Reducing drag of the obstacle in the channel by boundary control: Theory and numerics


Fluid structure interaction comprising of an elastic body immersed in the moving fluid is considered. The fluid is modeled by an incompressible Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions. The goal is to reduce a drag of the obstacle by changing the flow profile on the inlet. This leads to a boundary control problem with a minimization of a hydro-elastic pressure on the interface between the solid and the fluid. The latter is expressed in a form of shape functional. The interface is "free" and it is itself an unknown variable. The problem is reformulated as a quasilinear PDE-control with a free boundary. A distinct feature of the model is that the boundary conditions imposed on the fluid domain are mixed [change from Dirichlet to Neumann]. This feature is well known to cause singularities in elliptic solutions. Handling of these requires a careful analysis of local singularities which depend on the geometry of the domain. The final result provides an existence of optimal control [with volume constraints] which minimizes the drag of the obstacle, under the assumption of sufficiently small strains. The obtained results are illustrated by numerical simulations which confirm and interpret the theoretical findings.

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