Numerical Solution of a Uniform Flow over a Sphere at Intermediate Reynolds Numbers

Numerical solutions of the transient uniform flow around a sphere are obtained. The transition takes place between an initial potential flow and a fully developed viscous field. The fluid is incompressible, homogeneous, and its flow is governed by the complete Navier-Stokes equations. The range of Reynolds number studied is Re = 1–1000 where a recirculatory wake appears and the nonlinear terms are essential, that is, they cannot be neglected or approximated. The flow is assumed to be axisymmetric throughout this range. A time-dependent stream function-vorticity formulation is adopted. The solution is obtained by constructing a finite difference approximation to the vorticity transport equation on an expanding spherical polar grid system. Central differencing of second-order accuracy both in time (Dufort-Frankel) and space is utilized. Experiments with numerical stability show an appreciable deviation from linearized stability analysis due to the large gradients of vorticity in the field. Quantitative physical results are obtained. The geometrical parameters characterizing the recirculatory wake compare favorably with those recorded in physical experiments. The detailed distribution of the vorticity on the sphere agrees with results obtained via the steady-state approach at Re = 10, 40, and 100. The computed drag coefficient CD agrees well with the standard drag curve over the range of Reynolds numbers investigated.