Higher-Order Compact Scheme for the Incompressible Navier-Stokes Equations in Spherical Geometry (Communications in Computational Physics, 2012)
A higher-order compact scheme on the nine point 2-D stencil is developedfor the steady stream-function vorticity form of the incompressible Navier-Stokes (NS) equations in spherical polar coordinates, which was used earlier only for the cartesian and cylindrical geometries. The steady, incompressible, viscous and axially symmetric flow past a sphere is used as a model problem. The non-linearity in the N-Sequations is handled in a comprehensive manner avoiding complications in calculations. The scheme is combined with the multigrid method to enhance the convergencerate. The solutions are obtained over a non-uniform grid generated using the transformation r = eξ while maintaining a uniform grid in the computational plane. Thesuperiority of the higher order compact scheme is clearly illustrated in comparisonwith upwind scheme and defect correction technique at high Reynolds numbers bytaking a large domain. This is a pioneering effort, because for the first time, the fourthorder accurate solutions for the problem of viscous flow past a sphere are presentedhere. The drag coefficient and surface pressures are calculated and compared withavailable experimental and theoretical results. It is observed that these values simulated over coarser grids using the present scheme are more accurate when compared toother conventional schemes. It has also been observed that the flow separation initiallyoccurred at Re=21.