# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 2 Validation of CFD Model  
The utilisation of VoF CFD within the study requires accurate, high-fidelity, mesh independent simulations. Specifically, the accuracy with regards to the predicted slosh force is critical as this is the key uncertainty when quantifying slosh induced energy dissipation in this work (see Equation (12)). This is due to this work involving prescribed tank motions (obtained from measurements). The VoF CFD code Elemental(r) 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].  
The validation of Elemental(r) 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 \(t=0\)s followed by free vibration [3].  
Using the measured tank accelerations from the experiment as input, CFD simulations were performed on a range of different mesh spacings. For the coarsest mesh considered (\(\Delta x=10^{-3}\)), 2D and 3D simulations resulted in near identical computed slosh forces and therefore energy dissipation due to the definition of \(E_{\mathit{Disp}}\) (see Equation (12)). In the interest of computational efficiency therefore, 2D was employed for all subsequent simulations. The 2D CFD predicted free-surface interface at various times is depicted in Figure 2. These may be compared to experimental footage (Figure 1). As shown, a range of different slosh regimes are present. These include liquid-solid impact (Figures 1b and 2b), aerated flow Figures 1c and 2c) and weakly linear slosh (Figures 1d and 2d). Further, a reasonable agreement exists between predicted and experimental footage.  
The use of LES to model turbulence required estimation of the Taylor micro-scale \((\lambda)\)[21] as this informs mesh resolution. This was done as part of the SLOWD project [22] which yielded  
\[\lambda\approx\sqrt{\frac{10 vk}{\epsilon}}=4\times 10^{-4}m \tag{1}\]  
where \(\
u\) is the fluid kinematic viscosity, \(\epsilon\) is kinetic energy dissipation rate and \(k\) is the turbulent kinetic energy. The above aided in informing suitable CFD mesh spacing as detailed next.  
Figure 3 compares the measured and CFD predicted vertical slosh loads (See Equation (11)) for Cartesian meshes, namely \(\Delta x=1\times 10^{-3}\) and \(\Delta x=5\times 10^{-4}\). Completing an error analysis and mesh sensitivity study provides Table 1 and Figure 4. This highlights engineering precision convergence for \(\Delta x\leq 6.67\times 10^{-4}\) as the computed dissipated energy (\(E_{Disp}\) in Figure 5 computed via Equation (12)) is found to lie within 2% of the experimental result. This validates the use of Elemental to predict violent slosh induced slosh loads and damping, provided that suitably fine meshes are employed. To ensure direct applicability, the SDOF slosh model used for our non-dimensional characterization will consist of one of  
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.