# 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
#### 3.2.2 Peak Discharge of Outflow Hydrograph  
In this section, peak discharge at dam site, produced by 18 models of partial dam collapse indicated in Table 3, is carefully investigated using the presented 3D CFD model.  
Figure 14: Near\(-\)bed iso\(-\)surface map of q\(-\)criterion at \(t=0.5\) s and \(t=2.0\) s in cases 8A and 8B.  
Figure 15: Contour maps of \(q-\)criterion at \(t=0.5\) s at different water depths.  
Both the spike-shaped impulsive peak and the long-duration normal one are observed when breach size is enlarged (Figure 16). The latter is more significant than the former. This point shows an opposite conclusion to that of total dam collapse mentioned in [10]. The larger the breach size is, the greater the peak discharge generated. Moreover, regarding dam gate locations A or B, the significant difference between peak discharge values yielded by case A and case B is larger when broken width is greater. This point can be explained by water depth and \(u\) velocity profiles in Figures 9 and 10, because discharge is an integral production of water depth and \(u\) velocity. Although minor change occurs between the two time series of \(u\) velocity of cases A and B (Figure 10), the major difference between the peaks of water depth yielded by cases A and B (Figure 9) causes dominant differences between peak flow rates.  
[PERSON] (1917) proposed an empirical equation to predict peak discharge in cases of full depth partial width breaches in rectangular channels [11]:  
\[\frac{Q_{p}}{b\sqrt{g}l_{o}^{3/2}}=\left(\frac{8}{27}\right)^{b/B} \tag{7}\]  
Table 4 shows the dependence of ratio \(\eta=\frac{Q_{p}}{b\sqrt{g}l_{o}^{3/2}}\) with b/B and initial water depth \(h_{0}\) (m); \(Q_{p}\) (m\({}^{3}\)/s) is peak discharge. The range of b/B in this study is from 0.08 to 0.25, which has rarely been studied in previous literature. When \(h_{0}\) remains constant, \(\eta\) calculated by [PERSON]'s formula is also constant. However, this value is varied and increased when water stage increased. Most numerical results are greater than the analytical solution of [PERSON] (1917) and symmetrical location of dam gate gives values of \(\eta\) greater than the asymmetrical one. Ascending initial water stage is, significantly, the difference between \(\eta-_{\text{Schoklitsch}}\) and \(\eta-_{\text{3D}\text{-B}}\). Regarding breach width b = 75 cm, 4.6%, 8.3% and 10.8% are the absolute error of \(\eta-_{\text{Schoklitsch}}\) and \(\eta-_{\text{3D}\text{,B}}\) corresponding with 3 values of water depth 20 cm, 40 cm and 60 cm, respectively. The scatter plot of comparison between peak discharge of models A and models B, with the analytical result of [PERSON], is reported in Figure 17. The former is closer with a diagonal line, than the latter, and all of them are greater than the analytical solution. Therefore, the VOF-LES model is straightforward in exactly capturing 3D hydraulic features of dam-break wave propagation.  
Figure 16: Discharge hydrographs taken with output interval 0.1 s. Solid line–cases B. Marker line–cases A. (**a**) \(h_{0}\) = 40 cm, (**b**) \(h_{0}\) = 60 cm.