiles for density, pressure and temperature. As seen in figures 3.12 and 3.13,

these are significantly improved using the modified formulation.

The formulation works by decreasing the magnitude of bulk artificial viscosity at

all scales as seen in the plot of spectra in figure 3.11. Since the bulk viscosity acts on

the dilatational motions only, it has no effect on vortical motions or overall turbulent

kinetic energy. The improvement in prediction of the acoustic field, is quite dramatic,

3.6. STRONG CONVECTING VORTEX 41

as can be seen in figures 3.11 and 3.13. The latter compares contours of dilatation

and density for the two formulations. An eddy shocklet, which can be clearly seen in

the figure on the right is completely obliterated by the constant coefficient formula-

tion. Significant improvement is also seen in the thermodynamic quantities, density,

temperature and pressure. This can be seen in the plot of RMS quantities (figure

3.12) and also in the density contours.

We also compare improvements to artificial bulk viscosity suggested by various

authors.

Figure 3.14 plots the dilatation spectrum for the compressible isotropic turbulence

problem considered in a previous section for the models of Mani et. al [56], Bhagat-

wala & Lele [4] and Kawai et. al [32], in addition to the original method of Cook

[12]. The dilatation-based artificial bulk viscosity methods show significant improve-

ment over the original method of Cook [12] for a large range of wavenumbers. The

implications of employing a Ducros-type switch in artificial diffusivity methods is not

included here, as a detailed discussion on the use of different switching functions is

provided in Kawai et. al [32]. The effect of physical bulk viscosity is also not discussed

here. As discussed before, physical bulk viscosity would have a significant effect on

the dilatation spectrum if it was comparable to artificial bulk viscosity around shocks.

However, for air, the fluid used here, physical β is very low compared to artificial β

and therefore dilatation spectrum would not be affected significantly.

3.6 Strong Convecting vortex

We present here, tests to verify that windowing treatment for boundaries is able to

absorb both vortical and dilatational fluctuations propagating out of the simulation

domain. In both cases, we plot contours of the most sensitive quantities, namely vor-

ticity and dilatation respectively to establish confidence that the boundary conditions

are performing satisfactorily.


The first case is a compressible convecting vortex, specified by the following nondi-

mensional initial conditions

U1(x, y) = U0 + Cvy

Rvexp

(

− r2

2R2v

)

U2(x, y) = −Cvx

Rvexp

(

− r2

2R2v

)

ρ(x, y) = ρ0 −C2

v

2exp

(

− r2

R2v

)

P (x, y) =1

γρ(x, y)

where U0 = 0.575, Rv = 0.05, Cv = 1.15×10−2. Velocities are nondimensionalised

by the freestream speed of sound, c0 and lengths by L, length of the (unwindowed)

domain. Figure 3.15 shows contours of vorticity for the convecting vortex. As can be

seen, the vortex is successfully absorbed at the boundary. The only perturbation in

the flow is that due to the vortex, and if the vortex is indeed absorbed at the boundary,

there should be no residual perturbation velocity in the domain. The reflection error,

therefore, is measured by comparing maximum perturbation velocity in the interior

of the flow domain after vortex passage with maximum swirl (perturbation) velocity

associated with the vortex was found to be Vmax|t>tconv/Vmax,swirl < 3%, where tconv

is the time taken for the vortex to completely exit the flow domain.

A limited parametric variation with respect to buffer parameters was done to es-

tablish the optimal paramters for the buffer. The buffer size, ǫ = Nbf/Ntot, and slope

paramter, s were varied. Parameter values ǫ = 0.1, corresponding to a buffer width of

10% of the domain size and s = 2 yielded the best results at minimal computational

expense.

Figure 3.16 plots traces of dilatation along the midplane of the flow domain. The

initial conditions chosen, set up, in addition to the convecting vortex, a convecting

gaussian acoustic pulse. Based on pressure perturbation ∆p/pambient, this pulse has

a 1.5% amplitude. This pulse can be seen to be diverging radially outward while

convecting from the left of the domain to the right due to mean velocity, as would be

3.7. BLAST WAVE 43

expected. Interestingly, the windowed boundary is unable to fully absorb this small

amplitude perturbation. While the wave amplitude attenuates by a factor of about 2,

small amplitude acoustic waves (0.5%−0.7%) continue to bounce around the simula-

tion domain at late times. This is seen in figure 3.16 where the amplitude of the blue

dashed curve is comparable to the red and black dashed curves, instead of decreasing

as would be the case if boundary absorption was perfect. This was also observed by

Colonius and Ran [11], where they note the small amplitude acoustic reflections at the

boundaries. However, for a large propagating disturbances, the magnitudes of these

reflections are at least two orders of magnitude lower and hence can be considered

tolerable. In the next section, we show that the windowing treatment is very effective

at absorbing large amplitude dilatational waves, such as a shock wave.

3.7 Blast Wave

An outward propagating blast wave is simulated, similar to the one described in

Bhagatwala & Lele [5], decsribed by the following nondimensional conditions. The

purpose of this test is to ensure that outgoing shock wave is absorbed by the windowed

boundary conditions without significant reflections corrupting solution in the flow

domain. In this case, we are interested in dilatation as it is the most sensitive quantity

for these types of problems.

ρ = ρ0

u = 0

ρe =P0

γ − 1+ ρ∆e0

e−(r2/ǫ2)

ǫ3

where ρ0 = 1, P0 = 0.71463, ∆e0 = 15.28415 and ǫ = 0.1. The parameter ∆e0

controls the energy added, and hence, the strength of the resulting shock. The im-

pulse is only added in the energy equation, the mass and momentum equations are

unaffected. The large energy deposition acts via pressure, to change momenta, and

through momenta, affects the mass distribution.


We look at slices of dilatation contours as the shock evolves and is eventually ab-

sorbed at the boundary. As seen in figure 3.17, the outgoing blast wave is absorbed at

the windowed boundary. Time is non-dimensionalised by τs, the time it takes for the

shock to first reach the domain boundaries. If the shock wave is successfully absorbed,

there should be no more compression waves in the domain. Compression waves are

characterized by negative dilatation. The reflection error, therefore, is measured by

comparing maximum negative dilatation in the domain after the shock has passed

with the maximum negative dilatation associated with the shock wave, just before it

hits the boundary. This error, θmin|t>tshock/θmin,shock < 4%, where tshock is the time

taken for the blast wave to completely exit the flow domain.

Figure 3.18 shows dilatation and pressure profiles for the blast wave problem. In

contrast with the convecting vortex problem, in this case, the outgoing shock wave is

effectively absorbed by the windowing treatment. This suggests that the windowing

treatment is well suited for absorbing strong outgoing disturbances rather than weak

ones. This is a desirable trait for our application of interest, where we have strong

vortical and dilatational disturbances propagating out of the domain.

At early times, when the shock is very strong, wiggles can be noticed around

the foot of the outward propagating shock. Figure 3.19 shows a zoomed in view

around this region. The maximum amplitude of the wiggles is 8% relative to ambient

pressure, P0 and 1% relative to pressure jump across the shock, ∆P .

3.7. BLAST WAVE 45

−0.5 0 0.5 1 1.5 2 2.5

1.8

2

2.2

2.4

2.6

2.8

3

k2(x1 − xs)/π

〈uiu

i〉/〈u

iui〉 u

pstrea

m

(a)

−0.5 0 0.5 1 1.5 2 2.5

4.95

5

5.05

5.1

5.15

5.2

5.25

5.3

k2(x1 − xs)/π〈ω

3ω3〉/〈ω

3ω3〉 u

pstrea

m

(b)

−0.5 0 0.5 1 1.5 2 2.5

1.8

2

2.2

2.4

2.6

2.8

3

k2(x1 − xs)/π

〈uiu

i〉/〈u

iui〉 u

pstrea

m

(c)

−0.5 0 0.5 1 1.5 2 2.5

4.95

5

5.05

5.1

5.15

5.2

5.25

5.3

k2(x1 − xs)/π

〈ω3ω3〉/〈ω

3ω3〉 u

pstrea

m

(d)

Figure 3.9: Mean quantities for the shock-vorticity/entropy wave interaction (ψ =450). Shock location: xs = 3π/2. The reference is the linear analysis solution ofMahesh et. al. [55]. (a) Mean-square kinetic energy (k2 = 1) (b) Mean-squarevorticity (k2 = 1). (c) Mean-square kinetic energy (k2 = 2). (d) Mean-square vorticity(k2 = 2). Figure reproduced from Johnsen et. al. [31]


−0.5 0 0.5 1 1.5 2 2.51.5

2

2.5

3

3.5

4

4.5

k2(x1 − xs)/π

〈uiu

i〉/〈u

iui〉 u

pstrea

m

(a)

−0.5 0 0.5 1 1.5 2 2.54.4

4.6

4.8

5

5.2

5.4

k2(x1 − xs)/π

〈ω3ω3〉/〈ω

3ω3〉 u

pstrea

m

(b)

−0.5 0 0.5 1 1.5 2 2.51.5

2

2.5

3

3.5

4

4.5

k2(x1 − xs)/π

〈uiu

i〉/〈u

iui〉 u

pstrea

m

(c)

−0.5 0 0.5 1 1.5 2 2.54.4

4.6

4.8

5

5.2

5.4

k2(x1 − xs)/π

〈ω3ω3〉/〈ω

3ω3〉 u

pstrea

m

(d)

Figure 3.10: Mean quantities for the shock-vorticity/entropy wave interaction (ψ =750). Shock location: xs = 3π/2. The reference is the linear analysis solution ofMahesh et. al.[55].(a) Mean-square kinetic energy (k2 = 1) (b) Mean-square vorticity(k2 = 1). (c) Mean-square kinetic energy (k2 = 2). (d) Mean-square vorticity (k2 =2). Figure reproduced from Johnsen et. al. [31]

3.7. BLAST WAVE 47

Figure 3.11: Comparison of Dilatation and artificial bulk viscosity spectra for 643

isotropic turbulence. Plus: 2563 DNS. Square: Constant coefficient model. Circle:Modified coefficient model

Figure 3.12: Comparison of RMS quantities for 643 isotropic turbulence. Solid line -2563 DNS, Dashed line - 643 Modified coefficient model. Dotted line - 643 Constantcoefficient model


Figure 3.13: Comparison of slices of dilatation and density contours: Dilatation onthe left and Density to the right. Top to bottom: Constant Cβ, 64

3 grid, ModifiedCβ, 64

3 grid and 2563 grid DNS. Same contour and grayscale levels are used in bothplots.

3.7. BLAST WAVE 49

101

10−4

10−3

10−2

10−1

k

Eθ

Figure 3.14: Dilatation spectrum at t/τ = 4 for the isotropic turbulence problem on a643 grid for different artificial diffusivity methods. Black solid: 2563 DNS; Red dash:Cook [12]; black dash-dot: Mani et. al. [56]; magenta dash (thin): Bhagatwala &Lele [4].


t/τv = 0 t/τv = 0.2 t/τv = 0.4

t/τv = 0.6 t/τv = 0.8 t/τv = 1.2

Figure 3.15: Absorption of a compressible, convecting strong vortex at boundarytreated with windowing. Plots shows contours of vorticity. Same contour levels havebeen used in all plots.

3.7. BLAST WAVE 51

−3 −2 −1 0 1 2 3−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

x

θ

(a)

−3 −2 −1 0 1 2 30.695

0.7

0.705

0.71

0.715

0.72

0.725

0.73

x

P

(b)

Figure 3.16: Traces of (a) Dilatation and (b) Pressure along one of the principal axesfor the convecting vortex problem. t/τv = 0.2 (black-solid), t/τv = 0.5 (red-solid),t/τv = 0.7 (blue-solid), t/τv = 1.2 (black-dashed), t/τv = 1.5 (red-dashed), t/τv = 2.2(blue-dashed)


t/τs = 0.2 t/τs = 0.5 t/τs = 0.7

t/τs = 1.2 t/τs = 1.5 t/τs = 2.2

Figure 3.17: Absorption of a spherical shock wave at a boundary treated with win-dowing. Plots shows contours of dilatation. Same contour levels have been used inall plots.

3.7. BLAST WAVE 53

−3 −2 −1 0 1 2 3

−30

−20

−10

0

10

x

θ

(a)

−3 −2 −1 0 1 2 30

1

2

3

4

5

6

x

P

(b)

Figure 3.18: Traces of (a) Dilatation and (b) Pressure along one of the principalaxes for the blast wave problem. t/τs = 0.2 (black-solid), t/τs = 0.5 (red-solid),t/τs = 0.7 (blue-solid), t/τs = 1.2 (black-dashed), t/τs = 1.5 (red-dashed), t/τs = 2.2(blue-dashed)

−1.8 −1.6 −1.4 −1.2 −1

0.6

0.8

1

1.2

1.4

1.6

x

P

Figure 3.19: Zoomed view of figure 3.18(b) around the foot of the shock at early time.

Chapter 4

Interaction of a Taylor Blast Wave

with Isotropic Turbulence

The goal of the present study is to do an analysis of a spherical shock-turbulence

interaction problem from a purely hydrodynamic standpoint. We wish to understand

how turbulence is affected by the passage of the shock and how the shock is modi-

fied by the field of turbulence. To that end, we neglect ionization and dissociation.

Also, the explosion is modeled as originating from a point source, approximated by a

delta function, rather than being distributed over a significant fraction of the domain.

The schematic in figure 4.1 illustrates the various elements of the problem. Periodic

boundaries are used for this simulation, as the shock is the fastest traveling wave in

the domain and until it reaches the boundaries of the simulation domain, there is no

contamination of the solution from neighboring periodic images.

The chapter is organised as follows. Section 2 describes the simulation details

which includes the set of equations being solved, the numerical scheme employed in

the code and the shock capturing algorithm used. Section 3 discusses the background,

isotropic turbulence, and the Taylor blast wave in absence of turbulence. The shock

properties are compared with the analytical solutions of Taylor [84] and Sedov [78].

Section 4 discusses shock-turbulence interactions for the Taylor blast wave. We look

54

4.1. BASE FLOWS 55

Figure 4.1: Schematic of the simulation domain for Taylor blast wave-turbulenceinteraction problem.

at the effect of the shock on turbulence through radial profiles and Lagrangian par-

ticle statistics and the effect of turbulence on the shocks through time evolution of

shock properties. Section 5 offers a comparison of the observed behavior of turbulent

vorticity with the linear interaction analysis (LIA) for a planar shock. Lastly, section

6 ends the chapter with conclusions.

4.1 Base flows

The characteristics of baseline compressible isotropic turbulence are first described.

This is followed by a characterization of the Taylor blast wave in absence of turbulence.

4.1.1 Isotropic turbulence

The turbulent field is one of decaying isotropic turbulence with eddy shocklets and

has been generated using the methodology proposed by Ristorcelli and Blaisdell [70].

The parameters characterizing this field are the turbulence Mach number Mt,0 and

56 CHAPTER 4. TAYLOR BLAST WAVE - TURBULENCE INTERACTION

0 1 2 3 4−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

t/τ0

S

tshock

(a)

0 1 2 3 430

35

40

45

50

55

60

65

70

t/τ0

En

str

op

hy

tshock

(b)

10−1

100

101

10−15

10−10

10−5

100

105

kη

E(k)/(ǫν

5)1/4

(c)

Figure 4.2: (a) Evolution of velocity derivative skewness and (b) Enstrophy with thetime at which the shock is launched, marked. (c) Converged 3D kinetic energy spectraat time of shock launch in Kolmogorov variables. 1283 (dashed); 2563 (dash-dotted);5123 (solid)

the Taylor scale Reynolds number Reλ,0, which are defined as follows.

Mt,0 =

√3urms,0

〈c0〉, Reλ,0 =

〈ρ0〉urms,0λ0〈µ0〉

(4.1)

where

urms =√

uiui/3, λ2 =〈u21〉

〈(∂1u1)2〉(4.2)

The energy spectrum satisfies

E(k) ∼ k4exp(−2(k/k0)2) (4.3)

3u2rms,0

2=

3〈u0〉22

=〈ui,0ui,0〉

2=

∫ ∞

0

E(k)dk (4.4)

This choice for the energy spectrum gives λ0 = 2/k0, where k0 is the most energetic

wavenumber, taken as k0 = 4. The energy containing eddy scale is l0 = 2π/k0. All

relevant length scales are non-dimensionalized by l0. The relevant time scale is defined

by τ0 = λ0/〈u0〉. As shown in Lumley [52], this scale is about 4 times the Kolmogorov

time scale for isotropic turbulence.

Before launching the shock, the turbulence is first allowed to evolve to a state

4.1. BASE FLOWS 57

100

101

102

10−15

10−10

10−5

100

k/k0

E(k

)/u

2 0

(a)

100

101

102

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

k/k0

ρ(k

)/ρ

0

(b)

100

101

102

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

k/k0

Ω(k

)/u

2 0k

2 0

(c)

100

101

102

10−12

10−10

10−8

10−6

10−4

10−2

k/k0

Θ(k

)/u

2 0k

2 0

(d)

Figure 4.3: Converged spectra for (a) Energy (b) Density fluctuations (c) Vorticity(d) Dilatation. 1283 (dashed); 2563 (dash-dotted); 5123 (solid) at t = tshock

where the nonlinear energy transfer mechanism becomes fully active. This is indi-

cated by the velocity derivative skewness attaining a steady negative value. The time

at which the enstrophy is at its peak, indicating that the vortical fluctuations are at

their greatest before viscosity starts dissipating them is selected as the time to launch

the shock. This occurs at about t = 1.5τ0. Figure 4.2 shows the evolution of the

velocity derivative skewness and enstrophy.

Figure 4.3 shows plots of spectra normalized by large scale variables, u0 and k0.

They show that as the grid resolution is increased a larger range of fine-scale motions


Time Reλ Mt ρrms/〈ρ0〉 tke/tke0 tkedil/tke

t0 100 0.6 0.34 1.0 0.2tshock 45 0.4 0.27 0.7 0.2

Table 4.1: Comparison of turbulence parameters at time of shock launch with thoseat time of turbulence launch.

100

101

10−10

10−8

10−6

10−4

10−2

k/k0

E(k

)/u

2 0

Dilatational KE

Solenoidal KE

Figure 4.4: Energy spectrum showing decomposition into dilatational and solenoidalmodes for turbulence field at t = tshock. Dilatational TKE (dashed) and total TKE(solid). Thatched region shows solenoidal component.

are captured. The turbulent kinetic energy (TKE) and density variance are well cap-

tured on 2563 grid; the spectra at this resolution drop by more than five decades

before any departure from the 5123 resolution can be noticed. A limited inertial sub-

range (due to low Reλ) can be noted in the spectra. Vorticity and dilatation spectra

reaffirm that dissipative scale motions are well captured in the finest grid calculation.

There is approximately two decades of overlap between the 5123 and 2563 calculations

for dissipative scales.

Table 4.1 lists the various turbulence quantities of interest as they evolve from the

4.1. BASE FLOWS 59

initial state of turbulence to the state at which the shock is launched. At this point,

the Taylor scale Reynolds number of the flow Reλ has decreased by more than half

from 100 to 45 and the turbulent Mach number Mt has decreased to 0.4 from 0.6.

So the turbulence can be considered to be weakly compressible. This value of Mt is

however high enough to form eddy shocklets that persist upto late times. As discussed

in more detail in Johnsen et al [31], these shocklets are captured crisply by the

improved artificial bulk viscosity method of Bhagatwala and Lele [4], even at modest

resolutions. Thermodynamic fluctuations, represented by ρrms here have declined

by about 30%, in line with the decay in turbulent kinetic energy. Interestingly,

the fraction of turbulent kinetic energy in dilatational or non-solenoidal modes as a

fraction of the total turbulent kinetic energy remains the same at 20%. Dilatational

and solenoidal components of TKE are also shown in figure 4.4. Dilatational TKE is

approximately 20% of total TKE at all energy containing scales almost all the way

down to dissipation. This suggests that dilatational dissipation is also close to 20%

of the solenoidal dissipation.

Figure 4.5 further characterizes the turbulent field in absence of shocks associated

with mean flow (there are still shocklets present), showing turbulent kinetic energy

(TKE) and RMS of thermodynamic fluctuations. All these quantities decay in absence

of any forcing.

4.1.2 Taylor Blast Wave

This section describes the evolution of the spherical ’pure’ blast wave, i.e the blast

wave in the absence of turbulence. The Taylor blast wave is launched, in a manner

similar to Ghosh & Mahesh [18] by depositing energy in a small volume around the

center of the domain, such that the local pressure increases to about three orders of

magnitude above that of the ambient field. This increased pressure drives an outward

propagating shock. Barenblatt [2] describes a very fast thermal pulse that precedes

the hydrodynamic shock. The simulations reported here use a time step that is too

large to resolve such fast waves and consequently are not observed here.


