# SPH Simulation of Sediment Movement from Dam Breaks

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental SPH Formulations

## 3. Standard SPH Fluid Model

^{3}is the initial density; the constant $B=\frac{{c}^{2}{\rho}_{o}}{\gamma}$, of which $c=10\times V$max; $V$max is the maximum flow velocity.

## 4. Two-Phase SPH Water–Sediment Model

#### 4.1. Lagrangian Form of Governing Equations

- (i)
- Equation for conservation of mass of water flow:$$\frac{D({\alpha}^{f}{\rho}^{f})}{Dt}=-{\alpha}^{f}{\rho}^{f}\nabla \cdot {\overrightarrow{u}}^{f}$$
- (ii)
- Conservation of momentum equation for water flow:$$\frac{D{\overrightarrow{u}}^{f}}{Dt}=-\frac{1}{{\rho}^{f}}\nabla {P}^{f}+\frac{1}{{\alpha}^{f}{\rho}^{f}}\nabla \cdot {\overrightarrow{\tau}}^{f}+\overrightarrow{g}-\frac{{\alpha}^{s}}{{\rho}^{f}}K({\overrightarrow{u}}^{f}-{\overrightarrow{u}}^{s})+\frac{1}{{\rho}^{f}}{S}_{US}K{\upsilon}_{t}^{f}\nabla {\alpha}^{s}$$
- (iii)
- Sediment mass conservation equation:$$\frac{D{\alpha}^{s}}{Dt}=({\overrightarrow{u}}^{f}-{\overrightarrow{u}}^{s})\cdot \nabla {\alpha}^{s}-{\alpha}^{s}\nabla \cdot {\overrightarrow{u}}^{s}$$
- (iv)
- Conservation of sediment momentum equation:

#### 4.2. SPH Discretizations in Two-Phase Equations

- (i)
- Conservation of water mass:$$\frac{{\left({\alpha}_{a}^{f}\right|}^{(t+\Delta t)}{\left({\rho}_{a}^{f}\right|}^{(t+\Delta t)}-{\left({\alpha}_{a}^{f}\right|}^{(t)}{\left({\rho}_{a}^{f}\right|}^{(t)}}{\Delta t}={\alpha}_{a}^{f}{\rho}_{a}^{f}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{b}^{f})}\cdot {\nabla}_{a}{W}_{ab}$$
- (ii)
- Conservation of water momentum:$$\begin{array}{l}\frac{{\left({\overrightarrow{u}}_{a}^{f}\right|}^{(t+\Delta t)}-{\left({\overrightarrow{u}}_{a}^{f}\right|}^{(t)}}{\Delta t}=-\frac{1}{{\rho}_{a}{}^{f}}{\displaystyle \sum _{b}\Delta {V}_{b}({P}_{a}^{f}+{P}_{b}^{f})}{\nabla}_{a}{W}_{ab}+\frac{1}{{\alpha}_{a}{}^{f}{\rho}_{a}{}^{f}}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{\tau}}_{a}^{f}+{\overrightarrow{\tau}}_{b}^{f})}\cdot {\nabla}_{a}{W}_{ab}\\ +\overrightarrow{g}-\frac{{\alpha}_{a}{}^{s}}{{\rho}_{a}{}^{f}}{K}_{a}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})-\frac{1}{{\rho}_{a}{}^{f}}{S}_{US}{K}_{a}{\left({\upsilon}_{t}^{f}\right|}_{a}{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}{}^{s}-{\alpha}_{b}{}^{s})}{\nabla}_{a}{W}_{ab}\end{array}$$
- (iii)
- Sediment mass conservation:$$\begin{array}{l}\frac{{\left({\alpha}_{a}{}^{s}\right|}^{(t+\Delta t)}-{\left({\alpha}_{a}{}^{s}\right|}^{(t)}}{\Delta t}=\\ {\alpha}_{a}^{s}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{s}{{}_{}}^{}-{\overrightarrow{u}}_{b}^{s})}\cdot {\nabla}_{a}{W}_{ab}-{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}^{s}-{\alpha}_{b}^{s})({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})}\cdot {\nabla}_{a}{W}_{ab}\end{array}$$
- (iv)
- Sediment momentum conservation:$$\begin{array}{l}\frac{{\left({\overrightarrow{u}}_{a}^{s}\right|}^{(t+\Delta t)}-{\left({\overrightarrow{u}}_{a}^{s}\right|}^{(t)}}{\Delta t}=-\frac{1}{{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({P}_{a}^{f}+{P}_{b}^{f})}{\nabla}_{a}{W}_{ab}-\frac{1}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({\tilde{P}}_{a}^{s}+{\tilde{P}}_{b}^{s})}{\nabla}_{a}{W}_{ab}\\ +\frac{{\alpha}_{a}^{f}}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{S}_{US}{K}_{a}{\left({\upsilon}_{t}^{f}\right|}_{a}{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}^{s}-{\alpha}_{b}^{s})}{\nabla}_{a}{W}_{ab}-{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{s}-{\overrightarrow{u}}_{b}^{s})({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})}\cdot {\nabla}_{a}{W}_{ab}\\ +\frac{1}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{\tau}}_{a}^{s}+{\overrightarrow{\tau}}_{b}^{s})}\cdot {\nabla}_{a}{W}_{ab}+\overrightarrow{g}+\frac{{\alpha}_{a}^{f}}{{\rho}_{a}^{s}}{K}_{a}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})\end{array}$$

## 5. Model Applications on Different Cases

#### 5.1. Model Application I

^{3}; the average particle size is ${d}_{s}=0.3$ mm; the angle of internal friction is 25° ± 0.4°; the roughness of the flume side walls is neglected. The experimental results of the initial sediment phase volume fraction α

^{s}= 0.53 ± 0.005 were selected for validation; the initial moment of the accumulation body is submerged in 0.1 m deep water; fluid density is ${\mathsf{\rho}}^{f}=$1000 kg/m

^{3}; kinematic viscosity coefficient is ${\upsilon}_{o}^{f}=1\times {10}^{-6}$ m

^{2}/s. After the test starts, the baffle that maintains the initial shape of the accumulation body is released instantaneously, and the effect of the baffle is neglected, and the accumulation body starts to collapse and reaches the final stable state within a few seconds.

#### 5.2. Model Application II

#### 5.3. Model Application III

## 6. Discussion

#### 6.1. Summary of the Study

- -
- The initial collapse;
- -
- The mixing of water and sediments released;
- -
- The flow velocity distribution of the upper flow layer, which is significantly larger than that of the lower sediment layer;
- -
- The flow velocities of the two layers when water and sediments are fully mixed.

#### 6.2. Limitations of the Approach

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\alpha}^{f}$ | volume fraction of water phase | - |

${\alpha}^{s}$ | volume fraction of sand phase | - |

${c}_{0}$ | speed of sound | m/s |

$\rho $ | density | Kg/m^{3} |

${\rho}_{0}$ | SPH particle’s reference density | Kg/m^{3} |

$\Delta t$ | time step size | s |

$\overrightarrow{g}$ | acceleration of gravity | m/s^{2} |

$K$ | drag force coefficient | - |

${m}_{0}{}^{s}$ | sediment mass | Kg |

${m}_{0}{}^{f}$ | fluid mass | Kg |

${P}^{f}$ | pressure of water | N/m^{2} |

$\tilde{{P}^{s}}$ | positive stress of the sediment phase | N/m^{2} |

${S}_{US}$ | reciprocal of Schmidt’s number | - |

t | time | s |

$\overrightarrow{{\tau}^{f}}$ | shear stress of the water flow | N/m^{2} |

$\overrightarrow{{\tau}^{s}}$ | shear stress of the sediment phase | N/m^{2} |

$\overrightarrow{{u}^{f}}$ | velocity of water phase | m/s |

$\overrightarrow{{u}^{s}}$ | velocity of sand phase | m/s |

${\upsilon}_{t}^{f}$ | turbulent viscosity coefficient | m^{2}/s |

$r$ | position vector of SPH particle | m |

$h$ | smoothing length | m |

$W$ | kernel function | - |

${\upsilon}_{o}$ | coefficient of laminar motion viscosity | m^{2}/s |

$\Delta V$ | particle’s volume | m^{3} |

${H}_{o}$ | initial water depth | m |

CFL | Courant–Friedrichs–Lewy coefficient | - |

${d}_{s}$ | Average sediment grain size | m |

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**Figure 4.**Validation of accumulation body collapse process [42].

**Figure 6.**Simulation results of a dam break of a reservoir silting (sediment phase volume fraction map).

**Figure 7.**Simulation results of a dam break of a reservoir silting (flow velocity distribution map).

**Figure 8.**Dam break validation results of a reservoir silting (experiment, VOF model, and Euler model results from literature Vosoughi et al. [39]). Adapted with permission from Elsevier, Journal of Hydrology, License ID 1383966-1.

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**MDPI and ACS Style**

Zheng, X.; Rubinato, M.; Liu, X.; Ding, Y.; Chen, R.; Kazemi, E.
SPH Simulation of Sediment Movement from Dam Breaks. *Water* **2023**, *15*, 3033.
https://doi.org/10.3390/w15173033

**AMA Style**

Zheng X, Rubinato M, Liu X, Ding Y, Chen R, Kazemi E.
SPH Simulation of Sediment Movement from Dam Breaks. *Water*. 2023; 15(17):3033.
https://doi.org/10.3390/w15173033

**Chicago/Turabian Style**

Zheng, Xiaogang, Matteo Rubinato, Xingnian Liu, Yufei Ding, Ridong Chen, and Ehsan Kazemi.
2023. "SPH Simulation of Sediment Movement from Dam Breaks" *Water* 15, no. 17: 3033.
https://doi.org/10.3390/w15173033