# On the Efficacy of Turbulence Modelling for Sloshing
## 4 Results and Discussion
### Assessment of Turbulence Models
Firstly, turbulent viscosity applied by the standard \(k\)-\(e\) is the highest, and also affects most of the fluid domain, and the calculated values are significantly lower for \(k\)-\(\omega\) SST and RNG \(k\)-\(e\). Secondly, looking at the shape of the sloshing liquid, the distribution of the liquid is the least complex for the standard \(k\)-\(e\) model, followed by \(k\)-\(\omega\) SST and then RNG \(k\)-\(e\). The no-model simulation, on the other hand, produces the most complex sloshing behaviour. This suggests the existence of a reverse correlation between the turbulent viscosity and the strength of the predicted sloshing, i.e., the higher the turbulent viscosity, the less complex the sloshing dynamics becomes and hence the low estimation of the dissipated energy. This is consistent with the results above, where the lowest dissipation is obtained by the standard \(k\)-\(e\) model and the highest and most accurate dissipation is obtained by no-model simulations. It is worth noting that multiple snapshots were checked throughout the life-cycle of each simulation, and the same conclusion can be drawn. The figure also shows that turbulent viscosity is highest at the interface due to the apparent overproduction of turbulent effects, which has a detrimental damping effect on the sloshing dynamics and leads to underestimation of the damping energy overall. The tested models suffer from this problem to different degrees, with the standard \(k\)-\(e\) model being the worst and RNG \(k\)-\(e\) the least affected. The phenomenon seems to impact the majority of two-equation closing models [50, 58]. Although this problem has been known for over two decades [49], the standard RANS models have been used successfully in many studies [51, 52, 53, 54, 55, 56, 57], and the underlying cause of this problem is not fully understood [50]. [PERSON] and [PERSON] [49] proved that \(k\)-\(\omega\) and \(k\)-\(e\) models are unstable in the region of potential flow beneath the surface, where the models can suffer from exponential growth in turbulent kinetic energy. Following [PERSON] and [PERSON] [49], [PERSON] and [PERSON] [50] performed stability analysis of a number of RANS turbulence closure models and showed that the models are unconditionally unstable. Understanding the underlying problem and more specifically, the differences between the models, is not trivial and requires detailed analyses of the design of each model. Although this is out of the scope of the present study, it will form the basis of what comes next. To overcome this issue, [PERSON] et al. [30], [PERSON] and [PERSON] [50], [PERSON] et al. [68] proposed damping methods for turbulent viscosity at the interface, which improved the accuracy of the simulations. The aim of the current work is to investigate the capacity of the standard and the widely used RANS models to provide an accurate representation of violent sloshing. Modified version of the models with turbulent viscosity damping at the interface will be addressed in future work with the aim of capturing similarly complex fluid dynamics as the no-model method is able to reproduce whilst avoiding the obvious downside of this problem, requiring turbulence modelling in order to correctly capture its physics.