A numerical method for study of the unsteady viscous flow between two concentric rotating spheres

A simple but very efficient numerical method based on the finite difference technique has been developed for solving time-dependent non-linear flow problems. The governing equations of motion are discretized by a backward-time central-space scheme, whereby only the variables in the non-linear terms other than the main variable of the transport equation are replaced with the corresponding values calculated previously. The resulting algebraic equations for each main variable are solved by directly applying the Gaussian algorithm altered by considering the sparse structure of the diagonal-banded coefficient matrix adequately.The method is applied here to study the unsteady axisymmetric isotherm flow of an incompressible viscous fluid in a spherical shell with a stationary inner sphere and a rotating outer sphere. The description given in literature of the flow under consideration concentrates analytically on the asymptotic behaviour for very large Reynolds number Re starting from the almost rigid rotation. The case of small or moderate Reynolds numbers could be studied numerically only for Re<-3000 because of certain numerical difficulties, which already lead to discrepancies for Re>1000. Therefore, no data are available for the large intermediate region at high Reynolds numbers. In contrast to literature, consistent solutions for a large range of Reynolds number from 10 to 20000 are obtained with the method described here. A comparison of the results with those in literature shows a good agreenment up to Re=1000. At high Re the flow field confirms certain features such as the Stewartson shear layers as predicted by the asymptotic theory. With the results presented, a contribution is made for filling the gap between the asymptotic theory and numerical results in literature.