Two-dimensional Numerical Modelling of Large Motions of Floating Bodies in Steep Waves

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T w o - d i m e n i i o n a l N u m e r i c a l o f Floating B o d i i Debabraia-Scii

Graduate Student, Ocean Engineering Memorial University of Newfoundland St. John's, Newfoundland, Canada

M o d e l l i n g o f Large M o t i o n s s i n Sleep. Waves

J.S. Pawlowski

Associate Visiting Professor

Memorial University of Newfoundland St. John's, Newfoundland, Canada

The paper presents a continuation of the earlier work 11,2] on the development of a relatively simple algorithm for the numerical modelling of large motions of floating or tub-merged bodies in steep waves. The problem is considered in two dimensions and the fluid flow is assumed to be irrotational and acyclic. A boundary element method, based on an application of Green's second identity in a finite control domain, is used t o solve for the unknown values of the velocity potential (^) or its normal derivative ( Ö ^ / ö n ) at the domain boundaries. The evolution ofthe free surface in time (t) is followed by means of an Eulerian method in which motions of the collocation points arc restricted to vertical displacements. Earlier results (1,2] have shown that with thc application of Airy or Stoke's 2nd order wave potentials on the upstream control boundary, and by adopting a form of Orlanski's radiation condition at the downstream boundary, propagation of steep waves in the control domain can be modelled effectively. Good results have also been obtained for the interaction of steep waves with vertical walls, { i j .

In the present work the method is extended by introducing a floating rigid body in the control domain. The impermeability boundary condition is imposed on the wetted surface of thc body. The generalized hydrodynamic forces . exerted by the fluid on the surface of the body arc derived by direct integration of pressure (p) determined from Bernoulli's equation:

The motions of the body arc obtained by integration in time of Newton's equations of moi)on

^ ^ . ^ = ^ . (2) for I = 1,2,3, with Mi denoting the mass of the body for t = 1,2, and its moment of inertia

about C.G. for i = 3. ƒ • represent generalized forces. At coUocation points which are fixed on the body surface, d<f>/dt in (1) is determined from:

dt ^ di~^" dx (3) where d<(>/dt denotes the rate of change in time of the potential at the collocation point

moving with velocity v.

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-vicinity of the intersection of t^e J J s u X T ^ h tV K T «^r^^'^'-o" grid in the introduce destabiUzing force elT^ts through T ''^ ^ ^ ' ^ f ° " " < »

is rectified by n^^ntainin^run^o ^ ' ^ " ^ ^ ^ ^ ^ ^ ^ (3). The difficulty

the free surface and wetted surf Ve of tTe bo ^ u " ' ^ ü r ^ " * » ' '«gridding on on the new grid is obtained by s p a r i L . r n 1V- r^ ' ^ " ' ' ^ information also occur in the application^f' ^ r

rordTr

t^^^^^^ *°

"^/''^

integration of equations of m o l ^ J w w \ i^^^^t . ^ ' ^ ' ^ ^ '^^^"'^

the trajectories of free s u r f a r ^ l o c a t L oofnt " " « r f ' ^ y the computation of determined by means of a centVa^ H ^ I , ^ " eliminated if d<i/rfr values are scheme. ^ " " ^ ' ^ l * *t the corrector ievel of the A-B-M

a

^^lsv^óS

;:i;^s

- - ^ ^ ^ ^ out

m

which wave excitations in a channel. The

^d^oTf^e

**d^rT*'"

^^^^

"^j'^'^»«<^ ^° the body, and displacements of ïhe body rLtt^^^^^^^^^^^ *° appropriate mounting device Oncoming ^ ?^ *° ^odes by an body, were measured'at semal

**.tatir V^^^^ '° Y^*'^"^**

of t h e -displacements in heave and^oU m o i « ^""^^ "^^'orf.es included is being used to replicate

thfe^jrm'e^f,

^

'^fl-

'^«orithm described above between experimental and como^ed K i ^ «'^iputations and comparisons

obtained for two wave l ^ T ^ ^ t ^ - ^ 19 ' T n 21 ^^'-^^^^ steepness of ^ / A = 0.04 ^ both cies "he h l l ^ corresponding to the wave

and higher wave steepness could notTe l^l^eveAl"'"*'''" 'f, ' ^ ^ ^ heave resonance matching between e x p e i i ^ ^

«rw^vTh^hr^^^

1

The synchronization between the experiment!?

.„7 ^'''«^^V

' ° 3a. by referencing the time histories

t o T « T a ^ u T J '^7T"t

' ' " ' ^ ^ ' ^ « ^ ' « - « d

figs. 2b and 3b, at the location of b o d y G t «^^^^^^ *^*^«o"«. «bown in behaviour of the body, good agreement

iflS'

^Tu ' *^^ount the resonant values in the region

of

comparnTnlatelr^^

' 7 ^

t " T " ' * * ^ ^^^"^^^

expenmental and computed values orr^were^^^ils ^ a n ' / ^ fig. 4. a sample of a numerical simulat^n Tn wh^K ' ^ o ^ " ^'^^^^^ I " displayed. simulation in which large roll amplitudes was achieved is

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-e.

F i g . 1 The body geometry

D _40riH. DID_{e 1.0} diD

1=

0.5 rID_{e 0.0625} «

fw.

• t v .

"air

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( « ) Comparison of measured and computed wave

elevatlnnt-(c) Sway force

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I

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W V U l i ' l / U l / .

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F i g . 3 Comparison for iJ/A = 0.21,i//A = 0.04 (for the

F i g , 4 A numerical simulation of roll motion of the rectan-gular body for B/d - 1.0; / / / A = 0.05 and Ü/X = 1/G