# Numerical Calculation of Slosh Dissipation
## 2 SPH Method: The SPH-Flow Solver
### Brief Recall of the \(\delta\)-LES-SPH Scheme  
In the present section, we briefly introduce the SPH scheme defined in [29; 30]. The governing Equation (1) is discretized as a set of fluid particles whose masses \(m_{i}\) remain constant during the motion. The particles are set initially on a lattice with homogeneous spacing \(\Delta r\), and hence, the particles' volumes \(V_{i}\) are evaluated initially as \(\Delta r^{n}\) (where \(n\) is the number of spatial dimensions). The particle masses \(m_{i}\) are calculated through the initial density field using the equation of state and the initial pressure field. During the time evolution, volumes \(V_{i}\) change in time according to the particle density \(\rho_{i}\). For the sake of brevity, in the following text the notation \(\mathbf{r}_{ji}\) indicates the differences in the particles positions \((\mathbf{r}_{j}-\mathbf{r}_{i})\), and the same holds for the velocity fields \(\mathbf{u}_{ji}\) and \(\delta\mathbf{u}_{ji}\), while, for the generic scalar field the notation \(f_{ij}\) just indicates the dependency of the field \(f\) on the indices \(i\) and \(j\). The spatial gradients are approximated through convolution summations with a kernel function \(W_{ij}\). This function has a compact support whose reference length is denoted by \(h\) and referred to as the smoothing length. As in [29], a C2-Wendland kernel is adopted in the present work. For this kernel, the radius of the support is \(2h\). In the numerical simulations that follow, the number of neighboring particles is chosen as \(2h=4\Delta r\) in two dimensions and \(2h=3\Delta r\) in three dimensions. The governing Equation (1) is discretized within the \(\delta\)-LES-SPH scheme as:  
\[\left\{ \begin{array}{rl}\frac{\mathrm{d}\rho_{i}}{\mathrm{d}t}&=-\rho_{i} \sum_{j}\left(\mathbf{u}_{ji}+\delta\mathbf{u}_{ji}\right)\cdot\
abla_{i}\,W_{ij}\,V_{ j}\,+\,\sum_{j}\left(\rho_{j}\,\delta\mathbf{u}_{j}+\rho_{i}\,\delta\mathbf{u}_{i} \right)\cdot\
abla_{i}\,W_{ij}\,V_{j}\,+\,\mathcal{D}_{i}^{\rho}\\ \rho_{i}\frac{\mathrm{d}\mathbf{u}_{i}}{\mathrm{d}t}&=\mathbf{F}_{i}^{p}\,+\,\mathbf{F}_{ i}^{v}\,+\,\rho_{i}(\mathbf{g}\,+\,\mathbf{f}_{NI})\,+\,\sum_{j}\left(\rho_{j}\,\mathbf{u}_{j} \otimes\delta\mathbf{u}_{j}+\rho_{i}\,\mathbf{u}_{i}\otimes\delta\mathbf{u}_{i}\right) \cdot\
abla_{i}\,W_{ij}\,V_{j}\\ \frac{\mathrm{d}\mathbf{r}_{i}}{\mathrm{d}t}&=\mathbf{u}_{i}+\delta\mathbf{u}_{i},\quad V _{i}=m_{i}\,\big{/}\rho_{i},\quad p=c_{0}^{2}(\rho-\rho_{0}),\end{array} \right. \tag{3}\]  
where the indexes \(i\) and \(j\) refer to generic \(i\)-th and \(j\)-th particles, \(\mathbf{F}_{i}^{p}\) and \(\mathbf{F}_{i}^{v}\) are the pressure and viscous forces acting on the particle \(i\). The vector \(\delta\mathbf{u}\) is the particle shifting velocity adopted to regularize the particles' spatial distribution during their motion. The time derivative \(d/dt\) used in (3) indicates a quasi-Lagrangian derivative since the particles are moving with the modified velocity \((\mathbf{u}+\delta\mathbf{u})\), and the above equations are written in an Arbitrary Lagrangian-Eulerian framework. For this reason, the continuity and the momentum equations contain terms with spatial derivatives of \(\delta\mathbf{u}\) (for details, see [31]). The term \(\mathcal{D}_{i}^{\rho}\) is the numerical diffusive term introduced by [32] to filter out the spurious high-frequency noise in the pressure field:  
\[\mathcal{D}_{i}^{\rho}:=\sum_{j}\,\delta_{ij}\,\mathbf{\psi}_{ji}\,\cdot\
abla_{i} W_{ij}\,V_{j} \tag{4}\]  
The law adopted in the present work for the particle shifting velocity \(\delta\mathbf{u}\) term, as well as the functions \(\delta_{ij}\) and \(\mathbf{\psi}_{ij}\), are given in [12] and not reported here for the sake of brevity.