Numerical studies are described of the flows generated by a sphere moving vertically in
a uniformly stratified fluid. It is found that the axisymmetric standing vortex usually
found in homogeneous fluids at moderate Reynolds numbers (25 [les ] Re [les ] 200) is
completely collapsed by stable stratification, generating a strong vertical jet. This is
consistent with our experimental visualizations. For Re = 200 the complete collapse
of the vortex occurs at Froude number F ≃ 19, and the critical Froude number
decreases slowly as Re increases. The Froude number and the Reynolds number are
here defined by F = W/Na and Re = 2Wa/v,
with W being the descent velocity
of the sphere, N the Brunt–Väisälä frequency, a
the radius of the sphere and v the
kinematic viscosity coefficient. The inviscid processes, including the generation of the
vertical jet, have been investigated by Eames & Hunt (1997) in the context of weak
stratification without buoyancy effects. They showed the existence of a singularity 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 diffusion, all the isopycnal surfaces
which existed initially in front of the sphere accumulate very near the front surface
because of density conservation 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 diffusivity and the buoyancy effects also exist. The density diffusion prevents
the extreme piling up of the isopycnal surfaces and allows the existence of a steady
solution, preventing the generation of a singularity or a jet. On the other hand, the
buoyancy effect works to increase the vertical velocity 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 the collapse of the vortex and the generation of the jet
occurs at much weaker stratifications than those necessary for the generation of strong
lee waves, showing that jet formation is independent of the internal waves. At low
Froude numbers (F [les ] 2) the lee wave patterns showed good agreement with the linear
wave theory and the previous experiments by Mowbray & Rarity (1967). At very low
Froude numbers (F [les ] 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 reduction of
the pressure on the rear surface of the sphere, which leads to the increase of pressure
drag. High velocity is induced also just outside the boundary layer of the sphere so
that the frictional drag increases even more significantly than the pressure drag.
THE BUOYANT TWO-DIMENSIONAL LAMINAR VERTICAL JET
