^{1}and Takahiro Tamaki

^{1}

### Abstract

The subgrid-scale (SGS) modeling in large-eddy simulation (LES) which accounts for the effect of unsteadiness and nonequilibrium state in the SGS is considered. Unsteadiness is incorporated by considering the spectral evolution in the forced homogeneous isotropic turbulence using the transport equation for the SGS energy. As for the unfiltered spectrum, perturbative expansion of the Kovasnay spectral model about the Kolmogorov −5/3 energy spectrum which constitutes a base equilibrium state in the inertial subrange, yields the extra components with −7/3 and −9/3 powers. It is shown that these spectra are actually extracted in the direct numerical simulation (DNS) data and these components govern the unsteady energy transfer. As for the SGS real-space representation of the spectral model, we consider the SGS one-equation model. The perturbation expansion is applied to the one-equation model by setting the base SGS energy as the standard Smagorinsky model, which assumes the equilibrium state in the SGS and its spectral counterpart is the Kolmogorov −5/3 spectrum. The solution yields the terms whose spectral counterparts are the components with −7/3 and −9/3 powers. These additional terms are induced by temporal variations of the base SGS energy. In the temporal variations of the grid-scale energy, SGS energy, SGS production term, and SGS dissipation which are obtained by applying the filter to the DNS data, it is shown that these quantities lag in time in this order. This time-lag is not realized in the standard Smagorinsky model and the one-equation model because the SGS dissipation is defined so that it instantaneously adjusts to the SGS energy. In the one-equation model, the direction of the energy cascade in the initial period is opposite to that obtained in the DNS data. To retrieve correct time-lag and direction of energy transfer, we relax this instantaneous adjustment and propose the nonequilibrium Smagorinsky model. In this nonequilibrium model, the SGS energy incurred by the −7/3 spectrum is added to the base Smagorinsky energy. Assessment in actual LES shows that the time-lag predicted using the standard Smagorinsky and the one-equation models is inaccurate, whereas good agreement with the DNS data is achieved in the nonequilibrium Smagorinsky model. Extraction of the grid-scale nonequilibrium energy spectrum yields the −7/3 and −9/3 components in addition to the base −5/3 spectrum. In the nonequilibrium Smagorinsky model, continuation of the grid-scale spectra into the SGS is established for the −5/3 and −7/3 components. As a result, the unsteady energy transfer is more accurately predicted, whereas the standard Smagorinsky model does not have the SGS counterpart for the −7/3 component. Feasibility of employing the eddy-viscosity approximation to account for the transfer in the period in which −9/3 spectrum prevails is discussed.

We are grateful to K. Kawamura for valuable discussions and assistance in development of the codes, and the referees for valuable comments. This work is partially supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 20560148). Main computations are performed at Cybermedia Centre, Osaka University.

I. INTRODUCTION

II. NONEQUILIBRIUM ENERGY SPECTRUM IN THE ONE-EQUATION MODEL

A. Nonequilibrium energy spectrum

B. Perturbation analysis of the one-equation model

C. Proposal of the nonequilibrium Smagorinsky model

III. PROFILES OF DNS DATA AND SPECTRUM EXTRACTION

A. Details of the DNS data

B. Extraction of nonequilibrium energy spectrum in DNS data

C. Energy transfer function in the DNS data

IV. ASSESSMENT OF THE NONEQUILIBRIUM SMAGORINSKY MODEL

A. Setup of the parameters in the SGS models

B. Profiles of the LES results

V. EXTRACTION OF GRID-SCALE ENERGY SPECTRUM

VI. CONCLUSIONS

### Key Topics

- Energy transfer
- 30.0
- Large eddy simulations
- 29.0
- Diffusion
- 14.0
- Isotropic turbulence
- 14.0
- Eddies
- 13.0

## Figures

Temporal variations in the average turbulent energy (⟨K⟩, shown using the solid line (red)) and the dissipation rate (⟨ɛ⟩, the solid line with circles (blue)) obtained from DNS.

Temporal variations in the average turbulent energy (⟨K⟩, shown using the solid line (red)) and the dissipation rate (⟨ɛ⟩, the solid line with circles (blue)) obtained from DNS.

Energy spectra normalized by (⟨⟨ɛ⟩⟩^{1/4}ν^{5/4}) obtained from the DNS data are shown. (a) ⟨E 0(k)⟩ and are plotted versus k⟨⟨η⟩⟩ using the solid line with circles (red) and the solid line (blue), respectively, and |⟨E 2(k)⟩| is plotted versus k⟨⟨η⟩⟩ using the black filled circles. The dotted lines indicate scaling with k ^{−5/3}, k ^{−7/3}, and k ^{−9/3}. (b) is plotted versus k⟨⟨η⟩⟩ using the solid line (blue) and is plotted using the solid line with circles (red). The dotted lines indicate scaling with k ^{−7/3}.

