The Proceedings Fifth International Conference on Numerical Ship Hydrodynamics (1990) / Chapter Skim
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Numerical Prediction of Semi-Submersible Non-Linear Motions in Irregular Waves
Numerical Prediction of Semi-Submersible Non-Linear Motions in Irregular Waves
Pages 391-402
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From page 391... ...
Non-linear second-order forces are evaluated by using Molin's method which permits to obtain them without explicitely solving for the second-order potential. Non-linear low frequency motions being large in magnitude, variations of second-order forces with the motion of the body mean position are considered in the time simulation.
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From page 392... ...
, in the inviscid, irrotational and incompressible fluid, that the two potential problems relating these two-scales motions can be separated and that the first-order problem is the same as if no low frequency motions exist and high frequency first-order forces and motions are only parametrically dependent on the low frequency motions. We shall emphasize this 'parametrical' dependence by representing the low frequency forces as a function of the body mean position.
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From page 393... ...
indicates that mean displacement of a floating body in regular wave is not strictely proportional to the square of the wave amplitude and that it depends also on the derivative of the steady forces. For example in the case of heave motion, the vertical steady force and its derivative being usually of the same sign, the mean displacement predicted by (17)
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From page 394... ...
(41) As we see the above expressions of oscillatory part of low frequency forces, two remarks have to be pointed out: one is that the number N of element wave frequencies should be large enough to represent correctly irregular waves and to evaluate the low frequency forces at the difference frequencies (Wj—Wj_~)
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From page 395... ...
and supposing G = ¢~.MoM, the Haskind integral on the body surface is transformed into integrals which don't contain any more the double spatial derivatives and can so be evaluated accurately. In order to evaluate the Haskind integral on the free surface, we divide the unlimited free surface into two regions: an inner region Sinn which is limited by the waterline and a boundary circle line some distance far away from the body and an outer region SOut which is the surface extending from the boundary circle line to infinity.
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From page 396... ...
The computing time for the low frequency forces is dominated by the evaluation of the Haskind integral on flee free surface, since other contributions are just in form of simple additions of the first-order quantities which are obtained once the first-order problem is solved completely. In flee same case of a vertical cylinder whose one quarter surface is divided into 108 panels, the first-order solution costs about 3 ~ni~utes on Vax8700 computer while the free surface Haskind integral is obtained after 10 minutes calculus for one frequency.
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From page 397... ...
The computer times on the series of Vax8000 computer, for one frequency, are about 9 minutes for the first-order problem and 30 minutes for the evaluation of the low frequency forces. Figure 4 Convergence with discretisation In o 0.30— ._ IS _ ~ 0 400 800 1200 1600 Panel Number In the regular waves, the second-order steady forces of model No.1 are evaluated for 16 frequencies from 2 rad./sec.
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From page 398... ...
The first derivatives of the low frequency forces and Else steady forces are obtained by the same procedure as those of flee steady forces in regular waves, applying the finite difference Clod to the results at three different mean positions. The obtained low frequency forces and the steady forces and their first derivatives are used in equation (24)
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From page 399... ...
The heave amplitude of the motion sillllllatioll is in good agreement with the experimental results. The classical method underestimates the heave motion for frequencies lower than that of resonance while it over-estimates the heave motion at larger
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From page 400... ...
The numerical results show that for floating semi-submersible bodies, the Haskind integral on flee free surface can be negligible. A simulation model for the prediction of low frequency motions taking account of the variation of the second-order low frequency forces with regard to the mean position has been presented.
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From page 401... ...
Author's Reply 1. The contribution of the second-order diffraction potential to the second-order loads consists of two Haskind integrals: one on the free surface that involves the secondorder correction to the free surface equation, the other over the hull that involves the second-order correction of the body surface equation.
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From page 402... ...
Author's Reply In an earlier paper[A3] of the author computed the vertical drift forces on the same models, using both the momentum method and the direct pressure integral method.
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