A stochastic jump diffusion particle-tracking model (SJD-PTM) for sediment transport in open channel flows

Authors


Abstract

[1] A stochastic jump diffusion (SJD) process is used to describe movement of sediment particles in open channel flows. The stochastic jump diffusion particle-tracking model (SJD-PTM) is governed by the stochastic differential equation (SDE). The SDE consists of three main terms including a mean drift motion term, a Wiener process representing random turbulent motion, and a Poisson process describing the abrupt movement of particles caused by probabilistic occurrences of the extreme flow perturbations. The proposed SJD-PTM can characterize the probabilistic properties of sediment particle movement by simulating the most probable trajectory and ensemble variances of particle movement associated with both flow turbulence and random occurrences of extreme flow perturbations. For the particle movement in response to extreme flow occurrences, we have introduced the particle relaxation time (temporal lag) to quantify the delayed response of sediment particles to large flow perturbations because of the particle inertia effect. The particle relaxation time is derived from a consideration of the forces exerted on particles under the effect of flow accelerations. The particle relaxation time is demonstrated to be dependent on particle size and density as well as the particle Reynolds number. The impact of drag and added mass forces on the ensemble mean and variance of particle trajectory is also evaluated. The particle relaxation time is found to be a factor that diminishes the impact of the extreme flow perturbations. It is shown that heavier and larger particles normally have a larger particle relaxation time, a decreased particle jump magnitude and a smaller ensemble variance of particle trajectories in the occurrences of extreme flow perturbations. It is concluded that the proposed SJD-PTM with the particle relaxation time can better describe the movement of sediment particles of nonnegligible size in response to extreme flow perturbations. The ability to quantify the variances of particle movement more comprehensively is one of the major advantages associated with the SJD-PTM for longer-term sediment transport modeling and predictions. The proposed SJD-PTM is verified against two distinct experimental data.

1. Introduction

[2] Sediment transport involves the movement of particles as suspended load or bed load by flows such as rivers, estuaries, or ocean currents. Extreme flow events, resulting from heavy storms, breach of dams, overflows from drainage ditches or large vortices associated with scour holes, increase the unsteadiness of flows and thus have major geomorphological effects. Accordingly, sediment transport because of extreme flow events could do great harm to water quality, structures, and land resources and thus cause immense economic and environmental damage. Despite the significant impacts of extreme flow events on sediment transport, movement of sediment particles in response to extreme flow events has not been fully understood and well characterized [e.g., De Sutter et al., 2001].

[3] Existing sediment transport models can be classified into suspended load transport models and bed load transport models according to the transport mode of particles. Bed load is transported primarily near the bed and is in contact with the channel bed for a considerable fraction of the time. In contrast, particles of suspended load supported by the turbulence of the mean flow reside primarily in the water column [Dade and Friend, 1998]. In principle, these models have employed deterministic differential equations or algebraic equations appropriate to the particle transport mode to estimate sediment concentrations or fluxes [Syvitski and Alcott, 1995; Arnold et al., 1998; Moulin and Slama, 1998]. The stochastic approach in the analysis of bed load sediment movement was initiated by Einstein [1937, 1950] and Kalinske [1947]. In recent years, several studies have also attempted stochastic approaches. Among others, most of studies used the Monte Carlo simulations to model sediment movement with morphological responses of the channels [Singer and Dunne, 2004; Van Vuren et al., 2005]. Lisle et al. [1998] have proposed to simulate sediment particle movement using a two-state Markov process. Cheong and Shen [1976] and Hung and Shen [1976] have established stochastic descriptions of sediment particles due to turbulence using probability density functions. Nevertheless, to our knowledge, the physically-based stochastic particle transport modeling remains at the developmental stage due to the complexity of the stochastic model formulations based on stochastic differential equations (SDEs) or stochastic partial differential equations and a lack of appropriate data for model validation.

[4] Individual particle transport in water flow has recently been described by the particle tracking models (PTMs). PTMs typically employ a Lagrangian approach whose observation system moves with fluid particles. In groundwater modeling, the earlier PTMs such as those of Ahlstrom et al. [1977] and Prickett et al. [1981] have been applied to particle random motion due to turbulence considering mainly the mean flow velocity. Thereafter further work on the PTMs has been accomplished considering the drift term because of the variability of the dispersion coefficient [e.g., Uffink, 1985; Tompson and Dougherty, 1988; Tompson and Gelhar, 1990; Bagtzoglou et al., 1992a, 1992b; Tchelepi et al., 1993; Fabriol et al., 1993; Tompson, 1993; Kitanidis, 1994]. More recently, Dimou and Adams [1993] applied the PTM to pollutant transport in coastal waters. In surface water modeling, Prakash [2004] and Kuchikulla [2006] applied PTMs to algal transport modeling in the Great Lakes. Recently, a few studies such as those of Man and Tsai [2007] and Iso and Kamemoto [2008] attempted to simulate solid particle trajectories in flows by the particle-tracking approach. The governing equations of more recent PTMs normally employ two terms including a drift term due to mean flow and a random term due to turbulence, which represent deterministic and stochastic properties of particle movement, respectively. The PTMs are Lagrangian based and focused on the movement of individual particles so that the stochastic concept can be easily applied to each particle.

[5] Recently, the measuring techniques using optical methods in multiphase flows have been advanced so that the velocity lag of sediment particles can be observed in laboratory [Phillips and Sutherland, 1990; Muste and Patel, 1997; Best et al., 1997]. For instance, the discriminator laser-Doppler velocimetry can be used to separately measure both particle and fluid velocity. Greimann et al. [1999] in an earlier study presented a theoretical analysis associated with observed particle velocity lag in response to flows. Cheng [2004] and Jiang et al. [2004] evaluated the velocity lag due to the drag force in sediment-laden regular open channel flow by experiments and numerical simulations. Although it is assumed that the sediment particle velocity is the same as the flow velocity in the majority of sediment transport models, the lag of sediment velocity has been considered in some recent attempts of sediment transport modeling. For example, Wu et al. [2006] attempted to simulate sediment transport considering the lag effects with the suspended and bed sediment transport equations. To explain the temporal lag, Wu et al. [2006] introduced a correction factor to suspended load concentrations, i.e., the ratio of the sediment velocity to flow velocity, into the storage term of the depth-averaged advection-diffusion suspended load transport equation. As a result, they estimated the reduced concentration of the mixture flow using a correction factor not the temporal lag itself. If the temporal lag of the particle velocity can be evaluated, we can then consider the effect of larger particles that are more affected by the time lag in response to flow changes.

[6] In this study the proposed model takes the temporal lag of sediment particles in response to extreme flow perturbations into consideration. In this paper extreme flow perturbations are referred to as particular extreme flow events with a time scale that is comparable to that of flow turbulence. The proposed model is then able to account for the delayed response of a particle in sediment transport modeling and to test the validity of a commonly adopted assumption in suspended load transport that sediment particles move simultaneously with fluid. The particle relaxation time, i.e., a time lag between sediment and fluid particles due to the inertia effect of sediment particles, is often observed during an extreme flow event. Such a time lag results in a delayed instantaneous particle velocity and a decreased mean particle velocity in a given time period. Moreover, since the temporal lag is due to the inertia effect of particles, the extent of particle retardation is dependent on the particle size and density. However, there are limited studies to date on the delayed response of sediment particles due to extreme flow conditions (including extreme flow events and large flow perturbations). The proposed model has an added advantage to simulate the particle movement in extreme flows while accounting for the delayed response of particles.

[7] To describe the stochastic properties of entities, researchers in various fields have employed the stochastic differential equation (SDE) such as an Ito process or a Langevin equation. The Ito process or the Langevin equation is a mathematical expression of stochastic processes associated with random phenomena such as suspended particle movement in a fluid or stock price fluctuations. Since the movement of particles is stochastic under the influence of turbulence, the SDE is appropriate to depict the movement of particles. There are several studies on sediment transport from the stochastic viewpoint in the area of hydraulics and hydrology. Most of the studies have focused primarily on statistical properties of inputs [Kleinhans and van Rijn, 2002; Sharma and Kavvas, 2005]. On the other hand, Man [2008] has attempted to develop a stochastic differential equation-based particle-tracking model for trajectory and concentration of suspended sediment particles without consideration of the particle relaxation time. Some studies have profoundly advanced the particle stochastic model for turbulent two-phase flows using the probability density function of particle location or velocity [Pope, 1994; Pope, 2000; Minier and Peirano, 2001; Peirano et al., 2006].

[8] This paper is one of the early efforts in developing a stochastic jump diffusion model for sediment particle movement in open channel flows. The proposed stochastic particle-tracking model is able to account for the delayed response of particles in response to extreme flow perturbations and to test the validity of a commonly adopted assumption that sediment moves along with flow in the case of extreme flow perturbations. The main objectives of this paper are to develop a physically based stochastic approach that better describes the essential stochastic characteristics of the sediment particle movement in response to probabilistic occurrences of extreme flows and to quantify the impact of particle relaxation time (temporal lag) on the sediment particle movement during extreme flows.

2. Stochastic Diffusion Model

[9] Particle movement in a flow can be delineated by a stochastic diffusion model. The Langevin equation is a stochastic diffusion equation describing Brownian motion. The Langevin equation of particle displacement is

equation image

where Xt is the position or trajectory of a particle = [X(t) Y(t) Z(t)]T, equation image(t, Xt) is the drift velocity vector, σ(t, Xt) is the diffusion coefficient tensor (3 × 3 diagonal matrix in flow), and Bt is a three-dimensional vector of the Wiener process at time t, i.e., the second term, σ(BtBs), has a normal distribution with a zero mean and a variance as (σσT)(ts) for st, which is independent of Xt. dBt/ dt (= W(t)) is the Gaussian White noise [Gardiner, 1985]. The first term on the right-hand side of equation (1) describes the mean drift motion of particles, whereas the second term represents the random motion due to turbulence. It has been shown that turbulent flows can be described by a Wiener process [Hanson, 2007].

[10] The exact probability density function equation for particle position is

equation image

The classical assumption of a turbulent diffusivity can be written in sample space as

equation image

The corresponding Fokker-Planck equation (or the forward Kolmogorov equation), which describes the time evolution of the probability density function of the position of a particle [Pope, 2000; Minier and Peirano, 2001], is

equation image

where f(x;t) denotes the probability density function of particle locations, equation imagei is the mean, and σij2 is the variance. Using equivalence between Fokker-Planck equations and stochastic Langevin equations, equation (1) can be rewritten as

equation image

Thus, the relationship between the diffusion coefficient and the turbulent diffusivities is shown as

equation image
equation image
equation image

where equation image is the mean streamwise fluid velocity; equation image is the mean transverse fluid velocity; equation image is the mean normal fluid velocity; ws is the particle settling velocity; and Dx, Dy, and Dz are the turbulent diffusivity in the longitudinal, transverse, and vertical directions, respectively. σij is the diffusion coefficient parallel to the ith axis in the jth component of a flow. Equation (1) can be further coupled with the advection-diffusion equation to estimate the particle concentrations [Man and Tsai, 2007].

3. Stochastic Jump Diffusion Model

3.1. Extreme Flow Events and Extreme Flow Perturbations

[11] An extreme event is defined as a catastrophic event that occurs with a low probability and high consequences. In particular, extreme flow events are considered as event-based flows that are statistical outliers as opposed to regular flows. Such flows can result from heavy precipitations associated with typhoons, hurricanes, tornados, tsunamis, or breach of dams. The time scale of extreme flow events varies, ranging from seconds or minutes (such as vortex induced large flow perturbations) to hours or days (such as flash floods). Extreme flow events normally cause significant flow velocity changes from ordinary flow conditions. As a result, particles in such flows may experience an abrupt large displacement within a short period of time. During extreme flows, particles may be subject to a sudden or abrupt transition from a regular flow state to an extreme flow state. “Extreme (or large) flow perturbations” in this study are extreme flow events, the duration of which is comparable to that of flow turbulence.

