# Numerical Study of Partial Dam-Break Flow with Arbitrary Dam Gate Location Using VOF Method
## 3 Results and Discussions  
### Mesh Sensitivity and Comparison 2D and 3D Numerical Solutions  
An experimental test case, taken from Shige-eda, is reproduced to assess mesh sensitivity and the robustness and effectiveness of the proposed 3D model [16]. The configuration of the physical model, and six gauge points, are illustrated in Figure 1. The initial reservoir water depth (\(h_{0}\)) was 0.4 m, and downstream was dry. A 50 cm-broken block of dam was placed asymmetrically with the central line of the reservoir and removed instantaneously. Two mesh blocks were set up: block 1 was inside the reservoir and had 3 wall boundaries (W); block 2 was in the flood plain and had 3 outflow boundaries (O). The processing time and mesh quality play important roles in computational costs and accuracy of modeling results, respectively [19]. Sensitivity mesh analysis of the modeling results is carried out on mesh cell dimensions. In this regard, three cell sizes 0.025 m, 0.015 m and 0.012 m were selected. The accuracy of the models in predicting temporal variation of water depth and velocity components were evaluated employing RMSE. All simulations were implemented on the Intel i7 2.9 GHz processor with 8 cores (Table 1). Figure 2 shows very good matching between the numerical results of water hydrograph yielded by the fine mesh of 0.015 m and the finest mesh of 0.012 m and observed data. While underestimation and strong spurious diffusion is observed in results obtained by coarse mesh of 0.025 m, especially at further gauges d, e, f. In comparison with 2D results taken from [18], 3D resolution of shock wave in this study is much better. Although arrival time of flood wave front is more or less the same at all studied points. However, in the early stage of dam collapse, the 3D CFD model captured peak flow very well. Dominant displacement is witnessed at two points, a and c at the center line of the dam gate, where the peak flow obtained by Flow 3D near the dam site is equal to about twice the value yielded by SWEs. The reason is that the 3D CFD method accounts for turbulent phenomena in rapid flow, while 2D SWEs do not. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model Mesh Size** & **Mesh Quality** & **Number of Cells** & **Time Step** & **Dam–Break Total** \\
**\(\Delta\mathbf{x}\times\Delta\mathbf{y}\times\Delta\mathbf{z}\) (m\({}^{3}\))** & **(-)** & **(s)** & **Time (s)** & **CPU Time(h)** \\ \hline
0.025 \(\times\) 0.025 \(\times\) 0.025 & Coarse & 510,328 & 0.0020 & 30 & 2.2 \\
0.015 \(\times\) 0.015 \(\times\) 0.015 & Fine & 2,183,656 & 0.0013 & 30 & 4.9 \\
0.012 \(\times\) 0.012 \(\times\) 0.012 & Finest & 4,239,769 & 0.0010 & 30 & 8.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulation properties. Figure 1: Configuration of experimental model. Additionally, Root Mean Square Error (RMSE) is the chosen index to validate the model employed. It is expressed by the following equation:  
\[RMSE=\frac{\sqrt{\frac{\sum\limits_{i=1}^{n}\left(X_{i,exp}-X_{i,sim}\right)^{2} }{n}}}{\left(X_{exp,max}-X_{exp,min}\right)} \tag{5}\]  
In Equation (5), \(X_{i,exp}\), \(X_{i,sim}\), \(X_{exp,max}\) and \(X_{exp,min}\) represented the empirical, numerical, maximum and minimum values of the \(X\) variables, respectively. If the value of RMSE is from 0 to 0.1, the model has excellent accuracy, and it is adequately accurate if RMSE is from 0.1 to 0.2 [26]. Table 2 illustrates value of RMSE of water depth and \(u\), \(v\) velocity components, corresponding with 3D results of three resolution meshes and 2D at 6 studied points. Certainly, 3D solutions more nearly coincide with empirical data than 2D solutions in almost all gauges. The selected CFD model performs horizontal \(u\)-velocity better than the transverse \(v\) component. Two fine grid cell sizes 0.015 m and 0.012 m give adequate accuracy in several gauges when value of RMSE of all studied factors is smaller than 0.2 (water depth at gauges a, d; \(u\)-velocity at gauges b, c, e, f).