Abstract
A stochastic multivariate framework was utilized to characterize the random properties associated with discrete sediment particle movement in an open channel flow. Difficulties in modeling sediment transport with a high degree of accuracy are mainly attributed to the complexity associated with flow randomness and interactions between the flow and sediment particles. In particular, flow turbulence is the major factor in assessing sediment particle movement. To analyze the random properties of particle movement, we adopted a stochastic particle-tracking approach (i.e., the Lagrangian approach) that is effective for tracking discrete, randomly moving particles as a function of time. A set of stochastic differential equations that reflects the effect of forces exerted on fluid and sediment particles on average and the randomly fluctuating motion by turbulence was formulated. The fluctuation motion is described by the Wiener process. The proposed stochastic multivariate model is expected to present a more comprehensive evaluation of fluid velocity, particle velocity and position under the influence of flow turbulence, as opposed to the stochastic univariate model that focuses primarily on the sediment particle position. The outcome of the stochastic multivariate particle tracking model can provide a probabilistic description for fluid and discrete suspended sediment particles through ensemble statistics of kinematic variables in a random system (e.g., the ensemble mean and ensemble variance of particle trajectories and velocity). The proposed multivariate model was validated by comparing the results with available experimental flume data. The applicability of the proposed stochastic multivariate particle tracking model was also evaluated.
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References
American Society of Civil Engineers (ASCE) (2008) Sedimentation engineering: processes, management, modeling, and practice. In: García MH (eds) ASCE manuals and reports on engineering practice no. 110
Ancey C, Davison AC, BoHm T, Jodeau M, Frey P (2008) Entrainment and motion of coarse particles in a shallow water stream down a steep slope. J Fluid Mech 595:83–114. doi:10.1017/S0022112007008774
Boughton BA, Delaurentis JM, Dunn WE (1987) A stochastic model of particle dispersion in the atmosphere. Bound Layer Meteorol 40:147–163
Bradley DN, Tucker GE, Benson DA (2010) Fractional dispersion in a sand bed river. J Geophys Res Earth Surf 115:F00A09
Broyda S, Dentz M, Tartakovsky DM (2010) Probability density functions for advective–reactive transport in radial flow. Stoch Environ Res Risk Assess 24:985–992
Cheong H-F, Shen H-W (1983) Statistical properties of sediment movement. J Hydraul Eng 109(12):1577–1588
Chepil WS (1958) Use of evenly spaced hemispheres to evaluate aerodynamic forces on soil surfaces. Eos Trans AGU 39(3):397–404
Chibbaro S, Minier J-P (2008) Langevin PDF simulation of particle deposition in a turbulent pipe flow. Aerosol Sci 39:555–571
Cuthbertson AJS, Ervine DA (2007) Experimental study of fine sand particle settling in turbulent open channel flows over rough porous beds. J Hydraul Eng 133(8):905–916
Da Prato G, Debussche A (2008) On the martingale problem associated to the 2D and 3D Stochastic Navier-Stokes equations. Atti Accad Naz Lincei Cl Sci Fis Mat Natur 19(2008):247–264. doi:10.4171/RLM/523
Dixon NS, Tomlin AS (2007) A Lagrangian stochastic model for predicting concentration fluctuations in urban areas. Atmos Environ 41:8114–8127
Einstein HA (1950) The bedload function for sediment transportation in open channels. U.S. Dept. of Agriculture, Technical Bulletin, 1026
Einstein HA, El-Samni EA (1949) Hydrodynamics forces on a rough wall. Rev Mod Phys 21:520–524
Furbish DJ, Haff PK, Dietrich WE, Heimsath AM (2009) Statistical description of slope-dependent soil transport and the diffusion-like coefficient. J Geophys Res Earth Surf 114:F00A05
Gard TC (1988) Introduction to stochastic differential equations. Marcell Dekker Inc, New York, pp 66–75
Hanson FB (2007) Applied stochastic processes and control for jump diffusions: modeling, analysis, and computation. Society for Industrial & Applied Mathematics, US
Hung CS, Shen HW (1976) Stochastic-models of sediment motion on flat bed. J Hydraul Div 102(12):1745–1759
Kennedy CA, Lennox WC (2001) A stochastic interpretation of the tailing effect in solute transport. Stoch Environ Res Risk Assess 15:325–340
Kennedy C, Ericsson H, Wong RLR (2005) Gaussian plume modeling of contaminant transport. Stoch Environ Res Risk Assess 20:119–125
Lauer JW, Parker G (2008) Modeling framework for sediment deposition, storage, and evacuation in the floodplain of a meandering river: theory. Water Resour Res 44:W04425. doi:10.1029/2006WR005528
Lavezzo V, Soldati A, Gerashchenko S, Warhaft Z, Collins LR (2010) On the role of gravity and shear on inertial particle accelerations in near-wall turbulence. J Fluid Mech 658:229–246
Lee H, Balachandar S (2010) Drag and lift forces on a spherical particle moving on a wall in a shear flow at finite Re. J Fluid Mech 657:89–125
Lee CY, Rozovskii BL (2012) On the stochastic Navier–Stokes equations driven by stationary white noise
Li R-M, Shen HW (1975) Solid particle settlement in open-channel flow. J Hydraul Div 101(7):917–931
Malmon DV, Dunne T, Reneau SL (2003) Stochastic theory of particle trajectories through alluvial valley floors. J Geol 111:525–542
Man C, Tsai CW (2007) Stochastic partial differential equation based model for suspended sediment transport in surface water flows. J Eng Mech ASCE 133(4):422–430
Martin RL, Jerolmack DJ, Schumer R (2012) The physical basis for anomalous diffusion in bed load transport. J Geophys Res 117
Mikulevicius R, Rozovskii BL (2004) Stochastic Navier–Stokes equations for turbulent flows. SIAM J Math Anal 35(5):1250–1310
Minier J-P, Peirano E (2001) The PDF approach to turbulent polydispersed two-phase flows. Phys Rep 352(1–3):1–214
Nezu I, Nakagawa H (1993) Turbulence in open channel flows. IAHR Monograph, Balkema, Rotterdam, The Netherlands
Noguchi K, Nezu I (2009) Particle-turbulence interaction and local particle concentration in sediment-laden open-channel flows. J Hydro-Environ Res 3:54–68
Oh J, Tsai CW (2010) A stochastic jump diffusion particle-tracking model (SJD-PTM) for sediment transport in open channel flows. Water Resour Res 46:W10508. doi:10.1029/2009WR008443
Ortiz de Zárate JM, Sengers JV (2009) Nonequilibrium velocity fluctuations and energy amplification in planar Couette flow. Phys Rev E 79:046308. doi:10.1103/PhysRevE.79.046308
Patnaik PC, Vittal N, Pande PK (1994) Lift coefficient of a stationary sphere in gradient flow. J Hydraul Res IAHR 32(3):471–480
Peirano E, Chibbaro S, Pozorski J, Minier J-P (2006) Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Prog Energy Combust Sci 32:315–371
Pinsky M, Karlin S (2011) An introduction to stochastic modeling, 4th edn. Academic Press, Cambridge
Pope SB (1994) Lagrangian PDF methods for turbulent flows. Annu Rev Fluid Mech 26:23–63
Sawford BL, Borgas MS (1994) On the continuity of stochastic models for the Lagrangian velocity in turbulence. Phys D 76:297–311
Schumer R, Meerschaert MM, Baeumer B (2009) Fractional advection-dispersion equations for modeling transport at the Earth surface. J Geophys Res Earth Surf 114:07
Spivakovskaya D, Heemink AW, Schoenmakers JGM (2007) Two-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters. Stoch Environ Res Risk Assess 21:235–251
Suh S-W (2006) A hybrid approach to particle tracking and Eulerian–Lagrangian models in the simulation of coastal dispersion. Environ Model Softw 21:234–242
Sumer BM, Oguz B (1978) Particle motions near the bottom in turbulent flow in an open channel. J Fluid Mech 86(1):109–127
Sweeney LG, Finlay WH (2007) Lift and drag forces on a sphere attached to a wall in a Blasius boundary layer. Aerosol Sci 38:131–135
Thomson DJ (1987) Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J Fluid Mech 180:529–556
Tsai CW, Yang F-N (2013) Modeling bedload transport using a three-state continuous-time Markov chain model. ASCE J Hydraul Eng 139(12):1265–1276
Wilson JD, Sawford BL (1996) Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Bound Layer Meteorol 78:191–210
Wilson JD, Yee E (2007) A critical examination of the random displacement model of turbulent dispersion. Bound Layer Meteorol 125:399–416
Wu F-C, Yang K-H (2004) A stochastic partial transport model for mixed-size sediment: application to assessment of fractional mobility. Water Resour Res 40:W04501. doi:10.1029/2003WR002256
Zhou ZY, Kuang SB, Chu KW, Yu AB (2010) Discrete particle simulation of particle–fluid flow: model formulations and their applicability. J Fluid Mech 661:482–510
Acknowledgements
The authors gratefully acknowledge the financial support from National Science Foundation under grant contract number EAR-0748787. Funding was provided by Ministry of Science and Technology, Taiwan (Grant No. 104-2628-E-002-011-MY3).
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Oh, J., Tsai, C.W. A stochastic multivariate framework for modeling movement of discrete sediment particles in open channel flows. Stoch Environ Res Risk Assess 32, 385–399 (2018). https://doi.org/10.1007/s00477-017-1410-3
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DOI: https://doi.org/10.1007/s00477-017-1410-3