The hydrodynamics of arrays of truncated elliptical cylinders

European Journal of Mechanics B/Fluids 37 (2013) 153–164

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The hydrodynamics of arrays of truncated elliptical cylinders Ioannis K. Chatjigeorgiou ∗ School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroön Polytechniou Ave, GR157-73, Zografos Campus, Athens, Greece

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Article history: Received 3 May 2012 Received in revised form 31 July 2012 Accepted 8 September 2012 Available online 19 September 2012 Keywords: Hydrodynamics Truncated elliptical cylinders Mathieu functions Negative Mathieu parameter

abstract The hydrodynamic scattering problem by arrays of elliptical truncated cylinders subjected to regular waves is investigated. Linear potential theory is employed that is based on the assumption of inviscid/incompressible fluid and irrotational flow. Effort is made to provide an analytical solution of the associated hydrodynamic boundary value problem. The solution is based on the semi-analytical formulation of the diffraction potentials and the application of the so-called Mathieu functions addition theorem. The latter is decomposed into even and odd Mathieu functions whereas it considers arbitrary angular orientations of the semi-major axes of the bodies. It was also extended to account for negative Mathieu parameters which correspond to the infinite real roots of the transcendental equation. The details of the interaction phenomena are investigated with the aid of the transfer functions of the exciting forces on two-body configurations. Indicative results for the free surface elevation around the bodies are also provided. © 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction A vast literature exists on wave/structure interaction problems. For a review on the employed methods, see e.g. [1]. The study of wave scattering by elliptical bodies is important in various scientific fields such as hydrodynamics, electromagnetics, optics and acoustics. Indicative examples of reported works in areas other than hydrodynamics are those due to Burke [2], Van Den Berg and Van Schaik [3], Sebak [4], Nigsch [5], Mao and Wu [6] and Lee [7,8]. Admittedly, most of the published articles concern single embodiments while only a small amount of works investigate the wave scattering by multiple elliptical cylinders and only in two dimensions [4,5,7,8]. The three dimensional hydrodynamic diffraction and radiation problems by isolated elliptical cylinders were investigated by Williams and his group in a series of articles using an analytical approach [9–14]. The authors provided semi-analytical formulations for the diffraction and radiation potentials induced by bottom mounted and truncated cylinders [9–12] as well as by submerged elliptical disks [13,14]. The study on the water wave scattering by the submerged elliptical disk was recently restored by Bao et al. [15]. The pure analytic solution for the hydrodynamic diffraction problem by arrays of bottom mounted elliptical bodies was provided by Chatjigeorgiou and Mavrakos [16,17]. Further, Chatjigeorgiou [18] extended the method developed in Chatjigeorgiou and Mavrakos [16,17] to tackle the scattering problem by bottom

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mounted elliptical and circular cylinders. The employed theory was based on the addition theorems which were originally derived by Schäfke [19] and Særmark [20], properly adapted to even and odd periodic and modified Mathieu functions. Recently, Chen et al. [21] and Chen et al. [22] have used a different line to study the hydrodynamics of arrays of bottom mounted elliptical and circular cylinders. These studies employed the approach called by the authors as the ‘‘null-field boundary integral equations’’ method. In fact, Chen et al. [22] validated the results presented in Chatjigeorgiou and Mavrakos [17] and reported an extremely favorable coincidence. Clearly what it is missing from the literature is the analytic solution for the hydrodynamic diffraction problem by arrays of truncated elliptical cylinders, which constitutes the subject of investigation of the present contribution. The solution method employs the technique of matched eigenfunction expansions, the way that is applied for circular or rectangular truncated bodies (see e.g. [23,24]). The theory that was developed in the context of the relevant studies [16–18] facilitates the necessary effort although the equations which are eventually derived are extremely more complicated, and apparently hard to be treated numerically. For truncated elliptical cylinders, the analytical formulation of the governing velocity potentials requires the definition of different velocity fields in order to satisfy the conditions of the boundary value problem. In addition, the series expansions of the diffraction potentials must incorporate the evanescent modes which, as discussed in Section 3.1.2, result in negative Mathieu parameters. The existence of negative Mathieu parameters unavoidably leads to significant numerical complications as the modified Mathieu functions need to be computed by additional series of

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I.K. Chatjigeorgiou / European Journal of Mechanics B/Fluids 37 (2013) 153–164

transformation from elliptic to Cartesian coordinates is x = c cosh u cos v

(1)

y = c sinh u sin v

(2) 2 1/2

where c = (a − b ) = ae with e being the elliptic eccentricity of the cylinder given by e = (1 − (b/a)2 )1/2 . Higher order effects are neglected and the assumption is made that the fluid is inviscid and incompressible and the flow is irrotational. Thus, linear potential theory can be employed, meaning that the fluid motion can be described by the first-order velocity potential, which in elliptic coordinates is expressed as 2

φ(u, v, z ; t ) = ℜ{ϕ(u, v, z ) exp(−iωt )}.

(3)

It follows that the velocity potential must satisfy the Laplace equation

∇ 2ϕ = 0

(4)

in the infinite region (I), u0 ≤ u and 0 ≤ z ≤ d, and in the lower field (II) 0 ≤ u ≤ u0 and 0 ≤ z ≤ h, where u0 = tanh−1 (b/a). The velocity potential must also satisfy the kinematical condition on the bottom

 Fig. 1. General arrangement of the array of elliptical cylinders: coordinate systems and geometrical definitions.

