# Simulation of Hydraulic Structures in 2D High-Resolution Urban Flood Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Finite Volume Godunov-Type SWE Model

**q**,

**f**,

**g**, and

**s**represent the vectors containing the flow variables, fluxes in the x- and y-directions, and source terms. Neglecting the Coriolis effect and the surface stresses, which are not significant for flood modeling, the vector terms may be given by [30]

_{x}and q

_{y}are the corresponding unit-width discharges in the x- and y-directions; z

_{b}is defined as the bed elevation above the datum; g is the gravity acceleration; r and f represent the rainfall intensity and infiltration rate, respectively; −∂z

_{b}/∂x and −∂z

_{b}/∂y define the bed slopes in the two Cartesian directions; and c

_{f = gn}

^{2}

_{/h}

^{1/3}is the bed roughness coefficient, with n being the Manning coefficient.

#### 2.2. Gate Model

_{u}are the gate opening and upstream flow depth, respectively; and ε = h

_{c}/e is the contraction coefficient, in which h

_{c}is the flow depth at vena contraction.

_{t}is the downstream tailwater depth.

#### 2.3. Model Coupling

**q**at cell i

#### 2.3.1. Flux Term Coupling Approach

- (1)
- Under the free-surface flow condition$$\mathit{F}{\left({\mathit{q}}_{i}^{n}\right)}_{k}=\left[\begin{array}{c}{q}_{sx}^{u}-{q}_{sx}^{t}\\ {u}_{u}{q}_{s}x+g\left({\eta}_{u}^{2}-2{\eta}_{u}{z}_{bu}\right)-{u}_{c}{q}_{sx}-g\left({\eta}_{c}^{2}-2{\eta}_{c}{z}_{bc}\right)\\ {u}_{u}{q}_{sy}-{u}_{c}{q}_{sy}\end{array}\right]$$
- (2)
- Under the submerged flow condition$$\mathit{F}{\left({\mathit{q}}_{i}^{n}\right)}_{k}=\left[\begin{array}{c}{q}_{sx}^{u}-{q}_{sx}^{t}\\ {u}_{u}{q}_{s}x+g\left({\eta}_{u}^{2}-2{\eta}_{u}{z}_{bu}\right)-{u}_{c}{q}_{sx}-g\left({\eta}_{H}^{2}-2{\eta}_{H}{z}_{bH}\right)\\ {u}_{u}{q}_{sy}-{u}_{H}{q}_{sy}\end{array}\right]$$

_{sx}and q

_{sy}are the unit-width discharges in the x- and y-directions calculated from Equations (3) and (4), respectively; u

_{u}, u

_{c}, and u

_{H}are the flow velocities obtained by dividing the discharges defined in Equations (3) and (4) by the corresponding flow depths.

#### 2.3.2. Source Term Coupling Approach

_{x}and n

_{y}define the x- and y-direction outward unit vector normal to the edge/boundary under consideration. This is then integrated into the source terms of 2D SWE model as

## 3. Model Validation

^{2}. The root-mean-squared error (RMSE) defined against water level is calculated to indicate the accuracy of the simulation results, which is defined as

_{i}and O

_{i}represent the simulated and analytical/observed data, respectively; and N is the number of observations available for comparison.

#### 3.1. Analytical Tests

_{u}, downstream water depth h

_{d}, and gate opening height e). In all of the simulations, the water was initially at rest, and the gate was rapidly opened to the specified height. The contraction coefficient was set to ε = 0.611.

#### 3.2. Flume Experiments

_{u}increases, the discharge simulated by the numerical model is slightly underestimated. This may be because of the existence of the piers of sluice gates, which create localized three-dimensional contraction effects that cannot be captured by the SWE model. Generally, the numerical simulations capture reasonably well the stage–discharge relationships in different flow conditions, which confirms the capability of the current model in predicting the highly transient waves through sluice gates.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The numerical and analytical solutions (at t = 5 s) for the four test cases defined in Table 1: (

**a**) Test 1. (

**b**) Test 2. (

**c**) Test 3. (

**d**) Test 4.

**Figure 7.**Comparison between numerical results and experimental measurements at t = 900 s: (

**a**) Orifice free flow; (

**b**) Orifice submerged flow.

**Table 1.**Initial conditions of the four numerical tests [33].

Test | h_{u} (m) | h_{d} (m) | e (m) |
---|---|---|---|

1 | 1 | 0.002 | 0.2 |

2 | 1 | 0.2 | 0.2 |

3 | 1 | 0.6 | 0.2 |

4 | 1 | 0.5 | 0.8 |

Test | 1 | 2 | 3 | 4 |
---|---|---|---|---|

RMSE (flux) | 0.0208 | 0.0233 | 0.0177 | 0.0221 |

RMSE (source) | 0.0675 | 0.0828 | 0.0469 | 0.0749 |

Case | e = 2.0 m (free flow) | e = 1.0 m (free flow) | e = 2.0 m; Q = 400 m ^{3}/s | e = 2.0 m; Q = 600 m ^{3}/s |
---|---|---|---|---|

RMSE | 0.0467 | 0.0590 | 0.0619 | 0.0851 |

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**MDPI and ACS Style**

Cui, Y.; Liang, Q.; Wang, G.; Zhao, J.; Hu, J.; Wang, Y.; Xia, X. Simulation of Hydraulic Structures in 2D High-Resolution Urban Flood Modeling. *Water* **2019**, *11*, 2139.
https://doi.org/10.3390/w11102139

**AMA Style**

Cui Y, Liang Q, Wang G, Zhao J, Hu J, Wang Y, Xia X. Simulation of Hydraulic Structures in 2D High-Resolution Urban Flood Modeling. *Water*. 2019; 11(10):2139.
https://doi.org/10.3390/w11102139

**Chicago/Turabian Style**

Cui, Yunsong, Qiuhua Liang, Gang Wang, Jiaheng Zhao, Jinchun Hu, Yuehua Wang, and Xilin Xia. 2019. "Simulation of Hydraulic Structures in 2D High-Resolution Urban Flood Modeling" *Water* 11, no. 10: 2139.
https://doi.org/10.3390/w11102139