The Smoothed Particle Hydrodynamics (SPH)method is revisited within a Large Eddy Simulation (LES)perspective following the recent work of [1]. To this aim, LESfiltering procedure is recast in a Lagrangian framework bydefining a filter centred at the particle position that moves withthe filtered fluid velocity. The Lagrangian formulation of LES isthen used to re-interpret the SPH approximation of differentialoperators as a specific model based on the decomposition of theLES filter into a spatial and time filter.The derived equations represent a general LES-SPH schemeand contain terms that in part come from LES filtering and inpart derive from SPH kernels. The last ones lead to additionalterms (with respect to LES filtering) that contain fluctuations inspace, requiring adequate modelling. Further, since the adoptedLES filter differs from the classical Favre averaging for thedensity field, fluctuation terms also appear in the continuityequation.In the paper, a closure model for all the terms is suggested andsome simplifications with respect to the full LES-SPH model areproposed. The simplified LES model is formulated in a fashionsimilar to the diffusive SPH scheme of Molteni & Colagrossi[2] and the diffusive parameter is reinterpreted as a turbulentdiffusive coefficient, namely ? ? . In analogy with the turbulentkinetic viscosity ? T , the diffusive coefficient is modelled througha Smagorinsky-like model and both ? T and ? ? are assumed todepend on the magnitude of the local strain rate tensor D.Some examples of the simplified model are reported forboth 2D and 3D free-decaying homogeneous turbulence andcomparisons with the full LES-SPH model are provided.

The Smoothed Particle Hydrodynamics (SPH) method, often used for the modelling of the Navier- Stokes equations by a meshless Lagrangian approach, is revisited from the point of view of Large Eddy Simulation (LES). To this aim, the LES filtering procedure is recast in a Lagrangian framework by defining a filter that moves with the positions of the fluid particles at the filtered velocity. It is shown that the SPH smoothing procedure can be reinterpreted as a sort of LES Lagrangian filter- ing, and that, besides the terms coming from the LES convolution, additional contributions (never accounted for in the SPH literature) appear in the equations when formulated in a filtered fashion. Appropriate closure formulas are derived for the additional terms and a preliminary numerical test is provided to show the main features of the proposed LES-SPH model.

The study of energetic free-surface flows is challenging because of the large range of interface scales involved due to multiple fragmentations and reconnections of the air-water interface with the formation of drops and bubbles. Because of their complexity the investigation of such phenomena through numerical simulation largely increased during recent years. Actually, in the last decades different numerical models have been developed to study these flows, especially in the context of particle methods. In the latter a single-phase approximation is usually adopted to reduce the computational costs and the model complexity. While it is well known that the role of air largely affects the local flow evolution, it is still not clear whether this single-phase approximation is able to predict global flow features like the evolution of the global mechanical energy dissipation. The present work is dedicated to this topic through the study of a selected problem simulated with both single-phase and two-phase models. It is shown that, interestingly, even though flow evolutions are different, energy evolutions can be similar when including or not the presence of air. This is remarkable since, in the problem considered, with the two-phase model about half of the energy is lost in the air phase while in the one-phase model the energy is mainly dissipated by cavity collapses.

An algorithm for coupling a classical Finite Volume (FV) approach, that discretize the Navier-Stokes equations on a block structured Eulerian grid, with the weakly-compressible SPH is presented. The coupling procedure aims at applying each solver in the region where its intrinsic characteristics can beexploited in the most efficient and accurate way: the FV solver is used to resolve the bulk flow and the wall regions, whereas the SPH solver is implemented in the free surface region to capture details of the front evolution. In order to avoid the difficulties connected with inhomogeneous domain decomposition, in both SPH and FV regions a weakly compressible flow model was firstly tested. However, the coupling procedure has beenimplemented in order to allow the adoption of different time steps between the two solvers. Thanks to this feature, it will be shown that the proposed technique is also able to reproduce an inhomogeneous coupling. Indeed, the FV solver, because of the space discretization adopted and the implicit time integration, naturally tends to an incompressible discrete solver, wherever the time step is much larger of what required to capture compressibility effects. The different coupling strategies as well as convergence studies have been carried out in the 2D framework.The reported results clearly prove that the combined use of the two solvers is convenient from the point of view of both accuracy and computing time.

