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Proceedings of OMAE’02
of OMAE2002
Offshore Mechanics
21 International Conference onProceedings
and Arctic Engineering
OFFSHORE MECHANICS AND ARCTIC ENGINEERING
JuneOslo,
23-28,
2002,Oslo,
Norway
NORWAY,
23-28 June,
2002
st
OMAE2002-28442
OMAE2002-28442
WAVE-DRIFT ADDED MASS OF A CYLINDER ARRAY
FREE TO RESPOND TO THE INCIDENT WAVES
Takeshi Kinoshita, Weiguang Bao,
Motoki Yoshida and Kazuko Ishibashi
Institute of Industrial Science, University of Tokyo
4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan
Phone & fax: 03.5452.6169
E-mail: kinoshit@iis.u-tokyo.ac.j
ABSTRACT
Conventional linear added mass and damping can be obtained
when a floating body is forced to oscillate in the calm water.
However, with the presence of the incident waves, there exists
an alternative source of added mass and damping caused by the
nonlinear interactions between waves and low-frequency
oscillations. Proportional to the square of the wave amplitude,
they are called the wave drift added mass and the wave drift
damping. The problem of a circular cylinder array slowly
oscillating in both diffraction and radiation wave fields is
considered in the present work. The frequency of the lowfrequency oscillation is assumed to be much smaller than the
wave frequency. Perturbation expansion based on two time
scales is performed to simplify the problem. Wave loads
including the wave drift added mass are formulated by
integration of the hydrodynamic pressure over the instantaneous
wetted body surface.
1. INTRODUCTION
As well known, structures and vessels moored in ocean waves
may undergo oscillations with low frequencies in the horizontal
plane, caused by the slowly varying nonlinear wave excitation.
To simulate this slow drift oscillation accurately, which is
important for the design of the mooring system, much effort has
been made in the investigation of nonlinear wave loads
especially the wave-drift damping in recent decades. As a result
of linear radiation problem, the conventional wave-radiating
damping is asymptotically small when the frequency of the
oscillation tends to vanish. Therefore, it is now commonly
accepted that wave-drift damping plays a key role in the
determination of the amplitude of low-frequency drift
oscillations at resonance. On the other hand, wave-drift added
mass is considered less important since conventional added
mass is of order of the displaced mass when the frequency of
oscillation tends to zero. Nevertheless, it has been reported that
the added mass increases significantly when measured in waves
[1, 2]
. In our previous work [3], a cylinder array slowly oscillating
in waves is considered and wave–drift added mass is calculated.
The cylinder array is restrained from the linear responses to the
incident waves. It has been found that the effects of the wavedrift added mass on the low-frequency drift motions, e.g. the
resonant frequency, are not negligible. In practical world, ocean
structures usually oscillate in response to the linear wave
excitation. Therefore, as an extension of the previous work, we
are going to consider the interaction of the low-frequency
oscillations with both the diffraction and radiation wave fields
of a circular cylinder array. Wave-drift added mass will be
calculated together with other nonlinear wave loads to make a
better understand of the physical mechanism of the lowfrequency drift motions. Experimental measurements are carried
out in a wave tank as well to validate the calculated results.
The problem of a circular cylinder array slowly oscillating in
regular waves is considered in the present work. The frequency
of low-frequency oscillations is σ, which is assumed to be much
smaller than the incident wave frequency ω. The linear
responses to the incident waves are not restrained, i.e. the
cylinder array is free to oscillate at the wave frequency. Similar
to the approach used by Newman [4] to evaluate wave drift
damping, two time scales are adopted in the perturbation
analysis. However, a coordinate frame following the low
frequency oscillation is adopted in the present work so that it is
not necessary to assume the amplitude of the low-frequency
oscillations to be small. The resulting boundary value problems
for each order of potentials are solved by the method of eigenfunction expansion in the limiting case of σ tends to zero.
1
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Forces acting on the body are evaluated by integration of
hydrodynamic pressure along the instantaneous wetted surface
of the body. From the quadratic nonlinear forces in terms of the
wave amplitude, the wave drift added mass is picked out by the
component contract to the phase of the acceleration of the lowfrequency oscillations. It has been found that potentials
associated with the acceleration of the low-frequency
oscillations make contribution to the wave drift added mass.
Experiment arrangement is also described. Measured data are
compared with the calculated ones to verify the present theory.
The boundary problems are formulated in the next section
followed by a discussion on the limiting case of asymptotically
small low frequency. The methods to solve the potentials are
briefly discussed in section four. Then formulae to evaluate
wave loads are presented. The equation of motion to solve the
amplitude of high-frequency responses is deduced in section
