Bayesian uncertainty quantification of turbulence models based on high-order adjoint

The uncertainties in the parameters of turbulence models employed in computational fluid dynamics simulations are quantified using the Bayesian inference framework and analytical approximations. The posterior distribution of the parameters is approximated by a Gaussian distribution with the most probable value obtained by minimizing the objective function defined by the minus of the logarithm of the posterior distribution. The gradient and the Hessian of the objective function with respect to the parameters are computed using the direct differentiation and the adjoint approach to the flow equations including the turbulence model ones. The Hessian matrix is used both to compute the covariance matrix of the posterior distribution and to initialize the quasi-Newton optimization algorithm used to minimize the objective function. The propagation of uncertainties in output quantities of interest is also presented based on Laplace asymptotic approximations and the adjoint formulation. The proposed method is demonstrated using the Spalart−Allmaras turbulence model parameters in the case of the flat plate flow using DNS data for velocities and the flow through a backward facing step using experimental data for velocities and Reynolds stresses.