Bessel functions [25,26]. Also, the Mathieu functions addition theorem must be accordingly extended to transform the products of Mathieu functions in terms of the real roots of the transcendental equation. Finally, particular attention must be given to the derivation and the solution of the complex linear system that provides the unknown expansion coefficients after applying the matching relations of the pressures and the fluxes. The afore-mentioned tasks are properly outlined and tackled in the following sections. 2. Formulation of the hydrodynamic problem The arrangement of the elliptical cylinders depicted in Fig. 1 is investigated. All cylinders are truncated. The main dimensions and the fluid regions defined by the geometry of each cylinder are shown in Fig. 2. The bodies are exposed to the action of monochromatic incident waves of frequency ω and linear amplitude H /2, propagating at angle α to the positive x direction. The semi-major and semi-minor axes and of the kth body are denoted by ak and bk respectively. The draught of the kth cylinder is d − hk , where d is the water depth. Elliptic cylindrical coordinates for each cylinder are employed (u, v , z), u = constant, v = constant being orthogonally intersecting families of confocal ellipses and hyperbolae, respectively. The z-axis is fixed on the bottom, pointing vertically upwards. The

∂ϕ ∂z

 =0

(5)

z =0

as well as the linearized condition on the free surface



−K ϕ +

∂ϕ ∂z

 =0

(6)

z =d

where K = ω2 /g and g is gravitational acceleration. The velocity potential must also satisfy the kinematical conditions on the wetted surface of all bodies





∂ϕ ∂u



∂ϕ ∂z



= 0,

h≤z≤d

(7)

0 ≤ u ≤ u0 .

(8)

u=u0

= 0, z =h

3. Velocity potentials 3.1. Infinite region (I) The total velocity potential in the infinite region (I) ϕ (I ) must satisfy Eqs. (4)–(6). In the context of the linear theory ϕ (I ) is decomposed into the incident wave potential ϕI and the total diffraction potential that involves the scattering of waves by all (I ) bodies ϕD . Thus

ϕ (I ) = ϕI + ϕD(I ) .

Fig. 2. An elliptical truncated cylinder: main dimensions and fluid regions.

(9)

I.K. Chatjigeorgiou / European Journal of Mechanics B/Fluids 37 (2013) 153–164

3.1.1. Incident wave potential When dealing with a multi body arrangement it is convenient to express the involved velocity potentials with respect to body-fixed coordinates of each constituent body. Let (xk , yk , z) be the Cartesian coordinates of any point in the reference field with respect to the body-fixed Cartesian coordinate system of body k. Then, the incident wave potential, normalized by −iω(H /2)d, will be given by

ϕI =

1 Z0 (z ) Kd Z0 (d)

Λk exp[ik0 (xk cos α + yk sin α)].

(10)

The same normalization form is adopted for all velocity potentials involved in the present study. In Eq. (10) −1/2

Z0 (z ) = N0 1

N0 =



2

1+

cosh(k0 z ), sinh(k0 d)

(11) (12)

and

Λk = exp [ik0 (Xk cos α + Yk sin α)]

(13)

where Xk , Yk are the Cartesian coordinates of the center of body k with respect to the global Cartesian coordinate system (see Fig. 1). In Eqs. (10)–(13), k0 is the wave number which is obtained through the celebrated dispersion relation k0 tanh(k0 d) = K .

(14)

Next, the incident wave potential is expressed in terms of the local elliptic coordinate system of body k(u(k) , v (k) , z ). 2 Z0 (z )

ϕI =

Kd Z0 (d)

 ×

∞ 

Λk (k)

(k)

+

(k)

(k)

(k)

im Ms(m1) (u(k) ; q0 ) sem (v (k) ; q0 ) sem (α; q0 )

(15)

(k)

where cem (v (k) ; q0 ) and sem (v (k) ; q0 ) are the even and the (1)

(k)

k) (k) (3) (k) (k) im A(mp Kcmp Mcm (u ; qp ) cem (v (k) ; q(pk) )Zp (z )

m=0 p=0

+

∞  ∞ 

k) k) im B(mp Ks(mp Ms(m3) (u(k) ; q(pk) )sem (v (k) ; q(pk) )Zp (z )

m=1 p=0

(19) where (k) Kcmp =

k) Ks(mp =

′(1)

(k)

(k)

′(3)

(k)

(k)

′(1)

(k)

(k)

′(3)

(k)

(k)

Mcm (u0 ; qp )

(20)

Mcm (u0 ; qp ) Msm (u0 ; qp )

(21)

Msm (u0 ; qp ) (3)

(k)

(3)

(k)

while Mcm (u(k) ; qp ) and Msm (u(k) ; qp ) denote the even and the odd modified Mathieu functions of the third kind, in the notation of Abramowitz and Stegun [27]. The primes denote differentiation with respect to the argument u. In the diffraction potential, these functions, together with the even and the odd periodic Mathieu (k) (k) functions cem (v (k) ; qp ) and sem (v (k) ; qp ), are calculated for the infinite number of Mathieu parameters q(pk) = −

 σ a e 2 p k k

(22) 2 that correspond to the infinite roots of the transcendental Eq. (17) including the imaginary solution. The negative Mathieu parameters correspond to the infinite evanescent modes of the associated hydrodynamic diffraction problem. The orthogonal eigenfunctions Zp (z ) are given by

Np =



m=1

(k)

∞  ∞ 

ϕD(I ),(k) =

Zp (z ) = Np−1/2 cos(σp z ),

(k)

(1) (k) im Mcm (u ; q0 ) cem (v (k) ; q0 ) cem (α; q0 )

m=0

∞ 

its local elliptic coordinate system (u(k) , v (k) , z ) as



2k0 d

155

1

 1+

2

sin(σp d)



2σp d

(k)

(23)

.

(24)

(k)

Finally, Amp and Bmp , are the unknown complex expansion coefficients which need to be determined by combining the pressures and the velocities at the adjacent boundary of regions (I) and (II).

(1)

odd periodic Mathieu functions while Mcm (u(k) ; q0 ) and Msm

(u(k) ; q(0k) ) are the even and odd modified Mathieu functions of the

3.2. Lower region (II)

first kind. The series expansion given by Eq. (15) is due originally to McLachlan [26] (p. 359, Eq. (3)) and it is expressed herein in a more compact form using the notations of Abramowitz and Stegun [27]. Also,

The velocity potential in the lower region (II) shall include both the incident and the diffraction components. The associated function, which must satisfy Eqs. (4), (5) and (8) is expressed in terms of the local elliptic coordinate system of body k as

(k)



q0 =

k0 ak ek

2 (16)

2

denotes the zero order Mathieu parameter that corresponds to the wave number k0 , or in other words to the imaginary root σ0 = ik0 of the transcendental equation

σp tan(σp d) + K = 0.

(17)

(25)

In Eq. (16) ek denotes the elliptic eccentricity of body k.

where ε0 = 1 and εn = 2, n ≥ 1. Also,

3.1.2. The diffraction potential in the infinite region (I) (I )

The total diffraction potential in the infinite region (I) ϕD must satisfy, in addition to Eqs. (4)–(6), the appropriate radiation condition at infinity, which in elliptic coordinates is expressed as lim (c cosh u)1/2

u→∞

(I ),(k)



1

∂

c sinh u ∂ u

(k)

Qn

 =−

(18)

Let ϕD denote the potential in region (I) due to the diffraction of the incoming waves by the body k. This is expressed in terms of

nπ ak e k 2hk

2

.

(26)

(k)

= 0, while for the zero Mathieu parameter it holds (k) (1) (k) that Mcm (u ; Q0 ) = cosh(mu(k) ) and Msm (u(k) ; Q0 ) = sinh (k) (k) (k) ˜ (mu ). Finally Fmn and Fmn are the unknown expansion coefClearly Q0 (1)



− ik0 ϕD(I ) = 0.

ϕ (II ),(k) (1) (k) (k) ∞  ∞  nπ z   (k) Mcm (u ; Qn ) = im εn Fmn cem (v (k) ; Qn(k) ) cos ′(1) (k) (k) h Mcm (u0 ; Qn ) m=0 n=0 ( 1 ) ( k ) ∞ ∞  ( k )  nπ z  (k) Msm (u ; Qn ) (k) (k) + im εn F˜mn se (v ; Q ) cos m n ′(1) (k) (k) h Msm (u0 ; Qn ) m=1 n=0

(k)

ficients associated with the lower region (II). Eq. (25) does not include the parts of the solution of the associated boundary value problem which involve the modified Mathieu function of the third

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I.K. Chatjigeorgiou / European Journal of Mechanics B/Fluids 37 (2013) 153–164

kind. In fact these terms were omitted as the modified Mathieu function of the third kind for negative Mathieu parameters is calculated in terms of the modified Bessel function of the second kind which tends to infinity for zero argument.

Ms(m3) (u(j) ; q(pj) )sem (v (j) ; q(pj) )

=i

r =0

3.3. Total diffraction potential in the infinite region (I)

+

The total diffraction potential in the infinite region (I) is composed by the individual diffraction potentials of all bodies, namely

ϕD(I ) =

N 

∞  (−1)m B(r3,−) m,p Mcr(1) (u(k) ; q(pk) )cer (v (k) ; q(pk) )

ϕD(I ),(j)

(27)

∞  (−1)m−r B(−3r),−m,p Ms(r1) (u(k) ; q(pk) )ser (v (k) ; q(pk) )

where ) B(r3,m ,p =

j =1

where N is number of bodies in the arrangement. Using Eq. (19), Eq. (27) becomes ∞  ∞ 

(I )

ϕD =

k) (k) (3) im A(mp Kcmp Mcm

(k)

(k)

(k)

(k)

(u ; qp ) cem (v ; qp )Zp (z )

m=0 p=0

+

∞  ∞ 

k) k) im B(mp Ks(mp Ms(m3) (u(k) ; qp(k) )sem (v (k) ; q(pk) )Zp (z )

m=1 p=0

+

N  ∞  ∞ 

(j) (j) (3) (j) (j) im Amp Kcmp Mcm (u ; qp ) cem (v (j) ; q(pj) )Zp (z )

+

∞ N  ∞  

∞  s,l=−∞

Hm denotes the mth order Hankel function of the first kind. For p = 0, namely the imaginary root of the transcendental Eq. (17), λ = k0 whereas for the real roots (p ≥ 1) for which the Mathieu parameter (k) can be written as qp = (iσp ak ek )2 (see Eq. (22)), λ = iσp . Thus, using the well-known relation that transforms the Hankel function with an imaginary argument to the modified Bessel function of the second kind, Eq. (31) is decomposed according to ∞ 

(3)

(j) (j) im Bmp Ksmp Ms(m3) (u(j) ; q(pj) ) sem (v (j) ; qp(j) )Zp (z ).

Eq. (28) expresses the total diffraction potential in region (I). Nevertheless various terms are still expressed in terms of the local elliptic coordinate systems of the bodies. In order to be able to employ the zero velocity condition (7) as well as the continuity relations of the velocities and the pressures at the common boundary of regions (I) and (II), Eq. (28) must be recast to a suitable expression written in terms of a single local elliptic coordinate system. Let us assume that this system must be the local elliptic coordinate system of the arbitrarily selected body k. Then, the last two sums of Eq. (28) must be properly transformed to be expressed in terms of the components of the elliptic system (u(k) , v (k) , z ). To this end the so-called addition theorem for Mathieu functions must be applied, and especially the forms that account separately for even and odd Mathieu functions. The existence of an addition theorem for Mathieu functions was shown by Særmark [20] who provided an expression in terms of the complex modified and periodic Mathieu functions in the notation of Meixner and Schäfke [25]. The relation due to Særmark [20] is a relatively simplified formula as the semi-major axis of one of the bodies was assumed to be horizontal and accordingly it cannot be employed for arbitrary placements of the involved elliptic coordinate systems. Here, the procedure is extended to accommodate the even and the odd periodic and modified Mathieu functions whereas it also accounts for random rotations of the semi-major axes of all bodies with respect to the horizontal. The associated analysis is outlined in detail in Appendix A. Using the relations provided in Appendix A and taking into account the angle notations of Fig. 1, the addition theorems in terms of the even and odd periodic and modified Mathieu functions of the third kind are expressed as (3) (j) (j) Mcm (u ; qp )cem (v (j) ; q(pj) )

=

∞ 

s,l=−∞

r =1

(32)

d∗r −l,l q(pk) Kl−s s,l=−∞ i(s−l)(π /2+ψjk ) −is(βj −βk )

(

π

×e

)

(σp Rjk )ds−m,m (q(pj) )

e

(33)

where Km is the mth order modified Bessel function of the second kind. Clearly Eq. (33) is valid for p ≥ 1. The coefficients d and d∗ are determined in terms of the expansion coefficients of the even and the odd periodic Mathieu functions [25–27]. The relevant relations are also given in Appendix A where the interested reader is referenced. Next, using the relations (27), (28), (29) and (30), the total diffraction potential in the infinite region (I) takes the following extended form:

