Mechanics of Sediment Transport: Particle Scale of Entrainment to Continuum Scale of Bedload Flux
Publication: Journal of Engineering Mechanics
Volume 143, Issue 11
Abstract
A mechanistic-cum-stochastic theory describing the sediment transport phenomenon from the particle scale of entrainment to the continuum scale of bedload flux under a steady unidirectional flow over a compact sediment bed is developed. The sediment particles, considered as discrete spherical particles, are subjected to hydraulically smooth, transitional, and rough wall-shear flows. At the particle scale, the hydrodynamic forces acting on a sediment particle resting over three closely packed bed particles are analyzed. The criteria for entrainment threshold in rolling, sliding, and lifting modes are determined introducing the velocity fluctuations. Comparison of the computed threshold curves with the experimental data reveals that the entrainment threshold primarily belongs to the transition of the sliding to the lifting modes. Then, at the particle scale, the entrainment probabilities in rolling, sliding, and lifting modes for a given median size of sediment particles are obtained. The rolling and sliding probabilities increase with an increase in Shields parameter, and after attaining their individual peak values, they decrease, while the lifting probability increases. Finally, the bedload flux, in a continuum scale, is obtained using the lifting probability.
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References
Abbott, J. E., and Francis, J. R. D. (1977). “Saltation and suspension trajectories of solid grains in a water stream.” Philos. Trans. R. Soc. London Ser. A, 284(1321), 225–254.
Ali, S. Z., and Dey, S. (2016). “Hydrodynamics of sediment threshold.” Phys. Fluids, 28(7), 075103.
Ali, S. Z., and Dey, S. (2017). “Origin of the scaling laws of sediment transport.” Proc. R. Soc. London Ser. A, 473(2197), 20160785.
Bagnold, R. A. (1956). “The flow of cohesionless grains in fluids.” Philos. Trans. R. Soc. London Ser. A, 249(964), 235–297.
Bagnold, R. A. (1973). “The nature of saltation and of ‘bed-load’ transport in water.” Proc. R. Soc. London Ser. A, 332(1591), 473–504.
Bagnold, R. A. (1974). “Fluid forces on a body in shear flow; experimental use of ‘stationary flow’.” Proc. R. Soc. London Ser. A, 340(1621), 147–171.
Best, J., Bennett, S., Bridge, J., and Leeder, M. (1997). “Turbulence modulation and particle velocities over flat sand beds at low transport rates.” J. Hydraul. Eng., 1118–1129.
Bose, S. K., and Dey, S. (2013). “Sediment entrainment probability and threshold of sediment suspension: Exponential-based approach.” J. Hydraul. Eng., 1099–1106.
Buffington, J. M., and 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(8), 1993–2029.
Casey, H. J. (1935). “Über geschiebebewegung.” Mitteilungen der Preussischen Versuchsanstalt für Wasserbau und Schiffbau, Das Anstalt, Berlin.
Charru, F., Mouilleron, H., and Eiff, O. (1999). “Erosion and deposition of particles on a bed sheared by a viscous flow.” J. Fluid Mech., 519, 55–80.
Cheng, N. S. (2016). “Representative grain size and equivalent roughness height of a sediment bed.” J. Hydraul. Eng., 06015016.
Cheng, N. S., and Chiew, Y. M. (1998). “Pickup probability for sediment entrainment.” J. Hydraul. Eng., 232–235.
Chien, N., and Wan, Z. (1999). Mechanics of sediment transport, ASCE, Reston, VA.
Clifford, N. J., McClatchey, J., and French, J. R. (1991). “Measurements of turbulence in the benthic boundary layer over a gravel bed and comparison between acoustic measurements and predictions of the bedload transport of marine gravels.” Sedimentology, 38(1), 161–166.
Dey, S. (1999). “Sediment threshold.” Appl. Math. Model., 23(5), 399–417.
Dey, S. (2014). Fluvial hydrodynamics: Hydrodynamic and sediment transport phenomena, Springer, Berlin.
Dey, S., and Ali, S. Z. (2016). “Hydrodynamics of sediment transport: Grain scale to continuum scale.” Scour and erosion, J. Harris, R. Whitehouse, and S. Maxon, eds., CRC Press, Boca Raton, FL, 29–40.
Dey, S., and Ali, S. Z. (2017). “Stochastie mechanics of loose boundary particle transport in turbulent flow.” Phys. Fluids, 29(5), 055103.
