Machine-learning-based feedback control for drag reduction in a turbulent channel flow

One of the successful feedback controls for skin-friction drag reduction designed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110), called ‘opposition control’, has a limitation in application because the sensors need to be placed slightly away from the wall, i.e. at $y^+ = 10$, and measure the instantaneous wall-normal velocity. In the present study we train convolutional neural networks using the database of uncontrolled turbulent channel flow at $Re_{\tau } = 178$ to extract the spatial distributions of the wall shear stresses and pressure that closely represent the wall-normal velocity at $y^+ = 10$. The correlations between the predicted wall-normal velocities at $y^+ = 10$ from the wall-variable distributions and true ones are very high, and they are 0.92, 0.96 and 0.96 for the streamwise and spanwise wall shear stresses and pressure, respectively. We perform feedback controls of turbulent channel flow with instantaneous blowing and suction determined by the trained convolutional neural networks from the measured wall-variable distributions. The predicted wall-normal velocities during the controls have higher energy at small to intermediate scales than the true ones, which degrades the control performance in skin-friction drag reduction. By applying a low-pass filter to the predicted wall-normal velocities to remove those scales, we reduce skin-friction drag by up to 18 % whose amount is comparable to that by opposition control. The convolutional neural networks trained at $Re_{\tau } = 178$ are also applied to a higher Reynolds number flow ($Re_{\tau } = 578$), and provide a successful skin-friction drag reduction of 15 %.