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OCR for page 375
The Effect of the Steady Perturbation Potential on the
Motions of a Ship Sailing in Random Seas
R. H. M. Huijsmans
Mantime Research Institute Netherlands
Wageningen, The Netherlands
A. J. Hermans
Technical University of Delft
Delft, The Netherlands
Abstract
In this paper results will be presented of model
tests and calculations of the wave drift force on a 200
kDWT tanker and half immersed sphere. The the-
ory of small forward speed motion computations is ex-
tended in order to allow larger horizontal distances in
the Green's functions by deriving a proper asymptotic
expansion of the low speed Green's function. Also an
alternative formulation for the wave drift forces has
been derived, based on the momentum balance as e.g.
derived by Newman or Marno. This alternative for-
mulation is derived for the small forward speed case.
1. . Introduction
Recently we derived a formulation for the descrip-
tion of the motions of a floating body with a small
velocity. The reason for such a formulation is related
to the wave drift damping phenomena. Large moored
tankers offshore exhibit low frequency resonance be-
haviour. These resonant forces are associated with
the slowly varying wave drift forces. These forces can
be computed with the help of linear diffraction the-
ory and taking into account the second order effects
of the pressure and the wave height. An extensive
study among many other studies was published by
Pinkster (14. Also second order wave excitation may
be taken into account in an approximative way, see
Benshop et al. t24. In an early paper Remery and
Hermans t3] indicated that for an accurate descrip-
tion of the low frequency motions not only the drift
forces are important but also the accurate prediction
of the damping coefficient near resonance. In a later
study by Wichers t4] he showed that this damping co-
e~cient was quadratic with respect to the wave height'
thus leading to the concept of wave drift damping co-
e~cient, which have been shown by Wichers et al. L5]
to be related with the forward speed dependency of
the wave drift forces. The effect of the varying wave
drift forces with speed has been described by a num-
ber of authors nowadays, beginning from Hermans and
Huijsmans t64 to the more recent publication by Nossen
et al. [7], Sclavonous [8] and Hu and Eatock Taylor
A. In the paper of Hermans and Huijsmans the speed
was restricted to be low, due to non-uniform character
of the asymptotic expansion scheme. In the paper of
Sclavonous the problem was solved by deriving explicit
Green's functions for the wave drift damping, with a
proper account of the disturbance of the steady poten-
tial Us In order to solve the forward speed problerr~
Zhao and Faltinsen t10] showed that tl~e treatment of
the speed dependent boundary conditions (depending
on the steady potential ¢~) have to be handled care-
fully. As soon as one tries to use the expansion scheme
in t111 for the unsteady potential with respect to for-
ward speed at a small but fixed forward speed one
is confronted with the non-u~iformities in the asymp-
totic expansion. In short ogle fields for a point source
that the second order results behave like (~2 where
~ _ up is the small parameter and R is the distance
g
to the point source.
In former studies only the speed effect due to a
uniform flow has been attempted. However the in-
fluence of the steady perturbation potential resulting
from the stationary fluid flow around the ship, on
the ship motion problem is not well understood. For
the case the current is head on or the ship's course
is at zero drift angle, then the influence can be ne-
glected. In case of a ship moving at a certain drift an-
gle it then appears to be of considerable influence, see
Huijsmans et al. (124. In our study we have incorpo-
rated the steady foreyard speed perturbation potential
into the ship motion problem. The forward speed ship
motion problem is solved using an efficient algorithm
375
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for the Green's functions involved. The solution of the
boundary integral equations comes from a very effec-
tive iterative solver. The calculation of the wave drift
forces is done in two alternative ways, one is the inte-
gration of the pressures over the mean wetted surface
of the body and secondly using the Maruo expression
for the wave drift forces corrected for the small forward
speed parameter. The nature of the non-uniformity in
the asymptotic expansions will be studied in this pa-
per and extended to uniform expansions. We keep in
mind that one of our goals is to arrive at a formulation
that makes use of the zero speed source potential.
In this paper we present a uniform approximation
valid for small values of the small parameter but also
for fixed finite values of OR.
2. Mathematical Formulation
We first derive the equations for the potential func-
tion 4~,t), such that the fluid velocity n(z,t) is de-
fined as u~x,t) = grad ¢(x,t). The total potential
function will be split up in a steady and a non-steady
part in a well-known way:
¢~2:, t) = Up + Add; U) + ¢~:e, i; U) (1)
In this formulation U is the incoming unperturbed ve-
locity field, obtained by considering a coordinate sys-
tem fixed to the ship moving under a drift angle ct. In
our approach this angle need not be small. The time
dependent part of the potential consists of a incoming
wave at frequency ~ an a diffracted and/or radiated
wave contribution. To compute the wave drift forces
all these components will be taken into account.
7
MY
At the free surface we have the dynamic and kinematic
boundary condition
9( + At + EVE · Vie = constant ~
at z = ~ (3)
~z-~-¢yc-G = 0 '
We assume that the waves are high compared to the
Kelvin wave pattern, but that they are both small in
nature, hence the free surface boundary condition can
be expanded at z = 0. Elimination of ~ leads to the
following non-linear condition:
Bt2~ + 9oZ¢ + ot(V~ V¢)
+V~ V [ ¢2 ¢| = 0 at z = 0
(4)
To compute the wave resistance at low speed the free
surface elevation must be treated more carefully, be-
cause the wave height is of asymptotically smaller or-
der. This problem has been studied extensively by
Eggers [13], Baba [14], Hermans t154 and Brandsma
[164. The velocity field is well described by the double
body potential with a small wave pattern.Therefore
we take the double body potential into account and
we neglect the stationary wave pattern. For the wave
potential Aid, t; U) the free surface condition now be-
comes:
¢~t + 9¢z + Oust +
+2V¢. V¢t + (U: + 2U¢= + ¢
+2(u+~¢y¢2.y+¢y¢egy
+~3U¢~ + ¢,.~= + ~y~ry)¢r +
+~2U¢~v + my + (y~yy)~ + {(2) {be}
at z = 0
= ~
(5)
The boundary conditions on the hull can be written in
a similar way for all radiating and diffracted modes.
We therefore treat the following general form, keeping
in mind that the actual form has to be used in the
computations. Generally we have the condition:
Fig. 1. Axis of coordinate system (Vet. n) = V(~)e ibex ~ S (6)
The equations for the total potential ¢' can be writ-
ten as:
/\f = 0 in the fluid domain De (2)
where S is the mean wetted area of the ship hull.
The non-linear operator {(2) on ~ win be neglected
as well. The first term in equation (5 ~ contains linear
terms in U.
376
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Representative terms from entire chapter:
drift force
Our Ansatz is that in order to obtain the first or-
der approximation with respect to U the higher order
terms in U may be neglected in the free surface con-
dition. In the next section we show that in general
this is true, but first we discuss the construction of
the regular part of the perturbation problem, with the
complete linear free surface condition.
We assume ¢(x, t; U) to be oscillatory.
f~x,t; U) = `~(x; u~e-iw'
The free surface condition is then written as:
_~~72~_ 2i~U¢~ + 2+
gig = D(U; ¢) {~) at z= 0 (8)
where D(U;~) is a linear differential operator acting
on ~ as defined in equation (5~.
We apply Green's theorem to a problem in Di in-
side S and to the problem in De outside S. where S is
the ship's hull. The potential function inside S obeys
condition (8) with D = 0, while the Green's function
fulfills the homogeneous adjoins free surface condition:
-~2G + 2i~UG`+
+U2G`` + gG: = 0 at ~ = 0 (9)
This Green's function has the form
Gf x, g j U) = - - +,—¢(x, A; U) (10)
where r = |x—Al and r' = |x—A'|, where A' is the
image of g with respect to the free surface.
Combining the formulation inside and outside the
ship we obtain a description of the potential function
defined outside S by means of a source and a vortex
distribution of the following form:
47r¢(x) =
//s^/(g)~3nGtX'<)dS~— 9 /T! L-rtg)Gfx,g~dr'+
//S (g)G~X'()ds: + 9 /T! ~ [~(g)0gG(2,(~+
{ t)~(g) + ~T^IT(g)} Gtx, g)] do/ +
+ 9 /T~ ~ ~na~g)Gtx' (ids +
9 //s Gin, ()D {(a) dSt (11)
with c', = cos(O2, t), AT — COs(OX,_) and ~x,,
COS(OX,7~), where n is the normal and t is the tan-
gent to the waterline and T = t x ~ the binormal. It
is clear that the choice of high = 0 for the integral
along the waterline will give no contribution up to or-
der U. The source distribution we obtain in this way
is not a proper distribution, because it expresses the
function ¢, in a source distribution along the free sur-
face with a strength proportional to derivatives of the
same function ¢. However this formulation is linear
in U and moreover the integrand tends rapidly to zero
for increasing distances R. So finally we arrive at the
formulation:
2xa (A //s a (g) ~ G (x, () dSf
+U / ``na (a) an G (2,g) do +
g / /FS ;3~Z G (x'() D {
¢(X'<;U) - T' lo do| dkF(8,k)+
~ 9 | do | dkF (8, k)
~ 7r/2 L2
where:
F(f' k' kexp(k(z + ~ +~(~—() cos8])
·cos ik(y—'7)sin8] (17)
The contours {~ and (2 are given as follows:
kl k2
I ~
O k3 k4
O <~_L2
find:
Fig. 2. Contours of integration ( -)
The contours are chosen such that the radiation
conditions are satisfied. The radiated waves are out-
going and the Kelvin pattern is behind the ship. For
small values of T the poles of equation (17 ~ behave as:
4;:,~ ~ ~ + 0(T) as ~ ~ O (18)
/,—A/ ~ ~ ~ C) (1) as T ~ O (19)
A careful analysis of the asymptotic behaviour of
Ax, (; U) for small values of U leads to a regular part
and an irregular part:
¢(X,<; U) = Jo (X, () + ¢] (—~ ()
...~t, (~,() + ~ ¢' (x,6) +
where
(20)
¢t, (x, () -
2k ~ k exp Katz + () J .`kR'~dk (21)
¢'1 (z,<)
4ik A, ~ k2 exp Katz + i) J ~kR)dk (22)
where R2 - (~ -- (12 + (y -- 9)'> and b' - arc tan
and
¢0 (x,<) =
(16) —4u f exp iL,tz + () sect 8] sin Ax—() sec 84 ~
· cos tidy—71) sin ~ sec2 8] sec2 Ode (23)
The expression in equation (23) gives the interaction
of the translating part of the Green's function with
the oscillatory part. In Hermans and Huijsmans t6] it
is shown that due to the highly oscillatory nature the
influence of equation (23) may be neglected in our first
order correction for small values of a.
The non-uniformity character of equation (20) for
large values of R becomes clear, if we analyse ¢~ (x, A)
a little bit further. The contour of integration L2 is
chosen well underneath the singularity k`' = w2/g and
performs a partial integration of equation (22~.
The end points give zero contribution. Hence we
4ik,,cos8/ Ok k ~dk [k2ek(Z+~)J~(kR)] dk (24)
We are mainly concerned with small values of
T(Z+~) because the pressure is calculated at the ship's
hull and we assume the horizontal length scales large
compared to the vertical length scale.
To get more insight in the structure of the source
function we deform the contour L in the complex plane.
ko
, ~ - -- k
iL
Fig. 3. Contours in complex plane
378
We use the relation Jn(z) = ,~ {H(l)(z) + H,(2)(z)}.
For large values of z the Hankel functions behave like
Half ~ jei(z-:n~-~/~) d
H(2) ~ I/ 2 e-i(Z-:n=-'r/~) (25)
We find as an approximation of equation (20)
~6 ~ ¢0 + r~6~ = 27ri {kOek° (a+) Ha) +
+2ricos8'd,k t~k2ek(Z+~) H(~)(kR)] k k ~ +
2 too (eik(Z+~) e-ik(z+~) ~
+ - / kit, + ~ Ko(kR)dk+
Trio Ok +zko k—Lo J
4r cos 8' t°° k2 | eik(Z+~) e—ik(z+~)
fir
·Kl(kR)dk
Jo 1~ (k + iko)2 (k—ko)2 J
(26)
This expression is then studied for large values of R.
The two integrals can be expanded with the help
of the following integral representation of the function
Kn (z)
Kn(z) = / e-ZC"Sit~coshntdt fly)
Jo
We first apply the method of steepest descend with
respect to the t integral and perform a partial inte-
gration with respect to k. It then turns out that both
integrals behave like (~(R-3/2), hence they lead to uni-
form expansions with respect to T. The integrals are
(~(1) as T ~ O. ~RE[O.~) -
The terms that follow from the residues give rise to
the expected non-uniform behaviour.The second term
in equation (26 ) may be written in the form:
T¢] (X, () |R =
2,ri * 2iT COS 8' [koH(~) (koR) +
+i,z + ¢,)k`2,H(~)(k,-'R! + Rk~2H!(~)(k,., + R)] ek°(Z+~)
(28)
The second term gives rise to non-uniform behaviour
of large values of z + A, however we restrict ourselves
to finite values of z + A.
Our main concern is the last term. We compare
this term with the first one in equation (26)
?,6~ie9 ~ 2~ik~-,ek° (my) H('t) ~ kO R)
{1 + 2iTkoR COS ~ ~ + 0(T ), ~[0.~) (29)
and for large values of R we have:
~ ~ 2~i~koek°(Z+~)ei(koR - =/~)
{1 + 2rikoRcos8'3 + (~(R-3/2) + (A r2) (30)
The origin of the non-uniformity is now clear. It is
the well-known phase shift of the wave numbers of the
PLK method. The residue of the exact source func-
tions leads to the exact phase shift, in our case we
have approximated:
exp(2ikoT(x—A)) by 1 + 2ikoT(x—A) (31)
This term originates from the a, t derivatives in the free
surface boundary condition. Before treating methods
to obtain uniform expansions we must keep in mind
the way we like to use the Green's function. This leads
to the insight that we need two different approaches.
One for the computation of the far field wave and one
for the computation of the integral equation. In the
far field the exact value of the wave number has to be
taken into account, while in the latter case a first order
correction of the wave number is sufficient to arrive at
solutions valid up to second order.
4. Expansion of Source Strength
. . .
In this section an approximate solution of equation
(12) will be derived. Inserting (14) and (15) into (20)
one obtains for like powers of T the following set of
equations:
—27raO (a)—i/s JO (it) ,9~ Go (id, A) dS~
= 4~Vo(~), x ~ S (32)
and
—2~i (I) - //s ~i (I) '~ Go (it, it) dSf
.//s ° (I) ^~ (ala) dS: + 41rV~(x)+
2 i~ ~Go (x,<) V¢~(~) V¢o (it) dSt. (33)
here Go (id 6) =—r + r'—160 (X, () iS the zero speed
pulsating wave source and
V(X) = Vo(X) + TVI (X) + 0(T ~ (34)
379
The potential functions in (14) now become: F(8, k) =
To (x) = - 4,r ils so (I) Go (x, A) dSf (35)
¢)1 (a) = - 4 |/ crO (g) Al (a, () dSk
—41r || ~1 (a) Go (it, A) do +
21rg ||FS Go (xi A) V¢(~) ¢0 (a) do (36)
In principle the solution of the problem is now
solving ¢0 (x) and ¢1 (:~) using the steady perturba-
tion potential. The steady perturbation potential
is determined using a boundary integral technique for
the steady double body flow, which originally comes
from a Hess and Smith type of algorithm. The steady
double body flow is calculated separately and is then
incorporated into the free surface integral.
In reference t6] it is shown that the non-uniform
term with respect to z' in the Green's function leads to
contributions that are asymptotically small compared
to the terms we have taken into account.
5. Uniform Asymptotic Expansions
In principle we have to solve (32) and (33) where
the source function suffers non-uniform behaviour. If
the size of the ship is order one with respect to T it is
sufficient to use (10) with (20). The question remains
how to compute the far field. This will be dealt with
in part 1 of this section.
If the size of the vessel becomes large with respect
to T. TR = o(1), we have to modify (20) in order to
obtain proper approximations of the source strength
from (32) and (33~. This problem is stated in part 2
of this section.
5.1 The Far Field
In the case where art) + Tin iS known, the far field
may be computed with the aid of equation (13~. Be-
cause for R large we cannot use (10) with (16~. As
explained before the contribution of the "Kelvin" resi-
dues from kit and kit are neglected. Contributions of
the wave residues are dominant but first we apply the
method of stationary phase to the integral with respect
to 8. The integrand F(8, k) for large values of R with
the notation x—R cos 8, y = R sin 8:
380
with
k exp~k~z + (~)
2 gk—(w + kU COS 8~2
{exp ~—ik~g cos ~ + ,7 sin 8~] · exp [ikR cost—8~] +
exp ~—it cos ~ ~ sin 8~: exp [ikR cost + 8~] }
(37)
We have to distinguish between the four quadrants at
infinity. We choose O < ~ < ~r/2. AD other quadrants
can be treated in a similar way, the results are the
same. We obtain:
~(x, (; U) ~
9 1~_7r2
e 4
[/ ~ exp {kid + ()—it cos ~ + ~ sin 8) + ikR)
'dk
gk—(w + Uk cos 8)'2
;- Ad; exp {kid + () + it cos ~ + ~ sin 8)—ikR} d
(2 gk—(w—Uk cos 8~2 ~
(38)
The first integral may be closed in the upper quarter
plane whilst the second one may be closed in the lower
quarter plane.The integrals along the imaginary axis
are of C'( R--:~/2 ).
We now finally obtain:
Ace, (; U) ~
/; ~ ei(kl (OR A/ ]) ~
27r! ~ / c . 1 .
V 1rR | (1 _ 2U COS 8~ + Kit MU cos 8))
exp {k~)(z + ()—ik~)~
Hence, to obtain the potential in the far field we ?,6~, =
use (13) with
=
Large Vessels
—- ~ ,—¢(x, A; U) and (41)
= too (id) + TO (I) (42)
In the case of large vessels (41) is not a good ap-
proximation for the source strength. We now have to
take care of the non-uniformity as described in (30).
A practical requirement is that we want to make use
of the zero speed oscillatory Green's function and its
derivatives. The function ~6, (it,<) can be computed
for the major part using algorithms as e.g. developed
by Newman t184 or Noblesse t194. One minor con-
tribution has to be evaluted separately. Keeping in
mind that the PLK method requires the omission of ~6 ~ To + Tip + (~(T2R) (48)
the most severe secular term to obtain uniform expan-
sions we may conclude that the procedure only needs
to avoid approximations as (31~. The following proce-
dure makes it possible to use the zero speed algorithms
with a slight modification. A proof of the validity can
be given rigorously with the same analytic manipu-
lations as described in Section 3. For instance the
following correction may be performed:
160—2~koek° (z+~)
~ [e ( ° / ) ~1 —exp~2ik('r(~ —(~3] (46)
and
~~ =
?,6'—2,rk,,ek°(Z+~) .
~ [e ( ° / ) 12ik~(x—Alp] (47)
The correction of TO can easily be performed in the
zero speed Green's function algorithm, while the cor-
rection of ¢~ can be performed either analytically or
numerically. It can be shown by inspection also that:
Hence the region of validity is extended in a proper
way.
If one wants to higher order approximations the
procedure has to be reconsidered. Corrections can be
obtained along the same line. However the advantage
of reduction to the zero speed algorithms is not avail-
able anymore. One has to devise a fast algorithm for
~72-
~JRes ~ (~°rcs + ~i ~ P~ °(Z (~) (43)
It can be shown by inspection that this multiplicative
correction yields the correct uniform asymptotic ex-
pansion up to (~2R) as ~ ~ 0 t204. The interval
of validity is properly extended. It is also possible to
apply the correction to Mores alone and to show that:
~rr.s = (¢orc:~(l — 2ikOT(~ Air) A- T¢lr'!~)
exp(2irkO(~—(~) + (27~2R) (44)
The correction is only needed for large values of KoR,
therefore the correction may be performed at asymp-
totic level. This leads to the following simplified re-
sults for the total ¢. This result is rewritten in asymp-
totic form where we made use of the explicit form of
the residues
(45)
where
6. Wave Drift Forces
In Hermans and Huijsmans t6] we described a way
to compute the first order forces and the second order
wave drift forces. The method we used there was based
on a direct pressure integration of the first and second
order pressures respectively. It has been shown before
(e.g. see Pinkster t14) that this method works well
and is even necessary in order to compute the slowly
varying wave drift forces.
At this moment we are mainly interested in the
constant component of the wave drift force. In this
section we recapitulate a method that leads to results
that possibly are more accurate numerically, because
when using the pressure integration technique one has
to use derivatives of the potential function over the
mean wetted surface. This is even more the case if one
uses pressure integration in the case of ship motions
with forward speed (see Huijsmans t21] ). Newman t224
and Marno [23] have derived an expression for the wave
drift forces and moments.
381
The mean forces and moments may be expressed ,¢,( t)=
as t22]:
F —
1' —
- || (p cos ~ + pVI~(Vl' cos ~—Vo sin 8)] Rd~dz (49)
F -
v
—|| [psin ~ + PVK(VK sin ~ + Ve cos 8)] Rd~dz (50)
sop
US + —e{kl (~)Z+i(kl (~)(~:'s~+9sirl~)—W[)}
+F(~)eiS`~1) ~/ie(k1 (fl)Z+i(kl (a)—At))
here (a iS the amplitude of the incoming wave and
F(~)eis(~) results from the asymptotic expansion of the
far field potentials in (53) with
M: = 4(j) (x; U) =
—fly VRVcR2dEdz (51 ) fls ~ (I) G (I, A) dSe
where p is the first order hydrodynamic pressure, V
is the fluid velocity with radial and tangential com-
ponents V'2,Vc and SO is a large cylindrical control
surface with radius R in the ship-fixed coordinate sys-
tem. Faltinsen and Michelsen t24] derive from these
formulas expressions in terms of the source densities
of the first order potentials. From:
6
{or (if) = ~(7) (if) + ~ (I) (if) HI (52)
j=l
where CYj = c~je-iWt j=1~1~6 are the six modes of mo-
tion and superscript 7 refers to the diffraction com-
ponent of the source strength. In our case we follow
the same reasoning to obtain similar results for the
slow forward speed case. Our velocity potential has
the form:
flex, t) =
Us + ¢(xi U) + Act, U)e is
9 /IFS V; V¢(i)G (X () dS'` (55)
where G (x, A) is approximated by (30~. Due to the
fact that the function Vow) decays rapidly as |~| ~ °°
we assume that R = Axe is large enough to take the
asymptotic expansion of G (x,() in the last term as
well. The function V¢(i) may be replaced by V¢~(-,,).
This leads to
-
|~ e-31ri/4
27r 1 _ 2U COS 8~ + kl(~)U COS 8)
· [us ~ (a) exp {k1 (~(—ibl (~< cos ~ + ,7 sin 8~} dS~
2i~' || V¢' V<~'oT)exp{—ibl(~)
9 FS
(< cos ~ + ~ sin 8~} do] (56)
= Ux+~;U)+
6 ~ where:
~ ¢(0) (X; U) + ¢(7) Em; U) + ~ ¢(i) (X; U) JO ~ e-iwt
¢(T) (I) = ¢~() ) (I) + ~ If) (I) j
j=1
(53)
where the potentials ¢(j) (x; U), j=1,7 have the form
(13) and are the potentials due to the motions and
diffraction effects. We assume that they are all deter-
mined by means of a source distribution where cr(j) (x) =
(I) ( ~ ) + Ta(i) (~ ).
In the far field R ~ 1 we neglect the influence
of the stationary potential ¢(x; U) in (53) hence we
approximate (53) by:
382
(57)
This result is of (CAT ~ as T ~ O VRe[O,=)-
The upper integration boundary in (49), (50) and
(51) is the free surface
~ -- -Re [(iw¢>(x, y, O)—U¢~.(x, y, O))e-i=~] (58)
It follows from the pressure term that we may write:
-
1: d Pg (, ~ _ P| (I VI 2—U2) dz (59)
We find the following expression for Fit.:
- Fx =
—P ~ {kooky + 2TIm(~)) R cos add
+ 4 / ~ t ( R2 ~°~8—~R¢R + Liz f z) R cos 8+ Fit = F( " + F( 2)
+ R (¢R¢8 + BROW) R sin 8] dEdz
and for Fy
F.', =
—4 / {kudos + 2TIm(~)) R sin Ode
4 Jo ~—~ ~(R2 ~ jR¢R + ~zfz) Rsin8+
—R (¢R¢8 + BROW) R cos 85 dude (61 )
and for M::
Mz =
+4 1 | R (¢R¢8 + BRIE) dude (62)
The integration with respect to z needs some ex-
tra attention due to the fact that the exponential be-
haviour of the incident wave and the radiated or dif-
fracted wave is different, due to the dependence of the
wave number on ~ and 8, respectively see (54).
The asteriks in equation (60) denote the complex
conjugate and
A=
go exp(k1 (,l3)z + ik~ (~)(~ cos ~ + y sink))
+F(~)eis(~)4 exp {kl (~)z + iEl (WRY (63)
We formally write (63) as sum of two components:
~ = HI + SIB (64)
For the wave number kit (a) we can write neglecting U
terms in the wave number depending on b:
k~(~) = k~(1 + 2~ cos 8) (65)
The first term is related to the incoming wave field
while the second one describes the diffraction and ra-
diation ellects.
A closer look at (63) and (60) shows that the con-
tribution to Fc consists of two parts. The first one,
F(.~), originates from those parts of the cross products
that behave like R-1/2 while the second one, F(2), orig-
inates from those square terms of the second part of
(63) they behave like Ret. We formally write:
(66)
(60) We now can write:
F(1)
_ plan Ri/2 /; F(~)(cos ~ + cos if) ~
· cos {(k1 (it) cos(8—,B)—kl (~))R - S(~)) do +
+rP 2(n R1/2 | F(~) [(cos ~ + cos JIB) cos
+- (cos2 ~—sin2 8))—sin ~ sin p~
· cos {(k~(/3) cos(8—,0)—k~(~))R - S(~)3 do
+(2) (67)
We now apply the method of stationary phase for large
values of R. To do so we have to determine the sta-
tionary points of the integrands. We determine the
point where:
~~; {k~(~) cos(8—it) -- k~(~)) -- O (68)
We find two points,
8~ = ~ + 2r sink and 82 =- 3 + ~ + 2r sin 3 (69)
The second point gives a zero contribution, where we
remark that the last term in (67) is cancelled by the T-
dependent contribution of the last term. It is obvious
that the other terms give zero contribution due to the
factor or in 82. We use the asymptotic result:
r2=
J l(~) cos {(Al (I) cos(8—p) -- kit (~))R—S(~)) do
tRko ] ( 1 - 2T COS 3)1/2 {1(3 + 2T cos 3)
· COS ( S (d + 2T cos ~ ) + ~ /4 } ( 70 )
As mentioned before the second contribution is zero.