six. A discussion on the comparison between theory and
experiment is given in the last section.
2. FORMULATION OF THE PROBLEM
The problem considered here is that an assembly of circular
cylinders is oscillating slowly in a train of regular waves. The
depth of the calm water is assumed to be h. The radius of the
cylinders is a. The draught is d. The cylinder array is free to the
linear responses to the incident waves. The frequency of the
slow oscillation is designated by σ while the wave frequency is
given by ω. It is assumed that σ << ω. The low-frequency
oscillation is restricted in the horizontal plane, i.e. in the mode
of surge, sway or yaw designated by j=1, 2 or 6 respectively. Its
displacement and velocity is expressed as follows respectively:
ξ j (t ) = Re iξ j e −iσt
( j = 1, 2 or 6)
(2-1)
ξ&j (t ) = Re σξ j e −iσt
{
{
}
}
where an over dot denotes the time differentiation. In (2-1),
ξ represents the amplitude of the slow oscillation, which is
assumed real without losing generality.
A Cartesian coordinate system following the low-frequency
oscillations, but not the high-frequency responses to the
incident waves, is adopted to describe the problem. The oxy
plane coincides with the undisturbed free surface while the zaxis is pointing upward. The coordinates of moving frame is
related to a space-fixed frame, say OXYZ, as follows:
X = x + δ j1ξ 1 (t )
( j = 1 or 2 )
Y = y + δ j 2ξ 2 (t )
(2-2)
X = x cos ξ 6 (t ) − y sin ξ 6 (t )
( j = 6)
Y = x sin ξ 6 (t ) + y cos ξ 6 (t )
The time derivative in the space-fixed frame can be transferred
to the moving frame by chain-rule differentiation:
d dt = ∂ ∂t − ξ& j (t ) ∂ ∂x j ( j = 1, 2 or 6)
(2-3)
where x1=x, x2=y and x6=θ, i.e. the azimuth angle.
2.1 Perturbation Expansion of the Potential.
The fluid is assumed to be inviscous and the flow to be
irrotational. Therefore, there exists a velocity potentialΦ(x, t). It
is natural to use two time scales to describe these two kinds of
motions with low and high frequency respectively. Following
the approach of Newman’s [4], the velocity potential can be
expressed by a perturbation expansion up to the quadratic order
in wave amplitude ζa as:
− iS (t )
Φ (x , t ) = Re φ1 (x )e j + φ2(0 ) (x ) + ... + σξ φ0 j (x )e − iσt
(2-4)
− i (S (t )+ σt )
− i (S (t )−σt )
+ φ1(+j ) (x )e j
+ φ1(−j ) (x )e j
+ φ2(0j ) (x )e − iσt + ...
{
[
]}
The potentials on the right-hand side of (2-4) depends only on
the space position x. The number in the subscript indicates the
order in wave amplitude while the letter j =1, 2 or 6 denotes that
the potential is related to the slow surge, sway or yaw motion
respectively. Superscripts are used if needed to denote harmonic
time dependence on the wave frequency. Here, potentials with
double wave frequency are omitted since they will not
contribute to the wave-drift added mass and damping.
The definition of the phase function Sj(t) comes from the
incident wave potential Φ10(x, t), which is the only specified
component in the first order potential. Expressed in the moving
frame, the incident wave potential is given by
− iS (t )
Φ10 (x, t ) = Re φ10e j
(2-5a)
{
withφ10 (x ) =
}
ζ a g cosh k0 (z + h )
exp[ik0 (x cos β + y sin β )] (2-5b)
iω cosh k0h
Here, the phase function is defined as
S j (t ) = ωt − (δ j1 + δ j 2 )ξ j (t )κ j
( j = 1, 2 or 6)
(2-6)
where κ 1 = k 0 cos β and κ 2 = k 0 sin β with k0 to be the wave
number of the incident waves. The so-called encountering
frequency ωe is obtained from the time derivative of Sj(t)
ω e = S& j (t ) = ω − (δ j1 + δ j 2 )ξ& j (t )κ j
( j = 1, 2 or 6)
(2-7)
In the case of low-frequency yaw motion, referring to the
moving frame, the incident wave angle β changes with time, i.e.
β = β 0-ξ6(t). Therefore, when a time derivative is taken, a term
of differentiation with respect to β should be added, e.g. for the
incident wave potential,
∂[Φ10 (x, t )] ∂t = Re − i ω − iξ&6 (t ) ∂ ∂β φ10 (x )e − iωt
(2-8)
Comparing with the case of j=1 or 2, it is convenient for the
later discussion to define κ 6 = i ∂ ∂β .
The high-frequency oscillations, i.e., the linear responses of the
body, are denoted by ηs(t) (s=1 ~ 6) where s=1, 2, 3 indicates
the translation and s =4, 5, 6 represents the rotation of the body.
Those responses will be affected by the low-frequency
oscillations. Hence, they can be expanded in a similar way as
the potential:
ηs (t ) = η0 s (t ) + σξ j ω g η (js+ ) (t ) + η (js− ) (t )
(2-9)
= Re iη0 s e − iS (t ) + iσξ j ω g[η js(+ )e − i ( S ( t ) +σt ) + η js(− )e − i ( S ( t ) −σt ) ]
{ (
{
(
)
}
)
}
The velocity of the high-frequency oscillations is obtained from
the time derivative of the corresponding displacement
2
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{
[
η&s (t ) = Re ωη0 s e −iS (t ) + σξ j (− 12 κ jη0 s + νη js( + ) )e − i (S (t )+σt )
(
)
]} ( )
+ − 12 κ jη0 s + νη js( − ) e − i (S (t )−σt ) + O σ 2
(2-10)
6
φ1 = φ10 + φ17 + ω ∑η0 sφ1s
2
with s = 1~6, j = 1, 2 or 6 and ν=ω /g in (2-9, 10).
up to the second order in wave amplitude. Here, ej denotes the
unit vector in x-direction or y-direction for j=1 or 2 while
e 6 = e 3 × x. η(t ) = (η1 (t ),η 2 (t ),η 3 (t ) ) denotes the vector of
high-frequency translation oscillations and the vector of the
rotational oscillations is given by α (t ) = (η 4 (t ),η 5 (t ),η 6 (t ) ) . H
represents a matrix of quadratic terms of rotational oscillations,
which is defined as (see Ogilvie’s work [5]):
 − 12 η 52 + η 62

