This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Due to the special hydrographic and physiographic conditions in Taiwan, flooding is likely to occur in the middle and lower reaches of a plain whenever serious rainstorm events occurred. Note worthily, the loss of lives and property caused by flooding are always most considerable in a metropolitan area, and the densely distributed buildings would, not only increase the impervious area, but also decrease the water storage area. Furthermore, a large number of intensive buildings have changed the original land flow conditions, resulting in a beam shrinking flow and the additional form drag phenomenon, which makes the flooding phenomenon more serious. The main purpose of this research is to find the correlation between building coverage and the Manning’s coefficient n through a water flume model experiment. To probe into this issue, the Manning’s roughness adjustment is further divided into a part caused by the surface impedance and a part caused by the building impedance. Thus, building coverage can be added to the general computing grid to reflect the flooding situation with buildings. The two-dimensional inundation model, based on this research, was applied to Taichung City for an actual case simulation. The simulation result of Typhoon Kalmaegi showed that the presented model can obtain a more accurate flooding situation in urban area by considering the blockage effects of buildings and adjusting the surface roughness.

Recently, higher, or more refined, resolution topographic data have been widely used in urban inundation modeling to represent the complicated flow patterns around buildings [

Many studies adopted the porosity treatment of the coarse grid in sub-grid models to simulate real flow conditions. Yu and Lane [

Although accurate information on porosity improves the performance in urban inundation simulation, there is a growing concern in the drag effect of buildings from bed roughness modification. The IMPACT project [

In this paper, steady-flow experiments are presented and performed in a rectangular channel to simulate the backwater effect and drag resistance due to the presence of buildings. Building drag resistance was discussed and expressed as Manning’s roughness, and the relationship between different building occupancy and adjusted Manning’s roughness was analyzed from experiment results and applied to the two-dimensional inundation model by considering building occupancy and Manning’s roughness adjustment in the coarse grid.

Assuming that the acceleration of water flow on land surface is small compared with the gravitation and friction, the two-dimensional (2D) depth-averaged shallow water equations on land surface can be written as [^{1/3}).

The flow conditions in urban areas differ from in rural plains due to the densely distributed buildings. Buildings may occupy a partial area of a coarse grid cell, which reduces the storage capacity of the cell when the flood depth is below the threshold value. In this study, buildings were conceptualized as a square region located at the cell’s center, as shown in _{0}), such that the flood water can only distribute to 1 − α_{0} of the area. Equation (1) is therefore modified as:

Schematic of simulation grid containing buildings.

The total resistance

The surface resistance is defined as the friction shearing stress on the ground surface area. The existence of buildings restricts the flow between buildings, which results in reduced friction forces due to a lower surface area (1 − α_{0})·_{w}_{0})·_{w}_{f}_{f}_{0} is Manning’s roughness without considering the building effect, and substituting τ_{w}^{1/3}). The building drag resistance can be converted to the same form as the shear resistance applied to the total area:
^{1/3}).

Substituting Equations (7) and (8) into Equation (5), we obtain the total friction force as follows:

In Equation (13), the building drag resistance,

In this study, the non-inertia wave model which neglects the acceleration terms of water flow has been adopted for simulation, but the acceleration of water flow due to restriction of the buildings should be considered in building block. Therefore, the modified Manning’s roughness according was applied to reflect the effect of acceleration terms.

The building drag resistance leads to an energy loss due to the obstruction of flow. An experiment was established to quantify building drag resistance. The experiment was installed in a horizontal flume at Hydrotech Research Institute, National Taiwan University (

Functional flume.

Building model represented by plastic pillars.

The flow in the flume was controlled as a constant discharge by an electric valve. Uniform flow conditions were established before the building block and a free overflow at the end of the flume was used to regulate the flow depth. The flow velocity was measured by a magnetic flow velocity meter, and velocity profiles were used to determine the mean velocity. The constant discharge of the flume can be calculated by flow velocity and water depth while no pillars are installed.

The upstream velocity and water depth were measured in the position where

Settlement of experiment.

Measure positions of flow depth and velocity.

The flume was built in a smooth boundary channel so that the surface resistance is small compared with the drag resistance in the experiment. Assuming that the effect of surface resistance can be neglected, the experimental slope of the energy-grade line (_{1} − _{2})/Δ_{1} and _{2} are the upstream and downstream water depth; whereas Δ

From Manning Equation, the experimental slope of the energy-grade line (

Considering the oscillation of water depth in the building block, _{1} + _{2})/2, and _{1} + _{2})/2.

_{1} increases with BCR significantly, the downstream depth _{2} remains unchanged. The results show that the downstream water depths remain unchanged, and there are only a few differences in downstream velocities. It indicates that the wave celerity is larger than the flow velocity in subcritical flow,

Water depths (unit: cm) and velocities (unit: cm/s) in different building coverage ratio (BCRs).

BCR (α_{0}) |
0.00 | 0.04 | 0.16 | 0.25 | 0.36 | 0.49 | 0.64 |
---|---|---|---|---|---|---|---|

Upstream mean water depth, Y_{1} |
8.50 | 8.80 | 9.00 | 9.30 | 9.55 | 10.20 | 11.70 |

Downstream mean water depth, Y_{2} |
8.50 | 8.50 | 8.50 | 8.50 | 8.50 | 8.50 | 8.50 |

Upstream mean water velocity, V_{1} |
13.74 | 13.33 | 13.25 | 12.59 | 12.29 | 11.53 | 9.85 |

Downstream mean water velocity, V_{2} |
13.74 | 13.60 | 13.60 | 13.61 | 13.62 | 13.65 | 13.84 |

We can calculate

According to Equations (10) and (13), the roughness corresponding to the blockage effect can be obtained.

Different value of _{0} and BCR were inputted into Equation (14) to observe the relationship of BCR and

Water depths in different BCRs (unit: cm).

BCR (α_{0}) |
0.00 | 0.04 | 0.16 | 0.25 | 0.36 | 0.49 | 0.64 |
---|---|---|---|---|---|---|---|

13.74 | 13.46 | 13.43 | 13.10 | 12.95 | 12.59 | 11.85 | |

8.50 | 8.65 | 8.75 | 8.90 | 9.03 | 9.35 | 10.10 | |

0.00000 | 0.00226 | 0.00376 | 0.00602 | 0.00789 | 0.01278 | 0.02406 | |

0.000 | 0.069 | 0.090 | 0.118 | 0.138 | 0.185 | 0.284 |

Relationship between

Relationship between _{0}.

Due to scale differences between the experiment and urban areas, roughness corresponding to the blockage effect in Equation (14) was modified by scale adjustment to allow applications in urban inundation modeling.

The ratio of simulation roughness of the urban area scale to the experimental scale (_{fr}_{r}_{r}_{r}

While water flow passes through buildings, systematic and sudden widening and narrowing may occur. The slope of the energy-grade line in the longitudinal direction of the flow can be determined by Borda’s formula [

Thus the ratio of the slope of energy-grade line is:
_{r}_{r}

By substituting _{fr}

Experimental roughness

To discuss the application of experiment results of adjusted Manning’s roughness, a numerical simulation of 2D inundation model was employed to simulate the blockage effect. In the 2D inundation model, BCR was used in continuity Equation (4) to account for building blockage in overland flow, and the adjusted Manning’s roughness was applied in momentum Equations (2) and (3) to reflect the flow resistance result from buildings.

The conveyance of cross section and building shape resistance caused by buildings in channel will induce a backwater effect. This research established a 0.99 m × 6.6 m area, as shown in _{0} was 0.05, and this value was also the same as the experiment setting of the hydraulic flume. The inflow was uniformly inputted in upstream cells as upper boundary conditions, and the downstream boundary was considered as a free weir flow.

Simulation grid of 2D inundation model in experiment scale.