Dey, S., Das, R., Gaudio, R., and Bose, S. K. (2012). “Turbulence in mobile-bed streams.” Acta Geophys., 60(6), 1547–1588.
Dey, S., Dey Sarker, H. K., and Debnath, K. (1999). “Sediment threshold under stream flow on horizontal and sloping beds.” J. Eng. Mech., 545–553.
Dey, S., and Papanicolaou, A. (2008). “Sediment threshold under stream flow: A state-of-the-art review.” KSCE J. Civil Eng., 12(1), 45–60.
Dey, S., Sarkar, S., and Solari, L. (2011). “Near-bed turbulence characteristics at the entrainment threshold of sediment beds.” J. Hydraul. Eng., 945–958.
Einstein, H. A. (1942). “Formulas for the transportation of bed load.” Trans. Am. Soc. Civil Eng., 107(1), 561–577.
Einstein, H. A. (1950). “The bed-load function for sediment transportation in open channel flows.”, U.S. Dept. of Agriculture, Soil Conservation Service, Washington, DC.
Einstein, H. A., and El-Samni, E. A. (1949). “Hydrodynamic forces on a rough wall.” Rev. Mod. Phys., 21(3), 520–524.
Engelund, F., and Fredsøe, J. (1976). “A sediment transport model for straight alluvial channels.” Nord. Hydrol., 7(5), 293–306.
Fenton, J. D., and Abbott, J. E. (1977). “Initial movement of grains on a stream bed: The effect of relative protrusion.” Proc. R. Soc. London Ser. A, 352(1671), 523–537.
Fernandez Luque, R. (1974). “Erosion and transport of bed-load sediment.” Ph.D. thesis, Delft Univ. of Technology, Meppel, Netherlands.
Fernandez Luque, R., and van Beek, R. (2010). “Erosion and transport of bed-load sediment.” J. Hydraul. Res., 14(2), 127–144.
Francis, J. R. D. (1973). “Experiments on the motion of solitary grains along the bed of a water-stream.” Proc. R. Soc. London Ser. A, 332(1591), 443–471.
Fredsøe, J., and Deigaard, R. (1992). “Mechanics of coastal sediment transport.” Advanced series on ocean engineering, Vol. 3, World Scientific, Singapore.
Garcia, M. H. (2008). Sedimentation engineering: Processes, measurements, modeling, and practice, ASCE, Reston, VA.
Gaudio, R., Miglio, A., and Dey, S. (2010). “Non-universality of von Kármán’s κ in fluvial streams.” J. Hydraul. Res., 48(5), 658–663.
Gilbert, G. K. (1914). “The transportation of debris by running water.”, U.S. Geological Survey, Washington, DC.
Grass, A. J. (1970). “Initial instability of fine bed sand.” J. Hydraul. Div., 96(3), 619–632.
Grass, A. J. (1971). “Structural features of turbulent flow over smooth and rough boundaries.” J. Fluid Mech., 50(2), 233–255.
Guy, H. P., Simons, D. B., and Richardson, E. V. (1966). “Summary of alluvial channel data from flume experiments, 1956–1961.”, U.S. Geological Survey, Washington, DC.
Heald, J., McEwan, I., and Tait, S. (2004). “Sediment transport over a flat bed in a unidirectional flow: Simulations and validation.” Philos. Trans. R. Soc. London Ser. A, 362(1822), 1973–1986.
Heathershaw, A. D., and Thorne, P. D. (1985). “Sea-bed noises reveal role of turbulent bursting phenomenon in sediment transport by tidal currents.” Nature, 316(6026), 339–342.
Hu, C., and Guo, Q. (2011). “Near-bed sediment concentration distribution and basic probability of sediment movement.” J. Hydraul. Eng., 1269–1275.
Hu, C. H., and Hui, Y. J. (1996). “Bed-load transport. I: Mechanical characteristics.” J. Hydraul. Eng., 245–254.
Iwagaki, Y. (1956). “Fundamental study on critical tractive force.” Trans. Japan Soc. Civil Eng., 41, 1–21.
Jain, S. (1992). “Note on lag in bedload discharge.” J. Hydraul. Eng., 904–917.
Jeffreys, H. (1929). “On the transport of sediments by streams.” Proc. Cambridge Phil. Soc., 25(3), 272–276.
Karahan, E. (1975). “Initiation of motion for uniform and non-uniform materials.” Ph.D. thesis, Technical Univ., Istanbul, Turkey.
