Различные подходы к моделированию морских волн

Кратко рассмотрены главные подходы в прямом моделировании поверхностных волн, основанные на полных уравнениях динамики невязкой жидкости со свободной поверхностью. Большая часть моделей предназначена для исследования прикладных и инженерных проблем. Предполагается, что основной является модель, записанная в криволинейной системе координат, в которой высота отсчитывается от волновой поверхности. В двумерной периодической формулировке при использовании конформной системы задача сводится к системе одномерных уравнений, легко решаемых с использованием Фурье метода. Для трёхмерных волн такие упрощения не существуют, и вертикальная скорость на поверхности рассчитывается путём решения трёхмерного уравнения Пуассона или методом поверхностного интеграла. Рассматривается приближённая схема, основанная на двухмерных уравнениях. Схема позволяет воспроизводить статистический режим волн с высокой точностью согласующийся с аналогичными результатами, полученными по точной трёхмерной модели.

1. Dysthe K.B. Note on a modification to the nonlinear Schrӧdinger equation for application to deep water waves // Proc. R. Soc. Lond. 1979. A 369. P. 105–114.

2. Chalikov D., Sheinin D. Direct modeling of one-dimensional nonlinear potential waves. Nonlinear ocean waves // Advances in Fluid Mechanics (Perrie W. (ed)). 1998. Vol. 17. P. 207–258.

3. Engsig-Karup A.P., Harry B., Bingham H.B., Lindberg O. An efficient flexible-order model for 3D nonlinear water waves // J. Comput. Physics. 2009. 228(6). P. 2100–2118. doi: 10.1016/j.jcp.2008.11.028

4. Engsig-Karup A., Madsen M., Glimberg S. A massively parallel GPU‐accelerated mode for analysis of fully nonlinear free surface waves // International Journal for Numerical Methods in Fluids. 2012. 2675. doi: 10.1002/fld.2675

5. Causon D.M., Mingham C.G., Qian L. Developments in multi-fluid finite volume free surface capturing methods // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 397–427. doi: 10.1142/9789812836502_0011

6. Ma Q.W., Yan S. Qale-FEM method and its application to the simulation of free responses of floating bodies and overturning waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 165–202. doi: 10.1142/9789812836502_0005

7. Greaves D. Application of the finite volume method to the simulation of nonlinear water waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 357–396.

8. Grue J., Fructus D. Model for fully nonlinear ocean wave simulations derived using fourier inversion of integral equations in 3D // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 1–42.

9. Ducrozet G., Bonnefoy F., Le Touzé D., Ferrant P. 3-D HOS simulations of extreme waves in open seas // Nat. Hazards Earth Syst. Sci. 2007. Vol. 7. P. 109–122. doi: 10.5194/nhess-7-109-2007

10. Ducrozet G., Bingham H.B., Engsig-Karup A.P., Bonnefoy F., Ferrant P. A comparative study of two fast nonlinear freesurface water wave models // Int. J. Numer. Meth. Fluids. 2012. Vol. 69. P. 1818–1834.

11. Ducrozet G., Bonnefoy F., Le Touzé D., Ferrant P. HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method // Comp. Phys. Comm. 2016. Vol. 203. P. 245–254. doi: 10.1016/j.cpc.2016.02.017

12. Touboul J., Kharif C. Two-dimensional direct numerical simulations of the dynamics of rogue waves under wind action // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 43–74.

13. Dalrymple R.A., Gómez-Gesteira M., Rogers B.D., Panizzo A., Zou S., Crespo A.J., Cuomo G., Narayanaswamy M. Smoothed particle hydrodynamics for water waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 465– 495.

14. Issa R, Violeau D, Lee E.-S., Flament H. Modelling nonlinear water waves with RANS and LES SPH models // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 497–537.

15. Lubin P., Caltagirone J.-P. Large eddy simulation of the hydrodynamics generated by breaking waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 575–604.

16. Kim K.S., Kim M.H., Park J.C. Development of MPS (Moving Particle Simulation) method for Multi-liquidlayer Sloshing // Journal of Mathematical Problems in Engineering. 2014. Vol. 2014. Article ID350165, 13 p. doi: 10.1155/2014/350165

17. Zhao X., Liu B.-J., Liang S.-X., Sun Z.-C. Constrained interpolation profile (CIP) method and its application // Chuan Bo Li Xue/Journal of Ship Mechanics. 2016. Vol. 20(4). P. 393–402. doi: 10.3969/j.issn.1007–7294.2016.04.002

18. Young D.-L., Wu N.-J., Tsay T.-K. Method of fundamental solutions for fully nonlinear water waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 325–355.

19. Ma Q.W., Yan S. Qale-FEM method and its application to the simulation of free responses of floating bodies and overturning waves // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 165–202.

20. Harlow F.H., Welch E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface // Phys. Fluids. 1965. Vol. 8. P. 2182–2189.

21. Noh W.F., Woodward P. SLIC (simple line interface calculation) // Lecture Notes in Physics. Springer-Verlag, New York, 1967. Vol. 59. P. 330–340.

22. Hirt C.W., Nichols B.D. Volume of fluid method for the dynamics of free surface // J. Comput. Phys. 1981. Vol. 39. P. 201–225.

23. Prosperetti A., Jacobs J.W. A numerical method for potential flow with free surface // J. Comp. Phys. 1983. Vol. 51. P. 365–386.

24. Miyata H. Finite-difference simulation of breaking waves // 1986. J. Comput. Phys. Vol. 5. P. 179–214.

25. Longuet-Higgins M.S., Cokelet E.D. The deformation of steep surface waves on water. I. A numerical method of computation // Proc. R. Soc. Lond. 1976. Vol. 350. P. 1–26.

26. Chalikov D.V. Numerical simulation of wind-wave interaction // Journal of Fluid Mechanics. 1978. Vol. 87. P. 561–582.

27. Dommermuth D.G. The laminar interactions of a pair of vortex tubes with a free surface // Journal of Fluid Mechanics. 1993. Vol. 246. P. 91–115.

28. Thompson J.F., Warsi Z.U., Mastin C.W. Boundary-fitted coordinate systems for numerical solution of partial differential equations. A review // J. Comput. Phys. 1982. Vol. 47. P. 1–108.

29. Fritts M.J., Meinhold M.J., von Kerczek C.H. The calculation of nonlinear bow waves // Proceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, Netherlands, 1988, P. 485–497.

30. Farmer J., Martinelli L., Jameson A. A Fourier method for solving nonlinear water-wave problems: application to solitarywave interactions // Journal of Fluid Mechanics. 1993. Vol. 118. P. 411–443.

31. Whitney J.C. The numerical solution of unsteady free-surface flows by conformal mapping // Proc. Second Inter. Conf. on Numer. Fluid Dynamics (ed. M. Holt), Springer-Verlag, 1971, P. 458–462.

32. Ovsyannikov L.V. Dynamics of liquid continuum. Trudy Inst. of Fluid Mechanics. Siberian Branch of Russian Academy of Science. 15, 104–125, 1973

33. Kano T., Nishida T. Sur le ondes de surface de l’eau avec une justification mathematique des equations des ondes en eau peu profonde // J. Math. Kyoto Univ. (JMKYAZ), 1979. 19–2. P. 335–370.

34. Fornberg B. A numerical method for conformal mapping. SIAM // J. Sci. Comput. 1980. Vol. 1. P. 386–400.

35. Tanveer S. Singularities in water waves and Rayleigh-Taylor instability // Proc. R. Soc. Lond. 1991. A435. P. 137–158.

36. Tanveer S. Singularities in the classical Rayleigh-Taylor flow: formation and subsequent motion // Proc. R. Soc. Lond. 1993. A441. P. 501–525.

37. Chalikov D., Sheinin D. Numerical simulation of surface waves based on equations of potential wave dynamics // Proc. ONR, Ocean Waves Workshop. 1994. Tucson, Arizona, 1994.

38. Chalikov D., Sheinin D. The Numerical Investigation of Wavenumber-Frequency Spectrum for 1-D Nonlinear Waves // WMO/ISCU, CAS/JSC Working Group on Numerical Experimentation, Report #19. WMO/TD-No.592, 1994.

39. Chalikov D., Sheinin D. Numerical investigation of wavenumber-frequency spectrum for 1-D nonlinear waves // Ocean Sciences Meeting, Feb.21–25, San-Diego, CA. Abstract in: EOS, transactions (supplement), AGU, 75, #3, p.100, 1994.

40. Chalikov D., Sheinin D. Numerical modeling of surface waves based on principal equations of potential wave dynamics // Technical Note. NOAA/NCEP/OMB, 1996. 54 p

41. Chalikov D., Sheinin D. Direct modeling of one-dimensional nonlinear potential waves // Nonlinear Ocean Waves, ed. W. Perrie, Advances in Fluid Mechanics. 1998. Vol. 17. P. 207–258.

42. Sheinin D., Chalikov D. Hydrodynamical modeling of potential surface waves // Problems of hydrometeorology and environment on the eve of XXI century. Proceedings of international theoretical conference, St. Petersburg, June 24– 25, 1999. St.-Petersburg, Hydrometeoizdat, 2001, 305–337.

