2.8　A bed-load function based on kinetic theory(10)

Zhong Deyu, Wang Guangqian, Zhang Lei

Abstract: Bed-load transport plays a fundamental role in morphological processes of natural rivers and human-made channels. This paper presents bed-load function derived on the basis of kinetic theory. The bed-load function is obtained by integrating the pick-up rate of bed sediment with respect to its longitudinal travel distance, following the basic concept on bed-load put forward by Einstein. The pick-up rate is expressed as an upwards flux of bed sediment and determined by invoking the particle velocity distribution function derived by solving the Boltzmann equation of kinetic theory. Comparisons of the present formula with six other bed-load formulas and the experimental data are also made in this paper. The results show that the present bed-load formula agrees well with the experimental data. In addition, the influences due to collision between particles on bed-load is discussed which shows that an appreciable damping of the intensities of bed-load is observed only when the shear acting on particles is large enough to increase the concentration and intensity of random motion of bed load particles so that the collisions between sediment particles can occur.

2.8.1　Introduction

Bed-load refers to the sediment particles moving on a river bed by sliding, rolling, or saltation. The mechanism involved in bed-load transport is essential for obtaining a better understanding of morphological processes in rivers and man-made fluvial channels. Knowledge of bed-load is also substantially important to river engineering. Prevention of bridge scour, exclusion of sediment from canal headworks, and improvement of waterways for navigation need detailed information on bed-load transport. As a result, bed-load transport has long been an essential topic investigated in fluvial hydraulics.

A reliable bed-load formula is necessary for the quantitative determination of solid discharge. A large number of bed-load formulas have been established. Those formulas are established based on quite different theories due to the absence of a universally recognized theoretical approach in studying bed-load transport. Moreover, theoretical analyses can only be performed under simple or simplified cases due to the complexity of bed-load transport processes. Therefore, allexisting bed-load formulas invariably depend on calibration using data from laboratory experiments or field observations. Consequently, the formulas are applied to the conditions under which they are established, and appreciable discrepancies between formulas are observed.

Well-known formulas include those suggested by Meyer-Peter and Mueller (1948), Einstein (1950), Yalin (1963), Bagnold (1973), Engelund and Fredsoe (1976), among others. Detailed reviews of these formulas can be found in literature, such as those by Chien and Wan (1999) and Wang et al. (2001). Other formulas, such as those presented by Parker et al. (1982), van Rijn (1984a), Wang and Zhang (1995), and Karim (1998), are also widely adopted in predicting bed-load discharge. Recently, a wide variety of problems associated with bed-load transport have been extensively examined. For instance, stochastic properties of bed-load transport were studied by Sun and Donahue (2000), Kleinhans and van Rijn (2002), and also Duan and Barkdoll (2008). The influence on bedload transport due to sediment-supply conditions of river basins was investigated by Bravo-Espinosa et al. (2003). The effect of the non-uniformity of bed material on bed-load transport was discussed by Wu et at. (2004). The disturbance depth of gravel river lied due to bed-load transport was studied by De Vries (2002). A probabilistic approach was developed to model the composition of bed-load by McEwan et al. (2004). The probabilistic properties of entrainment of bed sediment and the travel distance of bed-load particles is theoretically investigated by Hunt (1999) and Hunt et al. (2006). Formula for bed-load transported under the circumstances of oscillatory flows was proposed by van Rijn (2007) by introducing a concept of instantaneous shear streets.

Previous studies show that the differences between bed-load and suspended load are apparent. On the one hand, turbulence in the proximity of a river bed affects the motion of particles to a considerable extent；therefore the particles transported as bed-load possess a remarkable degree of randomness. The length, height, and velocity of saltation particles, considered to constitute the main body of bed-load (Yalin, 1972；van Rijn, 1984a, 1986；Wang and Zhang, 1987, 1995), are all random variables (Hu and Hui, 1996a, b). On the other hand, bed-load particles are relatively large in size compared with suspended particles；therefore, hydrodynamic effects arising from average flows together with gravitational force play a predominant role in bed-load transport. Accordingly, a distinct path of a bed-load particle in its successive saltations can be observed in most situations. Owing to these two important differences, the theory applied to bed-load transport is expected to be remarkably different from that applied to suspended load transport.

Researchers had made great efforts to seek a suitable theory to characterize bed-load transport. Two representative theories were presented in literature. One was developed by Einstein (1950) on the basis of statistical theory, and the other was proposed by Bagnold (1973) based on energy conservation principle. Besides these two theories prevailing in the studies of bed-load, the Lagrangian method is also popular, in which the equations of motion of an individual sediment particle are solved to obtain thetransport parameters associated with bed-load (e.g., Yalin, 1972；van Rijn, 1984a；Wang and Zhang, 1987, 1995). Among these theories, of particular importance is the one proposed by Einstein (1950), which has a profound influence on later research due to its elaborate probabilistic consideration. However, the complicated processes involved in bed-load transport result in a few less-rigorous premises in his theory that may be improved further. For instance, Einstein (1950) assumed that the ratio of particle travel distance to particle diameter is a constant；however, both experimental observations and theoretical studies offer no support to this assumption (Yalin, 1972).

Several modifications to the Einstein bed-load function have been made (Yalin, 1972；Wang and Zhang, 1985；Sun and Zhu, 1991；and most recently, Wang et al., 2008). Although these modifications result in notable improvement to Einstein's bed-load function, some issues remain open. The most significant one is how to determine the pick-up rate of bed sediment, a key parameter involved in deriving bed-load formula. In the majority of the reported formulations following Einstein (1950) and subsequent modifications (Yalin, 1972；Wang and Zhang, 1985；Wang et al., 2008), the pick-up rate is formulated semi-empirically or empirically by introducing a parameter referred to as “exchange time”, which is thought to be one of the main reasons responsible for the errors in predicting sediment discharge.

In fact, the aforementioned problems in previous studies arise largely from the fact that the statistical theory of Einstein (1950) fails to fully consider the dynamic properties of moving particles and carrier fluid in his statistical description of bed-load transport. Taking this fact into consideration, the theory applied to bed-load is expected to possess following features：firstly, it can provide a microscopic description of the motion of bed-load particles, in which the forces acting on particles as well as the effects due to turbulence of carrier fluid are fully included. Secondly, it can provide a macroscopic statistical description of bed-load transport, in which the statistical properties, for example, the bed-load discharge, can be obtained. Fortunately, previous investigation shows that kinetic theory meets the expectation. Actually, kinetic theory has been widely adopted in studying multiphase flows widely found in industries. It is believed that kinetic theory is more suitable to study multiphase flows compared with other theories (see detailed review by Gidaspow, 1994). By introducing a particle velocity distribution function and its governing equation, i.e. the Boltzmann equation, it is natural and convenient in kinetic theory to obtain the statistical properties of solid particles moving in turbulent flows. Moreover, as the motion equations of particles are also employed as basic governing equations, the combined effects of external forces and turbulence of carrier fluid on the motion of particles can be fully considered. Kinetic theory has been employed by Wang and Ni (1990, 1991), Ni et al. (2000), Zhong et al. (2001), and Zhong and Zhang (2006) to explore suspended load transport, and the results are found to be reasonable compared with observations.

In this paper kinetic theory is used to study bed-load transport. Instead of a comprehensive examination of every aspect of bed-load, we focus our attention exclusively on establishing a bed-load formula with a solid theoretical foundation. The rest of this paper is organized as follows. First, following the fundamental concept of Einstein (1950) on bed-load transport, a theoretical expression for the transport rate of bed-load is formulated based on kinetic theory. Second, the parameters associated with the newly derived bed-load formula are determined. Finally, the results given by the present formula are compared with those given by six well-known formulas and laboratory data sets, and a brief discussion on the effect on bed-load due to collisions between particles is presented.

