OCR for page 515
By using the dynamic and kinematic free sur-
face conditions, it follows that Ski k = 1,6
satisfy correctly to O(U)
a2¢S a2¢S
k sieves V.k ~ id ~ 2 ~ - ]
atk
+ gaz = 0, on z = 0
all)
Also (~0~67)e;~t satisfies equation (10). Far
away from the body equation (10) becomes
2 ask ask
ok ~ 2ieU (cos a a + sin a ay )
O. on z = 0
(11)
Equation (10) expresses the fact that the
waves interact with the local steady flow As
around the body. Equation ( 11 ) resembles the
classical free-surface condition with forward
speed, which for a = 0 can be written as
2 ark 2 amok
ax
ask
gaz = 0 , on z = 0
(12)
The difference between equation (11) and (12)
is that terms of o(U2) are neglected. Zhao &
Faltinsen [9] found for two-dimensional flow
that it is appropriate to use equation (11)
when T = WU/9 ~ ~ O. 15. In the three-dimen-
sional flow case we expect a similar limita-
tion. An example on calculations of the real
part of the velocity potential due to a har-
monically oscillating source satisfying
either free surface condition (11) or (12)
is shown in Fig. 2. The curves in the upper
half plane correspond to calculations with
equation (11). The T-value is 0.1,
s ~/9 = 0.6773 where zs is the z-
coordinate of the source point. The calcula-
tions in Fig. 2 are for points on the mean
free surface. We note a small phase dif-
ference between the wave systems when con-
dition (11) or (12) is used. This is more
evident on the upstream side of the source.
The amplitudes are in good agreement. The
consequence of neglecting terms of o(U2) in
the free surface condition is that fewer wave
systems occur far away from the body.
However, the consequence of this is of no
practical importance for small T-values below
0.15. A further simplification of the source
potential G is sometimes used. One writes
G = G. ~ T aG | +
U
Fig. 2 Calculated values of the real part of the Green's function on the
free surface. In the upper half part of figure equation (11)
is used. In the lower half part the classical free surface con-
dition (12) is used (T ' 0.1, me; ~ 0.6773)
( The ca l cu l a t i ons i n the l over ha l f p l ane has been prov i deaf by
J. Hoff).
This simplification can lead to large errors
at some distance from the source point. This
is illustrated in Fig. 3. The conditions are
the same as in Fig. 2. The curves in the
upper half plane correspond to that equation
(13) has been used, while the curves in the
lower half plane corresponds to that the free
surface conditions (11) has been used.
Fig. 3 Calculated values of the real part of the Green's function on the
free surface. In ache upper half part of the figure equation (13)
has been used. In the lower half part free surface condition (11}
has been use ~ T ~ 0.1, ~j - O.6773 ~ .
515
OCR for page 516
The body boundary conditions can be written 4~.k(Xo'Yo,Zo)
as
ark
-
an
lank + ok ' k = 1,6
~ an , k -
(14)
Equation (14) applies on the mean wetted body
surface Sg. The nk- and ink-components are
defined by
-
n = (n1'n2,n3)
ran = (n4,n5,n6) (15)
_ ~
m = - n ~ VW = (m1,m2'm3)
~ _ ~
-n ~V(rxW) = (m4,m5,m6)
~ ~ ~ ~ _
where W = V¢s and r = xi + yj + zk.
Positive normal direction is into the fluid
domain.
The ink-terms in equation (14) arise because
the steady motion potential does not satisfy
the body boundary condition on the instan-
taneous body surface correct to °((a) The
derivation is based on a Taylor expansion,
which means the formulation is breaks down
at sharp corners. This will lead to dif-
ficulties which will be further discussed
later in the text.
It is also necessary to specify a radiation
condition. When the free surface condition
(11) is used, it means that the waves are
propagating away from the body.
A solution to the boundary value problems for
ok can be found by applying Green's second
identity to the functions ok and 1/R, in a
fluid domain enclosed by the boundary S
defined by Sg U SF U Sc U SO (see Fig. 4).
Then we obtain the following expression
~ DOMAIN
' HI SC 'HI!
DOMAIN I
550
Fig. 4 Oefinit10n of surfaces used in the integration of equation (16).
516
1l (ok an R ~ ank R) ds (x,y,z) (16)
where R = ~ x-xO) + (y-yO) ~ (z-zO) , Sc
is a vertical cylindrical control surface, SF
is the mean free surface between Sc and Sg,
SO is a bottom surface inside Sc.
We separate the total fluid domain into two
parts. Part I is the fluid domain inside the
boundary S while part II is outside S. In the
outer domain (far-field) the free surface
condition (11) is assumed valid, while in the
inner domain (near-field) the complete free
surface condition expressed by equation (10)
is used.
We assume that the velocity potential ok in
the outer domain can be represented by a sum
of multipoles (including sources) with singu-
larities inside the body (see Fig. 4).
A m + A m
am ax am By
m m
aG(x;Xm) a G(x;x
A+ at A
4m az Am 2
m ax
m
a G(X;xm) a G(X;
AT + A —
em 2 A7m ax By
By m m
m
a G(x;xm) a G(x;x
+ A8m ax az ~ Age By a
m m m m
(17)
where Xm = (xm,ym,zm) is the coordinate of a
singularity point. The Green's function G and
its multipoles satisfy the far-field free
surface condition (11) and the radiation con-
dition.
