Author:
CALVO DAVID C.,AKYLAS T. R.
Abstract
The stability of steep gravity–capillary solitary waves in deep water is numerically
investigated using the full nonlinear water-wave equations with surface tension. Out
of the two solution branches that bifurcate at the minimum gravity–capillary phase
speed, solitary waves of depression are found to be stable both in the small-amplitude
limit when they are in the form of wavepackets and at finite steepness when they
consist of a single trough, consistent with observations. The elevation-wave solution
branch, on the other hand, is unstable close to the bifurcation point but becomes
stable at finite steepness as a limit point is passed and the wave profile features two
well-separated troughs. Motivated by the experiments of Longuet-Higgins & Zhang
(1997), we also consider the forced problem of a localized pressure distribution applied
to the free surface of a stream with speed below the minimum gravity–capillary
phase speed. We find that the finite-amplitude forced solitary-wave solution branch
computed by Vanden-Broeck & Dias (1992) is unstable but the branch corresponding
to Rayleigh’s linearized solution is stable, in agreement also with a weakly nonlinear
analysis based on a forced nonlinear Schrödinger equation. The significance of viscous
effects is assessed using the approach proposed by Longuet-Higgins (1997): while for
free elevation waves the instability predicted on the basis of potential-flow theory is
relatively weak compared with viscous damping, the opposite turns out to be the case
in the forced problem when the forcing is strong. In this régime, which is relevant
to the experiments of Longuet-Higgins & Zhang (1997), the effects of instability can
easily dominate viscous effects, and the results of the stability analysis are used to
propose a theoretical explanation for the persistent unsteadiness of the forced wave
profiles observed in the experiments.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
30 articles.
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