Abstract
The present paper aims to investigate the transport of suspended sediment in an open channel turbulent flow. The method of moments is used to solve the unsteady two-dimensional suspended sediment transport equation where the moment equations are evaluated using a standard finite difference implicit scheme. The distribution of concentration is obtained using the Hermite polynomial representation of central moments. The solution exists for arbitrary form of eddy diffusivity and in contrast to other existing works, the present model considers the most general boundary condition at the channel bed which may be absorbing, reflecting or both. According to such nature of channel bed, the transport process has been distinguished into two phases, viz., suspension phase and deposition phase together with re-entrainment of particles. The model in its present form can be well applicable for sediment transportation in environmental processes.
Similar content being viewed by others
References
Rouse H (1937) Modern conceptions of the mechanics of fluid turbulence. Trans ASCE 102:463–505
Hunt JN (1954) The turbulent transport of suspended sediment in open channels. Proc R Soc Lond Ser A 224(1158):322–335
Ni JR, Qian Wang G (1991) Vertical sediment distribution. J Hydraul Eng 117(9):1184–1194
Umeyama M (1992) Vertical distribution of suspended sediment in uniform open-channel flow. J Hydraul Eng 118(6):936–941
Graf WH, Cellino M (1998) Suspension flow in open channels. Stochastic hydraulics, vol 2000. CRC Press, Boca Raton, pp 85–93
Mazumder BS, Ghoshal K (2006) Velocity and concentration profiles in uniform sediment-laden flow. Appl. Math. Model. 30(2):164–176
Kundu S, Ghoshal K (2014) Effects of secondary current and stratification on suspension concentration in an open channel flow. Environ Fluid Mech 14(6):1357–1380
Kundu S, Ghoshal K (2017) A mathematical model for type ii profile of concentration distribution in turbulent flows. Environ Fluid Mech 17(3):449–472
Mei CC (1969) Nonuniform diffusion of suspended sediment. J Hydraul Div 95(1):581–584
Hjelmfelt AT, Lenau CW (1970) Nonequilibrium transport of suspended sediment. J Hydraul Div HY7
Cheng KJ (1984) Bottom-boundary condition for nonequilibrium transport of sediment. J Geophys Res Oceans 89(C5):8209–8214
van Rijn LC (1986) Mathematical modeling of suspended sediment in nonuniform flows. J Hydraul Eng 112(6):433–455
Coles D (1956) The law of the wake in the turbulent boundary layer. J Fluid Mech 1(2):191–226
Celik I, Rodi W (1988) Modeling suspended sediment transport in nonequilibrium situations. J Hydraul Eng 114(10):1157–1191
Pritchard D (2006) Rate of deposition of fine sediment from suspension. J Hydraul Eng 132(5):533–536
Dorrell RM, Hogg AJ (2011) Length and time scales of response of sediment suspensions to changing flow conditions. J Hydraul Eng 138(5):430–439
Ashida K, Okabe T (1982) On the calculation method of the concentration of suspended sediment under non-equilibrium condition. Proc Jpn Conf Hydraul 26:153–158
Ouillon S, Le Guennec B (1996) Modelling non-cohesive suspended sediment transport in 2d vertical free surface flows. J Hydraul Res 34(2):219–236
Wu W, Wang SS (2007) One-dimensional modeling of dam-break flow over movable beds. J Hydraul Eng 133(1):48–58
Brown GL (2008) Approximate profile for nonequilibrium suspended sediment. J Hydraul Eng 134(7):1010–1014
Zhang S, Duan JG, Strelkoff TS (2013) Grain-scale nonequilibrium sediment-transport model for unsteady flow. J Hydraul Eng 139(1):22–36
Liu X, Nayamatullah M (2014) Semianalytical solutions for one-dimensional unsteady nonequilibrium suspended sediment transport in channels with arbitrary eddy viscosity distributions and realistic boundary conditions. J Hydraul Eng 140(5):04014011
Liu X (2016) Analytical solutions for steady two-dimensional suspended sediment transport in channels with arbitrary advection velocity and eddy diffusivity distributions. J Hydraul Res 54(4):389–398
Jing H, Chen G, Wang W, Li G (2018) Effects of concentration-dependent settling velocity on non-equilibrium transport of suspended sediment. Environ Earth Sci 77(15):549
Claudin P, Charru F, Andreotti B (2011) Transport relaxation time and length scales in turbulent suspensions. J Fluid Mech 671:491–506
Einstein HA (1950) The bed-load function for sediment transportation in open channel flows. 1026. US Government Printing Office
Yalin MS (1977) Mechanics of sediment transport, 2nd edn. Pergamon Press, Oxford
Van Rijn LC (1984) Sediment pick-up functions. J Hydraul Eng 110(10):1494–1502
Parker G, Garcia M, Fukushima Y, Yu W (1987) Experiments on turbidity currents over an erodible bed. J Hydraul Res 25(1):123–147
