2.1　A new equation of bedload transport⁽²⁾

Wang Shiqiang, Zhang Ren

Abstract: A set of new equations of bedload transport are presented in this paper. Hydrodynamic and stochastic analyses have been adopted to formulate the model. The saltation equations of grain motion are derived. The equation of bedload transport rate gives a fine agreement both with the experimental data of sand and nylon in flumes and pipes.

2.1.1　Introduction

Numbers of bedload transport formulae have been proposed by various authors since Du Boys. Among those, the works of H. A. Einstein (1942), Meyer-Peter and Müller (1948), R. A. Bagnold (1966), M. S. Yalin (1972), F. Engelund and J. Fredsφe (1976), H. Nakagawa and T. Tsujimoto (1986) merit more attention. So far, there are a number of divergence amony those theories. For instance, the calculated results from them are quite different with each other particularly for high transport rate. The results from Einstein's equation only give a good agreement with nylon grain data while the results from Bagnold's or Yalin's equation only agree well with sand data for high flow condition in the pipe by Wilson (1966). The difference between them is more than 100% (see Fig. 5). Some important problems in the processes of formulating the models still need further effort on research work. They have been discussed in this article.

2.1.2　The model of bedload transport

In this model, the bed meterial is suggested as uniform particles. The bed surface is treated as a flat bed and only grain resistance is considered. The saltation is taken into account because more attention is paid on higher flow condition. Both drag force and lift force are considered in the model, but the probability of detachment of the partiels from bed surface depends only on the lift force. According to experiment results, the probability density distribution of lift force follows the normal error law and that of jump length is taken as exponentisl distribution suggested by Nakagawa and Tsujimoto (1986).

in which, k_∗=k/k₀, k is variability factor of lift, k₀ is standard deviation of k, L is the jump length, is the average Jump length.

The bedload transport rate in unit width at the cross section (x=0) equals the weight of the grains passing through the section in unit time. Taking an infinitesimal reach dx with unit width at x, the weight of grains picked up from bed surface in the area dW=A₀r_sDPdx, where D is the diameter of grains, r_s is the unit weight of grain, P is the probability of detachment of grains from bed surface.

Considering that only those grains with a jump length more than x could pass through section x=0, their weight dW_x=d WP_∗, where P_∗ is the probability of the jump length. It is determined by

those grains are picked up from bedsurface in a time duration of T_∗ which is called as the exchange time in Einstein's formula. Taking account of the effect of the surrounding particles on the grain going to be picked up, T_∗ should be expressed as

in which u_b is the initial velocity or the velocity of grain at section x=0. The coefficient A_∗ must be larger than unity. Hance, the bedload transport rate in unit width at section x=0 should be

because the change of the value u_b with x is quite limited, the u_b can be substituted by the average velocity of grain u_b () from x=0 to approximately. Thus

in which P can be determined by following condition,

where, lift force F_L=C_LπrD²a²u²_∗ (1+k₀k_∗)/8/g, submerged weight W′=(r_s-r)πD³/6, u_∗ is shear velocity, C_L is the coefficient of lift force, a is the ratio of local flow velocity and shear velocity, r is the unit weight of fluid and g is gravitational acceleration. As a result, we obtain

where k₀ is taken as 0.5, the parameter of flow condition M=1/ =(r_s-r) D/(rR′s), s is the energy slope, R′ is the hydraulic radius with respect to the grain, B is the coefficient for detachment condition of grains.

2.1.3　The equation of the saltation of a grain in water

The trajectory of a grain picked up from bed surface can be divided into three stages (refer to Fig. 1). The first stage is from point 1 to 2, a grain just jumps over the bed surface. The second stage is from point 2 to 3 which is the highest point of the trajectory. The third stage is from point 3 to 4. The differential equation of the saltation of grain in y dircction can be expressed by

in which, F_L and W′ are expressed by Equ. (5). The weight of grain W=r_sπD³/6, the flow resistance F_y=C_DπrD²²/8/g, where C_D is the resistance coefficient, and express the vertical component of the velocity and acceleration of grain. Assume that F_L=0 except the first stage. The directions of W′and F_y are opposite in the third stage.

In the first stage, . In the second stage, t=0：=₀, t=T_b：=0 and y=y_m. In the third stage, t=0：=0, t=T_C：y=0. According to above conditions, the following solution can be obtained from Equ. (7)

Where

where c=3rC_DCy_m/ (4r_sD), W_c=(4 (r_s-r) gD/(3C_DCr))^1/2. Neglecting T₀, the total duration of the jump of grain T=T_b+T_c.

The differential equation of the saltation of grain in x direction can be expressed by

where, the drag force F_x=C_DπrD² (au_∗-)²/8/g, and express the component of the velocity and acceleration of grain in x direction. The jump length of grain is obtained as

The average velocity of grain in a jump length L in x direction can be determined by

where

L and u_b (L) are random variables. The randomness of them should mainly depend on that of ₀. According to the Equ. (2), the probability of the condition that the jump length L is larger than average value is 0.368.k_∗ corresponding to should be substituted for k_∗ in Equs. (8) and (9) to determine and u_b (). can be calculated by following identity

where P is determined by equ. (6) in which B is taken as 0.03.

