Unsteady optimal control of shear flows based on the discrete and continuous adjoint Navier-Stokes equations.

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Optimal flow control based on the continuous adjoint has been very successful so far. Nevertheless, this method shows deficiencies for unsteady flow cases at high Reynolds numbers and when models for the non-represented scales are applied. These deficiencies lead to wrong gradient directions during the determination of the minimum of a cost functional and are caused by inconsistencies of the continuous adjoint of the numerically approximated (primal) flow state. As a consequence, only limited control horizons are possible, additionally, all terms in the flow equations of the primal simulation need to be differentiable, which is certainly not the case for all models. Making use of the discrete adjoint offers the possibility to reach machine precision in determining the gradient numerically, based on the calculated primal flow state. For unsteady turbulent flows at high Reynolds numbers this issue has not been investigated in depth. Although the discrete approach is exact based on the numerical (i.e. discrete primal) flow solution, it still contains the modelling and discretization errors when compared to the `exact` flow solution.The aim of this proposal is to compare the continuous and discrete approach, by minimizing the sound emission to the far-field over very long time horizons for several defined flow cases. The comparison is performed using different resolutions and Reynolds numbers, by making use of DNS and Large Eddy Simulation. The discrete adjoint is developed using automatic differentiation tools (AD-Tools) applied on the same flow solver, which serves as a basis for the continuous adjoint control. The comparison tries to identify strengths and weaknesses of the respective approaches and intends to determine successfull approaches to control turbulent flows at high Reynolds numbers.