PHYSICAL REVIEW E 93, 053113 (2016) Analysis of free-surface flows through energy considerations: Single-phase versus two-phase modeling Salvatore Marrone CNR-INSEAN, Marine Technology Research Institute, Rome, Italy and ´ Ecole Centrale Nantes, LHEEA Laboratoire (ECN / CNRS), Nantes, France Andrea Colagrossi * CNR-INSEAN, Marine Technology Research Institute, Rome, Italy Andrea Di Mascio CNR IAC, Istituto per le Applicazioni del Calcolo “Mauro Picone,” Rome, Italy David Le Touz´ e ´ Ecole Centrale Nantes, LHEEA Laboratoire (ECN / CNRS), Nantes, France (Received 17 December 2015; revised manuscript received 22 April 2016; published 20 May 2016) 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. DOI: 10.1103/PhysRevE.93.053113 I. INTRODUCTION In the last decades, the interest in free-surface flows has notably grown from both the scientific and the engineering points of view, the involved problems ranging from marine and coastal engineering fields. As is well known, this class of flows involves challenging phenomena to model and simulate because of the multiple fragmentations and reconnections of the air-water interface. Several numerical methods have been developed to date to tackle this problem; among them, two classes of methods gained vast popularity in the numerical community: mesh-based computational fluid dynamics (CFD) solvers coupled with interface capturing techniques (finite volumes, differences, elements with level set, or volume of fluid approaches) and particle methods (SPH, MPS). The former class, based on a Eulerian grid, is currently applied in most of naval or coastal and marine engineering applications. On the other side, particle methods, thanks to their meshless Lagrangian character, have proven to be remarkably effective when dealing with large deformation and fragmentations or reconnections of the air-water interface. Besides the general numerical scheme adopted, the pre- liminary choice that must be made when solving free-surface * andrea.colagrossi@cnr.it flows is related to the modeling of the gaseous phase. The density ratio between water and air being large (∼820), in order to save computational resources and to simplify the numerical modeling, only the liquid phase is often modeled, the role of the gas being assumed negligible. Similarly, as the flow velocity is much smaller than the sound speed in water, the chosen model is often the incompressible approximation, density variation being therefore neglected. Nevertheless, when simulating complex free-surface flows, these choices are not at all obvious nor easily justified. The replacement of air with vacuum deprives the flow of cushioning mechanisms in cavities, which can significantly alter the dynamics of the flow and energy transfer processes. Moreover, when a vacuum cavity collapses, a discontinuous drop of mechanical energy occurs (see, e.g., [1]) when the incompressible model is assumed. Conversely, in a weakly compressible model the cavity collapse induces rapid exchanges between mechanical energy and internal (elastic) energy [2], which are dissipated in few cycles by numerical viscosity. The objective of the present study is to address the implications of single-phase approximation on the simulation of energetic free-surface flows, with particular focus on energy evolution, transfer, and dissipation. To this end, a smoothed particle hydrodynamics (hereinafter SPH) solver is used. This solver was chosen because of its intrinsic conserva- tion properties (mass, momenta, and energy), the absence 2470-0045/2016/93(5)/053113(13) 053113-1 ©2016 American Physical Society