Stochastic particle tracking modeling for sediment transport in open channel flows
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Sediment transport in flow has a practical impact on environmental and economic aspects of human society, for instance, water quality, hydraulic structures and land resources. A systematic understanding of the sediment transport processes is of critical significance to establish proper water resources and sediment management plans. Both random properties of flows and varying properties of sediment particles can induce stochastic nature of sediment particle movement in the flows. Thus, stochastic approaches or analyses are beneficial to analyzing the variability associated with the movement of sediment particles. In this context, the focus of this study is to model various features of sediment transport in open channel flow with stochastic approaches. The scope of the study includes the following main issues: the movement of sediment particles in turbulent open channel flows in the occurrences of extreme flows, the deposition and resuspension processes of sediment particles, sediment concentrations and its uncertainty, and various modeling framework of stochastic particle tracking models. Turbulence in a flow is a primary source of stochastic property of particle movement in the flow. Furthermore, extreme flows that might occur occasionally in a random manner reinforce the randomness of the movement of sediment particles in the flow. The volatile flow velocity of extreme flows will not only affect the mean trend of particle movement but also intensify the uncertainty of particle movement. Specifically, since extreme flow events randomly occur per se, the random manner of the occurrences generates the stochastic property that affects the movement of sediment particles. Thus, it is effective to employ stochastic approaches for describing sediment transport processes associated with uncertainty. Herein, both a ‘stochastic diffusion process’ and a ‘stochastic jump diffusion process’ are introduced to describe stochastic particle movement in open channel flows. The ‘stochastic jump diffusion process’ represents the particle movement in response to extreme flow events that randomly occur in a turbulent open channel flow, whereas the ‘stochastic diffusion process’ characterizes the particle movement in a turbulent open channel flow. As a result, both the stochastic diffusion particle tracking model (SD-PTM) and the stochastic jump diffusion particle tracking model (SJD-PTM) can present particle trajectories, and roughly estimated instantaneous velocities. The ensemble statistics of the particle trajectories and velocities radically contain information on the stochastic characteristics of sediment particle movement. The SD-PTM and SJD-PTM to estimate particle trajectory and velocity is verified with data of Sumer and Oguz (1978), Muste and Patel (1997), Cuthberson and Ervine (2007) and Muste et al. (2009). The sediment concentration and sediment flux are highly-sought, practical variables in that the existence and amount of suspended sediments in surface waters have a direct influence on water quality and its suitability for drinking and industrial purposes. Especially, the estimation of sediment concentrations demonstrates the transporting process of suspended sediment through its spatial and temporal distributions. The sediment concentrations play a significant role as a pragmatic indicator in the decision making process. Thus, the previous-stated particle-based stochastic approach for sediment transport is enhanced to predict the suspended sediment concentration, and to quantify the uncertainty of the sediment concentrations. The method also allows for particle entrainment into flows and particle settlement on the bed as main processes in open channel flows. Through multiple realizations of the particle movement with stochastic properties, the SD-PTM shows not only sediment concentrations at a specific location and time but also uncertainty for the estimated sediment concentrations. The proposed method, in this context, is a more straightforward method to evaluate uncertainty due to stochastic properties in the particle movement and a unique way to present the uncertainty of sediment concentrations. The proposed stochastic particle tracking model for sediment concentrations is verified with data of Coleman (1986). The final goal of this study is to pursue further investigation into two different types of stochastic particle tracking approaches describing sediment particle movement associated with randomness. The different types of approaches are classified into the ‘univariate’ and the ‘multivariate’ stochastic particle tracking models according to the selection of key stochastic variables that describe the randomness of natural phenomena. In the ‘univariate’ stochastic particle tracking model, one state vector (e.g., particle position) is regarded as a targeted variable. The above proposed models can be thought of as the ‘univariate’ stochastic particle tracking model. In the ‘multivariate’ stochastic particle tracking model, the sediment particle velocity and position are joint Markovian state variables, since the flow velocity evolves in time according to a generalized stochastic differential equation. Model comparisons are performed and both models are verified with data of Sumer and Oguz (1978).