# Simulating Flow of An Urban River Course with Complex Cross Sections Based on the MIKE21 FM Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. River Planning Scheme

^{2}rectangular cross section, with a bottom longitudinal slope of 0.001), along the south side of Road X, the west side of Road Y, and the north side of Road Z; reroute River A to the east side of College B, and merge the new river course into the downstream of current River A on the south side of College B.

^{3}/s, and the planned river flow rate for once-in-50-years floods is 34.5 m

^{3}/s.

#### 2.2. Model Description

^{2}; $f$ = 2Ωsinφ is the Coriolis parameter (Ω is the angular rate of revolution and φ is the geographic latitude) in s

^{−1}; $\mathsf{\rho}$ is the density of water in kg/m

^{3}; ${\mathsf{\rho}}_{0}$ is the reference density of water in kg/m

^{3}; ${s}_{xx}$, ${s}_{xy}$, ${s}_{yx}$ and ${s}_{yy}$ are components of the radiation stress tensor in kg/s

^{2}; ${\mathsf{\tau}}_{sx}$ and ${\mathsf{\tau}}_{sy}$ are the surface stress in kg/s

^{2}$\xb7$m; ${\mathsf{\tau}}_{bx}$ and ${\mathsf{\tau}}_{by}$ are the bottom stress in kg/s

^{2}$\xb7$m; ${p}_{a}$ is atmospheric pressure in kg/s

^{2}$\xb7$m; $s$ is the magnitude of the discharge due to point sources in s

^{−1}; ${u}_{s}$ and ${v}_{s}$ are the velocity by which the water is discharged into the ambient water in m/s; ${T}_{xx}$, ${T}_{xy}$ and ${T}_{yy}$ are the lateral stresses in m

^{2}/s

^{2}, which are estimated using an eddy viscosity formulation based on the depth average velocity gradients:

^{2}/s; ${c}_{s}^{}$ is a constant, which should be chosen within the range from 0.25 to 1.0; $l$ is a characteristic length in m.

#### 2.3. Calculation Conditions

#### 2.3.1. Model Parameters

#### 2.3.2. Terrain Processing and Meshing

#### 2.3.3. Boundary Conditions

#### 2.4. Scenario Settings

## 3. Results and Discussion

#### 3.1. Downstream Trapezoidal Stretch

#### 3.2. Transition Stretch (Horn Mouth Stretch 2).

#### 3.3. Culverts and Open Channels

_{left}< V

_{right}< V

_{middle}, the water depths are in the sequence H

_{left}< H

_{right}< H

_{middle}, and the flow rates are in the sequence Q

_{left}< Q

_{right}< Q

_{middle}. The middle culvert channel in this stretch has relatively larger flow velocity and flow rate, which should be protected from scouring; the left culvert channel has lower flow velocity and flow rate, which should be dredged in time; the horizontal flow velocity (relative flow velocity between the two sides) in the culvert suddenly increases and the vertical flow velocity suddenly decreases, which indicates that the horizontal circulation effects in all culverts are enhanced and the height difference between the two sides at the turns of culvert increases.

_{left}< V

_{middle}< V

_{right}and the flow rates are in the sequence Q

_{left}< Q

_{middle}< Q

_{right}. Therefore, dredging in the two open channels need to be enhanced, especially in the left channel of the culvert between two open channels; the current flow velocity in the open channel suddenly increases, and the average backwater level is about 0.04 m, which meets the planning requirements of safe super elevation for height.

_{left}< V

_{right}≈ V

_{middle}, and the flow rates are in the sequence Q

_{left}< Q

_{middle}< Q

_{right}; therefore, in-time dredging is still necessary in the left culvert channel. The horizontal flow velocity in the section suddenly decreases and the vertical flow velocity (relative flow velocity between the both sides) suddenly increases, which indicates that the horizontal circulation effects in all culverts are enhanced and that the height difference between the two sides at the turns of culverts increases.

_{right}< V

_{left}≈ V

_{middle}, and the water depths are in the sequence H

_{middle}≈ H

_{left}< H

_{right.}Therefore, the right culvert channel needs to be dredged in time. The horizontal flow velocity (relative flow velocity between the both sides) suddenly increases, and the vertical flow velocity suddenly decreases, which indicates that the horizontal circulation effects in all culverts are enhanced and that the height difference between the two sides at the turns of culvert increases.

^{3}; and the water depth change in the 3-channel culverts basically remains the same, with the maximum change of 0.34 m and the maximum water depth of 2.24 m, Therefore, the water depth does not meet the planning requirement of the safe super elevation (at least 0.3 m). Hence, regular dredging is rather necessary. It is also shown that the setting of the six open channels is not only convenient for dredging, but also can be used to adjust the flow rate in the culverts.

^{3}. Therefore, when the roughness coefficient of the downstream stretch changes from 0.025 to 0.017, the water level in the culverts decreases slightly, but the flow rate, water depth and flow rate in the culverts and open channels do not change significantly. From the drainage function perspective, the concrete lining in the downstream stretch is beneficial to rapid discharge; however, from the overall appearance and return of investment perspective, this planning study still recommends the ecological slope lining for the downstream stretch.

#### 3.4. Upstream Stretch (Horn Mouth Stretch 1 and the Upstream Trapezoidal Stretch)

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Hydraulic factors map of the downstream trapezoidal stretch (simulation scenario 1). (

**a**) Water depth map; (

**b**) Horizontal flow velocity; (

**c**) Vertical flow velocity; (

**d**) Current flow velocity (From bottom to top and left to right, the flow velocity was set to be positive).

**Figure 3.**Comparison of hydraulic factors between scenario 4 and scenario 2. (

**a**) Water depth difference; (

**b**) Current flow velocity difference.

**Figure 4.**Water depth differences and current flow velocity differences in scenario 4 and scenario 2. (

**a**) Water depth difference; (

**b**) Current flow velocity difference.

**Figure 5.**Hydraulic factors map in the culverts and open channels (simulation scenario 1). (

**a**) Water depth map; (

**b**) Horizontal flow velocity; (

**c**) Vertical flow velocity; (

**d**) Current flow velocity.

**Figure 6.**Enlarged water depth map of the four turns of 3-channel culverts (simulation scenario 1). (

**a**) Enlarged water depth map of the locations marked with red circles 1 in Figure 5a; (

**b**) Enlarged water depth map of the locations marked with red circles 2 in Figure 5a; (

**c**) Enlarged water depth map of the locations marked with red circles 3 in Figure 5a; (

**d**) Enlarged water depth map of the locations marked with red circles 4 in Figure 5a.

**Table 1.**Model parameters table [31].

Model Parameters | Recommended Range | Module Default Value | Value in This Study |
---|---|---|---|

Time step | 0.01–30 s | 0.01–30 s | 0.01–0.03 s |

Wet and dry boundaries | - | h_{dry} = 0.005h _{flood} = 0.05h _{wet} = 0.1 | h_{dry} = 0.01h _{flood} = 0.1h _{wet} = 0.3 |

Manning coefficient (n) | 0.01–0.05 m^{1/3}/s | 0.03 m^{1/3}/s | Roughness coefficient of ecological slope section (n): 0.025; Roughness coefficient of 3-channel culverts (n): 0.017 |

Vortex viscosity coefficient | 0.25–1.0 | 0.28 | 0.28 |

Structures | - | - | Described in the terrain DEM |

**Table 2.**Roughness coefficients table [41].

No. | Boundary TYPE and Conditions | n |
---|---|---|

1 | Thoroughly planed wood boards and freshly cleaned pig iron pipes and cast-iron pipes with smooth lining and joints | 0.01 |

2 | Dirty water supply and drainage pipes; ordinary concrete surfaces; ordinary brickworks | 0.014 |

3 | Old brickworks; very rough concrete surfaces; carefully excavated smooth rock faces | 0.017 |

4 | Canals in solid clay; loess with continuous silt layers; well-maintained large canals in earth | 0.0225 |

5 | Ordinary large earth canals; well-maintained small canals in earth; natural rivers under excellent conditions | 0.025 |

6 | Earth drains under particularly bad conditions; natural rivers under poor conditions (with much wild grasses and stones, irregular and curved riverbeds and many collapses and deep pools, etc.) | 0.04 |

Scenario | Roughness Coefficient | Water Level Boundary | Flow Rate Boundary |
---|---|---|---|

1 | The roughness coefficient of the ecological revetment section was taken as 0.025 and the roughness coefficient of the culvert was taken as 0.017 | Water depth for once-in-20-years floods (1.46 m) | Flow rate for once-in-20-years floods (29.4 m^{3}/s) |

2 | Water depth for once-in-50-years floods (1.58 m) | Flow rate for once-in-50-years floods (34.5 m^{3}/s) | |

3 | 0.025 | Water depth for once-in-50-years floods | Flow rate for once-in-50-years floods |

4 | 0.017 | Water depth for once-in-50-years floods | Flow rate for once-in-50-years floods |

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**MDPI and ACS Style**

Wang, Q.; Peng, W.; Dong, F.; Liu, X.; Ou, N. Simulating Flow of An Urban River Course with Complex Cross Sections Based on the MIKE21 FM Model. *Water* **2020**, *12*, 761.
https://doi.org/10.3390/w12030761

**AMA Style**

Wang Q, Peng W, Dong F, Liu X, Ou N. Simulating Flow of An Urban River Course with Complex Cross Sections Based on the MIKE21 FM Model. *Water*. 2020; 12(3):761.
https://doi.org/10.3390/w12030761

**Chicago/Turabian Style**

Wang, Qianxun, Wenqi Peng, Fei Dong, Xiaobo Liu, and Nan Ou. 2020. "Simulating Flow of An Urban River Course with Complex Cross Sections Based on the MIKE21 FM Model" *Water* 12, no. 3: 761.
https://doi.org/10.3390/w12030761