^{1,a)}and Daniel Livescu

^{1,b)}

### Abstract

Linear forcing has been proposed as a useful method for forced isotropic turbulence simulations because it is a physically realistic forcing method with a straightforward implementation in physical-space numerical codes [T. S. Lundgren, “Linearly forced isotropic turbulence,”Annual Research Briefs (Center for Turbulence Research, Stanford, CA, 2003), p. 461; C. Rosales and C. Meneveau, “Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,” Phys. Fluids17, 095106 (2005)]. Here, extensions to the compressible case are discussed. It is shown that, unlike the incompressible case, separate solenoidal and dilatational parts for the forcing term are necessary for controlling the stationary state of the compressible case. In addition, the forcing coefficients can be cast in a form that allows the control of the stationary state values of the total dissipation (and thus the Kolmogorov microscale) and the ratio of dilatational to solenoidal dissipation. Linear full spectrum forcing is also compared to its low wavenumber restriction. Low wavenumber forcing achieves much larger Taylor Reynolds number at the same resolution. Thus, high Reynolds number asymptotics can be more readily probed with low wavenumber forced simulations. Since, in both cases, a solenoidal/dilatational decomposition of the velocity field is required, the simplicity of the full spectrum linear forcing implementation in physical-space numerical codes is lost. Nevertheless, low wavenumber forcing can be implemented without using a full Fourier transform, and so is computationally less demanding.

This work was performed under the auspices of DOE. We are grateful for the use of the Los Alamos National Laboratory Roadrunner system during the open science allocations, San Diego Supercomputer Center’s Blue Gene system, and Lawrence Livermore National Laboratory’s ASC Purple, where these simulations were performed. We appreciate the assistance of Mahidhar Tatineni of SDSC. Finally, we thank Mark Taylor of Sandia National Laboratories for providing the code and assistance to compute angle-averaged structure functions.

I. INTRODUCTION

II. GOVERNING EQUATIONS AND NUMERICAL METHOD

III. LINEAR FORCING FOR STATIONARY COMPRESSIBLE ISOTROPIC SIMULATION

IV. FULL SPECTRUM VERSUS LOW WAVENUMBER FORCING

V. SINGLE VERSUS ENSEMBLE REALIZATIONS

VI. CONCLUSIONS

### Key Topics

- Turbulent flows
- 48.0
- Reynolds stress modeling
- 25.0
- Isotropic turbulence
- 24.0
- Mach numbers
- 17.0
- Compressible flows
- 11.0

## Figures

Evolution of , which is the dilatational to solenoidal dissipation ratio, for unsplit and split forcing. When a single forcing term is used [Eq. (7)] there is no control over , and it often continues to grow throughout the simulation. With a split solenoidal/dilatational forcing [Eq. (17)] quickly adjusts to the imposed value (solid horizontal lines). For clarity, single forcing term data are averaged over a 5 s window. The light gray is unaveraged data, and shows high variability in the single term forcing results.

Evolution of , which is the dilatational to solenoidal dissipation ratio, for unsplit and split forcing. When a single forcing term is used [Eq. (7)] there is no control over , and it often continues to grow throughout the simulation. With a split solenoidal/dilatational forcing [Eq. (17)] quickly adjusts to the imposed value (solid horizontal lines). For clarity, single forcing term data are averaged over a 5 s window. The light gray is unaveraged data, and shows high variability in the single term forcing results.

The ratio of dilatational to solenoidal kinetic energy reaches the equilibrium value about the same time as ; however, the specific value is different than and depends on the turbulent Mach number.

The ratio of dilatational to solenoidal kinetic energy reaches the equilibrium value about the same time as ; however, the specific value is different than and depends on the turbulent Mach number.

Evolution of the total dissipation for unsplit and split forcing. The forcing coefficients can be cast in a form that allows the specification of the dissipation (solid horizontal lines) and, thus, the Kolmogorov microscale. The split forcing method adheres closely to the imposed dissipation (solid horizontal lines) unlike the single term forcing method.

Evolution of the total dissipation for unsplit and split forcing. The forcing coefficients can be cast in a form that allows the specification of the dissipation (solid horizontal lines) and, thus, the Kolmogorov microscale. The split forcing method adheres closely to the imposed dissipation (solid horizontal lines) unlike the single term forcing method.

Time variation of the turbulent kinetic energy for unsplit and split forcing. The increase in for unsplit forcing causes the turbulent kinetic energy to increase as well.

Time variation of the turbulent kinetic energy for unsplit and split forcing. The increase in for unsplit forcing causes the turbulent kinetic energy to increase as well.

The energy flux is equal to the dissipation for a range of wavenumbers for low- forcing but not for full spectrum linear forcing, which requires larger resolutions to develop an inertial range. The results correspond to the parameters from run 1c except as noted.

The energy flux is equal to the dissipation for a range of wavenumbers for low- forcing but not for full spectrum linear forcing, which requires larger resolutions to develop an inertial range. The results correspond to the parameters from run 1c except as noted.

The energy content at low wavenumbers is lower when full spectrum linear forcing is used, compared to low- forcing, which results in a lower overall Reynolds number. The results correspond to the parameters from run 1c.

The energy content at low wavenumbers is lower when full spectrum linear forcing is used, compared to low- forcing, which results in a lower overall Reynolds number. The results correspond to the parameters from run 1c.

Compensated kinetic energy spectra obtained from low- split forced simulations at different and values: (a) total kinetic energy, (b) solenoidal, and (c) dilatational parts of the kinetic energy.

Compensated kinetic energy spectra obtained from low- split forced simulations at different and values: (a) total kinetic energy, (b) solenoidal, and (c) dilatational parts of the kinetic energy.

Third-order structure function from low-wavenumber split linear forcing (thick lines, black on-line) and full spectrum split linear forcing (thin lines, red on-line) for series 1c. At a particular resolution, low- forcing produces a higher peak than full spectrum forcing.

Third-order structure function from low-wavenumber split linear forcing (thick lines, black on-line) and full spectrum split linear forcing (thin lines, red on-line) for series 1c. At a particular resolution, low- forcing produces a higher peak than full spectrum forcing.

Structure function curves for the high resolution, nearly incompressible simulation 2a, with grid cells and , using low- split forcing. All three curves peak near the theoretically expected values (horizontal lines). Bottom: 4/15 law, ; middle: 4/5 law, ; and top: 4/3 law, .

Structure function curves for the high resolution, nearly incompressible simulation 2a, with grid cells and , using low- split forcing. All three curves peak near the theoretically expected values (horizontal lines). Bottom: 4/15 law, ; middle: 4/5 law, ; and top: 4/3 law, .

Isotropy relation at third order for simulation 2a, with grid cells and using low- split forcing. Solid line (red on-line): , dashed line (blue on-line): .

Isotropy relation at third order for simulation 2a, with grid cells and using low- split forcing. Solid line (red on-line): , dashed line (blue on-line): .

Error in average statistics when using separately each of three realizations (thin lines) or the average of the three (thick line). The horizontal axis is the total model time required to compute the average. This comparison shows that computing time is better spent measuring statistics over one simulation, rather than taking the average of an ensemble of simulations.

Error in average statistics when using separately each of three realizations (thin lines) or the average of the three (thick line). The horizontal axis is the total model time required to compute the average. This comparison shows that computing time is better spent measuring statistics over one simulation, rather than taking the average of an ensemble of simulations.

Error in the modified wavenumber when the sixth-order compact finite difference scheme is on the same grid as a spectral method (a) and on a finer grid (b) with grid spacing contracted by a factor of .

Error in the modified wavenumber when the sixth-order compact finite difference scheme is on the same grid as a spectral method (a) and on a finer grid (b) with grid spacing contracted by a factor of .

## Tables

Parameter values of simulations, where series 1 was used for long-time simulations on a mesh (Figs. 1–6) and series 2 was used for high-resolution simulations on a mesh (Figs. 5–10). All simulations use and . For the split forcing described below, the initial values of and target are close to their values in the stationary state.

Parameter values of simulations, where series 1 was used for long-time simulations on a mesh (Figs. 1–6) and series 2 was used for high-resolution simulations on a mesh (Figs. 5–10). All simulations use and . For the split forcing described below, the initial values of and target are close to their values in the stationary state.

Low- forcing achieves nearly double the Taylor Reynolds number as that of full spectrum forcing at the same resolution with the same criterion for all runs.

Low- forcing achieves nearly double the Taylor Reynolds number as that of full spectrum forcing at the same resolution with the same criterion for all runs.

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