A Fourier spectral solver for confined Navier-Stokes flow

No-slip boundaries have an important effect on forced and decaying two-dimensional turbulence due to their role as vorticity source. During intensive vortex-wall interactions high-amplitude vorticity filaments are produced. Most of these filaments roll up and form small-scale vortices that are advected into the interior by larger-scale vortices. From a computational point of view, it is a challenge to resolve the multiple temporal and spatial scales. Another challenge is to solve 2D turbulence in different geometries, e.g. square, triangle, circle, or ellipse. In this study we use a fast Fourier spectral technique to simulate the Navier-Stokes equations with no-slip boundary conditions. This is enforced by an immersed boundary technique called volume penalization. The approach has been justified by analytical proofs of the convergence with respect to the penalization parameter. However, the solution of the penalized Navier-Stokes equations is not smooth on the surface of the penalized volume. Therefore, it is not a priori known whether it is possible to actually perform accurate fast Fourier spectral computations. Convergence checks are reported using a recently revived, and unexpectedly difficult, dipole-wall collision as a test case. It is found that Gibbs oscillations have a negligible effect on the flow evolution, also for 2D flows without the presence of reflection symmetry. Convergence results are reported of the angular momentum production by intensive flow-wall interaction.