Control of streamwise vortices developing in compressible boundary layers

We derive and test an optimal control algorithm in the context of compressible boundary layers, in an attempt to suppress or at least limit the growth of streamwise vortices caused by high-amplitude freestream disturbances. We aim to reduce the vortex energy and ultimately delay the transition to turbulent flow. We introduce flow instabilities to the flow either through roughness elements equally separated in the spanwise direction or via freestream disturbances. We analytically reduce the compressible Navier-Stokes equations to the compressible boundary region equations (CBRE) in a high Reynolds number asymptotic framework, based on the assumption that the streamwise wavenumber of the streaks is much smaller than the cross-flow wavenumbers. We employ Lagrange multipliers to derive the adjoint compressible boundary region equations, and the associated optimality conditions. The wall transpiration velocity represents the control variable, whereas the wall shear stress or the vortex energy designates the cost functional. We report and discuss results for different Mach numbers, wall conditions, and spanwise separations.