A random matrix approach for quantifying model-form uncertainties in turbulence modeling

With the ever-increasing use of Reynolds-Averaged Navier-Stokes (RANS) simulations in mission-critical applications, the quantification of model-form uncertainty in RANS models has attracted attention in the turbulence modeling community. Recently, a physics-based nonparametric approach for quantifying model-form uncertainty in RANS simulations has been proposed, where Reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits (Xiao et al., 2016). However, a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints. In this work we propose a random matrix approach for quantifying model-form uncertainties in RANS simulations with the realizability of the Reynolds stress guaranteed, which is achieved by construction from the Cholesky factorization of the normalized Reynolds stress tensor. Furthermore, the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints. We demonstrate that the proposed approach is able to ensure the realizability of the Reynolds stress, albeit in a different manner from the physics-based approach. Monte Carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent Gaussian random fields to the Reynolds stress random field with the marginal distributions and correlation structures as specified. Numerical simulations on a typical flow with separation have shown physically reasonable results, which verify the proposed approach. Therefore, the proposed method is a promising alternative to the physics-based approach for model-form uncertainty quantification of RANS simulations. The method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering.