Maximizing dissipation in a turbulent shear flow by optimal control of its initial state

Abstract

Turbulent shear flows are known to evolve self-similar at sufficiently high Reynolds numbers. In this regime, flow properties remain constant when normalized with local large-scale reference values. Dissipation of turbulent energy occurs at small scales. Thus, controls which increase dissipation levels in a shear flow may lose their effectiveness once the shear flow attains self-similar behavior governed by large-scale motion. In the current work, we investigate the optimal control of kinetic-energy dissipation in a temporally evolving turbulent mixing layer, with controls acting on its initial condition. Gradient-based optimization is used, relying on direct numerical simulations, and adjoint formulations for the determination of the gradients. Focus is on long optimization time windows T up to 160 nondimensional time units. First we investigate the optimal controls which maximize the total energy dissipation over a simulation time window. For increasing T, optimal controls are found to become also optimal for preceding time horizons. Results indicate that these controls facilitate a fast transition of the mixing layer to a self-similar state, which occurs at nondimensional time t≈160-200. We find that the transition Reynolds number at which self-similarity emerges is a factor of 4-5 lower than transition Reynolds numbers reported for uncontrolled temporal mixing layers. Second, we investigate the maximization of the rate of dissipation at the end of the simulation time window. It is observed that dissipation rate can be increased up to two times higher than self-similar dissipation levels. However, at long optimization time windows this leads to a delay of transition to self-similarity. We further find that optimization is strongly nonconvex, and for the second case, we identify two local optima, which are further discussed in terms of differences in coherent structures emerging during the mixing-layer evolution. © 2011 American Institute of Physics.