# Hydrodynamics of a Free Floating Vertical Axisymmetric Oscillating Water Column Device

Authors: S. A. Mavrakos; D. N. Konispoliatis
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper aims at presenting a general formulation of the hydrodynamic problem of a floating or restrained oscillating water column device. Three types of first-order boundary value problems are investigated in order to calculate the velocity potential of the flow field around the device. The horizontal and vertical exciting wave forces, the rolling moment, the hydrodynamic parameters, the volume flows, and the drift forces are obtained in order to find the loads on the structure. The efficiency rate of the device is calculated in connection with the absorbed power and the capture length of energy absorption. Finally, the resulting wave motion inside and outside the device and the inner air pressure are examined.


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## Body

## 1. Introduction

In the last years considerable efforts and advances have been made worldwide in exploiting the energy of ocean waves due to the rise of the world’s energy consumption and the requirement for “green” energy production. Among several classes of designs proposed for the wave energy conversion, the oscillating water column device (OWC) has received considerable theoretical attention. The OWC device is a partially submerged, hollow structure open to the seabed below the water line. The vertical motion of the sea surface alternatively pressurizes and depressurizes the air inside the structure generating a reciprocating flow through a self-rectifying turbine which is installed beneath the roof of the device. Full sized fixed and floating prototypes were built in Norway (in Toftestallen, near Bergen, 1985), Japan (Sakata, 1990), India (Vizhinjam, near Trivandrum, Kerala State, 1990), Portugal (Pico, Azores, 1999), UK (Scotland, 2000), and Japan (1998) [1, 2].A nearshore or onshore OWC device was analysed among others by McCormick [3, 4], Evans [5], Sarmento et al. [6], Evans and Porter [7], and Kokkinowrachos et al. [8], for 2D and 3D geometries of OWC devices with infinitesimal wall thickness. Hydrodynamic analysis on a freely floating oscillating water column device could benefit from previous worldwide studies on the dynamics of ships in wavy seas. The first theoretical analysis of a floating oscillating water column device has been presented by Masuda [9]. He developed a navigation buoy powered by wave energy, and equipped with an air turbine, while later promoted the construction of a floating barge (80 m × 12 m), named Kaimei, which housed several OWCs equipped with different types of air turbines [10]. Recently, Martins-Rivas and Mei [11] presented an analysis on an OWC device mounded at the tip of a breakwater with infinitesimal wall thickness and Falcão et al. [12] studied the dynamics of an OWC spar buoy.In the present contribution, a vertical axisymmetric oscillating water column device with finite wall thickness is examined that is free to move in finite depth waters. Numerical results are given from the solution of three boundary value problems, namely, the diffraction problem (body fixed in waves, atmospheric pressure on the OWC), the radiation problem resulting from the forced oscillations of the body in otherwise still water, also under atmospheric conditions above the OWC, and the radiation problem resulting from an oscillating pressure head acting on the inner free surface of the OWC. Particularly, numerically evaluated linear exciting wave forces in the horizontal and vertical directions along with the volume flow, the added mass and wave damping coefficients are calculated for the freely floating device. In view of evaluating the rigid body motions of the device, the motion equations are solved in the frequency domain, for various values of turbine parameters related to the pressure drop inside the oscillating chamber. The developed method is based on matched axisymmetric eigenfunction expansions of the velocity potential in properly defined ring-shaped fluid regions around the device and could be considered as an extension of the methods employed by Miles and Gilbert [13], Garrett [14], and Yeung [15] as far as a circular dock is concerned, by Mavrakos [16, 17] for bottomless cylinders, by Kokkinowrachos et al. [18] for the hydrodynamic analysis (diffraction and radiation problems) of arbitrary shaped vertical bodies of revolution. In particular, the method extends the formulation given by Evans and Porter [7] to account for finite wall thickness of the vertical bottomless cylindrical chamber and the related 2D formulation by Kokkinowrachos et al. [8] to the case of freely moved vertical cylindrical ducts. The first-order numerical results for the oscillating water column device are supplemented with corresponding ones concerning the wave elevation around the OWC configuration. Finally, the paper investigates the time-independent part of the second-order wave forces (drift forces) on the device’s wall and the absorbed wave energy and capture length from the OWC device. Among several types of self-rectifying air turbines that have been developed to equip OWCs [19, 20], it is assumed that a “Wells” turbine, named after its inventor, is installed beneath the roof of the device.


## 2. Formulation of the Hydrodynamic Problem

We consider a freely floating vertical axisymmetric cylindrical OWC device (Figure1) of internal and external radius b and a, respectively, excited by regular waves of amplitude H/2, frequency ω, and wave number k in constant water depth, d. The draught of the device’s chamber is (d-h). We assume small amplitude waves and motions and inviscid incompressible irrotational flow. Cylindrical coordinates (r,θ,z) are introduced with the vertical axis Oz directed upwards and origin at the sea bed. For the OWC device, we expect the internal free surface of the device to be subjected to an oscillating pressure head Pin with Pin(t)=Re[pin0·e-iωt], having the same frequency, ω, as the incident wave.Figure 1
Typical floating oscillating water column device with finite wall thickness. Ring elements definition.The time harmonic complex velocity potential of the flow field around the structure can be expressed as(2.1)Φ(r,θ,z;t)=Re[ϕ(r,θ,z)·e-iωt].
We decompose the above velocity potential as
(2.2)ϕ=ϕ0+ϕ7+∑j=1,3,5x.j0·ϕj+ϕP.
Here, ϕ0 is the velocity potential of the undisturbed incident harmonic wave; ϕ7 is the scattered potential for the body fixed in the waves with the duct open to the atmosphere, that is, for a pressure in the chamber equal to the atmospheric one; ϕj, (j=1,3,5), is the radiation potential resulting from the forced body motion in the jth mode of motion with unit velocity amplitude x.j0 with the duct open to the atmosphere; ϕP is the radiation potential resulting from the oscillating pressure head, Pin, in the chamber for the body fixed in otherwise calm water.The undisturbed incident wave potential,ϕ0, can be expressed in cylindrical co-ordinates as [21]
(2.3)ϕ0(r,θ,z)=-iω(H2)∑m=0∞εmimΨ0,m(r,z)cos(mθ),
where
(2.4)1dΨ0,m(r,z)=Z0(z)dZ0′(d)Jm(kr).
Here, Jm is the mth order Bessel function of the first kind, εm is the Neumann’s symbol defined as ε0=1 and εm=2 for m≥1 and Z0(z):
(2.5)Z0(z)=[12[1+sinh(2kd)2kd]]-1/2cosh(kz),
with Z0′(d) being its derivative at z=d. Frequency ω and wave number k are related by the dispersion equation:
(2.6)ω2=k·g·tanh(kd).
The diffracted potential, ϕD, around the restrained structure is described by the velocity potential:
(2.7)ϕD=ϕ0+ϕ7.
In accordance to (2.3) the diffracted velocity potential of the flow field around the OWC device can be written in the form:
(2.8)ϕD(r,θ,z)=-iω(H2)∑m=0∞εmimΨD,m(r,z)cos(mθ).
The fluid flow caused by the forced oscillation of a vertical axisymmetric body in otherwise still water is symmetric about the plane θ=0 and antisymmetric about plane θ=π/2 for surge, (j=1), and pitch, (j=5), whereas it is symmetric about both these planes for heave, (j=3). Thus the corresponding velocity potentials for these modes of motion can be expressed as
(2.9)ϕ1(r,θ,z)=Ψ1,1(r,z)cos(θ),(2.10)ϕ3(r,θ,z)=Ψ3,0(r,z),(2.11)ϕ5(r,θ,z)=Ψ5,1(r,z)cos(θ).
As for the radiation potential ϕP, we note that due to the high sound speed in air, the low frequency of sea waves, and the relatively small per unit volume air kinetic energy, the air pressure can be assumed to be spatially uniform throughout the axisymmetric chamber [11]. Thus, since the forcing of the internal free surface is independent of θ, the radiation potential ϕP will be axially symmetric and so it will include only the m=0 angular mode, that is, it holds:
(2.12)ϕP(r,θ,z)=pin0iωρΨP,0(r,z).
In the functions Ψj,m of (2.8)–(2.12) the first subscript j=D,1,3,5,P denotes the respective boundary value problem each time considered, whereas the second one indicates the value of m which must be taken into account for the solution of the corresponding problem. Thus the functions ΨD,m,(m=0,1,2,…), Ψ1,1, Ψ3,0, Ψ5,1, and ΨP,0 remain the principal unknowns of the problem.The velocity potentialsϕj, (j=D,1,3,5,P), have to satisfy the Laplace equation
(2.13)∇2ϕj=∂2ϕj∂r2+1r∂ϕj∂r+1r2∂2ϕj∂θ2+∂2ϕj∂z2=0
within the entire fluid domain, the linearized boundary condition at the outer and inner free sea surface (z=d), that is,
(2.14)ω2ϕj-g∂ϕj∂z={0forr≥a,j=D,1,3,5,P,0for0≤r≤b,j=D,1,3,5,-iωρpin0for0≤r≤b,j=P,
and the zero normal velocity on the sea bed, that is,
(2.15)[∂ϕj(r,θ,z)dz]z=0=0,j=1,3,5,7,P.
Furthermore, the velocity potentials ϕj, (j=D,1,3,5,P), have to fulfill following kinematic conditions on the mean device’s wetted surface S0:
(2.16)∂ϕi∂n→=0,i=D,P,∂ϕj∂n→=nj,j=1,3,5,
where in (2.16) ∂()/∂n→ denotes the derivative in the direction of the outward unit normal vector n→, to the wetted surface S0 of the device and nj are its generalized components defined as (n1,n2,n3)=n→ and (n4,n5,n6)=r→×n→, where r→ is the position vector with respect to the origin of the coordinate system.We also require the scattered potential to satisfy an appropriate radiation condition asr→∞, which has the form:
(2.17)limr→∞kr(∂ϕj∂r-ikϕj)=0,j=1,3,5,7,P.
Moreover, the velocity potentials, ϕj, and their derivatives, ∂ϕj/∂r, (j=D,1,3,5,P) must be continuous at the vertical boundaries of neighbouring fluid regions (Figure 1). This results in:
(2.18)Ψj,mIII(b,z)=Ψj,mM(b,z)for0≤z≤h,(2.19)∂Ψj,mIII∂r|r=b=∂Ψj,mM∂r|r=bfor0≤z≤h,(2.20)Ψj,mI(a,z)=Ψj,mIII(a,z)for0≤z≤h,(2.21)∂Ψj,mI∂r|r=a=∂Ψj,mIII∂r|r=afor0≤z≤h.
The superscripts I, III, M imply quantities corresponding to respective types of ring elements.Starting with the method of separation of variables for the Laplace differential equation, appropriate expressions for the velocity potentials,Ψj,mi, in each fluid domain (Figure 1; i=I,III,M) can be established [14, 16, 21]. These expressions satisfy the corresponding conditions at the horizontal boundaries of each fluid region and, in addition, the radiation condition at infinity in the outer fluid domain. As a result, the velocity potentials in each fluid domain fulfil a priori the kinematical boundary conditions at the horizontal walls of the device, the linearized condition at the free surface, the kinematical one on the sea bed, and the radiation condition at infinity.


