The Euler-type Description Of
Lagrangian Water Waves
Author(s)
E. V. Buldakov, P. H. Taylor & R. Eatock Taylor
Abstract
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.
1 Introduction
The Lagrangian description by its nature is more appropriate for representing violent
free surface motion, and a perturbation technique applied to the equations
in the Lagrangian form may be able to describe effects that cannot be captured
by classical Eulerian expansions. For example, such phenomena as overturning
waves or jet-like fluid behaviour on a rigid surface can in principle be described
by low order perturbation methods if applied to equations of motion in Lagrangian
form. There are just a few published examples where a perturbation technique has
been applied to water wave problems in the Lagrangian formulation. The early
attempt of Pierson [1] can be mentioned, where the Lagrangian displacement of
fluid particles was represented as a regular expansion with respect to powers of
Keywords
Related Book
Other papers in this volume
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