^{1,2}, Nicholas P. Malaya

^{2}, Henry Chang

^{1}, Paulo S. Zandonade

^{3}, Prakash Vedula

^{4}, Amitabh Bhattacharya

^{5}and Andreas Haselbacher

^{6}

### Abstract

Large eddy simulation(LES), in which the large scales of turbulence are simulated while the effects of the small scales are modeled, is an attractive approach for predicting the behavior of turbulent flows. However, there are a number of modeling and formulation challenges that need to be addressed for LES to become a robust and reliable engineering analysis tool. Optimal LES is a LESmodeling approach developed to address these challenges. It requires multipoint correlation data as input to the modeling, and to date these data have been obtained from direct numerical simulations (DNSs). If optimal LES is to be generally useful, this need for DNS statistical data must be overcome. In this paper, it is shown that the Kolmogorov inertial range theory, along with an assumption of small-scale isotropy, the application of the quasinormal approximation and a mild modeling assumption regarding the three-point third-order correlation are sufficient to determine all the correlation data required for optimal LESmodeling. The models resulting from these theoretically determined correlations are found to perform well in isotropic turbulence, with better high-wavenumber behavior than the dynamic Smagorinsky model. It is expected that these theory-based optimal models will be applicable to a wide range of turbulent flows, in which the small scales can be modeled as isotropic and inertial. The optimal models developed here are expressed as generalized quadratic and linear finite-volume operators. There are significant quantitative differences between these optimal LES operators and standard finite-volume operators, and these differences can be interpreted as the model of the subgrid effects. As with most other LESmodels, these theory-based optimal models are expected to break down near walls and other strong inhomogeneities.

The research reported here was supported under NSF Grant Nos. CTS-001435, CTS-03-52552, and OCI-07-49286 and AFOSR Grant Nos. F49620-01-1-0181, FA9550-04-1-0032, and FA9550-07-1-0197, and by the Center for Simulation of Advanced Rockets, which is funded by the Department of Energy through University of California Grant No. B341494. Partial support for R.D.M. was provided by the MEC in Spain through their Sabbatical Program. This support is gratefully acknowledged. We also benefited from many discussions with Professor Javier Jiménez and Professor Ron Adrian.

I. INTRODUCTION

II. OPTIMAL LES

A. Finite-volume optimal LES

B. Treatment of the mean

III. THEORETICAL DETERMINATION OF CORRELATIONS

A. LES correlations and multipoint velocity correlations

B. Scaling

IV. APPLICATION OF THEORETICAL OPTIMAL LES

A. Dynamic optimal LES

B. Asymptotic optimal LESmodels

C. LES results

V. CHARACTERIZATION OF THE ASYMPTOTIC KERNELS

A. Operator contributions to energy transfer

B. Spectral analysis of the operators

C. Role of the three-point third-order correlation

VI. CONCLUSIONS

### Key Topics

- Large eddy simulations
- 80.0
- Turbulence simulations
- 60.0
- Turbulent flows
- 43.0
- Finite volume methods
- 30.0
- Numerical modeling
- 22.0

## Figures

Relative error in the quasinormal approximation to the nonzero elements of the fourth-order correlation , with separation in the direction. To obtain , the error in is normalized by . Due to isotropy 2 and 3 indices can be swapped with identical results.

Relative error in the quasinormal approximation to the nonzero elements of the fourth-order correlation , with separation in the direction. To obtain , the error in is normalized by . Due to isotropy 2 and 3 indices can be swapped with identical results.

Generic stencil geometries for the stencil on a staggered grid, for use when estimating fluxes on the faces shown in gray. The stencil used here is obtained by adding a cell above and below the stencil shown on the left (in the direction).

Generic stencil geometries for the stencil on a staggered grid, for use when estimating fluxes on the faces shown in gray. The stencil used here is obtained by adding a cell above and below the stencil shown on the left (in the direction).

Relative error in estimating the dissipation from the longitudinal third-order structure function of a finite-volume filtered velocity field using Eq. (57). Filtered structure functions were computed by filtering the DNS at of Ref. 8. The filter was defined on a cubical finite-volume grid of the size noted, on a periodic domain of size . Each finite volume is of size , and is the offset between the volumes used to compute the structure function.