In the numerical solution SB, SF, Sc and So
are divided into plane quadrilateral ele-
ments. The velocity potential is assumed
constant over each element. At SB the term
ask/an is replaced by the body boundary con-
dition (14). At SF the term B.k/8n is
replaced by the free surface condition (10)
which includes the velocity potential and its
first order derivatives along the free sur-
face as unknowns. The first order derivatives
of ok are numerically approximated by a
Taylor expansion which are only function of
ok on the free surface. The approximation is
OCR for page 517
correct to 0(~), where ~ is a characteristic
length of the elements. At SO and SO the term
3tk/an is replaced by equation (17).
By letting the point (xO,yO'zO) in equation
(16) approach the mid-points of each element
on the boundary surface S. we obtain a
Fredholm integral equation of the second
kind. This results in N number of equat,ions.
The total number of unknowns is N+NII, where
NII is the number of terms used in the multi-
pole expansion (17). The NII additional
equations are obtained by matching the inner
and outer solutions at the control surfaces
SO and SO. This is done by a least square
condition. It means we require the
aEr aE
( ( nary)) _r _
where
Er = 2 ~ [Re(.kI _ ~k)]2
+ ~Im(.II ~I)]2~
0 (18)
Further ok is the inner domain solution, And
and ok are defined by equation (17) and No
is the number of elements on ScUSO which is
going to match the outer solution. This leads
to the following condition
N
~ {-i~k(xj) G I(xj;xM:)
+ i ~ 2 2 AnmG (xj;xm)] GNI(Xj;xMI)} = 0
where GO is defined by writing equation (17)
as
k 21 2 And G (x;xm)
. .
and GNI(xj;xMI) is the conjugate of
G (xj;xMI). In the following equations the
sign means the time average.
Equation (19)is satisfied for NI = 1,L and MI
= 1,K. This means a total of NII = LxK number
of equations. Number of multipole terms have
to be much smaller than number of control
surface elements. As an example with the
results presenting in Fig. 10, NII=10 and
NC=56 .
surface condition ( 11 ) and the radiation con-
dition. The radiation condition is taken care
of by introducing a Rayleigh viscosity p. The
Green's function Gent can be written as
G(x,y,z;xO,yO,zO) =
(x - xO ) + (Y - Yo ) + (Z - Zo ) ]
(20)
- [(x-xO) + (y-yO) + (Z+zo)2] 36
A(z+zO+ircos(U-~+a) )
+ ~ f du r dA Ae1 2 ~
-a O A~g(~ +2~UAcosu-2ip(~+UAcosu))
where x = Jocose and y = rsin8. Expression
(20) can be simplified similarly as Grekas
[1] did by using the residue theorem and
introducing exponentional integrals. The
derivatives of the Green's function
(Multipoles) were obtained by numerically
evaluating the analytic expressions for the
derivatives.
Having found the velocity potential by the
method described in the previous text, the
added mass, damping and first order excita-
tion forces can be obtained by integrating
the fluid pressure over the mean wetted body
surface correctly to °((a) and O(U). When we
have solved the equations of the first order
motions, we can find the mean wave forces and
moments correctly to o((a2) and O(U) either
by directly integrating the pressure or by
using the equations for conservation of
momentum. We will show how this can be done
if the equation for conservation of momentum
is used.
We start with expressions given by Newman
[7]. The rate of change of momentum M(t)
in the fluid volume Q inside S = SBUSFUScUSO
(see Fig. 5) is
dt M = - P iit(p + gz) n + V (Vn-Un)] ds (21)
Here V is the fluid velocity vector, Un is
the normal velocity of surface S. n is the
normal vector to S (positive direction out of
the fluid) and Vn = V.n. The total fluid
pressure is given by
awl
P = -PsZ - P at ~ PV¢S veil
- Q EVES ~2 _ p 8~2 (22)
- pV. ·V. + 0(~3) + o(U2)
The Green's function (source function) that s 2 a
we need should satisfy the far-field free
517
OCR for page 518
at'
~ ~cc
' $,
n
''
\~ So
\ disc
Fig. 5 Def initions used of surfaces in the calculation of mean wave
f orces.
By time averaging equation (21), assuming
that SO is a horizontal plane at great depth
and using boundary condition on S it follows
that the mean wave drift forces can be writ-
ten
F = -p || ( ~ n + V V ) ds
m S P m m n
c
+ 1
O , m=1,2
- | | pgzn3 ds, m=3
SB+SF
The integration is over the instantaneous
surface S. We have to be careful when ana-
lysing the problem and keep all contributions
which are correctly to second order in the
incident wave amplitude and first order in
the current velocity. For the first term of
the first integral in equation (23), we
should first integrate up to the mean free
surface, in which only the terms -up ~V¢1~2
- pV¢s.V¢2 in equation (22) have contribu-
tions. In the integration from the mean free
surface to the instantaneous free surface A,
the first three terms of equation (22) have
contributions. In the second term of the
first integral, we can write Vm =
8(.s+~1+~2)/8Xm and Vn = a(¢S+~1~2)/8n. In
the integration up to the mean free surface,
the terms a~s/an.~2/axm' a~l/an~a~l/axm and
a¢2/an~a~s/axm have contributions. In the
integral from the mean free surface to the
instantaneous free surface A, the terms which
have contributions are B¢S/an.~1/3xm and
a¢S/axm. a~l/an.
