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S. A. Mavrakos, D. N. Konispoliatis, "Hydrodynamics of a Free Floating Vertical Axisymmetric Oscillating Water Column Device", Journal of Applied Mathematics, vol. 2012, Article ID 142850, 27 pages, 2012. https://doi.org/10.1155/2012/142850
Hydrodynamics of a Free Floating Vertical Axisymmetric Oscillating Water Column Device
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.
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 , Sarmento et al. , Evans and Porter , and Kokkinowrachos et al. , 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 . 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 . Recently, Martins-Rivas and Mei  presented an analysis on an OWC device mounded at the tip of a breakwater with infinitesimal wall thickness and Falcão et al.  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 , Garrett , and Yeung  as far as a circular dock is concerned, by Mavrakos [16, 17] for bottomless cylinders, by Kokkinowrachos et al.  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  to account for finite wall thickness of the vertical bottomless cylindrical chamber and the related 2D formulation by Kokkinowrachos et al.  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 (Figure 1) of internal and external radius and , respectively, excited by regular waves of amplitude , frequency , and wave number in constant water depth, . The draught of the device’s chamber is . We assume small amplitude waves and motions and inviscid incompressible irrotational flow. