The implicit large-eddy simulation (ILES) has been utilized as an effective approach for calculating many complex flows at high Reynolds number flows. Richtmyer–Meshkov instability (RMI) induced flow can be viewed as a homogeneous decaying turbulence (HDT) after the passage of the shock. In this article, a critical evaluation of three methods for estimating the effective Reynolds number and the effective kinematic viscosity is undertaken utilizing high-resolution ILES data. Effective Reynolds numbers based on the vorticity and dissipation rate, or the integral and inner-viscous length scales, are found to be the most self-consistent when compared to the expected phenomenology and wind tunnel experiments.

References

References
1.
Richtmyer
,
R. D.
,
1960
, “
Taylor Instability in Shock Acceleration of Compressible Fluids
,”
Commun. Pure Appl. Math.
,
13
(
2
), pp.
297
319
.
2.
Meshkov
,
E. E.
,
1969
, “
Instability of the Interface of Two Gas Accelerated by a Shock Wave
,”
Izv., Acad. Sci., USSR Fluid Dyn.
,
4
, pp.
101
104
.
3.
Muller
,
E.
,
Fryxell
,
B.
, and
Arnett
,
D.
,
1991
, “
Instability and Clumping in SN 1987A
,”
Astron. Astrophys.
,
251
, pp.
505
514
.
4.
Lindl
,
J.
,
1998
,
Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive
,
AIP
,
New York
.
5.
Kifonidis
,
K.
,
Plewa
,
T.
,
Scheck
,
L.
,
Janka
,
H. T.
, and
Müller
,
E.
,
2006
, “Non-Spherical Core Collapse Supernovae-II. The Late-Time Evolution of Globally Anisotropic Neutrino-Driven Explosions and Their Implications for SN 1987 A,”
Astronom. Astrophys.
,
453
(
2
), pp. 661–678.
6.
Haan
,
S. W.
,
Lindl
,
J. D.
,
Callahan
,
D. A.
,
Clark
,
D. S.
,
Salmonson,
J. D.
,
Hammel
,
B. A.
,
Atherton
,
L. J.
,
Cook
,
R. C.
,
Edwards,
M. J.
,
Glenzer
,
S.
, and
Hamza
,
A. V.
,
2011
, “Point Design Targets, Specifications, and Requirements for the 2010 Ignition Campaign on the National Ignition Facility,”
Phys. Plasmas
,
18
(5), p.
051001
.
7.
Grinstein
,
F. F.
,
Margolin
,
L. G.
, and
Rider
,
W. G.
, eds.,
2010
,
Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics
, 2nd printing,
Cambridge University
,
New York
.
8.
Zhou
,
Y.
,
Grinstein
,
F. F.
,
Wachtor
,
A. J.
, and
Haines
,
B. M.
,
2014
, “
Estimating the Effective Reynolds Number in Implicit Large Eddy Simulation
,”
Phys. Rev. E
,
89
(
1
), p.
013303
.
9.
Sagaut
,
P.
,
2006
,
Large Eddy Simulation for Incompressible Flows
,
3rd ed.
,
Springer
,
Berlin
.
10.
Zhou
,
Y.
,
2001
, “
A Scaling Analysis of Turbulent Flows Driven by Rayleigh–Taylor and Richtmyer–Meshkov Instabilities
,”
Phys. Fluids
,
13
(
2
), pp.
538
543
.
11.
Clark
,
T. T.
, and
Zhou
,
Y.
,
2006
, “
Growth Rate Exponents of Richtmyer–Meshkov Mixing Layers
,”
ASME J. Appl. Mech.
,
73
(
3
), pp.
461
468
.
12.
Tritschler
,
V. K.
,
Zubel
,
M.
,
Hickel
,
S.
, and
Adams
,
N. A.
,
2014
, “
Evolution of Length Scales and Statistics of Richtmyer–Meshkov Instability From Direct Numerical Simulations
,”
Phys. Rev. E
,
90
(
6
), p.
063001
.
13.
Cohen
,
R. H.
,
Dannevik
,
W. P.
,
Dimits
,
A. M.
,
Eliason
,
D. E.
,
Mirin
,
A. A.
,
Zhou
,
Y.
,
Porter
,
D. H.
, and
Woodward
,
P. R.
,
2002
, “
Three-Dimensional Simulation of a Richtmyer–Meshkov Instability With a Two-Scale Initial Perturbation
,”
Phys. Fluids
,
14
(
10
), pp.
3692
3709
.
14.
Lombardini
,
M.
,
Pullin
,
D. I.
, and
Meiron
,
D. I.
,
2012
, “
Transition to Turbulence in Shock-Driven Mixing: A Mach Number Study
,”
J. Fluid Mech.
,
690
, pp.
203
226
.
15.
Batchelor
,
G. K.
, and
Proudman
,
I.
,
1956
, “
The Large-Scale Structure of Homogeneous Turbulence
,”
Philos. Trans. R. Soc. London, Ser. A
,
248
(
949
), pp.
369
405
.
16.
Zhou
,
Y.
,
Matthaeus
,
W. H.
, and
Dmitruk
,
P.
,
2004
, “
Colloquium: Magnetohydrodynamic Turbulence and Time Scales in Astrophysical and Space Plasmas
,”
Rev. Mod. Phys.
,
76
, pp.
1065
1035
.
17.
Zhou
,
Y.
, and
Oughton
,
S.
,
2011
, “
Nonlocality and the Critical Reynolds Numbers of the Minimum State Magnetohydrodynamic Turbulence
,”
Phys. Plasmas
,
18
(
7
), p.
072304
.
18.
Zhou
,
Y.
,
1995
, “
A Phenomenological Treatment of Rotating Turbulence
,”
Phys. Fluids
,
7
(
8
), pp.
2092
2094
.
19.
Thornber
,
B.
,
Mosedale
,
A.
, and
Drikakis
,
D.
,
2007
, “
On the Implicit Large Eddy Simulation of Homogeneous Decaying Turbulence
,”
J. Comput. Phys.
,
226
(
2
), pp.
1902
1929
.
20.
Thornber
,
B.
, and
Zhou
,
Y.
,
2012
, “
Energy Transfer in the Richtmyer–Meshkov Instability
,”
Phys. Rev. E
,
86
(
5
), p.
056302
.
21.
Garcia-Uceda Juarez
,
A.
,
Raimo
,
A.
,
Shapiro
,
E.
, and
Thornber
,
B.
,
2014
, “
Steady Turbulent Flow Computations Using a Low Mach Fully Compressible Scheme
,”
AIAA J.
,
52
(
11
), pp.
2559
2575
.
22.
Shanmuganathan
,
S.
,
Youngs
,
D.
,
Griffond
,
J.
,
Thornber
,
B.
, and
Williams
,
R.
,
2014
, “
Accuracy of High-Order Density-Based Compressible Methods in Low Mach Vortical Flows
,”
Int. J. Numer. Methods Fluids
,
74
(
5
), pp.
335
358
.
23.
Probyn
,
M.
,
Thornber
,
B.
,
Drikakis
,
D.
,
Youngs
,
D.
, and
Williams
,
R.
,
2014
, “
An Investigation Into Non-Linear Growth Rate of Two-Dimensional and Three-Dimensional Single-Mode Richtmyer–Meshkov Instability Using an Arbitrary-Lagrangian–Eulerian Algorithm
,”
ASME J. Fluids Eng.
,
136
(
9
), p.
091208
.
24.
Thornber
,
B.
,
Mosedale
,
A.
,
Drikakis
,
D.
,
Williams
,
R. J. R.
, and
Youngs
,
D.
,
2008
, “
An Improved Reconstruction Method of Compressible Flows With Low Mach Number Features
,”
J. Comput. Phys.
,
227
(
10
), pp.
4873
4894
.
25.
Thornber
,
B.
,
Drikakis
,
D.
,
Youngs
,
D. L.
, and
Williams
,
R. J. R.
,
2010
, “
The Influence of Initial Conditions on Turbulent Mixing Due to Richtmyer–Meshkov Instability
,”
J. Fluid Mech.
,
654
, pp.
99
139
.
26.
Thornber
,
B.
,
Drikakis
,
D.
,
Youngs
,
D. L.
, and
Williams
,
R. J. R.
,
2011
, “
Growth of a Richtmyer–Meshkov Turbulent Layer After Reshock
,”
Phys. Fluids
,
23
(
9
), p.
095107
.
27.
Aspden
,
A.
,
Nikiforakis
,
N.
,
Dalziel
,
S.
