A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles

An efficient numerical scheme to compute flows past rigid solid bodies moving through viscous incompressible fluid is presented. Solid obstacles of arbitrary shape are taken into account using the volume penalization method to impose no-slip boundary condition. The 2D Navier–Stokes equations, written in the vorticity-streamfunction formulation, are discretized using a Fourier pseudo-spectral scheme. Four different time discretization schemes of the penalization term are proposed and compared. The originality of the present work lies in the implementation of time-dependent penalization, which makes the above method capable of solving problems where the obstacle follows an arbitrary motion. Fluid–solid coupling for freely falling bodies is also implemented. The numerical method is validated for different test cases: the flow past a cylinder, Couette flow between rotating cylinders, sedimentation of a cylinder and a falling leaf with elliptical shape.