KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
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Development of 3-D Nonlinear Wave Driver Using SPH

SPH을 활용한 3차원 비선형 파랑모형 개발

  • 조용준 (서울시립대학교 토목공학과) ;
  • 김권수 (서울시립대학교 대학원 토목공학과)
  • Received : 2008.06.04
  • Accepted : 2008.08.12
  • Published : 2008.09.30
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Abstract

In this study, we newly proposed 3-D nonlinear wave driver utilizing the Navier-Stokes Eq. the numerical integration of which is carried out using SPH (Smoothed Particle Hydrodynamics), an internal wave generation with the source function of Gaussian distribution and an energy absorbing layer. For the verification of new 3-D nonlinear wave driver, we numerically simulate the sloshing problem within a parabolic water basin triggered by a Gaussian hump and uniformly inclined water surface by Thacker (1981). It turns out that the qualitative behavior of sloshing caused by relaxing the external force which makes a free surface convex or uniformly inclined is successfully simulated even though phase error is visible and an inundation height shrinks as numerical simulation more proceeds. For the more severe test, we also simulate the nonlinear shoaling and refraction over uniform beach of wedge shape. It is shown that numerically simulated waves are less refracted than the linear counterpart by Hamiltonian ray theory due to nonlinearity, energy dissipation at the bottom and side walls, energy loss induced by breaking, and the hydraulic jump occurring when breaking waves encounter a down-rush by the preceding wave.

Navier-Stokes식, Gaussian 분포형 용출함수를 이용한 내부조파, energy absorbing layer로 삼차원 파랑모형을 새롭게 구성하였다. Navier-Stokes식의 수치적분에는 정교한 수치기법인 SPH(Smoothed Particle Hydrodynamics)가 활용된다. 제안된 파랑모형의 검증은 삼차원 포물형 용기에서의 sloshing현상과 Thacker(1981)의 해석해를 토대로 수행되었다. 초기 수면 형상이 Gaussian hump인 경우와 일방향으로 경사진 경우에 대해 수치모의 하였다. 수치모의 결과 수면이 융기되도록 구속한 외부조건이 해제되면서 시작되는 자유진동의 정성적 거동은 비교적 정확히 모의되었으나 시간이 경과될수록 위상차, 침수선이 퇴각하는 등 초기 수면과는 상당히 다른 결과를 보였다. 최종적인 검증은 쐐기모양 해안에서의 비선형 천수, 굴절거동의 수치모의를 토대로 진행되었다. 수치모의 결과 굴절되는 양이 Hamiltonian ray theory가 제공하는 수치보다 전반적으로 작게 나타났다. 이러한 현상은 이상유체와 선형 이론에 기초한 Hamiltonian ray theory에서 간과된 비선형성, 점성으로 인한 양안과 저면에서의 에너지 감쇄, 쇄파 과정에 유동계에 도입되는 에너지 감쇄, 선행파랑에 의한 down-rush와 조우시 발생하는 도수 등에 기인하는 것으로 판단된다.

