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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use { [ 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ~ ∂ 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ~ ~ ~ ~ ~ ~ ~ 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 7 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright 2002 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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