0 1 2 3 40.02

0.04

0.06

0.08

0.1

0.12

t/τ0

(1/2)u

′2/u

2 0

(a)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

t/τ0

ρ′2/ρ

2 0,P

′2/ρ

0c2 0,T

′2/T

2 0

(b)

Figure 4.5: Evolution of (a) Turbulent kinetic energy (TKE) and (b) Thermodynamicquantities (density, pressure, temperature) for pure turbulence (without shocks) inDNS mode on a 2563 grid. Legend for plot (b): Density (black-solid), Pressure(blue-solid), Temperature (red-solid). Pressure and Temperature offset by 0.2 and0.4 respectively.

ρ = ρ0

u = 0

ρe =P0

γ − 1+ ρ∆e0

e−(r2/ǫ2)

ǫ3

where ρ0 = 1, P0 = 0.71463, ∆e0 = 15.28415 and ǫ = 0.1. The parameter

∆e0 controls the energy added, and hence, the strength of the resulting shock. The

impulse is only added in the energy equation, the mass and momentum equations are

unaffected. The large energy deposition acts via pressure, to change momenta, and

through momenta, affects the mass distribution.

Figure 4.6 shows the wave diagram for a Taylor blast wave. This was obtained by

plotting the trace along an axis of the quantity 10log|∇ρ| for several time steps. This

is similar to Lombardini [48] who also uses the logarithm of the magnitude of the

4.1. BASE FLOWS 61

x/lo

t/τ 0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.6: Density schlieren wave diagram for expanding pure shock. Quantityplotted is 10log|∇ρ|

density gradient to generate wave diagrams for 1D simulations of planar, cylindrical

and spherical converging shock waves with material interfaces. The diagram shows

a single shock wave propagating outwards. Since the wave diagram is based on the

density profile, which, unlike pressure, is intially homogeneous, it takes approximately

t = 0.025τ0 to attain its maximum post shock value, after which it decays. Hence

the wave diagram appears to follow a different power law behavior before and after

this time. Eventually, at far enough distances, the shock weakens to an acoustic wave

and Taylor’s power law scaling no longer holds. However, in the Taylor blast case

described in this chapter, the domain is small enough that the shock remains strong

throughout the simulation and strong shock conditions are valid.

Taylor [84] first derived the equation for the propagation of a strong shock in a

quiescent medium,

Rs(t) = S(γ)ρ−1/50 E1/5t2/5 (4.5)

where Rs is the shock radius at any time, ρ0 is the density of the undisturbed fluid,

E is the energy deposited at initial time and S(γ) is a function of γ.


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

t/τ0

Rs/l 0

(a)

10−4

10−3

10−2

10−1

10−1

100

t/τ0

Rs/l 0

(b)

Figure 4.7: Comparison of shock radius for expanding pure shock with Taylor’s 2/5thpower law scaling (best-fit with respect to S(γ)). (a) Linear plot (b) Log plot. (×- symbol) - Best fit of a 2/5th power law curve. (o - symbol) - Radius from 2563

simulation.

The value of artificial bulk viscosity β is used to find the shock radius as fol-

lows. The region where β is greater than a threshold is identified as the shell that

encompasses the shock. This shell is bound by radii Rin and Rout.

Rin = minr : β(r) ≥ 0.1βmax (4.6)

Rout = maxr : β(r) ≥ 0.1βmax (4.7)

where βmax is the maximum value of artificial bulk viscosity in the domain. The

centroid of all points that define the shell Rin ≤ Ri ≤ Rout, is then used to calculate

the shock radius.

Rs =1

N

i=N∑

i=1

Ri (4.8)

where Rs is the shock radius at any time and N is the number of points in the shell.

Figure 4.7 compares the radius obtained from a 3D pure shock simulation with

4.2. SPHERICAL SHOCK-TURBULENCE INTERACTION 63

Shock αtheoretical αfit

Taylor blast wave 0.400 0.405

Table 4.2: Comparison of theoretical vs. best fit value of the exponent for pureTaylor blast wave flow. The theoretical values are Taylor’s exponent for the Taylorblast wave.

Taylor’s theory. The theoretical curve is plotted with an exponent of 2/5 and a best-

fit value of the pre-factor S(γ). The log plot should be a straight line according to

theory and plotting the data from theory and simulation on a log-log scale gives a

good idea of the departure from 2/5th power law behavior. As seen in the figure,

the agreement is good, except at early times, while the shock is still in the formation

phase. This phase is marked by a vertical dotted line in the plots. Table 4.2, which

compares the theoretical value of the shock exponent with the best fit value obtained

from simulation data, shows good agreement. Data points after the formation phase

are considered for the comparison.

4.2 Spherical shock-turbulence interaction

We now superimpose the turbulent fluctuations on the initial field for the blast wave

to generate a spherical shock propagating through a field of isotropic turbulence. Ta-

ble 4.3 shows details of the cases presented here. The Taylor blast wave starts out at

a very high Mach number, but decays rapidly as it propagates outwards. A weaker

blast wave is also considered, in which the starting Mach number is much lower. This

shock also decays rapidly and soon attains a Mach number comparable to the back-

ground turbulent Mach number. In both cases the starting state of the turbulence

field is the same, the Taylor scale Reynolds number Reλ is 45 and the turbulent Mach

number Mt is 0.4 as seen in table 4.1. In all cases, the shock is captured by 4 grid

points, a characteristic of the artificial viscosity method employed here (see Cook

[12]), giving a shock thickness ls = 3∆ and a shock thickness to turbulence length

scale ratio ls/l0 of 0.05, i.e the shock length scale is much smaller than turbulence.

All shocks in this chapter including eddy shocklets in the isotropic turbulence field


are captured and not resolved.

A shock time scale is computed based on the time it takes for its pressure ratio

to decay to 1/2 its value and this is compared with the turbulence time scale τ0. For

the strong Taylor blast wave, this ratio is very small, indicating that the turbulence

is almost “frozen” as the shock propagates through it. The weak Taylor blast wave

has a time scale comparable to the background turbulence, so the frozen turbulence

assumption is only partly valid. These estimates of the shock time scale have been

computed in the early stages of the shock, and τs would naturally change as the

shocks propagate. The purpose for including these here is to give an order of mag-

nitude estimate of the range of time scales involved in the problem. The decay rate

of the Taylor blast wave decreases as it propagates outwards, but for the duration

considered in this chapter, it retains its order of magnitude. This duration is shown

in column 6 of table 4.3. In all cases, it is less than τ0.

The last column in table 4.3 compares the grid resolution for all the cases. In

order to do a DNS, one needs to resolve the finest viscous scales, the Kolmogorov

scales, given by η = (ν3/ǫ)1/4, where ν is the kinematic viscosity and ǫ = 2ν〈sijsij〉 isthe kinetic energy dissipation rate. This results in the condition kmaxη ≥ 1.5 (Pope

[64]), where kmax is the highest resolved waveumber on a given grid. The Kolmogorov

scale, as defined above, initially decreases behind the shock. The expansion region

behind causes η to eventually rise. As seen in table 4.3, these scales are slightly un-

derresolved in our 2563 simulations. However, for the finest grid simulations at 5123

the Kolmogorov scale is well resolved. We include a grid-resolution study to establish

that the numerical results for turbulence statistics are essentially grid-independent.

The weak Taylor blast wave is similar in behavior to the strong Taylor blast wave,

except that the magnitudes of changes in various flow quantities are much smaller.

Hence, these are only tabulated in table 4.4 while detailed plots are only shown for

the strong shock case. The weak shock however, shows interesting behavior in the

context of the time evolution of shock properties and is discussed in section 4.9.


Case Grid Ms Ms/Mt ls/l0 τs/τ0 (tfinal − tshock)/τ0 kmaxη

M60G128 1283 60 150 0.1 0.01 0.2 0.5M60G256 2563 60 150 0.05 0.01 0.2 1.28M16G256 2563 16 40 0.05 0.17 0.5 1.2M7G256 2563 7 12 0.05 0.32 0.8 1.02M60G512 5123 60 150 0.025 0.01 0.2 2.51M7G512 5123 7 12 0.025 0.32 0.8 2.01

Table 4.3: Comparison of initial conditions for Taylor blast wave shock-turbulenceinteraction cases.

4.2.1 Tangentially averaged radial profiles

Here we describe the methodology followed to obtain the radial profile plots that are

presented in the following sections and represented by the 〈.〉 symbol. The averages

are taken in both tangential directions and calculated as functions of the radius. A

binning procedure is adopted, whereby data values within a shell are averaged and

taken as representative of the average within each radial shell.

f(rN) =i=N+1∑

i=N

(∑

θ

∑

φ

f(ri))

(4.9)

rN =

√32∆, N = 1

rN−1 + 2√3∆ 2 ≤ N ≤ 4

rN−1 +√3∆, N > 4

(4.10)

where ∆ is the grid spacing. A few caveats need to be kept in mind while analysing

these profiles. The thickness of the shock as suggested by the radial profiles is not

indicative of its true numerical thickness, which is much smaller than is suggested by

these profiles. This inaccuracy is due to finite bin size which is slightly larger than the

grid spacing. Also, when the shock is corrugated by interaction with the turbulence,

it will appear thicker in an averaged profile.

Another trend seen in the tangentially averaged variance data is a dip at low radii

even when no shock is present. This is due to the fact that the peak wavenumber for


t/τ0 Ms 〈ρ〉ps/ρ0 〈ω′2〉ps/ω′20 〈η〉ps/η0 Ret

Strong 0.02 7.0 0.12 0.004 33.8 0.12shock 0.06 3.6 0.09 0.006 22.7 0.09

0.18 2.1 0.07 0.006 23.6 0.07

Weak 0.08 1.4 0.33 0.04 1.8 0.2shock 0.3 1.2 0.41 0.07 3.5 1.5

0.6 1.0 0.65 0.11 2.6 6.2

Table 4.4: Evolution of flow parameters shock with time for the strong (caseM60G512) and weak (case M7G512) Taylor blast wave-turbulence interaction. Sub-script ‘ ps’ stands for post-shock values and are computed at r = Rs − 0.3l0 behindthe shock. Turbulent Reynolds number is defined as Ret = 〈urms〉psl0/〈ν〉ps. Retbefore the shock is 140.

all turbulence spectra k0 has a value of 4. This means that for spatial scales close

to this scale, the field is somewhat correlated. This in turn implies that, for bins

with a small radius, the deviation from the local tangential mean is smaller compared

to larger shells which use data spread over larger length scales. This hypothesis was

confirmed by increasing k0 to a much higher value of 32 which gave almost completely

flat profiles for variance at all radii. This confirms that the observed trend has valid

physical basis and is not a post processing artifact.

4.2.2 Interaction of turbulence with a Taylor blast wave

Table 4.4 provides a quick glance at the evolution of the various turbulence and shock

parameters for strong and weak Taylor blast wave-turbulence interaction cases. One

interesting quantity is the turbulent Reynolds number, which, for the strong shock

case decreases to almost zero behind the shock, having a value of Ret = 140 in pre-

shock turbulence. This is on account of the expansion wave which greatly attenuates

velocity fluctuations in both radial and tangential directions. The weak shock case

shows a similar qualitative behavior, except that the jump magnitudes are much

smaller than the strong shock case. Other quantities are discussed in greater detail

in the following sections.


0 0.5 1 1.5 20

10

20

30

40

50

60

r/l0

〈p〉/p

0

(a)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

r/l0〈ρ〉/ρ

0

(b)

0 0.5 1 1.5 2−1

0

1

2

3

4

5

6

7

r/l0

〈U

r〉/u

0

(c)

Figure 4.8: (a) Pressure (b) Density (c) Radial velocity profiles for strong Taylorblast wave (case M60G512). t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18(dash-dotted).

4.2.3 Mean flow fields

Figure 4.8 shows tangentially averaged radial profiles of pressure, density and radial

velocities for the strong Taylor blast wave. An excellent discussion on flow resulting

from intense explosions can be found in Thompson [85]. intThese are similar to a

quiescent pure shock simulation (i.e without turbulence). The post shock region is

the one of interest. Both pressure and density decrease, as material is forced out

by the flow behind the shock wave. The radial velocity gradually decreases behind

the shock, going to zero at the origin. As this velocity pushes the fluid away from

the center, the density there decreases to almost zero. The pressure also decreases,

but settles down to about 40% of the peak value (Bach and Lee [1]). Almost all the

added energy is carried away by the flow generated by the shock wave to increasingly

large distances. Figure 4.8 shows the shock intensification and propagation stages.

The post-shock density initially increases, as the shock wave attains its full intensity

during the formation phase, then decreases as it enters the propagation phase. These

formation and propagation phases are described in detail in Ghosh and Mahesh [18].


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

r/l0

〈ω

′2

r〉τ

2 0

(a)

0 0.5 1 1.5 20

2

4

6

8

10

12

14

r/l0

〈ω′2 t〉τ

2 0

(b)

Figure 4.9: Comparison of tangentially averaged radial profiles from 1283, 2563 and5123 simulations (cases M60G128, M60G256 and M60G512) at t/τ0 = 0.04 to showconvergence. (a) Radial vorticity and (b) Tangential vorticity. 1283 (+ -symbol);2563 (× -symbol); 5123 (∗ -symbol)

4.2.4 Numerical convergence

We conduct simulations at different resolutions to establish numerical convergence.

This is shown for tangentially averaged radial and tangential vorticity fields. There

are two reasons for choosing vorticity as the quantity for establishing convergence.

First, vortical disturbances are the defining characteristic of any turbulent flow and

hence the most important diagnostic. Second, it is a derived quantity and is therefore

more sensitive to changes in grid size. Note that pre-shock and post-shock fields are

converged at higher resolutions. Flow fields at the shock location cannot converge

because shock capturing is necessarily first order for any numerical scheme. Because

of shock corrugations, this lack of convergence is seen at radial locations affected by

shock corrugations.

Figure 4.10 shows probability density functions (PDF) for density, turbulent ki-

netic energy and vorticity variance magnitude for the strong blast case at different

resolutions. Details regarding the method used for generating these plots can be

found in chapter 6 on spherical Richtmyer-Meshkov instability simulations. Good


10−1

100

10−4

10−2

100

ρ

P(ρ)

(a)

101

102

103

10−8

10−6

10−4

10−2

ΩP

(Ω

)

(b)

100

101

10−6

10−4

10−2

100

k

P(k)

(c)

Figure 4.10: Probability density function (PDF) for strong Taylor blast wave from1283, 2563 and 5123 simulations (cases M60G128, M60G256 and M60G512) at t/τ0 =0.04 to show convergence. (a) Density (b) Vorticity magnitude (c) Turbulent kineticenergy. 1283 (+ -symbol); 2563 (× -symbol); 5123 (∗ -symbol)

convergence can be observed between the two highest resolution cases. This gives

further confidence in the results discussed in this chapter.

4.2.5 Kolmogorov scales

In the paper by Larsson and Lele [40], the authors conduct a study of planar shock-

turbulence interaction and show that downstream of the shock, the Kolmogorov scale

decreases on account of the shock compressing the turbulence and decreasing the

diameter of the smallest eddies. In our study, we would expect that the Taylor blast

wave should increase the size of smallest eddies. The length scale of these eddies,

which represent the smallest scales in the flow and where the kinetic energy of the

flow dissipates is given by

η =

(〈ν〉3〈ǫ〉

) 1

4

(4.11)

where 〈ǫ〉 = 2〈ν〉〈sijsij〉 is the tangentially averaged dissipation and ν is the kinematic

viscosity.

Figure 4.11 shows the Kolmogorov scales for the strong Taylor blast wave case.


0 0.5 1 1.5 210

−3

10−2

10−1

100

r/l0

〈η〉/l 0

(a)

0 0.5 1 1.5 2

100

101

102

103

r/l0

〈ν〉/ν

0

(b)

0 0.5 1 1.5 2

100

101

102

103

104

r/l0

〈ǫ〉/ǫ0

(c)

Figure 4.11: (a) Kolmogorov scale η, (b) Kinematic viscosity ν (c) Dissipation rate ǫfor strong Taylor blast wave (case M60G512). ν and ǫ are plotted in log scale. t/τ0 = 0(dotted), t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted).

The blast wave first decreases the Kolmogorov scales immediately behind the shock

due the shock-induced compression, whereas further downstream, in the expansion

region, it is seen to increase them. To understand the behavior in the expansion

region, we plot the kinematic viscosity ν and the dissipation rate ǫ. Both ν and ǫ

increase by around 2 orders of magnitude upon shock passage. The increase in ν is

due to an increase in temperature in the post-shock zone, which increases the dynamic

viscosity µ, and a decrease in density ρ which further contributes to an increase in ν,

since ν = µ/ρ. In the expression for η, ν is raised to 3/4th power whereas ǫ is raised

to 1/4th power. The kinematic viscosity is thus seen to dictate the behavior of the

Kolmogorov scale, which increases downstream of the shock.

4.2.6 Vorticity

Table 5.5 shows at a quick glance, the quantities plotted in figure 5.10 and later

in the chapter. Both tangential and radial vorticity variance components decrease

considerably in the post-shock region. This effect is more pronounced for the strong

blast wave, where the attenuation is much higher (around 0.5% of pre-shock value)

than for the weak one. These observations are now discussed in more detail.


t/τ0 〈ω′2r 〉ps/ω

′20 〈ω′2

t 〉s/ω′20 (〈ω′2

t 〉/〈ω′2r 〉)ps

Strong 0.02 0.034 0.033 1.1Blast 0.06 0.015 0.008 1.2

0.18 0.004 0.002 1.3

Weak 0.08 0.11 0.09 0.8Blast 0.23 0.09 0.12 0.9

0.31 0.14 0.08 0.9

Table 4.5: Evolution of vorticity variance components and anistropy with time forblast wave-turbulence interaction (Cases M60G256 and M7G256). The subscript ‘ps’ stands for quantities evaluated downstream of the shock.

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

r/l0

〈ω

′2

r〉τ

2 0

(a)

0 0.5 1 1.5 20

5

10

15

20

r/l0

〈ω

′2

t〉τ

2 0

(b)

0 0.5 1 1.5 20

5

10

15

20

25

r/l0

〈ω

′2

t〉/〈ω

′2

r〉

(c)

Figure 4.12: Evolution of (a) Radial vorticity (b) Tangential vorticity and (c) Vorticityanisotropy for the strong Taylor blast wave (Case M60G512). t/τ0 = 0 (dotted),t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted)


0 0.5 1 1.5 2 2.5 3−40

−20

0

20

40

60

80

100

120

r/l0

(a)

0 0.5 1 1.5 2 2.5 310

−4

10−2

100

102

r/l0

(b)

Figure 4.13: Vorticity sources and sinks for the strong Taylor blast wave (CaseM60G512) at t/τ0 = 0.06. (a) Linear plot (b) Log plot. Absolute values are usedfor the log plot. Baroclinic generation: (∇p × ∇ρ)τ 20 /ρ2 ( - symbol), Vorticitydilatation: −ω(∇ · u)τ 20 (♦ - symbol)

The evolution equation for vorticity in a compressible flow is given by

∂ω

∂t= −(u ·∇)ω

︸︷︷︸

convection

+ (ω ·∇)u︸︷︷︸

stretching/tilting

− ω(∇ · u)︸︷︷︸

vorticity−dilatation

+∇p×∇ρ

ρ2︸︷︷︸

Baroclinic

+1

Reref

(

∇×(1

ρ∇ · τ

))

︸︷︷︸

V iscous

(4.12)

The two key terms for this problem are the vorticity-dilatation term and the baro-

clinic generation term. The vorticity-dilatation term only appears in the compressible

version of the equation. It acts as a sink for vorticity in regions of positive dilatation

and as a source in regions of negative dilatation. The attenuation of vorticity seen

in the Taylor blast wave case is due almost entirely to this term. The other critical

term for this problem is the familiar baroclinic generation term.

Figure 4.12 shows profiles of radial and tangential vorticity variances for the strong

Taylor blast wave case. At the shock, the radial vorticity is initially unaffected but


eventually decreases smoothly behind the shock due to the expansion wave as dis-

cussed above. The tangential vorticity undergoes a sharp increase at the shock and

then decreases immediately behind it. However, both components come into equi-

librium and there is a return to isotropy at about r ∼ Rs − 0.5l0 behind the shock.

So the total vorticity is sharply attenuated by the flow following the Taylor blast

wave. This is in contrast with planar shock-turbulence interaction, where vorticity

is amplified after passage through the shock as was shown by Larsson and Lele [40].

To understand this behavior, we plot the source and sink terms in figure 4.13. The

baroclinic generation term in the background is about four orders of magnitude lower

than at the shock. In this problem, the primary pressure and density gradients in

the flow are due to the shock and are more or less aligned, even in the presence of

background turbulence. Consequently, at their peak near the shock, the baroclinic

generation term is much smaller than the vorticity-dilatation term, (by about one

order of magnitude, as can be seen from the log plot). The post-shock region is one

of large positive dilatation, which contributes as a sink of vorticity.

