Abstract
This paper proposes a 3-D non-hydrostatic free surface flow model with a newly proposed general boundary-fitted grid system to simulate the nonlinear interactions of the bi-chromatic deep-water gravity waves. First, the monochromatic bidirectional and bi-chromatic bidirectional waves of small wave steepness are successively simulated to verify the abilities of the numerical model. Then, a series of bi-chromatic progressive waves of moderate wave steepness and different crossing angles are simulated and analyzed in detail. It is found that if the crossing angle is close to or smaller than the resonant angle, apparent discrepancies are observed among the numerical results, the linear wave theory, and the steady third-order theory. Otherwise, the three solutions coincide well. Through analysis, it is concluded that the discrepancies are caused by the third-order quasi-resonant interactions between the bi-chromatic progressive waves. Such interactions not only could modify the wave spectrum, but could also change the wave shape patterns.
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References
Kimmoun O., Branger H., Kharif C. On short-crested waves: Experimental and analytical investigations [J]. European Journal of Mechanics-B/Fluids, 1999, 18(5): 889–930.
Fuchs R. A. On the theory of short-crested oscillatory waves. Gravity waves [R]. National Bureau of Standards Circular 521, Washington DC, USA: U. S. Government Printing Office, 1952, 187–200.
Hsu J. R. C., Tsuchiya Y., Silvester R. Third-order approximation to short-crested waves [J]. Journal of Fluid Mechanics, 1979, 90: 179–196.
Craig W., Nicholls D. P. Traveling gravity water waves in two and three dimensions [J]. European Journal of Mechanics-B/Fluids, 2002, 21(6): 615–641.
Nicholls D. P., Reitich F. Stable, high-order computation of traveling water waves in three dimensions [J]. European Journal of Mechanics-B/Fluids, 2006, 25(4): 406–424.
Fuhrman D. R., Madsen P. A. Short-crested waves in deep water: A numerical investigation of recent laboratory experiments [J]. Journal of Fluid Mechanics, 2006, 559: 391–411.
Hammack J. L., Henderson D. M., Segur H. Progressive waves with persistent two-dimensional surface patterns in deep water [J]. Journal of Fluid Mechanics, 2005, 532: 1–52.
Henderson D. M., Patterson M. S., Segur H. On the laboratory generation of two-dimensional, progressive, surface waves of nearly permanent form on deep water [J]. Journal of Fluid Mechanics, 2006, 559: 413–427.
Le Mehaute B. On the highest periodic short-crested wave [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering, 1986, 112: 320–330.
Longuet-Higgins M. S., Phillips O. M. Phase velocity effects in tertiary wave interactions [J]. Journal of Fluid Mechanics, 1962, 12: 333–336.
Sharma N., Dean R. G. Second-order directional seas and associated wave forces [J]. Journal of Petroleum Science and Engineering, 1981, 21(1): 129–140.
Madsen P. A., Fuhrman D. R. Third-order theory for bichromatic bi-directional water waves [J]. Journal of Fluid Mechanics, 2006, 557: 369–397.
Phillips O. M. On the dynamics of unsteady gravity waves of finite amplitude I. The elementary interactions [J]. Journal of Fluid Mechanics, 1960, 9: 193–217.
Longuet-Higgins M. S. Resonant interactions between two trains of gravity waves [J]. Journal of Fluid Mechanics, 1962, 12: 321–332.
Longuet-Higgins M. S., Smith N. D. An experiment on third-order wave interactions [J]. Journal of Fluid Mechanics, 1966, 25: 417–435.
Mcgoldrick L. F., Phillips O. M., Huang N. E. et al. Measurements of third-order resonant wave interactions [J]. Journal of Fluid Mechanics, 1966, 25: 437–456.
Bonnefoy F., Haudin F., Michel G. et al. Observation of resonant interactions among surface gravity waves [J]. Journal of Fluid Mechanics, 2016, 805: R3.
Madsen P. A., Fuhrman D. R. Third-order theory for multi-directional irregular waves [J]. Journal of Fluid Mechanics, 2012, 698: 304–334.
Stelling G., Zijlema M. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation [J]. International Journal for Numerical Methods in Fluids, 2003, 43(1): 1–23.
Ai C., Jin S., Lv B. A new fully non-hydrostatic 3D free surface flow model for water wave motions [J]. International Journal for Numerical Methods in Fluids, 2011, 66(11): 1354–1370.
Ai C., Ding W., Jin S. A general boundary-fitted 3D non-hydrostatic model for nonlinear focusing wave groups [J]. Ocean Engineering, 2014, 89: 134–145.
Mori N., Onorato M., Janssen P. et al. On the extreme statistics of long-crested deep water waves: Theory and experiments [J]. Journal of Geophysical Research, 2007, 112: C09011.
Waseda T., Kinoshita T., Tamura H. Evolution of a random directional wave and freak wave occurrence [J]. Journal of Physical Oceanography, 2009, 39(3): 621–639.
Waseda T., Kinoshita T., Tamura H. Interplay of resonant and quasi-resonant interaction of the directional ocean waves [J]. Journal of Physical Oceanography, 2009, 39(9): 2351–2362.
Beji S. Nonlinear wave transforms and randomness [J]. Coastal Engineering Journal, 2019, 61: 590–598.
Acknowledgements
This work was supported by the Liaoning Revitalization Talents Program (Grant No. XLYC1807010), the Fundamental Research Funds for the Central Universities (Grant No. DUT2019TB02).
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Project supported by the National Natural Science Foundation of China (Grant Nos. 51720105010, 51979029 and 51679031).
Biography: Jian-jian Xie (1983-), Female, Ph. D. Candidate
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Xie, Jj., Ma, Yx., Dong, Gh. et al. Numerical study of nonlinear interactions of bi-chromatic progressive deep-water waves. J Hydrodyn 33, 602–620 (2021). https://doi.org/10.1007/s42241-021-0047-3
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DOI: https://doi.org/10.1007/s42241-021-0047-3