4.1. Model Performance with Different Building Representations, Manning Coefficients, and Mesh Resolutions
The Manning coefficient is an extremely significant parameter in numerical models with the BR method. When the Manning coefficient is greater than 104 , numerical model performance does not improve, which is of great practical importance in such models. While establishing an urban flood numerical model, collecting accurate terrain and other urban data is difficult. Therefore, the resistance effect of buildings is expressed by the Manning coefficient. The conclusions of this study provide further support for the use of the Manning coefficient. However, the 104 of Manning coefficient has no physical meaning in the real world, and is only useful in the numerical model. Because there are windows, basement, square and so on in building area, the flood could in fact flow through the building area partly. Therefore, the set of Manning coefficient not only depends on the collection of basic hydrologic data, but also depends on the calibration and validation of the numerical model, which should be agree with the actual states.
In terms of model performance, the BB, BH, and BR methods yielded results that were similar to the measurements, as shown in the hydrographs in
Figure 9 and
Figure 10. However, the water depth predictions form more uniform and smoother curves than the measurements. Dottori et al. also reported this phenomenon [
31]. An interesting observation in the hydrographs is that the BH method exhibits greater data oscillation than the other two methods, indicating that the BH method has better performance in capturing the complex interaction between buildings and water flow than the other two methods. Additionally, for nearly all hydrographs, peak depth occurs a few seconds earlier in the predictions than it does in the measurements. This may be due to flow front celerity being influenced by mesh resolution; as observed in
Figure 6 and
Figure 7, the peak time for 1 cm mesh resolution is a little behind that for 5 cm mesh resolution, which is closer to the measurements. Furthermore, the time spent for water to move differs from reality when the grids located in front of the buildings are changed from dry to wet [
25]. An et al. also reported this phenomenon, citing the following potential reason: as the inflow rate of the experiment was calculated by gauging the tank upstream of the flow domain, a time gap occurred between the simulated and numerical models [
16].
Mesh resolution influences model performance. The runtime computational cost of the 1, 2, and 5 cm mesh resolutions are 2.5 h, 0.6 h, and 0.2 h respectively, which means that the finer is the mesh resolution, the greater is the central processing unit (CPU) runtime. The water depth peak value and peak time also showed differences across the mesh resolutions. For certain gauges, e.g., P3, P4, and P5, which were stroke by flow directly, the predicted peak depth values are greater than the measured values at the different mesh resolutions. For other gauges, e.g., P7, P9, and P10, which were located at the back of the building zones, the predicted peak depth values are slightly smaller than the measured values at different mesh resolutions. Although different mesh resolutions were implemented, the phenomenon of the predicted time of water depth peak being ahead of the measurements did not change, as shown in
Figure 6 and
Figure 7. However, the peak depth time for 1 cm mesh resolution showed slight improvement compared with the results of the 5 cm mesh resolution, but this is not very subtle. Data oscillation is more apparent for the results of 1 cm mesh resolution compared with that of 2 and 5 cm mesh resolutions. This oscillation is more noticeable for the staggered layout than the aligned layout, particularly in gauges P6, P8, and P9. This observation indicates that reducing the mesh resolution may help the numerical model capture data oscillation.
The aforementioned phenomenon should be noted and considered in numerical models in the future.
4.2. Abnormal Results Analysis
Explaining the results of gauge P5 in the aligned case and those of gauge P8 in the staggered case is difficult. These results may be related to the potential for complex flow fields at these two locations, which might cause unusual water flow. Some possible explanations are as follows.
For gauge P5 in the aligned case, the flood passed through the middle of two buildings in the first row; therefore, the P5 location was impacted by water shock at about 12.4 s, before the water jump occurred. An important finding was that the simulated values of the water jump’s depth were greater than the observed values. Similar water jump values can be found at gauges P3 and P4 in the aligned and staggered cases, respectively. After several seconds, the simulated values are consistent with the measurements (see
Figure 9 and
Figure 10). For example, after ~20 s, the simulated results of gauges P3 and P4 are very accurate, and after about 50 s, the simulated result of gauge P5 also becomes accurate. This phenomenon indicates that numerical models have some issues in dealing with abruptly changing values. Additionally, for gauge P5, which was located in the middle of four buildings and also faced the strike of the flood, the numerical model has some difficulties in simulating the oscillation of the values in the complex flow field. However, the discrepancies in water depth variation between the predictions and measurements were corrected over time; eventually, they will be eliminated.
Because P7 and P8 were located in symmetrical positions in the staggered case, comparing the results obtained at these two locations reveals that the predicted values for two sites are similar; however, the measurement values are obviously different. A possible explanation for this is that a concealed channel groove exists in the terrain to the north of the buildings (see
Figure 2).
Figure 11 shows that water velocity is faster in the channel groove. Therefore, the channel groove has some influence on velocity and depth at gauge P8 (see
Figure 11). According to Bernoulli’s equation, higher flow speed occurs at the point where pressure is lower. Therefore, dragging due to the high-speed flow in the channel groove causes more water to flood gauge P8 in the northeast direction; i.e., greater water flow occurs at gauge P8 and the water depth therein is slightly higher than that at gauge P7. In addition,
Figure 11 shows that the channel groove has a greater influence in laboratory measurements, resulting in a higher water depth at gauge P8. The hydrographs of predicted and measured results at gauge P8 are almost parallel after 30 s (
Figure 10), suggesting that the influence of the channel groove is underestimated in the numerical model when compared with reality.
The results of the Toce River Valley case demonstrate that certain challenges exist when using numerical models to simulate resistance under complex flow conditions. These results might be explained in part by the assumptions of classical SWEs, e.g., the pressure distribution is approximately hydrostatic and the vertical velocity is ignored. Owing to the existence of the water jump, the channel groove, and other complex flow conditions, the limitations of the SWEs must be recognized [
32]. Thus, further research is required to deal with complex flow.