4.2.7 Particle statistics

Three quantities are tracked along particle trajectories, density ρ, vorticity ω and po-

tential vorticity ω/ρ. Three phases can be identified in the particle statistics for the

strong Taylor blast wave. The first phase is the pre-shock phase, where the particles

are influenced only by the background turbulence and the shock has not yet reached

them. The second phase is when they are compressed as the shock passes through

them. The third phase is the post-shock phase when the particles are in the expan-

sion region generated by the shock behind it. Distribution functions of particle fields

are plotted in figure 4.14. The three stages can be clearly demarcated in the Taylor

blast wave case. The density starts out at its average value for isotropic turbulence,

increases when the shock passes through it, and then decreases as the expansion takes

over.


(a) (b) (c)

Figure 4.14: Distribution functions of (a) Particle density, (b) Vorticity and (c) Po-tential vorticity for the strong Taylor blast wave (Case M60G256). t/τ0 = 0 (solid),t/τ0 = 0.02 (dashed), t/τ0 = 0.2 (dash-dotted)

Vorticity also shows the effect of shock compression as it increases from its pre-

shock value and then decreases below it in the expansion zone.

As further evidence of the role of the vorticity-dilataion in vorticity evolution, we

plot potential vorticity which is simply ω/ρ and evolves according to

D

Dt

(ω

ρ

)

=1

ρ((ω ·∇)u+∇T ×∇s) (4.13)

The Lagrangian evolution equation for potential vorticity eliminates the vorticity-

dilatation source/sink term from the vorticity evolution equation. If the decreased

vorticity observed in the post-shock region is entirely due to this term, the potential

vorticity should be almost unchanged in the pre-shock and post-shock environments.

This is in fact, what is observed as can be seen in the third plot of figure 4.14. The

Lagrangian statistics thus reaffirm that the decrease in vorticity observed for a Taylor

blast wave through a region of turbulence is primarily due to the vorticity-dilatation

term in the vorticity evolution equation.


t/τ0 〈u′2r 〉ps/u

′20 〈u′2

t 〉s/u′20 (〈u′2

t 〉/〈u′2r 〉)ps

Strong 0.02 0.675 0.627 1.01Blast 0.06 0.296 0.319 1.03

0.18 0.057 0.041 1.04

Weak 0.08 0.16 0.22 0.7Blast 0.23 0.15 0.21 0.8

0.31 0.16 0.19 0.9

Table 4.6: Evolution of velocity variance components and anistropy with time forblast wave-turbulence interaction (Cases M60G256 and M7G256). The subscript ‘ps’ stands for quantities evaluated downstream of the shock.

4.2.8 Velocity variances

Table 4.6 shows at a quick glance, quantities plotted in figures 4.15 and 4.16. Similar

to vorticity variances, velocity variances are also attenuated downstream of the shock,

but by a smaller degree. The velocity variances are plotted in figures 4.15 and 4.16.

As expected, much larger jumps are seen in the strong shock case as compared to

the weak shock one. In contrast to vorticity, the tangential component of velocity

variance decreases smoothly at the shock, whereas the radial component first shows

a sharp increase at the shock and then a sharp decline. This behavior is in line with

planar shock-turbulence interaction as seen in Lee et. al. [42, 43] and Larsson &

Lele [40]. The difference in time scales between the two shocks is also apparent, as in

figure 4.15, the turbulence can be considered stationary as the shock passes through,

whereas in figure 4.16, the decay is clearly visible.

4.2.9 Thermodynamic fluctuations

Figure 5.14 plots radial profiles of variances of pressure and density for the strong blast

wave-turbulence interaction case. The scale on the y-axis was taken to be logarithmic

due to the large magnitude of the change in pre-shock, shock and post-shock scales.

The fluctuations rise at the shock location and then decay behind it. This behavior is

similar to velocity and vorticity, in line with the attenuation of turbulent fluctuations

in the post-shock expansion zone. These plots show that attenuation of turbulent


0 0.5 1 1.5 20

0.5

1

1.5

2

r/l0

〈u

′2 r〉/u

2 0

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

r/l0

〈u

′2 t〉/u

2 0

0 0.5 1 1.5 20

5

10

15

20

25

r/l0

〈u

′2 r〉/〈u

′2 t〉

Figure 4.15: Evolution of radial and tangential velocity variances and Reynolds stressanisotropy for strong shock case M60G512. t/τ0 = 0 (dotted), t/τ0 = 0.02 (solid),t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r/l0

〈u

′2 r〉/u

2 0

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

r/l0

〈u

′2 t〉/u

2 0

0 0.5 1 1.5 20

1

2

3

4

5

6

r/l0

〈u

′2 r〉/〈u

′2 t〉

Figure 4.16: Evolution of radial and tangential velocity variances and Reynolds stressanisotropy for weak shock case M7G512. t/τ0 = 0 (dotted), t/τ0 = 0.08 (solid),t/τ0 = 0.3 (dashed), t/τ0 = 0.6 (dash-dotted)


0.2 0.4 0.6 0.8 1

10−2

100

102

r/l0

〈P′2〉/

P2 0

(a)

0.2 0.4 0.6 0.8 1

10−6

10−4

10−2

r/l0

〈ρ′2〉/ρ

2 0

(b)

Figure 4.17: (a) Pressure and (b) Density RMS plots for strong blast wave caseM60G256

fluctuations is, in a sense, across the board i.e it is seen in all turbulent quantities.

4.2.10 Instantaneous profiles

Figures 4.18 and 4.19 show instantaneous traces of pressure, density and radial ve-

locity for the strong and weak blast wave-turbulence interaction cases. There are

discernable differences compared to the averaged radial profiles shown in figure 4.8.

The shock is much thinner in the instantaneous profiles than the averaged ones. A

characteristic of artificial viscosity appears to be that shocks are captured by approx-

imately 4 grid points. This difference in thickness is indicative of the distortion in

shock structure caused by turbulence. Another feature visible in the instantaneous

plots are turbulent fluctuations, which are smoothed out in the averaged profiles. The

effect of shock Mach number, a function of the initial strength of the blast wave is

also visible on comparing figures 4.18 and 4.19. The weaker shock is highly distorted

and “broken” at late times, as also seen later in figures 4.22 and 4.25.


−2 −1 0 1 20

10

20

30

40

50

60

70

x/l0

P/P

0

(a)

−2 −1 0 1 20

0.5

1

1.5

2

2.5

3

3.5

x/l0

ρ/ρ

0

(b)

−2 −1 0 1 2−2

0

2

4

6

8

x/l0

Ur/u

0

(c)

Figure 4.18: Traces of (a) Pressure (b) Density and (c) Radial velocity for strong blastwave-turbulence interaction case M60G256. t/τ0 = 0 (dotted), t/τ0 = 0.02 (solid),t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted)

−2 −1 0 1 20.5

1

1.5

2

2.5

3

3.5

4

4.5

x/l0

P/P

0

(a)

−2 −1 0 1 20

0.5

1

1.5

2

2.5

x/l0

ρ/ρ

0

(b)

−2 −1 0 1 2−0.5

0

0.5

1

1.5

2

x/l0

Ur/u

0

(c)

Figure 4.19: Traces of (a) Pressure (b) Density and (c) Radial velocity for weak blastwave-turbulence interaction case M7G256. t/τ0 = 0 (dotted), t/τ0 = 0.08 (solid),t/τ0 = 0.3 (dashed), t/τ0 = 0.6 (dash-dotted)


0 0.05 0.1 0.15 0.2 0.2510

−4

10−3

10−2

10−1

100

t/τ0

Pra

tio

(a)

0 0.05 0.1 0.15 0.2 0.251

2

3

4

5

6

7

t/τ0

Rs/l

0

(b)

0 0.05 0.1 0.15 0.210

−5

10−4

10−3

10−2

t/τ0

χ

(c)

Figure 4.20: Evolution of (a) Shock pressure ratio, (b) Shock radius and (c) Shockasphericity, for the strong Taylor blast wave (Case M60G256). Shock with Turbulence(+ symbol), Pure shock (o symbol)

4.2.11 Time evolution of the shock

In this section, we seek to understand the effect of the turbulence on shock evolu-

tion and compare it with the case with no background turbulence. In absence of

turbulence, the shock is nominally spherical. Turbulence distorts this sphericity de-

pending on its relative strength compared to the shock. We define a shock asphericity

parameter χ based on the shock radius Rs as follows

χ =

i=N∑

i=1

√

(Ri − Rs)2

NRs(4.14)

where N is the number of points forming the shell and Ri is the radius of any point

in the shell Rin ≤ Ri ≤ Rout as defined eariler. In addition, the pressure ratio across

the shock is also computed for the cases with and without turbulence. Figure 4.20

shows all these quantities for the expanding strong shock. The shock radius does not

deviate much from the pure shock case, which in turn follows Taylor’s t2/5 scaling for

a strong shock. At early times, the shock completely overwhelmes the turbulence and

is unaffected by it. At late times, however, as it weakens it starts feeling the effects

of the turbulence. The pressure ratio declines more rapidly than in the pure shock


0 0.2 0.4 0.6 0.810

−2

10−1

100

t/τ0

Pra

tio

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

t/τ0

Rs/l

0

(b)

0 0.2 0.4 0.6 0.8 110

−5

10−4

10−3

10−2

t/τ0

χ

(c)

Figure 4.21: Evolution of (a) Shock pressure ratio, (b) Shock radius and (c) Shockasphericity, for the weak Taylor blast wave (Case M7G256). Shock with Turbulence(+ symbol), Pure shock (o symbol)

case. At early times, the asphericity starts out somewhat higher, because of lower

sampling. As the shock progresses through the domain, the sampling improves and

consequently, χ decreases. At this point, for the pure shock, χ continues to decrease,

whereas for the case with turbulence, it is seen to level out, indicating that shock

corrugation is keeping pace with the increasing radius, further indication that the

shock has started to weaken and is being influenced by the turbulence.

4.2.12 Effect of shock strength: Weak Taylor blast wave

Figure 4.21 shows time evolution plots for the weak Taylor blast wave. An interesting

feature to note here is that the shock radius does not follow the 2/5th power scaling

law as the shock is only of moderate to weak strength. Also the deviations from the

corresponding pure shock case are much greater than for the strong shock case. The

pressure ratio decays and the shock distorts rapidly, as seen in the plot for χ. In fact,

as early as t/τ0 = 0.3 it is not even a contiguous sphere. Parts of the shock have

merged into the background turbulence.

This can be clearly seen in figure 4.22, which shows slices taken along the x − y

plane of pressure contours for strong and weak Taylor blast wave cases. The contour


(a) (b)

Figure 4.22: Slices along x − y plane at z = L/2 of pressure contours for (a) Weak(Case M7G256) and (b) Strong (Case M60G256) Taylor blast waves with a linearcolormap. Same mean shock radius in both plots, however contour levels for bothplots are not the same.

levels in both images are not same due to large difference in pressure magnitudes.

They are however, at the same mean shock radius. The strong shock is seen to

maintain a nearly circular profile, whereas the weak shock is highly distorted.

In the weak shock, the background turbulence and the shock are of comparable

strength, as is evident from the turbulence contours which are clearly visible. The

strong shock on the other hand, is much stronger than the background turbulence, as

is evident from the almost uniform looking field outside the shock. A linear colormap

was used for both plots.

Figure 4.23 compares rescaled transverse vorticity profiles between the weak and

strong shock cases as they evolve in time. The horizontal axes have been translated so

that the origin coincides with the shock location and scaled by the shock corrugation

amplitude. The vertical axes have been scaled by the magnitude of the upstream

pre-shock transverse vorticity. The unscaled plots, e.g. figure 4.12, show a widening

trend for the radial profiles for both cases. In the scaled versions the peaks line up,

as would be expected since the horizontal axes are shifted to the shock location, and

the rising portion of the distributions maintains its steepness. This confirms that

the smearing of the profile observed in figure 4.12, etc. is associated with the shock


0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

3

(r − Rs)/R′

s

〈ω

′2

t〉/ω

′2

t,u

(a)

−1 −0.5 0 0.5 1 1.5

x 104

0

1

2

3

4

5

6

7

8

9

(r − Rs)/R′

s

〈ω

′2

t〉/ω

′2

t,u

(b)

Figure 4.23: Transverse vorticity profiles with horizontal axes translated to shockradius and rescaled to shock corrugation amplitude. (a) Weak case (M7G256), t/τ0 =0.08 (solid), t/τ0 = 0.2 (dashed), t/τ0 = 0.3 (dash-dotted) (b) Strong case (M25G256),t/τ0 = 0.02 (solid), t/τ0 = 0.06 (dashed), t/τ0 = 0.18 (dash-dotted)

corrugations. The decrease in vorticity fluctuations from the peak towards smaller

radii is due to the expansion wave and is not expected to be scaled by the shock

corrugation and the profile widening trend is maintained. The data for the weak

blast wave case also illustrates a limitation of shock corrugation based scaling; when

the shock is substantially deformed, a simple statistic of shock corrugation (variance)

is insufficient to scale profiles of other physical quantities.

4.3 Comparison with linear theory

In this section, we compare our results with existing linear theory for planar shock-

turbulence interaction. In particular, we compare the transverse vorticity amplifi-

cation ratio from Lee et al [43] with our Taylor blast wave shock-turbulence cases.

Figure 4.24(a) shows this comparison. For the blast wave cases, we plot amplification

against the shock pressure ratio. The reason for this choice of parameters is that the

4.3. COMPARISON WITH LINEAR THEORY 83

101

101

Shock pressure ratio

Tra

nsv

ers

e V

ort

icity

am

plif

ica

tion

(a)

101

100

101

Shock Pressure ratio

Sh

ock

De

nsi

ty r

atio

(b)

Figure 4.24: Comparison of transverse vorticity amplification ratios with linear theory.(a) Taylor blast wave. Case M7G256 (x symbol), Case M16G256 (o symbol), CaseM60G256 (+ symbol). Solid line in (a) refers to the linear theory result from Lee etal [43] and (b) refers to the Rankine-Hugoniot relation

(a) (b)

Figure 4.25: Contours of dilatation superimposed on slices along x − y plane atz = L/2 of Mach number contours for (a) Weak (Case M7G256) and (b) Strong (CaseM60G256) Taylor blast waves. Plots are at different times and different colormaps,but same mean radius.


simulation data were not saved at sufficient temporal resolution to accurately com-

pute the shock Mach number at a given time. The pressure and density ratios were

taken from the tangentially averaged radial profiles of density and pressure.

A clear trend is discernable. The vorticity amplification shows behavior consistent

with linear theory after the blast wave has sufficiently expanded. The pressure ratio

is moderate in this regime and the shock radius is large enough that the shock front

is almost planar. At high Mach numbers, the amplification is less than that predicted

by (planar shock) linear theory on account of the high curvature of the shock, its 3D

nature and the geometric effect of expansion.

An additional observation can be made regarding the data from the weak blast

wave case. As noted earlier, the background turbulence is strong enough in this case

to severely distort the shock and as figure 4.22 shows the shock surface appears to

have ‘holes’. This is reminiscent of the ‘broken’ shock regime of Larsson and Lele [40].

These holes can be seen in figure 4.25 which plots slices of Mach number con-

tours at nearly same shock radius for strong and weak blast wave cases. The broken

shock can be seen in the weak case, where part of flow around the shock is subsonic

(Ms < 1). The strong shock on the other hand, maintains its integrity, being well

above Mach 1.

Figure 4.24(b) shows a comparison of pressure and its corresponding density ratio

for the planar linear theory and from data taken from spherical blast wave simulations.

The curve for the planar case asymptotes to a value of 6, which is predicted by

the Rankine-Hugoniot conditions for a γ value of 1.4. The blast wave data show

conformance to linear theory at low pressure ratios, associated with larger radii as

the blast wave expands outwards. Curvature effects are less important in that regime.

At high pressure ratios, associated with smaller radii, curvature effects are significant

and the density ratios are lower than predicted by planar theory. Also, curvature

effects appear to be more important for stronger initial shocks, compared to weaker

4.4. CONCLUSIONS 85

ones, since data points for the weak blast wave case lie almost on the planar theory

curve.

4.4 Conclusions

Simulations of a Taylor blast wave through a field of compressible decaying isotropic

turbulence have been carried out. Effect of the shocks on the background turbu-

lence has been quantified primarily through variances of vorticity components, its

anisotropy and evolution of Kolmogorov scales. The Taylor blast wave is found to

increase Kolmogorov scales after a local reduction close to the blast wave and attenu-

ate vorticity behind the shock. It was shown that the Kolmogorov scales are primar-

ily affected by the kinematic viscosity field and vorticity evolution by the vorticity-

dilatation term in the vorticity evolution equation.

Lagrangian tracking of particles was used to provide further corroboration for the

reason behind vorticity attenutation/amplification. The particle statistics also show

that the vorticity-dilatation term in the vorticity evolution equation is primarily re-

sponsible for the vorticity field behavior.

The effects of turbulence on shock structure and evolution are examined by com-

puting the shock pressure ratio and shock radius and comparing them to the pure

shock case. A new parameter for characterising the asphericity of the shock has been

proposed and shown to work quite well. Through a simple rescaling, it is shown that

shock corrugation causes a significant smearing of profiles in the region around the

shock.

Transverse vorticity amplification ratios for the Taylor blast wave shock-turbulence

interaction problem are compared with results from linear theory. In both cases, the

spherical shock-turbulence interaction shows linear behavior when the shock is far

from the origin. When the shock is closer to the origin, geometric effects predomi-

nate, manifesting as lower amplification ratios.

Chapter 5

Interaction of a Converging

Spherical Shock Wave with

Isotropic Turbulence

The goal of this study is to do an analysis of the spherical shock-turbulence interac-

tion problem from a purely hydrodynamic standpoint. We seek to understand how

turbulence is affected by the passage of the shock and how the shock is modified by

the field of turbulence. To that end, we neglect any effects associated with thermo-

chemical non-equilibrium, ionization and dissociation, processes which may arise in

a more realistic model of spherical compression of a gas by a converging shock. We

consider the interaction of the shock launched by an initial pressure and density im-

balance across a spherical surface with isotropic homogeneous turbulence in a triply

periodic domain. This setup is analogous to a spherical Sod shock tube problem. The

traditional shock tube setup results in a shock wave and a contact interface propa-

gating into the low pressure side and an expansion wave propagating into the high

pressure side. In the converging shock regime, the post shock flow is affected by the

presence of the contact surface. Also, when the shock reflects back from the origin,

there ensues a complex interaction between the outward propagating shock and in

the inward propagating contact. Payne [61] first suggested that the contact surface

can be eliminated by appropriately choosing pressure and density jump ratios accross

86

87

Figure 5.1: Schematic of the simulation domain for Converging shock-turbulenceinteraction problem.

the shock tube diaphragm. Starting from the shock tube governing equations, we

have derived pressure and density jump conditions that achieve a theoretically zero

strength contact interface for any desired shock Mach number.

The schematic in figure 6.1 illustrates the various elements of the problem. The

use of a periodic domain would contaminate the solution at late times as the outward

propagating expansion waves from neighboring domains enter the simulation domain.

To overcome this, we use a technique called windowing as formulated by Colonius and

Ran [11]. In this technique, a buffer layer is set up at the boundaries in order to damp

and absorb outgoing waves, while still maintaining periodicity.

This chapter is organised as follows. Section 5.2 discusses the background isotropic

turbulence, and the converging and reflected shocks in absence of turbulence. The

shock properties are compared with the analytical solutions of Guderley. Section 5.3

discusses shock-turbulence interactions for converging and reflected spherical shocks.

88 CHAPTER 5. CONVERGING SHOCK - TURBULENCE INTERACTION

Time Reλ Mt ρrms/〈ρ0〉 tke/tke0 tkedil/tke

t0 100 0.6 0.34 1.0 0.2tshock 45 0.4 0.27 0.7 0.2

Table 5.1: Comparison of turbulence parameters at time of shock launch with thoseat time of turbulence launch.

We look at the effect of the shock on turbulence through radial profiles and Lagrangian

particle statistics and the effect of turbulence on the shocks through time evolution of

shock properties. Section 5.4 offers a comparison of the observed behavior of turbulent

vorticity with the linear interaction analysis (LIA) for a planar shock. Section 5.5

ends the chapter with conclusions from the work.

5.1 Base flows

5.2 Isotropic turbulence

As described in detail in the previous chapter, the field used in these converging shock

simulations is also decaying isotropic turbulence with eddy shocklets. Turbulence has

evolved to a state where the nonlinear energy transfer mechanism becomes fully active

before the shock is launched. The properties of this field have been discussed in detail

in Bhagatwala & Lele [5], Johnsen et al [31] and the chapter on blast wave-turbulence

interaction. The salient features of the turbulent field evolution from initial conditions

to the point of shock launch are reiterated in table 5.1.

Reλ has decreased from 100 to 45 and Mt has decreased to 0.4 from 0.6. Mt is

high enough to form eddy shocklets. k0, the most energetic wavenumber is taken

to be 4. Length scales in this problem have been nondimensionalized by l0, where

l0 = 2π/k0, k0 being the peak wavenumber of the energy spectrum. Time scales

are nondimensionalised by τ0 ∼ 4τη, where τ0 = λ0/u0, λ0 being the Taylor scale

Reynolds number, u0 being the RMS velocity at t0 and τη is the Kolmogorov time

scale.

