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OCR for page 391
Numerical Prediction of Semi-Submersible Non-Linear Motions
in Irregular Waves
X. B. Chen and B. Molin
Institut Fransa~s du Petrole
Rueil-Malmaison, France
Abstract
A numerical model to predict non-linear vertical motions
of a semi-submersible in regular and irregular waves is presented.
Non-linear second-order forces are evaluated by using Molin's
method which permits to obtain them without explicitely solving
for the second-order potential. Non-linear low frequency motions
being large in magnitude, variations of second-order forces with
the motion of the body mean position are considered in the time
simulation. As application cases, two floating bodies with small
waterplane areas are taken into account. Comparisons of calcu-
lations with experimental results show that this theory correctly
predicts the low frequency motions.
1. Introduction
The nonlinear behavior of floating offshore structures in
regular and irregular waves is an interesting and important topic
in ocean engineering. The problem of dynamic interaction of
ocean waves and a floating structure is intrinsically nonlinear.
Even though potential flow is assumed, direct solution to the
three-dimensional problem has not been achieved yet. For small
amplitude waves, the perturbation method is often used to sep-
arate it into first- (linear) and second-order problems. The first-
order problem can be solved by using a singularity method. The
second-order problem is quite more complicated. The second-
order problem is important as it yields the second-order loads
whose frequencies, disjointed from those of the first-order, may
be close to the resonant frequencies of the fluid-structure sys-
tem. These exciting loads are not very large in magnitude but
significant amplification may be induced due to the small system
damping.
The large low-frequency motions of floating bodies ex-
cited by second-order low frequency loads are dealt with in this
paper. In recent years, numerous contributions (Newman[02]i
1974, Pinkster and Huijsmans[03] 1982, Molin[04] 1983 etc.) have
been published. Although horizontal low-frequency motions for
a moored floating structure are often evaluated and in reason-
able agreement with experiments, vertical resonant motions have
been little studied and poorly predicted (Pinkster et DiJk[05]
1985~. A time simulation model based on separation of mean
large slow motions of bodies and their first-order motions by
two time scales is presented. Triantafyllou's[06] (1982) theory on
the horizontal surge, sway and yaw motions is extended, in this
model, to full six degrees low frequency motions. Specifically the
variation of the second-order loads with body mean position is
Reference number at end of paper
391
studied and some interesting remarks are deduced.
Exact second-order loads are obtained by applying Mo-
lin's[°~] (1979) method and using two identities to transform
Haskind integrals (Chen[~°] 1989) for second-order potential con-
tributions, besides first-order quantities contributions which are
evaluated straightforwardly after solving the first-order problem.
The system damping being an important factor is composed of
radiation part and viscous part. At low frequencies as the radi-
ation damping is negligible, the viscous part presented by linear
and quadratic model is used in time simulation.
The validation of numerical model is made by comparison
with experiment results. Two floating bodies having low vertical
resonant frequency idealizing characteristic offshore structures
which have large submerged volumes with comparatively small
waterplane areas, are used in experiments undertaken at the Ship
Research Institute of Japan by Molin[~] (1982~. Comparisons
of numerical calculations with experimental results show that
this theory predicts correctly the large low frequency vertical
motions.
2. Low Frequency Motions Theory
A floating body is assumed to be submitted to bichro-
matic waves of amplitudes al, a2 and frequencies we, w2. It re-
sponds oscillatorilly with small amplitudes at the frequencies we
and w2 around its mean postion which moves with large am-
plitude at the low frequency two—way. As illustrated by figure
1, three reference systems are defined: (1) A three-dimensional
Cartesian coordinate system Oo~oyozo fixed in espace with the z-
axis vertically upwards. (2) System ozyz following the large low
frequency motion which has its origin at the body gravity center
G. and (3) Another coordinate system OXYZ tied to the body
which coincides with ozyz in the absence of small high frequency
motions. The three systems coincide at the initial time.
Assuming a vector rO(t) with six components of surge,
sway, heave, roll, pitch and yaw which designate the six degrees
of freedom of the body:
r0(t) = tzoft), yO(t), Soft), joft)'coft), joft)3T (1)
Two small parameters ~ and ~ are introduced to express this
vector into two parts:
r0(t) = routs + ~ · {(t, the) (2)
Where a, a perturbation parameter, is proportional to the ratio
of wave amplitude to wave length while ~ represents the ratio
between the two time scales of motions: a high frequency mo-
OCR for page 392
tion with small amplitude around its mean position and a low
frequency motion of the body mean position of large amplitude
because of low resonant frequencies of the system. (More strictly
Figure 1 Reference Systems
so
~YO z
/ \1 1 /` and
o
'it 1~
IVY ~
we should take six ratios of time scales since the six low resonant
frequencies are not the same).
It was shown by Triantafyllou[06], in the inviscid, irro-
tational and incompressible fluid, that the two potential prob-
lems relating these two-scales motions can be separated and that
the first-order problem is the same as if no low frequency mo-
tions exist and high frequency first-order forces and motions are
only parametrically dependent on the low frequency motions. We
shall emphasize this 'parametrical' dependence by representing
the low frequency forces as a function of the body mean position.
The high frequency motions excited by first-order forces
can easily be derived by classical means of using motion transfer
function and those from second-order high frequency excitation
are of order 52 and negligible. As these high frequency motions
are supposed to be small as compared to the low frequency mo-
tions, our efforts are concentrated on the evaluation of the large
mean position motions:
rG(t) = [xG(t), yG(t), zG(t), ~G(t), ~G(t), (G(t)]T (3)
We shall first establish the low frequency motion equation, in
the following, assuming the dependence on the mean position
of the second-order low frequency forces and proceed the time
simulation and frequential analyses. The complete second-order
forces are evaluated in the last paragraph of this section.
