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POTENTIAL FLOW WITH FREESURFACE

Fig.1

The equation for the free surface is:

z = h(x,y)

The disturbance generated by the hull and the waes will be:

+= uUUThe following assu!"tions will be considered:

# $niscid# $rrotational# (%teady flow)# ($nco!"ressible)

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$n this conditions, the elocity "otential for the total elocityis:

zyxU

,,=

The &a"lace equation:

02

2

2

2

2

2

=

+

+

yyx

'oundary conditions for the hull:

1. 0=hsn

. 0lim =

r

Fig.

The additional boundary conditions needed for the freesurface are:

1.The flow in the nor!al direction is zero:

0=fsn

or 0=

fsnU

here:

zyxU

,,=

and

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+

+

=

22

,

y

h

x

h

y

h

x

h

nfs

The *ine!atic free surface '+:

0=+zy

h

yx

h

x

2. ill assu!e that the static "ressure on the free surface is

constant.

""lying the 'ernoulli equation far away and on a "oint onthe waes will hae:

ghUpUp ++=+

The dyna!ic free-surface '+ will be:

05,0 2

222

=+++ Uzyx

gh

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3. radiation condition that "reents u"strea! waes

Two additional difficulties are introduced with the free-

surface:

# The free-surface '+ are non-linear# The free-surface '+ !ust be a""lied on the unnown

way free-surface

&inearization of the free-surface '+

$ntroduce a nown a""roxi!ate (basic) solution

( )H,

and a disturbance with res"ect to the basic solution:( )',' h

ill assu!e that the exact solution can be obtained byadding the basic solution and the disturbance:

+= and 'hHh +=$ntroducing the linearization to the free surface '+ will

obtain:

0=

+

++y

h

yx

h

xzy

H

yx

H

x

Hzzyyxxyx

Ug

h

+

+

= 2

2

1 22

2

Three ty"es of basic solutions:1. /ndisturbed flow

. 0ouble-!odel flow. %olution fro! a "reious iteration

1. &inearization about the undisturbed flow

The *elin free-surface '+ will be:

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0''

2

2

2

22

=

+

+

zxg

U

. &inearization about the double-!odel flow. The 0awson!ethod

yand

x

are nown fro! the double-!odel solution, and

0,0 =

=z

H due to sy!!etry

=2 0=

+

+y

h

yx

h

xz

""lied at z=3:

yyxxyxU

gh

+

= 2

2

1 222

. &inearization about the "reious iteration

. 0ouble-!odel solution'. %ole the "roble! using the 0awson free-surface '++. ""ly the full linearized free-surface '+ on the way free-

surface0. $terate

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Numerical method

The free-surface is a""roxi!ated by flat "anels haing aconstant source strength.

&inear equations can be added to the syste! of equations:[ ][ ] [ ]BA =

i4 and 'i are !ore co!"licated for the free-surface "anels.

The free-surface '+ contains elocity deriaties.The x-deriates are co!"uted by a four "oint u"strea!

o"erator.5ne additional effect of the four "oint u"strea! o"erator is to

introduce the free-surface radiation condition of no u"strea! waes(0awson).

ill sole the linear syste! of equations by 6aussianeli!ination or by using an iteratie soler

+o!"ute elocity co!"onents:

x

NE

j

jijxi UXU =

+=1

yNE

j

jijyi UYU =

+=1

z

NE

j

jijzi UZU =

+=1

The "ressure fro! 'ernoulli equation:

22

21

5,

=

=U

gz

Uo

ppCpi

The wae height is co!"uted fro! the dyna!ic free-surface '+

zzyyxxyxU

gh ++= 2

2

1 222

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