A locally refined cut-cell method with exact conservation for the incompressible Navier-Stokes equations

We present a mimetic discretization of the incompressible Navier-Stokes equations for general polygonal meshes. The discretization employs staggered velocity variables and results in discrete equations that exactly conserve mass, momentum and kinetic energy (in the inviscid limit) up to and including the boundaries where Dirichlet conditions apply. Moreover, the discrete equations give rise to a discrete global vorticity that is consistent with the Dirichlet boundary conditions. As the method retains all its favorable properties on general meshes, it can be perfectly applied as a locally refined Cartesian mesh cut-cell method. We numerically verify the conservation properties for the lid-driven cavity flow and demonstrate the method for the unsteady flow around a circular cylinder.