Bimodal Bed Load Transport Characteristics under the Influence of Mixture Ratio

Abstract

The transport of a non-uniform bed load in a river is a complicated process and has enormous implications on the sediment flux and anomalous riverbed evolution. To investigate the transport characteristics of the non-uniform bed load and the related particle interactions, a real-time monitoring system of the bed load transport was developed to determine the instant transport rate and grain composition of the bed load. Doppler Velocimetry was used to synchronously measure the fluctuating velocity in high frequency. A total of 211 cases of flume experiments were conducted, focusing on non-uniform sediment with a bimodal pattern. The experimental results indicate that the random fluctuation of the bed load transport amount closely depends on the flow-intensity fluctuation caused by the turbulence burst near the bed. When the value of the flow-fluctuation peak is bigger than 2.5 σ, the coarse sands tend to incipient motion in high probability but are mostly fine sand transport when the peak is less than 1.5 σ. The transport rate of fine particles remains continuous throughout the process, while that of coarse particles is intermittent because the incipient motion mechanism of bed load sands mainly follows three modes. If the difference in diameter between the coarse and fine particles is large, the transport of coarse particles may undergo supernormal transport because of the effect of the fine particles on the coarse particles. The bed load flux of total, fine, and coarse sand present different trends with changes in the bed material composition, in which the transport rate of coarse sands and total bed load sands presents a humped curve in terms of the mixture ratio, and the optimal corresponding mixture ratio η_c is about 3:7. The optimal mixture ratio is not fixed, and it depends on the grain composition and size differential of bed material. With a proper mixture ratio, the transport rate of a non-uniform bed load is higher than the uniform bed load of related size. These findings might provide valuable support for predicting bed load transport and bed evolution in rivers.

1. Introduction

Non-uniform bed load movement is an important form of sediment transport in alluvial rivers and differs significantly from the transport of suspended sediment concerning the movement mechanism and transport laws [1,2,3]. Many pieces of research have focused on the uniformity caused by grain size variability but have rarely referred to the influence of the mixture ratio (coarse grain/fine grain), even though the composition of non-uniform sediments is one of the important factors influencing the transport rates of the bed load [4].

Einstein and Bagnold established the original theory for the movement of the non-uniform bed load [2,3], which provided a valuable basis for the following research. Hu and Hui analyzed bed load transport mechanisms via observations of the movement of jumping particles near the bed using high-speed photography [5]. The transport rate of the non-uniform bed load was studied according to the stochastic theory [6], and the formula of Einstein was extended. The relationship between the transport-layer thickness and the equivalent grain size was established under non-uniform sediment conditions [7]. Then, research on the effects of the microstructure of the bed surface on the bed load transport proceeded by means of laboratory experiments and modeling [8,9,10,11,12,13,14].

The bed load Einstein formulas were amended and corresponded greatly to experimental data and natural observations [6,8,15,16]. By analyzing the particle velocity during downward trajectories, the correlation between the particles and the turbulent structures was analyzed, and the movement of the particles was strongly correlated with flow ejection events. Particles fall back toward the bed as the correlation with the turbulent structures decreases [17,18,19]. Considering their complicated motion, bed load particles may also depend on turbulent structures, and this relationship warrants further investigation. The kinetic theory, suitable for investigating complex physical processes regarding bed loads, was introduced to sediment transport research [20,21,22]. Based on the tracer experiments of bed load, the random walk model of individual bedload trajectories was developed and related the multi-range diffusion characteristics to the timescales of these processes [23].

Concerning a non-uniform bed load with different grain sizes on the bed surface, the grains are affected by the near-wall laminar layer, including both the exposure effect of the coarse particles and the hiding effect of the fine particles [24,25,26,27]. The representative grain size and the key factor of the flow intensity were introduced, and the transport-rate formula for the non-uniform bed load was separately established using test data obtained under certain conditions [28,29,30]. The original flow and sediment conditions of the data used for establishing these formulas are limited. Therefore, these formulas are mostly empirical or semi-empirical, and their applications are defined for specific conditions. Nevertheless, there are a few formulas with general applicability. Parameters reflecting the interactions among particles with different diameters were introduced after the non-uniform sediments were grouped [31,32,33,34,35], and then the non-uniform sediment transport rates were obtained using the weighted-average method [36,37,38]. A larger proportion of coarse particles in the bed material yields more powerful hiding effects on the fine particles in the bed.

To understand the role of episodic sediment supply on the threshold of motion, streambed state, and stability in non-uniform gravel bed channels, flume experiments were conducted, and the importance of bed surface evolution on grain entrainment for different diameters was demonstrated [39]. In fact, the transport intensity of the bed load is closely related to the incipient motion of the grain on the bed along the stream. The incipient motion of non-uniform sediments is related to not only the flow intensity and the grain size but also the non-uniformity of the bed material. The equivalent diameter and the relationship between the exposure degree and the non-uniform grain size were introduced, and the incipient-motion velocity formulas for individual fractions of non-uniform sediments were derived [40]. Considering the influences of the bed-material properties and the particle-exposure degree, the incipient motion laws for non-uniform particles were investigated from the perspective of the critical shields stress in sediment incipient motion [41]. In addition, bed load particle motions were researched using PIV (particle image velocimetry) technology [42,43]. The segregation of non-uniform particles occurs because of the friction and collisions among them. The gap-filling mechanism in the vertical segregation of particles was proposed, and it was reported that the concentration and the velocity of particles influence the segregation [44].

