Control of Görtler vortices in high-speed boundary layer flows using nonlinear boundary region equations

We formulate a mathematical framework for the optimal control of compressible boundary layers to suppress the growth rate of the streamwise vortex system before breakdown occurs. We introduce flow instabilities to the flow either through roughness elements equally separated in the spanwise direction or via freestream disturbances. We reduce the compressible Navier-Stokes equations to the boundary region equations (BRE) in a high Reynolds number asymptotic framework wherein the streamwise wavelengths of the disturbances are assumed to be much larger than the spanwise and wall-normal counterparts. We apply the method of Lagrange multipliers to derive the adjoint compressible boundary region equations and the associated optimality conditions. The wall transpiration velocity represents the control variable while the wall shear stress or the vortex energy designates the cost functional. The control approach induces a significant reduction in the kinetic energy and wall shear stress of the boundary layer flow. Contour plots visually demonstrate how the primary instabilities gradually flatten out as more control iterations are applied.