Kirchner, J. W., Dietrich, W. E., Iseya, F., and Ikeda, H. (1990). “The variability of critical shear stress, friction angle, and grain protrusion in water-worked sediments.” Sedimentology, 37(4), 647–672.
Kline, S. J., Reynolds, W. C., Schraub, F. A., and Runstadler, P. W. (1967). “The structure of turbulent boundary layers.” J. Fluid Mech., 30(4), 741–773.
Komar, P. D., and Carling, P. A. (1991). “Grain sorting in gravel-bed streams and the choice of particle sizes for flow-competence evaluations.” Sedimentology, 38(3), 489–502.
Kramer, H. (1935). “Sand mixtures and sand movement in fluvial model.” Trans. Am. Soc. Civil Eng., 100, 798–838.
Lajeunesse, E., Malverti, L., and Charru, F. (2010). “Bed load transport in turbulent flow at the grain scale: Experiments and modeling.” J. Geophys. Res., 115(F4), F04001.
Lane, E. W., and Kalinske, A. A. (1939). “The relation of suspended to bed materials in rivers.” Trans. Am. Geophys. Union, 20(4), 637–641.
Lee, H. Y., and Hsu, I. S. (1994). “Investigation of saltating particle motions.” J. Hydraul. Eng., 831–845.
Li, Z., and Komar, P. D. (1986). “Laboratory measurements of pivoting angles for applications to selective entrainment of gravel in current.” Sedimentology, 33(3), 413–423.
Ling, C. H. (1995). “Criteria for incipient motion of spherical sediment particles.” J. Hydraul. Eng., 472–478.
Lopez, F., and Garcia, M. H. (2001). “Risk of sediment erosion and suspension in turbulent flows.” J. Hydraul. Eng., 231–235.
Mantz, P. A. (1977). “Incipient transport of fine grains and flakes by fluids-extended Shields diagram.” J. Hydraul. Div., 103(6), 601–615.
Meyer-Peter, E., Favre, H., and Einstein, H. A. (1934). “Neuere versuchsresultate über den geschiebetrieb.” Schweizerische Bauzeitung, 103(13), 147–150.
Meyer-Peter, E., and Müller, R. (1948). “Formulas for bed-load transport.” Proc., 2nd Meeting of Int. Association for Hydraulic Research, Vol. 3, Stockholm, Sweden, 39–64.
Miller, M. C., McCave, I. N., and Komar, P. D. (1977). “Threshold of sediment motion under unidirectional currents.” Sedimentology, 24(4), 507–527.
Miller, R. L., and Byrne, R. J. (1966). “The angle of repose for a single grain on a fixed rough bed.” Sedimentology, 6(4), 303–314.
Morsi, S. A., and Alexander, A. J. (1972). “An investigation of particle trajectories in two-phase flow systems.” J. Fluid Mech., 55(2), 193–208.
Neill, C. R. (1967). “Mean velocity criterion for scour of course uniform bed material.” Proc., 12th Congress of Int. Association for Hydraulic Research, Vol. 3, Fort Collins, CO, 46–54.
Nelson, J., Shreve, R. L., McLean, S. R., and Drake, T. G. (1995). “Role of near-bed turbulence structure in bed load transport and bed form mechanics.” Water Resour. Res., 31(8), 2071–2086.
Nezu, I. (1977). “Turbulent structure in open channel flow.” Ph.D. thesis, Kyoto Univ., Kyoto, Japan.
Niño, Y., García, M., and Ayala, L. (1994). “Gravel saltation. 1: Experiments.” Water Resour. Res., 30(6), 1907–1914.
Paintal, A. (2010). “Concept of critical shear stress in loose boundary open channels.” J. Hydraul. Res., 9(1), 91–113.
Papanicolaou, A. N. (1997). “The role of turbulence on the initiation of sediment motion.” Ph.D. thesis, Virginia Institute of Technology, Blacksburg, VA.
Papanicolaou, A. N., Diplas, P., Dancey, C. L., and Balakrishnan, M. (2001). “Surface roughness effects in near-bed turbulence: Implications to sediment entrainment.” J. Eng. Mech., 211–218.
Papanicolaou, A. N., Diplas, P., Evaggelopoulos, N., and Fotopoulos, S. (2002). “Stochastic incipient motion criterion for spheres under various bed packing conditions.” J. Hydraul. Eng., 369–380.
