On two-dimensional moonpool resonance for twin bodies in a two-layer fluid

Applied Ocean Research 40 (2013) 1-13

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Applied Ocean Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o n n / l o c a t e / a p o r

On two-dimensional moonpool resonance for twin bodies in a two-layer fluid

Xinshu Zhang', Piotr Bandyk

Departmem of Naval Architecture and [Marine Engineering. University of Michigan. Ann Arbor. MI 48109, United States

A R T I C L E I N F O A B S T R A C T

Article history:

Received 6 November 2011

Received in revised form 6 November 2012 Accepted 8 November 2012 Keywords: Moonpool Resonance Surface wave Internal wave Hydrodynamic coefficients

This paper studies the moonpool resonance of two heaving rectangular bodies in a t w o l a y e r fluid. A m a t h e -matical model is proposed based on an eigenfunction matching approach. The motion of the t w o - d i m e n s i o n a l bodies is a s s u m e d to be vertical and h a r m o n i c . Heave added m a s s and d a m p i n g coefficients are computed to examine the h y d r o d y n a m i c behavior of the t w i n bodies. The free surface and internal w a v e elevations are obtained near the resonant frequencies. T h e presented results and analyses reveal that there exist both Helmholtz and higher-order resonances in the t w o - l a y e r fluid system, w h i c h is s i m i l a r to the single-layer fluid case. It is also found that the resonances are closely associated w i t h the free surface elevation inside moonpool gap, not the w a v e elevation at the interfacial surface. In addition, p a r a m e t r i c studies have been performed to identify the dependencies of h y d r o d y n a m i c behavior on geometry and density stratification.

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1. Introduction

This paper aims to study the moonpool wave resonance phe-nomenon in a two-layer fluid system due to the oscillating heave motion of two identical rectangular bodies (or called twin bodies). The moonpool studied in the present paper models the opening/gap between two floating bodies, such as a liquified natural gas (ING) carrier and LNG terminal in the case of side-by-side arrangements, or between the individual hulls of a multi-hull vessel. In offshore opera-tion, the fluid motion inside the moonpool and the dynamic behavior of the hull are very critical to the design and analysis of ships, and also very important to the development of an efficient and reliable methodology or procedure for offshore operations such as pipeline laying and equipments/cargo transferring. Moreover, the motion dy-namics of the LNG ship during offloading is also crucial for tank slosh-ing analysis of the liquified gas. Molin [1 ] and Mclver [2] demonstrated a resonance inside the moonpool using linearized water wave theory. Yeung and Sean [3] studied moonpool resonance for two symmetric rectangular bodies in finite water depth by using an eigenfunction matching method. Faltinsen et al. [4] studied the two-dimensional wave sloshing inside a moonpool based on a domain decomposition scheme and the Galerkin method, and compared numerical results with experiments. However, most of the previous moonpool hydro-dynamic studies are focused only on a single-layer fluid; very few studies have been performed for a two-layer fluid case, which is very common in the ocean environment.

When the wave-structure interacdon is treated using potential

' Corresponding author. Tel.: +1 281 721 2369.

E-mall address: xinshuz@umich.edu (X. Zhang).

0141-1187/S - see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.apor.2012.ll.004

theory, the boundary value problem may have an eigenvalue to sat-isfy the condition of no radiated waves exisring at infinity. The corre-sponding eigenfuncdon is called a 'trapping structure'. At the trapped mode frequency, the added mass coefficients can be infinite. Much re-search has been conducted to invesdgate on the possibility of 'wave trapping' phenomena for both fixed and floadng bodies. The so-called 'trapped mode' has been researched by Mclver [5-7] and Kuznetsov et al. [8] for two-dimensional and three-dimensional cases, respec-rively. Newman [9] also studied the trapped wave resonance in a floating torus using WAMIT. Kuznetsov'et al. [10] investigated the wave trapping modes for two-dimensional bodies in a two-layer fluid. However, the existence of wave trapping may be a rare exception for actual moonpools, including the presently studied dual heaving rect-angular bodies.

Linton and Mclver [11] studied the wave radiation and scattering of a horizontal cylinder in a two-layer fluid. Yeung and Nguyen [12,13] derived a Green function for a steadily translating source and a two-dimensional transient Green function for an oscillating source. The latter can be used for the computation of wave-structure interaction in the time domain. Alam et al. [14] investigated using the three-dimensional Green function for an oscillation source translating with steady speed and compared the far-field radiated waves w i t h results using direct numerical simulation.

An eigenfunction matching method is developed in the present study to solve the boundary value problem for a two-layer fluid sys-tem. The eigenfunction matching method has been widely used to study wave-body interaction problems (Yeung [15], Shipway and Evans [16]) for either two-dimensional or three-dimensional cases. Mavrakos [17] also applied the method to compute the hydrody-namic coefficients for two concentric surface-piercing truncated cir-cular cylinders. More recently, Mavrakos and Chatiigeorgiou [18] used the method to study the second-order wave diffraction problem for

2 X. Zhang. P. Bandyk /Applied Ocean Research 40(2013) 1-13

li

Fig. 1. Definition of the problem and coordinate systems.

two concentric circular cylinders.

In the present paper, we focus on the wave radiation due to heave excitations of the twin bodies in the upper-layer fluid. We first give the mathematical formulation, followed by a numerical scheme proposed to solve the discretized linear system. The hydrodynamic behavior near resonant modes is examined. The outer region radiated free surface and internal waves are computed and discussed. In order to identify the effects of geometry and fluid stratification on the resonant fluid motion, parametnc studies are performed for both intermediate-draft and shallow-intermediate-draft floating bodies.

2. Mathematical formulation

The surface and internal waves caused by small amplitude heave motion of two rectangular hulls with identical geometry in a two-layer fluid are studied. The problem sketch is shown in Fig. 1. The width of each rectangular body is 26. The distance between the two centers of the bodies is 2W. The draft of each floating cylinder is d. The depth of upper and lower fluid is / i j and hj, respectively, w i t h a total water depth h = hi + \\2. The fluid is assumed to be ideal and the flow irrotational. The fluid density for the upper and lower fluid is P I and P2. respectively. Since only heave motion is studied here, the hydrodynamic problem is symmetric aboutx = 0 and only the domain X > 0 is considered. Note that the present study only considers the case of d < h i , which assumes the twin bodies do not penetrate the interfacial surface at z = - h i .

Let the heave motion of the two identical bodies be f cos(cut), where w is the angular frequency, and f is the motion amplitude. We assume ^ « 0(1). The velocity potential within the two-layer fluid can be written as

(1)

where (?i("')(x,z) is the spatial velocity potential, m = 1 represents the solution in the upper fluid and m = 2 represents the solution in the lower fluid. The governing equation for (/>("'' is the Laplace equation:

;)x2 ;)z2 ' (2)

The linearized free surface boundary condition is written as

^ n ) - ; < 0 ( i ) = O a t z = 0 (3) where Y. = co^/g, and g is the gravitadonal acceleration. (j)z = i)4>/'óz

The boundary conditions at the interfacial surface can be written as (see [13] and (19]):

at Z :

Y {4'^ - l(4>0)) = .^P> - at z = - h i where y = p\/pi is the density ratio.

(4)

(5)

Because only synchronized piston type motion of the twin bodies is considered, the hydrodynamic problem is symmetric about the z-axis, implying:

X = 0, z) = 0

The boundary condition at the seabed is given by z'^' = 0 a t z = - h i - h 2 = - h

(6)

(7)

In addition, (/>("'' must satisfy the no-flux condition on the solid body boundaries:

<^S^^ = Q a t x = Wq:B, - d < z < 0 (8)

The body boundary condition at the bottom of the body gives:

( ^ r ' ( W - B < x < W + B , z = - d ) = 1 (9)

2.\. Division of computational domain

The right side of the fluid domain (x > 0) can be decomposed into three rectangular sub-domains which include I, II, and III, as shown in Fig. 1. The velocity potentials in those regions are denoted by (?!)'""'' and </)'"("'', respectively. </>"('"> is velocity potential of the subdomain underneath the body. Hence, the fluid within region II should satisfy the following governing equation:

w i t h the following boundary conditions: ! . i ' ( ' ' = l for W - B < x < W - h B , z = - d i ' ' ( 2 ) = 0 for W - B < x < M / - h B , z = - h

(10)

(11) (12)

The velocity potential <;l)''"'' and ^A'"'"'' for regions I and III should satisfy:

,;.j/;(m) , -;,Hi(m).

with the following boundary conditions: ! , i " ' ( i ' - K , / . ' " ' 0 ) = o a t z = 0 !.,' " " 2 ) = 0 at z = - h ^i.wi(m) ^ ^ i / ( m ) at X = W T fi. - h < z < - d !>,i " " " " = at X = W T B, - h < z < - d ! ) i - " ' ' " " = 0 a t x = W q = B , - d < z < 0 (13) (14) (15) (16) (17) (18)

Eqs. (16) and (17) are needed to ensure the continuity of the fluid velocity and potential at the common boundary of the neighboring regions. The symmetry condition about z-axis at x = 0 for region I gives:

0 i ' ' " ' = O f o r x = 0 , - h < z < 0 (19)

In region III, the radiation condition for the wave elevation implies that:

where Y(x,t) is the outgoing free surface wave elevation. Note that all the solutions in the three regions have to be matched at the ad-joining boundaries.

2.2. Solution in subdomain I

The velocity potential in region 1 can be written as

<j>'^'"\x, z) = £ B]A'{I<0\x)Z('"\li<-/\z)+

X. Zhang. P. Bandyk/Applied Ocean Research 40(2013) 1-13

The eigenfunction in the vertical direction Zt^^J is written as

j=0 Y,B]K'{i<f.x)Z^<"\iif\z)

j=o

(21)

where A' and Z^"") are the eigenfunctions in the horizontal and vertical directions, respectively Bj and B? are the coefficients which will be determined by matching fluid velocity and potential at the juncture boundary between regions I and II. fej"', J = 0,1 oo are the eigen-values which will be calculated by solving the dispersion relations (22) and (26).

For J = 0, the dispersion relation can be shown to be

g 2(1-Fy tits) " f l + t 2 - f ( - l ) " + ' v/(ti - h t 2 ) ^ - 4 e t i t 2 ( l - l - y t , t 2 ;

(22)

where n = 1 represents the dispersion relation for surface wave mode and n = 2 represents dispersion relation for internal wave mode. Yeung and Nguyen [13] derived the dispersion relation shown in (22) for outgoing waves using the method of separation of vanables.

with 6 = l - y

tl = tanh(fe/!i)

t2 =tanh(;</i2)

For J > 1, the dispersion relation can be shown to be

(23) (24) (25) g 2 ( l - y t i t 3 with tl =tan(fchi) f2 = tan(;</]2) ; [ t i + t 2 - K - l - 4 e f i t 2 ( l - y t i f j ) (26) (27) (28) 2 ( ' ) ( e z ) ^ sinh fc^")z+cosh/<[,"'z §><[) ail^ li") ,(11) ^ ^ s i n fcfz+cos / . f ' z for J = 0 for j > 1 (31) co^ coshl<^"'{z+li) gfe<")sinh/4'')h2 w2 COS fe^"'(Z-t-/l) g f e f s i n f c f / i 2 for j = 0 for j > 1 (32)

where a;(fej"') is the amplitude ratio defined by

cosh (/<("5hi) ( l - ^ tanh/<("'/,i ) for j = 0 .(")

5 ( ' < f ' > i ) f - l - ^ t a n ; 4 % ] f o r j > l (33)

The eigenfunction in the horizontal direction A' for region 1 is written as \ ' ( / < f , x ) = cos cos ( A ^ " ' ( W - B ) ) c o s h ( f c f x ) cosh {kf\w~B)) for J = 0 for j > 1 (34)

2.3. Solution in subdomain III

The solution in region III can be written as

0"'('")(x, z) = £ ^ ] A " ' ( ; < ( " , X)Z("')(/<('), Z ) + J=0

'£AjA"'{l<f\x)Z('"\k^j .(2)

(35) J=0

A] and Aj are the coefficients which w i l l be determined by matching

the boundary condition at the juncture boundary between regions II and HI.

A"' is the eigenfunction in the horizontal direcrion for region III,

which can be shown to be

g-/<5"'(MW+B))f^^ . ^ ^ (36)

By selecting y = 1, the two-layer fluid system becomes a single-layer fluid, and the dispersion relation (22) reduces to

=/s:tanh (/</!), a>| = 0; (29)

and the other dispersion relation (26) reduces to:

^ = -fetan(/<h), wj = 0: (30)

which are consistent with the dispersion relations for single-layer fluid system (see Yeung and Sean [3] and Wehausen and Laitone [20]).

2.4. Solution in subdomain II

The solution in region II can be decomposed into a homogeneous solution (f)'"^ and a particular solution (/)"f, which is constructed to satisfy the inhomogeneous boundary condition described in Eq. (9). Thus the total solution tp" is written as

The homogeneous solution can be expressed as: cc

(l>"''{x.z) = ^H(Xi,x)Y{Xi.z) (38)

X. Zhang. P. Bandyk/Applied Ocean Research 40 (2013) 1-13 w i t h HiXi.X): where Co + Do x-W B for ! = 0 cosHx-W)^ sinXiix-W) ^^^.^^ cos Xi B sm Ai B

^ cosh Ai ( x - W ) ^ sinh Ai(x - W) . ^ ^ cosh Ai B sinh Ai B

where Ai are the eigenvalues; AQ = 0 and A,- for i > 1 are obtained by solving the following dispersion relation (40) and (41):

(40) (41) Yh + Tl ^ 2 , - ( l - y fori ^ 2 w i t h „ ^. , _ I coth Ai (fii - d) f o r i = 1 i i ( A i , x ) _ I cotAi(hi - d ) f o r i > 2 T n v ^ - c o t h ( A i h 2 ) f o r i = l i 2 ( A i , x ) - cot(Aifi2) f o r i > 2 (42) (43)

The eigenfunction in the vertical direction V'(Ai, z) is wntten as For 1 = 0: Y{Xi.z Fori= YiXi.z F o r i > 2: y(Ai.z) = 1 for - /ll < z < - d Y for - /i < z < - / l l cosh Ai(z + d) sinhAi(d - / i i ) cosh Ai(z + /?) sinhAi/i2 f o r - / ! i < z < - d for - /i < z < - / l l (44) (45) cosAi(z + d) sin Ai(d - hi) cos Xi{z+h)

f o r - / i i < z < - d f o r - / ] < z < - / i i

(46) sinAi/i2

The particular solution (t>"P subject to boundary conditions (7) and (9) can be shown to be 1 2S for 4>"P{x,z) = p -y for with /3 = [ l - ( ^ " y)h:

s

By matching the velocity potential at x = W - B and x = W + B, and using the orthogonality condition for Yu we obtain:

(50) (51) I V - B Ci - Di = ^ [ < <^'. Vi > - < Yi > ] at X Di = [ < (p"'. Yi>-< cp"". Yi > ] at X = W + -. f '^w(z)Y\Xi.z)dz-. J -h

y^hi + ylhi-d) for i = 0

/i2 + sinh(2Ai/i2)/2A,- ^ 2sinh^Ai/i2 /ll - d - s i n h ( 2 A i ( d - / n ) ) / 2 A i . ^ ^ ^ 2 s i n h 2 A i ( d - / i i ) /i2 + sin(2Ai/i2)/2Ai ^ 2sin2Ai/i2 / l l - d - s i n ( 2 A i ( d - / i i ) ) / 2 A i y -(52) for ! > 2 2 s i n 2 A i ( d - / i i )

with the weight function w(z) defined by

w(z)

_ I

(53)

and the first inner product at the right side of Eqs. (50) and (51) calculated by ' . y i > •• j ^ w{z)cl) {W-B.z)Yi(^Xi.z)dz :Y,B}A'{lé/\W-B]L^^h (54) J2Bj^'{kf\W-B)L\j j = 0 (2)

•kin _{Yi>= j |'w(z)<^"'(W+ B.z)Yi{Xi.z)dz}

= Y,A]A'"{kf\w+B)L^^^+

f ] A ] A " ' ( / < f , W + B ) L ( ; ' j = 0

The coefficients L\p are written as

(55)

: ƒ ''w(z)Z("" (/</"', z ) y ( A i , z ) d z , n = l , : (56)

w i t h the detailed expression of L^^^ is given inAppendix A. By matching the fluid velocity at x = W - B, it yields:

f^B]A''{k^/\x)Z^"Hk\'\z) k=0

+ Y,B]A'\kf.x)Z^"%kf.z) = f ] H ' ( A i , x ) y ( X i , z ) + <^;^'P(x,z)

i=0

By using the orthogonality of Z("''(/<^-^', z), we obtain:

B]A'\k^P.x)M^P

= r \ z ] V H'( Ai. x)y(Xi, z)Z(""(/<yz) dz +|"||w(z)</>i'P(x,z)Z('")(/<y',z)dz

= f ; H ' ( A i . x ) L y ' + f'w{z)4>'i'{x.z)Z^"\k^p.z)dz 1=0

(57)

X. Zhang, P. Bandyk/AppUed Ocean Research 40 (2013) 1-13

where the derivative of eigenfunction H for region II can be written in: as H'{Xi,W-B)--H ' ( A i , W + B ) = with A'\kf\W-B): Do - f for I = 0

Ciki tanXjB + D,A; cotA,B for / = 1 (59) -CiXi tanhXiB + D,Xi coth A,B for i > 2

for I = 0

-CiXi tanA,B + DjA,- cotX,B for i = 1 (60)

