Advertisement

Simulation of breaking waves on slope beaches integrating the MPS method into Iwagaki wave theory

  • L. E. Jaime-Ledezma
  • J. Sanchez-MondragonEmail author
  • A. O. Vazquez-Hernandez
  • J. A. Morales-Viscaya
  • G. Ochoa-Ruiz
Technical Paper
  • 9 Downloads

Abstract

Breaking waves are a natural phenomenon that can affect infrastructure located near the coast and cause soil erosion problems as well. For this reason, it has been investigated through experiments and numerical simulations for decades. With the advancement of new numerical approaches such as the moving particle semi-implicit (MPS) method that allows a better modeling of the fragmentation process, it is possible now to adequately represent its complex behavior on breaking waves. In this work we apply the MPS method to simulate breaking waves in a slope beach, which is simulated as a two-dimensional numerical tank, where cnoidal waves are generated by using Iwagaki wave theory. To achieve numerical wave generation, we first used an approximation through linear wave theory using piston, flap and a hyperbolic wavemakers with a 1.25-s wave period. Then, a numerical tank was modeled using nonlinear wave theory to simulate a series of different wave periods. Our results show that through the MPS method it is possible to represent the type of wave breaking defined analytically, as well as to determine the predicted wave profile and the momentum for the breaking wave, in the same way wave velocity and pressure under the wave can be approximated to their theoretical values.

Keywords

Wave breaking MPS method Slope beach Iwagaki wave theory 

Notes

Acknowledgements

The author J. Sanchez-Mondragon thanks to Dirección de Cátedras CONACYT for the financial support granted during the research included in this manuscript.

References

  1. 1.
    Ataie-Ashtiani B, Shobeyri G, Farhadi L (2006) Modified incompressible SPH method for simulating free surface problems. Fluid Dyn Res 40(9):637–661.  https://doi.org/10.1016/j.fluiddyn.2007.12.001 CrossRefzbMATHGoogle Scholar
  2. 2.
    Battjes JA (1974) Surf similarity. In: Proceedings of the 14th international conference on coastal engineering, vol 1.  https://doi.org/10.1061/9780872621138.029
  3. 3.
    Bonet J, Kulasegaram S, Rodriguez-Paz MX, Profit M (2004) Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems. Comput Methods Appl Mech Eng 193(12–14):1245–1256.  https://doi.org/10.1016/j.cma.2003.12.018 CrossRefzbMATHGoogle Scholar
  4. 4.
    Bonet J, Lok TS (1999) Variational and momentum preservation aspects of smooth particle hydrodynamic formulation. Comput Methods Appl Mech Eng 180(1–2):97–115.  https://doi.org/10.1016/S0045-7825(99)00051-1 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen X, Xi G, Sun ZG (2014) Improving stability of MPS method by computational scheme based on conceptual particles. Comput Methods Appl Mech Eng 278:254–271.  https://doi.org/10.1016/j.cma.2014.05.023 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christensen ED, Deigaard R (2001) Large eddy simulation of breaking waves. Coast Eng 42:53–86.  https://doi.org/10.1016/S0378-3839(00)00049-1 CrossRefGoogle Scholar
  7. 7.
    Dalrymple RA, Rogers BD (2006) Numerical modeling of water waves with the SPH method. Coast Eng 53(2):141–147.  https://doi.org/10.1016/j.coastaleng.2005.10.004 CrossRefGoogle Scholar
  8. 8.
    Edmond YML, Shao S (2002) Simulation of near-shore solitary wave mechanics by an incompressible SPH method. App Ocean Res 24(5):275–286.  https://doi.org/10.1016/S0141-1187(03)00002-6 CrossRefGoogle Scholar
  9. 9.
    Fadafan MA, Hessami-Kermani MR (2017) Moving particle semi-implicit method with improved pressures stability properties. J Hydroinf 20(6):1268–1285.  https://doi.org/10.2166/hydro.2017.121 CrossRefGoogle Scholar
  10. 10.
    Galvin CJ Jr (1968) Breaker type classification on three laboratory beaches. J Geophys Res 73(12):3651–3659.  https://doi.org/10.1029/JB073i012p03651 CrossRefGoogle Scholar
  11. 11.
    Gómez-Gesteira M, Cerqueiro D, Crespo C, Dalrymple RA (2005) Green water overtopping analyzed with a SPH model. Ocean Eng 32(2):223–238.  https://doi.org/10.1016/j.oceaneng.2004.08.003 CrossRefGoogle Scholar
  12. 12.
    Gotoh H (2009) Lagrangian particle method as advanced technology for numerical wave flume. Int J Offshore Polar Eng 19(3):161–167Google Scholar
  13. 13.
    Gotoh H, Ikari H, Memita T, Sakai T (2005) Lagrangian particle method for simulation of wave overtopping on a vertical seawall. Coast Eng J 47(2–3):157–181.  https://doi.org/10.1142/S0578563405001239 CrossRefGoogle Scholar
  14. 14.
    Gotoh H, Khayyer A (2016) Current achievements and future perspectives for projection-based particle methods with applications in ocean engineering. J Ocean Eng Mar Energy 2(3):251–278.  https://doi.org/10.1007/s40722-016-0049-3 CrossRefGoogle Scholar
  15. 15.
    Gotoh H, Khayyer A (2018) On the state-of-the-art of particle methods for coastal and ocean engineering. Coast Eng J 60(1):79–103.  https://doi.org/10.1080/21664250.2018.1436243 CrossRefGoogle Scholar
  16. 16.
    Gotoh H, Khayyer A, Ikari H, Arikawa T, Shimosako K (2014) On enhancement of Incompressible SPH method for simulation of violent sloshing flows. Appl Ocean Res 46:104–115.  https://doi.org/10.1016/j.apor.2014.02.005 CrossRefGoogle Scholar
  17. 17.
    Gotoh H, Okayasu A (2017) Computational wave dynamics for innovative design of coastal structures. Proc Jpn Acad Ser B 93(8):525–546.  https://doi.org/10.2183/pjab.93.034 CrossRefGoogle Scholar
  18. 18.
    Gotoh H, Sakai T (2006) Key issues in the particle method for computation of wave breaking. Coast Eng 53(2):171–179.  https://doi.org/10.1016/j.coastaleng.2005.10.007 CrossRefGoogle Scholar
  19. 19.
    Grilli ST, Guyenne P, Dias F (2001) A fully nonlinear model for three-dimensional overturning waves over an arbitrary bottom. Int J Numer Methods Fluids 35(7):829–867.  https://doi.org/10.1002/1097-0363(20010415)35:7<829::AID-FLD115>3.0.CO;2-2 CrossRefzbMATHGoogle Scholar
  20. 20.
    Gui Q, Dong P, Shao S (2015) Numerical study of PPE source term errors in the incompressible SPH models. Int J Numer Methods Fluids 77(6):358–379.  https://doi.org/10.1002/fld.3985 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hedges TS, Kirkgöz MS (1981) An experimental study of the transformation zone of plunging breakers. Coast Eng 4:319–333.  https://doi.org/10.1016/0378-3839(80)90026-5 CrossRefGoogle Scholar
  22. 22.
    Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. J Comput Phys 227(1):264–278.  https://doi.org/10.1016/j.jcp.2007.07.013 CrossRefzbMATHGoogle Scholar
  23. 23.
    Ikeda H, Koshizuka S, Oka Y, Park HS, Sugimoto J (2001) Numerical analysis of jet injection behavior for fuel-coolant interaction using particle method. J Nucl Sci Technol 38(3):174–182.  https://doi.org/10.1080/18811248.2001.9715019 CrossRefGoogle Scholar
  24. 24.
    Iwagaki Y (1965) Studies on cnoidal waves (second report)—on the wave velocity and wave length. Disaster Prev Res Inst Annu 8:343–351 (in Japanese)Google Scholar
  25. 25.
    Iwagaki Y (1967) Studies on cnoidal waves (fourth report)—on hyperbolic waves(l). Disaster Prev Res Inst Annu 10B:283–294 (in Japanese)Google Scholar
  26. 26.
    Iwagaki Y (1968) Hyperbolic waves and their shoaling. In: Proceedings of the 11th international conference on coastal engineering, London, United Kingdom, September.  https://doi.org/10.1061/9780872620131.009
  27. 27.
    Iwagaki Y, Sakai T (1967) On the shoaling of finite amplitude waves. In: Proceedings of the 14th international conference on coastal engineering, Japan (in Japanese) Google Scholar
  28. 28.
    Khayyer A, Gotoh H (2008) Development of CMPS method for accurate water-surface tracking in breaking waves. Coast Eng J 50(2):179–207.  https://doi.org/10.1142/S0578563408001788 CrossRefGoogle Scholar
  29. 29.
    Khayyer A, Gotoh H (2009) Modified moving particle semi-implicit methods for the prediction of 2D wave impact pressure. Coast Eng 56(4):419–440.  https://doi.org/10.1016/j.coastaleng.2008.10.004 CrossRefGoogle Scholar
  30. 30.
    Khayyer A, Gotoh H (2010) A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method. App Ocean Res 32(1):124–131.  https://doi.org/10.1016/j.apor.2010.01.001 CrossRefGoogle Scholar
  31. 31.
    Khayyer A, Gotoh H (2010) On particle-based simulation of a dam break over a wet bed. J Hydraul Res 48(2):238–249.  https://doi.org/10.1080/00221681003726361 CrossRefGoogle Scholar
  32. 32.
    Khayyer A, Gotoh H (2011) Enhancement of stability and accuracy of the moving particle semi-implicit method. J Comput Phys 230(8):3093–3118.  https://doi.org/10.1016/j.jcp.2011.01.009 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Khayyer A, Gotoh H, Shao SD (2008) Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coast Eng 55(3):236–250.  https://doi.org/10.1016/j.coastaleng.2007.10.001 CrossRefGoogle Scholar
  34. 34.
    Khayyer A, Gotoh H, Shao S (2009) Enhanced predictions of wave impact pressure by improved Incompressible SPH methods. App Ocean Res 31(2):111–131.  https://doi.org/10.1016/j.apor.2009.06.003 CrossRefGoogle Scholar
  35. 35.
    Khayyer A, Gotoh H, Shimizu Y (2019) A projection-based particle method with optimized particle shifting for multiphase flows with large density ratios and discontinuous density fields. Comput Fluids 179:356–371.  https://doi.org/10.1016/j.compfluid.2018.10.018 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Khayyer A, Gotoh H, Shimizu Y, Gotoh K (2017) On enhancement of energy conservation properties of projection-based particle methods. Eur J Mech B Fluids 66:20–37.  https://doi.org/10.1016/j.euromechflu.2017.01.014 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kolahdoozan M, Ahadi MS, Shirazpoor S (2014) Effect of turbulence closer models on the accuracy of MPS method for the viscous free surface flow. Sci Iran 21(4):1217–1230Google Scholar
  38. 38.
