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Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows

Published online by Cambridge University Press:  25 September 2009

MALIKA OURIEMI ,
PASCALE AUSSILLOUS  and
ÉLISABETH GUAZZELLI
MALIKA OURIEMI
Affiliation:
IUSTI CNRS UMR 6595 – Polytech'Marseille – Aix-Marseille Université (U1) 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France
PASCALE AUSSILLOUS
Affiliation:
IUSTI CNRS UMR 6595 – Polytech'Marseille – Aix-Marseille Université (U1) 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France
ÉLISABETH GUAZZELLI*
Affiliation:
IUSTI CNRS UMR 6595 – Polytech'Marseille – Aix-Marseille Université (U1) 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France
*
‡Email address for correspondence: [email protected]
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Abstract

We propose a two-phase model having a Newtonian rheology for the fluid phase and friction for the particle phase to describe bed-load transport in the laminar viscous regime. We have applied this continuum model to sediment transport by viscous shearing flows. The equations are shown to reduce to the momentum equation for the mixture and the Brinkman equation for the fluid velocity. This modelling is able to provide a description of the flow of the mobile granular layer. At some distance from threshold of particle motion, where the continuum approach is more realistic as the mobile layer is larger than one particle diameter, there is very little slip between the two phases and the velocities inside the mobile bed have approximately a parabolic profile. When the Poiseuille (or Couette) flow is not significantly perturbed, simple analytical results of the particle flux varying cubically with the Shields number and of the bed-load thickness varying linearly with it can then be obtained. These predictions compare favourably with experimental observations of bed-load transport in pipe flows.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 295 - 319
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

†

Present address: IFP-Lyon, Rond-Point de I'échangeur de Solaize, BP3, 69360 Solaize, France.

References

REFERENCES

Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds – Equation of motion. Ind. Engng Chem. Fundam. 6, 527–539.CrossRefGoogle Scholar
Bagnold, R. A. 1956 The flow of cohesionless grains in fluids. Phil. Trans. R. Soc. Lond. A 249, 235–297.Google Scholar
Brinkman, H. C. 1947 a A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.CrossRefGoogle Scholar
Brinkman, H. C. 1947 b On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 81–86.CrossRefGoogle Scholar
Buffington, J. M. & Montgomery, D. R. 1997 A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers. Water Resour. Res. 33, 1993–2029.CrossRefGoogle Scholar
Camemen, B. & Larson, M. 2005 A general formula for non-cohesive bed load sediment transport. Estuarine Coastal 63, 249–260.CrossRefGoogle Scholar
Cassar, C., Nicolas & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.CrossRefGoogle 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, 111–121.CrossRefGoogle Scholar
Charru, F. & Mouilleron-Arnould, H. 2002 Instability of a bed of particles sheared by a viscous flow. J. Fluid Mech. 452, 303–323.CrossRefGoogle 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, 55–80.CrossRefGoogle Scholar
Cheng, N. S. 2004 Analysis of bed load transport in laminar flows. Adv. Water Resour. 27, 937–942.CrossRefGoogle Scholar
Dancey, C. L., Diplas, P., Papanicolaou, A. N. & Diplas, M. 2002 Probability of individual grain movement and threshold condition. J. Hydraul. Engng 128, 1069–1075.CrossRefGoogle Scholar
Einstein, H. A. 1942 Formulas for the transportation of bed load. Transactions 2140, 561–597.Google Scholar
Einstein, H. A. 1950 The bed load function for sedimentation in open channel flows. Tech. Rep. 1026, 1–69.Google Scholar
Goharzadeh, A., Khalili, A. & Jørgensen, B. B. 2005 Transition layer thickness at a fluid-porous interface. Phys. Fluids 17, 057102.CrossRefGoogle Scholar
GDR Midi 2004 On dense granular flows. Eur. Phys. J. 14, 341–365.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Martinus Nijhoff Publishers.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 a The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–241.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 b Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
Jackson, R. 1997 Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid. Chem. Engng Sci. 52, 2457–2469.CrossRefGoogle 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, 727–730.CrossRefGoogle ScholarPubMed
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for homogeneous monodisperse gas fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229–263.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1986 Viscous resuspension. Chem. Engng Sci. 41, 1377–1384.CrossRefGoogle Scholar
Lobkovsky, A. E., Orpe, A. V., Molloy, R., Kudrolli, A. & Rothman, D. H. 2008 Erosion of granular bed driven by laminar fluid flow. J. Fluid Mech. 605, 47–58.CrossRefGoogle Scholar
Loiseleux, T., Gondret, P., Rabaud, M. & Doppler, D. 2005 Onset of erosion and avalanches for an inclined granular bed sheared by a continuous laminar flow. Phys. Fluids 17, 103304.CrossRefGoogle Scholar
Meyer-Peter, E. & Muller, R. 1948 Formulas for bed-load transport. Report of 2nd Meeting of the International Association of Hydraulic and Structural Research, 34–64.Google Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
Ouriemi, M., Aussillous, P., Medale, M., Peysson, Y. & Guazzelli, E. 2007 Determination of the critical Shields number for particle erosion in laminar flow. Phys. Fluids 19, 061706.CrossRefGoogle Scholar
Paintal, A. S. 1971 Concept of critical shear stress in loose boundary open channels. J. Hydraul. Res. 9, 91–113CrossRefGoogle Scholar
Pouliquen, O., Cassar, C., Jop, P., Forterre, Y. & Nicolas, M. 2006 Flow of dense granular material: towards simple constitutive laws. J. Stat. Mech. P07020.CrossRefGoogle Scholar
Ribberink, J. S. 1998 Bed-load transport for steady flows and unsteady oscillatory flows. Coastal Engng 34, 58–82.CrossRefGoogle Scholar
Stickel, J. J. & Powel, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Ann. Rev. Fluid Mech. 37, 129–149.CrossRefGoogle Scholar
Van der Hoef, M. A., Beetstra, R. & Kuipers, J. A. M. 2005 Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254.CrossRefGoogle Scholar
Vanoni, V. A. 1966 Sediment transportation mechanics: initiation of motion. J. Hydraul. Div. Am. Soc. Civ. Engng 92, 291–314.Google Scholar
White, C. M. 1940 The equilibrium of grains on the bed of a steam. Proc. R. Soc. Lond. 174, 322–338.Google Scholar
Wong, M. & Parker, G. 2006 Reanalysis and correction of bed-load relation of Meyer–Peter and Muller using their own database. J. Hydr. Engng 132, 1159–1168.CrossRefGoogle Scholar
Yalin, S. 1963 An expression for bed-load transportation. J. Hydraul. Division HY3, 221–250.CrossRefGoogle Scholar
Zhang, D. Z. & Properetti, A. 1997 Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions. Intl J. Multiphase Flow 23, 425–453.CrossRefGoogle Scholar