# Subgrid Model of Fluid Force Acting on Buildings for Three-Dimensional Flood Inundation Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fundamental Concept of SG Model for Building Fluid Force

#### 2.2. Fundamental Equations of SG Model

_{3D}is divided into Δt

_{3D1}and Δt

_{3D2}, and within Δt

_{3D1}, horizontal 2D and 3D calculations are performed, and the results of both calculations are exchanged. However, within Δt

_{3D2}, only horizontal 2D calculations are performed without 3D calculations, and the results of the 3D calculations performed within Δt

_{3D1}are continuously reflected in the horizontal 2D calculations. Thus, Δt

_{3D2}, which does not perform 3D calculations, is not restricted by the CFL condition and Δt

_{3D2}can be set to a large value, resulting in improved calculation efficiency. It should be noted that the computation time interval Δt

_{2D}for a horizontal 2D analysis does not necessarily have to match that of Δt

_{3D1}(Figure 3b). To reflect the results of the 3D calculation on the horizontal 2D calculation, the difference between the depth-averaged terms in 3D equations of motion and each term in the horizontal 2D equations of motion is calculated as a correction term at Δt

_{3D1}. The correction term is incorporated in the horizontal 2D equations of motion. In contrast, to reflect the results of a horizontal 2D calculation in a 3D calculation, the depth-averaged velocity in the previous 3D calculation is replaced by the result of the horizontal 2D calculation. In other words, the vertical velocity distribution in the previous 3D calculation is retained, but the depth-averaged velocity is updated.

_{s}, u

_{n}, and w* are the velocities in the s, n, and σ directions, respectively; R is the radius of curvature in the s coordinate; N = n/R; g is the acceleration owing to gravity; $\rho $ is the density of water; A

_{H}and A

_{V}are the horizontal and vertical eddy viscosity coefficients, respectively; and F

_{bs}, F

_{bn}and F

_{gs}, F

_{gn}are the building and bridge girder fluid forces per unit mass in the s and n directions, respectively. In the Hy2-3D model, the vertical eddy viscosity coefficient A

_{V}is expressed by the zero-equation model, which is one of the turbulence models, and the horizontal eddy viscosity coefficient A

_{H}is expressed in a simple form proportional to A

_{V}:

_{bs}and F

_{bn}obtained using the SG model, the fluid force is obtained for each building and distributed to each grid according to the fraction α occupied by the building, as depicted in Figure 1. In particular, if the number of buildings in each grid is M

_{max}, the fluid force in the s direction acting on building m (=1 − M

_{max}) is f

_{bs}(m), and the volume of the building with the volume V

_{b}(m) in the grid is V

_{b}′(m), and the fluid force distributed in this grid is f

_{bs}(m)V

_{b}′(m)/V

_{b}(m). When deriving the momentum equations, both sides are divided by the mass of the control volume ($=\rho \mathsf{\Delta}V$, $\mathsf{\Delta}V$: grid volume), so that F

_{bs}is expressed as follows:

_{b}′(m) to the grid volume ΔV. Similarly, if the fluid force in the n direction acting on building m is f

_{bn}(m), F

_{bn}is expressed as follows:

_{bs}, f

_{bn}in the s, n directions acting on individual buildings are expressed using the general drag formula as follows:

_{Db}is the drag coefficient of the building, and ${\widehat{u}}_{s},{\widehat{u}}_{n}$ are the velocities in the s, n directions interpolated at the building center. The available building information includes the building width and building plane area A

_{b}. However, because it is complicated to calculate the building width perpendicular to the flow direction data obtained from this calculation, we assume that the building is square and obtain the building mean width B using the following equation:

_{Db}in Equations (8) and (9) are set to 1.2 based on the experimental results reported by Kuwahara [51]. The fluid forces on the bridge, F

_{gs}and F

_{gn}, are also expressed in the same way as the fluid forces on the building but are omitted here.