## 3. Diffraction and Radiation Potentials for Various Fluid Regions

For each type of fluid domain the following expressions for the functionsΨj,mi involved in (2.8)–(2.12) can be derived.(
a
) Infinite Fluid Domain. Type I (r≥a,0≤z≤d)
One has(3.1)1δjΨj,mI(r,z)=gjmI(r,z)+Fj,m,0IHm(kr)Hm(ka)Z0(z)+∑i=1∞Fj,m,iIKm(air)Km(aia)Zi(z),
where j=D,1,3,5,P and
(3.2)gDmI(r,z)={Jm(kr)-Jm(ka)Hm(ka)Hm(kr)}Z0(z)dZ0′(d),g11I(r,z)=g30I(r,z)=g51I(r,z)=gP0I(r,z)=0,δD=δ1=δ3=d,δ5=d2,δP=1.Hm and Km are the mth order Hankel function of the first kind and the modified Bessel function of the second type, respectively, and Fj,m,iI are unknown Fourier coefficients to be determined by the solution procedure. Moreover, Zi(z) are orthonormal functions in [0,d] defined by (2.5) for i=0 and by
(3.3)Zi(z)=[12[1+sin(2aid)2aid]]-1/2cos(aiz),i≥1.
The eigenvalues αi are roots of the transcendental equation:
(3.4)ω2+g·ai·tan(aid)=0,
which possesses one imaginary, a0=-ik,k>0, and infinite number of real roots. Substituting the value of a0 in (3.3) and (3.4), (2.5) and (2.6) can directly be obtained.(
b
) Fluid Region of Type III (b≤r≤a,0≤z≤h)
One has(3.5)1δjΨj,mIII(r,z)=gjmIII(r,z)+∑q=0∞εq[RmqIII(r)Fj,m,qIII+R*mqIII(r)F*j,m,qIII]cos(qπzh),
where j=D,1,3,5,P and
(3.6)gDmIII(r,z)=g11III(r,z)=gP0III(r,z)=0,(3.7)g30III(r,z)=z2-(1/2)r22hd,g51III(r,z)=-r[z2-(1/4)r2]2hd2,RmqIII(r)=Km(qπb/h)Im(qπr/h)-Im(qπb/h)Km(qπr/h)Im(qπa/h)Km(qπb/h)-Im(qπb/h)Km(qπa/h),m,q≠0,(3.8)Rm0III(r)=(r/b)m-(b/r)m(a/b)m-(b/a)m,m≠0,q=0,R00III(r)=ln(r/b)ln(a/b),m,q=0,R*mqIII(r)=Im(qπa/h)Km(qπr/h)-Km(qπa/h)Im(qπr/h)Im(qπa/h)Km(qπb/h)-Im(qπb/h)Km(qπa/h),m,q≠0,(3.9)Rm0*III(r)=(a/r)m-(r/a)m(a/b)m-(b/a)m,m≠0,q=0,R00*III(r)=ln(a/r)ln(a/b),m,q=0.Im is the mth order modified Bessel function of the first kind and Fj,m,qIII, Fj,m,q*III are Fourier coefficients to be determined by the solution procedure. The functions gjmIII(r,z) in (3.6) and (3.7) above represent harmonic particular solutions for the surge, heave, and pitch modes of motion [15, 17, 18] that fulfill the inhomogeneous boundary condition (2.16) for the heave and pitch motions at the bottom surface of the OWC device.(
c
) Interior Fluid Region of Type M  (r≤b,0≤z≤d)
One has(3.10)1δjΨj,mM(r,z)=gjmM(r,z)+∑i=0∞Fj,m,iMIm(air)Im(aib)Zi(z),
where j=D,1,3,5,P and
(3.11)gDmM(r,z)=g11M(r,z)=g30M(r,z)=g51M(r,z)=0,gP0M(r,z)=1.
Here, the orthonormal functions Zi(z) are defined by (2.5) and (3.3), for i=0 and i≥1, respectively, and ai are given by (3.4). Moreover, Fj,m,iM are unknown Fourier coefficients in the Mth fluid region. The function gP0M(r,z) has been introduced in (3.10) in order for the inhomogeneous boundary condition on the water surface in the interior fluid region, (2.14) to be fulfilled in the case of the pressure radiation problem.The solutions for the potential functionsΨj,mi,(i=I,III,M and j=D,1,3,5,P), expressed through (3.1), (3.5), and (3.10), are constructed in such a way that the conditions at all the horizontal boundaries of each fluid region and, in addition, the radiation condition at infinity in the outer fluid domain are a priori satisfied. Moreover, the potential functions Ψj,mi have been constructed in such a way that their homogeneous parts can be reduced to simple Fourier series at the vertical boundaries of the various fluid regions, that is, r=a,b. This feature of the velocity potential representations facilitates essentially the solution procedure. The kinematic conditions at the body’s vertical walls, (2.16), as well as the requirement for continuity of the potential and its radial derivative, (2.18)–(2.21), at the vertical boundaries of neighboring fluid domains remain to be fulfilled. Expressing these conditions, the system of equations for the unknown Fourier coefficients is obtained. Once these coefficients have been calculated, the functions Ψj,mi and hence the velocity potential for all fluid regions can be evaluated. Methods for the solution of the diffraction and the radiation problem originated from the forced body motion in otherwise still water have been extensively reported in previous works [16, 17] and, so, they are no further elaborated here. Only the solution procedure for the pressure radiation problem will be outlined in the next section.


## 4. Solution Procedure for the Pressure Radiation Problem

The condition for continuity of the potential function atr=b and r=a is expressed by (2.18) and (2.20), respectively. Multiplying both sides of equations by (1/h)cos(νπz/h) and integrating over their region of validity, that is, 0≤z≤h the following set of equations can be obtained:
(4.1)∑q=0∞FP,0,q*III=QP,0,q*+∑i=0∞Lq,iFP,0,iM,for0≤z≤h,r=b,(4.2)∑q=0∞FP,0,qIII=∑i=0∞Lq,iFP,0,iI,for0≤z≤h,r=a,
where
(4.3)Lq,i=1h∫0hZi(z)cos(qπzh)dz,(4.4)QP,0,q*=1h∫0hgP0M(b,z)cos(qπzh)dz,
are defined in Appendix A.Now, the condition for the continuity of the first derivative of the potential atr=b and r=a as expressed by (2.19) and (2.21), respectively, as well as the kinematic condition on the vertical boundaries of the device as described by (2.16) must be fulfilled too. Multiplying both sides of (2.19), (2.21), and (2.16) with the weight function (1/d)Zμ(z), integrating over the region of their validity, that is, h≤z≤d and 0≤z≤d, respectively, and adding the resulting expressions, the following set of equations is obtained:
(4.5)∑i=0∞FP,0,iMA0,iM=hd∑q=0∞εqLq,iD0qIIIFP,0,qIII+hd∑q=0∞εqLq,iD0q*IIIFP,0,q*III,atr=b,(4.6)∑i=0∞FP,0,iIA0,iI=hd∑q=0∞εqLq,i(A0qIIIFP,0,qIII+A0q*IIIFP,0,q*III),atr=a,
where
(4.7)A0,iM=aib∂I0(air)∂r|r=b1I0(aib),A0,iI=aia∂K0(air)∂r|r=a1K0(aia),(4.8)D0qIII=b∂RqIII∂r|r=b,D0q*III=b∂R0q*III∂r|r=b,A0qIII=a∂R0qIII∂r|r=a,A0q*III=a∂R0q*III∂r|r=a,
and R0qIII, R0q*III are defined in (3.8) and (3.9), respectively. For the numerical implementation of the method, series (3.1), (3.5), and (3.10) expressing the potential in I, III, M fluid region will be truncated after Q, M, and N terms, respectively.Substituting (4.1) into (4.5) will provide the unknown Fourier coefficients, FP,0,iM, in the M fluid domain in relation with the Fourier coefficients, FP,0,qIII, in the III fluid domain, in the following matrix form, that is,
(4.9)([A0,iM]-hd∑q=0∞[Li,qM,III]·[εq]·[D0q*III]·[Lq,iIII,M])·{FP,0,iM}=hd[Li,qM,III]·[εq]·[D0qIII]·{FP,0,qIII}+hd[Li,qM,III]·[εq]·[D0q*III]·{QP,0,q*},
where, {FP,0,iM} and {FP,0,qIII} are both complex vectors, the elements of which are the unknown Fourier coefficients in the Mth and IIIth fluid domain, respectively, [A0,iM] is a (N×N) diagonal matrix given by (4.7), [Li,qM,III] is a (N×M) matrix given by (4.3), [εq] is the Neumann’s symbol (M×M) diagonal matrix with ε11=1 and εkk=2 for k≥2, [D0qIII] and [D0q*III] are (M×M) diagonal matrices defined in (4.8), and {QP,0,q*} is a column vector (M×1) defined in (4.4).From (4.9) and (4.1), the Fourier coefficients, FP,0,qIII and FP,0,q*III, in the IIIth fluid domain are connected with the bellow relation:
(4.10){FP,0,q*III}=hd[Lq,iIII,M]·[Gi,i]-1·[Li,qM,III]·[εq]·[D0qIII]·{FP,0,qIII}+([I]+hd[Lq,iIII,M]·[Gi,i]-1·[Li,qM,III]·[εq]·[D0q*III])·{QP,0,q*},
where [I] is the unit matrix and
(4.11)[Gi,i]=[A0,iM]-hd∑q=0∞[Li,qM,III]·[εq]·[D0q*III]·[Lq,iIII,M],
where, {FP,0,q*III} is a complex vector, the elements of which are the unknown Fourier coefficients in the IIIth region.Substituting (4.2) into (4.6) will provide the unknown Fourier coefficients, FP,0,iI, in the Ith fluid domain in relation with the Fourier coefficients, FP,0,qIII,FP,0,q*III, in the III fluid domain, in the following matrix form, that is,
(4.12)[A0,iI]·{FP,0,iI}=hd[Li,qI,III]·[εq]·[A0qIII]·{FP,0,qIII}+hd[Li,qI,III]·[εq]·[A0q*III]·{FP,0,q*III},
where, {FP,0,iI} is a complex vector, the elements of which are the unknown Fourier coefficients in the I fluid domain, [A0,iI] is a (Q×Q) diagonal matrix given by (4.7), and [Li,qI,III] is a (Q×M) matrix given by (4.3).After determining the Fourier coefficients for the pressure radiation problem in all fluid domains, by solving the system equations (4.9), (4.10), and (4.12), and the Fourier coefficients for the diffraction and motion radiation problem from previous work [16, 17] and substituting them into (3.1), (3.5), (3.10), the velocity potential of the flow field in each fluid region around the free floating body can be determined.