Energy spectra normalized by (⟨⟨ɛ⟩⟩^{1/4}ν^{5/4}) obtained from the DNS data are shown. (a) ⟨E 0(k)⟩ and are plotted versus k⟨⟨η⟩⟩ using the solid line with circles (red) and the solid line (blue), respectively, and |⟨E 2(k)⟩| is plotted versus k⟨⟨η⟩⟩ using the black filled circles. The dotted lines indicate scaling with k ^{−5/3}, k ^{−7/3}, and k ^{−9/3}. (b) is plotted versus k⟨⟨η⟩⟩ using the solid line (blue) and is plotted using the solid line with circles (red). The dotted lines indicate scaling with k ^{−7/3}.

Isocontours of deviatric spectra normalized by (⟨ɛ⟩^{1/4}ν^{5/4}) obtained from DNS are shown as functions of the wavenumber k⟨η⟩ and t. Scaling is logarithmic, allowing for clearer display of the structure. The small frame shows the temporal variations of ⟨K⟩ (solid line/red) and ⟨ɛ⟩ (solid line with circles/blue).

Isocontours of deviatric spectra normalized by (⟨ɛ⟩^{1/4}ν^{5/4}) obtained from DNS are shown as functions of the wavenumber k⟨η⟩ and t. Scaling is logarithmic, allowing for clearer display of the structure. The small frame shows the temporal variations of ⟨K⟩ (solid line/red) and ⟨ɛ⟩ (solid line with circles/blue).

Isocontours of the energy flux Π(k) normalized by ⟨ɛ⟩ obtained from DNS are shown as functions of the wavelength k⟨η⟩ and t. Scaling is logarithmic. The small frame shows the temporal variations of ⟨K⟩ (solid line/red) and ⟨ɛ⟩ (solid line with circles/blue).

Isocontours of the energy flux Π(k) normalized by ⟨ɛ⟩ obtained from DNS are shown as functions of the wavelength k⟨η⟩ and t. Scaling is logarithmic. The small frame shows the temporal variations of ⟨K⟩ (solid line/red) and ⟨ɛ⟩ (solid line with circles/blue).

Distributions of the energy transfer function T 1(k) normalized by ⟨⟨η⟩⟩⟨⟨ɛ⟩⟩ and flux Π1(k) normalized by ⟨⟨ɛ⟩⟩ obtained from the DNS data are shown as functions of the wavelength k⟨⟨η⟩⟩. (a): Average of the deviation from the long term average in Phase 1; (b): Phase 2. The transfer function T 1(k) is shown using the solid line (blue), and the flux Π1(k) using the solid line with circles (red). The dotted lines show the distributions of T 1(k) and Π1(k) in Eq. (11) . Scaling is logarithmic.

Distributions of the energy transfer function T 1(k) normalized by ⟨⟨η⟩⟩⟨⟨ɛ⟩⟩ and flux Π1(k) normalized by ⟨⟨ɛ⟩⟩ obtained from the DNS data are shown as functions of the wavelength k⟨⟨η⟩⟩. (a): Average of the deviation from the long term average in Phase 1; (b): Phase 2. The transfer function T 1(k) is shown using the solid line (blue), and the flux Π1(k) using the solid line with circles (red). The dotted lines show the distributions of T 1(k) and Π1(k) in Eq. (11) . Scaling is logarithmic.

Temporal variations in the grid-scale and SGS quantities obtained from the filtered DNS data. The average grid-scale energy ⟨K gs ⟩ (shown using the black filled circles), SGS energy ⟨K sgs ⟩ (solid line/blue), SGS dissipation term ⟨ɛ sgs ⟩ (dashed-dotted line/red), and the SGS dissipation approximated using the standard Smagorinsky model ⟨ɛ0⟩ (open circles/red) are shown.

Temporal variations in the grid-scale and SGS quantities obtained from the filtered DNS data. The average grid-scale energy ⟨K gs ⟩ (shown using the black filled circles), SGS energy ⟨K sgs ⟩ (solid line/blue), SGS dissipation term ⟨ɛ sgs ⟩ (dashed-dotted line/red), and the SGS dissipation approximated using the standard Smagorinsky model ⟨ɛ0⟩ (open circles/red) are shown.

Distributions of temporal cross correlation functions obtained using the filtered DNS data. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

Distributions of temporal cross correlation functions obtained using the filtered DNS data. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

Temporal variations in the average grid-scale energy ⟨K gs ⟩ and SGS energy ⟨K sgs ⟩. Filtered DNS data (shown using the black filled circles) and the results obtained using the standard Smagorinsky model (dashed line/red), the one-equation model (solid line with circles/green), and the nonequilibrium Smagorinsky model (solid line/blue) are shown. Nonequilibrium denotes the nonequilibrium Smagorinsky model: (a) ⟨K gs ⟩; (b) ⟨K sgs ⟩.

Temporal variations in the average grid-scale energy ⟨K gs ⟩ and SGS energy ⟨K sgs ⟩. Filtered DNS data (shown using the black filled circles) and the results obtained using the standard Smagorinsky model (dashed line/red), the one-equation model (solid line with circles/green), and the nonequilibrium Smagorinsky model (solid line/blue) are shown. Nonequilibrium denotes the nonequilibrium Smagorinsky model: (a) ⟨K gs ⟩; (b) ⟨K sgs ⟩.