In the present work the simulation of water impacts is discussed. The investigation is mainly focused on the energy dissipation involved in liquid impacts in both the frameworks of the weakly compressible and incompressible models. A detailed analysis is performed using a weakly compressible Smoothed Particle Hydrodynamics (SPH) solver and the results are compared with the solutions computed by an incompressible mesh-based Level-Set Finite Volume Method (LS-FVM). Impacts are numerically studied using single-phase models through prototypical problems in 1D and 2D frameworks. These problems were selected for the conclusions to be of interest for, e.g., the numerical computation of the flow around plunging breaking waves. The conclusions drawn are useful not only to SPH or LS-FVM users but also for other numerical models, for which accurate results on benchmark test-cases are provided.

In questo rapporto viene presentato un nuovo algoritmo per il calcolo di flussi a superficie libera con forte deformazione e frammentazione. L'algoritmo è ottenuto accoppiando un classico schema ai Volumi Finiti (FV), in cui le equazioni di Navier-Stokes sono discretizzate su una griglia Euleriana, con un approccio basato su un metodo Lagrangiano, basato sulla Smoothed Particle Hydrodynamics (SPH). L'algoritmo è formulato in modo da sfruttare al meglio le caratteristiche di ogni schema nella maniera più efficiente e accurata: lo schema ai Volumi Finiti è usato per risolvere la zona del flusso lontana dalla superficie libera e i flussi a parete, mentre il solutore SPH è implementato solo nella regione con superficie libera, al fine di catturare i dettagli della evoluzione del fronte. I risultati discussi provano che l'uso combinato dei due solutori è conveniente sia dal punto di vista dell'accuratezza che da quello del tempo di calcolo.

In the present work the numerical simulation of breaking wave processes is discussed. A detailed analysis is performed using Smoothing Particle Hydrodynamics (SPH) models as well as a mesh-based Level-Set Finite Volume Method (LS-FVM). Considerations on the numerical dissipation involved in such models are discussed within the frameworks of weakly compressible and incompressible ssumptions. The breaking wave processes are simulated using both mono- and two-phases models. Due to the extensive test-cases discussed, the present analysis is limited to a bi-dimensional framework. Test-cases with increasing complexities are considered starting from a simple 1D impact of two water-jet up to complex shallow water breaking waves. The analyses presented in this article are not only useful to weakly compressible SPH or LS-FVM users but can be extended to other numerical models for which accurate (convergent) results on benchmark test-cases are provided.

An algorithm for coupling SPH with an externalsolution is presented. The external solution can be either anotherSPH solution (possibly with different discretization) or a differentnumerical solver or an analytical solution.The interaction between the SPH solver and the externalsolution is achieved through an interface region. The interfaceregion is defined as a fixed portion of the computational domainthat provides a boundary condition for the SPH solver. A ghostfluid, composed by fully lagrangian particles (i.e. ghost particles)covering the interface region, is used to impose the boundarycondition. The ghost particle evolution, including its position, isintegrated in time according to the field of the external solution.The physical quantities of the ghost particles needed in theintegration scheme are obtained through an MLS interpolationon the field of the external solution. When a ghost particle crossesthe boundary of the interface region, entering in the SPH domain, it evolves according to the SPH governing equation. The spatial distribution of the ghost particles can become largely non-uniform due to the forcing by the external solution. Thus, a packing algorithm is applied on the ghost particles in the interface region, to guarantee a particle distribution suitable for SPH operators. Since the ghost particles can exit from the interface region, a seeding algorithm is needed to introduce new ghost-particles. The algorithm is tested on several benchmarks and with the external solutions given by other SPH solvers with different discretizations and by analytical solutions. The technique is deeply investigated in terms of accuracy, efficiency and possible applications. Finally a coupled simulation involving a finite volume solver is presented.