Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T07:40:39.393Z Has data issue: false hasContentIssue false
  • English
  • Français

Einstein conjecture and resting-time statistics in the bed-load transport of monodispersed particles

Published online by Cambridge University Press:  14 August 2019

Luigi Fraccarollo Open the ORCID record for Luigi Fraccarollo [Opens in a new window]  and
Marwan A. Hassan Open the ORCID record for Marwan A. Hassan [Opens in a new window]
Luigi Fraccarollo*
Affiliation:
Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano, 77, 38123, Trento, Italy
Marwan A. Hassan
Affiliation:
Department of Geography, The University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Sediment transport in rivers consists, at moderate discharge stage, of individual grains that undergo a series of step movements and rest periods (bed-load). Following a large number of grain trajectories in time and space is difficult and the results are affected by bias due to censorship of the time-spatial window. Therefore, the data sets available for the description of the statistics of resting-times, travel-time and lengths of the steps, are still insufficient. In this paper, an innovative experimental methodology has been designed and applied to get data representing the evolution of a bed surface and to support a robust statistical analysis of sediment transport. The methodology is based on image sequences taken of a flat bed made of well-sorted (mono-dispersed) particles. The acquired data are interpreted analytically through equations that describe the effects of grain entrainment and deposition. We show that grains’ displacement have a mean value independent of bed-load rate under low to moderate transport intensity for a given sediment type and bed-slope. Hence, we provide a strong validation of the seminal conjecture of Einstein in his theoretical statistical description of sediment transport. Finally, we describe the probability density functions of the resting-time for a few values of the sediment discharge.

JFM classification

Geophysical and Geological Flows: Sediment transport Geophysical and Geological Flows: River dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 1077 - 1089
Copyright
© 2019 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.)