## 5. Volume Flow

During the water oscillation inside the chamber the dry air above the free surface is being pushed through a Wells turbine. The volume flow produced by the oscillating internal water surface is denoted byQ(t)=Re{q·e-iωt} where
(5.1)q=∬SiuzdSi=∬Siu(r,θ,z=d)rdrdθ=∬Si∂ϕ∂zrdrdθ.
Here uz denotes the vertical velocity of the water surface, and Si the cross-sectional area of the inner water surface.By substituting (2.2) into (5.1) it proves convenient to decompose the total volume flow, q, into three terms associated with the diffraction, qD, the motion-dependent-, qR, and the pressure-dependent-radiation problems, qP, as follows:
(5.2)q=qD+qR+qP.Assuming uniform pressure distribution inside the chamber, only the pumping mode form=0 affects the volumetric oscillations in evaluating qP. Moreover, by substituting (2.8) into (5.1) it can be shown that only modes with m=0 contribute to qD. Finally, the volume flow rate due to motion-dependent radiation potentials qR can be expressed on the basis of the relative displacement between the internal free surface elevation and the motions of the device. Due to the fact that the motions in the horizontal plane (surge) and the pitch mode for a vertical axisymmetric OWC device do not affect the volume of air in its chamber, only the velocity potential due to the vertical displacement x30 of the device will contribute to the volume flow rate, that is,
(5.3)qR=q3-x.30Si.
By accounting (5.2), (5.1), and (5.3) it can be obtained that
(5.4)qD=∬Si∂∂z(ϕ0+ϕ7)dSi,qP=∬Si∂ϕP∂zdSi,qR=∬Six.30∂ϕ3∂zdSi-x.30Si.
Next, by substituting (3.10) into (2.8), (2.10), and (2.12), we have, respectively,
(5.5)qD=(-iω)2ω2gH2dπb(FD,0,0MJ1(kb)kJ0(kb)N0-1/2cosh(kd)+∑j=1∞FD,0,jMI1(ajb)ajI0(ajb)Nj-1/2cos(ajd)),(5.6)qp=(-iω)2pin0gρπb(FP,0,0MJ1(kb)kJ0(kb)N0-1/2cosh(kd)+∑j=1∞FP,0,jMI1(ajb)ajI0(ajb)Nj-1/2cos(ajd)),(5.7)q3=x˙302ω2gdπb(F3,0,0MJ1(kb)kJ0(kb)N0-1/2cosh(kd)+∑j=1∞F3,0,jMI1(ajb)ajI0(ajb)Nj-1/2cos(ajd)).
The volume flow q3, induced by the heave mode forced oscillation, can be written as
(5.8)q3=(-e3P+iωd3P)·x˙30,
where e3P,d3P are the real and imaginary part of (5.7). The volume flow associated with the pressure radiation problem qP, (5.6), can be decomposed into real and imaginary part, fPP, gPP, respectively, as
(5.9)qP=-(fPP-igPP)·pin0,
where gPP is in phase with the flow acceleration and amounts to the added hydrodynamic inertia (i.e., radiation susceptance) and fPP is in phase with the flow velocity amounting to radiation damping (i.e., radiation conductance) [7, 11].We assume that the Wells turbine is placed in a duct between the chamber and the outer atmosphere and the total volume flowQ is proportional to the chamber air pressure [7, 22]:
(5.10)Q(t)=Λ·Pin(t),Λ representing a pneumatic complex admittance.The mass flow rate of air though the turbine can be written as [5]
(5.11)m.(t)=-d(ρin(t)·Vin(t))dt.
Here Vin is the air chamber volume, ρin is the air density, which can be related to the atmospheric density ρa, by assuming isentropic density-pressure relations, small air density variation and linearization with respect to the pressure [5], that is,
(5.12)ρin(t)=ρa[1+Pin(t)γ·Pa].
whith γ=1.4 being the adiabatic constant and Pa the atmospheric pressure. By inserting (5.11), (5.12) into (5.10), the air volume flow rate through the turbine is [5]
(5.13)m.ρa=Q(t)-V0γ·PadPin(t)dt,
where V0 is the value of Vin in undisturbed conditions. The turbine aerodynamic performance can be written in dimensionless form as [23, 24]
(5.14)Φ=m.ρaND3,Ψ=PinρaN2D2,Π=EρaN3D5,
where N is the rotational speed (radians per unit time) of turbine blades, E the turbine power output, and D the turbine rotor diameter. It was found [25] that for the Wells turbine:
(5.15)Φ=K·Ψ,
where K is constant for a given turbine geometry (independent of turbine size or rotational speed). Substitution of (5.14), (5.15) into (5.13) gives [26]
(5.16)q=[KDρaN+(-iω)V0γPa]pin0.
From (5.10) and (5.16):
(5.17)Λ=gT+(-iω)V0γPa,
where gT=KD/ρaN.


## 6. Wave Forces

The various forces on the oscillating water column device can be calculated from the pressure distribution given by the linearised Bernoulli’s equation:(6.1)P(r,θ,z;t)=-ρ∂Φ∂t=(-iω)ρϕ·e-iωt,
where ϕ is the velocity potential in each fluid domain I, III, M.The horizontal wave force,Fx=Re(fx·e-iωt), on the device is given by
(6.2)fx=fxOut-fxIn=-iωρa∫z=hz=d∫0θ=2πϕIcosθdθdz+iωρb∫z=hz=d∫0θ=2πϕMcosθdθdz,
where fxOut is the horizontal force on the device’s external wall, and fxIn is the horizontal force on the chamber’s wall.The oscillating pressure inside the chamber does not affect the horizontal exciting force on the floating device since the radiation potential includes only them=0 angular mode. So, the horizontal force on the OWC device is the same as on an open vertical axisymmetric bottomless duct with finite wall thickness [16].On the contrary, the oscillating pressure inside the chamber does affect the vertical force due to unit internal pressure head, on the floating device. The total vertical force on the OWC device,Fz=Re(fz·e-iωt), is equal to the sum of the vertical wave exciting force, Fzopen=Re(fzopen·e-iωt), on the device when the duct is open to the atmosphere (diffraction load) and the vertical force, Fzclose=Re(fzclose·e-iωt), due to the internal pressure head when the duct is closed:
(6.3)fz=fzopen+fzclose=-iωρ∫r=br=a∫0θ=2πϕIII(-1)rdθdr=-iωρ∫r=br=a∫0θ=2π(ϕDIII+ϕPIII)(-1)rdθdr.
Substituting (2.8) and (2.12) into (6.3) we extract
(6.4)fzopenB=2kdtanh(kd){12FD,0,0III(1-1-(b/a)22ln(a/b))-12FD,0,0*III(b2a2-1-(b/a)22ln(a/b))+2∑n=1∞(-1)n(hnπa)2[FD,0,nIII(A0nIII-D0nIII)+FD,0,n*III(A0n*III-D0n*III)]},(6.5)fzcloseπa2pin0=2{12FP,0,0III(1-1-(b/a)22ln(a/b))-12FP,0,0*III(b2a2-1-(b/a)22ln(a/b))+2∑n=1∞(-1)n(hnπa)2[FP,0,nIII(A0nIII-D0nIII)+FP,0,n*III(A0n*III-D0n*III)]},
where B=πρga2(H/2) and FD,0,nIII, FD,0,n*III, FP,0,nIII, and FP,0,n*III are the diffraction and pressure radiation Fourier coefficients for the IIIth type of ring element and A0nIII, A0n*III, D0nIII, and D0n*III are defined at (4.8).The vertical forcefzclose induced by the oscillating pressure head pin0 is related [8, 22] to the volume flow q3, induced by the heave mode forced oscillation, (see (5.8)). The above force, fzclose, can be written as
(6.6)fzclose=(-eP3+iωdP3)pin0=(e3P-iωd3P)pin0,
where e3P,d3P are the real and imaginary part of (6.5)The moment on the device about a horizontal axis lying at an arbitrary distancez=e from the sea bed is the real part of M·e-iωt, where M is made up of Ms and Mb arising from the pressure distribution on the device’s vertical walls and on its bottom, respectively, [16]:
(6.7)Ms=-iωρa∫z=hz=d∫0θ=2πϕI(z-e)cosθdθdz+iωρb∫z=hz=d∫0θ=2πϕM(z-e)cosθdθdz,Mb=-iωρ∫r=br=a∫0θ=2πϕIII(-1)r2dθdr=-iωρ∫r=br=a∫0θ=2π(ϕDIII+ϕPIII)(-1)r2dθdr.