[12] Mathematically, extreme flow events can be defined with two properties such as the discontinuity and the shift. The discontinuity that occurs at t0 can be mathematically defined as equation image The shift due to jump can be written as [X(t0)] = equation imageX(t0 + Δt) − equation imageX(t0 − Δt). In addition to the discontinuity and the shift, “extreme flow perturbations” should also satisfy the requirement of the time scale. In terms of the time scale, the duration of extreme flow perturbation T is assumed to be very short compared to the time scale associated with the ensemble of the deposition and erosion processes. We have presented that the extreme flow perturbations can be treated as a stochastic jump process when the extreme event duration T is at least 1 order of magnitude smaller than that of the time scale of sediment transport process T0, i.e., O(T) ≪ O(T0). In this paper we plan to investigate and focus on the sediment transport of from fine particles to coarse suspended particles in response to (short time) large flow perturbations. For example, large vortex shedding may appear in a scour hole resulting from frequent local scouring around bridge piers; thus, the vortices can induce a rapid change of particle displacement near or at the scour hole in a similar way as particles in large flow perturbations. For long-term prediction, in the applications of the proposed model to sediment transport modeling during floods, the occurrences of flood events (typically with a time scale of hours or days) can be treated as a Poisson jump process during the entire target long-term period (in a longer term such as weeks or months).

[13] The stochastic jump diffusion processes have been used for modeling random abrupt changes in time with stochastic differential equations in various fields, including the optimal economics of biological populations in uncertain and disastrous environments, and financial engineering with applications in option pricing and optimal portfolios [Hanson, 2007]. Sediment particles in an open channel are subject to sudden abrupt movement because of the passing of extreme flow events such as flash floods, storm surges, or dam break waves. The stochastic diffusion process previously stated (equation (1)) describes particle movement in regular flow states. The discontinuous change of particle movement occurs during the extreme flow events. Movement of sediment particles in surface flows follows a continuous time stochastic jump diffusion process in situations where it allows for abrupt discharges resulting from an extreme flow event [Man, 2008]. Therefore, the governing equation that describes the particle jumps due to extreme events can be developed by adding a jump term to the Langevin equation or an Ito process. The governing equation for a stochastic variable Xt can be written as

equation image

where h(t, Xt) is the jump amplification factor and Pt denotes a Poisson process with jump rate λ so that the interarrival time between extreme events is exponentially distributed. It is assumed that two random processes Bt and Pt are independent.

3.2. Poisson Jump Processes

[14] In this paper we use the Poisson process to model the random occurrences of extreme flow perturbations. A Poisson process is a counting process used as a mathematical representation for random occurrences of events such as arrivals of customers to the cafeteria. Herein large flow perturbations meet the following conditions for Poisson process: Large flow perturbations (1) do not occur before t = 0 (N(0) = 0) and (2) have independent and stationary increment. In other words, the Poisson process is a counting process of events arriving in the interval (0, t] in which the interarrival times of successive events are independently and identically distributed exponential random variables [Kao, 1996]. The Poisson process is thus considered appropriate to represent an arrival process of random occurrences in a real-world system. Thus, we simulate the random occurrences of large flow perturbations in a period by using the Poisson process with the average rate of occurrences. On the basis of the stationary property of the incremental Poisson process, the probability distribution is

equation image
equation image

where Pr denotes the probability and λ is the mean frequency of occurrence of events. In modeling, the mean frequency of extreme flow occurrences can be determined and retrieved from historical record or data. In other words, λ is the ratio of the number of the extreme flow event occurrences to a selected period of time. According to the zero-one jump law, as Δt → 0+ (λ > 0), the probability distribution for the differential Poisson process is presented as follows [Hanson, 2007].

equation image
equation image
equation image

We herein employ the probabilities in modeling to estimate the probability of occurrence of extreme flow events.

[15] Furthermore, to quantify the jump amplification factor h(t, Xt), we can express

equation image

where Fr denotes forces exerted on the particle in regular flow, and Fe denotes additional forces incurred during the extreme flow events. As Fe moves the particle from one flow state to another, Fe can then be expressed as

equation image

where equation imagef is the increased mean drift flow velocity due to an extreme event, equation imagefo is the mean drift flow velocity before the extreme event, and τp is the particle relaxation time.

[16] On the other hand, the jump amplification factor h(t, Xt) is represented by the distance caused by the velocity change due to the extreme flow perturbations. Thus, h(t, Xt) can be rewritten as

equation image

where a is the acceleration of a particle. Since the acceleration can be expressed by the additional forces due to extreme flow events, equation (15) can be written as equation (16).

equation image

Substituting equation (14) into equation (16), the jump amplification over the extreme event duration T can then be expressed as equation (17).

equation image

It is noted that dPt = 1 within a time interval [t, t + dt] when there is only one extreme event occurrence [Man, 2008]. Therefore, the stochastic jump diffusion process that can describe discontinuous change of particle movement can accommodate the delayed response of sediment particles to extreme flows by the particle relaxation time.

4. Particle Relaxation Time

[17] Since the size of sediment particles is normally larger than that of fluid particles, there should be a lag in time for a particle to respond to extreme flow perturbations because of the inertia effect of the sediment particles. In aerodynamic engineering, according to Owen [1969], the particle relaxation time is the time taken by a particle to adjust to the change in the ambient flow. From the force balance equation, mpequation image = mpgFD = mpg − 3πμdpVp with Vp(t = 0) = 0 the particle velocity and relaxation time can be expressed as Vp = {ρpdp2g(1 − image and τp = (ρpdp2)/(18μ). For sediment particle movement in response to an extreme flow event, the relaxation time can be regarded as the time needed for a particle to move from a regular flow state to an extreme flow state. It is necessary to define the particle relaxation time of sediment particles appropriate to the situation of change from the regular flow state to the extreme flow state. It is assumed that sediment particles adjust to an ambient flow after the particle relaxation time. Herein we derive the particle relaxation time by considering predominant forces on the sediment particle in open channel flow such as drag and added mass. The general force balance equation is

equation image

where i = 1, 2, and 3. Gi is the gravitational force, Bi is the buoyancy force, Di is the drag force, and Ai is the added mass.

