## 5 Theory Analysis and Discussion
### Effect of Velocity Skewness on the Travel Distance  
In order to analyze the movement behavior of the sediment particles in details, the interaction between the particles and the state of the force were applied to study the contact rolling motion of the particles on the bed surface, and the expression of the moving distance was derived to reflect the sediment transport behavior under unsteady flow. As seen from Figure 13, the contact rolling distance \(x\) is given by:  
\[x=\sqrt{R^{2}-\left(R-\Delta\right)^{2}} \tag{26}\]  
When the particle reaches the critical rolling state from the still, the angular acceleration \(\alpha=\frac{d^{2}\theta}{dt^{2}}=0\), the angular velocity \(\omega=\frac{d\theta}{dt}=0\), \(u_{b}=u_{b0}\), \(\theta=\theta_{0}\), which is brought into Equation (20):  
\[ \begin{split}\frac{d^{2}\theta}{dt^{2}}=\frac{5}{28D^{2}}\frac{ \rho}{(\rho_{s}+\rho/2)}\Bigg{[}\bigg{(}u_{b}-D\frac{d\theta}{dt}sin\theta \bigg{)}^{2}(3C_{D}sin\theta+3C_{L}cos\theta+C_{D}+C_{L})+4C_{a}D\frac{du_{b}} {dt}\bigg{(}sin\theta+\frac{1}{3}\bigg{)}\\ -\frac{{u_{b0}}^{2}(3C_{D}sin\theta_{0}+3C_{L}cos\theta_{0}+C_{D }+C_{L})}{4 gDcos\theta_{0}}4 gDcos\theta-3 CD^{2}\bigg{(}\frac{d\theta}{dt} \bigg{)}^{2}cos^{3}\theta\Bigg{]}\end{split}  \tag{27}\]  
For accelerated motion, \(a=\frac{du_{b}}{dt}\) and \(u_{b}-u_{x}\)\(>\)0, based on the premise that the above assumption is true, there is:  
\[\frac{d(u_{b}-u_{x})}{dt}=\frac{85}{112D}\frac{\rho C_{D}}{(\rho_{s}+\rho/2)} \bigg{[}{u_{b0}}^{2}-(u_{b}-u_{x})^{2}-\frac{64C_{a}D}{51C_{D}}a\bigg{]} \tag{28}\]  
When the initial condition is \(u_{x}=u_{0}\), \(t=0\), Equation (28) is solved by:  
\[\frac{dx}{dt}=u_{b}-m+\frac{2m}{\Big{(}\frac{m+u_{b}-u_{0}}{m-u_{b}+u_{0}} \Big{)}e^{mpt}+1} \tag{29}\]  
where \(m=\sqrt{{u_{b0}}^{2}-\frac{64C_{a}D}{51C_{D}}}a\), \(p=\frac{85\rho C_{D}}{56D(\rho_{s}+\rho/2)}\), bring the initial condition \(t=0\), \(x=0\) to solve the equation:  
\[x=(m+u_{b})t-\frac{2}{p}\bigg{[}ln\bigg{(}e^{mpt}+\frac{m-u_{b}+u_{0}}{m+u_{ b}-u_{0}}\bigg{)}-ln\bigg{(}\frac{2m}{m+u_{b}-u_{0}}\bigg{)}\bigg{]} \tag{30}\]  
In order to study the relationship between the travel distance of particles and the skewness of the unsteady flow velocity, bring the conditions that \(a=0\)-0.45 m/s\({}^{2}\), \(C_{D}=0.4\), \(C_{L}=0.1\), \(\rho=1000\) kg/m\({}^{3}\), \(\rho_{s}=2650\) kg/m\({}^{3}\), \(t=1\) s, \(D=2R=1\)\(\sim\)2 mm, \(\Delta=1\) mm, \(u_{b}=0.5\) m/s, and \(u_{0}=0.2\) m/s to solve Equation (25), and bring the data into Equation (30) to get the relationship between the velocity skewness and the travel distance of the sediment particles, as shown in Figure 14. The experimental data are also given in the figure for comparison. It clearly shows that the theoretical results have good agreements with experimental data. It was suggested that in the same period of time, the rolling motion distance of the sediment particles increased as the flow velocity skewness increased. The travel distance of the particles can effectively characterize the length of the moving distance of particles on the bed surface. Therefore, the travel distance of the sediment particles can effectively reflect the sediment transport rate of the horizontal bed surface within a certain period of time, and thus the velocity skewness had a positive relation with the sediment transport rate of the bed load, and the sediment transport rate increased with the increase of the velocity skewness. Since the unsteady flow had velocity skewness compared to the steady flow, it is inferred that the unsteady flow produce a larger sediment transport rate than the steady flow in the same period, as shown in Figures 6 and 8. The sediment transport rate under the unsteady flow was significantly larger than the steady flow through area integration of the unsteady flow, which was consistent with the phenomenon observed during the experiment. During the deceleration motion, \(u_{b}-u_{x}<0\). When \(u_{x}=u_{0}\), \(t=0\), the solution of Equation (28) is obtained.