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# A. Papanikolaou and H. Nowacki. Second-Order Theory of Oscillating Cylinders on a Regular Steep Wave

```III.4
Second-Order Theory of Oscillating
Cylinders in a Regular Steep Wave
A. Papanikolaou and H. Nowacki
Technical lJniversity af Berlin
Berlin. Germany
Thirteenth Symposium on
NAVAL HYDRODYNAMICS
OctoberO
-
10, 1980
Sasakawa Hall, Tokyo
Office of Naval Research
National Academy of Sciences
Shipbuilding Research Association of Japan
III-4
Second-Order Theory of Oscillating
Cylinders in a Regular Steep Wave
A. Papanikolaou and H. Nowacki
Technical University of Berlin
Betlin, Germany
(n)
l
ABSTRACT
r-J<
The nonlinear two-dimensional hydrodynamic problem of a cylinder with arbitrary
cross-section shape performing finite ampli-
tude oscillations in the free surface of a
regular steep wave j_s treated on the basis
of nonlinear system dynamicsr potential and
perturbation theory. A complete second order transfer model is developed from the assumptions of a second-order incident wave,
a quadratic dynamic model of the system, and
a second-order perturbation expansion of the
hydrodynamic nonlinearities in potential
flow.
The given nonlinear wave-body flow
system 1s decomposed into six second-order
subsystems to which a perturbation expansion in several small parameters is applied.
This results in a set of linear boundary
value subproblems of uniform type, which
can be solved by close-fitting methods.
On this basis second-order expresslons
for the hydrodynamic pressures, forces, and
moments are obtained and introduced in the
equations of the body motions to obtain the
motion response.
Numerlcal examples illustrate the soIution procedure and several physical aspects of the second-order mode1.
NOUENCLATURE
jt
(t) (t)
,n
g
c(t) (r,
h
, (n)
i
?
l_
r44
j
srrmbo,l
Description
a
wave amplitude
b
maximum half-beam
w
A
-(1) -(2)
j'k
'jtik
j
area of cylinder cross section
reference quantity for motion
i, eg. l2l
waterline beam, Fig. 3
k
+
k
1
III-4.1
generalized restoring force
coefficient of order n for
(i,k), eq. (106)
hydrodynamic force or moment
amplitude of order n in direction j caused by problem k or
by interaction between probIems i and k, as applicable,
eqs. (90) to (921 , rt = 1 or 2
hydrodynamic force or moment
of order n in direction j, eq.
(88) , with ,F (') H (t) .
= (.r)
(n)
(n) -nr (,)=
. (2t
rik
6)
=v
,
u
acceleration of gravity
Green's function for order n,
eq. (s3)
water depth
hcrizontal hydrodynamic force
of order n
lmaginary unit (space), e.9.
eq. (54)
unit vector in positive
x-di-rection
moment of inertia (ro11), eq.
(ee)
imagi-nary unit (time), e.g.
eq. (55)
unit vector in positive
y-dlrection
wave numberr €9. (58)
unit vector in positive
ro11 axis directlon
free sqrface inhomog. for sec.order problem (i,k), (451 ,(4gl
Symbol
Description
,ftll)
aJ(
M
iu
(
t')
,{")
+
n
."1.")
llK
o-x-y
5-;-y
p (n)
(r,i, t)
nl') (r,i) ,
nji' (*,tr
i
,.(1)_
_(2)
K ', f-ik
i
s
(t)
So
t
V
vn
x, (t)
"j")v :{n,:,ll)
o1
Yc
-
Y
(x,
^R
v-R(n)
t)
,
B
6
d
'k
I
1.r,
+ (n)
aI<
^"
body
p
(n)
uik
lnhomogeneities
ilt.4 -2
incident wave phase angle,
eq.
(57), 6r-o to. regular
sinewave (t-0) in posltive x-direction
perturbation parameter for
rnctctent wave , eq. (21
Eff::::ii:;: 3;:"Ui"- for
perturbation parameter
motj.on in direction i, for (2)
eq.
perturbation parameter in general
couplex variable source
Point,
eq. (54)
'
incident wave length, eg. (5g)
di mensionles s hydrodynamic
coefricieni-6f *;;;.. ,,,,
lppils.
caused by (i,k), ee. (1Og)
di."r=iorrless hydrodynamic
mass coefficieni or
caused by {i,k;, .n. 6.J..*.,,
(1Og)
frequency parameter, eq. (5g)
fluid density
body mass density, eq. (99)
velocity potential, eg. (1:y
space potential functions
orders (1) or (2), eq. (13)of
decomlrcsed space
of orders (1) or potentials
(2), eqs .116) ,
(171
1{
I
ot,Problems
appli_
!i,k), as(79)
to
position vector l-n o-x-Y-sYstem, eq. (g1)
draft, Fig.3
velocity vector of body motion
normal component of body velo_
city, eq. (9)
body.motion in direction 1,
(111.,_for sway
S?:
neave (i=3), and roll-l:.=2),
(i= )
body motion amplitudes of or_
oer n, eg. (141
vertical coordinate of body
center of gravity
horizontal inertia force of
order n, eg. (100)
free surface elevatj_on profile
vertical inertia force of or_
der n, eg. (1OO)
variable, field poj_nt,
:".P1:T.
(s4)
eq.
(n)
"7
hydrodynamic pressure ampli_
t.ude of order n caused by
problem, k or by interaction
-boundary
ror, tirst
(k) or second_order
(ar,<) problems, eqs. (30) (44),
,
(47) , (48)
arc length on cylinder contour
unit tangent vector, Fig. 3
wetted cyllnder contour at
,time t, Fig. 3
wetted cylinder contour at
time t=0, Fig. 3
between 3 ana pos.
-1"?+.
x-direction,
Fig. 3
section area coefficient
Phase angle of order Dr
9€D€_
ral, eg. (93)
to
unit normal vector, positive
outward, Fig. 3
directional cosine components
of order n for directlln
il-eqs. (21) to (24)
"
inertiaf right-handed Carte_
sran coordj-nate system, Fig.3
body-fixed right-handed Carte_
sran coordinate system, Fig.3
hydrodynamic pressure of
order n
time
T
(n)
c
generalized body
coefficlent of order n mass
(i,k) ,
for
eq. (106)
cylinder mass per un j_t length,
eq.
- (99)
hydrodynamic rolI moment of
order n
moment of inerti_a of order
n,
eg. (100)
PB
@(x,y;t)
oJ1)t*,vl
"0ffr)rx,y1
t,j^'),*,"1
^dli)t
"''r
frequency of incident wave
and
of first-order motio" ;;;;";;;
body boundary differential
ope_
rator, eq. (19)
Laplace operator
free surface differential
ope_
rator, eq. (19)
tor, eq. (2ol
Eapiltonrs nabla operator
OJ
B{}
A{
F(n
R(n
v{
]
2r)
2r)
{
}
{
}
}
Subscript Conventions :
The paper contains several
multiply sub_
scripted guantities rn symbol
expressions
of the general form
.i t tjl)
,
where some or
gf- the subscripts may be
present. These .11
symbols denote:
a) Superscript n_j-n_ parentheses
the order
n
to
of the quantity inrefers
ttre per_
gfrq
otre
that
.i.o 4l
shii[r
for
reoil
r
x.mdilni
hry
ir
sl!rE
ilrN
eridl:
lhaGiG
@ooul.f
aEE I
co-.d
,f .d
ucull
mnrfu I
PI!!.\$ilI
paeEtl
@
!imqrr
*rl
i
[email protected]{t,
surfro
toal
qu"td
tcd dq
try ILq
EEoer
ran
I
nrnl
seamld
-tr-aild
dercti
I
l)
)Lnt,
r8)
E
rlr
,
turbatLoh expansion.
bl I€ft subscript m denotes the dependence
, upon the motions of first (m=1 ) and second order (m=2) of the followlng quantities: potentials, eq. (17), associated
disturbances, e.9. eqs. (441 to (49) ,
pressures, eq. (65), and forces,
eq. (87).
cl l,eft subscript j (j = 2, 3, 4l is used
with direction cosines, eqs. (21) Lo
l24l , or \dith forces, eqs. (90) to (92).
It denotes the reference direction j.
For the forces both subscripts m and j
may occur, eq. (87).
dl Right subscripts i,k (i,k=0,2,3,4,71
relate to the cause of the subscripted
quantity. First-order quantities have a
single subscript k, second-order quantlties require at least two subscripts
i, k; second-order steady-state quantities carry the additional subscript o.
Other Conventions:
Lengths are made dimensionless by b or
la l, ancrleebv maximum wave slor:e or with
'rELpu.i t6 i raaian, time hv ,-t, and
mass per unit length by p bz.
INTRODUCTION
B)
I
E
16),
4nd
nse
ope-
Pe-
a-
o
r-
The past decade has witnessed a rapid
growth of j-nterest in the nonlj-near aspects
of ship motj-ons. It is well established
that linear theory succeeds extremely well
in predicting many important phenomena in
ship motions within the accuracy required
for desj-gn purposes; but it has also been
recognized all along that the range of validity of the linear model is restricted
by its fundamental assumptions of small
w_avq heights and small motion amplitudes
t t ] rfrere has been growing experimental
evidence in recent years, accompanied by
basic theoretical results, suggesting that
nonlinearities 1n waves and ship motions
are also a matter of important practical
conseguence. This holds for the prediction
of any large amplitude motj-on effectsr p&rticularly extreme values, and has received
much attention in connectlon with large amplitude roll- (capsizing), hydrodynamic j-mpacts (slamming, section flare effects),
wave-induced bending moments and other nonlinear phenomena.
The present paper addresses itself to
one speclfic aspect of nonlinear ship motions, that is, the problem of an infinitely long cylinder oscillating in the free
surface in shray, heave and roII in response
to a regular incident beam \$rave of finite
amplitude (on deep water or water of limited depth). Ihis case is an important reference poj-nt for solving the much more general problem of ship motions in an i.rregular
seaway, although a comprehensive nonlinear
theory for the latter purpose remains to be
developed.
III.4-3
The current two-dimensional hydrodynamic problem is approached by means of nonlinear system dynamics, perturbation theory
and potential flow methods. This implies
that several, but not aII, of the nonlinearities present in the physical problem can
be taken into account and that thb results
may be valid for moderate wave heights and
moderate motion amplitudes, in other words,
for nonlj-nearlties of such degree that the
second-order perturbation approxi-mation
will remain sufficiently realistic.
Specifically the approach will a11ow
for nonlinearities in the
- I'ree surface boundary condition for incident and motion generated waves (by
including second-order effects and satisfying this condition in the true free
surface - in a perturbation sense),
- Body boundary condition
(by satisfying it over the rea1Iy wetted
regime, approxJ-mating the shape of the
latter by a Taylor expansion around the
rest position of the body in the spirit
of perturbation theory) .
A11 derived responses such as pressures, forces, motions are also evaluated
to the second order. However, nonlinearities of viscous orlgin (Iike separation at
sharp corners) are dlsregarded, although
subsequent, empirically based corrections
of the system model remain feasible.
Much of the previous work on the subject of this paper has been devoted to the
incident wave and forced oscillation subproblems. Numerous higher-order wave theoiies have been developed by Stokes [2] t
Levi-Civitd, Sielbreia and others. Papanikolaou has reviewed several of these theories, including a third order solution of
hj-s own, in [:I . For the present purpose
a Stokes second-order wave wiII be sufficient to represent the j-ncident wave flow.
