Flow past a sphere moving vertically in a stratified diffusive fluid

Numerical studies are described of the flows generated by a sphere moving v ertically in a uniformly stratified fluid. It is found that the axisymmetri c standing vortex usually found in homogeneous fluids at moderate Reynolds numbers (25 less than or equal to Re less than or equal to 200) is complete ly collapsed by stable stratification, generating a strong vertical jet. Th is is consistent with our experimental visualizations. For Re = 200 the com plete collapse of the vortex occurs at Froude number F similar or equal to 19, and the critical Froude number decreases slowly as Re increases. The Fr oude number and the Reynolds number are here defined by F = W/Na and Re = 2 Wa/v, with W being the descent velocity of the sphere, N the Brunt-Vaisala frequency, a the radius of the sphere and v the kinematic viscosity coeffic ient. The inviscid processes, including the generation of the vertical jet, have been investigated by Eames & Hunt (1997) in the context of weak strat ification without buoyancy effects. They showed the existence of a singular ity of vorticity and density gradient on the rear axis of the flow and also the impossibility of realizing a steady state. When there is no density di ffusion, all the isopycnal surfaces which existed initially in front of the sphere accumulate very near the front surface because of density conservat ion and the fluid in those thin layers generates a rear jet when returning to its original position. In the present study, however, the fluid has diff usivity and the buoyancy effects also exist. The density diffusion prevents the extreme piling up of the isopycnal surfaces and allows the existence o f a steady solution, preventing the generation of a singularity or a jet. O n the other hand, the buoyancy effect works to increase the vertical veloci ty to the rear of the sphere by converting the potential energy to vertical kinetic energy, leading to the formation of a strong jet. We found that th e collapse of the vortex and the generation of the jet occurs at much weake r stratifications than those necessary for the generation of strong lee wav es, showing that jet formation is independent of the internal waves. At low Froude numbers (F less than or equal to 2) the lee wave patterns showed go od agreement with the linear wave theory and the previous experiments by Mo wbray & Rarity (1967). At very low Froude numbers (F less than or equal to 1) the drag on a sphere increases rapidly, partly due to the lee wave drag but mainly due to the large velocity of the jet. The: jet causes a reductio n of the pressure on the rear surface of the sphere, which leads to the inc rease of pressure drag. High velocity is induced also just outside the boun dary layer of the sphere so that the frictional drag increases even more si gnificantly than the pressure drag.