# Study of the Sloshing Dynamics in Partially Filled Rectangular Tanks with Submerged Baffles Using VOF and LES Turbulence Methods for Different Impact Angles

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## Abstract

**:**

^{®}FLUENT. The study aims to characterize the effect of changing the baffle’s angle; hence, 10 simulations have been performed: without a baffle, 90°, 30°, 60°, 120° and 150°, either maintaining the baffle’s length or the projected height constant. The computational fluid dynamics (CFD) method using volume of fluid (VOF) and large eddy simulation (LES) are used to predict the movement of the fluid in two dimensions, which have been benchmarked against experimental data with excellent agreement. The motion is sinusoidal in the +X direction, with a frequency of oscillation equal to its first vibration mode. The parameters studied have been the free surface elevation, values at three different points and maximum; the center of gravity’s position, velocity, and acceleration; and the forces against the tank’s walls. It has been found that the 90° angle has the most significant damping effect, stabilizing the free-surface elevation, reducing the center of gravity dispersion, and leveling the impacting forces. Smaller angles also tame the sloshing and stabilize it.

## 1. Introduction

## 2. Numerical Model

#### 2.1. Governing Equations

#### 2.2. Volume of Fluid (VOF)

#### 2.3. Sloshing Natural Frequency

_{n}is the frequency of oscillation in the vibration mode, g is gravity, L is the tank length, and h is the fluid height.

## 3. Simulation Setup

#### 3.1. Tank Schematic and Mesh

_{b}) or the projected height (H

_{b}) is maintained constant throughout the different simulations and to a value of 75 mm, half of the water’s depth, as defined by Joshi et al. [19], to see how both parameters affect the results. Three wave gauges (WG) have been placed to measure the free surface elevation, as by Liu and Lin [1].

^{3}and viscosity μ = 1.003E−3 kg/(m·s); and air with ρ = 1.225 kg/m

^{3}and viscosity μ = 1.7894E−5 kg/(m·s). The tank undergoes a sinusoidal displacement x(t) = dsin(wt), where d = 0.005 m, as defined by Liu and Lin [1], Saravanan et al. [20], and Pandit and Chandra Biswal [21] under its first mode natural frequency w = w

_{n}= 6.736 rad/s (period T = 0.9328 s, velocity 0.005 × 6.736 = 0.03368 m/s much less than impacts from Khezzar et al. [11]). Time (t) has been discretized with time-steps of 0.0001 s for a total of 7.47 s, equal to 8 acceleration periods. As for the spatial domain, a mesh grid of 5 mm elements has been used, as by Liu and Lin [1] and Xue et al. [2,22]. A mesh convergence study was carried out to check that, with this mesh size, results were providing the right benchmark response. Concerning Reynolds number Re = LU/ν (with L being the gap between baffle and free surface 0.075 m, U the wave, x velocity, and ν the kinematic viscosity of 0.0001 m

^{2}/s) Re transition from laminar occurs at 0.03068 m/s with Re = 2301, and turbulence should start for 0.05335 m/s with Re = 4001.

#### 3.2. Method of Solution

^{®}Fluent R2 2020 with a pressure-based segregated solved technology with no skewness neighbor correction. The VOF model is used to capture the interface between the two fluids. The LES model is adopted to model turbulence, using the wall-adapting local eddy viscosity (WALE) as the sub-grid scale (SGS) model.

^{−6}.

#### 3.3. Model Validation

_{n}= 5.315 rad/s (T = 1.182 s, velocity 0.0093 × 5.315 = 0.04943 m/s much less than impacts from Khezzar et al. [11] but similar to Liu and Lin [1], Saravanan et al. [20], and Pandit and Chandra Biswal [21]). Figure 2 tracks the free surface elevation along the tank length. Reynolds-averaged Navier–Stokes (RANS) models such as k-e or k-w underestimate the sloshing, whereas LES perfectly tracks the free surface elevation compared to experimental data.

_{n}= 6.058 rad/s (velocity 0.005 × 6.058 = 0.03029 m/s much less than impacts from Khezzar et al. [11] but very similar to Liu and Lin [1], Saravanan et al. [20], and Pandit and Chandra Biswal [21]). Three wave gauges measure the free surface elevation at each time step: one in the center of the free surface 285 mm and two 20 mm away from the walls, similar to those in Figure 1. These measurements compared with experimental data and simulations can be seen in Figure 3. Results from this research show even greater accuracy than the previous work and, therefore, validate the LES turbulence model’s use and the discretization methods.

## 4. Results and Discussion

_{b}’s importance is signaled over any other parameter. H

_{b}is crucial for the fluid’s overall stability, being directly proportional to the reduction of the standard deviation of the CG position, velocity, and acceleration, and the forces acting on the walls. H

_{b}is inversely proportional to the standard deviation, maximum, and RMS values of the free surface elevation at the three Wave Gauges. On the other hand, H

_{b}is directly proportional to the minimum values; this reinforces the importance of H

_{b}when it comes to dampening the motion over a mean value.

## 5. Conclusions

_{b}constantly signals a linear correlation with the study’s parameters to reduce or increase them in favor of the fluid’s stability.

_{b}has the most significant importance when it comes to taming capabilities. This also is true when comparing baffles with the same angle. Baffles with H

_{b}= 75 mm perform better than those with the same angle but L

_{b}= 75 mm. This is related to the previous and next conclusion because baffles with a constant L

_{b}have a smaller H

_{b}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Tank schematic for (

**a**) h = 150 mm, L = 500 mm and (

**b**) baffle configuration for fixed H

_{b}= 75 mm and (

**c**) for fixed L

_{b}= 75 mms.

**Figure 2.**Free surface elevation along the tank length at t = 3.55 s (3 periods T). The dotted line represents the initial free surface elevation.

**Figure 3.**Comparison of the time series of the free surface elevation h at probe 1, probe 2, and probe 3, respectively.

**Figure 4.**Free surface elevation at WG1. The solid line represents the simulations where H

_{b}= 75 mm, the dotted line where L

_{b}= 75 mm.

**Figure 5.**Free surface elevation at WG3. The solid line represents the simulations where H

_{b}= 75 mm, the dotted line where L

_{b}= 75 mm.

**Figure 6.**Maximum free surface elevation through time, that is, the height of the wave through the simulation time.

**Figure 8.**X and Y scattered plot for each angle. Left column for fixed H

_{b}= 75 mm (

**a**) CG position, (

**b**) velocity and (

**c**) acceleration and right column for fixed L

_{b}= 75 mm (

**d**) CG position, (

**e**) velocity and (

**f**) acceleration.

**Figure 9.**Kernel density distribution. Left column for fixed H

_{b}= 75 mm (

**a**) CG position, (

**b**) velocity and (

**c**) acceleration in X direction and right column for fixed L

_{b}= 75 mm (

**d**) CG position, (

**e**) velocity and (

**f**) acceleration in Y direction.

**Figure 10.**Forces for left column for fixed H

_{b}= 75 mm for (

**a**) left wall, (

**b**) right wall and right column for fixed L

_{b}= 75 mm for (

**c**) left wall, (

**d**) right wall.

h_{max} RMS | WG_{1} RMS | WG_{2} RMS | WG_{3} RMS | x_{max} RMS | ||
---|---|---|---|---|---|---|

L_{b} | −0.46 | −0.58 | 0.61 | −0.49 | 0.098 | |

H_{b} | −0.92 | −0.96 | 0.94 | −0.92 | 0.92 | |

C_{b} | −0.014 | −0.16 | 0.15 | −0.05 | −0.3 | |

CG_{x} min | CG_{x} max | CG_{x} RMS | CG_{y} min | CG_{y} max | CG_{y} RMS | |

L_{b} | 0.56 | −0.58 | −0.38 | 0.94 | −0.66 | −0.51 |

H_{b} | 0.88 | −0.97 | −0.73 | 0.62 | −0.96 | −0.94 |

C_{b} | 0.18 | −0.15 | −0.036 | 0.89 | −0.27 | −0.073 |

V_{x} min | V_{x} max | V_{x} RMS | V_{y} min | V_{y} max | V_{y} RMS | |

L_{b} | 0.6 | −0.6 | −0.54 | 0.66 | −0.66 | −0.63 |

H_{b} | 0.97 | −0.97 | −0.96 | 0.96 | −0.96 | −0.97 |

C_{b} | 0.17 | −0.17 | −0.091 | 0.27 | −0.26 | −0.22 |

A_{x} min | A_{x} max | A_{x} RMS | A_{y} min | A_{y} max | A_{y} RMS | |

L_{b} | 0.56 | −0.56 | −0.54 | 0.7 | −0.57 | −0.61 |

H_{b} | 0.96 | −0.96 | −0.96 | 0.98 | −0.93 | −0.96 |

C_{b} | 0.12 | −0.12 | −0.093 | 0.3 | −0.16 | −0.2 |

s CG_{x} | s CG_{y} | s V_{x} | s V_{y} | s A_{x} | s A_{y} | |

L_{b} | −0.53 | −0.65 | −0.54 | −0.63 | −0.55 | −0.61 |

H_{b} | −0.96 | −0.97 | −0.96 | −0.97 | −0.87 | −0.96 |

C_{b} | −0.086 | −0.25 | −0.088 | −0.22 | −0.14 | −0.2 |

F_{LW} min | F_{LW} max | F_{LW} RMS | F_{RW} min | F_{RW} max | F_{RW} RMS | |

L_{b} | 0.61 | −0.59 | 0.43 | 0.56 | −0.57 | −0.71 |

H_{b} | 0.97 | −0.94 | 0.88 | 0.96 | −0.96 | −0.98 |

C_{b} | 0.19 | −0.19 | 0.014 | 0.12 | −0.14 | −0.31 |

s F_{LW} | s F_{RW} | |||||

L_{b} | −0.5 | −0.5 | ||||

H_{b} | −0.94 | −0.94 | ||||

C_{b} | −0.059 | −0.47 |

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**MDPI and ACS Style**

Vallés Rebollo, X.; Sadeghi, E.; Kusano, I.; García-Granada, A.-A.
Study of the Sloshing Dynamics in Partially Filled Rectangular Tanks with Submerged Baffles Using VOF and LES Turbulence Methods for Different Impact Angles. *Computation* **2022**, *10*, 225.
https://doi.org/10.3390/computation10120225

**AMA Style**

Vallés Rebollo X, Sadeghi E, Kusano I, García-Granada A-A.
Study of the Sloshing Dynamics in Partially Filled Rectangular Tanks with Submerged Baffles Using VOF and LES Turbulence Methods for Different Impact Angles. *Computation*. 2022; 10(12):225.
https://doi.org/10.3390/computation10120225

**Chicago/Turabian Style**

Vallés Rebollo, Xavier, Ehsan Sadeghi, Ibuki Kusano, and Andrés-Amador García-Granada.
2022. "Study of the Sloshing Dynamics in Partially Filled Rectangular Tanks with Submerged Baffles Using VOF and LES Turbulence Methods for Different Impact Angles" *Computation* 10, no. 12: 225.
https://doi.org/10.3390/computation10120225