# Numerical Study of Partial Dam-Break Flow with Arbitrary Dam Gate Location Using VOF Method
## 3 Results and Discussions
### Effect of Initial Water Stage, Breach Size and Position of Dam Gate on Hydraulic Characteristics  
#### 3.2.1 Dynamics and Structure of Dam-Break Wave Inside Reservoir  
Eighteen different models are implemented to numerically investigate the effect of position of dam breach and its width (\(b\)) or breach ratio (\(b/B\), where \(B\) is total length of dam) and initial water depth (\(l_{0}\)) on hydraulic characteristics of both rarefaction and shock waves (Table 3). Initial stage \(h_{0}\) at the reservoir are taken as 20 cm, 40 cm and 60 cm, respectively, while downstream is dry. Two positions of reservoir outlet, asymmetrical (A) and symmetrical (B) with the central line of the dam, are involved. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Gauge** & \multicolumn{3}{c}{**a**} & \multicolumn{3}{c}{**b**} & \multicolumn{3}{c}{**c} \\ \hline
**Model** & \multicolumn{3}{c}{**3D**} & \multicolumn{3}{c}{**2D**} & \multicolumn{3}{c}{**3D**} & \multicolumn{3}{c}{**2D**} & \multicolumn{3}{c}{**3D**} & \multicolumn{3}{c}{**2D**} \\ \hline
**Mesh** & **0.025** & **0.015** & **0.012** & **0.015** & **0.025** & **0.015** & **0.012** & **0.015** & **0.025** & **0.015** & **0.012** & **0.015** \\ \hline \(h\) & 0.129 & 0.0  
To carry out the formation and structure of a rarefaction wave on the reservoir, 6 cases, 7A, 7B, 8A, 8B, 9A, 9B, corresponding with \(h_{0}=60\) cm, are selected. The simulation time for all models was about 2 h of total dam-break time of 5.0 s, exceeding 3.02 million cells. Streamlines show the direction of fluid elements that will travel assuming the flow at that point in time is steady [33]. The sudden overturning of reservoir water led to the formation and propagation of the rarefaction wave on water shortage and shock wave at the flood plain (Figures 4 and 5). The free surface oscillates slightly within the reservoir in the early stage. In turn, the wave front velocity is increased by increasing breach size, because kinetic energy dissipation increases at the dam site in early-stage \(t=0.5\) s. In case B, streamline is performed symmetrically (see Figure 4), while the result in Figure 5 affirms the strong asymmetrical streamline, due to the effect of asymmetric location of the dam gate. Moreover, when \(t=2.0\) s, return flow within the reservoir is split into two parts, the small part goes to both the upper and lower corners of the reservoir and the larger one goes through the mouth of the reservoir in all cases 7B, 8B, 9B. At the same time, a small portion of water volume travels along the upstream bank of the reservoir and the larger one goes through the tank's outlet in all models 7A, 8A, 9A. Later on, at \(t=4.0\)s, all fluid elements within the reservoir move on towards the tank's outlet, while the wetted area at flood plain generated by the smallest gate's width is the largest (Figure 5). Figures 6 and 7 present water depth and velocity profiles at the center of the reservoir's mouth when \(t=1.5\) s. The dam-break wave propagation is reduced by decreased breach width \(b\) and depends slightly on location of dam gate (Figure 6). Conversely, the wave front of velocity downstream decreases from \(b=0.25\) m to \(b=0.75\) m, and is independent of the reservoir outlet's position. The numerical result of depth-averaged velocity gets its maximum value at \(x=1.0\) m in both 7A and 7B, then is gradually disregarded. While the value generated by larger breach size increases near the dam site and remains constant for the rest of the floodable area. On the other hand, the intersection point of three water depth profiles is after the dam site's place in both cases of dam gate's location, while the conjunction of 3 velocity profiles is placed at \(x=0\). Figure 5: Streamline of magnitude velocity at different time in 3 models: (**a**) 7A, (**b**) 8A, (**c**) 9A. Figure 6: Water depth profiles at central axis of partial dam collapsed. Transverse \(v\) velocity symmetrically distributes at the cross section near reservoir outlet in the 3 models 7B, 8B, 9B (Figure 8).