# Visual Simulation of Detailed Turbulent Water by Preserving the Thin Sheets of Fluid

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## Abstract

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## 1. Introduction

**Problem Statement.**In previous methods for preserving fluid sheets [21,22], every attribute except for the mass is linearly interpolated when the particles split. When splitting occurs to obtain a pair of particles, $(i,j)$, the new mass is given by $m=({m}_{i}+{m}_{j})/3$. The new masses of particles i and j decrease to ${m}_{i}^{new}=\frac{2}{3}{m}_{i}$ and ${m}_{j}^{new}=\frac{2}{3}{m}_{j}$ after the split. The fluid velocity, u, and particle density, $\rho $, on the grid are affected by the changes in mass.

## 2. Related Work

**Turbulence Details.**Stam [24] proposed an unconditionally stable semi-Lagrangian method for turbulence details that is used widely in the fluid simulation despite numerical dissipation. To express the small-scale features of smoke and overcome numerical dissipation, Fedkiw et al. [25] proposed the vorticity confinement method. However, it cannot calculate the amount of confinement force to be injected without causing instability. To ensure the stability, He et al. [26] proposed the adaptive vorticity confinement method by adjusting the coefficients for vorticity according to the vorticity magnitude.

**Thin Sheet Details.**The level-set method [32] is frequently used in the field of fluid simulation. In this method, the distance to the nearest surface, which is defined implicitly as a zero contour, is saved for each node. This method has been refined to obtain many other novel approaches; for example, Enright et al. [33], Wang et al. [34], and Mihalef et al. [35] attempted to reduce the numerical loss using Lagrangian particle approaches.

## 3. Preserving the Thin Sheet and Turbulence Details of Water

- 1.
- Water particles are advected and their density is calculated.
- 2.
- Singular value decomposition (SVD) is used to extract thin particles from water particles.
- 3.
- The distances and relative velocities between thin particle pairs are used to find candidate positions.
- 4.
- The candidate positions are used to insert and break new water particles.
- 5.
- To restore the missing turbulence, the ghost density is calculated for all of the water particles, including the newly inserted water particles. The ghost mass is then calculated based on the density to ensure the conservation of mass.
- 6.
- The ghost mass is used to rasterize the velocities of particles on an Eulerian grid and advect the water particles using a FLIP solver.
- 7.
- The fluid surfaces are reconstructed.

#### 3.1. Preserving Thin Sheets

#### 3.1.1. Extracting Thin Particles

#### 3.1.2. Finding Candidate Positions

#### 3.1.3. Inserting and Removing Particles

#### 3.2. Synthesizing Turbulence Details

## 4. Implementation

## 5. Results

## 6. Discussion and Additional Explanation

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Streams of falling water represented by our method. The fluid sheets of water are expressed visually and the turbulent details are captured around the fluid sheets, thereby obtaining improved results compared with existing methods. (

**a**) Existing method [21,22]; (

**b**) Proposed method; and (

**c**) Proposed method—animation sequence.

**Figure 3.**Selected thin particles from water particles (thin particles are colored violet and lie on the outer edge of the fluid body).

**Figure 4.**Visualization of stretching thin sheet particles and their masses. (

**a**) Thin sheet particles (orange: Inserted particles); (

**b**) Mass of particles (red: High; blue: Low).

**Figure 9.**Visualization of particle masses during water splashing using proposed method (inset image: Particle view, particles are colored according to their mass).

**Table 1.**Parameters used in this paper. Specific parameters generate the example animations presented in this study.

Symbol | Description | Value |
---|---|---|

${\alpha}_{1}$ | Density radius scale | 4.0 |

${\alpha}_{2}$ | Thin sheet rate | 0.2 |

${\alpha}_{3}$ | Minimum insertion distance | 0.8 |

${\alpha}_{4}$ | Maximum insertion distance | 3.5 |

${\alpha}_{5}$ | Velocity radius scale | 1.0 |

${\alpha}_{6}$ | Maximum thin particle density | 0.2 |

${\alpha}_{7}$ | Maximum thin particle distance | 0.2 |

$\Delta t$ | Time-step | 0.006 |

**Table 2.**Sizes of the example scenes presented in this study (Num.: Number, Res.: Resolution, dx : Grid point spacing).

Figure | Num. of Water Particles | FLIP Solver Grid Res. | Surface Reconstruction Grid Res. | Pressure Solver Grid Res. | dx |
---|---|---|---|---|---|

1 | 322,454 | 100 × 100 × 100 | 256 × 256 × 256 | 100 × 100 × 100 | 0.01 |

4 | 45,200 | 150 × 150 × 150 | – | 150 × 150 × 150 | 0.006 |

7,8 | 1,802,626 | 128 × 128 × 128 | 256 × 256 × 256 | 128 × 128 × 128 | 0.007 |

9 | 843,512 | 128 × 128 × 128 | 256 × 256 × 256 | 128 × 128 × 128 | 0.007 |

10 | 3,079,593 | 180 × 180 × 180 | 180 × 180 × 180 | 180 × 180 × 180 | 0.005 |

11 | 498,143 | 100 × 100 × 100 | 256 × 256 × 256 | 100 × 100 × 100 | 0.01 |

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**MDPI and ACS Style**

Kim, J.-H.; Kim, W.; Kim, Y.B.; Lee, J.
Visual Simulation of Detailed Turbulent Water by Preserving the Thin Sheets of Fluid. *Symmetry* **2018**, *10*, 502.
https://doi.org/10.3390/sym10100502

**AMA Style**

Kim J-H, Kim W, Kim YB, Lee J.
Visual Simulation of Detailed Turbulent Water by Preserving the Thin Sheets of Fluid. *Symmetry*. 2018; 10(10):502.
https://doi.org/10.3390/sym10100502

**Chicago/Turabian Style**

Kim, Jong-Hyun, Wook Kim, Young Bin Kim, and Jung Lee.
2018. "Visual Simulation of Detailed Turbulent Water by Preserving the Thin Sheets of Fluid" *Symmetry* 10, no. 10: 502.
https://doi.org/10.3390/sym10100502