An Hamiltonian interface SPH formulation for multi-fluid and free surface flows N. Grenier a , M. Antuono b , A. Colagrossi b,c, * , D. Le Touzé a , B. Alessandrini a a Fluid Mechanics Laboratory, École Centrale Nantes, Nantes, France b INSEAN, The Italian Ship Model Basin, 00128 Roma, Italy c Centre of Excellence for Ship and Ocean Structures, NTNU, Trondheim, Norway article info Article history: Received 2 March 2009 Received in revised form 13 August 2009 Accepted 15 August 2009 Available online 21 August 2009 Keywords: SPH Hamiltonian particle system Multi-fluid Interfacial flows Free surface flows Non-diffusive interface abstract In the present work a new SPH model for simulating interface and free surface flows is pre- sented. This formulation is an extension of the one discussed in Colagrossi and Landrini (2003) and is related to the one proposed by Hu and Adams (2006) to study multi-fluid flows. The new SPH scheme allows an accurate treatment of the discontinuity of quantities at the interface (such as the density), and permits to model flows where both interfaces and a free surface are present. The governing equations are derived following a Lagrangian variational principle leading to an Hamiltonian system of particles. The proposed formula- tion is validated on test cases for which reference solutions are available in the literature. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Multi-fluid flows play a significant role in numerous engineering applications characterized by strong dynamics of the flow (e.g. flows involved in mixing/separation devices, engines, propellers with cavitation, etc.). With respect to this, the SPH scheme has proved to be a valuable candidate as simulation method (see for example [2,3]). Even for that flows (i.e. jets, sprays, impacts, free surface reconnections, etc.) which are generally modeled by using one-fluid SPH scheme (see e.g. [4]), the air phase can have a large influence on the flow evolution and on the subsequent loads on structures. In this context, the main advantage of the SPH model is that fluid elementary volumes are followed in their Lagrangian motion and, conse- quently, the interface between two fluids will remain sharply described. Hence, the interface will not be diffused like in stan- dard mesh-based methods (Volume Of Fluids, Level-Set, Constrained Interpolation Profile, etc.). Nonetheless, although the classical SPH formulation succeeds in correctly simulating one-fluid flows, the presence of an interface and the physical conditions associated make a stable two fluid formulation more difficult to derive. The main issue is the estimation of the ratio between the pressure gradient and the density inside the momentum equation, since the den- sity is discontinuous when crossing the interface. Since the SPH scheme relies on a smoothing procedure (namely, each par- ticle is associated to a compact support on which the smoothing is made), the accuracy in modeling sharp discontinuities worsens when the compact support intersects the interface. Indeed, in this eventuality, the density of the fluid on the other side of the interface spuriously influences both the local density and pressure fields and, consequently, the acceleration of the concerned particle. 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.08.009 * Corresponding author. Address: INSEAN, The Italian Ship Model Basin, 00128 Roma, Italy. Tel.: +39 0650299343; fax: +39 065070619. E-mail address: a.colagrossi@insean.it (A. Colagrossi). Journal of Computational Physics 228 (2009) 8380–8393 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp