Stochastic particle based models for suspended particle movement in surface flows

Modeling of suspended sediment particle movement in surface water can be achieved by stochastic particle tracking model approaches.In this paper,different mathematical forms of particle tracking models are introduced to describe particle movement under various flow conditions,i.e.,the stochastic diffusion process,stochastic jump process,and stochastic jump diffusion process.While the stochastic diffusion process can be used to represent the stochastic movement of suspended particles in turbulent flows,the stochastic jump and the stochastic jump diffusion processes can be used to describe suspended particle movement in the occurrences of a sequence of extreme flows.An extreme flow herein is defined as a hydrologic flow event or a hydrodynamic flow phenomenon with a low probability of occurrence and a high impact on its ambient flow environment.In this paper,the suspended sediment particle is assumed to immediately follow the extreme flows in the jump process（i.e.the time lag between the flow particle and the sediment particle in extreme flows is considered negligible）.In the proposed particle tracking models,a random term mainly caused by fluid eddy motions is modeled as a Wiener process,while the random occurrences of a sequence of extreme flows can be modeled as a Poisson process.The frequency of occurrence of the extreme flows in the proposed particle tracking model can be explicitly accounted for by the Poisson process when evaluating particle movement.The ensemble mean and variance of particle trajectory can be obtained from the proposed stochastic models via simulations.The ensemble mean and variance of particle velocity are verified with available data.Applicability of the proposed stochastic particle tracking models for sediment transport modeling is also discussed.     

STOCHASTIC THEORY OF THE DISTRIBUTION OF SEDIMENT PARTICLES SUSPENDED IN A TURBULENT FLOW

The random motion of sediment particles suspended in a turbulent flow is studied by means of stochastic process. Results of analysis of particle's frequency response to the random force exerted on the particle due to fluid turbulence suggest that only the lower part of the whole frequency range of the eddy motion will govern the motion of the particle. The mean values of particle velocity and displacement in the vertical direc- tion are calculated and it is found that particle velocity vp- can be decomposed into a mean motion and a velocity fluctuation vp- , where is equal to the fall velocity in tranquil fluid. An Ito^ random differential equation for particle dis- placement Yp is developed, from which a Fokker-Planck equation for the probability density function p(y,t) is derived on the basis of the theory of Markov process, where y denotes the vertical coordinate. The vertical distribution of the particle is thus interrelated to the random motion of the particle. The an effect that a particle will be subject to in the neighborhood or the bottom boundary is taken into consideration and a corresponding Fokker-Planck equation is developed. Analytical solution of the Fok- ker-Planck equation including the lift force effect shows that probability density p(y,t) for the particle displacement has a maximum value at y = H where the perpen- dicular component of the lift force balances the particle gravity. This theoretical result agrees with experimental observations as reported in literature.     

An analysis of the characteristics of rough bed turbulent shear stresses in an open channel

Entrainment of sediment particles from channel beds into the channel flow is influenced by the characteristics of the flow turbulence which produces stochastic shear stress fluctuations at the bed. Recent studies of the structure of turbulent flow has recognized the importance of bursting processes as important mechanisms for the transfer of momentum into the laminar boundary layer. Of these processes, the sweep event has been recognized as the most important bursting event for entrainment of sediment particles as it imposes forces in the direction of the flow resulting in movement of particles by rolling, sliding and occasionally saltating. Similarly, the ejection event has been recognized as important for sediment transport since these events maintain the sediment particles in suspension. In this study, the characteristics of bursting processes and, in particular, the sweep event were investigated in a flume with a rough bed. The instantaneous velocity fluctuations of the flow were measured in two-dimensions using a small electromagnetic velocity meter and the turbulent shear stresses were determined from these velocity fluctuations. It was found that the shear stress applied to the sediment particles on the bed resulting from sweep events depends on the magnitude of the turbulent shear stress and its probability distribution. A statistical analysis of the experimental data was undertaken and it was found necessary to apply a Box-Cox transformation to transform the data into a normally distributed sample. This enabled determination of the mean shear stress, angle of action and standard error of estimate for sweep and ejection events. These instantaneous shear stresses were found to be greater than the mean flow shear stress and for the sweep event to be approximately 40 percent greater near the channel bed. Results from this analysis suggest that the critical shear stress determined from Shield's diagram is not sufficient to predict the initiation of motion due to its use of the temporal mean shear stress. It is suggested that initiation of particle motion, but not continuous motion, can occur earlier than suggested by Shield's diagram due to the higher shear stresses imposed on the particles by the stochastic shear stresses resulting from turbulence within the flow.     

