Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-07T12:54:20.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Core mechanisms of drag enhancement on bodies settling in a stratified fluid

Published online by Cambridge University Press:  22 July 2019

Jie Zhang Open the ORCID record for Jie Zhang [Opens in a new window] ,
Matthieu J. Mercier Open the ORCID record for Matthieu J. Mercier [Opens in a new window]  and
Jacques Magnaudet Open the ORCID record for Jacques Magnaudet [Opens in a new window]
Jie Zhang
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Shaanxi 710049, China Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
Matthieu J. Mercier
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
Jacques Magnaudet*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
*
†Email address for correspondence: magnau@imft.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Stratification due to salt or heat gradients greatly affects the distribution of inert particles and living organisms in the ocean and the lower atmosphere. Laboratory studies considering the settling of a sphere in a linearly stratified fluid confirmed that stratification may dramatically enhance the drag on the body, but failed to identify the generic physical mechanism responsible for this increase. We present a rigorous splitting scheme of the various contributions to the drag on a settling body, which allows them to be properly disentangled whatever the relative magnitude of inertial, viscous, diffusive and buoyancy effects. We apply this splitting procedure to data obtained via direct numerical simulation of the flow past a settling sphere over a range of parameters covering a variety of situations of laboratory and geophysical interest. Contrary to widespread belief, we show that, in the parameter range covered by the simulations, the drag enhancement is generally not primarily due to the extra buoyancy force resulting from the dragging of light fluid by the body, but rather to the specific structure of the vorticity field set in by buoyancy effects. Simulations also reveal how the different buoyancy-induced contributions to the drag vary with the flow parameters. To unravel the origin of these variations, we analyse the different possible leading-order balances in the governing equations. Thanks to this procedure, we identify several distinct regimes which differ by the relative magnitude of length scales associated with stratification, viscosity and diffusivity. We derive the scaling laws of the buoyancy-induced drag contributions in each of these regimes. Considering tangible examples, we show how these scaling laws combined with numerical results may be used to obtain reliable predictions beyond the range of parameters covered by the simulations.

