Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed

Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed Ioannis K. Chatjigeorgiou a,n, Touvia Miloh b a School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroö n Polytechniou Ave, Zografos Campus, Athens 15773, Greece b Faculty of Engineering, Tel-Aviv University, Ramat Aviv 69978, Israel

a r t i c l e i n f o

abstract

Article history: Received 5 April 2014 Accepted 21 April 2014

The present study treats the hydrodynamic diffraction problem including forward speed of a fully submerged prolate spheroid advancing rectilinearly under a monochromatic wave field in water of infinite depth. The analytic method explicitly satisfies the Kelvin– Neumann boundary conditions. The formulation is based on employing spheroidal harmonics and expressing the ultimate image singularity system as a series of multipoles distributed along the major axis of the spheroid between the two foci. The outlined procedure results in compact closed-form expressions for the six Kirchhoff velocity potentials as well as for the various components of the hydrodynamic loads exerted on the rigid body moving under waves. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Green's function Multipole expansion Image singularities Wave resistance Wave diffraction Spheroidal harmonics

1. Introduction The hydrodynamics of immersed spheroids moving under waves in deep water have attracted the attention of many notable researchers without however admitting a generally acceptable analytic solution. With regard to the hydrodynamic applications, spheroidal geometries (oblates and prolates) are designated as axisymmetric and non-axisymmetric depending on whether their longitudinal symmetrical axis is perpendicular or parallel to the undisturbed free-surface respectively. Among the various common arrangements, the most important practical applications are those related to the nonaxisymmetric case of prolate spheroids (stationary or moving in sea waves) and for that reason they have been the focus of investigation of many significant contributions in the past. The ultimate task is obtaining a solution of the coupled forward speed–wave diffraction problem which must account for the transformation of the diffraction potential due to the flow lines which are induced by the steady forward speed of the spheroid. An analytic linearized solution for the combined case is presented here for the first time. The reported works on the subject adequately address and solve either the Wave Resistance or the Wave Diffraction problems, but not the coupled problem. The wave resistance problem of a non-axisymmetric prolate spheroid, moving rectilinearly in an otherwise calm sea, was originally tackled by Havelock (1931) using a slender body approach (namely, without satisfying the Neumann body boundary condition). For calculating the wave resistance, Havelock (1931) employed the Lagally (1922) theorem. An exact solution for the concerned problem was first suggested by Farell (1973) who sought the solution in a form of a source distribution over the surface of the spheroid (Farell, 1971, 1973). The employed method resulted in a Fredholm

n

Corresponding author. Tel.: þ30 210 772 1105; fax. þ 30 210 772 1412. E-mail address: [email protected] (I.K. Chatjigeorgiou).

http://dx.doi.org/10.1016/j.jfluidstructs.2014.04.012 0889-9746/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

integral equation of the second kind for the source distribution. The wave resistance problem was also tackled numerically by Doctors and Beck (1987), using the Kelvin–Neumann formulation. Farell's (1973) formulation was accordingly implemented in a series of studies by Wu and Eatock Taylor (1987, 1988, 1989) and Wu (1995) which investigated the wave diffraction – radiation problems of submerged rigid spheroids. The same authors also considered the coupled diffraction and forward speed problem but its numerical implementation was only restricted to spherical shapes (Wu and Eatock Taylor, 1989; Wu, 1995). Recently Chatjigeorgiou (2012, 2013) adopted a different line of approach to solve the wave diffraction problem by oblate (Chatjigeorgiou, 2012) and prolate (Chatjigeorgiou, 2013) spheroids. In particular he based his solution on Thorne's (1953) method of multipole expansions, originally given in spherical and polar coordinates. By using a proper addition theorem (Cooke, 1956), he succeeded to obtain a similar transformation in oblate and prolate spheroidal coordinates. Both studies however (Chatjigeorgiou, 2012, 2013) are dedicated to axisymmetric spheroids. The most comprehensive study available on the combined forward speed–wave diffraction problem by non-axisymmetric spheroids, again assuming a slender body advancing under waves, appears to be that of Havelock's (1954). Nevertheless, Havelock's (1954) solution for the uniform stream potential does not actually satisfy the free surface boundary condition and it only accounts for the body's boundary condition in an otherwise infinite medium. The present paper aims to fill in the gap towards the quest of a complete and efficient analytical solution for the general diffraction problem of a non-axisymmetric prolate spheroid moving with a constant forward speed under a monochromatic wave field of arbitrary wave heading. The solution is valid for any depth of submergence and it satisfies both the linearized free-surface and the Neumann body boundary conditions. The present study extends a recent effort by the same authors, related only to the diffraction problem of stationary prolate spheroids (Chatjigeorgiou and Miloh, 2013). Here again we employ Miloh's theorem (Miloh, 1974) for external spheroidal harmonics, which appears to simplify the analysis. The final solution is distinguished for its relative versatility and elegance. Its accuracy and efficacy is demonstrated through comparative calculations against reported numerical data of the diffraction problem. We also present results of several numerical simulations for different spheroids and wave heading angles. Our final goal was to highlight the effect of the forward speed on the exciting pressure forces and moments acting on the spheroid during the steady course of its motion and to demonstrate that these wave loads are indeed significant and clearly non-negligible.

2. Formulation of the hydrodynamic problem The prolate spheroid (see Fig. 1) is considered submerged at a distance f below the undisturbed free surface in a fluid of infinite depth. The semi-major and the semi-minor axes of the spheroid are denoted respectively by a and b, whilst its qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eccentricity is e ¼ 1 ðb=aÞ2 . The linearized analysis is dedicated to the non-axisymmetric case (i.e., major axis parallel to the free-surface), which has significant practical interest for naval applications. We start from the general loading condition in which the spheroid advances with steady velocity U and is simultaneously exposed to the action of monochromatic incident waves propagating at an arbitrary heading angle β relatively to the longitudinal axis of the spheroid. Left-handed Cartesian (x, y, z) coordinates are employed, fixed on the undisturbed free surface, with the vertical z-axis pointing downwards. The coordinate system that is fixed on the center of the body (x, y, zn) is obtained from the former with the simple transformation z¼ zn þf. Within the realm of the linear theory the velocity potential is written as n o Φ ¼  Ux þ Ucϕðx; y; zÞ þRe ½ϕI ðx; y; zÞ þ ϕD ðx; y; zÞeiωt ;

ð1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where c ¼ a2  b ¼ ae is the half distance between the foci of the spheroid, ϕI and ϕD denote the incident and the diffraction potentials respectively, ϕ is the disturbance due to the motion of the body and ω is the frequency of encounter. Apparently ϕ, ϕI and ϕD are harmonic functions, which satisfy the linearized free-surface boundary condition. For the time-

Fig. 1. 3-D image of a prolate spheroid below the free surface (non-axisymmetric case) with a/b ¼6.

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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varying components of the velocity potential, the latter reads   ∂ϕj ∂ 2  iω þU ¼ 0; ϕj g ∂x ∂z

3

ð2Þ

for j¼I and D, whilst for the steady component ϕ one gets ∂2 ∂ϕ ϕ  ξ ¼ 0; ∂z ∂x2

ð3Þ

where ξ ¼ g=U 2 and g denotes the gravitation acceleration. Eqs. (2) and (3) are applied on z¼0 and it is understood that all potentials satisfy a proper radiation condition at infinity. The incident wave potential ϕI ðx; y; zÞ is taken in the form ϕI ðx; y; zÞ ¼

igA  K 0 z  iK 0 ðx e ω0

cos β þ y sin βÞ

;

ð4Þ

where ω0 is the wave frequency, A is the linear amplitude and K 0 ¼ ω20 =g. Following Eq. (2) the encounter frequency is connected to the incident wave frequency ω0 by the following relation: ω ¼ ω0  K 0 U cos β;

ð5Þ

whilst the encounter wave number is K ¼ ω2 =g. In Eq. (4) β¼π denotes head seas and β¼ 0 denotes following seas. Finally in order to satisfy the Neumann body boundary condition, the following must hold on the body surface: ∂ϕD ∂ϕ ¼  I; ∂n ∂n

ð6Þ

∂ ð  x þ cϕÞ ¼ 0; ∂n

ð7Þ

where n is the normal on the body surface S0 directed into the fluid. 3. The steady and the diffraction potentials 3.1. Green's function in Cartesian coordinates Let us first rederive following Wehausen and Laitone (1960, Section 13), the corresponding Green's function for the linearized 3-D free-surface boundary condition [see Eq. (2)]. Thus, using the well-known integral relation Z 1Z π 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e  kjz  f j þ ikðx cos a þ y sin aÞ da dk; ð8Þ 2π 0 2 π 2 2 x þ y þ ðz  f Þ and taking the (x, y) Fourier transform of Eq. (2), one gets ^ pffiffiffiffi pffiffiffiffiffiffiffiffi ^ þ ∂ϕ ¼ 0; ð9Þ ð K k 1=ξ cos aÞ2 ϕ ∂z ^ denotes the Fourier Transform of ϕ. Green's function, satisfying the linear free-surface condition, can then be where ϕ simply written as Z 1Z π 1 1 Zðk; aÞe  kðz þ f Þ þ ikðx cos a þ y sin aÞ da dk; ð10Þ Gðx; y; zÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2π 0 π x2 þy2 þ ðz f Þ2 where

pffiffiffiffi pffiffiffiffiffiffiffiffi ð K  k 1=ξ cos aÞ2 þ k pffiffiffiffiffiffiffiffi : Zðk; aÞ ¼ pffiffiffiffi ð K  k 1=ξ cos aÞ2  k

ð11Þ

The above 3-D relation contains both the case of a stationary pulsating source when U¼1/ξ¼0 [see Eq. (13.15) in Wehausen and Laitone, 1960], as well as that for a steadily moving source corresponding in Eq. (11) to K¼0 [see Eq. (13.36) in Wehausen and Laitone, 1960]. We note that the kernel (11) has a simple pole at a certain value of k (vanishing denominator) and thus the integral in (11) is in fact improper unless defined as a principal value (see for example, Gradshteyn and Ryzhik, 2007, Section 3.05, p. 252). The frequency-domain kernel (11) could be accordingly recast using the Laplace transform to incorporate the effects of water of finite depth, or even some time-dependent (time domain) linear free-surface effects (see for example Wehausen and Laitone, 1960). Clearly for 1/ξ¼0 we recover the diffraction problem whereas for K¼0 we get the forward speed (wave resistance) problem. It is also possible to formulate (using the same framework) the corresponding radiation problem by considering the spheroid to be freely floating and executing small oscillations about mean (equilibrium) position in response to the incident regular wave. However, the detailed analysis of the above “ship motion” problem will be deterred to a future study. Thus, the present work is focused on determining the hydrodynamic pressure loads exerted on a fully submerged prolate restrained spheroid moving rectilinearly with a constant (surge) velocity parallel to the undisturbed free surface in the presence of regular waves. Analytic Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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solutions for both classical problems of “wave resistance” (calm sea) as well as “pure diffraction” of a fixed body due to an incident wave-field can be obtained as limiting cases of the presented combined “diffraction with forward speed” problem. 3.2. Expansion in prolate spheroidal coordinates Green's function given by Eq. (10) can be effectively transformed into spheroidal coordinates. Hence the non-axisymmetric prolate spheroidal coordinate system (u, ϑ, ψ), 0 ru o1, 0 r ϑ r π and 0 rψ o2π is introduced and it is assumed to be fixed on the center of the body. The transformation between the concerned system and the Cartesian one is x ¼ c cosh u cos ϑ;

ð12Þ

y ¼ c sinh u sin ϑ sin ψ;

ð13Þ

z ¼ c sinh u sin ϑ cos ψ:

ð14Þ

Using the notation originally introduced by Nicholson (1924) the above are written alternatively as x ¼ ζμ and z þ iy ¼ ðζ 2 1Þ1=2 ð1 μ2 Þ1=2 eiψ , where ζ ¼ cosh u and μ ¼ cos ϑ. We employ throughout the paper dimensionless representations where lengths are normalized with respect to the half distance between the foci c (equivalent to letting c ¼1). In order to express the original Green's function as a series of multipoles, we make use of Miloh's (1974) theorem for the ultimate image singularity system for spheroidal harmonics. In particular we employ the following fundamental relation that expresses any exterior spheroidal harmonic in terms of prescribed singularities distributed on the major axis of the spheroid between the two foci:   Z 1 ∂ ∂ m 1 ð1 λ2 Þm=2 P m m imψ n ðλÞ ffidλ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm þi ¼ ð15Þ n ðμÞQ n ðζÞe 2 ∂z ∂y 2 1 2 ðx  λÞ þ y þ z2 m where P m n and Q n denote the associate Legendre functions of the first and the second kind respectively. Eq. (15) was first suggested by Havelock (1952) without a proof and later rigorously obtained by Miloh (1974). Also, we choose to reverse the orientation of the z-axis whereas after manipulating Eq. (10) with the aid of Eq. (15) the former admits the following form: Z 1 1 m m m m imψ Gm  k e  kðf  zÞ f n ðx; y; k; aÞdk; ð16Þ n ðx; y; zÞ ¼ P n ðμÞQ n ðζÞe 4π 0

where Z

m

f n ðx; y; k; aÞ ¼

π π

Zðk; aÞð1  sin aÞm eikðx

cos a þ y sin aÞ

Z

1 1

 ikλ ð1  λ2 Þm=2 P m n ðλÞe

cos a

ð17Þ

dλ da:

To simplify Eq. (17) and accordingly Green's function (16), we must find an appropriate closed form expression for the second definite integral. This is Z

1 1

 ikλ ð1  λ2 Þm=2 P m n ðλÞe

¼ 2ð  1Þn in þ m

cos a

dλ

 1=2 ðn þ mÞ! 1 π J n þ 1=2 ðk cos aÞ; m ðn  mÞ!ðk cos aÞ 2k cos a

ð18Þ

where J denotes the Bessel function of the first kind. Eq. (18) is obtained using the following relation given in Gradshteyn and Ryzhik (2007, 7.321, p. 797): Z 1 π21  ν in Γð2ν þnÞ  ν α J ν þ n ðαÞ; ð1  λ2 Þν  1=2 eiαλ C νn ðλÞdλ ¼ n!ΓðνÞ 1

ð19Þ

which is valid for Re(ν)4  1/2 while n is an integer. Here C νn ðλÞ denotes the Gegenbauer polynomial given by 1=2  ν

C νn ðλÞ ¼ 2ν  1=2 Γðn þ2νÞΓðν þ 1=2Þð1 λ2 Þ1=4  1=2ν P n þ ν  1=2 ðλÞ=Γð2νÞΓðn þ1Þ:

ð20Þ

After substituting Eq. (20) into the fundamental relation (19) and performing several tedious mathematical manipulations, the latter yields the sought transformation (18). Next, we introduce Eq. (18) into Eqs. (17) and (16) which yields 1 ðn þ mÞ! m m imψ þ ð  1Þn þ 1 in þ m Gm n ðx; y; zÞ ¼ P n ðμÞQ n ðζÞe 2π ðn  mÞ! Z 1Z π 1=2 ð1  sin aÞm  2kf  π n  Zðk; aÞ J n þ 1=2 ðk cos aÞekz þ ikðx m e 2k cos a ð cos aÞ 0 π

cos a þ y sin aÞ

da dk:

ð21Þ

Note that we have also changed the origin of the z-axis letting z ¼zn f, where zn denotes the vertical axis fixed on the center of the spheroid and pointing upwards. Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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Eq. (21) is manipulated further by using a useful relation, originally derived by Havelock (1954): ekz

n

þ ikðx cos a þ y sin aÞ

1

s

¼ ∑ ∑ ½F ts cos tψ þ iF~ ts sintψP ts ðμÞP ts ðζÞ;

ð22Þ

s¼0t ¼0

where  1=2 εt s þ t ðs  tÞ! π ð2s þ1ÞNt ðaÞ i J s þ 1=2 ðk cos aÞ; ðs þ tÞ! 2k cos a 2

ð23Þ

 1=2 t εt ðs  tÞ! π ð2s þ 1ÞN~ t ðaÞ J s þ 1=2 ðk cos aÞ; F~ s ¼ is þ t ðs þ tÞ! 2k cos a 2

ð24Þ

F ts ¼

ð1 þ sin aÞt þ ð1  sin aÞt ; ð cos aÞt

ð25Þ

ð1 þ sin aÞt ð1  sin aÞt ; N~ t ðaÞ ¼ ð cos aÞt

ð26Þ

Nt ðaÞ ¼

whilst εt ¼1 for t¼0, εt ¼2 for t¼1,2,… Next, Eqs. (22)-(26) are introduced into Eq. (21) which is finally expressed as i 1 s h m m t t imψ ~ mt Gm þ ∑ ∑ C mt ð27Þ n ðx; y; zÞ ¼ P n ðμÞQ n ðζÞe ns cos tψ þiC ns sin tψ P s ðμÞP s ðζÞ; s¼0t ¼0

where εt ðn þ mÞ!ðs tÞ! ð2s þ 1Þ ð  1Þn þ 1 in þ m þ s þ t ðn  mÞ!ðsþ tÞ! 8 Z 1  2kf Z π e N m ðaÞNt ðaÞ J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞda dk;  Zðk; aÞ cos a k 0 0 C mt ns ¼

ð28Þ

mt εt ðn þ mÞ!ðs  tÞ! ð2s þ1Þ C~ ns ¼ ð 1Þn þ 1 in þ m þ s þ t ðn  mÞ!ðs þ tÞ! 8 Z 1  2kf Z π e N~ m ðaÞN~ t ðaÞ J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞda dk: Zðk; aÞ  cos a k 0 0

ð29Þ

~ The discussion that follows concerns explicitly the first coefficient C mt ns whilst the coefficient C ns is obtained from the former by simply replacing N m ðaÞ by N~ m ðaÞ [see Eqs. (25) and (26)]. The main goal is to calculate efficiently the double integral in (28) using the kernel Z(k, a) [see Eq. (11)]. Here this integral is denoted by I mt ns . Observing that the denominator in Z(k, a) must have two roots, the double integral is split into two parts reducing simultaneously the integration range over a. Thus, pffiffiffiffiffiffiffiffi   Z 1 Z π=2  2kf pffiffiffiffi ð K  k 1=ξ cos aÞ2 þk e 1 1 Nm ðaÞN t ðaÞ  I mt ¼ ns k  k1 k  k2 cos a k ð1=ξÞ cos 2 aðk1 k2 Þ 0 0 mt

J n þ 1=2 ðk cos aÞJs þ 1=2 ðk cos aÞdadk pffiffiffiffiffiffiffiffi   Z 1 Z π=2  2kf pffiffiffiffi ð K þ k 1=ξ cos aÞ2 þk e 1 1 N m ðaÞNt ðaÞ  þ ð  1Þn þ m þ s þ t 2 k  k3 k  k4 cos a k ð1=ξÞ cos aðk3 k4 Þ 0 0 J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞdadk; where

ð30Þ

k1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 2τ cos þ 1 þ 4τ cos aÞK ; 2τ2 cos 2 a

k2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ2τ cos  1 þ 4τ cos aÞK ; 2τ2 cos 2 a

ð31Þ

k3 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  2τ cos þ 1  4τ cos aÞK ; 2τ2 cos 2 a

k4 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2τ cos  1  4τ cos aÞK ; 2τ2 cos 2 a

ð32Þ

pffiffiffiffiffiffiffiffi with τ ¼ K=ξ ¼ Uω=g. The limits of integration over a in the second double integral in Eq. (30) were originally [π/2, π] and they were changed after changing the argument of integration, letting a ¼ a0 þ π. That explains the occurrence of the factor ð 1Þn þ m þ s þ t in Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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Eq. (30) as it holds that Nm ða þ πÞNt ða þ πÞ J n þ 1=2 ðk cos ða þ πÞÞJ s þ 1=2 ðk cos ða þ πÞÞ cos ða þ πÞ Nm ðaÞN t ðaÞ ðn þ s þ 1Þiπ e J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞ: ¼ ð 1Þm þ t þ 1 cos ðaÞ

ð33Þ

Clearly, for pure diffraction when 1=ξ ¼ τ ¼ 0 the expression for I mt ns obtains nonzero values only when n þm þsþt is even. So far Green's function was constructed satisfying the Laplace equation, and the free surface and bottom boundary conditions. The complex kernel in Eq. (30) taken over real values of k is interpreted here as a contour integral in the Cauchy principal value sense (e.g. Whittaker and Watson, 1990, p. 117), by including the residue contributions from the poles on the real axis. The poles (singularities) are symmetrically winnowing around an infinitely small semi-circle such that the sign of the residue term (plus in the present case), is chosen to specifically satisfy the radiation condition of outgoing waves at infinity. Hence the integral in Eq. (30) is expanded as Z 1 Z 1 FðkÞ FðkÞ dk ¼ PV dk þ iπFðsÞ; ð34Þ k  s k s 0 0 where the acronym PV is used to denote the usual Cauchy's principal value integral and the last term represents the residue contribution [e.g. Eq. (13.17) in Wehausen and Laitone, 1960]. With the aforementioned remarks the sought coefficients C mt ns obtain their final form which is εt n þ 1 n þ m þ s þ t ðn þmÞ!ðs tÞ! ð2s þ 1Þ C mt i ns ¼ ð  1Þ ðn mÞ!ðs þtÞ! 8 ( Z Z p ffiffiffi ffi p ffiffiffiffiffiffiffi ffi   1 π=2  2kf ð K  k 1=ξ cos aÞ2 þ k e 1 1 Nm ðaÞN t ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :  PV  k  k1 k  k2 cos a k 1 þ 4τ cos a 0 0 J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞda dk Z π=2 Nm ðaÞNt ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe  2k1 f J n þ 1=2 ðk1 cos aÞJ s þ 1=2 ðk1 cos aÞ þ 2iπ cos a 1 þ 4τ cos a 0  e  2k2 f J n þ 1=2 ðk2 cos aÞJ s þ 1=2 ðk2 cos aÞÞda " Z Z pffiffiffiffi pffiffiffiffiffiffiffiffi   1 π=2  2kf ð K þk 1=ξ cos aÞ2 þ k e 1 1 N m ðaÞNt ðaÞ nþmþsþt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PV  þ ð  1Þ k  k3 k  k4 cos a k 1  4τ cos a 0 0 J n þ 1=2 ðk cos aÞJ s þ 1=2 ðk cos aÞda dk Z π=2 Nm ðaÞNt ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe  2k3 f J n þ 1=2 ðk3 cos aÞJ s þ 1=2 ðk3 cos aÞ þ 2iπ cos a 1  4τ cos a 0   e  2k4 f J n þ 1=2 ðk4 cos aÞJ s þ 1=2 ðk4 cos aÞÞda :

ð35Þ

It is worth noting that the problem has no solution when τ ¼1/4, namely when the encounter frequency coincides with the critical frequency ωc ¼g/(4U) (see Dagan and Miloh, 1982), where the classical linearized solution fails (i.e. exhibiting resonance conditions with infinitely large wave amplitudes). Eq. (35) requires further treatment when τ41/4, as for a 4arccosð1=4τÞ the poles k3 and k4 become complex. Nevertheless, we will not proceed to further manipulation of Eq. (35), as we restrict our attention in the present case to the range of τ o1/4, which is realistic for common ship speeds and ocean wave frequencies. Thus the most demanding issue as far as the calculation of the coefficients C mt ns is concerned, arises from the Cauchy principal value integrals the elaboration of which is always a challenge (see e.g. Tassin et al., 2013). Here, we have chosen to perform the integration numerically relying on the efficiency of the Sinc function series approximation developed by Bialecki and Keast (1999). Numerical problems associated with the integrands (e.g. oscillations due to large arguments of the Bessel functions) and the upper limit (infinity) of the semi-infinite Cauchy integrals, have been surmounted by refining at each run the integration mesh until the relative error between successive runs is of the order of 10  10. 4. The diffraction and the steady potentials First, we consider the diffraction potential. This is expressed in terms of a multipole series eigen-expansion of spheroidal harmonics according to ϕD ¼

n igA  K 0 f 1 m e ∑ ∑ Am n Gn ðx; y; zÞ; ω0 n¼0m¼0

ð36Þ

m where Gm n ðx; y; zÞ is given by Eq. (27) and An are the unknown expansion coefficients to be obtained by invoking the Neumann-type body boundary condition (6). The incident wave potential must also be expressed in terms of an eigenexpansion of spheroidal harmonics [see Eq. (4)]. Thus, after reversing the orientation of the z-axis in Eq. (4) to denote upwards and using an expansion similar to Eq. (22) (see Wu and Eatock Taylor, 1989), the incident wave potential is

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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expressed in series of multipoles as i n h igA  K 0 f 1 m m ~m ϕI ¼ e ∑ ∑ Bm n cos mψ þ iB n sinmψ P n ðμÞP n ðζÞ; ω0 n¼0m¼0

7

ð37Þ

where nþm Bm n ¼ ð  1Þ

 1=2 εm n þ m ðn  mÞ! π ð2n þ 1ÞN m ðβÞ i J n þ 1=2 ðK 0 cos βÞ; ðn þ mÞ! 2K 0 cos β 2

 1=2 m εm ðn mÞ! π ð2n þ 1ÞN~ m ðβÞ J n þ 1=2 ðK 0 cos βÞ: B~ n ¼ ð  1Þn þ m þ 1 in þ m ðn þmÞ! 2K 0 cos β 2 Hence, after substituting Eqs. (36) and (37) into Eq. (6), which is applied on the body's surface S0    ∂ϕD ∂ϕ 1 b ; ¼  I ; ζ ¼ ζ 0 ¼ cosh u0 ¼ cosh tanh a ∂ζ ∂ζ

ð38Þ

ð39Þ

ð40Þ

one gets the following linear systems that determine the unknown expansion coefficients involved in the diffraction potential: Arl þ

r _r n P_ l ðζ 0 Þ 1 mr r P l ðζ 0 Þ ; ∑ ∑ Am r r n C nl ¼  Bl Q_ ðζ Þ n ¼ 0 m ¼ 0 Q_ ðζ Þ 0

l

r _ r ðζ Þ n r m mr r P P_ ðζ Þ 1 A~ l þ lr 0 ∑ ∑ A~ n C~ nl ¼  B~ l lr 0 : Q_ ðζ Þ n ¼ 0 m ¼ 0 Q_ ðζ Þ 0

l

ð41Þ

0

l

l

ð42Þ

0

Here the dots denote differentiation with respect to the argument. Eqs. (41) and (42) were derived by invoking the orthogonality relations of the trigonometric and the associated Legendre functions of the first kind [Eq. (8.14.13) in Abramowitz and Stegun, 1970]. Apparently, we have distinguished between the expansion coefficients obtained by applying the cosine and sine orthogonality relations. The employment of the appropriate coefficients depends on the loading component that is considered. The derivation of Eqs. (41) and (42) virtually completes the solution for the diffraction component. Next we consider the steady potential ϕ. It is noted that in the study of Havelock (1954) that term was expressed as (see also Lamb, 1916) ϕ ¼ P 01 ðμÞ

Q 01 ðζÞ : 0 Q_ ðζ Þ 1

ð43Þ

0

Namely, it satisfies the body boundary condition but not the condition on the free surface. To remedy this glitch we express the steady potential in terms of our Green's function, which by default satisfies all conditions of the investigated boundary value problem including the condition on the free surface. Thus, ϕ is taken in the form 1

n

m ∑ Dm n Gn ðx; y; zÞ;

ϕ¼ ∑

ð44Þ

n¼0m¼0

where Dm n are unknown expansion coefficients which are obtained by requiring that [see also Eq. (7)] ∂ ð  cμζ þ cϕÞ ¼ 0; ∂ζ

ð45Þ

to be applied on ζ ¼ ζ 0 . Introducing Eq. (44) into Eq. (45) and invoking the orthogonality properties of the trigonometric and the associate Legendre functions of the first kind, the following linear system is finally derived in terms of the expansion coefficients Dm n Dm n þ

m δn1 δm0 P_ n ðζ 0 Þ 1 s ; ∑ ∑ Dts C tm m sn ¼ m _ Q ðζ Þ s ¼ 0 t ¼ 0 Q_ ðζ Þ 0

n

n

ð46Þ

0

~ mt m ~m where δij denotes Kroneker's delta function. Having calculated the expansion coefficients C mt ns , C ns , Bn and B n , the unknown m m coefficients An for the diffraction component and Dn for the steady component are obtained using trivial methods of linear algebra. 5. Hydrodynamic loading on the spheroid For the coupled forward speed and wave diffraction problem, the hydrodynamic exciting forces and moments can be obtained from Newman (1978), Wu and Eatock Taylor (1988) and Wu (1995) as Z F ¼  ρ ½iωðϕI þ ϕD Þ þ W U ∇ðϕI þ ϕD Þn dS; ð47Þ S0

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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8

Z M ¼ ρ ½iωðϕI þϕD Þ þ W U∇ðϕI þ ϕD Þðr  nÞdS; S0

ð48Þ

where we define (see also Das and Cheung, 2012) W ¼ U∇ðcϕ  xÞ;

ð49Þ

whilst n is the (outward) normal to the body surface with " # ðζ 20  1Þ1=2 μ ζ 0 ð1  μ2 Þ1=2 sin ψ ζ 0 ð1  μ2 Þ1=2 cos ψ ; ; : n¼ ðζ 20  μ2 Þ1=2 ðζ 20  μ2 Þ1=2 ðζ 20  μ2 Þ1=2

ð50Þ

In addition, the differential area on the spheroidal body is expressed as dS ¼ c2 ðζ 20  1Þ1=2 ðζ 20 μ2 Þ1=2 dμ dψ:

ð51Þ

The quadratic term in Eqs. (47) and (48) is calculated on the body surface ζ ¼ ζ 0 . Hence, the first term of the dot product is zero and accordingly can be reduced to   Ucð1 μ2 Þ ∂ðϕI þ ϕD Þ ∂ϕ ζ 0 þ W U ∇ðϕI þ ϕD Þ ¼ 2 ∂μ ∂μ c2 ðζ 0  μ2 Þ þ

Uc ∂ðϕI þ ϕD Þ ∂ϕ : ∂ψ ∂ψ c2 ðζ 20  1Þð1  μ2 Þ

ð52Þ

Hence, the final relation that provide the hydrodynamic loading on the moving spheroid under waves will read  2

 1 n 4 b m0 0 fx ¼ π e  K 0 f ð1  τ0 cos βÞ B01 P 01 ðζ 0 Þ þ A01 Q 01 ðζ 0 Þ þ ∑ ∑ Am n C n1 P 1 ðζ 0 Þ 3 a n ¼ 0m ¼ 0  2   Z Z b τ0  K 0 f 2π 1 ð1 μ2 Þμ ∂ðϕI þϕD Þ ∂ϕ ζ dμ dψ e þ i 0 2 2 a K0c ∂μ ∂μ 0  1 ζ 0 μ Z Z τ0 c  K 0 f 2π 1 μ ∂ðϕI þ ϕD Þ ∂ϕ dμ dψ; i e 2 2 ∂ψ ∂ψ K0a 0 1 1μ

 1 n 1 m m1 4 b 1 f y ¼ i π e  K 0 f ð1  τ0 cos βÞ B~ 1 P 11 ðζ 0 Þ þ A~ 1 Q 11 ðζ 0 Þ þ ∑ ∑ A~ n C~ n1 P 11 ðζ 0 Þ 3 a n¼0m¼0   Z Z b τ0  K 0 f 2π 1 ð1  μ2 Þ3=2 ∂ðϕI þ ϕD Þ ∂ϕ  ζ0 þ sin ψ dμ dψ e i 2 2 aK 0 c ∂μ ∂μ 0  1 ζ 0 μ Z Z τ0 c  K 0 f 2π 1 1 ∂ðϕI þ ϕD Þ ∂ϕ sin ψ dμ dψ; e i 1=2 2 K 0 ab ∂ψ ∂ψ ð1 μ Þ 0 1

ð53Þ

ð54Þ

Fig. 2. Real part of the surge exciting force acting on a prolate spheroid a/b ¼6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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9

 1 n 4 b m1 1 f z ¼  π e  K 0 f ð1 τ0 cos βÞ B11 P 11 ðζ 0 Þ þA11 Q 11 ðζ 0 Þ þ ∑ ∑ Am n C n1 P 1 ðζ 0 Þ 3 a n¼0m¼0 b τ0  K 0 f e i aK 0 c τ0 c  K 0 f i e K 0 ab

Z

2π

Z

2π

Z

0

Z 0

ð1  μ2 Þ3=2 ∂ðϕI þ ϕD Þ 2 2 ∂μ  1 ζ0  μ 1

1

1

1

ð1  μ2 Þ1=2

  ζ0 þ

 ∂ϕ cos ψ dμ dψ ∂μ

∂ðϕI þ ϕD Þ ∂ϕ cos ψ dμ dψ; ∂ψ ∂ψ

ð55Þ

Fig. 3. Imaginary part of the surge exciting force acting on a prolate spheroid a/b ¼ 6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Fig. 4. Real part of the sway exciting force acting on a prolate spheroid a/b¼ 6 immersed at f ¼2b below the free surface (Fn ¼0.0001).

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

10

!