0
0


2
2
1
H =  η 4η 5
0
(2-13)
− 2 η4 + η6

2
2 
1
 η 4η 6
η
η
−
η
+
η
5 6
4
5 
2

This boundary condition should be expanded about the mean
body surface S0.
2
∇Φ ⋅ n = ξ&(t )e j + B& − B ⋅ ∇(∇Φ ) − 12 (B ⋅ ∇ ) (∇Φ )
)
(
)
(
)
]
[
⋅ (n + α × n + Hn) on S 0
(2-14)
with
B = η (t ) + α (t ) × x + Hx
Then, the perturbation expansions of (2-4) are substituted into
the above boundary condition, together with the expressions for
the displacement and velocity of high-frequency oscillations
expressed in (2-9) and (2-10). Resorting terms according to
their orders in wave amplitude and time dependence, the
boundary condition satisfied by each potential on the mean
body surface is obtained. The boundary conditions for the first
three potentials in the perturbation expansion of (2-4), i.e. φ1,
φ2(0 ) and φ0j, are well known. They are stated as:
∂φ1 ∂n = ωB0 ⋅ n
(2-15a)
where B0 = η0 + α 0 × x, η0 = (η01 ,η02 ,η03 )and α 0 = (η04 ,η05 ,η06 )
φ2(0n) = Vn
{ [
(
) (
)
with Vn = 12 Re − n ⋅ iB0 ⋅ ∇ ∇φ 1* + i∇φ1* − iωB0* × α 0
∂φ 0 j ∂n = n j
( j = 1, 2 or 6)
(2-16)
s =1
2.2 Boundary Value Problems.
The velocity potential is governed by the Laplace equation in
the fluid domain and satisfy an impermeable condition on the
sea bottom z=-h. A brief discussion on the boundary condition
at the body surface and the free surface will be given here.
The total potential satisfies an impermeable condition on the
~
instantaneous wetted body surface S ,
~
∇Φ (x ,t ) ⋅ N = U ⋅ N
on S
(2-11)
where N is the unit normal vector on the instantaneous body
surface and U is the velocity of the body given by
U = ξ& j (t )e j + η& (t ) + α& (t ) × x + H& x + L ( j = 1, 2 or 6) (2-12)
(
responses of the body just in the same way as in the linear wave
problems:
]}
(2-15b)
(2-15c)
The incident wave potential φ10 is specified in (2-5b). The
boundary conditions satisfied by the diffraction potential and
each radiation potential are simplified to:
∂φ1s ∂n = ns (s = 1 ~ 6),
∂φ17 ∂n = − ∂φ10 ∂n
(2-17)
with (n1 , n2 , n3 ) = n and (n4 , n5 , n6 ) = x × n
On the other hand, collecting terms associated with the time
factor of exp[ −i ( S j (t ) ± σt )] gives the boundary condition for
the interaction potentials φ 1(±j ) :
∂φ1(±j ) ∂n =
1
2
[iη ⋅ (− (n ⋅ ∇)w
0
+ δ j 6 2n × e 3 )
±
j
+ iα 0 ⋅ (− ( n ⋅ ∇)( x × w j )+ δ j 6 2 x × ( n × e 3 ) ±
]
+ ( −κ j Β0 + 2νΒ j(± ) ) ⋅ n
)
(2-18)
with w j = ∇φ0 j − e j ( j = 1, 2 or 6), Β j(± ) = η j(± ) + α j(± ) × x,
η j(± ) = (η j(1± ),η j(2± ) ,η j(3± ) ), α j(± ) = (η j(4± ) ,η j(5± ),η j(6± ) )
Here, the superscript ± following a function in parentheses on
the right-hand side of (2-18) denotes the function itself or its
complex conjugate respectively. According to this boundary
condition, the potential φ 1(±j ) is further divided into a diffraction
part and a radiation part in a similar way as the linear potential
φ1 but the form is more complicated:
[
φ (±j ) = ω g φ1(±j 7) + ∑ νη 0 sφ1(±js) + (− 12 κ jη0 s + νη js(± ) )φ1s
6
1
s =1
]
(2-19)
Then, the boundary condition for each potential component is
given by
∂φ1(±js) ∂n = 12 i (m js ) ± ν ( s = 1 ~ 6),
∂φ1(±j 7) ∂n = 0 ( j = 1, 2 or 6)
with ( m j1 , m j 2 , m j 3 ) = −( n ⋅ ∇) w j + δ j 6 2n × e 3 ,
(2-20)
( m j 4 , m j 5 , m j 6 ) = −(n ⋅ ∇ )( x × w j ) + δ j 6 2 x × ( n × e 3 )
For the potential φ2(0 ) , the boundary condition is obtained by
collecting terms of second order in wave amplitude and with a
time factor of e-iσt:
∂φ 2(0j ) ∂n = V jn
(2-21)
Here, the normal velocity Vjn on the mean body surface in (219e) is a complicated combination of lower order displacement,
potentials and their derivatives. Since we are interested in the
limiting case of σ to be asymptotically small, the expression of
it will be given later.
When the free surface condition is considered in the moving
frame, it is stated as follows on the exact elevation of the free
surface z=ζ(x, y, t):
The first order potential φ1 is further divided into an incident
wave potential φ10, a diffraction potential φ17 and radiation
potentials φ1s (s=1 ~ 6) corresponding to six modes of the linear
3
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~
∂ 2Φ ∂t 2 + g ∂Φ ∂z − 2ξ& j (t )∂ 2Φ ∂t∂x j − ξ&&j (t )∂Φ ∂x j
+ 2∇(∂Φ ∂t ) ⋅ ∇Φ − 2ξ& j (t )∇(∂Φ ∂x j )⋅ ∇Φ
+
1
2
(∇Φ ⋅ ∇ )(∇Φ ⋅ ∇Φ ) = 0
η js(± ) = 12 (ηˆ js ± ση js )
(2-22)
on z = ζ (x, y, t )
divided into a diffraction part and a radiation part as follows:
where the wave elevation ζ of the free surface is evaluated by
ζ = − 1 g ∂Φ ∂t − ξ& j (t )∂Φ ∂x j + 12 ∇Φ ⋅ ∇Φ z =ζ
(
)
= − 1 g [∂Φ ∂t − ξ& (t )∂Φ ∂x + ∇Φ ⋅ ∇Φ
− 1 g (∂Φ ∂t ∂ Φ ∂t∂z − ξ& (t )(∂Φ ∂t ∂ Φ ∂x ∂z
+ ∂Φ ∂x ∂ Φ ∂t∂z )] + O (Φ )
j
1
2
j
2
2
j
∂φ 2
∂z = f 2
∂φ 0 j ∂z − σ
2
∂φ1 j ∂z − (ω ± σ )
[
*
1 1 zz
}
g φ0 j = 0
(± )
(± )
{
where f 2 = Re − iω (2 g )φ φ
(0 )
2
(
±
j 1
2
2
2
∂φ 2 j ∂z − σ
2
(0 )
g ∂φ 0 j ∂z ) ± σ (2ω )
(0 )
g φ2 j = f 2 j
(2-24d)
)]
(2-24e)
For the same reason as the normal velocity on the body surface,
