Stability of the flow past a sphere

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Abstract
Experiment shows that the steady axisymmetric flow past a sphere becomes unstable in the range 120 < Re < 300. The resulting time-dependent non-axi-symmetric flow gives rise to non-axisymmetric vortex shedding at higher Reynolds numbers. The present work reports a computational investigation of the linear stability of the steady axisymmetric base flow. We use a spectral technique to represent the base flow. We then perform a linear stability analysis with respect to axisymmetric and non-axisymmetric disturbances. A spectral technique similar to that employed in the base-flow calculation is used to solve the linear-disturbance equations in stream-function form (for axisymmetric disturbances), and in a modified primitive variables form (for non-axisymmetric disturbances). 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, and dimensionless frequency (non-dimensionalized as a Strouhal number, St) 0.0955. The critical Re, azimuthal mode number, and St are favourably compared to previous experimental work.