Relative error in estimating the dissipation from the longitudinal third-order structure function of a finite-volume filtered velocity field using Eq. (57). Filtered structure functions were computed by filtering the DNS at of Ref. 8. The filter was defined on a cubical finite-volume grid of the size noted, on a periodic domain of size . Each finite volume is of size , and is the offset between the volumes used to compute the structure function.

Volumes (and their numbering) in a staggered grid stencil to estimate the flux through the surface between volumes −1 and 1. For the normal component flux , the volumes with solid outlines are on the -component mesh, and the dashed-outline volumes are not used. For the tangential-component flux , the solid volumes are on the -component mesh, and the dashed volumes are on the -component mesh. In Tables I, II, and V–VIII, the volumes are referred to by the numbers shown here, but no distinction is made between and because the values associated with these volumes are the same.

Volumes (and their numbering) in a staggered grid stencil to estimate the flux through the surface between volumes −1 and 1. For the normal component flux , the volumes with solid outlines are on the -component mesh, and the dashed-outline volumes are not used. For the tangential-component flux , the solid volumes are on the -component mesh, and the dashed volumes are on the -component mesh. In Tables I, II, and V–VIII, the volumes are referred to by the numbers shown here, but no distinction is made between and because the values associated with these volumes are the same.

The matrix norms and of the difference between the asymptotic kernels and the kernels at finite , as a function of .

The matrix norms and of the difference between the asymptotic kernels and the kernels at finite , as a function of .

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds number using the finite- kernels, with resolutions ranging from to ( to , respectively). The solid lines in both plots are determined from the Kolmogorov theory. In (a), the two solid lines are a slope (shallow), and the result of filtering a spectrum. In (b) the straight line is , and the other solid line is the structure function of the filtered velocity determined from .

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds number using the finite- kernels, with resolutions ranging from to ( to , respectively). The solid lines in both plots are determined from the Kolmogorov theory. In (a), the two solid lines are a slope (shallow), and the result of filtering a spectrum. In (b) the straight line is , and the other solid line is the structure function of the filtered velocity determined from .

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds number using both finite and asymptotic kernels (signified with “A”), with resolution and ( and , respectively). Solid lines in (a) and (b) are as in Fig. 6.

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds number using both finite and asymptotic kernels (signified with “A”), with resolution and ( and , respectively). Solid lines in (a) and (b) are as in Fig. 6.

Three-dimensional energy spectra (a) and third-order structure function (b) from LES of isotropic turbulence on a finite-volume grid at using the dynamic Smagorinsky model and finite- optimal models, with and stencils. Also shown is a filtered DNS. Numbers on the curve labels indicate the stencil size, 2 signifies the stencil, and 4 is the stencil.

Three-dimensional energy spectra (a) and third-order structure function (b) from LES of isotropic turbulence on a finite-volume grid at using the dynamic Smagorinsky model and finite- optimal models, with and stencils. Also shown is a filtered DNS. Numbers on the curve labels indicate the stencil size, 2 signifies the stencil, and 4 is the stencil.

Spectra of the linear operators. Spectra have been scaled so that they are consistent approximations to the (negative) second derivative. Shown are the spectra for the asymptotic operators and those determined for a grid . and are the spectra for the and operators, respectively. Note that these operators are formulated to represent the flux in Eq. (2), which appear with a minus sign so a positive is dissipative.

Spectra of the linear operators. Spectra have been scaled so that they are consistent approximations to the (negative) second derivative. Shown are the spectra for the asymptotic operators and those determined for a grid . and are the spectra for the and operators, respectively. Note that these operators are formulated to represent the flux in Eq. (2), which appear with a minus sign so a positive is dissipative.

Bilinear spectra [(a)–(c)] and [(d)–(f)] . The spectra are for asymptotic optimal [(a) and (d)], optimal [(b) and (e)], and fourth-order operators [(c) and (f)]. Black contours are positive and gray are negative. The heavy contours indicate the zero level. Contour values are incremented by 0.3.