_{s}and U

_{n}are the depth-averaged velocities in the s and n directions, respectively:

_{H}

_{2D}is the depth-averaged horizontal eddy viscosity coefficient, C

_{f}is the bottom friction coefficient, and G

_{s}and G

_{n}are correction terms in the s and n directions, respectively, reflecting the results of the 3D calculation in the horizontal 2D calculation. The bottom friction coefficient C

_{f}and friction velocity ${U}_{*}$ in Equation (4) are expressed as follows:

_{s}and G

_{n}used to account for these effects are expressed by the following equations, which represent the differences between the depth-averaged 3D and horizontal 2D results:

_{s}and G

_{n}do not include a vertical diffusion term. For details on the calculation procedure for the Hy2-3D model, please refer to Nihei et al. [47]. Additionally, it is noted that the density of the fluid does not vary, and the proposed model does not apply to saline water.

#### 2.3. Study Site

^{2}, and a population of approximately 140,000 within the basin. It is a first-class river managed by the national government [53]. As depicted in the elevation contour (Figure 4a), the Kuma River Basin is surrounded by steep mountains, and the river is one of the three most rapid rivers in Japan. The topographical features of the Kuma River Basin include the Yatsushiro Plain in the lower reaches (0–10 km point (kp)), a narrow mountain channel in the middle reaches (10–52 kp), and the Kuma Basin in the upper reaches (52 kp). The entrance to the middle reaches becomes constricted during floods, and the Hitoyoshi urban area, located upstream of the constricted area, tends to become vulnerable to inundation damage [53].

#### 2.4. Computational Conditions

^{−1/3}s in the 58–64 kp section of the river channel and to 0.035 m

^{−1/3}s in the other sections. To verify the fundamental performance and effectiveness of the SG model, we compared the case of the SG model (Case 1) with the case where the roughness coefficient had an equivalent roughness value in the flooded area, as in the BR model (Case 2) as well as the case where the roughness coefficient was constant (Case 3-1, n = 0.06 m

^{−1/3}s; Case 3-2, n = 0.03 m

^{−1/3}s). The following equation was used for the equivalent roughness n in Case 2 [50]:

_{0}is the roughness coefficient on the ground (=0.03 m

^{−1/3}s) and θ is the occupancy of the building in the grid. In Case 1, the roughness coefficient was required to evaluate the bottom friction in the flooded area and was set uniformly to n = 0.030 m

^{−1/3}s.

_{b}and the location of the building center. A histogram of building width B (Figure S3) indicated that most of the building widths measured between 8 and 12 m, which was approximately the same as the grid resolution. Some buildings were smaller than the grid resolution, whereas others spanned several grids. In this analysis, the CPU time was approximately 12 h when we used an Intel(R) Xeon(R) W-2245 CPU @ 3.90GHz with RAM of 64.0 GB computer for numerical analysis.

## 3. Results and Discussion

#### 3.1. Validation of Hy2-3D Model

^{−1/3}s) and 3-2 (n = 0.03 m

^{−1/3}s) with constant roughness coefficients, the maximum absolute values of the peak water-level difference were 0.15 and 0.45 m, respectively. This result indicates that even if n is kept constant, the results do not change significantly from those of the equivalent roughness model if an appropriate value (Case 3-1 in this case) is used. In Case 3-2, n = 0.03 m

^{−1/3}s, the peak water level difference in the river is roughly within 0.25 m except at 61 kp, and the impact of the resistance evaluation of the flooded area on the river water level is very small because the river and inundation flows are combined. It is concluded that the accuracy of the Hy2-3D model is generally good, regardless of the building resistance model used.

^{2}= 0.990, indicating that the calculated values were generally in good agreement with the observed values. Similarly, for the peak water depth, the RMSE of the difference between the observed and calculated values was 0.45 m, and the slope of the regression line was 0.930 and R

^{2}= 0.937, indicating that the calculated values were in good agreement with the observed values. The RMSE of the peak water depth was larger than that of the peak water level because it reflected the spatial variation in the ground height data.