In regard to the nonllnear forced motion problem Ogilvie [4] derived the secondorder steady force on a submerged circular
cylinder, which could be achieved without
explicitly solving the second-order boundary
value problem. This was done later by Parissis [ 5] for the circular cylinder heav-ing
in the free surface and by C.M. f,ee [5] for
Lewis forms. Lee's solution to the secondorder set of boundary value problems is
based on multipole expansions and conformal
mapping. Potash [7] , working with close-fit
techniques instead, extended the problem to
sway, heave, and rol1, including their coup1ing, and.to erbitrary secti-on shapes. Papanikolaou [8,9J ,on the basis of a similar
approach, reexamined the foregolng results
for the heaving cylinder and removed some
j-nconsistencies in the analytical expressions and numerical results. Fina11y, Masumoto [tO] developed an approach to Lhe complete second-order oscillation problem using
multipole expansions in analogy to C.M.Leers
treatment of the forced oscillation case.
Several approximate solutions to the
second-order forced motion problem have been
derived without soLving the corresponding
boundary vaLue problem completely and rlgo-
rously. C.H. Kim hll dealt with forced heaving of triangular cyli-nders by an iteration
which departed from a zero-frequency solutlon. Grim llZl derived approximate solutions based ol low-frequency assumptions
for larqe amplitudes
of roll. Salvesen's
-[t:]
approacf, in
to nonlinear heave (and
pitch) is also based on low frequency approximations. Sdding [ta] used Greents theorem to derive the second-order force on an
oscillatlng cylinder from first-ord-er potentials of tfie problem. Yamashita [tS] developed an approximate solutlon for "thj-n"
oscillatlng cylinders with results up to
the third orderseveral authors, €.9. [to-ta] , in contrast to the perturbation school of thought,
have pursued direct time-domain solutions
of the complete nonllnear problem, using
initial value formulations and numerical integration schemes. These methods do not depend on linearizing assumptions and are attractive, in fact, sometimes perhaps the on1y available recourse, for dealing with very
large motion amplitudes and waves. However,
aside from some unresolved questions regarding the treatment of the radiation conditj-on 1n the complete nonl-inear case there
are also some practical limitations: Computer time requirements tend to be heavy, and
validation and generalization of results are
difficult
to perform for lack of frequency-
dependent submodels.
Experimental results for several aspects of the nonllnear problem have been
presented by Vugts [tS] .- ,^ Tasai and Kote-rayama [20i , Yamashita t1 5J . The agneement
between indi-vidual test results and secondorder theories is encouraging, although the
available evidence is far from systematic
and complete.
The research reported in this paper has
the alm of defining a complete second-order
modeL for a cylinder of arbitrary shape oscillating in the free surface in three degrees of freedom. This model combines a quadratic dynamic system moclel- with a secondorder hydrodynamic model. The incident beam
wave 1s thus a second-order regular wave,
that is, a "steep" rvave in the sense of
Stokes. The system model is subclivj-ded into
several separate, but interacting flow systems:Incldent wave, forced motion, and diffraction are three primary subsystems familiar from linear theory. In second-order
theory it is not sufficient lo model these
individual flows to the second order, but
it is al-so necessary to account for the mutual- second-order interactions in these
f1ows. This requires introdueing three further second-order subsystems.
A perturbation expansion j-n five sma11
parameters is performed on the nonLinear
system in order to recognize the "smallness
parameters" characterlstic of each subprobIem. The expansion yie3-ds a corresponding
set of.20 linear boundary value problems.
The great majority of these can be regarded
as radiation problems and represented by
Freclholm integral equations of the second
kind. This permits uniform numerical treat-
I[t-4-4
I
ment of all these subproblems by close-fittj-ng methods. The standardized hydrodynamic
coefficients obtained from the submodels are
assembled into the complete transfer model
via the equations of motion to solve for the
motion response and other responses. The paper describes the soluti-on procedure in detai1. Some numerical examples are g'iven to
illustrate the physical properties of the
transfer model and its components.
2.
i
t
a mol
of fr
of fr
stanl
flow
tage
set (
flow
amena
FORMULATION
will
of 1r
tion
2. 1 Dynamic System ltodel-
The nonLinear system of an incident
wave and a body oscillating in response to
this wave on the free surface of a fl-uid
domain may be clescribecl by two sets of assumptions, cal-l-ed the dlmamic system model
and the hydrodynamic modeL. The former is
related to the general- dlmamic behavior of
the system in terms of its input and output,
the latter consists of the hydrodynamic characteristics assumed for the fl-ow system.
The two models must' of course, be chosen
in accordance wittr eactr ottrer.
Motions
NONLI NEAR
WAVE/SHI
Forces
P
SYSTEM
!'ig. 1: Dlmanic System Model
The dynamic system model of the nonlinear wave-ship systen (Fig. 1) is characterized by the relation between its input signal i(t) antl its output signaLs oi(t),j=1,
...,J. In the foll-wing we are interested
ln the steady-state betravior of the system,
that is, in the time-periodic response of
the ship to excitation by a regular steep
wave. Initial- transient phenomena of the
response are disregarded. Por this purpose
we assume a nonLinear reLationship between
input and outPut of the form:
N
o.(r)
,
= x a1"){ttr)}n
J
J
(1)
,.=6
and i.n particular a quactratic time response
model (N=2)..,_-fhis modeL is introduced l:lere
(as in earl-ier work by r.ee [Zt] and ottrers)
as an a priori working trlpotlresis, chiefly
because it can be demonstrated. to be comPatible with the second-order hydrodynamic
model to be deveIopedl by perturbation methods.
intre
sionl
chari
pertr
IIe fu
I
.fit-
lamic
[s are
odel
Dr the
[e patr de-
Ito
Lhe
nt
eto
Ld
asodel
is
rof
utput,
c cha-
It Is a particular ProPertY of this
qoadratic model that the response oi (t) to
aL ronochromatic harmonlc input si9n61 i(t)
of frequency o will contain harmonic terms
of frequencies t: and 2rrr , as well as a constant "d.c." shift term.
In vie\,{ of Lhe complexity of the total
flow process it is now'of practlcal advantage to decompose the system model into a
set of separate, but interacting nonlinear
flow subsystems, each individually better
amenable to hydrodynamic analysis. This
riI1 be done much in analogy to the concepts
of incident flow, forced motion anil iliffraction f1ow, familiar from linear theory.
At the same time it is convenient to
introduce a system of several small dimensionless parameters e1 suitable for defining
characteristic "smallness ratios" in the
perturbation expansions of the subprobl-ems.
lle introduce:
.o
em.
=
Ztr
it la,l= k
I
., =1":to-'
sen
lu,ul
= .o
(xry-1
r (1)r
l*i
.:- = -Ul , 1 = 2' 3,
(2)
of the motion amplitudes 1n each degree of
freedom. Thus we have available a set of
physically relevant smallness parameters for
the major subsystems of the flow mode1.
The use of five small parameters does
not imply that they are meant to be lndependent of each other, but only that their physical interdependence need not be considered
until after the subsystem flow problems are
solved. In fact, eo and a7 are both dependent on wave height. and e7(e6) and ei are
physically linked by body dimensions. However, at least two physically independent
smallness assumptions can be made, for examp1e, the traditj-ona1 "small wave steepness"
and "smaI1 motion amplitude" assumptions.
To this extent our approach parallels New:
man's in IZZ) who used three perturbatlon
parameters in a thin shi-p oscillation problem to characterize the orders of beamlength ratio (thinness), wave steepness,
and motion amplitudes.
The decomposj-tion of the dynamic system model S into a set of second-order subsystem models iS, assuming for the time
being that all- ek are of equal order of magnitude, can now be expressed in terms of the
characteristic sma11 paramelers present 1n
each subsystem:
I
4
S(x,y;t;eO,er,e.)
+
L
,S(xry;t;eO) + 2s(x'yiL;er) +
3S(x,y;t;er) + 45(x,Y;t;e.eO)
with .*
->
= amPlitude of regular beam wave
(a, in general complex)
I
= wave length
w
k = 2r = wave number
vb = r{'1
- 5=dimensionless
frequency parameter
onliactet sigj=1,
ted
kb
here
hers)
efty
rolnPa-
ric
me-
Further:
= maximum half-beam of section
= 2, 3, 4 for sway, heave. roll,
respectiveLy
*11)
= complex first-order amplitude in
rdr-rect]-on L
b,l_ = reference quantity for motion i
bZ = b3 = b = maximum half-beam
be = 1 rad = unit reference angle
The parameter e0 is a measure of wave
steepness in the incident flow, e7 is characteristic of the magnitude of the diffraction ftow, and the ei define the smallness
b
i
TII.4.5
I-
* o(.;)
A )
= (8c'-Bc-+9) /'8"4
= s/c ,
"1
s = sinh (kh), c = cosh (kh)
he
,Ponse
65(x,y;t;eOer)
CO
of
(1)
*
= (vb){co{ r + efr ct)} -1 + orel)
eep
Eeen
55(*,y;E;eie7)
(3)
+
and physically for a second-order wave on
water depth h [ 3] :
stem,
POse
=
FrG,
2 :
SECoND-0RDEn DYilAMlc sYsTEfl HoDEL
Fig. 2 illustrates the mutual interactlons
of the six basic nonlinear subsystems:
Nonlinear
Nonlinear
Nonlinear
Nonlinear
US: Nonlinear
US: Nonlinear
,S:
,S:
,S:
nS:
i-ncident
wave
di-f f raction
forced motlon
interaction of ., S and ,S
interaction of ,S and ,S
interaction of ., S and ,S
1
The second-order model- of (3) differs
form linear theory in two ways: The famil-1ar basic flow systems 1s, 25,. 35 have to
be extended to second-order leve1, and their
second-order interactions have to be taken
into account, which is done by the subsystem 45, 53, 65. Each subsystem iS corresponds to a nonlinear boundary falue problem of potential f1ow, which by perturbaof
tion methods can be reduced to classesIinear boundary value subproblems is (n) .
Some of these classes are further Eubilivided into erements-oi types is{n) ana is{p),
that is, linear subproblems-to be derived
in cletail from the hydroilynamic model.
Fig. 2 shows the complete scheme of nonlinear flow subsystems iS and their subprob1em classes and elemefits, connected by so1id lines, whereas the dashed l-ines with
arrows indicate how the nonlinear subsystems +s interact with each other via their
lineal boundary value subproblems.
* = *2+ x cos *4 - Y sin xn
(4)
y=x3+!cosxn*isinxn
The motion is assumed to have existed long
enough for alL transient effects to have decayed. Further, we assume inviscid, irrota-
tional flow, which ensures the existence of
a velocity potential- 6(x,y,t) satisfying
Laplace's equati-on for an incompressible
f1uid.
Combining the ki-nematic and dynami-c
boundary conditions on the free surface
y=y(x,t), extending to infinity-on both
sides of the body, one obtains [8] :
+
Ott(x,Y(1;t);t)
o +o o. ) - o2o 2(o
-xExytyxxx
q
v
20
(s
o o -o2o
y vv
xyxy
If the fluid has a horizontal bottom at
y=-h, then
2.2 Boundary Condltions
A cylinder of arbitrary cross section
in or just below the
free liquid surface in response to a beam,
regular, steep wave of amplitude aw and
frequency to.
