## 2 Inception Similarity Optimization Analysis  
The movement modes of sediment under the action of shear flow are divided into three types: suspension, jumping and transport [19; 20]. The transport of sediment refers to the sliding and rolling of sediment particles along the bottom of the riverbed. The phenomenon of sediment transport has little to do with the type of flow. The formation of the sediment transport phenomenon is due to the coaction of the drag force generated by the shear flow and the internal frictional resistance of sediment. At the same time, the uplift force of the shear flow is not enough to make the sediment suspend. The transport motion of sediment is the most common movement mode besides suspension. Taking the critical incipient state of the tailings on the dam surface as the research object, similarity analysis was conducted for the overtopping dam in transport mode. Based on the inception similarity analysis in [PERSON] et al. [18], tailings particles can achieve force equilibrium in both the horizontal and vertical directions at the critical condition of inception (Figure 1), assuming that tailings particles are spherical in shape. The drag force due to the waterflow \(F_{D}\), resistance \(F_{f}\), uplift force \(F_{L}\) and underwater gravity \(W\) can be represented as follows:  
\[F_{D}=C_{D}\frac{\pi d^{2}}{4}\frac{\rho u_{b}^{2}}{2} \tag{1}\]  
\[F_{f}=\tau_{f}A_{f}=\tau_{f}\frac{\pi d^{2}}{4} \tag{2}\]  
\[W=\frac{\pi}{6}(\gamma_{s}-\gamma)d^{3} \tag{3}\]  
\[F_{L}=C_{L}\frac{\pi d^{2}}{4}\frac{\rho u_{b}^{2}}{2} \tag{4}\]  
where \(C_{D}\) is the drag coefficient of waterflow; \(C_{L}\) is the uplift coefficient; \(d\) is the diameter of sediment particles (m); \(\rho\) is the density of waterflow (kg/m\({}^{3}\)); \(u_{b}\) is the instantaneous velocity of waterflow acting on the surface of bedload particles (m/s); \(\tau_{f}\) is the shear strength of mud bed (Pa); \(A_{f}\) is the area of particles (m\({}^{2}\)); \(\gamma_{s}\) is the bulk density of sediment (kg/m\({}^{3}\)); \(\gamma\) is the bulk density of waterflow (N/m\({}^{3}\)). It can be seen from the sediment movement mechanics and the turbulent inception test that the shear stress of the flow on the riverbed sediment is related to the Reynolds number [21]. The actual flow in the process of overtopping dam break of the tailings pond is mainly turbulent flow; that is, the Reynolds number is very large. The surface flow velocity of sediment particles reaching mechanical equilibrium and just initiating motion under turbulent flow conditions (also known as the critical shear velocity) can reflect the magnitude of bed shear stress in open channel flow. The horizontal drag force exerted on sediment particles by water flow can be expressed as:  
\[\tau_{cr}=\rho u_{b}^{2} \tag{5}\]  
where \(\tau_{cr}\) is the critical inception shear resistance. In the horizontal force direction of sediment particles, the equilibrium equation is established:  
\[F_{D}=F_{f} \tag{6}\]  
The relationship between critical shear stress and sediment surface velocity can be derived from Equations (1), (3) and (6):  
\[u_{b}^{2}=\frac{2\tau_{cr}}{C_{D}\rho} \tag{7}\]  
An equilibrium equation is established in the vertical stress direction of the sediment particles:  
\[F_{n}=W-F_{L} \tag{8}\]  
According to the soil mechanics mentioned, considering the non-viscosity between the particles, the expression of shear strength can be obtained:  
\[\tau_{f}=\sigma_{n}\cdot tan\phi \tag{9}\]  
where \(\tau_{f}\) is the shear strength (Pa); \(\sigma_{n}\) is the stress acting on sediment particles (Pa); \(\phi\) is the Internal friction angle of sediment (rad/\({}^{\circ}\)). The normal stress of tailings particles acting on the subsoil can be expressed by the supporting force:  
\[\sigma_{n}=\frac{F_{n}}{A}=\frac{W-F_{L}}{\frac{\pi d^{2}}{4}} \tag{10}\]  
\[F_{f}=\tau_{f}A_{f}=\tau_{f}\frac{\pi d^{2}}{4} \tag{2}\]0 Schematic diagram depicting the force analysis for initiating sediment particle movement.