Optimal state feedback control of streaks and Gortler vortices induced by free-stream vortical disturbances

A framework for active wall-transpiration control of spatially developing, pre-transitional, boundary-layer flows over flat and concave walls (in the streamwise direction) is developed. The controller is designed to suppress the energy growth of streaks and Gortler vortices (with emphasis on the latter) induced by free-stream vortical disturbances within an incompressible boundary layer. The control framework uses the primitive variables, velocity and pressure. The flow model is based on the linearised unsteady boundary-region (LUBR) equations, which are the rigorous asymptotic form of the Navier-Stokes equations in the limit of low frequency and long-streamwise wavelength. At a particular wavenumber, the effect of free-stream turbulence appears as explicit forcing of these equations which is obtained by asymptotic matching with the far field conditions. An optimal control problem that accounts for this explicit forcing is formulated and solved. The objective cost function that is minimised comprises the weighted energy of the streaks/Gortler vortices and of the actuation. Results show the effectiveness of the developed control approach.