The simulated velocity profiles were analyzed using the three building representation methods and the two building layouts. We present the velocity profiles along the line segment between gauges P5 and P8 at 15 s in
Figure 12. In the aligned layout, no building existed on this line; however, two buildings existed on this line in the staggered layout. Interestingly, the velocity of the BB method is lowest among the three methods at gauge P5 in the aligned case and at gauge P8 in the staggered case.
Figure 9 and
Figure 10 show that the water depth is greater at the P5 and P8 locations, partially confirming the law of conservation of mass. In addition, for the BB method, the flow velocities are not zero in building zones, indicating that the water overflowed the building zones (although the velocities are relatively small). These results demonstrate that in contrast to the BH and BR methods, the BB method could not prevent the water from entering the building zone. In order to simulate the urban buildings realistically, setting an artificially high elevation value could prevent the water from entering the building zone in the BB method and future studies on this phenomenon are recommended. Based on the velocities along the line segment, some differences exist among the three building representation methods. Owing to the absence of observed velocity data, the simulated velocities could not be assessed in relation to the measurements.
The Froude number is used herein to further analyze the interactions between buildings and water flow.
Maps of the Froude number were drawn for an instant of time (at about 15 s) at the peak water depth with three building representation methods and two layouts. As shown in
Figure 13, there are complex flow regimes around the buildings. Because of the difference between the inflow discharges in two layouts (see
Figure 2c), we could find the flood flow of aligned layout has a lag time of 1.4 s to the staggered layout. In the same building layout, it is clear that the maps of Froude number are pretty similar, such as
Figure 13a–c or
Figure 13d–f, and the flow regimes around the building area are same under the different building representation methods, especially the interactions of subcritical and supercritical flows. Behind the buildings and along the flow direction, there is subcritical flow, whereas in the middle of the buildings and along the flow direction, there are subcritical and supercritical flows around the buildings. When the flow field is stable after 15 s, the distributions of subcritical and supercritical flows are also stable. Furthermore, TELEMAC-2D provides good performance during the transition between subcritical and supercritical flows. Additionally, the results of Froude number maps demonstrate once again that TELEMAC-2D with the three building representations is sufficiently accurate for urban flood simulations.
Figure 14 shows the Froude numbers at gauges P5 and P8 in the aligned and staggered cases, respectively. Notably, the Froude numbers range mostly between 0.8 and 1.1 when the flow field attains stability after 15 s. Critical flow has a complex flow regime and strong sensitivity toward water depth [
33]. Thus, accurately simulating water flow depth is difficult when the modeled flow regime is critical.
The different mesh resolutions have some influence on the two abnormal results outlined above, as shown in
Figure 6 and
Figure 7. For gauge P5 in the aligned layout, the finer mesh resolution may help reduce the discrepancy between the predictions and measurements. For gauge P8 in the staggered layout, the finer mesh resolution has greater capacity to capture the data oscillation and thereby reflect reality. A finer grid resolution is better than a coarse resolution in terms of the depth value and peak time. However, a coarse grid with a classical SWE model would somewhat neglect water storage reduction and flow energy loss [
34]. Therefore, mesh resolution has some sensitivity toward water depth value.
4.3. Further Considerations and Suggestions
This paper used the TELEMAC-2D model to analyze urban flooding by taking a classical experiment test Toce case and compared with previous researches. This paper has provided a great contribution to the problem related to urban flooding model benchmarking, focusing on the effect of different building representation. However, this study does not study detailed real-world events with numerical model, because the research of this paper is basic, and concentrates on the performance and accuracy of this model, and adopts different accuracy indicators to evaluate the simulated results. This study provides some support for using the TELEMAC-2D model to simulate real-world situations. There are many researches focus on the real-world urban flood. Segura-Beltran et al. studied a detailed case using surveys and geomorphologic map to evaluate the performance of the numerical model [
35]. Macchione et al. adopted non-conventional information (such as amateur videos, photographs, news reports, etc.) to establish a past urban flood events model [
36]. Martin-Vide et al. analyzed an extreme flash flood in Spain [
37]. However, Neal et al. mentioned that it is difficult to assess the accuracy of the numerical simulation due to the lack of detailed inundation extent or water surface elevations [
38]. Although there many difficulties to simulate a real-world urban flood, it is vital importance for urban management and development, also it points out the next research direction for the future.
In addition, more consideration can be given to the urban flood in the numerical model. Flood risk management is an important future research field. To this end, numerical modeling is a significant method owing to its efficiency and precision. In general, land-use maps showing impermeable surfaces, particularly gray infrastructure, in urban areas could be collected via refined aerial remote sensing. In urban flood models, some gray infrastructure cannot be infiltrated but water can pass through these surfaces; e.g., roads. Other infrastructure such as buildings cannot be simultaneously infiltrated and overflowed. Therefore, further details, particularly relating to topography and the height of gray infrastructure, need to be considered in actual urban flood models. The Manning coefficient is also highly significant in relation to roads because it represents the flood velocity. In the context of flood risk prediction and damage assessment, further research into model buildings and setup-related parameters in urban districts is warranted.
Finally, as reported previously [
39,
40], several advanced techniques such as 3D flood hazard visualization have been implemented to better present numerical results. Numerical model results could provide support to urban flood management. The TELEMAC-2D model can simulate urban floods, as shown in this study, and as it is an open source code model, it is flexible and adaptive, which is advantageous to developing an urban flood model for risk management. Consequently, related urban flood research can be developed in the future, the results of which could be helpful in terms of flood management.