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In this paper, we present a method for performing uniformly high-order direct numerical simulations of high-speed flows over arbitrary geometries. The method was developed with the goal of simulating and studying the effects of complex isolated roughness elements on the stability of hypersonic boundary layers. The simulations are carried out on Cartesian grids with the geometries imposed by a third-order cut-stencil method. A fifth-order hybrid weighted essentially non-oscillatory scheme was implemented to capture any steep gradients in the flow created by the geometries and a third-order Runge-Kutta method is used for time advancement. A multi-zone refinement method was also utilized to provide extra resolution at locations with expected complex physics. The combination results in a globally fourth-order scheme in space and third order in time. Results confirming the method's high order of convergence are shown. Two-dimensional and three-dimensional test cases are presented and show good agreement with previous results. A simulation of Mach 3 flow over the logo of the Ubuntu Linux distribution is shown to demonstrate the method's capabilities for handling complex geometries. Results for Mach 6 wall-bounded flow over a three-dimensional cylindrical roughness element are also presented. The results demonstrate that the method is a promising tool for the study of hypersonic roughness-induced transition.

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The effects of surface roughness on the stability of hypersonic flow are of great importance to hypersonic vehicles. The overall goal of our research is to provide a better understanding of the effects of surface roughness on transitional and turbulent hypersonic boundary layers. Direct numerical simulations have been performed of flow in the Boeing/AFOSR Mach 6 quiet wind tunnel at Purdue University. Three-dimensional simulations were performed of the nozzle without any roughness and 2D simulations were performed with an isolated roughness element located on the wall of the nozzle. Two different roughness heights were simulated, both of which were less then the local undisturbed boundary layer thickness. The roughness elements produced a combination of Mach wave and expansion fan in the nozzle. The influence of the roughness elements on the flow is investigated in this paper. It is found that the presence of the elements has a substantial effect on the boundary layer and stability profiles downstream of the element.