# Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach

## Abstract

Despite their well-known limitations, Reynolds-Averaged Navier–Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in today's engineering analysis, design and optimization. While the predictive capability of RANS models depends on many factors, for many practical flows the turbulence models are by far the largest source of uncertainty. As RANS models are used in the design and safety evaluation of many mission-critical systems such as airplanes and nuclear power plants, quantifying their model-form uncertainties has significant implications in enabling risk-informed decision-making. In this work we develop a data-driven, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., realizability, smoothness, and symmetry). An iterative ensemble Kalman method is used to assimilate the prior knowledge and observation data in a Bayesian framework, and to propagate them to posterior distributions of velocities and other Quantities of Interest (QoIs). We use two representative cases, the flow over periodic hills and the flow in a square duct, to evaluate the performance of the proposed framework. Both cases are challenging for standard RANS turbulence models. Simulation results suggest that, evenmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22622205

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 324; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ACCOUNTING; BENCHMARKS; CAPTURE; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DECISION MAKING; DISTRIBUTION; FILTERS; ITERATIVE METHODS; LIMITING VALUES; NAVIER-STOKES EQUATIONS; OPTIMIZATION; PERFORMANCE; PERIODICITY; REYNOLDS NUMBER; ROUGHNESS; STRESSES; SYMMETRY; TURBULENT FLOW

### Citation Formats

```
Xiao, H., E-mail: hengxiao@vt.edu, Wu, J. -L., Wang, J. -X., Sun, R., and Roy, C. J.
```*Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.07.038.

```
Xiao, H., E-mail: hengxiao@vt.edu, Wu, J. -L., Wang, J. -X., Sun, R., & Roy, C. J.
```*Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach*. United States. doi:10.1016/J.JCP.2016.07.038.

```
Xiao, H., E-mail: hengxiao@vt.edu, Wu, J. -L., Wang, J. -X., Sun, R., and Roy, C. J. Tue .
"Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach". United States. doi:10.1016/J.JCP.2016.07.038.
```

```
@article{osti_22622205,
```

title = {Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach},

author = {Xiao, H., E-mail: hengxiao@vt.edu and Wu, J. -L. and Wang, J. -X. and Sun, R. and Roy, C. J.},

abstractNote = {Despite their well-known limitations, Reynolds-Averaged Navier–Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in today's engineering analysis, design and optimization. While the predictive capability of RANS models depends on many factors, for many practical flows the turbulence models are by far the largest source of uncertainty. As RANS models are used in the design and safety evaluation of many mission-critical systems such as airplanes and nuclear power plants, quantifying their model-form uncertainties has significant implications in enabling risk-informed decision-making. In this work we develop a data-driven, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., realizability, smoothness, and symmetry). An iterative ensemble Kalman method is used to assimilate the prior knowledge and observation data in a Bayesian framework, and to propagate them to posterior distributions of velocities and other Quantities of Interest (QoIs). We use two representative cases, the flow over periodic hills and the flow in a square duct, to evaluate the performance of the proposed framework. Both cases are challenging for standard RANS turbulence models. Simulation results suggest that, even with very sparse observations, the obtained posterior mean velocities and other QoIs have significantly better agreement with the benchmark data compared to the baseline results. At most locations the posterior distribution adequately captures the true model error within the developed model form uncertainty bounds. The framework is a major improvement over existing black-box, physics-neutral methods for model-form uncertainty quantification, where prior knowledge and details of the models are not exploited. This approach has potential implications in many fields in which the governing equations are well understood but the model uncertainty comes from unresolved physical processes. - Highlights: • Proposed a physics–informed framework to quantify uncertainty in RANS simulations. • Framework incorporates physical prior knowledge and observation data. • Based on a rigorous Bayesian framework yet fully utilizes physical model. • Applicable for many complex physical systems beyond turbulent flows.},

doi = {10.1016/J.JCP.2016.07.038},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 324,

place = {United States},

year = {2016},

month = {11}

}