Flow past a sphere moving vertically in a stratified diffusive fluid

Abstract

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.

Mixed convection from an isolated spherical particle

“…8c shows that the strength of the upward jet increases as the surface temperature is increased. The collapse of rear vortex due to density stratification was observed by Torres et al [31].…”

Numerical Study on the Mixed Convection Heat Transfer between a Sphere Particle and High Pressure Water in Pseudocritical Zone

“…The rear vortex will vanish if buoyancy force is strong enough. The collapse of rear vortex due to density stratification was also observed by Torres et al (2000) and Bhattacharyya and Singh (2008) for an assisting convective flow [18,32]. The buoyancy force is determined by the density difference which results from the nonuniform temperature field.…”

Low-Reynolds-number swimming at pycnoclines

“…For salt stratifications Pr ≈ 700; for temperature stratifications Pr ≈ 7. Diffusion of the stratifying agent is important because it prevents the extreme compression of isopycnals (surfaces of constant density) as particles or organisms traverse them (23,24). Here we focus on Pr ¼ 700.…”

“…The effect of buoyancy is often quantified by means of the Froude number (23,25,26), Fr ¼ U∕ðNaÞ, where N ¼ ffiffiffiffiffiffiffiffiffiffiffi ffi γg∕ρ 0 p is the Brunt-Väisälä frequency, the natural frequency of oscillation of a vertically displaced particle in a stratified fluid, and γ ¼ −dρ∕dz is the background density gradient. However, Fr measures the relative importance of inertial and buoyancy forces, whereas in the inertialess world of microorganisms it is more appropriate to compare viscous and buoyancy forces.…”