Stochastic modeling of random roughness in shock scattering problems: Theory and simulations

Random roughness is omnipresent in engineering applications and may often affect performance in unexpected way. Here, we employ synergistically stochastic simulations and second-order stochastic perturbation analysis to study supersonic flow past a wedge with random rough surface. The roughness (of length d) starting at the wedge apex is modeled as stochastic process (with zero mean and correlation length A) obtained from a new stochastic differential equation. A multi-element probabilistic collocation method (ME-PCM) on sparse grids is employed to solve the stochastic Euler equations while a WENO scheme is used to discretize the equations in two spatial dimensions. The perturbation analysis is used to verify the stochastic simulations and to provide insight for small values of A  , where stochastic simulations become prohibitively expensive. We show that the random roughness enhances the lift and drag forces on the wedge beyond the rough region, and this enhancement is proportional to (d/A)2(d/A)2. The effects become more pronounced as the Mach number increases. These results can be used in designing smart rough skins for airfoils for maximum lift enhancement at a minimum drag penalty.