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Settling disks in a linearly stratified fluid

Published online by Cambridge University Press:  17 December 2019

M. J. Mercier Open the ORCID record for M. J. Mercier [Opens in a new window] ,
S. Wang Open the ORCID record for S. Wang [Opens in a new window] ,
J. Péméja ,
P. Ern Open the ORCID record for P. Ern [Opens in a new window]  and
A. M. Ardekani Open the ORCID record for A. M. Ardekani [Opens in a new window]
M. J. Mercier*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, 31400 Toulouse, France
S. Wang*
Affiliation:
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
J. Péméja
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, 31400 Toulouse, France
P. Ern
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, 31400 Toulouse, France
A. M. Ardekani
Affiliation:
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
*
†Email address for correspondence: [email protected]
‡Present address: Davidson School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, IN 47907, USA.
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Abstract

We consider the unbounded settling dynamics of a circular disk of diameter $d$ and finite thickness $h$ evolving with a vertical speed $U$ in a linearly stratified fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ and diffusivity $\unicode[STIX]{x1D705}$ of the stratifying agent, at moderate Reynolds numbers ($Re=Ud/\unicode[STIX]{x1D708}$). The influence of the disk geometry (diameter $d$ and aspect ratio $\unicode[STIX]{x1D712}=d/h$) and of the stratified environment (buoyancy frequency $N$, viscosity and diffusivity) are experimentally and numerically investigated. Three regimes for the settling dynamics have been identified for a disk reaching its gravitational equilibrium level. The disk first falls broadside-on, experiencing an enhanced drag force that can be linked to the stratification. A second regime corresponds to a change of stability for the disk orientation, from broadside-on to edgewise settling. This occurs when the non-dimensional velocity $U/\sqrt{\unicode[STIX]{x1D708}N}$ becomes smaller than some threshold value. Uncertainties in identifying the threshold value is discussed in terms of disk quality. It differs from the same problem in a homogeneous fluid which is associated with a fixed orientation (at its initial value) in the Stokes regime and a broadside-on settling orientation at low, but finite Reynolds numbers. Finally, the third regime corresponds to the disk returning to its broadside orientation after stopping at its neutrally buoyant level.

JFM classification

Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A2
Copyright
© 2019 Cambridge University Press

