# Probability density function method for variable-density pressure-gradient-driven turbulence and mixing

## Abstract

Probability density function (PDF) methods are extended to variable-density pressure-gradient-driven turbulence. We apply the new method to compute the joint PDF of density and velocity in a non-premixed binary mixture of different-density molecularly mixing fluids under gravity. The full time-evolution of the joint PDF is captured in the highly non-equilibrium flow: starting from a quiescent state, transitioning to fully developed turbulence and finally dissipated by molecular diffusion. High-Atwood-number effects (as distinguished from the Boussinesq case) are accounted for: both hydrodynamic turbulence and material mixing are treated at arbitrary density ratios, with the specific volume, mass flux and all their correlations in closed form. An extension of the generalized Langevin model, originally developed for the Lagrangian fluid particle velocity in constant-density shear-driven turbulence, is constructed for variable-density pressure-gradient-driven flows. The persistent small-scale anisotropy, a fundamentally 'non-Kolmogorovian' feature of flows under external acceleration forces, is captured by a tensorial diffusion term based on the external body force. The material mixing model for the fluid density, an active scalar, is developed based on the beta distribution. The beta-PDF is shown to be capable of capturing the mixing asymmetry and that it can accurately represent the density through transition, in fully developed turbulence andmore »

- Authors:

- Los Alamos National Laboratory

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1014451

- Report Number(s):
- LA-UR-10-03443; LA-UR-10-3443

TRN: US1102611

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Conference

- Resource Relation:
- Conference: 49th AIAA Aerospace Sciences Meeting and Aerospace Exposition ; January 4, 2011 ; Orlando, FL

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75; 97; ACCELERATION; ANISOTROPY; ASYMMETRY; BINARY MIXTURES; DECAY; DENSITY; DIFFUSION; DISTRIBUTION; HYDRODYNAMICS; KINETIC ENERGY; LAGRANGIAN FUNCTION; PROBABILITY DENSITY FUNCTIONS; RAYLEIGH-TAYLOR INSTABILITY; TURBULENCE; VALIDATION; VELOCITY

### Citation Formats

```
Bakosi, Jozsef, and Ristorcelli, Raymond J.
```*Probability density function method for variable-density pressure-gradient-driven turbulence and mixing*. United States: N. p., 2010.
Web.

```
Bakosi, Jozsef, & Ristorcelli, Raymond J.
```*Probability density function method for variable-density pressure-gradient-driven turbulence and mixing*. United States.

```
Bakosi, Jozsef, and Ristorcelli, Raymond J. Fri .
"Probability density function method for variable-density pressure-gradient-driven turbulence and mixing". United States. https://www.osti.gov/servlets/purl/1014451.
```

```
@article{osti_1014451,
```

title = {Probability density function method for variable-density pressure-gradient-driven turbulence and mixing},

author = {Bakosi, Jozsef and Ristorcelli, Raymond J},

abstractNote = {Probability density function (PDF) methods are extended to variable-density pressure-gradient-driven turbulence. We apply the new method to compute the joint PDF of density and velocity in a non-premixed binary mixture of different-density molecularly mixing fluids under gravity. The full time-evolution of the joint PDF is captured in the highly non-equilibrium flow: starting from a quiescent state, transitioning to fully developed turbulence and finally dissipated by molecular diffusion. High-Atwood-number effects (as distinguished from the Boussinesq case) are accounted for: both hydrodynamic turbulence and material mixing are treated at arbitrary density ratios, with the specific volume, mass flux and all their correlations in closed form. An extension of the generalized Langevin model, originally developed for the Lagrangian fluid particle velocity in constant-density shear-driven turbulence, is constructed for variable-density pressure-gradient-driven flows. The persistent small-scale anisotropy, a fundamentally 'non-Kolmogorovian' feature of flows under external acceleration forces, is captured by a tensorial diffusion term based on the external body force. The material mixing model for the fluid density, an active scalar, is developed based on the beta distribution. The beta-PDF is shown to be capable of capturing the mixing asymmetry and that it can accurately represent the density through transition, in fully developed turbulence and in the decay process. The joint model for hydrodynamics and active material mixing yields a time-accurate evolution of the turbulent kinetic energy and Reynolds stress anisotropy without resorting to gradient diffusion hypotheses, and represents the mixing state by the density PDF itself, eliminating the need for dubious mixing measures. Direct numerical simulations of the homogeneous Rayleigh-Taylor instability are used for model validation.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri Jan 01 00:00:00 EST 2010},

month = {Fri Jan 01 00:00:00 EST 2010}

}