In calculation, C_D is taken as a constant of 0.45. In the third Stage, as the grain falls downward, considering the effect of the bed surface on C_DC and of the on F_y, the resistance coefficient C_DC is taken as 25[1+1.2/(_m/D)]²C_D to get a good fitting with observed data.

The calculated results of /D, _m/D and u_b ()/u_∗ obtained by above equations are shown in Figs. 2 and 3. They are functions of flow condition and specific weight of grain r_s/r. In Fig. 2, the theoretical results are in fine agreement with Francis’data (1973) in high flow condition and with Engelund's equation (1976) in low flow condition. So far, the experiment data of /D from different authors (Lugue et al, 1976；Yalin, 1972；Nakagawa et al, 1986) are much divergent. The results from our equations are in the range of them. The tendency of change of /D with flow condition is in agreement with observed data (Yalin, 1972) and other theoretical results (Rijn, 1984). From Fig. 3, it can be seen that, the two assumptions made by H. A. Einstein, such as the average jump length =100D and thickness of bed load y_m=2D, are suitable only to a very narrow range of the flow condition.

2.1.4　The equation of bedload transport rate

Substituting above equations of saltation into Equ. (4), we obtain the equation of dimensionless bedload transport rate

Where,

in which, a, k_t and f (H) are determined by Equ. (9) to (16), P is determined by Equ. (6), substituting B_∗ for B to consider the effect of the different condition of pick up of grains on the bed surface. Taking A=0.0032 and B_∗=0.08 to get a good fitting with the experimental data, the results are shown in Fig. 4. Changing Bagnold's and Yalin's equations to the form of N-M curve together with Einstein's equation in Fig. 5 (Ning Chien, 1983), and comparing it with Fig. 4, we can see that, the result of Einstein's formula is merely a coincidence with the nylon data, and its transport rate is too small in comparison with the sand data in high flow condition. On the other hand, the resuit of Bagnold's equation is only suitable for the natufal sand. It is noted that the N-M curves obtained from Equ. (17) give a good agreement both with the sand data and nylon particle data in high flow condition.

2.1.5　Conclusions

(1) The prime mover of the bedload is the lift force and drag force of the flow. Equations (4) and (17) show that the bedload transport rate is directly proportional to the probability of detachment, the average jump length and velocity of grain of bedload.

(2) The jump Iength and velocity of grain are stochastic variables due to randomness of the initial condition of the jump. The dimensionless average values of them /D and u_b ()/u_∗ depend on the flow condition and specific weight of grain. The effect of r_s/r on /D and N is more and more distinct with the increase of the flow strength.

(3) The fine agreement between Equ. (17) and observed data proves that besides the flow condition the specific weight of grain is an important factor in determination of transport rate especially in the region of high flow condition.

References

[1] Bagnold R A. An approach to the sediment transport problem from general physics[R]. U. S. Geol. Survey, Prof. paper 422-I, 1966.

[2] Chien Ning, et. al. Mechanics of sediment movement[M]. Science press, Beijing, China. 1983.

[3] Einstein H A. Formula for the transportation of bedload[J]. Trans., ASCE, Vol 107. 1942.

[4] Engelund F, Fredsφe J. A sediment transport model for straight alluvial channels[J]. Nordic Hydrology, Vol 7. 1976.

[5] Francis J R D. Experiments on the motion of solitary grains along the bed of a water stream. Proc. R. Soc. London, A332. 1973.

[6] Luque R F, Beek R V. Erosion and transport of bedload sediment[J]. Journal of Hydraulic Research, Vol 14, No. 2. 1976.

[7] Meyer-peter E, R. Müller. Formula for bedload transport[C]. Proc. 2nd congress I.A.H.R., Stockholm, 1948.

[8] Nakagawa H, Tsujimoto T. Applicability of stochastic approach to alluvial hydraulics[C]. Proc. Third Intern. Symp. on River Sedim., printed in USA, 1986.

[9] Rijn L C Van. Sediment transport, part I：Bed load transport[J]. J. Hyd. Div., proc. ASCE, Vol. 110, HY10. 1984.

[10] Wilson K C. Bed load transport at high shear stress[J]. J. Hyd. Div., Proc. ASCE, 1966, Vol 92, HY6.

[11] Yalin M S. Mechanics of sediment transport[M]. Pergamon press. 1972.

Notation

D　diameter of grain

　dimensionless flow strength

　average jump length of grain

M　parameter of flow condition (=1/)

N　dimensionless bedload transport rate

r_s/r　specific weight of grain

u_b ()　average velocity of grain from x=0 to

u_∗　shear velocity

_m　average jump height of grain