Stokes Waves at Finite Depth
Abstract
The main properties of the Stokes waves are considered. Several methods of numerical investigation of wave dynamics are discussed. A conformal surface-following coordinate system is defined. Stationary potential waves equation in this coordinate system are represented. Algorithm of very fast numerical solving of stationary one-dimensional potential equations for the case of optional depth is described. The characteristics of numerical runs for the deep depth case are investigated: time of experiments, iteration number, potential, kinetic and total energy, asymmetry, excess. Area in coordinates of depth and steepness where solution exists is specificated. The geometrical characteristics of the Stokes waves as function of steepness and depth are investigated: asymmetry, maximum of the local steepness, maximum of the local second derivate and also phase velocity. The forms of waves for steepness 0.01 are shown. Possible application of obtained results is considered.
About the Authors
D. V. Chalikov
Department of the P. P. Shirshov Institute of Oceanology of RAS
Russian Federation
Russian Federation
Saint-Petersburg
K. Yu. Bulgakov
Department of the P. P. Shirshov Institute of Oceanology of RAS
Russian Federation
Russian Federation
Saint-Petersburg
References
1. Stokes G. G. On the theory of oscillatory waves. Trans. Cambridge Philos. Soc. 1847, 8, 441—445.
2. Crapper G. D. An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 1957, 96, 417—445.
3. Clamond D., Grue J. A fast method for fully nonlinear water wave dynamics. J. Fluid Mech. 2001, 447, 337—355.
4. Grilli S., Guyenne P., Dias F. A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom. Int. J. Num. Methods Fluids. 2001, 35, 829—867.
5. Clamond D., Fructus D., Grue J., Krisitiansen O. An efficient method for three-dimensional surface wave simulations. Part II: Generation and absorption. Journ. Comp. Physics. 2005, 205, 686—705.
6. Fochesato C., Dias F., Grill S. Wave energy focusing in a three-dimensional numerical wave tank. Proc. R. Soc. 2006, 462, 2066, 2715—2735.
7. Zakharov V. E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 1968, 9, 190—194.
8. Dommermuth D., Yue D. A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 1987, 184, 267—288.
9. West B., Brueckner K., Janda R., Milder M., Milton R. A New Numerical Method for Surface Hydrodynamics. J. Geophys. Res. 1987, 92, 11803—11824.
10. Tanaka M. A method of studying nonlinear random field of surface gravity waves by direct numerical simulation. Fluid Dyn. Res. 2001, 28, 41—60.
11. Tanaka M. Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech. 2001, 444, 199—221.
12. Ducrozet G., Bonnefoy F., LeTouze D., Ferrant P. 3-DHOS simulations of extreme waves in open seas. Nat. Hazards Earth System Sci. 2007, 7, 109—122.
13. Whitney J. C. The numerical solution of unsteady free-surface flows by conformal mapping. Proc. Second Inter. Conf. on Numer. Fluid Dynamics. Springer-Verlag, 1971. 458—462.
14. Ovsyannikov L. V. Float problem of non-stationary movement of fluid with the free boundaries. Dinamika sploshnoy sredy. 1971, 8, 22—26. (in Russian)
15. 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. 1979, 19, 2, 335—370.
16. Fornberg B. A numerical method for conformal mapping. SIAM, J. Sci. Comput. 1980, 1, 3. 386—400.
17. Tanveer S. Singularities in water waves and Rayleigh-Taylor instability. Proc. R. Soc. 1991, 435, 1893. 137—158.
18. Maklakov D. V. Almost-highest gravity waves on water of finite depth. Eur. J. Appl. Math. 2002, 13, 67.
19. 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.
20. Chalikov D., Sheinin D. Direct Modeling of One-dimensional Nonlinear Potential Waves. Advances in Fluid Mechanics. 1998, 17, 207—258.
21. Chalikov D., Sheinin D. Modeling of Extreme Waves Based on Equations of Potential Flow with a Free Surface. Journ. Comp. Phys. 2005, 210, 247—273.
22. Sheinin D., Chalikov D. Hydrodynamic modeling of potential surface waves. Problem of hydrometeorology and environment on the eve of XXI century, Proceeding of international theoretical conference. St.- Petersburg, Gidrometeoizdat. 2000, 305—337. (in Russian)
23. Chalikov D. Statistical properties of nonlinear one-dimensional wave fields. Nonlinear processes in geophysics. 2005, 12, 1—19.
24. Chalikov D. V. Harmonic Wave Deep Water Transformation. Fundamentalnaya i prikladnaya gidrofizika. 2010, 9, 3, 14—21. (in Russian)
25. Drennan W. M., Hui W. H., Tenti G. Accurate calculation of Stokes wave near breaking. Eds C. Graham, S. K. Malik. Continuum Mechanics and its Applications. Washington DC, Hemisphere Publishing, 1988. 463—473.
26. Longuet-Higgins M.S., Tanaka M. On the crest instabilities of steep surface waves // J Fluid. Mech. 1997. V. 336. P. 51—68.
27. Longuet-Higgins M. S. The Instabilities of Gravity Waves of Finite Amplitude in Deep Water. II Subharmonics. Proc. Roy. Soc. Lond. 1978, 360, 489—505.
28. Benjamin T. B., Feir J. E. The Disintegration of Wave Trains in Deep Water. J. Fluid. Mech. 1967, 27, 417— 430.
29. Chalikov D. Simulation of Benjamin-Feir instability and its consequences. Phys of Fluid. 2009, 19, 016602- 15. doi: 10.1063/1.2432303.
30. Kurkin A. A., Pelinovskij E. N. Freak-waves: facts, theory and modelling. Nizhnij Novgorod, NNGU, 2004. 157 p. (in Russian)
31. Chalikov D. Freak waves: their occurrence and probability. Phys of Fluid. 2009, 21, 076602-18. doi: 10.1063/1.3175713.
32. Chalikov D. V. Statistics of Extreme Wind Waves. Fundamentalnaya i prikladnaya gidrofizika. 2009, 5, 3, 4—24. (in Russian)
33. 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. Eur. J. Mech. B/Fluids. 2002, 21, 283—291.
34. Zakharov V. E., Dyachenko A. I., Prokofiev A. O. Freak waves as nonlinear stage of Stokes wave modulation instability. Eur. J. Mech. B/Fluids. 2006, 25, 677—692.
35. Dyachenko A. I., Zakharov V. E. Modulation instability of Stokes wave – freak wave. JETP Letters. 2005, 81, 255—259.
36. Dyachenko A. I., Zakharov V. E. On the Formation of Freak Waves on the Surface of Deep Water. JETP Lett. 2008, 88, 307—311.
37. Shamin R. V. On the Existence of Smooth Solutions to the Dyachenko Equations Governing Free-Surface Unsteady Ideal Fluid Flow. Doklady Akademii Nauk. 2006, 73, 1, 112—113. (in Russian)
38. Orszag S. A. Transform method for calculation of vector coupled sums. Application to the spectral form of vorticity equation. J. Аtmos. Sci. 1970, 27, 890—895.
39. Prosperetti A., Jacobs J. W. A numerical method for potential flow with a free surface. J. Comput. Phys. 1983, 51, 365—386.
Review
For citations:
Chalikov D.V., Bulgakov K.Yu. Stokes Waves at Finite Depth. Fundamental and Applied Hydrophysics. 2014;7(4):3-15. (In Russ.)
Views:
93
Email this article 