# Effect of Fuel Sloshing on the Damping of a Scaled Wing Model—Experimental Testing and Numerical Simulations

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Methodology

- Ground vibration tests with and without tank attached, part of which was presented in [13],
- Step release tests without liquid (dry tests), with and without additional mass to account for the “frozen” liquid,
- Step release tests with tap water as the working liquid, with and without added dye
- Studies with and without the baffles, with the baffles of different solidity ratios,
- Studies with varying filling levels.

#### 2.1.1. Nonlinear System Identification

- The amplitude of each component is limited at $2y\left(0\right)$.
- The frequency of each component is limited in the interval $(1\pm x)\xb7{\omega}_{j}\left(0\right)$, where ${\omega}_{j}\left(0\right)$ is the linear estimate of the frequency component and x is the frequency deviation allowed for.
- If the damping ratio of one component ${\zeta}_{j}$ exceeds a threshold value (set here at 0.9, or $90\%$ of critical damping), the respective component is eliminated from the signal estimate ${y}_{e}$ for all subsequent bins.

#### 2.1.2. Example of Miniwot Data Curve Fitting

#### 2.2. Numerical Methodology

#### 2.2.1. Structural Model

#### 2.2.2. Quasi-Steady Model of the Transversal Sloshing Force

- Scaling between the experimental and numerical (MiniWoT) tank dimensions.
- Producing a continuous and smooth sloshing force for numerical analysis, which can not be directly extracted from the experimental data. A result of the finite sampling frequency producing discrete data points in each cycle, plus discrete amplitudes and frequencies of the harmonic cycles considered.
- Identified forces assume the fluid has reached a quasi-steady state under the excitation and, therefore, can not account for sharp transients (i.e., free-release of the structure). A mechanism must be introduced to account for the initial energy input into the fluid to excite the identified motions within the data-set.

## 3. Results

#### 3.1. Large Amplitude Vibrations of the Dry System

#### 3.2. Large Amplitude Vibrations of the Wet System

**A**represents the step release event, and point

**B**represents the point in time when the wet system reaches ${A}_{1}=2.5$ g.

**B**in Figure 13 has a different position in the dry and wet cases since the different decay rates imply the reach of distinct amplitudes at the same time instant. In the 50% fill case, between points

**A**and

**B**, ${\zeta}_{1}$ occupies a relatively narrow region, while after point

**B**it decreases with a near-linear trend. Similar damping trends were observed in 1DOF experiments [5], with regions of almost constant damping ratio characterising the violent vertical sloshing region denoted as R1. Single degree of freedom experiments also indicated that the damping ratio starts to decrease at even larger amplitudes of motion [6]. The same phenomenon will be investigated here using the numerical model that was developed based on the 1DOF experiments.

#### 3.3. Numerical Results

## 4. Discussion

**B**in Figure 12 and Figure 13, it was mentioned in Section 2 that the choice of the surrogate model’s frequency is not straightforward. The reason behind this is that different tank geometries at different scales can excite different sloshing patterns. Figure 17 shows the spanwise variation of the three sloshing mode frequencies (for modes (2,0), (2,1) and (1,2)) depending on tank cell geometry. For example, Figure 18 shows the high amplitude sloshing patterns developed from the first symmetric sloshing mode (2,0). Since the tank width (dimension in the chordwise direction) is variable, the sloshing frequencies were approximated considering an average tank width for each cell. The direction of excitation is significantly vertical, and the sloshing patterns will be excited when the frequency of excitation (F1 in this case) is double the sloshing modal frequency. This parametric excitation regime generates Faraday waves [28]. Such effects were extensively studied before in the controlled high-amplitude vertical sloshing experiments and were shown to be responsible for energy dissipation [5,7]. As seen in Figure 17, the sloshing mode (2,0) has its frequency in the vicinity of one-half of the first bending mode frequency of the MiniWot demonstrator for most of the tank cells. Furthermore, especially in the cells closer to the tip of the wing model (6,7,8), there are two additional modes (2,1) and (1,2) that may be excited. This suggests that in-plane sloshing influences can be expected at the tip of the wing. Referring to Figure 16, this behaviour can indeed be observed in the recorded footage. This discussion shows that the insights based on the linear modal analysis can still be used to support the analysis of these highly nonlinear and transient sloshing regimes.

- The spanwise location of the fuel influences the vertical sloshing regimes. Previous single degree-of-freedom studies indicate substantial sloshing-induced damping effects that depend on the sloshing regime [5,7]. There is thus an indication that depending on the available amounts of fuel, the net damping following the excitation similar to the step release (such as a discrete gust) may be increased by moving the liquid to the specific spanwise positions, taking into account the filling level as well. Supplementary investigations are planned in order to further assess such effects.
- Liquid sloshing patterns dominated by different sloshing modes are prominent at different spanwise locations and persistent even at the lowest amplitudes of excitation. Such sloshing modes are highly dependent on the geometry of the tank and filling level, and they have previously been observed to add substantial additional damping at lower amplitudes by promoting liquid impacts with the top of the tank. Therefore, there exists the possibility that the geometry of the fuel tank could be tailored to promote such sloshing patterns and, hence, achieve an increased sloshing-induced damping.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

