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.

Unsteady turbulence in uniformly stratified unsheared flow is analysed using rapid distortion theory (RDT). For inviscid flow with no molecular diffusion the theory shows how the initial conditions, such as the initial turbulent kinetic energy KE0 and potential energy PE0, determine the partition of energy between the potential energy associated with density fluctuation and the kinetic energy associated with each of the velocity components during the subsequent development of the turbulence. One parameter is an exception to this sensitivity to initial conditions, namely the limit at large time of the ratio of potential energy to vertical kinetic energy. In the linear theory, this ratio depends neither on the Reynolds number Re, nor the Prandtl number Pr nor the Froude number Fr. This is consistent with turbulence measurements in the atmosphere, wind tunnel and water tank experiments, and with large-eddy simulations, where similar values of the ratio are found. The RDT results are extended to show the effects of viscosity and diffusion where Re is not very large, explaining the sensitivity of the spectra and the fluxes to the value of the Prandtl number Pr. When Pr is larger than 1, the high-wavenumber components of the three-dimensional spectra induce a vertical flux of temperature (density) that is positive (negative), and therefore ‘countergradient.’ On the other hand, when the thermal diffusivity is stronger and Pr is less than 1, lower-wavenumber components become countergradient sooner since the high-wavenumber components are prevented from becoming countergradient. When all the wavenumber components are integrated to derive the total vertical density flux, it becomes countergradient more quickly and more strongly in high-Pr than in low-Pr turbulence. All these theoretically derived differences between high-Pr and low-Pr turbulence are consistent with the experimental measurements in water tank and wind tunnel experiments and numerical simulations. It is shown that the initial kinetic and potential energy spectrum forms E(k) and S(k) near k = 0 determine the long-time limit values of the variances and the covariances, including their decay rate with time. In the special case of Pr = 1, the oscillation time period of the three-dimensional spectrum function is independent of the wavenumber and is the same as that of an inviscid fluid with the effect of viscosity/diffusion being limited to the damping of all the wavenumber components in-phase with each other. Furthermore, the non-dimensional ratios of the covariances, including the normalized vertical density flux and the anisotropy tensor, agree with the inviscid results if S(k) is proportional to E(k), or if either S(k) or E(k) is identically zero. However, even when Pr = 1, in the ‘one-dimensional spectrum’ in the x-direction, there is a transitory countergradient flux for high wavenumbers; only in this case is there a qualitative difference with the three-dimensioanl spectrum. This paper shows that the characteristic differences in the behaviour of stably st...

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.