2.1 Low Frequency Motion Equation
The complex notations are afterward admitted in follow-
ing analytical expressions. For exemple, a real harmonic function
H(t) of frequency w having an amplitude h and a phase ,(] is rep-
resented by:
H(t) = h ~ costed +,l])
and so:
= h cos,l} · costed)—h sin l] · singlet)
=~{h~e~iWt} (4)
h=hcos,B—i~hsin,0=h~e~i~ (5)
where i = ~/=[ and ~{F) designates the real part of the complex
function F.
Assuming the mass-inertia matrix of system and added
mass-inertia matrix M + m, damping matrix B. restoring ma-
trix K and exciting low frequency forces F(ro, t), the motion
equation can be written:
(M + m) d`2 rG(t) + B do rG(t) + KrG(t) = F(rG, t) (6)
In this equation, the exciting low frequency forces can be
caused by wind, current and waves. For the sake of simplicity,
F(rG, t) only includes the wave effects. Using Taylor's series,
these low frequency forces can be expressed by:
F(rG, t) = F(roG' t) + t~rG—TOG) V] F(rOG, t) + (7)
By denoting:
Fott) = F(roG, t) (8)
F:(t) = VF(rOG, t) (9)
noting that F~(t) is a 6 * 6 matrix, the equation (7) can be then
written as:
F(rG, t) = Forth + F:(t) (rG—TOG) + (10)
Substituting the equation (10) in which the first two terms are
used and the vector TOG, representing the origin position of the
body, is taken as zero vector into the equation (6), the latter
can be rewritten as:
(M+m~d'2rG(t)+Bd~rG(t)+[K—F:(t)]rG(t) = Fott) (11)
This equation ressembles Mathieu's equation if Fat (t) is periodic.
The solutions to this equation will be discussed in detail for the
cases of regular, Dichromatic and irregular waves.
Motion Id Regular Waves
A regular wave with an amplitude a and a phase ,B of
frequency w, in the direction of angle ~ with the axis x is repre-
sented by the following form:
7~(t) = a cos~kx cos ~ + by sin ~—at—l])
with:
= JR {a · emit } (12)
a = a . eik(= cOs e+y sin 8)-i~ (13)
The low frequency forces contain only the steady forces
which are function of wave frequency and mean position:
F(ro, t) = Foe + Fat ro (14)
in which Foe is the steady forces at the origin position and F.`:
their first derivatives with regard to mean position displacement.
They are expressed in complex form:
and:
Foe = ~ { 2 an F20(w,—w) } (15)
Fat = at{ {aa*F (w —w)} (16)
where a* is complex conjugate of a, the complex amplitude of
wave at the original point COG, YG) and F20(wj,—we) is the trans-
fer function of the second-order low frequency forces in bichro-
matic wave of frequencies wj and we (here we impose Wj = we =
w). F2~(wj,—we) is the gradient matrix of Frown,—we). Intro-
ducing these equations (1~15 16) into (11), we obtained a new
formula to calculate the mean displacement of the body:
392
OCR for page 393
roO = [K—37 { 2 aa* F2l (w ~—Cal) }] 37 { 2 aa* F20(w,—w) }
(17)
instead of the classical formula:
rGoc = K-i ~ { 2 aa F20(w,—w)} (18)
The formula (17) indicates that mean displacement of a
floating body in regular wave is not strictely proportional to the
square of the wave amplitude and that it depends also on the
derivative of the steady forces. For example in the case of heave
motion, the vertical steady force and its derivative being usually
of the same sign, the mean displacement predicted by (17) may
be much larger than (18) which is classically used. Another in-
teresting remark can be obtained by considering the free motion
of the system represented by the equations (ll). Tl~e natural res-
onant frequency of heave motion, assumed to be decoupled from
the other degrees of freedom, is given by:
WRz = ~ (= ~{2 aa F2l:(~'—w)} (19)
This shows that the natural frequency in still water is diflfer-
ent from that in a regular wave. The natural frequency may be
smaller when wave amplitude increases and it depends in addi-
tion on wave frequency. In fact, the derivatives of steady forces
play a role like a supplement (or decrease) of the restoring forces
in the system.
Motion In Bichromatic Waves
Free surface elevation of Dichromatic wave is represented,
at first-order form, as:
t7(t) = 9~(8,~t,4:,t)+772(8,~2,42,t)
= ~ {at e~i~l'} + ~ {a2 · e~i~t } (20,
here al and a2 are complex amplitudes of waves including the
directional angle ~ and their phases §~ and ,02, and corresponding
frequencies w~ and w2.
In this Dichromatic wave system, the low frequency forces
are composed of a steady part and an oscillatory part which has
a difference frequency (~:—and:
and they are written by:
F(rG, t) = Fa(rG) + F2(rG, t) (21)
Fa(rG) = F`o AFRO rG (22,
F2(lo,t) = F20(t) + F21(t) lo (23)
Introducing above identities into the equation (11), we
have the motion equation as:
(M + m)d~2rG(t) + Bd~rG(t) + [K—Fat—F2~(~)]rG(t)
= Fdo + F20(t) (24)
in which the low frequency forces can be rewritten in complex
form including their transfer functions:
F&o = ~ { 2a~a;F20(~t,—Am) + 2a2a2F20(~2,—~2)3 (25)
Fat --~{2a~a;F2~(~:,—we) + 2a2a2F2~((~,—w2)} (26)
F20(t) = ~ {ala~F20(~l,—W2) · e-i(~l-W2)~} (27)
F2~ (t) = ~ {at a2 Fat (ma,—w2) · e -i(~t ~W9)t } (28)
The evaluation of the transfer function of second-order low fre-
quency forces will be considered in the next paragraph (2.2~. It
is assumed to be known now.
The exact solution to the equation (2~1) can be obtained by
time simulation method. We integrate this equation, supposing a
starting point of rG(O) = `& rG(O) = 0, by a stantard fourth-order
Runge-Kutta method. The motion simulate;] becomes stable af-
ter some transient periods.
The frequency domain analysis, in Dichromatic wave case,
is possible if the following assumption is admitted:
rG(t) = rGO + 37 [rG~e~i(~~~)~} + ~ {ro ze~i2(~l~w2)~} (29)
Introducing above equation into the equation (24), each mode of
motion can be approximatively obtained by below identities:
rGO = [K—F`~-~ F`o (30)
rot = [ - (M ~ m)(w:—ways—iB(~—w2) + K—F,`~] ~
Alar [F20(~,—w2) ~ F2i (at,—W2)rGO] (31)
rG2 = [ - 4(M + m)(w:—ways—i2B((~:—w2) + K—Fat]
· 2 alas [Fat (at,—w2)rG~] (32,
The formula (30) for mean displacment is the same as (17~.
The oscillatory amplitude of frequency (ad—w2~) iS determined
by (31~. It depends not only on the low frequency forces but
also on the mean displacement (rGo) and derivatives of steady
forces (F`~. Since the second-order forces are proportional to
the square of the wave amplitudes, the amplitude of the low
frequency is thought to be the same too. It may not be true be-
cause the 'supplement' stiffness (Fat) increases with the square
of wave amplitudes. For instance, in case of heave motion, Fizz
being positive, the motion amplitude does not increase paraboli-
cally as wave amplitude increases. The motion amplitude should
be less than proportional to wave amplitude square when the
difference frequency is larger than that of resonance and more
when the difference frequency is smaller. We keep the double dif-
ference frequency 2(~1—w2) term because this frequency may be
closer to the resonant frequency of the system when the different
frequency (we—w2) is much lower.
Motion In Irregular Waves
The unidirectional irregular wave is often represented by
a finite series of elementary Airy components whose amplitudes
are deduced from the wave power spectrum S,7,7(W):
N
7~(t) = ~ aj cos(kjx cos ~ + kjy Sill ~—wjt—dj) (33)
j=
and:
aj = ~/2S,1,7 (A )tWj (34)
where ~ is the wave direction angle. Phase angles Al (j = 1, 2, · · ·,
N) are determined at random between (0, 27r) and {Wj is the jth
discretization step of power spectrum frequency width. If we use
the complex presentation, the wave system is rewitten by:
393
OCR for page 394
t7(t) = 32 {I ad · e }
with:
ad = `72S,7,7 (wj )~j . e$ki(~ cOs 8+y sin 8)-~§J (36)
If all discretization steps bwj(j = 1, · · · ,11) are equal (=
bw) and the largest low frequency considered am is:
Wm < n · bw n < N (37)
we can then express the second-order steady part and the oscil-
latory part of the low frequency forces by following simple and
double summation forms:
F`o = ~ { ~ 2 as a; F20 (wj,—wj ) } (38)
Fat = FIR { Am, 2 a} ad* Fat (wj ~—no ) } (39)
~ n N ~
F20(t) = ~ ~ ~ ~ ajaj_eF20(wj,—Wj_~) · e~i(~j~~j-~)t
t=t j=~+i )
(40)
n N ~
Fat (t) = SIR }~ Am, Am, ajaj*_~F2~ (Wj,—Wj_~) · e~i(~j~~j~~)t
1~=1 j=l+1 )
(41)
As we see the above expressions of oscillatory part of low
frequency forces, two remarks have to be pointed out: one is
that the number N of element wave frequencies should be large
enough to represent correctly irregular waves and to evaluate the
low frequency forces at the difference frequencies (Wj—Wj_~) for
= 1,.~.,n and j = 1,---,N (N is typically between 100 and
200). Another is that the double summations of low frequency
forces at each time step need more computing tinge as the num-
ber N is large. Prior to doing the simulation by same equation
(24) as in Dichromatic wave case, two special files have to be built
up. First file containing the second-order force transfer function
at origin mean position can be obtained by running repeately
a first-order diffraction-radiation code for a number (m) of fre-
quencies (m may be equal to N/10 or N/5) which cover the
whole frequency range and successively a second-order code for
repeating m(m+1~/2 times The first- and second-order solutions
for other frequencies can then be obtained by an interpolation
method. Second file containing the derivatives of second-order
force transfer function can be achieved by etablishing two files
at two or three different mean positions and using the finite dif-
ference method.
2.2 Low Frequency Forces
The low frequency forces, obtained by integrating the
second-order hydrodynamic pressure on the body wetted surface,
consist of one part which depends only on first-order quantities
(potential, velocity and motions etc.) and another part which
depends on the second-order velocity potential. They can then
be written as:
Frown,—w') = F21(Wj,—w') + Fb2(Wj,—w`) (42)
The first part forces F2~(Wj,—we) are formulated by integrat-
ing the hydrodynamic pressure which corresponds to quadratic
terms in Bernoulli's equation and corrective terms due to the in-
tegration over the average surface instead of instantaneous sur-
face (supposing a given mean position). These forces consisting
of four terms are presented by:
2 Afro
+P2~/ v~ji).v¢(~) NdS
2 ./. ./s.Co(MoMj~'v¢( )—MoAl'~ivit)'Nds
2(Rj Fens + R' Fog) (43)
where p is water density and 9 is the acceleration due to gravity.
The indices j and e represent the first-order quantities corre-
sponding the wave frequencies Hi and we and the sign * des-
ignates the conjugate complex. N is the general normal vector
of body surface towards the fluid. The first contribution is an
integral of relative water elevation (difference of water eleva-
tion ~ and vertical displacement () on the average waterline I'o.
The second contribution is the integral of pressure due to the
fluid quadratic velocity (~(~) being first-order velocity potential).
The third one comes from the corrective term of the first-order
pressure on the average surface instead of instantaneous surface
(MoM being translation of one point Mo on body surface) and
the fourth one is a correction of first-order inertia forces Fin due
to the first-order rotations R.
