Flow control using shape optimization for unsteady flows

摘要

We aim to show how to use a gradient-based shape optimization tool, first designed for steady configuration, as a tool for aerodynamical flow control. The gradients are provided, in discret level, using automatic differentiation by program. The key idea is that for cost functions lying on the shape, their sensitivity with respect to the state can be neglected in the gradient evaluation. This being often the case in applications, as cost functions are often based on boundary integrals, a cheap evaluation of the gradient avoiding the flow solver differentiation is possible. Hence, we obtain a control law to be applied through existing control devices (e.g. injection or piezzo-electric). This is therefore a cheap alternative to feedback laws obtained through control theory. One advantage is that this approach does not require any evaluation of the flow to build the transfer function. Indeed, the control law is build in real time. In the past, we have used this approach and the approximate gradient for shape optimization on various 2 and 3D, incompressible and compressible configurations of inviscid and viscous turbulent flows [1, 2, 3]. In control problems, the aim is to minimize some unsteady cost function using the same approximation of the gradient we use for steady flows. This gives a control law to be applied by available control devices: piezzo-electric or injection/suction devices. However, for a control law to be efficient and realizable by such control mechanism, the amount of the required deformation or injection/suction velocity has to be as small as possible. For this reason we use transpiration boundary conditions rather than an ALE formulation to simulate the equivalent shape deformation. Our unsteady control approach comes from a generalization of our shape optimization strategy for steady flows. We use instantaneous sensitivities of an unsteady cost function with respect to control parameters to provide control laws.