A critical reader will notice that formally we have to
deal with a non-uniformity due to the fact that we have
a first order 'uniform' solution. The phase correction
is of first order. Hence in our asymptotic result of (70)
we neglect the second order phase correction. The only
383
way to do the analysis properly is to use the exact value
of the wave number in (40) instead of our asymptotic
approximation.
Summarizing we now have:
Fat) ~
A ilk ~ cos(S(~*) + x/4)
·F(~*) {cos if* + cos id+
-2r cos(2 p))
where A =—PA and A* = ~ ~ 2T sin it.
(71)
A similar analysis can be followed for the sway force
and leads to the following result:
Fat) ~
A ilk ~ cos(S(~*) + ~/4)
·F(~*) {sin it* + sin 3+
—2T sin(2 if))
(72)
The second part of the wave drift force may now
be written as:
F(2) =
—4ko/ F2~{cos~—2r cos(2 8~) do (73)
For the sway force this leads to:
F(2) =
—4koJr F2~{sin~—2T sin(2 8~) do (74)
7. Numerical Results
For the validation of the mathematical model pre-
sented in the previous sections computations have been
performed on a fully loaded 200 kDWT tanker and a
half immersed sphere.
The description of the tanker can be found in
Huijsmans et al. t124. The numerical results of the
wave drift forces on the tanker have been validated
against results of model test experiments. The cal-
culations for the 200 kDWT tanker are applicable to
the situation of a tanker in both head-on current and
waves as well as the case of the waves and current co-
parallel under 135 degrees, as displayed in the next
figure.
w
c: ~ tow
Fig. 4. Definition of waves and current
The calculated steady double body flow using the
Hess and Smith algorithm apply to the situation of a
tanker under a drift angle of 135 degrees (in principle
the solution can be made applicable to any drift angle
Oc)
The boundary conditions on the mean wetted hull
for the radiation problem have been applied with the
steady perturbation potential set to zero, which then
results in the well-known expression for the m-terms
of the radiation boundary condition. A more proper
way of handling the radiation boundary condition as
published by Zhao et al. t10] was not attempted here
due to the difficulty of the splitting the boundary con-
ditions into linear forward speed dependent boundary
conditions.
For the results of computation for the wave drift
force on the sphere a comparison was made with results
of computations as was obtained by Nossen et al. A.
The panel discretization of the tanker amounted
to 238, 720 and 1610 panels in order to check the nu-
merical accuracy of the pressure integration technique.
The result of the computation are depicted in Fig. 15.
The number of panels for the sphere amounted to 744.
To solve the integral equations for the source
strength a(j) (x) and cr(i) (x) large algebraic equations
have to be solved.
In the present solver use is made of a newly devel-
oped iteration scheme. Details of this new solver for
Boundary Integral Equations will be published in the
near future. To solve a large system of 1610 unknowns
for the radiated and diffracted modes only required 4
to 6 iteration steps to gain an accuracy of 10-4 in the
Euclidian norm. The timing was approximately 28
CPU seconds per wave frequency for 1610 panels on
an ETA lOP mini supercomputer from ETA systems.
The analysis of the results of the wave drift forces us-
ing the pressure distribution integration, as shown in
384
Figs. 9 to 12 leads to the observation that the numer-
ical accuracy of the pressure distribution integration
greatly depends on the panel discretization. Detailed
results are presented in Table 1 for the frequencies of
0.5 and 0.7 rad/s.
Table 1
Fir for 180 deg. waves and current
238 720
0.5 15.5 17.3
0.7 11.2 15.3
1610
l
14.9
15.1
7.1 Half Immersed Sphere
to 7.
The results of computation for the sphere are pre-
sented for the mean wave drift force. The mean drift
force on the sphere has been calculated in two ways,
i.e. one describing the correct influence of the steady
perturbation potential and the other is using the slen-
derness approximation in which the influence of the
steady perturbation potential has been neglected.
The region of integration over the free surface was
from ~ = R to r = 3R, where R is the radius of the
sphere. The number of free surface panels amounted
to 810 panels.
These results are presented in the following Figs. 5
Cat 40
1
.C 20
X
Lie
Drift force on sphere
V = 0.0 m s—1
4' _
~ ~rcssur~f Hi\
0.00 0.25 V 0.50 0.75 tOO
Frequency in rod s—1
Fig. 5. Zero speed wave drift forces
L
8.oo
~ . . .
0.25 ~ 0.50 0.75 1.00
Frequency in rod s—1
60
E 40
o
· _
X 20
Drift force on sphere
Pressure integration
~ v _ o.o ms—1 1
it,
L J
8.1 )O Q25 0.! SO 0.75 1.00
Frequency in rod s—1
Fig. 6. Forward speed wave drift forces
60
Drift force on sphere
Influence stationary potential
. . .
1F~ __ _
11 ~ stat.po eq 0.0
D E} stat.pot ne 0.0
Fig. 7. Influence of steady potential
Drift force on sphere
V = 1.0 m s—1
nor 1 1 1
E 40
· _
X 20
1_
r-1~
mpuls
Nossen
~1
lo_
8. 0 0.25 0.50 0.75 1.00
Frequency in rod s—1
in,
~ _
Fig. 8. Forward speed results Nossen
385
In Fig. 8 the results of computation are presented
for En = 0.032 against other results as obtained by
Nossen et al. t74.
The observed difference between the results of
Nossen and ours is due to the fact that the results
of Nossen apply to a fixed cylinder in waves whereas
our results apply to the free floating case. In the range
where diffraction dominates, the results tend to be the
same.
The zero speed wave drift forces on the sphere are
calculated in two ways; i.e. one using a pressure dis-
tribution integration technique and the other one uses
the momentum flux analysis (Maruo [23~.
Drift force in 135 degrees
V = 2.0 m s—1
30
a)
~ 20
C) modeltest
pressure
1
It
8. 0 o. 25 0 iO 0. 75 t00
Frequency in rod s—1
Fig. 9. Mean surge drift force
Drift force In 135 degrees
V = 2.0 m s—1
100. ~ ' ' ~ '
~ modeltest
-
pressure
50
~1
' '5a~b' 1 1
Too 0.2 0.50 0. 75 t00
Frequency in rod s—1
Fig. 10. Mean sway drift force
For the forward speed case this was also done, how-
ever now the equations (71) and (73) have to be used
for the momentum flux analysis.
One can see from Figs. 5 to 7 that the calculations
using the momentum balance are not so much influ-
enced by the panel discretization. This can be made
plausable by observing that the expression for the wave
drift forces from the momentum balance only simple
source distribution integration over the mean wetted
hull is required. It is our opinion that due to the in-
tegration of higher order derivatives in the pressure
distribution for the forward speed case the results are
more sensitive for the way the integration is performed.
Drift force in 135 degrees
v = 2.0 m s—1
30
l
20
o
c
L,= 10
m
1
- 1
~ N
: _~__
8. lo o. o. 0 o. '5 t' 10
Frequency in rod s—1
Fig. 11. Mean surge drift force
Drift force in 135 degrees
v = 20 m ,:—1
~Q5 1 1 1 1
. 100 _
_ ;~e
moddtest
presto
Cal
E 75 1
o
~ 1
__
3.00 025 OSO- Q75 t00
Frequency in rod s—1
Fig. 12. Mean sway drift force
386
7.2 Tanker
At MARIN a number of tests have been conducted.
The test comprise in short regular wave test on a sta-
tionary moored 200 kDWT tanker in 82.5 metres of
water depth with waves and current parallel to each
other. The results of model tests and computations
for head waves and 2 knots current have been exten-
sively described by Wichers [25] and Huijsmans [21~.
The results of model tests for the tanker with the waves
and current coming from 135 degrees are presented in
Figs. 9 and 12 for the mean surge and sway drift force
respectively.
30
r
20
o
· _
x lo
Drift force In 135 degrees
V = 2.0 m s—1
~ stat~ot eq 0.0
Cl stat .pot ne 0.0 r
_~_-
8.1 )O 0.' 2 o' sO 0. 5 tt ~0
Frequency in rod s—1
Fig. 13. Influence of steady potential
Drift force In 135 degrees
V = 2 0 m ~—1
_!
E
a so
c
11
Ho 1 1
STAT.POT ED 0. ~
Ci STATFOT NE 0. to_ _
/ ~
8. so o. 25~ 0.! SO 0. r5 t O
Frequency in rod s—1
Fig. 14. Influence of steady potential
The resulting drift forces are calculated in two ways;
i.e. one using the pressure integration technique and
one using the momentum flux analysis. The results
have been presented in Figs. 11 and 12 for the surge
and sway wave drift force respectively. The influence
of the stationary potential on the wave drift forces has
been studied.The results are displayed in Figs. 13 and
14. From these results one is tempted to conclude
that the influence of the stationary potential on the
drift forces in this case is restricted to the sway drift
force results, however, previous observations also in-
dicate that the use of a pressure integration for the
calculation of the wave drift forces with forward speed
are somewhat doubtful, due to the large dependency
on the panel discretization of the body.
Drift force in 135 degrees
Influence of discretization
V = 2.0 m s—1
30
1
20
c
· _
x 10
o
_ x 1610 panels - __
C) 720 panels A
1B 238 panels ~ ~
0.00 0.2~ 0.50 0.75 1.00
Frequency in rod s—1
Fig. 15. Influence of panel discretization
References
[1] Pinkster, J.A., Low frequency second order wave
exciting forces on Boating structures, PhD thesis,
University of Delft ( 1980).
[2] Benschop, A., Hermans, A.J. and Huijsmans,
R.H.M., "Second order diffraction forces on a ship
in irregular waves", Journal of Applied Ocean Re-
search, Vol. 9 (1987).
[3] Remery, G.F.M. and Hermans, A.J., "Slow drift
oscillations of a moored object in random seas",
Proceedings of the Offshore Technology Confer-
ence, Houston, Paper 1500 (1971).
387
[4] Wichers, J.E.W., `'On the low frequency surge
motions of vessels moored in high seas", Off-
shore Technology Conference, Houston, Paper
4437 (1982~.
t54 Wichers, J.E.W. and Huijsmans, R.H.M., "On
the low frequency hydrodynamic damping force
acting on offshore moored vessels", Offshore
Technology Conference, Houston, Paper 4813
(1984~.
t64 Hermans, A.J. and Huijsmans, R.H.M., "The ef-
fect of moderate speed on the motions of floating
bodies", Schiffstechnik (March 1987~.
t74 Nossen, J., Palm, E. and Grue, J., "On the solu-
tion of the radiation and diffraction problems for a
floating body with a small forward speed", Inter-
national Workshop on Water Waves and Floating
Bodies, Oslo (1989~.
[8] Sclavonous, P., "The slow drift wave damping
of floating bodies", International Workshop on
Water Waves and Floating Bodies, Oslo (1989~.
t9] Hu, C.S. and Eatock Taylor, R., "A small forward
speed pertubation method for wave-body prob-
lems", International Workshop on Water Waves
and Floating Bodies, Oslo (1989~. t20]
t104 Zhao, R. and Faltinsen, O., "A discussion of
the m-terms in the wave-body interaction prob-
lem", International Workshop on Water Waves
and Floating Bodies, Oslo (1989~.
t11] Huijsmans, R.H.M. and Hermans, A.J., "A fast
algorithm for the calculation of 3-D ship motions
at moderate forward speed", Proceedings of 4th
Numerical Ship Hydrodynamic Conference, Wash-
ington (1985~.
[12] Huijsmans, R.H.M. and Wichers, J.E.W., "Con-
siderations on wave drift damping of a moored
tanker for zero and non-zero drift angles", Pro-
ceedings of the PRADS Symposium, Trondheim
(1987~.
Eggers, K., "Non-Kelvin dispersive waves around
non slender ships", Schiffstechnik, Vol. 8, (1981~.
t144 Baba, E., Wave resistance of ships in low speed,
Technical Report 109, Mitsubishi Technical Bul-
letin ( 1976).
t15] Hermans, A.J., The wave pattern of a ship sailing
at low speed, Technical Report 84A, University of
Delaware (1980~.
t164 Brandsma, F.J. and Hermans, A.J., "A quasi-
linear free surface condition in slow ship theory",
Schiffstechnik (April 1985~.
t17] Wehausen, J. and Laitone, E., Surface Waves,
Volume 9 of Handbuch der Physik, Springer Ver-
lag (1960~.
t184 Newman, J.N., "The evaluation of free-surface
Green's functions", Proceedings of the 4th Inter-
national Conference on Numerical Ship Hydrody-
namics, Washington ( 1985~.
t19; Telste, J. and Noblesse, F., "Numerical evalua-
tion of Green's functions of water wave diffrac-
tion and radiation", Journal of Ship Research,
Vol. 1111 (1987).
Hearn, G., Priv. Communication ( 1989 ).
[21] Huijsmans, R.H.M., "Wave drift forces in cur-
rent", 1 6th Conference on Naval Hydrodynamics,
Berkeley~ 1986~.
t22] Newman, J.N., "The drift force and moment on
a ship in waves", Journal of Ship Research, Vol.
10 (1967~.
t23] Marno, H., "The drift of a body floating on
waves", Journal of Ship Research, Vol. 4 (1960~.
t24] Faltinsen, O. and Michelsen, F., "The motions of
large structures in waves at zero Froude number",
Symposium on the Dynamics of Marine Vehicles
and Structures in Waves, London (1974~.
t25] Wichers, J.E.W., "Progress in computer simula-
tions of SPM moored vessels", Offshore Technol-
ogy Conference, Houston, Paper 5175 (1986~.
388
DISCUSSION
by R. Zhao
( 1 ) Discuss about direct pressure integration
method
(2) Discuss about wave drift damping
Author's Reply
The direct pressure integration method is
based on the integration of local hydrodynamic
quantities like water velocities and gradients
of pressures over the mean wetted hull. This
procedure i s demons bated in our papers
presented at the ONR Conference in 1986, [Al ]
and at the UTAM symposium, [A2 ].
One of the main problems in using the
direct pressure integration method i s based on
the accuracy of the underlying hydrodynamic
data. In the zero speed case there is
sufficient evidence that this procedure, using
3-D diffraction theory results, will lead to
meaningful results. In the non-zero speed
case however this evidence is lacking and to
the author's opinion the accuracy of the f irst
order results based on 3-D diffraction results
with forward speed must be even more accurate
than the zero speed results due to the fact
that in the evaluation of the pressure
distribution, gradients of the water velocity
distribution are required. In pursuing the
direct pressure integration one has to resort
to e.g. higher order panel methods to obtain
the required accuracy.
The question regarding the wave drift
damping has not been addressed in this paper,
however information can easily be obtained
using the results of our paper. At the moment
we only calculated the drift force at two
speeds, which is in principle enough to obtain
wave drift damping data. However, if one
wants to obtain reliable results for the wave
drift damping coefficient several more speeds
(also negative speeds) have to be calculated.
This will be done in the near future.
[A1 ] R. H. M. Hui jsmans, "Wave Drift Forces in
Current", 15th ONR Symposium, Berkeley,
1986.
[A2] R. H. M. Hui jsmans, "Slowly Varying Wave
Drift Forces in Current", IUTAM
Symposium on non-linear water waves,
Tokyo, 1987.
389
