# BHR equations re-derived with immiscible particle effects

## Abstract

Compressible and variable density turbulent flows with dispersed phase effects are found in many applications ranging from combustion to cloud formation. These types of flows are among the most challenging to simulate. While the exact equations governing a system of particles and fluid are known, computational resources limit the scale and detail that can be simulated in this type of problem. Therefore, a common method is to simulate averaged versions of the flow equations, which still capture salient physics and is relatively less computationally expensive. Besnard developed such a model for variable density miscible turbulence, where ensemble-averaging was applied to the flow equations to yield a set of filtered equations. Besnard further derived transport equations for the Reynolds stresses, the turbulent mass flux, and the density-specific volume covariance, to help close the filtered momentum and continuity equations. We re-derive the exact BHR closure equations which include integral terms owing to immiscible effects. Physical interpretations of the additional terms are proposed along with simple models. The goal of this work is to extend the BHR model to allow for the simulation of turbulent flows where an immiscible dispersed phase is non-trivially coupled with the carrier phase.

- Authors:

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

- Publication Date:

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

- Sponsoring Org.:
- DOE/LANL

- OSTI Identifier:
- 1179062

- Report Number(s):
- LA-UR-15-23316

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 33 ADVANCED PROPULSION SYSTEMS; 79 ASTRONOMY AND ASTROPHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; 42 ENGINEERING; 20 FOSSIL-FUELED POWER PLANTS; 22 GENERAL STUDIES OF NUCLEAR REACTORS; 97 MATHEMATICS AND COMPUTING; 77 NANOSCIENCE AND NANOTECHNOLOGY; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; multiphase, compressible, turbulence, variable density, particle, BHR, Reynolds stress, volume averaging, Favre averaging

### Citation Formats

```
Schwarzkopf, John Dennis, and Horwitz, Jeremy A.
```*BHR equations re-derived with immiscible particle effects*. United States: N. p., 2015.
Web. doi:10.2172/1179062.

```
Schwarzkopf, John Dennis, & Horwitz, Jeremy A.
```*BHR equations re-derived with immiscible particle effects*. United States. doi:10.2172/1179062.

```
Schwarzkopf, John Dennis, and Horwitz, Jeremy A. Fri .
"BHR equations re-derived with immiscible particle effects". United States. doi:10.2172/1179062. https://www.osti.gov/servlets/purl/1179062.
```

```
@article{osti_1179062,
```

title = {BHR equations re-derived with immiscible particle effects},

author = {Schwarzkopf, John Dennis and Horwitz, Jeremy A.},

abstractNote = {Compressible and variable density turbulent flows with dispersed phase effects are found in many applications ranging from combustion to cloud formation. These types of flows are among the most challenging to simulate. While the exact equations governing a system of particles and fluid are known, computational resources limit the scale and detail that can be simulated in this type of problem. Therefore, a common method is to simulate averaged versions of the flow equations, which still capture salient physics and is relatively less computationally expensive. Besnard developed such a model for variable density miscible turbulence, where ensemble-averaging was applied to the flow equations to yield a set of filtered equations. Besnard further derived transport equations for the Reynolds stresses, the turbulent mass flux, and the density-specific volume covariance, to help close the filtered momentum and continuity equations. We re-derive the exact BHR closure equations which include integral terms owing to immiscible effects. Physical interpretations of the additional terms are proposed along with simple models. The goal of this work is to extend the BHR model to allow for the simulation of turbulent flows where an immiscible dispersed phase is non-trivially coupled with the carrier phase.},

doi = {10.2172/1179062},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2015},

month = {5}

}