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Resonantly forced gravity–capillary lumps on deep water. Part 2. Theoretical model

Published online by Cambridge University Press:  31 March 2011

YEUNWOO CHO ,
JAMES D. DIORIO ,
T. R. AKYLAS  and
JAMES H. DUNCAN
YEUNWOO CHO
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
JAMES D. DIORIO
Affiliation:
Department of Mechanical Engineering, University of Maryland, College Park, MD 20740, USA
T. R. AKYLAS*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
JAMES H. DUNCAN
Affiliation:
Department of Mechanical Engineering, University of Maryland, College Park, MD 20740, USA
*
†Email address for correspondence: trakylas@mit.edu
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Abstract

A theoretical model is presented for the generation of waves by a localized pressure distribution moving on the surface of deep water with speed near the minimum gravity–capillary phase speed, cmin. The model employs a simple forced–damped nonlinear dispersive equation. Even though it is not formally derived from the full governing equations, the proposed model equation combines the main effects controlling the response and captures the salient features of the experimental results reported in Diorio et al. (J. Fluid Mech., vol. 672, 2011, pp. 268–287 – Part 1 of this work). Specifically, as the speed of the pressure disturbance is increased towards cmin, three distinct responses arise: state I is confined beneath the applied pressure and corresponds to the linear subcritical steady solution; state II is steady, too, but features a steep gravity–capillary lump downstream of the pressure source; and state III is time-periodic, involving continuous shedding of lumps downstream. The transitions from states I to II and from states II to III, observed experimentally, are associated with certain limit points in the steady-state response diagram computed via numerical continuation. Moreover, within the speed range that state II is reached, the maximum response amplitude turns out to be virtually independent of the strength of the pressure disturbance, in agreement with the experiment. The proposed model equation, while ad hoc, brings out the delicate interplay between dispersive, nonlinear and viscous effects that takes place near cmin, and may also prove useful in other physical settings where a phase-speed minimum at non-zero wavenumber occurs.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 288 - 306
Copyright
Copyright © Cambridge University Press 2011

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References

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