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Revisiting slope influence in turbulent bedload transport: consequences for vertical flow structure and transport rate scaling

  • Raphael Maurin (a1), Julien Chauchat (a2) (a3) and Philippe Frey (a4)
  • Please note a correction has been issued for this article.


Gravity-driven turbulent bedload transport has been extensively studied over the past century in regard to its importance for Earth surface processes such as natural riverbed morphological evolution. In the present contribution, the influence of the longitudinal channel inclination angle on gravity-driven turbulent bedload transport is studied in an idealised framework considering steady and uniform flow conditions. From an analytical analysis based on the two-phase continuous equations, it is shown that: (i) the classical slope correction of the critical Shields number is based on an erroneous formulation of the buoyancy force, (ii) the influence of the slope is not restricted to the critical Shields number but affects the whole transport formula and (iii) pressure-driven and gravity-driven turbulent bedload transport are not equivalent from the slope influence standpoint. Analysing further the granular flow driving mechanisms, the longitudinal slope is shown to not only influence the fluid bed shear stress and the resistance of the granular bed, but also to affect the fluid flow inside the granular bed – responsible for the transition from bedload transport to debris flow. The relative influence of these coupled mechanisms allows us to understand the evolution of the vertical structure of the granular flow and to predict the transport rate scaling law as a function of a rescaled Shields number. The theoretical analysis is validated with coupled fluid–discrete element simulations of idealised gravity-driven turbulent bedload transport, performed over a wide range of Shields number values, density ratios and channel inclination angles. In particular, all the data are shown to collapse onto a master curve when considering the sediment transport rate as a function of the proposed rescaled Shields number.


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Since original publication we have corrected the author affiliations by adding affiliation no. 4. See doi:10.1017/jfm.2018.121.



