# On the Efficacy of Turbulence Modelling for Sloshing
## 4 Results and Discussion
### Assessment of Turbulence Models  
To assess the performance of different turbulence models, energy dissipation estimations via three-dimensional simulations are amalgamated in Figure 6. All the presented models are able to produce good estimation of the energy dissipation for the low and medium excitation \(Fr<1.5\) levels, except for the no-model simulation, which overestimates the dissipation for \(Fr=1.35\). However, as the excitation increases, the no-model matches far better with the reference data, while the other turbulence models deviate. In all the high \(Fr\) number cases, the standard \(K\)-\(\epsilon\) underestimated the dissipation, while the dissipation estimated with \(k\)-\(\omega\) SST is slightly lower than the reference data, but still within reasonable uncertainty bounds. Considering the whole range of \(Fr\) tested here, RNG \(K\)-\(\epsilon\) appears to have the best estimation of dissipated energy. A good comparison between the models is to examine how well each depicts the damping force behaviour. The hysteresis of the damping forces as function of the displacement computed with the different models are shown in Figure 7. At \(Fr=0.7\), the computed forces with the no-model, \(k\)-\(\omega\) SST and RNG \(k\)-\(\epsilon\) models show inconsiderable differences, but the symmetric hysteresis is not obtained. The predicted forces accurately match the reference data during the negative stroke, but the damping forces are underestimated during the positive stroke. On the other hand, the standard \(k\)-\(\epsilon\) model preserved the symmetry of the damping force hysteresis, but significantly underestimated the peak force. For \(Fr=1.07\), all models managed to demonstrate precise representation of the damping force for excitation. Note that the experimental case defines \(Fr=1.035\) (not 1.07 as the simulations), which can explain the slight underestimation of the lower side of  
At \(Fr=1.35\), all models fail to depict an accurate damping force throughout the whole cycle, in spite of the accurate estimation of the dissipation energy of some models. The no-model simulation produces a relatively accurate estimation of the damping force during the downward stroke, but significantly overestimates the magnitude of the damping force in the upward stroke, leading to an increase in the overall dissipated energy. On the other hand, the RNG \(k\)-\(e\) model underestimates the force in the downward stroke, but accurately computes the force in the upward stroke. The damping forces computed by the standard \(k\)-\(e\) model are lower than the reference values throughout the cycle, albeit the confined area of hysteresis agrees with the reference data, resulting in a good estimation of the dissipation energy. Looking at the force hysteresis of the case with \(Fr=1.85\), it is clear that the no-model simulation is the most accurate. The behaviour of the damping force, its peak values and locations agree very well with the experimental data. The damping force obtained by the turbulence models \(k\)-\(\omega\) SST and RNG \(k\)-\(e\) follows the experimental data in the upward stroke, but in the downward stroke the peak force is underestimated. The magnitude of the damping force computed with the standard \(k\)-\(e\) is lower than the reference during the whole cycle. It appears that in the case of strong sloshing, the turbulent viscosity introduced by the turbulence models has a non-physical damping effect on the dynamics of the sloshing. This leads to a reduction in the liquid movement and reduction in the calculated damping effect of the sloshing itself. Figure 8 shows four snapshots of the liquid sloshing with \(Fr=1.85\). The left-side figures are three-dimensional solid contours of \(\alpha>=0.5\) and coloured with the turbulent viscosity. The right figures are middle-plane slices of the left figures. There are two main observations to be made.