Homogenization-based optimization and design of microstructured membranes: flow past a circular cylindrical shell

A formal framework to characterize and optimize the flow past permeable membranes through homogenization is proposed and applied to the two-dimensional wake flow past a permeable cylindrical shell. An effective stress jump condition is employed to model the presence of the membrane, where the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip coefficient. The characterization of the steady flow solution for several filtrability and slip values, kept uniform over the membrane, shows that the flow is dominantly influenced by the filtrability and exhibits a recirculation region which moves downstream of the body and eventually disappears, and the suppression of the vortex shedding as long as large values of the filtrability are employed. An inverse procedure to obtain the microscopic geometry is implemented and verified in the case of uniform distributions of filtrability and slip over the membrane. Variations of the filtrability and slip along the membrane are then considered and their distributions are optimized to fulfill a given objective in the context of adjoint-based optimization. The optimal distributions are finally linked back to the microscopic geometry via a homogenization-based inverse procedure.