^{1,a)}, P. Givi

^{1}and S. B. Pope

^{2}

### Abstract

A methodology termed “frequency-velocity-scalar filtered mass density function” (FVS-FMDF) is developed for large eddy simulation(LES) of turbulent flows. The FVS-FMDF takes account of unresolved subgrid scales (SGSs) by considering the joint probability density function (PDF) of the frequency, the velocity, and the scalar fields. An exact transport equation is derived for the FVS-FMDF in which the effects of convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to PDF methods in Reynolds-averaged Navier–Stokes simulations. The FVS-FMDF transport is modeled via a set of stochastic differential equations (SDEs). The numerical solution procedure is based on a hybrid finite-difference (FD)/Monte Carlo (MC) method in which the LES filtered transport equations are solved by the FD, and the set of SDEs is solved by a LagrangianMC procedure. LES of a temporally developing mixing layer is conducted via the FVS-FMDF, and the results are compared with those via the Smagorinsky SGS closure. All these results are also assessed by comparison with those obtained by direct numerical simulation (DNS). The FVS-FMDF predictions show favorable agreements with DNS data.

This work is sponsored by NASA under Grant No. NNX08AB36A and the U.S. Air Force Office of Scientific Research with Dr. Julian M. Tishkoff as the Program Manager under Grant No. FA9550-06-1-0015 to University of Pittsburgh and Grant No. FA9550-09-1-0047 to Cornell University. The computational support is provided by the National Science Foundation through TeraGrid resources at Pittsburgh Supercomputing Center.

I. INTRODUCTION

II. FORMULATION

III. THE FREQUENCY-VELOCITY-SCALAR FILTERED MASS DENSITY FUNCTION

A. Definitions

B. Exact FVS-FMDF transport equation

C. Modeled FVS-FMDF transport equation

IV. NUMERICAL SOLUTION PROCEDURE

V. RESULTS

A. Flow simulated and numerical specifications

B. Model assessment

C. Comparative assessments of the FVS-FMDF

VI. SUMMARY AND CONCLUDING REMARKS

### Key Topics

- Monte Carlo methods
- 25.0
- Large eddy simulations
- 23.0
- Turbulent flows
- 13.0
- Lagrangian mechanics
- 11.0
- Turbulence simulations
- 8.0

## Figures

The instantaneous isosurface of the filtered frequency at time .

The instantaneous isosurface of the filtered frequency at time .

Consistency of instantaneous filtered frequency obtained from FD and MC at time . The solid and dashed lines denote the linear regression and the 45° lines, respectively. denotes the correlation coefficient. The circles denote the filtered frequency values on the FD grid points.

Consistency of instantaneous filtered frequency obtained from FD and MC at time . The solid and dashed lines denote the linear regression and the 45° lines, respectively. denotes the correlation coefficient. The circles denote the filtered frequency values on the FD grid points.

Cross-stream variation of particle frequency values near the plane at time . The white and black circles denote particles originated in low-speed and high-speed streams, respectively. The squares denote the frequency [Eq. (15)] values on the FD grid points.

Cross-stream variation of particle frequency values near the plane at time . The white and black circles denote particles originated in low-speed and high-speed streams, respectively. The squares denote the frequency [Eq. (15)] values on the FD grid points.

Reynolds-averaged filtered (solid line) and SGS mixing (dashed line) frequency values obtained from FVS-FMDF simulations at time .

Reynolds-averaged filtered (solid line) and SGS mixing (dashed line) frequency values obtained from FVS-FMDF simulations at time .

Particle streamwise velocity values obtained from simulations with the SGS mixing frequency defined as (a) conditional filtered frequency , (b) , and (c) [Eq. (15)]. These particles are collected near the plane at time .

Particle streamwise velocity values obtained from simulations with the SGS mixing frequency defined as (a) conditional filtered frequency , (b) , and (c) [Eq. (15)]. These particles are collected near the plane at time .

Reynolds-averaged (a) filtered streamwise velocity and (b) SGS kinetic energy at time . The results obtained from simulations with the SGS mixing frequency defined as (solid line), (dashed line), and (symbol ◇). The black circles denote filtered DNS data.

Reynolds-averaged (a) filtered streamwise velocity and (b) SGS kinetic energy at time . The results obtained from simulations with the SGS mixing frequency defined as (solid line), (dashed line), and (symbol ◇). The black circles denote filtered DNS data.

Cross-stream variation of Reynolds-averaged filtered (a) streamwise velocity and (b) passive scalar; temporal variation of (c) momentum thickness and (d) scalar thickness of the shear layer. The predictions are obtained from DNS, the FVS-FMDF, and the Smagorinsky model. The dashed lines denote the Smagorinsky predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Cross-stream variation of Reynolds-averaged filtered (a) streamwise velocity and (b) passive scalar; temporal variation of (c) momentum thickness and (d) scalar thickness of the shear layer. The predictions are obtained from DNS, the FVS-FMDF, and the Smagorinsky model. The dashed lines denote the Smagorinsky predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Temporal variation of (a) resolved kinetic energy, (b) SGS dissipation, (c) resolved dissipation, and (d) backscatter predicted by DNS and LES using the FVS-FMDF and the Smagorinsky model. The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF (, ), (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Temporal variation of (a) resolved kinetic energy, (b) SGS dissipation, (c) resolved dissipation, and (d) backscatter predicted by DNS and LES using the FVS-FMDF and the Smagorinsky model. The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF (, ), (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged SGS kinetic energy and components of SGS stress and SGS scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF (, ), (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged SGS kinetic energy and components of SGS stress and SGS scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF (, ), (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged SGS kinetic energy and components of SGS stress and SGS scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged SGS kinetic energy and components of SGS stress and SGS scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged resolved kinetic energy and components of resolved stress and resolved scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged resolved kinetic energy and components of resolved stress and resolved scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged resolved kinetic energy and components of resolved stress and resolved scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged total kinetic energy and components of total stress and total scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged total kinetic energy and components of total stress and total scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Reynolds-averaged total kinetic energy and components of total stress and total scalar flux tensors obtained at time . The dashed lines denote the Smagorinsky model predictions. The symbols denote (▽) FVS-FMDF , (◇) FVS-FMDF , (◻) FVS-FMDF , (△) FVS-FMDF , and (●) filtered DNS data.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content