2.3　Sediment transport rate for non-uniform sand⁽⁵⁾

Wang Shiqiang, Zhang Ren

Abstract: Based on the equations of bedload transport by authors, the functional relation of total Sediment transport capacity of flow has been found. The modifying coefficient due to non-uniformity of the sand grain size on the transport capacity has been analysed and summrised from the flume and field data. The equations presented in the paper can be used to predict the transport rate of bedload, suspended load and bed material load for various grain grading. The predicted results provide a good agreement with the observed data of a great number of natural alluvial streams.

2.3.1　Introduction

The sediment in natural streams is generelly non-uniform in grain size, so the composition of grain size of the moving sediment and the bed material would be changed, that will affect the sediment transport capacity and the frictional characteristic of the alluvial channels. So far, most of equations for sediment transport rate are suitable to uniform sediment, they are not able to predict correctly the transport rate of non-uniform sediment for various grain grading particularly under the condition of rapid sorting.

H. A. Einstein and Ning Chien started the studies on the transport rate for nonuniform sediment in early fifties, (H. A. Einstein, 1950, H. A. Einstein and Ning Chien, 1953) which greatly affects the research works in this fieid. But Einstein did not make enough hydrodynamic analyses on the grain saltation in water, some hypotheses in his theory are unreasonable. The predicted results by Einstein equations often greatly deviate from observed data.

2.3.2　The equation of bedload transport

Based on the hydrodynamic and probabilistic analyses on grain saltation, the equations of dimensionless transport rate N. relative thickness y_m/D and velocity u_b/u_∗ of the bedload in function of M and r_s/r have been obtained by authors (Wang Shiqiang and Zhang Ren, 1987). Fig. (1) shows the functional relationship between N and M, where, N=q_b/D_g, D_g=D^3/2g^1/2r_s×[ (r_s—r)/r]^1/2, 1/M is the parameter of flow intensity in terms of rRS/[ (r_s-r) D], q_b is bedload transport rate per width, D is diameter of grain, u_b is the average velocity of bedload, u_∗ is shear velocity, S is energy slope, g is gravity acceleration, r and r_s are specific weight of water and grain respectively, R is hydraulic radius.

The following simplified equations are equivelent to above mentioned equations under the condition of M<26 and r_s/r=2.68.

in which, m₁=-0.1018-9.7707·10^-3 M+6.752·10^-4 M²-2.93·10^-5 M³

in which, m₂=-0.0394-0.01452 M+6.024·10^-4 M²-2.76·10^-5 M³

in which, m₃=-0.0941-0.02 M+1.045·10^-3 M²-3.8·10^-5 M³

A bove equations of bedload with r_s/r equal to 2.68 are the basis of equation of total sediment transport capacity.

2.3.3　The equations of total sediment transport rate

H. A. Einstein (1950) suggested that the transport rate of suspended load is determined by the distribution of velocity and sediment coneentration along the water depth. He made the hypothesis that the bottom concentration of suspended load equals the top concentration of bedload. The observed data proved that it is reasoneble if the sediment concentration is not too high.

Based on the diffusion theory given by H. Rouse, the vertical distribution of sediment concentration in suspension can be got if the top concentration S_a of bed lood is determined. According to the experiments (Fu Xiao, 1988), the distribution of the probability density of jump heights is close to an exponential function. It means that the concentration in bedlood decreases with the increase of distance y from the bed. If the velocity of grains u_b (y) is proportional to the flow velocity u (y), thus, u_b (y)/u_b=1.234 (y/y_m)^0.234. With the assumption of constant transport rate along depth in bedload, S_a=0.81 where is the average bedlood concentration which equals to q_b/ (r_su_by_m), y_m can be calculted by Eq. (3). Thus, the transport rate q_T of total bed material load can be obtained by the following equation

The transport rate of bed material load for specific grain size group is

where i_bq_b=i₀N D_g, N can be got by the equation (1).i₀, i_T, i_b and i_s are the fractions of given size group for bed material, total bed material load, bedload and suspended load respectively.P, I₁ and I₂ are parameters which are the same relation as H. A. Einstein's equation.

2.3.4　The modifying coefficient on sediment transport due to non-uniformity of the sediment

In this paper 26 sets of observed data published and 29 sets of data unpublished by H. A. Einstein (I.R.T.C., 1987) for the total transport rate of non-uniform sediment in flume have been analysed. Table 1 shows the comparison between the observed and predicted results by the Eq. (5) where is the ratio of measured total sediment transport rate to the predicted one and K_A is the average value of K_A. The calculated results provide quite good agreement with the observed data. It shows that the equations suggested by authors are reasonable and could be used for predicting the total sediment transport rate.

