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Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid

  • H. Hanazaki (a1), S. Nakamura (a1) and H. Yoshikawa (a1)


The flow past a sphere moving vertically at constant speeds in a salt-stratified fluid is investigated numerically at moderate Reynolds numbers $\mathit{Re}$ . Time development of the flow shows that the violation of density conservation is the key process for the generation of the buoyant jet observed in the experiments. For example, if the sphere moves downward, isopycnal surfaces are simply deformed and dragged down by the sphere while the density is conserved along the flow. (The flow pattern is inverted if the sphere moves upward. Some explanations are given in the introduction.) Then, the flow will never become steady. As density diffusion becomes effective around the sphere surface and generates a horizontal hole in the isopycnal surface, fluid with non-conserved density is detached from the isopycnal surface and moves upward to generate a buoyant jet. These processes will constitute a steady state near the sphere. With lengths scaled by the sphere diameter and velocities by the downward sphere velocity, the duration of density conservation at the rear/upper stagnation point, or the maximum distance that the isopycnal surface is dragged downward, is proportional to the Froude number $\mathit{Fr}$ , and estimated well by ${\rm\pi}\mathit{Fr}$ for $\mathit{Fr}\gtrsim 1$ and $\mathit{Re}\gtrsim 200$ , corresponding to a constant potential energy.  The radius of a jet defined by the density and velocity distributions, which would have correlations with the density and velocity boundary layers on the sphere, is estimated well by $\sqrt{\mathit{Fr }/2\mathit{Re }\mathit{ Sc}}$ and $\sqrt{\mathit{Fr }/2\mathit{Re}}$ respectively for $\mathit{Fr}\lesssim 1$ , where $\mathit{Sc}$ is the Schmidt number. Numerical results agree well with the previous experiments, and the origin of the conspicuous bell-shaped structure observed by the shadowgraph method is identified as an internal wave.


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Abaid, N., Adalsteinsson, D., Agyapong, A. & McLaughlin, R. M. 2004 An internal splash: levitation of falling spheres in stratified fluids. Phys. Fluids 16, 15671580.
Barry, M. E., Ivey, G. N., Winters, K. R. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267291.
Camassa, R., Falcon, C., Lin, J., McLaughlin, R. M. & Parker, R. 2009 Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime. Phys. Fluids 21, 031702.
Camassa, R., Falcon, C., Lin, J., McLaughlin, R. M. & Mykins, N. 2010 A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number. J. Fluid Mech. 664, 436465.
Castro, I. P., Snyder, W. H. & Marsh, G. L. 1983 Stratified flow over three-dimensional ridges. J. Fluid Mech. 135, 261282.
Chashechkin, Yu. D. & Levitskii, V. V. 2003 Pattern of flow around a sphere oscillating a neutrally buoyancy horizon in a continuously stratified fluid. J. Vis. 6, 5965.
D’Asaro, E. A. 2003 Performance of autonomous Lagrangian floats. J. Atmos. Ocean. Technol. 20, 896911.
Gibson, C. H. 1980 Fossil temperature, salinity, and vorticity turbulence in the ocean. In Marine Turbulence (ed. Nihoul, J.), pp. 221257. Elsevier.
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.
Hanazaki, H. 1988 A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech. 192, 393419.
Hanazaki, H., Kashimoto, K. & Okamura, T. 2009a Jets generated by a sphere moving vertically in a stratified fluid. J. Fluid Mech. 638, 173197.
Hanazaki, H., Konishi, K. & Okamura, T. 2009b Schmidt number effects on the flow past a sphere moving vertically in a stratified diffusive fluid. Phys. Fluids 21, 026602.
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 21822189.
Higginson, R. C., Dalziel, S. B. & Linden, P. F. 2003 The drag on a vertically moving grid of bars in a linearly stratified fluid. Exp. Fluids 34, 678686.
Hunt, J. C. R. & Snyder, W. H. 1980 Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech. 96, 671704.
Levitskii, V. V. & Chashechkin, Yu. D. 1999 Natural oscillations of a neutrally buoyant body in a continuously stratified fluid. Fluid Dyn. 34, 641651.
Mowbray, D. E. & Rarity, B. S. H. 1967 The internal wave pattern produced by a sphere moving vertically in a density stratified liquid. J. Fluid Mech. 30, 489495.
Ochoa, J. L. & Van Woert, M. L.1977 Flow visualisation of boundary layer separation in a stratified fluid. Unpublished Report, Scripps Institute of Oceanography, 28 pp.
Pearson, H. J., Puttock, J. S. & Hunt, J. C. R. 1983 A statistical model of fluid-element motions and vertical diffusion in a homogeneous stratified turbulent flow. J. Fluid Mech. 129, 219249.
Srdić-Mitrović, A. N., Mohamed, N. A. & Fernando, H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175198.
Taneda, S. 1956 Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Japan 11, 11041108.
Torres, C. R., Hanazaki, H., Ochoa, J., Castillo, J. & Van Woert, M. 2000 Flow past a sphere moving vertically in a stratified diffusive fluid. J. Fluid Mech. 417, 217236.
Yick, K. Y., Torres, C. R., Peacock, T. & Stocker, R. 2009 Enhanced drag of a sphere settling in a stratified fluid at small Reynolds numbers. J. Fluid Mech. 632, 4968.
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Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid

  • H. Hanazaki (a1), S. Nakamura (a1) and H. Yoshikawa (a1)


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