A Divergence-free Velocity Reconstruction for Incompressible Flows

In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretization methods for incompressible flows violate this property. The origin of the problem is that discrete velocity approximations of incompressible flows are usually not divergence-free. Hence, the use of divergencefree velocity reconstructions is proposed wherever an L scalar product appears in the discrete variational formulation. The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids. It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.