2.8.2　Theory

1. Basic relation

Consider a steady and uniform two-dimensional turbulent open channel flow passing over an erodible bed composed of non-cohesive uniform spherical particles (Fig. 1). The particles start to move when the shear stress acting on them exceeds a certain threshold. The moving sediment is referred to as bed-load if it consists of those particles moving on or near the river bed by sliding, rolling, or saltation. According to Einstein (1950), the transport rate of bed-load is the quantity of sediment particles picked up from the upstream bed surface passing through the cross section I-I per unit time (Fig. 1). It is usually interpreted as follows (Einstein, 1950；Yalin, 1972；van Rijn, 1986).

where qb=volumetric transport rate of bed-load per unit width；E=pick-up rate of bed material；λ=travel distance of bed-load particles, which is often identified by the saltation length of sediment particles (Yalin, 1972；van Rijn, 1986). In Einstein's (1950) derivation, the travel distance λ of a bed-load particle is considered to be deterministic and proportional to its diameter. However, laboratory observations (Hu and Hui, 1996a, b) show that λ is significantly affected by the turbulence of the carrier fluid, resulting in a high degree of uncertainty. Apparently, the assumption by Einstein (1950) is not compatible with the experimental observations of bed-load transport. To improve Einstein's derivation, Wang and Zhang (1987) suggested a different version of Eg. (1) to include the effects of the randomness of λ through introducing a probability distribution of travel distance. They obtained

where W=weight of sediment particles passing through the section I-I (Fig. 1)；T=exchange time, and PL=probability distribution of the travel distance.

We also employ Eq. (1) as a starting point to develop a theoretical relation for the bed-load transport rate, but it is re-formulated in line with kinetic theory in this paper with the consideration of the randomness of saltation particles. In the following subsections, first, the pick-up rate E, the most important parameter, is determined analytically by invoking kinetic theory (Zhong et al., 2010), rather than a semi-empirical or empirical relationship (van Rijn, 1984b)；then, an analytical expression for bed-load transport is obtained under the framework of kinetic theory.

2. Pick-up rate E

The rate at which bed sediment particles are picked up to perform suspension was investigated analytical recently by Zhong et at. (2010) using kinetic theory. In this paper, the pick-up rate of bed sediment performing saltation is also obtained by employing kinetic theory.

For simplicity, the river bed is assumed to be flat. A Cartesian coordinate system is used, of which the origin is located on the bed surface, the x-axis is directed along the main stream, and the y-axis is perpendicular to the river bed and directed upward (as shown in Fig. 1). In kinetic theory, if f (v, r, t) denotes the particle velocity distribution function in phase space (v, r, t), then f (v, r, t) dvdr represents the number of particles with the velocity v found in a small control volume dr at the position r. It should be noticed that v and r are independent of each other in phase space. Figure 1 shows an individual sediment particle being picked up from the bed surface and transported downstream from point r0 to r0, with a travel distance λ and a saltation height δ. In kinetic theory, the rate at which the sediment particles are picked-up from the point r0 at t0 is written in terms of upwards flux as follows：

For an arbitrary flow intensity, a sediment particle with a given diameter picked up from the bed surface can be transported either as bed-load or as suspended load. Because our attention is focused on bed-load transport, a restriction should be imposed on Eg. (3) to exclude the bed sediment entrained into suspension. According to Einstein (1950), bed-load consists of the particles which can be picked up from upstream bed surface and at the same time are carried downstream within bed-load layer. Consequently, we introduce a joint probability density distribution function f (v, r0, t0；r1, t1) in phase space (v, r, t) to specify those bed-load particles being picking-up at point r0 at time t0 and passing through point r1 at time t1 (Fig. 1). Thus the pick-up rate corresponding to the bed-load particles can be expressed as

f (v0, r0, t0；r1, t1) also can be exprcssed in terms of conditional probabilities as follows

Where f (r0, t0|r1, t1)=probability density function that a particle is at r0 at t0, given that it is at r1 at t1. This probability density function can also be considered as the probability of transition from state (r0, t0) to state (r1, t1).

