## 4 Wind-Driven Rain Load on Conductor  
After the mesoscale WRF simulation of Lekima, local wind field and rainfall intensity in Zhoushan Island was obtained, and the subsequent procedure of our simulation system was to solve the coupled wind and rain field around the transmission line and find out the WDR load on conductors. The WDR simulation was carried out at 0000 UTC on 10 Aug with a mean wind speed of \(U=20.3\) m/s, a wind angle of \(\beta=12.1^{\circ}\) and a rainfall intensity of \(R_{\rm h}=39\) mm/h for the targeted transmission line in Zhoushan Island, as shown in Figure 10. Two-dimensional computational domains and meshes of the fluid region around the cross section of the conductor are established, as presented in Figure 11. Based on the drag coupling between rain phases and wind phase, the horizontal raindrop velocities are assumed to be equal to the inflow wind velocity, and the vertical velocities of rain phases are set by Equation (10). The phase volume fraction \(\alpha_{k}\) at the inlet boundary can be calculated as  
\[\alpha_{k}=\frac{R_{\rm h}f_{\rm h}(R_{\rm h},D)}{V_{\rm t}(D)}, \tag{19}\]  
where \(R_{\rm h}\) is the rainfall intensity; \(f_{\rm h}(R_{\rm h},D)\) represents the volume fraction of raindrops with diameter \(D\) in all raindrops passing through a unit area horizontal plane in unit time, i.e., flux fraction, can be calculated as  
\[f_{\rm h}(R_{\rm h},D)=\frac{D^{3}N(R_{\rm h},D)V_{\rm t}(D)}{\int_{0}^{\infty} D^{3}N(R_{\rm h},D)V_{\rm t}(D)dD}, \tag{20}\]  
Figure 9: Six-hour averaged rainfall intensity at six meteorological stations during Lekima. (**a**) Station Honglia; (**b**) Station Shangyu; (**c**) Station Hangzhou; (**d**) Station Huzhou; (**e**) Station Linan; (**f**) Station Yuhuan. where \(N(R_{\rm h},D)\) is the raindrop size distribution function which follows the modified \(\Lambda\) distribution proposed by [PERSON] [43], and is expressed as  
\[N(R_{\rm h},D)=N_{0}(R_{\rm h})D^{2.93}\exp(-\Lambda(R_{\rm h})D), \tag{21}\]  
\[N_{0}(R_{\rm h})=1.98\times 10^{-5}{R_{\rm h}}^{-0.384} \tag{22}\]  
\[\Lambda(R_{\rm h})=5.38{R_{\rm h}}^{-0.186} \tag{23}\]  
Figure 12 presents the rain phase flux fraction with different rainfall intensities. According to the illustrated raindrop size distribution, the discretized raindrop sizes are set to be \(D=0.5\sim 6.0\) mm with the increment \(\Delta D=0.5\) mm as inlet rain condition, for the WDR simulation of rainfall intensity \(R_{\rm h}=39\) mm/h. Figure 11: Two-dimensional mesh of computational domain for WDR. Figure 10: Location of the target transmission line. The volume fraction and raindrop velocities at the top boundary are defined in the same way as the inlet boundary condition. The boundary conditions at the bottom, outflow boundary, and the conductor surface are defined as  
\[\left\{ \begin{array}{lcl}\frac{\partial\alpha_{k}}{\partial n}=0,&\frac{\partial u _{k}}{\partial n}=0&\text{for }\mathbf{u}_{k}\cdot\mathbf{n}>0\\ \alpha_{k}=0,&\mathbf{u}_{k}=0&\text{for }\mathbf{u}_{k}\cdot\mathbf{n}<0^{\prime} \end{array} \right. \tag{24}\]  
where \(\mathbf{n}\) is the unit normal vector pointing out of the computational domain. The first expression in Equation (24) indicates that the raindrops will impact the solid surface just like no blocking. Therefore, based on the momentum theorem, the rain load caused by raindrop impact can be calculated as [44]  
\[p_{rain}=\int_{0}^{\infty}\rho_{l}\alpha_{k}V_{n}(D_{k})^{2}dD_{k}, \tag{25}\]  
\[\tau_{rain}=\int_{0}^{\infty}\rho_{l}\alpha_{k}V_{n}(D_{k})V_{\tau}(D_{k})dD_{k} \tag{26}\]  
where \(p_{rain}\) and \(\tau_{rain}\) refers to the rain phase pressure produced by the momentum loss in the normal and tangential direction, respectively. \(V_{n}(D_{k})\) and \(V_{\tau}(D_{k})\) are the velocity of rain phase with diameter \(D_{k}\) towards the solid surface along normal and tangential directions, respectively.