 2 1 n 4 b m1 1 my ¼ π 1  2 e  K 0 f ð1  τ0 cos βÞ B12 P 12 ðζ 0 Þ þ A12 Q 12 ðζ 0 Þ þ ∑ ∑ Am C P ðζ Þ n n2 2 0 5 a n¼0m¼0 i

i

!   Z 2π Z 1 2 τ0 b ð1  μ2 Þ3=2 μ ∂ðϕI þϕD Þ ∂ϕ  ζ cos ψ dμ dψ 1  2 e  K0f þ 0 ∂μ ∂μ K0c a ζ 20  μ2 0 1 τ0 c 2

K0b

! Z 2π Z 1 2 b μ ∂ðϕI þϕD Þ ∂ϕ cos ψ dμ dψ; 1  2 e  K0f 1=2 ∂ψ ∂ψ a 0  1 ð1  μ2 Þ

ð56Þ

Fig. 5. Imaginary part of the sway exciting force acting on a prolate spheroid a/b ¼6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Fig. 6. Real part of the heave exciting force acting on a prolate spheroid a/b¼ 6 immersed at f ¼2b below the free surface (Fn ¼0.0001).

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

!

 2 1 n 1 m m1 4 b 1 mz ¼  i π 1  2 e  K 0 f ð1  τ0 cos βÞ B~ 2 P 12 ðζ 0 Þ þ A~ 2 Q 12 ðζ 0 Þ þ ∑ ∑ A~ n C~ n2 P 12 ðζ 0 Þ 5 a n¼0m¼0 !   Z 2π Z 1 2 2 3=2 τ0 b ð1 μ Þ μ ∂ðϕI þϕD Þ ∂ϕ  ζ sin ψ dμ dψ 1  2 e  K0f þ i 0 ∂μ ∂μ K 0c a ζ 20  μ2 0 1 ! Z 2π Z 1 2 τ0 c b μ ∂ðϕI þϕD Þ ∂ϕ sin ψ dμ dψ: i 1  e  K0f 2 2 1=2 ∂ψ ∂ψ a 0  1 ð1 μ2 Þ K 0b

11

ð57Þ

Eqs. (53)–(57) provide respectively the surge, sway and heave forces, as well as the pitch and yaw moments. Obviously, the roll moment for a non-axisymmetric prolate spheroid is always zero. The loading expressions (53)–(57) have been normalized by ρgAa2 for the forces [Eqs. (53)–(55)] and by ρgAa2b for the moments [Eqs. (56) and

(57)]. Finally, it is noted that ϕI and ϕD denote the incident and the diffraction components normalized by igA=ω0 e  K 0 f . The formulae for

Fig. 7. Imaginary part of the heave exciting force acting on a prolate spheroid a/b¼ 6 immersed at f ¼2b below the free surface (Fn ¼0.0001).

Fig. 8. Real part of the pitch exciting moment acting on a prolate spheroid a/b ¼6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

12

calculating the hydrodynamic loading [Eqs. (53)–(57)] are given in their complete form incorporating all second-order effects; though they could be omitted due to their small contribution under linearization.

6. Numerical results and discussion 6.1. Validation The derivation of reliable numerical predictions also requires the accurate computation of the associate Legendre functions and the Bessel functions with fractional order, which are practically represented by the spherical Bessel functions.

Fig. 9. Imaginary part of the pitch exciting moment acting on a prolate spheroid a/b ¼6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Fig. 10. Real part of the yaw exciting moment acting on a prolate spheroid a/b¼ 6 immersed at f ¼2b below the free surface (Fn ¼0.0001).

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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13

Fig. 11. Imaginary part of the yaw exciting moment acting on a prolate spheroid a/b ¼6 immersed at f¼ 2b below the free surface (Fn ¼ 0.0001).

Table 1 pffiffiffiffiffiffi Real part of the exciting forces acting on submerged prolate spheroid simulating a sphere (a/b ¼1.0001) at forward speed with Fn ¼ U= ag ¼ 0:4 and immersed at f¼ 2a; the results have been normalized by ρgπa3 KA; heading angle β ¼π/2; column A: present method results; column B: Wu and Eatock Taylor (1988). K0 a

Surge force

0.1 0.2 0.3

Sway force

Heave force

A

B

A

B

A

B

0.0027 0.0216 0.0560

0.0028 0.0233 0.0643

0.0022 0.0113 0.0239

0.0022 0.0115 0.0248

 1.6984  1.4218  1.1662

 1.7038  1.4269  1.1718

Table 2 pffiffiffiffiffiffi Imaginary part of the exciting forces acting on submerged prolate spheroid simulating a sphere (a/b¼1.0001) at forward speed with Fn ¼ U= ag ¼ 0:4 and immersed at f ¼2a; the results have been normalized by ρgπa3 KA; heading angle β ¼ π/2; column A: present method results; column B: Wu and Eatock Taylor (1988). K0a

Surge force

0.1 0.2 0.3

Sway force

Heave force

A

B

A

B

A

B

0.0312 0.0454 0.0312

0.0253 0.0426 0.0357

1.6627 1.3709 1.1264

1.6633 1.3722 1.1270

0.0055 0.0347 0.0821

0.0056 0.0367 0.0913

Table 3 Convergence sequence of the real and the imaginary part of the surge exciting force acting on a moving prolate spheroid (a/b¼4; f¼ 2b) with pffiffiffiffiffiffi Fn ¼ U= ag ¼ 0:15 and subjected to the action of monochromatic incident waves propagating at angle β ¼ π; the listed data have been normalized by 2

4=3πK 0 ab Aρgexpð  K 0 f Þ. K0a

0.1 0.2 0.3 0.4 0.5 0.6

Re[fx]

Im[fx]

N ¼3

N ¼5

N¼8

N ¼10

N¼3

N¼5

N¼8

N ¼ 10

0.0000  0.0001  0.0008  0.0030  0.0071  0.0125

0.0000  0.0001  0.0008  0.0030  0.0070  0.0123

0.0000  0.0001  0.0008  0.0030  0.0070  0.0123

0.0000  0.0001  0.0008  0.0030  0.0070  0.0123

 1.0746  1.0771  1.0742  1.0685  1.0615  1.0553

 1.0770  1.0790  1.0762  1.0706  1.0640  1.0583

 1.0768  1.0789  1.0760  1.0705  1.0639  1.0582

 1.0768  1.0789  1.0760  1.0705  1.0639  1.0582

Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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14

Table 3 (continued ) K0a

Re[fx]

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Im[fx]

N ¼3

N ¼5

N¼8

N ¼10

N¼3

N¼5

N¼8

N ¼ 10

 0.0173  0.0184  0.0127 0.0004 0.0182 0.0364 0.0515 0.0624 0.0700 0.0758 0.0805 0.0837 0.0855

 0.0170  0.0178  0.0118 0.0015 0.0197 0.0382 0.0531 0.0635 0.0707 0.0768 0.0821 0.0860 0.0883

