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OCR for page 103

A Numerical Research of
Nonlinear Body-Wave Interactions
Z. Zhou, M. Gu (China Ship Scientific Research Center, China)
ABSTRACT
This paper presents numerical research
results of nonlinear body-wave interactions.
The body-wave interaction is treated as a
transient problem with known initial condi-
tions. The development of the flow can be
obtained by a time-stepping procedure, in which
the velocity potential of the flow at any
instant is obtained by utilizing a source
density distribution on all boundary surfaces.
The Orlanski's method is used to implement the
boundary condition at the open boundary. The
position of intersection points are determined
by a direct method.
The contents of this paper are: (1)
A study of the nonlinear radiation problem of a
floating body; (2) A study of nonlinear wave
diffraction problem around large structures;
(3) Interaction of nonlinear waves with a free
floating body; (4) An attempt in generating a
numerical wave tank. Pretty good agreement is
met between the numerical results and the
analytical solution.
NO, lEt~CLATUR`E
a cylinder radius
d still water depth
F. six components of force/moment acting
~ on the body.
9 the acceleration due
H incident wave heisht
, Id, Iz moment of inertia about
x,y and z
K wave number
K wave number of the
m mass of the body
t] number of elements on Sb+Sc+Sd+Sf
~I~,Nc number of elements on So, Sc
Nd,Nf number of elements on Sd,Sf
SO wetted body surface
Sc outer boundary or
SO bottom surface
Sf free surface
Zig vertical position
wave elevation
p fluid density
~ total wave potential
Zhenquan thou and flaoxiang Gu,
China Ship Scientific Research Center, P.O. Box 115, Audi 214082, Jiangsu, China
to gravity
axes
velocity potential of incident waves
(s scattered wave potential
I . INTRODUCTION
The interactions between large floating
structures and the sea wave are generally
predicted on the basis of linear ship motion
theories, which are formally valid for small--
amplitude sinusoidal
forces, the methods which have been applied so
successfully to the linear predictions are no
longer available. Therefore much attention has
been paid to perturbations involving second
order potential in the past few years. However,
when the wave is very steep and the nonlinear
effects more serious, it may not be appropriate
to consider only second order forces.
waves. For nonlinear wave
wave envelope
control surface
of the body centre
An alternative approach to the nonlinear
body-wave interaction problem has been adopted
by Isaacson [1], in his model, the nonlinear
wave-body interaction is solved numerically by
a time stepping procedure. In this field, Lin,
Newman and Yue [2] have presented a method of
matching the finite computational domain to a
linear outer solution for the transient heaving
motion of an axisymmetric cylinder. Liu and
Yangt3] have presented the study results for
nonlinear three-dimensional but axisymmetric
free-surface problems using a mixed Eulerian--
Lagrangian scheme.
The main difficulties in simulating fully
three-dimensional nonlinear interactions
between a free-surface and a body are as
follows: (1) The treatment of the body and
free-surface interface. A confluence of
boundary conditions exists at the line of
intersection of the free-surface and the body
surface. As a result, the panel solution
exhibits a singularity at points on that line,
which is the one of the main causes of diver-
gence. (2) The selection of suitable outer
boundary condition or radiation condition. The
outer boundary in a numerical model is a
control surface surrounding the body and waves
which reflect the requirement at the far field
in guaranteeing for the uniqueness of solution.
To accomplish this the outer boundary condition
03

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is made to fulfil the function of allowing all
the outgoing radiated and scattered waves to
pass through without any reflections into the
inner wave field. (3) Stability of the free
surface. In the case of nonlinear waves
the wave slope is large, which is conducive to
unsteadiness, and hence disrupt the computa-
tional effort to reach a steadystate solution.
(4) The interaction of the body motion and wave
motion. For each time step, it is necessary to
re-determine the relative positions of the body
and the wave, which in turn needs high computa-
tional accuracy.
It is realized from the very beginning
that in dealing with nonlinear computations of
body-wave problems, the method adopted at the
present state-of-the-art depends on the
target aimed for investigation, if it were the
local phenomenon such as the nature of singu-
larity at the body-wave intersection and of
wave breaking and/or spray formation were aimed
at, then full attention should be paid to
the treatment of the local singularity.
However, if it were the global forces and
moments acting on the body that is aimed at,
then the method should adopt a pragmatic
treatment of local singularities, which paying
attention to the rest of the problems to keep
the computational treatment robust and effic-
ient. The method outlined in the present paper
follows the latter philosophy. In the present
method' the body wave interaction is treated as
a transient problem. The development of
the flow is obtained by a time stepping
procedure, the velocity potential is obtained
by utilizing a panel method, in which simple
source are distributed over all boundary
surfaces, including the free surface, the
immersed surface of the body and the bottom
surface. Plane quadrilateral surface elements
are used to approximate those surfaces,
and the physical variables are assumed to be
constant at each element. The center of each
element is used as the pivotal point. The
integral equation for the source density is
replaced by a set of algebraic equations. When
this set of equations are solved, velocities
at and off the surfaces are obtained.
The contents of this paper are:
(l) A study of the nonlinear radiation problem
of a floating body. (2) A study of the non-
linear wave diffraction problem around large
structures. (3) Interactions of nonlinear waves
with a free floating body. (4) An attempt in
generating a 3-D numerical wave tank.
2. MATHEMATICAL FORMULATION
As in Fig.l, an arbitrary body is shown
floating at the free surface and moving in
large amplitude in waves.
Let x,y,z from a Cartesian coordinate
system as indicated in Fig.l, with x measured
in the direction of incident wave propagation
and z measured upwards from the still water
level. The fluid motion is described by a
velocity potential ~ which satisfies the lap
,;~/'/ BY
5! I' ~/~7
Scan ~
I (T ' 1 - t- -- if'
1
~ ~1
Fig.l Coordinates systems
lace's equation within the fluid domain,
V2~(Xt ye it t) = ~
the boundary conditions are as follow:
a¢(x,y,z,t) = van on sb (2)
it = at ~ at 30 + ii 30 on Sf (3)
3$ + go + 1(v$.v¢) =o on Sf (4)
3¢ (x,y,z,t) = 0 at z = -d (5)
also satisfies a suitable radiation condi-
tion to be discussed later and a proper initial
condition.
In order to solve nonlinear interaction of
the Body and wave, The development of the flow
is obtained by a time stepping procedure, with
the velocity potential field at any one
instant obtained by a panel source method.
In this method, ~ is represented as the
velocity potential of the field point which is
evaluated by
¢(x,y,z) = i:~(q~r(l rids
am
where rid p , q) is the distance between the
points p and q. The normal derivative of the
integral in (6) at p of the surface 2Q is
Vn = -~2~(p)+n~p)~(q) -,qyds]
am
where the unit vector n is normal to the body
surface and pointing outwards into the fluid.
In order to make the computer program
extensible to arbitrary bodies of 3 dimensions,
the surface So, Sc, Sf are divided into finite
number of elements. Assuming that the source
104

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where
density is taken as constant over each element
the potential at p of Eq. (6) may be written in
a discretized form as:
N
hi = i. tjjcj i=1 2 rat (8)
~ i j = | J s j [ ( x-~; ~ Z+( yen ) 2+z2 ] 2
The normal induced velocity is then
N
where
ni j-1 ii i i=1,2, ,~1
(-2n j=
i ~Jsjr3
ilk
In conducting the time stepping procedure the
boundary conditions at each time interval t+6t
are established in explicit forms containning
quantities at times At and t- At which are
known from previous iterations e.g:
(3n~t+At F ($t ~t-At~3n~t' On~t-At) on
fit ~ FC(¢t, ~t-At ad t al~t-At) on
Calf At = Fort (t-At 3¢ t 3n~t-At) onS
f 3.1 Formulation
(an~t+At = F\t ~t-At An t an~t-At) on
(12)
Takinginto account of (8) and (10) above and
the boundary conditions (12) a set of surface
integrals on source density cite At) may be
up-dated to the advanced time interval t+At
thus:
N
j-1 id ~ Y Z t+At) = Gj .t+~t i 1 (13)
in which the matrix of coefficients are
and
K..
id
Mjj tij
K..
id
r.
~j =
~d+1~-~'Nd+Nc+Nf (14)
i Nd+~c+Nf+1' ~ · ·'N
~d
an i t+At
. c
Fuji t+At
(~) tat
(3n) t+At i=Nd+NC+Nf+1 N
the velocity of a point in the field may be
expressed as
i=Nd+1, . . ., Nd+NC
=Nd+NC+1, · · ·, Nd+NC+~'lf
j =
Whenever the velocity potential ~ and the
velocity v as well as the body motions are
known at time t the quantities of these
variables at the advanced time interval t+At
may be evaluated according to the following
process. Update the wave elevation over
the free surface Sf at time t+ At; according to
the motion of the body determining the new
position of the body' which modifies the
immersed body surface at interval t+6t under
the new free surface Sf~t+At); in the new fluid
domain Q(t+ At) re-calculate the coefficient
matrix Mij' and update the matrix of boundary
conditions Gj t+At;finally find the solution of
source density ott+At) at the interval t+At
from (13~. The pressure p over the body surface
is obtained by the unsteady Bernoulli s
(11) equation:
p=_p tgz+$t+2V¢.V~ ~ (17)
The force components Fk may be calculated as
appropriate integrations of pressure
ok =-J:PnkdS
(18)
,.
where subscript KC1 2 ... 6 and Fk correspond
to fores in and moments about x y z direction
respectively.
3. THE NONLINEAR RADIATION PROBLEM OF A
FLOATI NO BODY
As indicated in Fig.1 consider the forced
heaving motion of an arbitrary floating body on
the free surface a Lagrangian s description is
used the kinematic and dynamic boundary condi-
tions (3) (4) are rewritten as:
Ox _ a`. Dy an Dz _ arm.
Dt ~ ax Dt ~ ay Dt Liz `19'
Dt = 2V$.V~ _ 9D
The above free-surface conditions are discre-
tized by an explicit time-step scheme as
follows:
xp~t+At) = xp~t)+26t{3(at~t-(D8~)t-At}P
yp~t+At) = yp~t)+2At{ 3`a¢'t~`aa¢'t-~t} P
zp~t+At) = zp~t)+26t{3(a$)t~(a$)t-At}P
On the pivotal point the potential
is:
(20)
. ~ t+lit )=¢j ~ t )+2At{ 3( 2V¢.V~-go it
_( 27~9rl~t-~t} ~ 21
j~l~ciixel+cijye2+0idze3~¢i (16)
Let the body be surrounded by a verti Cal
control surface Sc . Ili thi n the immedi ate
vicinity of Sc . the scattered wave element may
be approxi mated as a pi ane wave, propagati ng
outwards with celerity along a direction 1.
Assumi ng that the scattered wave near the outer
105

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boundary satisfies the Qrlanski's [4] condi-
tion, one have
an + cR`ty3¢ = 0 (22)
where CR is the phase velocity for potential
function. Let R denote the intersection point
of the free surface and the outer boundary, and
point R-1 a neighbouring point AR distant from
point R along the radial direction, the phase
velocity CR is then obtained from ~ of the
free-surface as:
t -¢tR l+¢tR 1
RC = 4~_1
26t(Vr)~-1
(23)
Considering that the scattered wave should be
outgoing waves, Cp > O. In practical computa-
tions, the limits on OR may be set as [~]
fat if tR-At
C~(t) = c(t) if O-CR At (24)
lo if CR<0
where CR(t) is obtained from Eq. (23).
The solution of the radiation problem may be
obtained from (13), together with the boundary
conditions listed in (19)-(24).
7.2 The Panelling of The Instantaneous '1etted
Surface Near The Free Surface.
At intersection line of the body and the
free-surface a confluence of boundary condition
exists. As a result, the solution exhibits a
singularity at the line. To accommodate this,
we determine the position of intersection
points by a direct method. From a physical
point of view, the fluid particle may not
penetrate into the body , it may only slide
along the body surface. Therefore, in the
numerical model, the vertical position of the
intersection point is obtained by interpolation with the body draft
along the radius form wave elevations. The
horizontal positions (x,y) of the intersection
point may be obtained by setting their absolute
increments as zero. Another advantage of this
method is in using the centre point of the
panel as the pivotal point, this avoiding
singularity at the intersection point of the
body and the free surface.
At the intersection of the freesurface and
the outer boundary, the vertical position of
the intersection point is determined by
allowing it to be the same as the wave eleva
tion calculated at the outer boundary surface.
For each time step, the immersed body
surface changes in accordance to the motion of
the body and of the wave surface. There are
several configurations which the panel on the
body may be intersected by the free surface
Anti. Fig.2.
?
Y:,5
Fig. 2: Nodal points on the intersected
body panel.
Let (x1 AYE ~Zl ~ and (x2 ~Y2 ~Z2 ~ be two
nodes on a panel line, which are located under
and above the wave surface acts respectively,
thus, the new node on the wave surface nuts is:
z0 = Anti
z -anti
Yo = Y1 + (Y2-Y1 ~ ~
z~-~(t)
X0 = X1 + (X2-X1)
(25)
3.3 ('umerical Examples of Large Amplitude
Radiation Problems and Discussion
In this section, results are presented for
the case of forced heaving motion of a floating
truncated vertical circular cylinder of radius
a and mean draft a/2 . The length, time
and mass units are so chosen that the radius a,
gravity 9 and density p all equal unity. The
vertical velocity of the body is prescribed to
be
V(t) = Hmsin~t
(?6)
in(t) = -2 - Scout (27)
The total number of panels is 232, with
tid=26, '''calls' 'Jf=84' "!b=74
Let the radius of outer boundary be P=5.0,
amplitude be ',4 =0.0S, frequency w=~n, time
increment be ~t=O.l, and water depth be d=4,
T=24, i.e. a total of 6 periods.
The comparison between Lin's resultst2]
and our results for the above problem is shown
in Fig. 3, it seems agreement is good.
3.3.1 The Effect of Different Radiation
Co-nditions Adopted for The "onlinear
Radiation Solution.
Two groups of computations are carried out
to investigate the effect of different radia
106

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.~ gin i.? 'i't 9-\ . '. ~ (- c,
4
/
Fig.3 Comparison of 'neave force 'nistGry
- present result; *8 Lin's result
r24.
tiGn conditions. flare, the time interval T=l6.G
. In group hi, radiation condition as proposed
by Isaacsontl] is adopted.
~ = 0, am = ~
In group A, the radiation condition of A. (2?)
i s adopted.
278
/39
-739
-27~
~.~-
Fig. 4: Co~,parison of 'neave fGrce for different
radiation conditions. ~ v v
group A; - group B.
to comparison of vertical force acting on
the cylinder in both cases is shown in F j g. ~ .
In the first one and half period, the result
of group ~ is consistent ,,~it'n that of group B.
But as the time i ncreases, there begins a
difference between them; the period of group A
becomes gradually less than that of group .'9.
Thi s phenomenon ari ses from the ref 1 ecti or of
scattered saves in group Hi. fit the beginning,
the scat-tared Haves d i d not reach the outer
boundary Sc , therefore results of t'ne two
groups are consistent. 'hen tine scattered -eves
reach the boundary Sc, in the case of group ,$
the condition of (~) induces a ingoing
reflection of the scattered slaves. It is the
interference of the scattered waves that causes
the chage in the period of the wave forces.
3.3.2 The Effect of Changing The Distance of
The ~u or ~ ~,~ ,~7~ = t~,~ Ike
()rigin.
A comparison of three groups o, calcula-
tion using different radii of the outer
boundary surface are carried out to investigate
the effect of changing distances of outer
boundary surface. 'here (a) 0=5.O, (b) R=3.0,
(c) R=lO.~. Fig. 5 shows the time history of
wave elevation at an intersection point of the
body and free surface. here, the result of
R=Q.Q is consistent with that of R-lO.O, but
the result for P=5.0 is different. For the
first cycle, (a) is consistent with (~)
and (c), but difference of amplitude appear
when time interval increases.
(as) Am/
107
Fig. 5 History of an intersection point. ---
group a; -** group b; 0 oo group c.
The free surface profiles in the radial
direction corresponding to groups (a), (b),
(c), for t=6.0, S.0, 14~6r · are shown in Fig.
6-~. For t=5.0' groups (a), (bland (c) are
consistent. For t=3.O, a slight reflection
causes the difference between group (a) and
group (b), (c). For t=14. 4 . inconsistencies
Bevel oped even between c)~=0, and 9=1(3. Lear the
body surface, the wave prof i 1 es are consi stent,
but near the boundary S c t',ey are different.
Fi g. ~ i s t'ne pressure on the Cottons center
panel . Fi g. 10 i s the verti Cal forces acti ng
on the cyl i nder of group ( a), ( ~ ), (c ) .
It i s shown that by changi ng the di stance of Sc
form, ha origin little effect is introduced as
far as the hydrodynami c forces are concerned.
Fro,-.n the above co~-`,pari sons a concl usi ve
rears may be - drawn: For the Orlanski's
condition, chant the radius of the outer
boundary causes 1 i ttl ~ i n f 1 uence to the wave
force but may have a l arge i nf l fiance to the
wave profiles. For coruputatiolial stablity it
is advi sable that t'ne outer boundary surface 'ce
place'! at a distance sufficient!: far from t'ne
'cocky.

OCR for page 103

~E-3 E-2
/440+
933
467
00t
A
-4~1
_~
-/4lX1
~F,_~
Mp,''-
~~l~v~A~' At, R
9.33
4.67
000
-4.67
-933
20.~ .
fly
667
0.00
-6.47
,3
_
Fig. 7 Free surface profi les (tag. 0~.
_*_group a;-O_grou? b;_ ^~-
group c.
F_3
~-_
-~ ~00 -ado too
, E-l
Fig.8 Free surface profiles (t=14.4)
- *- group a; - 0 - group b; -6 -
group c.
108
Future
_/~ - ~, R
/0~#
E-J
00 ''28.~ 4~ ~00 92~0
in'
,/
;,!
Fig. 6 Free surface profi les (t=6)
_ *_group a; _~- group b; -~-
group c.
32-~
Ad/
49
4
47~9
.02
A
f4
i.76
/3&
0.00
is too' SOLO 6~00 0200 NATO -f38
[-1 -2.76
_~4
i
t' 3.,/' 6.~i 9~ I`?.72 Of..
Fig. 9 Pressure on the central element
*** group a; ~ 0 Q group b,
Err, ~,.'rc~
Fig.10 Heave forces for different
radii. *~*** group a;
~ 0 0 group b; - group c.
3.3.3 The Effect of Chanqinq the Amplitude of
Body's Oscillation.
Generally, large-amplitude motions result
in nonlinear phenomena. Four cases of large
amplitude forced heaving motions are investi-
gated in which R=8.0, w-ln, d=4.0, T=16.0,
t=Q.1 and in case (a) H =0.05, da=H'=lQZ, h is
the mean draft. (b) t] =0.10, ~ b=20/o' (c)
H=0.15, dc-30% (d) H=0.25, Gd=50%.Fig.11-14
show the radial free surface profiles for the
above four cases. With increasing amplitude,
the wave profiles change drastically as
steeper waves appear and greater amount
of wave -energy are transported outwards. Fig.15
shows wave surface profiles of four cases of I]
(at t= 9.0 ~ respectively. In each
figure, configuration and phase of wave
profiles for these four cases are the same, but
the greater the oscillation amplitude, the
steeper the wave and the stronger the non-
linearity.

