New asymptotic description of nonlinear water waves in Lagrangian coordinates
A new description of two-dimensional continuos free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in R². Components of the transformation function satisfy the linear Euler-t...
| المؤلفون الرئيسيون: | , , |
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| التنسيق: | Journal article |
| اللغة: | English |
| منشور في: |
Cambridge University Press
2006
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| الموضوعات: |
| الملخص: | A new description of two-dimensional continuos free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in R². Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is contructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times. |
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