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Sediment-laden fresh water above salt water: linear stability analysis

Published online by Cambridge University Press:  05 December 2011

P. Burns
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
E. Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
*
Email address for correspondence: meiburg@engineering.ucsb.edu

Abstract

When a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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References

1. Aliseda, A., Cartellier, A., Hainaux, F. & Lasheras, J. C. 2002 Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 468, 77105.CrossRefGoogle Scholar
2. Batchelor, G. K. & Nitsche, J. M. 1991 Instability of stationary unbounded stratified fluid. J. Fluid Mech. 227, 357391.CrossRefGoogle Scholar
3. Berrut, J. P. & Mittelmann, H. D. 2004 Adaptive point shifts in rational approximation with optimized denominator. J. Comput. Appl. Maths 164–165, 8192.CrossRefGoogle Scholar
4. Berrut, J. P. & Trefethen, L. N. 2004 Barycentric Lagrange interpolation. SIAM Rev. 46, 501517.CrossRefGoogle Scholar
5. 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
6. Bosse, T., Kleiser, L. & Meiburg, E. 2006 Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling. Phys. Fluids 18, 027102.CrossRefGoogle Scholar
7. Chandler, F. 1998 The convective instability of fluid interfaces. PhD thesis, University of Southern California.Google Scholar
8. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
9. Cueto-Felgueroso, L. & Juanes, R. 2009 Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems. J. Comput. Phys. 228, 65366552.CrossRefGoogle Scholar
10. Graf, F., Meiburg, E. & Härtel, C. 2002 Density-driven instabilites of miscible fluids in a Hele–Shaw cell: linear stability analysis of the three-dimensional Stokes equations. J. Fluid Mech. 451, 261282.CrossRefGoogle Scholar
11. Green, T. 1987 The importance of double diffusion to the settling of suspended material. Sedimentology 34, 319331.CrossRefGoogle Scholar
12. Henniger, R., Kleiser, L. & Meiburg, E. 2010 Direct numerical simulations of particle transport in a model estuary. J. Turbul. 11, 131.CrossRefGoogle Scholar
13. Hill, P., Syvitski, J. P. M., Cowan, E. & Powell, R. 1998 In situ observations of floc settling velocities in Glacier Bay, Alaska. Mar. Geol. 145, 8594.CrossRefGoogle Scholar
14. Hoyal, D. C., Bursik, M. I. & Atkinson, J. F. 1999a The influence of diffusive convection on sedimentation from buoyant plumes. Mar. Geol. 159, 205220.CrossRefGoogle Scholar
15. Hoyal, D. C., Bursik, M. I. & Atkinson, J. F. 1999b Settling-driven convection: a mechanism of sedimentation from stratified fluids. J. Geophys. Res. 104, 79537966.CrossRefGoogle Scholar
16. Huppert, H. E. & Manins, P. C. 1973 Limiting conditions for salt-fingering at an interface. Deep-Sea Res. 20, 315323.Google Scholar
17. Huppert, H. E. & Turner, J. S. 1981 Double-diffusve convection. J. Fluid Mech. 106, 299329.CrossRefGoogle Scholar
18. Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 1642.CrossRefGoogle Scholar
19. Linden, P. F. & Shirtcliffe, T. G. L. 1978 The diffusive interface in double-diffusive convection. J. Fluid Mech. 87, 417432.CrossRefGoogle Scholar
20. Maxworthy, T. 1999 The dynamics of sedimenting surface gravity currents. J. Fluid Mech. 392, 2744.CrossRefGoogle Scholar
21. McCool, W. W. & Parsons, J. D. 2004 Sedimentation from buoyant fine-grained suspensions. Cont. Shelf Res. 24, 11291142s.CrossRefGoogle Scholar
22. Meiburg, E. & Kneller, B. 2010 Turbidity currents and their deposits. Annu. Rev. Fluid Mech. 42, 135156.CrossRefGoogle Scholar
23. Milliman, J. D. & Syvitski, J. P. M. 1992 Geomorphic/tectonic control of sediment discharge to the ocean: the importance of small mountainous rivers. J. Geol. 100, 525544.CrossRefGoogle Scholar
24. Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2002 High resolution simulations of particle-driven gravity currents. Intl J. Multiphase Flow 28, 279300.CrossRefGoogle Scholar
25. 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.CrossRefGoogle Scholar
26. Parsons, J. D. & Garcia, M. H. 2000 Enhanced sediment scavenging due to double-diffusive convection. J. Sedim. Res. 70, 4752.CrossRefGoogle Scholar
27. Pharo, C. H. & Carmack, E. C. 1979 Sedimentation processes in a short residence-time intermontane lake, Kamloops Lake, British Columbia. Sedimentology 26, 523541.CrossRefGoogle Scholar
28. Radko, T. & Stern, M. 2000 Finite-amplitude salt fingers in a vertically bounded layer. J. Fluid Mech. 425, 133160.CrossRefGoogle Scholar
29. Redekopp, L. 2002 Elements of instability theory for environmental flows. In Environmental Stratified Flows (ed. Grimshaw, R. ), pp. 223281. Kluwer Academic.Google Scholar
30. Schmitt, R. 1979 The growth rate of super-critical salt fingers. Deep-Sea Res. 26A, 2340.CrossRefGoogle Scholar
31. Sreenivas, K., Singh, O. & Srinivasan, J. 2009 On the relationship between finger width, velocity, and fluxes in thermohaline convection. Phys. Fluids 21, 026601.CrossRefGoogle Scholar
32. Stern, M. 1969 Collective instability of salt fingers. J. Fluid Mech. 35, 209218.CrossRefGoogle Scholar
33. Syvitski, J. P. M., Asprey, K., Clattenburg, D. & Hodge, G. 1985 The prodelta environment of a fjord: suspended particle dynamics. Sedimentology 32, 83107.CrossRefGoogle Scholar
34. Tee, T. W. & Trefethen, L. N. 2006 A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28 (5), 17981811.CrossRefGoogle Scholar
35. Turner, J. S. 1967 Salt fingers across a density interface. Deep-Sea Res. 14, 599611.Google Scholar
36. Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
37. Weimer, P. & Slatt, R. M. 2007 Introduction to the Petroleum Geology of Deepwater Setting. AAPG Studies in Geology , American Association of Petroleum Geologists.Google Scholar
38. Wright, L. D. & Coleman, J. M. 1974 Mississippi River mouth processes: effluent dynamics and morphologic development. J. Geol. 82 (6), 751778.CrossRefGoogle Scholar