3. Velocity distribution functionf (v, r, t) of sediment particles

The key to deriving the pick-up rate is to find the particle velocity distribution function f (v, r, t). In kinetic theory, the variation of f (v, r, t) in the phase space (v, r, t) is given by the Boltzmann equation. It is expressed as below：

where m=ρpπD3/6=mass of a sediment particle；∑F=resultant of all external forces acting on a sediment particle； (∂f/∂t) coll=collision effect on the transport of velocity distribution function. Since Eq. (6) is a highly nonlinear partial differential equation, especially due to the term on the right hand side implying collisions among particles, only asymptotic solutions in some simple cases can be obtained. However, for a steady and uniform two-dimensional flow in an open channel, if the concentration of bed-load is not so high that the collisions among particles are negligible, the solution to Eq. (6) can be determined as follows (Wang and Ni, 1990, 1991；Ni et al., 2000；Zhong and Zhang, 2006)：

where n0=number density of sediment patricles at the reference point r=r0；, with Ci=vi-Vi (i=x, y, and z), in which vi and Vi are respectively the instantancous and ensemble-averaged velocities of sediment particles in the i-direction；K=/3=turbulence intensity of sediment particles, in which the Einstein repeated-suffix summation convention is used. The potential function ϕ (r) in Eq. (7) is

where Fx, Fy=components of F in the x-and y-direction, respectively. Because a uniform flow in the main stream direction, i.c., x-direction, isassumed, the integration ∫∑Fx/mdx is a constant. Without loss of generality, the integration is set to 0.

Equation (7) implies that the velocity distribution function of sediment particles carried by turbulent flows consists of two parts：one corresponding to the random turbulence field, and the other the external force field. Once the two fields are found, the velocity distribution function can be determined. In this section, the external force field is studied, whereas the turbulence field represented by the turbulence intensity K is discussed in a later section.

Among the forces acting on bed-load sediment particles perpendicular to the river bed, gravitational force, drag force, lift force, and buoyant force are significant to the motion of bed-load (Yalin, 1972). In this paper, ∑Fy is written as

where CD and CL=drag and lift cocfficients, respectively；u∗=shear velocity. In Eq. (9), the expression for the lift forcc proposed by Sumer (1984) was adopted. Substituting Eq. (9) into Eq. (8) lcads to

By substituting Eq. (10) into Eq. (7), we obtain the velocity distribution function as

In the integration of Eq. (10) with respect to y, it is assumed that the lift coefficient CL is constant. In fact, reported studies on CL (e.g., Yalin, 1972；Sumer, 1984) show that it varies with y. But the present state of knowledge does not permit a reliable analytical expression for CL, and therefore it is simply considered as a constant in performing the integration in Eq. (10).

4. Transition probability densityf (r0, t0|r1, t1)

Determination of the transition probability density f (r0, t0|r1, t1) of a sediment particle moving in turbulent flows remains extremely challenging. But for the present specific situation, we infer that f (r0, t0|r1, t1) is the conditional probability of a sediment particle that is picked up by a river flow from a possible position r0 at time t0, given that it is transported downstream passing through point r1 in cross section I-I at time t1 as bed-load. Basically, the larger the distance from r0 to r1, the smaller the possibility of a sediment particle picked-up at r0 to arrive at r1 in a single step when the particle moves as bed-load (Einstein, 1950). As a first approximation, in this paper we assume that f (r0, t0|r1, t1) has an exponential distribution as follows：

where L=longitudinal characteristic length of bed-load transport. In fact, if L is considered identical to thetravel distance λ of bed load particles, Eq. (12) is simply the probability density distribution function of the travel distance suggested by Nakagawa and Tsujimoto (1986) and Wang and Zhang (1987).

