Chapter 3. Modeling suspended sediment transport under the influence of turbulent
3.3 Results and discussion
Figure 3.5 plots the fitting results for semi-theoretical curves of turbulence intensities and the fitting result for Reynolds stress. Statistical properties of velocity time series are demonstrated to agree reasonably with the semi-theoretical curves. The statistical properties of diffusion and the jump term time series alone are also presented.
Thus, the contributions of different terms to turbulence intensities and Reynolds stress can be demonstrated. In the experiments, ejection and sweep events are responsible for most of the Reynolds stress in the near-wall region. It is interesting, on the other hand, in the outer region, the jump terms cannot explain most of the Reynolds stress in the proposed model (concentrations). However, the ejection and sweep events considered in this model are the primary events related to the hairpin vortices in the near wall region.
The contributions of the jump terms to the turbulence intensities and the Reynolds stress in the proposed model are similar to those contributed by the hairpin vortices, or the so-called Type A eddies as in the work of Perry & Marusic (1995). Accordingly, this result is consistent with the findings of Perry & Marusic (1995).
Figure 3.6, 3.7 and 3.8 display histograms of velocity fluctuations and instantaneous Reynolds stress. Nakagawa & Nezu (1977) used Gaussian-based Gram-Charlier distributions, and Bose & Dey (2010) used exponential-based Gram-Charlier distributions as the distributions of turbulence velocity fluctuations. Relative to the normal distribution, the velocity fluctuation distributions that are obtained from experimental data are more concentrated at small values, and are spread more widely over
large values. The results of this model are consistent with the above observations and demonstrate that the complex distributions of velocity fluctuations may be caused by the superposition of two velocity components whose magnitudes follow normal distributions.
The histogram of instantaneous Reynolds stress also agrees closely with data from Nakagawa & Nezu (1977) and Bose & Dey (2010).
FIGURE 3.5 Fitting results for semi-theoretical curves of turbulence intensities and Reynolds stress. Bottom right part of figure displays velocity fluctuations in J′¥′ plan at
? = 0.1 ë.
The proposed model can provide velocity fluctuation distributions that are similar to the experimental findings, possibly showing that the assumptions regarding ejection and sweep events are acceptable. However, the model cannot reflect the skewness of distributions of velocity fluctuations, perhaps because of the assumption that the ejection and sweep events are similarly distributed in space. In fact, the distributions of ejection and sweep events are similar but not identical. The number of weak sweep events exceeds that of ejection events in the near-wall region. From the near-wall region to the water surface, ejection events have increasing conditional average turbulence intensities but not the sweep events (Nakagawa & Nezu 1977; Cellino & Lemmin 2004; Noguchi & Nezu 2009). The difference between ejection and sweep events can be considered to capture the asymmetry of distributions of velocity fluctuations.
The most important results concerning the construction of flow configuration are the magnitudes of π in the diffusion terms. The value of parameters π in the fitting of semi-theoretical curves of turbulence intensities varies with ti. Figure 3.9 displays the fitting results for the diffusion term in vertical direction À∆. In the proposed model, À∆ is of the same order of magnitude as the Rouse diffusivity when ti larger than 0.5í, indicating that the turbulence intensities can be used to explain particle diffusivity on a time scale of around 0.5í.
FIGURE 3.6 Histograms of stream-wise velocity fluctuation at ? = 0.2 ë.
FIGURE 3.7 Histograms of wall-normal velocity fluctuation at ? = 0.2 ë.
FIGURE 3.8 Histogram of instantaneous Reynolds stress at ? = 0.2 ë.
Tsujimoto (2010) derived the relationship between particle diffusivity and particle displacement — within a certain Δi as
@= =“ ”mÉGä . (3.9)
He recommended determining Δi as follows.
‘≤ ≡©∗’ÉG= ñm`^∗
÷◊ÿ ä, (3.10)
where FG∗ = FG J∗ℎ; FG is turbulent eddy viscosity; ⁄¤ is the ratio of sediment particle velocity standard deviation to vertical turbulence intensity ≈∆; ‹∆ is normalized vertical turbulence intensity ‹∆ = ≈∆ J∗; For finer sediment ⁄¤ ≅ 1. Depth average turbulence eddy viscosity is FG = IJ∗ℎ 6, and depth average normalized turbulence intensity in the
experiment of Noguchi & Nezu (2009) for clear water is ‹∆ ≈ 0.7849. Thus, Δi ≅ 0.7867í under the flow conditions in the experiments of Noguchi & Nezu (2009).
If sediment particles exhibit motion that is very close to flow motion, then the distance moved by a fluid particle within Δi should be close to —. Turbulence intensity has the information of fluid particle movement. Therefore, particle diffusivity can be expressed in terms of turbulence intensity if a suitable Δi is chosen. In this section, the temporal scale of turbulent diffusion is assessed by considering the time step that is used to characterize the turbulence intensities.
Figure 3.10 and 3.11 present the example of a sample path and the results of suspended sediment transport simulations for sediment particles with different diameter t=, respectively. In figure 3.10, from one of the sample paths one should notice that when an ejection event occurs, a particle can still move downward, as the impact of Brownian motion on particle movement might be in the opposite direction. It should be remembered that the study of this section characterizes the ejections and sweeps as the jump events.
The remaining flow structures are represented by the advection and the Brownian motion terms when appropriate.
FIGURE 3.9 Magnitudes of fitted D∆ fitted based on various values of ti.
In the simulation, the sediment concentration profile is shown to fluctuate around a mean concentration profile. As such, this model might have the potential to reflect the uncertainty of sediment concentration, as demonstrated in the experiment of Cellino &
Lemmin (2004). With a long-term average, the contributions of ejection and sweep events to sediment concentration profiles can be seen. Such events mainly affect particles at a depth of less than 20% of the boundary layer thickness ë. This result is consistent with the discussion of Muste et al. (2009).
FIGURE 3.10 One of the sample path in the simulation for particles with fi= = 1.2 and t= = 0.25mm.
FIGURE 3.11 Results of sediment transport simulations for sediment particles with fi= = 1.2, different diameters t= and settling velocity <=.
The proposed model focuses primarily on the ejection and sweep events in the near wall region. Although a full description of the sediment concentration profile for the whole region is yet to be achieved, an important concept related to the time scale of suspended sediment transport can be discussed when figure 3.11 is compared with figure 3.12. Figure 3.12 plots the result of simulation in which all flow motions are represented by the diffusion terms that fit turbulence intensities with ti = 0.01í. Neglecting the jump terms, the pickup probability proposed by Bose & Day (2013) was used to re-suspend sediment particles. In this simulation, the strength of flow was not underestimated; however, the particles were not in suspension. The particles accumulated at the bottom of the channel, where they were picked up temporarily by the re-suspension mechanism. The result reveals that for the given turbulence intensities, flow events of selected characteristic time scales including ejection and sweep events or others of similar time scales would affect sediment particle carrying capacity. It also implies that the time scale of suspended sediment transport, or so-called turbulent diffusion, may be similar to the time scale of ejection and sweep events, which is of the order of “seconds” (The duration distribution of ejection and sweep events has been shown in figure 3.1). The duration of the events may have the same physical meaning with Δi that is proposed by Tsujimoto (2010). Under this time scale, the transport phenomenon can be more precisely characterized. Since the duration of the ejection and sweep event would not be a constant, Δi in theory should be modeled as a random variable.
FIGURE 3.12 Result of simulation for particles with fi= = 1.2 and t= = 0.25mm in which all flow motions are represented by the diffusion terms with ti = 0.01í.
Memorylessness is an important property of a Markov process. It refers to the fact that the future state of the system depends only on its most recent state. The Markov process is the mathematical foundation for the link between the stochastic particle tracking model and the advection-diffusion equation (or the Fokker-Planck equation).
However, in the proposed model, memorylessness may be affected by the ejection and sweep events that last for several time steps. The mathematical linkage between the proposed model and the relevant equation with the Eulerian framework should be verified in detail, because the memory in the proposed model may lead to a new coupled equation other than the advection-diffusion equation. Physically, the advection-diffusion equation cannot describe the fluctuations of sediment concentrations that were described by
experimental reports such as Cellino & Lemmin (2004); however, it fits long-term average profiles fairly well. Accordingly, any equation with Eulerian framework coupled with the proposed model should give similar results to the advection-diffusion equation with respect to the long-term average concentrations. If the coupled equation can be derived, then the physical interpretations of turbulent diffusion should have a clearer picture than the interpretations that are given by the advection-diffusion equation.
3.4 Conclusions
Focusing of the impact of ejection and sweep events in the near wall region on suspended sediment transport, a two-dimensional stochastic diffusion jump particle tracking model was proposed to illustrate suspended sediment particle trajectory. Despite a full explanation of suspended sediment transport in the outer region has yet to be completed owing to the insufficient knowledge about the wall bounded turbulence structure, the model can provide new physical insights about turbulent diffusion when it verifies the two hypotheses proposed in §1.1.2.
First of all, the ejection and sweep events in the near wall region not only contribute to the conditional average sediment flux but also play a role in long term average sediment concentration profile. Most of the previous studies emphasized the impact of ejection and sweep events on the bed load transport and the entrainment/deposition processes. This model confined the influence of those events on suspended sediment transport under the
20% of the boundary layer thickness. To develop a delicate model of suspended sediment transport, the ejection and sweep events in the near wall region need to be considered.
With the verification of the second hypothesis, the temporal scale of suspended sediment transport has been discussed. The duration of the flow event is the key for turbulence to suspend a particle. It implies that the coherence of the turbulence plays a role in sediment transport mechanism. The simulation of sediment concentration may encounter problems if a ”pure random” flow field is used. And the time scale of the coherent structures, the scale of the order of “seconds” in this study, has been recommended. Nevertheless, the diffusion coefficient used in the conventional analysis of the suspended sediment transport is mainly gained in the view point of the momentum transfer, as seen in the majority of previous studies and applications. The study of this section provides a possible way to access needed information to estimate diffusion coefficient from the kinematic property of the turbulence, or turbulence intensity. It is more straightforward than the previous methods when it comes to explaining the microscopic particle behavior.
The study of this section is the first to consider the impact of near-wall turbulent coherent structures on the suspended sediment transport using random walk based model.
With simple concepts and assumptions, several complex flow features are captured. More in-depth knowledge about flow structures will lead to a more refined stochastic particle tracking model for the modeling of suspended sediment particles.