A Meshless Lagrangian Method for Free-Surface and Interface Flows with Fragmentation

A meshless method based on the Smoothed Particle Hydrodynamics (SPH) algorithm has been developed for the analysis of two-dimensional unsteady incompressible inviscid rotational flows. The method simulates the evolution of both single-phase and multi-phase fluids separated by sharp interfaces. Moreover it can model large deformations and fragmentation of free surfaces and interfaces, as well as the presence and motions of solid boundaries. Fluid-fluid and/or fluid-solid impacts and the resulting flow and pressure evolutions can be described. In principle the method is easy to code, but a robust, accurate and efficient code requires appropriate numerical choices. The features of the numerical solver have been detailed described. The mass conservation and Euler equations are solved and the fluid properties are evaluated in the domain by following the motion of a finite number of fluid particles. The latter move according to the forces exerted by the surrounding particles. The neighbors of any particle are identified through a Cartesian-grid algorithm periodically applied during the simulations. This proved to be a good compromise between efficiency and effectiveness. The standard SPH formulation has been generalized to treat particles distributed non-uniformly and with different sizes (i.e. local refinements) crucial when dealing with large interface deformations and breaking phenomena. The fluid is modeled as weakly compressible by using an equation of state between density and pressure fields. This avoids the need to solve a Poisson equation for the pressure. Two alternative time evolution schemes have been combined with the SPH technique: a modified Euler method and a fourth-order Runge-Kutta scheme. The choice between them depends on the time scale involved in the problem of interest and the desired accuracy. The second scheme is more suitable for long-time simulations but is computationally more expensive. For the former time scheme dynamic time step within a multiple time scale technique is adopted. Concerning the numerical features, an isotropic modified Gaussian kernel with uniform smoothing length is used as interpolant. The tensile instability is avoided using the technique proposed by Monaghan (2000). A fictitious viscosity term is introduced in the Euler equations to enlarge the stability limits of the method. The particle interpenetration is prevented by modifying the motion particle equation with a XSPH velocity correction. This regularizes also the weakly-compressible treatment of liquids. A periodical re-initialization of the density field keeps the consistency between mass, density and occupied area of the particles, and works against unphysical oscillations near the deformable boundaries. For multi-phase flows, the state equation is altered to ensure the interface sharpness. The resulting method conserves correctly mass, linear momentum and angular momentum. It showed also to preserve satisfactorily the energy. The stability and convergence properties are discussed theoretically from a general point of view, and quantified heuristically through the study of many test cases. In particular, the solver is applied to problems with single- and two-phase fluids, with and without rigid boundaries. All of them are characterized by relevant interface deformations. Some of them experience impact events, phase entrainment phenomena, vorticity generation. The present results are successfully validated by experiments and verified by analytical and other numerical results available in literature. Both single-phase and two-phase SPH proved to be convergent and accurate. The two-phase formulation needs to be improved in terms of efficiency. The method is used to study some problems of practical interest in many fields: (i) the bre