For the aim of computing compressible turbulent flowfield involving shock waves, an implicit large eddy simulation (LES) code has been developed based on the idea of monotonically integrated LES. We employ the weighted compact nonlinear scheme (WCNS) not only for capturing possible shock waves but also for attaining highly accurate resolution required for implicit LES. In order to show that WCNS is a proper choice for implicit LES, a two-dimensional homogeneous turbulence is first obtained by solving the Navier–Stokes equations for incompressible flow. We compare the inertial range in the computed energy spectrum with that obtained by the direct numerical simulation (DNS) and also those given by the different LES approaches. We then obtain the same homogeneous turbulence by solving the equations for compressible flow. It is shown that the present implicit LES can reproduce the inertial range in the energy spectrum given by DNS fairly well. A truncation of energy spectrum occurs naturally at high wavenumber limit indicating that dissipative effect is included properly in the present approach. A linear stability analysis for WCNS indicates that the third order interpolation determined in the upwind stencil introduces a large amount of numerical viscosity to stabilize the scheme, but the same interpolation makes the scheme weakly unstable for waves satisfying $kΔx≈1$. This weak instability results in a slight increase in the energy spectrum at high wavenumber limit. In the computed result of homogeneous turbulence, a fair correlation is shown to exist between the locations where the magnitude of $∇×ω$ becomes large and where the weighted combination of the third order interpolations in WCNS deviates from the optimum ratio to increase the amount of numerical viscosity. Therefore, the numerical viscosity involved in WCNS becomes large only at the locations where the subgrid-scale viscosity can arise in ordinary LES. This suggests the reason why the present implicit LES code using WCNS can resolve turbulent flowfield reasonably well.

1.
Daimon
,
Y.
,
,
T.
,
Takaki
,
R.
,
Fujita
,
K.
, and
Takekawa
,
K.
, 2006, “
Evaluation of Ablation and Longitudinal Vortices in Solid Rocket Motor by Computational Fluid Dynamics
,” AIAA Paper No. 2006-5243.
2.
Lesieur
,
M.
,
Metais
,
O.
, and
Comte
,
P.
, 2005,
Large-Eddy Simulations of Turbulence
,
Cambridge University
,
Cambridge
.
3.
Grinstein
,
F. F.
,
Margolin
,
L. G.
, and
Rider
,
W. J.
, 2007,
Implicit Large Eddy Simulation
,
Cambridge University
,
Cambridge
.
4.
Boris
,
J. P.
,
Grinstein
,
F. F.
,
Oran
,
E. S.
, and
Kolbe
,
R. L.
, 1992, “
New Insights Into Large Eddy Simulation
,”
Fluid Dyn. Res.
0169-5983,
10
, pp.
199
288
.
5.
Grinstein
,
F. F.
,
Fureby
,
C.
, and
DeVore
,
C. R.
, 2005, “
On MILES Based on Flux-Limiting Algorithms
,”
Int. J. Numer. Methods Fluids
0271-2091,
47
, pp.
1043
1051
.
6.
van Leer
,
B.
, 1979, “
Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method
,”
J. Comput. Phys.
0021-9991,
32
, pp.
101
136
.
7.
Shu
,
C. W.
, and
Osher
,
S.
, 1988, “
Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes
,”
J. Comput. Phys.
0021-9991,
77
, pp.
439
471
.
8.
Shu
,
C. W.
, and
Osher
,
S.
, 1989, “
Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, II
,”
J. Comput. Phys.
0021-9991,
83
, pp.
32
78
.
9.
Liu
,
X. D.
,
Osher
,
S.
, and
Chan
,
T.
, 1994, “
Weighted Essentially Non-Oscillatory Schemes
,”
J. Comput. Phys.
0021-9991,
115
, pp.
200
212
.
10.
Jiang
,
G. S.
, and
Shu
,
C. W.
, 1996, “
Efficient Implementation of Weighted ENO Schemes
,”
J. Comput. Phys.
0021-9991,
126
, pp.
