A Multigrid Solver for the Steady Incompressible Navier-Stokes Equations on Curvilinear Coordinate Systems

Abstract

A multigrid solver for the steady incompressible Navier-Stokes equations on a curvilinear grid is constructed. The Cartesian velocity components are used in the discretization of the momentum equations. A staggered, geometrically symmetric distribution of velocity components is adopted which eliminates spurious pressure oscillations and facilitates the transformation between Cartesian and co-or contravariant velocity components. The SCGS (symmetrical collective Gauss-Seidel) relaxation scheme proposed by Vanka on a Cartesian grid is extended to this case to serve as the smoothing procedure of the multigrid solver, in both "box" and "box-line" versions. Due to the symmetric distribution of velocity components of this scheme, the convergence rate and numerical accuracy are not affected by grid orientation, in contrast to a scheme proposed in the literature in which difficulties arise when the grid lines turn 90° from the Cartesian coordinates. Some preliminary numerical experiences with this scheme are presented.