## 7. Added Mass and Damping Coefficients

The hydrodynamic reaction forces and pitching momentFi,j acting on the oscillating water column device in the ith direction due to its sinusoidal motion with frequency ω and unit amplitude in the jth direction can be obtained as the real part of fi,j·e-iωt, from (6.1):
(7.1)fi,j=-ρω2∬SΨj,m(r,z)cos(mθ)nidS.
Herein ni is defined at (2.16) and Ψj,m is obtained from (3.1), (3.5), and (3.10) in each fluid domain I, III, M.The complex forcefi,j may be written in the form [27]:
(7.2)fij=ω2aij+iωbij,
where aij and bij are the added mass and damping coefficients, respectively, both real and dependent on frequency ω. Substituting the appropriate expressions for Ψj,m in (7.1) and using (7.2), the relations for nondimensionalized hydrodynamic coefficients can be found [17].


## 8. Motion and Air Pressure Calculation

Assuming that the waves and the device oscillations are described by small amplitude motions, the hydrodynamic problem is well characterized by a linear approach. The 6 degrees of freedom system of motion equations, in the frequency domain, are written as(8.1)∑j=16(mkj+akj)x¨j+bkjx˙j+ckjxj=Fk(t),k=1,2,…,6,
where xj is the 6-degree displacement vector, mkj is the mass matrix, akj is the frequency-dependent added mass matrix, bkj is the frequency-dependent damping matrix, ckj is the stiffness matrix, and Fk represent the total force (diffraction and pressure induced radiation) on the device.Under the assumption of symmetrical mass distribution, a vertical body of revolution performs under the action of a regular incident wave three degrees of freedom motion, that is, two translations (x1: surge, x3: heave) and one rotation (x5: pitch). So the (8.1) can be reduced to
(8.2)(m+a33)x¨30+b33x˙30+c33x30=f3,(8.3)(m+a11)x¨10+b11x˙10+c11x10+(mXg3(0)+a15)x¨50+b15x˙50+c15x50=f1,(8.4)(I55+a55)x¨50+b55x˙50+c55x50+(mXg3(0)+a51)x¨10+b51x˙10+c51x10=f5,
where
(8.5)c11=c15=c51=0,c33=ρgAwp,c55=ρgV(GM¯).
Also, Awp is the water plane area, GM¯ is the metacentric height, Xg3(0) is the vertical distance of the centre of gravity from the reference point of motions, and I55 is the pitch mass moment of inertia of the device.Since the oscillating pressure inside the chamber does affect only the vertical exciting force on the floating device, (8.2)  can be recast as follows:
(8.6)(-iω(m+a33)+b33+iωc33)x˙30=fzopen+fzclose+fMP
with fMP being the force on the chamber’s wall due to inner pressure; it is equal to
(8.7)fMP=Si·pin0.
Combining (8.7), (8.6) we obtain
(8.8)(-iω(m+a33)+b33+iωc33)x˙30+(-fzclosepin0-πb2)pin0=fzopen.
From (5.16) and (5.2) it is obtained that
(8.9)[gT+(-iω)V0γPa]pin0=qD+qR+qP
which in turn by considering (5.3) is reduced to
(8.10)[-q3x.30+πb2]x.30+[gT+(-iω)V0γPa-qPpin0]pin0=qD.
Solving the system of (8.3), (8.4), (8.8), and (8.10), we calculate the unknown motion components of the device xj0j=1,3,5 and the internal chamber’s pressure pin0. When the body is considered restrained in the waves, the air chamber pressure is deduced from (8.10) for x˙30=0.


## 9. Wave Runup

The free surface elevation outside and inside the device is given by the linearized boundary condition at the outer and inner free sea surface. It holds:(9.1)Z(r,θ,d;t)=Re{ζ(r,θ,d)·e-iωt}=Re{(iωgφI(r,θ,d))(cos(ωt)-isin(ωt))},(9.2)Z(r,θ,d;t)=Re{ζ(r,θ,d)e-iωt}=Re{(iωgφM(r,θ,d)-pin0ρg)(cos(ωt)-isin(ωt))},
for the outer and inner fluid domains, respectively.


## 10. Drift Forces

The time-independent part of the second-order wave forces (drift forces) on the floating OWC device, has been evaluated using the direct integration of all pressure terms contributing to second-order loads components over the instantaneous body’s wetted surface. It holds [28]:
(10.1)Fx¯T=-∫WL12·ρ·g·ζr¯Tnxdℓ+m·R·Xg¯..¯T+∬S012·ρ·|∇Φ|2¯TnxdS0+∬S0ρ·X¯·∇Φt¯TnxdS0,(10.2)Fz¯T=m·R·Xg¯..¯T+∬S012·ρ·|∇Φ|2¯TnzdS0+∬S0ρ·X¯·∇Φt¯TnzdS0,
where m is the device’s mass, R is the transformation matrix, containing only first-order angular displacement (see, Appendix B), X¨g is the acceleration matrix of the body’s centre of gravity (see, Appendix B), ρ is the water density, g is the acceleration due to gravity, ζr is the relative wave height (see, Appendix B), Φ is the time harmonic complex velocity potential of the flow field around the structure, and X¯ is the displacement vector of a point on the device (see, Appendix B).


## 11. Absorbed Power

The time-averaged power absorbed by the device from the waves,Pout, is obtained from [11]
(11.1)Pout=12Re[Λ·|pin0|2],
where Λ is the turbine’s pneumatic admittance, (5.17). By inserting (5.2), (5.16), and (5.17), into (11.1), we obtain
(11.2)Pout=KD2ρaN|pin0|2=KD2ρaNRe[(qD+qR)·(qD¯+qR¯)](fPP+KD/ρaN)2+(ωV0/γPa+gPP)2.
Alternatively, (11.1) can be expressed as:
(11.3)Pout=12Re[(qD+qR)pin0¯]-12fPP|pin0|2=|qD+qR|28·fPP-fPP2|pin0-qD+qR2·fPP|2,
where fPP, gPP are defined in (5.9). Using (11.3), the maximum absorbed power can be obtained as
(11.4)Pmax=|qD+qR|28·fPP
corresponding to an optimum inner pressure head pin0opt, equal to
(11.5)pin0opt=qD+qR2·fPP.
The capture width ℓ [7, 11] is the ratio of the power absorbed by the device to the available power per unit crest length of the incident wave, that is,
(11.6)ℓ=2·Poutρ·g·(H/2)2·Cg.
where Cg is the group velocity of the incident wave, given by
(11.7)Cg=12ωk(1+2kdsinh(2kd)),
and k is the wave number.In order to achieve maximum capture width, it is not always practical to change the device’s geometry, but it is likely easier to control the rotational speedN and the turbines diameter D. The optimum value of gT, (5.17), is obtained from ∂ℓ/∂gT=0 and from (11.6) it is equal to
(11.8)gTopt=fPP2+(gPP+ωV0γPa)2.
When gT takes the above optimum value, the maximum value of the absorbed power can be written in the bellow form:
(11.9)Pmax=|qD+qR|28·fPP{1-2·gTopt·(gTopt-fPP)2·gTopt·(gTopt+fPP)-4·ωV0γPa·gPP}.