All these contributions can be evaluated directely once
the first-order problem is solved. In fact, the first-orecr veloc-
ity potential obtained by a singularity method is used to have
the first-order excitation forces and motions by mechanical equa-
tions. A source distribution which is kinemalica]ly equivalent to
the body responding in waves can be obtained by considering
first-order motions and used to evaluate first-order quantities
(potential and its spatial derivatives on bocly's surface and the
water elevation on the waterline) that are needed for the first
part of the second-order forces.
The second-order velocity potential ~(2)~,y,z,t) is as-
sumed to have incident, diffraction and radiation parts in the
same manner as the first-order potential:
4~(2)(X ye Z t) = ~ {all (2) + ¢(2) + ~~(2)) · e~~(Wj~~)~} (44)
in which the incident potential ¢(2) is known:
2
~9
where:
in which H is the waterdepth.
y,, _
1 g~kj—key tank—k'~H)—(Wj—W')2
cosh[l kj—k' (A + H)] A- i(kj - k')(~ cos 8+y sin 8) (45)
A- = j ~ kjk' t1 + tanh~kjH) tanh~k'H)]
- [wj cosh2(kj0) we cosll2(k'H)] (46)
The second-order radiation potential is supposed to sat-
isfy the same equations as the first-order radiation potential. The
only difference corresponds to low frequency (wj—~c) instead
of the frequencies Wj or we. The solution by using the same sin-
gularity method gives us the potential damping and added mass
for one low frequency.
394
All nonhomogeneous properties of the second-order prob-
OCR for page 395
lem are tied with the second-order diffraction potential which
satisfies not only Laplace's equation, the sea bed condition equa-
tion and a proper radiation condition but also the nonhomoge-
neous free surface condition and the nonhomogeneous body sur-
face condition. These two latter conditions are written as:
on the free surface and:
on the body surface.
—God—W'y2¢D2) + 9,' ¢(2) = a(2) (47)
~ IDA) =—~ ¢(2) + a(2) (48)
The nonhomogeneous terms a(2) arid a(2) being functions
of the first-order potentials are expressed by:
a(2) = it —W`~(Vit) · V¢( ) + Vest · V¢pj )
_u~ [¢j~)~-w' 5~(~)*+g5 ¢(~)*)
integral in fluid domain including the Laplace operator on ODE)
and (l into boundary surface integrals. By analysing asymptotic
expressions of the velocity potentials and using the stationary
phase theorem for the surface integral at infirmity, Molill[°~] (1979)
has shown that the following identity is true:
I ISCO D ||SCO [ knit + c ] jIdS
9 | .l o aL D . bids (54'
where:
1 = 1, 2, · · , 6
Introducing this identity in the expression (51), we have the com-
plete formula for the Ith component of the low frequency forces:
F221(Wj,—Wl) = F2II(Wj,—WI) + F2CI(Wj,—WC) + F231(Wj,—W')
(55)
in which we name the incident integral:
+(—Aid I +§Oz2~' )¢Pj] F2,,(~j,—w') =—it's—w'Jpll off Mods (56)
and:
+ 2 ~ [() (_Wj2 By ¢( j) + g 5z2 ¢( ))
+~-wj2,9 ¢(j)+g5 2~(~j))~(~)*] (49)
a(2) = (VEj—V¢ji))R'* · nO + (VE'!—V¢(~) JRj · nO
—(MoMj · V)V¢(~)*nO—(MoM' · V)V¢ji)nO (50)
in which ¢() is the first-order incident potential and ¢(~) is the
sum of the first-order diffraction and radiation potentials. VE is
the velocity of the point Me on the body surface.
It is important to note in the above expressions the dou-
ble spatial derivatives of the first-order velocity potential. For
a three dimensional body of arbitrary form, because of the sin-
gularity of Green function as shown by t09], the double spatial
derivatives near the body surface are not possible to be evaluated
accurately by using a numerical method. The nonhomogeneous
terms can not be then obtained directely. Belt we are going to
try to calculate the integrals containing Close terms.
by:
The second part of the low frequency forces are written
F22(h~j,—Ad) = i~h~j—W.C)p|| (~(2) + fD2))NdS (51)
sea
The contribution of the incident potential is easily obtained as
the second-order incident potential is known (see equation 45~.
In order to evaluate the contribution of the diffraction potential,
additional potentials ',61 (I = 1,···,6) are introduced. These ad-
ditional potentials satisfy Laplace's equation, the sea bed and the
radiation conditions and the two following boundary conditions:
—(Wj—w')2?~' + g,90 ~,b1 = 0 z = 0 (52)
¢' = Nl on ScO (53)
1= 1,2,.~.,6
these equations being the same as in the first-order radiation
problem, these additional potentials are obtained in the same
way.
Green's second identity is applied to transform the volume
395
the Haskind integral on the body surface:
F2CI(~j'—Ad) = i(Wj—W')P Jl [~( )—a( ] ?/)ldS (57)
and the Haskind integral on the free surface:
F231~j,—me) = it—~e)§ | /, aLD¢IdS (58)
With these three integrals including only terms function of the
first-order velocity potential, the complete low frequency forces
may be evaluated without explicitely knowing the second-order
velocity potential which is more difficult to have, on condition
that the difficulty of the double derivatives of the first-order ve-
locity potential can be overcome.
An identity deduced from Stokes' theorem can be derived
by making the vectorial analysis:
| | (G V)V~ · no dS = /(V~ ~ G) dI'
+ | IS [(V~ V)G + Vat div(G>)] · no dS (59)
By using this identity into (57) and supposing G = ¢~.MoM, the
Haskind integral on the body surface is transformed into integrals
which don't contain any more the double spatial derivatives and
can so be evaluated accurately.
In order to evaluate the Haskind integral on the free sur-
face, we divide the unlimited free surface into two regions: an in-
ner region Sinn which is limited by the waterline and a boundary
circle line some distance far away from the body and an outer
region SOut which is the surface extending from the boundary
circle line to infinity.
In the inner region, since the Haskind integral contains
the double spatial derivative of the first-order potential, another
identity derived from Riemann's theorem is developped as:
s. LIZ Jar [0Y 02 ]
~/IS [02 62~+ ,~yG- Gyp] dS (60)
Using this identity, the Haskind integral on the inner region can
be transfomed into integrals in which double derivative does not
appear any more and may be so calculated numerically by using
OCR for page 396
classical quadratic method.
In the outer region, the asymptotical form of the non-
homogeneous term a(2), in the cylindrical reference (r,E,z), is
derived by using the developpment of Kocl~in's function in the
presentation of the first-order velocity potential. This asymptotic
expression of a(2) as form of a product of Fourier series in ~ and
oscillatory radial functions in r and the similar expression for FEZ
are introduced into the Haskind integral. This integral is then
separated into a product of two line integrals: first integral in
which is not difficult to be calculated by applying the orthogo-
nality properties of Fourier series product and another integral
in the radial distance r which is easily obtained by analytical
results of Fresnel integral.
The value of the Haskind integral on the free surface is
the sum of integral results on the inner region and on the outer
region. This value varies oscillatorily with the radial distance
of the boundary circle line which separates Else inner and outer
regions. As shown in [10i, the oscillatory value converges much
more rapidly than that without the outer region integral and
the final result is very close to the average value of oscillation.
In a test case of a vertical cylinder (height/radius=5.) for which
we have analytical results, the inner region is discretized by 36
points in circumference direction and 10 points in radial direction
per composite wave length Ac which is defined lay:
2,T
=
c kj—k' + kj'
where kj' satisfies:
(61)
Gus _ w')2 = gkj' · tanh~kj'H) (62)
A good agreement between numerical and analytical results is
obtained when the radial distance is more than three Ac. The
average value of integral results for the radial distances between
4Ac and 5Ac is taken as the final result of flee lIaskind integral.
The error between the numerical result and the analytical result
is less than five per cent.
The computing time for the low frequency forces is dom-
inated by the evaluation of the Haskind integral on flee free sur-
face, since other contributions are just in form of simple addi-
tions of the first-order quantities which are obtained once the
first-order problem is solved completely. In flee same case of a
vertical cylinder whose one quarter surface is divided into 108
panels, the first-order solution costs about 3 ~ni~utes on Vax8700
computer while the free surface Haskind integral is obtained after
10 minutes calculus for one frequency. Ashen the system responds
to only some frequencies, it is necessary to evaluate completely
the low frequency forces for these frequencies. But it is also rea-
sonable, when the frequencies considered are numerous, to take
the approximation which consists of neglecting purely and sim-
ply the free surface integral. Allis economical approximation is
adequate when the diffraction effects are weal;. For in.stance, flee
submersible or semi-submersible with small waterplane areas are
the cases where the approximation can lie used. This is shown by
Matsui[°~] who considered the ITTC semi-subn~ersil~le platform.
The work in [09] has the same conclusion.
3. Model Test Presentation
The model tests were carried out at the Ocean Basin of
the Ship Research Institute of Japan, while the second author
was on sabbatical leave in 1981-1982. Two l~ottle-sl~aped models
were considered, with different neck diameters. Cable 1 presents
some characteristics of both models.
Table 1 Characteristics of the Models
Designation Model No.1 Model No.2
R1 Neck diameter 11.5 cm 21.6 cm
HI Neck height 20.0 cm 20.0 cm
R2 Bottle diameter 63.0 cm 63.0 cm
H2 Bottle height 70.0 cm 70.0 cm
M Displacement 220.3 Kg 225.5 Kg
Cg Center of gravity 33.8 cm 34.7 cm
Cb Center of body 35.4 cm 36.4 cm
Kz Vertical Stiffness 113.3 N/m 370.6 N/m
The test set-up is discribed on figure 2. The mooring sys-
tem was constituted by two linear springs. The ballasts of the
models were adjusted to ensure large natural periods in roll and
pitch. In all tests it appeared that the pitcl, response was neg-
ligibly small. The heave motion of the models was measured by
a potentiometer. It was checked during the tests that the surge
motion was always small enough to ensure an accurate measure-
ment.
Figure 2 Test set-up
,// ~ ~ r
~ .
Potentiometer
T
R1
1
wave ~
R2
Fir
. . 11
H
~ ~ A/ ~ ' ~~ ~1 ~ ~ / r
Tests in regular waves were first undertaken for tile pur-
pose to evaluate the vertical steady force. The mean vertical dis-
placements were measured for wave periods ranging from 0.6 to
2.0 seconds and double wave amplitudes of 4. and 6. centimeters.
Extinction tests were made in still water and in regular
waves, in order to estimate the heave damping, which is an im-
portant quantity in the evaluation of low frequency motion. In
this series of tests, the model was given a vertical downward dis-
placement (10 to 20 cm) and released. The leave damping divas
classically estimated from the record of the decaying motion. Tl~e
tests showed that the damping increases antler the effect of wave
superimposition. Another feature observed in regular waves, is
that the zero down-crossing intervals decreased as tl,' heave mo-
tion decayed to zero. This variation of ~~at~'ral periods is tal;en
into account by the equation (19~.
A large proportion of the tests undert akel1 were in bichro-
396
OCR for page 397
matic waves. In these experiments, two regular waves of equal
amplitudes were superimposed. One pulsation was kept constant
and different beat frequencies were achieved by varying the other.
Another kind of tests was aslo made at constant beat frequency
and varying wave amplitude. Tests in irregular waves plus cur-
rent were aslo undertaken.
In next section, the tests results and analysis are pre-
sented in the comparison with the numerical predictions. 0.40 -
4. Numerical Predictions and Comparisons
-
For the purpose of the numerical solution to the diffrac-
tion-radiation problems that is needed to evaluate the low fre-
quency forces, the body wetted surface is n~csl~ed by quadri-
lateral panels. The principle of discretization is that the panels
close to the free surface and near the corners of body surface are
finer than the others. Figure 3 presents the mesh used in the full
numerical calculus. The total number of panels is 880.
Figure 3 Mesh of the body surface
The numerical accuracy depends directly on the repre-
sentation of the body form, that is, the size of the discretiza-
tion. A mesh giving an accurate solution to the first-order prob-
lem may not be refined enough for the second-order problem.
The convergence with discretisation was so investigated first.
Model No.1 is chosen to evaluate the first-order vertical force
and the second-order steady force in a regular wave of frequency
(w = 6.283 rad./sec.) and the low frequency vertical force in
Dichromatic waves of frequencies (~ = 7.222 rad./sec. and
w2 = 6.283 rad./sec.~. Figure 4 presents the numerical results
with respect to the panel number in which the first-order verti-
cal force Few) with the markers ~ ~ ~ ~ is adimensionalized
by pgLLa (here the reference length L = 1 m), the second-
order steady force F2z~w,—w) with the markers (-. ~ ~ by
(pgLaa/2) and the low frequency force Fame,—w2) with the
markers ~ ~ ~ ) by (pgLa~a2~.
From the figure 4 we see that the first-order forces have
converged with 100 panels, while the second-order forces need
over 800 panels. If the criterion of panel size for the first-order
is one sixth of wave length, that of the second-order should be
one fifteenth of the average wave length (about 800 panels on
the whole surface). In order to achieve a good accuracy for the
low frequency forces, we have then chosen the discretisation of
880 panels presented by the figure 3. The computer times on
the series of Vax8000 computer, for one frequency, are about
9 minutes for the first-order problem and 30 minutes for the
evaluation of the low frequency forces.
Figure 4 Convergence with discretisation
In
o
0.30—
._
IS
_ ~
0 400 800 1200 1600
Panel Number
In the regular waves, the second-order steady forces of
model No.1 are evaluated for 16 frequencies from 2 rad./sec. to
14 rad./sec. at three vertical position of the body: the origin po-
sition (neck height H1=20 cm), the higher position (neck height
lI1=18 cm) and the lower position (neck height H1=22 cm). The
results are presented by the figure 5 in which the curve ~ ~
describes the steady forces as a function of the wave frequencies
at the origin position, the curve ~—— —jet the higher position
and the curve (- - - -) at the lower position.
Figure 5 Vertical drift forces at three reference positions
0.30 -
cn
~ 0.20 -
.=
0.10 -
.
at'
i.~\
~'~
0.0 3.0 6.0 9.0
Frequency (Hz)
12.0 15.0
As the first-order vertical body motions are small, the
drift forces dominated by the contribution of the quadratic term
of the fluid velocity (see the equation 43) are always positive
upward because the fluid motion is larger above the model than
underneath. For the same reason the drift forces are higher at
the higher position and lower at the lower position, as compared
to the origin position, and they have a peak for a frequency equal
to about 6.5 rad./sec.
Figure 6 presents the derivatives of vertical drift forces
with respect to the mean vertical position. These derivatives are
obtained by finite differences. The adimenionalized vertical drift
forces Fdo: at the origin mean position are drawn by the curve
~ ~ and their first derivatives Fd1z by the curve ~—— —~
which are adimensionalized by (`pgLaa/2 * L/3cm). The second
derivatives adimensionalized by (`pgLaa/2 * 2L2/9cm2) are also
presented by the curve (- - - -I.
397
OCR for page 398
Figure 6 Derivatives of the vertical drift forces
0.40 —
0.30 —
u'
a)
o
0.20
0.10
0.00 —
l"\
o.o 3.0 6.0 9.0 12.0 15.0
Frequency (Hz)
The derivatives shown by the figure have similar shapes (a
peak for a frequency equal to 6.5 rad./sec.) as the drift forces at
the origin position. The values of the first derivatives are smaller
than those of the drift forces and the second derivatives are much
smaller than the first derivatives (with the adimensionalizations
given above). So it is legitimate to use only the first term in the
Taylor series of the steady forces (see the equation 7~.
Using the vertical drift forces and their first derivatives
evaluated numerically, the vertical mean displacements of the
model in regular waves are obtained by the equation (24~. Figure
7 presents this result as the curve ~ ~ and the comparison
with the model tests undertaken at the Ship Research Institute
of Japan (markers ~ ~ ~ ~ and with the results calculated by
the classical method of the equation (18) (the curve- - - -I.
Figure 7—Vertical mean displacements
0.201 1
0.15
c`i
*E :
cot
~ 0.10—
c'
-
o
0.05 —
0.0 3.0 6.0 9.0
Frequency (Hz)
12.0 1 5.0
The vertical mean displacements are adimensionalized by
the square of the wave amplitude. The comparison from the fig-
ure 7 shows a good agreement between the experiments and the
numerical predictions. But the classical method underestimates
the vertical mean displacement.
In the Dichromatic waves tests for model No.2 the second
wave frequency ~2 = 5.712 rad./sec. is kept constant while the
first frequency we varies from 6.283 rad./sec. t.o 7.140 rad./sec.,
which covers the whole frequency range used in the model tests.
The second-order low frequency forces at the origin mean po-
sition are presented on figure 8. The top curve describes the
amplitude of total low frequency forces which consist of the first
398
part (see equation 42) shown in the middle, flee seco~-~d-order in-
cident contribution and the Haskind integral on the body surface
shown underneath and the contribution of the IIaskind integral
over the free surface (bottom curve).
Figure 8 Low frequency forces
cat 0.40-
~L
o
0.20
F2s 1
0.50 0.75 1.00 1.25 1.50
Low Frequency (Hz)
The second-order low frequency forces are dominated by
the first part. But it is not correct to take al ly this part as an ap-
proximation of the low frequency forces because flee contribution
of second-order potential (set-down effect) is important and in-
creases with the difference frequency. Nevertheless the contribu-
tion of the Haskind free surface integral is very small ~ compared
to the other contributions. The approximation which consists in
neglecting this small but expensive contribution is justified.
The steady forces are evaluated as the sum of the steady
forces in two regular waves of two different frequencies. The first
derivatives of the low frequency forces and Else steady forces are
obtained by the same procedure as those of flee steady forces in
regular waves, applying the finite difference Clod to the results
at three different mean positions. The obtained low frequency
forces and the steady forces and their first derivatives are used
in equation (24) for the motion simulation. Figure 9 presents
one of the Dichromatic wave which was used ill Else model tests
(co: = 6.756 rad./sec. and w2 = 5.712 r(l~l./.sec.) with a beat
amplitude equal to 4 cm fat = a2 = 2cm.~. The low frequency
motion simulation is shown on figure 10.
In the equation (17), the added mass is fallen as the nu-
merical result at the frequency equal to 1.044 ~ ad./sec. and the
linear heave damping is derived from the extinction tests. The
simulation is started at an initial point such that the displace-
ment and the velocity are zero. The simulate<] Notion is stable
after less than ten periods. The low frequency motion a~npli-
tude and the mean displacement are measured from the stable
simulation.
In the same Dichromatic wave the simulations are worked
out for double beat amplitudes from 2 cm to 8 cm. The results
are shown by the curve ~ ~ in figure 11. The markers ~ ~ ~ · ~
are the experimental points and the curve (- - - -) designates the
classical calculus without the first derivatives of the second-order
low frequency forces.
According to the frequency domain analysis in tile above
section 2, the amplitude curve of classical method is parabolic
because the second-order motions are simply proportional to the
square of the wave amplitude. But the motion silllulation which
takes account of the variation of the second-order forces with
respect to the mean position is not the same. When the beat
frequency is larger than the natural frequency the simulation
OCR for page 399
Figure 9—Bichron~atic wave
0 10 20 30 40 50 60 70 80 90 100 It0 120 130 140 150
Time {second)
Figure 10—Low frequency motion
0.10—
0.05 -
a, -O. 05 —
0 10 20
30 40 50 60
amplitude is less than a square function of the wave amplitude.
The difference between the two increases with the wave ampli-
tude. The comparison shows that the simulation amplitude curve
is much closer to the experimental results than that from using
the classical method.
Figure 11 Heave amplitude as a function of wave amplitude
20-1 1
16—
c'
cat
-
12
8
I
/'
0 2 4 6 8 10
Bich. Wave Hight (cm)
The whole low frequency motions are obtained for the
frequency range of the resonance by the same way. The figure 12
presents the results of the double heave aml>lit~lde with respect
399
70 80
Time (second)
90 100 110
to the beat frequency. The markers ( ~ · ~ · ) are the pOilltS
of the model tests results. The continuous line (—) is derived
from the motion simulation while the dotte(l line is obtained by
using the classical method.
Figure 12—Heave amplitude of model No.2
12.0-
*
I 9.0-
~ 6.0 -
ce
a)
3.0 -
0.50 0.75 1.00 1.25 1.50
Low Frequency (Hz)
The heave amplitude of the motion sillllllatioll is in good
agreement with the experimental results. The classical method
underestimates the heave motion for frequencies lower than that
of resonance while it over-estimates the heave motion at larger
OCR for page 400
frequencies. Moreover the motion simulation predicts correctly
the resonant frequency of the system.
For model No.1 in dichromatic waves, the numerical com-
putations are carried out for frequencies w~ varying from 6.347
rad./sec. to 7.222 rad./sec. and fixed w2 = 6.283 rad./sec. The
double beat amplitude is 5.7 cm (a, = a2 = 2.85 cm). Same pro-
cedures are used as for the model No.2. The results are presented
on figure 13.
Figure 13 Heave amplitude of model No.1
~ 20
.o
o
cats 10 -
~D
I
___,__
0.00 0.20 0.40 0.60 0.80 1.00
Low Frequency (Hz)
Again the classical method does not predict well the low
frequency heave motions. The motion simulation predicts cor-
rectly the resonant frequency of the body in Dichromatic waves.
Table 2 Resonant frequency prediction
~ hi_ ~ S ~ ~ Experiment I
Model No.1 0.57 rad/sec 0.47 rad/sec 0.45 rad/sec
Model No.2 1.04 rad/sec 1.00 rad/sec 1.00 rad/sec
Although the numerical prediction gives the correct fre-
quency of resonance, the agreement with the experimental results
for model No.1 is not excellent. It is necessary to note that the
heave damping in the simulation using the equation (24) takes
the form as:
B = Bo + Bi · | d'z0(t)|
in which the linear damping Be is derived from to extinction
tests and the quadratic damping Be is computed by the following
formula:
Bi = ~ pSCa
where S is the bottom area and the drag coefficient C`` is taken
equal to 3. It is possible that the unsatisfactory comparison with
the model tests arises from the unclear determination of the
heave damping. Further investigation on the viscous damping
of the system is necessary.
f. Conclusions
Using the Haskind integral relations, the second-order
low frequency forces can be completely evaluated by Else aid of
two transformation identities, without explicitely solving for Else
second-order potential. The numerical results show that for float-
ing semi-submersible bodies, the Haskind integral on flee free
surface can be negligible.
A simulation model for the prediction of low frequency
motions taking account of the variation of the second-order low
frequency forces with regard to the mean position has been pre-
sented. Even though satisfactory comparisons with the model
tests results are obtained for the heave motions in Dichromatic
waves, the system damping for low frequency notion is needed
for further investigations.
Reference
1. B. Molin 1979 "Second order diffraction loads upon three-
dimensional bodies" Applied Ocean Research. Vol:l No:4 ppl97-
202.
2. J.N. Newman 1974 "Second-order, slowly-varing forces on
vessels in irregular waves" Proceedings of International Sym-
posium on the Dynamics of Marine Vehicles and Structures in
Waves. ppl9~197.
