Results and discussions

Figure 4.9 shows the simulated sediment concentration profile of the particles with J_@ = 1.2 and diameter F_@ = 0.025mm . The range of concentrations after the simulation reaches the steady state is shown. Figure 4.10 displays example sample paths from SD-PTM and the RTIB-based model for comparison. Although both models provide the same average sediment concentration over the long term, they provide different simulated particle trajectories.

FIGURE 4.9 Simulated and measured sediment concentration profiles in experiment of Noguchi & Nezu (2009). Properties of particles are J_@ = 1.2 and F_@ = 0.025mm.

FIGURE 4.10 Sample paths of particles with J_@ = 1.2 and F_@ = 0.25mm.

Calculate the sediment diffusion coefficient from the vertical concentration profile, using the method of Coleman (1970). Recall the advection-diffusion equation in steady state,

d_@e^fg(e)_fe − h_@i j = 0. (4.28)

The sediment diffusion coefficient in the vertical direction can be calculated as

d_@e = ω_@i j ^{fg e}_fe . (4.29)

The 2^nd-order central finite difference scheme is used to estimate the derivative of i j :

d_@e = g enme Xg eXme^{)klg e me} (4.30)

Figure 4.11 presents the moving average of fitted diffusion coefficient using (4.30) throughout the depth of water. Interestingly, the effective diffusion coefficient that is calculated using (4.18) does not equal that fitted from the simulated concentration profile. The difference indicates that the governing equation of the proposed model does not directly correspond to the advection-diffusion equation. If the governing equation that is based on particle movement can be written in the form of the advection-diffusion equation, then the determination and modification of the sediment advection-diffusion coefficient under different hydraulic conditions will be clearer. The proposed model should be derived using Eulerian concept in the future. Thereafter, the theoretical sediment diffusion coefficients in three directions may be determined.

FIGURE 4.11 The diffusion coefficient calculated using (4.18) and the diffusion coefficient fitted from simulated sediment concentration profile.

4.4 Conclusions

To incorporate the temporal scale of the turbulence structures in a random walk-based model, a novel stochastic process, called random time interval Brownian motion, is proposed. Two properties of such a stochastic process are elucidated. The first two time-averaged statistical moments of particle velocity are unaffected by the distribution of event durations, and the proposed model leads to normal/Fickian diffusion on average in the long term in spite of the local super-diffusion behavior. The relationship between two random variables $′ and 1₂, and the effective diffusion coefficient is

derived. The structure of the model that is based on RTIB is elucidated, simply implemented and the results seen for the first time.

A random walk considering event duration leads to Fickian diffusion on average in the long term and this fact may explain why the advection-diffusion equation fairly well approximates turbulent diffusion; however, an improved model is needed to describe the particle behavior at a time scale smaller than the temporal scale of turbulence structures. As such, RTIB should be useful, as it describes the super-diffusion behavior at a relatively small time scale.

In the advection-diffusion-related random walk model, the Wiener process that is used to simulate Brownian motion is scaled by the time step; therefore, the issue of time step determination is merely one of the numerical stability and accuracy if the time step is much smaller than the time scale of the simulated physical phenomena. The scalar spreading is determined by the diffusion coefficient. Problems are encountered only when the process is used to represent the bulk spreading phenomena and the statistical properties of a single particle velocity simultaneously.

The event duration 1₂ in RTIB is the duration of the local trend in turbulence velocity. Even though the definition of local trend seems arbitrary, RTIB fortunately exhibits the same standard deviation of magnitude independently of the distribution of event duration, and the ensemble effective diffusion coefficient is not sensitive to the type of event duration distribution. (The effective diffusion coefficient depended on the

first two moments of event duration). The validity of (4.18) for the random walk with controllable persistence reveals that the type of distribution that is followed by event duration is unconstrained. Notice that the distribution may not include negative values because a negative duration is meaningless in a physical problem and if the event duration has a heavy-tailed distribution, then events of long duration may cause super-diffusion in a short term simulation, as seen at the beginning of the top figure of figure 4.3.

Again, the purpose of the proposed stochastic process is not to represent a specific turbulence structure but to incorporate the temporal scale of turbulence in a random walk-based model. To quantify the contribution of a targeted turbulence structure to suspended sediment transport, a model similar to that proposed in §3 can be used. At this time, the diffusion velocity components can be simulated using RTIB, and the jump terms represent the targeted flow structure.

Since the event duration is already accounted for in the properties of turbulence, the truncation error of the event duration is the major consideration in the determination of the time step in the proposed model. As suggested in §3, a time step that is smaller than the Kolmogorov time scale is recommended to simulate the finest flow motions.

In summary, as a candidate method for simulating the randomness of turbulence velocity fluctuations, RTIB and its related sediment transport model have two advantages. First, the first two time-averaged moments of an RTIB time series are

independent of the distribution of event durations. The turbulence intensity can be used as the standard deviation of RTIB magnitude without adjustment. Second, the effective diffusion coefficient is related only to the magnitude of the process and the first two statistical moments of the duration distribution. It is not sensitive to the type of distribution of event durations. These benefits make the simulation controllable and easy to perform.

Difficulties will be encountered in setting the parameters in RTIB. First, the flow event must be defined before its distribution is obtained. The physical meaning of event duration has been discussed, but a reasonable definition of such an event has not yet been conclusively established. Second, the model is sensitive to the turbulence intensity, and turbulence modulation must be considered if sediment particles are present in the flow. Although further information about turbulence properties is required using the proposed model, the model fills the gap between single particle velocity and bulk diffusion phenomena.

As a first attempt to incorporate the temporal scale of turbulence structures in a random walk-based model, flow velocity is represented in a simplified manner by RTIB with a constant value in every time interval. However, the proposed stochastic process can be used to simulate local trend of flow velocity fluctuations, preserving operability and flexibility of the random walk model. The model has potential to be developed for both research and implementation.

4.5 Recommendations for future work

First of all, the temporal scale of the turbulence structure has been demonstrated to be crucial in the random walk-based model to explain bulk diffusion phenomena and single sediment particle velocity at the same time; nevertheless, the distribution of the duration of the flow events is typically unavailable. A standard procedure for determining the distribution of event durations at various flow elevations from the experimental and simulation data is sought.

Second, the first two time-averaged statistical moments of the RTIB time series are not sensitive to the type of event duration distribution so the turbulence intensity can be used as the standard deviation of RTIB magnitude without adjustment; however, the influence of the event duration distribution on the magnitude distribution of the RTIB time series is not yet clear. In wall-bounded turbulence, the probabilities that fluid particles move up and down are not equal, and the distribution of vertical turbulence velocity fluctuation is skewed. To represent in detail the statistical properties of turbulence velocity fluctuations, the relationship between their distribution and the velocity distribution in the RTIB time series must be known.

Third, the properties of the proposed stochastic process are analyzed over the long term. The process exhibits local trends of turbulence velocity fluctuation, and the analysis of the process over a relatively short simulation time may provide further information about the transport of suspended sediment in open channel flow. The

physical meaning of the process behavior within such a short period is also an issue that warrants discussion.

Last but not least, unlike the advection-diffusion equation, the governing equation of the proposed model is a Lagrangian formulation. The effects of the memory in the proposed model on the bulk diffusion phenomena are not fully understood. The derivation of (4.18) sheds only some light on this question. An equation that is equivalent to RTIB and is a Eulerian formulation is required. Unlike the complex governing equation of the SDJ-PTM in §3 that contains multiple stochastic processes, the formulation is simplified and it contains only a single stochastic process, which can be mathematically described. Based on the definition of RTIB, the Eulerian formulation of the proposed model may be derived.

Chapter 5. Conclusions

When a random walk-based model is used to model the transport of suspended sediment in an open channel flow, the inconsistency between sediment particle movement and simulated particle motion that is implied by the advection-diffusion equation is revealed. This inconsistency indicates that the advection-diffusion equation fits only the overall phenomena of turbulent diffusion and makes the determination of the sediment diffusion coefficient a black box. The temporal scale of the turbulence structure and the local trend in flow velocity are suggested to be essential in filling the gap between the bulk diffusion phenomena of sediment concentration and the movement of a single sediment particle.

A new stochastic process, called the random time interval Brownian motion, is developed to incorporate the temporal scale of turbulence structures into a random walk model. Using this model, the relationship between diffusion coefficient and particle movement is derived. In this scenario, the local trend of flow velocity fluctuations does not change the average diffusion behavior over the long term; however, it does change the rate of scalar spreading and therefore the diffusion coefficient. The derivation justifies the applicability of the advection-diffusion equation to turbulent diffusion. The proposed stochastic process provides a way to simulate turbulent diffusion based on single particle movement that is consistent with the statistical properties of flow velocity.

Finally, a preliminary structure of a suspended sediment transport model is suggested. According to the results of a numerical simulations, the model that is based on the RTIB is not directly related to the advection-diffusion equation, and the sediment diffusion coefficient must be regressed from the simulated sediment concentration profile. A future investigation should focus on the relationship between RIBM and an equivalent equation that is a Eulerian formulation.

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