Nonlinear optimal control of bypass transition in a boundary layer flow

The central aim of the paper is to apply and assess a nonlinear optimal control strategy to suppress bypass transition, due to bimodal interactions [T. A. Zaki and P. A. Durbin, "Mode interaction and the bypass route to transition," J. Fluid Mech. 531, 85 (2005)] in a zero-pressure-gradient boundary layer. To this end, a Lagrange variational formulation is employed that results in a set of adjoint equations. The optimal wall actuation (blowing and suction from a control slot) is found by solving iteratively the nonlinear Navier-Stokes and the adjoint equations in a forward/backward loop using direct numerical simulation. The optimization is performed in a finite time horizon. Large values of optimization horizon result in the instability of the adjoint equations. The control slot is located exactly in the region of transition. The results show that the control is able to significantly reduce the objective function, which is defined as the spatial and temporal integral of the quadratic deviation from the Blasius profile plus a term that quantifies the control cost. The physical mechanism with which the actuation interacts with the flow field is investigated and analysed in relation to the objective function employed. Examination of the joint probability density function shows that the control velocity is correlated with the streamwise velocity in the near wall region but this correlation is reduced as time elapses. The spanwise averaged velocity is distorted by the control action, resulting in a significant reduction of the skin friction coefficient. Results are presented with and without zero-net mass flow constraint of the actuation velocity. The skin friction coefficient drops below the laminar value if there is no mass constraint; it remains however larger than laminar when this constraint is imposed. Results are also compared with uniform blowing using the same time-average velocity obtained from the nonlinear optimal algorithm.