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Research Papers: Techniques and Procedures

# Implicit Large Eddy Simulation of Two-Dimensional Homogeneous Turbulence Using Weighted Compact Nonlinear Scheme

[+] Author and Article Information
Keiichi Ishiko

Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japank.ishiko@cfd.mech.tohoku.ac.jp

Naofumi Ohnishi, Kazuyuki Ueno, Keisuke Sawada

Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan

J. Fluids Eng 131(6), 061401 (May 13, 2009) (14 pages) doi:10.1115/1.3077141 History: Received April 18, 2008; Revised December 17, 2008; Published May 13, 2009

## Abstract

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.

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## Figures

Figure 3

Initial energy spectrum. The maximum value in the energy spectrum appears at the wavenumber of kp. Eddies with this size are dominant in the initial vorticity distribution.

Figure 4

Initial vorticity distribution

Figure 5

Computed vorticity distribution at t=2.0. (a) ILES-WCNS, (b) ILES-CS5, (c) LES-DSM, and (d) DNS.

Figure 8

Comparison of (a) kinetic energy and (b) enstrophy

Figure 10

Contours of (a) |∇×ω| and (b) weights (ui,jres)2+(vi,jres)2 for interpolation in WCNS

Figure 14

Comparison of energy spectra. (a) t=2.0 and (b) t=6.0.

Figure 17

Comparison of temporal variation of kinetic energy with the initial kinetic energy KE0 minus TΔS for three different Mach numbers, M0=0.1, 0.05, and 0.01. (a) CS5 and (b) WCNS.

Figure 1

The modified wavenumber of fifth order compact schemes with various interpolations. The upper figure shows the real part and the lower one the imaginary part. A closeup view of the imaginary part near kΔx≈1 is shown together.

Figure 2

L2-norms of the difference between the exact solution for a linear advection problem and that given by fifth order compact scheme with various interpolations

Figure 6

Computed vorticity distribution at t=6.0. (a) ILES-WCNS, (b) ILES-CS5, (c) LES-DSM, and (d) DNS.

Figure 7

Comparison of energy spectra. (a) t=2.0 and (b) t=6.0.

Figure 9

Comparison of (a) Reynolds number and (b) enstrophy in logarithmic scale. The solid lines indicate the fitted power laws.

Figure 11

Computed vorticity distribution at t=6.0. (a) ILES-WCNS-512 and (b) ILES-WCNS-2048.

Figure 12

Comparison of energy spectra at t=6.0

Figure 13

Comparison of (a) kinetic energy and (b) enstrophy

Figure 15

Comparison of (a) kinetic energy and (b) enstrophy

Figure 16

Temporal history of kinetic energy with the predicted change using the initial kinetic energy KE0 minus TΔS

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