Effect of sediment particles on the velocity profile of sediment-water mixtures in open-channel flow

The presence of sediment particles in open-channel flow has an important effect on turbulence; thus, an empirical, turbulent eddy viscosity formula was established for application in the limit for low concentrations. The current study establishes a theoretical relation for the mixture viscosity based on the two-phase mixture model. The percentage contribution of the three mechanisms of mixture viscosity,namely, fluid turbulence(FT), particle turbulence(PT), and inter-particle collisions(IPCs), w...     

A probabilistic model for sediment entrainment:The role of bed irregularity

A generalized probabilistic model is developed in this study to predict sediment entrainment under the incipient motion, rolling, and pickup modes. A novelty of the proposed model is that it incorporates in its formulation the probability density function of the bed shear stress, instead of the near-bed velocity fluctuations, to account for the effects of both flow turbulence and bed surface irregularity on sediment entrainment. The proposed model incorporates in its formulation the collective effects of three para-meters describing bed surface irregularity, namely the relative roughness, the volumetric fraction and relative position of sediment particles within the active layer. Another key feature of the model is that it provides a criterion for estimating the lift and drag coefficients jointly based on the recognition that lift and drag forces acting on sediment particles are interdependent and vary with particle protrusion and packing density. The model was validated using laboratory data of both fine and coarse sediment and was compared with previously published models. The study results show that all the examined models perform adequately for the fine sediment data, where the sediment particles have more uniform gra-dation and relative roughness is not a factor. The proposed model was particularly suited for the coarse sediment data, where the increased bed irregularity was captured by the new parameters introduced in the model formulation. As a result, the proposed model yielded smaller prediction errors and physically acceptable values for the lift coefficient compared to the other models in case of the coarse sediment data.     

On the mechanisms of the saltating motion of bedload

Saltation of sediment particles is an important pattern of bedload transport.Based on force analysis for sediment particles,a Lagrangian model was proposed for the saltating motion of bedload in river flows,which was then solved with numerical method.Simulation results on the saltating trajectories neglecting particle rotation and turbulence effects compare fairly well with experimental observations.The mean values of the saltation parameters (saltation height,length and velocity) also agree well with the previous experimental data.Based on the numerical results,regression equations for the dimensionless saltation height,length and velocity were presented.Using the numerically achieved characteristics of the sediment saltation,we also obtained mathematical expression for the sediment transport rate.The studies in this paper are significant for its contribution to mechanism of the bedload motion and the computation of sediment transport rate.     

An improved formula for incipient sediment motion in vegetated open channel flows

Quantifying incipient sediment motion in vegetated open channel flow is pivotal for estimating bed load transport and the aquatic ecological environment in rivers.A new formula is developed to predict the critical flow velocity for incipient sediment motion in the presence of emergent vegetation,by incorporating the influence of vegetation drag that characterizes the effects of mean flow and turbulence on sediment movement.The proposed formula is shown to agree with existing experimental data.Mo...     

Numerical simulation of flow and aquaculture organic waste dispersion in a curved channel

The dispersion and deposition of particulate organic matter from a fish cage located in an idealized curved channel with a 90° bend are studied for different horizontal grid resolutions. The model system consists of a three-dimensional, random-walk particle tracking model coupled to a terrain-following ocean model. The particle tracking model is a Lagrangian particle tracking simulator which uses the local flow field, simulated by the ocean model, for advection of the particles and random walk to simulate the turbulent diffusion. The sinking of particles is modeled by imposing an individual particle settling velocity. As the homogeneous water flows through the bend in the channel, the results show that a cross-channel secondary circulation is developed. The motion of this flow is similar to a helical motion where the water in the upper layers moves towards the outer bank and towards the inner bank in the lower layers. The intensity of the secondary circulation will depend on the viscosity scheme and increases as the horizontal grid resolution decreases which significantly affects the distribution of the particles on the seabed. The presence of the secondary circulation leads to that most of the particles that settle, settle close to the inner bank of the channel.     

RAPTOR‐UAV: Real‐time particle tracking in rivers using an unmanned aerial vehicle

River system measurement and mapping using UAVs is both lean and agile, with the added advantage of increased safety for the surveying crew. A common parameter of fluvial geomorphological studies is the flow velocity, which is a major driver of sediment behavior. Advances in fluid mechanics now include metrics describing the presence and interaction of coherent structures within a flow field and along its boundaries. These metrics have proven to be useful in studying the complex turbulent flows but require time‐resolved flow field data, which is normally unavailable in geomorphological studies. Contactless UAV‐based velocity measurement provides a new source of velocity field data for measurements of extreme hydrological events at a safe distance, and could allow for measurements of inaccessible areas. Recent works have successfully applied large‐scale particle image velocimetry (LSPIV) using UAVs in rivers, focusing predominantly on surficial flow estimation by tracking intensity differences between georeferenced images. The objective of this work is to introduce a methodology for UAV based real‐time particle tracking in rivers (RAPTOR) in a case study along a short test reach of the Brigach River in the German Black Forest. This methodology allows for large‐scale particle tracking velocimetry (LSPTV) using a combination of floating, infrared light‐emitting particles and a programmable embedded color vision sensor in order to simultaneously detect and track the positions of objects. The main advantage of this approach is its ability to rapidly collect and process the position data, which can be done in real time. The disadvantages are that the method requires the use of specialized light‐emitting particles, which in some cases cannot be retrieved from the investigation area, and that the method returns velocity data in unscaled units of px/s. This work introduces the RAPTOR system with its hardware, data processing workflow, and provides an example of unscaled velocity field estimation using the proposed method. First experiences with the method show that the tracking rate of 50 Hz allows for position estimation with sub‐pixel accuracy, even considering UAV self‐motion. A comparison of the unscaled tracks after Savitzky–Golay filtering shows that although the time‐averaged velocities remain virtually the same, the filter reduces the standard deviation by more than 40% and the maxima by 20%. Copyright © 2017 John Wiley & Sons, Ltd.     

Hindered settling with an apparent particle diameter concept

In this work a new expression has been developed to predict the settling velocity of a sediment particle which is dispersed in a sediment-fluid mixture during a turbulent flow. A concept of apparent particle diameter has been introduced and is defined by the diameter of the spherical volume in which the particle can move randomly after collision with other particles in suspension. The effect of suspension concentration is studied on the mass density of the sediment-fluid mixture. It has been shown that the settling velocity of sediment particle in a sediment-fluid mixture is a function of different characteristics of the sediment particle such as settling velocity in clear fluid, suspension concentration, relative mass density and Reynolds number. The model has shown good agreement when compared with previously published experimental data and it’s prediction accuracy is superior than the other existing models.     

NUMERICAL STUDY ON THE MIXING OF UNSORTED SEDIMENT PARTICLES DISCHARGED INTO A CROSS-FLOW BY MULTIPHASE PARTICLE-IN-CELL （MP-PIC） METHOD

The mixing characteristics of dredged sediments of variable size discharged into cross-flow are studied by an Eulerian-Lagrangian method. A three-dimensional (3D) numerical model has been developed by using the modified k-ε parameterization for the turbulence in fluid phase/water and a Lagrangian method for the solid phase/sediments. In the model the wake turbulence induced by sediments has been included as additional source and sink terms in the k-ε model; and the trajectories of the sediments are tracked by the Lagrangian method in which the sediment drift velocities in cross-flow are computed by a multiphase particle-in-cell (MP-PIC) method and the diffusion process is approximated by a random walk model. The hydrodynamic behavior of dumped sediment cloud is governed by the total buoyancy on the cloud, the drag force on each particle and velocity of cross-flow. The cross-flow destroys more or less the double vortices occurred in stagnant ambience and dominates the longitudinal movement of sediment cloud. The computed results suggest satisfactory agreement by comparison with the experimental results of laboratory.     