5.3. SPHERICAL SHOCKS 89

5.3 Spherical shocks

This section describes the evolution of spherical ’pure’ shocks, i.e shocks in the absence

of turbulence. These pure shock cases are computed on the same mesh as the one

used later for shock-turbulence interaction calculations.

5.3.1 Converging shock

The converging shock is launched by increasing the pressure and density in a zone

outside a specific radius Rs0 and letting the flow evolve. This is equivalent to a

spherical shock tube with a spherical diaphragm located at the specific radius. In

a typical shock tube set-up, pressure and density ratios across the diaphragm are

increased by the same factor. In this case, in addition to the shock and expansion

waves, one also obtains a contact surface which propagates in the same direction as

the shock. This contact surface creates a complex series of wave interactions when

the shock wave passes through it on its way out after reflecting off of the origin. The

problem then becomes one of shock-turbulence, shock-contact surface and contact

surface-turbulence interaction. In keeping with our goal of isolating only the shock-

turbulence interaction aspect of the flow, the contact surface is a feature we would

like to avoid.

We therefore choose pressure and density ratios so as to generate an initial shock

wave of desired Mach number Ms0 with no contact surface. With slight modifications

to the analysis in Liepmann and Roshko [45], one can derive the following set of

implicit equations for jump conditions across the diaphragm for a shock with Mach

number Ms0 and no contact surface.

ρ2ρ1

=(γ + 1)M2

s0

(γ − 1)M2s0 + 2

P2

P1= 1 +

2γ

γ + 1(M2

s0 − 1)


ρ4ρ1

=ρ2ρ1

(

1− a1a4

(γ − 1)

(ρ2ρ1

− 1

)√

1

2(ρ2/ρ1)(γ + 1− (γ − 1)(ρ2/ρ1))

)− 2

γ−1

P4

P1=

P2

P1

(

1− a1a4

(γ − 1)

(P2

P1− 1

)√

1

2γ(2γ + (γ + 1)(P2/P1 − 1))

)− 2γ

γ−1

where φ4/φ1 and φ2/φ1 (φ ∈ ρ, P) correspond to diaphragm and shock jump

ratios respectively. The ratio of specific heats γ is the same across the diaphragm

and a is the speed of sound given by√

γP/ρ. The converging shock is then specified

according to the following (smoothened) jump conditions,

ρ = ρ1 +ρ12

(

1 + tanh

(r − Rs0

ǫ′′

))(ρ4ρ1

− 1

)

(5.1)

p = p1 +p12

(

1 + tanh

(r − Rs0

ǫ′′

))(p4p1

− 1

)

(5.2)

u = 0 (5.3)

For the results shown in this chapter we used Rs0 = 0.7(L/2) and ǫ′′

= 0.03.

A schlieren wave diagram for the converging pure shock problem elucidates the

driving flow. This is shown in figure 5.2. At the time of shock launch, one can

observe two fronts, the shock wave propagating radially inwards and the expansion

wave travelling outwards. Note the absence of a contact surface due to appropriately

chosen shock tube parameters. The shock gets progressively stronger as it approaches

the origin at which point, it is at its strongest. It then reflects off of the origin and

propagates outwards. The expansion wave propagates outwards and interacts with

the buffer layer where windowing is applied. This layer absorbs the wave with no

observable reflections or transmission from neighboring periodic replicas. As discussed

in later sections, pressure and density acheive very large values as the converging shock

nears the origin which leads to large gradients. Additionally, the shock is relatively

poorly resolved near the origin compared to the rest of the domain. This results in

higher numerical dissipation, which is seen as a diffuse patch in the schlieren plot

near the origin. As discussed in the section 5.6.1, radial velocity decays linearly to


x/Rs0

t/t s

−1 0 10

0.5

1

1.5

2

10

20

30

40

50

60

Figure 5.2: Density schlieren wave diagram for converging pure shock. Quantityplotted is 10log|∇ρ|. Shock initial Mach number Ms0 = 1.4. ts is time taken forshock to reach the origin, i.e. ts = tRs=0.

zero at the origin in the reflected shock regime.

Guderley [19] first derived the scaling law for converging shocks,

Rs(t) = A(γ)(−t)α (5.4)

where Rs(t) is the radius of the shock, A(γ) is a constant and time is measured

backwards from the point when the shock reaches the origin. The constant α is called

the Guderley exponent and in the strong shock limit, depends only on γ. Interestingly,

Shock αtheoretical αfit α′fit

Converging(inward) 0.7172 0.7179 0.7272Converging(outward) 0.7172 0.7177 0.7342

Table 5.2: Comparison of theoretical vs. best fit value of exponents for convergingpure shock. The theoretical values are the Guderley exponents for a converging shock.αfit represents a two-parameter fit with respect to Rs0 and ts. α

′fit represents a one

parameter fit with respect to Rs0.


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

t/ts

Rs/R

s0

(a)

10−2

10−1

100

10−2

10−1

100

t/ts

Rs/R

s0

(b)

Figure 5.3: Comparison of shock radius for converging pure shock with an assumedbest-fit for Guderley’s power law scaling (best-fit with respect to Rs0). Points belowthe dashed line are excluded when making the fit. (a) Linear plot (b) Log plot. (×- symbol) - Best fit of Guderley’s power law curve. (o - symbol) - Radius from 2883

simulation. Line closer to x− axis in the log plot corresponds to reflected shock whilethe line further from x− axis represents the converging shock.


the same exponent also holds when the shock reflects off of the origin and propagates

outwards. Thus, assuming time is measured from the instant the shock is launched,

we get the following relations for inward and outward propagating shocks.

Rs(t) =

Rs0(1− t/ts)α inward propagating shock

Rs0(t/ts − 1)α outward propagating shock(5.5)

where Rs(t) is the shock radius at any instant, Rs0 is the initial shock radius and

ts is the time taken for the shock to reach the origin. This time is related to τ0 by

ts = 0.53τ0. The value of ts used for computing αfit, where both ts and Rs0 are varied

is ts/τ0 = 0.55 for the reflected shock and ts/τ0 = 0.53 for the converging shock. The

value of Rs0 used for αfit is 0.68(L/2) for the reflected shock and 0.7(L/2) for the

converging shock. The shock radius as a function of time is plotted in figure 5.3 and

compared with theory. The log plot should be a straight line in order to conform

with the assumed best-fit solution. It is seen that, except near the origin, the shock

radius compares well with a best fit curve of the theoretical solution. Note that the

log plot exaggerates the region near the point where the shock is close to the origin

and so the discrepancy is somewhat magnified. Away from the origin, denoted by

the dashed line, the 3D simulation agrees well with the theory. Table 5.2 shows the

comparison of the theoretical value of the Guderley exponent with the best fit value

from simulation. The theoretical curve is plotted with Guderley’s exponent and a

best-fit value of the pre-factor Rs0. The agreement is slightly better for the inward

propagating shock than for the outward propagating one. A two parameter fit with

respect to Rs0 and ts gives a better agreement than a one parameter with respect to

Rs0 alone. Data points very close to the origin, have been excluded for the purpose of

this comparison, as the numerical errors become significant as the shock approaches

a theoretical singularity at the origin, which can only be crudely approximated by

any discretised version of the equations.


5.4 Spherical shock-turbulence interaction

We now add the two components described in the last section to generate a spherical

shock propagating through a field of isotropic turbulence. Table 5.3 shows details

of the cases presented here. The grid sizes include the buffer region needed for the

windowing technique described before. Therefore a 2883 grid is equivalent to a 2563

grid for just the flow part. The converging shock starts out relatively weak, but gains

strength as it propagates inwards. In both cases the starting state of the turbulence

field is the same, the Taylor scale Reynolds number Reλ is 45 and the turbulent Mach

number Mt is 0.4 as seen in table 5.1. In all cases, the shock is captured by 4 grid

points, a characteristic of the artificial viscosity method employed here (see Cook

[12]), giving a shock thickness ls = 3∆ and a shock thickness to turbulence length

scale ratio ls/l0 ∼ 0.05 and ls/η ∼ 3, i.e the shock length scale is much smaller than

energetic scales of turbulence and comparable to the dissipation length scale. All

shocks in this chapter including eddy shocklets in the isotropic turbulence field are

captured, and not resolved.

A shock time scale, τs, is computed based on the time it takes for its pressure

ratio to increase by a factor 2 and this is compared with the turbulence time scale

τ0. For the weakest converging shock, τs/τ0 ∼ 1, while for the strongest, τs/τ0 << 1,

so the assumption of frozen turbulence is only valid for the strong shock case. These

estimates of the shock time scale have been computed in the early stages of the shock,

and τs would naturally change as the shocks propagate. The purpose for including

these here is to give an order of magnitude estimate of the range of time scales in-

volved in the problem.

The last column in table 5.3 compares the grid resolution for all the cases. In

order to do a DNS, one needs to resolve the finest viscous scales, the Kolmogorov

scales, given by η = (ν3/ǫ)1/4, where ν is the kinematic viscosity and ǫ = 2ν〈sijsij〉 isthe kinetic energy dissipation rate. This results in the condition kmaxη ≥ 1.5 (Pope

5.5. TANGENTIALLY AVERAGED RADIAL PROFILES 95

Case Grid Ms0 Ms0/Mt ls/l0 τs/τ0 (tfinal − tshock)/τ0 kmaxη

C1G288 2883 1.4 3.5 0.05 0.68 1.46 1.69C2G288 2883 2.4 6.0 0.05 0.25 0.72 0.85C3G288 2883 4.2 10.5 0.05 0.15 0.38 0.81C2G144 1443 2.4 6.0 0.12 0.25 0.85 0.43C2G576 5763 2.4 6.0 0.02 0.25 0.67 1.63

Table 5.3: Comparison of initial conditions for converging shock-turbulence interac-tion cases. Grid sizes include buffer zone needed for windowing.

[64]), where kmax is the highest resolved waveumber on a given grid. For a converg-

ing shock, the Kolmogorov scale, as defined above, decreases behind the shock and

remains lower than the pre-shock value. As seen in table 5.3, these scales are underre-

solved in one of the simulations, two are borderline and two exceed Pope’s resolution

criteria. In light of these numbers, this study can be considered an exploratory or

partial DNS. To ensure that the resolution in this study is sufficient for reliable statis-

tics, one of the higher Mach number cases was computed at twice the resolution, i.e

on a 5763 grid. The last column shows that this simulation qualifies as DNS based

on the criterion above. The smallest value of η from both sets of calculations are

reported in the table. The difference is only about 2%, which provides confidence

that the dissipation range is sufficiently resolved in the 2883 simulations.

5.5 Tangentially averaged radial profiles

Here we describe the methodology followed to obtain the radial profile plots that are

presented in the following sections and represented by the 〈.〉 symbol. The averages

are taken in both tangential directions and calculated as functions of the radius. A

binning procedure is adopted, whereby data values within a shell are averaged and


taken as representative of the average within each radial shell.

〈f(rN)〉 =i=N+1∑

i=N

(∑

θ

∑

φ

f(ri, θ, φ))

(5.6)

rN =

√32∆, N = 1

rN−1 + 2√3∆ 2 ≤ N ≤ 4

rN−1 +√3∆, N > 4

(5.7)

where ∆ is the grid spacing. A few caveats need to be kept in mind while analysing

these profiles. The thickness of the shock as suggested by the radial profiles is not

indicative of its true numerical thickness, which is much smaller than is suggested by

these profiles. This inaccuracy is due to finite bin size which is slightly larger than the

grid spacing. Also, when the shock is corrugated by interaction with the turbulence,

it will appear thicker in an averaged profile.

Another trend seen in the tangentially averaged variance data is a dip at low radii

even when no shock is present. This is due to the fact that the peak wavenumber for

all turbulence spectra k0 has a value of 4. This means that for sufficiently small rN ,

rN ≤ 1/k0. Consequently, for spatial scales close to this limit, f(r, θ, φ)/〈f(rN)〉 ∼1 and hence 〈f ′2(rN)〉/〈f(rN)〉2 ∼ 0, i.e. the field is somewhat correlated. This

hypothesis was confirmed by increasing k0 to a much higher value of 32, so that

rN > 1/k0 ∀ N which gave almost completely flat profiles for variance at all radii.

This confirms that the observed trend is has valid physical basis and is not a post

processing artifact.

5.6 Converging shock-turbulence interaction

Table 5.4 shows at a quick glance the evolution of the various turbulence and shock

parameters for converging shock-turbulence interaction. Ret increases behind the

shock for the converging shock and decreases for the reflected shock. The compression

behind the shock amplifies the velocity fluctuations and as the shock increases in

5.6. CONVERGING SHOCK-TURBULENCE INTERACTION 97

t/τ0 Ms 〈ρ〉s/ρ0 〈ω′2〉s/ω′20 〈u′2〉s/u20 〈η〉s/η0 Ret

0.04 2.6 2.7 2.4 3.8 0.36 3040Converging 0.16 2.9 3.8 7.4 5.0 0.37 3900

Phase 0.31 3.6 5.2 10.8 5.6 0.32 49500.42 4.8 7.3 23.4 7.6 0.28 6920

0.52 1.8 29.5 6.4 82.5 0.14 14980Reflected 0.60 1.5 23.2 5.1 52.5 0.16 12660Phase 0.65 1.4 20.4 4.0 36.8 0.18 10820

0.71 1.3 18.8 3.1 27.8 0.22 9210

Table 5.4: Evolution of flow parameters with time for converging shock-turbulenceinteraction (Case C2G288). The subscript ‘ s’ stands for shock values and are com-puted at r = Rs(t). Turbulent Reynolds number is defined as Ret = 〈urms〉sl0/〈ν〉s.Ret before the shock is 140.

strength as it propagates inwards, so do the velocity fluctuations behind it. An

interesting observation is that M reflecteds < M converging

s if the shocks are compared at

the same radius. This is because the compression from the converging shock causes

ccompressed > cunshocked, and so even though overall pressure levels are quite high, as

seen in figure 5.4, Mach number of the reflected shock is quite low. Other quantities

in the table are discussed in greater detail in the following sections.

5.6.1 Mean flow fields

The converging shock can be thought of as a spherical analogue of the planar shock

tube problem with a hypothetical diaphragm at a specific radius of Rs0 = 0.7(L/2).

As the diaphragm ruptures, a shock wave forms and propagates towards the low pres-

sure side. The imploding shock wave increases in strength as it moves towards the

origin. This is seen in figure 5.4. The initial conditions set up a radially inward prop-

agating shock wave and an outward propagating expansion wave, which is absorbed

by the buffer layer at the outer edge of the simulation domain. It may be recalled that

these features were seen in the wave diagram in figure 5.2. In the converging phase,

pressure and density increase behind the shock as the shocked fluid is compressed.


0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

250

r/l0

〈p〉/

p0

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

r/l0

〈ρ〉/

ρ0

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/l0

〈S〉

(c)

0.2 0.4 0.6 0.8 1 1.2 1.4−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

r/l0

〈Ur〉/

u0

(d)

Figure 5.4: (a) Pressure, (b) Density and (c) Entropy and (d) Radial velocity pro-files for converging shock (Case C2G288). Converging phase: t/τ0 = 0.04 (black-dotted), t/τ0 = 0.16 (black-solid), t/τ0 = 0.31 (black-dashed), t/τ0 = 0.42 (black-dash/dotted). Reflected phase: t/τ0 = 0.52 (blue-dotted), t/τ0 = 0.60 (blue-solid),t/τ0 = 0.65 (blue-dashed), t/τ0 = 0.71 (blue-dash/dotted).


0.2 0.4 0.6 0.8 1 1.2 1.4−10

−8

−6

−4

−2

0

2

4

6

r/l0

u·∇

s

Figure 5.5: Plot of convective term in entropy equation u · ∇s (Case C2G288). Con-verging phase: t/τ0 = 0.04 (black-dotted), t/τ0 = 0.16 (black-solid), t/τ0 = 0.31(black-dashed), t/τ0 = 0.42 (black-dash/dotted). Reflected phase: t/τ0 = 0.52(blue-dotted), t/τ0 = 0.60 (blue-solid), t/τ0 = 0.65 (blue-dashed), t/τ0 = 0.71 (blue-dash/dotted).

During the reflected phase, they increase just behind the shock, then gradually de-

crease away from the shock in the post-shock expansion zone.

Entropy, defined as S = log(p/ργ) shows an interesting behavior. During the

converging phase, it undergoes a jump at the shock as expected from the Rankine-

Hugonoit conditions. In the expansion phase however, it continues to remain concen-

trated at the origin. This can be explained from the behavior of the entropy jump,

which is much smaller for the expanding shock than the converging one. The entropy

transport equation is given by,

∂S

∂t+ uj

∂S

∂xj= − 1

ρT

∂qj∂xj

︸︷︷︸

Thermal diffusion

+1

ρTΦ

︸︷︷︸

Momentum diffusion

(5.8)

Entropy increases to a very large value at the origin when the converging shock


0 0.5 1 1.5 20

1

2

3

4

5

6

7

r/l0

〈ω

′2

r〉τ

2 0

(a)

0 0.5 1 1.5 20

2

4

6

8

10

12

14

16

r/l0

〈ω

′2

t〉τ

2 0

(b)

Figure 5.6: Comparison of tangentially averaged radial profiles from 1443, 2883 and5763 simulations (cases C2G144, C2G288 and C2G576) at t/τ0 = 0.24 to show con-vergence. (a) Radial vorticity and (b) Tangential vorticity. 1443 (+ -symbol); 2883

(× -symbol); 5763 (∗ -symbol)

is at its strongest. As the shock then expands outwards, it propagates into an al-

ready highly compressed fluid from the converging shock. As a result Ms,reflected <<

Ms,converging, i.e. the Mach number of this shock is much lower than that of the con-

verging shock, as seen in column two of table 5.4. Since entropy jump across a shock

is highly dependent on Mach number, ∆s ∝ (M2s − 1)3, a weaker shock results in

much smaller entropy gradients, which results in the observed behavior. This was

confirmed by plotting uj∂S∂xj

for converging and reflected phases. Indeed, the pro-

files of this quantity were found to be flatter for the reflected phase compared to the

converging phase as seen in figure 5.5.


We conduct simulations at different resolutions to establish numerical convergence.

This is shown for tangentially averaged radial and tangential vorticity fields in figure

5.10. There are two reasons for choosing vorticity as the quantity for establishing con-

vergence. First, vortical disturbances are the defining characteristic of any turbulent


100

101

10−5

10−4

10−3

10−2

10−1

ρ

P(ρ)

(a)

102

103

10−10

10−8

10−6

10−4

ΩP

(Ω

)

(b)

10−1

100

101

10−6

10−4

10−2

100

k

P(k)

(c)

Figure 5.7: Probability density function (PDF) for converging shock-turbulence in-teraction from 1443, 2883 and 5763 simulations at t/τ0 = 0.24 to show convergence.(a) Density (b) Vorticity magnitude (c) Turbulent kinetic energy. 1443 (+ -symbol);2883 (× -symbol); 5763 (∗ -symbol)

flow and hence the most important diagnostic. Second, it is a derived quantity and is

therefore more sensitive to changes in grid size. Note that pre-shock and post-shock

fields show trend towards numerical convergence at higher resolutions. Flow fields at

the shock location cannot converge because shock capturing is necessarily first order

for any numerical scheme. Because of shock corrugations, this lack of convergence is

also seen at radial locations affected by shock corrugations.

Figure 5.7 shows probability density functions (PDF) for density, turbulent kinetic

energy and vorticity variance magnitude for the converging shock interacting with

isotropic turbulence at different resolutions. Details regarding the generation of these

plots can be found in the following chapter on spherical Richtmyer-Meshkov instability

simulations. All three quantities show good convergence between the 2883 and 5763

cases, giving us confidence in the results presented in the remainder of the chapter.

5.6.3 Kolmogorov scales

The η profiles for the converging shock, defined as


0.2 0.4 0.6 0.8 1 1.2 1.40

0.005

0.01

0.015

0.02

0.025

r/l0

〈η〉/l 0

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5x 10

−3

r/l0

〈ν〉/ν

0

(b)

0.2 0.4 0.6 0.8 1 1.2 1.410

−1

100

101

102

r/l0

〈ǫ〉/ǫ0

(c)

Figure 5.8: (a) Kolmogorov scale η, (b) Kinematic viscosity ν and (c) Dissipationrate ǫ for converging shock (Case C2G288). ν and ǫ are plotted in log scale. Forlegend please refer to figure 5.4.

〈η〉 =(〈ν〉3

〈ǫ〉

)1/4

(5.9)

are shown in figure 5.8(a). As discussed in Larsson & Lele [40], compression of the

smallest fluid eddies by the shock decreases their diameter, consequently decreasing

η. To understand this behavior, we plot the kinematic viscosity ν, defined as ν = µ/ρ

and dissipation rate ǫ, defined as 〈ǫ〉 = 2〈ν〉〈sijsij〉. The dissipation rate increases

downstream of the shock, but the kinematic viscosity decreases. The temperature

increases behind the shock as we would expect, but so does the density. As a result,

µ, defined as µ = T n according to the power law definition increases behind the shock,

but ν decreases due to increasing ρ. Across the shock, ν decreases and ǫ increases.

