## 4 Bed-Load Transport under Non-Uniform Sand Condition
### Effect of Particle Size on Sediment Transport Rate  
Figure 10 shows the variation of the transport rate among the different groups of non-uniform sand. It was demonstrated that in the same composed grains the sediment transport rate under unsteady flow was greater than that of the steady flow. The unsteady flow also had a stronger transport capacity for the large-size particle than that of the steady flows. For the small-sized particles, such as \(\varphi_{0.296\text{:}}\varphi_{0.664\text{:}}\varphi_{1.333}\), the transport capacity(\(d=0.296\) mm) under the steady flow condition was stronger than that under the unsteady flow, and the transport rate difference was less than 23.4%. Whereas in large-sized particles (\(d=1.333\) mm), the transport capacity under the steady flow condition was weaker than that under the unsteady flow, and the transport rate difference is less than 12.1%. For the same non-uniform sand, there existed certain difference in the sediment transport rate of the same  
Figure 9: Transport rate comparison between steady flow obtained through unsteady flow area integration and unsteady flow under four non-uniform sand conditions. particle size between steady flow and unsteady flow, which was determined by the hydrodynamic characteristics of the unsteady flow. In the process of motion, the unsteady flow would produce acceleration and deceleration, and the generated inertial force should not be negligible. Meanwhile, the velocity changed sharply in a short time, and then the particle impulsive motion was strong which would cause the particles a sharp change in force. The movement of the particles was greatly different from that of the steady flow. The experimental results showed that the particle size ratio and flow conditions had important influences on the mass ratio of coarse and fine sediment transport, which could be described by dimensionless parameters, as follows:  
\[\eta=f\bigg{(}\frac{\delta_{1}+\delta_{2}}{2},\frac{2(U_{p}-U_{b})}{(t-t_{0})g }\bigg{)} \tag{3}\]  
where \(\eta\) is the mass ratio of the coarse and fine sediment transported in the results; \(\delta_{1}\) and \(\delta_{2}\) refer to the diameter ratio before movement starting between the previous and latter particle size in the same group respectively; \(\delta=(\delta_{1}+\delta_{2})/2\) represents the average grain of the original bed sediment diameter ratio; and \(2(U_{p}-U_{b})/[(t-t_{0})g]=a/g\) characterizes unsteady flow properties-relative velocity skewness. Therefore, the relationship between the mass ratio of different grain sizes that collected in the box, the original sediment particle size distribution and the relative velocity skewness of the flow can be  
Figure 10: Sediment transport rate of different non-uniform sand groups under unsteady flow (solid scatters denote unsteady flow, and open scatters indicate steady flow corresponding to the same transport). (**a**) \(d_{50}=0.664\) mm; (**b**) \(d_{50}=1.333\) mm; (**c**) \(d_{50}=2.639\) mm; and (**d**) \(d_{50}=1.333\) mm. obtained based on the following Equation (4). The \(R^{2}\) was 0.80, and the statistical significance \(P\) was less than 0.05, showing a good power function. The relationship is shown in Figure 11.  
\[\eta=10^{4}\Big{(}17.9976^{-21.44}+0.377(a/g)^{1.31\times 10^{-4}}-0.376\Big{)} \tag{4}\]  
It can be seen from Equation (4) that when the size ratio increases, the size ratio of the coarse and fine sediment obtained by transporting decreases and tend to be uniform. The increase of the flow velocity skewness increases the ratio of the coarse and fine sediment, and the amount of coarse sediment transport is greater than that of fine sediment. The reason is that when the ratio of the sediment particle size increases, the smaller size counterparts are easier to start to move under the condition of the certain flow. However, the initiation and transport of the larger sediment particles became more and more difficult, resulting in a decrease in the ratio of the transported sediment.