4.1. Effect of Drag Force

[18] Because the longitudinal velocity change mainly occurs during an extreme flow event, the force balance equation in the x direction can take into account the effect of external forces due to such extreme event as follows:

equation image

where ρs is the particle density, ∀p is the particle volume (= (4/3)π(dp/2)3), Ap is the projected area of the particle (= π(dp/2)2), Vp is the particle velocity, uf is the fluid velocity, Vr is the particle relative velocity (= Vpuf), and CD is the drag coefficient (= (24/Rep)(1 + 0.152Rep0.5 + 0.0151Rep)) where Rep = (ρdpVpuf∣)/μ. Representing ψ(Rep) by (1 + 0.152Rep0.5 + 0.0151Rep), we can rearrange the equation as a partial differential equation of Vr.

equation image

The general solution to equation (19) is

equation image

Given the initial condition, Vr = ufo (t) − uf (0) at t = 0, we can obtain

equation image

The particle relaxation time can then be defined as equation (22).

equation image

Likewise, the force balance equation in the z direction is

equation image

The particular solution to the above equation can be listed as follows.

equation image

The particle relaxation time in the z direction can be defined as

equation image

Thus, the particle relaxation time derived in the z direction is the same as that in the x direction. We also find that the particle relaxation time is affected by the particle properties such as particle size, density, and the particle Reynolds number.

4.2. Effect of Added Mass

[19] We can also derive the relaxation time considering the added mass as well as the drag force. The added mass is the force of sheer flow surrounding particles, which is thought of as a resistance force. The added mass (or virtual mass) force arises from the difference in acceleration of the reference fluid and the sediment particle. The added mass was early defined with the mass of the displaced fluid and the relative acceleration [e.g., Wiberg and Smith, 1985]. Maxey and Riley [1983], Auton et al. [1988], and Schmeeckle et al. [2007] have presented the corrected formula of the added mass by pointing out a distinction between the derivatives. Herein, we have employed the formula ρCmp (Duf/Dt − dVp/dt) instead of the formula −ρCmp (dVr/dt) in the added mass force. Similarly, we have

equation image

Representing ψ(Rep) by (1 + 0.152Rep0.5 + 0.0151Rep), we can rearrange the equation as a partial differential equation of Vr.

equation image

The general solution to the partial differential equation is

equation image

With the initial condition, Vr = ufo(t) − uf(0) at t = 0, one can obtain

equation image

The particle relaxation time can then be defined as equation (27).

equation image

Likewise, in the z direction, we have

equation image

The particular solution can then obtained as follows:

equation image

The particle relaxation time in the z direction can be defined as

equation image

Thus, the particle relaxation time derived in the z direction is the same as that in the x direction.

[20] As shown in equations (22) and (27), the relaxation time is determined by the particle properties (size and density) and the particle Reynolds number. It is consistent with the theoretical and experimental work of Cheng [2004], which states that the dimensionless velocity lag distribution varies with three dimensionless parameters, i.e., particle Reynolds number, dimensionless particle diameter, and specific gravity. The results of Cheng [2004] convey that the velocity lag becomes larger as particle diameter or specific gravity is larger. It also shows that a smaller particle Reynolds number is normally associated with larger velocity lags. Figures 1 and 2 display the dimensionless particle relaxation time calculated from equations (22) and (27) according to particle properties. Above all, as particle size or density increases, the relaxation time increases proportionally as in Figure 1 and Figure 2. The Φ scale is used for the characterization of the grain size distribution of sediment as suggested by Krumbein [1936]. As the definition of Φ scale, i.e., Φ = −log2(dp) where dp is the diameter of the sediment (mm), it is noted that larger particles have smaller Φ values. Furthermore, the extent of the effect of each force on the particle time lag can be compared with Figure 1 and Figure 2. It is found that the drag force is critical to the relaxation of sediment particles. Added mass also plays a secondary role in the relaxation of sediment particles.

Figure 1.

Dimensionless relaxation time (only drag force).

Figure 2.

Dimensionless relaxation time (drag force and added mass).

[21] The particle relaxation time previously derived is applicable to both the regular flow and extreme flows. In the following applications of the proposed SJD-PTM, it is assumed that sediment particles move simultaneously with flow particles in regular flows and that the time lag of particles primarily occurs during the extreme flows. Thus, the particle relaxation time hereafter indicates the time needed for a particle to move from a regular flow state to an extreme flow state.

5. Applications and Simulation Results

[22] Figure 3 delineates schematically the modeling process of the proposed stochastic jump diffusion particle tracking model. As shown in Figure 3, stochastic modeling of particle movement includes two major random processes that require information from hydrodynamic properties and particle characteristics. The Euler-Maruyama method [Kloeden et al., 1994] is applied to integrate a linear SDE.

equation image

where Xn+1 is the particle position in the longitudinal direction at tn+1, Xn is the particle position in the longitudinal direction at tn, and Δt is time step. Zn+1 and Zn are the particle positions in the vertical direction at tn+1 and tn, respectively. equation imagecan be approximated as the logarithmic profile (= (u*/κ)ln(30Z/kc)), where κ is von Karman constant. For open channel flows, the turbulent diffusivity can be estimated from various ways (e.g., Dz = κu*z(1 − z/H)). ΔBt (= equation imageN(0,1) and ΔPt (equations (10), (11), and (12)) are the Wiener and Poisson distributed random variables generated by the respective random number generator. Stepwise particle velocity for each time step can be roughly estimated from the particle displacement as

equation image

where Vn is the average particle velocity during a time period from tn to tn+1. The estimated stepwise particle velocity from the proposed model is used for the comparison with the measured particle velocity in model verification.

Figure 3.

Schematic modeling process with SJD-PTM.

[23] Sediment particle trajectories in a flow are simulated in the following examples using the proposed stochastic jump diffusion particle-tracking model. As the proposed stochastic jump diffusion model considers the effect of the relaxation time, we can compare the results of the proposed model in the following scenarios: (1) in the absence of the relaxation time and (2) in the absence of extreme flow perturbations. Besides, the resultant trajectories of the stochastic diffusion model are also included for comparison. The governing equation of the stochastic diffusion model is a Langevin equation or an Ito process, which is equivalent to the equation used in random walk particle-tracking models. In the following examples, the duration of a jump is assumed short enough to be considered equivalent to the computational time step so that the jump can be considered extreme flow perturbations. Also, the examples are focused on an infinite spatial domain without boundary effects.

5.1. Case 1: One-Dimensional Open Channel Flow

[24] The flow conditions in this example are given in Table 1. The total simulation time is 100 s, and the time step is 0.01 s. To show the stochastic results effectively, we performed 1000 iterative simulations and computed the respective ensemble means and variances of these scenarios. Figure 4 shows the particle trajectories in the one-dimensional flow field. The relative time in Figure 4 is defined as the time normalized by the total simulation time. Results are displayed from the stochastic diffusion model without extreme flow perturbations (SD, -·-), the stochastic jump diffusion model with extreme flow perturbations (SJD, ⋯), the stochastic jump diffusion model considering relaxation time from drag force (SJD-R, —) and the stochastic jump diffusion model considering relaxation time from drag force and added mass (SJD-RA, ---). The ensemble means represented by bold lines have reflected the stochastic scenarios represented by fine lines.

Figure 4.

Sediment particle trajectories and ensemble mean in a 1-D flow (relative time: normalized time by the total simulation time).

Table 1. Flow Conditions and Particle Properties for Case 1
Mean drift velocityequation image = 0.1 m/s
Diffusivityσ = 0.1 m/(s1/2)
Mean rate of occurrence of extreme flow perturbations (mean frequency of occurrences)0.1/s (λ = 10)
Magnitude of the extreme flow perturbations2 m/s
Particle propertiesparticle diameter2 mm
particle density4000 kg/m3

[25] Figure 4 presents the particle trajectories by various models. Figures 4a and 4d demonstrate the differences in the mean particle trajectories among the SD, SJD, SJD-R, and SJD-RA models. In particular, the differences are well explained by the ensemble mean lines. The SD model (-·-) shows how well the stochastic equation describes the diffusion term due to turbulence as random fluctuations in a regular flow. The SJD model (⋯) demonstrates that the additional term for jumps due to extreme flow perturbations can be modeled by introducing the concept of a stochastic process. As the ensemble means of the SJD model (⋯) simulation manifest, the overall particle velocity during the same time period is increasing because of the random occurrences of extreme flow perturbations. It can mainly affect the erosion/deposition of sediment such as downstream bridge scouring. The SJD-R (—) and SJD-RA (---) models illustrate the effect of a time lag due to the inertia effect of the particle and shows the modified result of the SJD model (⋯). As demonstrated in the simulations, the proposed stochastic jump diffusion model can reckon the probabilistic properties in sediment transport, which could not be explicitly included in most of the sediment transport models. Besides, Figures 4b and 4c show that the occurrences of extreme flow perturbations are modeled as a random process in the proposed SJD-PTM.

[26] The ensemble variances of iterative 1000 simulation results of particle trajectories are delineated in Figure 5. As shown in Figure 5, the variances of particle trajectories (i.e., the uncertainties of the trajectories) are increasing when the likelihood of occurrences of extreme flow perturbations is considered. As the jump term is added as part of the stochastic process, the variances of particle movement have increased owing to the uncertainty associated with random occurrences of extreme flow perturbations. Moreover, if the particle temporal lag due to the drag force is considered in the jump term, the variances of particle trajectories decrease. The uncertainties due to the occurrences of extreme flow perturbations decrease when considering the particle relaxation time, as a delayed response of the particle reduces the immediate impact of the occurrence of extreme flow events. An increased particle relaxation time due to added mass will lead to a decreased jump amplification factor and thus a smaller ensemble variance of the particle trajectories.

Figure 5.

Variances of sediment particle trajectories in a 1-D flow (relative time: normalized time by the total simulation time).

5.2. Case 2: Two-Dimensional Open Channel Flow

[27] The flow condition of the second example is given as in Table 2. The total simulation time is 120 s, and the time step is 0.025 s. We performed 1000 iterative simulations and computed the respective ensemble means and variances of those scenarios.

Table 2. Flow Conditions and Particle Properties for Case 2
Mean drift velocitylongitudinal directionequation image = 0.286 m/s
vertical directionequation image = 0.0 m/s
Diffusivitylongitudinal diffusivityσx = 0.1 m/(s1/2)
vertical diffusivityσz = 0.008 m/(s1/2)
Mean rate of occurrence of extreme flow perturbations (mean frequency of occurrences)0.15/s (λ=15)
Magnitude of the extreme flow perturbationslongitudinal direction2.5 m/s
vertical direction0.5 m/s
Particle propertiesparticle diameter5 mm
particle density4000 kg/m3

[28] Figure 6 shows the particle trajectories in a two-dimensional flow field in simulation. Similar to case 1, results are displayed from the stochastic diffusion model without extreme flow perturbations (SD, -·-), the stochastic jump diffusion model with extreme flow perturbations (SJD, ⋯), the stochastic jump diffusion model considering relaxation time from drag force (SJD-R, —), and the stochastic jump diffusion model considering relaxation time from drag force and added mass (SJD-RA, ---). The results are also presented in plane plot of the particle position as in Figure 7.

Figure 6.

Sediment particle trajectories and ensemble mean in a 2-D flow (relative time: normalized time by the total simulation time).

Figure 7.

Longitudinal and vertical mean particle position in a 2-D flow.

[29] Figures 8 and 9 display the variances of stochastic trajectories in the longitudinal and vertical direction, respectively. Variances of particle trajectories can be attributed to both the random turbulent term and a Poisson jump term (equation (9)), i.e., uncertainties of the sediment particle movement are due to both the turbulence and the occurrence of extreme events.

Figure 8.

Variances of longitudinal particle position in a 2-D flow.

Figure 9.

Variances of vertical particle position in a 2-D flow.

[30] On one hand, variances of particle trajectories due to flow turbulence can be evaluated by comparing Figure 8 and Figure 9. The given longitudinal diffusivity in Figure 8 is 0.1 m/(s1/2) and the vertical diffusivity in Figure 9 is 0.008 m/(s1/2). Thus, the diffusivity in the longitudinal direction (Figure 8) is approximately an order of magnitude greater than the diffusivity in the vertical direction (Figure 9). Thus, the scenarios based on different values of diffusivity result in differences of more than an order of magnitude in the variance of the trajectories in both the longitudinal and vertical direction. The variances due to larger diffusivity in Figure 8 are approximately 2 orders of magnitude greater than those due to a smaller value of diffusivity in Figure 9. The slope of particle position variance plotted against time is determined by the square of the diffusion coefficient σij2 when the jump term is not considered. It is confirmed that a smaller value of the diffusivity produces smaller variances of the trajectories of sediment particle movement.

[31] On the other hand, variances due to random occurrences of extreme flow perturbations can be demonstrated by comparing case 1 (λ = 10) and case 2 (λ = 15). A larger λ value corresponds to more frequent occurrences of extreme flow perturbations in a given time period. The ensemble mean of the stochastic jump diffusion model in case 1 is smaller than that in case 2, whether or not the particle relaxation time is being considered; thus, the λ value affects both the mean trajectories and the overall velocities. In the aspect of the variance, variances of the stochastic jump diffusion model are higher with a larger λ value. Hence, by modeling the occurrences of extreme flow events as a random process, we can estimate the variance of particle trajectory in response to the likelihood of extreme flow event occurrences.

6. Data Verification

[32] The existing data of Sumer and Oguz [1978] are used for verifying the proposed model with which the particle relaxation time is associated. The experimental system of Sumer and Oguz [1978] was designed to observe the movement of particles of different size in a laboratory flume. Among others, Sumer and Oguz [1978] provided velocity measurement for one type of particles (2.8 mm diameter). We can verify the proposed model of the particle movement by comparing with the mean and standard deviation of their measured particle velocities. Sumer and Oguz [1978] categorized their measured particle velocities according to the vertical component of particle velocity (v = 0, v > 0, and v < 0) to predict the probability density function. Thus, we use the data set of longitudinal particle velocities when v = 0 for model calibration, and those when v > 0 or v < 0 for model validation. Since their measured mean streamwise velocity of the particle follows the logarithmic velocity profile, the logarithmic profile is used for the determination of the mean drift of the proposed model. The diffusion coefficient is calibrated using the standard deviation of particle velocities when v = 0. The stochastic diffusion process conveys that the distance a particle moves by (the) flow turbulence is normally distributed with a zero mean and a variance of σ2 Δt (i.e., ΔX by flow turbulence ∼ equation imageN(0,1). The variance of particle position can thus be written with parameters of the proposed model as

equation image

From the above relation between the variance of particle position and diffusivity, the variance of particle velocity can be also linked with parameters of the proposed model as

equation image

The value of diffusion coefficient (σ) can be calibrated with the standard deviation (sv) of one set of the measured particle velocity. The time step can be properly chosen from a characteristic time scale of turbulent motion. Among scales of turbulent motion, since the Taylor microscale is related to the turbulence fluctuations, the time scale for the turbulence fluctuations can be estimated as tλ (=lλ/u′). The Taylor-scale Reynolds number Rλ (=ulλ/v) can be related to the turbulence Reynolds number as Rλ = ((20/3)Re)1/2. From the relationship, Rλ can be estimated to have an order of magnitude of 102. Consequently, a proper order of magnitude of the Taylor length scale lλ can be presented as 10−1 cm (i.e., [ulλ] ∼ 1 cm2/s where u′ = 2 – 2.5 cm/s), and the time scale for the turbulence fluctuations has an order of magnitude of 10−1s (approximately 0.2–0.25 s). As a result, the diffusion coefficient, σ ( = equation image), can be determined as 0.0128 m/(s1/2). We estimate the standard deviations of longitudinal velocities of 1000 particles and compare the standard deviations to validate the proposed model. The mean and the standard deviations of longitudinal particle velocities agree reasonably well with those of the experimental data in Figure 10. The mean velocity, in the form of a logarithmic profile, closely agrees with the mean streamwise velocity of experimental data, as shown in Figure 10a. The standard deviation of particle velocity in Figure 10b can be considered to be the error for the experimental velocity data. The order of magnitude of the standard deviation falls into the similar range as in Figure 10b. Although Sumer and Oguz [1978] measured and presented only the particle velocity in the near-bottom layer of the channel, the standard deviation of particle velocity can be concluded to have the almost same trend with that from the experimental data by the Kolmogorov-Smirnov (K-S) test. The K-S test is a nonparametric test that does not require any implicit assumptions about the distribution of data, which herein makes it a pertinent way to decide whether the measured and estimated particle velocities agree well. The method is based on the empirical cumulative distribution function (Figure 11). The Kolmogorov-Smirnov statistic (D) quantifies the largest vertical distance between the empirical distribution functions of two samples. As the result of the K-S test, the null hypothesis that the samples are drawn from the same distribution cannot be rejected because the P value (P = 0.066) is not smaller than the significance level (α = 0.05). Therefore, the test provides sufficient evidence that the estimated standard deviation of particle velocity and the measure one show a similar trend. Figure 10 also shows that the particle velocities can be realistically estimated by subtracting the particle response time on each time step. In particular, the results coupled with the particle relaxation time may slightly better represent the experimental data (open circle) than the results without the particle relaxation time. This is attributed to the fact that the particle lag can reduce the immediate effect of flow turbulence on variance of particle velocity. The greater the particle size or density, the larger the difference is between different models. The standard deviations of particle trajectory are calculated from 1000 realizations of particle trajectory associated with four different particle-tracking models. The slight difference in the standard deviation of particle trajectory demonstrates the effect of delayed response of particles on the standard deviation. The curves of the empirical cumulative distribution function in Figure 11 demonstrate that the curve for the case considering the particle relaxation time is the closest one to the curve for the experimental data. Hence, the SJD-PTM is able to be reinforced with the particle relaxation time. Such comparison supports that the delayed response of particles can cause a slightly smaller mean and standard deviation of particle velocities.

Figure 10.

Comparison of particle velocities between the experimental data [Sumer and Oguz, 1978] and modeling results.

Figure 11.

Empirical cumulative distribution function of the standard deviation of particle velocity (experimental data [Sumer and Oguz, 1978] and modeling results).

[33] Furthermore, we have also verified the proposed model with more recent experimental data of Cuthbertson and Ervine [2007]. Data of Cuthbertson and Ervine [2007] used for model validation includes flow and particle velocities in a regular turbulent open channel flow measured by the particle-tracking techniques. Cuthbertson and Ervine [2007] measured the instantaneous flow and particle velocities for two similar flow conditions. Thus, we use one data set for calibration of the diffusion coefficient and the other set for model validation. It is found in calibration that the diffusion coefficient can be better represented by the standard deviation of particle velocities at the center of the depth. Figure 12a shows the good agreement between the estimated mean particle velocity and the measured mean particle velocity. Figure 12b presents the comparison between the estimated and measured standard deviation of particle velocity. The particle velocity estimated by the proposed model (dash-dotted line) is in good agreement with experiment data (open circle) over the range. The estimated standard deviation of particle velocity falls into the same range of the measured standard deviation of particle velocity. Also, both can be told to have the similar trend by the K-S test as shown in Figure 13. The K-S test results in a greater P value (P = 0.147) than the significance level (α = 0.05), i.e., the null hypothesis that the samples are drawn from the same distribution cannot be rejected. The result shows a great potential of proposed model to realize the stochastic property of particle movement in regular open channel flows. Therefore, the foregoing analysis has demonstrated the validity of the proposed model applied to particle movement in a regular open channel flow.

Figure 12.

Comparison of particle velocities between the experimental data [Cuthbertson and Ervine, 2007] and modeling results.

Figure 13.

Empirical cumulative distribution function of the standard deviation of particle velocity (experimental data [Cuthbertson and Ervine, 2007] and modeling results).

7. Discussions

[34] On the basis of the previous result of case 1, several implications can be made from Figure 4. Results of the SD model with those of the SJD model are first compared. The difference between these two models is the particle jump term. Since the jump process stands for the extreme flow events, the results show that the extreme flows increase the velocities of particles during the extreme flow events and ultimately the trajectories of sediment particles. By comparing the results of the SJD model with the SJD-R model, we can conclude that the traveling distance of a particle considering the relaxation time is less than that without the relaxation time. It can also be deduced from Figure 4 that the particle relaxation time can be thought of as a factor that diminishes the impact of the extreme flow perturbations. It can be reasonably explained that the overall velocity of sediment particles is reduced by the temporal lag of particles during a given period. The drag force exerted on particles mainly yields the time lag and thus brings resistance to slow down the particles. The SJD-R model enables us to more directly accounts for the varying effect of particle size and density. When forces acting on particles are reckoned in the derivation of the particle relaxation time, the results can show the influence of other forces but drag force on the delayed response of particles. Here the result of two models considering the particle relaxation time demonstrates the effect of the added mass on particles. As the added mass is the force of sheer (=filmy) flow surrounding particles, it acts as a resistance force and reduces the impact of flood as the drag force. However, the impact of the added mass on the particle relaxation time is relatively smaller than that of the drag force. Figure 4d illustrates the fact that the mean trajectory differences between the SJD-R and SJD-RA models are smaller than those between the SJD and SJD-R models.

[35] On the basis of the previous result of case 2, it is suggested that the stochastic jump diffusion model without the particle relaxation time may be sufficient if the sediment particles are as small as a fluid particle (as the sediment particle moves concurrently with the fluid particle). However, if the sediment particles have a nonnegligible particle size and mass, it is more precise to employ the stochastic jump diffusion model considering the particle relaxation time. The proposed SJD-PTM can simulate random movement of various sediment particles in response to probabilistic occurrences of extreme flow events.

[36] The stochastic jump diffusion particle tracking model (SJD-PTM) proposed in this study can be distinguished from most of the existing sediment transport models as follows. First, a stochastic hydraulic process is defined as a spatial and/or temporal process involving probability in hydraulics. Flow and particle transport processes are a sequence of random variables indexed by space and time and therefore are intrinsically stochastic. To fully describe such stochastic processes, development of flow and sediment transport models based on stochastic governing equations is essential. While existing sediment transport models mostly employ deterministic governing equations for sediment concentrations, the proposed SJD-PTM takes a fully stochastic approach. The proposed SJD-PTM can be defined by a stochastic modeling system that consists of stochastic governing equations incorporated with stochastic input variables and parameters. On the other hand, the stochastic governing equations are built upon physically based stochastic principles of particles, not merely on statistical properties of input variables or parameters. The above feature enables a model to simulate the stochastic trajectory of sediment particles and is expected to estimate the resulting sediment concentrations more closely. Second, in the SJD-PTM, the occurrence of extreme flow events is considered as a random process. In many cases, sediment transport models have focused on bed evolution in the respect of geomorphological impact of floods or unsteady flows [Liu et al., 2004; Martini et al., 2004; Cao et al., 2006]. Most of the sediment transport models simulate the particle movement in response to a given extreme flow. The proposed SJD-PTM, on the other hand, considers the probabilistic occurrences of extreme flow events as a random process accounting for the likelihood of particle movement by the frequency of extreme flow events. Consequently, the proposed SJD-PTM is expected to reflect a longer lasting influence of sediment movement on bed evolution. In other words, a long-term prediction of sediment movement in channels is achievable by considering a sequence of extreme flow event occurrences rather than a particular extreme flow event. Third, most of the sediment transport models assume that sediment particles simultaneously move along with flow. This assumption is fairly reasonable for suspended load transport when describing the movement of sufficiently fine sediment particles in a regular flow with a negligible time lag. However, such an assumption might be crude for particles that have larger size or density especially during extreme flows, including large flow perturbations as well as large floods. Extreme flow events including large flow perturbations may cause a significant change in particle velocity and thus a sudden large displacement of particles. Provided that the time lag of sediment particles in response to extreme flows can be taken into account, the sediment transport model is then able to more accurately simulate the movement of particles of varying size and density.

8. Conclusions

[37] In this paper we have proposed a stochastic model as an alternative approach for sediment particle movement modeling in open channel flows, defined as the stochastic jump diffusion particle tracking model (SJD-PTM). The proposed SJD-PTM is governed by a stochastic differential equation composed of continuous time random variables and two random processes, and also requires inputs (e.g., particle properties, flow condition, extreme flow condition, and parameters). The stochastic differential equation (SDE) in the proposed SJD-PTM describes the particle movement with three major physical terms including a mean drift term, a random turbulent term, and a Poisson jump term caused by the probabilistic occurrences of extreme flows. The jump term enables the proposed SJD-PTM to model the abrupt sediment particle movement in extreme event flows. Both the random turbulent term and Poisson jump term represent the stochastic properties of sediment particle movement in extreme flows. Specifically, the random turbulent term is modeled as a Wiener process and the jump term due to random occurrences of extreme flow events is modeled as a Poisson process. In addition, the proposed SJD-PTM takes the particle relaxation time in the jump term into consideration so that it can simulate the delayed movement of particles in response to the extreme flows. It is shown that the particle relaxation time is dependent on particle properties such as particle size and density and also the particle Reynolds number. To demonstrate the capability of the proposed SJD-PTM, we have compared the proposed SJD-PTM with a stochastic diffusion model and a stochastic jump diffusion model without considering the particle relaxation time.

[38] The sediment particle trajectories resulted from the proposed SJD-PTM demonstrate the following features. The jump term in the SDE capacitates the proposed SJD-PTM to evaluate the effect of extreme flows on sediment particle movement. In the examples, the particle velocity is increasing during extreme flow perturbations due to flow acceleration. It can mainly affect the sediment erosion/deposition such as downstream bridge scouring. By modeling the occurrences of extreme events as a Poisson process, the proposed SJD-PTM can also demonstrate the probabilistic movement of sediment particles in response to extreme flow perturbations. Furthermore, particles have a delayed response to extreme flows because of the inertia effect of particles, defined as the particle relaxation time. It is found in this study that the overall particle velocity of the SJD-PTM with the particle relaxation time is lower than that of the SJD-PTM without the particle relaxation time; thus, the particle relaxation time can be regarded as a factor that diminishes the impact of the extreme flow perturbations. It can be explained that the overall velocity of sediment particles is reduced by the delayed response of particles during a given period. The proposed SJD-PTM enables us to simulate the varying effect of particle size and density on the particle relaxation time. Therefore, the jump term including the particle relaxation time capacitates the proposed SJD-PTM to generate more realistic results such as particle trajectories for particles of varying size.

[39] On the other hand, the variance of the stochastic sediment particle trajectories is due to flow turbulence and the probabilistic occurrences of extreme events. The uncertainty of stochastic sediment particle trajectories can be quantified by the variances. As the diffusion coefficient of the turbulence becomes greater, the variances (uncertainties) of particle trajectory increase. As the frequency of extreme flow occurrences becomes larger, the variances (uncertainties) of particle trajectory also increase. On this account, quantifying the uncertainty of particle movement more comprehensively is one of major advantages of the SJD-PTM when modeling extreme events as a random process for longer term predictions. It is demonstrated that the ensemble variances of particle movement can be attributed to the random occurrences of the particle jumps during the extreme events in a long-term prediction. When the temporal lag of particles due to the drag force and/or added mass is considered in the jump amplification factor, the variances decrease because the time lag of sediment particles reduces the uncertainty and delays the impact associated with the occurrence of extreme events.

[40] It is concluded that the proposed SJD-PTM can characterize the probabilistic properties of sediment transport and the most probable pathline in addition to the ensemble average particle movement. We have validated the proposed SJD-PTM against available experimental data by Sumer and Oguz [1978] and Cuthbertson and Ervine [2007]. However, more work is needed to refine the proposed method such as consideration of particle movement under the nonequilibrium condition and inclusion of extreme flow events of varying intensity.

Notation
Ap

projected area of the particle (m2).

Bt

vector of the Wiener process (s1/2).

CD

drag coefficient.

dp

particle diameter (m).

Dx

longitudinal turbulent diffusivities (m2/s).

Dy

transverse turbulent diffusivities (m2/s).

Dz

vertical turbulent diffusivities (m2/s).

f(x;t)

probability density function of particle locations.

Fr

forces exerted on the particle in regular flow (N).

Fe

additional forces incurred during the extreme flow events (N).

h(t, Xt)

jump amplification factor (m).

lλ

Taylor length scale (cm).

Pt

Poisson process.

Rλ

Taylor-scale Reynolds number.

Rep

particle Reynolds number.

tλ (=lλ/u′)

Taylor time scale for the turbulence fluctuations (s).

uf or uf

fluid velocity (including extreme flow) (m/s).

equation image (t, Xt) or equation imagei

drift velocity vector (m/s).

equation imagef

increased mean drift flow velocity due to an extreme flow (m/s).

equation imagefo

mean drift flow velocity before the extreme flow (m/s).

u

Taylor velocity scale (cm/s).

equation image

mean streamwise fluid velocity (m/s).

V, Vp, or Vp

particle velocity (m/s).

Vn

stepwise particle velocity (m/s).

equation imagen

ensemble mean of stepwise particle velocity (m/s).

Vr

particle relative velocity (m/s).

equation image

mean transverse fluid velocity (m/s).

p

particle volume (m3).

ws

particle settling velocity (m/s).

equation image

mean normal fluid velocity (m/s).

Xt or (X, Y, Z)

particle position (m).

λ

mean frequency of occurrence of events

ρs

particle density (kg/m3).

σ(t, Xt) or σij

diffusion coefficient tensor (m/(s1/2)).

τp

particle relaxation time (s).

Acknowledgments

[41] The authors gratefully acknowledge the financial support from National Science Foundation under grant contract number EAR-0748787. We also would like to express our gratitude for the laboratory data provided by Cuthbertson and Professor Ervine.

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