## 12. Numerical Results

The calculation of the Fourier coefficientsFj,m,iI, Fj,m,nIII, Fj,m,n*III, Fj,m,iM, j=D,1,3,5, and P is the most significant part of the numerical procedure, because of their influence on the accuracy of solution. For the case of the first and the Mth ring element, i=20, terms were used, while for the third ring element N=50, since it was found that the results obtained for those values were correct to an accuracy of within 1%.Since the pressure contribution inside the chamber affects only the vertical forces exerted on the device, the horizontal exciting force coincides with the one computed by Mavrakos [16] for a device open to the atmosphere. So, at first for the sake of computer code’s verification, a geometric configuration investigated in [16] has been chosen. It is a body with immersion h/b=3.46 and water depth d/b=4 for three cases of wall thickness a/b=4, a/b=2, and a/b=1.5. In Figure 2 the horizontal and vertical exciting force on the OWC device is compared with the results by Mavrakos [16] for the moon pool body when the inner pressure is equal to the atmospheric.Figure 2
Horizontal and vertical exciting force on the OWC device for different cases of wall thickness.Then, a three-dimensional oscillating water column device is investigated, which is floating in a water depthd=15 m with chamber’s draught equal to 5 m. The OWC’s internal and external radii are 2 m and 4 m, respectively. The vertical force on the restrained device due to a unit inner pressure head, expressed though (6.5), is plotted in Figure 3.Figure 3
Vertical exciting force on the OWC device due to a unit inner pressure head.Next, we examine how the results’ accuracy is affected by the number of terms considered in the potentials’ series expansions. In Figures4 and 5 the values of fxN-fx10/fx10 and fzN-fz10/fz10 are plotted for various numbers of series terms N. Here, fxN and fzN are the modulus of the horizontal and the total vertical exciting force, respectively, exerted on the same as the above OWC device with a turbine parameter gT=1 m5/(N·s).Figure 4
Graphics forfxN-fx10/fx10 versus N element terms.Figure 5
Graphics forfzN-fz10/fz10 versus N element terms.In Tables1 and 2, the module of the air pressure trapped in the OWC chamber is presented for the various values of gT and ka. The same geometry of the OWC is assumed as the one previously examined. In Table 1 results for the restrained device, while in Table 2 for the free floating one, are given. It is worthwhile to mention that when gT≫0, the device’s chamber is approximating the open condition, so the inner pressure head tends to zero. On the contrary, by letting gT→0 the turbine at the top of the chamber is approximating the closed condition, thus the air chamber pressure is increased.Table 1
Modulus of the inner pressurepin0/(H/2) for various values of gT andka, for a restrained OWC device.
g
T
=
0.5
g
T
=
1
g
T
=
2
g
T
=
3
g
T
=
6
g
T
=
10
k
·
a
=
0.4
4.94
E
+
00
4.71
E
+
00
4.05
E
+
00
3.39
E
+
00
2.11
E
+
00
1.35
E
+
00
k
·
a
=
1.0
1.30
E
+
00
1.26
E
+
00
1.12
E
+
00
9.71
E
-
01
6.33
E
-
01
4.15
E
-
01
k
·
a
=
1.5
4.85
E
-
01
4.36
E
-
01
3.30
E
-
01
2.50
E
-
01
1.39
E
-
01
8.60
E
-
02
k
·
a
=
2.0
1.93
E
-
01
1.62
E
-
01
1.11
E
-
01
7.98
E
-
02
4.24
E
-
02
2.61
E
-
02Table 2
Modulus of the inner pressurepin0/(H/2) for various values of gT and ka, for a floating OWC device with mass moment of inertia I=1738.87tn·m2.
g
T
=
0.5
g
T
=
1
g
T
=
2
g
T
=
3
g
T
=
6
g
T
=
10
k
·
a
=
0.4
5.87
E
-
01
5.68
E
-
01
5.07
E
-
01
4.38
E
-
01
2.85
E
-
01
1.85
E
-
01
k
·
a
=
1
.
0
4.02
E
-
01
3.98
E
-
01
3.84
E
-
01
3.63
E
-
01
2.82
E
-
01
2.13
E
-
01
k
·
a
=
1.5
1.52
E
-
01
1.45
E
-
01
1.26
E
-
01
1.06
E
-
01
6.37
E
-
02
4.18
E
-
02
k
·
a
=
2
.
0
6.73
E
-
02
6.21
E
-
02
4.91
E
-
02
3.86
E
-
02
2.13
E
-
02
1.36
E
-
02Next, comparisons are presented with the results given by Martins-Rivas and Mei [11]. To this end, we assume a vertical cylindrical oscillating water column device, having a wall thickness (a-b)/d=0.005, external radius a/d=0.5 and floating with a draught (d-h)/d=0.2. Figure 6 concerns the conductance, B, and susceptance, C, as defined by B=Re[(qp/pin0)/(a/ωρ)] and C=Im[(qp/pin0)/(a/ωρ)], respectively, while in Figure 7 the nondimensional optimum turbine parameter gT·ρ/a is plotted against the wave number parameter kd. In calculating the optimum turbine parameter though (11.8) since isentropy is assumed γPa=ca2, ca being the sound velocity in air. The comparisons with the results of Martins-Rivas and Mei [11] are excellent.Figure 6
ConductanceB and susceptance C for an isolated OWC device in the open sea.Figure 7
Optimum turbine parameter admittancegT·(ρ/a) versus kd.Next, in Table3 the RAOs of the heave motion are given against ω2b/g for an OWC device with dimensions d/b=2, d-h/b=1/3 and a/b=1.8, a/b=1.2, and a/b=1.034. The results are compared with those by Mavrakos and Bardis [29] that correspond to the open moon pool case. For the present calculations, a value of gT=1000 m5/(N·s) is assumed in order to approximate the open chamber case.Table 3
RAOs of heave motionx30/(H/2) versus ω2b/g for a floating OWC device with turbine parameter gT=1000m5/(N·s). Comparisons with an open moon pool cylinder.
ω
2
b/g
Open moon pool cylinder [29]
OWC device withgT=1000m5/(N·s)
a/b = 1.8
a/b = 1.2
a/b = 1.034
a/b = 1.8
a/b = 1.2
a/b = 1.034
0.037
0.9549
0.9733
0.9716
0.9548
0.9733
0.9718
0.883
0.6458
0.9582
0.9462
0.6440
0.9576
0.9489
1.223
0.0124
0.5061
0.7706
0.0191
0.5038
0.7864
1.348
0.2244
0.2075
0.5358
0.2201
0.2094
0.5665
1.761
0.6407
0.3720
0.1550
0.6118
0.3639
0.2022
1.926
0.3234
0.6149
0.2998
0.3196
0.5862
0.3011Further comparisons concerning the free surface elevation inside and outside of the device are presented in Figures8 and 9 when gT≫0. The results are compared to those of Mavrakos [16] for an open moonpool cylinder. The device’s immersion is h=10 m, inner and outer radius are b=2 m, a=20 m, respectively, and the water depth is d=20 m. In Figures 8 and 9 the wave profile inside and outside of the structure is being presented for wave periods T=6.3 s and T=6.0 s, wave lengths λ=59.8 m and λ=54.8 m, respectively, and gT=100 m5/(N·s). The angle of wave attack is θ=0°.Figure 8
Wave profile inside and outside OWC device for a wave periodT=6.3 s.Figure 9
Wave profile inside and outside OWC device for a wave periodT=6.0 s.Next, the OWC device examined in Figure3 is further investigated here with the aid of Table 4 as far the wave runup (see (9.2) and (9.1)) inside and outside the structure is concerned. For the calculations three values of turbine parameter Λ=gT=1,10,100 (m5/(N·s)) are assumed for t=0 s.Table 4
Wave runup inside and outside an OWC device versuska. Angle of wave attack ϑ=0.
k
·
a
Wave runup on the outer OWC’s surface
Wave runup on the inner OWC’s surface
g
T
=
1
g
T
=
10
g
T
=
100
g
T
=
1
g
T
=
10
g
T
=
100
0.4
0.8313
0.8305
0.8313
1.5166
1.6172
1.6361
0.8
0.3555
0.3507
0.3502
−0.4638
−0.4492
−0.0742
1.0
0.0825
0.0811
0.0821
−0.1741
−0.1374
−0.0988
1.3
−0.4048
−0.4061
−0.4049
−0.0814
−0.0691
−0.0622
1.6
−0.8494
−0.8507
−0.8494
−0.0807
−0.0765
−0.0740
2.0
−1.3019
−1.3029
−1.3019
−0.1256
−0.1249
−0.1238We can observe that the wave runup on the structure’s external wall tends to almost the same values independent of whether the duct is closed or open to the atmosphere (expressed though the variousgT values of turbine parameters; values of gT tending to zero correspond to almost closed chamber, while large values to the open duct case). Contrary, the wave runup on the inner wall is affected by the gT values.The maximum absorbed energy (see (11.4)) for the same as the above OWC device is presented in Table 5 versus ka.Table 5
Comparison between the maximum energyPmax/(H/2)2 of a restrained and a floating OWC device versus ka. The mass moment of inertia I=1738.87tn·m2.
k
·
a
Restrained OWC device
Floating OWC device
0.2
956.983
0.3219
0.4
308.105
5.7967
0.6
147.688
139.779
0.8
90.6602
31.8176
1.0
63.7096
16.7789
1.5
34.5299
8.1777
2.0
22.5046
6.1624
2.5
16.1687
5.2424
3.0
12.3529
4.6839Next, in Figure10 the maximum capture width is plotted for a restrained OWC device with various external radius to water depth ratios a/d=1/8,1/4,1/2, and 1, wall thickness (a-b)/d=0.005, and floating in a water depth d/h=2. These results are compared very well with the ones given in Evans and Porter [7] work.Figure 10
The maximum capture width againstω2·d/g in various geometries of a restrained OWC device.Finally, in Figure11 results concerning the horizontal mean second-order wave drift forces exerted on the OWC device that was investigated in Figure 3 of the present work are given for three different values of the turbine admittance characteristics gT.Figure 11
Horizontal mean second-order wave drift forces againstk·a for various turbine parameters gT=1,10,1000 (m5/(N·s)).


## 13. Conclusion

An analytical method has been developed to solve the diffraction and radiation problems around a floating vertical axisymmetric OWC device. This method provides an efficient tool for the complete hydrodynamic analysis of this important type of wave energy conversion devices, including the evaluation of the first-order wave exciting forces, the mean second-order wave drift forces, the hydrodynamic parameters, the pressure head inside the chamber, the air flow flux, and the maximum absorbed power. The results of this analysis are of particular importance for the design of such types of wave energy converters, for which the hydrodynamic parameters and the turbine characteristics have to be combined in order to improve the wave energy conversion.



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*Source: 142850-2012-11-20.xml*