43. Zakharov V.E., Dyachenko A.I., Vasilyev O.A. New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface // European Journ. of Mech, B/Fluids. 2002. V. 21. P. 283–291.

44. Zakharov V.E., Dyachenko A.I., Prokofiev A.O. Freak waves as nonlinear stage of Stokes wave modulation instability // European Journal of Mechanics, B/Fluids. 2006. Vol.

45. P. 677–692. 45. Dold J.W. An efficient surface-integral algorithm applied to unsteady gravity waves // Journal of Comp. Phys. 1992. Vol. 103. P. 90–115.

46. Crapper G.D. Introduction to water waves. Wiley, Chichester, 1984. 224 p.

47. Yuen H.C., Lake B.M. Nonlinear dynamics of deep-water gravity waves // Adv. in Appl. Mech. 1982. Vol. 22. P. 67–229. doi: 10.1016/S0065-2156(08)70066-8

48. Chalikov D., Sheinin D. Modeling of Extreme Waves Based on Equations of Potential Flow with a Free Surface // J. Comp. Phys. 2005. Vol. 210. P. 247–273.

49. Chalikov D. Statistical properties of nonlinear one-dimensional wave fields // Nonlinear Processes in Geophysics. 2005. Vol. 12. P. 1–19.

50. Agnon Y., Babanin A.V., Young I.R., Chalikov D. Fine scale inhomogeneity of wind-wave energy input, skewness, and asymmetry // Geophys. Res. Lett. 2005. Vol. 32. doi: 10.1029/2005GL022701

51. Chalikov D. Numerical simulation of Benjamin-Feir instability and its consequences // Physics of Fluid. 2007. Vol. 19. 016602–15. doi: 10.1063/1.2432303

52. Чаликов Д.В. Статистика экстремальных ветровых волнs // Фундаментальная и прикладная гидрофизика. 2009. № 3(5). С. 4–24.

53. Chalikov D. Freak waves: their occurrence and probability // Physics of Fluids. 2009. Vol. 21. 076602. doi: 10.1063/1.3175713

54. Babanin A.V., Chalikov D., Young I.R., Saveliev I. 2010 Numerical and laboratory investigation of breaking of steep two-dimensional waves in deep water // Journal of Fluid Mechanics. 2010. Vol. 644. P. 433–463. doi: 10.1017/S002211200999245X

55. Galchenko A., Alexander V. Babanin, Dmitry Chalikov, I.R. Young. Modulational instabilities and breaking strength for deep-water wave groups // J. Phys. Oceanogr. 2010. 40, No. 10, P. 2313–2324. doi: 10.1175/2010JPO4405.1

56. Chalikov D., Babanin A.V. Simulation of wave breaking in one-dimensional spectral environment // Journal Phys. Ocean. 2012. Vol. 42, N11. P. 1745–1761.

57. Chalikov D. Numerical modeling of sea waves (eBook). Springer, 2016. 330 p. doi: 10.1007/978-3-319-32916-1

58. Beale J.T. A convergent boundary integral method for three-dimensional water waves // Math. Comput. 2001. Vol. 70. P. 977–1029.

59. Xue M., Xu H., Liu Y., Yue D.K.P. Computations of fully nonlinear three-dimensional wave and wave–body interactions. I. Dynamics of steep three-dimensional waves // Journal of Fluid Mechanics. 2001. Vol. 438. P. 11–39.

60. Grilli S., Guyenne P., Dias F. A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom // Int. J. Num. Methods Fluids. 2001. Vol. 35. P. 829–867.

61. Clamond D., Grue J. A fast method for fully nonlinear water wave dynamics // Journal of Fluid Mechanics. 2001. Vol. 447. P. 337–355.

62. Clamond D., Fructus D., Grue J., Krisitiansen O. An efficient method for three-dimensional surface wave simulations. Part II: Generation and absorption // J. Comp. Physics. 2005. Vol. 205. P. 686–705. doi:10.1016/j.jcp.2004.11.038

63. Fructus D., Clamond D., Grue J, Kristiansen Ǿ. An efficient model for three-dimensional surface wave simulations. Part I: Free space problems // J. Comp. Phys. 2005. Vol. 205. P. 665–685.

64. Guyenne P., Grilli S.T. Numerical study of three-dimensional overturning waves in shallow water // Journal of Fluid Mechanics. 2006. Vol. 547. P. 361–388.