C,>.( tanhAfB + D,Ai cothAfB for i > 2

2(g;<W)' (sinh 1^%)

h2 + 21^ (n)

ft)" / / I , sinh2/<:^"'/)i

hi_ sinh 2/<[)"^/ii

^ 2 + 4/<('"

,(")

-/(<"^tan(fc|,"'(M/~B))for i = 0

kfhanhikf\w-B)) for i > 1 (61)

Taking the derivative of Eq. (47) yields:

( X , Z ) : 1 -( i - y ) / i 2 -x/S f o r - / ) i < z < - d - y x / S f o r ~ h < z < -h] (62) 2g(/4"')" w ' ' 1 ,-(i-cosh2;<[,"'h,) for j = 0 2(g;<f^)' (sin kf^hj , y h2 + . 2M"'/l2 2k' .(") (66) ft,4 / /!, sin 2kt">/i, ,(„).2 2 4j^(n) h, sin 2/<("'h, H — H ^ 2 + 4,,(n) ft)" r ( l - c o s 2 f c f / ! , ) 2g{kf)

By matching the velocity at x = W + B, it yields:

for j > 1

Therefore, the second term on the right side of Eq. (58) can be ^ A}A"''{kf\ x)Z'-'"\kf\ z)+

written as k=o r d o TAjA"''{kf\x)Z^"'\kf\z)^ jyz)r^^Z^^\lf.z)üz = J-^ ' w'^jzoshkfd-coshkfhi) )=0 (67) y ^ +

L

yft)2 y ( caHcos kf^d-cos k f h i )

j=0 By taking the advantage of the orthogonality of Z('")(;<(^', z), i t yields:

(63)

g i t )

sin kf^d - sin fcf'/i,

g{kf^) J > 1 AjA"''(/<(.",x)Mj') = / w(z) W'(Ai. x)i'(Ai, z)Z^"'W, z) dz+ ^d '•=° ƒ ^w(z)0i'P(x,z)Z('")(/<y',z)dz £ H'(X, , X ) L ( " + r\z)cj>'^{x,z)Z^'^W, z)dz (68)

Similarly, by using the orthogonality condition for Z'^'^^ky',z), Eq. (57) yields:

3jA'\kf\x)Mf

r-d ^

i=0

with

A™' ffet"' : f ' , I V + B ) = i/<^"' for ; = 0

-kf^ for j > 1 (69)

: [ w{z)J^H'{Xi,x)Y{Xi.z)Z('"\kf\z)dz

ƒ ^w(z)^*,f''(x,z)Zf"')(/<f ,z)dz

f2HXXi.x)Lf^+ r'w{z)ct>'Hx,z)Z^'^\kf\z)dz

Similarly by using the orthogonality of Z('")(fej.^', z), it yields: (64) A]A'"'[\f,x)Mf _-d

f w(z) H'iXi, x)Y{Xi, z)Z("')(;<f, z) dz

z)dz

The iwj."' in Eqs. (58) and (64) is defined by

M j = r w(z)rz(""(/</"),z)l^dz

J —h

+/j^w<z)</.i'P(x,z)z('")(;<(-= £ H U i . x ) L | , ' ' + r\z)4>'^{x,z)Z^'^\kf\z)dz

(65) 2.5. £va/[(ation o/wove e/evafions and hydrodynamic coej?icients (70)

By substituting (31) and (32) into (65) and integrating, it results

The form of the free surface elevation at z = 0 is given by

6 X. Zhang. P. Bandyk/Applied Ocean Research 40 (2013) 1-13

-OB

-0,8

x/(W-B)

Fig. 2. Fluid velocity continuity on the interfacial surface, W = 5.0, B = 1.0, tf = 1.0, /ii = 2.0, hi = 2.0, y = 0.7, and KB = 0.25. -;~ .•\,J<l< rl >i,a.v..-. KIJ K n.Sdl ~>~ . - M il 7,.<,.-s. KI3 i = 0..-iO» OaiDjihui. KB " Cl..r)UJ .1 nmn,,ht!), KB = li.SnO X , k,K(A')

Fig.3. Convergence tests for hydrodynamic coefficients-heave added mass and damp-ing, W = 5.0, B = 1.0, hi = 2.0, /i2 = 2.0, d = 1.0, y = 0.7, K = ccP-/g, N is number of terms.

where K = (rP-fg.

The internal wave elevation atz = - / i i is evaluated by

1 - y

(72)

The non-dimensional heave added mass and damping coefficients of the twin bodies can be computed using the velocity potential on the wetted body surface through

033+1033 = ^ <p"{X.-d)dX =

/

where 033 and 033 represent the heave added mass and damping coefficients of the body, respectively, due to heave motion.

By substituting (38) and (47) into Eq. (73) and integrating:

033 + iÖ33 = 62 N ,-W+ y H{Xi.x)Y[Xi.-d)dx + P , , | ( h - d ) ^ ^ ( l - y ) ; i 2 d \ 2B(3M/2 + B2; 2S 6S (74)

Note that the dimensional added mass and damping coefficients are non-dimensionalized b y p i B ^ andpicuB^, respectively.

2Ü 1 20

Added mass-present results 1 Dam ping-present results r 0 Added mass(Yeung 2006) , a Damping(Yeung 2006) 1 1 -1 \ J . 0.2 0,3 0.4 KB

Fig. 4. Hydrodynamic coefficients of the twin floating bodies in heave motion 033 and 633 with W = 5 . 0 , B = 1.0, hi =20.0,h2 = 2 . 0 , d = l.O.y = 0.7,;<: = &)Vg•

80 60 f -20 - Added nia.ss D a m p i n g 0.2 0.4 0.6 0.8 1 K U 1.2 1.4 1.6 l.i

Fig. 5. Hydrodynamic coefficients of the twin floating bodies in heave moUon 033 and h33, VV = 5.0, B = 1.0, hi = 2.0, hi = 2.0, d = 1.0, y = 0.7, Sw = 0.002.