Experimental study on three dimensional movements of particles Ⅱ Effects of particle diameter on turbulence characteristics

Based on the 3D PTV （Particle Tracking Velocimetry） measuring system, the 3D movement characteristics of particles with four different diameters were investigated. Under specific flow conditions, the impact of particle diameters on 3D motion of particles was studied, and the turbulence characteristics were analyzed by different statistical methods. The results showed that the turbulence intensity of coarse particle decreased as the diameter increased. In near wall region, the probability density distributions of longitudinal and vertical fluctuation velocities both deviated from the normal distribution; while in the outer region, the probability density distribution of vertical fluctuation velocity approximately agreed with the normal distribution.     

Incompressible SPH scour model for movable bed dam break flows

In this study an incompressible smoothed particle hydrodynamics (ISPH) approach coupled with the sediment erosion model is developed to investigate the sediment bed scour and grain movement under the dam break flows. Two-phase formulations are used in the ISPH numerical algorithms to examine the free surface and bed evolution profiles, in which the entrained sediments are treated as a different fluid component as compared with the water. The sediment bed erosion model is based on the concept of pick-up flow velocity and the sediment is initiated when the local flow velocity exceeds a critical value. The proposed model is used to reproduce the sediment erosion and follow-on entrainment process under an instantaneous dam break flow and the results are compared with those from the weakly compressible moving particle semi-implicit (WCMPS) method as well as the experimental data. It has been demonstrated that the two-phase ISPH model performed well with the experimental data. The study shows that the ISPH modelling approach can accurately predict the dynamic sediment scouring process without the need to use empirical sediment transport formulas.     

Simulation of sediment motions using a discrete particle model in the inner surf and swash-zones

A volume-of-fluid Navier–Stokes solver (RIPPLE) was used to simulate inner surf and swash zone flow with a 3 s wave period and wave height of 0.14 m on a planar, 1:10 sloping beach (Iribarren number of 1.0). In addition to other hydrodynamic information, RIPPLE was used to provide high-resolution predictions of the pressure gradient and fluid velocity in the horizontal and vertical dimensions that served as forcing to a discrete particle model (DPM). Sediment transport processes in the inner surf and swash zones were simulated for a thin veneer of sediment particles over a 5 m test section in the DPM. Coupling between RIPPLE and the DPM was one-way such that particle–particle and fluid–particle interactions in the DPM did not provide feedback to alter the flow predicted by RIPPLE. The numerical simulation showed strong sediment suspension localized under vortices that reach the bed. Interestingly, the bulk of the sediment located in the small-scale vortex originated from locations nearly 0.2 m landward. These findings suggest that (1) sediment motion for a single swash event can be significant, (2) that sediment measured in suspension likely originates from locations other than the bed directly below the suspension plume suggesting the importance of sediment advection and (3) that sparse cross-shore measurements in the field will only sporadically capture localized suspension events.     

Computational modeling of flow and sediment transport and deposition in meandering rivers

A computational modeling analysis of the flow and sediment transport, and deposition in meandering-river models was performed. The Reynolds stress transport model of the FLUENT^TM code was used for evaluating the river flow characteristics, including the mean velocity field and the Reynolds stress components. The simulation results were compared with the available experimental data of the river model and discussed. The Lagrangian tracking of individual particles was performed, and the transport and deposition of particles of various sizes in the meandering river were analyzed. Particular attention was given to the sedimentation patterns of different size particles in the river-bend model. The flow patterns in a physical river were also studied. A Froude number based scale ratio of 1:100 was used, and the flow patterns in the physical and river models are compared. The result shows that the mean-flow quantities exhibit dynamic similarity, but the turbulence parameters of the physical river are different from the model. More strikingly, the particle sedimentation features in the physical and river models do not obey the expected similarity scaling.     

3D numerical simulation of particle-particle collisions in saltation mode near stream beds

The importance of particle-particle collisions in sediment saltation in the bed-load layer is analyzed herein by means of numerical simulation. The particle saltation theoretical/numerical model follows a Lagrangian approach, and addresses the motion of sediment particles in an open channel flow described by a logarithmic velocity profile. The model is validated with experimental data obtained from the literature. In order to evaluate the importance of the phenomenon, simulations with and without particle-particle collisions were carried out. Results for two different sediment concentrations are presented, namely 0.13% and 2.33%. For each concentration of particles, three different flow intensities were considered, and trajectories of two different particle sizes, within the sand range were computed. Changes in particle rotation, particle velocity, and angle of trajectory before and after particle-particle collisions appear to be relatively important at lower shear stresses, whereas they decrease in significance with increasing flow intensities. Analyses of the evolution in time of the second order moment of particle location suggest that inter-particle collisions introduce transverse diffusion in saltating particles in the span-wise direction.     

