Prediction of optimal control input in a fully developed turbulent channel flow by machine learning

抄録

In the present study, we optimize the spatio-temporal distribution of an artificial body force for reducing skin friction drag in a fully developed turbulent channel flow at a low frictional Reynolds number of 110. Specifically, by applying the optimal control theory, the optimal body force distributions for minimizing the kinetic energy at the end of a prescribed time horizon are obtained for fifty independent uncontrolled initial fields. Two different time horizons of T⁺ = 10.9 and 109 are considered. A comparison of the optimal control inputs for the two time horizons reveals that the optimal control input for T⁺ = 10.9 is smoother and applied so as to oppose the local velocity fluctuation, while the optimal control input for T⁺ = 109 is more intermittent and localized around low-speed streaks. The optimal body forces become maximal around y⁺ = 20, where near-wall turbulent structures are dominant. Using the obtained dataset of the instantaneous velocity fields and the corresponding optimal control inputs, we train a machine learning model which learns the relationship between them. It is demonstrated that the present machine learning model predicts quite well the optimal control input for the short time horizon of T⁺ = 10.9, while the prediction performance tends to deteriorate when the time horizon increases to T⁺ = 109. Nonetheless, the essential intermittent and localized features of the optimal control input are well predicted even for the longer time horizon by the present machine learning model. The present results suggest that the developed machine learning model could be used to establish an on-line feedback controller without conducting expensive forward and adjoint looping for determining the control input. Furthermore, it is also shown that employing a non-linear activation function significantly improves the prediction accuracy. This indicates that the relationship between the instantaneous flow field and the optimal control input is essentially non-linear.