An adjoint-based approach to the optimal control of separated flows

Flow separation is relevant to many industrial applications, since it is detrimental to the aerodynamic performance of vehicles, induces vibrations in mechanical structures, but can also contribute to improve mixing in combustion devices. In this thesis, the fundamental problem of separated flow control is addressed using adjoint-based methods applied to sensitivity analysis. Regions where steady control is the most effective to alter the flow are identified from so-called sensitivity maps. These maps are obtained by solving adjoint equations, at the same computational cost as that of solving the uncontrolled flow. The effect of any small-amplitude control being predicted from these maps, they provide useful information to find efficient control strategies, without the need to actually compute the controlled flow. Sensitivity information can also be used as a building block in iterative optimization algorithms aimed at designing optimal control configurations. The problem is tackled from various angles by targeting several characteristic quantities of separated flows: amplification of external forcing (time-harmonic) or external noise (stochastic), geometric properties (position of separation and reattachment points, angle of the dividing streamline at the wall, area of backflow region and recirculation region), linear stability properties, and aerodynamic forces (drag, lift). The sensitivity of these quantities to control is systematically assessed. In particular, passive control by means of a small device introduced in the flow (e.g. a wire), and active control by means of blowing/suction at a solid wall are considered. The amplification of external perturbations is typically large in globally stable but convectively unstable flows, because of non-normal effects, which might trigger bifurcation to unsteadiness and turbulence. It is observed that such amplification can be drastically reduced using appropriate control. In addition, in a backward-facing step flow the sensitivity of white noise amplification is found to be well captured when considering only the optimal perturbation at the most amplified frequency, thus simplifying control design. In the steady flow past a wall-mounted bump, the position of the reattachment point is found to be very sensitive to control, with a sensitivity map largely similar to that of the backflow area and the recirculation area. In contrast, the separation point and the separatrix angles appear robust. In the flow past a typical bluff body, namely a circular cylinder, the sensitivity of the recirculation length is observed to be strongly correlated to that of the leading eigenmode’s growth rate close to the linear instability threshold. Finally, lift and drag sensitivities to steady actuation are presented for the steady flow past another bluff body, the square cylinder, at subcritical Reynolds number. It appears that lift and drag can be modified independently with wall control if its location and orientation are chosen carefully. More generally, the sensitivities of individual pressure and viscous forces can be obtained from a modified adjoint problem, while the sensitivity of any combination of lift and drag is readily obtained at no additional cost. As evidenced by this thesis, a wide variety of steady flow properties can be analysed in terms of sensitivity to steady control, opening interesting prospects for the control of separated flows. This calls for extensions to unsteady flows and unsteady control configurations, either with adjoint-looping to take into account the full flow dynamics, or with a suitable modelling of time-averaged characteristics.

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