## 3 Numerical Modelling
### Motion Equations  
The gate vertical vibration in the closing process was mostly the entire vibration. It was feasible and reliable to regard the emergency gate as a rigid body to describe its fluid-structure interaction [17; 18]. In this study, the deformation stiffness of steel rope in the vertical direction was simplified to the equivalent system stiffness of a single degree of freedom, which can be expressed as \(k=EA/L\) (where \(k\) is the system equivalent stiffness; \(E\) is the elastic modulus of steel rope; \(A\) is the steel rope area; and \(L\) is the steel rope length). Considering the size and material properties of the steel rope, \(EA\) was 3.17E5. The steel rope length \(L\) would increase gradually with the emergency gate dropped. The equivalent system damping was obtained by the Rayleigh relation, and the coefficient was 3% [19]. The gate-closing speed was controlled by the macro through the user-defined function (UDF) in Fluent. Thus, the vertical vibration with a single degree of freedom of the gate-closing process can be expressed as:  
\[(m+m^{\prime})y^{\prime}+(c+c^{\prime})y^{\prime}+(k+k^{\prime})y=F(t) \tag{1}\]  
where \(y^{\prime}\), \(y^{\prime}\) and \(y\) are the vertical vibration acceleration, velocity and displacement, respectively; \(m\) is the system mass; \(c\) is the system damping; \(k\) is the system spring; \(m^{\prime}\) is the added mass; \(c^{\prime}\) is the added damping; \(k^{\prime}\) is the added spring; and \(F(t)\) is the external excitation. The added coefficient loads derived from the mesh deformation of fluid. Nevertheless, the loads can be transmitted by the macro in Fluent. Thus, the gate vertical vibration process can be presented as:  
\[my^{\prime\prime}+cy^{\prime}+ky=f(t) \tag{2}\]  
\[k=m\omega_{0}^{2},c=2m\omega_{0}\xi \tag{3}\]  
where \(\xi\) is the system damping ratio; \(\omega_{0}\) is the system natural frequency; and \(f(t)\) represents the gate loads with fluid-structure coupling, and it is equal to the holding force of the emergency gate. Based on the current standard of hydraulic steel gates [20], the holding force of the emergency gate can be presented as:  
\[f(t)=n_{G}G+G_{j}+W_{S}+P_{X}-P_{t}-(T_{zd}+T_{zs}) \tag{4}\]  
where \(T_{zd}\) is the panel friction force; \(T_{zs}\) is the seal friction force; \(G\) is the gate mass; \(W\)s is the water pressure at the gate top; \(G_{j}\) is the gate counterweight; \(P_{X}-P_{t}\) is the hydrodynamic loads at the gate bottom (\(P_{X}\) is the suction force and \(P_{t}\) is the lifting force); and \(n_{G}\) is the gate weight correction factor. The instantaneous displacement of the system was addressed by solving the motion equations utilizing an improved fourth order Runge-Kutta algorithm [21]. Thus, Equation (2) was discretized and rearranged as:  
\[y(t_{n+1})=y(t_{n})+y^{\prime}(t_{n})\Delta t+\frac{\Delta t^{2}}{6}(K_{1}+2K_ {2}+2K_{3}+K_{4}) \tag{5}\]  
in which:  
\[\left\{ \begin{array}{c}K_{1}=\frac{f(t_{n})}{m}-\frac{c}{m}y^{\prime}(t_{n} )-\frac{k}{m}y(t_{n})\\ K_{2}=\frac{f(t_{n})}{m}-\frac{c}{m}\left[y^{\prime}(t_{n})+\frac{\Delta t}{2} K_{1}\right]-\frac{k}{m}\left[y(t_{n})+\frac{\Delta t}{2}y^{\prime}(t_{n}) \right]\\ K_{3}=\frac{f(t_{n})}{m}-\frac{c}{m}\left[y^{\prime}(t_{n})+\frac{\Delta t}{2} K_{2}\right]-\frac{k}{m}\left[y(t_{n})+\frac{\Delta t}{2}y^{\prime}(t_{n})+ \frac{\Delta t^{2}}{8}K_{1}\right]\\ K_{4}=\frac{f(t_{n})}{m}-\frac{c}{m}\left[y^{\prime}(t_{n})+K_{3}\Delta t \right]-\frac{k}{m}\left[y(t_{n})+y^{\prime}(t_{n})\Delta t+\frac{\Delta t^{2 }}{2}K_{1}\right]\end{array} \right. \tag{6}\]  
where \(K_{1}\), \(K_{2}\), \(K_{3}\) and \(K_{4}\) are intermediate functions, and \(\Delta t\) is the time step in the fluid zone. In the numerical simulation, two operation modes of direct closing and vibratory closing were carried out. The direct closing had a fixed speed \(v_{0}\) (\(v_{0}\) is the speed at which the hoist equipment releases the steel rope). The speed of vibratory closing was required to take the effect of fluid-structure interaction into account.