Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-25T11:08:39.177Z Has data issue: false hasContentIssue false

A two-phase flow model of sediment transport: transition from bedload to suspended load

Published online by Cambridge University Press:  22 August 2014

Filippo Chiodi
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, (PMMH UMR 7636 ESPCI – CNRS – University Paris Diderot – University P. M. Curie) 10 rue Vauquelin, 75005 Paris, France
Philippe Claudin
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, (PMMH UMR 7636 ESPCI – CNRS – University Paris Diderot – University P. M. Curie) 10 rue Vauquelin, 75005 Paris, France
Bruno Andreotti*
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, (PMMH UMR 7636 ESPCI – CNRS – University Paris Diderot – University P. M. Curie) 10 rue Vauquelin, 75005 Paris, France
Email address for correspondence:


The transport of dense particles by a turbulent flow depends on two dimensionless numbers. Depending on the ratio of the shear velocity of the flow to the settling velocity of the particles (or the Rouse number), sediment transport takes place in a thin layer localized at the surface of the sediment bed (bedload) or over the whole water depth (suspended load). Moreover, depending on the sedimentation Reynolds number, the bedload layer is embedded in the viscous sublayer or is larger. We propose here a two-phase flow model able to describe both viscous and turbulent shear flows. Particle migration is described as resulting from normal stresses, but is limited by turbulent mixing and shear-induced diffusion of particles. Using this framework, we theoretically investigate the transition between bedload and suspended load.

© 2014 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.)



Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds – equation of motion. Ind. Engng Chem. Fundam. 6, 527539.Google Scholar
Andreotti, B., Barrat, J.-L. & Heussinger, C. 2012 Shear flow of non-Brownian suspensions close to jamming. Phys. Rev. Lett. 109, 105901.Google Scholar
Andreotti, B. & Claudin, P. 2013 Aeolian and subaqueous bedforms in shear flows. Phil. Trans. R. Soc. Lond. A 371, 20120364.Google Scholar
Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media, Between Fluid and Solid. Cambridge University Press.Google Scholar
Aussillous, P., Chauchat, J., Pailha, M., Médale, M. & Guazzelli, E. 2013 Investigation of the mobile granular layer in bedload transport by laminar shearing flows. J. Fluid Mech. 736, 594615.Google Scholar
Bagnold, R. A. 1956 The flow of cohesionless grains in fluids. Phil. Trans. R. Soc. Lond. 249, 235297.Google Scholar
Berzi, D. 2011 Analytical solution of collisional sheet flows. J. Hydraul. Engng 137, 12001207.Google Scholar
Berzi, D. 2013 Transport formula for collisional sheet flows with turbulent suspension. J. Hydraul. Engng 139, 359363.CrossRefGoogle Scholar
Bonnoit, C., Darnige, T., Clément, E. & Lindner, A. 2010 Inclined plane rheometry of a dense granular suspension. J. Rheol. 54, 6579.Google Scholar
Boyer, F., Guazzelli, E. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107, 188301.Google Scholar
Camemen, B. & Larson, M. 2005 A general formula for non-cohesive bed-load sediment transport. Estuar. Coast. 63, 249260.Google Scholar
Capart, H. & Fraccarollo, L. 2011 Transport layer structure in intense bed-load. Geophys. Res. Lett. 38, L20402.Google Scholar
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.CrossRefGoogle Scholar
Celik, I. & Rodi, W. 1988 Modeling suspended sediment transport in non-equilibrium situations. J. Hydraul. Engng 114, 11571191.Google Scholar
Charru, F. 2006 Selection of the ripple length on a granular bed. Phys. Fluids 18, 121508.Google Scholar
Charru, F., Andreotti, B. & Claudin, P. 2013 Sand ripples and dunes. Annu. Rev. Fluid Mech. 45, 469493.Google Scholar
Charru, F. & Hinch, E. J. 2006 Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow. J. Fluid Mech. 550, 111121.Google Scholar
Charru, F. & Mouilleron-Arnould, H. 2002 Instability of a bed of particles sheared by a viscous flow. J. Fluid Mech. 452, 303323.Google Scholar
Charru, F., Mouilleron-Arnould, H. & Eiff, O. 2004 Erosion and deposition of particles on a bed sheared by a viscous flow. J. Fluid Mech. 519, 5580.Google Scholar
Cheng, N. S. 2004 Analysis of bed load transport in laminar flows. Adv. Water Resour. 27, 937942.Google Scholar
Coleman, N. L. 1970 Flume studies of the sediment transfer coefficient. Water Resour. Res. 6, 801809.CrossRefGoogle Scholar
Cowen, E. A., Dudley, R. D., Liao, Q., Variano, E. A. & Liu, P. L.-F. 2010 An in situ borescopic quantitative imaging profiler for the measurement of high concentration sediment velocity. Exp. Fluids 49, 7788.Google Scholar
van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 10071011.Google Scholar
Durán, O., Andreotti, B. & Claudin, P. 2012 Numerical simulation of turbulent sediment transport, from bed load to saltation. Phys. Fluids 24, 103306.Google Scholar
Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191208.Google Scholar
Einstein, H. A.1950 The bed-load function for sedimentation transportation in open channel flows. Technical Bulletin 1026, pp. 1–71, US Department of Agriculture.Google Scholar
Fall, A., Lemaître, A., Bertrand, F., Bonn, D. & Ovarlez, G. 2010 Continuous and discontinuous shear thickening in granular suspension. Phys. Rev. Lett. 105, 268303.Google Scholar
Ferguson, R. I. & Church, M. 2004 A simple universal equation for grain settling velocity. J. Sedim. Res. 74, 933937.Google Scholar
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 124.Google Scholar
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion, and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407, 167200.Google Scholar
GDR MiDi,   2004 On dense granular flows. Eur. Phys. J. E 14, 341365.Google Scholar
van Hecke, M. 2010 Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys.: Condens. Matter 22, 033101.Google Scholar
Jackson, R. 1997 Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid. Chem. Engng Sci. 52, 24572469.Google Scholar
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441, 727730.Google Scholar
Lajeunesse, E., Malverti, L. & Charru, F. 2010 Bedload transport in turbulent flow at the grain scale: experiments and modeling. J. Geophys. Res. 115, F04001.Google Scholar
Leighton, D. & Acrivos, A. 1987 Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109131.Google Scholar
Le Louvetel-Poilly, J., Bigillon, F., Doppler, D., Vinkovic, I. & Champagne, J.-Y. 2009 Experimental investigation of ejections and sweeps involved in particle suspension. Water Resour. Res. 45, W02416.Google Scholar
Lerner, E., Düring, G. & Wyart, M. 2012 A unified framework for non-Brownian suspension flows and soft amorphous solids. Proc. Natl Acad. Sci. USA 109, 47984803.Google Scholar
Marchioli, C., Armenio, V., Salvetti, M. V. & Soldati, A. 2006 Mechanisms for deposition and resuspension of heavy particles in turbulent flow over wavy interfaces. Phys. Fluids 18, 025102.Google Scholar
Meyer-Peter, E. & Müller, R.1948 Formulas for bed load transport. In Proceedings 2nd Meeting, IAHR, Stockholm, Sweden, pp. 39–64.Google Scholar
Nielsen, P. 1992 Coastal Bottom Boundary Layers and Sediment Transport. World Scientific.Google Scholar
Nnadi, F. N. & Wilson, K. C. 1992 Motion of contact-load particles at high shear stress. J. Hydraul. Engng 118, 16701684.Google Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157199.Google Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, E. 2009 Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295319.Google Scholar
Ovarlez, G., Bertrand, F. & Rodts, S. 2006 Local determination of the constitutive law of a dense suspension of non-colloidal particles through magnetic resonance imaging. J. Rheol. 50, 259292.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Revil-Baudard, T. & Chauchat, J. 2013 A two-phase model for sheet flow regime based on dense granular flow rheology. J. Geophys. Res. Oceans 118, 619634.CrossRefGoogle Scholar
Ribberink, J. S. 1998 Bed-load transport for steady flows and unsteady oscillatory flows. Coast. Engng 34, 5882.Google Scholar
van Rijn, L. C. 1984 Sediment transport, part I: bed-load transport. J. Hydraul. Engng 110, 14311456.Google Scholar
Sumer, B. M., Kozakiewicz, A., Fredsøe, J. & Deigaard, R. 1996 Velocity and concentration profiles in sheet-flow layer of movable bed. J. Hydraul. Engng 122, 549558.Google Scholar
Trulsson, M., Andreotti, B. & Claudin, P. 2012 Transition from viscous to inertial regime in dense suspensions. Phys. Rev. Lett. 109, 118305.Google Scholar
Wong, M. & Parker, G. 2006 Reanalysis and correction of bed-load relation Meyer-Peter and Müller using their own database. J. Hydraul. Engng 132, 11591168.Google Scholar
Wyart, M., Nagel, S. R. & Witten, T. A. 2005 Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids. Europhys. Lett. 72, 486492.Google Scholar
Yalin, S. 1963 An expression for bed-load transportation. J. Hydraul. Div. ASCE 89, 221250.Google Scholar