Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Fourier-Kochin Theory of Free-Surface Flows
Fourier-Kochin Theory of Free-Surface Flows
Pages 120-135
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From page 120... ...
Two applications and extensions of the theory, to the coupling between an inner viscous flow and an outer potential flow and the representation of timeharmonic ship waves generated by an arbitrary singularity distribution, are also presented. INTRODUCTION For most practical purposes, the flow due to a ship advancing in waves is effectively potential and linear beyond a small distance from the ship hull, although viscous and nonlinear effects can be significant near the hull.
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From page 121... ...
The velocity representation of the kernel spectrum function M is shown in t3] to provide a substantially more solid mathematical basis than the potential representation for the purpose of numerically evaluating influence coefficients.
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From page 122... ...
A significant property of this approach is that the Fourier integrals (13a) are not singular because the spectrum functions [J and M vanish in the large-wavenumber limit k ~ x for a singularity distribution, whereas (4)
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From page 123... ...
inner nonlinear viscous flow => outer linear potential flow coupling, can be useful to determine far-field waves and wave drag (which can be evaluated more accurately via Havelock's formula for the energy radiated in the trailing wave pattern than by integration of the hull pressure) , and to analyze the far-field influence (notably upon the wave pattern and the drag)
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From page 124... ...
in terms of the velocity u at a control surface. FOURIER REPRESENTATION OF GENERIC DISPERSIVE WAVES The most difficult aspect of the Fourier-Kochin formulation, indeed of any approach based on a Green function satisfying the free-surface condition (1)
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From page 125... ...
The wave component Em is given by a single Fourier integral along the dispersion curves defined in the Fourier plane by the dispersion relation D=O. Specifically, (31a,b)
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From page 126... ...
and steady flows, and to the Green function of wave di~raction-radiation at small forward speed, are examined in 4,5. Another application, to timeharmonic ship waves characterized by the dispersion function (5)
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From page 127... ...
, ~0~N ~ _ 4 -3 -2 - 1 0 a/f 2 aF ~ ~ Fig.3 Dispersion curves in the Fourier planes (a,§~/f2 arid (a,§~'F2 ~ Fig.5 Comparison of wavelengths A`, Au, As, A rat 1 \ n En O.25 \ of/ \d 0.00 ~ , , 0.00 0.25 0.50 T 127
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From page 128... ...
defines several distinct dispersion curves and related distinct wave compm nents, which include systems of inner and outer V waves, complete or partial ring waves and fan waves. The dispersion curves defined by (28a)
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From page 129... ...
Thus, if T > ~7~ a ship only generates trailing waves, whereas waves are radiated both ahead and behind a ship if O < T < it. The upstream waves are ring waves if O < T < 1/4 or incomplete ring waves and outer fan waves if 1/4 ~ T < ~7~ .
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From page 130... ...
are now successively considered. Fourier representations of wave components The inner V wave component Din exists for all values of T and corresponds to the dispersion curve in the region a+ ~ car < oo where cat+ is defined by (32a)
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From page 131... ...
Illustrative calculations The foregoing Fourier representations of the wave components yiv , Rev, (pRF and OR in (35) provide simple analytical representations of the wave components generated by an arbitrary (volume, surface, and/or line)
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From page 133... ...
Fig.6g Ring-Fan and Inner ~ waves for ~ = 0.255 and f = 1 (-22.5 <5,S7.5, -12<~ <0 |
From page 134... ...
Illustrative calculations of the real and imaginary parts of the V waves TV+ REV, the ring waves OR, and the ring-fan and inner V waves SERF+ TV associated with the foregoing spectrum function are depicted in the upper and lower halves of Figs 6a-l for ~ = 0 at several values of the Strouhal number. Specifically, the inner and outer V waves and the ring waves are depicted in Figs 6a,c,e and Figs 6b,d,f for T = 0, 0.2, 0.245; and the ring-fa~ and inner V waves are depicted in Figs 6~-l for T = 0.255,0.272,0.5, 1 ,2,4.
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From page 135... ...
The spectrum function S typically associated with a continous or discrete distribution of sources and/or dipoles over the hull of a ship, is arbitrary in these Fourier representations, which therefore provide simple explicit analytical representations of the wave components radiated by an arbitrary (volume, surface, and/or line)
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