# Comparative Study on Water Temperature Stratified Flow under Different Vertical Coordinate Systems in Delft3D

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{3}/s.

## 3. Results

#### 3.1. Comparison of the σ-Coordinate System and the z-Coordinate System Model

#### 3.2. Effect of Grid Resolution on the z-Coordinate System Model

#### 3.3. Effect of Bottom Slope on Simulation Accuracy

- When the bottom slope is the same, the outlet water temperature of the σ-coordinate system model is always lower than that of the z-coordinate system model. This means that the z-coordinate system model causes more vertical diffusion and leads to a higher water temperature of the outlet;
- In the σ-coordinate system model, with the increase in the bottom slope, the arrival time of cold water at the outlet is gradually advanced. This suggests that the larger the bottom slope of a reservoir, the higher the velocity of the cold-water underflow. When the bottom slope is more than 10‰ (conditions C4~C7), the outlet water temperature at the end time increases with the increasing bottom slope. This can be explained by the larger bottom slope, which contains more static water with a temperature of 21.44 °C. Compared to a large amount of initial hot water, the cold-water underflow is relatively small, which means that the outlet water temperature is more difficult to decrease;
- In the z-coordinate system model, for condition C2, as the bottom slope increases, the arrival time of cold water at the outlet is advanced; for condition C3, the simulation result is similar to condition C2; for conditions C4~C7, the arrival time of cold water at the outlet is gradually delayed with the increasing bottom slope. This suggests that when the bottom slope is more than 10‰ (conditions C4~C7), the simulation result of the z-coordinate system model may be incorrect;
- Since the σ-coordinate system model can simulate the formation process of cold-water underflow in the reservoir accurately, the simulation result of the σ-coordinate system model can be regarded as a reference to study the effect of bottom slopes on the simulation accuracy of the z-coordinate system model. As shown in Figure 8, when the bottom slope increases, the simulation deviation between the σ-coordinate system model and the z-coordinate system model also increases, which means that the numerical errors caused by the z-coordinate system model continue to increase. It can be concluded that the simulation accuracy of the z-coordinate system model decreases with the increasing bottom slope.

## 4. Discussion

## 5. Conclusions

- When simulating cold water flowing into a reservoir, the σ-coordinate system model has high simulation accuracy and is not affected by the truncation errors of the horizontal baroclinic gradient force. In contrast, the z-coordinate system model is inaccurate due to artificial vertical diffusion;
- The numerical errors in the z-coordinate system cold-water underflow reservoir model mainly consist of the following two parts: one is caused by the staircase representation of the bottom boundary, and one is caused by artificial vertical diffusion as the vertical grid lines are not parallel with the flow direction;
- To reduce the numerical error caused by the staircase representation of boundaries in the z-coordinate system model, the methods of remapping of near-bottom layers and increasing the vertical grid resolution have little influence on the simulation results. However, the method of increasing the horizontal grid resolution can decrease this numerical error, although there is still a large deviation from the observed data;
- To reduce artificial vertical diffusion in the z-coordinate system cold-water underflow reservoir model, the vertical grid lines should be parallel with the flow direction. The F-test shows that when the bottom slope is less than 18‰, the z-coordinate system model can simulate cold water flowing into a reservoir accurately.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the generalized reservoir hydrodynamics (GRH) flume. (

**a**) Plan view. (

**b**) Side view.

**Figure 3.**Sketch of reservoir models for different bottom slopes. (

**a**) C1. (

**b**) C2. (

**c**) C3. (

**d**) C4. (

**e**) C5. (

**f**) C6. (

**g**) C7.

**Figure 5.**Velocity profile at position 11.43 m from upstream at 11 min for different vertical coordinate models.

**Figure 6.**Time variation of outlet water temperature for the z-coordinate system model with different grid resolutions.

**Figure 7.**Velocity profile at position 11.43 m from upstream at 11 min of the z-coordinate system model with different grid resolutions.

**Figure 8.**Comparison of outlet water temperature of the σ-coordinate system and the z-coordinate system models under different bottom slopes conditions.

Simulation Conditions | Vertical Coordinate System | Horizontal Grid | Vertical Grid Number | |
---|---|---|---|---|

Longitudinal Grid Number | Lateral Grid Number | |||

A1 | σ-coordinate | 80 | 9 | 36 |

A2 | σ-coordinate (anti-creep) | 80 | 9 | 36 |

A3 | z-coordinate | 80 | 9 | 36 |

A4 | z-coordinate (remapping) | 80 | 9 | 36 |

Simulation Conditions | Time Step (min) | Manning Number (m ^{−1/3}s) | Background Horizontal Eddy Viscosity Coefficient (m ^{2}/s) | Background Horizontal Eddy Diffusion Coefficient (m ^{2}/s) | Turbulence Model |
---|---|---|---|---|---|

A1~A4 | 0.01 | 0.009 | 0.01 | 0.01 | k-ε model |

Simulation Conditions | Vertical Coordinate System | Horizontal Grid | Vertical Grid Number | |
---|---|---|---|---|

Longitudinal Grid Number | Lateral Grid Number | |||

B1 | z-coordinate | 80 | 9 | 36 |

B2 | z-coordinate | 80 | 9 | 72 |

B3 | z-coordinate | 80 | 9 | 100 |

B4 | z-coordinate | 160 | 9 | 36 |

B5 | z-coordinate | 240 | 9 | 36 |

Simulation Conditions | C1 | C2 | C3 | C4 | C5 | C6 | C7 |
---|---|---|---|---|---|---|---|

Bottom slope | 0‰ | 5‰ | 10‰ | 15‰ | 20‰ | 25‰ | 30‰ |

Depth of the downstream section (m) | 0.300 | 0.391 | 0.483 | 0.574 | 0.666 | 0.757 | 0.849 |

Height of the outlet hole (m) | 0.050 | 0.065 | 0.080 | 0.096 | 0.111 | 0.126 | 0.141 |

Simulation Conditions | C1 | C2 | C3 | C4 | C5 | C6 | C7 |
---|---|---|---|---|---|---|---|

α | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |

F | 1.02 | 1.04 | 1.30 | 1.91 | 3.62 | 19.12 | 43.26 |

P(F < = f) | 0.48 | 0.48 | 0.33 | 0.14 | 0.02 | 0.00 | 0.00 |

F critical value | 2.69 | 2.69 | 2.69 | 2.69 | 2.69 | 2.69 | 2.69 |

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**MDPI and ACS Style**

Lang, Y.; Hu, Z.; Hao, R.; Li, Y.; Han, L. Comparative Study on Water Temperature Stratified Flow under Different Vertical Coordinate Systems in Delft3D. *Water* **2022**, *14*, 2737.
https://doi.org/10.3390/w14172737

**AMA Style**

Lang Y, Hu Z, Hao R, Li Y, Han L. Comparative Study on Water Temperature Stratified Flow under Different Vertical Coordinate Systems in Delft3D. *Water*. 2022; 14(17):2737.
https://doi.org/10.3390/w14172737

**Chicago/Turabian Style**

Lang, Yun, Zijun Hu, Ruixia Hao, Yafei Li, and Lijuan Han. 2022. "Comparative Study on Water Temperature Stratified Flow under Different Vertical Coordinate Systems in Delft3D" *Water* 14, no. 17: 2737.
https://doi.org/10.3390/w14172737