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References

Ardekani, A. M., Dabiri, S. & Rangel, R. H. 2008 Collision of multi-particle and general shape objects in a viscous fluid. J. Comput. Phys. 227 (24), 10094–10107.CrossRefGoogle Scholar
Ardekani, A. M. & Stocker, R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105, 084502.CrossRefGoogle Scholar
Auguste, F., Fabre, D. & Magnaudet, J. 2010 Bifurcations in the wake of a thick circular disk. Theor. Comput. Fluid Dyn. 24 (1–4), 305–313.CrossRefGoogle Scholar
Bayareh, M., Doostmohammadi, A., Dabiri, S. & Ardekani, A. M. 2013 On the rising motion of a drop in stratified fluids. Phys. Fluids 25 (10), 103302.CrossRefGoogle Scholar
Biró, I., Gábor Szabó, K., Gyüre, B., Jánosi, I. M. & Tél, T. 2008a Power-law decaying oscillations of neutrally buoyant spheres in continuously stratified fluid. Phys. Fluids 20 (5), 051705.CrossRefGoogle Scholar
Birò, I., Gyüre, B., Jànosi, I. M., Szabo, K. G. & Tél, T.2008b Oscillation and levitation of balls in continuously stratified fluids. arXiv:physics/0702208 [physics.flu-dyn].Google Scholar
Blanchette, F. & Shapiro, A. M. 2012 Drops settling in sharp stratification with and without marangoni effects. Phys. Fluids 24 (4), 042104.CrossRefGoogle 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.CrossRefGoogle 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 (8), 081701.CrossRefGoogle 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.CrossRefGoogle Scholar
Carazzo, G. & Jellinek, A. M. 2012 A new view of the dynamics, stability and longevity of volcanic clouds. Earth Planet. Sci. Lett. 325–326, 39–51.CrossRefGoogle Scholar
Chrust, M., Bouchet, G. & Dušek, J. 2013 Numerical simulation of the dynamics of freely falling discs. Phys. Fluids 25 (4), 044102.CrossRefGoogle Scholar
Clavano, W. R., Boss, E. & Karp-Boss, L. 2007 Inherent optical properties of non-spherical marine-like particles – from theory to observation. Oceanogr. Mar. Biol. 45, 1–38.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic Press.Google Scholar
Dalziel, S. B., Carr, M., Sveen, J. K. & Davies, P. A. 2007 Simultaneous synthetic schlieren and PIV measurements for internal solitary waves. Meas. Sci. Technol. 18, 533–547.CrossRefGoogle Scholar
Doostmohammadi, A. & Ardekani, A. M. 2013 Interaction between a pair of particles settling in a stratified fluid. Phys. Rev. E 88, 023029.Google Scholar
Doostmohammadi, A. & Ardekani, A. M. 2014 Reorientation of elongated particles at density interfaces. Phys. Rev. E 90, 033013.Google ScholarPubMed
Doostmohammadi, A. & Ardekani, A. M. 2015 Suspension of solid particles in a density stratified fluid. Phys. Fluids 27 (2), 023302.CrossRefGoogle Scholar
Doostmohammadi, A., Dabiri, S. & Ardekani, A. M. 2014 A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratified fluid. J. Fluid Mech. 750, 5–32.CrossRefGoogle Scholar
Economidou, M. & Hunt, G. R. 2009 Density stratified environments: the double-tank method. Exp. Fluids 46, 453–466.CrossRefGoogle Scholar
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121.CrossRefGoogle Scholar
Fabre, D., Auguste, F. & Magnaudet, J. 2008 Bifurcations and symmetry breaking in the wake of axisymmetric bodies. Phys. Fluids 20 (5), 051702.CrossRefGoogle Scholar
Fabre, D., Tchoufag, J. & Magnaudet, J. 2012 The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 24–36.CrossRefGoogle Scholar
Fernandes, P. C., Risso, F., Ern, P. & Magnaudet, J. 2007 Oscillatory motion and wake instability of freely rising axisymmetric bodies. J. Fluid Mech. 573, 479–502.CrossRefGoogle Scholar
Fernando, H. J. S., Lee, S. M., Anderson, J., Princevac, M., Pardyjak, E. & Grossman-Clarke, S. 2001 Urban fluid mechanics: air circulation and contaminant dispersion in cities. Environ. Fluid Mech. 1, 107–164.CrossRefGoogle Scholar
Field, S. B., Klaus, M., Moore, M. G. & Nori, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252–254.CrossRefGoogle 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.CrossRefGoogle 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 (2), 026602.CrossRefGoogle 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.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
Hillebrand, H., Dürselen, C.-D., Kirschtel, D., Pollingher, U. & Zohary, T. 1999 Biovolume calculation for pelagic and benthic microalgae. J. Phycol. 35, 403–424.CrossRefGoogle Scholar
Houghton, I. A., Koseff, J. R., Monismith, S. G. & Dabiri, J. O. 2018 Vertically migrating swimmers generate aggregation-scale eddies in a stratified colum. Nature 556 (7702), 497–500.CrossRefGoogle Scholar
Kindler, K., Khalili, A. & Stocker, R. 2010 Diffusion-limipted retention of porous particles at density interfaces. Proc. Natl Acad. Sci. 107 (51), 22163–22168.CrossRefGoogle ScholarPubMed
Koh, R. C. Y. & Brooks, N. H. 1975 Fluid mechanics of waste-water disposal in the ocean. Annu. Rev. Fluid Mech. 7, 187–211.CrossRefGoogle Scholar
Leonard, B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Engng 19.1, 59–98.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 (3), 449–468.CrossRefGoogle Scholar
Martin, D. W. & Blanchette, F. Ç. 2017 Simulations of surfactant-laden drops rising in a density-stratified medium. Phys. Rev. Fluids 2, 023602.CrossRefGoogle Scholar
McNown, J. S. & Malaika, J. 1950 Effects of particle shape on settling velocity at low Reynolds numbers. EOS Trans. AGU 31 (1), 74–82.CrossRefGoogle 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 (3), 489–495.CrossRefGoogle Scholar
Mrokowska, M. M. 2018 Stratification-induced reorientation of disk settling through ambient density transition. Sci. Rep. 8, 412.CrossRefGoogle ScholarPubMed
Ochoa, J. L. & Van Woert, M. L.1977 Flow visualization of boundary layer separation in a stratified fluid. Unpublished Rep. Scripps Institute of Oceanography.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.CrossRefGoogle Scholar
Oster, C. 1965 Density gradients. Sci. Am. 213, 70–76.CrossRefGoogle Scholar
Pitter, R. L., Pruppacher, H. R. & Hamielec, A. E. 1973 A numerical study of viscous flow past a thin oblate spheroid at low and intermediate Reynolds numbers. J. Atmos. Sci. 30, 125–134.2.0.CO;2>CrossRefGoogle Scholar
Prairie, J. C., Ziervogel, K., Arnosti, C., Camassa, R., Falcon, C., Khatri, S., McLaughlin, R. M., White, B. L. & Yu, S. 2013 Delayed settling of marine snow at sharp density transitions driven by fluid entrainment and diffusion-limited retention. Mar. Ecol. Progr. Ser. 487, 185–200.CrossRefGoogle Scholar
Srdić-Mitrović, A. N., Mohamed, N. A. & Fernando, H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198.CrossRefGoogle Scholar
Tchoufag, J., Fabre, D. & Magnaudet, J. 2014 Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders. J. Fluid Mech. 740, 278–311.CrossRefGoogle Scholar
Torres, C. R., Hanazaki, H., Ochoa, J., Castillo, J. & Woert, M. V. 2000 Flow past a sphere moving vertically in a stratified diffusive fluid. J. Fluid Mech. 417, 211–236.CrossRefGoogle Scholar
Towns, J., Cockerill, T., Dahan, M., Foster, I., Gaither, K., Grimshaw, A., Hazlewood, V., Lathrop, S., Lifka, D., Peterson, G. D. et al. 2014 XSEDE: Accelerating Scientific Discovery. Comput. Sci. Engng 16 (5), 62–74.CrossRefGoogle Scholar
Wagner, G. L., Young, W. R. & Lauga, E. 2014 Mixing by microorganisms in stratified fluids. J. Mar. Res. 72 (2), 47–72.CrossRefGoogle Scholar
Wang, S. & Ardekani, A. M. 2015 Biogenic mixing induced by intermediate Reynolds number swimming in stratified fluid. Sci. Rep. 5, 17448.Google Scholar
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7 (2), 197–208.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Yick, K.-Y., Stocker, R. & Peacock, T. 2007 Microscale synthetic schlieren. Exp. Fluids 42, 41–48.CrossRefGoogle 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.CrossRefGoogle Scholar
Zvirin, Y. & Chadwick, R. S. 1975 Settling of an axially symmetric body in a viscous stratified fluid. Intl J. Multiphase Flow 1 (6), 743–752.CrossRefGoogle Scholar