202
228
.
11.
Drikakis
,
D.
, 2002, “
Embedded Turbulence Model in Numerical Methods for Hyperbolic Conservation Laws
,”
Int. J. Numer. Methods Fluids
0271-2091,
39
, pp.
763
781
.
12.
Drikakis
,
D.
, 2003, “
Advances in Turbulent Flow Computations Using High-Resolution Methods
,”
Prog. Aerosp. Sci.
0376-0421,
39
, pp.
405
424
.
13.
Garnier
,
E.
,
Mossi
,
M.
,
Sagaut
,
P.
,
Comte
,
P.
, and
Deville
,
M.
, 1999, “
On the Use of Shock-Capturing Schemes for Large-Eddy Simulation
,”
J. Comput. Phys.
0021-9991,
153
, pp.
273
311
.
14.
Lele
,
S. K.
, 1992, “
Compact Finite Difference Schemes With Spectral-Like Resolution
,”
J. Comput. Phys.
0021-9991,
103
, pp.
16
42
.
15.
Deng
,
X. G.
, and
Mao
,
M. L.
, 1997, “
Weighted Compact High-Order Nonlinear Schemes for the Euler Equations
,” AIAA Paper No. 97-1941.
16.
Deng
,
X. G.
, and
Zhang
,
H. X.
, 2000, “
Developing High-Order Weighted Compact Nonlinear Schemes
,”
J. Comput. Phys.
0021-9991,
165
, pp.
22
44
.
17.
Ishiko
,
K.
,
Ohnishi
,
N.
, and
,
K.
, 2005, “
Implicit LES of Turbulence Using Weighted Compact Scheme
,”
Nagare
0286-3154,
24
(
5
), pp.
515
523
.
18.
Das
,
A.
, and
Moser
,
R. D.
, 2002, “
Optimal Large-Eddy Simulation of Forced Burgers Equation
,”
Phys. Fluids
1070-6631,
14
(
12
), pp.
4344
4351
.
19.
Aprovitola
,
A.
, and
Denaro
,
F. M.
, 2004, “
On the Application of Congruent Upwind Discretizations for Large Eddy Simulations
,”
J. Comput. Phys.
,
194
, pp.
329
343
. 0021-9991
20.
Kraichnan
,
R. H.
, 1971, “
Inertial-Range Transfer in Two- and Three-Dimensional Turbulence
,”
J. Fluid Mech.
0022-1120,
47
, pp.
525
535
.
21.
,
Y.
, and
Liu
,
M. S.
, 1994, “
A Flux Splitting Scheme With High-Resolution and Robustness for Discontinuities
,” AIAA Paper No. 94-0083.
22.
Chasnov
,
J. R.
, 1997, “
On the Decay of Two-Dimensional Homogeneous Turbulence
,”
Phys. Fluids
1070-6631,
9
(
1
), pp.
171
180
.
23.
Das
,
C.
,
Kida
,
S.
, and
Goto
,
S.
, 2001, “
Overall Self-Similar Decay of Two-Dimensional Turbulence
,”
J. Phys. Soc. Jpn.
0031-9015,
70
(
4
), pp.
966
976
.
24.
Batchelor
,
G. K.
, 1969, “
Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence
,”
Phys. Fluids
0031-9171,
12
, pp.
233
239
.
25.
van Bokhoven
,
L. J. A.
,
Trieling
,
R. R.
,
Clercx
,
H. J. J.
, and
van Heijst
,
G. J. F.
, 2007, “
Influence of Initial Conditions on Decaying Two-Dimensional Turbulence
,”
Phys. Fluids
1070-6631,
19
, p.
046601
.
26.
Thornber
,
B.
,
Drikakis
,
D.
,
Williams
,
R. J. R.
, and
Youngs
,
D.
, 2008, “
On Entropy Generation and Dissipation of Kinetic Energy in High-Resolution Shock-Capturing Schemes
,”
J. Comput. Phys.
,
227
, pp.
4853
4872
. 0021-9991