, and
Bell
,
J. B.
,
2008
, “
Analysis of Implicit LES Methods
,”
Commun. Appl. Math. Comput. Sci.
,
3
(
1
), pp.
103
126
.
28.
Moin
,
P.
, and
Mahesh
,
K.
,
1998
, “
Direct Numerical Simulation: A Tool in Turbulence Research
,”
Annu. Rev. Fluid Mech.
,
30
(
1
), pp.
539
578
.
29.
Saddoughi
,
S. G.
, and
Veeravalli
,
S. V.
,
1994
, “
Local Isotropy in Turbulent Boundary Layers at High Reynolds Number
,”
J. Fluid Mech.
,
268
, pp.
333
372
.
30.
Tennekes
,
H.
, and
Lumley
,
J. L.
,
1972
,
First Course in Turbulence
,
MIT
,
Cambridge, MA
.
31.
Batchelor
,
G. K.
,
1953
,
The Theory of Homogeneous Turbulence
,
Cambridge University
,
Cambridge, UK
.
32.
Zhou
,
Y.
,
1993
, “
Degree of Locality of Energy Transfer in the Inertial Range
,”
Phys. Fluids A
,
5
(
5
), pp.
1092
1094
.
33.
Zhou
,
Y.
,
1993
, “
Interacting Scales and Energy Transfer in Isotropic Turbulence
,”
Phys. Fluids A
,
5
(
10
), pp.
2511
2524
.
34.
Fureby
,
C.
, and
Grinstein
,
F. F.
,
1999
, “
Monotonically Integrated Large Eddy Simulation of Free Shear Flows
,”
AIAA J.
,
37
(
5
), pp.
544
556
.
35.
Zhou
,
Y.
, and
Speziale
,
C. G.
,
1998
, “
Advances in the Fundamental Aspects of Turbulence: Energy Transfer, Interacting Scales, and Self-Preservation in Isotropic Decay
,”
ASME Appl. Mech. Rev.
,
51
(
4
), pp.
267
301
.
36.
Sreenivasan
,
K. R.
,
1998
, “
An Update on the Energy Dissipation Rate in Isotropic Turbulence
,”
Phys. Fluids
,
10
(
2
), pp.
528
529
.
37.
Kaneda
,
Y.
,
Ishihara
,
T.
,
Yokokawa
,
M.
,
Itakura
,
K.
, and
Uno
,
A.
,
2003
, “
Energy Dissipation Rate and Energy Spectrum in High Resolution Direct Numerical Simulations of Turbulence in a Periodic Box
,”
Phys. Fluids
,
15
(
2
), pp.
L21
L24
.
38.
Monin
,
A. S.
, and
Yaglom
,
A. M.
,
1975
,
Statistical Fluid Mechanics: Mechanics of Turbulence
, Vol.
2
,
MIT
,
Cambridge, MA
.
39.
Bataille
,
F.
,
Rubinstein
,
R.
, and
Hussaini
,
M. Y.
,
2005
, “
Eddy Viscosity and Diffusivity Modeling
,”
Phys. Lett. A
,
346
(1–3), pp.
168
173
.
40.
Chollet
,
J.
,
1984
, “
Two-Point Closures as a Subgrid-Scale Modelling Tool for Large Eddy Simulations
,”
Turbulent Shear Flows IV
,
Springer-Verlag
, Berlin/Heidelberg, pp.
62
72
.
41.
Zhou
,
Y.
,
2010
, “
Renormalization Group Theory for Fluid and Plasma Turbulence
,”
Phys. Rep.
,
488
(
1
), p.
1
.
42.
Dimontakis
,
P. E.
,
2000
, “
The Mixing Transition in Turbulent Flows
,”
J. Fluid Mech.
,
409
, pp.
69
98
.
43.
Zhou
,
Y.
,
2007
, “
Unification and Extension of the Concepts of Similarity Criteria and Mixing Transition for Studying Astrophysics Using High Energy Density Laboratory Experiments or Numerical Simulations
,”
Phys. Plasmas
,
14
(
8
), p.
082701
.
44.
Zhou
,
Y.
,
Robey
,
H. F.
, and
Buckingham
,
A. C.
,
2003
, “
Onset of Turbulence in Accelerated High-Reynolds-Number Flow
,”
Phys. Rev. E
,
67
(
5
), p.