^{2}for all cases. The RMSEs in Cases 2 and 3-1 were similar to those in Case 1, and the slope of the regression line was almost unity. In Case 3-2, the RMSE was larger than those in the other three cases for both the peak water level and depth. A significance test between Case 1 and the other three cases for the difference between the calculated results and the observed data indicated a statistically significant difference (p < 0.05) only in Case 3-2 but not in Case 2 or 3-1. Thus, it is quantitatively clear that there is no statistically significant difference between Case 1 of the SG model, Case 2 of the equivalent roughness model, and Case 3-1 of the appropriate constant roughness coefficient (=0.06 m

^{−1/3}s) with respect to the reproducibility of water level and depth in the river and inundation flow analysis. It is clear that the impact of the building resistance evaluation model is small. The validity of the Hy2-3D model, which is the basis of the analysis, was also verified. The high reproducibility of the water level and depth distribution in the Hy2-3D model, despite the simple and almost uncalibrated setting of the roughness coefficient in the river channel, may be attributed to the good reproduction of the complex flow distribution and the appropriate introduction of resistances, such as bridge girders.

#### 3.2. Horizontal Map of Velocity Distribution

^{−1/3}s) indicates a high flow velocity and low water depth across the entire cross-section. In Case 3-1 (with n = 0.06 m

^{−1/3}s), the velocity levels are similar to those in Cases 1 and 2, but the fluctuations in the velocity distribution are smaller, and there is no indication of a decrease in velocity near the buildings or an increase in the velocity on the road without buildings. However, in Cases 1 and 2, the contrast in velocity fluctuation was larger than those in Cases 3-1 and 3-2, with lower velocities in the building area and higher velocities on the road. However, a closer look reveals that the flow velocity in Case 1 is higher than that in Case 2 on the road and in areas without buildings and that the flow velocity in the grid where buildings exist is often larger for Case 1 than for Case 2. The RMS of the flow velocity in the A-A′ cross section is 1.20 and 1.16 m for Cases 1 and 2, respectively, indicating that the fluctuation of flow velocity for Case 1 is larger than that for Case 2. This reflects the fact that the difference in flow velocity between the grids with and without buildings is larger in Case 1, as described above. Case 1, which uses the SG model, indicates low velocities on the building grid and high velocities on the grid without buildings, for example, on the road, owing to low resistance, suggesting that the SG model adequately evaluates the fluid force acting on buildings. The values of velocity on the grid with buildings were in the order Case 1 > Case 2 because the high velocities on the nonbuilding grid, such as roads, diffused horizontally and caused the velocities on the building grid to be relatively large. In addition, because equivalent roughness is used in Case 2, the roughness coefficient affects the vertical and horizontal eddy viscosity coefficients (Equations (4) and (5)) as well as the bottom friction force in this model. Therefore, the spatial variation in the velocity distribution is expected to be less sharp than that of the SG model (Case 1) because of the effect of the increased roughness on the area around the building grid. It is also noted that the water level decreased and increased near the lateral distance of 200–250 m and 250–400 m, respectively. This is because the higher ground elevation in the lateral distance of 200–250 m results in lower water levels due to high drag and inadequate water supply from the upstream side.

#### 3.3. Vertical Distribution of Streamwise Velocity

^{−1/3}s, which is a significantly large value (Figure 10b). This results in a significant roughness height k

_{s}, which is considered responsible for the negative velocity near the bottom. This is similar to the zero-plane displacement in atmospheric turbulence fields over urban canopies [58]. These results indicate that the equivalent roughness model has limitations in accurately reproducing 3D flow velocity distributions around buildings with large roughness coefficients. It was also suggested that the SG model can reproduce the vertical velocity distribution based on the vertical structure of a building.

