# On the Efficacy of Turbulence Modelling for Sloshing
## 4 Results and Discussion
### Assessment of Two-Dimensional Simulations  
Regarding the numerical simulations, the first test was to check whether two-dimensional simulations are sufficient to capture the dynamics of the sloshing. Without a turbulence model, two types of simulations were conducted. The first was a set of three-dimensional simulations at four excitation amplitudes corresponding to \(Fr=0.7,1.07,1.35\) and \(1.85\). The second was a  
Figure 5: Comparison between the two-dimensional and the three-dimensional simulations using the phase average of the damping forces for different \(Fr\) numbers. (**a**) \(Fr=0.7\), (**b**) \(Fr=1.07\), (**c**) \(Fr=1.35\), (**d**) \(Fr=1.85\).  
set of two-dimensional simulations with the same \(Fr\) numbers, where the breadth of the tank, \(L_{y}\), is omitted. Figure 3 also compares these based on the estimated dissipated energy. The three-dimensional simulations can accurately estimate the levels and the trend of dissipation within the experimental error bounds. The two-dimensional simulations failed to produce reasonable results, except for in the case of \(Fr=1.35\). Considering the overall behaviour of the two-dimensional simulations, it is likely that the result achieved at \(Fr=1.35\) is actually a coincidence. This is further confirmed by examining the hysteresis of the damping force with the displacement. Figure 5 also compares the damping forces hysteresis computed by the two-dimensional and three-dimensional simulations. It is clear that the two-dimensional simulations do not reproduce the correct damping forces for all the cases, including the case with \(Fr=1.35\) which bears a good estimation of the damping energy. The shape, peak values, and locations of the damping force are incorrectly estimated by the two-dimensional simulations. This further shows that relying on a single case or a single measured variable for validation of the efficacy of turbulent modelling can be misleading.  
On the contrary, good agreement between computed damping forces and the reference data is achieved by the three-dimensional simulations. A detailed discussion of the three-dimensional no-model simulation results is shown later alongside a discussion of the simulations using turbulent modelling.  
For high \(Fr\) numbers, the deviation in results of the two-dimensional simulations is expected as the impact of the liquid onto the tank's wall results in three-dimensional turbulent sloshing, and the sloshing is not dominated by a single sloshing mode in this case. At low excitation (\(Fr<=1\)), the fluid movement is dominated by the single and symmetric slowing mode, and hence the main fluid movements are symmetrical and two-dimensional. However, the two-dimensional simulation underestimated the dissipation. This can be attributed to the missing viscous force on the front and back walls (the walls parallel to the x-z plane), and in the case of low \(Fr\), the viscous force accounts for around 20% of the total damping forces. As the excitation increases, the surface starts to break, creating three-dimensional patterns, and the symmetric sloshing mode becomes less dominant and the viscous force on the vertical wall also becomes less important. Therefore, where possible, it is crucial to validate the simulation setup under different conditions. For the excitation levels considered in the paper, three-dimensional simulations are adopted.