A time-spectral approximate Jacobian based linearized compressible Navier-Stokes solver for high-speed boundary-layer receptivity and stability

A numerical method for conducting linear receptivity and stability investigations of high-speed wall-bounded flows based on the linearized compressible Navier-Stokes equations is presented. The current approach is directly applicable for stability investigations of arbitrarily complex geometries. The left-hand-side operator for the linear system of equations is obtained by computing numerical right-hand-side Jacobians while the right-hand-side is build from exact flux Jacobians. Utilizing the numerical right-hand-side Jacobian approach avoids lengthy, error prone derivation of the stability equations in the context of generalized curvilinear coordinates. The governing equations are solved using either time-stepping or time-spectral discretizations. Three different time-spectral approaches, i.e., direct inversion, unfactored and factored schemes, are presented and their numerical characteristics for the solution of the linearized Navier-Stokes equations for linear receptivity and stability analysis for large-scale transition problems are explored. Linear receptivity and stability calculation results are provided for different solver options. Performance comparison of the three schemes are presented for a wide range of test cases: An incompressible cross-flow for a swept flat plate boundary layer, a supersonic shock-boundary-layer interaction, hypersonic boundary layers on a flat plate and a flared cone, and, finally, for the receptivity of a hypersonic boundary layer for a right sharp cone.