(5)
=0
The kinematic body boundary condition implies that the normal velocity of the fluid
on equals that of the body Vrr(Fi9. 3):
0r(xr-h;t)
shape is osciltating
S(t):
wi th
0
n =
+
n ={
(nV)Q(x,y;t)
sino, -coso } ,
0n = sino
Vn- = n V(x,v;t)
s ={coso,sinc}
coso
0
Vn = sino
1 = tx/ax =
i=avlat=i:*
:r.2
-
x4
(y cos
0
v
cos0 v
xn
+ i sin x4)
in{* .o" xn - i stn
sin c(t) = ?yl0s = y cos x,q + i-sin
cos o(t) =
Fig. 3: Coordinate
Ex,/0s = x cos x
4
- y sr"n
xn
x4
thL.s
ar
llE L
br
leqnatll
,-3
EeI
IJ
dary rt
aectaq
asguirto a',
batam
the sq
| (x,t;l
rith
{
o
tie fq
mif6a
tfte ts
gest tl
are of,
firstDitude
vergeo
clMY Il
lEtese'
lthysic
cial. c
sectio
to er.i
a1thf,
rios
yetI
&1 lt
riU g
frequ
fact t
lnteut
Gorret
teoti,r
lnsitl
ry cd
tlat I
secd
exf,)r'e!
where the prime denotes the derivative with
Systems
Cartesian coordinate systems are
enployed (Fig. 3): The right-handed system
o-x-y is fixed in space so that o-x corresponds to the undisturbed fluid surface and
y is positive upward; the right-handed system 6-x-y is fixed in the moving body anil
coincides with o-x-y when the body is in its
equilibrium position. The displ-acements of
sway, heave and ro11 are denoted by x2, x3
and x4, respectively. Due to these motions
a point (f,y) moving with the body has the
following coordinates measured in the stationary system:
Two
respect to arc length s, the dot indicates
a time derivative, o(t) is the angle between
unit tangent vector 3 and the positive xdirection, and ri the unlt normal vector
which is positive out\$/ard.
At large dj-stance from the body a suitable Sommerfeld radiation condition is imposed. Physically thj-s corresponds to the
fact that the incident wave and the motion
generated by the ship are the only disturbances present. Mathematically this condltion ensures uniqueness of the solution potentials.
In determining the potentials the motions of the body xi (t) will be assumed to
be known. The normalized potentials and corresponding hydrodynamic forces determined on
a
o
(-'?;l
,
!
i
Sis G
tials,
cooed
are o{
i
presstr
protrtI
tifiei
that I
ard r
fit-4-6
/
placements xi(t) maY be.correspondingly expanded into i perturbation series in terms
Lt tn. small plrameters e1, characterizing
itre magnitude of the motions j-n each direc-
aasumPtlon l,ili be substitutecl into the
of motion l-ater to derive the acbody motions in a given wave.
tion i, eq.
urbation
Tn order to reduce the nonlinear boun."to" problem defined in- the- preceding
to I set of linear subProblems, Ye
ttat tne potentiat o can be expanded
a power series in terms of five perturt"
n iarameters..k (k=0,- 2, 3:,1:_ Jl,Yn
r."6"a order inKaccordance with (3):
!,;t; e*) = t* ,1"
(t2l
22
(14)
.-jkt'tt
r e:r xjl)
=r
Kr
x=o n=o
Neglecting trivial terms:
2
jnurt - -:?) l rat)
x.t (t) = , *.(") "or
"=1^i
where we have introduced the abbreviated
notation to be used from now on:
t(o) = o , i,k =
(ruestions concerning the convergence of
firegoing expansion, or particularly its
iotm convergence, must be left open for
roment. Cuirent calculation results -999: that the second-order potentials ai['
the
arAar ^f maonitude
to
to the
^-..r..^1^h+
order
-G equivalent
of
-oflTagnilude.
mag0
the
that
potentlals
t-order
1[,''"o
con-de of the ei, e1 will govern the should
e-of the 6*p.n'iiott. The series of
fot suffilientlY smal1 values
"
but how this l-imits the
e farameters,
secondi"ii t.rg. oi validity of thisout'
Sper theory has yet to be found
eautioir is i-n place for non-vertical
in tire waterline with regard
tiot
"frap"s
uniqueness of solutions '
and
existence
in our exPerience to date no sehave arisen
practical difficulties
7
According to the quadratic response
(1) a regular steep wave of frequency ut
1'produce-physical output effects with
ies , ana Zo . AlLowing for this
i" tt separati-on of variables forof the
iiars " in itz) bY including terms
esponding frequencies, expanding the poi-ais for-sma1i perturbations about the
itions at rest, and treating the boundasrrow [8]
conditlons .."otairrgivl-o""-.""
the relevant potentlal terms up to the
J-oraet are iircluded in the simplified
,y;t;e*)
=i.*rlt'
rl*"'*
-'i ot
e'+
z
x.(r)
"i'-'
i
oli)r*'v'tl r ot.ll
rff i'*
o, 2, 3, 4,
(21
.l ,.ll'* *1"' , kto, ,', *!1' - .i?)
lrar)
This change of notation is eguivalent toall
consider
will initially
="Vi"S thit we
problems and their responses
ny&rodynamic
way, namely for e i=1 ' Howii-, . t6t*.tizea
-ti
ieintroiluced later in solbe
will
"rr"i,
of motions and determi"i"g'tL.^"quations
ning the actual resPonse
Bv substituting these perturbation exana (i4), and-corresponding
p..rsioi=-iir)
its
Lnes for the wave profile y=y(x,t) and and
equation
taplace
Lhe
into
aeiivatives,
in secthe boundary conditions fgrmulated
reduce the
iion z .2, i-:t is possible [8] to boundary
qirr.r, nonllnear time-ilependent
to a set of onl-v
;;i;; pi-ure* rot o (x,v,t)
boundary value problinear
=""".-h"""ndent
tlms tor- * (n) 1x,y). The nonlinear problem
contains boundary- conditlons on free and
,""i"g boundariei, the linear subproblems
the
i"""f"" only fixed boundaries, namely
of the liquid and body
""ai=t"tf"d-positions
rhis is in the spirit of the per"rif.."".
where the conditions at the
method
t"if"ii""
approxtrue positions of these surfaces are
imateh by Taylor series expansions about
the positions at rest.
The linear boundary value problems resulting from this perturbation development
we
are deicribed in the followi-ng section'
restrict ourselves to the deep water caseand
iio* tt"r" on. Details of the derivation
iEqaraj-ng the conseque1gg.s of limlted water
aeptn can be found in [231
(13)
olfl)t*'vl e-2jort + orel)
expressj-on omits some trj-vial Potenls as welL as some time-indePendent seterms whose hydrodynamic effects
of fourth order.
By analogous reasonj-ng one obtains exi-ons of equivalent form for the-wave
file y=y(x,t) and for all physical quanies t-o be derived from the potential '
forces
t is, for example the pressures, dj-smoment on the body. The unknown
tlt-4-7
2.4 Boundary Value Problems
itn order to obtain a formulation for
potenthe boundary value problems for theunknown
the
of
independent
is
tials which
motion amptitudes it is convenj-ent to norof the dismalize thL potentials in terms velocity
oii".m.trt.v6Iocity and angular
Lomponents of the body or the exciting wave'
are for
;h;;. applicable. These components(14)
and
potentials, from
lr',"-ilrll-order
the incictent wave velocitY:
{
I
a
2
v1r-1
=i =r
n= I
ti
L v.atnl
vln) = ?
I
{
-jnot
n=1
(n)
n,rl xl"',
(n)
(r)
v.
e
=-
UJK
-r 'a,
i= 2, 3,
iA
";"w
r L = o,
(1s)
4
1
!
7
.!
measures the distance of
The phase angle 6w"crest
d{
from the origin at
the incident wave
time zero.
In terms of these velocity components
the first order potentlals may be expressed
by introducing the normalized potentials
tf,") a" follows:
'* ol')* ol')
odl)r",vv = ,*
tl",
.il
r.j.g. 4: Geometry of the BoundaryVal-ue Problems
The time-dependent.direction cosines of
the unit normal vector fr, aetinea in(8) ,
ana (f1) , Bdy be approximated by means of a
perturbation expansion about the body position at rest and, thus expressed in terms of
body-f ixed coordinates :
for
k= 0, 2, 3, 4, 7
The second-order potentlals may be
split up into two sets of normalized terms,
those due to flrst-order displacem"ents and
veloclties (l-eft subscript equal to one),
and those resulting from second=order veloclties (1eft subscript two):
. (2)
. (2)
. (2) * . (21
titk 9ik
9ik = tQik t 29ik
' (21
tQix
o.s *11)
for i, k= 0, 2,
3,
29ii
for i=k= 2, 3,
4
' (2)
4,7'
\;(2j
v.L(2) 2'LL
^o.
d
(1G)
a
,.1n=ot
= i .In) "-inrot . ,jl' ,
(til
"l",tli' ,
'
Nl2)
a
n(?)
o.1
For the sake of brevity we introduce
the following differential 6perators ( [ 6l
lal ):
Free-Surface Differential Operator
Ftvl i F(x,y) i =( ry - vr' ), (x,y) e sr
Body-Surface Differential
B { r(x,y) } .(7'rx-f-Fy),
R(v){ F(x,Y)} =Rer(F* * JvF },
,
r.to' , "1" = *l').n(1)
k;2) .":22' + o.25 *;"'r"11) t
t,jlir"(])
(r'
+ o.2s1*j1)12r"j?)
(;'
.n(o)={_;,"rl={;.
Lo
tf*"'*vv-)
(18)
Operator
(x,v) e so (19)
rator
I
I
I
k=I
with
nlo)
=
a
.
i'= 2,3,4 (2r\
,
Ope-
,*,r, d* (24 I
L
(22)
)
, i--2
, i=3
{
l23l
(2)
(1)
(21
12r
Q)
' j,n2r = itot
L")r' = ino2 = it
(0) .,; tzal
(2) f-in
'Lra
itz r =Lo, i=4
A similar expansion is performed for
the dirgction cosines of the unit tangent
vector s.
Co1lectinq terms whose conimon factor is
.. .3-1
ln=1 , 2; i, k=0 , 2, 3, 4, 7l
the following well-posed linear
we'obtain"-jnot
boundary value problems, generally of mixed
forn (third kind, Robin problem), for the
unknown potential functiirns fi(n| [z:] :
I
I
:
I
I
HI
1
1
I
I
I
First-Order Boundarv Value Problems
(
The boundaries
trated in Fig. iT'
So, SR and S" are illus-
,l
fl(
trr-4-8
otl'
with right-hand sides
a) i = k = 2, 3, 4
O, (x,y) e DUDi for k=0,
(xry)e D, fot k*O
l2s)
k=0
(x,y)
esousi,for
F(v){?11)} = O,
tx,y) esi ,'fot klO (26 )
(27 )
stt{1)r= rl" , (x,y) eso , k{o
tiu 3.t t ) O , ,*ry)eoSB ( Y* --) 128)
[email protected] K y
'
nt"rttll)
t=
o , t*,vl iR , klo
sr.
(2)
1
129)
right-hand sides
tk(1)
(0)
kn'-'
(1)
,tli) = u(zt
, k= 2, 3,
x=r
144)
=ltI'),
I rxs
.v?jil . ril,,, k=4
[,rrj]l
'*,' * rr'tl',' * tl',t,ll], (4s)
(1
k:r
m
4
= k-1 in Eq. (43)
ient to evaluate
ing potentials
For this reason
b) i*
sfrnmetry conditions
t-*,rt = (-r)*-'tJ1)
ljl)
tk=z,3,ai^xlo,7
146'.l,
(30)
- attjl)l
,7
rkk
[,;l: , k=2
(x,v),
!1".r1Iflt+
EN 1< K,
(31)
a, 2, 3,4, 7.
right-hand sides
Second-Order Boundary Va1ue Problems
1. Potentials caused uy t{2) , ee.
, 3, 4z
o,
tli'
= 0, (x,y) e
rra"rqfilfl)t =
0r
arrtlfl)r =.n
K
+ie- ,tliI
R(4,)!tli) ) = 0r
(
,tff)t-*,yr
k
E-SB
( y+
f (avr {,tli,
,=
at,tjf)
r=
ilg- ,tlil,
R(,!v) t rt
,ill)
jf
t-*,r1
)
=
r=
,'ji',
,'ji',
y1 = 12 ,41
-1,(l)
*Q3*" - -'v(1)
Y93y= ,
(4?')
11=(1,4)
,t--21
(2)
(35)
-6)
(x,y)E1st"R
0. (xry)
tik
(34)
so
(
36
)
(
37
)
t
,r=3 )k=0,7
lrik
I
t[])r,t=nJ
L
= (-1 f-'rtli)(*,y)
3, 4
=
l
(33)
2. Potentials caused uv *j.1
'rr,l' '
, 2, 3, 4, 7z
o,?li'
(2)
(32)
D
(x,y) e Sa
o), (x,y)e
= 0, (x,y)
k)=(2,3)
-!(1)
*Q2*" - -1,(1)
!Q2r",
117) ,
,
(x,v)
e
s
(x,y)e
s
F
o
(3e)
(40)
0, (x,y) eS", ( y+ --)
(41)
Or,*rrrd*
"
(4tll,
tll) * o ,2tj1)t(1)
'y(1)'v(1)
9k**
Ii
(42t
. ?l"tlll
+
,
in Eq. (43),k/O,7
(49)
(50)
The first and seconcl order potentials for
k=0, 7 (incident wave and diffraction problems) are in general nonsymmetric about the
potentials can
f-axis. But the first order (k
odd) and antibe spl-it up into symmetric
symmetric (k even) parts. Then, for k=0, 7r
the first-order problem must be solved twice
for each k, modifying eq. (31) to apply to
both the odd and the even case. This requlis
in four complex first-order potentials U rl''
L
r-D',tjl)r*, y)
(21
m = i+k
L1
(38)
eD
-
f -ik
(48)
(43)
III.4 .9
(k=0, 7). Accordingly.four second-order po-
.t.,gPt?il:
obtained,
tential
Eential- functions fi? {ft (ft=0,
{t=0, 7)
71 are
(a:),
;;";;.;i";-i,itn
replacins eq. tsol',rtr,
by relations for the odd and even cases.
It may be of interest to note i-n passing
that previously pubLished resuLts for the
forceh oscillalion case (r,ee [6] , Potash
t Z] ) regarding the second-order free sur-
face inhomogenous terms, corresponding to
our eqs. (45) and (49) (both complex), are
not consistent with each other nor with our
results. However, there is fu1l agreement
with Lee's later (corrected) problem formutation [21] . For more detait!, see [8] ,
I z:]
3.
PROBLEM SOLUTION
3.1 Boundary Value Problems Type
In the preceding section a set of linear boundary value problems was introduced
by eqs. l?.tr\, to (31) for the first-order potentials \$,!'' and by eqs. (32) to (50) for
the secondlorder potentials O'Ji). aff potential functions are in general*Complex (with
respect to time) according to eqs. (13) to
(17). Referring back to ?ig. 2 for orlentation, one can now distinguish:
- First-order incident wave potential
(k=0; n='l ) , solution known.
- First-order forced oscillation potentials
(k--2, 3, 4i n=l ), three unknown functions.
- First-order diffraction potentlal (k=7;
n=1), symmetrj-c and antisymmetric parts,
two unknown potential functions.
- Second-order forced oscillation potentials lj- , k=2, 3, 4; i
n=2), total
of six unknown potential=k;functions.
- Second-order incident wave and diffraction potentials (k=0, 7i n=2). On deep
water these potentJ-a1s are trj-vial and
need not be cons-idered.
- Second-order interaction potentials
(i=2, 31 4i k=01 1i n=21, symmetric
and antisymmetric parts, basically 12
unknown potentj-aI functions, but these
may be reduced back to 6 numerical evaluations by combining the symmetrlc and
antisymmetric parts, respectively, of the
disturbances resulting from the first-order free wave and tliffraction potentials.
- Second-order interaction potential between free wave and diffraction flow (i,
k=0, 7 i n=21, this vanishes trivially
on deep water.
Aside from the lncident \^/ave problem,
whose solution to the second order is well
known, the boundary value problems for all
of the potentials mentioned in the precedlng are of the same type, whose general
form is:
a6(n) = o, (x,y)e D
(x,y)e S"
F(r2v){o(n)1=
"(t'),
(x,y)e so
(51)
Bto("))=
"(t),
(n)
.o=
O, (x,y)e"sr(y+ --1
v
R(n2v){o(")}= O,
,*,t,{rl
The only essential dj-fference bet\,/een
and the second order lies in the
the first
f?TT=3ff:l'.i:.;,;\$:.::uTuf3.I i.tt+r:"3,
for the second order.
Boundary value problems of type (51)
according to Sommerfeld, summarizing under
this name a1l those problems which may be
d
described by pulsating sources j-n a finite
fluid domain.
The uniform format of the boundary vaIue problem type describing the linear submodels of the nonlinear flow system has several practical advantages. Questions of existence and unl-queness of solutlons may be discussed in a very general way once and for
all. Above all, however, it is of immediate
benefit that a sj-ng1e numerical solution
approach may be applied to all subproblems
under discussion.
3.2 Integral Equation
rffili,
@
fted
tdt
@u
ffi
aII(
h
d
ail
tril
Uet
tEo
Method
Green's third theorem of potentiaf theory, applied to the 6(n)-boundary value
problem (51), yields the following inhomogeneous integral equation of Fredholm type of
the second kind (Helmholtz integral equation), tgl :
mil
mEn
@M
mud
affi
lkr
'UnSt
p,J[tdl,t
!tuba
- 2ro(n)(*,r) *
s
[email protected]
. aG(n) (*,y iL,nl
.ro(n) tE, n)-----an;--dsQ
o
.
(52)
(*,r;E,0)L(ntqnE
= tc(n)*(n) (E,n)dso - {
"tn)
to-t"
cryllt
,@@
[
@ffi6
[email protected]
Trr
@ilufr
In order to apply thj-s formulation,
suitable Green's functions must be avaitable
i.e., functions which are solutions to a
boundary value problem similar to (51), but
with homogeneous boundary conditions [8]
On deep water the Greents function of
n'th order is, [ 8] :
(t) (z;
E) = R"i {Log(z-e) - rog (z-E)
)
r - ir (z-L)
*2+}-------_u*j 2ne-in-v (z-e)
\
c
@-
trmrmt
[email protected]
ffi
fforr
I
arflfrl
brnr
I
+
(53)
1 n v-K
U
with space complex variables for field point
P (zl
and source point Q ( 6 ) :
z=x+lY
e = E + ln , E = € - in
(s4)
In the complex time domain the time-dependent Greenis function of nrth order is represented in accordance with eq. (13):
[email protected]
*{
@rryl
III-4-r0
gln) lrr4ral
(zrr)e-inurt}
= R€.{"(n)
)
proper treatment of this effect is even more
lmportant than in 11near theory.
Proposed cures-to this problem may be
analytical (Ufse11 [ 27] , ogitvie-Shin [28] ,
Papanikotaou [ 8]), semi-analyticat and -numerical (pau1ling ana wood [29] , ohmatsu [:O]),
or- purely. numerical (f'altinsen [:t] , papanj-kolaou l. I, 23) ). fn the present context
numerical methods, based on interpolation
of regular frequency results, were preferred
to others up to moderately high frequencies
because they were convenient to use and gave
reliable results, though not always without
substantial expense of computer time [ 2S]
But it must be mentioned that purely analytical methods for this purpose have not yet
been extended to second-order situations
with inhomogeneous free surface boundary conditions.
(55)
The integral eguation (52) represents
Iy a pair of coupled integral eqqa:
because the unknown functions q (n)
complex. A physical interpretation of
fogUrplation is as follows: The poten0(nl (x,y) in a field point p(xly) is
of contributions from a double
of (unknown) intensity
f potenLial
produced
dipoles arraiged
)
E,
n
'( Se, from a slnglebyl.aygF
potential of
source intensity -Rtrr, ( E, n ) on S
in the event of second-order
potentia?l
second-order potentia
f,urther single lEyer potential of krror.,
rr.a e- lntensi-ty
in{-anci
+,, Lrzt
r 12) (
t E,
r
O)
^r on Sp.
The
: Helmholtz integral
inteqral eguatlon
equatlon formuformu(52), which was apparently first ininto ship motion theory by potash
fl zl ,, 4p
is
analytilally
srrqrJ
Lruolry
g"rr6.ui tItdll
*ore geIIeIctl.
[rutc
th.r,
Is
source-sink
-weIl-Kno\^rn conventional source-well-kno\^rn
[zq) by including boundary value
lems of inhomogeneous mixed Lype. In
tion, even when applied strictly
to a
p6oblem, it has shown certain advannumerically, especially for more comsectlon shapes, and with regard to
sensitivity to the well-known irregular
uency phenomenon [ 8] , lzl) , lzsl - .
The discretization of the integral
tion system (52) into an algebraic set
equatlons is conventlonal in the sense
close-flt methods and will not be dishere. Details can be found in [8] ,
231 . In contrast to Frank's close-fit
the normal derivatives of Greenrs
tion in (52) are taken with respect to
e point coordinates, which simplifies
integral expressions to be evaluated
the discrete panels [ 8]
Some particular analytical difficulties
i. evaluating the integral expressJ_ons
?!
(52). 91" of these pertains to Lhe line
Gegrlt over Sp, which contains a singulalry where Sp intersects with Ss. This caui numerical difficulties
in evaluatlng the
r:ntia1 and its derivatives in the viciniof the slngular point. Further the intetion along Sp, which should be extended
infinity, muit be truncated, so that some
ytical criterion for the truncation Iit is.reguired. Our approach to these quesons is discussed in section 3.3.3.
rt is well-known [ 26] that integral
ion formulations like (52), or is used
Frank, fail to provicte finite, unique,
sically meaningful solutions at or near
in "irregular" frequencies corresponto eigenvalues of the adjoint lnterior
ial boundary value problem. This phen is known as "irregularity problem,'.
second-order theory this type oi effect
ins at much l-ower frequencies, the first
rder irregularity occurring at about
quarter the frequency of the correspon'first-order vaLue. This suggests that
IrI.4
3.3 Solution Potentials
3.3.1 Incident Wave problem
The derivation of second-order potentials for a regular, steep (Stokes) wave
may be assumed to be known, e.g. [ 3] . rt
will be presented in a special, normalized
form in the present context
For water
fiJz)
"r i"ii"ite-a"ptn
is trivial and need
not be consideied. For
a second-order regular, steep \$rave propagating in the positive x-direction on deep water (beamwise), the first-order potentiil
for the wave with amplitude aw and wave
length tr, must satisfy eqs. (251, (26) and
(28) and is of the form
tJ" = (kb) "-1 .k(Y+jx)
This
(16)
may
(
s6
)
be normalized according to (15),
*J"
=-s ur 1 1.,;.kY uj (kx+6r)
(57)
where
(vb) = (kb) (1 +
+ otej),eo=kl.,l{sa)
'3r
with the di-mensi-onless frequency parameter
)
-1
(vb) = r,rg'b
(59)
Note that the wave corresponding to
this potential is, in fact, exact to the
third order although here nonlinear effects
are visible only in the relation between
wave frequency and !,rave length, eg. (59) .
The phase angle 6, measures the position
of the wave crest retative to the origin
at time zero.