If K_D is the ratio between observed and predicted bedload transport rate for a given grain size and D_k is the ratio between the diameter D_i of the given grain size and D₆₅, the relationship between K_D and D_k can be found in Fig. 2 where K_D increases with the increase of D_k. The curveⅠin Fig. 2 shows the relationship between K_D and D_k for 55 sets of Einstein's flume data. It can be seen that if the non-uniformity of the sediment is not considered the transport rate of predicted bedload and suspended load will be larger than measured results for grains finer than D₆₅ and smaller for grains coarser than D₆₅.

H. A. Einstein and other scientists have early suggested to introduce a“Hiding factor”of grains in the calculation of non-uniform sediment. They pointed out that the finer grains in a mixcure are protected by coarser grains on the bed that makes the lift force on finer grains less than that in uniform condition. According to the analyses by authors, the finer grains in a mixture are also subjected to the retarding effect of the coarser grains due to collision between both of them with different velocities in water besides hiding effect on the bed. In the mean time, the coarser grains are subjected to more flow tractive force and accelerating effect from the finer grains due to exposure on the bed and collision in water. These effects make the transport rate of finer grains decreased and that of coarser grains increased in a mixture. The hiding or exposing effects change with flow intensity. They decrease with the increase of flow intensity because the rest time of coarser grains is reduced. The analyses of the data of flume and natural rivers show that the curveⅠin Fig. 2 is only suitable for the upper regime of flow. According to field data of natural rivers the lower the flow intensity or the larger the ratio H/D is, the more the deviatinn of the K_D^-D_k curve from the curveⅠwill be.

The hiding or exposing effects on grains with same size are different due to the different relative position between coarser and finer grains, so it is better to reflect the effects by modifying i₀ as a whole. In this way, the modifying process is much simplified. In addition, it can reflect all the the effects due to the non-uniformity of the sediment in movement. Considering the modifying coefficient K_D due to the non-uniformity of the sediment, i_bq_b in the Eq. (5) should be calculated by Eq. (6))

The other equations remain unchanged. According to the analyses of observed data, K_D can be approximately predicted by Fig. 2 or following

when the flow is in upper regime, U_E>0.011,

when the flow is in lower regime, U_E≤0.011,

where, D_k=D_i/D₆₅, U_E=VS/[ (gν)^1/3D_gr], D_gr=[g (r_s-r)/r/ν²]^1/3 D₅₀, ν is the kinematic viscosity. The demarcation of U_E for distinguishing the flow regime was explained by White et al. (White, W. R., Bettess, R. and Wang Shiqiang 1987).

2.3.5　Verification of the equations

Based on the calculated or observed hydraulic parameters, the transport rate of bedload, suspended load and bed material load for given grain size and the total transport rate of the mixture can be obtained by authors computing program. Table 2 shows the comparison between the observed and the predicted bed material load in suspension by authors’equations. The predicted results provide a good agreement with the observed data. Table 3 shows the comparison between the observed and the predicted sediment loads by H. A. Einstein's equation. The results predicted by anthor's equations are much better than that obtained by Einstein's equation.

2.3.6　Conclusions

(1) The bedload transport equations by authors are the basis in deriving the functional relationship of the total sediment transport rate. The thickness and velocity of the bedload are changable with various flow intensities. The bedload concentration increases with the decrease of distance from the bed.

(2) For non-uniform sediment the coarser grains in a mixture provide hiding and retarding effects on the finer grains and are subjected to more flow drag and accelerating effect from fine grain simultaneously. These effects decrease with the increase of flow intensity. The modifying coefficient K_D due to non-uniformity of sediment can be predicted by Fig. 2 or the Eq. (7).

(3) The verification shows that the sediment load predicted by authous' equations has a good agreement with the observed load, and they are much better than the results obtained by Einstein's equations.

References

[1] Einstein H A. The bed load function for sediment transportation in open channel flows[R]. US Dept. Agri., Tech. Bull. 1026, 1950.

[2] Einstein H A, Ning Chien. Transport of sediment mixtures with large ranges of grain size[J]. Missouri river Div. Sediment series No. 2, US Corps Engrs., 1953.

[3] Fu Xiao. The experimental study of saltation of grain of bed load in water[D]. Beijing：Tsinghua University,1988.

[4] Intern. Research and training centre on erosion and sedim.[R], Sediment transport data in laboratorial flumes. 1987.

[5] Wang Shiqiang, Zhang Ren. A new equation of bedload transport[C]. Proc. 22 congress IAHR, 1987.

[6] White W R, Bettess R, Wang Shiqiang. Frictional characteristics of alluvial streams in the lower and upper regimes[J]. Proc. ICE, Part 2, Vol. 83, 1987.