3. Numerical scheme and convergence tests

The inhomogeneous system of the six integral equations including (50), (51), (58), (64), (68) and (70) couple the six groups of coefficients Q, Di, A ] , Aj, B j , and Bj. The associated eigenvalue series Xi is trun-cated using Nj terms and kfK n = 1,2 are truntrun-cated using Nj terms. Hence, the six integral equations are discretized into a linear system of rank 2Ni + 4Nj. The linear system is solved using LU decomposition for each forced motion frequency. Once the six groups of coefficients are determined, the hydrodynamic coefficients and wave elevations can be computed using Eqs. (71), (72) and (74).

The continuity of fluid velocity at the interfacial boundary is i l -lustrated and verified in Fig. 2. As seen, both the real and imaginary parts of velocity due to upper layer fluid 0p' match the velocity of the lower layer fluid at the interfacial surface. The symmetry con-dition about z-axis is automatically satisfied by using the symmetric horizontal function given in Eq. (34).

Convergence tests are presented in Fig. 3 for a typical moonpool configuration. Two forced motion frequencies are chosen for the tests.

KB = 0.501 corresponds to a normal wave frequency and KB = 0.809 is

the one close to the first higher-order resonant frequency. The 'Error' shown in the flgure is defined as (Vs - Vnmax)/{VNmax). where N is the number of terms for i/j and 'V represents added mass/damping. The largest number of terms Nmax = 70. As illustrated i n the log-log plot, the prediction of the hydrodynamic coefficients has achieved good

X. Zhang, P. Bandyk/Applied Ocean Research 40 (2013) 1-13 ₇ -A"/J = 0.:L<),re-n/ KB = 0:2.i,rciil KO = 0.2!), / e.i/ 0 ' 0 3 0 3 0.4 0 5 0 6 0.7 0,8 0,9 N / ( W - B ) -0,1 - 0 . 1 5 - 0 - 2 - 0 , E 5 , - A'B = 0.19,ii)i«(/ - KB = i).2'i,iina,) - 7i'B =t 0.20, i/iif/,/ 0 1 0 ? OA 0-5 0 6 0 7 . - < / ( \ V - B ) 0 0 0 9

Fig. 6. Real and imaginary components of free surface elevation across the Helmholtz resonant frequency KB = 0.216.

—/i'.ZÏ 0.7S0, »Y «/ - KB -•= 0.825. real • A'yj=-- 0.,SaO, 0.Ü KB ~ - KJ3 - • KB U.78{), ijim<} — 0.825, iniar/ = 0.830, t.tiitiff : 0.4 -0 2 -X ' 0 - 0 . 3

/

Fig. 7. Real and imaginary components of free surface elevation r\ across the first resonant frequency KB = 0.808.

KB =. l.rA\0.rcül ^ - - KB = 1.67(Krcal KB =^ l . ö 7 ö , r e f ( / ,. -AO -P.Q / ' \ 0 -yu \ • •• , . --10 y - 6 0 r ^ 0,2 0.3 O.a 0.5 0.6 0.7 x / ( W - B ) 0,C6 Ü 0 4 0.02 -O.tM - 0 . 0 6 KB = , - KB = = 1.5f)0.lmrt<7 l..">76, 7>/(f7// 1.578, iiitiLtj N \ \ X _ _ _ • 0.1 0,2 0 3 ÜA 0 6 0-7 0 8 x/(AV-B) Fig. 8. Real and imaginary components of free surface e l e v a t i o n a c r o s s the second resonant frequency KB = 1.575.

convergence, even around ttie resonant frequencies.

4. Prediction of hydrodynamic coefficients

In order to validate the present model, a limiting case is selected to compare with the results for a single-layer fluid by Yeung and Sean [3], The depths of the upper and lower layers are h i = 20.0 and /12 = 2.0, respectively The draft of the twin bodies is d = 1.0. Since / i i jh^ = 10, the present results are expected to approach those of the single-layer

fluid with depth h i . As illustrated in Fig. 4, the present computational results compare well with the predictions for the single-layer fluid case.

Fig. 5 shows the heave added mass and damping coefficients of the twin bodies as a function of KR for a typical moonpool configu-ration. The hydrodynamic coefficients are also listed in Table 1. We seek the critical frequencies where there is minimal wave radiation damping in the outer region. As can be seen in Fig. 5, three critical frequencies are found at /CB = 0.216, /CB = 0.808, and fCB = 1.575.

8 X. Zhang, P. Bandyk /Applied Ocean Research 40(2013) 1-13 — KJ3 Q.ni.rciil - KB 0 . 2 1 . real KB = i).'2'J.rtal ' K B 0 . 1 9 , « / ( « , < • / •KB = ().2>l,-irna(] KB = Ci:2<:).lrivui . /

-/

Fig. 9. Real and imaginary components of internal wave elevation ) / across the Helmholtz resonant frequency KB = 0.216.

- / a , ; = 0.780. rc<;/ KB = 0.S25. real -KB = 0.7S0.ima,/ K B = 0 . 8 2 - 5 , / J f i ( i , 9 0,1 0,2 0,3 0,4 t>,5 Q,G 0,7 0 8 0.9 x/{\V-B) 0 0,1 0,2 0.3 0 6 0 7 0 8 >c/(\V-B)

Fig. 10. Real and Imaginary components of internal wave elevation ;/ across the first resonant frequency KB = 0.Ï

K B l . . ' - ) ( ) 0 . 7 - ( : . ( i ; - KB •••= l.r>7H, rani 0.1 0.2 0,3 0.5 0 8 0-? 0,8 0,9 ! x/(W-B) 0,08 o.oei-Q.04 0.02 U -O.02 - 0 , 0 4 - 0 . 0 8 - 0 , 0 8 - 0 1 K B i . 5 0 0 , / ; , ) « . < ; KB = 1 . 5 7 8 , iiuafi 0 0,1 0,2 0,3 0,4 O S 0,6 0 , / [18 .v/(\V-B)

Fig. 11. Real and imaginary components of internal wave elevation i;' across the second resonant frequency KS = 1.575.

The Helmholtz mode can be identified around KB = 0.216, where the added mass changes sign from negative to positive and damping drop down approaching zero. At the other two resonant frequencies, thus called higher-order resonances, the added mass and damping both show spikes. The added mass changes sign near those regions and the damping coefficients suddenly increase and then drop, approach-ing zero. This phenomenon is very similar to the sapproach-ingle-layer fluid situation.

5. Free surface and interfacial surface elevations

Both free surface and interfacial wave elevations have been com-puted using Eqs. (71) and (72). The moonpool conflguration and strat-ification parameters are same as those in Fig. 5. Fig. 6 shows the real and imaginary components of the complex moonpool free surface wave elevation if inside the moonpool near the Helmholtz frequency

KB = 0.216. As observed in these figures, the imaginary component

X. Zhang, P. Bandyk /Applied Ocean Research 40 (2013) 1-13 ₉ • Rürtt c o m p o t u m t • i ' ï i -0.5'-4 5 6 x / ( W 4 - B ) — • — - H o a l coniponctit-• coniponctit-• coniponctit-• ' > coniponctit-• : imaj-^iiiary ccjiiipimi:]!)! X / ( W h - B )

Fig. 12. Radiated wave elevation at the free surface and interfacial surface, W = 5.0, B = 1.0, hi = 2.0, h2 = 2.0, d = 1.0, y = 0.7, and KB = 0.19.

Table 1

Hydrodynamic coefficients of twin bodies in a two-layer fluid, W = 2.0, hi = 2.0, d = 1.0, and y = 0.7. 5.0, B = 1.0,h, ft) KB 033 b33 0.4 0.016 -9.859 31.243 0.6 0.036 -3.660 17.238 0.8 0.065 -1.101 14.322 1.0 0.102 -7.067 13.204 1.2 0.147 -3.620 1.171 1.4 0.199 -0.113 1.989 1.6 0.261 1.853 0.469 1.8 0.330 3.002 1.165 2 0 0,408 3.449 1.098 2.2 0.494 3.689 1.024 2.4 0.587 3.983 0.881 2.6 0.689 4.489 0.761 2.8 0.799 13.046 3.671 3.0 0.918 3.385 0.210 3.2 1.044 3.969 0.175 3.4 1.179 4.258 0.125 3.6 1.322 4.469 0.086 3.8 1.473 4.683 0.058 4.0 1.632 4.522 0.035 4 2 1.799 4.762 0.023

- Surface, wave .'unplituclo

l u t o n i a l wave a i i i p l i t i K l e

K B

Fig. 13. Outer region(lll) radiated wave amplitudes on free surface and interfacial surface, intermediate draft body, W = 5 . 0 , B = 1.0, hi = 2.0, =2.0, d = 1.0, and y = 0.7.

the added mass and damping characteristics shown in Fig. 5. Also, there is a nonzero mean wave elevadon in the Helmholtz resonant modes. Fig. 7 shows the real and imaginary parts of free surface ele-vadon near the first higher-order resonant frequency KB = 0.808. Fig. 8 shows the real and imaginary parts of free surface elevation near the second higher-order resonant frequency KB = 1.575. As shown in Fig. 7, the real component of the free surface changes sign around KB = 0.805, while the imaginary part changes sign a little beyond KB = 0,825. It is also interesting to find that the behavior of the higher-order resonance is characterized by a standing free surface wave existing in the moonpool gap.

Figs. 9-11 show the internal wave elevations at the resonant fre-quencies. These plots suggest that resonance is not directly associ-ated with the internal wave on the interfacial surface. Also seen in those figures, the magnitude of both free surface and internal waves are significantiy decreased from the first higher-order to the second

higher-order resonant mode.

The radiated wave elevations at the free surface and interfacial surface, outside the moonpool in region HI, are illustrated in Fig. 12 for forced motion frequency KB = 0.19. Both the real and imaginary components are presented in the figure. The dependence of outer re-gion radiated wave amplitude on wave frequency is illustrated in Fig. 13. Near the Helmholtz resonant frequency, the surface wave ampli-tude decreases to nearly zero. This is consistent w i t h the low wave damping values near Helmholtz resonant frequency, as illustrated in Fig, 5. At the higher-order resonant frequencies, both the surface and internal waves show resonance charactenstics. In order to examine the case where the bottom of the body is close to the interfacial sur-face, we present the radiated wave amplitudes for a 'shallow-draft body' (depth of the upper fluid layer does not exceed the draft of the body d significantly, i.e. (/ii - d)//ii < < 1.0) in Fig. 14. As shown.

10 X. Zhang, P. Bandyk /Apphed Ocean Research 40(2013) 1-13

401 ^ 1 1 .

Surfi>ce wave amplitude - ~ lutenial wave am])litu(le

"2 35 - i 3 30 • i' •ri 1

1 • > "

Fig. 14. Outer region(lll) radiated wave amplitudes on free surface and interfacial surface, shallow-draft body, IV = 4.0, B = 1.0, in = 0.3, /12 = 1.0, d = 0.2, and y = 0.7.

for a single body case. Also, the resonant frequencies decrease with increasing W/B.

6.2. Shallow-draft body

Additional analysis has been performed for a shallow-draft body, where the depth of the upper layer fluid hi does not exceed the draft of the body d significantly, i.e. (/ii - d)/fii < < 1.0, Fig. 18 shows the de-pendence of hydrodynamic coefficients on the oscillation frequency for different depths of the lower layer fluid /12. The depth of the upper layer fluid is /ii = 0.3. The draft of the body is d = 0.2. The density ratio is y = 0.7. The body spacing is W = 4.0 while the width of the body B = 1.0. As can be seen in these figures, the depth of the lower layer fluid drastically alters the higher-order resonant frequencies.

Fig. 19 illustrates the dependence of hydrodynamic coefficients as a function of oscillation frequency for different density ratios y. The other physical parameters are the same as in Fig. 18, except a fixed hj = 0.5 and varying / . The comparison of results with different density ratio Y suggests that density stratification affects the higher-order resonant modes for the present shallow-draft body.

both the surface wave and internal wave amplitudes are reduced near the Helmholtz mode, which is different from the intermediate-draft body case as shown in Fig. 13.