By using the body boundary conditions and
Stokes theorem (Ogilvie & Tuck [5]) it can be
shown that the following terms
a. a¢2 a¢2 a.
| | ( ~pv~s ~ V¢2 em + P an aX + P an ax )
m m
cO
(24)
l
+ f ax an ( ~ 1 atl) do
cc m
will partly cancel. ScO means the control
surface up to the mean free surface and Cc is
the water line between the mean free surface
and the control surface (see Fig. 5). For m =
1,2 expression (24) is zero. For m = 3 it is
equal to p8~5/8n ~ ¢2dl. However, this will
Cc
be cancelled by a similar term in the last
integral in equation (23). When we integrate
over SB in equation (23) we follow a similar
procedure as outlined by Ogilvie to] for
zero current velocity. The final expressions
for mean wave drift forces are
m 29 f ( at + v~sev~l ) nmdQ
c
2 r I lV¢1 I no ds
(23) cO
a¢1 a¢1 a¢1 a.
P J J axm an ds ~ P f as ~ an do
cO c
O , m = 1,2
| ~ Jr P9znm ds ~
(25)
P llt~(atl)+v~s aZ(V¢l)~+\ I V4)ll2]nmds
SO
to
- P f (at + v~s~v~l)(n + a x r) e n do
C
r 111 = 3
where SFo is the mean free surface. CB is
defined in Fig. 5. In the integral over CB r n
is the normal vector to C8 in the horizontal
plane with positive direction out of the body
volume. Further ~ = (A ~2 r 03 ) and a = (~4'
q5rq6). The derivations is based on the body
surface is wallsided at the intersection bet-
ween the mean free surface and the body sur-
518
OCR for page 519
face. We note that the second order potential
does not contribute in equation (25).
The mean wave forces can also be obtained
by using direct pressure integration. By
following a similar analysis as outlined by
Ogilvie [6] for zero current velocity we
find that
. .
Fm = - P I I {\ I V¢1 I +
SB
o
.
(r1 + a X r).V(at} + V. ·V¢1)+V. ·V¢2]n
. _ . _ . .
at
+ (a x n)m (at ~ V¢s.V¢1)} ds
(26)
Fig. 6a Fixed Circe
flow.
2 f r1(rl3 + y,74 - xr'5) ] n dl U
- || pgznmds, m = 1,2,3
where See is the mean wetted body surface.
When the body is fixed it is possible to show
that the second order potential does not I
contribute. where r = I/(x-xl)2+(y-y )2 and Sg is the body
surface. We use plane panels with constant
singularity density over each panel.
One possible source of large inaccuracies in
the procedure outlined above as well in other
procedures is the presence of the mj-terms in
the body boundary conditions (see equation
(14)). This will be further discussed in the
following section.
Discussion of the mj-terms
We will illustrate the difficulties with the
mj-terms by giving same simple examples with
two-dimensional bodies in infinite fluid. We
will start with studying the detail of the
behaviour of the first and second order deri-
vatives of the velocity potential at the body
boundary. This will be done by a similar
panel method as we used in the three-
dimensional flow case. We write the velocity
potential as
2,r¢S ( X1 ~ Y1 )
I (an log r ~ Us an log r) ds(x,y)
(27)
519
U=1 (< W'(h
lar cylinder in infinite fluid and in steady incident
by
FIR
Ma
' I' x
We
-:
~2
Fig. 6b Definition of parameters in the analysis of local two-dl~ensional
f low around a sharp corner with an laterlor angle 8.
We will choose a simple case with uniform
current past a two-dimensional circular
cylinder (¢s = Ux+¢sB with radius 1 and
U = 1 in an infinite fluid domain (see Fig.
6a). The potential due to the body ~sB is
cosO/r and the normal derivative B¢sg/8n is
-cosO/r2 at the body boundary. We can then
divide the boundary into line elements and
for each element assume ~sB and 3¢SB/3n are
constant with values which are equal to the
correct values at the mid-point of the ele-
ment. The potential and its derivatives out-
side the body boundary can be obtained by eq.
(27) and derivatives of eq. (27). Fig. 7
shows the result of ~sB' B¢SB/3n, r~18¢sB/aO,
a ~SB/an2 as a function of the distance
along the normal vector to the body boundary
at the mid-point of the element. The results
are for ~ = 45° (see Fig. 5). The effect of
different number of elements NB is investi-
gated. The horizontal axis is the ratio bet-
ween the distance Al from the boundary and
the length As of the elements. The results
show that we get convergence and correct
results Of tsb and B¢SB/8r at the boundary.
However, for r~l8¢SB/aO and 32¢SB/Br2 we can-
not obtain correct results at the boundary.
The reason is that we are not integrating
OCR for page 520
8
~ -
8
_~
A LIZ
^ ~ ~ _
_~ i.
++
. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~^ ~ ~
~ ~ ~ ~ 3 ~ Hi ~ ~ ~ B ~ ~ ~ ~ ~ ;' ~
0 AL
'n ~' 0.250 O.500 0.750 Moo
N8 =*°
+ I_ + ~ ~ ~ + + + ~ +- 1 - 1
+
+
U) I
C~_ ~ · · · ~ *__,& ~~h ~
O. ~ ~ . ~ `_, ~ ~ ~ ~ ~ . ~ at. . ~ ~ ~ ~ ~ ~
o
8_
o
.r
8
ID
;- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ E E ~ ~ ~ ~ ~ ~ ~
D I I I —
'0 =0 0.250 0.500 o.?So to AS
Fig. 7 Calculated values of velocity potential and its derivatives for
the case presented in fig. 6. (The steady incident flow is exclu-
ded). Analytical solution.
Numerical values: Us: 6, 8.S/Br: A, r~1 B`pS/ar: x,
a2.S/3r2 : +. (AL and as are def ined in Fig. 6) NO . number of
pane l s .
with correct curvature and with correct
variation of tsB and a¢SB/8r over the ele-
ments. From Fig.7 we can see that r~18¢SB/aB'
82~5~/8r2 are satisfactorily estimated at a
distance of O(As) along the normal vector of
the element. That means we may use an extra-
polation method to calculate the velocity
along the body and the second order derivati-
ves of the velocity potential. After some
tests with a circular cylinder and a sphere
and an ellipsoid in the three-dimensional
case we found that the velocity along the
body, the second order derivatives and the
mj-terms are in good agreement with the ana-
lytical solutions. The other way to integrate
the term mjlogr(or 1/R) over the body boun-
dary is to apply the formula given by Ogilvie
& Tuck (1969).
, ~
JJ m. log r(orl/R)ds =
~-44 SO J
- || V¢sVlog r(or 1/R)nj ds
(28)
This formula is valid for a body without
sharp corners, wall-sided at the free surface
and when As satisfies the rigid free surface
condition. From a numerical point of view
this formula is more simple to calculate
because it only includes first order deriva-
tives of the steady potential. It is expected
to give more accurate numerical results than
by direct integration of the mj-term.
For a body with sharp corners the mj-terms
are singular. The consequence is that eqs.
(27) and (28) are not integrable. For example
in the case of uniform current past a two
dimensional section with a sharp corner the
complex potential W(z) in the vicinity of the
edge can be approximated as (see. Fig. fib).
W(z) = ClZl/A + C
(29)
where A = 2 - 8/~ and C1 and C are constants.
For a rectangular section the first order and
second order derivatives of the potential are
o(|z |-1/3) and o(|z ~~4/3). In the vicinity
of the corner it is possible to show m2 =
C2 R-4/3 for x = 0 and m2 = 0 for y = 0,
where C2 is a real constant (see Fig. fib).
This means egs. (27) and (28) are not
integrable. Actually, this is true for all
corners with internal angles less than a.
However, if we solve the wave-current-body
interaction problem in the time domain by
using for instance Green's second identity
and satisfy the body boundary condition on
the exact body boundary, the expressions are
integrable. The reasons why the integrals are
not integrable when the body boundary con-
dition is satisfied on the mean position of
the body boundary is that the formulation of
the body boundary condition is wrong. The mj_
-terms have been derived by a Taylor
expression. This is not valid at a corner. We
will show how we can avoid the difficulty
with the mj-terms. We divide then the velo-
city potential Ok into two parts
k k ok
where
(30)
elk atkb
an k ' an ilk (31)
The following solutions of Ok satisfy the
body boundary conditions and Laplace equation
a 8¢S a 8¢S a a¢S
.1 = ~ ax ' t2 = ~ By ' .3 = ~ az
520
OCR for page 521
a 8¢s ads
t4 = ~ Y az + Z By
a. a.
.5 = xazs - z aS
a ads a¢S
.6 = ~ x - ~ Y ax
(32)
By using Green's second identity we obtain
tahe following expression for Ok = ~k-.k and
ok (see Fig. 8).
( k ok) |x=x.
Al (ok ~ Ok) an R) - an R]ds(x,y,z)
4~ Ma 1 C =
k x=x1
( t [ok Ban R ~ an R] ds(x,y,z)
(33)
(34)
where x1=(xl~yl~zl) is inside S~=SBUSFUSCySo'
C=1 when x1 is inside S2=(SF-SF )USCUSoUS
and C = 0 when x1 is outside S2. INF
The integration surface is closed. S is part
of the free surface and does not need to
coincide with SF. By subtracting equation
( 34 ) frOm equations (33) we find that
( k k)| X X IJ ~ ( k ok) an R
an (.k ~ Ok) R] ds
I +l +S (ok an R an R) ds
~ I J ( k as 1 _ k l) ds
SF+SINF n R an R
(35)
where C=1 when x1 is outside S2 and C = 0 x1
is inside S2.
The last integralais a known quantity. The
unknowns are Ok-. on the body and ok on SF,
Sc and So. By writing the integral over SB
like it is shown in equation (35) the
integrand of the integral over SB is
integrable. The solution procedure to find
SF
\7 ~
SINF
I tsO I
Fig. 8 Def 4nition of surfaces used in the integration of equation
(33) and (34).
the unknowns can be done similarly as in the
previous section. What we have done now is to
analytically isolate the difficulties with
the mj-terms. This procedure is also valid
when ship motions at forward speed is eva-
luated. The same procedure can also be used
to solve the second order potential problem
where a similar difficulty occur.
We also have to be careful when we find the
added mass and damping coefficient. We will
illustrate this by a simple example. Fig. 9
presents results which shows the effect of
bilge radius r on sway added mass for a two-
dimensional body in infinite fluid. The para-
meters of the body is given in the figure.
From Fig. 9 we can see how the added mass is
dependent on the bilge radius in the cases
with and without current. The added mass with
current is going to infinity when the bilge
radius r ~ O. This is an unphysical result.
The reason to this behaviour can be found by
studying the dynamic pressure part used in
finding added mass and damping. Correct to
°((a) and o(U2) we can write
A22
p{B/2)2
1nD
7S
An
T ~ B=H
Current _: _ x
u L 1 Err;
MOB ~
y
1
. .
, - g
id, ~
Do --
0 00 0 25 0.50 0 75 1 00 r/( Bl2 J
-
_~
'I ~
~~ _
Fin. 9 Illustrati.^n .^.~ .^~1AII1O.4~
X X X U =0 167
Bw
-~G~ U=0000
ficients for a body in a current can lead to unphysical results.
The calculations are done for infinite fluid. (A22 ~ two-
d i pens i one 1 added mass 4. n sway ) .
521
OCR for page 522
P = ~ P[ Bt Ilk + V¢s V.k~k ~~g~z~
(36)
2 (a V (V.s) )k]
where
a= (rat ~ zrl5 - yr16)i
(37)
(~2 ~ Z~4 + xq6)~ + (03 + yq4 x05)
The index k in the last term in equation (36) ~
means that we only consider displacement in --
mode k.
What we have done in the calculations pre-
sented in Fig. 9 is to use the two first
terms in equation (36). In this way we have
included singular terms of o(U2), which are
cancelled by the last term in equation (36).
Actually we can write equation (36) as
P = ~ P ( at Ilk+ V¢s · V.k Ilk) (38)
This means that V. ·V.(a)pk cancels the last
term in equation (56). If we use equation
(38) we will find that the results for added
mass at U ~ O is the same as for U = 0.
However, this is not generally true when a
free surface is present. What is true then is
that the singular corner behaviour when the
radius of curvature goes to zero is can-
celled. Since our theory is currect to O(U)
we can also write
P = ~ P ( at Ilk ~ V¢s.(V.k)~u O ok ( )
This discussion illustrates that false
effects can be created due to the singular
corner behaviour if we are not careful in
analysing the results.
Numerical results for mean wave loads
Calculations of mean wave loads require in
general higher accuracy than computations of
linear wave loads. We will therefore con-
centrate our numerical studies on mean wave
loads. Both a direct pressure integration
method and the equations for conservation of
momentum have been used. When the current
effect is incorporated, the procedure is
correct to O(U). Calculations have been per-
formed both for horizontal and vertical mean
wave loads.
The first case we will discuss is incident
regular waves on a fixed vertical cylinder
that is penetrating the free surface. The
draught of the cylinder is 0.25 R where R is
522
— ~ = 0.000 ~ based on
\/9 ~ momentum and
OFF = 0 0479 J energy relations
-D-> OFF =~°°°° 1 direct pressure
integration
JO- ~ = 0 0479 _
1
¢~ W. ~ ~
1~/,~
=~ 0 3~ 0~ 0~ 11~ 9
Fig. 10 Numerical results of horizontal driftforces F2 for U/~i ~ 0.000
U/~ ~ 0.0479 with direct pressure integration method and a
method based on conservation of momentum' and energy. The body is a
fixed vertical cylinder with draugth-radius ratio 0.25. Element
distribution: NN1 ~ 16. NN2 ~ 12. NN3 ~ 14, NN4 - 4 NN5 ~ 8
R ~ 1.0, R1 ~ 3.0, H = 1.2A (see Fig. 11). Element lengths on the
body are nearly constant.
~1
Hi=
Side view
~ {opv~ew
Fig. 11 Def inition of number of elements and dimensions of control sur-
faces used in the numerical solution of flow around a vertical
cylinder of finite length.
the cylinder radius. The cylinder bottom is
impermeable. The current direction coincides
with the wave propagation direction.
An example on results for horzontal mean wave
loads are presented in Fig. 10 as a function
of ~02R/g where o0 is the circular frequency
of oscillation of the waves without current
present. Both for zero and non-zero current
speed we note an important difference in the
calculations based on direct pressure
integration and the results based on the
equations for conservations of momentum in
OCR for page 523
the fluid. The panel distribution used in the
calculations can be illustrated by means of
Fig. 11. By referring to the nomenclature in
the figure, NN1 = 16, NN2 = 12, NN3 = 14, NN4
= 4, NN5 = 8, R = 1.0, R1 = 3.0 and H = 1.2 A
(A = incident wave-length). The panel dimen-
sions on the body were of nearly constant
equal length. The reason to the differences
in the results is that the direct pressure
integration method is sensitive to the
distribution of the element in the vicinity
of the corner at the bottom of the cylinder.
This can be illustrated by Fig. 12 where the
calculations are presented as a function of
R/AL when e02R/g = 0.8. AL means the length
of the element nearest to the corner on the
vertical side (see Fig. 11 ) . NN1, NN2 and NN3
were the same as used in the calculations
presented in Fig. 10 while NN4 varied from 4
to 12 and NN5 from 8-12. This means that the
total number of elements were quite similar
in the calculations presented in Fig. 10. It
is the size of the elements that differes
significantly. The height of the elements on
the vertical side were selected so that
Ln'1/Ln is a constant, there n = 1 means the
element closest to the corner, n = 2 the ele-
ment next closest and so forth. The constant
ratio was always below 1.5. On the horizontal
bottom the length of the element in the
radial direction was selected in a similar
manner, starting with an element closest to
the corner. The length of the element on the
bottom closest to the corner was the same
order of magnitude as the height of an ele-
ment on the vertical side closest to the
corner. However, the most important parameter
in the calculations by the direct pressure
integration method was the distribution of
the elements on the vertical side close to
the corner. The reason was associated with
the contribution from the velocity square
~2
2 P9:o {2R)
CD
1
~ _
o
term in Bernoulli's equation, which is singu-
lar, but integrable at the corner.
In Fig. 13 are shown numerical results for
vertical drift forces on a vertical cylinder
that is free to oscillate in surge and heave
and restrained from oscillating in pitch. The
incident wave propagation direction is in the
positive x-direction. The draught h of the
cylinder is equal to the cylinder radius. The
Fit
9`a (2Rl
4. see ~
3°°°1
1.500 -
aom
-1.soo -
~ (~ )
.
Direct pressure integration
Based en momentum
and energy relations
1Q 000 Q300 0600 O900 1200
Fig. 13 Numerical results of vertical mean wave force F3 with direct
pressure integration method and a method based on conservation of
momentum and energy. The body is a vertical cyl Under that is free
to oscillate in surge and heave and restrained from oscillating in
roll. The draught-radius ratio is 1.0. Element distribution:
NN1 ~ 16, NN2 ~ 10, NN3 ~ 14, NN4 . 8, NNS - 8, R ~ 1.0, R1 ~ 2.5,
H ~ 1.2A (see Fig. 11). Element length on the body is nearly
constant. Zero current velocity.
F3
2 PI {2Rl
u = o.ooo ~ based on
) momentum and
~ = Q.0479 J energy relations
At-> OFF =°°°°°1 direct pressure
~ intenmtion
2.000
>0- ~~ = 0.047g J loon
Fig. 12 Numerical results of horizontal drift forces F2 with direct
pressure integration method and a method based on conservation of
momentum and energy. Data presented as a function of R/6L
(AL defined in the figure 11) NN1 - 16, NN2 - 12, NN3 ~ id, NNd
4-12, NN5 ~ 8-12, R ~ 1.0, R1 ~ 3.0, H - 1.2A (see Fig. 11).
High density of elements close to the cylinder corner. The body is
the same as used in Fig. 11.
523
-1.000
-2 000
U N N - FIR =0.70 based on
Aid ~1 momentum
~R=0.58 J and energy
O 0 _ relQtlons
~ FOR - 070 l direct
¢0-> ~ R = 0.58 J integration
-~
- i ~ I 1 —
10.000 200.000 400.000 60Q000 800.000 ~ L
Fig. 14 Numerical results of vertical mean wave forces F3 with direct
pressure integration method and a method based on conservation of
momentum and energy Data presented as a function of R/6L (AL
defined in the figure 11). NN1 - 16, NN2 ~ 10, NN3 ~ 1d, NN4 - 8-12,
NN5 . 8 - 12, R ~ 1.0, R1 ~ 2.5, H ~ 1.2A (see Fig. 13). High den-
sity of elements close to the cyl inder corner. The body 15 the
same as used in Fig. 13. Zero current velocity.
OCR for page 524
current velocity is zero. The panel dimen-
sions of the body were of nearly equal
length. By referring to the nomenclature in
Fig. 11, NN1 = 16, NN2 = 10, NN3 = 14, NN4 =
8, NN5 = 8, R = 1.0, R1 = 2.5, H = 1.2 A. The
large differences between the two different
methods occur in the vicinity of heave reso-
nance. The reason to the differences is again
that the direct pressure integration method
is sensitive to distribution of the elements
in the vicinity of the corner between the
bottom and the vertical side of the cylinder.
