2.6　New equations of sediment transport rate⁽⁸⁾

Wang Shiqiang, Zhang Ren, Hui Yujia

Abstract: A set of new equations of height, length and velocity of grain saltation and bed load transport rate are presented in this paper. They all depend on shear stress of flow, diameter and specific gravity of sediment grain. Considering the effect of bed surface on the fall velocity of grain near the bed, the non-uniform distribution of bed load concentration and the link with the concentration of bed load and suspended load, new equations of suspended load and total sediment transport rate are obtained. These equations agree fairly well with observed data and are more suitable for wide range of sediment transport. Comparison among six important sediment transport equations by using the observed data shows that the proposed equation by the authors is the better one. The relationship of the comprehensive modified coefficient which reflects the effect of non-uniformity of a mixture on the sediment transport rate for each specific size group is suggested preliminarily.

2.6.1　Introduction

H. A. Einstein (1950) pioneered studies of bed load and suspended load by combining stochastic and mechanical analyses and linking bed load concentration with the bottom concentration of suspended load. This research is significant, but one of its important shortcomings is lack of the analyses of grain saltation. The saltation variables, such as dimensionless length, height and velocity of grain jump, are assumed to be constants in Einstein's equations. In fact, these saltation parameters vary in a large range with the changing flow condition.

M. S. Yalin (1972), R. A. Bagnold (1973), F. R Luque (1976), L. C. van Rijn (1984), P. L. Wiberg & J. D. Smith (1989) and others studied the phenomenon of bedload transport using models of grain saltation. They made hydrodynamic analyses or experiments on grain saltation and got alot of useful results. Most of the bedload transport equations as mentioned above include the velocity u_b of grain saltation in flow direction and the sediment concentration c_s in the bed load layer, and exclude the jump length L and the probability P of detachment of a grain from the bed. Because the parameter c_s is very complex, it is difficult to predict directly through theoretical analysis. So far, all the approaches to predict c_s are empirical or semi-empirical.

S. Wang and R. Zhang (1987, 1989) proposed a set of equations through hydrodynamic and stochastic analyses for predicting the saltation parameters and the sediment transport rate of bed load, suspended load and total load, The latest findings of the effect of specific gravity of grain on the saltation parameters and the bedload transport rate are much different from all the other existing results. The authors’equation of the bedload transport rate includes u_b, L and P, and avoids using bedload concentration c_s directly. The decrease of fall velocitv of grains due to the effect of the bed surface is considered, which affects the profile of the sediment concentration not only near the bed but also along all the water depth.

Under the guidance of authors, a series of flume experiments and analyses of grain saltation and sediment transport rate for sediment grains with different specific gravities have been carried out in the Sediment Research Laboratory, Tsinghua University since 1987 (X. Fu, 1988；C. Hu, 1989, S. Wang and R. Zhang, 1990. L. Wang, 1991. Z. Cheng, 1993). A lot of meaningful experimental and analytical results have been obtained. Supplemented by these results, further verification, improvement and expansion of the originally proposed equations were made and will be described in following section.

2.6.2　Equations of grain saltation and bedload transport rate

1. Equations of Grain Saltation

According to experimental results, grains in saltation account for most of the bedload transport under ordinary flow conditions for sandy bed. Hence, the grain saltation is considered as the main pattern of bed load transport in the author's research.

The forces acting on a saltation grain near bed are very complex which include the submerged weight W′, the lift force F_L, the drag force F_D, the Saffman force, the Magnus force, the Basset force, the added mass force and so on. Considering that the force W′, F_D and F_L are much more important than other forces, only these three forces are included in the saltation analysis by authors.

The differential equation of grain saltation in y and x direction can be expressed respectively as

where W is the weight of a grain；g is gravitational acceleration； and express the acceleration component of a grain in the flow direction and its vertical direction respectively；F_Dx and F_Dy are the flow resistance in x and y direction respectively. The solutions were obtained by simplifying and integrating above diffirential equations. The theoretical equations to predict the saltation parameters were obtained by the stochastic and hydrodynamic analyses, but they are quite complicate and are not convenient to be used (Wang & Zhang, 1987). The following new simplified equations are summarized as the equivalent to them.

where

Where a is the mean jump height of grains or thickness of bedload layer；L_m is the mean jump length；u_b-mean velocity of grain jump in flow direction；u_∗=; J-energy slope；R′-hydraulic radius due to grain roughness；R′ equals the water depth H for flat bed；D-grain diameter; γ_s and γ-specific gravity of sediment grain and water, respectively.

