Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-17T04:53:45.073Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Lagrangian description of steady surface gravity waves

Published online by Cambridge University Press:  08 October 2007

DIDIER CLAMOND
DIDIER CLAMOND*
Affiliation:
Institutt For Energiteknikk, P O Box 40, 2027 Kjeller, Norwaydidier.clamond@ife.no
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 589 , 25 October 2007 , pp. 433 - 454
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abrashkin, A. A. 1996 Three-dimensional Gouyon waves. Fluid Dyn. 31, 583–587.CrossRefGoogle Scholar
Abrashkin, A. A. & Zen'kovich, D. A. 1990 Vortex stationary waves on a shear flow. Izv. Akad. Nauk SSSR, Fiz. Aanos. Okeana 26, 35–46. (In Russian.)Google Scholar
Aris, R. 1962 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Buldakov, E. V., Taylor, P. H. & Eatock-Taylor, R. 2006 New asymptotic description of nonlinear water waves in Lagrangian coordinates. J. Fluid Mech. 562, 431–444.CrossRefGoogle Scholar
Chang, H. K., Liou, J. C. & Su, M. Y. 2007 Particle trajectory and mass transport of finite-amplitude waves in water of uniform depth. Eur. J. Mech. B/Fluids 26, 385–403.CrossRefGoogle Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. 286, 183–230.Google Scholar
Constantin, A. 2001 On deep water wave motion. J. Phys. A: 34, 1405–1417.Google Scholar
Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535.CrossRefGoogle Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. & Knuth, D. E. 1996 On the Lambert W function. Adv. Comput. Math. 5, 329–359.CrossRefGoogle Scholar
Craik, A. D. D. 2005 George Gabriel Stokes on water wave theory. Annu. Rev. Fluid Mech. 37, 23–42.CrossRefGoogle Scholar
Dias, F. & Bridges, T. J. 2006 The numerical computation of freely propagating time-dependent irrotational water waves. Fluid Dyn. Res. 38, 803–830.CrossRefGoogle Scholar
Drennan, W. M., Hui, W. H. & Tenti, G. 1992 Accurate calculations of Stokes water waves of large amplitude. Z. Angew. Math. Phys. 43, 367–384.CrossRefGoogle Scholar
Dubreil-Jacotin, M.-L. 1932 Sur les ondes de type permanent dans les liquides hétérogènes. Accad. Naz. Lincei (6) 15, 814–819. (In French.)Google Scholar
Dubreil-Jacotin, M.-L. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie. J. Math. 13, 217–291. (In French.)Google Scholar
Dubreil-Jacotin, M.-L. 1935 Complément à une note antérieure sur les ondes de type permanent dans les liquides hétérogènes. Rend. Accad. Naz. Lincei (6) 21, 344–346. (In French.)Google Scholar
Fenton, J. D. 1988 The numerical solution of steady water wave problems. Comput. Geosci. 143, 357–368.CrossRefGoogle Scholar
Fenton, J. D. 1990 Nonlinear wave theories. The Sea vol. 9, pp. 3–25. Wiley.Google Scholar
Fenton, J. D. 1999 Numerical methods for nonlinear waves. Adv. Coastal Ocean Engng 5, 241–324. World Scientific.CrossRefGoogle Scholar
Gerstner, F. 1802 Theorie der Wellen. Abhand. Kön. Böhmischen Gesel. Wiss. Prague (in German).Google Scholar
Gouyon, R. 1958 Contribution à la théorie des houles. Thèse, Université de Toulouse, France (in French).CrossRefGoogle Scholar
Kalisch, H. 2004 Periodic traveling water waves with isobaric streamlines. J. Nonlinear Math. Phys. 11, 461–471.CrossRefGoogle Scholar
Kharif, C. & Pelinovsky, E. 2003 Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/Fluids 22, 603–634.CrossRefGoogle Scholar
Kiebel, I. A. 1933 On some two-dimensional motions of a heavy compressible fluid. Prikl. Math. Mech. 1, 51–55. (In Russian.)Google Scholar
Kravtchenko, J. & Daubert, A. 1957 Closed trajectory waves in a finite depth. La Houille Blanche 3, 408–429. (In French.)CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Leblanc, S. 2004 Local stability of Gerstner's waves. J. Fluid Mech. 506, 245–254.CrossRefGoogle Scholar
Levi-Civita, T. 1912 Sulle onde di canale. Rend. Accad. Naz. Lincei (5) 21, 3–14. (In Italian.)Google Scholar
Longuet-Higgins, M. S. 1986 Eulerian and Lagrangian aspects of surface waves. J. Fluid Mech. 173, 683–707.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1987 Lagrangian moments and mass transport in Stokes waves. J. Fluid Mech. 179, 547–555.CrossRefGoogle Scholar
Miche, M. 1944 Mouvements ondulatoires de la mer en profondeur constante ou décroissante. Ann. Ponts Chaussées 114, 25–78, 131–164, 270–292, 369–406. (English translation: Univ. Calif. Wave Res. Lab. 3, 363, 1954.)Google Scholar
Schwartz, L. W. 1974 Computer extended and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. G.G. Stokes Math. & Phys. Papers, vol. 1, pp. 314–326. Cambridge University Press.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. Handbuch Phys. 9, 446–778. Springer.Google Scholar
Weniger, E. J. 1989 Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent deries. Comput. Phys. Rep. 10, 189–371.CrossRefGoogle Scholar
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond A 302 (1466), 139–188.Google Scholar
Wilton, J. R. 1914 On deep water waves. Phil. Mag. S. VI 27 (158), 385–394.CrossRefGoogle Scholar