3. J.A. Pinkster and R.H.M. Huusmans 1982 "The low fre-
quency motions of a semi-submersible in waves" Proceeding of
Conference on Behaviour of Offshore Structure. pp447-466.
4. B. Molin 1983 "Three-year experience in flee numerical pre-
diction of the slow-drift motion of noosed tankers" Proceeding
of 15th OTC. ppl87-191.
5. J.A. Pinkster and A.W. van DUk 1982 and 1985 "Low
frequency behavior of semi-submersibles" and "Wave drift forces
on large semi-submersibles" Joint Industry Projects NSMB.
6. M.S. ~i~tafyllou 1982 "A consistent hydrodynamic theory
for moored and positioned vessels" Journal of Ship Research.
Vol:26 No:2 pp97-105.
7. B. Molin 1989 "Comportements no~-lineaires des plate-
formes semi-submersibles" Actes des 2emes Jo~rnees do l'Hydro-
dynamiques. ppl2~140.
8. T. Matsui 1987 "Second-order hydrody~.~mic forces on n~oor-
ed vessels in random waves" IUTAM symposium pp292-300.
9. X.B. Chen 1988 "Etude des reponses du second-ordre d'une
structure soumise a une houle aleatoire" These de Doctorat de
ENSM.
10. X.B. Chen 1989 "Evaluation des efforts de basse frequence
sur une structure soumise a une houle irregl~licre: con~parai-
son de differences approximations" Actes des Scenes Journees de
l'Hydrodynamique. ppl41-156.
11. B. Molin 1982 "Some experiments on the low-frequency
heave Notion of floating bodies with sill<`ll waterl~la~le areas"
IFP report. Reference number: 30 166.
400
OCR for page 401
DISCUSSION
by R. H. Huijsmans
I like to congratulate the authors with
their fine paper and their treatment of the
mean position dependency of the second order
forces, which resembles the way the wavedrift
damping concept has been introduced.
I have a question regarding the low-
frequency drift forces depending on the second
order potential F22. In using Lighthill's
transformation to the Haskind integral we
showed [A1] that the simplified analysis of
this integral as was proposed by Pinkster [A2]
was valid for a wide frequency range. This
simplified analysis has the benefit of small
computational efforts. Would the authors
comment on this?
A next question concerns the Fig.8 of your
paper. It seems to me that for practical
irregular waves notably JONSWAP type wave
spectra, these only exists a small frequency
band where the envelope spectrum has some
significance. Therefore larger difference
frequencies need not to be regarded
extensively. Can the authors comment on this.
[A1] A. Bens chop, A.J. Hermans, R.H.M.
Huijsmans, "Second order diffraction
forces on a ship in irregular waves",
Applied Ocean Research, 9, 1987.
[A2] J.A. Pinkster, "The low frequency
excitation forces on ships", PhD Thesis,
University Delft, 1980.
Author's Reply
1. The contribution of the second-order
diffraction potential to the second-order
loads consists of two Haskind integrals: one
on the free surface that involves the second-
order correction to the free surface equation,
the other over the hull that involves the
second-order correction of the body surface
equation. For difference frequency problems
it is generally accepted that the free-surface
integral is small, which has been confirmed
both by the discusser's results [A1] and by
ours. However, if the body is allowed to
respond to waves, the body surface integral is
not small, even for the difference frequency
case. Thus Pinkster's approximate method,
which only takes account of the second-order
incident potential on the body surface, may
not be valid.
2. We are thankful to the discusser that
this second comment has made us aware that all
figures in the paper are incorrectly labeled.
The low frequencies on the horizontal axes are
not expressed in Hertz but in radian per
second (as it is written in the text). We
apologize for that error.
DISCUSSION
l
by R. Eatock Taylor
It appears that the second order
correction due to the body motions involves
the large low frequency motions (eq.7) and the
small wave frequency motions (eq.43). This
presumably results from the two time scale
approach, although the details are not given.
Is this consistent with including terms from
the second order potential for the low
frequency forces? My memory of
Triantafyllou's argument is that the
corresponding forces are of third order,
because of the small parameter which results
from taking the derivative of the potential
with respect to the slow time [i.e., the
factor ( ~j-~L)].
This leads to a second question. The
expression for the second order incident
potential given in eq.45 does not appear to be
valid-at very small difference frequencies. I
suggested this at the Water Waves Workshop in
Norway earlier this year, and I would like to
ask the authors whether their formulation
leads to a discontinuity in the vertical force
as ~j+WL ; i.e. do they reach the regular
wave result in the limit?
Author's Reply
We agree that our method can be rigorously
justified only in the case when the time scale
of the low frequency motion is very long
compared to the wave motion time scale.
Obviously this is not the case for the
vertical motion of more traditional semi-
submersible platform. but still we contend
that variation of the low frequency exciting
forces with the low frequency vertical motion
should be account for, just as the low
frequency horizontal velocity affects the low
frequency horizontal loads.
The discrepancy between the vertical drift
force and the ~j ' AL limit of difference
frequency vertical force had already been
noted in [9]. In the present case it does not
appear as our numerical model is restricted to
the infinite water depth case.
401
OCR for page 402
DISCUSSION
by R. Zhao
(1) Discuss about vertical drift force by
using direct pressure integration or based on
momentum and energy relations.
Author's Reply
In an earlier paper[A3] of the author
computed the vertical drift forces on the same
models, using both the momentum method and the
direct pressure integral method. Numerical
results showed excellent agreements. This
success however may have been partially due to
the fact that the diffraction-radiation
problem was solved with a fluid finite
elements technique, which allows for an
accurate evaluation of the fluid particle
velocities.
[A3] B. Molin and J. P. Hairault, "On Second-
Order Motion and Vertical Drift
Forces for Three-Dimensional Bodies in
Regular Waves", Proc. Int. Workshop on
Ship and Platform Motions, Berkeley, 1983
402
Representative terms from entire chapter:
frequency forces
