The problem of boundary-layer flow encountering a three-dimensional hump revisited

The receptivity of boundary-layer flow to a three-dimensional hump (an array of humps) at a finite Reynolds number is solved with the help of an expansion of the solution of linearized Navier-Stokes equations into the biorthogonal eigenfunction system. Results of the triple-deck theory are revisited, and it is shown that there is qualitative agreement of the flow-field structure with the finite Reynolds number case. There are a pair of counter-rotating vortices behind the roughness element that bring the high-speed fluid down into the wake region. To analyze the flow-field far downstream of the roughness elements, the receptivity problem solution is used as inflow into a marching procedure of solving the linearized boundary-layer equations. The results reveal that there is a reversal of the streamwise velocity perturbation. Downstream from the point of the reversal, the perturbations possess transient growth. Comparison with the optimal disturbances originating at the point of reversal demonstrates that the velocity field in the wake is close to the optimal one. This means that the optimal perturbations are realizable in experiment with an array of humps placed on the wall.