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.
This study proposes a novel approach for the wall-boundary immersed flow simulation, wherein the Navier-Stokes equation is modified to include a level-set definition of a solid body in fluid flow. The proposed numerical model is defined via a system of differential equations based on the law of conservation and has a continuous approximate profile near the solid body. It yields a stable viscosity solution using a simple algorithm and scheme without any upwind schemes, numerical limiters, or addition filters. The model is numerically validated via solutions of flow around a cylinder, which are consistent with theoretical and experimental results for both steady and unsteady cases based on the wide Reynolds number (Re=8–160) of laminar flow condition.
The Bingham fluid simulation model was constructed and validated using a physically consistent particle method, i.e., the Moving Particle Hydrodynamics (MPH) method. When a discrete particle system satisfies the fundamental laws of physics, the method is asserted as physically consistent. Since Bingham fluids sometimes show solid-like behaviors, linear and angular momentum conservation is especially important. These features are naturally satisfied in the MPH method. To model the Bingham feature, the viscosity of the fluid was varied to express the stress-strain rate relation. Since the solid-like part, where the stress does not exceed the yield stress, was modeled with very large viscosity, the implicit velocity calculation was introduced so as to avoid the restriction of the time step width with respect to the diffusion number. As a result, the present model could express the stopping and solid-like behaviors, which are characteristics of Bingham fluids. The proposed method was verified and validated, and its capability was demonstrated through calculations of the two-dimensional Poiseuille flow of a Bingham plastic fluid and the three-dimensional dam-break flow of a Bingham pseudoplastic fluid by comparing those computed results to theory and experiment.