This can be illustrated by Fig. 14 where the
calculations are presented as a function of
R/AL when e02R/g = 0.58 and 0.7. AL means in
this case the length in radial direction of
the element closest to the corner on the
horizontal side (see Fig. 11 ) . The distribu-
tion of elements were selected similarity as
in the previous example. Total number of ele-
ments are nearly the same for all calcula-
tions presented. When the direct pressure
integration method is used to calculate the
vertical mean wave force, around heave reso-
nance, the contribution from the velocity
square term in equation (26) is large and of
opposite sign to the other contributions in
the integral over SBo. The absolute values
of these terms are nearly equal to the velo-
city square term. This means a high accuracy
is needed in the integration over SBo.
In Figs. 15 and 16 are presented numerical
results for horisontal drift forces on a ver-
tical cylinder with draught-radius ratio 3.0.
The method based on conservation of momentum
was used. The effect of using higher density
of elements close to corner between the bot-
tom and the vertical side of the cylinder was
investigated. There was a maximum of 1% dif-
ference in drift forces. The influence of
number of singular points inside the body
(see equation (17)) was investigated. Also
the effect of number of multipoles was
studied (see equation (17)). In the calcula-
tions presented in Figs. 14 and 15 number of
singular points is two and number of multipo-
les for each singular point is 10. If only
one singular point was used there was a maxi-
mum of 1% diffrence in drift forces. If
number of multipoles was increased to 16
there was a maximum of 0.2% differences in
driftforces. The effect of increasing R1 (see
Fig. 11) from 3.0 to 5.0 was studied. The
difference in results were less than 2%.
CONCLUSIONS
A theoretical method to analyse current-wave-
body interactions is presented. It is demon-
strated that the mj-terms arising in the body
boundary conditions can cause large errors in
the numerical result. A theoretical way to
provide stable numerical solutions is pre-
sented.
A method to calculate mean wave forces based
on conservation of fluid momentum is pre-
sented. It is demonstrated that a direct
pressure integration method can lead to large
errors in prediction of mean wave forces when
the body has sharp corners.
-0-0- ~ = o.oooo
_O_O_ u = 0 0319
~ = Q0638
7_~ ~ =-0.0319
Vim =-ao63s
O O
F:
2 P9~ (2RI
._
._
o
A_
o
A_
n
Aid
8t · ~ R
on 000 3.~ 0.6m o.soo t.200 9
Fig. 15 Numerical results for horizontal wave drift forces on a fixed ver-
tical cylinder with draugth-radius ratio 3.0. Current velocity U
is in the same direction as the wave propagation direction. Ele-
ment distribution: NN1 ~ 16, NN2 ~ 16, NN3 ~ 14. NN4 ~ 12.
NNS . 5, R1 ~ 1.0, R1 ~ 4.0, H ~ 1.2A (see Fig. 11).
Element length on the body is nearly constant.
- > ~ = 0.0000
_O_O_ u = 0 0319
O ~ U = ao63a
_ ~
3952 211 `°~ ~~~ ~~ 0.0319
c,
. _
o
A_
o
To
._
o
~ a- ,,~ =-ao63s ma
car
~ ~0
/~/'~W
§.A ~ W~ ,1
~02 R
Fig. 16 Numerical for horizontal wave drift forces on a vertical cylinder
with draught-radius ratio 3.0. The cylinder is free to oscillate in
surge only. Element distribution is the same as used in Fig. Is.
524
OCR for page 525
References
1. Grekas, A. 1981, Contribution a ['etude
Theorique et Experimentale des Efforts
du Second Ordre et du Comportement Dyna-
mic d'une Structure Marine Sollicitee
par une Haul Reguliere et un Courant,
These de Docteur Ingenieur (Ecole
Nationale Superieure de Mechanique).
2. Hess, J.L., Smith, A.M.O. 1962, Calcula-
tion of Non-lifting Potential Flow about
Arbitrary Three-dimensional Bodies,
Report No. E.S. 40622, Douglas Aircraft
Division, Long Beach, California.
3. Mavrakos, S.A. 1988, The Vertical Drift
Force and Pitch Moment on Axisymmetric
Bodies in Regular Waves, Applied Ocean
Research, Vol. 10, No. 4.
4. Molin, B., 1983, On Second-Order Motion
and Vertical Drift Forces for Three-
dimensional Bodies in Regular Waves,
Proc. Int. Workshop on Ship and Platform
Motion, Berkeley, pp. 344-357.
Ogilvie, T.F., Tuck, E.O., 1969, A
Rational Strip Theory of Ship Motion:
Part I, Report No. 013. The Department of
Naval Architecture and Main Engineering,
The University of Michigan, College of
Engineering.
6. Ogilvie, T., 1983, Second-Order Hydrody-
namic Effects on Ocean Platforms, Proc.
Int. Workshop on Ship and Platform
Motions. Berkeley, pp . 205-265 .
Newman, J.N., 1967, The Orift Force and
Moment on Ships in Waves, Journal of Ship
Research, Vol. 11.
8. Newman, J.N., 1978, The Theory of Ship
Motions, Advances in Applied Mechanics,
Vol. 18.
9. Zhao, R., Faltinsen, O.M., 1988, Interac-
tion Between Waves and Current on a Two-
dimensional Body in the Free Surface,
Applied Ocean Research. Vol. 10, No.