The hydrodynamics action of a natural river leads to a bed material composition with bimodal characteristics, which have two significantly different grains, coarse and fine. A formula for determining the bed load flux by separating the coarse and fine particles and considering the critical shear stress was proposed [31]. The effects of bed armoring on the flux of a bimodal bed load were analyzed from a view of the relationship between the typical bed load formulas and the shear stress [45]. Methods for estimating the incipient drag stress for particles of different sizes were proposed by conducting bimodal bed load transport experiments [28]. However, in this research, measurement instruments and the sediment conditions in the experiment need to be improved to deeply understand the mechanism of bed load random transport.

Because of the complexity of the particle motion near the bed, the intricate mechanism of the coupling between the fluid and the particles, as well as the interaction between the bed load movement and the near-wall turbulence, are not fully understood. The interaction among the particles, the hiding, and exposure effect particularly the role played by the fine particles during non-uniform sediment movement and the influence of the particle composition on the bed load transport have not been sufficiently investigated [9]. The non-uniform particle interactions are a physical process and may not be easy to characterize directly or be quantified. However, its influence can be indicated by the increase in the bed load flux. The effect of particle interaction on the bed load transport has been found by the flume experiment. Therefore, it is necessary to study the effect of the non-uniformity of the particles on the bed load transport. This paper fully discusses the influence of particle non-uniformity (i.e., the mixture ratio of coarse to fine particles) on bed load transport, focusing on the movement of bimodal non-uniform sediments, according to an abundant series of flume experiments. The mixing ratio would be defined as a characteristic index and used to describe the effect of particle interactions (between coarse particles and fine particles) on the bed load flux.

2. Methods

The experiments were conducted in a 20 m × 0.25 m flume system, which included a flume with an adjustable slope, a control system for the self-circulation flow, and special measurement instruments. The overall layout of the system is shown in Figure 1 and Figure 2.

2.1. Flume System Used in the Experiment

The side walls and bottom of the flume were made of highly transparent glass. A water pump with a frequency converter was adopted in the water-circulation system to ensure a constant flow in the flume. The flow was measured by an E-mag electromagnetic flow meter. To ensure the flow was steady, a parallel grid composed of several rows of parallel pipes was set at the start of the flume (see Figure 1). The main test segment included an entrance transition section Section AB, an erodible area Section BC (where some of the static bed sands start to move in the form of bed load), an observing area, Section CD (where the movement and exchange of the bed load are observed), an export transition zone Section DE, and the dynamic sampling area of the bed load transport rate, Section EF. In order to make the condition of bed load particle incipient similar to the bed load in the natural riverbed, the observing area is a sanded bed with natural sand. The bed material comprised bimodal non-uniform sand, and the thickness of the bed was about 8.5 cm. Acoustic doppler velocimetry (ADV) for three-dimensional (3D) velocity measurement and image-capturing instruments for obtaining particle-motion images were set separately in the test section. A sampling monitor for the transported material (SMTM) was installed at the end of the sink to indicate the real-time bed load transport rate (see Figure 2), and a louvered rear door was used to control the water depth in the flume (see Figure 1).

2.2. Sediment Used in Experiments

To investigate the mechanism behind the particle interaction between the fine particles and the coarse particles during the movement of the non-uniform bed load, natural sand, and gravel of different colors, which are easy to track, were chosen as sediment particles in all experiments. The densities of the particles ranged from 2600 to 2800 kg/m³. The composition of the model sands was bimodal, and the main test sands used in the experiments are shown in

Table 1. Sediment diameter and differential.

No.	Fine Sand Diameter df/mm	Coarse Sand Diameter dc/mm	dc/df
1	0.6~0.9	3.0~4.0	4.67
2	0.6~0.9	5.0~6.0	7.33
3	1.2~2.0	6.0~8.0	4.35
4	2.0~3.0	6.0~8.0	2.80
5	2.0~3.0	8.0~10.0	3.60

The sediment diameter and the differential between the coarse sand and the fine sand. The maximum diameter differential is 4.67, while the minimum is 2.80, and the sand diameter differential of various groups may have an effect on the experiment results. The data in Table 1 are necessary to analyze the interactions between the particles.

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The diameters of the particle were in the range of 0.6–10 mm, and the diameter ratio of the coarse particle to the fine particle was in the range of 2.8–7.33. Sediments with different mixture ratios of coarse particles to fine particles are shown in Figure 3. To characterize the composition characteristics of the particles, non-uniformity indices of the generalized bed materials were introduced, considering the mixture ratio of coarse particles to fine particles η, and the generalized equivalent diameter $\overline{D}$ was defined as the following equation:

$$\eta =\frac{M_c}{M_f}, \overline{D}=d_c\cdot P_c+d_f\cdot P_f$$

(1)

where M_c (M_f) denotes the mass of particles of one size, P_c(P_f) is the corresponding mass percentage of particles of one size, and the subscripts c and f represent the coarse and fine particles, respectively. The mixture ratio of coarse to fine is used as a dynamic symbol of the non-uniformity of the bed material.

2.3. Monitoring System of the Amount of Bed Load

To investigate the movement property of the bed load, a dynamic monitoring system (SMTM) was developed and used to monitor the amount of bed load. The system comprised a suspended sample collector, adjustment devices for hoisting and siphoning, an electronic balance with the functions of automatic measurement and data output, an output data line with an EIA –RS-232C port (EIA is the abbreviation of the Electronic Industry Association, and RS is the abbreviation of the Recommended Standard), a computer, and other instruments, as shown in Figure 4, Figure 5 and Figure 6.