ϕD(I ) =

∞  ∞ 

k) (k) (3) (k) (k) im A(mp Kcmp Mcm (u ; qp ) cem (v (k) ; q(pk) )Zp (z )

m=0 p=0

+

∞  ∞ 

k) k) im B(mp Ks(mp Ms(m3) (u(k) ; q(pk) )sem (v (k) ; q(pk) )Zp (z )

m=1 p=0

+

N  ∞  ∞ 

j) (j) im A(mp Kcmp Zp (z )

∞ 

j̸=k m=0 p=0

) (1) (k) (k) B(r3,m ,p Mcr (u ; qp )

r =0

× cer (v (k) ; q(pk) ) −i

N  ∞  ∞ 

j) (j) im A(mp Kcmp Zp (z )

∞ 

j̸=k m=0 p=0

(3) (−1)r B− r ,m,p

r =1

× Ms(r1) (u(k) ; q(pk) ) ser (v (k) ; q(pk) ) +i

N  ∞  ∞ 

j) j) im B(mp Ks(mp Zp (z )

(k)

∞  (−1)m B(r3,−) m,p r =0

(k)

(k)

(k)

× Mcr (u ; qp ) cer (v ; qp ) +

(−1)r B(−3r),m,p Ms(r1) (u(k) ; qp(k) )ser (v (k) ; qp(k) )

2i

(1)

r =0

−i

) B(r3,m ,p = −

∞ 

j̸=k m=1 p=0

) (1) (k) (k) (k) (k) B(r3,m ,p Mcr (u ; qp )cer (v ; qp )

∞ 

(j)

(k)

d∗r −l,l (q0 )Hl−s (k0 Rjk )ds−m,m (q0 )

× ei(s−l)ψjk e−is(βj −βk )

j̸=k m=1 p=0

(28)

d∗r −l,l (q(pk) )Hl−s (λRjk )ds−m,m (q(pj) )ei(s−l)ψjk e−is(βj −βk ) (31)

Br ,m,0 =

j̸=k m=0 p=0

(30)

r =1

N  ∞  ∞ 

j) j) im B(mp Ks(mp Zp (z )

j̸=k m=1 p=0

(29)

(1)

(k)

∞ 

(−1)m−r B(−3r),−m,p

r =1

(k)

(k)

(k)

× Msr (u ; qp ) ser (v ; qp ).

(34)

I.K. Chatjigeorgiou / European Journal of Mechanics B/Fluids 37 (2013) 153–164

Although Eq. (34) is quite complicated, its derivation is of paramount importance as it describes the total diffraction potential which is induced due to the scattering of waves by all bodies, with respect to the local elliptic coordinate system of the arbitrarily selected body k.

where

4. Expansion coefficients

Vps =

The derivation of Eq. (34) completes the mathematical formulation of the involved velocity potentials associated with the hydrodynamic problem at hand. The next step is the calculation of (k) (k) (k) (k) the expansion coefficients Amp , Bmp and Fmn , F˜mn associated with regions (I) and (II) respectively. These will be obtained by applying the body boundary condition (7) as well as the continuity relations at the common boundary between the fluid regions (I) and (II), namely

(ϕD(I ) + ϕI )u=u(k) = (ϕ (II ) )u=u(k) , 0 ≤ z ≤ hk 0 0   (I ) ∂ϕI ∂ϕD + = 0, hk ≤ z ≤ d ∂u ∂u (k)

(35)

(36)

+

(I )

∂ϕD ∂ϕI + ∂u ∂u



 = (k) u=u0

∂ϕ ∂u

(k) u=u0

,

0 ≤ z ≤ hk .

(37)

First, Eq. (35) will be applied. After introducing Eqs. (15), (25) and (34) into Eq. (35) and using the orthogonality relations of the even and odd periodic Mathieu functions as well as the orthogonality relation of the vertical eigenfunctions cos(sπ z /h), the following are derived: (k) in Fns

=

(1)

(k)

′(1)

(k)

1 h

Anl +

+i =−

+

(k)

=− (k) j) j) (3) (−1)m im B(mp Ks(mp Br ,−m,p Mcr(1) (u0 ; q(pk) )Vps

+

Msn (u0 ; Qs ) Msn (u0 ; Qs )

∞  ∞ 

(k)

(k) (k) im Bmp Ksmp Ms(m3) (u0 ; q(pk) )Vps Lsnm (Qs(k) ; q(pk) )

m=1 p=0

−i

N  ∞  ∞  ∞  j) (j) (3) (−1)r im A(mp Kcmp B−r ,m,p j̸=k m=0 r =1 p=0

× Ms(r1) (u0(k) ; qp(k) )Vps Lsnr (Qs(k) ; q(pk) ) +

(41)

2π



sen (v (k) ; Qs(k) )ser (v (k) ; q(pk) )dv (k) .

(42)

0

(j)

(j) (3)

) (−1)m im−n B(mlj) Ks(mlj) B(n3,− m,l

1

(k)

Kd Z0 (d)

Λk δ0l cen (α; q0 ) +

hk

1 ′(1)

d Mcn (u(k) ; q(k) ) l 0 (k)

(k) im−n εs Fms Vls Lc mn (Qs(k) ; ql )

(43)

N  ∞  ∞  ∞  j) j) (3) (−1)m−r im B(mp Ks(mp B−r ,−m,p j̸=k m=1 r =1 p=0

× Ms(r1) (u0(k) ; qp(k) )Vps Lsnr (Qs(k) ; q(pk) )

N  ∞ 

(j) (3) (−1)n im−n A(mlj) Kcml B−n,m,l

N  ∞ 

(−1)m−n im−n B(mlj) Ks(mlj) B(−3n)−m,l

2

1

Kd Z0 (d) hk

(k)

Λk δ0l sen (α; q0 ) 1

∞  ∞ 

1) (k) (k) d Ms′( n (u0 ; ql ) m=1 s=0

(k)

(k) im−n εs F˜ms Vls Lcmn (Qs(k) ; ql ).

(44)

(38)

=

π

cen (v (k) ; Qs(k) )cer (v (k) ; q(pk) )dv (k)

0

j̸=k m=1

×Lcnr (Qs(k) ; q(pk) ) ∞ 2 Λk V0s  m (1) (k) (k) (k) (k) i Mcm (u0 ; q0 )cem (α; q0 )Lcnm (Qs(k) ; q0 ) + Kd Z0 (d) m=0

(k)

π 1

2π



(40)

j̸=k m=0

j̸=k m=1 r =0 p=0

(k)

1

(−1)s σp h sin(σp h) (sπ )2 − (σp h)2

im−n Aml Kcml Bn,m,l

∞  ∞ 

(k)

j) (j) (3) im A(mp Kcmp Br ,m,p Mcr(1) (u0 ; q(pk) )Vps

∞ N  ∞  ∞  

′(1)

(k)

N  ∞ 

2

Bnl − i

×Lcnr (Qs(k) ; q(pk) )

(k) in F˜ns

h

dz = Np−1/2

j̸=k m=1

j̸=k m=0 r =0 p=0

(k)

(k)

m=0 s=0

(k)

(k)

(k)

j̸=k m=0

×

k) (k) (3) im A(mp Kcmp Mcm (u0 ; q(pk) )Vps Lcnm (Qs(k) ; q(pk) )

(1)

 sπ z 

0

N  ∞ 

m=0 p=0

+i

(k)

Next, Eqs. (15), (25) and (34) are introduced into Eqs. (36) and (37). Again the orthogonality relations for the even and the odd periodic Mathieu functions as well as the orthogonality relation of the vertical eigenfunctions Zp (z ) must be applied. It is important to note that the orthogonality relation for Zp (z ) should be employed first. After extensive mathematical manipulations we result in the following relations that combine the unknown expansion coefficients of the problem.

(k)

N  ∞  ∞  ∞ 

Zp (z ) cos

Lsnr (Qs(k) ; q(pk) ) =

(k)

Mcn (u0 ; Qs )

+

(k)

im Ms(m1) (u0 ; q0 )sem (α; q0 )Lsnm (Qs(k) ; q0 )

h



Lcnr (Qs ; qp ) =

Mcn (u0 ; Qs )

∞  ∞ 

m=1

(k)

(k)

(II ) 

Kd Z0 (d)

157

(39)

u=u0



2 Λk V0s

∞ 

Eqs. (43) and (44) can also be used for calculating the unknown expansion coefficients for the total diffraction potential in the simplified case of bottom mounted elliptical cylinders. In this case, the last terms of the right hand side of Eqs. (43) and (44) must be set equal to zero while the simplified expressions are employed only for l = 0. For more information on the subject the interested reader is referenced to the work of Chatjigeorgiou and Mavrakos [17]. Generally, Eqs. (38), (39), (43) and (44) construct a linear sys(k) (k) (k) tem which must be solved in terms of the unknowns Fns , F˜ns , Anl , (k)

Bnl . That system can be reduced to two only linear equations by introducing Eqs. (38)–(39) into the two others, yielding a strongly complicated system. The associated equations are quite lengthy and their detailed writing is omitted. After calculating the coeffi(k) (k) (k) cients Anl and Bnl the corresponding unknown coefficients Fns and (k)

F˜ns for the total velocity potential in the lower field (II) (see Fig. 2) are determined through Eqs. (38) and (39). In fact, the derivation (k) (k) (k) (k) of Fns , F˜ns , Anl , Bnl completes the solution of the problem. The

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diffraction potential in the infinite region (I) is subsequently obtained through Eq. (28) whereas the total velocity potential in the lower region (II) is provided by Eq. (25). For the numerical implementation which starts from the calculation of the unknown ex(k) (k) (k) (k) pansion coefficients Fns , F˜ns , Anl , Bnl and results in the derivation of the velocity potentials as well as the hydrodynamic pressure distributions, the hydrodynamic loading and the wave elevation, it is necessary to truncate the sums that provide the components of the velocity potentials, in terms of the number of eigenmodes m and the number of eigenfunctions n and p (according to the notation adopted in Eqs. (25) and (28)). 5. Pressure distributions, exciting forces and wave elevation The linear hydrodynamic pressure referring to wetted surfaces of any elliptical cylinder k of the array and normalized by ρ g (H /2) will be written as (I )

P (I ),(k) = Kd(ϕI + ϕD )u(k) =u(k)

(45)

P (II ),(k) = Kd(ϕ (II ) )z =hk .

(46)

0

The indices (I) and (II) in the above relations imply that the pressure is calculated on the lateral and the bottom surfaces respectively. The hydrodynamic exciting forces in surge, sway and heave directions are obtained by integrating the linear hydrodynamic pressures on the wetted surfaces of body k. These forces are given in the following nondimensional form after they have been normalized by ρ ga2k (H /2) Fx(k) = −Kd Fy(k) = −Kd

Fz(k) = −

bk



a2k 1





d

2π



hk

ak

Kd

d

(47)

(ϕI + ϕD(I ) )u(k) =u(k) sin v (k) dv (k) dz

(48)

0

0 2π



(ϕI + ϕD(I ) )u(k) =u(k) cos v (k) dv (k) dz

Fig. 3. Geometrical definitions for addition theorems.

hk

0

ck

2 

0

(k) u0



2π

(ϕ (II ) )z =h

k 2 ak 0 0 (k) (k) (k) × (cosh 2u − cos 2v )du dv (k) .