2. Equation of Bed Load Transport Rate

The bed load transport rate per unit width q_b should be equal to the weight of the grains in saltation passing through a cross section of unit width in unit time. It can be determined by the following equation (Wang and Zhang, 1987)

Where A_∗ is a coefficient. The probability p of detachment of grains from the bed can be got by the condition F_L/W′>1. The following simplified equation can be used to get a good approximation to the results computed by the complicated theoretical equations,

where

The equations (3)～(10) show that the values of the saltation parameters and the bedload transport rate for Θ=Θ_c all are larger than zero. It is the result of stochastic analyses and in agreement with observed data.

3. Verification, Comparison and Analyses

The relationships between the saltation parameters a/D, L_m/D, u_b/u_∗ and Ψ(=1/Θ) with various γ_s/γ from the above Wang-Zhang equations are shown in Figs. 1, 2 and 3, respectively. These parameters from authors' equations increase with the increase of Θ for a given γ_s/γ, but their increasing rates reduce. The changing rate and the value of the parameters for a given Θ decrease with decreasing γ_s/γ.

Fig. 4 shows the relationships between a/D and HJ/D with various γ_s/γ where the curves A, B and C correspond to the curves 1_a, 1_b and 1_c in Fig. 1, respectively. If a/D were related only to Θ, for example, the curve 1_b should be unique for various γ_s/γ in Fig. 1 as it was generally accepted in the past, then the curve 1_b for various γ_s/γ values in Fig. 1 can be transformed into the curves A_b, B and C_b for γ_s/γ=4.22, 2.65, and 1.14 respectively in Fig. 4. Comparing the curves A, B, C in Fig. 4 it is shown that a/D increases with the decrease of γ_s/γ for a given HJ/D, but the values of a/D for different γ_s/γ are very close when HJ/D is larger than 1. It is quite different from the past conception, for instance, the difference between two values of a/D for the curves A_b and C_b still is quite large as HJ/D is larger than 1 in Fig. 4. The tendency of variation in curves of L_m/D～HJ/D and u_b/u_∗～γ_s/γ and u_b/u_∗～HJ/D, γ_s/s is similar to that mentioned above.

Comparison of some observed data and formulae of various authors is also shown in Figs. 1, 2 and 3. Two curves marked 1c in Figs. 1 and 2 by the authors accord well with the observed data from X. Fu (1988). Bagnold. (1973, 1977) presented the empirical relation of jump height, a/D=k_b (Θ/Θ_c)^0.3, where k_b is taken as 1.4 based on the flume data and 2.8 based on the field data from rivers. Considering the fact that Θ is small for the flume data and large for natural rivers. two curves 2_a and 2_b after Bagnold are plotted in Fig. 1 for two different ranges of Θ, respectively. The curves 3 after van Rijn (1984) for D=0.5mm almost coincides with 1_b.C. Hu (1988) carried out experiments on grain saltation for various γ_s/γ and summarized the relations of a/D, L_m/D and u_b/u_∗. The results show that a/D and L_m/D all are the function of not only Θ but also γ_s/γ. This conclusion is consistent with the theoretica1 analysis by the authors (1987). The curves 5_a and 5_b using Hu′s relation for γ_s/γ=2.65 and 1.14 for smooth bed in Fig. 1 are basically consistent with the authors' curve 1_b and 1_c respectively. In Fig. 2, the curves 2_a and 2_b after Van Rijn for D=1mm and 0.5mm are on both sides of curve 1_b by authors；the curves 3_a and 3_b from Hu for γ_s/γ=2.65 and 1.14 show that L_m/D decreases with decreasing γ_s/γ for a given Θ. In Fig. 3, the curves from the equation developed by the authors are quite consistent with the observed data of J. E. Abbott and J. R. D. Francis (1977) and the curve 2 from F. Engelund's relation (1976) which is verified by Luque's data with smaller flow intensity. The curves 4_c and 4_b from Yalin's equation (1972) also show similar effect of γ_s/γ on u_b/u, as pointed out by the authors. The curve from van Rijn is close to that from the author.