6. Parametric Studies

Parametric studies have been performed to examine the depen-dence of moonpool resonance on geometry and stratification param-eters, including the relative depths of the fluid layers, density ratio, and body spacing. Both the cases of intermediate-draft body and deep-draft body are investigated.

6.1. hitermediate-draft body

First, we select twin bodies with a draft smaller than the depth of upper fluid. The depth of the upper fluid is fixed at hi = 2.0 and the depth of the lower layer fluid /12 is varied from 0.2 to 5.0. Fig. 15 shows the hydrodynamic coefficients of the twin bodies for different depths of the lower layer fluid. The density ratio y is chosen as 0.7 and the drat of body d is f .0. Note that by setting h] = 2.0 and fi2 = 0.2, the first case is very similar to a single-layer fluid situation. To study the effect of /i2, we compare the solutions with different hj. As illustrated in the figure, the Helmholtz frequency has been shifted due to different depths of the lower layer fluid. As also can be observed, the first higher-order resonant frequency is shifted from KB = 0.77 to KB = 0.82. However, the second higher-order resonant frequency has changed negligibly. It may be concluded, for the present moon-pool configuration, though the fluid stratiflcation can change both the Helmholtz and higher-order resonant frequencies, as higher-order resonant frequency increases, the impact is gradually decreasing. In addition, as the depth of the lower layer fluid increases, the resonant frequencies also increase.

Fig. 16 shows the variation of the added mass and damping coeffi-cients with the oscillation frequencies for different density ratios y = 0.4,0.5,0.6,0.7. As illustrated in those figures, the density ratio varia-tion only affects the Helmholtz mode slightly, while the higher-order resonant modes are nearly unaffected.

In order to identify the dependence of the resonance on the width of the moonpool. Fig. 17 presents the added mass and damping coef-ficients versus frequency for different relative moonpool width ratios

W/B. Both the Helmholtz and higher-order resonant modes have been

significantly changed as the the width of moonpool varies. In particu-lar, for the case of W/B = 1.0, there is no resonance, which is expected

7. Concluding remarlfs

In this paper, the hydrodynamic problem of the twin heaving bod-ies in a two-layer fluid has been investigated using the eigenfunction matching method. The heave added mass and damping coefficients have been computed for different configurations. By examining the computational results, it is found that there are both Helmholtz and higher-order resonances for a moonpool in a two-layer fluid, which is very similar to the case of a single-layer fluid system. Near the Helmholtz resonant frequency, the added mass changes sign and the damping approaches zero. At the higher-order resonant modes, the added mass and damping show spikes. The higher-order resonant mode is also characterized by a standing free surface wave inside the moonpool gap. The parametric studies for intermediate-draft body indicate that there are signiflcant differences in the characteristics of the Helmholtz resonant mode between two-layer and single-layer fluid cases due to different lower layer fluid depths and density ra-tios. Though the higher-order resonant modes are also affected by the relative depth of upper and lower layers, the effect is not signif-icant for intermediate-draft body. However, for shallow-draft body with a draft close to the depth of upper layer fluid, the stratiflcation has strong impacts on the hydrodynamic behavior and resonances over a wide range of frequencies. The hydrodynamic resonances are also reflected in the outer region (III) radiated free surface and i n -ternal wave elevations. For intermediate-draft body case, both free surface and internal wave amplitudes show resonant characteristics near higher-order resonances, while only free surface wave shows resonance near Helmholtz mode. However, for shallow-draft body case, both the outer region (III) radiated surface and internal waves show resonant behavior near Helmholtz resonant frequency.

Appendix A. Expression of L) ( " )

Z.J"' can be written as follows, for six different cases of i and j: For 1 = 0 , j = 0:

sinh kh

a

sinh

ft)^(cosh khi - cosh fed)

X. Zhang. P. Bandyk /Apphed Ocean Research 40 (2013) 1-13 ₁₁ /i-7 =0.2 - - / l 2 = 1.0 • • • / i 2 = 2 . 0 h-2 = 5.0

• J

,r-f

Fig. 15. Added mass and damping coefficients of the twin bodies in heave motion with different depths of the lower layer fluid fc, W = 5.0, B = 1.0, h, = 2.0, d = 1.0, and y = 0.7.

- b i - l i s i l y l u l i o 7 = 0.4 DeiKsit.v ratio 7 = 0 . . 5 Di^usHy ratio y—O.i) Dcur^ily rntio ')~().7 KB -.Dcn.sity ratio 7-={),4 Dcnisity ratio 7~l).5 • Density r-M'io 7—0.6 Density r a t i n 7^=0.7 0.2 O.-l Ü.6 0.8 K B

Fig. 16. Added mass and damping coefficients of the twin bodies in heave motion with different density ratios y , I V = 5,0, B = 1.0, hi = 2.0, fc = 2.0, and d = 1.0.

2 - U ' / i ? = 1.0 - W/B = 2.0 •W/B = i.o w/B = r>.o 0.2 OA 0.6 O.B I 1.2 1.4 1.6 l.( K B M 7 B - - = : 1.0 - - 1 1 7 0 = 2.0 • l l V B = :i.O I I 7 B = G.O

12 X. ZImng. P. Bandyk /Applied Ocean Researcli 40(2013) 1-13 4 r - c : k J / f t / - . -

_7:'

/ / •

p

/

1

Fig. 18. Added mass and damping of the twin bodies in heave motion with different depths of the lower layer fluid I12, hi = 0.3, d = 0.2, W = 4.0, B = 1.0, and y = 0.7.

A / .

Ié

_A/

• D f i i s i l y ratio 7=0.1 - - Density rniio 7=^0.5 • Density rat io 7.---O.O

— - Density rat icj 7—0.1 — - Density ratio -y—O.ó

DeiLsity r a i i o 7 - : n . { ) 1 ; 1

Ï

1

Fig. 19. Added mass and damping of the twin bodies in heave motion with different density ratios y. b, = 0.3,112 = 0.5, d = 0.2, W = 4.0, and B = 1.0.

