Optimal initial perturbations in a boundary layer with wall actuation

Waves of span-wise velocity at the surface of the flow body, the wall, are known to be very effective in reducing the friction drag in turbulent channels and boundary layers. They can also delay the laminar-turbulent transition. To investigate this interesting property, in this work, we add velocity perturbations within a 3D Blasius boundary layer with wall actuation by means of a standing sinusoidal wave in the stream-wise direction. We look for the initial velocity perturbation pattern able to trigger the maximum energy gain in a given target time. The Navier-Stokes equations act as a constraint in the optimization problem. The results are strongly affected by the actuating parameters, namely the amplitude and wave-length of the sinusoidal profile, in terms of the energy gain and also of the space travelled by initial velocity perturbations during the target time. Opposite behaviours arise, such as an energy gain/loss whenever the actuating wave-length is greater/smaller of the space travelled by the perturbation.