Distributions of temporal cross correlation functions obtained using the one-equation model. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green)), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red)), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue)), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

Distributions of temporal cross correlation functions obtained using the one-equation model. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green)), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red)), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue)), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

(a) Temporal variations in the average grid-scale energy ⟨K gs ⟩ (shown using the black line with open circles), SGS energy ⟨K sgs ⟩ (solid line/blue), SGS dissipation term ⟨ɛ sgs ⟩ (filled circles/red), and the SGS production term ⟨P⟩ (solid line with filled circles/green) obtained using the nonequilibrium Smagorinsky model are shown. (b) Distributions of temporal cross correlation functions obtained using the nonequilibrium Smagorinsky model. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green)), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red)), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue)), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

(a) Temporal variations in the average grid-scale energy ⟨K gs ⟩ (shown using the black line with open circles), SGS energy ⟨K sgs ⟩ (solid line/blue), SGS dissipation term ⟨ɛ sgs ⟩ (filled circles/red), and the SGS production term ⟨P⟩ (solid line with filled circles/green) obtained using the nonequilibrium Smagorinsky model are shown. (b) Distributions of temporal cross correlation functions obtained using the nonequilibrium Smagorinsky model. The functions between the average grid-scale energy ⟨K gs ⟩ and the SGS energy ⟨K sgs ⟩ (C τ(⟨K gs ⟩, ⟨K sgs ⟩), shown using the solid line with circles (green)), SGS production term ⟨P⟩ (C τ(⟨K gs ⟩, ⟨P⟩), solid line (red)), and the SGS dissipation ⟨ɛ sgs ⟩ (C τ(⟨K gs ⟩, ⟨ɛ sgs ⟩), solid line with triangles (blue)), are shown. The small inset shows the cross correlation functions averaged in Phase 1 (solid lines) and Phase 2 (dashed lines).

Grid-scale portions of the energy spectra normalized by (⟨⟨ɛ⟩⟩^{1/4}ν^{5/4}) obtained using the standard and nonequilibrium Smagorinsky models are shown. ⟨E 0(k)⟩, , and are plotted versus k⟨⟨η⟩⟩ using the solid line with circles, the solid line (blue), and the solid line with triangles (red), respectively. The dotted lines indicate scaling with k ^{−5/3}, k ^{−7/3}, and k ^{−9/3}: (a) the standard Smagorinsky model; (b) the nonequilibrium Smagorinsky model.

Grid-scale portions of the energy spectra normalized by (⟨⟨ɛ⟩⟩^{1/4}ν^{5/4}) obtained using the standard and nonequilibrium Smagorinsky models are shown. ⟨E 0(k)⟩, , and are plotted versus k⟨⟨η⟩⟩ using the solid line with circles, the solid line (blue), and the solid line with triangles (red), respectively. The dotted lines indicate scaling with k ^{−5/3}, k ^{−7/3}, and k ^{−9/3}: (a) the standard Smagorinsky model; (b) the nonequilibrium Smagorinsky model.

## Tables

Parameters for the computed case: Taylor microscale Reynolds number Re λ; average kinetic energy K(K = u i u i /2); average dissipation rate ɛ; integral length scale L; Taylor microscale λ; average Kolmogorov length η(=(ν^{3}/ɛ)^{1/4}); eddy turnover time according to L, τ L (=L/u ^{′}); Kolmogorov time scale τ K (=(ν/ɛ)^{1/2}); characteristic time due to forcing ; eddy turnover time due to forcing T(=l f /u ^{′}); skewness of velocity fluctuation ; flatness of velocity fluctuation ; skewness of velocity derivative (u x = ∂u 1/∂x 1); flatness of velocity derivative . The second and third columns show the averages in Phase 1 and Phase 2, respectively (see Sec. III ).

Parameters for the computed case: Taylor microscale Reynolds number Re λ; average kinetic energy K(K = u i u i /2); average dissipation rate ɛ; integral length scale L; Taylor microscale λ; average Kolmogorov length η(=(ν^{3}/ɛ)^{1/4}); eddy turnover time according to L, τ L (=L/u ^{′}); Kolmogorov time scale τ K (=(ν/ɛ)^{1/2}); characteristic time due to forcing ; eddy turnover time due to forcing T(=l f /u ^{′}); skewness of velocity fluctuation ; flatness of velocity fluctuation ; skewness of velocity derivative (u x = ∂u 1/∂x 1); flatness of velocity derivative . The second and third columns show the averages in Phase 1 and Phase 2, respectively (see Sec. III ).

Volume and temporal averaged values of the grid-scale energy K gs and the SGS energy K sgs . DNS denotes the filtered DNS data, Smagorinsky denotes the standard Smagorinsky model, One-eq. denotes the one-equation model using the Adams-Bashforth method, and Nonequilibrium denotes the nonequilibrium Smagorinsky model.

Volume and temporal averaged values of the grid-scale energy K gs and the SGS energy K sgs . DNS denotes the filtered DNS data, Smagorinsky denotes the standard Smagorinsky model, One-eq. denotes the one-equation model using the Adams-Bashforth method, and Nonequilibrium denotes the nonequilibrium Smagorinsky model.

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