Application of compact schemes in the CUSP framework for strong shock–vortex interaction

Abstract

We present a methodology for application of compact schemes for unsteady compressible flow problems involving strong shocks and vortices. A compact interpolation formula is combined with a solver based on the ‘convective upwind and split pressure’ (CUSP) scheme [32] in conjunction with a limiter to restrict the numerical fluxes at the cell faces. The interpolation scheme is derived in a manner so that its use in a first derivative calculation results in high accuracy in the wave number space. This is established by the modified wavenumber and the numerical group velocity of the scheme. Use of compact schemes in compressible flow with shocks has mostly been limited to shocks at lower Mach number range. We test our solver by computing a Mach 7 shock interacting with a strong vortex. Results from the lower resolution version of the solver match well with existing data from an ENO scheme based solver. The high resolution scheme shows the break-up of the vortex into smaller vortices in great details, including roll-up of Kelvin–Helmholtz instability generated small scale vortices from slip lines. The presented method can be seamlessly integrated into existing finite volume or finite difference compressible solvers when such high resolution capability is required.