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OCR for page 341
TimeDomain Calculation of the Nonlinear Hydrodynamics of
WaveBody Interaction*
C. Yang, Y. Z. Liu
Shanghai Jiao Tong University,
Shanghai, China
N. Takagi
Nihon University
Chiba, Japan
Abstract S source point
The boundary element method coupled with
timemarchthg finite difference is adopted
and improved to calculate the nonlinear hy
drodynamics of wavebody interaction. The
radiation condition and initial condition
have been studied through specially chosen
examples such as the cylinder undergoing
forced heave motion or forced sway motion and
the body floating or standing in periodic
waves with wave front, and steady solution of
practical interest has been obtaind in a de
finite calculation domain by less computer
time. A few comparisons are made with avail
able solution and model test results. It is
concluded that the method is capable of pre
dicting forces due to nonlinear wave quite
accurately with requirement of median com
puter.
XG
ZG
H
C
T
a
DF
d
Nomenclature ~
horizontal velocity of the body
vertical velocity of the body
wave height
wave velocity
wave frequency or oscillating fre
quency of the body undergoing forced
motion
wave period
oscillating amplitude of the body
undergoing forced motion
radius of circular cylinder
mean water draft of the body
still water depth
radius of the outer open boundary
mass density of fluid
oxyz frame of reference, with z pointing 1. Introduction
upward and z=0 the still water ~~~
surface
In many design cases, the application of
linear diffraction theory is not entirely
appropriate for the prediction of wave forces
on large offshore structures of general form.
For 3D nonlinear free surface problem, basi
cally there are two approaches commonly used
in literature. One is based on finite dif
ference method, in which the solution of
NavierStokes equations by MAC and its various
modifications 5MAC,SUMMAC,ABMAC,IMP and re
cently TUMMAC (13~2) 0 ) are relatively po
pular. This aproach appears to have the capa
city tocope with large amplitude nonlinear
waves and even breaking, but considerable
further development will be necessary to be
realistically used due to its high cost and
need of supercomputer. The other approach is
based on boundary element method coupled with
finite difference timemarching method, first
introduced by LonguetHiggins and Cokelet (43
and then followed by Faltinsen (51, Vinje et
al. (6), Isaacson (7), Lin et al. (8), and
others. This approach is more suitable to
deal with the wave diffraction of large off
shore structures for engineering use and is
sb
Sf
n
V
n
t
At
velocity potential
immersed body surface
free surface
outer open boundary surface
unit normal vector directed outward
from the fluid region n=(n~,n,,nz)
normal velocity of the body surface
acceleration of gravity
time variable
time increment
(x,y,t) elevation of the
3D case
(x,t) elevation of the
2D case
field point
the Project Supported Partly by National
Natural Science Foundation of China.
341
OCR for page 341
adopted in this paper.
The work of this paper is to solve 2D and
3D nonlinear problems with free surface. The
wavebody interaction is treated as a tran
sient problem with known initial condition
and is solved by integralequation method
based on Green's theorem. It is emphasized to
deal with the following aspects through dif
ferent model problems in the paper: (a) the
radiation condition. (b) the point at the
junction of the body and the free surface,
(c) the initial condition for a body floating
or standing in periodic wave.Only after des
cribing these three aspects correctly is the
solution stable and practical.
Numerical calculation should be truncated
at finite distance and the smaller the domain
the better, while the physical domain is in
finite. Therefore a numerical radiation con
dition should be posed so as no reflected
waves from the truncated surface, ie. outer
open boundary. For an axisymmetrical cylinder
heaving in still water, several approaches
for the formulation of the radiation condi
tion have been tested, and it is found that
the usual onedimensional Sommerfeld condi
tion is the simplest and can give reasonable
results both for the wave pattern and wave
force. For the cylinder sway in still water,
the Som~erfeld condition is extended to 2D
case and the wave direction is determined
only by wave itself. The numerical results
show good agreement with the tested ones done
in Nihon University, Japan. For the diffrac
tion of a solitary wave upon a fixed vertical
circular cylinder, the Sommerfeld condition
is further extended as a radiation condition
by assuming that the scattering of the outer
going wave is small at the outer open boun
dary. The numerical results is appropriate
compared with Isaacson's t73 analytical solu
tion. The same works have been done for the
diffraction of a periodic wave upon a fixed
cylinder.
In the numerical calculation, Lagrangian
free surface condition is used. The point at
the junction of the body and the free surface
is determined by extrapolation for axisym
metrical flow and by satisfying both the con
ditions on the free surface and on the body
for general 3D flow,that is, according to
the body condition we can obtain the velocity
of the intersection point, by which the new
position of intersection point con be deter
mined through satisfying the free surface
condition.
In order to shorten initial transient pro
cess, appropriate initial boundary condition
should be posed. Numerical tank is good for
this purpose, and it can make the vicinity of
the body be still water.The another advantage
of the numerical tank is capable to produce
waves with any water depth and any water
bottom condition. The approach of numerical
tank is described in detail in the doctoral
thesis by C.Yang t93. A 2D nonlinear Stokes
wave with wave front is also suitable for the
initial condition. The details are described
in subsequent section.
342
2. Formulation of the Problem
2.1 Basic Equations
.
The basic assumptions are that the fluid
is inviscid, incompressible and the flow is
irrotational. Select the coordinate system
and computation domain as shown in Fig.1, the
velocity potential ¢~x,y,z,t) satisfies
V20(x,y,z?t)=0 in fluid domain (1)
00=0 on S (2)
an d
a0=v
On n
Do 30
Dt~a~ I
Dy 30 ~
Dt by j
Dz 30 1
Dt Liz )
on Sb (3)
t on Sf (4)
Dt ~ 0.570 V0 on S' (5)
Here the conditions on the free surface are
expressed in Lagrangian form. These equations
are solved with suitable initial and radia
tion conditions.
2.2 Timemarching Procedure
The free surface and the velocity poten
tial on it are calculated by finite dif
ference timemarching method and they can be
expressed in the form
xn+1=xn+O.56t(3(3a0~) ~(~0x) )
yn+1 yn+o 5^t(3(0a0)n~(~0y) )
n+1=zn+O 5At(3(0a0)n(~0) )
0 =0 +O.5At(3( ~+O.5V0 90)
( ~+O.5V0 V0) ) (7)
} (6)
Where superscript n denotes the value at
taunt. Eq. ( 7) can be rewritten as follows
1=F1(¢n,~n1,(30)n (~)n1)
on Sf ( 8 )
Where F. is a known function.
According to the mode in which body moves
we can obtain
( ~ 0 ) 1 = F2 ( An, An 1 , ( 00 ) n ( ~ ) n 1 And 1 )
on :+1
OCR for page 341
Where F is a known function for forced moron
and it c2an be obtained by solving Newton's
Law of momentum simultaneously with the solu
tion of the velocity potential 0 for freely
motion, because implicit scheme is usually
used in the Bernoulli's equation t9] for this
case.
With suitable radiation condition the
velocity potential on the outer open boundary
can be described as
0 =F3(0 ,0 '( ~ ~ ,ta0) )
c ( 1 0 )
Where F. is also a known function and can be
obtained in section 3.
Once the free surface Sf, immersed body
surface Sb, outer open boundary Sc and the
velocity potential On, normal velocity
(30/~n) on them are known (of course before
time t=n~t all of these quantities are also
known), Sf+1 and 0 1 on Sf+1 can be obtained
from Eqe.(6)~(7). ~ 1 and (30/3n~n+1 on <~1
can be determined in terms of the new points
at the junction of the body and the free sur
face and Eq.(9) separately. SC+1 and 0 1 on
Sc 1 can be determined in terms of the radia
tion condition Eq.(10). From Green's third
fomula, we have
¢(~=~ i(G(x,9 8~0(~)
~ ~ 0(~))ds (11)
Where S=Sf+Sb+Sc. Both points ~ and ~ are on
the boundary S. ~(~)=2~ for a smooth surface
at point x. Rankine source with suitable
images is used as the Green function G(x,8),
which is so chosen that the bottom condition
are satisfied automatically.
Now the integral equation (11) can be
solved numerically at time t=(~+1)At, and we
can obtain (30/~n)n+1 on Sf 1 and SO ~ 0
on ~ 1. In this way we can go further to
time t=(n+2) at and calculations can be ad
vanced over a sufficient duration.
Once the values of the velocity potetial
on the body surface are known, the pressure
on the body surface can be obtained from
Bernoulli equation represented in a frame of
reference fixed on the body.
P=p(~Ve V0+0.5V0 V0+gz) (12)
where ( a ~ 0/~ t)n+1=(0n+1_0n)/^t 0n+1 and On
are the values of the velocity potential on
the same point of the body at different time
steps and ve is the velocity of the body at
that point. The hydrodynamic forces on the
body are calculated from the formula
3.1 Forced Motion
3. Radiation Condition
(13)
Assuming that ~ cylinder starts to move in
still water, initial condition could be~atis
fied easily, so we specially choose the e~am
ples of forced heave motion and forced sway
motion as the calculation models to find out
the propriate radiation condition.
In numerical calculations we use a cylin
drical coordinate system (ores) on the outer
open boundary S and make r=R=const represent
S .
First let a circular cylinder undergo a
forced heave motion in still water, here the
flow is axisymmetric. Before the wave reaches
the S we have
c
0=0 on so (14)
With such radiation condition we have tried
different radius R. i.e. expanding the open
boundary gradually to find out how the scale
of the outer open boundary affects the nume
rical solution. Then we assume that the wave
near the outer boundary satisfies Sommerfeld
radiation condition t10)
3~0t+C03~00 on SO (15)
The phase velocity C0 varies only with time
on So, and it can be determined by 0 1 and
(30/3n) on Sf near Sc. Then 0 1 on SO can
be obtained according to C0. Eq.(15) can make
the wave on SO be propagating outerwerd along
r direction.
Next let a ci rcular cylinder undergo forced
sway mo ti on. Now the flow i ~ nonax) symm e t ri c
and Sommerfeld condition is described as
at+C03l° on So (16)
Here 1 is an unknown direction of outergoing
wave on the outer boundary, phase velo ci ty Cp
varies with time t and angle (3 and is equal
at the same vertical line. From Eq.( 16) we
have
where
343
0(kr ~ ~ R~a6)=0 (17)
kr=cos(l,])
~ =cos( 1,6)
(18)
OCR for page 341
The solution of Eq.(17) can be expressed as
0=0(krr+\ RDCpt) (19)
where
kr=d ~~(~0r)2+(~0) )%
~ =dR80oo/t taaO' 2+( R83Oo)2 )%
J1 arts
)1 if Trio
(20)
(21)
such ~ can make the wave at outer boundary be
outergoing propagating wave.
Substituting Eq.(20) into Eq.~17), we have
~ dC¢(( ~~2+(R~0)2~%=0 (22)
Up to now by Eq.~22) we can determine Cp
in teems of 0~+' and (30/an~n on Sf near So,
and obtain 0 +1 on So according to Cp.
3.2 Wave Diffraction Problem
For the diffraction of a solitary wave or
periodic wave upon a fixed or floating verti
cal cylinder, the Sommerfeld condition is
further extended as a radiation condition by
assuming that the scattering of the outergo
ing wave is mall on the outer open boundary.
Replacing O in Eq.(22) by Is, we can also
obtain 0n+1 on So, here
0~3=00W
and 0 is total velocity potential and Ow is
velocity potential of incoming wave.
4. Dencription of the Initial Condition
. _ .
4.1 Radiation Problem
.