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Anderson, T. B. & Jackson, R. 1967 Fluid mechanical description of fluidized beds. Equations of motion. Ind. Engng Chem. Fundam. 6 (4), 527539.
Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.
Armanini, A., Capart, H., Fraccarollo, L. & Larcher, M. 2005 Rheological stratification in experimental free-surface flows of granular-liquid mixtures. J. Fluid Mech. 532, 269319.
Armanini, A. & Gregoretti, C. 2005 Incipient sediment motion at high slopes in uniform flow condition. Water Resour. Res. 41 (12), w12431.
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.
Bagnold, R. A. 1956 The flow of cohesionless grains in fluids. Phil. Trans. R. Soc. Lond. A 249, 235297.
Capart, H. & Fraccarollo, L. 2011 Transport layer structure in intense bed-load. Geophys. Res. Lett. 38 (20), L20402.
Chauchat, J. 2017 A comprehensive two-phase flow model for unidirectional sheet-flows. J. Hydraul. Res. 0 (0), 114.
Cheng, N.-S. & Chen, X. 2014 Slope correction for calculation of bedload sediment transport rates in steep channels. J. Hydraul. Engng 140 (6), 04014018.
Chiew, Y.-M. & Parker, G. 1994 Incipient sediment motion on non-horizontal slopes. J. Hydraul. Res. 32 (5), 649660.
Christensen, B. A. 1995 Incipient sediment motion on non-horizontal slopes. J. Hydraul. Res. 33 (5), 725730.
Clark, A. H., Shattuck, M. D., Ouellette, N. T. & O’Hern, C. S. 2015 Onset and cessation of motion in hydrodynamically sheared granular beds. Phys. Rev. E 92, 042202.
DallaValle, J. M. 1948 Micrometrics: The Technology of Fine Particles, 2nd edn. Pitman.
Damgaard, J. S., Whitehouse, R. J. S. & Soulsby, R. L. 1997 Bed-load sediment transport on steep longitudinal slopes. J. Hydraul. Engng 123 (12), 11301138.
Dey, S. 2003 Threshold of sediment motion on combined transverse and longitudinal sloping beds. J. Hydraul. Res. 41 (4), 405415.
Diplas, P., Dancey, C. L., Celik, A. O., Valyrakis, M., Greer, K. & Akar, T. 2008 The role of impulse on the initiation of particle movement under turbulent flow conditions. Science 322 (5902), 717720.
Duran, O., Andreotti, B. & Claudin, P. 2012 Numerical simulation of turbulent sediment transport, from bed load to saltation. Phys. Fluids 24 (10), 103306.
Einstein, H. A. 1942 Formulas for the transport of bed sediment. Trans. Amer. Soc. Civil Engrs 107, 561574.
Fernandez Luque, R. & Van Beek, R. 1976 Erosion and transport of bed-load sediment. J. Hydraul. Res. 14 (2), 127144.
Fredsøe, J. & Deigaard, R. 1992 Mechanics of Coastal Sediment Transport. World Scientific.
Frey, P. 2014 Particle velocity and concentration profiles in bedload experiments on a steep slope. Earth Surf. Process. Landf. 39 (5), 646655.
Frey, P. & Church, M. 2011 Bedload: a granular phenomenon. Earth Surf. Process. Landf. 36, 5869.
Gilbert, G. K. 1914 The Transportation of Débris by Running Water. Government Printing Office.
Hsu, T. J., Jenkins, J. T. & Liu, P. L. F. 2004 On two-phase sediment transport: sheet flow of massive particles. Proc. R. Soc. Lond. A 460 (2048), 22232250.
Iversen, J. D. & Rasmussen, K. R. 1994 The effect of surface slope on saltation threshold. Sedimentology 41 (4), 721728.
Iversen, J. D. & Rasmussen, K. R. 1999 The effect of wind speed and bed slope on sand transport. Sedimentology 46 (4), 723731.
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.
Ji, C., Munjiza, A., Avital, E., Ma, J. & Williams, J. J. R. 2013 Direct numerical simulation of sediment entrainment in turbulent channel flow. Phys. Fluids 25 (5), 056601.
Karmaker, T. & Dutta, S. 2016 Prediction of short-term morphological change in large braided river using 2D numerical model. J. Hydraul. Engng 142 (10), 04016039.
Larcher, M., Fraccarollo, L., Armanini, A. & Capart, H. 2007 Set of measurement data from flume experiments on steady uniform debris flows. J. Hydraul. Res. 45, 5971.
Li, F. & Cheng, L. 1999 Numerical model for local scour under offshore pipelines. J. Hydraul. Engng 125 (4), 400406.
Li, L. & Sawamoto, M. 1995 Multi-phase model on sediment transport in sheet-flow regime under oscillatory flow. Coast. Engng Japan 38, 157178.
Maurin, R.2015 Investigation of granular behavior in bedload transport using an Eulerian–Lagrangian model. PhD thesis, Université Grenoble Alpes.
Maurin, R., Chauchat, J., Chareyre, B. & Frey, P. 2015 A minimal coupled fluid-discrete element model for bedload transport. Phys. Fluids 27 (11), 113302.
Maurin, R., Chauchat, J. & Frey, P. 2016 Dense granular flow rheology in turbulent bedload transport. J. Fluid Mech. 804, 490512.
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883889.
Meyer-Peter, E. & Müller, R. 1948 Formulas for bed-load transport. In Proceedings of the 2nd Meeting of the IAHSR, pp. 3964. IAHR.
Ni, W.-J. & Capart, H. 2015 Cross-sectional imaging of refractive-index-matched liquid-granular flows. Exp. Fluids 56 (8), 163.
Niño, Y. & García, M. 1998 Using Lagrangian particle saltation observations for bedload sediment transport modelling. Hydrol. Process. 12 (8), 11971218.
Nino, Y. & Garcia, M. 1994 Gravel saltation: 2. Modeling. Water Resour. Res. 30 (6), 19151924.
Ouriemi, M., Aussillous, P., Medale, M., Peysson, Y. & Guazzelli, É. 2007 Determination of the critical shields number for particle erosion in laminar flow. Phys. Fluids 19 (6), 061706.
Prandtl, L. 1926 Bericht über neuere Turbulenzforschung. Hydraulische Probleme. Vorträge Hydrauliktagung Göttingen 5, 113.
Recking, A., Degoutte, G., Camenen, B. & Frey, P. 2013 Dynamique et aménagement des torrents et rivières de montagne, chap. Hydraulique et transport solide, pp. 133199. Quae.
Revil-Baudard, T. & Chauchat, J. 2013 A two-phase model for sheet flow regime based on dense granular flow rheology. J. Geophys. Res. 118 (2), 619634.
Revil-Baudard, T., Chauchat, J., Hurther, D. & Barraud, P.-A. 2015 Investigation of sheet-flow processes based on novel acoustic high-resolution velocity and concentration measurements. J. Fluid Mech. 767, 130.
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidization: Part i. Trans. Inst. Chem. Engrs 32, 3553.
Rickenmann, D. 1991 Hyperconcentrated flow and sediment transport at steep slopes. J. Hydraul. Engng 117 (11), 14191439.
Rickenmann, D. 2001 Comparison of bed load transport in torrents and gravel bed streams. Water Resour. Res. 37 (12), 32953305.
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (2), 385400.
Schmeeckle, M. W., Nelson, J. M. & Shreve, R. L. 2007 Forces on stationary particles in near-bed turbulent flows. J. Geophys. Res. 112 (F2), F02003.
Schwager, T. & Pöschel, T. 2007 Coefficient of restitution and linear spring-dashpot model revisited. Granul. Matt. 9, 465469.
Seminara, G., Solari, L. & Parker, G. 2002 Bed load at low shields stress on arbitrarily sloping beds: failure of the bagnold hypothesis. Water Resour. Res. 38 (11), 1249.
Shields, A.1936 Anwendung der Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Doktor-Ingenieurs dissertation, Technischen Hochschule, Berlin.
Smart, G. M. 1984 Sediment transport formula for steep channels. J. Hydraul. Engng 110 (3), 267276.
Smart, G. M. & Jaeggi, M.1983 Sediment transport on steep slopes. Tech. Rep. 64. ETH Zurich.
Šmilauer, V. et al. 2015 Yade Documentation, 2nd edn. The Yade Project (
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 (10), 549558.
Takahashi, T. 1978 Mechanical characteristics of debris flow. J. Hydraul. Div. 104 (8), 11531169.
Takahashi, T. 2007 Debris Flow: Mechanics, Prediction and Countermeasures. Taylor & Francis.
Valyrakis, M., Diplas, P., Dancey, C. L., Greer, K. & Celik, A. O. 2010 Role of instantaneous force magnitude and duration on particle entrainment. J. Geophys. Res. 115 (F2), F02006.
Wiberg, P. L. & Smith, J. D. 1987 Calculations of the critical shear stress for motion of uniform and heterogeneous sediments. Water Resour. Res. 23 (8), 14711480.
Wilcock, P. R. & Crowe, J. C. 2003 Surface-based transport model for mixed-size sediment. J. Hydraul. Engng 129 (2), 120128.
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