# Numerical Study on Sloshing Characteristics with Reynolds Number Variation in a Rectangular Tank

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}to 1.6 × 10

^{4}. The results were analyzed by the FFT technique and compared quantitatively with the magnitude of amplitude spectrum. Birknes-Berg and Pedersen [14] conducted a study on the treatment of boundary layers on a solitary sloshing wave, and the results were analyzed with boundary layer thickness in the range of Re = 1 × 10

^{10}. However, this paper provides the specific boundary layer flow variation with Reynolds number changes on the sloshing flow.

## 2. Analysis Model and Methods

#### 2.1. Computational Domain and Boundary Conditions

_{n}is n-th wave number, nπ/B, and the sloshing simulation in the rectangular tank is carried out under the unsteady state and horizontal excitation for all cases. The ideal gas properties of air are considered and the density of water is 998 kg/m

^{3}. The front and back sides of the numerical model have symmetric boundary conditions and horizontal motion is considered for periodic excitation. The checking position of sloshing pressure on the wall, MP, is located at 0.15 m from the bottom of the tank, which is the same as the free surface level. Here, B = 1 m, h = 0.15 m and H = 0.6 m.

^{4}to 7.2 × 10

^{4}were considered, as shown in Table 1. The Reynolds number definition [16] is given by

_{w}and μ

_{w}are water density and dynamic viscosity of water, respectively.

#### 2.2. Governing Equations

**Volume Conservation equation:**

**Continuity equation:**

**Momentum equation:**

_{p}is the total number of phases, α and β are phases, r is the fixed volume fraction, U is the velocity, p is the pressure and i, j denotes the tensor. The third term in the right hand side of Equation (7) is the factor that distinguishes between the homogeneous and inhomogeneous models. Γ

^{+}indicates the positive mass flow rate per unit volume, and the Γ

_{αβ}is given by

_{αβ}is the interfacial area proportional to the volume fraction density. The interfacial area of the free surface model is given by

#### 2.3. Analytical Solution for Linear Sloshing

#### 2.4. Discussion of the Numerical Model with Inhomogeneous VOF

## 3. Results and Discussion

#### 3.1. Observing Nonlinearity of Sloshing Flows

^{4}). Although the magnitude of amplitudes between analytical and numerical results matches well initially, the phase difference gradually increases, as time goes by, due to the influence of nonlinearity and wall friction. Therefore, it seems to be that applying the analytical solution to estimate free-surface oscillation is unreasonable even in terms of rough criteria. Figure 3b,c show the results of Case 2 (Re = 2.5 × 10

^{4}) and Case 3 (Re = 3.6 × 10

^{4}), respectively. It can be observed that the discrepancy between the local maximum and local minimum values of free-surface elevation increases with time. From the above results, it can be inferred that the sloshing flow (over Re = 1.8 × 10

^{4}) has a strong nonlinearity and it seems to be very difficult to estimate the sloshing flow using the linear analytical solution.

#### 3.2. Sloshing Characteristics with Reynolds Number Variation

#### 3.3. Comparison of Sloshing Characteristics by FFT Analysis

#### 3.4. Visual Observation on the Sloshing Impact Motion

^{4}. The typical characteristics of violent sloshing such as overturning and breaking waves can be observed. In Figure 7e, it seems to be that the more violent sloshing flow appears due to the superposed strong nonlinear fluid flow, with the increases of the Reynolds number. Overall, the wave elevation increases in proportion to the Reynolds numbers, and also the sloshing flows in each case show developing irregular flow patterns as the Reynolds number increases.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison between numerical and experimental data [23].

**Figure 3.**Comparison of sloshing wave elevation between numerical and analytical solutions; (

**a**) Case 1; (

**b**) Case 2; and (

**c**) Case 3.

**Figure 5.**(

**A**) Pressure patterns of slosh for Case 1 (

**a**,

**a**

**′**), Case 2 (

**b**,

**b**

**′**) and Case 3 (

**c**,

**c**

**′**); (

**a**,

**b**,

**c**) are time histories of pressures from 0 to 30 s at MP for each cases; (

**a**

**′**,

**b**

**′**,

**c**

**′**) are the enlarged figures near the maximum pressures. (

**B**) Pressure patterns of slosh for Case 4 (

**d**,

**d**

**′**) and Case 5 (

**e**,

**e**

**′**); (

**d**,

**e**) are time histories of pressures from 0 to 30 s at MP for each cases; (

**d**

**′**,

**e**

**′**) are the enlarged figures near the maximum pressures.

**Figure 6.**FFT analysis of the time historical data of the sloshing pressure; (

**a**) Case 1, (

**b**) Case 2, (

**c**) Case 3, (

**d**) Case 4, and (

**e**) Case 5.

**Figure 7.**Visualization of sloshing flows in the rectangular tank at impact moments of each case; (

**a**) Case 1, (

**b**) Case 2, (

**c**) Case 3, (

**d**) Case 4, and (

**e**) Case 5.

Case | Reynolds No. | Natural Frequency (ω_{1}, s^{−1}) | Excited Frequency (Hz) |
---|---|---|---|

1 | 1.8 × 10^{4} | 3.68 | 0.586 |

2 | 2.5 × 10^{4} | ||

3 | 3.6 × 10^{4} | ||

4 | 5.0 × 10^{4} | ||

5 | 7.2 × 10^{4} |

Case | S |
---|---|

1 | 1.111 |

2 | 1.567 |

3 | 1.759 |

Case | Max. Pressure Fluctuation |
---|---|

No. | Pa |

1 (Re 1.8 × 10^{4}) | 15 |

2 (Re 2.5 × 10^{4}) | 64 |

3 (Re 3.6 × 10^{4}) | 162 |

4 (Re 5.0 × 10^{4}) | 315 |

5 (Re 7.2 × 10^{4}) | 789 |

Case | P_{Max} | Max. Amplitude Spectrum | |
---|---|---|---|

No. | Pa | Second | - |

1 (Re 1.8 × 10^{4}) | 602 | 18.5 | 1682 |

2 (Re 2.5 × 10^{4}) | 695 | 15.1 | 1994 |

3 (Re 3.6 × 10^{4}) | 829 | 11.7 | 2317 |

4 (Re 5.0 × 10^{4}) | 1020 | 11.4 | 2748 |

5 (Re 7.2 × 10^{4}) | 1520 | 9.7 | 3430 |

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**MDPI and ACS Style**

Kim, H.; Dey, M.K.; Oshima, N.; Lee, Y.W.
Numerical Study on Sloshing Characteristics with Reynolds Number Variation in a Rectangular Tank. *Computation* **2018**, *6*, 53.
https://doi.org/10.3390/computation6040053

**AMA Style**

Kim H, Dey MK, Oshima N, Lee YW.
Numerical Study on Sloshing Characteristics with Reynolds Number Variation in a Rectangular Tank. *Computation*. 2018; 6(4):53.
https://doi.org/10.3390/computation6040053

**Chicago/Turabian Style**

Kim, Hyunjong, Mohan Kumar Dey, Nobuyuki Oshima, and Yeon Won Lee.
2018. "Numerical Study on Sloshing Characteristics with Reynolds Number Variation in a Rectangular Tank" *Computation* 6, no. 4: 53.
https://doi.org/10.3390/computation6040053