# Three-Dimensional Numerical Method for Simulating Large-Scale Free Water Surface by Massive Parallel Computing on a GPU

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Lattice Boltzmann Method

_{i}. Through solution and statistics of particle distribution function f

_{i}corresponding to various discrete velocities, the macroscopic variables of the fluid such as density $\rho =\sum {f}_{i}$ and momentum $\rho u=\sum {f}_{i}{e}_{i}$ of the fluid are obtained.

**r**and time $t$; $\{{F}_{\alpha}:\alpha =0,1,\dots ,18\}$ refers to the forcing term; $\{{m}_{\alpha}(\mathit{r},t):\alpha =0,1,\dots ,18\}$ refers to the moment of distribution functions (see reference [17] for details). The relationship among m, transfer matrix M, and distribution function f can be expressed as:

#### 2.2. Single-Phase Free Surface Lattice Boltzmann Model

#### 2.2.1. Mass Flow Computation

_{i}and f

_{inv(i)}, where e

_{inv(i)}= −e

_{i}, and $\mathsf{\Delta}{m}_{i}$ denotes the mass variable quantity on $i$ direction along the particle velocity [11].

#### 2.2.2. Reconstruction of Interface Cells

**u**refer to pressure and velocity at position

**x**, respectively. In addition, to balance the forces given by the gas pressure and velocity at interface, the distribution functions whose discrete velocity direction

**e**

_{i}satisfies

**e**

_{i}

_{·}

**n**< 0 shall be reconstructed [13], where

**n**refers to the surface normal direction and can be obtained by the second-order central difference approximation $\mathit{n}=\sum {s}_{i}{\mathit{e}}_{i}\epsilon (\mathit{x}+{\mathit{e}}_{i})$.

#### 2.2.3. Mass Allocation

**x**distributed to the cell at

**x**+

**e**

_{i}is [13]:

#### 2.3. Large Eddy Simulation Based on the LB Method

## 3. Testing and Discussion

#### 3.1. Sudden Whole Dam-Break Flow

#### 3.2. Partial Dam-Break Flow

#### 3.3. Simulation of Dam-Break Flow over a Triangular Step

#### 3.4. Simulation of Flood Discharge of an Arch Dam

#### 3.4.1. Analysis of Flow Patterns

#### 3.4.2. Analysis of Lattice Sensitivity

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parallel Computing of the SPFS-LB Model on a GPU

#### Appendix A.1. Technology of Parallel Computing on a GPU

**Figure A1.**Parallel program of the SPFS-LB model based on Compute Unified Device Architecture (CUDA).

#### Appendix A.2. Analysis of Parallel Computing

Lattice | GPU | CPU | ||
---|---|---|---|---|

128 | 256 | 512 | MLUPS | |

96 × 128 × 48 | 41.19 (33.48) | 51.26 (41.67) | 59.00 (47.96) | 1.23 |

192 × 256 × 96 | 40.92 (33.82) | 52.00 (42.98) | 60.03 (49.61) | 1.21 |

384 × 512 × 192 | 41.31 (33.86) | 52.82 (43.30) | 60.92 (49.93) | 1.22 |

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**Figure 2.**Water surface profile of a 1-D dam-break wave (60 s after dam failure), (

**a**) h = 1.75 m, (

**b**) h = 5 m.

**Figure 10.**Water surface and velocity of flood discharge of an arch dam (reservoir water level: 450 m).

**Table 1.**Comparison of the discharge coefficients between the simulated results and the empirical formula results.

Reservoir Water Level (m) | 445 | 450 | 455 | |
---|---|---|---|---|

Empirical formula results [24] | Side contraction coefficients | 0.958 | 0.929 | 0.889 |

H/H_{d} | 0.307 | 0.820 | 1.332 | |

Discharge coefficients | 0.419 | 0.487 | 0.530 | |

Simulated results | Discharge coefficients | 0.402 | 0.477 | 0.543 |

Relative error (%) | 4.1 | 2.1 | 2.6 |

_{d}refers to the design head, which is taken as 9.76 m according to the design data of the arc dam.

**Table 2.**Relative error between the simulated results and empirical formula results with different lattices.

Lattice | Δx | Discharge Coefficients | Relative Deviation | |
---|---|---|---|---|

Simulated Results | Empirical Formula Results [24] | |||

96 × 128 × 48 | 0.52 m | 0.470 | 0.487 | 3.5% |

192 × 256 × 96 | 0.39 m | 0.475 | 0.487 | 2.4% |

384 × 512 × 192 | 0.26 m | 0.477 | 0.487 | 2.1% |

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**MDPI and ACS Style**

Peng, Y.; Diao, W.; Zhang, X.; Zhang, C.; Yang, S. Three-Dimensional Numerical Method for Simulating Large-Scale Free Water Surface by Massive Parallel Computing on a GPU. *Water* **2019**, *11*, 2121.
https://doi.org/10.3390/w11102121

**AMA Style**

Peng Y, Diao W, Zhang X, Zhang C, Yang S. Three-Dimensional Numerical Method for Simulating Large-Scale Free Water Surface by Massive Parallel Computing on a GPU. *Water*. 2019; 11(10):2121.
https://doi.org/10.3390/w11102121

**Chicago/Turabian Style**

Peng, Yongqin, Wei Diao, Xujin Zhang, Chunze Zhang, and Shuqing Yang. 2019. "Three-Dimensional Numerical Method for Simulating Large-Scale Free Water Surface by Massive Parallel Computing on a GPU" *Water* 11, no. 10: 2121.
https://doi.org/10.3390/w11102121