Abd El-Gawad S, Cantelli A, Pirmez C, Minisini D, Sylvester Z, Imran J (2012) Three-dimensional numerical simulation of turbidity currents in a submarine channel on the seafloor of the niger delta slope. J Geophys Res Oceans 117:C05026
Cantero-Chinchilla FN, Dey S, Castro-Orgaz O, Ali SZ (2015) Hydrodynamic analysis of fully developed turbidity currents over plane beds based on self-preserving velocity and concentration distributions. J Geophys Res Earth Surf 120(10):2176–2199
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(2):619–634
Chiodi F, Claudin P, Andreotti B (2014) A two-phase flow model of sediment transport: transition from bedload to suspended load. J Fluid Mech 755:561–581
Chauchat J, Cheng Z, Nagel T, Bonamy C, Hsu TJ (2017) Sedfoam-2.0: a 3-d two-phase flow numerical model for sediment transport. Geosci Model Dev 10(12):4367–4392
Cheng Z, Chauchat J, Hsu TJ, Calantoni J (2018) Eddy interaction model for turbulent suspension in reynolds-averaged euler-lagrange simulations of steady sheet flow. Adv Water Resour 111:435–451
Mazumder BS, Ghoshal K, Dalal DC (2005) Influence of bed roughness on sediment suspension: experimental and theoretical studies. J Hydraul Res 43(3):245–257
Dorrell RM, Hogg AJ, Pritchard D (2013) Polydisperse suspensions: Erosion, deposition, and flow capacity. J Geophys Res Earth Surf 118(3):1939–1955
Dorrell RM, Amy LA, Peakall J, McCaffrey WD (2018) Particle size distribution controls the threshold between net sediment erosion and deposition in suspended load dominated flows. Geophys Res Lett 45(3):1443–1452
Cao Z, Carling PA (2002) Mathematical modelling of alluvial rivers: reality and myth. part 2: Special issues. In: Proceedings of the Institution of Civil Engineers-Water and maritime engineering, vol 154, pp 297–307. Thomas Telford Ltd
Sayre WW (1967) Dispersion of mass in open-channel flow. Hydrology papers (Colorado State University); no. 75
Monin AS (1959) On the boundary condition on the earth surface for diffusing pollution. Adv Geophys 6:435–436
Dobbins WE (1943) Effect of turbulence on sedimentation. Proc Am Soc Civil Eng 69(2):235–262
Cui Y (2007) The unified gravel-sand (tugs) model: simulating sediment transport and gravel/sand grain size distributions in gravel-bedded rivers. Water Resour Res 43(10):W10436
Sulaiman M, Fujita M, Tsutsumi D (2007) Bed variation model considering porosity change in riverbed material. J Japan Soc Eros Control Eng 60(1):11–18
Horvat Z, Isic M, Spasojevic M (2015) Two dimensional river flow and sediment transport model. Environ Fluid Mech 15(3):595–625
Bui VH, Bui MD, Rutschmann P (2019) Advanced numerical modeling of sediment transport in gravel-bed rivers. Water 11(3):550
Aris R (1956) On the dispersion of a solute in a fluid flowing through a tube. Proc R Soc Lond Ser A 235(1200):67–77
Mehta RV, Merson RL, McCoy BJ (1974) Hermite polynomial representation of chromatography elution curves. J Chromatogr A 88(1):1–6
Wang P, Chen GQ (2016) Transverse concentration distribution in Taylor dispersion: Gill’s method of series expansion supported by concentration moments. Int J Heat Mass Transf 95:131–141
Debnath S, Saha AK, Mazumder BS, Roy AK (2019) Transport of a reactive solute in a pulsatile non-Newtonian liquid flowing through an annular pipe. J Eng Math 116(1):1–22
Debnath S, Saha AK, Mazumder BS, Roy AK (2017) Dispersion phenomena of reactive solute in a pulsatile flow of three-layer liquids. Phys Fluids 29(9):097107
Anderson DA, Tannehill JC, Pletcher RH (1984) Computational fluid mechanics and heat transfer. Hemisphere Publishing Corporation, New York
Van Rijn LC (1987) Mathematical modelling of morphological processes in the case of suspended sediment transport
Mazumder BS, Paul S (2012) Dispersion of settling particles in oscillatory turbulent flow subject to deposition and re-entrainment. Eur J Mech B 31:80–90
Håkanson L (2006) The relationship between salinity, suspended particulate matter and water clarity in aquatic systems. Ecol Res 21(1):75–90
Hathaway JC, Nelson BW (1973) Environmental framework of coastal plain estuaries, vol 133. Geological Society of America
Acknowledgements
The authors greatly acknowledge the valuable suggestions and comments of editor and reviewers. They are thankful to the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Govt. of India, for providing the funding through the research project with No. EMR/2015/002434. The authors are thankful to Prof. Subhasish Dey, Dept. of Civil Engineering, IIT Kharagpur, the advisor of this project, for his valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Debnath, S., Ghoshal, K. & Kumar, J. Unsteady two-dimensional suspended sediment transport in open channel flow subject to deposition and re-entrainment. J Eng Math 126, 6 (2021). https://doi.org/10.1007/s10665-020-10070-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10665-020-10070-7