(49)

Finally the wave elevation calculated with respect to the local elliptic coordinate system of body k(u(k) , v (k) ) and normalized by H /2 is written as

  η(k) . = Kd ϕI + ϕD(I ) z =d H /2

(50)

Clearly, when employing Eqs. (45)–(50), the corresponding velocity potentials must be expressed with respect to the local elliptic coordinate system of the elliptical cylinder k. 6. Numerical results and discussion The solution method outlined above can be used for a variety of hydrodynamic interaction problems. It is equally efficient for calculating the hydrodynamic loading, wave run-up and linear pressure distributions on bottom mounted interacting elliptical cylinders. This is achieved by letting the draught of the cylinder to be nearly equal to the water depth. It can be also applied for isolated bodies by extending the separation distances Rjk (see Fig. 1) between the centers of the cylinders. Finally, it can approximate simpler geometries such as circular cylinders or plates by adjusting the elliptic eccentricity to be nearly zero and unity respectively. In fact the latter case, i.e. an elliptical cylinder approximating a plate was used for validation purposes and the associated results are shown in Fig. 4. In particular this

Fig. 4. Linear run-up on a plate (on the half of its length, a = 5 m) approximated as an elliptical bottom mounted cylinder with e → 1. Deep water was assumed. The y axis in the figure coincides with semi-major axis of the body. Wave heading 90°, Ka = 19.725.

figure shows the linear wave run-up on the half length of a 10m plate that was approximated as an isolated bottom mounted elliptical cylinder with e → 1 and semi-major axis a = 5 m. The plate was subjected to monochromatic incident waves with period T = 1.01 s. The numerical results were compared with those reported by Molin et al. [28] demonstrating an extremely favorable coincidence. It should be mentioned herein that the comparison concerns only the linear case, namely the first impact of the incident waves to the plate. Molin et al. [28] extended their study to investigate the tertiary wave interactions which clearly cannot be captured by the linear theory employed in the present. It is interesting to observe that after the edge of the plate the amplitude of the waves is equal to unity as was expected to be. The solution method was also validated against the results reported by Chatjigeorgiou and Mavrakos [17] for two interacting bottom mounted elliptical cylinders showing again a perfect coincidence. No additional comparisons will be shown in the present as it would be a redundant repetition. On the

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159

Fig. 5. Top view of an array of two identical elliptical truncated cylinders: b/a = 0.25; d/a = 1.5; h/d = 1/3. Fig. 7. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 30° (first configuration).

Fig. 6. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 0° (first configuration).

contrary new results for interacting truncated elliptical cylinders are provided and the calculations are discussed in the following. Clearly, the convergence of the calculations depends on the order of truncation. For the case studies presented herein satisfactory convergence was achieved using m = 5 modes and n = p = 3 eigenfunctions for the lower and the infinite fields respectively according to the notations of Eqs. (25) and (28). The examined configurations are shown schematically in Fig. 5. The first configuration consists of two parallel identical truncated cylinders with b/a = 0.25, d/a = 1.5 and h/d = 1/3. The centers of the cylinders fall in the same perpendicular and the separation distance is determined by R/a = 3. In the second configuration, the parallel placement is retained whereas the top cylinder is moved horizontally by 2a. The results for the hydrodynamic exciting forces (normalized by ρ g (H /2)a2 ) are shown in Figs. 6–10 for the first configuration and in Figs. 11–15 for the second configuration. Five heading angles were considered namely 0◦ , 30◦ , 45◦ , 60◦ and 90◦ . For the first configuration the wave heading 0◦ is symmetrical for the two bodies leading to equal exciting forces (see Fig. 6). The heave force starts from a constant value at k0 a → 0 and decays progressively until it practically vanishes. Regarding the

Fig. 8. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 45° (first configuration).

surge forces the effect of the interaction is manifested through the variations in the associated curve at the high wave frequency range whereas the most profound impact occurs in the sway force where it is evident that the interacting waves induce large lateral loads and peaks at specific frequencies. The symmetry is lost for wave headings different than 0◦ (Figs. 7–10). Here the sway forces become dominant and exhibit strong variations. The curves depicting the wave loadings on the two bodies are detached meaning that the exciting forces are different. The surge forces follow a smoother variation compared to the sway forces whereas it is interesting to observe that the forces in heave exhibit apparent peaks along the investigated frequency range. The most interesting remarks however concern the sway loading the magnitude of which admits large values. Although the exposed cylinder (here denoted as body I) experiences the maximum of the hydrodynamic impact, the assumed ‘‘protected’’ cylinder (denoted as body II) is subjected to strong loading as well. In fact, at the high wave frequency range the magnitudes of the sway exciting forces are comparable. Moreover, the associated curves follow a wavy trend

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Fig. 9. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 60° (first configuration).

Fig. 10. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 90° (first configuration).

dominated by several sharp peaks. The situation becomes clearer for wave heading 90◦ (see Fig. 10). It should be mentioned that analogous phenomena are not induced by an array of similarly arranged bottom mounted cylinders (see e.g. [17]). Clearly the profound wavy variation of the sway forces is the outcome of the truncation of the cylinders. As expected, for perfectly beam seas the specific arrangement of the bodies results in zero surge forces. In the second configuration, i.e. when the upper cylinder is moved horizontally by 2a the symmetry is lost completely regardless the wave heading angle. This is immediately apparent in Figs. 11–15 which show that different exciting forces are applied on the two bodies. In fact, the variation of the hydrodynamic loading with respect to the wave frequency can be considered unpredictable. It is evident that the specific placement eliminates the sharp peaks in the sway loading curves but the hydrodynamic interaction phenomena are still determinative as can be deduced by the complicated variation depicted in the figures. With regard to the resulting wave pattern around the bodies, some indicative results for arbitrary selected cases are shown

Fig. 11. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 0° (second configuration).

Fig. 12. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 30° (second configuration).

in Figs. 16–19 which correspond to the first configuration. The body locations are also included schematically to provide a visual representation of the disturbances induced on the free surface due to the hydrodynamic interaction. In fact the angle of propagation is easily detected through the direction of the local elevations. Figs. 16 and 17 correspond to waves propagating along the semimajor axes of the bodies and demonstrate significant changes in the wave scattering in terms of the incident wave frequency. For the larger wavelength k0 a = 2.0 (Fig. 16) strong disturbances occur in a wide area around the bodies, which in addition exhibit a symmetry due to the symmetrical placement of the bodies with respect to the direction of the incoming waves. The maximum wave elevation occurs at symmetrical positions ahead and behind the bodies whereas it is evident that the diffraction phenomena decay away from the cylinders. For the shorter wavelength k0 a = 4.0 (Fig. 17) the stronger disturbances are concentrated around the bodies. The maximum elevation occurs at the downstream nose while the pattern of the free surface elevation behind the

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161

Fig. 13. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 45° (second configuration).