Keywords

References

  1. 조용준, 김권수, 유하상(2008) Swash 대역에서의 해빈표사 부유 거동에 관한 연구. 대한토목학회논문집, 대한토목학회, 제28권, 제1B호, pp. 95-109
  2. 조용준, 이헌(2007) Lagrangian Dynamic Smagorinsky 난류모형과 SPH를 이용한 쇄파역에서의 비선형천수거동에 관한 연구, 한국해안해양공학회지, 한국해안해양공학회, Vol. 19, No. 1, pp. 81-96
  3. 조용준, 김권수(2008) 제체의 갑작스런 붕괴로 인한 충격파 수치 해석 - SPH를 중심으로. 대한토목학회논문집, 대한토목학회, 제28권, 제3B호, pp. 261-270
  4. 조용준, 김민균(2007) Immersed water channel과 water chamber로 구성된 환경친화적 직립방파제의 해수교환효과와 파에너지 축출체로서의 가능성에 대하여. 대한토목학회논문집, 대한토목학회, 제27권, 제4B호, pp. 441-453
  5. Airy, G.B. (1845) Tides and waves. Encyclopedia Metropolitana, Art. 192, London
  6. Batchelor, G.K. (1967) An Introduction to fluid dynamics, Cambridge Univ. Press, Cambridge, UK
  7. Berkhoff, J.C.W. (1972) Computation of combined refraction-diffraction. Proceedings of the 13th International Conference on Coastal Engineering, Vancouver, ASCE , Vol. 1, pp. 472-490
  8. Boussinesq, J. (1871) Theorie de l'intumescence liquide, applelee onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus de l'Academie ded Sciences 72, pp. 755-759
  9. Chamberlain, P.G. and Porter, D. (1995) The modified mild-slope equation, J. Fluid Mech., Cambridge, U.K., Vol. 144, pp. 419-443 https://doi.org/10.1017/S0022112084001671
  10. Chandrasekara, C.N. and Cheung, K.F. (1997) Extended linear refraction-diffraction model, J. Waterway., Port, Coast., and Ocean Eng., Vol. 123, No. 5, pp. 280-286 https://doi.org/10.1061/(ASCE)0733-950X(1997)123:5(280)
  11. Dalrymple, R.A. and Knio, O. (2000) SPH Modelling of water waves. Proc. Coastal Dynm., Lund 2000
  12. Dalrymple, R.A. and Rogers, B.D. (2006) Numerical modeling of water waves with the SPH method, Coastal Engineering Vol. 53, pp. 141-147 https://doi.org/10.1016/j.coastaleng.2005.10.004
  13. 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, Vol. 181, pp. 375-389 https://doi.org/10.1093/mnras/181.3.375
  14. Gomez-Gesteira, M. and Dalrymple, R.A. (2004) Using a 3D SPH method for wave impact on a tall structure, J. Waterway, Port, Coastal, and Ocean Engineering, Ocean Engineering 130, pp. 63-69 https://doi.org/10.1061/(ASCE)0733-950X(2004)130:2(63)
  15. Guazzelli, E., Rey, V., and Belzons, M. (1992) Higher-order Bragg reflection of gravity surface waves by periodic beds, J. Fluid Mech., Vol. 245, pp. 301-317 https://doi.org/10.1017/S0022112092000478
  16. Israeli, M. and Orszag, S.A. (1981). Approximation of radiation boundary condition, J. Comp. Phys. Vol. 41, pp. 113-115
  17. Komar, P.D. (1976) Beach processs and sedimentation, Prentice Hall
  18. Korteweg, D.J. and de Vries, H. (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philosophical Magazine 39, pp. 422-443 https://doi.org/10.1080/14786449508620739
  19. Larsen, J. and Dancy, H. (1983) Open boundaries in short-wave simulations-a new approach, Coast. Eng., Vol. 7, pp 285-296 https://doi.org/10.1016/0378-3839(83)90022-4
  20. Lin, P.Z., Liu and Philip, L.F. (1998) A numerical study of breaking waves in the surf zone. J. Fluid Mech., Vol. 359, pp. 239-264 https://doi.org/10.1017/S002211209700846X
  21. Lucy, L.B. (1977) A numerical approach to testing the fission hypothesis. Astronomical Journal 82, pp. 1013-1024 https://doi.org/10.1086/112164
  22. Massel, S.R. (1993) Extended refraction-diffraction equation for surface waves, Coast. Eng., Vol. 19, pp. 97-126 https://doi.org/10.1016/0378-3839(93)90020-9
  23. Monaghan, J.J. (1992) Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics 30, pp. 543-574 https://doi.org/10.1146/annurev.aa.30.090192.002551
  24. Monaghan, J.J. (1994) Simulating free surface flows with SPH. Journal of Computational Physics 110, pp. 399-406 https://doi.org/10.1006/jcph.1994.1034
  25. Monaghan, J.J. and Kos, A. (1999) Solitary waves on a Cretan beach. Journal of Waterway Port, Coastal and Ocean Engineering 125, pp. 145-154 https://doi.org/10.1061/(ASCE)0733-950X(1999)125:3(145)
  26. Monaghan, J.J. and Kos, A. (2000) Scott Russell's wave generator. Phys. Fluids, 12, pp. 622-630 https://doi.org/10.1063/1.870269
  27. Morris, J.P., Fox, P.J., and Zhu, Y. (1997) Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics 136, pp. 214-226 https://doi.org/10.1006/jcph.1997.5776
  28. Nadaoka, K., Hino, M., and Koyano, Y. (1988) Structure of the turbulent flow field under breaking waves in the surf zone. J. Fluid Mech. 204, pp. 359-387
  29. Nwogu, O. (1993) An alternative form of the Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coast. and Ocean Eng. Vol. 119, pp. 618-638 https://doi.org/10.1061/(ASCE)0733-950X(1993)119:6(618)
  30. Peregrine, D.H. (1967) Long waves on beaches. J. Fluid Mach., Vol. 27, pp. 815-827 https://doi.org/10.1017/S0022112067002605
  31. Rayleigh (1876) On waves. Philosophical Magazine 1, pp. 257-279 https://doi.org/10.1080/14786447608639037
  32. Russel, J.S. (1844) Report on Waves. 14th meeting of the British Association for the Advancement of Science, pp. 311-390
  33. Smagorinsky, J. (1963) General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev., Vol. 91, pp. 99-164 https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
  34. Stokes, G.G. (1847) On the theory of oscillatory waves. Transactions of the Cambridge philosophical society 8, pp. 441-455
  35. Takeda, H., Shoken, M.M., and Minoru, Sekiya (1994) "Numerical simulation of viscous flow by smoothed particle hydrodynamics." Progress in Theoretical Physics 92, pp. 939-959 https://doi.org/10.1143/PTP.92.939
  36. Thacker, W.C. (1981) Some exact solutions to the nonlinear shallow water wave equations. Journal of Fluid Mechanics 107, pp. 499-508 https://doi.org/10.1017/S0022112081001882
  37. Ubbink, O. (1997) Numerical prediction of two fluid systems with sharp interfaces. Ph. D. Thesis, Imperial College of Science, Technology and Medicine, London
  38. Wei, G. and Kirby, J.T. (1995) Time-dependent numerical code for extended Boussinesq equations. J. Waterway, Port, Coast. and Ocean Eng. Vol. 121, pp. 251-261 https://doi.org/10.1061/(ASCE)0733-950X(1995)121:5(251)
  39. Wei, G., Kirby, J.T., and Sinha, A. (1999) Generation of waves in Boussinesq model using a source function method. Coastal Engineering 36, pp. 271-299 https://doi.org/10.1016/S0378-3839(99)00009-5
  40. Yoshizawa, A. (1986) Statistical theory for compressible turbulent shear flows with application to sub-grid modeling, Physics of Fluids A 29, pp. 2152-2164 https://doi.org/10.1063/1.865552