5. Theoretical bed-load formula

From the fundament relation between bed-load discharge and the entrainment rate of bed sediment[Eq. (2)], we found that the average flux passing through cross section I-I has the form as F=Eλ/H, where H is the characteristic height of bed-load, and thus we deduced from the relation that the rate at which bed-load particles possibly pass through a small interval of Dy1 at the point r1 as follows：

The integration with respect to the velocities is independent of x0；therefore, Eq. (13) can be rewritten as follows：

Because the integra (x1-x0)·f (r0, t0|r1, t1) dx0 is the mathematical expectation of the travel distance of bed-load particles, one can easily find that Eq. (14) is formally identical with Eq. (1). By substituting Eqs. (11) and (12) into Eq. (14), we arrived at

where Cm=n0πD3/6=volumetric concentration of movable sediment particles at the reference level y1=0, which is equal to pc0. The parameter p is the incipient probability of sediment particles resting on the surface of the river bed, and c0 is the volumetric concentration of bed material, usually equal to 0.6.θ=ρfu2∗/[gD (ρp-ρf)], the Shields parameter, and θL=4/(3CL). In the derivation of Eq. (15), the equality vy=Vy+Cy=0+Cy is used, which implies that the ensemble averaged vertical velocity of sediment particles is zero.

Integration of Eq. (15) with respect to y1 gives the bed-load discharge of per unit width as follows：

where Γ(v, z) =tv-1e-t dt is the incomplete Gamma function. The parameters f1 and f2are respectively expressed as

2.8.3　Determination of related parameters

1. Characteristic height and length of bed-load transport

It has been commonly accepted that saltation is the main form of bed-load transport (e.g., Yalin, 1972；van Rijn, 1984a, 1985；Wang and Zhang, 1995). We also base our derivation on this conclusion, and therefore the characteristic height H and length L are identified with the average saltation height δ and length λ of jumping particles, respectively.

Many studies on the saltation of bed-load particles have been reported (e.g., Yalin, 1972；van Rijn, 1984a；Wang and Zhang, 1987；Hu and Hui, 1996a, b). Most of the existing relationships proposed for δ and λ areempirical, except a few analytical investigations. In this study, the empirical relations for δ and λ suggested by Wang and Zhang (1995) obtained based on experimental observations which cover a wide range of flow intensities and particles sizes are used for the evaluation of δ and λ. They are

and

Where m1=0.74+0.1 (ρp/ρf)0.37θ-0.57 and m2=0.25+0.1 (ρp/ρf)0.20θ-0.50. But it is observed that Eq. (18) provides better agreement with experimental data when it is applied to calculate bed-load discharge by Eq. (16), if its coefficient of 180 is modified to 81.82.

2. Turbulence intensityKof bed-load particles

The turbulence intensity K must be determined before implementing Eq. (16). Theoretically, K is expected to be obtained through solving the conservation equations for turbulence energy of both solid and liquid phases derived in kinetic theory. However, numerical solutions are invariably associated with specific boundary conditions, and thus they are usually impossible to be generalized. To avoid this limitation, we follow Wang and Ni (1991) and assume that , where α is a dimensionless proportional coefficient.

Then, by substituting into Eq. (16) we obtained the bed-load transport rate as follows：

where

θL is a variable associated with the lift coefficient CL. As the lift coefficient deceases sharply to zero when the distance of a particle to the bed surface increases (Yalin, 1972), many researchers consider that the lift force only contributes to the detachment of a sediment particle from bed surface. Accordingly, θL=4/3 (3CL)≫1 holds for most part of the bed-load layer, and it is reasonably to deduce that 1/θL, in comparison with 1/θ, has a very small magnitude. Hence, in Eq. (21) the effects stemming from 1/θL become inconsequential and thus they are neglected in this paper.

