Uniform Weighted Compact / Non-compact Schemes For Shock / Boundary Layer Interaction

The critical problem of CFD is perhaps an accurate approximation of derivatives for a given discrete data set. Based on our previous work on the weighted compact scheme (WCS), a uniform weighted compact / non-compact scheme (UWCNC) has been developed. Similar to WENO, three high order candidates, left, right, and central, are constructed by using Hermite polynomials. According to the smoothness, three weights are derived and assigned to each candidate. The weights will lead the scheme to be upwind-biased when approaching the shock or other discontinuities but quickly becomes central, compact, and of high order just off the shock. Therefore, the new scheme can get a sharp shock without oscillation, but keep central, compact and of high resolution in the smooth area. This feature is particularly important to numerical simulation of the shock-boundary layer interaction, where both shock and small eddies are important. Comparing with 5th order WENO which has 5th order accuracy in the smooth area and 3rd order accuracy near the shock, UWCNC scheme is superior with smaller stencils and higher order of accuracy. The necessary dissipation is provided by weights and some high order upwind-biased scheme. The new scheme has been successfully applied to 1-Dimensional shock tube and shock-entropy interaction and 2-Dimensional incident shock reflection. The new scheme has obtained sharper shock, no deformation of expansion wave, and much higher resolution than 5th order WENO for small length scales. A variety of cases including shock-boundary interaction with incident shock and double angles has been tested. The preliminary numerical solution is encouraging.