Recking, A., Frey, P., Paquier, A., Belleudy, P., and Champagne, J. Y. (2008). “Feedback between bed load transport and flow resistance in gravel and cobble bed rivers.” Water Resour. Res., 44(5), W05412.
Reichardt, H. (1951). “Vollständige darstellung der turbulenten geschwindigkeitsverteilung in glatten leitungen [Complete representation of the turbulent velocity distribution in smooth pipes].” Z. Angew. Math. Mech., 31(7), 208–219 (in German).
Schmeeckle, M. W., and Nelson, J. M. (2003). “Direct numerical simulation of bedload transport using a local, dynamic boundary condition.” Sedimentology, 50(2), 279–301.
Shields, A. F. (1936). “Application of similarity principles and turbulence research to bed-load movement.” Mitteilungen der Preussischen Versuchsanstalt für Wasserbau und Schiffbau, Vol. 26, Das Anstalt, Berlin, 5–24.
Smart, G. M. (1984). “Sediment transport formula for steep channels.” J. Hydraul. Eng., 267–276.
Sun, Z., and Donahue, J. (2000). “Statistically derived bedload formula for any fraction of nonuniform sediment.” J. Hydraul. Eng., 105–111.
Sutherland, A. J. (1967). “Proposed mechanism for sediment entrainment by turbulent flows.” J. Geophys. Res., 72(24), 6183–6194.
Tregnaghi, M., Bottacin-Busolin, A., Marion, A., and Tait, S. (2012). “Stochastic determination of entrainment risk in uniformly sized sediment beds at low transport stages. 1: Theory.” J. Geophys. Res., 117(F4), F04004.
USWES (U.S. Waterways Experiment Station). (1936). “Flume tests made to develop a synthetic sand which will not form ripples when used in movable bed models.”, U.S. Waterways Experiment Station, Vicksburg, MS.
van Driest, E. R. (1956). “On turbulent flow near a wall.” J. Aeronaut. Sci., 23(11), 1007–1011.
Vanoni, V. A. (1946). “Transportation of suspended sediment by water.” Trans. Am. Soc. Civil Eng., 111(1), 67–102.
van Rijn, L. C. (1981). “Computation of bed-load concentration and bed-load transport.”, Delft Hydraulics Laboratory, Delft, Netherlands.
van Rijn, L. C. (1984). “Sediment transport. I: Bed load transport.” J. Hydraul. Eng., 1431–1456.
Wang, X., Zheng, J., Danxun, L., and Qu, Z. (2008). “Modification of the Einstein bed-load formula.” J. Hydraul. Eng., 1363–1369.
White, C. M. (1940). “The equilibrium of grains on the bed of a stream.” Proc. R. Soc. London Ser. A, 174(958), 322–338.
White, S. J. (1970). “Plane bed thresholds of fine grained sediments.” Nature, 228(5267), 152–153.
Wiberg, P. L., and Smith, J. D. (1987). “Calculations of the critical shear stress for motion of uniform and heterogeneous sediments.” Water Resour. Res., 23(8), 1471–1480.
Wilcock, P. R. (1988). “Methods for estimating the critical shear stress of individual fractions in mixed-size sediment.” Water Resour. Res., 24(7), 1127–1135.
Wu, F. C., and Chou, Y. J. (2003). “Rolling and lifting probabilities for sediment entrainment.” J. Hydraul. Eng., 110–119.
Wu, F. C., and Jiang, M. R. (2007). “Numerical investigation of the role of turbulent bursting in sediment entrainment.” J. Hydraul. Eng., 329–334.
Wu, F. C., and Lin, Y. C. (2002). “Pickup probability of sediment under log-normal velocity distribution.” J. Hydraul. Eng., 438–442.
Wu, F. C., and Yang, K. H. (2004). “A stochastic partial transport model for mixed-size sediment: Application to assessment of fractional mobility.” Water Resour. Res., 40(4), W04501, 1–18.
Yalin, M. S., and Karahan, E. (1979). “Inception of sediment transport.” J. Hydraul. Div., 105(11), 1433–1443.
Zanke, U. C. E. (2003). “On the influence of turbulence on the initiation of sediment motion.” Int. J. Sed. Res., 18(1), 17–31.
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Journal of Engineering Mechanics
Volume 143 • Issue 11 • November 2017
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©2017 American Society of Civil Engineers.
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Received: Jun 13, 2016
Accepted: Apr 27, 2017
Published online: Aug 28, 2017
Published in print: Nov 1, 2017
Discussion open until: Jan 28, 2018
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