## 2 Lagrangian and Eulerian Methods  
### SPH Formulation  
SPH is a Lagrangian meshfree particle method developed by [PERSON] [55] and [PERSON] and [PERSON] [56]; the reader is referred to [57; 58; 59; 60] for a general description of SPH. In the present paper, a WCSPH model, coupled with a sub-particle scale (SPS) approach to modeling turbulence [61] has been used. Description of the WCSPH can be found in [PERSON] [62] and [PERSON] et al. [63]. WCSPH has been widely used for many applications such as wave breaking [23; 25], dam break flows [64; 65], wave generation and absorption [66], hydraulics jumps [42] and the interaction between jets and waves [38]. The alternative approach, developed by [PERSON] and [PERSON] [67; 68], is defined as incompressible SPH. The reader can find a comparison between the two algorithms in [69; 70; 71]. The main difference between the two approaches exists in the way the pressures are calculated: the former explicitly uses an equation of state [23; 62], while the latter solves a Poisson pressure equation with a source term as a function of velocity divergence or time variation density. The motion is represented by the Navier-Stokes Equations for a weakly compressible SPH semi-discrete form:  
\[\left\{ \begin{array}{c}\frac{d\rho}{dt}=-\rho\
abla\mathbf{v}\\ \frac{d\rho}{dt}=-\frac{1}{\rho}\
abla P+\Gamma+\mathbf{f}\end{array} \right. \tag{1}\]  
where \(\rho\) is the density, \(v\) is the velocity vector, \(P\) is the pressure, \(\Gamma\) denotes the dissipation terms and \(\mathbf{f}\) represents accelerations due to external forces, such as gravity. In the SPH formalism, the discrete form of the continuity equation at a point \(i\) can be written as follows [72]:  
\[\frac{d\rho_{i}}{dt}=\rho_{i}\sum_{j}\frac{m_{j}}{\rho_{j}}\big{(}\mathbf{v}_{i}- \mathbf{v}_{j}\big{)}\cdot\
abla_{i}W\big{(}\mathbf{x}_{i}-\mathbf{x}_{j},h\big{)}+D_{i} \tag{2}\]  
The summations are extended to all of the particles \(j\) at a distance from \(i\) smaller than \(2h\) (\(h\) defined as the smoothing length), i.e., lying within the circle where the adopted cubic-spline kernel function \(W\)[60] is defined. The term \(D_{i}\) represents a numerical diffusive term [73], which can be introduced in order to reduce density fluctuation. The general form of a density diffusion term is:  
\[D_{i}=\delta hc_{i}\sum_{j}\psi_{ij}\cdot\
abla W\big{(}\mathbf{x}_{i}-\mathbf{x}_{j}, h\big{)}V_{j} \tag{3}\]  
where \(\delta\) is the Delta-SPH coefficient, which controls the magnitude of the diffusion term, \(c_{i}\) is the numerical speed of sound, \(V_{j}\) is the associated volume of the \(j\)-th particle and \(\psi_{ij}\) is the artificial dissipation term; in the present paper, the artificial dissipation term proposed by [PERSON] and [PERSON] [74] was chosen. The discrete form of the momentum equation, written in an SPH formalism, can be written as it follows [71]:  
\[\frac{dv_{i}}{dt}=-\sum_{j}m_{j}\left(\frac{P_{i}+P_{j}}{\rho_{i}\rho_{j}} \right)\
abla W\big{(}\mathbf{x}_{i}-\mathbf{x}_{j},h\big{)}+\langle\Gamma_{i}\rangle+ \mathbf{f} \tag{4}\]  
The momentum dissipation term \(\Gamma_{i}\) is obtained coupling the viscous dissipation in the laminar regime, as approximated by [PERSON] and [PERSON] [68], with a sub-particle scale (SPS) model [71]. The former can be expressed with the following formula:  
\[\upsilon_{o}\
abla^{2}\mathbf{v}_{i}=\sum_{j}m_{j}\frac{4\upsilon_{0}(\mathbf{x}_{i}- \mathbf{x}_{j})\cdot\
abla W\big{(}\mathbf{x}_{i}-\mathbf{x}_{j},h\big{)}}{\big{(}\rho_{i }+\rho_{j}\big{)}\left(\big{(}\mathbf{x}_{i}-\mathbf{x}_{j}\big{)}^{2}+\eta^{2}\right)} \big{(}\mathbf{v}_{i}-\mathbf{v}_{j}\big{)} \tag{5}\]  
where \(\upsilon_{0}\) is the kinematic viscosity and \(\eta\) is a parameter that guarantees a non-singular operator. In the DualSPHysics solver, \(\eta\) is equal to \(0.001h^{2}\), with \(\big{\{}\eta\ \in\ \mathbb{R};x_{i}-x_{j}>\eta\big{\}}\).