Skip to main content Accessibility help

Formation of drops and rings in double-diffusive sedimentation

  • Yi-Ju Chou (a1) (a2), Chen-Yen Hung (a1) and Chien-Fu Chen (a1)


We conduct numerical simulations to investigate the formation and evolution of drops and vortex rings of particle-laden fingers in double-diffusive convection in stably stratified environments. We show that the temporal evolution can be divided into double diffusion, acceleration and deceleration phases. The acceleration phase is a result of the vanishing temperature perturbation in the drop during the descent in the layer of uniform temperature. The drop decelerates because it transforms into a vortex ring. A theoretical drag model is presented to predict the speed of the spherical drop with the low drop Reynolds number. By formulating the boundary condition based on the vorticity, our drag model gives a more general form of the drag coefficient for small spherical drops and shows good agreement in predicting the drag coefficient. Drops with five particle sizes are compared, and it is found that although the greater vertical settling enhances vertical transport, the final state differs little among the various sizes. Comparison of our drag model with the simulation results under various bulk conditions and previous experimental results shows good model predictability. Finally, a comparison with the salt-finger case shows that the diffusive nature of the dissolved scalar field, along with the wake effect, can result in an apparent loss of mass from the drop and a permanent presence of the connection between the drop and its parent finger. This makes the observed detachment of the particle-laden drop much less likely in the salt-finger case.


Corresponding author

Email address for correspondence:


Hide All
Alldredge, A. & Cohen, Y. 1987 Can microscale chemical patches persist in the sea? Microelectrode study of marine snow, fecal pellets. Science 235, 689691.
Arthur, R. S. & Fringer, O. B. 2014 The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech. 761, 360398.
Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111133.
Boussinesq, J. 1913 Existence of a superficial viscosity in the thin transition layer separating one liquid from another contiguous fluid. C. R. Acad. Sci. USA 156, 983989.
Burns, P. & Meiburg, E. 2012 Sediment-laden fresh water above salt water: linear stability analysis. J. Fluid Mech. 691, 279314.
Burns, P. & Meiburg, E. 2015 Sediment-laden fresh water above salt water: nonlinear simulations. J. Fluid Mech. 762, 156195.
Bush, J. W., Thurber, B. A. & Blanchette, F. 2003 Particle clouds in homogeneous and stratified environments. J. Fluid Mech. 489, 2954.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.
Chang, Y.-C., Chiu, T.-Y., Hung, Y.-C. & Chou, Y.-J. 2019 Three-dimensional Eulerian–Lagrangian simulation of particle settling in inclined water columns. Powder Technol. 348, 8092.
Chen, C. F. 1997 Particle flux through sediment fingers. Deep-Sea Res. I 44 (9–10), 16451654.
Chester, W., Breach, D. R. & Proudman, I. 1969 On the flow past a sphere at low Reynolds number. J. Fluid Mech. 37, 751760.
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 133147.
Chou, Y.-J., Cheng, C.-J., Chen, R.-L. & Hung, C.-Y. 2019 Instabilities of particle-laden layers in the stably stratified environment. Phys. Fluids doi:10.1063/1.5123317.
Chou, Y. J. & Fringer, O. B. 2008 Modeling dilute sediment suspension using large-eddy simulation with a dynamic mixed model. Phys. Fluids 20, 11503.
Chou, Y. J. & Fringer, O. B. 2010 A model for the simulation of coupled flow-bedform evolution in turbulent flows. J. Geophys. Res. 115, C10041.
Chou, Y.-J., Gu, S.-H. & Shao, Y.-C. 2015 An Euler–Lagrange model for simulating fine particle suspension in liquid flows. J. Comput. Phys. 299, 955973.
Chou, Y.-J. & Shao, Y.-C. 2016 Numerical study of particle-induced Rayleigh–Taylor instability: effect of particle settling and entrainment. Phys. Fluids 28, 043302.
Chou, Y.-J., Wu, F.-C. & Shih, W.-R. 2014a Toward numerical modeling of fine particle suspension using a two-way coupled Euler–Euler model. Part 1. Theoretical formulation and implications. Intl J. Multiphase Flow 64, 3543.
Chou, Y.-J., Wu, F.-C. & Shih, W.-R. 2014b Toward numerical modeling of fine particle suspension using a two-way coupled Euler–Euler model. Part 2. Simulation of particle-induced Rayleigh–Taylor instability. Intl J. Multiphase Flow 64, 4454.
Cui, A. & Street, R. L. 2004 Large-eddy simulation of coastal upwelling flow. Environ. Fluid Mech. 4, 197223.
Cundall, R. & Strack, O. 1979 A discrete numerical model for granular assemblies. Geotechnique 29, 4765.
Elghobashi, S. & Truesdell, G. C. 1993 On the two-way interaction between homogeneous turbulence and dispersed solid particles. I. Turbulence modification. Phys. Fluids 5 (7), 1790.
Ferry, J. & Balachandar, S. 2001 A fast Eulerian method for disperse two-phase flow. Intl J. Multiphase Flow 27, 11991226.
Fringer, O. B. & Street, R. L. 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319353.
Green, T. 1987 The importance of double diffusion to the settling of suspended material. Sedimentology 34, 319331.
Green, T. & Schettle, J. W. 1986 Vortex rings associated with strong double-diffusive fingering. Phys. Fluids 29 (7), 21092112.
Hadamard, J. 1911 Slow permanent movement of a viscous liquid sphere in a viscous liquid. C. R. Acad. Sci. 152, 17351738.
Hamdhan, I. N. & Clarke, B. G. 2010 Determination of thermal conductivity of coarse and fine sand soils. In Proceedings World Geothermal Congress, Bali, Indonesia, pp. 17. International Geothermal Association.
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213245.
Hizzett, J. L., Hughes Clarke, J. E., Sumner, E. J., Cartigny, M. J. B., Talling, P. J. & Clare, M. A. 2018 Which triggers produce the most erosive, frequent, and longest runout turbidity currents on deltas? Geophys. Res. Lett. 45, 855863.
Houk, D. & Green, T. 1973 Descent rates of suspension fingers. Deep-Sea Res. 20, 757761.
Hoyal, D. C., Bursik, M. I. & Atkinson, J. F. 1999 The influence of diffusive convection on sedimentation from buoyant plumes. Mar. Geol. 159, 205220.
Huppert, H. E. & Turner, J. S. 1981 Double-diffusive convection. J. Fluid Mech. 106, 299329.
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308323.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Leonard, B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Engng 19, 5998.
Linden, P. F. 1973 On the structure of salt fingers. Deep-Sea Res. 20, 325340.
Maxey, M. R. & Riley, J. J. 1983 Equation of motion of a small sphere in linear shear flows. Phys. Fluids 26, 883889.
Moffatt, H. K. & Moore, D. W. 1978 The response of Hill’s spherical vortex to a small axisymmetric disturbance. J. Fluid Mech. 87 (4), 749760.
Parsons, J. D., Bush, J. W. M. & Syvitski, J. P. M. 2001 Hyperpycnal plume formation from riverine outflows with small sediment concentrations. Sedimentology 48, 465478.
Parsons, J. D. & Garcia, M. H. 2000 Enhanced sediment scavenging due to double-diffusive convection. J. Sedim. Res. 70, 4752.
Perng, C. Y. & Street, R. L. 1989 Three-dimensional unsteady flow simulations: alternative strategies for a volume-average calculation. Intl J. Numer. Fluids 9 (3), 341362.
Pozrikidis, C. 1986 The nonlinear instability of Hill’s vortex. J. Fluid Mech. 168, 337367.
Proudman, I. & Pearson, J. R. A. 1957 Expansion at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237262.
Radko, T. 2013 Double-Diffusive Convection. Cambridge University Press.
Reali, J. F., Garaud, P., Alsinan, A. & Meiburg, E. 2018 Layer formation in sedimentary fingering convection. J. Fluid Mech. 816, 268305.
Rybczynski, W. 1911 On the translatory motion of a fluid sphere in a viscous medium. Bull. Acad. Sci. Cracow Ser. A 2, 40.
Schiller, L. & Nauman, A. 1935 A drag coefficient correlation. VDI Zeitung 77, 318320.
Schmitt, R. W. 1979 Flux measurements on salt fingers at in interface. J. Mar. Res. 37, 419436.
Schmitt, R. W. 1994 Double-diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255285.
Schmitt, R. W. & Evane, D. L. 1978 An estimate of the vertical mixing due to salt fingers based on observations in the North Atlantic central water. J. Geophys. Res. 83, 29132919.
Schmitt, R. W., Ledwell, J. R., Montgomery, E. T., Polzin, K. L. & Toole, J. M. 2005 Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical atlantic. Science 308, 685688.
Segre, P. N., Liu, F., Umbanhowar, P. & Weitz, D. A. 2001 An effective gravitational temperature for sedimentation. Nature 409, 594597.
Shao, Y.-C., Hung, C.-Y. & Chou, Y.-J. 2017 Numerical study of convective sedimentation through a sharp density interface. J. Fluid Mech. 824, 513549.
Taylor, J. R. & Bucens, P. 1989 Laboratory experiments on the structure of salt fingers. Deep-Sea Res. A 36, 16751704.
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. Lond. A 239, 6175.
Turner, J. S. 1967 Salt fingers across a density interface. Deep-Sea Res. 14, 599611.
Venayagamoorthy, S. K. & Fringer, O. B. 2007 On the formation and propagation of nonlinear internal boluses across a shelf break. J. Fluid Mech. 577, 137159.
Wang, R.-Q., Law, A. W.-K., Adams, E. E. & Fringer, O. B. 2009 Buoyant formation number of a starting buoyant jet. Phys. Fluids 21 (12), 125104.
Yu, X., Hsu, T.-J. & Balachandar, S. 2013 Convective instability in sedimentation: linear stability analysis. J. Geophys. Res. 118, 256272.
Yu, X., Hsu, T.-J. & Balachandar, S. 2014 Convective instability in sedimentation: 3-D numerical study. J. Geophys. Res. 119, 81418161.
Zang, Y. & Street, R. L. 1995 Numerical simulation of coastal upwelling and interfacial instability of a rotational and stratified fluid. J. Fluid Mech. 305, 4775.
Zang, Y., Street, R. L. & Koseff, J. R. 1994 A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 1833.
Zedler, E. A. & Street, R. L. 2001 Large-eddy simulation of sediment transport: current over ripples. J. Hydraul. Engng. 127 (6), 444452.
Zedler, E. A. & Street, R. L. 2006 Sediment transport over ripples in oscillatory flow. J. Hydraul. Engng. 132 (2), 180193.
Zhao, W., Frankel, S. F. & Mongeau, L. G. 2000 Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12 (3), 589596.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Formation of drops and rings in double-diffusive sedimentation

  • Yi-Ju Chou (a1) (a2), Chen-Yen Hung (a1) and Chien-Fu Chen (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed