# On the Efficacy of Turbulence Modelling for Sloshing
## 2 Problem Description  
Simulations of a vertically oscillating tank with a single harmonic motion are presented. The simulations replicate the physical experiment of [PERSON] et al. [59], which is illustrated in the schematic diagram shown in Figure 1. This consists of a closed cuboid tank, half-filled with water, with the dimensions \(L_{x}=10\) cm, \(L_{y}=3\) cm and \(L_{z}=5\) cm, where, \(z\) represents the vertical direction. The tank is forced to oscillate vertically with a constant amplitude, \(A\), and a frequency, \(f\), and hence the displacement of the tank is expressed with a sinusoidal relationship as  
\[z=A\sin(\omega t), \tag{1}\]  
where, \(z\) is the displacement of the tank from the origin, \(\omega=2\pi f\), and \(t\) is the time. As can be seen in Equation (1), for a given tank, the dominant parameters determining the sloshing damping effect are the amplitude and the frequency of the oscillation. These two parameters are combined as a non-dimensional Froude number \(Fr\), that represents the ratio between the inertial and the gravitational forces, i.e.,  
\[Fr=\sqrt{\frac{A\omega^{2}}{g}} \tag{2}\]  
In their results, [PERSON] et al. [59] detailed the measurements of damping forces and an analysis of the recorded sloshing dissipation energy. This renders the case a good benchmark for testing modelling simulation setups and different turbulence models. The work presented in [59] is part of an active research project, and the authors provided a newer version of the experimental data than used in [59] that has less uncertainty, these are used within this work. The tests are conducted for a range of \(Fr=0.7\) to \(Fr=1.85\) and are compared directly against the experimental data. The Fr number is changed by keeping the frequency constant at \(\omega=52.15\), while varying the amplitude between \(A=1.72\) mm to \(12.46\) mm. This range covers two types of sloshing regimes, \(R_{1}\) and \(R_{2}\), as referred to in [7; 59]. In the \(R_{1}\) regime, sloshing is turbulent and associated with high damping. This is mainly caused by the liquid impacting the upper and lower walls that results from the high excitation. In the lower excitation \(R_{2}\) regime, damping is seemingly caused by typical sloshing modes without surface breaking [59]. Comparison with experiment is based on the damping force induced by the sloshing and the associated dissipated energy. Energy dissipation is quantified via hysteresis loop analysis of the hydrodynamic forces acting on the tank walls. Throughout the simulations, the vertical forces on the inner tank walls are recorded at every time step. The recorded forces are the sum of the pressure force acting on the horizontal walls, \(F_{p}\), and the sum of the viscous forces acting on the vertical walls, \(F_{v}\). The total force exerted on the tank walls in the direction of the motion can be expressed as [7; 15]  
\[F_{T}=F_{p}+F_{v}. \tag{3}\]  
Figure 1: Schematic diagram of oscillating tank setup and dimensions. The total force consists of the static loading of the water in the tank, inertia of the water as the tank accelerates and decelerates, and the damping effect of the fluid sloshing inside the tank, this can be expressed as  
\[F_{T}=m_{w}g+m_{w}\bar{z}+F_{s}, \tag{4}\]  
where \(m_{w}\) is the mass of the water (i.e., 75 g), \(g\) the gravitational acceleration and \(F_{s}\) is the damping force of the sloshing. The water mass and tank acceleration are predetermined, and the damping force can be computed as a function of time. Sloshing is an unsteady process and the damping forces can vary from cycle to cycle, especially during turbulent sloshing. All presented simulations start with stationary and undisturbed liquid, which go through a transition period before hitting a statistically steady state. The simulations run for a total time of 20 s, this corresponds to 166 cycles (the excitation frequency is \(\omega=52.15\) rad/s and the period is \(T_{p}=0.12\) s).