Studying the Effects of Compressibility in Planar Couette Flow using Resolvent Analysis

Abstract

Analysis of the resolvent operator is used to study the properties of compressible planar Couette flow. In particular, we study how changing the Mach number affects the shape and amplitude of responses to optimal disturbances across a range of spatial and temporal frequencies. We consider Mach numbers up to 5, and show that the dependence of the resolvent norm(leading singular value) on streamwise and spanwise wavenumbers follows similar trends to the incompressible case, with the amplitude of the resolvent norm typically decreasing with increasing Mach number. An exception to this occurs when acoustic eigenmodes (which are not present in the incompressible regime) have eigenvalues sufficiently close to the temporal frequency ω such that modal resonance with this mode is the dominant contributor to the resolvent gain. This occurs, for example, for streamwise-constant disturbances for sufficiently low spanwise wavenumber. In addition, the resolvent formulation of the governing equations allows us to study independently the effects due to an altered mean/equilibrium profile due to compressibility, and the effects due to the changing linearized Navier-Stokes equations. This approach provides a framework for the study of compressible turbulent wall-bounded flows.