TECHNICAL PAPER A three-dimensional SPH model for detailed study of free surface deformation, just behind a rectangular planing hull Abbas Dashtimanesh • Parviz Ghadimi Received: 21 January 2013 / Accepted: 15 May 2013 Ó The Brazilian Society of Mechanical Sciences and Engineering 2013 Abstract Transom wave behind planing hulls is a com- plicated physical phenomenon that has lead to computa- tional challenge for many researchers. On the other hand, smoothed particle hydrodynamics (SPH) which is known as a meshless Lagrangian approach can simulate free sur- face flows with strongly nonlinear physics. Therefore, effort has been made in the current study to develop a 3D-SPH code for three-dimensional simulation of transom stern flow behind a rectangular planing hull. It is also aimed to give some new physical insights into this highly nonlinear problem. Different techniques such as sub par- ticle scale turbulence model and moving least square density filter among others are also implemented. To val- idate the developed 3D-SPH code, the benchmark problem of dam breaking is investigated. Moreover, to verify the capability of the presented SPH model for transom flow simulation, previous experimental studies at low Froude numbers are considered. Comparisons display good agreement between the numerical results and experimental findings. Furthermore, a detailed discussion about rooster tail formation is presented. Keywords Rectangular planing hull  Free surface deformation  SPH model  3D model 1 Introduction Numerical study of high speed craft and specifically planing hulls is of major interest for many researchers. On the other hand, one of the most complex phenomena related to planing hull is transom stern flow. In reality, generation of strong wave field behind planing hull, which is known as rooster tail, leads to a severe computational challenge. Therefore, many researchers have tried to gain a better understanding of various characteristics of this highly nonlinear phenomenon. Some have used experi- mental setup while others have implemented numerical methods. Literature related to studies on this topic can be divided into two main parts: experimental work and numerical simulations. It is worth mentioning that there exist some analytical solutions which are out of the intended aim of this paper. O’Dea et al. [1] was one of the first researchers who studied the transom stern to obtain the flow field charac- teristics around a destroyer hull. A set of experiments was carried out by Robards and Doctors [2] on the rectangular transom stern to survey transom ventilation as well as longitudinal wake profile. The process of transom stern ventilation at low speed was also considered by Doctors [3, 4] and empirical formulas were derived. Fu et al. [5] performed a series of experiments on R/V Athena model and free surface profile behind the transom stern was measured. Maki et al. [6] considered a prismatic model with 20° deadrise angle. They studied ventilation process at transom stern, thoroughly. Yamano et al. [7, 8] also tried to present detailed experimental studies in the field of transom stern flow. Two series of experiments were also conducted by Maki et al. [9]. They studied transom flow for two different high speed destroyers. Transom wave was mea- sured and an empirical relation for transom drying was Technical Editor: Francisco Ricardo Cunha. A. Dashtimanesh  P. Ghadimi (&) Department of Marine Technology, Amirkabir University of Technology, Hafez Ave, No 424, P.O. Box 15875-4413, Tehran, Iran e-mail: pghadimi@aut.ac.ir 123 J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-013-0035-9