Recommendation for the next section and future work

In the current model, movements that are not directly associated with ejection and sweep events are represented using diffusion process. However, the conflict between the time step ti that provides numerical stability and the time scale îi that facilitates the observation of turbulent diffusion makes it difficult for the proposed model to give a complete sediment concentration profile. To solve this dilemma, the duration of flow events that are not directly related to ejection and sweep events need to be independent of ti.

Second, the representations of the turbulent structure in the outer region may be refined. Perry & Marusic (1995) classified eddies into three types - eddies that are/are not rooted in the near wall region, and eddies that are more isotropic than the other two types (Type-A, B, and C). Just as Type-A eddies, or hairpin vortices, are associated with ejection and sweep events, Type-B eddies should be associated with some kinds of flow motion. Representations of such motions may have similar forms to the jump terms that are used in this study. The diffusion term should be used to represent the motions that are caused by Type-C eddies, which are more isotropic and homogeneous than the motions that are associated with Type-A and B eddies. The time step of the order of the Kolmogorov time scale is recommended to capture the finest motions of turbulence.

Another necessary task in the refinement of the proposed model is to consider ejection and sweep events separately. As mentioned above, the properties of ejection and

sweep events are similar rather than identical. Difficulties may be encountered in determining the state of a particle given the complex relationships among flow motions that are caused by ejection and sweeps events. However, such determinations enable the constructed model to simulate flow field more realistically.

The developed model considers particle velocity as a superposition of flow velocity and particle settling velocity. Therefore, the drag force of flow and the gravitational force are considered, but the lifting force that is caused by the velocity gradient is not. Whether the lifting force is important in suspending particles in flow field is not yet clear. If it should be considered, information about the spatial correlation of flow velocity may be required to simulate the lifting force on the particles. The two probable ways in which the lifting force can be integrated into the model are to calculate the force from the state of a particle and to transform it into a velocity component; and to incorporate the average effect of the lifting force by adjusting the settling velocity of the particles.

Similarly, particles are picked up by not only the drag force but also the lifting force.

Particles are picked up by sweep events and outward interactions (Q1 events) (Dwivedi et al. 2011). In this model, only strong ejection events are considered to be able to pick up particles. Bose & Dey (2013) used the distribution of vertical velocity fluctuation to determine whether a particle is picked up. Cao (1999) used a semi-theoretical formula that considered ejection and sweep events to obtain the sediment concentrations at the reference height. These investigations simulated entrainment based on turbulent coherent

structures. If the entrainment process can be refined, then the sub-processes in the model can be more mutually consistent. However, it is not easy to integrate detailed entrainment mechanism to the model. The process of particle entrainment is extremely complex. Other than the direct pick up from the channel bed by the drag force or lifting force, particle entrainment classically involves the accumulation of particles by sweep events and their being lifted into the upper region by ejection events (Nino & Garcia 1996).

In the next section, a novel stochastic process is proposed to deal with the conflict in the chosen of the numerical time step ti, and finally gives a quantified relationship between turbulent velocity intensities and effective diffusion coefficient.

Chapter 4. Incorporating the particle memory effect under the impact of turbulence structures into suspended sediment transport modeling

4.1 Motivation

The advection-diffusion equation has been widely applied in simulations of suspended sediment transport, but the specification of the diffusion coefficient is always an issue. Most estimates of the diffusion coefficient rely on the relationship between the momentum transfer rate of turbulence and the mass transport rate of sediment particles, i.e. a combination of turbulent eddy viscosity and the turbulent Schmidt number is used to obtain the turbulent diffusion coefficient of suspended sediment particles. The eddy viscosity and Schmidt number can be set for various particle properties and transport rates (concentrations). In the implementation of hydraulic engineering, the Schmidt number is typically calibrated against the data, but the calibration of the Schmidt number is not based on the movement of particles, so the estimation of the diffusion coefficient is a

“black box”. To evaluate the sediment diffusion coefficient more precisely, its physical meaning with respect to particle movement must be understood.

Random walk theory provides a way to simulate the probabilistic trajectory of sediment particles in turbulent diffusion. The relationship between turbulent diffusion and particle trajectory elucidates the gap between the sediment diffusion coefficient and the turbulent intensity (i.e., the standard deviation of flow velocity fluctuation). The

Wiener process, which is used to simulate Brownian motion, is scaled by a time step.

Whatever time step is selected, the second statistical moment of particle displacement is proportional to time and the spreading rate depends on the diffusion coefficient. The provided sample path may not be consistent with other physical properties of interest, such as velocity fluctuations of the sediment particles if the time step is incorrectly chosen (Tsujimoto 2010).

