Uncertainty quantification of the transonic flow around the RAE 2822 airfoil

Uncertainty quantification (UQ) is particularly important in transonic flow problems owing to the amplification of input variability across shock waves. In this study we focus on the transonic flow over the RAE 2822 airfoil subject to a combination of uncertainties in the Mach number, angle of attack, and thickness–to–chord ratio. We represent the variability in the form of uniform probability distributions. This problem corresponds to the external flow test case of the Workshop on Quantification of CFD Uncertainties (Hirsch et al. 2009) organized by the European Sixth Framework Programme research project NODESIM–CFD on Non–Deterministic Simulation for CFD–Based Design Methodolo-gies (Hirsch et al. 2006). The test problem poses specific difficulties for UQ methods due to the presence of a discontinuity in the pressure field, although smooth response surfaces for integral quantities of lift, drag, and pitching moment are expected. Two UQ methods are compared to assess their ability to approximate smooth response surfaces in multi– dimensional probability spaces efficiently and to maintain robustness in the presence of discontinuities. The increased attention for UQ methodologies originates from the experience that conventional methods such as the Monte Carlo approach are too computationally intensive for application to computational fluid dynamics (CFD) problems. On the other hand, the Stochastic Collocation (SC) method (Babuška et al. 2007) based on Gauss quadrature sampling and Lagrangian polynomial interpolation in parameter space, although quite efficient, has been shown to have difficulty approximating higher–dimensional probability spaces and discontinuous responses. Also separated solution approximations have been developed to achieve a linear increase of computational costs with dimension (Doostan & Iaccarino 2009), but those are applied only to smooth problems. For robust approximation of discontinuous responses, multi–element SC (Foo et al. 2008) and Stochastic Galerkin (Le Maˆıtre et al. 2004) methods have been proposed. These approaches are usually based on discretizing the probability space and then using surface reconstruction techniques. For higher–order interpolations these methods can still result in local oscillations and overshoots. Often not all samples in an element can be reused after refinement, and tensor product extensions to higher dimensions are required, which compromises the efficiency of multi–element discretizations. Motivated by the RAE 2822 test case, we develop in this paper a Simplex Elements Stochastic Collocation (SESC) method that combines a robust approximation of discon-tinuous responses with an efficient discretization in multi–dimensional probability spaces. The SESC method is an extension of the simplex elements method with Newton–Cotes quadrature (Witteveen et …