# On the Efficacy of Turbulence Modelling for Sloshing
## 4 Results and Discussion  
Figure 3 shows the experimental measurements of energy dissipation against \(Fr\) number [59]. When comparing these results, it is important to consider that the dissipation is normalised by the amplitude squared. Thus, absolute dissipated energy always increases with the excitation amplitude, whereas the normalised values do not necessarily follow the same behaviour. The figure shows a clear trend and pattern of the damping as function of the excitation parameter, \(Fr\). The figure shows three regions \(R_{2}\) at low \(Fr\), \(R_{1}\) at high \(Fr\) and an intermediate region \(R_{2}\)&\(R_{1}\). To understand why these three regions appear, it is important to consider the sloshing behaviour of the liquid at these regions. Figure 4 shows a snapshot of the sloshing for a case at region \(R_{1}\) (\(Fr=0.7\)) and another one in region \(R_{2}\) (\(Fr=1.85\)). Two other cases at the start and the end of the intermediate region \(R_{2}\)&\(R_{1}\) (\(Fr=1.07\) and \(1.35\), respectively) are also shown. These snapshots are produced by three-dimensional and no-model simulations. In the region \(R_{2}\), the normalised dissipated energy is reasonably constant and small. The dissipation in the region induced by a single sloshing mode, can be seen in Figure 4a, with a small variation in the energy dissipation  
Figure 2: Illustration of the mesh resolution on the damping force as a function of the displacement obtained by no-model and \(k\)-\(\omega\) SST simulations. The dashed, solid and dotted lines correspond to Mesh-1, Mesh-2 and Mesh-3, respectively. between cycles. As the excitation amplitude increases and enters the region \(R_{2}\&R_{1}\), the liquid starts to impact the ceiling of the tank, and the liquid surface breaks. In this region, dissipation seems to grow linearly. Looking at Figure 4b,c it can be observed that the symmetric sloshing mode is still dominant. However, the dominance of a single sloshing mode in this region fades as the \(Fr\) number increases, since the impact with the upper wall becomes stronger leading to and inducing turbulence. With further increase in the excitation amplitude, the response region \(R_{1}\) is reached where the sloshing becomes very turbulent and the single sloshing mode vanishes. In \(R_{2}\) region, the increase in normalised energy dissipation with the excitation level is slow, and reaches a saturation limit before falling down again. Figure 4: Snapshots of the liquid sloshing at different \(Fr\) numbers, (**a**) \(Fr=0.7\), (**b**) \(Fr=1.07\), (**c**) \(Fr=1.35\) and (**d**) \(Fr=1.85\). The opacity is reduced to 50% to show the internal voids. Figure 3: The normalised dissipated energy as a function of \(Fr\) number computed by two-dimensional and three-dimensional no-model simulations and compared with experimental measurements [59]. The dissipated energy is computed from the hysteresis of the mean damping force cycle. For a detailed comparison between the simulations' setup, the hysteresis of the mean damping force cycle with displacement is shown in Figure 5 for four excitations, \(Fr=0.7,1.07,1.35\) and \(1.85\). Note that some simulations were conducted at \(Fr\) numbers that do not exactly match their experimental counterparts, as the full experimental data were not available during the run time. However, the \(Fr\) numbers are sufficiently close to produce a valid comparison between the cases, and the simulation cases are compared with the experimental case of the closest \(Fr\) number, i.e., the simulated cases of \(Fr=1.07\) are compared with the experimental case of \(Fr=1.035\). The experimental measurements of [59] show that at a low \(Fr\) number of 0.7, the damping force hysteresis is symmetric. The damping force also has two peaks, one at the positive displacement and the other negative. In this case, the two peak forces are exerted at the upper and lower ends of the stroke, which also correspond to the maximum acceleration and deceleration of the fluid mass.