Figs. 1, 2 and 3 show that the values of a/D, L_m/D and u_b/u_∗ for γ_s/γ=2.65 have ranges from 1.2 to 12, from 1.0 to 700 and from 4 to 12.5, respectively with Θ changing from 0.06 to 2. Therefore, the assumption of constant parameters in grain saltation would result in a great error for the prediction of bedload transport rate.

The curves from authors’equations of the bed load transport rate show good agreement with the experimental data plotted in Fig. 5. The curves 1_b and 1_c are in good agreement with the data from K. C. Wilson (1966) for γ_s/γ=2.67 and 1.138 respectively. After converting Meyer-Peter and Muller's (1948) and Yalin's equations (1972) into a form of Φ-Ψ relationship together with Einstein's equation (1950) and comparing them with the curve 1_b by authors for natural sand grain in Fig. 5, it shows that the curve 4 from Yalin is almost coincident with the curve 1_b, the curve 2 from Einstein is consistent with 1_b in the range from Ψ=2 to Ψ=17 and the curve 3 from Meyer-Peter lies between the curve 2 and 1_b or 4.

The curves 1_b and 1_c in Fig. 6 are plotted for γ_s/γ=2.65 and 1.14 respectively from authors’equation using recent experimental data from L. Wang (1991) and Low (1989). It shows that the authors’equation accords well with the data for different γ_s/γ values. The Φ values and the variation in Φ-Ψ diagram decrease with the decrease of γ_s/γ for given Ψ or Θ. The tendencies of variation of dimensionless bed load transport rate (excluding the factor γ_s/γ) with HJ/D and γ_s/γ are similar to those in Fig. 4. It can be explained from above equations of grain saltation that if the total forces acting on a grain and initial velocity of the grain are given, the acceleration of a grain will increases and the jump duration decreases with the decrease of γ_s/γ, resulting in a decrease of the values of saltation parameters and the bed load transport rate for a given Θ.

2.6.3　Equations of sediment transport rate of suspended load and total load

1. Model and Equations

According to the experimental data (Nakagawa, 1976；Fu, 1988), the distributions of probability density of jump lengths or heights of grains are close to an exponential function. i.e. the frequency of the jump lengths or heights of a grain in unit time decreases with the increase of jump length or height. It means that the number of jumping grains and the corresponding concentration at an elevation y from bed decrease with the increase of y. The local flow velocity u_y near bed is proportional to u_∗ (y/D)^0.234, so u_y/u_m=1.234 (y/a)^0.234, where u_m is the mean velocity of flow in the bed load layer with thickness a. Assuming u_b (y)～u_y and uniform bedload transport rate in depth, i.e., u_b (y) S (y) =u_bS_m, then u_b (y)/u_b=u_y/u_m and u_b (a) S_a=u_bS_m, S_a=S_mu_b/u_b (a) =S_mu_m/u_a=0.81S_m, thus

where S(y), S_a and S_m are the concentration at y, that at a and the average concentration of bed load respectively, u_b(y) and u_b(a) are the local velocity of bed load at y and at a respectively.

Wiberg and Smith (1989) adopted a quite meticulous approach to predict S(y) in bed load layer. According to their suggestion, S(y) is only related to the reciprocal of the vertical velocity u_L(y) of grain jump at same level y. It results in a maximum concentration S_a in the bedload layer. This implies a discrepancy between Wiberg and Smith's concept and the above mentioned result based on experiments. As a matter of fact, S(y) depends not only on 1/u_L but also on other more important factors, such as 1/u_b, probability p and particularly the distributions of probability density of jump lengths and heights.

Considering that the concentration of suspended load at the point y=a is the same as S_a, the suspended transport rateqs per unit width from y=a to y=H can be obtained by the integration,

in which u_y can be calculated by the equation after Einstein (1950),

where x₁ is a variable after Einstein (1950), D₆₅ is the diameter of bed material for which 65% by weight of the sample is finer, H is water depth.

