# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{®}[4,5,6,7,8].

^{®}volume-of-fluid (VoF) CFD software.

- We validate the accuracy of the CFD code Elemental
^{®}for modelling violent vertical slosh physics relevant to this article. - We detail the CFD based energy budget used to quantify effects of scaling non-dimensional properties on the system.
- We present a non-dimensional analysis of a SDOF tank under vertical slosh, isolating the functional relationship characterising slosh induced energy dissipation.
- We define the non-dimensional parameter space of interest for the problem under consideration (to include both experimental and full scales).
- Finally, we develop novel scaling-laws which correlate the slosh induced energy dissipation as a function of the identified non-dimensional parameters. This is done via curve fitting of CFD generated data.
- The developed novel scaling laws are finally applied to quantify ideal (representative of full scale aircraft) experimental slosh induced energy dissipation.

## 2. Validation of CFD Model

^{®}has been employed extensively to model violent slosh [17,18]. Turbulence is modelled via Large-eddy simulation (LES) employing the Smagorinsky-Lilly model [19] and a weakly compressible gas model [15,16] is employed while the liquid-gas interface initialisation is performed to machine precision accuracy via the AGI tool [20].

^{®}as an accurate tool for this work requires the comparison of the computed results against relevant experimental data. The modelling of the $50\%$ fill-level Protospace experiment [3] has been selected for this purpose as it was designed to be a small scale representation of wing-tank based slosh. The Protospace experiment involved the damped vibration (structural and slosh damping) of a cantilever beam mounted with a seven compartmental fuel tank, as shown in Figure 1. The experiment involved applying an initial deflection and then releasing the cantilever at $t=0s$ followed by free vibration [3].

## 3. Energy Analysis

## 4. Dimensional Analysis

## 5. Scaling of Violent Sloshing Systems

#### 5.1. Froude Scaling Rules

#### 5.2. Practical Considerations

## 6. Non-Dimensional Property Parameter Space

## 7. Non-Dimensional Study Results and Analysis

## 8. Dissipated Energy Scaling Laws

## 9. Scaling-Laws Application

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Footage of interface regimes for the Protospace Experiment. (

**a**) t = 0.0 s; (

**b**) t = 0.12 s, liquid–solid impact; (

**c**) t = 0.4 s, Aerated Flow Slosh; (

**d**) t = 3 s, Reconstituted Free Surface.

**Figure 2.**CFD predicted interface regimes for the Protospace experimental case. (

**a**) t = 0.0 s; (

**b**) t = 0.12 s; (

**c**) t = 0.4 s; (

**d**) t = 3 s.

**Figure 4.**Percentage difference between CFD predicted and experimentally derived slosh induced energy dissipation for various mesh resolutions.

**Figure 6.**Influence of flow physics on slosh induced damping, work and loads. (

**a**) Density Ratio and Reynolds Number Scaling; (

**b**) Fluid Surface Tension Properties.

**Figure 9.**Change in Dissipated Energy due to Scaling of Non-dimensional Properties for each oscillation n.

**Figure 10.**Response surface of $\left|\right|\Delta {E}_{Disp}\left|\right|$ as a function of $\overline{\rho}$ and $Fr$.