3. Incipient probability p

As one of the most important criteria in the mechanics of sediment transport, the initiation of sediment transport has been extensively studied. Einstein (1950) started the statistical approach in the studies of the incipient motion of sediment particles. Although it suffers from some drawbacks (Yalin, 1972；Wang, 1985；Sun and Zhu, 1991；Wang et al., 2008), the approach is important due to its creative idea on the detachment of bed sediment. Modifications have been made to Einstein's incipient probability p by many researchers (Yalin, 1972；Wang, 1985；Hu and Hui, 1995；Wang et al., 2008). We adopt the incipient probability suggested by Yalin (1972) for a stationary rough turbulent flow as follows：

where η0=0.5 and B∗=0.143 suggested by Einstein (1950).

2.8.4　Comparisons with previous formulas and experimental data

For the convenience of comparing the present formula with existing bed-load formulas, we divide both sides of Eq. (20) by and obtain

where Φ=dimensionless transport rate of bed-load suggested by Einstein (1950).

In this paper, the experimental data obtained by Gilbert (1914), Meyer-Peter and Mueller (1948), and Wilson (1966) with specific densities of 2.67～2.68 are used for the purpose of verification, and six formulas developed respectively by Meyer-Peter and Mueller (1948), Einstein (1950), Yalin (1963), Engelund and Fredsoe (1976), Wang and Zhang (1995), and Cheng (2002) are selected for comparison.

Before Eq. (23) can be applied to calculate the bed-load transport rate, the proportional constant α is the only empirical coefficient remaining to be determined. According to the experiments on bed-load transport conducted in a flume and observed via the 3D PTV technique by Tang et al. (2006), the turbulence intensity of bed-load particles has a close relation to the shear velocity of the carrier flow. The proportional coefficient α was found to be in the range of 0.91 to 2.23. We found that when α=0.85, Eq. (23) produces the least deviation from the experimental data.

Figure 2 compares Eq. (23) to other formulas and experimental data sets. It can be found that the transport rate of bed-toad calculated by Eq. (23) is in good agreement with experiment data for both moderate and intensive sediment transport. Figure 3 is a box chart which depicts the distribution of relative errors by different formulas. It shows that the relative errors produced by the formulas proposed respectively by Yalin (1972), Cheng (2002) and Eq. (23) are more likely to be normal distributions. It also indicates that the exponential formula of Cheng (2002) and Eq. (23) obtained presently have comparatively higher accuracy than the others, while Yalin's formula has the minimum scattering, so far as the experimental data used in this paper are considered.

The application of Eq. (23) to weak sediment transport is shown in Fig. 4. The results based on the exponential formula of Cheng (2002) are also plotted in this figure. It is clear that satisfactory accuracy can also be achieved when Eq. (23) is applied to the situation of weak sediment transport observed by Paintal (1971) and Taylor and Vanoni (1972). It is interesting that the results given by Eq. (23) are very close to that given by the exponential formula of Cheng (2002), except for the cases of very weak sediment transport (Fig. 4). In fact, it can be verified that the present formula is indeed an “exponential formula” (Eq. (23）)；therefore, the similarity in accuracy between these two equations is only natural.

It can be found from Eq. (23) that the dimensionless bed-load transport rate also explicitly depends on the specific density ρp/ρf, as has also been reported by Wang and Zhang (1990) and Hu and Hui (1995). In Fig. 5, the observed results of transport rate of sediment with specific densities of 1.138 and 1.26 are plotted. Evident deviations in Ψ-Φ relation can be observed due to the different specific densities. The lines produced by Eq. (23) for different specific densities are also drawn in Fig. 5, which are in line with the observed tendency that Φ is in direct proportion to the specific density for the same flow intensity Ψ. It should be pointed out that only two sets of experimental data with different specific densities are collected in this paper；therefore, the comparison shown in Fig. 5 is only a preliminary validation of the relationship between Φ and ρp/ρf.