the forcing term f 2(0j ) on the right-hand side of (2-24e) will be
specified later for the vanishing low frequency σ.
3. LIMITING CASE OF ASYMPTOTICALLY SMALL σ.
As a first step of approximation, it is appropriate to consider the
limiting case when σ tends to zero. In this limiting case,
neglecting terms with σ2, the free surface condition (2-24c) for
the potential φ0j becomes a rigid wall condition:
∂φ 0 j ∂z = 0
z=0
( j = 1, 2 or 6)
(3-1)
Together with the boundary condition (2-15c) on the body
surface, it can be observed that this potential is equivalent to the
steady disturbance potential when the body is moving or
rotating constantly at unit velocity in calm water with a ‘rigid’
free surface. It should be noted that the solution of φ0j is a real
function and vanishes at the far field with an order of O(1/r).
Here, r is the radial coordinate. It is implied by the free surface
condition (3.1) that the imaginary part of φ0j is of order σ2.
Next, the asymptotic form of the potential φ1( ±j ) is considered.
By inspection of the free surface condition, this potential,
together with the corresponding part of the linear body
responses η js( ± ) , is further expanded into a series of σ:
φ (± ) = 1 (ψˆ ± σψ~ )
(3-2a)
1j
2
j
j
js
0s
(3-3c)
s
Substituting (3-3a, b, c) and (2-16) into (2-24d) and neglecting
terms of order σ2 or higher, the free surface condition for the
auxiliary potentials ψˆ j , ψ~ j and ψ~ 'j is given by
∂ψˆ js ∂z − νψˆ js
= −2i ∂φ1s ∂x j − 2κ jφ1s + 2i∇φ 0 j ⋅ ∇(φ1s + δ s 7φ10 )
− i (φ1s + δ s 7φ10 ) ∂ φ 0 j ∂z
∂z − νψ~
2
js
2
(3-4a)
on z = 0
js
−1
±
− i ∂φ1 ∂x j − i (φ 0 j ) ± (∂ 2φ1 ∂z 2 − ν ∂φ1 ∂z )
(0 )
with φ1(±js) = 12 ψˆ js ± 12 σ [ψ~ js + (1 − δ s 7 )( − κ j ν ψ~' s
+ η η ψ~' )]
( s = 1 ~ 7, j = 1, 2 or 6)
∂ψ~
g φ1 j = f 1 j
⋅ ∇(φ 0 j ) − 12 iφ1 ∂ φ 0 j ∂z − σ
(− κ φ
(2-24b)
(± )
f 1 j = ω g − i ∂φ1 ∂x j − κ jφ1 + i (ω ± σ ) ω ∇φ1
(3-3b)
6
~
+ ∑ [νη 0 sψ~ js + νη jsφ1s + ( −κ jη 0 s + νηˆ js )ψ~' s ]
(2-24c)
(± )
(3-3a)
s =1
s =1
3
z =0
(0 )
ψ~ j = ω gψ~ j 7
(2-23)
In (2-23), the expansion about the mean free surface, i.e. z=0,
has been made. In the same way, the free surface condition in
(2-22) is also expanded about the mean free surface and the
perturbation expansions are substituted into it to yield the free
surface condition for each order of potential:
∂φ1 ∂z − νφ1 = 0
(2-24a)
(0 )
6
ψˆ j = ω g ψˆ j 7 + ∑ [νη 0 sψˆ js + ( −κ jη0 s + νηˆ js )φ1s ]
j
2
j
Combined with the expansion (2-19), ψˆ j
(3-2b)
~
and ψ j are also
= ω [2νψˆ js − i ∂φ1s ∂x j − κ jφ1s + 2i∇φ0 j ⋅ ∇(φ1s + δ s 7φ10 ) (3-4b)
(
)]
(
− iφ0 j ∂ 2 (φ1s + δ s 7φ10 ) ∂z 2 − ν 2 (φ1s + δ s 7φ10 ) on z = 0
∂ψ~' s ∂z − νψ~' s = 2ω / gφ1s on z = 0
( s = 1 ~ 6)
(3-4c)
Here, the fact that φ0j is real has been used.
According to (2-20), the boundary condition for these auxiliary
potentials on the mean body surface S0 is readily obtained. The
auxiliary potentials ψ~ j and ψ~ 'j satisfy a homogeneous
boundary condition on the mean body surface while the body
condition for ψˆ j is given by
∂ψˆ js ∂n = im js ν
( s = 1 ~ 6),
(3-5)
∂ψˆ j 7 ∂n = 0 on S 0 ( j = 1, 2 or 6)
As mentioned previously, the inhomogeneous term f 2(0j ) on the
right-hand side of the free surface condition (2-24e) is a
combination of the lower order potentials φ0 j , φ1 , φ 2(0 ) and φ1(±j )
according to the order in wave amplitude and time dependence.
In the limiting case of asymptotically small σ, by dropping
terms of order σ and higher, this forcing term in the free surface
condition is expressed as:
f 2(0j ) = Re 1 (2 g ) iω φ1*ψˆ jzz − φ1*zzψˆ j − 3ν 2 (φ1*φ1x
{ (
)
j
− φ 1 ∇φ1 ⋅ ∇φ j 0 ) − 2ν φ φ φ j 0 zz + 2∇φ ⋅ ∇φ1 x
*
2
− φ φ1zz + iκ φ φ
*
1x j
*
j 1 1 zz
(
*
1
+ φ ∇φ1 ⋅ ∇φ j 0 − ∇φ
*
1 zz
(φ ∇φ ⋅ ∇φ
))+ δ (∇φ ⋅ ∇φ −
⋅ ∇(∇φ1 ⋅ ∇φ j 0 ) +
⋅ ∇ ∇φ1 ⋅ ∇φ1*
*
1 1
1
2
j 0 zz
j6
*
1
1
1
*
1
− ∇φ j 0
3
2
j
*
1
ν 2φ1φ1* )β
(3-6)
}
In deduction of (3-6), the free surface condition of lower order
potentials has been used.
4
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In a similar way, when σ tends to zero, the normal velocity Vjn
in the body surface condition (2-21) can be written as
V jn = Re 12 {− w j ⋅ H 0 n − n ⋅ H 0 x ⋅ ∇ + 12 ( B0 ⋅ ∇)
]
[
[
( B0* ⋅ ∇) (∇φ 0 j ) + n ⋅ ( − B0 ⋅ ∇(∇φ 0 j ) + iκ j B0 )
× α − i ( − B ⋅ ∇(∇ψˆ j ) − ∇ψˆ j × α 0* ) − iω g
*
0
*
0
(3-7)
}
1
solution satisfying a homogeneous free surface condition and
will eliminate the normal velocity caused by the other two parts
of the solution on the body surface.
When the solution for ψ~ js and φ 2(0j ) is considered, difficulties
arise from the free surface condition. It can be observed that the
forcing term in the free surface condition contains ψˆ js , which is
( Bˆ j ⋅ ∇(∇φ * ) + ∇φ * × α j ) + iν ( − Bˆ j × α 0* + B0* × α j )]
1
(3-4a) except the above ones. The third part, ψˆ gjs , is a general
with Bˆ j = B j(+ ) + B j(− ) , αˆ j = α j(+ ) + α j(− ) , B j(± ) = η j(± ) + α j(± ) × x
secular at far field as mentioned before. Because, ψˆ js satisfies
− η 05η 05* − η 06η 06*

H 0 = 12  η 04η 05* + η 05η 04*
 η 04η 06* + η 06η 04*

an inhomogeneous free surface condition itself. It is not easy to
find a derivative operator to obtain a special solution like the
method to obtain ψˆ jsp . If the special solution is expressed in an
0
*
− η 04η 04 − η 06η 06*
η 05η 06* + η 06η 05*

0

0

− η 04η 04* − η 05η 05* 
According to its boundary conditions given in (3-6) and (3-7),
the solution of the potential φ 2( 0j ) is obviously a real function.
4. SOLUTIONS OF THE POTENTIALS
In the present work, only the limiting case of asymptotically
small σ is considered. The double-body solution is applied to
the linear radiation potential φ0j of the low-frequency oscillation
since it satisfies a rigid wall condition (see 3-1) on the free
surface. As mentioned previously, its solution is real.
It is now a routine work to solve the wave potentials φ1 and φ 2( 0 )
either by means of eigen-function expansion or by boundary
element method with a proper Green function. It should be
mentioned here, the solution of the potential φ 2( 0 ) , is a real
function and is not wavelike according to the boundary
conditions. By using proper integral identities and boundary
conditions, the contribution from the potential φ 2( 0 ) to the
quadratic wave forces can be transferred to an integral
involving the forcing term f 2( 0 ) over the free surface.
Therefore, it is not necessary to solve φ 2( 0 ) explicitly.
Next the interaction potentials ψˆ js , ψ~ js , ψj1s and φ 2( 0j ) are
considered for asymptotically small σ. The solution for
ψˆ js consists of three parts:
ψˆ js = ψˆ jsp + ψˆ sjs + ψˆ gjs
( s = 1 ~ 7, j = 1, 2 or 6)
(4-1)
The first part of the solution, ψˆ , is obtained by a derivative
p
js
operator with respect to ν applied to the first two forcing terms
on the right side of (3-4a):
ψˆ jsp = −2(i ∂ ∂x j + κ j ) ∂φ1s ∂ν
(4-2)
integral over the free surface by using a proper Green function,
the integral might be divergent due to the property of ψˆ js .
Therefore, the problems for ψ~ and φ ( 0 ) remain unsolved. We
5. CALCULATION OF THE WAVE LOADS
Once the potentials are solved, the hydrodynamic pressure p
can be obtained by the Bernoulli equation and the wave forces
are evaluated by the integration of the hydrodynamic pressure
along the instantaneous wetted body surface.
F (t ) = − ρ ∂Φ ∂t − ξ& (t ) ∂Φ ∂x + 1 ∇Φ ⋅ ∇Φ + gz Nds (5-1)
∫ (
~
S0
over the free surface involving the remaining forcing terms in
j
j
)
2
where ρ is the density of the fluid. Then the integral is
transferred to the mean position of the wetted body surface S0.
This will increase terms in the integrand caused by the deviation
of the instantaneous wetted body surface from its mean position
and the rotation of the normal vector N. For example, the
pressure is expanded as:
2
p S~ = p S + B ⋅ ∇p + 12 [B ⋅ ∇] p + L
(5-2)
0
0
In addition, the contribution from the wave elevation at the
intersection of the body surface and the free surface is expanded
to an integral along the mean water line C0 and an integral in the
vertical direction from mean water surface to the relative wave
elevation defined as
ζ r = ζ − B3
(5-3)
with B3 = (η 3 (t ) + η 4 (t ) y − η 5 (t ) x )
where wave elevation ζ is given in (2-23).
Summing up all these contributions, (5-1) becomes:
F (t ) = − ρ ∫ ∂Φ ∂t − ξ j (t )∂Φ ∂x j + 12 ∇Φ ⋅ ∇Φ + gz
S0
[
+ B ⋅ ∇(∂Φ ∂t − ξ j (t )∂Φ ∂x j
+ 12 ∇Φ ⋅ ∇Φ + gz )]( n + α × n + Hn) ni ds
It satisfies an inhomogeneous free surface condition with a
corresponding forcing term of −2(i ∂ ∂x j + κ j )φ1s . It should be
noticed that this part of solution is secular at far field. Hence it
is only valid near the body. By means of the Green function for