The distribution of suspended concentration S_y can be deduced from diffusion equation

Taking

And

where ε_y is diffusion coefficient, von Karman's constant κ=0.4, ω₀ and ω is the fall velocity of grain excluding and including the effect of bed surface, respectively. Eq. (16) is taken from H. A. Lorentz (1907).

Substituting eq. (15) and (16) into (14) and integrating yield

where

Eq. (17) can be converted into the following expression (20) with enough accuracy, verified by observed data (S. Wang, 1993),

where

F₁ (y)≥1, z₁≤z, so the effect of bed surface makes the distribution of sediment concentration more uniform and the sediment transport rate increased. For example, as z=4.3 and D=0.25mm the effect of the bed surface on the concentration profile increases the transport rate of suspended load by 40%. The transport rate increases with decreasing D for a given z and with increasing z for a given D.

Integrating Eq. (13) yields

where

where A=a/H, Replacing S_a with Eq. (12), then

The total sediment transport rate q_T can be predicted by

The transport rate for specific grain size group for non-uniform sediment is

in which

where i₀, i_T, i_b and i_s are the fractions of a given size group for bed material, total sediment load, bed load and suspended load, respectively.

2. Verification and Comparison

The total sediment transport rate can be computed by Eqs. (19)～(27b) together with the equations (3)～(11b) of grain saltation and bedload transport. The results of the verification of authors' equations with the flume data by Einstein (IRTCES. 1987) is shown in Table 1 where K_A is the ratio of predicted total sediment transport rate to the measured one and is the average value of K_A. The predicted value is the sum of calculated sediment transport rate for each grain size group in a mixture by Eq. (26). Computational results using the authors' equation provide quite good agreement with the observed data. The ratios K_A for 55 tests are between 0.5 and 2.3.

Comparison of the authors’equation (Wang-Zhang) and other five important equations for total sediment transport rate (Ackers-White (1973), Engelund (1972), van Rijn (1984), C. T. Yang (1982), R. J. Zhang (WHEE. 1981)) were made by using the observed data (Cheng 1993). The results are shown in Table 2. In the application of Wang-Zhang's equation. D₃₀ is adopted as the representative diameter for non-uniform bed material. In Table 2, P_k is the percentage of number of calculated data with K_A between 0.5 and 2.0. The data set 1 are taken from Guy et al,. Gilbert and Meyer-Peter for sand, the data set 2 are taken from U.S.W.S. for light-weighted grains and the data set 3 are taken from Einstein, Kennedy and Vanoni for sand (Brownlie, 1981).

It can be seen from Table 2 that the results predicted by the equation of Wang-Zhang, Ackers-White and van Rijn are more accurate for grains with various specific gravities and the equations of C. T. Yang and Engelund are not suitable for the light material.

3. Effect of Non-uniformity of Mixture on Sediment Transport

The mutual effect of fine and coarse grains in a mixture at bed surface and in water must be taken into account in order to predict the sediment transport rate for each specific grain size group accurately. The fine grains in a mixture are subject to the hiding effect on the bed and the retarding effect from coarser grains in flow due to the collision among grains with different velocity. Meanwhile, the coarser grains are subject to the lift and drag forces larger than that in uniform grain condition due to exposure on the bed and collision in the water. Therefore the transport rate of finer grains would decrease and that of coarser grains would increase in a mixture.

On the basis of analyzing lots of data from flumes and rivers, a comprehensive modifying coefficient K_D due to non-uniformity of the sediment mixture is introduced in Eq. (27a) in the form of

Replacing Eq. (27a) by Eq. (28), with other unchanged equations, prediction of the transport rate of each specific grain size group can be more accurate for graded sediment. The modifying coefficient K_D is a function of flow intensity, relative water depth H/D and D_k itself is the ratio of the diameter D of given size group to D₆₅ in a mixture. On the basis of analyses of the flume data from Einstein (IRTCES, 1987) and the river data from the Yangtze and the Yellow rivers, the following empirical relationships are preliminarily proposed for computing K_D.

