Finite difference solution to the flow between two rotating spheres

Abstract

Abstract This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.