The nonlinear transient growth of supersonic shear flows

The linear instability analysis for the flow transition to turbulence in supersonic flow has been studied extensively, including the exponential growth modes know as Mack instability and the linear transient growth. For supersonic flows, the density, velocity and temperature are coupled together to enhance the nonlinear interaction in the flow transition process. However, the fully nonlinear interactions and the related mechanism of the supersonic flows have not been investigated. In this study, we developed a framework to calculate the nonlinear optimal disturbance of supersonic shear flows, including the Couette flow and Poiseuille flow. The fifth order weighted essentially non-oscillatory scheme is employed in the solver of compressible flow in curvilinear coordinates.  The nonlinear optimal disturbance is calculated by optimizing the total perturbation energy including the perturbations of density and temperature. The proposed method is validated by the incompressible nonlinear transient growth and the linear transient growth of compressible shear flows. The Mach number effect on the nonlinear optimal disturbance and its nonlinear transient growth is investigated.