 0.0170  0.0178  0.0119 0.0014 0.0196 0.0381 0.0531 0.0635 0.0708 0.0768 0.0822 0.0860 0.0883

 0.0170  0.0178  0.0119 0.0014 0.0196 0.0381 0.0531 0.0635 0.0708 0.0769 0.0822 0.0860 0.0883

 1.0515  1.0500  1.0485  1.0427  1.0287  1.0056  0.9754  0.9413  0.9055  0.8687  0.8306  0.7911  0.7508

 1.0551  1.0544  1.0537  1.0487  1.0356  1.0132  0.9838  0.9510  0.9172  0.8828  0.8469  0.8094  0.7711

 1.0550  1.0542  1.0535  1.0485  1.0354  1.0131  0.9837  0.9507  0.9168  0.8823  0.8463  0.8086  0.7702

 1.0550  1.0542  1.0535  1.0485  1.0354  1.0131  0.9837  0.9507  0.9168  0.8823  0.8463  0.8086  0.7702

Table 4 Convergence sequence of the real and the imaginary part of the heave exciting force acting on a moving prolate spheroid (a/b¼4; f¼ 2b) with pffiffiffiffiffiffi Fn ¼ U= ag ¼ 0:15 and subjected to the action of monochromatic incident waves propagating at angle β ¼π; the listed data have been normalized by 2

4=3πK 0 ab Aρg expð  K 0 f Þ. K0a

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Re[fz]

Im[fz]

N ¼3

N ¼5

N¼8

N ¼10

N¼3

N ¼5

N¼ 8

N ¼ 10

 1.9955  2.0320  2.0642  2.0912  2.1093  2.1127  2.0961  2.0581  2.0034  1.9408  1.8786  1.8194  1.7608  1.6991  1.6322  1.5612  1.4882  1.4140  1.3394

 1.9967  2.0330  2.0652  2.0924  2.1108  2.1147  2.0987  2.0615  2.0077  1.9462  1.8854  1.8283  1.7724  1.7130  1.6476  1.5778  1.5065  1.4344  1.3621

 1.9966  2.0329  2.0652  2.0924  2.1108  2.1146  2.0986  2.0614  2.0075  1.9460  1.8851  1.8280  1.7720  1.7125  1.6471  1.5773  1.5059  1.4337  1.3612

 1.9966  2.0329  2.0652  2.0924  2.1108  2.1146  2.0986  2.0614  2.0075  1.9460  1.8851  1.8280  1.7720  1.7125  1.6471  1.5773  1.5059  1.4338  1.3612

 0.0006  0.0049  0.0167  0.0393  0.0742  0.1201  0.1715  0.2198  0.2566  0.2788  0.2895  0.2957  0.3023  0.3099  0.3164  0.3194  0.3185  0.3147  0.3085

 0.0006  0.0049  0.0167  0.0393  0.0742  0.1200  0.1714  0.2196  0.2563  0.2782  0.2886  0.2946  0.3017  0.3107  0.3186  0.3225  0.3220  0.3188  0.3136

 0.0006  0.0049  0.0167  0.0393  0.0742  0.1200  0.1714  0.2196  0.2564  0.2783  0.2887  0.2946  0.3016  0.3106  0.3185  0.3223  0.3219  0.3187  0.3135

 0.0006  0.0049  0.0167  0.0393  0.0742  0.1200  0.1714  0.2196  0.2564  0.2783  0.2887  0.2946  0.3016  0.3106  0.3185  0.3223  0.3219  0.3187  0.3135

To this the robust FORTRAN routines developed by Zhang and Jin (1996) were incorporated into the dedicated computer code. As a means for validation we have used the numerical computations reported by Wu and Eatock Taylor (1987, 1989) on the exciting forces and moments acting on stationary immersed prolate spheroids. It is reminded that in the associated studies the authors considered only the diffraction problem and based their approach on solving a Fredholm integral equation arising from the distribution of sources on the surface of the spheroid. Wu and Eatock Taylor (1989) investigated only one geometrical case, namely a rigid spheroid of slenderness a/b¼6 and immersion depth of f¼2b whereas in Wu and Eatock Taylor (1987) several other cases were considered but results were provided only for β¼0 and for restricted range of frequencies. Here, we have reproduced with excellent agreement the exciting loading components for the configuration (slenderness and immersion) heading angles and wave frequencies, investigated in Wu and Eatock Taylor (1989). 2 The associated calculations are depicted in Figs. 2–11 using the same normalization, namely 4=3πab AρgK 0 expð  K 0 f Þ for 3 the forces and 4=3πab AρgK 0 expð  K 0 f Þ for the moments. Our results have been obtained assuming Fn¼0.0001 (and as a result KE K0) that was proven sufficient to allow, for most of the wave frequencies, a perfect coincidence (up to the fourth significant digit) compared to the calculations listed in Wu and Eatock Taylor (1987). The present results were obtained by truncating the infinite summations of the linear systems (41), (42) and (46) up to degree N ¼8, which again, was proven Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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15

Fig. 12. Magnitudes of the surge force acting on a moving prolate spheroid a/b¼ 4 immersed at f ¼2b below the free surface in head and following seas; modes used N ¼8; the symbols denote the results of Wu and Eatock Taylor (1987) for a stationary spheroid and heading angle β ¼01.

Fig. 13. Magnitudes of the heave force acting on a moving prolate spheroid a/b ¼4 immersed at f¼ 2b below the free surface in head and following seas; modes used N ¼8; the symbols denote the results of Wu and Eatock Taylor (1987) for a stationary spheroid and heading angle β ¼01.

sufficient to provide adequate convergence up to the fourth significant digit. Accordingly, that was the truncation employed for all computations discussed in the present study for relatively large K0a. For small wave frequencies below K0a ¼1, the convergence is almost immediate using only a couple of modes. Our model was further validated against some reported results on the hydrodynamic loading exerted on a rigid body subjected to regular waves and a uniform stream. In particular, we refer to the numerical predictions of Wu and Eatock pffiffiffiffiffiffi Taylor (1988) concerning a sphere moving steadily at Fn ¼ U= ag ¼ 0:4 (where a herein denotes the sphere radius), which nearly coincides with the axes of a spheroid when a Eb. Throughout this section we examine encounter frequencies that are below the critical frequency (Dagan and Miloh, 1982) and accordingly we limit our comparisons to K0a ¼0.1, 0.2 and 0.3. The associated results normalized by the same factor suggested by Wu and Eatock Taylor (1988) are listed in Tables 1 and 2 and correspond to a sphere immersed at f¼2a, moving with constant speed and subjected to the action of regular waves propagating at angle β¼π/2 relatively to sphere's motion. Although the comparisons are clearly favorable it is important to Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

16

I.K. Chatjigeorgiou, T. Miloh / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

Fig. 14. Magnitudes of the pitch moment acting on a moving prolate spheroid a/b¼ 4 immersed at f ¼2b below the free surface in head and following seas; modes used N¼ 8.

Fig. 15. Magnitudes of the surge force acting on a moving prolate spheroid a/b ¼4 immersed at f¼ 2b below the free surface in beam seas (β ¼ 901); modes used N¼ 8.

note that the differences observed in Tables 1 and 2 arise from the diffraction component. It is also mentioned that the same rate of disagreement was observed for the ‘pure’ diffraction problem of a fixed sphere. Furthermore, as far as real spheroids are concerned, results obtained by the present method are practically identical with those given in and Wu and Eatock Taylor (1987, 1989). It is believed that that agreement between the two would be even better if a/b was even smaller. Nevertheless the present method was built explicitly for spheroids and there are obvious limitations for limiting cases such as spheres, mainly due to the fact that Legendre functions (and their derivatives) admit very large values for large degrees (and accordingly orders) at large arguments ζ0. Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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Fig. 16. Magnitudes of the sway force acting on a moving prolate spheroid a/b¼ 4 immersed at f¼ 2b below the free surface in beam seas (β ¼ 901); modes used N¼ 8.

Fig. 17. Magnitudes of the heave force acting on a moving prolate spheroid a/b¼ 4 immersed at f¼ 2b below the free surface in beam seas (β ¼901); modes used N ¼ 8.

6.2. The coupled forward speed and wave action problem for a prolate spheroid The hydrodynamic forces and moments exerted on the body for the coupled problem are obtained through the dedicated relations (47) and (48). Here we consider a prolate spheroid with a/b¼4, immersed at f¼2b below the free surface, moving rectilinearly along its longitudinal x-axis and subjected at the same time to the action of regular waves with pffiffiffiffiffiffi various heading angles. The steady velocities which are considered correspond to Fn ¼ U= ag equal to 0.1 and 0.15 and the results are given for τ o1/4. The specific geometry was taken to be the same as in the work of Wu and Eatock Taylor (1987), who presented results obtained through the diffraction problem for β¼0 and up to K0a¼1. The calculations presented in Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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Fig. 18. Magnitudes of the pitch and yaw moments acting on a moving prolate spheroid a/b¼ 4 immersed at f¼ 2b below the free surface in oblique seas; modes used N ¼ 8; heading angle 901.

Fig. 19. Magnitudes of the surge force acting on a moving prolate spheroid a/b¼ 4 immersed at f¼ 2b below the free surface in oblique seas; modes used N ¼ 8.

pffiffiffiffiffiffiffiffiffi 3 2 Tables 3 and 4 and Figs. 12–23 have been normalized by 4=3πK 0 ab Aρgexpð  K 0 f Þ for the forces and 4=3π K 0 ab Aρg expð  K 0 f Þ for the moments. The numerical predictions are convergent for the investigated range of wave frequencies K0a ¼[0.1–5.0], using a truncation up to degree N ¼8. Sufficient accuracy is obtained from N ¼5, but absolute coincidence, at least up to the depicted significant digit, is achieved for N ¼8. The occurrence of the coupled term W U∇ðϕI þ ϕD Þ in the associated relations, does not affect the truncation needed for getting the same level of convergence. Relevant convergence studies are illustrated in Tables 3 and 4 for τ o 1/4. Figs. 12–23 provide a graphical representation for the variation (with respect to the wave frequency K0a) of the exciting forces and moments for several heading angles and the two selected values of Fn. Figs. 12 and 13 contain also the predictions of the “pure” diffraction problem, due to Wu and Eatock Taylor (1987) for the same geometry and the same immersion. It is Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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Fig. 20. Magnitudes of the sway force acting on a moving prolate spheroid a/b¼ 4 immersed at f¼ 2b below the free surface in oblique seas; modes used N ¼ 8.

Fig. 21. Magnitudes of the heave force acting on a moving prolate spheroid a/b¼ 4 immersed at f¼2b below the free surface in oblique seas; modes used N ¼ 8.

again evident that the comparisons are generally favorable when the present method is employed with Fn-0. The most important conclusions that can be drawn based on the depicted calculations are outlined succinctly in the following: (i) the surge force is the less affected. In fact for low wave frequencies, head seas slightly reduce the surge loading and following seas slightly increase it (Figs. 12 and 19); (ii) the opposite condition is observed for the heave and sway forces for all heading angles considered (Figs. 13, 16, 17, 20 and 21); (iii) the surge force for beam seas with β¼π/2 is not actually zero (Fig. 15). This is due to the fact that the forward motion of the spheroid produces an asymmetric flow although the body is symmetric. For the same reason, the sway and heave forces are affected as well; (iv) the most affected components are the exciting moments. The pitch moments in particular, are decreased and increased for following and head seas respectively (Figs. 14 and 22). The situation for the yaw moment however (Fig. 23) is more complicated; (v) the pitching and yawing moments are developed when the spheroid advances parallel to wave crests (beam seas) (see Fig. 18). Using Havelock's words in his Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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Fig. 22. Magnitudes of the pitch moment acting on a moving prolate spheroid a/b¼ 4 immersed at f ¼2b below the free surface in oblique seas; modes used N¼ 8.

Fig. 23. Magnitudes of the yaw moment acting on a moving prolate spheroid a/b ¼4 immersed at f¼ 2b below the free surface in oblique seas; modes used N ¼ 8.

simplified formulation (Havelock, 1954) “this arises because when account is taken of the speed, the pressure distribution on the solid is altered and is no longer symmetrical fore and aft”. 7. Conclusions By employing Havelock's spheroid theorem and the method of image singularities, we presented a new solution for the coupled linearized hydrodynamic problem corresponding to the rectilinear motion of a fully submerged rigid prolate spheroid in the presence of ambient monochromatic wave train of arbitrary heading. As such, a compact semi-analytic solution is given here for the first time for the combined coupled forward speed–wave diffraction problem. The general formulation is valid for any spheroidal geometry and submergence as well as for any wave frequency and direction of Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i

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propagation with respect to the steady course of the body. Some numerical simulations are presented for the operation range of wave frequencies below the critical limit (where linearized theory fails) and for water of infinite depth. The last restriction is not a severe one as the present method can be also extended for cases of finite depth (including shallow water). In particular, we provided useful expressions for evaluating the hydrodynamic loads exerted on the spheroid in the course of its motion including pressure forces (surge, sway and heave) and moments (pitch and yaw). The numerical code was found to be very robust and efficient compared to the existing methods for evaluating the wave resistance OR wave diffraction, which are all based on solving a Fredholm integral equation of the second kind for the unknown source distribution over the surface of the wetted body satisfying the Neumann boundary conditions. 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Please cite this article as: Chatjigeorgiou, I.K., Miloh, T., Hydrodynamics of submerged prolate spheroids advancing under waves: Wave diffraction with forward speed. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j. jfluidstructs.2014.04.012i