056305
.
45.
Zhou
,
Y.
,
Remington
,
B. A.
,
Robey
,
H. F.
,
Cook
,
A. W.
,
Glendinning
,
S. G.
,
Dimits
,
A.
,
Buckingham
,
A. C.
,
Zimmerman
,
G. B.
,
Burke
,
E. W.
,
Peyser
,
T. A.
,
Cabot
,
W.
, and
Eliason
,
D.
,
2003
, “
Progress in Understanding Turbulent Mixing Induced by Rayleigh–Taylor and Richtmyer–Meshkov Instabilities
,”
Phys. Plasmas
,
10
(
5
), pp.
1883
1896
.
46.
Domaradzki
,
J. A.
,
Xiao
,
Z.
, and
Smolarkiewicz
,
P.
,
2003
, “
Effective Eddy Viscosities in Implicit Large Eddy Simulations of Turbulent Flows
,”
Phys. Fluids
,
15
(
12
), pp.
3890
3893
.
47.
Domaradzki
,
J. A.
, and
Radhakrishnan
,
S.
,
2005
, “
Effective Eddy Viscosities in Implicit Large Eddy Simulations of Decaying High Reynolds Number Turbulence With and Without Rotation
,”
Fluid Dyn. Res.
,
36
(4–6), pp.
385
406
.
48.
Castiglioni
,
G.
, and
Domaradzki
,
J. A.
,
2015
, “
A Numerical Dissipation Rate and Viscosity in Flow Simulations With Realistic Geometry Using Low-Order Compressible Navier–Stokes Solvers
,”
Comput. Fluids
,
119
, pp.
37
46
.
49.
Schranner
,
F. S.
,
Domaradzki
,
J. A.
,
Hickel
,
S.
, and
Adams
,
N. A.
,
2015
, “
Assessing the Numerical Dissipation Rate and Viscosity in Numerical Simulations of Fluid Flows
,”
Comput. Fluids
,
114
, pp.
84
97
.
50.
Olson
,
B. J.
, and
Greenough
,
J.
,
2014
, “
Large Eddy Simulation Requirements for the Richtmyer–Meshkov Instability
,”
Phys. Fluids
,
26
(
4
), p.
044103
.
51.
Kolmogorov
,
A. N.
,
1941
, “
Decay of Isotropic Turbulence in Incompressible Viscous Fluids
,”
Docl. Akad. Nauk SSSR A
,
31
, pp.
538
541
.
52.
Hinze
,
J.
,
1975
,
Turbulence
,
2nd ed.
,
McGraw-Hill
, New York.
53.
Oberlack
,
M.
,
2002
, “
On the Decay Exponent of Isotropic Turbulence
,”
Proc. Appl. Math. Mech.
,
1
(
1
), pp.
294
297
.
54.
Saffman
,
P. G.
,
1967
, “
The Large-Scale Structure of Homogeneous Turbulence
,”
J. Fluid Mech.
,
27
(
03
), pp.
581
593
.
55.
Sreenivasan
,
K. R.
, and
Antonia
,
R. A.
,
1997
, “
The Phenomenology of Small-Scale Turbulence
,”
Annu. Rev. Fluid Mech.
,
29
(
1
), pp.
435
472
.
56.
Falkovich
,
G.
,
1994
, “
Bottleneck Phenomenon in Developed Turbulence
,”
Phys. Fluids
,
6
(
4
), pp.
1411
1414
.
57.
Ishihara
,
T.
,
Gotoh
,
T.
, and
Kaneda
,
Y.
,
2008
, “
Study of High-Reynolds Number Isotropic Turbulence by Direct Numerical Simulation
,”
Annu. Rev. Fluid Mech.
,
41
, pp.
165
180
.
58.
Thormann
,
A.
, and
Meneveau
,
C.
,
2014
, “
Decay of Homogeneous, Nearly Isotropic Turbulence Behind Active Fractal Grids
,”
Phys. Fluids
,
26
(
2
), p.
025112
.
59.
Andrews
,
M. J.
,
Youngs
,
D. L.
,
Livescu
,
D.
, and
Wei
,
T.
,
2014
, “
Computational Studies of Two-Dimensional Rayleigh–Taylor Driven Mixing for a Tilted-Rig
,”
ASME J. Fluids Eng.
,
136
(
9
), p.
091212
.
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