# Statistical Analysis of Bed Load Transport over an Armored Bed Layer with Cluster Microforms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Backgrounds

^{2}, may be adopted to define the diffusivity of the bed load transport process, which can be expressed as [36]:

^{2}grows linearly with time; for subdiffusion, 0 < γ < 1; for superdiffusion, γ > 1; and for ballistic transport, γ = 2 and σ

^{2}grows the same as particles moving in different directions with no pauses or direction changes [36].

## 3. Experimental Techniques and Methodology

#### 3.1. Experimental Setup

#### 3.2. Image Processing

_{i}, y

_{i}), that are functions of time t

_{i}. The streamwise and transverse particle velocities, V

_{x}and V

_{y}, can be readily computed as:

## 4. Experimental Results

#### 4.1. MSD Growth and Particle Diffusion Regimes

#### 4.2. Particle Number Activity

#### 4.3. Particle Velocity

_{x}measurements and therefore cannot be neglected. Therefore, we adopted the (thin-tailed) Gaussian distribution with exponential tails and the (heavy-tailed) α-stable distribution with power-law tails to describe the two-sided V

_{x}distribution. Figure 4 shows that the probability distribution of the streamwise particle velocity significantly deviates from the Gaussian distribution as V

_{x}→+∞ and is well described by the α-stable distribution. The thick tails in the α-stable distribution imply that there is a higher fraction of high-velocity bed load particles over the armored bed layer than predicted by the thin-tailed distributions. The left tail in the α-stable distribution on the left indicates a noticeable amount of grains moved in the upstream direction, which has not been reported in other experiments.

_{y}, was plotted in Figure 5. Similarly, V

_{y}shows a symmetric bell-shaped distribution that has a narrow central peak and heavy tails on both sides that are much thicker than those of the Gaussian distribution. Thus, we proposed that the α-stable distribution also provides a better fit for V

_{y}with proper parameters. Our experimental results contradict the widely accepted view that the transverse particle velocity follows the thin-tailed exponential or Gaussian distributions based on previous experiments [10,13,17]. After comparing with the previous experiments, we attribute this inconsistency to the armored gravel bed configuration with cluster microforms, which was not commonly seen in previous experiments. This will be further discussed in Section 5.1.

#### 4.4. Particle Step Length and Rest Period

## 5. Discussion

#### 5.1. Comparison with Previous Experiments

#### 5.2. Particle Diffusion Regime and Particle Motion Statistics

#### 5.3. Limitations of the Present Study

## 6. Conclusions

- The scaling growth of the MSD with time showed that the particle diffusion regimes in this experiment varied at different time scales: the particle diffusion regime was superdiffusive at small time scales (T < 50 s), but became subdiffusive at larger time scales.
- The particle number activity followed a negative binomial distribution, which is consistent with the theoretical solution derived in Ancey et al. [22] by accounting for the collective entrainment processes. The power-law tails in the probability distributions of both the streamwise and transverse particle velocities indicated a significant amount of fast-travelling particles, whereas the negative tail in the streamwise particle velocity distribution indicated a noticeable amount of grains occasionally moved in the upstream direction, which has rarely been considered in previous experiments. This may be explained by the particle clusters over dual impact on bed load transport exerted by the armored bed layers: some particles are accelerated in the preferential paths between particle clusters, while others were obstructed by the clusters to produce negative streamwise particle velocities. The distributions of the transverse particle velocity, the step length and the rest period all displayed heavy tails that were well described by the α-stable and M-L distributions. Nonetheless, the effect of the armored bed layer with cluster microforms on the distributions of these parameters needs to be confirmed with more experimental data.
- The particle transport regime may be scale-dependent, i.e., the transport process may exhibit different diffusion regimes on different timescales. Meanwhile, the shapes of the distribution tails of the bed load motion parameters varied as observation time scale increases. We propose that the heavy tails in the distributions of the bed load motion parameters were associated with small time scales and may average out over longer observation periods. Nonetheless, the time scales associated with the transition of particle diffusion regimes and those with the changes in the statistical characteristics of bed load particle motions were different. Further investigation is needed in unraveling their connections.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic plot of the experimental flume (drawing not to scale). The flow was from right to left.

**Figure 3.**(

**a**) Discrete probability distribution of the particle number activity for the interval Δt = 1 s. The inset shows the same data with the fitted negative binomial and Poisson distributions on a semilog plot. (

**b**) Exceeding probability of the particle number activity.

**Figure 4.**(

**a**) Discrete probability distribution of the streamwise particle velocity (Δt = 0.1 s) with theoretical lines that fit the data; the inset shows the same data on a semilog plot. (

**b**) Exceedance probability of the particle streamwise velocity.

**Figure 5.**(

**a**) Discrete probability distribution of the transverse particle velocity (Δt = 0.1 s) with theoretical lines that fit the data; the inset shows the same data on a semilog plot. (

**b**) Exceedance probability of the particle transverse velocity.

**Figure 6.**(

**a**) Discrete probability distribution of the particle step length (Δt = 0.1 s) with theoretical lines that fit the data; the inset shows the same data on a semilog plot. (

**b**) Exceedance probability of the particle step length.

**Figure 7.**(

**a**) Discrete probability distribution of the particle rest period (Δt = 0.1 s) with theoretical lines that fit the data; the inset shows the same data on a semilog plot. (

**b**) Exceedance probability of the particle rest period.

**Figure 8.**Bed surface profile of armored bed with clusters and “flow-accelerating belts”. Red points represent the visible largest particles of d

_{m}= 17 mm (three typical shapes of clusters bounded by dotted yellow circles, and labeled by ①line, ②heap, ③ring, respectively), and the blue lines are drawn along flow belts (water flows from left to right).

**Figure 9.**Changes in the probability density function (PDF) of the particle step length for observation intervals of Δt = 0.1 s (

**a**), 0.3 s (

**b**), 0.5 s (

**c**) and 1 s (

**d**) on semilog plots with theoretical lines representing Gaussian and α-stable distributions.

Size Group | Median Size (mm) | Size Range (mm) | Weight Fraction (%) |
---|---|---|---|

1 | 0.45 | 0.25–0.8 | 10 |

2 | 1.0 | 0.6–1.25 | 15 |

3 | 3.0 | 1.0–5.0 | 25 |

4 | 7.0 | 5.0–10.0 | 25 |

5 | 17.0 | 15.0–20.0 | 25 |

Q (L/s) | H (cm) | Fr | Re | Θ | U* (m/s) |
---|---|---|---|---|---|

130 | 16.1 | 0.64 | 9.8 × 10^{4} | 0.13 | 0.08 |

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**MDPI and ACS Style**

Zhu, Z.; Zhang, S.; Chen, D.
Statistical Analysis of Bed Load Transport over an Armored Bed Layer with Cluster Microforms. *Water* **2019**, *11*, 2082.
https://doi.org/10.3390/w11102082

**AMA Style**

Zhu Z, Zhang S, Chen D.
Statistical Analysis of Bed Load Transport over an Armored Bed Layer with Cluster Microforms. *Water*. 2019; 11(10):2082.
https://doi.org/10.3390/w11102082

**Chicago/Turabian Style**

Zhu, Zhenhui, Shiyan Zhang, and Dong Chen.
2019. "Statistical Analysis of Bed Load Transport over an Armored Bed Layer with Cluster Microforms" *Water* 11, no. 10: 2082.
https://doi.org/10.3390/w11102082