Nonlinear optimal control of transition due to a pair of vortical perturbations using a receding horizon approach

Abstract

This paper considers the nonlinear optimal control of transition in a boundary layer flow subjected to a pair of free stream vortical perturbations using a receding horizon approach. The optimal control problem is solved using the Lagrange variational technique that results in a set of linearized adjoint equations, which are used to obtain the optimal wall actuation (blowing and suction from a control slot located in the transition region). The receding horizon approach enables the application of control action over a longer time period, and this allows the extraction of time-averaged statistics as well as investigation of the control effect downstream of the control slot. The results show that the controlled flow energy is initially reduced in the streamwise direction and then increased because transition still occurs. The distribution of the optimal control velocity responds to the flow activity above and upstream of the control slot. The control effect propagates downstream of the slot and the flow energy is reduced up to the exit of the computational domain. The mean drag reduction is $55\,\%$ and $10\,\%$ in the control region and downstream of the slot, respectively. The control mechanism is investigated by examining the second-order statistics and the two-point correlations. It is found that in the upstream (left) side of the slot, the controller counteracts the near-wall high-speed streaks and reduces the turbulent shear stress; this is akin to opposition control in channel flow, and because the time-average control velocity is positive, it is more similar to blowing-only opposition control. In the downstream (right) side of the slot, the controller reacts to the impingement of turbulent spots that have been produced upstream and inside the boundary layer (top–bottom mechanism). The control velocity is positive and increases in the streamwise direction, and the flow behaviour is similar to that of uniform blowing.