Significance of particle interaction to the modelling of cohesive sediment transport in rivers

Aggregation processes of fine sediments have rarely been integrated in numerical simulations of cohesive sediment transport in riverine systems. These processes, however, can significantly alter the hydrodynamic characteristics of suspended particulate matter (SPM), modifying the particle settling velocity, which is one of the most important parameters in modelling suspended sediment dynamics. The present paper presents data from field measurements and an approach to integrate particle aggregation in a hydrodynamic sediment transport model. The aggregation term used represents the interaction of multiple sediment classes (fractions) with corresponding multiple deposition behaviour. The k–ε–turbulence model was used to calculate the coefficient of vertical turbulent mixing needed for the two‐dimensional vertical‐plane simulations. The model has been applied to transport and deposition of tracer particles and natural SPM in a lake‐outlet lowland river (Spree River, Germany). The results of simulations were evaluated by comparison with field data obtained for two levels of river discharge. Experimental data for both discharge levels showed that under the prevailing uniform hydraulic conditions along the river reach, the settling velocity distribution did not change significantly downstream, whereas the amount of SPM declined. It was also shown that higher flow velocities (higher fluid shear) resulted in higher proportions of fast settling SPM fractions. We conclude that in accordance with the respective prevailing turbulence structures, typical aggregation mechanisms occur that continuously generate similar distribution patterns, including particles that settle toward the river bed and thus mainly contribute to the observed decline in the total SPM concentration. In order to determine time‐scales of aggregation and related mass fluxes between the settling velocity fractions, results of model simulations were fitted to experimental data for total SPM concentration and of settling velocity frequency distributions. The comparison with simulations for the case of non‐interacting fractions clearly demonstrated the practical significance of particle interaction for a more realistic modelling of cohesive sediment and contaminant transport. Copyright © 2004 John Wiley & Sons, Ltd.     

The movement of sediment in rivers

Sediment movement in rivers is a complex phenomenon. The rate of sediment transport is related to many variables such as water discharge, average flow velocity, stream power, energy slope, shear stress, water depth, particle size, water temperature, and strength of turbulence. Different theories of sediment transport were developed by assuming different independent variables as the dominant variables. This survey provides a comprehensive review of the important theories of incipient motion and sediment transport. It discusses basic concepts and findings upon which knowledge of sediment transport is based and presents mathematical derivations and equations only in sufficient detail to illustrate some basic concepts. Data collected from natural rivers and laboratory flumes are used to compare the accuracy and applicability of different sediment transport equations. Finally, procedures are suggested for selecting sediment transport equations under different flow and sediment conditions.     

Turbulence-particle interactions under surface gravity waves

The dispersion and transport of single inertial particles through an oscillatory turbulent aquatic environment are examined numerically by a Lagrangian particle tracking model using a series of idealised test cases. The turbulent mixing is incorporated into the Lagrangian model by the means of a stochastic scheme in which the inhomogeneous turbulent quantities are governed by a one-dimensional k- ε turbulence closure scheme. This vertical mixing model is further modified to include the effects of surface gravity waves including Coriolis-Stokes forcing, wave breaking, and Langmuir circulations. To simplify the complex interactions between the deterministic and the stochastic phases of flow, we assume a time-invariant turbulent flow field and exclude the hydrodynamic biases due to the effects of ambient mean current. The numerical results show that the inertial particles acquire perturbed oscillations traced out as time-varying sinking/rising orbits in the vicinity of the sea surface under linear and cnoidal waves and acquire a non-looping single arc superimposed with the high-frequency fluctuations beneath the nonlinear solitary waves. Furthermore, we briefly summarise some recipes through the course of this paper on the implementation of the stochastic particle tracking models to realistically describe the drift and suspension of inertial particles throughout the water column.