International Journal of Computational Methods Vol. 15, No. 3 (2018) 1846007 (19 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219876218460076 Viscous Flow Past a NACA0012 Foil Below a Free Surface Through the Delta-Plus-SPH Method P. N. Sun ∗ , F. R. Ming † and A. M. Zhang College of Shipbuilding Engineering Harbin Engineering University Harbin, P. R. China ∗ sunpengnan@yeah.net † mingfuren@hrbeu.edu.cn B. Wang National Key Laboratory of Shock Wave and Detonation Physics Institute of Fluid Physics CAEP, Mianyang, P. R. China Received 25 December 2017 Accepted 7 March 2018 Published 16 April 2018 The present work is dedicated to the modeling of viscous flow past a NACA0012 foil fixed in a current below a free surface. To this end, the δ + -smoothed-particle hydrody- namics (SPH) model has been adopted. This SPH model prevents the inception of the numerical tensile instability in the flow region characterized by negative pressure since a tensile instability control (TIC) has been included. In the TIC, a pressure differencing formulation (PDF) has been adopted for the momentum equation in the flow region characterized by negative pressure. In order to completely remove the numerical noise in the vorticity field, in this work, the PDF is also applied for the region with posi- tive pressure, but except for the free-surface region in order to ensure the free surface stability when wave breaking occurs. The mechanism of PDF being able to eliminate the numerical noise in the vorticity field is also briefly analyzed. In order to reduce the nonconservation of total momentum induced by the PDF, a particle-shifting tech- nique (PST) is implemented in each time step for regularizing the particle position. In the numerical results, δ + -SPH results are validated by the experimental data and other verified numerical results. Improvements of the results of δ + -SPH with PDF with respect to the ones without using PDF are demonstrated. Parametrical studies based on the δ + - SPH model regarding the breaking and non-breaking waves generated by the flow past a submerged foil are also carried out. Keywords : Smoothed-particle hydrodynamics; free-surface flow; tensile instability; par- ticle shifting; vorticity noise; viscous flow past a body; wave breaking. † Corresponding author. 1846007-1 Int. J. Comput. Methods Downloaded from www.worldscientific.com by UNIVERSITY OF NEW ENGLAND on 04/16/18. For personal use only.