    Kondo M, Koshizuka S (2011) Improvement of stability in moving particle semi-implicit method. Int J Numer Method Fluids 65(6):638–654.  https://doi.org/10.1002/fld.2207 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Koshizuka S, Nobe A, Oka Y (1998) Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J Numer Method Fluids 26:751–769.  https://doi.org/10.1002/(SICI)1097-0363(19980415)26:7<751::AID-FLD671>3.0.CO;2-C CrossRefzbMATHGoogle Scholar
  40. 40.
    Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434.  https://doi.org/10.13182/NSE96-A24205 CrossRefGoogle Scholar
  41. 41.
    Koshizuka S, Tamako H, Oka Y (1995) A particle method for incompressible viscous flow with fluid fragmentation. Comput Fluid Dyn J 4(1):29–46Google Scholar
  42. 42.
    Laitone EV (1960) The second approximation to cnoidal and solitary waves. J Fluid Mech 9:430–444.  https://doi.org/10.1017/S0022112060001201 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Laitone EV (1962) Limiting conditions for cnoidal and Stokes waves. J Geophys Res 67:1555–1564.  https://doi.org/10.1029/JZ067i004p01555 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lee BH, Park JC, Kim MH, Hwang SC (2011) Step-by-step improvement of MPS method in simulating violent free-surface motions and impact-loads. Comput Methods Appl Mech Eng 200:1113–1125.  https://doi.org/10.1016/j.cma.2010.12.001 CrossRefzbMATHGoogle Scholar
  45. 45.
    Li D, Sun Z, Chen X, Xi G, Liu L (2015) Analysis of wall boundary in moving particle semi-implicit method and a novel model of fluid-wall interaction. Int J Comput Fluid Dyn 29(3–5):199–214.  https://doi.org/10.1080/10618562.2015.1028924 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Lin P, Liu PLF (1998) A numerical study of breaking waves in the surf zone. J Fluid Mech 359:239–264.  https://doi.org/10.1017/S002211209700846X CrossRefzbMATHGoogle Scholar
  47. 47.
    Lind SJ, Xu R, Stansby PK, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231(4):1499–1523.  https://doi.org/10.1016/j.jcp.2011.10.027 MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Liu X, Lin P, Shao S (2014) ISPH wave simulation by using an internal wave maker. Coast Eng 95:160–170.  https://doi.org/10.1016/j.coastaleng.2014.10.007 CrossRefGoogle Scholar
  49. 49.
    Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30:543–574.  https://doi.org/10.1146/annurev.aa.30.090192.002551 CrossRefGoogle Scholar
  50. 50.
    Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399–406.  https://doi.org/10.1006/jcph.1994.1034 MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. J Comput Phys 136(1):214–226.  https://doi.org/10.1006/jcph.1997.5776 CrossRefzbMATHGoogle Scholar
  52. 52.
    Naito S, Minoura M (1994) Research on element wavemakers and wave field generated by their combination. In: Proceedings of the 4th international journal of offshore and polar engineering conference, Osaka, Japan, April 10–15. ISOPE-I-94-183Google Scholar
  53. 53.
    Ng KC, Hwang YH, Sheu TWH (2014) On the accuracy assessment of Laplacian models in MPS. Comput Phys Commun 185(10):2412–2426.  https://doi.org/10.1016/j.cpc.2014.05.012 CrossRefzbMATHGoogle Scholar
  54. 54.
    Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method: towards higher order convergence. J Comput Phys 225(2):1472–1492.  https://doi.org/10.1016/j.jcp.2007.01.039 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Pu JH, Shao S (2012) Smoothed particle hydrodynamics simulation of wave overtopping characteristics for different coastal structures. Sci World J Article ID 163613:1–10.  https://doi.org/10.1100/2012/163613
  56. 56.
    Sakakiyama T (1996) Numerical simulation of nonlinear wave over permeable submerged breakwater. In: Proceedings of the ASME fluids engineering division conference, vol 3Google Scholar
  57. 57.
    Sakakiyama T, Kajima R (1992) Numerical simulation of nonlinear wave interacting with permeable breakwaters. In: Proceedings of the 23rd international conference on coastal engineering, Venice, Italy, October 4–9.  https://doi.org/10.1061/9780872629332.115
  58. 58.
    Sanchez-Mondragon J (2016) On the stabilization of unphysical pressure oscillations in MPS method simulations. Int J Numer Methods Fluids 82:471–492.  https://doi.org/10.1002/fld.4227 MathSciNetCrossRefGoogle Scholar