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Fig.16 is the time history of a body wave
surface intersection point which is the same as
the wave elevation on the body surface. It is
shown that a larger wave motion appears for a
l arger body osci l l ati on ampl i tude, but the
peri od and trend are the same. I n the case of
the l arger ampl i tudes, there appears more
asymmetry in the curves which easily results in
i nstabi l i ty of the numeri Cal process. Fi 9. 17
shoals the pressure acti ng on the bottom centre
panel. A distortion at the peak of the pressure
'history as a result of strong nonlinearity
appears when the osc i 1 1 ati on arr~pl i tude becomes
too 1 arge, whi 1 e a drop i n pressure appears i n
the very begi nni ng. Si nce the body starts to
rise from rest and form its initially displaced
pos i ti on at the botoom of the stroke. The
compari son of the verti Cal hydrodynami c Force
acti ng on the body i n Fi 9. 18 shows l i ttl e
change for the four cases consi dered.
It i s shown that the vari ati ons i n the
radial free surface profi les and in the local
pressures are much more serf ous than i n i ts
vertical forces for different ampl itudes of
oscillation. Thus the nonlinear effect should
be considered when deal ing with a local
strength probl em i nvol vi ng i ntegrati on of l ocal
pressures.
300
Loot
zoo
J.oo
/
^~00
- ~
_ 2.40 3.~0
42 5/
3605
70~ I,, 7'
die ~
Fig. 15 Wave surface profiles for different
amplitudes (t=9.03. -~ - case a; 33
- - case b; -~- case c;
case d. 165
4. NONLINEAR WAVE DIFFRACTION AROUND LARGE
STRUCTURES l6S
4. 1 Formu l at i on
In approach to the diffraction problem,
the wave diffraction is treated as a transient
probl em wi th known i n i ti al cond i ti ons corre-
spondi ng to sti l l water i n the immedi ate
vi ci ni ty of the structure, and `'i th a pre-
scribed incident wave form approaching froin x=~
and propagati ng past the structure, Fi g. 19.
685
437
?2t
Lid
a
-20 2~4
-451
6~
5542
238~ ~ 9 S2 iIdb
4.7~Il4 ~It9
Fig. 1& History of an intersection
point for different arr,plitud.~s.
E_
~ ~ rat
~, .
00 2.3' 476 7/4 9~'
Fi 9. 17 Pressure on the bottom
central panel.
-3.'
~:_~\t~^~ t-0rGe
Time
1/ 2~\g, S.°,S :8.93 V~'
Fi g. 18 Heave Forces for di fferent ampl i tudes.
-case a; 0 0 0 case b;
666 case c; ~ ** case d.
109

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Fig. 19 An isometric view of body-wave
i nteracti on. H 2 /d=1 /3, t=1 . 3.
then incident waves exist, Lagrangian free
surface conditions of the form (10) clay easily
l ead to i nterpenetrati on of the slave surface
panel and the body surface. I n order to avoi d
the difficulty of shaping and rearranging, the
free surface panel, a set of Euleri an free--
surface cond i ti ons i s proposed here.
at ~ n = 0 ~ (29)
at + ire + 2`v¢~2 Q:
For the outer boundary condition provided
that i t i s lyi ng at a sui tabl e di stance from
the bod:', then i n the begi nni ng ,oeri od when the
scatterer] waves are sti 11 in the inner domain,
the outer boundary condition may be written as
the Isaacson ' s versi on, i . e.
an a¢O
an On ~ = to (30'
however, when T > Tc, where Tc is the tine when
the scattered waves travel l i no at i ts group
vel oci ty Yogi n to meet the outer boundary, the
Qrlansl i's condition is adopted:
at ~ C,p(KrarS + K~r¢9) = 0
K = 06.a~s /,/(a~s)2+(a~s)2
Kin = <~.a~s /l/~2~(3'~s)2 (32)
Here
ec
J1 if apse O
an
t-1 if a s ~ 0
I t i s assumed here that the scattered wave po-
tenti al
As = ~ - ¢°
It must be mentioned that a note of n~athemati-
cal i nconsi stency occurs here as for nom i near
wave dynamics, the principle of superposition
i s not stri ctl y appl i cabl e.
For the initial condition, as there is
i n i ti a l l y no i no i dent Slave moti on i mmed i atel y
ad j acent to the body, SO the scattered poten-
tial is initially zero.
I n our numeri Cal procedure, the corner
point q0 of the wave surface panel is deter-
mi ned by the area average of i ts four surroun~-
ing panels, let rl, s represent wave elevation
and area respect) vel y, then
~-~+~eDo+~+~Oa
No= (3~)
iSA4SB+~SD
For each tirre step, the adjusted wave surface
wi 11 truncate a part of the body panel s near
the free-surface. If the re~,ai ni ng part of the
body panel becomes seal ler then £, a predeter-
mi ned number, then i t i s del eted from the body
surface to avoid divergence.
4.2 humeri Cal ~xampl es
~ corn,outer program which incorporates the
method descri bed above has been used to
generate results for a few specific situations
i n order to establ i sh the practi Cal vi anti l i ty
of the method used. Suitable comparisons
with avai fable results may be made only for
relatively few restricted cases for which known
di ffracti on sol uti ons are avai l abl e. [ 1 ~ .
d.2.1 T'ne Diffraction of Small A.molitu de
'flames Past A Verti Cal Ci rcu l ar Cyl i nder
A surface-pi erci ng verti Cal cyl i nder i s
sub jected to a 1 i near _ i nci dent .lav=. here,
d/a=2, Fi/a=n. 1, Ka=1. 5, k/k=~. S.
Fig. 2~) shoes a) the incident wave
elevation at x-O, b) the time history of the
hori zontal force acti ng on the cyl i nder. The
resul ts obtai net! front our program i s shown i n
dotted l i nes, the pre;di ctec force vari ati on
agrees wi th that pi ven i n ~ 1].
The program has al so been tested for the
case of z surface piercing vertical cylinder
sub jected to a ,arescri bed sol i tary cave tGr`~`
(31) given by
~ = Hsech2 [R(xs-Ct) ~ (36)
w'nere T= /3~/4d~, Xs=X+a+3.~/?, and d/a=~.5,
H/d=~. 1.
Comparison of results with closed-forr~
solutions giver, by Isaacson t1 ] is presented in
Fig. 21. The solid line indicates the closed--
form solution given by Isaacson, the broken
line is t'ne numerical results calculated By
I saacson ' s model ~ 11 , and the dotted l i no i s
the numerical result calculated by the present
(33) model. Results evaluated by this report
serves to highl ight the 1 act that the outer
boundary condition presented in ~ hiis section is
capabl ~ of al 1 owi ng the outgo) ng scattered
`.'aves to pass through efrectiv-~ly wit'.. 1 iti;le
(34! reflection inwards. It also enables the
110

OCR for page 103

computation of wave forces to run on for a
sufficient duration of time so that the wave
motion is fully established at the vicinity of
the body.
4.2.2 The Diffreaction of Large Amplitude
Shallow Water leaves ~
Results are presented for a circular dock
fixed at the free surface subjected to a lrage
amplitude shallow water wave. let d/a-1.5,
-I.| / ~t/T
to F/pd~ad
W~
00 - ~
-/.0
Fig. 20: a) The incident wave elevation
at X=O. b) History of horizontal wave
force. --- closed-form solution [13;
o o ~ present results.
~2
F
.
-/ fit
1` 1
4 ~6
~\ ~em.
--2
quit
·~'
Fig. 21: Horizontal wave force. - closed-form
solutiont13; --a. -- numerical
resul ts[13; ~ ~ , present results.
h/a=~. 5, d/gT -0.018, and Hi /d=1/4, H2/d-1/S,
where h is the draft of the dock, T is the wave
period, ill, H2 are wave heights.
Fig. 22 shows a) the incident wave
elevation at x=O, b) comparison of horizontal
forces for cases of two wave heights. The
dot-chain line represents the result predicted
by liner wave theory [53. The broken line and
the solid line represent results given by
the present theory for H/d=1/4 and 1/3 respec-
tively. There is a 00° phase difference
between the wave force and the incident wave
form. The maximum force coefficient occurring
at t/T=Q.488 are respectively 15% and 217
greater than that of the linear-theory predic-
tion.
The total wave profiles occurring for
diffraction around the fixed dock are presented
in Fig. 23. It is noticeable that the steepness
is larege. In Fig. 23, it is seen that
the incident waves diffract around the body,
and propagate past the body, comparing the wave
crests at two sides of the body, a little delay
of the phase may be noticed found. It shows
that the wave behind the body may come from
diffraction of the wave in front of the body.
Since the wave is obstructed by the body,
the wave elevation in front of the body swells
up to a val ue, much hi gher than that behi nd the
body. As a result, a pressure difference is
formed and a periodic wave force is generated.
Fi g. 19 i s an i sometri c vi ew of the
body-wave interaction showing the mechanism of
di ffracti on wi th Li /d=1/3, t=1 . 3.
E -!
Jon
s.oo
o,oo
a
-moo
-10
-1
/
sot
ALL
moo
-1000
F/p~la2
, ~ +/r
~ . . ~ . .
no too moo 600 ~E ~
Fig. 22: a) Incident slave elevation at X=O. b)
Hori zontal wave forces. --. -
linear diffraction solutiont53; ---
H1/d=1/4; tl2/d=1/3.
5. IFITER;ACTIONS OF 51Q~ILIi'5~AR SURFACE '`lAVE
WITH A FREELY FLOATI FIG BODY
5.1 Formulation
In order to consider the effects of three
dimensional nonlinear interactions between slave
and body, the boundary conditions on the
immersed body surface are expressed in terms of
velocity Uk (k=1,2,...6) which are the six
velocity components in the moving co-ordinate
system fixed to the body. Thus
3$ 6
(37)
The six values of Uk may be determined from the
dynamical equations of motion:

OCR for page 103

F1
6=
or siniply,
F1 = m(Ul+U5U3-U6U2)
F2 = m(U2+U6Ul-U4U3)
F3 = m(u3+u4u2-u5u1)
F4 = Ixu4-( Iy~Iz)u5u6
F5 = Iyu5~( Iz-Ix)u6u4
Izu6 (Ix Iy)u4u5 w,,=. ~
(3S) may be expressed as jj Q-~¢Q;kzimknkQnkj/`s' (49)
which was neglected in his num,erical
computati ons.
Final ly, substituting Uk in (45) into the
body boundary cond i ti on ( 37 ), the body surface
equati on may be rewri tten as:
N 6
i2-1 (Ki j-Aj j )0i = k£1nk; Hk/mk
( 38)
where
i=1,2, ,N (48)
Fk mkuk + fk
llere the external force F k may be grouped into
the fol 1 owi ng subgroups
Fk FAk+FBk+FCk (4Q)
where FAk denotes the fluid dynamic forces and
moments of potent) al nature, FBki s that due to
gravi ty of the body and FCk that due to
external moori ng forces or vi scous darnpi ny
forces.
The f 1 u i d dynam,i c ~ orce components
are:
FAk -P j_3 n k j ~S j [ 9Z~ - vev+2 ( V~ ) 2 ]
substi~utin~q (,)~1 ), into (39), we have
+ p Zb a~ ~s
- p Z~ { nkjASj [9Z-Ve.V¢+2(V¢)2 ] j}
fk + F3k + FCk (42)
I n the coord i nate system f i xed to the body
center G. nkj6S do not vary v.'ith time for the
tin~e step consi~ered, thus, (42) may be written
as
aHk
at (t) = ink(t)
and for the next time ste,~ t+At.
where
Hk(t+At) = Hk(t)+-At{ 3hk(t)-hk(t-At)} )
H k ( t ) = m kU k+ P j _ 1 d~j n k; [S;
hk = - PjZb{0kj [Sj [gZ-Ve-V¢+2(V¢)2 ~ jI
fk + FBk + FCk
(46)
Ve is the velocity at the point considered on
the body.
V = V + ~xr
e 9
(47)
Vg i s the vel oci ty of the center C, r i s the
posi ti on vector rel ati ve to the ori si n 0.
The ter,n -n`tS j~e.v

Fig. 24: a) Incident wave elevation at X=O. b)
Heave vari ati on for a f 1 oati ng
doc k . --- H 1 /d=1 /4; H 2/d=1 /3 .
-/o. 00
E-1
A0,~ 1/d
§.o
000
G
-500
,/~OOIF~/~'0
i
.~o
o.oo ~,
-7§oo 00 y 4.00 6.00 &.
-/~oo . E-l
a fixed dock would easily satisfy the initial
condition as reguired by the present paper'
vi z. at t-O there i s no i nci dent wave immedi-
ately adjacent to the fixed dock.
The hi story of hori zontal wave force due
to diffraction by the fixed dock is shown in
Fig. 29. Here, d/a=1. 5. A maximun~ horizontal
force coefficient of F/ p g`1a2 =Q.65 seems
reasonabl e compared to Fi 9. 22.
~~[~ ~-Y~
1 ~ ~
1 ~ 1
~I ~ I
' ~ '' i
: R/a-8.-
Fig. 25: a) Incid~nt wave elevation at X=Q. b) -'
Hori zontal wave , orce. ',1 /d=1/~. '/0`
6. A~] ATTE`VlPT TO SIM~JLATE A t:!UMEnICAL '~'A'JE
TANK
A numeri cal ~nodel i s constructed for a
cyl i ndri cal wavernaker operati ng i n a seakeepi ny
tank of finite depth but of infinite dimen-
s i ons. Attenti on i s focussed on numeri cal 1 y
simulating the nonlinear waves with a wave
front and acti ng on another body i n a numeri-
cal seakeepi ng tank. The fi rst step i s
to generate numerically the waves simulating
3-D nonlinear waves generated by a wavemaker.
The second step i s to s i mu l ate the tan k
experdr,ient by requiring these numerical `~aves
wi th a wave front to act on cornputati onal
model s of l arge offshore structures or shi ps.
The attempt to justify the prediction of
nom i near ~`ave force on an offshore structure
nur`~eri cal l y i s a real i sti c proposi ti on.
For illustration, the results of simulat-
i ng the di ffracti on ~x,oeriment of a 1 arge
floating dock , ixed on the free surface and
subjected to an approaching nonlinear wave with
a wave front generated by a cyl i ndri cal wave
maker numeri cal l y are presented.
The computi onal model i s as Fi g. 27. The
cylindrical wavemaker is a circular floating
dock operati ng i n a forced heave mode, the
i nstantaneous draft of whi ch i s ex,urnssed as
h~t) - -DF+Hsi n~t ~ 50)
where the vel oci ty of i ts centre ~ i s accord-
i ngly:
Vg~t) ~H - . cos~t (51 )
l~lere, OF/a=l.O, t] /a=0.2, d/~=~.O. m=~
The generated wave time history ¢(t) and wave
potent) al hi story 0(t) at ~ poi nt ,~ ( x/a=-O. 8,
y/a=~) are shown in Fiy. 2~. The nun~erical ly
yenerated wave al ong x di recti on i s a propagat-
i ng wave with a front, which xYhen acted on
Fig. 27: Computional model of a numerical tank.
E -` ~
:~6.00
-Ao.oO
-fs,oo ,¢
10' i~-640
2.~0
0.00
-2.gO
-500
Fi 9. 28: a) The generated numeri cal wave; b)
The generated wave potent) al .
tE -1 F/egHa~
-5o
~,, C' ~ ~
i0. ~20 30 \40 / 50. E~]
~ ~ 5 0
_ _10
\J
Fi g. 29: F1ori zon ual wave force.

