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New asymptotic description of nonlinear water waves in Lagrangian coordinates

  • E. V. BULDAKOV (a1), P. H. TAYLOR (a2) and R. EATOCK TAYLOR (a2)
Abstract

A new description of two-dimensional 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 $\mathbb{R}^2$. 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 constructed, 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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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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