A Lagrangian meshless method for free-surface flows Part I: method and implementation. A. Colagrossi , M. Landrini and M.P. Tulin 1 Ocean Engineering Laboratory, UCSB. 2 INSEAN, The Italian Ship Model Basin Introduction A distinguishing feature of free-surface flows, and more in general of multi-phase flows, is that interfaces separat- ing media can break and fragment. These circumstances are by no means rare, and are of rather great importance in engineer- ing applications. Most of the CFD solvers proposed are usually based on the use of a grid (fixed or moving) to discretized the field equations, and then coupled to algorithms either to track or capture the interfaces. Here, we describe a different strategy based on a mesh-less method, named Smoothed Particle Hydrodynamics (SPH) [6], where a number of fluid particles is tracked in a Lagrangian fashion to determine the evolution of the flow field. Intra- and boundary-particles interactions are modeled by discretizing the field equations and the boundary conditions through the interpolation integral technique. The method has been already used successfully to highlight some mechanisms of wave break- ing [12]. Scope of the present paper and of the presentation is summarizing our most recent experience aimed to further improve accuracy, efficiency, and applicability of the method. Interpolation integral In meshless methods of the type here considered, the field of a generic quantity, say , is represented through ”interpolation integrals” of the form: (1) where is a weight function and is a measure of the support of , i.e. where differs from zero. Physically, is also representative of the domain of influence of . In the SPH framework, is called smoothing function or kernel, and has the following properties: for , and zero otherwise. decreases monotonously as increases. In the limit for , the kernel function becomes a Dirac delta function, and therefore Formally, we can deduce approximation to any derivative of the Influenced Point x P node x* Domain of influence Ω x* Kernel function W(x P - x*;h) Domain Ω Fig. 1: Sketch of the kernel function field by differentiating (1). For example: (2) Kernel functions In practical computations the smoothing func- tion affects both the CPU requirements and the stability proper- ties of the algorithm. We have experienced b-spline kernels of third and fifth order, as well as the guassian kernel: (3) where . This kernel has not compact support. Therefore, we introduced a cut-off limit and renormalized the kernel to match the property of unit integral. The use of (3) resulted in a more efficient code. Interpolation of scattered data If data samples of the field are known, a naive discrete approximation of (1) gives: (4) and a discrete approximation to the gradient (2) is: (5) where, for brevity, . The surface in- tegral contribution in (2) is usually neglected. We observe that the above approximations, when used on data arbitrarily scattered, do no allow to reproduce correctly very simple constant or bi-linear functions. In general we have (6) (7) Therefore, a constant field will not be reproduced correctly and it will introduce a spurious gradient, etc. The capability of ”re- producing” given functions is clearly related to the convergence properties of the method [1]. Here, we summarize some of the 1