-
u
3.3.2 First-order Radiation Problems
AIl remaining potential flow problems
are rafliEtion problems of the form (51 ) ,
with L(l)=g ge1 the first order. These probIems can all be solved by a uniform procedure: The integral equation formulation (52)
is discretized by introducing N straightline element panels of constant potential
value on.the body contour So and, in the
spirit of the close-fit method, setting up
an algebraic system of equations based on
the boundary conclitions at the midpoints of
the dj-screte panels. Details of the procedure and the system of influence coefficients and right-hand side terms in the ?Igebraj-c equation system are given in [23]
The forced motion first-order potentials (k=2, 3, 4l are determined in a single
calculation using synmetry or antisymmetrY,
whereas the diffraction problem (k=7) must
be solved twice for the svmmetrical and antisymmetrical parts of rjl^) .in (30).
once the potentials/'dr1tl ;." known
on So, eq. (52) may be used again to calculate this potential at any desired point
(x,y) in the fluid domain D. In particular
the first-order potentlals can be evaluated
on the boundary Sp (free surface) which is
required to obtain the lnhomogeneous terms
(45) and (491, for the second11{f), problems.
"q-":
orcter
3.3.3 Second-Order Radiation Problems
The general procedure in solving the
second-order radiation problems is the same
as for the first order, that is, eq. (52)
is again discretized and applied to points
on the body contour (x,y) e Se. In fact, the
problems associated with second-order onset
ilo*= "l2l Or=2, 3, 4), €9s. (32) to (37),
are comi)Ietely analogous in form to the
first-order problems and can be treated
accordingly without difficulty.
However, the remaining second-order
radiation problems, which are caused by
flrst-order disturbances proportional to
*.( 1). ,rJ1) ti,l=0, 2, 3, 4', 7i, d.o introduce
s6me spiicial questions regardlng the evaluation of the right-hand sides, €eS. (441 to
(49). These expressions involve first and
second partial derivatives of the potential
which are to be approximated numerically.
The accuracy of these approxlmations must
be examined carefully. In our computing
experience for a variety of different section shapes it has been found advantageous
to transform some of the second deri-vatives
with respect to x and y, where required, to
expressions with derivatives in the tangential clirection s, eqs. (441 to (49). This
has helped, in Befticular, to obtain stable
results for . r!(.t on 56.
Reoarclidq the numerical evaluation of
the terfr ,L!?l on Sp some further problems
arise. ThA 'f*prop.i integral over sp i-n
(521 involving this term in the lntegrand
requires integration to infinity, but in
practice integration must be truncated at
a "sufficient" distance "x- " from the body.
fll.4
This problem is also familiar from time-domaln and finite element formulations of the
present problem llll , lzzl , [::] , where
to
in addition it is fundamentally difficult
It
meet the radiation condition at infinity.
can be shown by theory that dpg.lo the harmonic asymptotic behavior of ,if, l,,
rim I.r1.')
t"l = o
at<
(60)
lI
(
r]D. ffi([
'!f
,{"
"{r,
lx1+-
Nevertheless an independent criterion must
rq!
be used to measure the truncation error. In
[email protected]
the present context the following indirect
[email protected]@
procedure was used: The first-order damping
[email protected]
coefficient was first calculated from near
[email protected]
field quantlties (by pressure integration)
ffi[r
and then compared to results derlved from
[email protected]
far field potentials (via radiated wave amplitudes), extending the range of [email protected]
ffielfl
tion on Sp step by step until suffj-cient
agreement was reached asymptotically. This
drfi
defined the truncation point x- . In prac_Wr
tice, x- was found to depend on frequency
( v b) and body shape.
Dil;"i
A particular difficulty
exists at tlre
intersection between S^ and Sp wfrere ,r{fl)
and 1r{f) are slngul.r? a==r^ing the sr-ngularity to be integrable, which cannot be
taken for granted for any section shape,
we treat this problem numerically by close1y approaching, but sti1l exempting the pole
in the integrations. However, the fundamen!
tal analytical problem, particularly for nonvertical sections, remalns unresolved desmli[@
pite John's valuable basic work lzaj
troM
il
,{
3.4 Pressures Forces
3.4. 1 Pressures
Moment
*p
' ,(I
[email protected]
The hydrodynamic pressure P (xryrt),
measured relative to atmospheric pressure
Ieve1, according to Bernoulli's equation
lii:rnrfflll
rfu
is
P(x,y;t)
1illlI
= - P9Y - P0t(x,Y;t)
1
- 2t'
lvoi2
(61)
Using the abbreviations from eqs. (13) to
117
)
(2\
o(1) = I *l',, ,0,', =.r. r . ix
(62)
'
I'k
.(2)-:
.(2)
i=
2,
31
4,
k=
,
O,
2, 3,4,7
2q' = | z+ii'
Q
,"
ora.i-r, the hydrodynamic pressure on S (t)
up to the second order:
{
nt
P(;(st,f t=);t)= - psi - {psr*j 1)
t1\(21
+
- jpo0"t*,y)- ]e-'i,dt
'*- - toe(x]-'
-o( 2)
o.25i x(l12) - 2)po(.0(2)+
,1"' *"ojt'', - o.s; ,^ri'*1"
- o.sip.,(;' *2(1) Y *3(1)
(2)
{ps(x o3
G;'- i;' r*j1)rrj"r e-2iu'lt
G;' *ri't*j1) I oj1)
+
+
-L2
+
( o]fl
uPuu,
* M-{[
-
o.2s;l *f,lt 12t
*
o.25ptlOj1) l2
- *'*1" - (xx'+yy-)
/t \
+ y , xi\L I'+(xy--yx )
*
't', * 0.5 j pro ,i'*1"
lTjr) * o.5jprrr ti'*)t\
rljr)
/l
I
(21
2Pi i
=
-L
(21
'
p(Ir29ii
il
(53)
t;e*) = p(o) rvl
.n ,1" ,i,t,.-Jot
t
k
n.l*'.'* ,nll) r*'Yl
f,
+
,r!?) I
(2t =
,nli'
rostatics
9g
.
. (1)
=l0o9t+
(2)
Eeg:ond
,nli' = -
tt
o
(72)
7
.25
pt,
,l:''. ,l:'')
(73)
4. Second Order (n=0, 1=1, m=1;
quasi-hydrostatics of second order)
al i=k=2, 3,
(2)
lPoii
4
rll,nli, . ,11, all,,
p,(i'to'ali' * r"to'ti:', *1"
*l fo.r, on tl*lt'l' i=4
= - 0.25
pt
+
L. , it4
o s *j3),
m='l
(1)
*3 , K=J
(2')
2P
oLL
pg
rn=1;
order)
oi1)
is
r"(o'rll', *1"
2r
b)i
,,jil
4
(21
z, p^rOii'
- o.2s pt Oin
:!)'*
,o(in(o)o(l) *
ojl)o{l' . ol:'r,11',
s) i=k=0,
(65)
Order (n=2, I=1r
a) i=k=2, 3,
+
i<k
k=0
o.s pt
lrik
o *1'), k=4 (67)
-0 , kl 3, 4
t
(70)
,
pgi
t=0
+
sl L=2, 3, 4i k=0, 7
= 2j p,r oli) - o. s o r oll'}o{l)
d)
- 0.sj
st Order
.(2\ _ o.s ot olr)0.(1)
ln Kn
9i1
(7
(65)
(n=0
p
ptor
4
oil'}o{l)r + 0.5i p,(i'to'r,ll' * r.(0)ol1))x(r)
"-2
Gmparing (64) with (63). yields the
1 pressure terms p(nl, dropping
fiactors er,.k,
X. Zeroth Order
zl
(6e)
,*1" * {k'to'ril'* *"to)oll)l*ll)t
jtot
.r"'
I;l,llll: ltjll'
,ri*l't,L=4
oll)of l'}r + 0.5 jpr.,i (. n to'0,1,1' * ,"(o'r,ll'
,l*'r'*
convention:
(68)
714
i< k
b) i,k=2
tPit<
!=4
pg x3(21 r r=J
, !* 3,
(2)
_2
^p
,njll t;'7r + ]'Ie.
. r l-011
(6,4)
2, 3, 4, k= 0, 2, 3, 4,'l
the left subscript is defined by the
=
,
\
lhese expressi-ons were,derived aftgr
ding the potentials 6(1) 3n6 6(2)
laylor series about the eguilibrium
of the body contour S- so that all
ls and theii derivativSs in (53)
to this position.
ES. (53) represents a complete hydrodytransfer model for the pressures on
rretted body contour S (t) . This transfer
corresponds to the dynamic system mol3l and can be decomposed into terms of
t orders (and frequencj-es) in analo(13):
3-
(1)2
25 pg V *i-'-,
.[:
*
+
Itr-4-13
= -0.5
k=2
l7 4)
1=3
r ori*!1',1=4
L;
3,
4;
,Ll 3,
i< k
4
,( oll'} ol:'} . *l:'all"
-0.5j pr,r{(.n(o)6(1)
+(.n(0);(t)
*."K
'lN
K
(7s)
. r"to'0,11)r *jl)*
to'tl:', *,1"1 06t
c,l i=2, 3, 4i k=0,
,r:?l
'
7
oll'} 6ll) - ollta[l)r
*1"
otr(rn(o) ol:) * r=(o'a,ll"
= -0.5 e(
-...,
,r(t;e*) =.F(o)+ i.* rf(l)e-J(!t +
-(2.\t -2.1' ot +
-(2)
*
itik
,l*tttk
,l*ttt* jtoik
(77')
d) i,k=0, 7; i< k
,"j?l
= -0.5
,( ol:)Tll' . ,ll'*,11"
s) i=k=0,
,rjil
=
where, in accordance with (65):
-o.2s
(78)
- (2)
rl2l
r(2', t
Jz'il
J^ii - j1-ii
_ (21
( l2l f(21 +
'j-oii
jl-oii
i2toii
The left subscriPts denote force and
components, resPectivelY:
7
e(+l:)oll'}
.
'l:'Tl:')
(7e)
denote
rn these expression" in(ol tt'a i=*(o)
the directibn cosines-of fr and 5 at time
t=g with in(o) from eq. Q3l and
{x' , t=2
t=:,"'o'={i-:
Ltiv'- Yx') ,
3
.4. 2 Forces,
moment
u
f)=z'
J( ) =t 1=:, vu
(88)
Li=4,
collectinq
ter\$s whose common factor is
--l
.,T ;-inry' (1, m, 11=e,722i t=2, 3, 4;
tlao,-f 3, 4, 7) the inclividual components
are defined by the fotlowing expressions:
(n=0, I=0, in=0)
1 . Ilydrostatics
(80)
''=4
Ivloment
P in (61)
Integration of the pressures
(t)
S
contour
body
over the ietted
the body
Lte nyaroaynamic foices acting on
and tire noirent about the origin o
ittl =-.r e i a",
ilrtt = - I n,;tlt* ) a', ? =
(r)
(87)
."(o) =-tp
l.rr(o)d"
Jg
2. First-Order HYlrodYnamics
ln=1 [=9, m=1)
(81)
(x-xz'Y-*3)
s
of Leibniz' integratlon-rule the
to an
iitegration over s(t) may be reduced
at rest'
i"teg.af over the wetted contour
s , i"a a few additional terms of second or-
By means
a8i tzl
t (S(t)
, lztl
;
.n'-'xl-'ds
.n(1)x(1)
-]7r,0,
s
I
'r,
:
eu) = i i(s;t;
^
rr{1)rr,(o)a"
so
t.-:_________-,/
E*) ds * IR
.(o)_-.._*
I* = R* i,,+
.rr(o)
i.._
(g3)
Rj = - 0.25 pst"jl)* i *1"-'.,:. o(t)li
ti.l;r"-r:,t - 0.25 psl*rl"* i ,1" -
-j ,r-'o(" l1 (t-);1
k=4
182)
3. Second-Orcler Eydrodynamics
ln=2 , L=1 , m=1 )
(0)
!?).r,(o)a.
--lrPii';
laK
so
- o.s ^I rl" i.(1)*j1),a.
.l2t
(84)
Ll- - '
i or k=4
where the subscripts denote
( )* = ( )l(;,f)=(o.5g,o)
( )- = ( )lt;,tt=(-o.sB,o)
*
(Bs)
i"".?lIr,*,r,,
i=k=4
rtl"
4. Second-Order Quasi-HYdr
Ti=O,
The transfer modet for the hydrodynamic
is deforces and moment to the second order(63)
pressures
rivea by substituting the(21)
to (24) into
and normal vectors fiom
is of anaexpression
resulting
iai f . ihe
logous form to (54):
(o)
*12)*
t.,,1?)
- _ l,p\ '"'(=ni"'x)-'t zz
,-------J + 0ls.n;;'x;-'-l
l=ffi=Tl
j'i?l = -r:,"i?l .n(o)a" -rrrri2,l ,,,(o)a"
IIr-4-14
o-so.r
G_
(2)xo4
(2)
*
nl":n(1);(1)u" -^,. ..P(0) (ino2
o,
+ 0.25.n j?'t,.j')f )u=
)
i or k=4
i=k=4
+
=
A
(n)
sin ( n,,:t +
6
from (14)
with xi(t)
+ joR
t(2)
ill';J,I,:=e
(t)
M_pB
7AA= Il(x'+y')
)
To determine the unknown motions x. (t),
€{I. (14), of the body in a given incideilt
mve the equations of motion must be taken
into consideration. These equations serve
to determine, in a second-order sense, the
actual motion amplitudes, hence the paraEters ei (L=2, 3, 4), and further any other
erplicit hydrodynamic quantities of interest.
Force components relate to the inertial
eordinate system o-X-y, the moment is taken
rith respect to the origin d of the coordioates fixed in the body. Equating hydrodyuanic pressure forces (moments), subscri_pt
F ' to inertia forces (moments), subscript
E. , in the equations of motion, we obtain:
i"=o
n= I
,_
.R =?
.R
.-inr,rr
"]r)
(roo)
i ,(n)
tR e--jnort
* .-l2l
tRo'
-rr1,
"'''R =
Substituting (100) and (86) to (88) lnto
(94) and separating terms of different order
(and frequency) the following sets of equations of motion are obtained for motion components of matching order (and frequency):
1. Hydrostatics (n=0)
r x n )ds (95)
+
x(gj
(99)
2
,R = t. *1"' .- jnurt
(94)
MR
It
="1 "
(n)
)
s(r)
A
The expressions in (98) can be developed into a perturbatj-on expansion to the second
order with respect to ei, using xi (t) from
(14) and a power series expansion-of the trigonometric functions. The resulting expressions are of the form
t-5 Body Motions
-/pids,MD=-Ip(
'
s(t)
an
-1
y^=M-IIydnl,
uA
n=u
F
xn)
and
_) _n
A(n) = lu(') l=(*.3{u(')} + r,?{u(')} )o.s
. (n)
6i"'
= arc ts( E": {u("1 i/ r*.{.(n))
(93)
Fp=FR,rr=
x, +tsi)I"sin
ru1"xrv".""
(e2)
fre expressions r{2) una fji) represenr
tfre contributions from the additional integrals (83) to (85) which approximate the
effects of the actual \$retted contour S (t)
deviating from So.
Thg lime-complex expressions for pressT-eP.p(n),
lsz) to (73), and forces
g(nl , eqs. "q=.
(SO1 , O1'), can readily be conrgrped to real notation for any quantltv
ampritude and phase ias a(nI ,
.
It* ='x'nrrn -
+ v)dm
(e6)
E(o) = o
(101)
;l:l :;r''
2. Hydrodynamics of Eirst Order
(n=1)
E(t) = L,(1)
= x(1)'1
Rl
k=0 k
v(1)
,Ju
="
k=0
=
"l"f-.11),ei
M(1) =',
= ,l,J
l(
k=o "_(1)
3. Hydrodynamics of. Segond OJder
(102)
(p=2)
rith il.
Mn, ir
g. fi^,
from (81), v from (10) and
A = body cioss sectional area.
The.hydrodynamlc pressure forces +En
_t were
deriv94, in 3.4 except for the
!\$l
.i or
ei
factors
fattors
or *,n,,
respectivLly.
respectiv6l
specEr-very.
"l.),forces
(moment are deThe inertiS
s (moment)
fined by:
!t =x*t *r"J,fi*=tu*
oo2I.=''ri; - x4yccos
xn + xiyosln
o
= x{g
+
oo
oo
H(2)
v(2) = ',
1
(97)
M(2)
xn)
o)_
_
x3 - x4ycsln
xn - xf,yocos x4)
I "ll)=
=i,k=Q
rk
(98)
III-4-15
ul-2)=
^ rl(
' '<=u
= ',
,1.2)=
i,k=Q rk
,l,l
"l'' f* *1"
,I',
(r03)
4. Quasi-Hydrostatics of Second_Order(n=2)
Exciting forces and moment are made dimen*
sionless by
J s(?l=
,1')
o =
i 'i=6--"it
o
1
,:2)
=. ;. ^v(?l=
o
v
or.K
+X
r ' X=U
(2)
fi1")
1
(104)
ol-
t II1)
7
.-.
M(2'.t
;
M(2)
=
= M(2)
o
RoI , k=o oik
t
-
,,2
wi
[fi
!(n)
ak
2pcl . d*
tu(n)
xik * l
il:)
].i<
1(n)
9k
2pd. d*nt.r
'utL'
, ,=
=-0.5o / t.("1,,(o)a=
K 1
rrt"
l,
,l=2
illi'
fljl)r.,
tr",.o
, i=4
, i=2 ,3
(114)
, i=4
drift components (of second ora.r) *1?)
result from the algebraic set of equatioH*
(104).,f,[e hgqizontal drift force is derive
from H1'' (xl!'remains indeterminate) and
may be"compaY6d to Maruo,s results [34] from
a momentum theory. From (86) and (87):
_(21
,;-'=
]
;(2)
;-;J -= r,i=o'r'k Hoik
with
_12)
no
H:"
=
o
z=en[2)"12
tr(zt
"o =
( 11
o. spg I arr
The nondi-mensional verti.cal drift
is
47
;(1)
= *t?)r-1
f,
E e.e.
=
oJ
crj
1=2 k=0 1 k ,viil (115
with Ij3' = -:3' l.,l-' = -(2)
xo3 e7-r
j'o''
,
(108)
=
trlaJern-1)
, i=2,3,n,-.
-1
The
eo
,n
tf "rf .rt-1)-1
ti
and, finally,
(a, i=2 ,3
. (n)
nik
(n)
and
E
The dinensionless hydrodynamic mass and damping in (105) are defined as
(
th
\
^x;{i tii'
=u,*
tu
0.5(k_i)
.33 = 1, d.*=(kb)
^22 = ,33 = 0.5 A, ^"
ttu
tu
--^-a tmnO=O .5t nn
^24 = ^42 =-O.5 Ic, " 44 = 0. 5cM
-tutu m.k
erse3
(to6)
= 0, cik = o, 3r. = I
=r!r, ]11,,-1 t(n)
ru(rr)
f-1"'
*i
=
tt
(n)
)
.
II"'
(l0s)
uit
(f rom (85)
i ttll', I,l") . tsjn)
4
o(n)
uik
(l)
k
k=2 lk
are obtained by matrix inversion
)
;r" ;ft='*jt
.?
1
, L=4' '7=
The solutions to ('105), which is of the
form
lji'. ftjl') + r lll ) a1;,
.}jl'r = ffjn),i= 2, 3,
wlth: f= a U-2
!(2)
r(1)
m..
mik'
IK = m.'.
].K =
E
= k=0,7
kb) '
47
a (2)
ilt
ikt from (86) , (8
i=2 k=0
(t
(,o)
I
2 pebd. e]
:- 2,3
BI2)
=ts
I
The first set of equations, ('101), concerns
hydrostatic effects due to zero1utl order
pressures, corresponding to the law of Archimedes. The set of eqs. (102) consists of
the first-order differential equations for
sway and roII (coupled), and for heave. The
unknown parameters eill--2, 3, 4) , defined
in eq. (2), can be derived from the solution
of (102). Eqs. (103), with ei substltuted,
comprise second-order equations of motion
for coupled sway-roll and for heave, and are
used to find second-order rnotion amplitudes
t(2) (frequency 20r ). In (104), fina1ly,
time-independent second-order effects,
called quasi-hydrostatic, are present and
can be determined using the s i from the solution of (102). T!!p yields the so-ca11ed
afift components *:!t, where the sway drift
x:4t is equated to"2ero, by virtue of an
aS5umed external force balancing the drift
force.
The dynannic equations (102) and (103)
can be written in dimensionless form as
follows:
*!r11"'
,1")
1
IrI-4-16
the roll drift
("hee1")
l:1' = -::r)irl.,tr-1 = {(klarl) . -r'I 4
^ i=2
.r'* ,iljil - 0.25(vb) ., in Ijtr;ttl,
{o.s d
A
) -t
7
r
k=0
lt17
l.
NUMERTCAL RESULTg
A computer program [35] has been developed to numerj.cally evaluate the boundary
value problems described in the precedlng
sections and to calculate the physical quantj-ties derived from the resulting integial
equations. The results presented in the following were generally obtained with N=25
discrete panels on the body surface So and
about 50 discrete elements in the free surface Sp (second-order). The free surface
ilisturbances for the second order on SF were
luated up to where the potentials reached
1imit, usually no more than 9
_asymptotic
If-beams away from the body. The frequenrange was 1O-5s (vb) 2.5 in steps of
=
=0.05.
The size of the program is 140 K words
a CYBER 175 computer. The program calcullates in one pass all pertineit iydrodynaa (vb)
ric quantities (potentials, pressirres, for, moments, and motions, where applicable)
any standard problem case of either
t order (k=0, 2, 3, 4, 7) or second or(i_2,3, 4, k=0t 2,3, 4, 7). Compilatime is around 1 0 seconds per standard
lem case. A complete evaluation up to
second order, comprising 13 standard
lem cases after suitable rearranqements,
about 1 minute of execution time per
ncy.
The results presented in Figs. 5 to 36
_ back
in
of the piper) with few 6xceptions
in only to second-order quantities.
are based on first-order results, which
t be included in the present paper. Nor
space permit a discussj_on of the ,'irrear frequency" phenomenon. Earlier publiions by papanikolaou [A] , [zz) , lzs]
be consulted for details on these is, including numerous first-order results
different seqtign shapes over a large
uency range [25J
less as indicated in the figures. The
ned heave amplitudes have been-standaril to. correspond to e3=1 o. xJl)=5. ,"lve phase anglesoare ilways pl5tted with
ancrement of 350
Fig. 5 for the triangle of B/T=0.8 ancl
. r/2 (finite flare)
demonstrates encoungly good agreement between theory aqfl,
riment for the hydrodynamic torce- VJj,
some allowance, for the phase an-Y+Fh The
51)4'.
appreciable phase snift, which
Dre abrupt in the measurements,
occurs
frequencies near where the force has a
, an observation made here for the
r although in the linear case simieffects are familiar from several_ other
. The caLcul-ated results.for the steate second-order force V"{3) in Fig.
5
are not in agreement with yamashitars mea_
surements regarding the absolute levet aI_
though both show a similar flat tendency;
we are uncertain of his definition of this
quantity.
In Fig. 6a fine Lewis form for B/T=0.g
and B=0.5 (sectj_on coefficient) is investigated, Fig. 7 shows the ellipse for the same
B/T. Hydrodynamic and steady state secondorder forces show excellent agreement with
measurements- The phase angles agree better
for the ellipse; for the Lewis form the agreement improves for decreasing motion am_
plitudes, proportlonal to e, as one must ex_
pect. fn Fig. 8 for the ellipse at B/T=1.4
all results are 1n very good agreement.
Figs. 9 to 1 1 are related to wider sec_
tions (B/T=2.0) of different fulIness. The
triangle shows the strongest, the U-shape
the weakest nonlinear effects. This appears
reasonable because the nonlinearities should
be responsive to how rapidly the section
shape changes near the waterli_ne. Eor the
circle in Fig. 10 the overall agreement in
al1 results with data from expeiiments appears extraordlnary. The minor deviations
that do exist increase with e, but remain
acceptable even at e=0.6.
The U-shape (Fig. 11) shows some, but
not much greater scatter.
It is of interest to compare the steady_
state forces in Figs. 9 to 12 in the limit
of (vb)+0. The tri_angle with large positive
flare in the water:Iine produces i net posi_
tive steady lift force, circle and U-sirape
have zero flare angles and a vanishing ,Zro_
frequency steady lift force, and the bulbous
form with negatlve flare causes a sma11 ne_
gative steady 1ift. At finite frequencies
the steady-state vertlcal force miy become
positlve or negative.
The results for the bulb (Fig. 12)
shoul-d be vlewed with caution due to the
very large e-values in the experiments. The
magnitude of the nonlinear effects is rather
smal-l, the scatter in the measurements con_
siderabl-e, and comparisons with the theory
for e up to 1.9'17 are of questionable valu6.
In Figs. 13 to 16, for a Lewls form
(B/I=2.0, B=Q.94), further comparlsons
of
forced motion results for pure sway, heave,
and roll_ as well as coupled sway-rb11 notions (with reference to standaidized para_
meters eir ek=1) are presented relative to
data from potashrs second-order theory [7J
(Simil-ar comparisons with theories Uv-oifrer
authors are also found in [B] , [S] ', lzZl ).
Fig. 13 relatgq,to the pure sway forced
motion.problem -(ZS)5t in Fig. 2) . fne agreement with Potash-in the vertj.cal forces, hy_
drodynamic and steady-state parts, is exceilent. These can be calcul_ated from first-order potentials exclug[vely. The second-order
horizontal_ forces frlj), winlcin involve second_
order potentials, differ appreciably from
Potash. This seems due to a devlation in
his second-order problem formulation Ig]
and the presence of irregular frequency
effects in his results
.Fgr heaving (proUrerq.3sjl)1 in Fig. 14
- hydrodynamic
the
force VtzJ-ae[enas on
tII-4-17
second-order potential-s, but the agreement
remains reasonabl-e, in part because the
heaving irregutarities arg^grilder. fn rol_l_the hydrodynaing (Fig. 15, problem 3Slfl
-ielatively
mic nonlinear effects aie
weak
for this section shape. Fig. 16 illustrates
the second-order effects in coupled forced
sway and roll motlons (problem 3slfr)y upon
the vertical force component. ThiS'emcompasses the second-order effects due to
first-order disturbances resuLting from
coupled *iay and ro11 motions. Comparisons
with Potash's results are problematic because it is not clear whether he was dealing with the same flow subsygfgm. (He ma
have included the problems 3S)22, ana 3Slfi
simultaneously) .
Figs. 17 to 19 present the steady-state
forces and moments actj-ng on a fixed body
1n a wave. These results stgqr. from the steady-state part of problem 5slzl Thesefnvestigations, and the ones that fo1Iow, were performed systematically for three section
shapes, the triangle (B=0.5), the circle
(\$=r/4), and the rectangle (B=1.o). Forces,
moments etc. are nondi_mensionalized with respect to wave amplitude I uro I , as customary
in the literature.
Fig. 17"c94gerns the horlzontal_ steadystate force Hn61). rn the limit of (vb )*oo
our results for all section shapes approach
unity in agreement with Manro,s [:a] analytically derived result frpm,qrgmentum considerations. The quantitlz fi.,61, ,plotted in
dashed lines, expresses th6-considerabl_e
contribution by the wetted part above water
(rest integrals in eqs. (83) , (84) ) upon
the total steady-state force. For ( vb ) * this term tends to the limit of 2 sin o_in
agreement with [36] . The contribution Hade
by the underwater part is smaller and negative, the sum of the two yields the net
force.
The corresponding vertical steady-state
forces (Eig. 'l8) are negative for circle
and rectangle (sinkage force), and positive
for the triangle (lj-ft). The zero frequency
limit of this force is zero for the wall-sided shapes^F,,ri r/2) and equal to 1 for the
triangle {trj[it + ct9 o, =
n / 4=11 .
"tS Fig. 19, are
negative, that is, heeling in the direction
of the lncident wave.
In order to determine the lnfluence of
the motions upon the hydrodynamic secondorder contributions, it 1s necessary to
solve the equations of motion to the first
order (eq. (105), n=1) to start wlth. The
results are reguired to solve for the sma1l
parameters ei (i=2, 3, 4) in terms of the
initially
asGumed e7 (e^), which measures
the relative slze of ttrE incident wave,
usi-ng eq. (114). Ihe first-order exci-tation
forces for a circle in a sine wave (6*=91
are shown in Fig. 20. Figs. 21 to 23 present
the corrgpponding first-order motion amplitudes *+', (i=2,3, 4), where the results
for sway-and roll stem, of course, from
their coupled equati-ons of motion. Near resonance all amplitudes are considerable because hydrodynamic damping is smal1, above
all in ro1l, unless empirical corrections
are made to allow for viscous damping. On
the basis of the amplitude results we may
now assign some bounds to the ei and limit
the wave heights accordingly viE e7 or w€
may assume some wave height, hence'e7, determine the corresponding values of ii, and
avoid frequencies where ei exceeds a specified limit, especially near resonance.
In Figs. 24 to 28 some first results
are presented descrj.bing the second-order
forces and moments which result from the second-order interactions between motion poten
tials and the combined J,qpident w4yp and dif
fraction potentials (4Stzt and 55tz,subproblems combined). These results tike into account the small parameters et=f(vb ) of the
motions, deduced from the lifrear response
ana1ysis (egs. (105), (114)). The curves exhibit some more or less pronounced resonant
peaks. These stem mainly from the behavior
of the ei nedr resonance, modi_fied in part
according to the basic frequency dependence
of the second-order hydrodynamic forces. Th
resul-ts presented here are stitl a function
of the initially
assumed parameter e7.
By solving all standardized subfrobl
of the system and associating them with the
pertinent small parameters .i.uk, it is pos
sibLe to assemble all contriEutions of second-order for the body freely oscillating
in the wave. The steady-state part of this
summati-on corresponds to the so-catled drif
tj-ng forces. Fig. 29 compares horizontal
drifting forces for three basic section
shapes. The results are credible, except
resonance. Comparisons based on Maru.ors famil-iar formulas [ga] give similar answers.
The asymptotic limit of this force for
( vb)r - shoulcl be one, as in Fig. 17 for
the fixecl body, because the motion amplit
go to zero at high frequencies.
Vertical drifting forces, eq. (1161 ,
are shown in Fig. 30, drifting moments,
eg. (117) , in Fig. 31. Rectangle and triang
have positive verticat drift (lift)
for mos
frequencies. The drifting moments are negatj-ve, they tend to heel toward the wave.
Masumotors results [tO] show similar tenden
cies.
It remains to solve the equations of
motlon of the seeond order, eq. (105), n=2.
The excitation forces of second order are
obtained from eg. (111) by summation of a1I
second-order hydrodynamic terms, which are
caused by the motions and velocities of fi
order. The verti.cal second-order excitation
forces for a semj--circle are plotted in
Fig. 32. According to eqs. (i07)r (108), n=
the second-order hyd.rodynamic mass and damping coefficients for heaving have been
dimensionless with the displaced fluid mass
of the semi-circle, as usual. The curves in
Fig. 32 are obtained by contractj_ng the fre
quency scale of the corresponding first-order curves, which are familiar, to one qua
ter so that the first-order results for
shifted to ( v b) =l .
The motion responses of the heavi-ng c
cular cylinder to the first and second
( v b) =4 .are
are presented in F19. 33. ft is interesting
III-4-l8
b-
t'
:
t
I
tit--
ll
i,
to note that two resonant peaks are present
in the second-order heave amplitudes, one
at the resonant frequency of the first-order
system which tends to excite the second-order
system, and one at the second-order sys1
'temrs own resonant frequency which lies at
about one quarter of the first-order resonance. The second-order amplitudes are relatively small compared to the first-order
values in this instance. This need not to be
so for other degrees of freedom.
To obtain an idea of the relative i:rportance of second-order effects in forced heaving motlons we may compare the curves in
Fig. 34. They represent the ratio of secondto first-order force amplitudes for a heaving circular cylinder as a function of the
amplitude parameter e3. The agreement betr,'reen calculated results and experimental
data from Tasai and Koterayama- [ZO] is very
good. Only at higher frequencies and amplitudes (e) do some differences develop. Viscous effects as a possible reason for part
of these dlfferences at hiqher frequencies
are discussea in I Zo]
A comparable diagram for heave excitation forces, second- to first-order ratio,
for a seml-circular cylinder oscillating in
a wave is given in Fig. 35. This ratio depends directly on e7, i.e., the relative
size of the incident wave to the body dlmensj-ons. The nonlinear influence on this quantity has a peak at some intermedlate frequency where larger motions are present due
to resonance.
The time-dependence of the heaving motion of a circular cylinder in a "standard"
wave (eZ=1), approximated to the first and
second order, is shown in Eig. 36 , together
with the second-order steady-state term. The
frequency (vb) of 0.25 corresponds to a
peak in the second-order effects (Fig. 33).
The nonlinear effects are not dramatic, but
noticeable. Amplitudes are about 158 greater than in the lj.near analysis and a steady lift effect is present. The result confirms why for a shape like the circl-e linear theory has been so successful in practice
whenever heaving motions and waves are reasonably small. That nonlinear effects upon
vertical loads (and, of course, local pressures) can be much more substantial in certain frequency ranges will be appreciated
from Figs. 34 and 35.
Although we have not yet examined motions in other degrees of freedom by the
current method, we would also expect significant hydrodynamlc nonlinearities in nearresonant ro1l, coupled with sway, in view
of the associated large ro11 amplitudes.
5. CONCLUSIONS
By means of an approach based on nonlinear system dynamics and nonlinear hydrodynamic theory it has been possibl-e to develop a complete second-order theoretical
model for the motions and hydrodynami-cs of
a cylindrical body of arbitrary cross section in a regular, second-order incident
hrave. A crucial first step is the decompo-
sitj-on of the total second-order flow system
into a set of nonl-inear subsyster-s comprising the second-order equivalents of the familiar forced rn-otion, incident r.^rave, and
diffraction flow systemsrbut also their mutual second-orcier interaction. Perturbatlon
theory, using several small parameters, has
then been applied systematically to derive
a complete set of first- and second-order
linear subsystems of the f1ow. These systems
together form the basis for developing a second-order transfer model of the dynamic
probl-em.
The equations of motion have been derived to the second order with all hydrodynamic couplings present. They include firstorder terms of incident wave frequency o,
and second-order terms of frequency 2 o as
well as a time-independent expression. A11
system responses are of the same basic form.
On the basis of this theoretical model
a numerical solution method has been deveIoped using an integral equation formulatlon
for a Robin type problem and a close-fit discretization. The fact that aIl problems,
with one trivial exception, were of the same
boundary value type, that is, radi-ation
problems, paved the way for a unified calcuLation procedure for all subproblems.
Numerical calculations were performed
for a variety of section shapes over a wlde
frequency range. These calculations included
some samples of each of the major subproblem
types. No insurmountable, fundamental obstacles were found in the path of the numerical
calculations.
In those relatlvely scarce cases
(forced motions) where comparison with experimental results was possible the agreement
in second-order effects was generally between good and excellent. Eor the heavlng
semi-circ1e, for which most experlmental
data are avallable, the agreement is outstanding, for sections with flare it is
slightl-y worse.
