^{1}, F. F. Grinstein

^{1}, C. R. DeVore

^{2}, J. R. Ristorcelli

^{1}and L. G. Margolin

^{1}

### Abstract

Turbulent mixing of a passive scalar by forced isotropic turbulence with a prescribed mean scalar gradient is studied in the context of implicit large-eddy simulation. The simulation strategy uses a multi-dimensional compressible flux-corrected transport algorithm, with low wavenumber momentum forcing imposed separately for the solenoidal and dilatational velocity components. Effects of grid resolution on the flow and scalar mixing are investigated at turbulent Mach numbers 0.13 and 0.27. Turbulence metrics are used to show that an implicit large-eddy simulation can accurately capture the mixing transition and asymptotic self-similar behaviors predicted by previous theoretical, laboratory, and direct numerical simulation studies, including asymptotically constant scalar variance and increasing velocity-to-scalar Taylor micro-scales ratio as function of effective Reynolds number determined by grid resolution. The results demonstrate the feasibility of predictive under-resolved simulations of high Reynolds number turbulent scalar mixing using implicit large-eddy simulation.

The authors thank Daniel Livescu for stimulating discussions and for sharing detailed information on his compressible turbulence forcing strategy. Los Alamos National Laboratory (LANL) is operated by the Los Alamos National Security, LLC for the U.S. Department of Energy NNSA under Contract No. DE-AC52-06NA25396. This work was made possible by funding from the LANL LDRD-ER on “LES Modeling for Predictive Simulations of Material Mixing,” through exploratory research Project No. 20100441ER.

I. INTRODUCTION

II. SIMULATION METHODOLOGY

A. Convergence issues

B. Governing equations

C. Simulation details

III. NUMERICAL RESULTS

A. Statistical analysis of under-resolved turbulent velocity field

B. Effects of compressibility on underresolved turbulent velocity field

C. Under-resolved passive scalar mixing by under-resolved turbulent velocity field

IV. CONCLUSIONS

### Key Topics

- Turbulent flows
- 35.0
- Large eddy simulations
- 24.0
- Isotropic turbulence
- 20.0
- Turbulence simulations
- 14.0
- Vortex dynamics
- 14.0

##### B01F3/00

## Figures

Time histories of the velocity variances for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563 cells; Case 1. (e) Time histories of the scalar variance for grid resolutions of 323, 643, 1283, and 2563.

Time histories of the velocity variances for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563 cells; Case 1. (e) Time histories of the scalar variance for grid resolutions of 323, 643, 1283, and 2563.

Isosurfaces of vorticity magnitude scaled by vorticity rms for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563, cells; Case 1.

Isosurfaces of vorticity magnitude scaled by vorticity rms for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563, cells; Case 1.

Compensated velocity spectra (black) plotted with the compensated scalar variance spectra (gray). Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; Case 1.

Compensated velocity spectra (black) plotted with the compensated scalar variance spectra (gray). Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; Case 1.

PDFs of velocity fluctuations scaled with rms of the velocity fluctuations u ′. Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; gray open-circle: Gaussian; Case 1.

PDFs of velocity fluctuations scaled with rms of the velocity fluctuations u ′. Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; gray open-circle: Gaussian; Case 1.

PDFs of (a) vorticity magnitude, (b) vorticity magnitude (from DNS data 39 ), (c) transverse derivatives, (d) strain rate magnitude, and (e) longitudinal derivatives scaled with the rms vorticity ω′; Case 1. ILES (black) - solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

PDFs of (a) vorticity magnitude, (b) vorticity magnitude (from DNS data 39 ), (c) transverse derivatives, (d) strain rate magnitude, and (e) longitudinal derivatives scaled with the rms vorticity ω′; Case 1. ILES (black) - solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

PDFs of (a) transverse velocity derivatives and (b) longitudinal velocity derivatives scaled with the rms vorticity ω′. Black dashed-dotted line: ILES Case 1 - 1283; black solid line: ILES Case 2 - 1283. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

PDFs of (a) transverse velocity derivatives and (b) longitudinal velocity derivatives scaled with the rms vorticity ω′. Black dashed-dotted line: ILES Case 1 - 1283; black solid line: ILES Case 2 - 1283. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

Compensated velocity energy spectra. Light gray: dilatational component; dark gray: solenoidal component; black: total. solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; Case 1.

Compensated velocity energy spectra. Light gray: dilatational component; dark gray: solenoidal component; black: total. solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; Case 1.

PDFs of (a) strain rate magnitude and (b) longitudinal velocity derivatives (based on the solenoidal velocity component) scaled with the rms vorticity ω′; Case 1. ILES (black) - solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

PDFs of (a) strain rate magnitude and (b) longitudinal velocity derivatives (based on the solenoidal velocity component) scaled with the rms vorticity ω′; Case 1. ILES (black) - solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563. DNS 39 (gray) - solid line: Re λ = 36; dashed line: Re λ = 60; dashed-dotted line: Re λ = 96; diamond: Re λ = 142; X: Re λ = 168.

Colormaps of fluctuating scalar field scaled by the rms of the fluctuating scalar field in mid-planes of the domain for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563 cells; Case 1. Superimposed on the lower left of each colormap is a box with side length equal to the scalar Taylor micro-scale λθ.

Colormaps of fluctuating scalar field scaled by the rms of the fluctuating scalar field in mid-planes of the domain for grid resolutions of (a) 323, (b) 643, (c) 1283, and (d) 2563 cells; Case 1. Superimposed on the lower left of each colormap is a box with side length equal to the scalar Taylor micro-scale λθ.

PDFs of (a) fluctuations of the passive scalar scaled with rms of the passive scalar fluctuations, (b) magnitude of fluctuating scalar derivative in mean scalar gradient direction scaled with its rms magnitude, and (c) magnitude of fluctuating scalar derivative in isotropic directions scaled with its rms magnitude; Case 1. Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; gray open-circle: Gaussian.

PDFs of (a) fluctuations of the passive scalar scaled with rms of the passive scalar fluctuations, (b) magnitude of fluctuating scalar derivative in mean scalar gradient direction scaled with its rms magnitude, and (c) magnitude of fluctuating scalar derivative in isotropic directions scaled with its rms magnitude; Case 1. Solid line: 323; dashed line: 643; dashed-dotted line: 1283; diamond: 2563; gray open-circle: Gaussian.

Non-dimensional scalar variance as a function of grid resolution determined Re λ . Gray open square/triangle/circle – line: LES; 3 gray open right -triangle – line: LES without scalar SGS model; 3 black full star – line: ILES Case 1; black open star – line: ILES Case 2.

Ratio of square of Taylor micro-scales as a function of grid resolution determined Re λ . Gray full circle – line: DNS; 5 gray full square – line: DNS; 8 gray full triangle – line: DNS; 9 gray open circle – line: LES; 3 gray open triangle – line: LES 3 without scalar SGS model; black full star – line: ILES Case 1; black open star – line: ILES Case 2.

Ratio of square of Taylor micro-scales as a function of grid resolution determined Re λ . Gray full circle – line: DNS; 5 gray full square – line: DNS; 8 gray full triangle – line: DNS; 9 gray open circle – line: LES; 3 gray open triangle – line: LES 3 without scalar SGS model; black full star – line: ILES Case 1; black open star – line: ILES Case 2.

## Tables

Time-averaged values of flow statistics and parameters. Case 1, Ma ∼ 0.27.

Time-averaged values of flow statistics and parameters. Case 1, Ma ∼ 0.27.

Time-averaged values of flow statistics and parameters. Case 2, Ma ∼ 0.13.

Time-averaged values of flow statistics and parameters. Case 2, Ma ∼ 0.13.

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