The Euler-type description of Lagrangian water waves
A new description of 2D continuous 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 a transformation function satisfy the linear Euler-type continuity...
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| Format: | Conference item |
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WITPress
2005
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| Summary: | A new description of 2D continuous 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 a 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 into 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 a classical problem of a regular wave traveling in deep water, and the fifth order Lagrangian asymptotic solution is constructed. In contrast with early attempts of Lagrangian regular-wave expansions, the presented asymptotic solution is uniformly-valid at large times. © 2005 WIT Press. |
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