Quantification of Uncertainties in Turbulence Modeling: A Comparison of Physics-Based and Random Matrix Theoretic Approaches

TL;DR

This paper compares physics-based and random matrix theory approaches for quantifying uncertainties in turbulence modeling, providing insights into prior specification and maximum entropy principles to improve RANS predictions.

Contribution

It introduces a method to assess and enhance prior selection in physics-based Bayesian turbulence uncertainty quantification using random matrix theory insights.

Findings

Perturbations of shape parameters are normally distributed for maximum entropy.

Turbulence kinetic energy perturbations should be log-normal.

Guidance on variance magnitudes for physical variables to achieve maximum entropy.

Abstract

Numerical models based on Reynolds-Averaged Navier-Stokes (RANS) equations are widely used in engineering turbulence modeling. However, the RANS predictions have large model-form uncertainties for many complex flows. Quantification of these large uncertainties originating from the modeled Reynolds stresses has attracted attention in turbulence modeling community. Recently, a physics-based Bayesian framework for quantifying model-form uncertainties has been proposed with successful applications to several flows. Nonetheless, how to specify proper priors without introducing unwarranted, artificial information remains challenging to the current form of the physics-based approach. Another recently proposed method based on random matrix theory provides the prior distributions with the maximum entropy, which is an alternative for model-form uncertainty quantification in RANS simulations. In this work, we utilize the random matrix theoretic approach to assess and possibly improve the specification of priors used in the physics-based approach. The numerical results show that, to achieve maximum entropy in the prior of Reynolds stresses, the perturbations of shape parameters in Barycentric coordinates are normally distributed. Moreover, the perturbations of the turbulence kinetic energy should conform to log-normal distributions. Finally, it sheds light on how large the variance of each physical variable should be compared with each other to achieve the approximate maximum entropy prior. The conclusion can be used as a guidance for specifying proper priors in the physics-based, Bayesian uncertainty quantification framework.