Energy reduction of streamwise streaks in high-speed boundary layer flows via optimally controlled wall deformations.

Previous work has shown that streamwise oriented vortices and accompanying streaks, originating from streamwise disturbances or surface imperfection, contribute to the transition to turbulence in boundary layer flows over curved surfaces and induce a significant increase in noise in supersonic and hypersonic wind tunnels, which can cause interference with the measurements in the test section. In this study, we develop a Lagrange multiplier-based wall deformation optimal control technique to suppress the growth rate of the streamwise vortex system in high-speed boundary layer flows. We formulate the method by first deriving the nonlinear compressible boundary region equations (NCBRE) from the full-compressible Navier-Stokes equations in a high Reynolds number asymptotic framework. We then utilize Lagrange multipliers to transform the constrained optimization problem into an unconstrained form. The transformation process results in another set of equations in the form of the adjoint compressible boundary region equations (ACBRE) and corresponding optimality conditions. In the present formalism, the wall displacement in the wall-normal direction and the wall shear stress designate the control variable and the cost functional, respectively. We report and discuss the results of the wall-deformation technique for a range of high-supersonic and hypersonic flows.