2.
10. Zhao, R., Faltinsen, O.M., Krokstad,
J.R., Aanesland, V., Wave-Current
Interaction Effects on Large-Volume
Structures", BOSS '88, Trondheim.
525
OCR for page 526
DISCUSSION
by R. Huijsmans
I first like to congratulate the authors
on their very concise treatment of the low
forward speed problem. I have a few questions.
1) Can the authors elaborate on how to
obtain the low frequency drift forces from
their "far field" expansion of the drift
force?
2) The authors identify the well known
problem in using double derivatives of the
stationary potential on the body. They used a
kind of extrapolation procedure to avoid the
problem. Have they now used the potential As
on B. described by a c2 function by using some
linear variation? (results of Fig.7 of their
paper for /&L/AS - 0)
3) The authors experienced some problems
even for the zero speed case in determining
the wave drift forces based on the pressure
integration, because of the presence of sharp
corners. Have they some experience on how
"sharp" these corners must be in order to get
into troubles.
4) The authors did not mention
actually solved the systems of equations
(directly?) and what the computational burden
of their method is with respect to the number
of panels. How much more expensive is the
treatment of the non-zero speed case with
respect to the zero speed case?
5) And the final question is regarding the
use of their method without the low forward
speed restriction. Can the authors give some
idea how to their method can be applied for
high speeds.
Author's Reply
We thank Huijsmans for his comments. The
replies are as follows:
1) Our "far" field expression is based on
conservation of momentum and energy which can
not be applied to calculate low frequency
drift forces. When we study low frequency
drift forces one should also include second
order potential. A discussion about this
problem is given by Faltinsen and Zhao[A1].
2) We have not applied high order panel
method to predict the second order derivatives
of stationary potential on the body. We think
for a body with sharp corners one will also
get numerical problems even we use high order
panel method. In our another formulation (see
eq.(35)) we can avoid to calculate the second
order derivatives on the body.
3) When one calculates wave drift force
based on pressure integration, it is difficult
to predict the contribution from u2-term,
because the velocity will be infinite at
sharp corners. We did not investigate how
"sharp" these corners must be to get into
serious problem. But from our experience we
think the most important thing is due to
cancellation effect of the contributions from
different terms. This will depend on the whole
body configuration the incident wave system as
well as the local sharpness of the corner. In
some cases only a few percent error in
predicting the contribution from u2-term will
give 100% error in wave drift forces.
4) The usual direct equation solver was
used to solve the systems of equation. The CPU
time is almost the same for the case with or
without current velocity.
5) Our method may apply to high speed
problems. In that case one should obviously
use another free surface condition and Greens
function.
[Al] Faltinsen, O. and Zhao, R.: Slow-Drift
Motion of a Moored Two-Dimensional Body
in Irregular Waves, J. of Ship
Research, Vol.33, No.2, June 1989,
pp.93-106.
DISCUSSION
by A. Hermans
I only shall make some remarks about
Figs.2 and 3 and the text just before those
figures. It looks like the authors are not
aware of a large amount of literatures about
the typical nonuniform behavior of that show
up in "both" figures. The way they think that
(13) has been applied leads to nonuniformities
at the distance of order I R. while the
application of (11) leads to a nonuniform
behavior at the distance of order 1 2R.
Already in 1882 Lindstedt noticed and remedied
this kind of nonuniform behavior in the
computation of the trajectories of planets.
In 1892 Poincare in his book on "Mecanique
Celeste" proved that the remedy that was given
is correct. In 1949 Lighthill extended their
theory to problems in fluid dynamics. The
approach suggested in (13) is uniformised
quite easily because the exact phase relation
can be used, while the approach according to
(11) always will have a phase error in the far
526
OCR for page 527
field. The uniform version of (13) and
application of (11) both lead to a correct
approximation of the amplitude.
Author's Reply
We would like to thank professor Hermans
for his comments. We think one has to have in
mind what one should calculate when we compare
the two different approaches. If one should
calculate the wave drift damping coefficient
that is proportional to the slow drift
velocity, the two approaches should be equal.
However if one want to study wave current
interaction and in particular the wave
elevations around the structure, the two
different approaches are different. The
approach that we are using, is then a better
approximation.
DISCUSSION
by H.J. Choi
On this occasion, I
would like to ask a
527
question which I have had for a long time. It
is a well-known fact that incident waves
deform significantly in amplitude and
propagation angle, depending on the magnitude
and incidence angle of current. As a
result, a considerable amount of radiation
stresses is to be built up in water, which
might contribute to the second-order forces on
marine structures, too. My question is if we
could discard the effect. If it is not the
case, how can you incorporate it into your
method?
Author's Reply
The effect of radiation stresses is
included when ye calculate mean wave forces. A
discussion of this is given by Longuet-
Higgins[Al].
[Al] Longuet-Higgins, M.S.: The Mean Force
Exerted by Waves on Floating or
Submerged Bodies with Applications to
Sand Bars and Power Machines, Proc.
R. Soc. Lond. A.352, pp.463-480.
OCR for page 528
Representative terms from entire chapter:
direct pressure
