# Progressive Dam-Failure Assessment by Smooth Particle Hydrodynamics (SPH) Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SPH Basic Theory

#### 2.1. SPH Basic Ideas

#### 2.2. SPH Basic Equation

#### 2.3. SPH Method for Solving the N-S Equation

#### 2.3.1. Navier–Stokes Equations

#### 2.3.2. Density Particle Approximation Method for Solving the N-S Equation

#### 2.4. Smooth Kernel Function Construction

#### 2.5. Other Key Technologies

_{1}is taken as 0.4, and λ

_{2}is taken as 0.25.

## 3. Calculus Analysis

#### 3.1. Transient Dam-Failure Arithmetic Simulation

_{1}for the experimental data points and N

_{2}for the numerical simulation data points by the free liquid level similarity equation:

_{1i}the experimental data vertical coordinate value, and y

_{2i}is the simulation result vertical coordinate value.

#### 3.2. Example Simulation of Gradual Collapse

- (1)
- The model has some improvements in the way the points are laid out, which can optimize the model to some extent for the traditional SPH dam-failure model;
- (2)
- The model facilitates the development of later studies, and the calculation of the height of the initial dam height versus the height of the moment of failure over time can be analyzed in comparison with Coleman’s theoretical solutions in the literature.

_{b}, is a function of time t with a dimensionless expression:

_{b}is the elevation of the bottom of the breach; H

_{b0}is the initial elevation of the dam; g is the acceleration of gravity. Equation (14) is mostly used in sea defense projects with an infinite water level in front of the dam. As shown in Figure 6, when the water level in front of the dam is constant, the trend of the elevation of the dam at the bottom can decrease indefinitely to approach 0. Due to the characteristics of the dam itself, it cannot eventually breach a height of 0 [32].

#### 3.2.1. Equally Spaced Collapse Simulation

#### 3.2.2. Progressive Collapse Simulation

#### 3.2.3. Analysis of Results

## 4. Conclusions

#### 4.1. Conclusions

- The simulation of dam-failure flow using the smooth particle hydrodynamics (SPH) method, including the simulation of the classical dam-failure mode and the simulation of the step-by-step dam-failure mode, as well as the further division of the two modes of study of the equal-spaced and progressive modes were investigated to explore the failure modes in line with the dam-failure flow;
- Compared to instantaneous dam failure, the calculation results of the breach development considering the progressive gradual dam-failure model are more consistent with the theoretical solution and closer to the actual dam-failure process;
- Under multiple progressive dam-failure modes, as the number of segments increases, the degree of agreement between the calculated results of the breach development and the theoretical solution increases, and progressive dam failure has a higher degree of agreement than equal-interval dam failure, while the total kinetic energy of the breaching flood decreases with the increase in the number of segments of the progressive dam failure.

#### 4.2. Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Comparison of the literature [26] and SPH simulation results.

**Figure 10.**Progressive dam-failure flow pattern (z/m, longitudinal displacement; x/m, lateral displacement).

Serial Number | Calculation Condition | Parameter Condition |
---|---|---|

1 | Total number of particles | 2812 |

2 | Boundary particle spacing | 0.0026 |

3 | SPH particle spacing | 0.0038 |

4 | Fluid density/(kg/m^{3}) | 1000 |

5 | Kernel function type | Cubic spline function |

6 | Density approximation methods | Approximation of the continuity equation |

7 | Time step | 0.0001 s |

8 | Smooth length | 1.25 particle spacing |

9 | Coefficient of viscosity | 1.2 |

10 | Time | 1.0 |

Time | T = 0.2 s | T = 0.4 s | T = 0.8 s | Average |
---|---|---|---|---|

Similarity | 92.1% | 90.7% | 82.3% | 88.4% |

Serial Number | Calculation Condition | Parameter Condition |
---|---|---|

1 | Total number of particles | 2911 |

2 | Boundary particle spacing/m | 1 |

3 | SPH particle spacing/m | 1 |

4 | Kernel function type | Cubic spline function |

5 | Density approximation methods | Approximation of the continuity equation |

6 | Time step | 0.0001 s |

7 | Smooth length h | 1.25 particle spacing |

8 | Fluid density/(kg/m^{3}) | 1000 |

9 | Coefficient of viscosity | 1.2 |

10 | Time/s | 2.0 |

**Table 4.**Comparison of similarity between simulation results and Coleman’s [31] theoretical solution Table.

Gradual Dam-Failure Condition | 2 Paragraph | 3 Paragraph | 4 Paragraph | 5 Paragraph |
---|---|---|---|---|

Equidistant dam failure | 72.2% | 85.9% | 92% | 95.4% |

Progressive dam failure | 79.1% | 95.2% | 96.9% | 97.1% |

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**MDPI and ACS Style**

Zhang, J.; Wang, B.; Li, H.; Zhang, F.; Wu, W.; Hu, Z.; Deng, C.
Progressive Dam-Failure Assessment by Smooth Particle Hydrodynamics (SPH) Method. *Water* **2023**, *15*, 3869.
https://doi.org/10.3390/w15213869

**AMA Style**

Zhang J, Wang B, Li H, Zhang F, Wu W, Hu Z, Deng C.
Progressive Dam-Failure Assessment by Smooth Particle Hydrodynamics (SPH) Method. *Water*. 2023; 15(21):3869.
https://doi.org/10.3390/w15213869

**Chicago/Turabian Style**

Zhang, Jianwei, Bingpeng Wang, Huokun Li, Fuhong Zhang, Weitao Wu, Zixu Hu, and Chengchi Deng.
2023. "Progressive Dam-Failure Assessment by Smooth Particle Hydrodynamics (SPH) Method" *Water* 15, no. 21: 3869.
https://doi.org/10.3390/w15213869