Moving particle method for modeling wave interaction with porous structures H. Akbari a, ⁎, M. Montazeri Namin b a School of Civil Engineering, College of Engineering, Tehran University, Tehran, Iran b Tehran University, Iran abstract article info Article history: Received 9 August 2012 Received in revised form 13 December 2012 Accepted 17 December 2012 Available online 12 January 2013 Keywords: ISPHP Porous flow Interface boundary Wave Free surface This paper presents a numerical model for simulating wave interaction with porous structures. Incompressible smoothed particle hydrodynamics in porous media (ISPHP) method is introduced in this study as a mesh free particle approach that is capable of efficiently tracking the large deformation of free surfaces in a Lagrangian coordinate system. The developed model solves two porous and pure fluid flows simultaneously by means of one equation that is equivalent to the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the extended Forchheimer equation for the flows inside the porous media. Interface boundary between pure fluid and porous media is effectively modeled by the SPH integration technique. A two-step semi-implicit scheme is also used to solve the fluid pressure satisfying the fluid incompressibility criterion. The developed ISPHP model is then validated via different experimental and numerical data. Fluid flow pattern through porous dam with different porosities is studied and regular wave attenuation over porous seabed is investigated. As a practical case, wave running up and overtopping on a caisson breakwater protected by a porous armor layer are modeled. The results show good agreements between numerical and laboratory data in terms of free surface displacement, overtopping rate and pressure distribution. Based on this study, ISPHP model is an efficient method for simulating the coastal applications with porous structures. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Porous materials are widely used in coastal structures and the study of wave interaction with these materials is a common field in coastal engineering. Rubble mound and submerged breakwaters are usually constructed with porous materials such as porous armor layers. Wave propagation and its dissipation over porous seabed are another example of wave interaction with porous media. In fact, wave energy is dissipated inside the porous structure because of flow friction. In order to know the functionality and stability of these structures, the flow motion inside and around the porous media must be determined. At the present, knowledge of wave interaction with porous media is not compatible with the wide application of porous media in coastal engineering. So, design guidelines are usually based on semi-empirical formulations extracted from flume or wave basin experiments with some laboratory limitations. In the recent years, there have been many developments in numerical modeling of wave interaction with the porous media. Huang et al. (2008) studied the propagation of a solitary wave over porous beds. Shao (2010) investigated the interac- tions between waves and submerged breakwater as well as the wave propagation over seabed. The numerical models developed so far try to represent accurately the behavior of the flow at macroscopic and microscopic (the grain/ pore) scales. Unlike macroscopic modeling, grain scale modeling suffers from huge computational costs in large scale domains and inaccuracies due to the complexity of grain configurations, shapes and dimensions. The early macroscopic equation that represents a simple linear relationship between flow rate and pressure drop in a porous media was presented by Darcy in the 19th century and many others added complementary terms to this equation afterwards. Forchheimer (1901) used an additional nonlinear term to account for the increased pressure drop at high velocities and Brinkman (1947) added a viscous shear stress term to the Darcy equation. After that, sev- eral attempts have been made for introducing a unified Navier–Stokes type equation for the description of flow through porous media. Wooding (1957) added the inertial force into the equation and Sollit and Cross (1972) proposed the momentum equations for the porous flow by including additional inertial and nonlinear resistance forces into the Darcy equations. In their analytical solution, they ignored the convection terms based on the assumption of quick dissipation of higher harmonics inside the coarse granular media. It was shown later by Losada et al. (1995) that this assumption is not accurate for modeling flows through very tortuous porous media where the second-order harmonics generated due to the convective fluctuation terms might be more significant than the first-order harmonics. Vafai and Tien (1981) added a convective term and Sakakiyama and Kajima (1992) extended Navier–Stokes equations to include the effect of inertial and convective Coastal Engineering 74 (2013) 59–73 ⁎ Corresponding author at: Tehran University, Iran. Tel.: +98 912 6840036; fax: +98 2166403808. E-mail addresses: hasanakbari@ut.ac.ir (H. Akbari), mnamin@ut.ac.ir (M.M. Namin). 0378-3839/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coastaleng.2012.12.002 Contents lists available at SciVerse ScienceDirect Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng