A TWO-PHASE CONSISTENT PARTICLE METHOD FOR WAVE IMPACT PROBLEMS WITH ENTRAPPED AIR POCKETS LUO MIN NATIONAL UNIVERSITY OF SINGAPORE 2015 A TWO-PHASE CONSISTENT PARTICLE METHOD FOR WAVE IMPACT PROBLEMS WITH ENTRAPPED AIR POCKETS LUO MIN (B.ENG, HARBIN INSTITUTE OF TECHNOLOGY, CHINA) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2015 DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Ph.D Candidate: Luo Min Signature: Date: Acknowledgement I am happy to take this opportunity to express my gratitude to my advisor, Professor Koh Chan Ghee for his invaluable guidance, support and encouragement throughout my study at NUS His critical attitude and rigorous scholarship in research has a great influence on me And I think this influence will accompany me in the study and work of the rest of my life Great thanks are also expressed to my co-supervisor Professor Bai Wei I am really appreciative of his kindness to arrange a meeting with me whenever I need a discussion on my research From these heuristic discussions, I have learned how to analyze a new problem and then think a way out to solve it Professor Lin Pengzhi in Sichuan University and Professor Shao Songdong in the University of Sheffield are acknowledged for the useful discussions and valuable comments from them In addition, the instructive comments and suggestions from Professors Khoo Boo Cheong and Vivien Chua in NUS are also appreciated Heartfelt gratitude is sent to Dr Gao Mimi, who guided me to learn the in-home developed numerical algorithm and laboratory experiments Without her patience and help, I could not have finished my Ph.D study in the tight time frame I would like to thank the staff in the Structural and Concrete Laboratory, especially Mr Koh Yian Kheng, Mr Ang Beng Oon and Mr Ow Weng Moon Without their help and assistance, I could not have finished my experimental study successfully I also would like to thank my friends: Dr Zhang Zhen, Dr Zhang Jian, Dr Zhang Mingqiang, Dr Zhang Yi, Dr Gao Ruiping, Ms Han Qinger, Ms Zhang Shanli, Ms I Zhang Hong, Mr Sun Gang, Mr Gao Qingfei, Mr Wang Yu, Dr Yu Chao and Mr Zhang Xiaodong, and all the other friends who have helped me It was the discussions with them that inspired my study and research And it was the recreational time spent with them that made my Ph.D journey relaxed and colorful Last but not least, I wish to express my love to my family: my parents, grandparent, younger sister and my girlfriend Thanks for their understanding, encouragement and love Without their supports, the completion of my thesis would not have been possible II Summary In many circumstances, violent fluid motions such as wave impacts on coastal/offshore structures generate air entrapment The entrapped air may affect the amplitude and duration of impact pressure because of the air cushion effect Numerical treatment of this problem remains a challenge because of its complexity, and most of the research findings were obtained from experiments In this context, the main objective of this thesis is to develop a new numerical method that can simulate violent wave impact processes with entrapped air pocket so as to achieve further insight into wave-impact processes and better prediction of impact pressures Most of the numerical methods developed for fluid dynamics problems can be classified into the mesh-based and meshless methods Among these methods, a Lagrangian meshless method (also called particle method) is adopted in this study because it is, in principle, capable of modelling large deformation, tracking fluid interface and avoiding the numerical diffusion induced by the discretization of the convection term of the Navier-Stokes Equations Among existing particle methods, the recently developed Consistent Particle Method (CPM) that computes the spatial derivatives in a way consistent with Taylor series expansion and eliminates the use of kernel function is selected because of its promising features to generate smooth fluid pressure without the use of artificial parameters such as the artificial viscosity The main challenges in modeling wave impact with entrapped air pocket include (a) approximations of gradient and Laplacian operators involving large density difference (three orders of magnitude for water-air flows) and (b) integrated modelling of incompressible water and compressible air To resolve the first issue, a new scheme is III proposed by dealing with the pressure gradient normalized by density Based on the generalized finite difference scheme, this approach uses all the neighbor particles (including those of another fluid) in the influence domain of a reference particle to compute the spatial derivatives with abrupt density discontinuity In addition, an adaptive particle selection scheme is proposed to overcome the problem of ill-conditioned coefficient matrix of pressure Poisson equation when particles are sparse and nonuniformly spaced These two improvements lead to the incompressible two-phase CPM (I-2P-CPM) for incompressible two-phase flows characterized by high density ratio To address the second challenge, a compressible solver is developed in the framework of thermodynamics In this way sound speed is not explicitly involved and thus the compressible solver avoids the problem as encountered by some other numerical methods in the determination of numerical or artificial sound speed In addition, this compressible solver can be easily integrated with the I-2P-CPM because they both use the same predictor-corrector scheme to solve the governing equation of primitive form This leads to the two-phase CPM (2P-CPM) that is applicable to incompressiblecompressible two-phase flows with abrupt density discontinuity The developed algorithms are validated by numerical examples in comparison with published results in the literature and the present experimental studies To demonstrate the performance of 2P-CPM, a new experiment is designed and conducted particularly for obtaining an air pocket and measuring its shape and pressure change under wave impact In all the cases considered, the numerical results agree well with the experimental results, including air pressure oscillation due to air cushion effect The results show that modelling of compressible air is crucial in wave impact scenarios with entrapped air pocket IV References Cooker, M.J and Peregrine, D.H (1995) Pressure-impulse theory for liquid impact problems, Journal of Fluid Mechanics, 297: 193-214 Cuomo, G., Allsop, W and Takahashi, S (2010) Scaling wave impact pressures on vertical walls, Coastal Engineering, 57(6): 604-609 Dalrymple, R.A and Rogers, B.D (2006) Numerical modeling of water waves with the SPH method, Coastal Engineering, 53(2-3): 141-147 Daru, V., Le Q , P., Duluc, M.C and Le M O (2010) A numerical method for ré tre, the simulation of low Mach number liquid–gas flows, Journal of Computational Physics, 229(23): 8844-8867 Ding, H., Spelt, P.D.M and Shu, C (2007) Diffuse interface model for incompressible two-phase flows with large density ratios, Journal of Computational Physics, 226(2): 2078-2095 Drazin, P.G (2002) Introduction to Hydrodynamic Stability New York: Cambridge University Press Emarat, N., Forehand, D.I.M., Christensen, E.D and Greated, C.A (2012) Experimental and numerical investigation of the internal kinematics of a surf-zone plunging breaker, European Journal of Mechanics - B/Fluids, 32: 1-16 Englmaier, P and Gerhard, O (1999) Gas dynamics and large-scale morphology of the Milky Way galaxy, Monthly Notices of the Royal Astronomical Society: Letters, 304(3): 512–534 Enright, D., Fedkiw, R., Ferziger, J and Mitchell, I (2002) A Hybrid Particle Level Set Method for Improved Interface Capturing, Journal of Computational Physics, 183(1): 83-116 Faltinsen, O.M., Landrini, M and Greco, M (2004) Slamming in marine applications, Journal of Engineering Mathematics, 48(3-4): 187-217 Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A and Timokha, A.N (2000) Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, Journal of Fluid Mechanics, 407(2000): 201-234 180 References Fang, J and Parriaux, A (2008) A regularized Lagrangian finite point method for the simulation of incompressible viscous flows, Journal of Computational Physics, 227(20): 8894-8908 Ferziger, J.H and Peric, M (1999) Computational methods for fluid dynamics Germany: Springer Fox, R.W., Mcdonald, A.T and Pritchard, P.J (2004) Introduction to Fluid Mechanics, 6th ed New York: Wiley Gao, M., 2011 Numerical simulation of liquid sloshing in rectangular tanks using Consistent Particle Method and experimental verification Ph.D Thesis National University of Singapore Ghasemi V, A., Firoozabadi, B and Mahdinia, M (2013) 2D numerical simulation of density currents using the SPH projection method, European Journal of Mechanics B/Fluids, 38: 38-46 Gingold, R.A and Monaghan, J.J (1977) Smoothed particle hydrodynamics-theory and application to non-spherical stars, Monthly Notices of the Royal Astronomical Society, 181: 375-389 Gó mez-Gesteira, M., Cerqueiro, D., Crespo, C and Dalrymple, R.A (2005) Green water overtopping analyzed with a SPH model, Ocean Engineering, 32(2): 223-238 Gotoh, H and Sakai, T (2006) Key issues in the particle method for computation of wave breaking, Coastal Engineering, 53(2-3): 171-179 Gray, J.P., Monaghan, J.J and Swift, R.P (2001) SPH elastic dynamics, Computer Methods in Applied Mechanics and Engineering, 190(49-50): 6641–6662 Grenier, N., Antuono, M., Colagrossi, A., Le Touzé D and Alessandrini, B (2009) An , Hamiltonian interface SPH formulation for multi-fluid and free surface flows, Journal of Computational Physics, 228(22): 8380-8393 Hao, Y and Prosperetti, A (2004) A numerical method for three-dimensional gas– liquid flow computations, Journal of Computational Physics, 196(1): 126-144 181 References Hattori, M., Arami, A and Yui, T (1994) Wave impact pressure on vertical walls under breaking waves of various types, Coastal Engineering, 22(1-2): 79-114 He, X., Chen, S and Zhang, R (1999) A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Taylor instability, Journal of Computational Physics, 152(2): 642-663 Heyns, J.A., 2012 Formulation of a weakly compressible two-fluid flow solver and the development of a compressive surface capturing scheme using the volume-of-fluid approach Ph.D Thesis Stellenbosch University Heyns, J.A., Malan, A.G., Harms, T.M and Oxtoby, O.F (2013) A weakly compressible free-surface flow solver for liquid–gas systems using the volume-of-fluid approach, Journal of Computational Physics, 240: 145-157 Hirt, C.W and Nichols, B.D (1981) Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics, 39(1): 201-225 Hu, C and Kashiwagi, M (2004) A CIP-based method for numerical simulations of violent free-surface flows, Journal of Marine Science and Technology, 9(4): 143-157 Hu, X.Y and Adams, N.A (2007) An incompressible multi-phase SPH method, Journal of Computational Physics, 227(1): 264-278 Hu, X.Y and Adams, N.A (2009) A constant-density approach for incompressible multi-phase SPH, Journal of Computational Physics, 228(6): 2082-2091 Ida, M (2000) An improved unified solver for compressible and incompressible fluids involving free surfaces Part Convection, Computer Physics Communications, 132(12): 44-65 Idelsohn, S.R., Oñ ate, E and Pin, F.D (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, International Journal for Numerical Methods in Engineering, 61(7): 964-989 Ikari, H., Gotoh, H and Sakai, T (2005) Simulation of wave run-up by liquid-gas twophase-flow MPS method The 3rd International Conference on Asian and Pacific Coasts (APAC 2005) Jeju, Korea: 1857-1867 182 References Johnson, G.R and Beissel, S.R (1996) Normalized smoothing functions for SPH impact computations, International Journal for Numerical Methods in Engineering, 39(16): 2725-2741 Kadioglu, S.Y., Sussman, M., Osher, S., Wright, J.P and Kang, M (2005) A second order primitive preconditioner for solving all speed multi-phase flows, Journal of Computational Physics, 209(2): 477-503 Kaplan, P (1992) Wave impact forces on offshore structures: Re-examination and new interpretations Offshore Technology Conference Kaplan, P., Murray, J.J and Yu, W.C (1995) Theoretical analysis of wave impact forces on platform deck structures 14th Intl Conf on Offshore Mechanics & Arctic Engng Copenhagen, Denmark Karni, S (1994) Multicomponent flow calculations by a consistent primitive algorithm, Journal of Computational Physics, 112(1): 31-43 Kees, C.E., Akkerman, I., Farthing, M.W and Bazilevs, Y (2011) A conservative level set method suitable for variable-order approximations and unstructured meshes, Journal of Computational Physics, 230(12): 4536-4558 Khayyer, A and Gotoh, H (2009) Modified Moving Particle Semi-implicit methods for the prediction of 2D wave impact pressure, Coastal Engineering, 56(4): 419-440 Khayyer, A and Gotoh, H (2010) A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method, Applied Ocean Research, 32(1): 124-131 Khayyer, A and Gotoh, H (2011) Enhancement of stability and accuracy of the moving particle semi-implicit method, Journal of Computational Physics, 230(8): 3093-3118 Khayyer, A and Gotoh, H (2013) Enhancement of performance and stability of MPS mesh-free particle method for multiphase flows characterized by high density ratios, Journal of Computational Physics, 242: 211-233 183 References Khayyer, A., Gotoh, H and Shao, S (2009) Enhanced predictions of wave impact pressure by improved incompressible SPH methods, Applied Ocean Research, 31(2): 111-131 Khayyer, A., Gotoh, H and Shao, S.D (2008) Corrected incompressible SPH method for accurate water-surface tracking in breaking waves, Coastal Engineering, 55(3): 236250 Kieffer, S.W (1977) Sound speed in liquid-gas mixtures: water-air and water-steam, Journal of Geophysical Research, 82(20): 2895–2904 Kim, Y (2001) Numerical simulation of sloshing flows with impact load, Applied Ocean Research, 23(1): 53 - 62 Kim, Y., Shin, Y.-S and Lee, K.H (2004) Numerical study on slosh-induced impact pressures on three-dimensional prismatic tanks, Applied Ocean Research, 26(5): 213-226 Kleefsman, K.M.T., Fekken, G., Veldman, A.E.P., Iwanowski, B and Buchner, B (2005) A Volume-of-Fluid based simulation method for wave impact problems, Journal of Computational Physics, 206(1): 363-393 Koh, C.G., Gao, M and Luo, C (2012) A new particle method for simulation of incompressible free surface flow problems, International Journal for Numerical Methods in Engineering, 89(12): 1582–1604 Koh, C.G., Luo, M., Gao, M and Bai, W (2013) Modelling of liquid sloshing with constrained floating baffle, Computers & Structures, 122: 270-279 Koren, B., Lewis, M.R., Van Brummelen, E.H and Van Leer, B (2002) Riemannproblem and Level-set approaches for homentropic two-fluid flow computations, Journal of Computational Physics, 181(2): 654-674 Koshizuka, S., Ikeda, H and Oka, Y (1999) Numerical analysis of fragmentation mechanisms in vapor explosions, Nuclear Engineering and Design, 189(1-3): 423–433 Koshizuka, S., Nobe, A and Oka, Y (1998) Numerical analysis of breaking waves using the moving particle semi-implicit method, International Journal for Numerical Methods in Fluids, 26(7): 751-769 184 References Koshizuka, S and Oka, Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid, Nuclear science and engineering, 123(3): 421434 Koshizuka, S., Tamako, H and Oka, Y (1995) A particle method for incompressible viscous flow with fluid fragmentation, Comput Fluid Dynamics J , 4: 29-46 Lee, B.H., Park, J.C., Kim, M.H., Jung, S.J., Ryu, M.C and Kim, Y.S (2010) Numerical simulation of impact loads using a particle method, Ocean Engineering, 37(23): 164-173 Li, S and Liu, W.K (2002) Meshfree and particle methods and their applications, Applied Mechanics Reviews, 55(1): Lin, P., 1998 Numerical modeling of breaking waves Ph.D Thesis Cornell University Lind, S.J., Xu, R., Stansby, P.K and Rogers, B.D (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves, Journal of Computational Physics, 231(4): 1499-1523 Liszka, T and Orkisz, J (1980) The finite difference method at arbitrary irregular grids and its application in applied mechanics, Computers & Structures, 11(1-2): 83-95 Liu, D., 2007 Numerical modeling of three-dimensional water waves and their interaction with structures Ph.D Thesis National University of Singapore Liu, D and Lin, P (2008) A numerical study of three-dimensional liquid sloshing in tanks, Journal of Computational Physics, 227(8): 3921-3939 Liu, G.R and Liu, M.B (2003) Smoothed Particle Hydrodynamics: A meshfree particle method Singapore: World Scientific Liu, M.B and Liu, G.R (2010) Smoothed Particle Hydrodynamics (SPH): An overview and recent developments, Archives of Computational Methods in Engineering, 17(1): 2576 185 References Liu, M.B., Liu, G.R and Lam, K.Y (2003) Constructing smoothing functions in smoothed particle hydrodynamics with applications, Journal of Computational and Applied Mathematics, 155(2): 263-284 Liu, X., Xu, H., Shao, S and Lin, P (2013) An improved incompressible SPH model for simulation of wave–structure interaction, Computers & Fluids, 71: 113-123 Lloyd's Register, 2005 Comparative sloshing analysis of LNG ship containment systems London Lloyd's Register, 2008 Guidance on the operation of membrane LNG ships to reduce the risk of damage due to sloshing London Lowe, R.J., Rottman, J.W and Linden, P.F (2005) The non-Boussinesq lock-exchange problem Part Theory and experiments, Journal of Fluid Mechanics, 537(-1): 101 Lucy, L.B (1977) A numerical approach to the testing of the fission hypothesis, Astronomical journal, 82(12): 1013-1024 Lugni, C., Bardazzi, A., Faltinsen, O.M and Graziani, G Fluid-structure interaction during wave-impact with air-entrapment in a sloshing tank Lugni, C., Bardazzi, A., Faltinsen, O.M and Graziani, G (2014) Hydroelastic slamming response in the evolution of a flip-through event during shallow-liquid sloshing, Physics of Fluids, 26(3): 032108 Lugni, C., Brocchini, M and Faltinsen, O.M (2006) Wave impact loads: The role of the flip-through, Physics of Fluids, 18(12): 122101 Lugni, C., Brocchini, M and Faltinsen, O.M (2010a) Evolution of the air cavity during a depressurized wave impact II The dynamic field, Physics of Fluids, 22(5): 056102 Lugni, C., Miozzi, M., Brocchini, M and Faltinsen, O.M (2010b) Evolution of the air cavity during a depressurized wave impact I The kinematic flow field, Physics of Fluids, 22(5): 056101 186 References Lv, X., Zou, Q., Zhao, Y and Reeve, D (2010) A novel coupled level set and volume of fluid method for sharp interface capturing on 3D tetrahedral grids, Journal of Computational Physics, 229(7): 2573-2604 Malan, A.G., Lewis, R.W and Nithiarasu, P (2002) An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I Theory and implementation, International Journal for Numerical Methods in Engineering, 54(5): 695-714 Marsooli, R and Wu, W (2014) 3-D finite-volume model of dam-break flow over uneven beds based on VOF method, Advances in Water Resources, 70: 104-117 Ming, P.-J and Duan, W.-Y (2010) Numerical simulation of sloshing in rectangular tank with VOF based on unstructured grids, Journal of Hydrodynamics, Ser B, 22(6): 856-864 Molteni, D and Colagrossi, A (2009) A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH, Computer Physics Communications, 180(6): 861-872 Monaghan, J.J (1994) Simulating free surface flows with SPH, Journal of Computational Physics, 110(2): 399–406 Monaghan, J.J (2000) SPH without a tensile instability, Journal of Computational Physics, 159(2): 290-311 Monaghan, J.J and Rafiee, A (2013) A simple SPH algorithm for multi-fluid flow with high density ratios, International Journal for Numerical Methods in Fluids, 71(5): 537561 Morris, J.P., Fox, P.J and Zhu, Y (1997) Modeling low Reynolds number incompressible flows using SPH, Journal of Computational Physics, 136(1): 214–226 Nithiarasu, P (2003) An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows, International Journal for Numerical Methods in Engineering, 56(13): 1815-1845 187 References Oger, G., Doring, M., Alessandrini, B and Ferrant, P (2007) An improved SPH method: Towards higher order convergence, Journal of Computational Physics, 225(2): 14721492 Osher, S and Sethian, J.A (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of computational physics, 79(1): 12-49 Peregrine, D.H (2003) Water-wave impact on walls, Annual Review of Fluid Mechanics, 35(1): 23-43 Peregrine, D.H and Thais, L (2006) The effect of entrained air in violent water wave impacts, Journal of Fluid Mechanics, 325: 377- 397 Perrone, N and Kao, R (1975) A general finite difference method for arbitrary meshes, Computers & Structures, 5(1): 45-57 Plumerault, L.R., Astruc, D and Maron, P (2012) The influence of air on the impact of a plunging breaking wave on a vertical wall using a multifluid model, Coastal Engineering, 62: 62-74 Price, D.J (2012) Smoothed particle hydrodynamics and magnetohydrodynamics, Journal of Computational Physics, 231(3): 759-794 Price, W.G and Chen, Y.G (2005) A simulation of free surface waves for incompressible two-phase flows using a curvilinear level set formulation, International Journal for Numerical Methods in Fluids, 51(3): 305–330 Ramkema, C (1978) A model law for wave impacts on coastal structures Coastal Engineering Proceedings Randles, P.W and Libersky, L.D (1996) Smoothed Particle Hydrodynamics: Some recent improvements and applications, Computer Methods in Applied Mechanics and Engineering, 139(1-4): 375 - 408 Schmidt, R., Oumeraci, H and Partenscky, H.W (1992) Impact loads induced by plunging breakers on vertical structures 23rd International Conference on Coastal Engineering, Venice, 1545-1558 188 References Sethian, J.A and Smereka, P (2003) Level set methods for fluid interfaces, Annual Review of Fluid Mechanics, 35(1): 341-372 Shadloo, M.S., Zainali, A., Sadek, S.H and Yildiz, M (2011) Improved Incompressible Smoothed Particle Hydrodynamics method for simulating flow around bluff bodies, Computer Methods in Applied Mechanics and Engineering, 200(9-12): 1008-1020 Shakibaeinia, A and Jin, Y.-C (2012) MPS mesh-free particle method for multiphase flows, Computer Methods in Applied Mechanics and Engineering, 229-232: 13-26 Shao, S (2012) Incompressible smoothed particle hydrodynamics simulation of multifluid flows, International Journal for Numerical Methods in Fluids, 69(11): 17151735 Shao, S., Ji, C., Graham, D.I., Reeve, D.E., James, P.W and Chadwick, A.J (2006) Simulation of wave overtopping by an incompressible SPH model, Coastal Engineering, 53(9): 723-735 Shao, S and Lo, E.Y.M (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface, Advances in Water Resources, 26(7): 787800 Shibata, K and Koshizuka, S (2007) Numerical analysis of shipping water impact on a deck using a particle method, Ocean Engineering, 34(3-4): 585-593 Souto-Iglesias, A., Delorme, L., Pé rez-Rojas, L and Abril-Pé rez, S (2006) Liquid moment amplitude assessment in sloshing type problems with smooth particle hydrodynamics, Ocean Engineering, 33(11-12): 1462-1484 Su, T.C., Lou, Y.K., Flipse, J.E and Bridges, T.J., 1982 A numerical analysis of large amplitude liquid sloshing in baffled containers Tanaka, M and Masunaga, T (2010) Stabilization and smoothing of pressure in MPS method by Quasi-Compressibility, Journal of Computational Physics, 229(11): 42794290 189 References Tofighi, N and Yildiz, M (2013) Numerical simulation of single droplet dynamics in three-phase flows using ISPH, Computers & Mathematics with Applications, 66(4): 525536 Topliss, M.E., Cooker, M.J and Peregrine, D.H (1992) Pressure oscillations during wave impact on vertical walls, Coastal Engineering Proceedings, 1(23) Toro, E.F (2009) Riemann solvers and numerical methods for fluid dynamics: a practical introduction 3rd ed Heidelberg: Springer Tsui, Y.-Y., Lin, S.-W., Cheng, T.-T and Wu, T.-C (2009) Flux-blending schemes for interface capture in two-fluid flows, International Journal of Heat and Mass Transfer, 52(23-24): 5547-5556 Wang, C.Z and Khoo, B.C (2005) Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations, Ocean Engineering, 32(2): 107-133 Wang, C.Z., Meng, Q.C., Huang, H.C and Khoo, B.C (2013) Finite element analysis of nonlinear wave resonance by multiple cylinders in vertical motions, Computers & Fluids, 88: 557-568 Wang, Y., Shu, C., Huang, H.B and Teo, C.J (2015) Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio, Journal of Computational Physics, 280: 404-423 Wemmenhove, R., 2008 Numerical simulation of two-phase flow in offshore environments Ph.D Thesis University of Groningen Wood, D.J., Peregrine, D.H and Bruce, T (2000) Wave impact on a wall using pressure-impulse theory I: trapped air, Journal of Waterway, Port, Coastal, and Ocean Engineering, 126(4): 182-190 Wu, G.X., Ma, Q.W and Eatock Taylor, R (1998) Numerical simulation of sloshing waves in a 3D tank based on a finite element method, Applied Ocean Research, 20(6): 337-355 Xu, H., 2013 Numercial simulation of breaking wave impact on structures Ph.D Thesis National University of Singapore 190 References Xu, R., Stansby, P and Laurence, D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, Journal of Computational Physics, 228(18): 6703-6725 Yabe, T and Wang, P.-Y (1991) Unified numerical procedure for compressible and incompressible fluid, Journal of the Physical Society of Japan, 60(7): 2105-2108 Yabe, T., Xiao, F and Utsumi, T (2001) The constrained interpolation profile method for multiphase analysis, Journal of Computational Physics, 169(2): 556-593 Yamasaki, J., Miyata, H and Kanai, A (2005) Finite-difference simulation of green water impact on fixed and moving bodies, Journal of Marine Science and Technology, 10(1): 1-10 Yildiz, M., Rook, R.A and Suleman, A (2009) SPH with the multiple boundary tangent method, International Journal for Numerical Methods in Engineering, 77(10): 14161438 Yoon, H.Y., Koshizuka, S and Oka, Y (1999) A particle-gridless hybrid method for incompressible flows, International Journal for Numerical Methods in Fluids, 30(4): 407-424 Yoon, S.Y and Yabe, T (1999) The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method, Computer Physics Communications, 119(2-3): 149-158 Zainali, A., Tofighi, N., Shadloo, M.S and Yildiz, M (2013) Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method, Computer Methods in Applied Mechanics and Engineering, 254: 99-113 Zhang, S., Yue, D.K.P and Tanizawa, K (1996) Simulation of plunging wave impact on a vertical wall, Journal of Fluid Mechanics, 327: 221 - 254 Zhang, Y., Zou, Q and Greaves, D (2009) Numerical simulation of free-surface flow using the level-set method with global mass correction, International Journal for Numerical Methods in Fluids, 63(6): 651–680 191 References Zhou, Z.Q., Kat, J.O.D and Buchner, B (1999) A nonlinear 3D approach to simulate green water dynamics on deck Proc 7th Int Conf Num Ship Hydrod, Nantes, France 192 Appendix Appendix: Derivation for the polytropic gas law based on thermodynamics and ideal gas law The entropy S of a thermodynamic system is defined as follows  Q  dS    ,  T rev (A-1) where Q is the heat transfer of the system with its surroundings, T the temperature of the system and the subscript „rev‟ means a reversible process The inequality of Clausius, deduced from the Second Law of Thermodynamics, states that  Q T  (A-2) As a consequence of the second law, we have dS  Q T (A-3) For a reversible process, the equality of Equation (A-3) holds That is,  Q  TdS (A-4) In addition, the work done by the system in an infinitesimal process can be computed by W  pdV (A-5) Substituting Equations (A-4) and (A-5) into the First Law of Thermodynamics, which can be represented as  Q  dU  W , gives 193 Appendix TdS  dU  pdV , (A-6) where U is the internal energy of the system This equation is frequently called the Gibbs equation and can be written for a unit mass as follows Tds  du  pdv , (A-7) where v is called the specific volume The ideal gas law as shown in Equation (3-4) also can be written in the form as pv  RT (A-8) where the ideal gas constant R  c p  cv , and c p and cv are respectively the specific heats at constant pressure and volume The ratio of specific heats is defined as   c p cv The change of internal energy of an ideal gas is du  cv dT (A-9) Further assuming an adiabatic process, since  Q  , we have Tds  (A-10) Substituting Equations (A-8) to (A-10) into Equation (A-7) gives  dv dp   v p (A-11) Integrating by part of Equation (A-11) and substituting v=1  leads to p   constant , which is the polytropic gas law for isentropic processes 194 (A-12) ... such as wave impacts on coastal/offshore structures generate air entrapment The entrapped air may affect the amplitude and duration of impact pressure because of the air cushion effect Numerical... such as breakwater, oil platform, tension leg platforms and ships can lead to serious structural damage and instability As a result, the loads generated by wave impacts are among the most important... develop a new numerical method that can simulate violent wave impact processes with entrapped air pocket so as to achieve further insight into wave- impact processes and better prediction of impact