The bed load sediments can be weighed in real time once they fall into the sand container in Figure 6, and the weight data are transmitted instantly to a computer terminal, which means that the weight data displayed by the computer are synchronized with the data measured by the electronic balance. Thus, the accumulation of the bed load sediments transported from the erodible bed is monitored dynamically. The measurement limit of the electronic balance is 15 kg, and its precision is 0.1 g. According to the transport rate of the bed load and the precision of the electronic balance, the sampling frequency of the balance was set to 1/3 Hz. The instant transport rate of the bed load can be calculated according to the accumulation weight of the bed load collected.

It should be noted that the dynamic Sampling Monitor for Transported Material (SMTM) was developed especially for the experiment and was used to monitor the transport rate of bed load. There were some problems with the sediment traps designed in former research [26,29,46], including very sensitive to the shape of the bed and to the flow pulsations, the possible errors caused by incorrect placement and touching walls, et al., and would affect the measurement of bed load. In order to improve the design of the sediments trap and avoid these problems to a great extent, the sand container (sand trap) was designed as the permeable, streamwise profile of an irregularly shaped trapezoid (keeping the sand container stable under the flow effect) and was hanged on the hook of the electronic balance, shown as Figure 6. Thus, we did not need to consider the influence of the water pressure on the container. The data measured by the SMTM were the submerged weight M_s of the bed load and needed to be converted into the actual weight M_n before calculating the transport rate of the bed load. When the experiment was finished, all the bed load sediments collected in the container were emptied and weighed, which was defined as M_t. M_t is used to calculate the average transport rate of the bed load and verify the reliability of M_n obtained from SMTM.

2.4. Image Collection of Particles and Turbulence Measurement

To obtain the instant areal distribution density of the bed load particles of different sizes, a dynamic collection system for the particle images was developed, as shown in Figure 7. The system comprised two sets of high-speed cameras, the sampling frequencies of which ranged from 20 to 200 Hz. Their resolution was up to 1280 × 1024 pixels. During the experiment’s progress, the images of particles moving near the bed were captured continually. A special program was adopted to process the collected images and identify the particles in the image. The whole process included gray disposal, threshold segmentation, and noise elimination. Finally, the instant compositions of the particles in the sampling area were obtained; in other words, the masses of coarse sands and fine particles in the bed load mixture were acquired. The original image was captured by a high-speed camera, and the recognized images are shown in Figure 8.

The instant transport rate of the coarse sands in the bed load mixture was calculated using Equation (2):

$$G_b{}_i=\frac{n_i{m}_i}{t_i}=\frac{n_i{m}_i}{\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$v_i$}\right.}=\raisebox{1ex}{$n_i{m}_i{v}_i$}\!\left/ \!\raisebox{-1ex}{$L$}\right.$$

(2)

where i denotes the particle size. G_bi is the instant transport rate of the particles of size i in the bed load mixture. t_i is the time taken by particles of size i to pass through the sample window of the camera. n_i is the number of particles of size i in an image, which indicates the instant number of particles of size i passing through the sample window. m_i is the average weight of one particle of size i. v_i is the average velocity of the particles of size i, which can be measured according to the video of the bed load movement in the sample window, and L is the downstream length of the sample window.

The near-bed velocity was measured using a 3D Doppler current meter at a sampling frequency of 10 Hz. Two sets of 3D Doppler current meters were arranged along the central axis of the flume, located separately in the erodible area and observation area, as shown in Figure 9. According to the instant 3D high-frequency velocity data and the synchronous sediment data of the bed load, the properties of the near-bed turbulence structure and their influence on the intermittent movement of the bed load were investigated.

2.5. Experimental Design

To investigate the influence of the sand composition and size on the bed load transport, multiple experiments were conducted. First, according to the sand size, a series of experiments was designed. Then, each set was subdivided into groups according to the inter-combinations of the various flow conditions and sand mixture ratio conditions. Several flow-intensity parameters were adopted, such as energy slope J, flow velocity U, and water depth H. The sand conditions of the bed refer to the mixture ratio of coarse particles to fine particles, bed materials, and recharge conditions. Our experiments were designed and conducted as 46 groups comprising 211 experimental cases, and the details are shown in

Table 2. The hydraulic and sediment parameters of experiments.

Sets	Groups	Mixture Ratio η	Flow Discharge Q/(m3s−1)	Fr	Sample Sand Diameter d/mm
					Coarse Sands	Fine Sands
Set 1	Case 1~3	6:4	0.016~0.027	0.79~0.95	6~7	2.5~4
	Case 4~6	6:4	0.016~0.027	0.69~0.8	8~10	2~3
	Case 7	6:4	0.027	0.8	8~10	2~3
Set 2	Case 1~3	4:6	0.027~0.030	0.72~0.76	8~10	2~3
	Case 4~6	4:6	0.025~0.034	0.70~0.78	6~7	2.5~4
Set 3~13	Each Set (Case1~6)	10:0 (9:1~0:10)	0.024~0.034	0.86~0.95	8~10	2~3
Set 14~24	Each Set (Case1~6)	10:0 (9:1~0:10)	0.024~0.034	0.86~0.95	6~8	2~3
Set 25~35	Each Set (Case1~3)	10:0 (9:1~0:10)	0.020~0.024	0.82~0.88	5~6	0.6~0.9
Set 36~46	Each Set (Case1~3)	10:0 (9:1~0:10)	0.016~0.020	0.92~0.96	3~4	0.6~0.9

Remark: The table shows the sediment diameter used in experiments and the differential between the coarse sand and the fine sand.

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Table 2 shows the experimental conditions of various groups, with abundant flow conditions and sand conditions. The abundant experiment conditions were valuable in ensuring the accuracy and reliability of the experimental data and results. The flume slope in the experiments was controlled at 0.1% to 0.3%. The water depth ranged between 0.08 m and 0.16 m. Six flow discharges were adopted according to the transport intensity of the bed load. The outflow pattern of the flume was a steady, uniform flow, and the experimental conditions did not strictly follow the limit for the ratio of the width to the depth for two-dimensional flow because the bed load transport in the core region of the flume was affected by the glass wall to a lesser extent. The duration and progress of each experiment were strictly controlled. The period that the bed load transport intensity could maintain the basic stable state was about 30 min. There were two methods of sediment supply from the upper reach used in the experiment. The first was natural supply. This requires a long enough sanded section using natural selection achieved by the flow, i.e., bed erosion. The selected sands were transported downstream and turned to the supply of the bed material of the lower reach. Using this, sediments were transported to the lower reaches to maintain the bed load transport intensity. The main purpose of this method was to investigate the change in the bed load transport intensity with the time elapsed in the experiment. The other method is sediment feeding, where sediments were supplied continuously artificially in the upper reaches, and the amount of feeding sands was slightly larger than that transported to the downstream reach. Using the aforementioned experimental design, the characteristics of the bed load motion under constant transport were examined.

3. Results

A total of 46 groups comprising 211 experimental cases experiments were established under various flow and sediment conditions. The experimental conditions are shown in Table 2. Experimental data referring to several major parameters were obtained, including the instant transport rates of the bed load, the particle compositions of the transported bed load sediments, and the key parameters of the turbulence near the bed.

3.1. Key Parameters

To analyze the experimental data, the introduction of the following analysis parameters was necessary.

The dimensionless time parameter T_s is defined as Equation (3):

$$T_s=\frac{T_t-T_0}{T_N-T_0}$$

(3)

where T_t is a certain natural time during the test, T₀ is the initial time of the test, and T_N is the terminal time. Every natural time can be described as a normalized time after a standardized treatment.

The instant transport rate of the bed load G_b(t) is defined as Equation (4):

$$G_b(t)=\frac{W_{st}-W_{s(t-1)}}{T_t-T_{t-1}}$$

(4)

where W_s represents the accumulation of the bed load (which is obtained using SMTM), W_st denotes the accumulation of bed load at moment t, and W_{s(t − 1)} (t − 1) is the accumulation at moment t − 1, which is one-time unit before t.

To examine the trend of the transport intensity of the coarse sands with the altering of the mixture ratio of coarse to fine η of the bed materials, the dimensionless relative transport intensity of coarse sands Φ’_c was introduced, defined as Equation (5):

$${{\Phi }^{\prime }}_{\mathrm{c}}=\frac{{\overline{\Phi }}_{\mathrm{ci}}}{{\overline{\Phi }}_{\mathrm{cm}}}$$

(5)

where ${\overline{\Phi }}_{\mathrm{ci}}$ denotes the average transport rate of the coarse sediments in the bed load mixture under the sand condition of the i^th mixture ratio of the bed materials (the mixture ratio of coarse to fine is in the range of 10:0, 9:1, …, and 0:10). ${\overline{\Phi }}_{\mathrm{cm}}$ is the maximum average transport rate selected from several ${\overline{\Phi }}_{\mathrm{ci}}$ for various mixture ratios under the same flow condition.

To describe the instant area distribution densities of the coarse and fine particles of the bed load mixture during the movement of non-uniform bed load, the instant selected ratio of the moving particles ${\phi }_{\mathrm{it}}$ was introduced and used to describe the proportion of i-size particles in the bed load mixture, as presented in Equation (6):

$${\phi }_{\mathrm{it}}=\frac{M_{\mathrm{it}}}{M_{\mathrm{st}}}$$

(6)

where M_st is the mass of the total bed load at moment t, and M_it is the mass of the particles of size i in the bed load at moment t, i.e., M_ct or M_ft. Using this parameter, the transport intensities of the coarse and fine particles can be indicated directly.

The near-bed turbulence plays a significant role in the movement of bed load sediments [17,28,29]. To indicate the influence on the bed load transport caused by the turbulence structures of the flow near the bed, the longitudinal turbulence intensity σ_x was defined, as shown in Equation (7):

$${\sigma }_{\mathrm{x}}=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{(u_{xi}-\overline{u_x})}^2}}{n}}$$

(7)

where $u_{\mathit{xi}}$ is the instant longitudinal velocity, $\overline{u_x}$ is the average velocity during the sampling period, and n is the number of instant-velocity samples measured during the sampling period.

3.2. Experimental Results

In the experiments, the instant areal distribution density of the coarse or fine particles in the observation zone, indicated by the instant selected ratio of the coarse particles in Equation (2), can be obtained using image-recognition technology. To investigate the effects of the flow intensity turbulence on the non-uniform particle motions, a 3D acoustic Doppler velocimeter (ADV) was used to collect the instant velocity data near the bed at high frequency. The curves in Figure 10 provide evidence of the association between the bed load sediments transported and the flow turbulence based on the simultaneous data under several level flow conditions indicated by the characteristic values.

To investigate the effect caused by the bed material composition on the amount of non-uniform bed load transported, a series of experiments was finished with various compositions of bed material and a fixed flow discharge, depth, and slope. The instant bed load transport rates were dynamically collected during the experiments. Figure 11 shows the transport rates changing with the bed-material compositions under the fixed-flow condition. Under different flow conditions, the changes in the bed load transport rates with the bed-material compositions, i.e., mixture ratios, were also obtained and are presented in Figure 12, Figure 13 and Figure 14.

4. Discussion

According to experimental data analysis, some interesting properties related to the non-uniform bed load movement are discussed in the following section.

4.1. Correlation of the Motion Characteristics of the Bed Load Particle with Turbulence Features

According to previous research and experimental observations of the mechanism of the intermittent bed load transport, the flow turbulence near the wall plays a significant role in the interval incipient motion of the bed material [2,24,26,29]. However, the experimental data show that intermittent transport of the bed load and the turbulence of the flow intensity near the bed are not synchronous in frequency, and the former is far lower than the latter. To avoid confusing the correlation between the high-frequency turbulence of the flow and the low-frequency fluctuations of the bed load transport amount, the sampling frequency of the ADV was set to 10 Hz based on tests and comparisons. The measurement point of the ADV was adjacent to the image-capturing section of the bed load movement. According to the constant steady flow theory, it can be assumed that the flow conditions at the measurement point are the same as the flow condition of the image-capturing section of the bed load movement in the contiguous flow fields.

The flow velocity near the bed exhibits a coherent feature, and the selected ratio of the coarse sediments also indicates the random property during the experiment. Therefore, both of them present an intermittent characteristic. The experimental data indicated that the intermittent property of the coarse bed load particles is associated with the coherent structures of the near-bed flow velocity. Although the transport of the coarse sediments is intermittent, the total amount of the transported sediments increases continuously, indicating that the fine sands were transported continuously during the experiment. The transport property of the coarse particles significantly differs from that of the fine particles. Therefore, the proportion of the coarse particles or fine particles in the mixture plays an important role in the total transport amount of the bed load mixture, and the influence of the non-uniformity of the particles must be considered [18,24,26,34].

In the experiments, both the selected ratio φ_c and the transport accumulation of the coarse bed load present a random fluctuation in response to the turbulence of the flow velocity near the wall, as shown in Figure 8 and Figure 10. Figure 8 indicates that the measurement range of ADV is exactly the active range of the bed load. The velocity data obtained by ADV are of high frequency and valid enough for collecting the turbulent characteristics of the near-bed flow, which has a critical effect on the incipient motion of bed load particles. The parameter σ_x is defined to indicate the turbulence intensity. The flow intensity and the selected ratio of the coarse sediments both have intermittent characteristics, and peaks and valleys were observed, as shown in Figure 10. The selected ratio curve of the coarse sediment is mostly zero and sometimes nonzero, indicating the intermittent transport of the coarse sediments. By comparing the two curves, it can be found that the nonzero points in the selected ratio curve correspond to the high peaks of the flow-intensity curve. When the flow intensity is low, the selected ratio of the coarse sediment is always zero. This means the coarse sand is not easily prompted to move by weak flow turbulence. The reason the transport rate of the non-uniform sediments is intermittent is mainly because the grain composition of the moving sediments fluctuates [28,31]. The intermittent gradation of the moving sediments was associated with the sweep in the high-speed strip and the coherent pulsation of the near-bed flow velocity [9,17,18], which were caused by the turbulence burst of the flow. The relevant mechanisms are analyzed in further detail in the following Section 4.2.

It should be noted that the velocimetry method (ADV) used in our experiment is not accurate enough for quantifying the exact association between the single particle and the flow pulsations nearby. In order to make the condition of bed load particle incipient similar to the bed load in the natural riverbed, the observing area is a sanded bed with natural sand, and there is quantity of bed load moving on the bed. It is hard to collect the actual velocimetry data around every single moving particle when there is a collective motion of sediments. Technically, there is not a suitable velocimetry method for these currently. Therefore, we focused on the general flow intensity of the flow field near the bed, which is the dynamic source of the particle incipient. We tried to associate the relationship between the general flow intensity of the flow field and the amount of the start-moving particles statistically in the observing area. Therefore, Doppler velocimetry was used in the experiments.

4.2. Three Incipient Motion Modes of Coarse Sediments in the Bed Load Mixture

The size of the coarse sediments in the mixture was much bigger than the fine sediments. According to our experiments, the critical flow condition of the incipient motion of coarse sediment was much lower than the mechanic calculation value. According to our research, the reasons for this are the special incipient motion modes of coarse sediments.

The first mode is the promoting mode. The frequency and intensity of flow velocity pulsation near the bed are attributed to the turbulence structure of the flow [47,48]. The intermittent burst events prompt the intermittent increase in the velocity at a higher frequency. The spectrum analysis of the instant selected ratio of the coarse sands and that of the flow fluctuation indicate that the turbulence burst near the bed is the main power prompting the transport of the coarse sands [24,25,29,49]. The turbulence kinetic energy is most powerful when the near-bed velocity reaches the positive fluctuation peaks. At this moment, the sweeps of the high-speed belt prompt the coarse sands near the bed to undergo synchronous burst transport [17,27,43,49]. The static particles are stimulated to the burst-motion state when the instant velocity reaches the fluctuation peak, named the promoting mode or burst-moving mode of the coarse particles.

According to H. Einstein, the instant lift force caused by the turbulence burst follows a normal error distribution [2]. Therefore, the probability distribution of the ultimate states of the turbulence intensity, which causes coarse particles to be lifted from the bed, should also follow a normal error distribution and can be expressed by a probability integral. As a result, as the pulsation intensities are different (σ–3σ), the incipient motion probabilities of the different-size grains near the bed are also different.

When the mean velocity of the flow approaches the incipient motion velocity of the sand mixture (the corresponding mean diameter), the coarse particles exhibit only intermittent movement due to the mixture structure and the interaction among particles. A series of experiments indicated that more than 70% of the time, the instant velocity causing the coarse particles to move is greater than u_c + 2σ but only slightly greater than u_c + 3σ, as shown in Figure 10. Therefore, u_c + 2σ was selected as the incipient motion condition for coarse particles in accordance with the incipient motion probability standard [4,37].

The second mode is the collision mode. As the turbulence initiates the pulsation of the near-bed velocity to reach a positive peak, the fine particles obtain a relatively high instant momentum, as shown in Figure 15. Then, the momentum transfer occurs when these fine particles collide with the adjacent coarse particles randomly [18,26,29] and once the momentum of the coarse particles obtained from the momentum transfer of collisions is sufficient. The coarse sediments start to move; in other words, the coarse sediments are prompted to move by particle-collision events. Therefore, this mode wherein the static sediments are prompted into motion is called the collision mode, which is the main incipient form of coarse sediments. The physical process of this mode can be simplified, as demonstrated in Figure 15. The coarse sands and the adjacent fine sands on the bed are simplified with two sizes of balls. Particle collisions with a boundary friction effect occur when the small balls are triggered by the flow pulsation and the burst events of near-bed turbulence.

Before the collision, the momentum of the smaller ball is denoted with M₁, where M₁ = m_s∙v_s1, and the momentum of the large static ball is M₂, where M₂ = m_b∙0. When the flow velocity reaches the peak pulsation burst v_max, the moving velocity of the small ball also reaches its peak v_smax. At this moment, as the momentum acquired by the large ball is sufficient, i.e., greater than the friction impulse from the bed, the large ball moves. Considering that the two balls move synchronously after the collision, the momentum transfer between the large ball and the small ball can be presented by the following equation: m_sv_smax = m_sv_s2+ m_bv_b2. Moreover, experimental data show that when the grain size differential is less than four, the random collision of the fine particles caused by the coherent pulsation is another main mechanism that triggers the intermittent motion of the coarse particles. It is important to note that not all collision events can motivate the motion of coarse sediments. Only fine particle collisions with sufficient momentum can effectively prompt coarse sediments to move. When the instant flow pulsation intensity is less than 1.5 σ, the momentum supplied by the fine-particle collisions is insufficient to trigger the coarse particles; therefore, only fine sediments are transported on the bed. The coarse sediments fail to move without the sufficient energy supplied by the fine-particle collisions.

The third mode is the collapsing mode, the core mechanism of which is mainly related to exposition and destabilization. Its physical process is simplified and presented in Figure 16. At first, the coarse and fine particles on the static bed are generally surrounded by each other, and they are interdependent in structure [10,24], as shown in Figure 16a. Then, the flow gradually reaches a certain intensity and sweeps away the fine particles around the coarse particles (Figure 16b). Therefore, the coarse particles gradually lose the surrounding protection and support from the circumambient fine particles, and when the exposure of the coarse particles is sufficient (Figure 16c), the surrounding protective structure comprising the coarse particles and fine particles (SPSCF) collapses, and the coarse particles lose stability and undergo transport (Figure 16d). The occurrence of the collapsing mode is one of the main incipient motion forms of the bigger particles on the non-uniform sand bed. The process of momentum transfer and accumulation in coarse particles requires an ideal size differential. The experimental data indicate that the collapsing-mode movement frequently occurs, especially when the grain-size differential is generally bigger than 4 and smaller than 10.

The physical process of the collapsing mode involves the collapse of the surrounding protective structure comprising the coarse particles and fine particles and is one of the main mechanisms that trigger the intermittent movement of the coarse sediments. It is the releasing effect of the fine particles in SPSCF that causes the collapsing-mode movement of the coarse sediments. The collapse of the surrounding protective structure mainly results from the following points. First, when the fine particles are washed away, the coarse particles, buried or half-buried in the bed, are gradually exposed to flow. When the coarse particles are affected by the flow pulsation and collision from the moving particles with high energy, the frequencies, and probability of the coarse sediments starting to move are increased significantly. This means that burst-mode moving and collision-mode moving both effectively motivate the coarse particles’ incipient motion and can directly prompt the coarse particles to move. Moreover, some coarse particles in SPSCF are initially supported by the surrounding fine particles in the determinate static structure. As the fine particles leave, it is easy for the coarse particles to lose stability, especially since they are affected continuously by the pulsation of the flow intensity and the collisions of the passing particles. The collapse of a structure composed of sediment clusters always involves a local area of the bed rather than one or two individual particles. Therefore, the collapsing-mode movement increases bed load flux once it happens, which makes the flux of the non-uniform bed load even more random.

4.3. Distinct Motion Properties of the Coarse and Fine Particles near the Bed

The association of the bed load motion with the flow structure and the incipient motion patterns of the coarse sands is necessary to reveal the mechanism of the random flux of the bimodal bed load. During this process, the role of coarse and fine sands in the random transport of the bimodal bed load and their transport characteristics must be determined. When the flow intensity is high enough, the coarse and fine particles on the bed both turn to a state of motion, as shown in Figure 10. However, their transport property varies a lot. The transport of the fine bed load is consistent, even if the transport flux of the fine particles fluctuates slightly due to being affected by the turbulent fluctuations near the wall. This means there are always static fine sediments moving and transporting on the bed. The transport of the fine sediments supplies the basic component for the transport flux of the bimodal bed load, and they play a dominant role in the incipient motion and the transport progress of the bimodal bed load.

However, the transport flux of the coarse sand is intermittent. Affected by the near-bed flow turbulence structure and collision-mode and collapsing-mode effects of the fine particles, the transport of the coarse sands in the bed load mixture is intermittent, i.e., there are not always coarse particles involved in the bed load mixture transported. The selected ratio of the coarse sand in the observation section is intermittently zero and sometimes varies from 0.7 to 0.9. This indicates that although non-uniform coarse sediment transport rarely occurs and is intermittent, once coarse particles participate in the movement, they play a prominent role in the bed load mixture in terms of quantity. If the size difference of the sediment mixture is large enough, there are large-scale fluctuations in the transport flux of bed load, which is mainly due to the intermittent motion of the coarse sands. Both the response of the sediments to the flow-turbulence structure near the bed and random particle effects, including the collision mode and collapsing mode shown in Figure 16, lead to the intermittence of the bed load flux.

4.4. Influence of Bed-Material Mixture Ratio on the Bed Load Transport Rate

Under the flow conditions of Q = 0.026 m³/s and Fr = 0.85, the instant transport rates at several typical normalized times, such as T’ = 0.2, 0.35, and 0.5, were chosen for comparison, as shown in Figure 11. An interesting phenomenon was found. First, based on the data collected at different moments, all the curves of the instant bed load transport rate with the mixture ratio of coarse to fine grains generally presented a similar trend. Moreover, the relationship between the non-uniform bed load transport rate and the particle mixture ratio exhibited a humped curve, as shown in Figure 11, and the mixture ratio of the coarse to fine sands in the bed material influences the amount of the bed load transported.

In Figure 11, the flow condition is fixed. The mixture ratios of the bed material range between 0:10 and 7:3. The coarse-particle size is 6–8 mm, and the fine-particle size is 2–3 mm. Experimental data indicated that when the bed material mixture ratio is 10:0, i.e., the bed material is totally composed of coarse sands, there is no bed load movement. This means the current flow intensity is lower than the critical condition of the coarse sands’ incipient motion. However, when increasing the proportion of fine sand in the bed material, both fine and coarse sands start to move gradually on the bed. Therefore, the threshold value of the mixture ratio of bed material means a lot to the transport of the non-uniform bed load. Our research indicates that there is no bed load transport until the proportion of the fine sand is greater than 0.3, i.e., the mixture ratio of coarse sands to fine sands is 7:3, which equals 2.33 (see Figure 11). When the mixture ratio is 7:3, only a little fine sand is transported. The curves indicate that as the mixture ratio of coarse sands to fine sands decreases from 10:0 to 7:3 or even less, i.e., 6:4, 5:5, 4:6, etc., the bed load transport rate increases significantly. This is mainly attributed to the coarse sands starting to move intermittently, which is due to three factors, i.e., the near-wall burst sweep effect and the particle triggering effect of the collision and collapsing modes, as shown in Figure 10, Figure 15 and Figure 16, respectively.

As the mixture ratio of the bed material decreases, e.g., the proportion of the fine sand in the bed material mixture increases, the transport rates do not continue to increase, as shown in Figure 11. When the mixture ratio is close to 3:7, the total transport intensity of the bed load increases extremely. However, the increasing gradient of the bed load proportion is reduced when the mixture ratio decreases. The related data are shown in

Table 3. The parameters of bed load under various mixture ratios conditions.

η	Φc	φ
7:3	0.3623	0.1097
6:4	0.6549	0.4874
5:5	0.6866	0.6080
4:6	0.8932	0.6999
3:7	1.0000	0.7791
2:8	0.4624	0.8146
1:9	0.3623	0.8706
0:10	0.0000	1.0000
Flow condition:	Q = 0.026m3/s, Fr = 0.85

. When the fine-sand proportion in the bed material is large enough, the mixture ratio reaches 3:7, i.e., 0.43, and the total transport rate of the bed load reaches its peak. Then, it decreases as the proportion of fine sand increases continuously. The whole curve presents a pattern of a hump (see Figure 11, Figure 12 and Figure 13). However, the increase in the proportion of fine particles has a different influence on the transport rates of the coarse and fine particles, as shown in Table 3. This indicates that the role played by fine particles in non-uniform sediment transport is critical and complicated.

A comparison of the experimental results under different flow-intensity conditions indicates that both the average and maximal value of the sediment-transport rates have a humped curve with respect to the bed-material mixture ratio of coarse to fine, as shown in Figure 12 and Figure 13. The non-uniform bed load transport property remains when the flow intensity is relatively low. However, the mixture ratio to which the humped-curve peak corresponds may vary from 3:7 to 2:8 according to the boundary conditions, as shown in Figure 14. Coincidentally, a similar characteristic was also observed when the coarse-sand transport intensities were analyzed. Figure 14 indicates that the accumulation of coarse sands is not only closely related to the flow discharge but also affected by the mixture ratio, and it presents a humped-curve relationship with a decrease in the mixture ratio. The corresponding mixture ratio of the peak is about 3:7, which depends on the size differential between the coarse and fine particles. The single peak curve property for coarse-sand transport causes the humped curve in the total bed load transport rate, as shown in Figure 11, Figure 12 and Figure 13, and the two-humped curves correspond to each other. By comparing Figure 14 and Figure 17, it is found that when the coarse-sand transport rates decrease sharply, the total transport rates also decline rapidly, although the fine transport rates increase, which indicates the influence of the reduced proportion of coarse sands.

According to the experimental result, under the same flow condition, the transport flux of a bimodal non-uniform bed load can be optimal as long as the size differential between the coarse and fine particles is significant and the mixture ratio of bed material is appropriate. The peak transport rate is larger than that of uniform bed material composed of pure fine sediments or coarse sediments. This indicates that the non-uniformity of the particles plays an important role in bed load transport.