OCR for page 103

Fig. 11 Nonlinear radial free surface profiles, case a, amplitude H=0.05, 6=107
(l

Fig. 13 Nonlinear radial free surface profiles~case c, amplitude H=0.15, 6=30%
(1<~<8, t=0.1, 0.3,...)
Fig. 14 Nonlinear radial free surface profiles' eases d@ amplitude H=0.25, 6=507
(l

Fig. 23 The wave profiles occurring for diffraction around a fixed dock. [/d-1/4
(-9.0

7. CONCLU S ~ 01~
A numerical study of body-wave interaction
as an approach to 3-D nonlinear sea!~eeping
problems has been attempted. For illustration,
several numerical examples are presented to
demonstrate the influences of the change of
different ,iJara,~.aeters, several i nteresti ng
remarks flay be dra``n.
~1 ) Regarding the
outer bounder: condition,
ecuat.i on i s adopted as
treatment of the
al though ncl anski ' s
, an open boundary
condition, the position of the outer boundary
sti 1 1 counts. It resul ts i n a more serf Gus
i no 1 thence on the wave surface prof i 1 es than on
the forces. Therefore for di fferent engi r`eer-
i ng probl ems' ~ di fferent radi us of outer
boundary may be adopted.
(2) General ly there is ~ difference
between the resul ts predi cted from nom i near
theory and that from 1 i near theory. So,i,cti,,ies
al though thi s di fference i s not 1 urge from a
gloha1 point of reviewer, it flay exhibit sir~nifi-
cant di f~erences at 1 cca1 areas.
(3) The blockage due to a large fixed body
to the wave wi 11 cause local swell up. The
pri nc i pal zone of i nf l uence relay be restri cted
to a sn!al 1 zone ad jack the boa>. If the
zones are ~ 1 ittle further from;, the body, the
cleave tori ons are rnai n 1 y i ncoIiii ng waves.
Thi s phenomena may al 1 ow us to treat the
bcdy-~'ave interaction by matc!-'ing tale nom inear
sol uti on i n an i nner dori,ai n near the body \~'i th
1 i near sol uti on i n the outer domai n.
(4) An unfavorabl e phasi ng of wave
and body ~;~oti on i s demonstrated i n whi ch
shi ppi ng of green water or sl ammi ng nay occur,
especi al 1 y i n the case of 1 arge all i tube
moti ons.
tan k of
novel idea o, ~ heaving cylinder is attempted,
and validated by cor~iputa' ion t'nat such waves
`~i th a wave front ;.~aiy be brought upon an
offshore structure wi th proper i ni ti ail condi-
tions and nu~.r,erical ly treated to Field non-
1 i near wave forces of the ri ght order.
(5) Difficulties associated with the;
i ntersecti on poi nts of the body ant! the
wave-surface has been ,;:ragmatical ly resolved by
the i nterpol ati on on the free-surface and
avoiding the singularity at the intersection
point. This may not be justified for the local
phenomena, but may be appl icabli~ for global
force eva1 uati ons.
An effort to sinful ate a nur~eri Cal wave
infinite horizontal diriiensio,ns by a;
.FFEPAl!C~
~1 ~ Isaacson, i;] de St. O., "~Ionlir'car
Slave Forces on Large Offshore Structures",
Coastal /Ocean Ens i petri no Report, ten i vers i ty of
r ri ti sh Col ui`,bi a, 1 Q,'31 .
t2] Lin, '-J. -'i,. . t'ewman, 3.'3. and Yue, O. it.,
Nonlinear Forced rlocior~s of Floating Bodies",
Proc. 1 5th O~/S,'t1, I-Ja,~bury, Siej<:t. 2-7, 1°~!.
-l ~ ~ Li u, Y. Z. and Yang, A,., "owl i near
Radi ati on Probl eta or An Axi symi~etri c
Cyl i nuder", Advances i n T~ydrodynar,~ics,
China. 'iol.Z, r'iJ.l, -1~
[4] Orlanski, L., ''A Simple Boundary Condition
For Unbounded Hyperbol i c F1 ows", J. Computa
ti onal Physi cs 21. 197~.
~] : saacson, i,. de St. O., "nonlinear Eve
Effects on sir; xed and F 1 oati ng !~`odi es'', J. F 1 u i c'
tech., `101. 2Q: pa ~G7-281 ~ 90?.
117

OCR for page 103

DISCUSSION
T. Francis Ogilvie
Massachusetts Institute of Technology, USA
You use the Orlanski condition for closure of the computational
domain. This is a condition for hyperbolic systems. Please explain
how you are able to use it in the water-wave problem, which is
elliptical.
AUTHORS' REPLY
Orlanski's condition is a numerical condition on the open boundary.
So far, the availability of Orlanski's condition cannot be proved in
mathematics, however, it is useful in numerical computation.
DISCUSSION
Choung M. Lee
Pohang Institute of Science and Technology, Korea
We always find difficulty in determining the point of intersection of
the free surface on a body under motion. It is mentioned in your
paper the free-surface intersection was obtained by interpolation of
the free-surface to the body. Would it not violate the kinematic body
boundary condition at the intersection point? Since your attempt is
to obtain an exact solution for nonlinear body-wave interaction, the
problem of finding the intersecting point should be addressed more
clearly.
AUTHORS' REPLY
As mentioned in the first section of this paper, the singularity of
intersection point is a local problem. In our paper, we adopt a
pragmatic treatment to local singularity. We have tested many
methods for treating the intersection points in 3-D problem, the direct
interpolation method may be the best one in these methods from our
numerical test; it keeps the computation robust and efficient.
118