A compressible boundary layer optimal control approach using nonlinear boundary region equations

High-amplitude free-stream turbulence and large surface roughness elements can excite a laminar boundary layer sufficiently enough to cause streamwise oriented vortices to form. The latter is accompanied by streaks of varying amplitudes that ‘wobble’ through inviscid secondary instabilities and, ultimately, transition to turbulence. In this paper, 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. This has a commensurate impact on the wall shear stress and heat transfer at the wall. Flow instabilities are introduced either through roughness elements equally separated in the spanwise direction or via free-stream disturbances. The compressible Navier-Stokes equations are reduced 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. The method of Lagrange multipliers is used to derive the adjoint compressible boundary region equations via an appropriate transformation of the original constrained optimization problem into an unconstrained form. In the present formulation, the wall transpiration velocity represents the control variable while the wall shear stress or the vortex energy represents the cost functional. Our study shows that this kind of 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.