A numerical study of three-dimensional stratified flow past a sphere

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Abstract
A numerical study is described of the Boussinesq flow past a sphere of a viscous, incompressible and non-diffusive stratified fluid. The approaching flow has uniform velocity and linear stratification. The Reynolds number Re (= 2ρ0Ua/μ) based on the sphere diameter is 200 and the internal Froude number F(= U/Na) is varied from 0.25 to 200. Here U is the velocity, N the Brunt-Väisälä frequency, a the radius of the sphere, μ the viscosity and ρ0 the mean density. The numerical results show changes in the flow pattern with Froude number that are in good agreement with earlier theoretical and experimental results. For F < 1, the calculations show the flow passing round rather than going over the obstacle, and confirm Sheppard's simple formula for the dividing-streamline height. When the Froude number is further reduced (F < 0.4), the flow becomes approximately two-dimensional and qualitative agreement with Drazin's three-dimensional low-Froude-number theory is obtained. The relation between the wavelength of the internal gravity wave and the position of laminar separation on the sphere is also investigated to obtain the suppression and induction of separation by the wave. It is also found that the lee waves are confined in the spanwise direction to a rather narrow strip just behind the obstacle as linear theory predicts. The calculated drag coefficient CD of the sphere shows an interesting Froude-number dependence, which is quite similar to the results given by experiments. In this study not only CD but also the pressure distribution which contributes to the change of CD are obtained and the mechanism of the change is closely examined.