4.5. Effects of Different Particle Sizes on the Non-Uniform Bed Load Transport

The relationships of the transport rate of fine sands, coarse sands, and whole bed load sands with the bed-material composition were analyzed under a certain flow condition, as shown in Figure 17. The results indicate that the transport rate of the fine sands, coarse sands, and whole sands present totally different trends when the mixture ratio of the bed material changes. However, the three curves are still correlated. The fine-sand accumulation increases continually as long as the flow intensity satisfies the incipient-motion condition of the fine sands and reaches the maximum value when the mixture ratio is 0:10. The fine sands play a leading role in the progress of the bed load transport. In contrast, the coarse sands do not transport until the proportion of fine sands reaches 30%, i.e., the mixture ratio is 7:3, when the accumulative effect of the fine sands is significant enough to prompt the coarse sands to move. The coarse sand plays a secondary and intermittent role in the transport of the bimodal bed load mixture, as shown in Figure 10.

The frequency of the collision and the releasing effect of the fine sands increases with the addition of the fine-sand proportion, i.e., as the mixture ratio decreases. This causes the area affected by the collapsing mode to gradually become enlarged. The coarse-sand transport rate presents an increasing trend, and the coarse-sand selected ratio in the bed load mixture also becomes considerable. The amounts of the coarse-sized bed load and the total bed load are both maximized when the mixture ratio is 3:7. Then, limited by the shortage of coarse sediment supply in the bed material, the coarse-sand transport rate declines gradually. Affected by the special transport properties of the fine and coarse particles and the significant size differential, the total bed load transport rate presents a humped curve as the mixture ratio changes, similar to coarse sands. The total transport rate is also maximized at the mixture ratio of 3:7. Then, influenced by the rapid decline in the coarse-sand transport rate, the total bed load transport rate also decreases and cannot surpass the peak value in the mixture ratio of 3:7, even if the fine-sediment transport rate still increases continuously because large-sized coarse sediment is weighty and dominant to the amount of the transported sands. It is important to note that, as shown in Figure 17, the total bed load increases again when the coarse-sand proportion is less than 15% and reaches a secondary peak when the coarse-sand proportion is 0%. There may be two reasons for this. First, when the coarse-sand proportion is small, the frequency of the collapsing mode is small or zero. The collapsing mode of SPSCF contributes significantly to the coarse-sand transport in the non-uniform sediment and affects the total transport rate of the non-uniform sediment. Moreover, the hidden effect of the coarse sands on the fine sands plays a dominant role in particle interactions and has a significant negative effect on the total transport rate.

In bimodal bed load transport, the releasing and motivating effects of the fine sand on the coarse sands are important. These effects make the coarse sediments supernormally active and easily move during incipient motion and transport progress. The more active the coarse sediment, the greater the amount of transported coarse sediments. Then, the proportion of the coarse sediments in the total bed load transported is also increased. The proportion of the coarse sands is biggest when the mixture ratio reaches the appropriate value η_c. That is why the hump curve emerges, and it is also a critical point to understand when considering the transportation property of a non-uniform bed load. Former research has indicated that sand content has a positive effect on the gravel transport rate [24,26]. However, their experimental data was incomplete and failed to present how non-uniformity and fine sediments affect the transport rate of the bed load, i.e., the humped-curve trend presented in our research.

5. Conclusions

The data of the instant transport rate and size distribution of the bimodal bed load were obtained using STMT and an image-recognition system. The transport property of the bimodal bed load was investigated, and we found that its mechanism was related to near-bed flow intensity turbulence, the non-uniformity of bed material, and particle-interaction effects. The amount of the bed load transported depended on the near-bed flow-fluctuation characteristic. Affected by coherent flow fluctuations and the fine-sand motivation caused by the collision effect and releasing effect, the coarse sands were transported intermittently on the bed. The mechanisms of the intermittent transport of the coarse sands were attributed to three modes of incipient motion. The factors influencing the three modes of particle incipient motion were different. The prompting mode was associated with the flow-fluctuation structure and the bed form. However, the collision mode and the collapsing mode were both mainly related to fine sand, size differential, and the proportion of fine sand in the bed material, as these factors affect the exposure of the coarse particles and the collision frequency of the fine particles. It was found that when the instant flow-fluctuation peak was greater than 2.5 σ, the coarse sands tended to incipient motion at a high probability. The criterion of the fluctuation intensity proposed above is relevant to the flow intensity and the bed material composition.

The non-uniformity factors of the bimodal particle, including the mixture ratio, size differential, and generalized equivalent diameter, have a significant influence on the bed load transport. The transport rates of fine sands, coarse sands, and total sands present different trends as the mixture ratio decreases. Fine sediments play a leading role in non-uniform transport. The role of coarse sands is intermittent and critical. Improving the probability and frequency of the collisions of fine sands helps to increase the collapsing area and prompt more coarse sands to move, which finally and significantly improves the transport rates of the coarse sands and total sands. For the bimodal bed load with an obvious size differential, the transport rates present a humped curve as the mixture ratio decreases. With the proper mixture ratio η_c, the transport rate of the bed load can reach its maximum. The transport rate of the bimodal non-uniform bed load is larger than that of uniform sediments comprising pure coarse or sand particles. The effect between particles is complicated and will likely reveal more details of particle collision in future work, including nearby flow fields, mechanics, momentum, and energy transfer in quantity. More accurate measurement instruments will be necessary for future work. The findings above may be valuable as reference data for improving the theory regarding non-uniform sediment motion. Offshore sediment transport plays an important role in coastal evolution, and the non-uniform bed load transport property presented in this paper may also benefit relevant research.