References

Ancey, C., Böhm, T., Jodeau, M. & Frey, P. 2006 Statistical description of sediment transport experiments. Phys. Rev. E 74 (1), 011302.Google Scholar
Ballio, F., Pokrajac, D., Radice, A. & Sadabadi, S. A. H. 2018 Lagrangian and Eulerian description of bed load transport. J. Geophys. Res.: Earth Surface 123 (2), 384–408.Google Scholar
Ballio, F., Radice, A., Fathel, S. L. & Furbish, D. J. 2019 Experimental censorship of bed load particle motions, and bias correction of the associated frequency distributions. J. Geophys. Res. 124 (1), 116–136.Google Scholar
Berzi, D. & Fraccarollo, L. 2013 Inclined, collisional sediment transport. Phys. Fluids 25 (10), 106601.Google Scholar
Campagnol, J., Radice, A. & Ballio, F. 2012 Insight on how bed configuration affects properties of bed load motion. In Proceedings of the 6th International Conference on Scour and Erosion (ICSE-6), Paris, France (ed. Fry, J.-J. & Chevalier, C.), pp. 27–34. Société Hydrotechnique De France.Google Scholar
Campagnol, J., Radice, A., Nokes, R., Bulankina, V., Lescova, A. & Ballio, F. 2013 Lagrangian analysis of bed-load sediment motion: database contribution. J. Hydraul Res. 51 (5), 589–596.Google Scholar
Capart, H. & Fraccarollo, L. 2011 Transport layer structure in intense bed-load. Geophys. Res. Lett. 38 (20), L20402.Google Scholar
Charru, F., Mouilleron, H. & Eiff, O. 2004 Erosion and deposition of particles on a bed sheared by a viscous flow. J. Fluid Mech. 519, 55–80.Google Scholar
Drake, T. G., Shreve, R. L., Dietrich, W. E., Whiting, P. J. & Leopold, L. B. 1988 Bedload transport of fine gravel observed by motion-picture photography. J. Fluid Mech. 192, 193–217.Google Scholar
Einstein, H. A. 1950 The Bed-Load Function for Sediment Transportation in Open Channel Flows, vol. 1026. US Department of Agriculture.Google Scholar
Fan, N., Zhong, D., Wu, B., Foufoula-Georgiou, E. & Guala, M. 2014 A mechanistic-stochastic formulation of bed load particle motions: from individual particle forces to the Fokker–Planck equation under low transport rates. J. Geophys. Res. 119 (3), 464–482.Google Scholar
Fathel, S. L., Furbish, D. J. & Schmeeckle, M. W. 2015 Experimental evidence of statistical ensemble behavior in bed load sediment transport. J. Geophys. Res. 120 (11), 2298–2317.Google Scholar
Ganti, V., Meerschaert, M. M., Foufoula-Georgiou, E., Viparelli, E. & Parker, G. 2010 Normal and anomalous diffusion of gravel tracer particles in rivers. J. Geophys. Res. 115 (F2), F00A12.Google Scholar
Habersack, H. M. 2001 Radio-tracking gravel particles in a large braided river in New Zealand: a field test of the stochastic theory of bed load transport proposed by Einstein. Hydrol. Process. 15 (3), 377–391.Google Scholar
Hassan, M. A. & Bradley, D. N. 2017 Geomorphic controls on tracer particle dispersion in gravel bed rivers. In Gravel-Bed Rivers: Process and Disasters (ed. Tsutsumi, D. & Laronne, J. B.), chap. 6, pp. 159–184. Wiley-Blackwell.Google Scholar
Heays, K. G., Friedrich, H. & Melville, B. W. 2014 Laboratory study of gravel-bed cluster formation and disintegration. Water Resour. Res. 50 (3), 2227–2241.Google Scholar
Hill, K. M., DellAngelo, L. & Meerschaert, M. M. 2010 Heavy-tailed travel distance in gravel bed transport: an exploratory enquiry. J. Geophys. Res. 115 (F2), F00A14.Google Scholar
Kasprak, A., Wheaton, J. M., Ashmore, P. E., Hensleigh, J. W. & Peirce, S. 2015 The relationship between particle travel distance and channel morphology: results from physical models of braided rivers. J. Geophys. Res. 120 (1), 55–74.Google Scholar
Lajeunesse, E., Malverti, L. & Charru, Fço 2010 Bed load transport in turbulent flow at the grain scale: experiments and modeling. J. Geophys. Res. 115 (F4), F04001.Google Scholar
Lee, H-Y., Lin, Y-T., Yunyou, J. & Wenwang, H. 2006 On three-dimensional continuous saltating process of sediment particles near the channel bed. J. Hydraul Res. 44 (3), 374–389.Google Scholar
Martin, R. L., Jerolmack, D. J. & Schumer, R. 2012 The physical basis for anomalous diffusion in bed load transport. J. Geophys. Res. 117 (F1), F01018.Google Scholar
McNamara, J. P. & Borden, C. 2004 Observations on the movement of coarse gravel using implanted motion-sensing radio transmitters. Hydrol. Process. 18 (10), 1871–1884.Google Scholar
Niño, Y. & Garcia, M. H. 1996 Experiments on particle-turbulence interactions in the near-wall region of an open channel flow: implications for sediment transport. J. Fluid Mech. 326, 285–319.Google Scholar
Nikora, V., Habersack, H., Huber, T. & McEwan, I. 2002 On bed particle diffusion in gravel bed flows under weak bed load transport. Water Resour. Res. 38 (6), 17-1–17-9.Google Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, E. 2009 Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295–319.Google Scholar
Pähtz, T. & Durán, O. 2018 The cessation threshold of nonsuspended sediment transport across aeolian and fluvial environments. J. Geophys. Res. 123 (8), 1638–1666.Google Scholar
Paintal, A. S. 1971 A stochastic model of bed load transport. J. Hydraul Res. 9 (4), 527–554.Google Scholar
Pyrce, R. S. & Ashmore, P. E. 2003 The relation between particle path length distributions and channel morphology in gravel-bed streams: a synthesis. Geomorphology 56 (1–2), 167–187.Google Scholar
Séchet, P. & Le Guennec, B. 1999 Bursting phenomenon and incipient motion of solid particles in bed-load transport. J. Hydraul Res. 37 (5), 683–696.Google Scholar
Seizilles, G., Lajeunesse, E., Devauchelle, O. & Bak, M. 2014 Cross-stream diffusion in bedload transport. Phys. Fluids 26 (1), 013302.Google Scholar
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.Google Scholar
Shields, A.1936 Anwendung der Äehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. PhD thesis, Technical University Berlin.Google Scholar
Voepel, H., Schumer, R. & Hassan, M. A. 2013 Sediment residence time distributions: theory and application from bed elevation measurements. J. Geophys. Res. 118 (4), 2557–2567.Google Scholar
Wong, M. & Parker, G. 2006 Reanalysis and correction of bed-load relation of Meyer–Peter and Müller using their own database. J. Hydraul. Engng 132 (11), 1159–1168.Google Scholar