The use of pseudo-spectral high order method HOSM for simulations of nonlinear waves on the surface of finite depth water
https://doi.org/10.59887/2073-6673.2025.18(4)-3
EDN: DULUYA
Abstract
Regimes and limitations of the numerical solution method for hydrodynamic equations using the approximation of the near-surface velocity potential by the high-order Taylor expansion (High Order Spectral Method, HOSM) are investigated. This approach is considered in the context of simulation of large ensembles of sea surface displacement fields under finite-depth conditions. The main attention is paid to the description of strongly nonlinear waves and waves with a wide frequency spectrum. The study is performed in planar geometry.
Keywords
potential gravity waves on the water surface,
sea surface waves,
nonlinear waves,
numerical simulation
Supporting agencies: The research is supported by the Russian Science Foundation under the grant № 22-17-00153.
About the Author
A. V. Slunyaev
A.V. Gaponov-Grekhov Institute of Applied Physics RAS ; HSE University
Russian Federation
Russian Federation
WoS ResearcherID: A-3272-2014, Scopus Author ID: 55957049100
46 Ulyanova Str., Nizhny Novgorod 603950
25/12 Str. Bolshaya Pecherskaya, Nizhny Novgorod, 603155
References
1. Tanaka M. A method of studying nonlinear random field of surface gravity waves by direct numerical simulation. Fluid Dynamics Research. 2001;28:41–60. https://doi.org/10.1016/S0169-5983(00)00011-3
2. Tanaka M. Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. Journal of Fluid Mechanics. 2001;444:199–221. https://doi.org/10.1017/S0022112001005389
3. Dommermuth D, Yue DKP. A high-order spectral method for the study of nonlinear gravity waves. Journal of Fluid Mechanics. 1987;184:267–288. https://10.1017/S002211208700288X
4. West BJ, Brueckner KA, Janda RS, Milder DM, Milton RL. A new numerical method for surface hydrodynamics. Journal of Geophysical Research. 1987;92:11803–11824. https://doi.org/10.1029/JC092iC11p11803
5. Onorato M, Osborne AR, Serio M. On the relation between two numerical methods for the computation of random surface gravity waves. European Journal of Mechanics — B/Fluids. 2007;26:43–48. https://doi.org/10.1016/j.euromechflu.2006.05.001
6. Zakharov V. Stability of periodic waves of finite amplitude on a surface of deep fluid. Journal of Applied Mechanics and Technical Physics. 1968;2:190–194. https://doi.org/10.1007/bf00913182
7. Chalikov D. Numerical modeling of sea waves. Springer; 2016. 330 p. URL: https://link.springer.com/book/10.1007/978-3-319-32916-1 (date of access: 18.03.2025).
8. Chalikov DV. Different approaches to numerical modeling of sea waves. Fundamental and Applied Hydrophysics. 2022;15(1):19–32. https://doi.org/10.59887/fpg/u1df-m1x7-1bxg
9. Klahn M, Madsen PA, Fuhrman DR. Simulation of three-dimensional nonlinear water waves using a pseudospectral volumetric method with an artificial boundary condition. International Journal for Numerical Methods in Fluids. 2021;93:1843–1870. https://doi.org/10.1002/fld.4956
10. Chalikov DV. Numerical modeling of three-dimensional potential waves. Fundamental and Applied Hydrophysics. 2014;7(1):7–31. (In Russ). URL: https://hydrophysics.spbrc.ru/jour/article/view/996 (date of access: 18.03.2025).
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. Computer Physics Communications. 2016;203:245–254. https://doi.org/10.1016/j.cpc.2016.02.017
12. Xiao W, Liu Y, Wu G, Yue DKP. Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. Journal of Fluid Mechanics. 2013;720:357–392. https://doi.org/10.1017/jfm.2013.37
13. Seiffert BR, Ducrozet G, Bonnefoy F. Simulation of breaking waves using the High-Order Spectral method with laboratory experiments: Wave-breaking onset. Ocean Modelling. 2017;119:94–104. https://doi.org/10.1016/j.ocemod.2017.09.006
14. Slunyaev A, Kokorina A. Account of occasional wave breaking in numerical simulations of irregular water waves in the focus of the rogue wave problem. Water Waves. 2019;2:243–262. https://doi.org/10.1007/s42286-019-00014-9
15. Canard M, Ducrozet G, Bouscasse B. Experimental reproduction of an extreme sea state in two wave tanks at various generation scales. In: OCEANS2022. Chennai, India; 2022. p. 1–6. https://doi.org/10.1109/OCEANSChennai45887.2022.9775216
16. Touboul J, Kharif C. Nonlinear evolution of the modulational instability under weak forcing and damping. Natural Hazards and Earth System Sciences. 2010;10:2589–2597. https://doi.org/10.5194/nhess-10-2589-2010
17. Fujimoto W, Waseda T, Webb A. Impact of the four-wave quasi-resonance on freak wave shapes in the ocean. Ocean Dynamics. 2019;69:101–121. https://doi.org/10.1007/s10236-018-1234-9
18. Slunyaev AV. Contributions of nonlinear spectral components to the probability distribution of rogue waves based on the results of numerical simulation of the Euler equations. Izvestiya, Atmospheric and Oceanic Physics. 2023;59:701–721. https://doi.org/10.1134/S0001433823060105
19. Slunyaev AV, Pelinovsky DE, Pelinovsky EN. Rogue waves in the sea: observations, physics, and mathematics. PhysicsUspekhi. 2023;66:148–172. https://doi.org/10.3367/UFNe.2021.08.039038
20. Fenton JD. Numerical Methods for Nonlinear Waves. In: Advances in Coastal and Ocean Engineering. Cornell University, USA. 1999;5:241–324. https://doi.org/10.1142/4086
21. Chalikov DV, Bulgakov KY. Stokes Waves at Finite Depth. Fundamental and Applied Hydrophysics. 2014;7(4):3–15 (In Russ). URL: https://hydrophysics.spbrc.ru/jour/article/view/964 (date of access: 18.03.2025)
22. Clamond D. Cnoidal-type surface waves in deep water. Journal of Fluid Mechanics. 2003;489:101–120. https://doi.org/10.1017/S0022112003005111
23. Lukomsky VP, Gandzha IS. Fractional Fourier approximations for potential gravity waves on deep water. Nonlinear Processes in Geophysics. 2003;10:599–614. https://doi.org/10.5194/npg-10-599-2003
24. Chang HK, Liou J-C. Fixed-frequency Stokes wave expansion. Ocean Engineering. 2006;33:417–424. https://doi.org/10.1016/j.oceaneng.2005.04.020
25. Rainey RCT, Longuet-Higgins MS. A close one-term approximation to the highest Stokes wave on deep water. Ocean Engineering. 2006;33:2012–2024. https://doi.org/10.1016/j.oceaneng.2005.09.014
26. Dyachenko SA, Lushnikov PM, Korotkevich AO. The complex singularity of a Stokes wave. JETP Letters. 2013;98:767–771.
27. Dyachenko SA, Lushnikov PM, Korotkevich AO. Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation. Studies in Applied Mathematics. 2016;137:419–472. https://doi.org/10.1111/sapm.12128
28. Shin JR. A Fourier series approximation for deep-water waves. Journal of Ocean Engineering and Technology. 2022;36:101–107. https://doi.org/10.26748/ksoe.2021.092
29. Roenby J. FentonWave — A stream function wave boundary condition for OpenFOAM [Computer software]. 2012. URL: https://github.com/roenby/fentonWave/tree/master (date of access: 01.07.2025)
30. Fenton JD. Nonlinear wave theories. In: The Sea. 1990;9:3–25.
31. Le Méhauté B. An Introduction to Hydrodynamics and Water Waves. Springer Science & Business Media; 1976.
32. Zhao K, Wang Y, Liu PL-F. A guide for selecting periodic water wave theories — Le Méhauté (1976)’s graph revisited. Coastal Engineering. 2024;188:104432. https://doi.org/10.1016/j.coastaleng.2023.104432
33. Watson KM, West BJ. A transport-equation description of nonlinear ocean surface wave interactions. Journal of Fluid Mechanics. 1975;70:815–826. https://doi.org/10.1017/S0022112075002364
34. Fenton JD, Rienecker MM. A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. Journal of Fluid Mechanics. 1982;118:411–443. https://doi.org/10.1017/S0022112082001141
35. Bateman WJD, Swan C, Taylor PH. On the calculation of the water particle kinematics arising in a directionally spread wavefield. Journal of Computational Physics. 2003;186:70–92. https://doi.org/10.1016/S0021-9991(03)00012-3
36. Ducrozet G. Modélisation des processus non-linéaires de génération et de propagation d’états de mer par une approche spectral. (In French). 2007.
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For citations:
Slunyaev A.V. The use of pseudo-spectral high order method HOSM for simulations of nonlinear waves on the surface of finite depth water. Fundamental and Applied Hydrophysics. 2025;18(4):28-49. (In Russ.) https://doi.org/10.59887/2073-6673.2025.18(4)-3. EDN: DULUYA
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