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Enhanced drag of a sphere settling in a stratified fluid at small Reynolds numbers

Published online by Cambridge University Press:  27 July 2009

KING YEUNG YICK ,
CARLOS R. TORRES ,
THOMAS PEACOCK  and
ROMAN STOCKER
KING YEUNG YICK
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
CARLOS R. TORRES
Affiliation:
Universidad Autónoma de Baja California, Km 107 Carretera Tijuana-Ensenada, Ensenada 22830, Baja California, México
THOMAS PEACOCK
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
ROMAN STOCKER*
Affiliation:
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
†Email address for correspondence: romans@mit.edu
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Abstract

We present a combined experimental and numerical investigation of a sphere settling in a linearly stratified fluid at small Reynolds numbers. Using time-lapse photography and numerical modelling, we observed and quantified an increase in drag due to stratification. For a salt stratification, the normalized added drag coefficient scales as Ri0.51, where Ri = a3N2/(νU) is the viscous Richardson number, a the particle radius, U its speed, ν the kinematic fluid viscosity and N the buoyancy frequency. Microscale synthetic schlieren revealed that a settling sphere draws lighter fluid downwards, resulting in a density wake extending tens of particle radii. Analysis of the flow and density fields shows that the added drag results from the buoyancy of the fluid in a region of size (ν/N)1/2 surrounding the sphere, while the bulk of the wake does not influence drag. A scaling argument is provided to rationalize the observations. The enhanced drag can increase settling times in natural aquatic environments, affecting retention of particles at density interfaces and vertical fluxes of organic matter.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 49 - 68
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Abaid, N., Adalsteinsson, D., Agyapong, A. & McLaughlin, R. M. 2004 An internal splash: levitation of falling spheres in a stratified fluid. Phys. Fluids 16, 1567.CrossRefGoogle Scholar
Barry, M. E., Ivey, G. N., Winters, K. B. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267–291.CrossRefGoogle Scholar
Basak, S. & Sarkar, S. 2006 Dynamics of a stratified shear layer with horizontal shear. J. Fluid Mech. 568, 19–54.CrossRefGoogle Scholar
Blanchette, F. & Bush, J. W. M. 2005 Particle concentration evolution and sedimentation-induced instabilities in a stably stratified environment. Phys. Fluids 17, 073302.CrossRefGoogle Scholar
Blanchette, F., Peacock, T. & Cousin, R. 2008 Stability of a stratified fluid with a vertically moving sidewall. J. Fluid Mech. 609, 305–317.CrossRefGoogle Scholar
Bush, J. W. M., Thurber, B. A. & Blanchette, F. 2003 Particle clouds in homogeneous and stratified environments. J. Fluid Mech. 489, 29–54.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic Press.Google Scholar
Dalziel, S. B. 1992 Decay of rotating turbulence: some particle tracking experiments. Appl. Sci. Res. 49, 217–244.CrossRefGoogle Scholar
Dalziel, S. B. 2006 DigiFlow, www.damtp.cam.ac.uk/lab/digiflow.Google Scholar
Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 2000 Whole-field density measurements by synthetic schlieren. Exp. Fluids 28, 322–335.CrossRefGoogle Scholar
Eames, I., Gobby, D. & Dalziel, S. B. 2003 Fluid displacement by Stokes flow past a spherical droplet. J. Fluid Mech. 485, 67–85.CrossRefGoogle Scholar
Farmer, D. & Armi, L. 1999 The generation and trapping of solitary waves over topography. Science 283, 188–190.CrossRefGoogle ScholarPubMed
Fofonoff, P. & Millard, R. C. Jr 1983 Algorithms for computation of fundamental properties of seawater. Unesco Tech. Pap. Mar. Sci. 44, 53.Google Scholar
Gargett, A. 1988 The scaling of turbulence in the presence of stable stratification. J. Geophys. Res. 93, 5021–5036.CrossRefGoogle Scholar
Greenslade, M. D. 1994 Strongly stratified airflow over and around mountains. In Stably Stratified Flows: Flow and Dispersion over Topography, Proceedings of the 4th IMA Conf. on Stably Stratified Flows, University of Surrey, September 1992 (ed. Castro, I. P. & Rockliff, N. J.), pp. 25–37. Oxford University Press.Google Scholar