65. Fochesato C., Dias F., Grilli S. Wave energy focusing in a three-dimensional numerical wave tank // Proc. R. Soc. A. 2006. 462. P. 2715–2735.

66. Craig W., Sulem C. Numerical simulation of gravity waves // J. Comput. Phys. 1993. Vol. 108. P. 73–83.

67. Zakharov V.E. Stability of periodic waves of finite amplitude on the surface of deep fluid // J. Appl. Mech. Tech. Phys. JETF. 1968. Vol. 2. P. 190–194.

68. Grue J., Fructus D. Model for fully nonlinear ocean wave simulations derived using fourier inversion of integral equations in 3D // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 1–42. doi: org/10.1142/9789812836502_0001

69. Liu Y., Gou Y., Bin Teng. B., Shigeo Yoshida S. An extremely efficient boundary element method for wave interaction with long cylindrical structures based on free-surface green’s function // Computation. 2016. 4(3):36. doi: 10.3390/computation4030036

70. Zheng Y.H., Shen Y.M., Ng C.O. Effective boundary element method for the interaction of oblique waves with long prismatic structures in water of finite depth // Ocean Engineering. 2008. Vol. 35, Iss. 5—6. P. 494—502. doi: 10.1016/J.OCEANENG.2007.12.003

71. Dommermuth D., Yue D. A high-order spectral method for the study of nonlinear gravity Waves // Journal of Fluid Mechanics. 1987. Vol. 184. P. 267–288.

72. West B., Brueckner K., Janda R., Milder M., Milton R. A new numerical method for surface hydrodynamics // Journal of Geophysical Research. 1987. Vol. 92. P. 11 803–11 824.

73. Tanaka M. Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations // Journal of Fluid Mechanics. 2001. Vol. 444. P. 199–221.

74. Toffoli A., Onorato M., Bitner-Gregersen E., Monbaliu J. Development of a bimodal structure in ocean wave spectra // Journal of Geophysical Research. 2010. Vol. 115, Iss. C3. C03006. doi: 10.1029/2009JC005495

75. Touboul J., Kharif C. Two-dimensional direct numerical simulations of the dynamics of rogue waves under wind action // Advances in Numerical Simulation of Nonlinear Water Waves. 2010. P. 43–74.

76. Ducrozet G., Bonnefoy F., Le Touzé D., Ferrant P. 3-D HOS simulations of extreme waves in open seas // Nat. Hazards Earth Syst. Sci. 2007. Vol. 7. P. 109–122.

77. Ducrozet G., Bingham H.B., Engsig-Karup A.P., Bonnefoy F., Ferrant P. A comparative study of two fast nonlinear free-surface water wave models // Int. J. Numer. Meth. Fluids. 2012. P. 1818–1834.

78. Chalikov D., Babanin A.V., Sanina E. Numerical modeling of three-dimensional fully nonlinear potential periodic waves // Ocean Dynamics. 2014. Vol. 64, 10. P. 1469–1486.

79. Chalikov D. High-resolution numerical simulation of surface wave development under the action of wind // Geophysics and Ocean Waves Studies / eds. K.S. Essa et al. London: IntechOpen, 2020. doi: 10.5772/intechopen.92262

80. Chalikov D. Accelerated reproduction of 2-D periodic waves // Ocean Dynamics. 2021. 71(4). doi: 10.1007/s10236-021-01450-3

81. Chalikov. D. A two-dimensional approach to the three-dimensional phase resolving wave modeling // Examines Mar. Biol. Oceanogr. 2021. 4(1). EIMBO.000576. doi: 10.31031/EIMBO.2021.04.000576

82. Chalikov D.A. 2D Model for 3D Periodic Deep-WaterWaves. J. Mar. Sci. Eng. 2022, 10, 410. doi: 10.3390/jmse10030410

83. Tolman H., Chalikov D. On the source terms in a third-generation wind wave model // J. Phys. Oceanogr. 1996. No 11. P. 2497–2518.

84. Chalikov D., Babanin A.V. Parameterization of wave boundary layer // Atmosphere. 2019.

85. Chalikov D., Rainchik S. Coupled Numerical Modelling of Wind and Waves and the Theory of the Wave Boundary Layer. Boundary-Layer Meteorol. 2011. Vol. 138. P. 1–141. doi: 10.1007/s10546-010-9543-7

86. Babanin A.V. Breaking and dissipation of ocean surface waves. Cambridge University Press, 2011. 463 p.

87. Babanin A., Chalikov D. Numerical investigation of turbulence generation in non-Breaking potential waves // Journal of Geophysical Research. 2012. Vol. 117. C00J17. doi: 10.1029/2012JC007929