# Fluid Structure Interaction of 2D Objects through a Coupled KBC-Free Surface Model

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Lattice Boltzmann Method

#### The Smagorinsky Subgrid Model

#### 2.2. The KBC Model

#### 2.3. Fluid Structure Interaction and Moving Boundary Treatment

#### 2.4. The Free-Surface Model

## 3. Model Validation

## 4. Results

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Obstacle definition into fluid domain. Empty squares represent the fluid nodes, filled squares belong to the obstacle; the red filled circle represents the exact wall position along the ith lattice direction. Continuous red arrows represent the known populations, while the dashed ones represent the ones to be reconstructed before streaming.

**Figure 2.**Schematic representation of refilling procedure. The obstacle, initially located at the dashed cyan line (new position is the solid cyan line) frees some nodes, the gray filled ones, which belonged to the solid. Their new fluid-dynamics properties are copied from both red filled fluid node ${\mathbf{x}}_{f}$ and solid one ${\mathbf{x}}_{s}$.

**Figure 6.**Slamming coefficient as a function of non-dimensional time for the sensitivity matrix reported in Table 3. Black curves are related to low Re, gray ones to high Re. Solid line refers to $D=50\mathrm{LU}$, dashed ones to $D=100\mathrm{LU}$, dotted ones to $D=150\mathrm{LU}$, and dash-dotted ones to $D=200\mathrm{LU}$.

**Figure 7.**Magnification of Figure 6 referring to the first impact phases. Black curves are related to low Re, gray ones to high Re. Solid line refers to $D=50\mathrm{LU}$, dashed ones to $D=100\mathrm{LU}$, dotted ones to $D=150\mathrm{LU}$ and dash-dotted ones to $D=200\mathrm{LU}$.

**Figure 8.**Vorticity plot for different resolution at ${t}^{\U0001f7c9}=1.0$ for Re = 3660: (

**a**–

**d**) the obstacle diameter from $50\mathrm{LU}$ to $200\mathrm{LU}$, respectively.

**Figure 9.**Comparison between experimental results presented by [32] and numerical model at $\mathrm{Fr}=0.4628$. Black dotted line represents experimental findings, solid red line the KBC results, magenta line the BGK without turbulence model, and the blue curve represents the BGK with Smagorinsky sub-grid model.

**Figure 10.**Comparison between experimental results presented by [32] and numerical model at $\mathrm{Fr}=0.5772$. Black dotted line represents experimental findings, solid red line the KBC results, magenta line the BGK without turbulence model, and the blue curve represents the BGK with Smagorinsky sub-grid model.

**Figure 11.**Comparison between experimental results presented by [32] and numerical model at $\mathrm{Fr}=0.6865$. Black dotted line represents experimental findings, solid red line the KBC results, magenta line the BGK without turbulence model, and the blue curve represents the BGK with Smagorinsky sub-grid model.

**Figure 12.**$\gamma $ parameter when the obstacle is completely submerged at ${t}^{\U0001f7c9}=2.4850$ for $\mathrm{Fr}=0.5772$.

**Figure 13.**$\omega $ parameter when the obstacle is completely submerged at ${t}^{\U0001f7c9}=2.4850$ for $\mathrm{Fr}=0.5772$.

**Figure 14.**Vorticity profile for KBC method when the obstacle is completely submerged at ${t}^{\U0001f7c9}=2.4850$ for $\mathrm{Fr}=0.5772$.

**Figure 15.**Vorticity profile for BGK method when the obstacle is completely submerged at ${t}^{\U0001f7c9}=2.4850$ for $\mathrm{Fr}=0.5772$.

**Figure 16.**Schematic representation of the ellipsoidal obstacle with the sensitivity analysis parameters.

**Figure 17.**Slamming coefficient as a function of the ellipse aspect ratio. Solid black line represents aspect ratio of 0.25, dotted black line 0.5, dashed black line the circular obstacle with ${R}_{y}/{R}_{x}=1.0$, solid light gray line the aspect ratio of 2.0, and dotted light gray line the largest aspect ratio of 4.0.

**Figure 18.**Magnification up to ${t}^{\U0001f7c9}=1$ of the slamming coefficient as a function of the ellipse aspect ratio. Solid black line represents aspect ratio of 0.25, dotted black line 0.5, dashed black line the circular obstacle with ${R}_{y}/{R}_{x}=1.0$, solid light gray line the aspect ratio of 2.0, and dotted light gray line the largest aspect ratio of 4.0.

**Figure 20.**Density evolution at different non-dimensional times for different ellipse geometry. The three snapshots for ever radii ratio represent half of wedge wetted, full wetted wedge, and three-quarters of wetted wedge, respectively. The three obstacle penetrations correspond to the same non-dimensional times, more specifically ${t}^{\U0001f7c9}=$ 1.05, 2.10, and 3.10, respectively.

**Figure 21.**Schematic representation of the theoretical instantaneous wetted diameter used in Equation (33).

**Figure 22.**Evolution of the new slamming coefficient as a function of the non-dimensional time. Solid black line represents aspect ratio of 0.25, dotted black line 0.5, dashed black line the circular obstacle with ${R}_{y}/{R}_{x}=1.0$, solid light gray line the aspect ratio of 2.0, and dotted light gray line the largest aspect ratio of 4.0.

$\mathit{D}\left[\mathbf{m}\right]$ | $\mathit{W}\left[\mathbf{m}\right]$ | $\mathit{H}\left[\mathbf{m}\right]$ | ${\mathit{h}}_{0}\left[\mathbf{m}\right]$ |
---|---|---|---|

0.125 | 28.0 | 2.5 | 1.0 |

Case | $\mathit{c}\left[\mathbf{m}/\mathbf{s}\right]$ | Re | $\mathbf{Fr}$ |
---|---|---|---|

1 | 0.5124 | 64050 | 0.4628 |

2 | 0.6390 | 79875 | 0.5772 |

3 | 0.7600 | 95000 | 0.6865 |

**Table 3.**Test matrix for sensitivity analysis at constant $\mathrm{Fr}=0.4628$. $\left[\mathrm{LU}\right]$ means that all the quantities are expressed in non-dimensional units characteristic of the LB framework.

Simlation No. | $\mathit{D}\left[\mathbf{LU}\right]$ | $\mathit{W}\times \mathit{H}\left[\mathbf{LU}\right]$ | ${\mathit{\nu}}_{0}\left[\mathbf{LU}\right]$ | $\mathbf{Re}$ |
---|---|---|---|---|

1 | 50 | 11,200 × 1000 | 0.00440 | 366 |

2 | 50 | 11,200 × 1000 | 0.00044 | 3660 |

3 | 100 | 22,400 × 2000 | 0.00880 | 366 |

4 | 100 | 22,400 × 2000 | 0.00088 | 3660 |

5 | 150 | 33,600 × 3000 | 0.01320 | 366 |

6 | 150 | 33,600 × 3000 | 0.00132 | 3660 |

7 | 200 | 40,000 × 4000 | 0.01760 | 366 |

8 | 200 | 40,000 × 4000 | 0.00176 | 3660 |

Simulation No. | ${\mathit{R}}_{\mathit{y}}/{\mathit{R}}_{\mathit{x}}$ |
---|---|

1 | 0.25 |

2 | 0.50 |

3 | 1.00 |

4 | 2.00 |

5 | 4.00 |

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**MDPI and ACS Style**

Chiappini, D.
Fluid Structure Interaction of 2D Objects through a Coupled KBC-Free Surface Model. *Water* **2020**, *12*, 1212.
https://doi.org/10.3390/w12041212

**AMA Style**

Chiappini D.
Fluid Structure Interaction of 2D Objects through a Coupled KBC-Free Surface Model. *Water*. 2020; 12(4):1212.
https://doi.org/10.3390/w12041212

**Chicago/Turabian Style**

Chiappini, Daniele.
2020. "Fluid Structure Interaction of 2D Objects through a Coupled KBC-Free Surface Model" *Water* 12, no. 4: 1212.
https://doi.org/10.3390/w12041212