## 3 Methodology Description
### Numerical Approach  
SPH formulation for free-surface flows can be based on a weakly compressible formulation, in which fluid pressure and density are linked through a barotropic law. Under this assumption, Navier-Stokes equations can be written as:  
\[\left\{ \begin{array}{l}\frac{D\rho}{Dt}=-\rho\;\
abla\cdot\mathbf{u} \\ \\ \frac{D\mathbf{u}}{Dt}=\mathbf{g}+\frac{\
abla\cdot\mathbb{T}}{\rho}+\mathbf{a}_{st} \\ \\ \mathbf{u}=\frac{D\mathbf{r}}{Dt},\qquad p=p(\rho)\end{array} \right. \tag{5}\]  
where \(\frac{D}{Dt}\) is the material derivative; \(\rho\) is the fluid density; \(p\) the pressure; \(\mathbf{u}\) the fluid velocity vector (bold letters indicate vector magnitudes) that is defined from the fluid particles position, \(\mathbf{r},\mathbf{g}\) a generic external volumetric force field F; and \(\mathbf{a}_{st}\) is a surface tension force field. \(\mathbb{T}\) indicates the stress tensor, which, for Newtonian fluid such as the ones used in the applications of this work, is defined as:  
\[\mathbb{T}=(-p+\eta\,\text{tr}\mathbb{D})\mathbb{I}+2\mu\mathbb{D}\,, \tag{6}\]  
with \(\mathbb{D}\) being the rate of strain tensor, i.e., \(\mathbb{D}=\left(\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\right)/2\); \(\mathbb{I}\) the identity matrix; and \(\mu\) and \(\eta\) the fluid viscosity coefficients, respectively. Relationship between pressure and density in (5) is established by means of a linear stiff equation of state (EOS). This allows for adjusting compressibility so that the speed of sound (\(c_{\text{s}}\)) can be artificially lowered [30]. In a weakly compressible context, the condition to be satisfied corresponds to Mach number \(\text{Ma}<0.1\). In this work, this condition is met if \(cs>10\cdot U\), with \(U\) being the maximum speed in the system, which is taken as the maximum speed of the tank. Hence, the set of Equations in (5) become in SPH  
\[\left\{ \begin{array}{l}\frac{D\rho_{i}}{Dt}\qquad\quad=-\rho_{i}\sum_{j}\mathbf{ u}_{ji}\cdot\
abla_{i}W_{ij}V_{j}+h\,c_{\text{s}}\,\delta\sum_{j}\mathbf{\psi}_{ji} \cdot\
abla_{i}W_{ij}V_{j}\\ \\ \frac{D\mathbf{u}_{i}}{Dt}\qquad\quad=-\frac{1}{\rho_{i}}\sum_{j}\left(p_{i}+p_{ j}\right)\
abla_{i}W_{ij}V_{j}+\mathbf{g}_{i}+\frac{\mu k}{\rho_{i}}\sum_{j}\,\pi_{ij} \
abla_{i}W_{ij}V_{j}+\mathbf{a}_{st_{i}}\\ \\ \frac{D\mathbf{r}_{i}}{Dt}=\mathbf{u}_{i}\qquad\quad p=p_{0}+c_{\text{s}}^{2}(\rho_{i}- \rho_{0})\end{array} \right. \tag{7}\]  
where \(\mathbf{u}_{ji}:=(\mathbf{u}_{j}-\mathbf{u}_{i})\), and \(V_{j}\) represents the volume of the neighbor particles \(j\), defined as \(V_{j}=m_{j}/\rho_{j}\). In the present work, the kernel function, defined as \(W_{ij}=W(\mathbf{r}_{j}-\mathbf{r}_{i},h)\), is a Wendland C2 kernel with \(h/dr=2\) being used, where \(h\) corresponds to the smoothing length and \(dr\) to the particle distance. The dissipation effects due to viscosity are modeled through \(\pi_{ij}\), as is the standard in the literature (see, e.g., [31]). This function is defined as:\[\pi_{ij}:=\frac{\mathbf{u}_{ji}\cdot\mathbf{r}_{ji}}{\parallel\mathbf{r}_{ji}\parallel^{2}}\,, \qquad\mathbf{r}_{ji}:=(\mathbf{r}_{j}\,-\,\mathbf{r}_{i}) \tag{8}\]  
An extra numerical term is added to the continuity equation in order to avoid spurious high-frequency pressure oscillations. The model used in this work corresponds to the one developed by [30], which is known as \(\delta\)-SPH. The function \(\mathbf{\psi}_{ji}\) from this term is defined as:  
\[\mathbf{\psi}_{ji}:=2\bigg{[}(\rho_{j}-\rho_{i})-\frac{1}{2}\Big{(}\langle\
abla \rho\rangle_{j}^{L}+\langle\
abla\rho\rangle_{i}^{L}\Big{)}\cdot\mathbf{r}_{ji} \bigg{]}\frac{\mathbf{r}_{ji}}{\parallel\mathbf{r}_{ji}\parallel^{2}} \tag{9}\]  
where \(\langle\
abla\rho\rangle_{i}^{L}\) represents the renormalized density gradient, as defined by [32]. The value for \(\delta\) is set always to 0.1. Boundaries are modeled according to the numerical boundary integrals approach, as described by [33]. They are discretized by elements that are defined by geometrical properties, such as their area, normal and tangent.