#### 3.4. Hydraulic Factors of Building Damage

^{2}are also shown. The p-values obtained by the t-test are also depicted to check for significant differences between the results of Cases 1 and 2. In Case 1, the fluid force F is obtained directly for each building, but not in Case 2. Therefore, the same method used in Case 1 was applied to calculate F in Case 2. The water depth h was plotted on y = x, and there was no significant difference between the cases (p > 0.10). This is because, as depicted in Section 3.1., the present inundation pattern is a flood in which the river and inundation flows are combined, and the water level of the river largely determines the water level in the inundated area. The variation in the flow velocity between the cases increased, particularly when the velocity exceeded 2.0 m/s. It was confirmed that the velocity in Case 1 was generally larger than that in Case 2. There was a statistically significant difference between Cases 1 and 2 (p < 0.10) at the 10% significance level. For the unit width discharge q and moment qh, the variation increased with v, and a significant difference was confirmed between the two cases (p < 0.05). Furthermore, for the fluid force F, the variation between the two cases was larger than that for the flow velocity, with the slope of the approximate line reaching 1.07. Because fluid force F is the product of h and v squared, the effect of the flow velocity was more pronounced. The difference between the two cases was significant at the 10% level (p < 0.10). The fluid force is a flood index that determines building damage, and the fact that this assessment differs significantly between the SG model and the conventional equivalent roughness model suggests that the assessment of building damage differs significantly depending on the difference in the models.

## 4. Conclusions

- In terms of the reproducibility of water levels and depths in river and inundation flow analyses, it was confirmed that the calculation accuracy of the Hy2-3D model was generally good. It was also quantitatively illustrated that there were no statistically significant differences in the water levels and depths among the cases for building resistance.
- In terms of the horizontal distribution of the velocity field, which is significant for building damage assessment, the contrast in the velocity difference between the building grid and the surrounding road grid was larger in the SG model (Case 1) than in the equivalent roughness model (Case 2). This is because, in the equivalent roughness model (Case 2), the roughness coefficient is larger even when a small number of buildings are included in the computational grid, and the roughness coefficient is reflected in the horizontal eddy viscosity coefficient; thus, the building effect is spread over a wider area.
- The SG model could reproduce the change in the vertical velocity distribution with the vertical structure of the building. However, the equivalent roughness model could not reproduce the flow velocity distribution with inflection points around the building. It also exhibited a limitation in reproducing the 3D flow velocity distribution around the building precisely because of the backflow near the bottom owing to the large roughness coefficient. Thus, it is clear that the SG model can accurately reproduce the horizontal and vertical structures of the flow velocity.
- A comparison of building loss indices, such as fluid forces acting on each building, revealed significant differences in flow velocity between Cases 1 and 2, particularly in the ranges of 0–3 m and >6 m inundation depths, where statistically significant differences were confirmed. Along with the results of the velocity analysis, similar statistically significant differences were also observed in the unit-width discharge q, moment qh, and fluid force F. These differences were attributed to the horizontal and vertical distribution of the flow velocity. These results suggest that the reproducibility of the vertical velocity distribution is a key factor and that the SG model incorporated into the 3D model can evaluate the inundation flow conditions in a manner that accurately reflects the fluid forces acting on the building, thus demonstrating the usefulness of the model.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic of fundamental concept of subgrid model for building fluid force. (

**a**) When medium grid resolutions are adopted, buildings of various heights are located in several computational grids. (

**b**) In Step 1, each building is divided horizontally and vertically for each computational grid. (

**c**) In Step 2, the flow velocity at each grid is interpolated at the center of each building. (

**d**) In Step 3, the fluid force obtained for each building is distributed to each grid.

**Figure 2.**Interpolation method of calculated velocities at the center of each building using IDW for evaluation of building fluid force. Velocities in s and n directions, u

_{s}and u

_{n}, respectively, are defined in staggered grids.

**Figure 3.**Time interval concept in 2D and 3D calculations in Hy2-3D model. (

**a**) Case when Δt

_{2D}= Δt

_{3D1}and (

**b**) case when Δt

_{2D}> Δt

_{3D1}.

**Figure 4.**(

**a**) Location and elevation map of the Kuma River Basin; (

**b**) computational domain from 51.8 kp to 68.6 kp along the Kuma River.

**Figure 5.**(

**a**) Temporal variations in basin-averaged precipitation and water level at Ohashi (61.5 kp) in the Kuma River. Precipitation and water level data were obtained from http://www.jmbsc.or.jp/en/index-e.html (accessed on 22 November 2022) and https://www.river.go.jp/index (accessed on 22 November 2022), respectively; (

**b**) boundary conditions of inflow discharge at upstream points and tributaries, and water level at the downstream point. River discharges in the Kuma River and 11 major tributaries were obtained from the runoff calculation results [49]. Water level at the downstream end was obtained from the computational results using 1D unsteady flow analysis performed by the authors.

**Figure 6.**(

**a**) Longitudinal distribution of calculated and observed water levels at various time points in the Kuma River; (

**b**) calculated and observed peak water levels; and (

**c**) difference in peak water levels between Case 1 and other cases. The calculated results for Case 1 are used in parts (

**a**,

**b**).

**Figure 7.**Temporal variation in calculated and observed water levels in the Kuma River. The calculated results for Case 1 are shown. The results at water-level observatories Ichibu (68.6 kp), Hitoyoshi (62.2 kp), Ohashi (61.5 kp), Nishizebashi (59.4 kp), Gogan (57.4 kp), and Watari (52.7 kp) are depicted.

**Figure 8.**Scatter plots of (

**a**) calculated and observed peak water levels and (

**b**) water depth in inundated area. The calculated results for Case 1 are used in the figure. Observed results are based on those reported by Ogata et al. [55].

**Figure 9.**(

**a**) Contour maps of calculated depth horizontal velocities at 10:00 a.m. on 4 July 2020, near the Hitoyoshi city area and (

**b**) cross-sectional distributions of calculated horizontal velocities and water levels with locations of buildings along section A-A′. Magnitude of depth-averaged horizontal velocities in all cases is depicted.

**Figure 10.**Vertical distribution of streamwise velocity at 11:30 a.m. on 4 July 2020 in Chaya District. (

**a**) Locations of four stations. (

**b**) Equivalent roughness n in this area. Calculated velocities for (

**c**) Case 1 and (

**d**) Case 2 are shown.

**Figure 11.**Correlation diagram of building loss indices for Cases 1 and 2 in lost buildings (160 buildings). p-value showing a statistically significant difference between Cases 1 and 2 is also illustrated (* p < 0.10).

**Figure 12.**Boxplot showing flood index by flood depth level in lost buildings for Cases 1 and 2. p-value indicating a statistically significant difference between Cases 1 and 2 is also shown (* p < 0.05).

**Table 1.**RMSE values, slopes, and R

^{2}in regression lines for calculated and observed peak water levels and depths for various cases.

Peak Water Level | Peak Water Depth | |||||
---|---|---|---|---|---|---|

RMSE [m] | Slope | R^{2} | RMSE [m] | Slope | R^{2} | |

Case 1 | 0.3815 | 1.0210 | 0.9898 | 0.4525 | 0.9300 | 0.9367 |

Case 2 | 0.3626 | 1.0200 | 0.9902 | 0.4421 | 0.9306 | 0.9390 |

Case 3-1 | 0.4178 | 1.0180 | 0.9875 | 0.4658 | 0.9339 | 0.9261 |

Case 3-2 | 0.5447 | 1.0060 | 0.9897 | 0.5480 | 0.9452 | 0.9343 |

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## Share and Cite

**MDPI and ACS Style**

Kubota, R.; Kashiwada, J.; Nihei, Y.
Subgrid Model of Fluid Force Acting on Buildings for Three-Dimensional Flood Inundation Simulations. *Water* **2023**, *15*, 3166.
https://doi.org/10.3390/w15173166

**AMA Style**

Kubota R, Kashiwada J, Nihei Y.
Subgrid Model of Fluid Force Acting on Buildings for Three-Dimensional Flood Inundation Simulations. *Water*. 2023; 15(17):3166.
https://doi.org/10.3390/w15173166

**Chicago/Turabian Style**

Kubota, Riku, Jin Kashiwada, and Yasuo Nihei.
2023. "Subgrid Model of Fluid Force Acting on Buildings for Three-Dimensional Flood Inundation Simulations" *Water* 15, no. 17: 3166.
https://doi.org/10.3390/w15173166