# Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

## Abstract

There exists significant demand for improved Reynolds-averaged Navier–Stokes (RANS) turbulence models that are informed by and can represent a richer set of turbulence physics. This paper presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data. A novel neural network architecture is proposed which uses a multiplicative layer with an invariant tensor basis to embed Galilean invariance into the predicted anisotropy tensor. It is demonstrated that this neural network architecture provides improved prediction accuracy compared with a generic neural network architecture that does not embed this invariance property. Furthermore, the Reynolds stress anisotropy predictions of this invariant neural network are propagated through to the velocity field for two test cases. For both test cases, significant improvement versus baseline RANS linear eddy viscosity and nonlinear eddy viscosity models is demonstrated.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Univ. of Texas at Austin, Austin, TX (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1333570

- Report Number(s):
- SAND-2016-7345J

Journal ID: ISSN 0022-1120; applab; PII: S0022112016006157; TRN: US1700127

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Fluid Mechanics

- Additional Journal Information:
- Journal Volume: 807; Journal ID: ISSN 0022-1120

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Ling, Julia, Kurzawski, Andrew, and Templeton, Jeremy.
```*Reynolds averaged turbulence modelling using deep neural networks with embedded invariance*. United States: N. p., 2016.
Web. doi:10.1017/jfm.2016.615.

```
Ling, Julia, Kurzawski, Andrew, & Templeton, Jeremy.
```*Reynolds averaged turbulence modelling using deep neural networks with embedded invariance*. United States. doi:10.1017/jfm.2016.615.

```
Ling, Julia, Kurzawski, Andrew, and Templeton, Jeremy. Tue .
"Reynolds averaged turbulence modelling using deep neural networks with embedded invariance". United States. doi:10.1017/jfm.2016.615. https://www.osti.gov/servlets/purl/1333570.
```

```
@article{osti_1333570,
```

title = {Reynolds averaged turbulence modelling using deep neural networks with embedded invariance},

author = {Ling, Julia and Kurzawski, Andrew and Templeton, Jeremy},

abstractNote = {There exists significant demand for improved Reynolds-averaged Navier–Stokes (RANS) turbulence models that are informed by and can represent a richer set of turbulence physics. This paper presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data. A novel neural network architecture is proposed which uses a multiplicative layer with an invariant tensor basis to embed Galilean invariance into the predicted anisotropy tensor. It is demonstrated that this neural network architecture provides improved prediction accuracy compared with a generic neural network architecture that does not embed this invariance property. Furthermore, the Reynolds stress anisotropy predictions of this invariant neural network are propagated through to the velocity field for two test cases. For both test cases, significant improvement versus baseline RANS linear eddy viscosity and nonlinear eddy viscosity models is demonstrated.},

doi = {10.1017/jfm.2016.615},

journal = {Journal of Fluid Mechanics},

issn = {0022-1120},

number = ,

volume = 807,

place = {United States},

year = {2016},

month = {10}

}

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