Numerical simulation of interfacial flows by smoothed particle hydrodynamics☆
Introduction
In many circumstances, violent fluid–structure interactions lead to air entrapment and multi-phase flows. In marine and coastal engineering applications [11], [31], the dynamics of the entrapped air at the impact may play a dominant role during the process and contribute to the high pressure maxima and pressure oscillations. Therefore, neglecting the air dynamics in impact flows may result in an incorrect approximation, particularly in predicting the short-time pressure characteristics (i.e., time-scales much shorter than the characteristic wave period) such as the pressure maxima, pressure rise times, and pressure oscillations.
A number of numerical techniques have been proposed to model flow fields with free surfaces or, more in general, with (sharp) interfaces separating immiscible fluids. Broad reviews are given in [34], [35]. Most of the proposed and successful methods are based on the use of an Eulerian grid spanning the whole domain (possibly including different fluids) where the fluid-flow equations are solved and coupled with a suitable technique to capture or track the interface.
A more limited number of approaches is based on the Lagrangian tracking of fluid elements, either distributed near the interface or all over the fluid domain. Among the latter, smoothed particle hydrodynamics (SPH) is a fully Lagrangian mesh-less technique originally developed to deal with astro-dynamical problems [22] and successfully extended to a variety of fluid-dynamic systems [19], [20]. Though it can be computationally more expensive than other Eulerian methods, SPH features a remarkable flexibility in handling complex flow fields and in including physical effects.
In particular, in a series of papers, Monaghan [21], [25] has shown the SPH capability to treat free-surface flows with breaking and multi-phase flows with small density differences between the considered media [23], [24]. When applying this approach to the air–water case, we found that severe instabilities develop along the fluid interface which prevented SPH to work.
In this paper, it is presented an original SPH implementation to handle two-dimensional interface flows with low density ratio. The key element of the present algorithm is a new form of the particle evolution equations, derived following [5], which improves the stability and removes fictitious surface-tension effects present in the standard SPH implementation [13]. Further improvements have been achieved by using: (i) a periodic re-initialization of the density field based on a moving-least-square interpolation [2] and (ii) a generalized form of Balsara’s correction [3], to the usual SPH artificial viscosity. The last treatments are beneficial also for SPH computations of free-surface flows.
In the following, first the general concepts of the SPH modelling for incompressible interfacial flows are given, and the standard SPH implementation is described. The difficulty in modelling two-phase flows with small density ratios is then discussed for the case of a gas bubble rising through water. On this ground, the present SPH-implementation is introduced and described in details. The effect of the many adaptations is discussed by considering the rising-bubble problem and the dam-break problem. The latter is finally studied, with more emphasis on the physics of the violent impact of water against a fixed structure and the effect of entrapped air on the resulting loads.
Section snippets
Basic aspects of the SPH method for incompressible flows
In SPH methods, the fluid field is represented as a collection of N particles interacting with each other through evolution equations of the general formThe interaction terms follow from mass- and momentum-conservation equations and contain the density ρi, the velocity and the pressure pi of the particles. The force can be any body force, though in the following only the action of the gravity will be considered. Finally, the last equation
Two-phase modelling of dam-break and impact problems
We discuss the flow originated by the break of a dam and the impact of the water front against a fixed vertical wall, located downstream the dam. In spite of its simplicity, this problem embeds several features related to practical hydrodynamic circumstances, e.g. slamming loads on ship hulls impacting on water surface, green-water loads on deck and deck-structures of marine structures, sloshing loads in tanks [11], or wave loads on coastal structures [31].
Conclusions
An implementation of the SPH method to deal with two-dimensional interface flows with low density-ratios has been presented. The method results stable and capable to easily treat a variety of air–water flows with interface breaking and air-entrapment. In particular, the proposed form of the interactions terms is the basic tool to keep the algorithm stable for small density ratios. Second, a density re-initialization procedure is introduced which improves the mass-area-density consistency [3]
Acknowledgements
This work has been partly supported as part of a program for the simulation of ship breaking waves by the Ship Hydrodynamics Program of ONR, managed by Dr. Pat Purtell. INSEAN research activity was supported by the Italian Ministero delle Infrastrutture e Trasporti through INSEAN Research Program 2000-02.
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This paper is dedicated to the memory of Maurizio Landrini who died only 40 years old in a tragic accident. Still very young he made significant scientific contributions to non-linear free-surface phenomena in hydrodynamics.