Numerical stability at r=0 in cylindrical coordinates

For physical accuracy, it is often desirable to solve the governing equations cast in the natural coordinate system of the geometry. In the case of axisymmetric flows, it is cylindrical coordinates. Although more physically realistic, the cylindrical coordinates introduce numerical challenges. The numerical stability of the convective term is characterized by the CFL condition, generally written as σ=u△t/△x<1. Near the cenerline (r=0), the CFL number in the azimuthal direction, σ_θ=u_θ△t/r△θ, becomes highly restrictive due to the small arc length and is one to two orders of magnitude larger than the radial CFL number, σ_r=u_r△t/△r. The goal is to relax the restriction imposed by the azimuthal CFL condition, such that only the radial CFL condition is limiting. To achieve this, we introduce a diffusion-like term, obtained through an explicit filtering of the flow variables, to remove oscillations which arise when σ_θ>1. The method is tested in inviscid and viscous flows and is shown to allow for larger timesteps while recovering the same solution.­­