2.8.5　Discussion on the influence due to collisions between particles

Collisions between particles can be an agent that imposes impacts on the transport rate of bed-load. First, the velocity distribution function has been proved to be influenced by collisions between particles. Second, two key parameters involved in present sandy, i.e., the saltation height and the length of bed-load particles are expected affected by the collisions. Because these two parameters used in this paper are empirically determined by Wang and Zhang (1995) by experimental observations, which have included the influences from the collisions among particles, the discussion presented in this paper is solely limited to the influences on velocity distribution and in turn its effects on bed-load transport.

In kinetic theory, the collision term can be approximated on the analogy of the method in the kinetic theory of gas molecules used by Ni et al. (2000), which is

where f (0) (v, r, t) is the zero order velocity distribution function, i.e., Eq. (11)；τ is relaxation factor with dimension of time which can be expressed as (Ni et al., 2000)

where l is the characteristic length and is estimated by the average free travelling length of particles (Gidaspow, 1994). Substituting the collision term [Eq. (24)] into Eq. (6), we finally obtained the velocity function as follows：

where

Substituting the new velocity distribution function instead of the zero order one into Eq. (14), we obtained the modified transport rate of bed load by considering collisional effect on the velocity distribution function. It is

where ϕ is a correction coefficient accounting for collisions among particles which equals

with

Figure 6 shows the comparison of the bed-load discharge with and without the consideration of the collisions. It can be found that it is advisable to neglect the effect of the collisions for weak and moderate sediment transport；whereas the collisional effect becomes apparent when bed load is in comparatively intensive transport. The decrease of the transport rate due to collisions (Fig. 6) is expected because the collisions between particles serve as an additional friction force which damps the motion intensity of bed load. It should be pointed out that only the collisions between particles are discussed here；the collisions occur between bed-load particles and bed surfaces are remained for further study.

2.8.6　Conclusions

In this paper the kinetic theory was employed to study the bed-load transport. Specifically, an analytical formula for bed-load transport was derived based on the kinetic theory. This formula is obtained by integrating the pick-up rate of bed sediment with respect to its longitudinal travel distance, following the basic concept on bedload put forward by Einstein (1950). The pickup rate, which is a key element in deriving the bed-load formula, is expressed as an upwards flux of bed sediment and determined by invoking the particle velocity distribution function. Under the assumption that the open channel flow is steady and uniform twodimensional, the particle velocity distribution function is derived by solving the Boltzmann equation of kinetic theory.

Comparisons of the present formula on bedload transport with six other bed-load formulas and the experimental data are also made in this paper. The results show that the present bed-load formula agrees well with the experimental data. It can be inferred that kinetic theory can achieve notable success while applied to bedload transport.

A simple discussion with respect to the roles played by the collisions between sediment particles in the bed load was presented by including the collision term in the Boltzmann equation in terms of a simple collision model. It showed that the collisions have an appreciable effect that damp the intensities of bed load is observed only when the shear acting on particles is large enough to increase the concentration and intensity of random motion of bed load particles so that the collisions between sediment particles can occur.

Admittedly, two problems remain unsolved in this paper. Firstly, the current study is limited to uniform sediment particles. Therefore, complicated hiding-exposure effects caused by non-uniformity of bed sediment (Parke et al., 1982；Wang and Zhang, 1995； Sun and Donahue, 2002；Kleinhans and van Rijn, 2002) are not involved in this formulation, and fractional transport rate of bed-load cannot be predicted by Eq. (23). Secondly, we limit our attention only to developing a bed-load formula, whereas other less important aspects associated with bed-load are not included in this paper.

Acknowledgements

The authors wish to express their most sincere appreciation to the anonymous reviewers for their insightful suggestions which lead us to further discussion on the roles played by collisions between particles in bed load. Financial support from the Natural Science Foundation of China (NSFC, Grant No. 51039004), the National Key Technologies R & D Program of China during the 12th Five-Year Plan Period (No. 2012BAB05B01), and the research funding from State Key Laboratory of Hydroscience and Engineering (Grant No. sklhse-2008-B-01) is also gratefully acknowledged.

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