Greenslade, M. D. 2000 Drag on a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 418, 339–350.CrossRefGoogle Scholar
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, 678–686.CrossRefGoogle Scholar
Kellogg, W. W. 1990 Aerosols and climate. In Interaction of Energy and Climate (ed. Bach, W., Pankrath, J. and Williams, J.), pp. 281–296. Reidel.Google Scholar
King, J. R., Shuter, B. J. & Zimmerman, A. P. 1999 Signals of climate trends and extreme events in the thermal stratification pattern of multibasin Lake Opeongo, Ontario. Can. J. Fish. Aquat. Sci. 56, 847–852.CrossRefGoogle Scholar
Koh, R. C. Y. 1966 Viscous stratified flow towards a sink. J. Fluid Mech. 24, 555–575.CrossRefGoogle Scholar
Larrazábal, G., Torres, C. R. & Castillo, J. 2003 An efficient and robust algorithm for 2D stratified fluid flow calculations. Appl. Numer. Math. 47, 493–502.CrossRefGoogle Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435–476.CrossRefGoogle Scholar
Lofquist, K. E. B. & Purtell, L. P. 1984 Drag on a sphere moving horizontally through a stratified liquid. J. Fluid Mech. 148, 271–284.CrossRefGoogle Scholar
MacIntyre, S., Alldredge, A. L. & Gotschalk, C. C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40, 449–468.CrossRefGoogle Scholar
Ochoa, J. L. & Van Woert, M. L. 1977 Flow visualization of boundary layer seperation in a stratified fluid. Unpublished report, Scripps Insititution of Oceanography 28.Google Scholar
Oster, G. 1965 Density gradients. Sci. Am. 213, 70–76.CrossRefGoogle Scholar
Patterson, J. C., Hamblin, P. F. & Imberger, J. 1984 Classification and dynamic simulation of the vertical density structure of lakes. Limnol. Oceanogr. 29, 845–861.CrossRefGoogle Scholar
Saad, Y. 2003 Iterative Methods for Sparse Linear Systems, 2nd ed. SIAM.CrossRefGoogle Scholar
Saggio, A. & Imberger, J. 2001 Mixing and turbulent fluxes in the metalimnion of a stratified lake. Limnol. Oceanogr. 46, 392–409.CrossRefGoogle Scholar
Scase, M. M. & Dalziel, S. B. 2004 Internal wave fields and drag generated by a translating body in a stratified fluid. J. Fluid Mech. 498, 289–313.CrossRefGoogle Scholar
Smith, R. B. 1979 The influence of mountains on the atmosphere. In Advances in Geophysics, vol. 21. Academic Press.Google Scholar
Smith, R. B. 1980 Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus 32, 348–364.CrossRefGoogle Scholar
Srdić-Mitrović, A. N., Mohamed, N. A. & Fernando, J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198.CrossRefGoogle Scholar
Stocker, R., Seymour, J. R., Samadani, A., Hunt, D. E. & Polz, M. F. 2008 Rapid chemotatic response enables marine bacteria to exploit ephemeral microscale nutrient patches. Proc. Natl Acad. Sci. 105, 4209–4214.CrossRefGoogle Scholar
Sutherland, B. R., Dalziel, S. B., Hughes, G. O. & Linden, P. F. 1999 Visualisation and measurement of internal waves by synthetic schlieren. Part 1. Vertically oscillating cylinder. J. Fluid Mech. 390, 93–126.CrossRefGoogle Scholar
Tenner, A. R. & Gebhart, B. 1971 Laminar and axisymmetric vertical jets in a stably stratified environment. Intl. J. Heat Mass Transfer 14, 2051–2062.CrossRefGoogle Scholar
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, 211–236.CrossRefGoogle Scholar
Tritton, D. J. 1988 Physical Fluid Mechanics, 2nd ed. Oxford University Press.Google Scholar
Turco, R. P., Toon, O. B., Ackerman, T. P., Pollack, J. B. & Sagan, C. 1990 Climate and smoke: an appraisal of nuclear winter. Science 247, 166–176.CrossRefGoogle ScholarPubMed
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Vosper, S. B., Castro, I. P., Snyder, W. H. & Mobbs, S. D. 1999 Experimental studies of strongly stratified flow past three-dimensional orography. J. Fluid Mech. 390, 223–249.CrossRefGoogle Scholar
Warren, F. W. G. 1960 Wave resistance to vertical motion in a stratified fluid. J. Fluid Mech. 7, 209–229.CrossRefGoogle Scholar
White, F. M. 2005 Viscous Fluid Flow, 3rd ed. McGraw-Hill.Google Scholar
Yick, K. Y., Stocker, R. & Peacock, T. 2006 Microscale synthetic schlieren. Exp. Fluids 42, 41–48.CrossRefGoogle Scholar
Zvirin, Y. & Chadwick, R. S. 1974 Settling of an axially symmetric body in a viscous stratified fluid. Intl. J. Multiph. Flow 1, 743–752.CrossRefGoogle Scholar