The increase in ǫ can be attributed to the amplification of velocity fluctuations as seen

in section 4.3.4 and since ǫ ∼ u′3/L. Both contribute to a decrease in η. However,

the dependence on ν is stronger than that on ǫ due that component raised to a higher

power in the definition of η. As a result, we observe that the η profile closely mimics

the profile of ν for both converging and reflected phases of the interaction. It decreases

immediately behind the shock, and increases as the post-shock flow gradually expands.

The overall effect of the converging and reflected shocks is a decrease in size of the


(a) (b)

Figure 5.9: Slices of total vorticity superimposed on dilatation contours showing shocklocation at mid plane for (a) Converging shock at t/τ0 = 0, (b) Reflected shock att/τ0 = 0.9 for case C2G288.

eddies, as can be seen in figure 5.9, which plots slices of vorticity magnitude along the

midplane. Plot 5.9(a) shows the vorticity field before shock launch and plot 5.9(b)

shows it at a late time, t/τ0 = 0.9 when Rs/Rs0 ∼ 1, i.e the reflected shock is almost

out of the domain. Smaller scale structures in the post reflection plot can be noted.

5.6.4 Vorticity

Table 5.5 shows at a quick glance, the quantities plotted in figure 5.10. Both radial and

tangential components of vorticity are seen to increase post-shock, however, similar to

the behavior of vorticity in planar shock-turbulence interaction, the radial component

increases gradually, whereas the tangential component jumps sharply at the shock

and then decays rapidly. This is seen in the column listing the anisotropy, where

(〈ω′2t 〉/〈ω

′2r 〉)s 6= 〈ω′2

t 〉s/〈ω′2r 〉ps, i.e 〈ω

′2r 〉ps is evaluated some distance behind the shock,

whereas 〈ω′2t 〉s is evaluated at the mean shock location. During the converging phase,

the vorticity field retains the anisotropy acquired during shock passage, on account of

the fluid being in a compressed state following the shock. During the reflected phase,

the post-shock field is seen to return back to isotropy as the fluid expands behind the


t/τ0 〈ω′2r 〉ps/ω

′20 〈ω′2

t 〉s/ω′20 (〈ω′2

t 〉/〈ω′2r 〉)s

0.04 1.5 4.6 4.2Converging 0.16 2.7 7.5 4.1

Phase 0.31 7.3 10.8 3.90.42 17.7 24.3 5.2

0.52 68.4 79.7 1.3Reflected 0.60 40.1 52.1 1.5Phase 0.65 25.8 34.9 1.4

0.71 21.8 22.7 1.3

Table 5.5: Evolution of vorticity components and anistropy with time for convergingshock-turbulence interaction (Case C2G288). The subscript ‘ s’ stands for shockvalues and are computed at r = Rs(t) and ‘ ps’ stands for quantities evaluateddownstream of the shock where they attain their maximum value.

shock.

The evolution equation for vorticity in a compressible flow, described in the pre-

vious chapter, but repeated here for convenience

∂ω

∂t= −(u ·∇)ω

︸︷︷︸

convection

+ (ω ·∇)u︸︷︷︸

stretching/tilting

− ω(∇ · u)︸︷︷︸


+∇p×∇ρ

ρ2︸︷︷︸

Baroclinic

+1

Re

(

∇×(1

ρ∇ · τ

))

︸︷︷︸

V iscous

(5.10)

As also seen in the previous chapter, the key terms for this problem are the

vorticity-dilatation term and the baroclinic generation term. In case of the Taylor

blast wave, vorticity-dilatation acts as a sink and baroclinic generation acts as a

source. In the converging phase in this problem, both vorticity-dilatation and baro-

clinic generation act as sources. In the reflected phase, the former is a sink while the

latter is a source. Figure 5.10 shows vorticity plots for the converging shock case.

Indeed, as the converging shock gets stronger, it amplifies the vorticity more as can

be seen in the larger jumps in both its radial and tangential components.

An interesting contrast with the Taylor Blast wave-turbulence interaction prob-

lem is during the expansion phase of the shock. In the former problem, post-shock


0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

120

140

r/l0

〈ω

′2

r〉τ

2 0

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

120

r/l0〈ω

′2

t〉τ

2 0

(b)

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

r/l0

〈ω

′2

t〉/〈ω

′2

r〉

(c)

Figure 5.10: Evolution of (a) Radial vorticity, (b) Tangential vorticity and (c) Vor-ticity anisotropy for the converging shock (Case C2G288). For legend please refer tofigure 5.4.

expansion drastically attenuates vorticity in its wake (Figure 8 in Bhagatwala & Lele

[5]). In the present problem, one can see a decreasing trend in the reflected shock

vorticity profiles in figure 5.10 at different times caused by the post-shock expansion

region, which can be seen in the diamond-blue curve in figure 5.11 (a). However, we

do not see the drastic attenuation as in the blast wave case. At a particular instance

of time, vorticity profiles show a plateau in the post-shock region instead of sharp

attenuation. This is due to the outward radial velocity, as seen in figure 5.4(d) which

advects the amplified vorticity away from the origin. Vorticity is, however, seen to

be considerably higher than the pre-shock, pure-turbulence value. Figure 5.9 shows

the overall intensification of vorticity and increase in the dynamic range of scales.

A look at the source and sink terms in figure 5.11 reveals that the vorticity-

dilatation term acts as a source of vorticity during the converging phase of the shock,

and a sink during the expansion phase. The baroclinic generation term is an order of

magnitude lower than the vorticity dilatation term during both phases.


0 0.5 1 1.5 2−20

−10

0

10

20

30

r/l0

(a)

0 0.5 1 1.5 210

−4

10−3

10−2

10−1

100

101

102

r/l0

(b)

Figure 5.11: Magitudes of vorticity sources and sinks for the converging phase (CaseC2G288) at t/τ0 = 0.16(black) and reflected phase at t/τ0 = 0.65 (blue). (a) Linearplot (b) Log scale. (∇P ×∇ρ)/ρ2 ( - symbol), −ω(∇ · u) (♦ - symbol).

5.6.5 Particle statistics

Two phases can be identified in the converging phase as seen in figure 5.12, pre-shock

and post-shock. The distribution functions of density, f(ρ) and vorticity, f(ω) show

the expected trend, µpostshockρ > µpreshock

ρ and µpostshockω > µpreshock

ω . µf denotes the

mean of the the quantity f , computed from the first moment of the distribution func-

tion f .

We plot a quantity called potential vorticity ω/ρ that evolves according to

D

Dt

(ω

ρ

)

=1

ρ((ω ·∇)u+∇T ×∇s) (5.11)

The potential vorticity distribution, f(ω/ρ) shows µpostshockω/ρ ∼ µpreshock

ω/ρ , demon-

strating that the vorticity-dilatation term is the dominant one in shaping the post-

shock vorticity field.

An interesting aspect of these plots is that σpostshockρ >> σpreshock

ρ and σpostshockω >>

σpreshockω . σf denotes variance of the quantity f , computed from the second moment


(a) (b) (c)

Figure 5.12: Distribution functions of (a) Particle density, f(ρ), (b) Vorticity, f(ω)and (c) Potential vorticity, f(ω/ρ) for converging shock (Case C2G144). t/τ0 = 0.03(solid), t/τ0 = 0.2 (dashed), t/τ0 = 0.3 (dash-dotted)

of the distribution function, f . This is on account of the significant amount of as-

phericity in the shock at late stage as seen in figure 5.16 in section 4.3.5. It should be

remembered that the shock started with a low Ms/Mt ratio which caused it to cor-

rugate and distort at a fairly stage in its evolution. This is evident in the asphericity

profile in figure 5.16. Although the shock is quite strong at late times, its distortion

remains sizable. Although the seed particles were clustered in a very thin shell at an

almost constant radius, they interact with the converging shock at different times, and

see a significantly varying shock strength and as a result, we see wider distributions

in these plots.

5.6.6 Velocity variances

Table 5.6 shows as a quick glance, quantities plotted in figure 5.13. As seen the

vorticity profiles, both radial and tangential components increase behind the shock,

for both converging and reflected shock phases. Velocity fluctuations, not being

gradient-based, do not show as sharp a jump as do vorticity components. Also, in

contrast with vorticity, it is the radial component of TKE, 〈u′2r 〉, that jumps sharply


t/τ0 〈u′2r 〉s/u20 〈u′2

t 〉ps/u20 (〈u′2r 〉/〈u

′2t 〉)s

0.04 5.4 1.2 4.1Converging 0.16 13.0 2.2 6.3

Phase 0.31 13.1 3.5 6.80.42 18.2 5.5 6.1

0.52 8.5 5.3 1.3Reflected 0.60 7.2 4.0 1.4Phase 0.65 5.6 3.3 1.4

0.71 3.9 2.7 1.2

Table 5.6: Evolution of fluctuating velocity components and Reynolds stress anistropywith time for converging shock-turbulence interaction (Case C2G288). The subscript ‘s’ stands for shock values and are computed at r = Rs(t) and ‘ ps’ stands for quantitiesevaluated downstream of the shock where they attain their maximum value.

at the shock, while the tangential component, 〈u′2r 〉 increases more gradually. Sim-

ilar to vorticity, the anisotropy (called Reynolds stress anisotropy in this case) is

retained by the compressed fluid behind the converging shock. Reynolds stress be-

comes isotropic in the expansion zone behind the reflected shock. Again, we notice

that (〈u′2r 〉/〈u

′2t 〉)s 6= 〈u′2

r 〉s/〈u′2t 〉ps, indicating that velocity fluctuations in the tan-

gential direction peak at some distance away from the shock, as was the case with

vorticity components. These trends were also noticed in streamwise and transverse

velocity variances in the planar shock-turbulence interaction study by Larsson and

Lele [40]. Velocity variance behaves in a similar manner to vorticity variance in the

reflected phase of the shock, in that it plateaus spatially while decaying temporally.

Similar to vorticity, this can be attributed to the outward radial velocity during the

reflected phase of the shock, which advects fluid with amplified fluctuations away

from the origin. These plots also corroborate with the increase in TKE dissipation

rate, ǫ as discussed in section 4.3.1.

5.6.7 Thermodynamic fluctuations

Figure 5.14 plots radial profiles of variances of pressure and density for the moder-

ate strength converging shock-turbulence interaction case. The scale on the y-axis


0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

r/l0

〈u

′2 r〉/u

2 0

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

7

r/l0〈u

′2 t〉/u

2 0

(b)

0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

7

r/l0

〈u

′2 r〉/〈u

′2 t〉

(c)

Figure 5.13: Evolution of (a) Radial velocity variance, (b) Tangential velocity varianceand (c) Reynolds stress anisotropy for the converging shock (Case C2G288). Forlegend please refer to figure 5.4.

was taken to be logarithmic due to the large magnitude of the change in pre-shock,

shock and post-shock scales and between converging and reflected shock phases. The

profiles show a familiar trend, sharp increase at the shock and then a gradual decay

away from it. The amplification at maximum compression is quite high, evident in

the considerably higher magnitudes of fluctuation amplification associated with the

reflected shock.

5.6.8 Instantaneous profiles

Figure 5.15 show instantaneous traces of pressure, density and radial velocity for the

converging and reflected phases of the converging shock-turbulence interaction cases.

Instantaneous profiles differ from the averaged radial profiles shown in figure 5.4 in

two important ways. The shock is much thinner in the instantaneous profiles than

the averaged ones. This difference in thickness is indicative of the distortion in shock

structure caused by turbulence as was also seen in the blast wave-turbulence interac-

tion problem in the previous chapter. Turbulent fluctuations, which were smoothed

out in the averaged profiles are also visible in the traces.


0.2 0.4 0.6 0.8 1

10−2

100

102

r/l0

〈P′2〉/

P2 0

(a)

0.2 0.4 0.6 0.810

−4

10−2

100

r/l0〈ρ′2〉/ρ

2 0

(b)

Figure 5.14: (a) Pressure and (b) Density RMS plots for moderate strength convergingshock case C2G288

−2 −1 0 1 210

−1

100

101

102

103

x/l0

P/P

0

(a)

−2 −1 0 1 20

5

10

15

20

25

30

35

x/l0

ρ/ρ

0

(b)

−2 −1 0 1 2−5

−4

−3

−2

−1

0

1

2

x/l0

Ur/u

0

(c)

Figure 5.15: Traces of (a) Pressure (b) Density and (c) Radial velocity for convergingshock-turbulence interaction case C2G288. Converging phase: t/τ0 = 0.04 (black-dotted), t/τ0 = 0.16 (black-solid), t/τ0 = 0.31 (black-dashed), t/τ0 = 0.42 (black-dash/dotted). Reflected phase: t/τ0 = 0.52 (blue-dotted), t/τ0 = 0.60 (blue-solid),t/τ0 = 0.65 (blue-dashed), t/τ0 = 0.71 (blue-dash/dotted).


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

t/ts

Rs/R

s0

(a)

0 0.5 1 1.5 210

−5

10−4

10−3

10−2

10−1

100

t/ts

χ

(b)

Figure 5.16: Evolution of (a) Shock radius and (c) Shock asphericity, for converging(Case C2G288). Shock with Turbulence (+ symbol), Pure shock (o symbol)

5.6.9 Time evolution of the shock

In this section, we seek to understand the effect of the turbulence on shock evolution

and compare it with the case with no background turbulence. In absence of turbu-

lence, the shock is nominally spherical. Turbulence distorts this sphericity depending

on its relative strength compared to the shock. As also described in Bhagatwala &

Lele [5], we define a shock asphericity parameter χ based on the shock radius Rs as

follows

χ =i=N∑

i=1

√

(Ri − Rs)2

NRs

(5.12)

where N is the number of points forming the shell and Ri is the radius of any

point in the shell Rin ≤ Ri ≤ Rout as defined eariler. Figure 5.16 shows these quan-

tities for the converging shock at Ms0 = 2.4. The shock radius deviates considerably

from the pure shock case, which as noted in section 3 follows Guderley’s scaling for

a converging shock. A significant part of this deviation in the ‘mean’ shock radius is

due to distortions of the shock surface.


This is seen in the asphericity which starts out higher than the pure shock case.

The asphericity of the pure shock is a measure of grid imprinting error. As the shock

converges, it is captured by fewer and fewer points, increasing the imprinting error

and showing up as increased χ in the asphericity plot. Plotting asphericity for both

pure shock and shock with turbulence then helps elucidate the asphercity arising

from shock distortions due to interactions with turbulence, over and above the grid

imprinting error.

The reason for initial asphericity for the shock-turbulence problem (over and above

the imprinting error) is that the shock is initialized to have the same Mach number

at all points that define the initial shock radius. However, due to fluctuations in local

density and pressure, the local speed of sound is different at these points. Conse-

quently, local shock speed is different and the shock therefore starts out aspherical.

A strong shock, as it propagates inward, strengthens rapidly and does not acquire

further asphericity due to interaction with turbulence. A weaker initial shock on

the other hand, acquires further asphericity until it becomes strong enough not to

be affected by surrounding turbulent fluctuations. This is seen in figure 5.17(a)-(c),

which compares slices of dilatation contours during the converging phase of the shock

at the same mean shock radius (Rs/Rs0 ∼ 1/2) for different initial shock Mach num-

bers. Figure 5.17(d) shows the pure shock case with Ms0 = 2.5 for comparison. The

stronger initial shock retains retains only the initial distortion, while the weaker initial

shocks are relatively more aspherical.

5.7 Comparison with linear theory

We compare our results with linear theory for planar shock-turbulence interaction. In

particular, we compare the transverse vorticity amplification ratio from Lee et al [43]

with the converging shock-turbulence cases during the converging phase of the shock.

Figure 5.18(a) shows this comparison. We plot it as a function of the shock density

ratio. The reason for this choice of parameter is that the simulation data were not

5.7. COMPARISON WITH LINEAR THEORY 113

(a) (b)

(c) (d)

Figure 5.17: Slices of dilatation contours. (a) Ms0 = 1.4 (Case C1G288), (b) Ms0 =2.5 (Case C2G288) and (c) Ms0 = 4.0 (Case C3G288) (d) Pure shock at Ms0 = 2.5 atdifferent times but same mean shock radius, Rs/Rs0 = 0.45. Contour levels are notsame across plots.


1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

Shock Density ratio

Tra

nsve

rse

Vo

rtic

ity a

mp

lific

atio

n

(a)

0 20 40 60 80 1000

2

4

6

8

10

Shock Pressure ratio

Sh

ock

De

nsi

ty r

atio

(b)

Figure 5.18: Comparison of transverse vorticity amplification ratios with linear theoryand shock pressure and density ratios with the Rankine-Hugonoit relation. CaseC1G288 (x symbol), Case C2G288 (o symbol), Case C3G288 (+ symbol). Solidline in (a) refers to the linear theory result from Lee et al [43] and in (b) refers toRankine-Hugoniot relation.

saved at sufficient temporal resolution to accurately compute the shock Mach number

at a given time.

Linear behavior is observed at low density ratios, when the shock is still far enough

away from the origin that it can be considered nearly planar. As it propagates in-

wards, its curvature increases and together with the geometric effect of increasing

compression, leads to a rate of increase of amplification ratio higher than that pre-

dicted by linear theory. This effect is more pronounced for higher shock Mach number

cases where variance of the shock radius 〈R′2s 〉 is lower and the shock focuses more

uniformly. Another effect of the geometry is that for lower initial Mach number (Ms0),

a higher amplification ratio is observed at the same stage of compression (same den-

sity ratio) compared to the higher initial Mach number. Though the density ratio is

the same in both cases, the shock has significantly higher curvature for lower Ms0,

therefore geometry effects predominate.

5.8. MAXIMUM COMPRESSION 115

Figure 5.18(b) compares the shock density and pressure ratios as defined by the

R-H relations for a planar shock with those obtained from simulations of converging

shocks in presence of turbulence. The maximum density ratio for the R-H curve

asymptotes to 6, which is the well known limit for a strong shock, limMs→∞ρ2/ρ1 =

(γ + 1)/(γ − 1), where Ms is the shock Mach number. For the converging shock

simulations, we do not see any such uniformly asymptotic behavior. As the shock gets

stronger, reflected in its pressure ratio, the density ratio continues to increase. The

decrease in slope we see at the highest pressure ratios is due to numerical viscosity

preventing the density from growing in an unbounded manner. The density ratio

appears to be strongly dependent on the initial Mach number of the converging shock,

with stronger initial shocks generating higher density ratios as the shock closes in on

the origin. It is worth noting that, in absence of viscosity, the density and pressure for

a converging shock exhibit singular behavior as the shock converges at the origin. As

the shock strength increases, other physical effects, such as dissociation, ionization

etc. would be expected to arise. In the present simulation, these are ignored and

the combination of numerical and artificial bulk viscosity acts to prevent singular

behavior.

5.8 Maximum compression

We also compare the maximum compression achieved by the converging shock in

presence and absence of turbulence. One would expect that for a shock propagating

through turbulence, the maximum compression achieved would be less than for a

pure shock with the same initial Mach number. We are interested in the magnitude

of this decrease in maximum compression and how is it affected by initial shock Mach

number.

Table 6.4 tabulates maximum density and pressure ratios at different Mach num-

bers for pure shock and shock with turbulence cases. As expected, these increase

with increasing shock strength. Figure 6.25 plots the maximum pressure and density

ratios as a percentage of their highest values for the corresponding pure shock cases.

It is seen that the decrease in maximum compression occurs is greatest for the lowest


Ms0 Pure shock Shock-Turbulenceρmax/ρ0 Pmax/P0 ρmax/ρ0 Pmax/P0

1.4 20 162 14 582.4 55 1490 48 8324.2 79 4712 83 2855

Table 5.7: Comparison of pressure and density ratios at maximum compression for apure shock with shock propagating through turbulence

0 1 2 3 4 5

0.4

0.5

0.6

0.7

0.8

0.9

1

Ms0

Figure 5.19: Pressure and density ratios at maximum compression in presence ofturbulence compared to pure shock case. Quantities plotted are (ρmax/ρ0)s−t /(ρmax/ρ0)p−s ( - symbol) and (Pmax/P0)s−t / (Pmax/P0)p−s (♦ - symbol). 2883

grid (solid line); 1443 grid (dashed line) s− t: shock-turbulence, p− s: pure-shock

5.8. MAXIMUM COMPRESSION 117

Ms0 ∆t/ts1.4 0.0922.4 0.0774.2 0.022

Table 5.8: Time delay for maximum compression with and without turbulence fordifferent cases. Time is taken at the point when shock radius is at its minimum andinstantaneous pressure is at its maximum.