the problem of pressure distribution on the free surface, the
second part, ψˆ sjs , of the solution can be expressed in an integral
2j
js
are still working on them.
+ 12 ρ g ∫
C0
[(∂Φ ∂t (∂Φ ∂t + 2 gB )
3
g + gB32
)
(5-4)
( g − ∂ Φ ∂t∂z ) + (∂Φ ∂t + gB3 )(∇Φ ⋅ ∇Φ
− 2ξ& j ∂Φ ∂x j + 2 B ⋅ ∇ ∂Φ ∂t ) ( n + α × n)dl
2
]
where wave elevation expressed in (2-23) has been applied to
the line integral.
The wave loads are then expanded in the same way as the
5
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velocity potential with two time scales, i.e.
Fi (t ) = Re F1i e −iωt + F2(i0 ) + ... + σξ j [ F0ij e −iσt
{
(+ )
+ F1ij e
− i (ω + σ )t
(− )
+ F1ij e
− i (ω −σ )t
(0 )
+ F2ij e
− iσt
6
~
~
~
F1 ji = X ji + ∑ [iνη 0 s ( µ~ jis + iλ jis ω )
}
+ ...]
s =1
(5-5)
~
+ i ( −κ jη 0 s + νηˆ js + ων η js )( µ 0is + iλ0is ω )
~
+ i ( −κ η + νηˆ )( µ~ ' + iλ ' ω )]
In, (5-5), i=1~6 for the linear loads in wave amplitude, i.e. F1i
and F1(ij± ) . Otherwise, only the horizontal components, i.e. i=1,
2 or 6, are considered. The expansion of the potential shown in
(2-4) is substituted into (5-4). By reorganizing terms, forces at
each order can be obtained.
The first term in (5-5) denotes the linear wave loads including
wave exciting forces X0i, conventional added mass µ0is and
damping λ0is for the linear responses of the body. The second
term represents the wave drift forces. These terms are well
known and further discussion will be omitted.
We are interested in the force components associated with low
frequency σ. The first one, F0ij is the linear force in i-th
direction per unit motion of ξj and is related to the linear added
mass A0ij and wave-radiating damping B0ij for the low-frequency
oscillations:
F0ij = −(− iσA0ij + B0ij ) = iσρ ∫ φ0 j ni ds
(5-6)
S0
In the limiting case that σ tends to zero, the linear waveradiating damping B0ij vanishes while the added mass tends to:
A0ij = ρ ∫ φ 0 j ni ds
(5-7)
S0
Collecting
terms
exp[ −i ( S j (t ) ± σt )] ,
associated
with
time
factor
force components relating to the
interaction between linear high-frequency responses and lowfrequency oscillations can be obtained:
F1(ji± ) = − ρ ∫ [ −i (ω ± σ )φ1( j± ) − 12 ∂φ1 ∂x j
S0
+ κ jφ1 + 12 (∇φ1 m iσB ) ⋅ ∇(φ0 j ) ± ]ni ds
i
2
± 2i ei ⋅ [α0 × ρσ ∫ (φ0 j ) ± nds ]
(5-8)
S0
± ρσ ∫ (iω g φ1 − B03 )(φ0 j ) ± ni dl
i
2
C0
with B03 = η03 + η04 y − η05 x
In the limiting case of asymptotically small σ, the force
component is expanded into a series of σ just like the way we
did for the potentials:
~
F1(ji± ) = Fˆ1 fi ± σF1 fi
(5-9)
j
[
Fˆ1 ji = Xˆ ji + ∑ iων η 0 s ( µˆ jis + iλˆ jis ω )
s =1
+ iω ( −κ jη 0 s + νηˆ js )( µ 0is + iλ0is ω )
]
js
jis
jis
The first term in (5-10a) and (5-10b) is analogous to the wave
exciting force in the linear wave problems and expressed as
Xˆ ji = ρ ∫ [iνψˆ j 7 + (∂ ∂x j − iκ j − ∇φ 0 j ⋅ ∇)
S
(5-11a)
(φ10 + φ17 )]ni ds
~
X = iρ [ (ω gψˆ + νψ~ + B ⋅ ∇φ )n ds
0
∫S
ji
j7
0
j7
0
0j
i
(
(5-
)
+ ei ⋅ ( α0 × ∫ φ0 j nds ) + ∫ iω g (φ10 + φ17 ) − B03 φ0 j ni dl ]
S0
C0
11b)
The remaining terms take a form of added mass and damping
but they are related to the interaction between the highfrequency responses and low-frequency oscillations. They are
calculated by the following integrals:
µˆ jis + λˆ jis ω
(5-12a)
= − iρ ν ∫ [iνψˆ js + (∂ ∂x j − iκ j − ∇φ 0 j ⋅ ∇)φ1s ]ni ds
S0
~
µ~ jis + iλ jis ω = ρ ∫ (ψˆ js + ωψ~ js )ni ds + iρ ∫ φ1sφ0 j ni dl
(5-12b)
~
µ~ jis' + iλ jis' ω = ρω ∫ ψ~ 'js ni ds
(5-12c)
S0
C0
S0
The last term in (5-5), F2ij(0 ) , is a force component in quadratic
order of wave amplitude, which can be separated into two parts
that is in phase with the acceleration and the velocity of the