  59. 59.
    Sanchez-Mondragon J, Vazquez-Hernandez O (2018) Solitary waves collisions by double-dam-broken simulations with the MPS method. Eng Comput 35(1):53–70.  https://doi.org/10.1108/EC-04-2016-0142 CrossRefGoogle Scholar
  60. 60.
    Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31(1):567–603.  https://doi.org/10.1146/annurev.fluid.31.1.567 MathSciNetCrossRefGoogle Scholar
  61. 61.
    Shadloo MS, Weiss R, Yildiz M, Dalrymple RA (2015) Numerical simulation of long wave run-up for breaking and non-breaking waves. Int J Offshore Polar Eng 25(1):1–7Google Scholar
  62. 62.
    Shakibaeinia A, Jin YC (2010) A weakly compressible MPS method for modeling of open-boundary free-surface flow. Int J Numer Methods Fluids 63(10):1208–1232.  https://doi.org/10.1002/fld.2132 MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Shao SD, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800.  https://doi.org/10.1016/S0309-1708(03)00030-7 CrossRefGoogle Scholar
  64. 64.
    Shao A, Ji C, Graham DI, Reeve DE, James PW, Chadwick AJ (2006) Simulation of wave overtopping by an incompressible SPH model. Coast Eng 53:723–735.  https://doi.org/10.1016/j.coastaleng.2006.02.005 CrossRefGoogle Scholar
  65. 65.
    Shibata K, Masaie I, Kondo M, Murotani K, Koshizuka S (2015) Improved pressure calculation for the moving particle semi-implicit method. Comput Part Mech 2(1):91–108.  https://doi.org/10.1007/s40571-015-0039-6 CrossRefGoogle Scholar
  66. 66.
    Skillen A, Lind S, Stansby PK, Rogers BD (2013) Incompressible smoothed particle hydrodynamics (SPH) with reduced temporal noise and generalised Fickian smoothing applied to body-water slam and efficient wave-body interaction. Comput Methods Appl Mech Eng 265:163–173.  https://doi.org/10.1016/j.cma.2013.05.017 MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Sueyoshi M, Kashiwagi M, Naito S (2008) Numerical simulation of wave-induced nonlinear motions of a two-dimensional floating body by the moving particle semi-implicit method. J Mar Sci Technol 13:85–94.  https://doi.org/10.1007/s00773-007-0260-y CrossRefGoogle Scholar
  68. 68.
    Suzuki Y, Koshizuka S, Oka Y (2007) Hamiltonian moving-particle semi-implicit (HMPS) method for incompressible fluid flows. Comput Methods Appl Mech Eng 196(29–30):2876–2894.  https://doi.org/10.1016/j.cma.2006.12.006 MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Tanaka M, Masunaga T (2010) Stabilization and smoothing of pressure in MPS method by quasi-compressibility. J Comput Phys 229:4279–4290.  https://doi.org/10.1016/j.jcp.2010.02.011 CrossRefzbMATHGoogle Scholar
  70. 70.
    Tsuruta N, Khayyer A, Gotoh H (2013) A short note on dynamic stabilization of moving particle semi-implicit method. Comput Fluids 82:158–164.  https://doi.org/10.1016/j.compfluid.2013.05.001 MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Watanabe Y, Saeki H (1999) Three-dimensional large eddy simulation of breaking waves. Coast Eng J 41:281–301.  https://doi.org/10.1142/S0578563499000176 CrossRefGoogle Scholar
  72. 72.
    Xu T, Jin YC (2016) Improvements for accuracy and stability in a weakly-compressible particle method. Comput Fluids 137:1–14.  https://doi.org/10.1016/j.compfluid.2016.07.014 MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Yoon HY, Koshizuka S, Oka Y (1999) A particle-gridless hybrid method for incompressible flows. Int J Numer Methods Fluids 30(4):407–424.  https://doi.org/10.1002/(SICI)1097-0363(19990630)30:4<407::AID-FLD846>3.0.CO;2-C CrossRefzbMATHGoogle Scholar
  74. 74.
    Zheng J, Soe MM, Zhang C, Hsu TW (2010) Numerical wave flume with improved smoothed particle hydrodynamics. J Hydrodyn Ser B 22(6):773–781.  https://doi.org/10.1016/S1001-6058(09)60115-3 CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.CONACYT Research Fellow - Instituto Mexicano del PetróleoEje Central Lázaro CárdenasMexico CityMexico
  2. 2.Instituto Tecnológico de la PazLa PazMexico
  3. 3.Lloyds RegisterMexico CityMexico
  4. 4.Facultad de IngenieríaUniversidad Autónoma de GuadalajaraZapopanMexico

Personalised recommendations