Mach number case (65% for density and 40% for pressure compared to pure shock

case) and tends to level off for stronger initial shocks (90% for density and 70% for

pressure). This is in line with expectation, since stronger shocks are less distorted

by turbulence, therefore remain more symmetric as they propagate inwards and con-

sequently are able to compress the fluid more effectively. The absolute values of the

various maxima shown in table 6.4 are somewhat grid dependent as they are affected

by numerical dissipation and resolution. However, the ratios plotted in figure 6.25 are

unlikely to be grid dependent, as both the pure-shock and shock-turbulence statistics

are computed on the same grids. To confirm this hypothesis, the ratios in figure 6.25

were computed for the different grids, and were found to not change by more than

5 − 6% for any of the quantities, even though quantities plotted in table 6.4 change

by 60− 80%.

Table 5.8 shows another way of assessing the impact of turbulence for the converg-

ing shock problem. Presence of turbulence causes the point of maximum compression

to occur later than without it. It is this delay, compared to the time it takes for the

corresponding pure shock (without turbulence) to reach maximum compression that

is tabulated in table 5.8. This time is taken when the shock radius is at its minimum

and instantaneous shock pressure is at its peak. Density peak occurs slightly later,

∼ t/ts = 0.01 after the pressure peak. It can be looked at as the degree to which

turbulence slows the converging shock down as it propagates inwards. As one would

expect intuitively, the stronger shock causes the least delay (∼ 2%) while the weakest

shock has the most delay (∼ 9%) as turbulence slows it down the most.


5.9 Conclusions

Simulations of converging shocks through a field of compressible decaying isotropic

turbulence have been carried out. Effect of the shocks on the background turbulence

has been quantified primarily through variances of vorticity components, velocity

components, their respective anisotropies and evolution of Kolmogorov scales. The

converging shock is seen to amplify vorticity and decrease the Kolmogorov scales in its

wake. It was shown that the vorticity evolution is affected primarily by the vorticity-

dilatation term in the vorticity evolution equation. The reflected shock phase of the

flow was also studied and differences between the two as well between the reflected

phase and a blast wave noted. The reflected shock continues to amplify vorticity

and velocity fluctuations, which are then attenuated by the expansion wave. The

overall effect on vorticity, however, compared to pre-shock, pure-turbulence state, is

amplification. This is in sharp contrast with blast wave-turbulence interaction where

the overall effect is a drastic attenuation of turbulent fluctuations. Also in contrast

with the blast wave case, is the behavior of the Kolmogorov scales, which decrease

compared to the pre-shock state.

Lagrangian tracking of particles was used to provide further corroboration for the

reason behind vorticity attenutation/amplification. The particle statistics also show

that the vorticity-dilatation term in the vorticity evolution equation is primarily re-

sponsible for the vorticity field behavior. The effects of turbulence on shock structure

and evolution are examined by computing the shock radius and asphericity, a normal-

ized measure of the variance in shock radius and comparing them to the pure shock

case. The pure shock results give an estimate of the grid imprinting error due to the

use of a Cartesian grid instead of a spherical one.

Transverse vorticity amplification ratios for the converging shocks are compared

with results from linear theory for a planar shock-turbulence interaction. The con-

verging shock-turbulence interaction shows linear behavior when the shock is far from

the origin. When the shock is closer to the origin, geometric effects predominate,

5.9. CONCLUSIONS 119

manifesting as higher rate of increase in amplification ratios for the converging cases.

Effect on maximum compression achieved due to presence of turbulence is quantified

as a function of initial shock Mach number. Stronger shocks are found to be less

affected by turbulence and are able to get close to maximum achievable compression

for the given Mach number. The time delay in reaching the point of maximum com-

pression was found to be the most for the weakest initial shock and the least for the

strongest initial shock.

Chapter 6

Richtmyer-Meshkov Instability in

Spherical Geometry

In this study, we take a detailed look at turbulence in spherical Richtmyer-Meshkov

Instability (RMI) in full 3D mode. The fluids considered in this study are air and

SF6. The shock is launched from the air (lighter) side of the interface. As the flow

evolves, a series of reflected and transmitted shocks are generated, which via baro-

clinic deposition of vorticity and its subsequent transport serve to mix the two fluids

in a turbulent mixing zone.

The most important parameter in this problem is the initial Mach number of the

shock MI . Variation with respect to this parameter is studied in this work. Three

initial Mach numbers corresponding to weak, moderate and strong initial shock are

studied. We study the effect of these parameters on growth of the mixing zone, vor-

ticity and turbulent kinetic energy. We also look at how interface perturbation causes

the peak compression amplitude to vary as a function of MI . This is an important

aspect in ICF, as a large decrease in maximum compression achieved compared to

the no perturbation case could affect viability of the target.

The grid is uniform Cartesian. Only an octant of the sphere is represented and

appropriate symmetry boundary conditions are used. The compact finite difference

120


Figure 6.1: Schematic of the simulation domain for Converging shock-turbulenceinteraction problem.

Shock αtheoretical αfit α′fit

Converging(inward) 0.7172 0.7174 0.7273Converging(outward) 0.7172 0.7178 0.7364

Table 6.1: Comparison of theoretical vs. best fit value of exponents for convergingpure shock. The theoretical values are the Guderley exponents for a converging shock.αfit represents a two parameter fit with respect to Rs0 and ts, while α

′fit represents a

fit with respect to Rs0 alone.

scheme described in Lui [51] and low dissipation and dispersion RK4 method described

in Hu et. al. [28] for time stepping, as described before has been used.

6.1 Spherical shocks

Guderley [19] first derived the scaling law for converging shocks,

Rs(t) = A(γ)(−t)α (6.1)

122 CHAPTER 6. SPHERICAL RMI

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

t/ts

Rs/R

s0

(a)

10−4

10−3

10−2

10−1

100

10−2

10−1

100

|1 − t/ts|

Rs/R

s0

(b)

Figure 6.2: Comparison of shock radius for converging pure shock with an assumedbest-fit for Guderley’s power law scaling. Line closer to x− axis corresponds to thereflected shock, line farther from x− axis corresponds to convverging shock. Plotshows the α′

fit fit. Points below the dashed line are excluded when making the fit. (a)Linear plot (b) Log plot (× - symbol) - Best fit of Guderley’s power law curve. (o -symbol) - Radius from 1283 simulation.


where Rs(t) is the radius of the shock, A(γ) is a constant and time is measured

backwards from the point when the shock reaches the origin. The constant α is called

the Guderley exponent and in the strong shock limit, depends only on γ. Interestingly,

the same exponent also holds when the shock reflects off of the origin and propagates

outwards. Thus, assuming time is measured from the instant the shock is launched,

we get the following relations for inward and outward propagating shocks.

Rs(t) =

Rs0(1− t/ts)α inward propagating shock

Rs0(t/ts − 1)α outward propagating shock(6.2)

where Rs(t) is the shock radius at any instant, Rs0 is the initial shock radius and

ts is the time taken for the shock to reach the origin. The shock radius as a function of

time is plotted in figure 6.2 and compared with theory. The value of ts used for com-

puting αfit, where both ts and Rs0 are varied is t′s/ts = 0.998 for the reflected shock

and t′s/ts = 1.000 for the converging shock. t′s is the value used for the fit and ts is

the value obtained from simulation. The value of Rs0 used for αfit is R′s0/Rs0 = 0.987

for the reflected shock and R′s0/Rs0 = 1.000 for the converging shock. R′

s0 is the

value used for the fit and Rs0 is the value obtained from simulation. The log plot

should be a straight line in order to conform with the assumed best-fit solution. For

computing the best-fit, points close to the origin, i.e when the shock is close to the

origin are ignored. This is because close to the origin when approximately Rs/ls < 5,

i.e the shock radius is somewhat comparable to the thickness of the captured shock,

the shock is not well resolved. However, as the reflected shock moves back out, it

again compares well with the theory.

It is seen that, except near the origin, the shock radius compares well with a

best fit curve of the theoretical solution. Note that the log plot exaggerates the

region near the point where the shock is close to the origin and so the discrepancy

is somewhat magnified. Away from the origin, the 3D simulation agrees well with

the theory. Table 6.1 shows the comparison of the theoretical value of the Guderley

exponent with the best fit value from simulation. The theoretical curve is plotted


with Guderley’s exponent and a best-fit value of the pre-factor A(γ). The agreement

is slightly better for the inward propagating shock than for the outward propagating

one. Data points very close to the origin, have been excluded for the purpose of

this comparison, as the numerical errors become significant as the shock approaches

a theoretical singularity at the origin which, as also mentioned in the chapter on

converging shock-turbulence interaction, can only be crudely approximated by any

discretised version of the equations.

6.2 Richtmyer-Meshkov Instability in spherical ge-

ometry

6.2.1 Initial conditions

The interface perturbation profile is given by h0 = A0cos(32θ)cos((32sinθ)φ) to obtain

an “egg carton” shaped disturbance, as seen in figure 6.3 which plots contours of

mass fraction at different times during the flow evolution. The reason for choosing

this particular type of profile is that it gives a homogeneous length scale for the

perturbation in θ−φ directions. This cannot be achieved by using spherical harmonics,

where in order to keep a constant wavenumber, the length scale decreases near the

poles. It may be noted that this is also the perturbation profile used extensively

in planar RMI simulations. Three simulations with initial shock Mach numbers MI

of 1.2, 1.8 and 3.0 have been carried out. Figure 6.6 shows density slices as the

flow evolves for a Mach 1.8 initial shock. We study the evolution of the mixing zone

width, vorticity and turbulent kinetic energy. Table 6.2 lists parameters for the initial

perturbation to the interface and quantities useful for scaling the computed results.

The initial shock is started very close to the interface on the air side with Rs0/R0 =

1.03. Interface amplitude relative to initial interface radius is A0/R0 = 0.01, while

that relative to perturbation wavelength is A0/λ0 = 0.03. The second-to-last and last

columns list the Mach number of the shock and normalized post-shock velocity jump

after it has reflected and hits the interface for a second time. Transition to turbulence

occurs soon after the reshock event. It is interesting to note that the Mach numbers

6.2. RICHTMYER-MESHKOV INSTABILITY IN SPHERICAL GEOMETRY 125

(a) (b)

(c) (d)

Figure 6.3: Evolution of spherical RMI for MI = 1.8. Contours of mass fraction,Converging shock regime: (a) t/ts = 0, (b) t/ts = 0.8. Post-reshock regime: t/ts =2.4, t/ts = 3.1


max max Reshock Reshock

Ms0 ∆v/Rs0 tsRs0/Rs0 (∑ω

′2i /k

20∆v

2) (∑u

′2i /Rs0

2) Ms ∆v/Rs

1.2 0.32 3.8 0.11 1.35 1.2 0.131.8 0.61 3.7 0.03 1.76 1.4 0.273.0 0.74 4.0 0.008 1.62 1.7 0.39

Table 6.2: Initial conditions. Initial radius of shock,Rs0, Initial radius of interface,R0, Initial wavenumber of perturbation, k0, k0R0 = 33, Atwood number, At = (ρ2 −ρ1)/(ρ2+ ρ1) = 0.67, Shock Mach number, Ms = Rs/c0, Velocity impulse to interfacedue to shock, ∆v/c0 = 2(M2

s − 1)/(γ+1)Ms. c0 is the speed of sound in air, ts is thetime taken for the shock to reach the origin.

Phase Converging Reflected Turbulent mixing

t/ts 0.0− 1.0 1.0− 2.3 2.3− 3.8

Table 6.3: Table of different regimes for spherical Richtmyer-Meshkov instability andtime ranges when they are applicable.

and consequently, the velocity jumps associated with reshock are much lower than

those for the initial shock. This is because, the reshock is propagating into an already

compressed fluid (SF6) from the initial shock. Consequently, the speed of sound is

higher, making the Mach number lower.

To help elucidate the figures and discussion in the sections below, it is useful to

define the different regimes of the flow. These are listed in table 6.3. Time is non-

dimensionalised by the time it takes for the shock from launch to reach the origin, ts.

Three broad regimes can be identified. The first is the converging shock regime, when

the shock is propagating inwards, not yet having reached the origin. The second is

the reflected shock regime, when the shock has rebounded off of the origin, but is

still within the simulation domain. The material interface is reshocked during this

regime. The third regime is that of turbulent mixing, when the baroclinic vorticity

deposited by the initial shock and reshock events, rises to a maximum (as seen in the

enstrophy plot in figure 6.16), and then decays as the two fluids mix.


(a) (b)

(c) (d)

Figure 6.4: Evolution of spherical RMI for MI = 1.8. Slices of density, ρ/ρunshockedair

(a) Converging shock regime, t/ts = 0.4, (b) t/ts = 0.8, Post-reshock regimes, (c)t/ts = 2.4, (d) t/ts = 3.1


(a) (b)

(c) (d)

Figure 6.5: Evolution of spherical RMI for MI = 1.8. Slices of air mass fraction, Y(a) Converging shock regime, t/ts = 0.4, (b) t/ts = 0.8, Post-reshock regimes, (c)t/ts = 2.4, (d) t/ts = 3.1


(a) (b)

(c) (d)

Figure 6.6: Evolution of spherical RMI for MI = 1.8. Slices of vorticity variancemagnitude, sqrt(ω

′2i ) (a) Converging shock regime, t/ts = 0.4, (b) t/ts = 0.8, Post-

reshock regimes, (c) t/ts = 2.4, (d) t/ts = 3.1


0 10 20 30 40 50 600

2

4

6

8

10

12

14

k0r

〈p〉/

p0

(a)

0 10 20 30 4010

0

101

102

k0r

〈p〉/p

0(b)

Figure 6.7: Tangentially averaged radial profiles of pressure for converging and re-flected shocks for MI = 1.8 case. (a) Converging, t/ts = 0.11 (dotted), t/ts = 0.35(solid), t/ts = 0.69 (dashed), t/ts = 0.92 (dash-dotted) (b) Reflected, t/ts = 1.33(dotted), t/ts = 1.52 (solid), t/ts = 1.71 (dashed), t/ts = 1.92 (dash-dotted). Arrowindicates location and propagation direction of shock.

6.2.2 Mean flow

To understand the driving flow, we plot tangentially averaged radial profiles of pres-

sure, density and radial velocity during the convering and reflected shock regimes.

There are shown in figures 6.7, 6.8 and 6.9 for the moderate strength shock with

initial Mach number MI = 1.8. The arrows on the curves indicate the location and

propagation direction of the shock front.

Pressure profiles during the converging phase show an increase in post-shock val-

ues as the shock propagates towards the origin. Pressure is continuous across the

material interface, therefore the interface signature is not visible in pressure profiles.

During the reflected shock phase, pressure decreases as shock expands radially out-

ward. Unlike the converging phase, the profile in the post-shock zone is not smooth,

due to interface corrugations, which grow after it has been reshocked. These corru-

gations, and the consequent mixing of fluids, modifies the local pressure. The shock,


0 10 20 30 40 50 600

10

20

30

40

50

60

70

k0r

〈ρ〉/

ρ0

(a)

0 10 20 30 4010

0

101

102

103

k0r

〈ρ〉/

ρ0

(b)

Figure 6.8: Tangentially averaged radial profiles of density for converging and reflectedshocks for MI = 1.8 case. (a) Converging, t/ts = 0.11 (dotted), t/ts = 0.35 (solid),t/ts = 0.69 (dashed), t/ts = 0.92 (dash-dotted) (b) Reflected, t/ts = 1.33 (dotted),t/ts = 1.52 (solid), t/ts = 1.71 (dashed), t/ts = 1.92 (dash-dotted). Arrow indicateslocation and propagation direction of shock.

at late times in its evolution, becomes quite weak as can be seen in the dash-dotted

curve in figure 6.7(b).

Density profiles for the converging shock regime clearly show both the shock and

the material interface. The arrow indicates the location and direction of propagation

of the shock. The second discontinuity shows the location of the perturbed material

interface. The perturbations grow in time as the shock propagates inwards, and this

is seen in the greater interface thickness compared to the shock front. The growing

interface is distributed across a larger number of grid points than the shock, and is

therefore distributed over a wider area. The density profiles during the reflected shock

phase are also instructive. Between t/ts = 1.33 (dotted line) and t/ts = 1.52 (solid

line), the shock crosses the interface, “reshocking” it. The tail of the arrows shows

the location of the shock. The second rise shows the position of the (considerably

smeared) interface. The shock is then seen to move away rapidly from the interface,

eventually exiting the domain at t/ts ∼ 2.3. After the primary shock has left the

simulation domain, there is a series of secondary shocks and expansion waves (not


0 10 20 30 40 50 60−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

k0r

〈Ur〉/

u0

(a)

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k0r

〈Ur〉/

u0

(b)

Figure 6.9: Tangentially averaged radial profiles of radial velocity for converging andreflected shocks for MI = 1.8 case. (a) Converging, t/ts = 0.11 (dotted), t/ts = 0.35(solid), t/ts = 0.69 (dashed), t/ts = 0.92 (dash-dotted) (b) Reflected, t/ts = 1.33(dotted), t/ts = 1.52 (solid), t/ts = 1.71 (dashed), t/ts = 1.92 (dash-dotted). Arrowindicates location and propagation direction of shock.

shown in this diagram) that persist upto late times, due to reflection/refraction at

the material interface that serve to further mix the two fluids.

Figure 6.9 shows profiles of radial velocity during the converging and reflected

shock phases of the problem. As before, the arrows indicate the location and direction

of propagation of the shock front. During the converging phase, the mean velocity is

radially inward, as indicated by 〈Ur〉 < 0. This is primarily reponsible for growth of

the interface perturbations, as seen in figure 6.6 which shows fingers of outer fluid (air)

growing into the inner fluid (SF6). As was the case with pressure, mean velocity is

continuous across the material interface, hence its signature is not visible in profiles

for radial velocity. During the reflected shock phase, the direction of post-shock

mean velocity switches to being radially outward. Shape of the profiles very close to

the origin is nearly linear. This is a standard feature of outward moving spherical

shocks and was also seen in radial velocity profiles for the blast wave and converging

shock problems in earlier chapters. Again, as was the case with the pressure profiles,

even though the location of the interface cannot be discerned, the amplification of


fluctuations and transition of the post-reshock field to turbulence can be observed in

the form of corrugations in the radial velocity profiles.


In this section, we show results from a limited grid convergence study. Due to the

very high cost of these calculations, a comprehensive grid convergence study is not

feasible, and the one presented is consequently of limited scope. The interface per-

turbation in this case is a single mode, given by h0 = A0cos(32θ). Figure 6.10 shows

traces of the primitive variables along one of the symmetry axes (x-axis) for the flow

at t/ts = 0.52, i.e when the shock is about half way between its initial location and

the origin. The passage of the shock from a light (air) to a heavier (SF6) medium

results in a reflected shock that propagates outwards in the lighter medium. This

reflected shock is seen in the profiles in figure 6.10 at x ∼ 1. As the flow progresses,

it travels in a radially outward direction and is absorbed at the outer boundaries.

The plots show that the 2563 and 5123 cases are converged, except at the shock and

the interface where convergence is necessarily first order. The higher resolution sim-

ulations also show fewer Gibbs’ oscillations around the shock and interface locations.

The level of this oscillations is 4 − 6%, with the lower end being observed for 5123

simulations and higher end for 1283. This gives some confidence in the 2563 results

presented in this chapter. Due to the prohibitively high cost of the highest resolution

simulations preclude their continuation into the post-reshock turbulent mixing phase.

However, from this limited study, we can observe that, post-shock flow is quite well

converged at the resolution used for other parts of this study.

Figure 6.11 plots artificial properties for the single mode case above on three

different grids. Both peak magnitude and width of all artificial properties decrease

with increasing number of grid points. This is in line with the construction of artificial

properties in Cook [12], where one of the requirements was to have the magnitudes

of artificial properties go to zero in the limit of zero grid spacing. The reason for


0 0.5 1 1.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Y

1283

2563

5123

(a)

0 0.5 1 1.50

5

10

15

20

25

30

35

40

45

xρ

1283

2563

5123

(b)

0 0.5 1 1.50

1

2

3

4

5

6

7

8

9

x

P

1283

2563

5123

(c)

0 0.5 1 1.5−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x

Ur

1283

2563

5123

(d)

Figure 6.10: Traces of flow variables along x-axis showing convergence for singlemode perturbation during converging phase on different grids at t/ts = 0.52. (a)Mass fraction (b) Density (c) Pressure and (d) Radial velocity. 1283 (black), 2563

(blue), 5123 (red).


0 10 20 30 40 50 600

1

2

x 10−4

k0r

µh

(a)

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

k0rβ

h

(b)

0 10 20 30 40 50 600

1

2

3

4

5

6x 10

−4

k0r

kh

(c)

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−4

k0r

Dh

(d)

Figure 6.11: Artificial properties on different grids at t/ts = 0.52. (a) Artificialshear viscosity (b) Artificial bulk viscosity (c) Artificial conductivity (d) Artificialdiffusivity. 1283 (black-solid); 2563 (blue-solid); 5123 (red-solid)


20 25 30 35

10−15

10−10

10−5

k0r

Dh

Figure 6.12: Zoomed in view of log of artificial diffusivity.

this requirement is that with a smaller grid spacing, there is less need for “capturing”

based methods, of which artificial viscosity is a part.

Figure 6.12 shows a zoomed view of figure 6.11(d) on a log scale, which shows

artificial diffusivity which is needed to capture the Air-SF6 interface.