low-frequency oscillation respectively, i.e.
F2(ij0 ) = −(− iσA2ij + B2ij )
(5-13)
The real part of it is involved in the calculation of the wavedrift damping B2ij, which is well discussed in the previous
works. We are interested in the imaginary part of the force
component F2ij(0 ) , which gives the wave drift added mass. In the
limiting case of vanishing σ, it can be calculated by:
A2ij = 14 [ −(η 05η 05* + η 06η 06* )( L x + δ 1i A01 j ) + (η 04η 05*
+ η 05η 04* ) ( L y + δ 2i A01 j ) − (η 04η 04* + η 06η 06* )( L z
+ δ 2i A02 j ) + (η 04η 06* + η 06η 04* ) L x
+ (η 05η 06* + η 06η 05* ) L y − (η 04η 04* + η 05η 05* ) Lz ]
~
~
+ Im{ε ism (η 0*( s + 3) F1 jm + ω g η j ( s + 3) F1*m )
− 12 ρ ∫ [−iφ 2(0j ) + ∇ψ~ j ⋅ ∇φ1* − iB0* ⋅ ∇(ψˆ j + ωψ~ j )
Substituting (3-2), (3-3) into (5-8) and rearranging terms in a
similar way as in the linear wave problem, we have
6
0s
(5-10b)
(5-10a)
(5-14)
S0
~
ρ
+ iνB1 j ⋅ ∇φ1* ]ni ds +
2g
~
∫C [(ψˆ j + ωψ j )(ωφ 1 + igB03 )
*
*
0
~
+ iν 2φ 0 jφ1φ1* − 2i φ 0 j ∇φ1 ⋅ ∇φ1* + gB j 3 (iνφ1* + ωB03* )
+ ωφ 0 j (νB03 + B0 ⋅ ∇)φ1* − igB03* B0 ⋅ ∇φ 0 j ]n i dl}
where
6
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~
~
~
~
~
~
~
B j 3 = η j 3 + η j 4 y − η j 5 x, B1 j = η1 j + α j × x,
~
~ ~ ~
~
~ ~ ~
η1 j = (η j1 ,η j 2 ,η j 3 ), α j = (η j 4 ,η j 5 ,η j 6 )
misc is the mass coefficient for the Coriolis force. This
Lx = ρ ∫ x ∂φ0 j ∂x ni ds, Ly = ρ ∫ y∂φ0 j ∂y ni ds,
S0
S0
Lz = ρ ∫ z ∂φ0 j ∂z ni ds
S0
It can be observed that the higher order potential ψ~ j and φ 2(0j )2
make a contribution to the wave-drift added mass, which was
missed in our previous work.
6. EQUATION OF MOTIONS
The remaining unknowns are the amplitude of the linear
response of the body to the incident waves. They are
determined from the equation of motions.
According to the Newton’s law, the movements of the origin of
the moving frame are governed by the following equations.
M [η&&(t ) + α&&(t ) × xG + α& (t ) × (α& (t ) × xG )]
= F − (δ + δ ) Mξ&& (t )e − δ M [ξ&& (t )e × x + ξ& (t )e (6-1a)
1j
2j
j
j
6j
6
3
G
6
3
× (ξ&6 (t )e3 × xG ) + 2ξ&6 (t )e3 × ( η& (t ) + α& (t ) × xG )]
Iα&&(t ) + α& (t ) × Iα& (t ) + MxG × η&&(t )
= K − (δ + δ ) Mx × ξ&& e − δ [ Iξ&& (t )e
1j
2j
G
j
6j
j
6
3
(6-1b)
+ ξ&6 e 3 × Iξ&6 (t )e 3 + MxG × ( 2ξ&6 (t )e 3 × η& (t ))
+ ξ&6 (t )e 3 × Iα& (t ) + α& (t ) × Iξ&6 (t )e 3 + I (ξ&6 (t )e 3 × α& (t ))]
where j=1, 2 or 6. In (6-1), xG is the position vector of the
gravity center. M denotes the mass of the body and I represents
the tensor of inertia moment about the origin of the coordinate
system. F and K are force and moment vectors, in which
restoring forces and moments supplied by the hydrostatic
pressure are also included in addition to those components
calculated in the previous section. The Coriolis force is also
included in these two equations.
Substituting expansions for the high-frequency responses and
forces, it is readily obtained the equation for the linear
amplitude η0 s (s=1~6), which is just the same as the
conventional linear motion equation for floating bodies.
The equation for the motion amplitude η js( ± ) (s=1~6 and j=1, 2
or 6) is obtained by picking up terms with corresponding time
factor exp[ −i ( S j (t ) ± οt )] . We are interested in the limiting case
of small. Using the expansion of (3-2b), the motion equation
can be expressed as:
6
ω g ∑ i ( −ω 2mis + ciz )ηˆ js
s =1
6
= Fˆiji − 2∑ (iωmisκ jη0 s + δ 6 jωmiscη0 s )
(6-2a)
s =1
6
~
~
6
ω g ∑ i ( −ω 2 mis + ciz )η js = Fiji + ∑ 2iνmisηˆ js
s =1
(6-2b)
s =1
where mis is the component of the conventional mass matrix and
coefficient matrix is defined as:
−M
0 Mz G
 0
 M
0
0
0


