# Numerical Simulation of Liquid Sloshing with Different Filling Levels Using OpenFOAM and Experimental Validation

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

**U**is the fluid velocity vector,

**τ**is the shear stress, C is the surface tension coefficient which is set to 0 in the current investigation, к is the interface curvature, α is the volume fraction, g is the acceleration of gravity,

**h**is the position vector of the mesh centre measured from the coordinates origin, p

_{rgh}is the dynamic pressure. The fluid density ρ and the viscosity coefficient μ are respectively calculated by densities (ρ

_{1}, ρ

_{2}) and viscosity (μ

_{1}, μ

_{2}) of two fluids using the volume fraction α,

**U**/∂t and ∂α/∂t. The Gauss linear discretization scheme is selected for gradient estimation, e.g., ∇·

**U**. Gauss linear corrected is considered for laplacian schemes such as ∇p

_{rgh}, ∇ρ. With regard to the divergence terms such as ∇·(

**UU**) and ∇·(

**U**α), the van Leer scheme is used. The PIMPLE algorithm, which is a combination of PISO (Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), is used for the velocity-pressure decoupling. More details of the InterDyMFoam can be found in the OpenFOAM website or other references. These will not be repeated here, only the solution process of the InterDyMFoam is illustrated in Figure 1 for completeness.

## 3. Experimental Setup

_{1}= 4.749 rad/s and ω

_{1}= 6.333 rad/s for the tank with the low (13.8%) and high (30.8%) filling level, respectively.

## 4. Model Validation

#### 4.1. Mesh Convergence Test

#### 4.2. Experimental Validation

_{1}) and the experimental data decays after t = 15 s. For the case shown in Figure 6b, the spectra shown in Figure 7b further confirm a satisfactory agreement. In addition, the free surface profiles are also compared. Some results in the case with a resonance excitation are shown in Figure 8, which compares the wave profiles at six instants between 8.55 s and 9.17 s. A good match has been observed.

#### 4.3. Three-Dimensional Results

_{1}, and A = 7 mm, and Case B, in which h = 200 mm, f = 0.96 f

_{1}, A = 7 mm.

## 5. Results and Discussions

#### 5.1. Effects of External Excitation Amplitudes

_{0}/ρgh, h is set as 0.2 m). Data obtained through physical model experiments (the amplitudes are 3 mm, 5 mm and 7 mm respectively) are also added to Figure 14 for validation. It can be noted that the impact pressure increases as the amplitude increases and that increase is basically linear in the non-resonance range. In order to further illustrate this relationship, Figure 15 shows the pressure-time histories in the cases with different excitation amplitudes. For the convenience of the comparison, the pressure results in the cases with A = 1 mm multiplied by a scaling factor of 3 is able to be comparable with the corresponding results with A = 3 mm. If two curves in Figure 15 match with each other (e.g., Figure 15a–d), the pressure is then linearly dependent on the motion amplitude; otherwise, the nonlinearity becomes more important (e.g., Figure 15e–f). This condition often occurs following the occurrence of wave resonance and breaking.

#### 5.2. Resonant Hysteresis and Resonance in Advance

_{1}to 0.92 ω

_{1}instead of keeping the first-mode natural frequency computed by the potential theory.

## 6. Conclusions

_{1}to 0.92 ω

_{1}moving away from the first-mode natural frequency ω

_{1}calculated by potential theory; (3) The main reason for this difference should be the nonlinear effect of wave breaking. The nonlinear effect of wave breaking was studied by a pressure-frequency response curve, the mean squared error-frequency response curve, the pressure-time history curve, and the spectral analysis at the maximum response frequency in the case of breaking and non-breaking. The result show that when at the same depth but without wave breaking, the phenomenon of resonant hysteresis disappears, and the resonance sloshing at a low filling level is more nonlinear than the resonance sloshing at a high filling level; (4) The pressure of the A = 0.5 mm case has been expanded 14 times to compare with the pressure of the A = 7 mm case to investigate how the sloshing state will be if wave cannot break. The result shows that the period of the shallow liquid sloshing decreases with wave breaking, therefore, the maximum response can be achieved at an external excitation frequency higher than the natural frequency. Combined with the fluid phenomenon in Figure 20 and Figure 21, the reasons may be the liquid climbing up the wall falls vertically and hits the free surface; this force accelerates the velocity of the sloshing wave. In the condition of deep water, the period of the liquid sloshing goes up with wave breaking, so the system resonance can be excited at an external excitation frequency lower than the natural frequency. One reasonable reason is that there is an obvious roof slamming and aerification which may dissipate energy during each slamming. Besides, a large amount of liquid splashing will reduce the depth of the water, which will lead to lower wave speed and a longer period of deepwater sloshing.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The experimental setup of liquid sloshing (

**a**). The motion simulation platform, (

**b**). The liquid tank.

**Figure 3.**The comparisons of both the actual motion platform displacement and theoretical displacement (amplitude is 7 mm, ω

_{A}= 1.425 rad/s, ω

_{B}= 4.749 rad/s, ω

_{C}= 14.249 rad/s).

**Figure 6.**The model validation under different filling levels, external frequencies and the same amplitude of A = 7 mm, (

**a**) h = 90 mm, ω = 0.5 ω

_{1}, (

**b**) h = 90 mm, ω = 0.8 ω

_{1}, (

**c**) h = 90 mm, ω = ω

_{1}, (

**d**) h = 200 mm, ω = 0.5 ω

_{1}, (

**e**) h = 200 mm, ω = 0.8 ω

_{1}, (

**f**) h = 200 mm, ω = ω

_{1}.

**Figure 7.**The Fast Fourier Transformation of the time history of the experimental and numerical pressure (amplitude is 7 mm, h = 90 mm, ω

_{a}= 0.5 ω

_{1}, ω

_{b}= 0.8 ω

_{1}).

**Figure 8.**The numerical wave profile and experimental snapshot in six moments between 8.55 s and 9.17 s (amplitude is 7 mm, h = 90 mm, ω = ω

_{1}).

**Figure 9.**The validation of the free surface time history curve, experimental data (Liu and Lin) from [12].

**Figure 10.**The validation of the pressure history curve, experimental data (et al.) from [27].

**Figure 12.**The comparison of impact pressure at different positions at the same height on the side wall. (

**a**) h = 90 mm, ω = 1.06 ω

_{1}, pressure on the bottom, (

**A**) h = 90 mm, ω = 1.06 ω

_{1}, pressure on the free surface, (

**b**) h = 200 mm, ω = 0.96 ω

_{1}, pressure on the bottom, (

**B**) h = 200 mm, ω = 0.96 ω

_{1}, pressure on the free surface.

**Figure 13.**The comparison of the 2D and 3D numerical results (

**a**) h = 90 mm, ω = 1.06 ω

_{1}, pressure on the bottom, (

**A**) h = 90 mm, ω = 1.06 ω

_{1}, pressure on the free surface, (

**b**) h = 200mm, ω = 0.96 ω

_{1}, pressure on the bottom, (

**B**) h = 200 mm, ω = 0.96 ω

_{1}, pressure on the free surface.

**Figure 15.**The comparison of pressure-time history curves at different amplitudes. (

**a**) h/L = 0.15, ω = 0.6 ω

_{1}, (

**b**) h/L = 0.33, ω = 0.6 ω

_{1}, (

**c**) h/L = 0.15, ω = 0.8 ω

_{1}, (

**d**) h/L = 0.33, ω = 0.8 ω

_{1}, (

**e**) h/L = 0.15, ω = ω

_{1}, (

**f**) h/L = 0.33, ω = ω

_{1}.

**Figure 16.**The pressure–frequency response curves at different filling levels. (

**a**) h/L = 0.054, (

**b**) h/L = 0.1, (

**c**) h/L = 0.15, (

**d**) h/L = 0.217, (

**e**) h/L = 0.25, (

**f**) h/L = 0.28, (g) h/L = 0.3, (h) h/L = 0.33, (

**i**) h/L = 0.433, (

**j**) h/L = 0.596.

**Figure 17.**The effects of wave breaking on shallow water sloshing. (

**left column**: wave breaking;

**right column**: no wave breaking).

**Figure 19.**The comparison of pressure-time history curve with wave breaking or not. (red line: wave breaking; black line: no wave breaking).

**Figure 20.**The fluid phenomenon in shallow water sloshing (h/L = 0.1, ω/ω

_{1}= 1.1, A = 7 mm, the falling liquid hits the free surface in the red circle).

**Figure 21.**The fluid phenomenon in deepwater sloshing (h/L = 0.596, ω/ω

_{1}= 0.92, A = 7 mm, the phenomena of roof slamming and aerification occurred in the red circle).

Two-dimensional Model | Three-dimensional Model | |
---|---|---|

Mesh Size | 5 × 5 mm | 5 × 5 × 5 mm |

Mesh Number | 15,600 | 936,000 |

Computing | CPU:AMD Ryzen 7 1700X Eight-Core Processor 3.40 GHz no parallel | |

Times | 1962 s | 394,971 s |

Analysis of Result | In agreement with the experimental data; Sometimes there are large pressure peaks. | Match well with the experimental value |

Case | h/L | ω_{1} (rad/s) | ω/ω_{1} | ω | A(m) |
---|---|---|---|---|---|

Case 1 | 0.15 | 4.749 | 0.5 | 2.3745 | 0.001 0.003 0.005 0.007 0.01 0.02 |

Case 2 | 0.6 | 2.8494 | |||

Case 3 | 0.7 | 3.3243 | |||

Case 4 | 0.8 | 3.7992 | |||

Case 5 | 0.9 | 4.2741 | |||

Case 6 | 1 | 4.749 | |||

Case 7 | 0.33 | 6.333 | 0.5 | 3.1665 | |

Case 8 | 0.6 | 3.7998 | |||

Case 9 | 0.7 | 4.4331 | |||

Case 10 | 0.8 | 5.0664 | |||

Case 11 | 0.9 | 5.6997 | |||

Case 12 | 1 | 6.333 |

Case | h/L | L(m) | h(m) | A(m) | ω_{1} (rad/s) | ω (rad/s) |
---|---|---|---|---|---|---|

Case 13 | 0.054 | 0.6 | 0.0324 | 0.007 | 2.942 | 0.8 ω_{1}–1.2 ω_{1} |

Case 14 | 0.1 | 0.06 | 3.953 | |||

Case 15 | 0.15 | 0.09 | 4.749 | |||

Case 16 | 0.217 | 0.1302 | 5.514 | |||

Case 17 | 0.25 | 0.15 | 5.804 | |||

Case 18 | 0.28 | 0.168 | 6.023 | |||

Case 19 | 0.3 | 0.18 | 6.15 | |||

Case 20 | 0.33 | 0.198 | 6.316 | |||

Case 21 | 0.433 | 0.2598 | 6.711 | |||

Case 22 | 0.596 | 0.3576 | 6.999 |

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**MDPI and ACS Style**

Chen, Y.; Xue, M.-A.
Numerical Simulation of Liquid Sloshing with Different Filling Levels Using OpenFOAM and Experimental Validation. *Water* **2018**, *10*, 1752.
https://doi.org/10.3390/w10121752

**AMA Style**

Chen Y, Xue M-A.
Numerical Simulation of Liquid Sloshing with Different Filling Levels Using OpenFOAM and Experimental Validation. *Water*. 2018; 10(12):1752.
https://doi.org/10.3390/w10121752

**Chicago/Turabian Style**

Chen, Yichao, and Mi-An Xue.
2018. "Numerical Simulation of Liquid Sloshing with Different Filling Levels Using OpenFOAM and Experimental Validation" *Water* 10, no. 12: 1752.
https://doi.org/10.3390/w10121752