Numerical study of spherical Couette flows for certain zenith-angle-dependent rotations of boundary spheres at low Reynolds numbers

B. V. Pal'tsev ; M. B. Solov'ev ; I. I. Chechel'

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 6, pp. 1095-1133

Résumé

The numerical method with splitting of boundary conditions developed previously by the first and third authors for solving the stationary Dirichlet boundary value problem for the Navier–Stokes equations in spherical layers in the axisymmetric case at low Reynolds numbers and a corresponding software package were used to study viscous incompressible steady flows between two con-centric spheres. Flow regimes depending on the zenith angle $\theta$ of coaxially rotating boundary spheres (admitting discontinuities in their angular velocities) were investigated. The orders of accuracy with respect to the mesh size of the numerical solutions (for velocity, pressure, and stream function in a meridional plane) in the max and $L_2$ norms were studied in the case when the velocity boundary data have jump discontinuities and when some procedures are used to smooth the latter. The capabilities of the Richardson extrapolation procedure used to improve the order of accuracy of the method were investigated. Error estimates were obtained. Due to the high accuracy of the numerical solutions, flow features were carefully analyzed that were not studied previously. A number of interesting phenomena in viscous incompressible flows were discovered in the cases under study.