# A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence

## Abstract

We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1569613

- Report Number(s):
- LA-UR-18-23839

Journal ID: ISSN 1468-5248

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Turbulence (Online)

- Additional Journal Information:
- Journal Name: Journal of Turbulence (Online); Journal Volume: 20; Journal Issue: 7; Journal ID: ISSN 1468-5248

- Publisher:
- Taylor & Francis

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Favre averages; mean turbulent reaction rates; variable density turbulence

### Citation Formats

```
Ristorcelli, J. R., and Bakosi, J. A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence. United States: N. p., 2019.
Web. doi:10.1080/14685248.2019.1662030.
```

```
Ristorcelli, J. R., & Bakosi, J. A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence. United States. doi:10.1080/14685248.2019.1662030.
```

```
Ristorcelli, J. R., and Bakosi, J. Mon .
"A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence". United States. doi:10.1080/14685248.2019.1662030. https://www.osti.gov/servlets/purl/1569613.
```

```
@article{osti_1569613,
```

title = {A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence},

author = {Ristorcelli, J. R. and Bakosi, J.},

abstractNote = {We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.},

doi = {10.1080/14685248.2019.1662030},

journal = {Journal of Turbulence (Online)},

number = 7,

volume = 20,

place = {United States},

year = {2019},

month = {9}

}

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