2006/08/14 by Eugeny Buldakov, Paul H. Taylor, R. Eatock Taylor · 41 citations
Earth and Planetary Sciences · Mathematics · #Ocean Waves and Remote Sensing #Coastal and Marine Dynamics #Oceanographic and Atmospheric Processes #Lagrangian and Eulerian specification of the flow field #Eulerian path #Mathematical analysis #Nonlinear system #Asymptotic expansion #Mathematics #Stream function #Lagrangian #Classical mechanics #Physics #Vortex #Mechanics
paper · doi:10.1017/s0022112006001443
published in Journal of Fluid Mechanics 562, 431-444 (Cambridge University Press)
openalex publication_date 2006/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 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.