Numerical study of the onset of instability in the flow past a sphere.

Abstract

Experiment shows that the steady axisymmetric flow past a sphere becomes unstable in the range 120 < Re < 300. The resulting time-dependent nonaxisymmetric flow gives rise to nonaxisymmetric vortex shedding at higher Reynolds numbers. The present work reports a computational investigation of the linear stability of the axisymmetric base flow. When the sphere is towed, fixed, or otherwise constrained, stability is determined solely by the Reynolds number. On the other hand, when the sphere falls due to gravity, the present work shows that a additional parameter, the ratio of fluid density to sphere density (β = ρ(f)/ρ(s)) is involved. We use a spectral technique to compute the steady axisymmetric flow, which is in closer agreement with experiment than previous calculations. We then perform a linear stability analysis of the base flow with respect to axisymmetric and nonaxisymmetric disturbances. A spectral technique similar to that employed in the base flow calculation is used to solve the linear disturbance equations in streamfunction form for axisymmetric disturbances, and in a modified primitive variable form for nonaxisymmetric disturbances. For the density ratio β = 0, which corresponds to a fixed sphere, the analysis shows that the axisymmetric base flow undergoes a Hopf bifurcation at Re = 175.1, with the critical disturbance having azimuthal wavenumber m = 1. The results are favorably compared to previous experimental work.