The evaluation of drift forces as a
second-order phenomenon is possible from
subproblems of the system. The results obtained show good agreement with other theories.
The response of the system can be evaluated by assembling the results of all
first- and second-order subproblems into the
equations of motion with small parameter
values dssigned to each case on a physical
basis. The overall transfer model is based
on frequency domain techniques because the
linear subproblems, of which the systen consists, all have frequency domain transfer
functions. This posslbility evidently has
significant practical advantages over tlmedomain solutions.
Investl-gations of the type reported
are, of course, only a prelude to developing a systematic understanding of nonlinear ship motlons in a nonlinear irregular
seaway. We feel, however, that the remaining
open questj-ons, despite thei-r great fundamental complexity, show a certain promj-se
today of gradually being amenable to higherorder analysis via the frequency domain by
rII-4.19
I
i.'
I
sultable extension of bi-spectral analysis
and generalized three-dimeirsional flow analysis
1 3. Salvesen,
N., "ship Motions in La
Waves", R. Timman Mernorial_, De1ft, 1 978.
methods.
14. S6ding, H.,
Forces on
Oscillating_Cylinders"Second_Order
in Waves", Schiffst
nik, vo1. 23, 1976, pp. 205*209.
15. yamashita, S., ,,Calcu1ations of the
Hydrodynamic Forces Acting
Cylin_
ders oscillating Verticaliy "p""-ffri"
irittr-i..g"
p1itude"l in.Japanese, Journ. Soc. Nava1Am_
alcl:[email protected], vol. 1 4l;-Jine-iffi
67-76.
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NONLfIIEAR,
1979.
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1
Discussion
F. Tasai and W. Koterayama{Kyushu tJniv.)
The
I
i.
writers would like to congratulate
the Authors on this very interesting and
valuable paper and would be most grateful
if they be kind enough to comment on a
few points.
1. Which boundary condition has the J-arger
effect on the second-order force, the
free-surface condition or the body-surface one ?
2. The results of the forced heaving tests20
indicate that the first-order forces and
the arnplitudes of progressive waves E3
depend on the-amplitude parameter e 3.
For example, E: tends to decrease lrith
an inerease in e:. can these phenomena
be
e4plaiqgd,wilh t_lris the.ory. ? - _
3. rn ria.'17 and 29, the horizontal dtifting f5rces fro(z) are greater than unity
at a certain frequencY. BY Maruors
theory, the nondimensionalized drifting
forces don't become larger than unity.
Did you comPare your calculated results
with those by Maruo's theory ?
H. Maio Uoxonama
N. u niv.)
In this paper, the auttrors have deveLa general method of computing secondorder hydrodynamic fofges on a cylindrical
body oscillating \rrith three degrees of freedom on waves of large amplitude. The computation prograrn seems to be very useful
for the assessment of the.Iateral oscillation of a shipTii b-earn seas where the motion
amplitude is liabLe to be large so that the
non-Iinearity in the motion is no longer
negligible. O;1e problenu_wli.ch I wish to
point out is the faCt that the theory upon
which the computation method is based ii a
perturbatton analysis in any case. The
boundary;yalue problem is formul-ated on the
body srrrface at its average position. It
is suspected that the perturbation theory
may present some_difficulty -in-6-rder to take
account of 'bhe change of shape of the wetted
portion ruith time which is liable to become
l-arge in the case of rolling. A question
is, to what extent, is the perturbation
anal.ysis .applicable at J-arge motion amplj.tude ?
oped.
As a previous pursuer of the same kind .rl
problem, I can ful1y appreciate tire pain!.I
taking patience the authors must have
required during the iourse of this work,
particularly, in keeping the orders of
magnitude consistent. I congratulate the
authors for this excellent milestone work
I have to confess that I did not
the patience to check every eguation inhav
this paper; however, with regird to the
passage concerning the irregular frequen_
cies, I have the following iuggestion.
This suggestion may only applt if the
h(n) = 6(n) - f (,),
such that
F(n2r){r,(n)} = o,
B{h(n)1 = p(n) - \$ rt"l,
(x,y)
e
D
(x,y) € sF
(x,y)e so
and the rest of the boundary conditions
remain the same as in Equation (51).
. ?-i{r.g the new boundary-value problem
- n\..,
ror
rs ]-dentical with the first-orde
problem, one of the techniques in t2gJ _
[30] should resolve the problem of indefi
nite solution at the iregular frequencie
f would think such an appioach wouid save
the computer time rnore significantly than
the interpol_ation method used in this
paPer
As to the solution for f (n) orre-can
find_the expression in Eguat-ions, (2I,22)
of "Surface Waves,' by wehausen a.rd Laiton
in whiclr.the expression should !g fe3 _-f (
When L(n) is nondecaying at lxl * - 6rg
behave as g)aztttzr , ih"-"oirrii"" i= !iu"n
in my earlj-er work [2.1]
1
A- Papanikolaou and H. Nowacki (Tech. Univ. Berlin)
We would like to thank all- discusse
for their valuabl-e commentsProfessors Tasai and Koterayama have
C.M. Lee (DINSFDC)
* Quote: But it must be mentioned that
purely analytical methods for this
have not yet been extended to second-orde
situations with j-nhomogeneous ffee surfa
boundary conditions.
The authors have completed the secondorder solution for freely floating twodimensionaL bodies subjeEt to beai waves.
-332-
I
l
In
Su?:tion?:
it is difficult
because-the rebody and free suraepl"ds on secti-on
three interestins
raised
tr:'=;;.;'*.I*tt'"
rit=t-"it'
ansv'er
;;-;r;; a general of
the
iltit"-i-p"rtance
face bound.ry "onai'lio"i
upon which
shape, frequency p"i"*tttt'-?"d
discussed'
is
subpioulem
ill"ir-"taer
the body boundary conii-i""v-"iioationsrt"t]=otracl
colrdition Bust
ii.il"' "ra-ttas" beins of- equal
Tr::?1:"'
;;-;;;.;a;d
exPansLon
6econd, our peiturbation
pathe-sma11
does not show any'eftect of
danping coheave
linear
tnL
ili"a".-.r-onA..
consistent with the
--ti.n-ii theorv'
-sria.ianr
we.apprecatit'""t
situation
different
tt'"t-"
:;;;'t;;;;"',
=;;;;i;";"or
results' where
uav exist ir, .,'tiy=ing test forces into
oi-*"i""t"d orders and
fi'. ;il;"ision rto*
aitttt"t't.
I.ittiu"tror,= do"'-d"P""a
on the assumed
;;:;;;;;il"
model'..For exevaluition
the
in
orders
evaluation model
ample, if a secoia:;;a"taclequate
to reprenot
t-.==*.a, but is effects'
show
may
thig
*""r*"d
sent the
Eg'
dependence
;;"i"-;-"false"
-"iltt-a"p"-t'd"t'"" mav"f 11-on
have
-also
;:*;;";,
in secondother reasons ""i-""""""ted--f9r
viscous
uotablv
ih"otv'
;;A;; polentiar
effects.
have been
Third, our drifting-forces
by
potentials
i""ilfiAld
calcurated r=ot
over
of second-order Pressures
il;;;;;;i;n
whereas
contour'
body
wetied
i't.-i.i""rly
lraruors rheory iilr ;; t"!+ "l.far-fierd
our
linear ineident'iit"
driftins
"ott=iderations'
tr'"-ii"tizonta:.
i"p"iiiL"t
;i;:r; an o,ttott=titined body' Fig'17
force on
the- former case
;;i;; io. " r:-xea bgdY:. rn
for
force
the nonilime.r"io"Ir-a;ift u"itv values
certain
a
in
the rectangr" t"i*"i"l"J i= due to- a resonant
freouency t..,g"l-*tti"tt
As stated i-n the PaPer'
;;;j;;";i.;;i";'
mtist be viewed
the result= ,'"tt't"=ottitt""
considerabl-e
the
of
oe""oi"
cautio,,
with
!.i''iri"a;-:i.i:;.:tl'J*i::"'i:"ffi::;"'
i" ti'i=
-i:r'l::"'"q"iil:";; fi=;;;-;;pri"ainvesti-
situation. rot-i'rtt-iixea uoaies
qated, only-tle i"tti"gf" Itl:-" nondj'men?I'"a-'iiiqhtlv srea'terthan
:1;;;i ili'rrils
one. It must be noted that the-reference
drirtins
;;;d i;'-pi"tti"g
iiil".rii
i;;;;; is o - sysl*ll
kinetic energy
tude aw- ln reality
the.
;:'"'"::*i:"3r'kl1?kinetic energy
the
of our second-order wave also- includes
.whichcomparr1:::-?:
second-orde. "o"tiinotiot'='
taken into account in any direct
from Maruol:--.
fr;";;;-results-derive&
some expera-that
noLed
be
ii"i.vl-'-ri-*"v arlo--inaicate the
possibilmental results
exceedrorces
driftins
H;:; illi""nttr
inc unitY-'^"
-'pio't"ssor Maruots remarks remind us
of the- perturbaor tn"-iill.-ri*it"tions
the nature of
in
rt-iies
tion method.that ihe boundarY.conditions
ii;; ;;fi;d or moving boundaries are met
on any free
I
il
iill
I
.pp'o*i*.t"1l-lY.T:Il":.";::;:';fli:=i:Tn,
p-p., that-a second-;:"H:":n:ll.EE=iX'aI
for moderi=."pI--i"'bL varid
llulii"Ir',""ivamPratirdes
and moderate wave
ate motion
are exceeded
heights- When.ti!="-fi*its
frequency paand
shape
depends on secEron
'=tcond-order extends
raireter.
The
ptl=""[
ti'rllit" to litt"tr -theorv
ii'Jl""ii*rt=for second-order hyilrodynamrc
We are grateful for Dr'L"9:-" the presexperience'with
based Jn hi"-lottg
ent problem. Iii=-;'"ggestion.to-transform
qo 3 homosene"
;ir; :;;;;;:order subPiobrems
rirstof
ttrat
to
-i"-"ia"t to-be the
;il ;;ii.';eoi""rt"i
to rely
able
order problems,
for
methods
on the well--knowtt--tt'tfyti"a1
i"
alleviating the'iti"soi"titY
-l:?:1"*
feabe
certainly
well conceivea ana-iiould
exthe
this-would-b:-i:
sible. However,
two boundary value
Dense of having to solvefor
every secondi"it""a-"r one
#:;;;;
our
under discussion' rn
5"tX!i=I"oi;;i;
for
schenes
pot"fv--ttumerical
experience,
wqrrea very reliabl-y
the same potpo"I-ittve
as for the second
*"ff
for the first ""l"-'"aa"a,that
we.have also
order. It mayt"iiyiltii
semi-analvtical
investigat"a
""a
the i-1leSr1laritv
methods ror aeali'ng- ryit\.lla'13
wolxed quite
i'r'itt'
ilIilffi rel-, izgil'
llley :nrv shift
fi:=i;;;iliaritv !o ?om?.
in the ranse or interii;";-fr;i-"t:'ri'lie
the second-ordernote
;:i;':p;;ticurairv for
important to phei='
ii
IiiirS*l--ii""iri
i-rr3g1-laritv
i1i.
ti
that the
the HeImusing
*"tt'-""duced'!Y
i=""t"'iii
with
f6rmuration
e;;;;i;;
holtz intesrar unknown runction'
I;"";";;;;l'ar
"i
-333 -
I
i
i,
j
I
I
I
1'
i
i
I
t
bv
allowing
the existeffects and, in i"iti."r"',-f9r
waterline'
the
in
il;=;i;';";ri;
nomenon
I
t
i
t
i
il
t
i
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