6.2.4 Mixing zone

The mixing zone thickness is defined as h =∫ s=Rmax

s=0〈Y 〉(1 − 〈Y 〉)ds where angled

brackets 〈.〉 indicate a tangential average as a function of radius. Figure 6.13 com-

pares mixing zone widths for the same initial perturbation, only the incident shock

Mach number is varied. For all temporal profiles, time has been normalized with time

taken for the shock to reach the origin so that the time of first shock and reshock are

similar for all cases. Note that they cannot be identical, as in the planar case, because

shock speed is a nonlinear function of time for a spherically converging shock. We

also plot spatial profiles of 〈Y 〉(1 − 〈Y 〉) for the MI = 1.8 case which corroborates

the trend observed in the temporal profiles.


0 1 2 3 4 50

2

4

6

8

10

12

14

16

t/ts

h/h

0

(a)

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

k0r

〈Y〉(

1−

〈Y〉)

(b)

Figure 6.13: (a) Comparison of mixing zone width. MI = 1.2 (red),MI = 1.8 (black),MI = 3.0 (blue). Width normalized by initial perturbation amplitude. (b) Profiles of〈Y 〉(1− 〈Y 〉) for MI = 1.8 case, Converging (black) and post-reshock (blue) phases.Y is the mass fraction of Air. (a) Converging, t/ts = 0.2 (dotted), t/ts = 0.36 (solid),t/ts = 0.52 (dashed), t/ts = 0.68 (dash-dotted). (b) Post-reshock regimes, t/ts = 2.23(dotted), t/ts = 2.88 (solid), t/ts = 3.1 (dashed), t/ts = 3.74 (dash-dotted).

The h profile has a shape similar to that observed in the planar case, with a

slight initial drop as the incident shock compresses the perturbation, then a rise

as the interface perturbation to grow into spikes and bubbles under the action of

baroclinically deposited vorticity. The compression due to the reshock causes another

drop in h, followed by a rise as the mixing zone grows in size and becomes turbulent.

The Mach number dependence is quite evident. In the linear growth phase, the profiles

line up at early times, except for the lowest Mach number, which flattens out earlier.

After reshock, the large peaks associated with the turbulent mixing zone occur at

slightly different times with the chosen time normalization. The shock speed based

scaling of time therefore is not accurate for the post reshock phase of the mixing zone

growth. It is not clear that simple scaling parameters exist for this highly nonlinear

process. The slope of the h/h0 curve when evaluated against the scaled time is

similar. This is an indication that the scale used in the plot captures an important

dependence. Simulations at other Atwood numbers and interface perturbations are

needed to judge if this is a robust scaling.


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

Y

P(Y

)

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

Y

P(Y

)

(b)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

Y

P(Y

)

(c)

Figure 6.14: Probability density functions (PDF) of species mass fraction in themixing zone during turbulent mixing regime. A value of 0 represents pure SF6 and avalue of 1 represents pure air. (a) MI = 1.2, (b) MI = 1.8, (c) MI = 3.0. t/ts = 2.8(solid), t/ts = 3.1 (dashed), t/ts = 3.4 (dash-dotted) for all three cases.

Another approach to understanding mixing behavior is through the probability

density functions (PDF). For any field f(x, t), one can generate a probability density

function P (f, t). For computing the PDF, the range [max(f),min(f)] is divided into

Nbin = 100 equal intervals. The data in the mixing zone, which is defined by the

threshold Y (1 − Y ) > 0.05 is binned into these intervals. If Nk is the number of

elements in bin k, the PDF is defined as

P (k, f, t) =Nk

∆f∑Nbin

k=1 Nk

(6.3)

where ∆f = [max(f)−min(f)]/Nbin is the grid spacing in f− space. This formu-

lation satisfies the PDF normalization condition:∫∞−∞ P (f, t)df = 1 in the discrete

sense.

Figure 6.14 shows PDF profiles for the three cases during the post-reshock phase.

This phase is the most relevant to mixing. Profiles for the higher Mach number cases

smooth out at late times, indicating the fluids are fairly well mixed, whereas the low-

est Mach number case continues to show two discernable peaks even at late times.

This behavior seems to indicate that higher shock Mach numbers serve to mix the


0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

t/ts

Y

(a)

0 0.5 1 1.5 2 2.5 30.06

0.08

0.1

0.12

0.14

0.16

t/ts

Y′2

(b)

Figure 6.15: Evolution of PDF based mean and variance of the mass fraction Y , withtime for different initial shock Mach numbers. (a) Mean (b) Variance. MI = 1.2(red), MI = 1.8 (black), MI = 3.0 (blue).

two fluids better.

We define mean and variance as follows,

Y =

∫ 1+

0−Y P (Y )dY (6.4)

Y ′2 =

∫ 1+

0−(Y − Y )2P (Y )dY (6.5)

Figure 6.15 plots the mean and variance based on the definitions above for the

mass fraction field. A well mixed fluid and unmixed fluids will both have a mean of

around 0.5. Initially, the fluids are not well mixed, and the mean is seen to be around

0.5. After the initial shock, the mean flow is radially inward, causing fingers of air

(Y = 1) to penetrate SF6 (Y = 0), increasing the concentration of air in the mixing

zone and driving the mean to a higher value. Post reshock, turblent mixing proceeds

at a brisk pace, decreasing the mean and bumping up the variance. As the fluids mix,

variance decreases again. This is seen in figure 6.15(b), which shows a high variance

at early times and low variance at late times. At a given instance of time, the higher


0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

t/ts

(1/2)∑

ω′ iω

′ i/k

2 0∆

v2

Figure 6.16: Comparison of Enstrophy and TKE. MI = 1.2 (red), MI = 1.8 (black),MI = 3.0 (blue).

Mach number cases show a lower variance. This corroborates what was seen in the

PDF plots, that is, higher incident shock Mach number serves to better mix the two

fluids.

6.2.5 Vorticity

Figure 6.16 shows the evolution of domain integrated vorticity variance i.e. enstrophy.

Similar to the planar case, it shows a double peak structure. The first rise and decay

corresponds to the linear growth phase, while the second corresponds to the post-

reshock turbulent mixing zone growth and eventual decay. Enstrophy is normalized

by a time scale based on the initial velocity impulse to the interface by the incident

shock. The vorticity profiles collapse quite well during the linear phase, but diverge

at late times, when the flow is nonlinear and turbulent.

To study the spatial nature of the vorticity field as it evolves over time, we plot


5 10 15 20 25 300

0.1

0.2

0.3

0.4

k0r

ω′ iω

′ i(r)

/k2 0∆

v2

(a)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k0r

ω′ iω

′ i(r)

/k2 0∆

v2

(b)

Figure 6.17: Radial profiles of vorticity magnitude for shock with MI = 1.8 (a)Converging, t/ts = 0.2 (dotted), t/ts = 0.36 (solid), t/ts = 0.52 (dashed), t/ts = 0.68(dash-dotted). (b) Post-reshock regimes, t/ts = 2.23 (dotted), t/ts = 2.88 (solid),t/ts = 3.1 (dashed), t/ts = 3.74 (dash-dotted).

it during the converging and reshocked phases of the shock in figure 6.17. The baro-

clinic vorticity deposited by the initial shock passage is amplified by the post-shock

compression. The plot for the post-reshock flow shows the turbulent phase of RMI.

The growth of the turbulent mixing zone can be clearly seen in the radial profiles.

To understand the mechanism behind evolution of vorticity and the resulting

enstrophy, we look at the vorticity evolution equation and plot the two most important

terms that make up the vorticity budget. We invoke again, the vorticity evolution

equation, repeated here from the previous two chapters for convenience

∂ω

∂t= −(u ·∇)ω

︸︷︷︸

convection

+ (ω ·∇)u︸︷︷︸

stretching/tilting

− ω(∇ · u)︸︷︷︸


+∇p×∇ρ

ρ2︸︷︷︸

Baroclinic

+1

Re

(

∇×(1

ρ∇ · τ

))

︸︷︷︸

V iscous

(6.6)

Since this is an inviscid simulation, the most important terms are the baroclinic

generation term and vorticity-dilatation term. The baroclinic term is a source of vor-

ticity, whereas the vorticity-dilatation terms can be a source or sink, depending on


10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k0r

(a)

0 10 20 30 40 5010

−4

10−3

10−2

10−1

100

k0r

(b)

Figure 6.18: Radial magnitude profiles of vorticity budget for shock with MI = 1.8during converging phase at t/ts = 0.36 (black) and turbulent mixing phase at t/ts =3.1 (blue). (a) Linear (b) Log scale. (∇P × ∇ρ)/ρ2 ( - symbol), −ω(∇ · u) (♦ -symbol).

whether is region is one of local compression or expansion, as discussed in previous

chapters. Figure 6.18 plots these. The magnitude of these two terms is plotted at at

one time instance during the converging phase of the shock and at one time instance

during the post-reshock, turbulent mixing phase. Vorticity is a vector quantity, so

here we plot only the magnitude of the source/sink terms.

Figure 6.18(a) is linear for both axes, while 6.18(b) has the ordinate in log-scale,

which gives a better appreciation of the difference in magnitude between the two.

The log plot also shows the turbulent nature of the field; the fluctuations are visible

in the radial profiles. In the linear plot, the fluctuations are “hidden” on account of

being much smaller in magnitude compared to the magnitudes of the corresponding

terms at the shock and interface locations. The baroclinic term is everywhere one to

two orders of magnitude larger than the vorticity-dilatation term, for both phases.

The baroclinic deposition caused by mis-aligned pressure and density gradients during

the initial shock impingement at the interface, drives the vorticity evolution as the

post-shock compression amplifies it. The reshock deposits further vorticity through

the baroclinic mechanism. Hence, we see here that the baroclinic terms is the main


102

103

104

10−8

10−6

10−4

Ω

P(Ω

)

Figure 6.19: PDF profiles of vorticity magnitude during the turbulent mixing phaseafter reshock during turbulent mixing regime. MI = 1.2 (red), MI = 1.8 (black),MI = 3.0 (blue). t/ts = 2.8 (solid), t/ts = 3.1 (dashed), t/ts = 3.4 (dash-dotted) forall three cases.

driver of the turbulent flow for the Richtmyer-Meshkov problem, in contrast to both

blast wave-turbulence and converging shock-turbulence problems, where the vorticity-

dilatation term was the primary driver of vorticity dynamics.

Another way to look at vorticity is through PDFs, as described before. Figure

6.19 plots PDFs of vorticity for all three cases at three different times during the

post-reshock turbulent mixing phase. All three cases show a wide distribution over

aproximately two orders of magnitudes. Predictably, the higher Mach number cases

are shifted by approximately one order of magnitude. Lower values of vorticity,

corresponding to large scales of turbulence have a much higher probability than large

values, corresponding to small scales and large gradients at which dissipation takes

place.


0 1 2 3 4 50

0.5

1

1.5

2

2.5x 10

−3

t/ts

1/2

∑u

′ iu′ i/

˙R

s0

2

(a)

0 1 2 30

1

2

3

4

5

6x 10

−4

t/ts

ǫ′

/k0∆

v˙

Rs0

2

(b)

Figure 6.20: Temporal evolution of turbulent kinetic energy (TKE) and dissipation.MI = 1.2 (red), MI = 1.8 (black), MI = 3.0 (blue).

6.2.6 Turbulent Kinetic Energy

Figure 6.20 shows the evolution of domain integrated perturbation kinetic energy,

which eventually becomes turbulent kinetic energy (TKE) as the flow becomes tur-

bulent and dissipation. For ease of nomenclature, both are referred to as TKE.

Pseudo-dissipation is defined in the same way as in Wilcox [90] and Latini et. al [39]

as

ǫ′

(t) = E(t)√

2Eω′ω′ (t) (6.7)

where Eω′ω′ = ω′2/2 is the turbulent enstrophy per unit mass averaged over the

mixing zone. Similar to enstrophy, TKE shows a two peak structure. The first

rise corresponds to the initial shock and decay, the second rise corresponds to the

post-reshock phase, after which the flow becomes turbulent, and eventually decays.

Pseudo-dissipation shows a similar profile, peaking following the initial incident shock,

then decreases. ǫ′

then rises by a factor of 3 to 4 as the reshock results in a rapid

deposition of energy into the mixing zone, and then decays as TKE also decreases.

Traces of pressure along one of the principal axes show an interesting feature of the

flow. As the reflected shock reshocks the interface, part of it refracts, and continuing


0 0.5 1 1.510

−2

10−1

100

101

102

103

Reflected shock

Secondaryreflections

Figure 6.21: Traces of pressure from t/ts = 1.1 to t/ts = 3.1 for MI = 1.8 case. Linesin blue show phase during which reflected shock is still within simulation domain.Lines in black are traces at later time showing formation of secondary shocks fromcoalescing waves moving inwards


5 10 15 20 25 30 350

2

4

6

8

x 10−3

k0r

u′ iu

′ i(r)

/˙

Rs0

2

(a)

0 10 20 30 40 50 600

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

k0r

u′ iu

′ i(r)

/˙

Rs0

2

(b)

Figure 6.22: Radial profiles of TKE for shock with MI = 1.8 (a) Converging (b)Post-reshock regimes. For legend, please refer to figure 6.17

to travel outwards, exits the simulation domain. This primary shock can be seen in

the blue colored lines in the plot. At the interface, it also sets up reflected shocks that

travel radially inwards. These are represented by black colored lines. Two secondary

reflected shocks can be seen forming through a coalescence of compression waves, one

stronger than the other. The stronger shock can be seen passing the weaker one on

its way towards the origin. These shocks cause the slight dip seen in temporal profiles

of TKE at t/ts ∼ 2.25.

Figure 6.22 plots tangentially averaged radial profiles of TKE for the converging

and post-reshock phases. During the converging phase, we observe two peaks, one

corresponding to the evolving mixing zone where the material interface is and the

other at the shock location due to shock corrugation. The peak at the shock grows

as the shock gets stronger as it propagates inwards, whereas the peak at the inter-

face decays as there is no fresh energy deposition and numerical viscosity damps the

fluctuations. Similar to vorticity profiles earlier, growth of the turbulent mixing zone

can be seen in the TKE profiles for the post-reshock flow as well.

Another way to look at TKE during the turbulent mixing phase is to plot PDFs,


10−2

10−1

100

10−4

10−2

100

k

P(k

)

Figure 6.23: PDF profiles of TKE during the turbulent mixing phase after reshockduring turbulent mixing regime. MI = 1.2 (red), MI = 1.8 (black), MI = 3.0 (blue).t/ts = 2.8 (solid), t/ts = 3.1 (dashed), t/ts = 3.4 (dash-dotted) for all three cases.

as we did for vorticity before. Figure 6.23 plots PDFs at three different times during

the mixing phase for all three simulation cases. We observe a structure similar to

vorticity PDFs. Lower values of TKE, corresponding to large scale flow structures

have a higher probability than small scale structures, where mixing takes place. All

three cases have show a range extending over at least two orders of magnitude, but

the higher Mach number cases have larger absolute values, due to more TKE being

deposited at the interface by the initial and reshock events.

Along with TKE and dissipation, we can compute an additional statistic, turbulent

Reynolds number Ret = q4/ǫν, where q2 is the TKE and ǫ is the pseudo-dissipation

defined above. Since we do not have a physical viscosity for these spherical RMI sim-

ulations, we use a representative value based on artificial shear viscosity. This value

is computed by averaging over the mixing zone during the turbulent mixing phase

for each case. The artificial viscosity is naturally higher for the higher Mach number

cases.


0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

1400

t/ts

Re t

Figure 6.24: Temporal profiles of turbulent Reynolds number, Ret = q4/ǫν where qdenotes turbulent kinetic energy. MI = 1.2 (red), MI = 1.8 (black), MI = 3.0 (blue).

Ret shows a peak after the initial shock as the instability is in the initial growth

phase. However, since the flow is not turbulent in this phase, this peak does not hold

much physical significance. The second and the more significant peak, occurs during

the turbulent mixing phase. The peak value of Ret decreases with increasing Mach

number. This is due to artificial viscosity being higher for the higher Mach number

cases, thereby decreasing Ret.

6.2.7 Maximum compression

We compare the maximum compression achieved by the converging shock in presence

and absence of interface perturbation. One would expect that for a shock propagating

through turbulence, the maximum compression achieved would be less than for a

pure shock with the same initial Mach number. We are interested in the magnitude

of change in maximum compression and how is it affected by initial shock Mach

number.


Ms0 RMI Unperturbedρmax/ρ0 Pmax/P0 ρmax/ρ0 Pmax/P0

1.2 86 113 86 1151.8 381 389 371 3863.0 841 471 1104 500

Table 6.4: Comparison of pressure and density ratios at maximum compression forspherical RMI with shock passing through an unperturbed interface

0.5 1 1.5 2 2.5 3 3.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Ms0

Figure 6.25: Pressure and density ratios at maximum compression in presence ofturbulence compared to pure shock case. Quantities plotted are (ρmax/ρ0)RMI /(ρmax/ρ0)unperturbed ( - symbol) and (Pmax/P0)RMI / (Pmax/P0)unperturbed (♦ - sym-bol)


Ms0 ∆t/ts1.2 0.0101.8 0.0083.0 0.018

Table 6.5: Time delay for maximum compression with and without interface pertur-bation for different cases.

Table 6.4 tabulates maximum density and pressure ratios at different Mach num-

bers for pure shock and spherical RMI with shock-unperturbed interface cases. They

exhibit an interesting behavior. For low to moderate initial shock Mach numbers,

interface perturbation has practically no effect on maximum compression achieved.

Only for the strong initial shock, do we see a tangible decrease in maximum com-

pression. Figure 6.25 plots the maximum pressure and density ratios as a per-

centage of their highest values for the corresponding unperturbed interface cases.

For the low and moderate Mach number cases, the maximum compression for the

RMI cases is between 98% − 103%, i.e almost unchanged. For the strong initial

shock however, there is a considerable decline in maximum compression as evidenced

by peak density ratio, (ρmax/ρ0)RMI/(ρmax/ρ0)unperturbed of 75% and pressure ratio,

(Pmax/P0)RMI/(Pmax/P0)unperturbed of 94%. These statistics, cannot be considered

conclusive however, since these numbers likely depend on parameters like perturba-

tion wavenumber and Atwood ratio as well.

Table 6.5 tabulates the time delay caused in reaching the point of maximum com-

pression due to interface perturbation. This delay is computed as a percentage of time

taken to reach maximum compression for the corresponding pure shock simulation.

This measure is similar to the one computed for the converging shock-turbulence

problem in table 5.8. The delay is seen to be negligible lying between 0.8% − 1.8%,

at least compared to the converging shock-turbulence problem, where it was between

2.5%− 10%.


6.2.8 Conclusions

Simulations of the Richtmyer-Meshkov instability for an Air-SF6 interface in spheri-

cal geometry have been carried out with three-dimensional ‘egg-carton’ type interface

perturbation. Parametric variation with respect to initial shock Mach number has

been studied. Effect of shock Mach number on growth rates of the mixing zone,

vorticity, turbulent kinetic energy and maximum compression achieved have been

studied. Different regimes for spherical RMI have been identified: converging shock,

reflected shock and turbulent mixing.

The growth of the mixing zone was studied through temporal and tangentially

averaged radial profiles of the mixing zone as well as PDFs and PDF based statis-

tics. Results shown here indicate that mixing is more efficient at higher initial shock

Mach numbers. Evolution of vorticity was studied through temporal profiles of en-

strophy, radial profiles of vorticity magnitude, vorticity budget, and PDFs of vorticity

magnitude. Baroclinic generation was found to be the driving force behind vorticity

dynamics. Attempt was made to scale enstrophy with shock induced velocity. This

was found to be good for scaling the converging phase of RMI up until the reshock,

after which the scaling does not appear to hold. PDFs show the same range of scales

for all initial Mach numbers, however, higher Mach numbers are seen to have higher

magnitudes, corresponding to smaller length scales.

Turbulent kinetic energy (TKE) was also studied through temporal and spatial

profiles and PDFs. Its behavior was found to be similar to that of vorticity. Tempo-

ral dissipation profiles were used to show that TKE dissipation was strongest during

the growth phase after the initial shock and turbulent mixing phase after reshock.

Profiles of dissipation appear to mimic those of TKE and vorticity. This is due to

numerical viscosity being large for the larger gradients that generate high magnitudes

of vorticity. Attempt was made to scale the data in different regimes. However, TKE

does not appear to follow any straightforward scaling behavior.


Maximum compression was studied with and without interface perturbation. In-

terface perturbation was found to not impact the maximum compression achieved

for low to moderate shock Mach numbers. However, for the strong shock case, the

maximum compression was much lower with the perturbation than without. The

time taken to reach the point of maximum compression was not seen to be affected

much due to presence of the perturbation.

Chapter 7

Conclusions

For simulations of a Taylor blast wave through a field of compressible decaying

isotropic turbulence, effect of the shocks on the background turbulence has been

quantified primarily through variances of vorticity components, its anisotropy and

evolution of Kolmogorov scales. The Taylor blast wave is found to increase Kol-

mogorov scales after a local reduction close to the blast wave and attenuate vorticity

behind the shock. It was shown that the Kolmogorov scales are primarily affected by

the kinematic viscosity field and vorticity evolution by the vorticity-dilatation term

in the vorticity evolution equation. Lagrangian tracking of particles was used to pro-

vide further corroboration for the reason behind vorticity attenutation/amplification.

The particle statistics also show that the vorticity-dilatation term in the vorticity

evolution equation is primarily responsible for the vorticity field behavior. The ef-

fects of turbulence on shock structure and evolution are examined by computing the

shock pressure ratio and shock radius and comparing them to the pure shock case.

A new parameter for characterising the asphericity of the shock has been proposed

and shown to work quite well. Through a simple rescaling, it is shown that shock

corrugation causes a significant smearing of profiles in the region around the shock.

Transverse vorticity amplification ratios for the Taylor blast wave shock-turbulence

interaction problem are compared with results from linear theory. In both cases, the

spherical shock-turbulence interaction shows linear behavior when the shock is far

153

154 CHAPTER 7. CONCLUSIONS

from the origin. When the shock is closer to the origin, geometric effects predomi-

nate, manifesting as lower amplification ratios.

In simulations of converging shocks through the same field of compressible decay-

ing isotropic turbulence, effect of the shocks on the background turbulence has been

quantified through variances of vorticity components, velocity components, their re-

spective anisotropies and evolution of Kolmogorov scales. The converging shock is

seen to amplify vorticity and decrease the Kolmogorov scales in its wake. It was

shown that the vorticity evolution is affected primarily by the vorticity-dilatation

term in the vorticity evolution equation and the Kolmogorov scales by the kinematic

viscosity field and TKE dissipation rate. The reflected shock phase of the flow was

also studied and differences between the two as well between the reflected phase and

a blast wave noted. The reflected shock continues to amplify vorticity and velocity

fluctuations, which are then attenuated by the expansion wave. The overall effect

on vorticity, however, compared to pre-shock, pure-turbulence state, is amplification.

This is in sharp contrast with blast wave-turbulence interaction where the overall ef-

fect is a drastic attenuation of turbulent fluctuations. Also in contrast with the blast

wave case, is the behavior of the Kolmogorov scales, which decrease compared to the

pre-shock state. Lagrangian tracking of particles was again used to provide further

corroboration for the reason behind vorticity attenutation/amplification. The particle

statistics also show that the vorticity-dilatation term in the vorticity evolution equa-

tion is primarily responsible for the vorticity field behavior. The effects of turbulence

on shock structure and evolution are examined by computing the shock radius and

asphericity, a normalized measure of the variance in shock radius and comparing them

to the pure shock case. Transverse vorticity amplification ratios for the converging

shocks are compared with results from linear theory for a planar shock-turbulence

interaction. The converging shock-turbulence interaction shows linear behavior when

the shock is far from the origin. When the shock is closer to the origin, geometric

effects predominate, manifesting as higher rate of increase in amplification ratios for

the converging cases. Effect on maximum compression achieved due to presence of

turbulence is quantified as a function of initial shock Mach number. Stronger shocks

155

are found to be less affected by turbulence and are able to get close to maximum

achievable compression for the given Mach number. The time delay in reaching the

point of maximum compression was found to be the most for the weakest initial shock

and the least for the strongest initial shock.

Simulations of the Richtmyer-Meshkov instability for an Air-SF6 interface in

spherical geometry have been carried out with three-dimensional ‘egg-carton’ type

interface perturbation. Parametric variation with respect to initial shock Mach num-

ber has been studied. Effect of shock Mach number on growth rates of the mixing

layer, vorticity, turbulent kinetic energy and maximum compression achieved have

been studied. Different regimes for spherical RMI have been identified: converg-

ing shock, reflected shock and turbulent mixing. The growth of the mixing layer

was studied through temporal and tangentially averaged radial profiles of the mixing

layer as well as PDFs and PDF based statistics. Results shown here indicate that

mixing is more efficient at higher initial shock Mach numbers. Evolution of vorticity

was studied through temporal profiles of enstrophy, radial profiles of vorticity mag-

nitude, vorticity budget, and PDFs of vorticity magnitude. Baroclinic generation

was found to be the driving force behind vorticity dynamics. Attempt was made to

scale enstrophy with shock induced velocity. This was found to be good for scaling

the converging phase of RMI up until the reshock, after which the scaling does not

appear to hold. PDFs show the same range of scales for all initial Mach numbers,

however, higher Mach numbers are seen to have higher magnitudes, corresponding to

smaller length scales. Turbulent kinetic energy (TKE) was also studied through tem-

poral and spatial profiles and PDFs. Its behavior was found to be similar to that of

vorticity. Temporal dissipation profiles were used to show that TKE dissipation was

strongest during the growth phase after the initial shock and turbulent mixing phase

after reshock. Profiles of dissipation appear to mimic those of TKE and vorticity.

This is due to numerical viscosity being large for the larger gradients that generate

high magnitudes of vorticity. Attempt was made to scale the data in different regimes.

However, TKE does not appear to follow any straightforward scaling behavior. Max-

imum compression was studied with and without interface perturbation. Interface

156 CHAPTER 7. CONCLUSIONS

perturbation was found to not impact the maximum compression achieved for low to

moderate shock Mach numbers. However, for the strong shock case, the maximum

compression was much lower with the perturbation than without. The time taken to

reach the point of maximum compression was not seen to be affected much due to

presence of the perturbation.

7.1 Recommendations for future work

The parametric variation in this study for all problems has been limited. For the blast

wave and converging shock with turbulence problems, only shock Mach number was

varied. While the relative Mach numbers of the shock and background turbulence

appear to be the most important parameters for this kind of study, a greater para-

metric variation is needed to fully understand these problems. In particular, varying

the turbulence Reynolds number would be the most important parametric variation

that might yield interesting results, especially in the weak shock regime. Due the

quasi-DNS nature of this study, the turbulence Reynolds number was low in order

to resolve the smallest scales. In order to see significantly different behavior, the

Reynolds number would need to be increased by a factor of 10 to 100. This would

necessarily put such simulations in the LES regime, requiring careful subgrid model-

ing.

The need for further parametric study is even more pertinent for spherical Richtmyer-

Meshkov instability (RMI) simulations. Effect of curvature and initial density ratio

are the two most important parameters that could potentially significantly influence

results. The former can be easily varied by changing the wavelength of the initial per-

turbation field. An increase in wavelength, i.e a smaller k0R0 would make curvature

more prominent, whereas a decrease in wavelength would bring it closer to planar,

at least during the initial phase and at large radii at late times. The latter can be

changed by changing the fluids used in the simulations. In this study, Air-SF6 have

been used. Other candidates are Air-He or He-Ar. The order of the two fluids can be

7.1. RECOMMENDATIONS FOR FUTURE WORK 157

switched, instead of light-heavy, a heavy-light combination can be used. Though this

has shown to not have much effect on statistics of planar RMI, presence of curvature

might be a factor in the spherical regime.

Appendix A

Symmetry boundary conditions for

derivatives and filters

For certain flows, there exist symmetries in one or more directions which can be

exploited to reduce the size of the computational domain by as much as 8 times.

For instance, in a simple one dimensional case a midpoint symmetry line can be

used to reduce the domain size by half, in a two dimensional case of a cylinder,

the flow is symmetric across the planes dividing the domain into quadrants. Only

one quadrant need be simulated along with symmetry boundary conditions on the

interior boundaries. For a problem with a spherical symmetry, only one octant need

be simulated, cutting down the problem size by an eighth. However, care needs to

be taken with treatment of derivatives and filters at the symmetry boundaries. Most

high order schemes use reduced order of accuracy and no filtering at the boundaries.

Some extrapolate the solution to ghost points on the other side. This can introduce

unwanted errors and instabilities in the simulation. However, we can take advantage

of the fact that we know the precise solution on the other side of the symmetry

boundary, depending on whether the function of interest is odd or even. This fact

can be used to reformulate the derivative and filter schemes so that the order of

accuracy of the interior scheme/filter is maintained at the symmetry boundaries as

well, making the simulation as accurate as it would have been with the entire domain.

158

A.1. DERIVATIVES 159

A.1 Derivatives

The derivatives considered here are high-order and compact with a five point stencil on

the left hand side and seven point stencil on the right hand side. This requires that a

matrix equation be solved to evaluate the derivatives. In the current implementation,

the LDU decomposition of this matrix is precomputed and stored. It would be easier

to implement these boundary conditions for an explicit scheme as only the right hand

side would need to be changed.

A.1.1 First derivative

In general, the compact scheme can be written as in [51]

β1f′i−2 + α1f

′i−1 + f ′

i + α1f′i+1 + β1f

′i+2 = (A.1)

c1fi+3 − fi−3

6h+ b1

fi+2 − fi−2

4h+ a1

fi+1 − fi−1

2h(A.2)

Considering i = 0 to be the boundary node, we introduce ghost points on the

other side. If function is even (symmetric) about the boundary, we have the following

relations (assuming here that symmetry is at the left boundary, the right bound-

ary is treated by usual boundary conditions appropriate for flow conditions there).

Note that although ghost points are used here, there is no extrapolation or other

approximation involved. The values at the ghost points are “exact”

Symmetry

f−3 = f3

f−2 = f2

f−1 = f1

f ′−2 = −f ′

2

f ′−1 = −f ′

1

160 APPENDIX A. SYMMETRY BOUNDARY CONDITIONS

Plugging these into above equations gives

f ′0 = 0 (A.3)

(1− β1)f′1 + α1f

′2 + β1f

′3 = c1

f4 − f26h

+ b1f3 − f14h

+ a1f2 − f02h

(A.4)

α1f′1 + f ′

2 + α1f′3 + β1f

′4 = c1

f5 − f16h

+ b1f4 − f04h

+ a1f3 − f12h

(A.5)

Antisymmetry

f−3 = −f3f−2 = −f2f−1 = −f1f ′−2 = f ′

2

f ′−1 = f ′

1

Plugging these into above equations gives

f ′0 + 2α1f

′1 + 2β1f

′2 = c1

2f36h

+ b12f24h

+ a12f12h

α1f′0 + (1 + β1)f

′1 + α1f

′2 + β1f

′3 = c1

f4 + f26h

+ b1f3 + f14h

+ a1f2 − f02h

β1f′0 + α1f

′1 + f ′

2 + α1f′3 + β1f

′4 = c1

f5 + f16h

+ b1f4 − f04h

+ a1f3 − f12h

A.1.2 Second derivative

The compact scheme for the second derivative can be written as

β2f′′i−2 + α2f

′′i−1 + f ′′

i + α2f′′i+1 + β2f

′′i+2 = (A.6)

c2fi+3 − 2fi + fi−3

9h2+ b2

fi+2 − 2fi + fi−2

4h2+ a2

fi+1 − 2fi + fi−1

h2(A.7)

Symmetry

Plugging in the symmetry conditions

A.1. DERIVATIVES 161

f−3 = f3

f−2 = f2

f−1 = f1

f ′′−2 = f ′′

2

f ′′−1 = f ′′

1

f ′′0 + 2α2f

′′1 + 2β2f

′′2 = c2

2(f3 − f0)

9h2+ b2

2(f2 − f0)

4h2+ a2

2(f1 − f0)

h2(A.8)

α2f′′0 + (1 + β2)f

′′1 + α2f

′′2 + β2f

′′3 = c2

f4 − 2f1 + f29h2

(A.9)

+b2f3 − f14h2

+ a2f2 − 2f1 + f0

h2(A.10)

β2f′′0 + α2f

′′1 + f ′′

2 + α2f′′3 + β2f

′′4 = c2

f5 − 2f2 + f19h2

(A.11)

+b2f4 − 2f2 + f0

4h2+ a2

f3 − 2f2 + f1h2

(A.12)

Antisymmetry

Plugging in antisymmetry conditions

f−3 = −f3f−2 = −f2f−1 = −f1f ′′−2 = −f ′′

2

f ′′−1 = −f ′′

1

We get,


f ′′0 = 0 (A.13)

α2f′′0 + (1− β2)f

′′1 + α2f

′′2 + β2f

′′3 = c2

f4 − 2f1 − f09h2

(A.14)

+b2f3 − 3f1

4h2+ a2

f2 − 2f1h2

(A.15)

α2f′′1 + f ′′

2 + α2f′′3 + β2f

′′4 = c2

f5 − 2f2 − f19h2

(A.16)

+b2f4 − 2f2

4h2+ a2

f3 − 2f2 + f1h2

(A.17)

A.2 Filters

Filter formulation follows the same procedure as for derivatives at the symmetry

boundaries. For the compact eighth order filter, we have

βfj−2 + αfj−1 + fj + αfj+1 + βfj+2 = afj +b

2(fj−1 + fj+1) (A.18)

+c

2(fj−2 + fj+2) +

d

2(fj−3 + fj+3) +

e

2(fj−4 + fj+4) (A.19)

where the hat symbol denotes a filtered quantity.

A.2.1 Symmetry

Following same treatment as for derivatives,

f0 + 2αf1 + 2βf2 = af0 + bf1 + cf2 + df3 + ef4 (A.20)

αf0 + (1 + β)f1 + αf2 + βf3 = af1 +b

2(f0 + f2) +

c

2(f1 + f3) (A.21)

+d

2(f2 + f4) +

e

2(f3 + f5) (A.22)

βf0 + αf1 + f2 + αf3 + βf4 = af2 +b

2(f1 + f3) +

c

2(f0 + f4) (A.23)

+d

2(f1 + f5) +

e

2(f2 + f6) (A.24)

A.2. FILTERS 163

βf1 + αf2 + f3 + αf4 + βf5 = af3 +b

2(f2 + f4) +

c

2(f1 + f5) (A.25)

+d

2(f0 + f6) +

e

2(f1 + f7) (A.26)

A.2.2 Antisymmetry

f0 = 0 (A.27)

αf0 + (1− β)f1 + αf2 + βf3 = af1 +b

2(f0 + f2) +

c

2(f3 − f1) (A.28)

+d

2(f4 − f2) +

e

2(f5 − f3) (A.29)

βf0 + αf1 + f2 + αf3 + βf4 = af2 +b

2(f1 + f3) +

c

2(f0 + f4) (A.30)

+d

2(f5 − f1) +

e

2(f6 − f2) (A.31)

βf1 + αf2 + f3 + αf4 + βf5 = af3 +b

2(f2 + f4) +

c

2(f1 + f5) (A.32)

+d

2(f0 + f6) +

e

2(f7 − f1) (A.33)

A.2.3 Gaussian filter

The gaussian filter is applied to smooth the hyperviscosity. It acts only on a symmetric

term, so only that is considered. The general form of the filter is

fj = a0fj + a1(fj−1 + fj+1) + a2(fj−2 + fj+2)

+a3(fj−3 + fj+3) + a4(fj−4 + fj+4)

With the symmetry boundary conditions this becomes

f0 = a0f0 + a1(2f1) + a2(2f2) + a3(2f3) + a4(2f4) (A.34)

f1 = a0f1 + a1(f0 + f2) + a2(f1 + f3) (A.35)

+a3(f2 + f4) + a4(f3 + f5) (A.36)


f2 = a0f2 + a1(f1 + f3) + a2(f0 + f4) (A.37)

+a3(f1 + f5) + a4(f2 + f6) (A.38)

f3 = a0f3 + a1(f2 + f4) + a2(f1 + f5) (A.39)

+a3(f0 + f6) + a4(f1 + f7) (A.40)

Appendix B

Outflow boundary conditions for

spherical R-M simulations

B.1 Characteristic-based boundary conditions

For an x-boundary, the hyperbolic terms in the Navier-Stokes equations corresponding

to waves propagating in the x direction can be rewritten in terms of characteristic

variables. The modified equations become (Poinsot & Lele [62])

∂ρ

∂t+ d1 +

∂

∂y(ρv) +

∂

∂z(ρw) = 0

∂ρet∂t

+1

2(ukuk)d1 +

d2γ − 1

+ (ρu)d3 + (ρv)d4 + (ρw)d5

+∂

∂y[(ρet + P )v] +

∂

∂z[(ρet + P )w]

=∂

∂xi(ujτij)−

∂qi∂xi

∂

∂t(ρu) + ud1 + ρd3 +

∂

∂y(ρuv) +

∂

∂z(ρuw) =

∂τ1j∂xj

∂

∂t(ρv) + vd1 + ρd4 +

∂

∂y(ρv2) +

∂

∂z(ρvw) +

∂P

∂y=

∂τ2j∂xj

∂

∂t(ρw) + wd1 + ρd5 +

∂

∂y(ρwv) +

∂

∂z(ρw2) +

∂P

∂z=

∂τ3j∂xj

165

166 APPENDIX B. OUTFLOW BOUNDARY CONDITIONS

where

L1

L2

L3

L4

L5

=

λ1(∂P∂x

− ρc∂u∂x)

λ2(c2 ∂ρ∂x

− ∂P∂x)

λ3∂v∂x

λ4∂w∂x

λ5(∂P∂x

+ ρc∂u∂x)

(B.1)

and

d1

d2

d3

d4

d5

=

1c2(L2 +

12(L1 + L5))

12(L5 + L1)

12ρc

(L5 −L1)

L3

L4

(B.2)

For the y-boundary,

∂ρ

∂t+ e1 +

∂

∂x(ρu) +

∂

∂z(ρw) = 0

∂ρet∂t

+1

2(ukuk)e1 +

e2γ − 1

+ (ρu)e3 + (ρv)e4 + (ρw)e5

+∂

∂x[(ρet + P )u] +

∂

∂z[(ρet + P )w]

=∂

∂xi(ujτij)−

∂qi∂xi

∂

∂t(ρv) + ve1 + ρe3 +

∂

∂x(ρvu) +

∂

∂z(ρvw) =

∂τ1j∂xj

∂

∂t(ρu) + ue1 + ρe4 +

∂

∂x(ρu2) +

∂

∂z(ρuw) +

∂P

∂x=

∂τ1j∂xj

∂

∂t(ρw) + we1 + ρe5 +

∂

∂x(ρwu) +

∂

∂z(ρw2) +

∂P

∂z=

∂τ3j∂xj

where

M1

M2

M3

M4

M5

=

µ1(∂P∂y

− ρc∂v∂y)

µ2(c2 ∂ρ∂y

− ∂P∂y)

µ3∂u∂y

µ4∂w∂y

µ5(∂P∂y

+ ρc∂v∂y)

(B.3)

B.1. CHARACTERISTIC-BASED BOUNDARY CONDITIONS 167

and

e1

e2

e3

e4

e5

=

1c2(M2 +

12(M1 +M5))

12(M5 +M1)

12ρc

(M5 −M1)

M3

M4

(B.4)

For the z-boundary

∂ρ

∂t+ f1 +

∂

∂x(ρu) +

∂

∂y(ρv) = 0

∂ρet∂t

+1

2(ukuk)f1 +

f2γ − 1

+ (ρw)f3 + (ρu)f4 + (ρv)f5

+∂

∂x[(ρet + P )u] +

∂

∂y[(ρet + P )v]

=∂

∂xi(ujτij)−

∂qi∂xi

∂

∂t(ρw) + wf1 + ρf3 +

∂

∂x(ρwu) +

∂

∂y(ρwv) =

∂τ3j∂xj

∂

∂t(ρu) + uf1 + ρf4 +

∂

∂x(ρu2) +

∂

∂y(ρuv) +

∂P

∂x=

∂τ1j∂xj

∂

∂t(ρv) + vf1 + ρf5 +

∂

∂x(ρvu) +

∂

∂y(ρv2) +

∂P

∂y=

∂τ2j∂xj

where

N1

N2

N3

N4

N5

=

ν1(∂P∂z

− ρc∂w∂z)

ν2(c2 ∂ρ∂z

− ∂P∂z)

ν3∂u∂z

ν4∂v∂z

ν5(∂P∂z

+ ρc∂w∂z)

(B.5)

168 APPENDIX B. OUTFLOW BOUNDARY CONDITIONS

and

f1

f2

f3

f4

f5

=

1c2(N2 +

12(N1 +N5))

12(N5 +N1)

12ρc

(N5 −N1)

N3

N4

(B.6)

B.2 Sponge treatment

In addition to a outflow boundary conditions described above, we also use a sponge

layer as described in Lui [51] to damp outgoing waves before they are absorbed by

the characteristic-based boundary conditions. In the damping sponge treatment, an

artificial absorbent terms is added in negative feedback form to the right hand side

of the Navier-Stokes equations.

∂Q

∂t= Navier-Stokes terms− σ(x)(Q−Qref)

︸︷︷︸

Sponge terms

(B.7)

The sponge strength σ is specified as an exponential,

σ(xi) = A(xend − xi)

n

(xend − xstart)n(B.8)

xstart and xend are the start and end points of the buffer. A = 4.0 and n = 3.0.

Buffer length L = xend − xstart is taken to be 10% of the domain length in each

direction.

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