(23)
For forced heave motion mentioned above we
have
Where
0(x,y,z,0)=0 on Sf It 0 (24) or,
n(~tY,z~o)=(zGnz)t=O+
on ~ It=o (25)
0~x,y,z,O)=O on Sct=o (26)
For forced sway motion, the formulas of
initial conditions on Sf and Sc are similar
with Bqe.(24),(26) and that on ~ becomes
an( ~ MY. Z'O)=( Sign,,) It o+
on ~ t=o
4.2 Body Standing or Floating in Waves
If the incoming wave is a solitary wave,
(27)
344
as used by Isaacson t7 ~ because it decays
rapidly away from its crest' the flow near
the body can be taken zero as initial condi
tion. If the incident wave is periodic Stokes
or Conoidal waves and the body is assumed to
be stationary at certain instant (as initial
time), there must be a transient period before
a steady state is approached. Sometimes as
reported by Vinje, Xie and Brevig 611], even
numerical troubles occur. In order to formu
late the initial condition properly, a
numerical tank is set up using the same pro
cedure. A 2D cylinder with different cross
section heaving can produce the required wave
profile, see Fig.2 and Fig.3. The details are
in doctoral thesis by C.Yang :9). With such 2
incident wave, the solution can approach
steady rapidly and initial condition can be
described as
0It=o=¢w It=o
0 lt=o=~w IT
on Sflt=0 (28)
on So t=o (29)
The numerical tank can be applied to any
water depth and any type of water bottom
boundary. Another expresion of the incident
wave is a 2D nonlinear Stokes wave with wave
front, it can be described as
(x,y,t)=A(x)(~ osO
16 sh3(kd)(2+ch(2kd))cos26) (30)
¢(x,y,t)=A(~)(2~ ch~k9d0~sinS
+ '42 ~ sin29)
6=k~~, s=z+d, x_~(xCgt) (kick) I
Cg_> 1+sh(2kd)), C=(k (kd)) ~ (32)
(31)
1exp(x+ka) if x+ka'O
A(x)= ~
O
if x+Ea~O
(33a)
1 if x+Ea`
A(x)=~0.5(1cos(x+ka)) if ~
for 3D case or linearly distributed for 2D
case, as in typical boundary method, the
above Green's fomula Eq.(11) becomes
~ ~j0~+ ~ Bii(~6n0)j+ ~ Ci;( ~ )~Di
where
i=1,2, ,NN
A iaG(x ~ d
Bij= ~ G(xi,~ ds
Cij=Bij
NB NF NC
D. =~ Bij(an)~ ~ Fiji ~ Aij0;
r1 i=.i
(34)
> (35)
i j No in j ) ~G=hOsinc~t
and NN=NB+NF+NC is total number of elements
(for 3D) or nodes (for 2D) on boundary
S=~+Sf+SC, and NB on ~ ,NF on Sf, NC on SO
respectively. These algebric equations can be
solved either by direct or iterative method.
The junction point of the body and free
surface is determined in the paper by extra
polation for axisymmetric flow and by satis
fying both the conditions on the body and on
the free surface for general 3D flow, that
is, according to the body condition we can
obtain the velocity of the intersection point,
by which the new position of intersection
point can be determined through satisfying
thefree surface condition.
6. Numerical Examples and Conclusion
6.1 Forced Heave Motion
As an example, we consider the forced
heave motion of a floating truncated vertical
gy~hder of radius a and mean draft a/2. The
vertical velocity of the body is prescribed
to be
ZG= housing
with body draft
H(t)= 2 hocos~t
(36)
(37)
In order to make our computation com
parable to Lin's (8) results, we also choose
that a=1, p=1 and g=1, the other initial
input data for calculation is ~ =~/2, h =0.05,
d=8, at=0.1.Besides the radius of the Outer
boundary and the radiation condition for
calculation are divided into following three
gI~Up8
345
(1) {Red. Cond. Eq.(14)
JR4 7
Grad. Cond. Eq.(14)
(3) {Red Cond. Eq.(15)
Fig.4, Fig.5 and Fig.6 show respectively the
time history of free surface profiles con
sisting with above three cases. From Fig.6 we
can observe the wave reflecting from the
outer boundary, Fig.7 shows the comparison of
heave force of case(3) with Lin's (8], and
Fig.8 shows the effect of water depth on
heave forces in which the calculation method
is similar to case(3).
6.2 Forced Sway Motion
Let a floating trancted cylinder undergo
forced sway motion. The radius of the cylin
der is a, the mean draft of it is a/2 and the
water depth is a. The gravity center of the
body can be described as
(38)
here h =0.05a and 4(a/g)%=0.8028. Comparison
has been made between the calculations and
experiments by Dr. N.Takaki in Nihon Univer
sity,Japan. Fig.9 shows good agreement of the
results of the free surface elevations at the
fixed point. Fig.10 gives the free surface
profiles at some fixed time.
6.3 Diffraction Problem
The diffraction problems of a vertical
circular cylinder standing on the seabed and
piercing the free surface by a solitary wave
has been calculated. Fig.11 shows the com
parison of hydrodynamic coefficients among
present results, Li's t12) difference solu
tion and Isaacson's (7] closedform solution.
Fig.12 gives numerical calculation model
of diffraction problem, in which the 2D in
coming wave is obtained by the forced heave
motion of a 2D cylinder, i.e. numerical tank.
Fig.13 is the time history of incoming wave
elevation Ed velocity potential of incoming
wave at point ~R, y=0 ( see Fig. 1 2 point A ) .
Fig. 14 is the time history of horizontal
wave fo roe.
The interaction of the 2D floating rectan
gular cylinder and the wave has been calcu
lated, where the cylinder is only with one
degree of freedom in z direction and the wave
is also produced by the numerical tank whi ch
ensure no wave in the vicinity of the cylind
er. Fig. 15 is the free surface elevations at
point ~R. Fig. 16 is the variations of hori
zontal wave force and vertical wave to roe .
Fig. 17 shows the variations of the body cen
ter aid the body velocity in z direction.
Fig. 18 arid Fig. t9 show the results of a
2D periodic Stokes wave (described by Eq.( I))
and Eq. ( 31 ) ) upon a fixed circular cylinder
and a truncated fixed circular cylinder at
OCR for page 341
the free surface.
6.4 Conclusions
From above examples the following conclu
sions are obtained:
(a) The boundary element method coupled
with timemarching finite difference shows
good prospect for practical use with rea
sonable cost and requirement of median com
puter.
(b) Sommerfeld condition with varying
wave speed used approximately as radiation
condition for radiation and diffraction po
tential in nonlinear case seems to be accep
table, at least for the case we have deft
with.
(c) Numerical tank is good for establish
ment of initial boundary condition with
shorter transient process, and it can be
applied to any water depth and any water
bottom condition. A 2D nonlinear Stokes wave
with wave front is also suitable for initial
condition.
(d) The determination of the location of
the intersection points at the free surface
and the body is serious problem, the approach
we used is succeded in our cases.
References
1. Bourianoff, G.I., Penumalli, B.R., "Nume
rical simulation of ship motion by DGerian
hydrodynamic techniques", Proc. of Second
Int'l Conf. on Numerical Ship Hydrodynamics
(1977).
2. Miyata, H. and Nishimura, S., "Finite
difference simulation of nonlinear ship
waves, J.Fluid Mech. Vol.157, pp.327357
(1985).
3. Nishimura, S., Miyata, H. and Kajitani,~.,
"Finitedifference simulation of ship waves
by the TUMMACIV method and its application
to hullform design", J. Soc. Nav. Archit.
Japan, Vol.157, pp.114 (1985).
4. Longuet_Higgins, M.S. and Cokelet, E.D.,
t'The defo Ration of steep surface waves on
water't, Proc. Roy. Soc. Series A, Vol.350
<1976).
,. Faltinsen, 0., ''Numerical solutions of
transient nonlinear free surface motion
outside or inside moving bodies"', Proc.
Second Int'1 Conf. on Numerical Ship Hydro
dynamics (1977).
6. Vinje, T. and Brevig, P., ''Nonlinear ship
motion't, Proc. Third Int'l Conf. on Numeri
cal Ship Hydrodynamics (1981?.
7. Isaacson, M. de st Q., "'Nonlinearwave
effects on fixed and floating bodies", J.
Squid Tech. Vo].120, pp.~67281 (t982).
346
8. Lin, V.M., Newman, J.N. and Yue, D.~.,
"Nonlinear forced motion of floating body",
Proc. of 15th Symp. on Naval Hydrodynamics
(1984).
9. Yang, C., "Time domain calculation of
three dimensional nonlinear wave forces",
Doctoral thesis, Shanghai Jiao Tong Univer
sity, China (1987~.
10. Olanski, L., "A Simple boundary condition
for unbounded hyperbolic flows", J. Comp.
Phys. Vol.21 (1976).
11. Vinje, T., lie, M.G. and Brevig, T.,"A
numerical approach to nonlinear ship motion
,Proc. of 14th Symp. on Naval Hydrodynamics
(1982).
12. Lin, B.Y. and Lu, Y.L., "A numerical
model for nonlinear wave diffraction around
large offshore structure", Proc. Second
Asian Congress on Fluid Mech. (1986).
SD i
Am
it
An= d

Fig.1 Frames of referenc" and integration
surface ( show with ~=0 )
o)~O. 3}45 DF/a1. O h,,/a.O. ~ d/a1 . 5 T.2~/u)
_ ~ ~ ~
0 20 40 60 80 ~ 00 X/e
Fig.2 Free surface elevation
at various times
a~fi77~0.50163 DF^~.0 hO/aO.1 d/a.~.5 'r2~)
Fig.3 Free surface elevation
at various times
OCR for page 341
Fig.d, .Noclicear free surface profiles for case (1)
(l~r<6.5. t. 0. Cat. 0.1, )
Fig. 6 .Noalisear free surface
profiles for c&se (3)
<~544.,, t~n, o.~, .
. .
_51 , , ~
~ 4 8 1~, TIKIE
Fig. 7 Heave fo rce va riationthO=0 .0 53
— case( 3)
Lin' s resul ts
~1
Fig.8 Heave force variation for
different water depths
 d= 1 +++ d= 2 d=8
0. 250 1 7(/hO ( x=4a, YO)
0.12` .
0.000
0.1 25
~0. 250
0.250
0.125
O. .000
0. 1 25
0. 250
347
 ~k~ 4;, ~ ~ t/T
' i/ho ( ~" ' 00 )
t/T
Fig.9 Free surface elevations
the fixed point
— model test results
 num e ri cal resul ts
t= 1. 5T
Fig. 10 Free surface profiles
OCR for page 341
n
1 .
2 .
Fig. 12 Numerical calculation model
0.1'
0.1]
0.osl
0.00f
o.oS~
0.10
0.15
Fig. 1 3a Incoming wave elevation for
3D fixed ci Ocular cylinder
at point A ( =R, y=O)
0.15~0/
o. 10. 
o.os
o An
_0.05
0.10 .
O. IS ~
A ~ ~ ~ oo
a:, \ / 1;0 ~ ~ 5 ~ 2 0 x lot
Fig. 1 3b Velocity potential of incoming
wave to r AD fixed ci Ocular
cylinder at point ~ (x—R,y=O)
lo~1 t
0.6~
0.41
° '1
n n
. _ _
0.2
 .4
0.6
o.a
2! f~Haa Boy /.5, H/~=o ~ 0.44
; ~ ~
Liz;\; 6~8oo'0~ 0~7
no
\ ·'
try
Fig. 11 Horizontal wave force variation
coo present results
Li ' s results
 Isaacson' s results
0.6
0.4
_ /y 0.2
/ X o.o
2a 0.4
, 0.6
.