The BCR of building grid was set as α_{0} in the inundation model, and Manning’s roughness of each building cell was calculated by mean of Equation (14) according to its BCR and _{0}. The adjusted Manning’s roughness of each BCR is listed in

Roughness under different BCRs.

BCR (α_{0}) |
0 | 0.04 | 0.16 | 0.25 | 0.36 | 0.49 | 0.64 |
---|---|---|---|---|---|---|---|

0.050 | 0.048 | 0.042 | 0.038 | 0.032 | 0.026 | 0.018 | |

0.000 | 0.035 | 0.099 | 0.122 | 0.140 | 0.177 | 0.286 | |

0.050 | 0.059 | 0.107 | 0.127 | 0.143 | 0.179 | 0.287 |

The inundation simulations were carried out with modified Manning’s roughness under different BCRs to show the backwater effect caused by buildings. The comparisons of differences between upstream and downstream water depth (_{1} − _{2}) of simulation results and experiment records are listed in

Comarisons of difference between upsream and downstream water depth of experiment and simulation results.

BCR (α_{0}) |
0 | 0.04 | 0.16 | 0.25 | 0.36 | 0.49 | 0.64 |
---|---|---|---|---|---|---|---|

_{1} − _{2} in Experiment (cm) |
0.0 | 0.3 | 0.5 | 0.7 | 1.05 | 1.7 | 3.2 |

_{1} − _{2} in Simulation (cm) |
0.08 | 0.17 | 0.82 | 0.99 | 1.18 | 1.73 | 3.42 |

Error (cm) | 0.08 | -0.13 | 0.32 | 0.29 | 0.13 | 0.03 | 0.22 |

Water profile of experiment and simulation results (α_{0} = 0.36).

Water profile of experiment and simulation results (α_{0} = 0.49).

Water profile of experiment and simulation results (α_{0} = 0.64).

Because the hydraulic experiment is unable to reveal the actual size of urban areas, thus, the 2D inundation model was also used to simulate inundation under roughness adjustment in urban areas. The simulation area was 330 m × 99 m, and the terrain slope 0.001 and Manning’s roughness _{0} = 0.05 were the same as in the flume experiment. The inflow was also a uniform flow and the outflow was a free weir flow.

_{r}_{r}

Simulation grid of 2D inundation model in real scale.

_{0} = 0.49) and 0.78 m α_{0} = 0.64), and instead of 0.54 m, these values were applied to the roughness adjustment in Equations (18) and (19). The underestimation of water depth has been improved when the simulated water depths were used to account for the roughness adjustment, as shown in

Taichung City in middle Taiwan has Dakeng Mountain on the east side and Dadu in the west side.

Water profile of experiment and simulation results in real scale (α_{0} = 0.36).

Water profile of experiment and simulation results in real scale (α_{0} = 0.49).

Water profile of experiment and simulation results in real scale (α_{0} = 0.64).

Elevation, water system, and sewer manholes of Taichung city.

There are 174 sewers and 2018 manholes in Taichung City.

Distribution of the sewer manholes of Taichung city.

Typhoon Kalmaegi in 2008 was adopted for model application. Typhoon Kalmaegi intruded Taichung on 17 July 2008.

Total amount and peak intensity of rainfall of Typhoon Kalmaegi in Taichung City.