Fig. 15. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 90° (second configuration).

Fig. 14. Exciting forces on an array of two identical elliptical truncated cylinders (see Fig. 5) for wave heading α = 60° (second configuration).

Fig. 16. Wave elevation |η|/(H /2) between the elliptical truncated cylinders of the first configuration of Fig. 5. Wave heading α = 0°; k0 a = 2.0. The arrow shows the direction of the incoming waves.

bodies obtains the form of V type waves. For both investigated wave frequencies the maximum elevations are comparable. The latter obtain significantly larger values for nonzero wave heading angles as shown in Figs. 18 and 19 which are dedicated to 45◦ and 90◦ of heading with respect to the horizontal. The wavelength in both cases is the same and corresponds to k0 a = 2.7. Indeed the angle of propagation is visually evident for both heading angles. Here the maximum elevation reaches up to 3.0–3.5 the amplitude of the incoming waves, it is localized, and occurs in the area between the bodies. It could be said that somehow the diffracted waves are trapped in the intermediate region. The most interesting remark however concerns the case of perfect beam seas (Fig. 19). Here apart from the localized magnifications along the wave propagation axis the depicted scattering pattern manifests that the lee side elliptical cylinder is not actually protected. In fact the maximum run-up occurs on its front side. This is definitely a crucial conclusion which demonstrates the determinative impact of the hydrodynamic interactions.

7. Conclusions The problem of hydrodynamic scattering by arrays of elliptical truncated cylinders was investigated. The present study extends the work of Chatjigeorgiou and Mavrakos [17] to consider truncated bodies. The solution of the hydrodynamic boundary value problem was achieved analytically using semi-analytical forms for the various velocity potentials in the fluid regions defined by the geometry of the interacting axisymmetric bodies. Also, the Mathieu functions addition theorem was employed to express the various diffraction potentials with respect to a global elliptic coordinate system. The developed methodology accounts for arbitrary angular placements of all bodies with respect to the horizontal axis of the global Cartesian coordinate system. It also accounts for negative Mathieu parameters that correspond to the infinite evanescent modes of the transcendental equation. The application of the zero velocity conditions on the wetted surfaces as well as the continuity relations of the velocities and

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Fig. 17. Wave elevation |η|/(H /2) between the elliptical truncated cylinders of the first configuration of Fig. 5. Wave heading α = 0°; k0 a = 4.0. The arrow shows the direction of the incoming waves.

Fig. 19. Wave elevation |η|/(H /2) between the elliptical truncated cylinders of the first configuration of Fig. 5. Wave heading α = 90°; k0 a = 2.7. The arrow shows the direction of the incoming waves.

system (r, γ ) fixed on the center of the cylinder at point O′ . Then, according to Meixner and Schäfke [25] (j) Mm (ˆu; qˆ )mem (ˆv ; qˆ ) =

∞ 

(j)

ds,m (ˆq)Zm+s (kr )ei(m+s)γ

(A.1)

s=−∞

(j)

where mem (ˆv ; qˆ ) and Mm (ˆu; qˆ ) are respectively the mth order periodic and modified Mathieu functions while (uˆ , vˆ ) denotes the local elliptic coordinate system fixed on point O′ . Furthermore, qˆ is the Mathieu parameter which is defined by the geometry of the upper side elliptical body. The index j here denotes the kind of the Mathieu function with j = 1, 2,3,4 and

(j) Zm (kr ) =

 Jm (kr )  Y (kr )  

Fig. 18. Wave elevation |η|/(H /2) between the elliptical truncated cylinders of the first configuration of Fig. 5. Wave heading α = 45°; k0 a = 2.7. The arrow shows the direction of the incoming waves.

the pressures at the adjacent elliptical boundary of the infinite and the lower fluid regions, provided a quite complicated complex linear system to be solved in terms of the unknown expansion coefficients of the involved diffraction potentials. The method was validated against reported numerical data. The numerical results presented in the study concerned two different two-body configurations. The hydrodynamic exciting forces and indicative results for the free surface elevation were provided. The calculations manifested the strong impact of the hydrodynamic interactions as well as evidence of cylinders’ truncation. Appendix A. Addition theorems for even and odd Mathieu functions

m (1) Hm (2) Hm

(kr ) (kr )

j j j j

=1 =2 =3 =4

(A.2)

where Jm and Ym are the Bessel functions of the first and the second (1),(2) = Jm ± iY m are the Hankel functions. kind respectively and Hm Finally the coefficients ds,m are given by Særmark [20] m d2s,m = (−1)s C2s (ˆq),

d2s+1,m = 0.

(A.3)

m Here C2s (ˆq) denotes the expansion coefficients of the periodic Mathieu functions

mem (ˆv ; qˆ ) =

∞ 

m C2s (ˆq)ei(m+2s)ˆv .

(A.4)

s=−∞

Eq. (A.1) is valid for r > aˆ , where aˆ is the semi-major axis of the upper side elliptical cylinder. Next we employ the celebrated Graf’s addition theorem for Bessel functions between the local polar coordinate systems (r, γ ) and (L, θ ). The second system is fixed on the center O of the lower body. Thus,

(−1)p Zp(j) (kr )e−ipξ =

∞ 

(j)

eisψ Js (kL)Z−p+s (kR).

(A.5)

s=−∞

With reference to the arrangement of Fig. 3, we first examine the upper side elliptical body together with the polar coordinate

Eq. (A.5) which is valid for L > R, is introduced into Eq. (A.1) and yields

I.K. Chatjigeorgiou / European Journal of Mechanics B/Fluids 37 (2013) 153–164

(j) Mm (ˆu; qˆ )mem (ˆv ; qˆ ) ∞ ∞  

=

(−1)p dp−m,m (ˆq)eipγ eisψ eipξ Js (kL)Z−(j)p+s (kR).

(A.6)

p=−∞ s=−∞

The transformation between polar and elliptic coordinate systems in the center of the lower point O is achieved through the following relation [25] Js (kL) = e

−isθ

∞ 

(1) d∗l,s (q)Ms+l (u; q)mes+l (v; q).