.  t/
1~2o~o  om 1
0.16 ~
0.22
0.22
0.44 .
,/Dr
I_._ \ ~ / \ ,1 ~'5 '
Fig. 15 Incoming wave elevation at ~R for
2D Moating rectangular cylinder
( d/DE 4, a/DF= 5. 6, R/DE~26 )
FX/p6D~d _
r /f~D?d—
~ ,
~ __ ~ \ \ 1
\J~
Fig. 16 Horizontal and vertical wave
force variations for 2D
floating rectagular cylinder
( d/DF=4, a/DF=5. 6, R/DF=26)
s0Io. sag—
so/D, , \~
''a An\ ' / '\ \ `~t
—~ 2.'5 " it \\ \ 7.S
~ _ ~
\
Fig.17 Variations of the body center and
the body velocity for 2D floating
rectangular cylinder in z direction
( d/DF= 4, a/DF= 5. 6, R/DF= 26 )
Id/  d _ 4/~2. /0.65, s/~.4
i/. — r' ~
r I
0.1t
owl
~~/: MAIL*
offs go gay lo ~szo
Fig. 18 Incoming wave elevation at ~0 and
horizontal waveforce for AD
fixed circular cylinder
rx/~~
\~//\ /,
2.0 x 10
Fig. 14 Horizontal waveforce variation
for 3D fixed circular cylinder
0.6 ! ,
o.. 
0.2 
o.o
0.2
_0.4
348
/~' 4/~1.S. /0.65, 8/_0.4, Dr/~o.s I''`
d'" — ~ \ , {K
Fig. 19 Incoming wave elevation at ~0 and
horizontal waveforce for 3D
truncated circular cylinder at the
free surface
OCR for page 341
DISCUSSION
by K.J. Bai
First of all, the authors should be
congratulated on the impressive numerical work
reported in the present paper. This paper is
a most welcome addition to the literature on
numerical computations for the nonlinear free
surface flow problem. In the following, I
would like to make three comments:
(l)the radiation condition given in
Eq.(15) is true only for linear (or nonlinear)
hyperbolictype wave. However, in the water
wave problem there exists local disturbance
term besides the propagating waves. Therefore
this radiation condition should be imposed at
a sufficient distance away from the heaving
vertical cylinder. Specifically, for this
heave motion, the local term in the potential
behaves like a pulsating freespace (Rankine)
source. In some cases, at the sufficient
distance away from the heaving cylinder where
the local disturbance term is negligible, the
propagating waves may be treated as linear.
This is because the nonlinear three
dimensional wave will be linear as it spreads
out. Recently we have made some numerical
tests on the matching of the Kelvin source
distribution and the local nonlinear numerical
scheme along the numerical radiation boundary,
which replaces the radiation condition in
Eq.(15). This matching procedure worked very
well in our numerical test. I wonder if you
have ever tested this scheme. I would like to
know how far one should take the radiation
boundary in order to use the radiation
condition Eq.(15).
(2) Similarly to the above question, I do
not understand the radiation condition given
in Eq.(22) for the sway motion of a vertical
cylinder. I think that for this asymmetric
motion for the swaying cylinder, the wave
number vector should be radial vector. It may
be seen from the fact that the potential for
the swaying vertical cylinder can be expressed
in a FourierBessel series in a sufficient
distance away from the cylinder.
(3)I do not understand the validity of
the equation in (23)' which is entirely based
on the linearity. Even though the diffracted
waves become small at the radiation boundary,
I do not see the logic behind the linear
superposition of the local nonlinear
part(i.e., the incoming wave, I guess) and the
linear part.
Author's Reply
(1) Although Olanski condition is just
only an approximation as radiation condition
for radiation and diffraction problems in
nonlinear case, the numerical results have
shown that this radiation condition can absorb
the reflected wave on the open boundary.
