HIgh order methods for numerical studies of receptivity in high-speed flows

When conducting numerical studies of high-speed Bows in which both low-energy phenomena and flow discontinuities are present, low dissipative methods which account for Gibbs phenomena, known as higher order shock capturing methods, must be considered. However, conventional higher order shock capturing methods are computationally very expensive and very dissipative when compared to linear higher order finite difference schemes. To overcome these limitations we have developed a simple hybrid finite difference scheme which simultaneously achieves low numerical dissipation in smooth flow regions and robust shock capturing near flow discontinuities. Shock capturing is carried out with a standard first, order upwind scheme, while a high-order upwind-biased scheme is used away from discontinuities. A smoothness indicator adapted from the previous work of Ren et al. 2003 is used to detect discontinuities, and the two schemes are smoothly blended with no adjustable parameters. This new methodology is simpler than most previous hybrid approaches, which generally rely on a weighted essentially non-oscillatory (WENO) scheme for shock capturing. By avoiding the expensive weighting of stencils of varying order of accuracy, the present method achieves considerable time savings, while retaining the robustness of the WEND method. The new met hod is demonstrated on several model problems, from inviscid one- dimensional to viscous two-dimensional examples. The fifth/first order hybrid scheme is then used on a hypersonic boundary layer receptivity study of a sharp leading edge Hat plate. The results are compared with a previous fifth-order WENO study with very good agreement. The effect of blunt nose is also considered on the hypersonic boundary layer receptivity by evaluating four blunt noses using the fifth/first order hybrid method. The wall pressure amplitude of the stable modes for a given frequency is found to increase with increasing bluntness. The location of the stable mode is also observed to shift upstream for up to a certain bluntness. These modes then start to shift downstream if the bluntness is further increased.