In molecular diffusion, particle spreading phenomena are consistent with Brownian motion because the particle changes its velocity and direction several times between measurements so memory of these properties is lost. As the resolution of measurement increases, a different behavior, called “ballistic motion”, will be observed. In this motion, the second moment of particle displacement is proportional to the time squared (Pusey 2011). Mathis, Hutchins & Marusic (2009) applied the Hilbert transform to the measured velocity time series of boundary layer flow and decomposed velocity fluctuations into large-scale and small-scale components. A large-scale fluctuation involves changing velocity over a particular period. When sediment particles exhibit flow motion, they have a memory of their velocity and direction and the diffusion behavior of sediment particles may deviate from Brownian motion within this time scale. To simulate suspended sediment transport and to describe sediment particle movement in a manner that is consistent with measured turbulence velocity fluctuations, the temporal scale of the flow structures should be considered in a random walk-based model.

Large flow motion in turbulence results in the temporal and spatial correlations of flow velocities in the flow field. It disturbs sediment particles for a particular period and carries particles over long distances (Cellino & Lemmin 2004; Okamoto, Nezu &

Katayama 2010). Chen, Sun & Zhang (2013) presented a model that is based on the fractional advection-diffusion equation to account for the long distances over which sediment particles are carried by large turbulent structures; however, the particle in the related equation moves by such a distance in a fixed time interval (time step). A particle effectively “jumps” from one position to another and its intermediate state has no impact on the system. The model cannot capture the complete process if the large flow motion carries particles for longer than a time step so it is not suitable for studying the temporal effect of turbulence structures/eddies. To model complete particle paths, another approach is required.

Turbulence structures/eddies disturb sediment particles for a period, in which the history of the system has to be considered to predict the future state. Hence, the temporal scale of turbulence structures/eddies may be mathematically represented considering the memory effect in the random walk model. This memory may arise from the conditional probability of system state that involves more than one previous time step or the autocorrelation in the probability of system state that influences more than a time step.

Restated, the memory mentioned here is the property that causes the system to deviate from the Markovian property.

Today’s random walk models cannot easily deal with the memory effect in a flexible way. Consider for example fractional Brownian motion and the stochastic multiplicative point process. Memory in the Brownian motion can be represented as an autocorrelation, and fractional Brownian motion is used to simulate such a situation. The increment of particles in fractional Brownian motion is not independent of time and all steps must be sampled at the beginning of the numerical simulation. In contrast, the stochastic multiplicative point process introduces a random walk in the waiting time and provides correlation between the durations of adjacent intervals of sharp pulses (Kaulakys, Gontis

& Alaburda 2005); nonetheless, it is still focused on the counting process of sharp pulses and the contribution of each pulse is averaged over time. The stochastic multiplicative point process cannot be used to simulate the impact of a signal that lasts for a specific duration. A new random walk model therefore be introduced to meet the needs of analysis for the influence of large flow motion on suspended sediment transport that happens in a specific duration.

Owing to the extreme complexity of suspended sediment transport, the considered properties related to the memory effect have to be abstracted from reality before an advanced analysis is performed. With respect to the temporal scale of flow events, the following characteristics of the targeted process for suspended sediment transport modeling are proposed.

1. Flow events always occur as part of the targeted stochastic process.

Whereas extreme large flow events occur sporadically, the targeted process

describes all kinds of flow motions/structures in turbulence. The flow event is a set of flow motions of various kinds that may happen rather than a specific flow structure.

2. The flow events last for a period, and the duration of each event is a random variable, which represents the temporal scale of the flow motions/structures.

3. The magnitude of a flow event may change and has a correlation in time or remain constant. In other words, the particle movement carried by the targeted process exhibits persistence within an event. The persistence in random walk theory is defined as the number of time steps of a walker in a particular direction before the walker changes its direction.

As indicated in the first point, the flow event may be referred to the development of a turbulence coherent structure, turbulence eddies or a disturbance, that dominates local turbulence velocity fluctuations (for example, the large-scale fluctuation in Mathis, Hutchins & Marusic (2009).) In each event, a particle may accelerate, decelerate and change direction at the end of the event. The purpose of proposing a new stochastic process is not to represent a specific flow structure/flow motion but to incorporate the

“temporal scale” of turbulence structures/flow motions into a random walk-based model.

The modeling of the targeted process using random walk theory is discussed below.