$\Delta x$ | ${\left|\right|\u03f5\left|\right|}_{2}$ (%) |
---|---|

$1\times {10}^{-3}$ | $1.22$ |

$6.67\times {10}^{-4}$ | $1.212$ |

$5\times {10}^{-4}$ | $1.210$ |

$4\times {10}^{-4}$ | $1.209$ |

Quantity | Units | ||||
---|---|---|---|---|---|

M | L | T | |||

$\mu $ | Liquid Viscosity | [Pa·s] | 1 | −1 | −1 |

$\rho $ | Liquid Density | [kg·m${}^{-3}$] | 1 | −3 | 0 |

${\rho}_{air}$ | Gas Density | [kg·m${}^{-3}$] | 1 | −3 | 0 |

$\gamma $ | Liquid Surface Tension | [N·m${}^{-1}$] | 1 | 0 | −2 |

$\theta $ | liquid–solid Contact Angle | [rad] | 0 | 0 | 0 |

v | Fluid speed | [m·s${}^{-1}$] | 0 | 1 | −1 |

g | Gravity | [m·s${}^{-2}$] | 0 | 1 | −2 |

m | Solid Mass | [kg] | 1 | 0 | 0 |

k | Structural Stiffness | [N·m${}^{-1}$] | 1 | 0 | −2 |

c | Structural Damping | [Ns·m${}^{-1}$] | 1 | 0 | −1 |

h | Tank Height | [m] | 0 | 1 | 0 |

l | Tank Length | [m] | 0 | 1 | 0 |

${\eta}_{0}$ | Height of fluid | [m] | 0 | 1 | 0 |

${y}_{0}$ | Initial tank offset (spring displacement) | [m] | 0 | 1 | 0 |

Quantity | Factor |
---|---|

Length | $\lambda $ |

Time | ${\lambda}^{1/2}$ |

Mass | ${\lambda}^{3}$ |

Velocity | ${\lambda}^{1/2}$ |

Acceleration | ${\lambda}^{0}=1$ |

Viscosity | ${\lambda}^{1.5}$ |

Surface Tension | ${\lambda}^{2}$ |

Force | ${\lambda}^{3}$ |

Energy | ${\lambda}^{4}$ |

f [Hz] | $\mathit{F}\mathit{r}$ | |
---|---|---|

Required | 3.35 | 1.65 |

Achieved | 7.0 | 3.44 |

Actual | Froude Scaled | Aircraft | |
---|---|---|---|

Protospace | Protospace | ||

$\theta $ | $1.05$ | 0 | 0 |

$\overline{\rho}$ | $1.21\times {10}^{-3}$ | $1.39\times {10}^{-3}$ | $1.39\times {10}^{-3}$ |

$Re$ | $161.7\times {10}^{3}$ | $4.67\times {10}^{3}$ | $52.2\times {10}^{3}$ |

$We$ | $5.73\times {10}^{3}$ | $2.78\times {10}^{3}$ | $69.4\times {10}^{3}$ |

$Fr$ | $3.44$ | $1.65$ | $1.65$ |

Parameter | Property Range |
---|---|

$\theta $$\in [0.52,1.51]$ | − |

$\overline{\rho}$$\in [6.49\times {10}^{-4},6.18\times {10}^{-3}]$ | $\therefore \rho \in [199,1894]$ kg·m${}^{-3}$ |

$Re$$\in [42.56\times {10}^{3},\infty )$ | $\therefore \mu \in [0,3.7\times {10}^{-3}]$ Pa·s |

$We$$\in [4.92\times {10}^{3},1.37\times {10}^{5}]$ | $\therefore \gamma \in [0.003,0.085]$ N·m${}^{-1}$ |

$Fr$$\in [0.69,5.16]$ | $\therefore f\in [1.4,10.5]$ Hz |

$\left|\right|{\mathit{E}}_{\mathbf{Disp}}\left|\right|$ | $\left|\right|{\mathit{W}}_{\mathit{s}}\left|\right|$ | $\left|\right|{\mathit{F}}_{\mathit{y}}\left|\right|$ | |
---|---|---|---|

$Re$ | $7.80$ | $4.85$ | $13.47$ |

$\overline{\rho}$ | $5.59$ | $3.24$ | $14.60$ |

$We$ | $9.23$ | $4.90$ | $12.99$ |

$\theta $ | $12.74$ | $6.43$ | $14.63$ |

$\theta ={90}^{\circ}$ | $94.06$ | $47.82$ | $34.27$ |

$Fr$ | $77.46$ | $59.08$ | $13.74$ |

**Table 8.**Fluid Properties due to Froude Scaling. (Nomenclature for scaling methods as defined above).

Protospace | Protospace${}^{\u2020}$ | Protospace * | Aircraft | |
---|---|---|---|---|

(Water) | (Cold Kerosene) | (Ideal) | (Cold Kerosene) | |

$\overline{\rho}$ | $1.21\times {10}^{-3}$ | $1.39\times {10}^{-3}$ | $1.39\times {10}^{-3}$ | $1.39\times {10}^{-3}$ |

$Re$ | $161.7\times {10}^{3}$ | $4.67\times {10}^{3}$ | $52.2\times {10}^{3}$ | $52.2\times {10}^{3}$ |

$We$ | $5.73\times {10}^{3}$ | $2.78\times {10}^{3}$ | $69.4\times {10}^{3}$ | $69.4\times {10}^{3}$ |

$Fr$ | $3.44$ | $1.65$ | $1.65$ | $1.65$ |

Protospace | Protospace${}^{\u2020}$ | Protospace * | |
---|---|---|---|

${E}_{Disp}\phantom{\rule{3.33333pt}{0ex}}\left(W\right)$ | $-37.21$ | $-5.32$ | $-5.59$ |

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**MDPI and ACS Style**

Wright, M.D.; Gambioli, F.; Malan, A.G.
CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh. *Appl. Sci.* **2021**, *11*, 10401.
https://doi.org/10.3390/app112110401

**AMA Style**

Wright MD, Gambioli F, Malan AG.
CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh. *Applied Sciences*. 2021; 11(21):10401.
https://doi.org/10.3390/app112110401

**Chicago/Turabian Style**

Wright, Michael Dennis, Francesco Gambioli, and Arnaud George Malan.
2021. "CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh" *Applied Sciences* 11, no. 21: 10401.
https://doi.org/10.3390/app112110401