Bilinear spectra [(a)–(c)] and [(d)–(f)] . The spectra are for asymptotic optimal [(a) and (d)], optimal [(b) and (e)], and fourth-order operators [(c) and (f)]. Black contours are positive and gray are negative. The heavy contours indicate the zero level. Contour values are incremented by 0.3.

The bilinear spectra for the normal quadratic operator along the diagonal . Shown are the spectra for the asymptotic operators , those determined for a grid , and those for a standard fourth-order operator. The vertical line at indicates the boundary between aliased (right) and unaliased (left) portions of the spectrum, and the dotted line is the spectrum of the exact Navier–Stokes operator.

The bilinear spectra for the normal quadratic operator along the diagonal . Shown are the spectra for the asymptotic operators , those determined for a grid , and those for a standard fourth-order operator. The vertical line at indicates the boundary between aliased (right) and unaliased (left) portions of the spectrum, and the dotted line is the spectrum of the exact Navier–Stokes operator.

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds with resolution of . Compared are results using the finite kernels, asymptotic kernels (signified with A), and asymptotic kernels generated by neglecting the terms in Eqs. (63) and (67) (signified with “N”). Solid lines in (a) and (b) are as in Fig. 6.

Three-dimensional energy spectra (a) and third-order structure functions (b) from OLES of isotropic turbulence at infinite Reynolds with resolution of . Compared are results using the finite kernels, asymptotic kernels (signified with A), and asymptotic kernels generated by neglecting the terms in Eqs. (63) and (67) (signified with “N”). Solid lines in (a) and (b) are as in Fig. 6.

## Tables

Values of the elements of as determined for the stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here are determined from the symmetry .

Values of the elements of as determined for the stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here are determined from the symmetry .

Values of the elements of as determined for the stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here are determined from the symmetry .

Contributions to dissipation of the quadratic and linear portions of the asymptotic LES operators for flux of the normal momentum (normal) and tangential momentum (tangential). The sum signifies three times the normal contribution plus six times the tangential, which represents the contributions to the total dissipation of kinetic energy .

Contributions to dissipation of the quadratic and linear portions of the asymptotic LES operators for flux of the normal momentum (normal) and tangential momentum (tangential). The sum signifies three times the normal contribution plus six times the tangential, which represents the contributions to the total dissipation of kinetic energy .

Contributions to dissipation of the quadratic and linear portions of the asymptotic LES operators determined by neglecting the terms in Eqs. (63) and (67) for flux of the normal momentum (normal) and tangential momentum (tangential). The sum signifies three times the normal contribution plus six times the tangential, which represents the contributions to the total dissipation of kinetic energy .

Contributions to dissipation of the quadratic and linear portions of the asymptotic LES operators determined by neglecting the terms in Eqs. (63) and (67) for flux of the normal momentum (normal) and tangential momentum (tangential). The sum signifies three times the normal contribution plus six times the tangential, which represents the contributions to the total dissipation of kinetic energy .

Values of the elements of for a stencil, with volume labels as defined in Fig. 4. The values of for volumes not listed are determined from the symmetry .

Values of the elements of for a stencil, with volume labels as defined in Fig. 4. The values of for volumes not listed are determined from the symmetry .

Values of the elements of and for a stencil. For and , volume label indicates a volume separated by volumes in the normal direction, with, for example, signifying and signifying the neighboring volume (in either direction). For , the volume labels are as defined in Fig. 4.

Values of the elements of and for a stencil. For and , volume label indicates a volume separated by volumes in the normal direction, with, for example, signifying and signifying the neighboring volume (in either direction). For , the volume labels are as defined in Fig. 4.

Values of the elements of for a stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here are determined from the symmetry , the fact that is invariant with respect to permutation of the volume order, and that is invariant to a swap of and .

Values of the elements of for a stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here are determined from the symmetry , the fact that is invariant with respect to permutation of the volume order, and that is invariant to a swap of and .

Values of the elements of , , and for a stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here is determined from the symmetry .

Values of the elements of , , and for a stencil, with volume labels as defined in Fig. 4. The value of for volumes not listed here is determined from the symmetry .

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