## 3 Comparison between Flow-3D, MIKE 3 FM, and 1D Exact Riemann Solver Predictions
### Free Surface
#### 3.1.1 Free Surface during the Early Stage
Significantly, as time passed, the error decreased from \(T=1.127\)-\(6.637\); the Flow-3D, MIKE 3 FM, and 1D approaches improved the forecast by reducing the RMSE values by \(\sim\)42.29%, \(\sim\)22.87%, and \(\sim\)54.84%, respectively, demonstrating that the differences between the values predicted by the three models to solve dam-break flows for dry beds were very small over time. Therefore, any of the three models can be selected as an appropriate model for predicting the free surface after the dam break for the dry-bed. Figure 7 shows the surface profiles with a depth ratio of \(\alpha=0.1\); in contrast to the observation of the dry-bed case, a jet was formed by the collision between moving and still water. A jet was observed after the dam break during the initial stage in [PERSON] et al. [15]. The formation of a wave-like vertical jet propagating downstream for the MIKE 3 FM was quite noticeable, which was consistent with the mushroom-jet results for the RANS (Flow-3D) calculations and experiments. The difference in wave shape, especially during the early stages (\(T=1.565\)-\(6.511\)), were attributed to the MIKE 3 FM and 1D analytical solution, and they were generally within 0.07 to 0.1 in the RMSE values, decreasing to 0.05 at \(T=8.891\) (Table 1). The reason for the more apparent difference between these two models and Flow-3D primarily lies in the assumption of a hydrostatic pressure distribution and the negligible vertical acceleration [30]. Despite significant deviations among the MIKE 3 FM simulation, analytical solution, and the experiments at the downstream wavefront, these two models successfully predicted the flow features after wave breaking (Figure 7). Although the differences between the calculated results compared with those of the experiments decreased when \(T>8.891\), the error was reduced. Among the three models, the Flow-3D had the best performance with the lowest RMSE, with a reduction to 0.04 from 0.09, which indicated that the estimation quality of the Flow-3D model was better than that of the MIKE 3 FM and 1D analytical solution for predicting the free surface during the initial stage in a wet bed with a depth ratio of \(\alpha=0.1\). Figure 8 shows a comparison of the water surface profiles in the wet-bed condition with \(\alpha=0.4\); the wavefront profiles during the initial stage were similar to those observed at the wet-bed case \(\alpha=0.1\). The enlargements shown in Figure 8 for \(T=1.565\) and \(2.379\) show the mushroom-jet formation. The RANS results obtained by Flow-3D nearly coincided with the experimental data at different times, and the RMSE values were \(<\)0.06. There were similar problems at the early stages at \(\alpha=0.1\) for the 3D SWE (MIKE 3 FM) and the analytical results. Specifically, there was poor consistency with the measured results, particularly at early times after the dam break, but the differences between the MIKE 3 FM, analytical solution and flow depth measurements for this case behaved more gently than at the water depth ratio \(\alpha=0.1\). The error analysis in Table 1 also shows smaller RMSE values than in the last case; the RMSE of the simulated free surface at \(T=1.565\)-\(8.891\) for the MIKE 3 FM ranged from 0.09 to 0.03 and that for the 1D analytical solution was from 0.09 to 0.04, but the error was generally reduced when \(T>4.007\). Thus, these results show that deviations decrease in the graphs when the depth ratios (\(\alpha\)) increase. To summarize, Figures 6-8 depict the evolution of the free surface profiles after the dam break. The consistency between the analytical solution and the MIKE 3 FM results was satisfactory for the dry-bed tests, but there was quite a noticeable difference in the formation of a wavefront propagating downstream for the wet-bed cases. In both the dry- and wet-bed cases, the measured profiles were very close to the corresponding ones calculated using the Flow-3D model. The comparisons demonstrated that the Flow-3D model gave more accurate results than the numerical results calculated using MIKE 3 FM.