For i = 0, J > 1:

sin fe/ii a

sin kcT'

£t)^(cos k/ii - cos fed)

For i > 2, J = 0:

, („) ct>^(X sin X/i2 cosh fe/12 + fe cos Xhj sinh feh2) (A.2)

'iO

gfe(7.2 + k2)sinX/i2 sinh khi a sin > . ( d - / i i

Fori = l , j = 0:

, ( i i ) _ M^i^ sinh/l/12 coshfe/i2 - k coshA/12 s'mhkhi) 10 " gfe(A2-fe2)sinhX/i2 sinhfe/i2

' « 2 + _{a sinhA(d - / i i}

A s i n h A ( d - / i i ) s i n n fe/ii - fe(cosh fed - cosh X(d - / i i ) c o s h f e h i ) (A.3)

A sin A(d - / i i ) sinh fehi +fe(cosh fed-cos X ( d - / i i ) c o s h fehi)

-X sinX(d - / l l ) c o s h fe/ii + f e ( - s i n h fed + cos A(d - hi)sinhfe/!i)

For i > 2, J > 1:

. (n) _ a)2(X sin Ah2 cos fe/12 - fe cos Xhj sin /(:/i2)

(A.5)

,(")

X2 - /(2

-X sinh A(d - / i i )cosh/</ii - /<(- sinh/<:d + cosh A(d - /!i)sinh/</!i)

For i = 1, J > 1:

(u2(x sinhX/i2 cos /</i2 + fe cosh A/12 sin Idii)

y

a s i n X ( d - / i i )

gfe - k^) sin X/12 sin khi

'a?

gk _(A.6)

y

gk (X2 + /^2) sinh X/12 sin /</i2

a s i n h A ( d - / i i )

X sinh A ( d - / i i ) s i n " / < / i i -/<(cos /<d - cosh A(d - / i i ) c o s / < / ! i ) -X s i n h X j d - h - i ) c o s /</ii + /c(-sin fed + coshA(d - / i i ) s i n /</ii;

A7Tfe2

(A.4)

X sin A(d - / i i ) sin /</!i + /<(cos lid - cos A(d - / i i ) cos /</ii

-X sinA.(d-/!i)cos /</ii - f e ( - s i n /<d + cos A ( d - f ! i ) s i n kh\]

where X and fe represent A,- and respectively, in order to save space.

Reference

[11 Molin B. On the piston sloshing modes in moonpools.Journal of Fluid Mechanics. 2001;430:27-50.

X. Zhang. P. Bandyk/Applied Ocean Research 40(2013) 1-13 ₁₃ |2] Mclver P. Complex resonances In the water-wave problem for a floating

struc-ture. Journal of Fluid Mechanics. 2005;536:423-443.

[3] Yeung RW.SeanRKM. On Helmholtzand higher-order resonance of twin floating bodies.Journal of Engineering Mathematics. 2006;58(l-4):251-265. |41 Faltinsen OM, Rognebakke OF, Timokha AN. Two-dimensional resonant

piston-like sloshing in a moonpool. Journal of Fluid Mechanics. 2007:575:359-397. [51 Mclver M. An example of non-uniqueness in the two-dimensinal linear water

wave problem. Journal of Fluid Mechanics. 1996;315:165-178.

[61 Mclver P, Mclver M, Zhang J. Exciting of trapped water waves by the forced motion of structures, journal of Fluid Mechanics. 2003 ;494:141 -162. [71 Mclver P. MclverM. Motion trappihgstnjctures in the three-dimensional

water-wave problem. Journal of Engineering Mathematics. 2007;58:67-75. [8| Kuznetsov N, Mclver P, Linton CM. On the uniqueness and trapped modes in the

water-wave problem for vertical barriers. Wave Motion. 2001 ;33:283-307. [91 Newman JN. Radiation and diffraction analysis of the Mclver toroid. Journal of

Engineering Mathematics. 1999;35:135-147.

[10| Kuznetsov N, Mclver M, Mclver P. Wave interaction with two-dimensional bod-ies floating in a two-layer fluid: uniqueness and trapped modes. Journal of Fluid Mechanics. 2005;536:423-443.

[11] Linton CM, Mclver M. The interaction of waves with horizontal cylinders in two-layer fluid. Journal of Fluid Mechanics. 1995;304:213-229.

[12] Yeung RW, Nguyen T Waves generated by a moving source in a two-layerocean of finite depth. Journal of Engineering Mathematics. 1999;35(l-2):85-107. [13] Yeung RW, NguyenT. Radiation and diffraction of waves in a two-layer fluid. In:

Proceedings of the 22nd symposium on naval hydrodynamics, 1999:875-887. [14] Alam MR, Liu Y, Yue DKP. Waves due to oscillating and translating disturbance

in a two-layer density-stratified fluid. Journal of Engineering Mathematics. 2009;65:179-200.

[15| Yeung RW. Added mass and damping ofa vertical cylinder in finite-depth waters. Applied Ocean Research. 1981;3:119-133.

[16] Shipway Bj, Evans DV. Wave trapping by axisymmetry concentric cylinders, journal of Offshore Mechanics and Arctic Engineenng. 2003;125:59-64. [17] Mavrakos SA. Hydrodynamic coefficients in heave of two concentric

surface-piercing truncated circular cylinders. Applied Ocean Research. 2004;26:84-97. [18] Mavrakos SA, Chatjigeorgiou IK, Second-order diffraction by two concentric

truncated cylinders. In: 21 st internatioanl workshop of water waves and floating bodies. Loughborough, UK; 2006.

[19] Whitham GB. Linear and nonlinear waves. New York: Wiley; 1974.

[20] Wehausen jV, Laitone EV. Surface waves. Handbuch der Physik, Springer Verlag 1960 vol. IX. p. 446-778.