JFM classification

Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Stratified flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 622 - 656
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abaid, N., Adalsteinsson, D., Agyapong, A. & McLaughlin, R. M. 2004 An internal splash: levitation of falling spheres in stratified fluids. Phys. Fluids 16, 1567–1580.Google Scholar
Acrivos, A. 1960 Solution of the laminar boundary layer equation at high Prandtl numbers. Phys. Fluids 3, 657–658.Google Scholar
Akiyama, S., Yusuke, W., Okino, S. & Hanazaki, H. 2019 Unstable jets generated by a sphere descending in a very strongly stratified fluid. J. Fluid Mech. 867, 26–44.Google Scholar
Alldredge, A. L., Cowles, T. J., MacIntyre, S., Rines, J. E. B., Donaghay, P. L., Greenlaw, C. F., Holliday, D. V., Dekshenieks, M. M., Sullivan, J. M. & Zaneveld, J. R. V. 2002 Occurrence and mechanisms of formation of a dramatic thin layer of marine snow in a shallow Pacific fjord. Mar. Ecol.-Prog. Ser. 233, 1–12.Google Scholar
Ardekani, A. M. & Stocker, R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105, 084502.Google Scholar
Auguste, F. & Magnaudet, J. 2018 Path oscillations and enhanced drag of light rising spheres. J. Fluid Mech. 841, 228–266.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. 1980 Mass transfer from small particles suspended in turbulent fluid. J. Fluid Mech. 98, 609–623.Google Scholar
Bergström, B. & Strömberg, J. O. 1997 Behavioural differences in relation to pycnoclines during vertical migration of the euphausiids. Meganyctiphanes norvegica (M. Sars) and Thysanoessa raschii (M. Sars). J. Plankton Res. 19, 255–261.Google Scholar
Bewley, T. & Meneghello, G. 2016 Efficient coordination of swarms of sensor-laden balloons for persistent, in situ, real-time measurement of hurricane development. Phys. Rev. Fluids 1, 060507.Google Scholar
Blanchette, F. & Shapiro, A. M. 2012 Drops settling in sharp stratification with and without Marangoni effects. Phys. Fluids 24, 042104.Google Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetrical high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7, 1265–1274.Google Scholar
Burns, P. & Chemel, C. 2015 Interactions between downslope flows and a developing cold-air pool. Boundary-Layer Meteorol. 154, 57–80.Google Scholar
Calmet, I. & Magnaudet, J. 1997 Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow. Phys. Fluids 9, 438–455.Google Scholar
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, 436–465.Google Scholar
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.Google Scholar
Camassa, R., Khatri, S., McLaughlin, R. M., Prairie, J. C., White, B. L. & Yu, S. 2013 Retention and entrainment effects: Experiments and theory for porous spheres settling in sharply stratified fluids. Phys. Fluids 25, 081701.Google Scholar
Candelier, F., Mehaddi, R. & Vauquelin, O. 2014 The history force on a small particle in a linearly stratified fluid. J. Fluid Mech. 749, 184–200.Google Scholar
Chadwick, R. S. & Zvirin, Y. 1974 Slow viscous flow of an incompressible stratified fluid past a sphere. J. Fluid Mech. 66, 377–383.Google Scholar
Chongsiripinyo, A. P. & Sarkar, S. 2017 On the vortex dynamics of flow past a sphere at Re = 3700 in a uniformly stratified fluid. Phys. Fluids 29, 020704.Google Scholar
Condie, S. A. & Bormans, M. 1997 The influence of density stratification on particle settling, dispersion and population growth. J. Theor. Biol. 187, 65–75.Google Scholar
D’Asaro 2003 Performance of autonomous Lagrangian floats. J. Atmos. Ocean. Technol. 20, 896–911.Google Scholar
Denman, K. L. & Gargett, A. E. 1995 Biological-physical interactions in the upper ocean: the role of vertical and small scale transport processes. Annu. Rev. Fluid Mech. 27, 225–256.Google Scholar
Farmer, D. & Armi, L. 1999 The generation and trapping of solitary waves over topography. Science 283, 188–190.Google Scholar
Hanazaki, H., Kashimoto, K. & Okamura, T. 2009a Jets generated by a sphere moving vertically in a stratified fluid. J. Fluid Mech. 638, 173–197.Google Scholar
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.Google Scholar
Hanazaki, H., Nakamura, S. & Yoshikawa, H. 2015 Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid. J. Fluid Mech. 765, 424–451.Google 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.Google Scholar
Kang, I. S. & Leal, L. G. 1988 The drag coefficient for a spherical bubble in a uniform streaming flow. Phys. Fluids 31, 233–237.Google Scholar
Kellogg, W. W. 1980 Aerosols and climate. In Interaction of Energy and Climate (ed. Bach, W., Pankrath, J. & Williams, J.), pp. 281–303. Reidel.Google Scholar
Kindler, K., Khalili, A. & Stocker, R. 2010 Diffusion-limited retention of porous particles at density interfaces. Proc. Natl Acad. Sci. USA 107, 22163–22168.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
List, E. J. 1971 Laminar momentum jets in a stratified fluid. J. Fluid Mech. 45, 561–574.Google 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.Google Scholar
Magnaudet, J. 2011 A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number. J. Fluid Mech. 689, 564–604.Google Scholar
Magnaudet, J., Rivero, M. & Fabre, J. 1995 Accelerated flows past a rigid sphere or a spherical bubble. 1. Steady straining flow. J. Fluid Mech. 284, 97–135.Google Scholar
Mehaddi, R., Candelier, F. & Mehling, B. 2018 Inertial drag on a sphere settling in a stratified fluid. J. Fluid Mech. 855, 1074–1087.Google Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.Google Scholar
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, 489–495.Google Scholar
Okino, S., Akiyama, S. & Hanazaki, H. 2017 Velocity distribution around a sphere descending in a linearly stratified fluid. J. Fluid Mech. 826, 759–780.Google Scholar
Pierson, J. L. & Magnaudet, J. 2018 Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics. J. Fluid Mech. 835, 808–851.Google Scholar
Ploug, H., Grossart, H. P., Azam, F. & Jorgensen, B. B. 1999 Photosynthesis, respiration, and carbon turnover in sinking marine snow from surface waters of Southern California Bight: implications for the carbon cycle in the ocean. Mar. Ecol. Prog. Ser. 179, 1–11.Google Scholar
Ploug, H., Iversen, M. H. & Fischer, G. 2008 Ballast, sinking velocity, and apparent diffusivity within marine snow and zooplankton fecal pellets: implications for substrate turnover by attached bacteria. Limnol. Oceanogr. 53, 1878–1886.Google Scholar
Riebesell, U. 1992 The formation of large marine snow and its sustained residence in surface waters. Limnol. Oceeanogr. 37, 63–67.Google Scholar
Rückenstein, E. 1959 On heat transfer between vapour bubbles in motion and the boiling liquid from which they are generated. Chem. Engng Sci. 10, 22–30.Google Scholar
Slinn, D. N. & Riley, J. J. 1998 A model for the simulation of turbulent boundary layers in an incompressible stratified flow. J. Comput. Phys. 144, 550–602.Google Scholar
Srdic-Mitrovic, A. N., Mohamed, N. A. & Fernando, H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198.Google Scholar
Sutor, M. M. & Dagg, M. J. 2008 The effects of vertical sampling resolution on estimates of plankton biomass and rate calculations in stratified water columns. Estuar. Coast. Shelf Sci. 78, 107–121.Google 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.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.Google Scholar
van Leer, B. 1977 Towards ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys. 23, 276–299.Google Scholar
Widder, E. A., Johnsen, S., Bernstein, S. A., Case, J. F. & Neilson, D. J. 1999 Thin layers of bioluminescent copepods found at density discontinuities in the water column. Mar. Biol. 134, 429–437.Google Scholar
Yajima, N., Imamura, T., Izutsu, N. & Abe, T. 2004 Scientific Ballooning. Springer.Google Scholar
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, 49–68.Google Scholar
Zvirin, Y. & Chadwick, R. S. 1975 Settling of an axially symmetric body in a viscous stratified fluid. Intl J. Multiphase Flow 1, 743–752.Google Scholar