Control of optimal transient growth on high AR trapezoidal wing boundary layers.*

We  performed control of optimal growth on a high aspect ratio trapezoidal wing.  The objective is to reduce the maximum transient growth. We chose NACA 2412 airfoil and base flow is computed using unstructured finite volume method. Stability computations are carried out at different streamwise locations and transient growth is computed using variational method. We use the LQR controller to attenuate the transient growth of instabilities. Application of  accurately computed wall control brings  reduction change in kinetic energy of the perturbations. In order to compute the control, we chose a cost function J  (integral sum of perturbation kinetic energy),  a function that minimizes the control effort with time. Feedback optimal control gain K is designed such a way that it minimizes the  perturbation kinetic energy. We find the optimal K by solving algebraic Ricaati equation. The  feedback -KX_f  is given to the plant and compute the growth rate again. We have computed growth and their control for a range of pressure gradients and Reynolds numbers with and without the application of controls,. A maximum reduction of  56% is observed. Further we could increase the Reynolds number by 30% that the flow can remain stable.