Where U_E=VJ/((g ν)^1/3D_gr), ν is the kinematic viscosity,

Twenty-seven sets of experiments on the transport rate of graded sediment were conducted in a 60m tilting flume at the Tsinghua University by S. Wang et al (1990). Good agreement is obtained in the comparsion between the experimental data and the predictions by Eqs. (28), (29) and other equations. The average ratio equals 1.25 and P_k=85%. The ratio of the bed material load for the size group (0.025～0.1mm) to the total is over 90%. The average ratio is 1.35 and 1.25 for the fine size group (0.025～0.05mm) and the coarse one (0.05～0.1mm) respectively. The percentage P_k is 85% and 82% for the two size group, respectively.

Comparing the prediction by Eq. (28) and other equations with the observed 91 sets of data from Mississippi, Rio Grande and other rivers in USA (Brownlie, 1981), the results are =0.84. P_k=83.4%. For the observed data of Yangtze River and lower Yellow River, =1～1.2, P_k=60%～77%. The discrepancy in average is quite small but the data points are more scattered. One of the main reasons for the latter is that the measured size distribution i₀ of bed material is often not representative. Of course, the K_D vs D_k relationship needs further study and improvement.

The equation of sediment transport rate proposed by authors has been used in some mathematical models for predicting the fluvial processes in river channel and reservoirs on the Yellow River with satisfactory results.

2.6.4　Conclusions

(1) The new equations are presented for predicting the saltation parameters and sediment transport rate of bedload, suspended load and total load. The bedload transport rate is directly proportional to the probability of detachment, the average jump length and the mean velocity of grain saltation. The transport rate of suspended load depends on the distribution of flow velocity and sediment concentration, the water depth and the concentration at the top of bedload layer. The effect of bed surface on increasing the resistance coefficient of falling grains in water near bed is considered that results in the increase of transport rate of bedload and suspended load. The equations are in good agreement with observed data.

(2) The parameters of grain saltation a/D, L_m/D, u_b/u_∗ and the dimensionless bed load transportrate Φ are functions of Θ and γ_s/γ. They all increase with the increase of Θ for given γ_s/γ and decrease with the decrease of γ_s/γ for given Θ or HJ/D. Their difference for different γ_s/γ is smaller than that obtained from the previous concept and the difference is quite small as HJ/D is larger than 1.

(3) Comparing the six important equations of total sediment transport rate with observed data shows that the authors', Ackers-White's and van Rijn's equation are relatively better and more suitable for material with various specific gravity.

(4) The mutual effect of fine and coarse grains in a mixture on the sediment transport rate for each specific grain size group is considered. The comprehensive modifying coefficient K_D is introduced and its relationships are proposed preliminarily.

Acknowledgements

The authors wish to sincerely thank the cooperation and assistance from C. Hu, X. Fu, L. Wang, Z. Cheng, J. Chen, J. Liu, X. Zhai and Y. Ren in experimental works. The authors also deeply appreciate the financial support provided by the National Natural Science Foundation of China and by the Ministry of Water Resources of China.

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Notation

a=thickness of bed load laver, or mean jump height of grains；

D=grain diameter；

H=water depth；

i₀, i_T, i_b, i_s=fraction of a given size group for bed material, total sediment load, bedload and suspended load in a mixture, respectively；

J=energy slope；

K_A, =ratio of calculated sediment transport rate to observed one and average value of K_A, respectively；

L_m=mean jump length of grain；

P=probability of detachment of grain；

P_k=percentage of number of calculated data between 0.5 and 2.0 for K_A；

S_y, S_a, S_m=sediment concentration at elevation y, y=a from bed and mean concentration of bedload, respectively；

u_∗=shear velocity of flow；

u_b=velocity of gram saltation in x direction；

z, z_l=power of concentration distribution of suspended load without and with the effect of bed surface. Respectively；

γ_s, γ=specific gravity of water and solid grain, respectively；

Φ=dimensionless sediment transport rate；

Θ, Θ_c=dimensionless shear stress due to grain and its critical value, respectively；

Ψ=1/Θ.