f | Frequency of the nonlinear signal component |

y | Original signal, Displacement |

${y}_{e}$ | Fitted function |

${y}_{e}^{k}$ | Fitted function in time bin k |

A | Amplitude of the nonlinear signal component |

F | Natural frequency of vibration |

M | Number of frequency components in the NLS analysis |

${N}_{S}$ | Bin sliding size in the NLS analysis |

${N}_{W}$ | Time bin size in the NLS analysis |

T | Natural period of vibration |

$\zeta $ | Damping ratio of the nonlinear signal component |

$\varphi $ | Phase of the nonlinear signal component, compact basis function |

$\mathit{q}$ | Vector of modal displacements |

${F}_{ext}$ | Loading force |

${F}_{s}$ | Sloshing force |

${F}_{t}$ | Total force |

${m}_{l}$, ${m}_{s}$ | Liquid and structural mass |

$\widehat{\xb7}$ | Non-dimensional quantity |

$\underline{\xb7}$ | Vector |

$\mathit{K}$ | Modal stiffness matrix |

$\mathit{C}$ | Modal damping matrix |

${\omega}_{c}$ | Characteristic frequency |

v | Velocity |

h | Height of fuel tank |

$\alpha $, $\beta $ | Nonlinear damping parameters |

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**Figure 1.**MiniWoT scaled test specimen. Top figure: (

**1**)—Cantilever to the strong wall connection, (

**2**)—scaled structure model, (

**3**)—scaled fuel tank model and (

**4**)—loading points for step release tests. Bottom figure: instrumentation layout.

**Figure 3.**Example of NLS identification result. Original signal is shown in black and the reconstructed signal using the fitted functions is shown in green.

**Figure 4.**Example of residuals and summed squared residuals for the fitted functions shown in Figure 3: (

**a**) residuals for the first bin; (

**b**) summed squared residuals for all bins.

**Figure 6.**Tank tip acceleration comparison between experiment and model. Deflection at the tip of the wing is 2.7% of the semispan: (

**a**) acceleration time series for the dry case; (

**b**) amplitude of the first bending component versus time.

**Figure 7.**Interpolated experimental response surfaces. Circular traces shown in the horizontal plane corresponding to the 3.3 Hz response surface.

**Figure 9.**Frequency response function for the dry and wet systems. The NLS-identified dry system is shown in green for comparison.

**Figure 10.**The amplitude, frequency and damping ratio of the first frequency component (component 1) of the acceleration time series measured throughout the fuel tank, dry case.

**Figure 11.**Typical sloshing response of the MiniWoT structure, $50\%$ filling level case compared to dynamically equivalent dry case.

**Figure 12.**Amplitude of component 1 vs. normalised time, 50% filling level case and dynamically equivalent dry case. Error bars show the variability between test repetitions.

**Figure 13.**Component 1 frequencies and damping ratios vs. component 1 acceleration amplitude, 50% filling level and dynamically equivalent dry case. Error bars show the variability between test repetitions. The amplitude decays following the step release event at

**A**.

**Figure 15.**Comparison between experimental and numerical amplitude, frequency and damping ratio of the first bending component: (

**a**) amplitude of first bending component versus time; (

**b**) damping ratio and frequency versus amplitude.

**Figure 16.**Selection of video segments for one representative 50% filling level test. Local acceleration for the first vibration component shown (left: tank tip).

**Figure 17.**Estimation of sloshing mode frequencies for the 50% filling level case for each MiniWoT tank cell, shown as a percentage of F1. Values coloured by their proximity to 50%.

**Figure 18.**Average liquid surface location in 1DOF harmonic excitation tests [7] (

**left**) and MiniWoT cells 3&4 sloshing pattern (

**right**).

Element Type | Nastran Element | Number of Elements |
---|---|---|

Plate | CQUAD4 | 10,147 |

Beam | CBEAM | 1200 |

Spring | CBUSH | 17 |

Spring | CELAS1 | 6 |

Mass | CONM2 | 55 |

MPC | RBE3 | 22 |

Number of nodes | 10,903 |

Parameter | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ |
---|---|---|---|---|---|

Value | 4.71 | 0.70 | 1.29 | 2.64 | 3.63 |

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**MDPI and ACS Style**

Constantin, L.; De Courcy, J.J.; Titurus, B.; Rendall, T.C.S.; Cooper, J.E.; Gambioli, F.
Effect of Fuel Sloshing on the Damping of a Scaled Wing Model—Experimental Testing and Numerical Simulations. *Appl. Sci.* **2022**, *12*, 7860.
https://doi.org/10.3390/app12157860

**AMA Style**

Constantin L, De Courcy JJ, Titurus B, Rendall TCS, Cooper JE, Gambioli F.
Effect of Fuel Sloshing on the Damping of a Scaled Wing Model—Experimental Testing and Numerical Simulations. *Applied Sciences*. 2022; 12(15):7860.
https://doi.org/10.3390/app12157860

**Chicago/Turabian Style**

Constantin, Lucian, Joe J. De Courcy, Branislav Titurus, Thomas C. S. Rendall, Jonathan E. Cooper, and Francesco Gambioli.
2022. "Effect of Fuel Sloshing on the Damping of a Scaled Wing Model—Experimental Testing and Numerical Simulations" *Applied Sciences* 12, no. 15: 7860.
https://doi.org/10.3390/app12157860