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On the time dependence of the wave resistance of a body accelerating from rest

Published online by Cambridge University Press:  26 April 2006

Yuming Liu  and
Dick K. P. Yue
Yuming Liu
Affiliation:
Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Dick K. P. Yue
Affiliation:
Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

We consider the large-time behaviour of the disturbances associated with an initial acceleration of a body in or near a free surface. As a canonical problem, we study the case of a body starting impulsively from rest to a constant velocity U. For a constant-strength translating point source (or dipole) started impulsively, it is known that the unsteady part of the Green function oscillates at the critical frequency ωc=g/4U (g is the gravitational acceleration) with an amplitude that decays with time, t, as t−1/2 and t−1 for t [Gt ] 1 in two (Havelock 1949) and three (Wehausen 1964) dimensions respectively. These classical results turn out to be non-realistic in that for an actual body, the associated source strengths are in general time dependent and a priori unknown, and must together satisfy the kinematic condition on the body boundary. We consider such an initial-boundary-value problem using a transient wave-source distribution on the body surface. Through asymptotic analyses, the unsteady behaviour of the solution at large time is obtained explicitly. Specifically, we show that for a general class of bodies satisfying a simple geometric condition (Γ ≠ 0), the decay rate of the transient oscillations (at frequency ωc) in the wave resistance and velocity potential is an order of magnitude faster: as t−3/2 and t−2, as t → ∞, in two and three dimensions respectively. For body geometries satisfying T = 0, for which the single source is a special case, the classical Green function results are recovered. These results are confirmed by an analysis in the frequency domain and substantiated by direct time-domain numerical simulations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 310 , 10 March 1996 , pp. 337 - 364
Copyright
© 1996 Cambridge University Press

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