(A.7)

For calculation purposes it is more convenient to employ the A and B expansion coefficients of the even and odd periodic Mathieu functions instead of using the coefficients ds−m,m and d∗r −l,l . The general forms that provide the periodic functions cem and sem are ce2m (v; q) =

∞ 

A2m 2r (q) cos 2r v

(A.15)

r =0

ce2m+1 (v; q) =

∞ 

2m+1 A2r +1 (q) cos(2r + 1)v

(A.16)

2m+1 B2r +1 (q) sin(2r + 1)v

(A.17)

2m+2 B2r +2 (q) sin(2r + 2)v.

(A.18)

r =0

l=−∞

It is noted that (u, v) are the elliptic coordinates of the local elliptic coordinate system of the lower body, the center of which coincides with the origin of that system at point O. In addition q herein denotes the Mathieu parameter that is defined by the particulars of the lower body. Also,

163

se2m+1 (v; q) =

∞  r =0

se2m+2 (v; q) =

∞  r =0

s+2l d∗2l,s = (−1)l C− 2l (q),

d∗2l+1,s = 0.

(A.8)

Eq. (A.7) is valid for u ≥ 0. Finally, after introducing Eq. (A.7) into Eq. (A.6) and rearranging the various indices we obtain the final form of the addition theorem of Mathieu functions (j) Mm (ˆu; qˆ )mem (ˆv ; qˆ ) ∞ ∞ ∞   

=

(−1)

s

r =−∞ s=−∞ l=−∞

(A.9)

Eq. (A.9) can be written in a more convenient form according to (j) Mm (ˆu; qˆ )mem (ˆv ; qˆ ) =

∞ 

Br(j,)m,p Mr(j) (u; q)mer (v; q).

d∗r −l,l (q) = 2−1/2 (−1)(r −l)/2 Arl (q) d∗−r −l,l (q) = −2−1/2 (−1)(−r −l)/2 Brl (q) d∗2r ,0 (q) = 21/2 (−1)r A2r 0 (q)

(A.10)

r =−∞

B(rj,)m,p =

(j)

(j)

ds−m,m (ˆq)d∗r −l,l (q)Z−s+l (kR)ei(s−l)ψjk e−isβj eilβk .

(j)

(A.11)

cem (ˆv ; qˆ ) = 2−1/2 mem (ˆv ; qˆ ) sem (ˆv ; qˆ ) = i2−1/2 me−m (ˆv ; qˆ ) (j) Mm (ˆu; qˆ ) = Mcm(j) (ˆu; qˆ )

(−1)m M−(j)m (ˆu; qˆ ) = Ms(mj) (ˆu; qˆ )

 (m = 0, 1, 2, . . .)   (m = 1, 2, 3, . . .) (m = 0, 1, 2, . . .)   (m = 1, 2, 3, . . .)

=

B(rj,)m,p Mcr(j) (u; q)cer (v; q)

∞  (−1)r B(−j)r ,m,p Ms(rj) (u; q)ser (v; q)

(A.13)

r =1

Ms(mj) (ˆu; qˆ )sem (ˆv ; qˆ )

=i

∞  ) (j) (−1)m B(rj,− m,p Mcr (u; q)cer (v; q) r =0

+

∞  (−1)r +m B(−j)r ,−m,p Ms(rj) (u; q)ser (v; q). r =1

(j)

∞ ∞ 1  

(−1)(−r −l+s−m)/2 Brl (q)Am q) s (ˆ 2 s=−∞ l=−∞ × Z−(j)s+l (kR)ei(s−l)ψjk e−isβj eilβk 1

∞ 

(j)

B−r ,−m,p =

1

∞ 

(A.22)

∞ 

(−1)(r −l+s+m)/2 Arl (q)Bm q) s (ˆ 2 s=−∞ l=−∞ × Z−(j)s+l (kR)ei(s−l)ψjk e−isβj eilβk

Br ,−m,p = −

(A.21)

(A.23)

∞ 

(−1)(−r −l+s+m)/2 Brl (q)Bm q) s (ˆ 2 s=−∞ l=−∞ × Z−(j)s+l (kR)ei(s−l)ψjk e−isβj eilβk .

(A.24)

References

r =0

−i

(−1)(r −l+s−m)/2 Arl (q)Am q) s (ˆ 2 s=−∞ l=−∞ × Z−(j)s+l (kR)ei(s−l)ψjk e−isβj eilβk

(A.12)

(j) Mcm (ˆu; qˆ )cem (ˆv ; qˆ ) ∞ 

∞ ∞ 1  

B−r ,m,p = −

s=−∞ l=−∞

In the above relation the index p was introduced to denote that is depending on the roots of the transcendental Eq. (17) which determine the values of the Mathieu parameters qˆ and q and accordingly the expansion coefficients ds−m,m (ˆq) and d∗r −l,l (q). Further, Eq. (A.10) is recast into even and odd forms using the following transformation formulas [25]

(m = 0, 1, 2, . . .) (m = 1, 2, 3, . . .) (A.20) (j)

Fig. 3) the coefficient Br ,m,p admits the following form ∞ ∞  

(A.19)

Consequently, the expressions for Br ,m,p which are involved in the addition theorems of even and odd Mathieu functions (see Eqs. (A.13) and (A.14)) become

Observing that θ = ψ + ψjk βk and γ + ξ = π + ψjk βj , (see

B(rj,)m,p =

(r = 0, 1, 2, . . .) (r = 1, 2, 3, . . .)

ds−m,m (q) = 2−1/2 (−1)(s−m)/2 Am s (q) ds+m,−m (q) = −2−1/2 (−1)(s+m)/2 Bm s (q) d−2m,2m (q) = 21/2 (−1)−m A2m ( q ). 0

(j) ds−m,m (ˆq)d∗r −l,l (q)Z−s+l (kR)eis(γ +ξ )

× eil(ψ−θ) Mr(j) (u; q)mer (v; q).

In order to transform Eq. (A.11) and achieving dependence from coefficients A and B, the following formulas will be employed [17]

(A.14)

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