0
0
0
0
Mc = 
−
Mz
0
0
0
G

 0
− Mz G 0 I z

My G 0 I 23
 Mx G
0
Mz G
0
− Iz
0
− I 13
− Mx G 
− My G 

0 

− I 23 
I 13 

0 
(6-3)
with Iz=1/2(I11+I22-I33).
Once the amplitudes of the high-frequency responses are
solved, they are substituted into (5-14) to calculate the wave
drift added mass or the other quadratic quantities in wave
amplitude.
7. DISCUSION
The interaction of the low-frequency oscillations with both the
diffraction and radiation wave fields is considered in the present
work based on the assumption that the frequency of the lowfrequency oscillation is much smaller than that of the incident
waves. Two time scales are adopted to describe these two kinds
of motions. In the limiting case, the velocity of the lowfrequency oscillations is also asymptotically small. It is used as
another perturbation parameter in addition to the wave
amplitude. In that sense, acceleration of the slow oscillations is
a second order quantity. To evaluate wave drift added mass, the
interaction terms, including potentials and wave loads as well as
the high-frequency responses affected by the low-frequency
oscillations, should be further expanded to the order of the lowfrequency acceleration. This is not a surprising result since the
wave drift added mass is linearly related to the acceleration of
the low-frequency oscillations although it is a quadratic force
component in wave amplitude.
Considering the complexity of the analysis, a numerical analysis
of fully nonlinear wave-body interaction might be a direct
method to deal with this problem. Nevertheless, the present
work gives some physical understanding of various
hydrodynamic mechanisms.
To compare with the calculated results, experiments are
performed in a towing tank, which is 54 meter long and 10
meter wide with a depth of 2 meters. The model is an array of
four cylinders. The radius of cylinder, a, is equal to 0.125 meter
and the draft changes from a to 3a. The cylinders are located at
the corners of a rectangular, with a length of 10a and a width of
5a. As shown in Fig. 1, the models are hung up by four wires,
which have an average length of about 4.5 meters. The length of
the wires is adjusted according to the draft of the models. The
weight of the wires is negligible when the natural frequency of
the whole system is evaluated. No other mooring device is used
in the experiments. Both the free decay test (denoted as FD) and
the forced oscillation test (designated as FO) are performed. In
the FD tests, the models are disconnected from the carriage and
the forces acting on the models are not measured. Only the
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Fig. 1 Setting-up of experiment equipments.
10
FO,d=2a,σ=0.929rad/s
FO,d=1a,σ=1.137rad/s
FD,d=1a
cal.,d=2a
cal.,d=1a
8
a
2
6
4
2
w
M /4ρπaζ
displacement of the models is measured by an optical position
sensor system. When FO tests are performed, the models are
connected to the carriage through soft springs and a pair of
cantilever load cells, which are used to measure the force acting
on the models. The carriage is driven to move along rails by a
servomotor so that it leads the models to oscillate slowly in
surge direction. Both the displacement and the forces are
measured in the FO tests. The frequency of the forced lowfrequency oscillation is set to be a little bit higher than the
natural frequency of the whole test system.
One of the experimental examples, i.e. the wave drift added
mass Mw normalized by Nρπaζa2 is plotted against wave
number k0L in Fig.2. Here N is the total number of cylinders.
The wave number is nondimensionalized by the longitudinal
distance L between cylinders. The frequency of the forced
oscillation σ =0.929 rad/sec when the draft d =2a and σ =1.137
rad/sec when d =a. Results of FD tests are also presented in this
figure. The agreement between FD and FO tests is good.
Calculated results, represented by lines, are shown in the figure
to compare with experimental ones. Only the interaction
between the low-frequency oscillation and the diffraction wave
field is considered in the calculation. It can be seen that they
agree fairly well with each other in general tendency although
departure between these two results can also be observed. Since
the linear response of the models to the incident waves and the
contribution of higher order potentials are not included in the
calculation as mentioned in the previous section, the difference
between these two results is expectable. Further work is needed
to include the contribution of the radiation wave field and
potentials related to the acceleration of the low-frequency
motions.
REFERENCES
1. Kinoshita T., Takaiwa K. (1990) Added mass increase due to
waves for slow drift oscillation of a moored semi-submersible,
Proc. of OMAE 1990, Houston.
2. Kinoshita T., Shoji K., Obama H. (1992) Low frequency
added mass of a semi-submersible influenced by incident
waves. Proc. of ISOPE 1992, San Francisco.
3. Bao W., Kinoshita T. (2001) Wave-drift added mass of a
cylinder array slowly oscillating in waves, Proc. of OMAE
2001 Rio de Nero.
4. Newman J.N. (1993) Wave-drift damping of floating bodies.
J. Fluid Mech. Vol.249: 241-259.
5. Ogilvie T.F. (1983) Second-order hydrodynamic effects on
ocean platforms. Proc. of International Workshop On Ship And
Platform Motions. P205-265.
0
-2
-4
-6
1
2
3
4
5
6
k L
7
8
9
10
11
0
Fig. 2 The wave drift added mass of a four-cylinder
array.
8
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