Besides because of the Sommerfeld radiation
condition with varying phase velocity every
time step, this condition is acceptable in
nonlinear case so long as the distance between
open boundary and body is large enough. Of
course that nonlinear solution matching with
linear Green's function on the open boundary
is also usable as an approximation, but
Sommerfeld condition is easier.
(2) If the vertical circular cylinder
undergoes forced sway motion, the flow would
be nonaxisymmetric in the vicinity of the
body, and we should solve this problem in 3D
flow, and it is different from the heave
motion of a circular cylinder. Because the
distance between the open boundary and body is
just only large enough, the direction of the
reflected wave on open boundary and body is
just only large enough, the direction of the
reflected wave on open boundary is an unknown
quantity which can be determined by Eq.(20).
If the open boundary is very far away from the
body, the direction of the reflected wave will
be along r direction, and it can be obtained
from Eq.(20).
(3) in the wave diffraction problem we use
~S=~~OW just only on the open boundary. On
the free surface we let ~ satisfy nonlinear
free surface condition, and Eq.(23) only used
as satisfying radiation condition on open
boundary.
DISCUSSION
by R.C. Ertekin
The authors should be commended for their
paper which initiates one of the first steps
in solving the exact nonlinear
diffraction/radiation problems governed by the
potential theory. I have a few questions on
the formulation and results.
1) What is the form of the Green function
which satisfy a~/an=0 on the sea floor? I
know of a way of placing image singularities
if the sea floor is horizontal so that
3~/3Z=0 there. But not the form of Green
function if it is arbitrary so that aO/3n=0
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OCR for page 341
2) How did you deal with the 3D problem
of neighboring panel compatibility, i.e. that
the surface elevation is continuous in passing
from one panel to another and having a common
boundary (without holes) between adjacent
elements? If the problem is axisymmetric, one
can use cubic spline interpolation, for
instance, to deal with this problem, as it was
done by Brown University researchers to solve
the problem of rain drop collision with a free
surface using the BEM (Symp. on Fluid
Dynamics, California Institute of Technology,
Pasadena, 19~39). I can see the resolution of
this difficulty in the case of a heaving
cylinder, but not so easily, in swaying
cylinder case in 3D.
3) Wang, Wu and Yates solved the solitary
wave diffraction by a vertical cylinder using
Boussinesq equations (17th ONR symp., The
Hague, 1988) and compared their results with
Isaacson's that you cited. They found
considerable discrepancy especially behind the
cylinder. Whereas your results agree well
with Isaacson's. Can you explain what you
think might be the reason for this?
4) Could you comment on the accuracy
efficiency of your time stepping method?
Author's Reply
and
1) Here Green Function is so chosen that
it can satisfy a9/an=0 on the sea floor when
sea floor is flat, that is G=1/r+1/r'. If it
is not flat, Green function can be described
as G=1/r. We must place singularities along
the sea floor. At this time SQ/3ni0 , and
the sea floor will become one part of the
integration surface.
2) In the numerical calculation we can
follow the fluid particles at every time step
by use of Lagrangian free surface condition
and time stepping scheme, then we can
determine the free surface elevations and the
shape of panel. If the shape of panel is very
different from the initial shape, we can
redivide the free surface element in terms of
some regulations.
3) The diffraction problem of a vertical
circular cylinder standing on the seabed and
piercing the free surface by a soliton wave
has been calculated. There are only small
differences between our results and
Isaacson's. I think that the different
methods by which the radiation condition and
the intersection points of the body and the
free surface can be described will affect the
numerical results.
4) The accuracy of time stepping method
will depend on the discretion of the elements,
the quantities of the time step and element
size, the scheme of finite difference and the
method of integration. If the radiation
condition and the determination of
intersection point are not appropriate, the
calculation will diverge after a few time
steps. If the free surface has very serious
nonlinear, for examples, the body enter into
the water suddenly the usual method to deal
with the free surface elevation could not be
very efficient.
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