Rain gauge | Total rainfall (mm) | Peak intensity (mm/h) |
---|---|---|

Taichung | 478.9 | 120.0 |

Dadu | 363.0 | 79.0 |

Shhigang | 341.0 | 74.5 |

Dakeng | 607.0 | 149.0 |

Chungchulin | 567.0 | 91.0 |

Tonlin | 409.5 | 82.5 |

Shengang | 225.5 | 49.5 |

Fenyuan | 377.5 | 78.5 |

Changhua | 248.0 | 52.0 |

Caotun | 295.0 | 71.0 |

The HEC-1 Model developed by the U.S. Army Corps of Engineers [

We compared two cases for reflecting building blockages in urban flood modeling with 80 m × 80 m cells. Case 1 only adopted the bare terrain elevation without roughness adjustment, and it is the traditional approach in urban flood modeling. The original Manning’s roughness of each grid cell was assumed according the land-use type of each cell and classified into the following three categories: 0.1 (waterway), 0.13 (agricultural, residential and traffic uses), and 0.20 (commercial and industrial uses). In Case 2, BCR values and adjusted roughness were used to represent the blockage effect. The BCR values were calculated from the occupied area of buildings in each cell. The Manning’s roughness of each cell was modified according to Equation (19) during the simulation. The average BCR values of all districts listed in

Inundation area of Taichung City (unit: ha).

District | Case 1 | Case 2 | Average BCR |
---|---|---|---|

Situn | 638.08 | 705.28 | 0.22 |

Nantun | 556.80 | 566.40 | 0.20 |

Beitun | 424.32 | 550.40 | 0.26 |

East | 186.24 | 255.36 | 0.36 |

South | 283.52 | 305.92 | 0.34 |

West | 222.72 | 246.30 | 0.41 |

North | 166.40 | 213.76 | 0.43 |

Central | 27.52 | 33.92 | 0.53 |

Total | 2505.60 | 2877.34 |

Simulated and investigated flooded areas of Typhoon Kalmaegi with building blockage effect (Case 2).

Inundation area of each districts of Taichung City (unit: ha).

Depth (m) | Situn | Nantun | Beitun | East | ||||
---|---|---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | Case 1 | Case 2 | Case 1 | Case 2 | |

0.5–1.0 | 453.76 | 500.48 | 339.20 | 349.44 | 345.60 | 437.76 | 131.84 | 139.52 |

1.0–2.0 | 126.08 | 144.00 | 176.00 | 176.00 | 32.64 | 66.56 | 52.48 | 89.60 |

2.0–3.0 | 34.56 | 36.48 | 8.96 | 8.32 | 28.80 | 26.88 | 1.28 | 11.52 |

above 3.0 | 23.68 | 24.32 | 32.64 | 32.64 | 17.28 | 19.20 | 0.64 | 14.72 |

Total | 638.08 | 705.28 | 556.80 | 566.40 | 424.32 | 550.40 | 186.24 | 255.36 |

0.5–1.0 | 201.60 | 216.32 | 128.00 | 140.47 | 159.36 | 198.40 | 22.40 | 20.48 |

1.0–2.0 | 61.44 | 69.12 | 69.76 | 72.23 | 6.40 | 14.72 | 5.12 | 13.44 |

2.0–3.0 | 18.56 | 17.92 | 10.24 | 17.28 | 0.64 | 0.64 | 0.00 | 0.00 |

above 3.0 | 1.92 | 2.56 | 14.72 | 16.32 | 0.00 | 0.00 | 0.00 | 0.00 |

Total | 283.52 | 305.92 | 222.72 | 246.30 | 166.40 | 213.76 | 27.52 | 33.92 |

The inundation model was developed to represent the effects of building blockage in urban areas. In this paper, we proposed a Manning’s roughness modification method according to the blockage effect of buildings. The simulation results from Kalmaegi Typhoon indicated that the proposed model, using the BCR and the modified Manning’s roughness, can produce more accurate results in urban inundation modeling.

The study was supported by the National Science Council, Taiwan, R.O.C. Valuable information and historical records were provided by the Central Weather Bureau and the Water Resources Agency. The authors are grateful for their considerable help.

The concept of this paper was developed by Ming-Hsi Hsu. The hydraulic experiment and data analysis were undertaken by Yen-Hsiang Wang. The simulation of inundation model was conducted by Yen-Hsiang Wang and Chen-Jia Huang with the supervision of Ming-His Hsu and Wei-Hsien Teng. The first draft of the article was written by Chen-Jia Huang and was revised by the other authors.

The authors declare no conflict of interest.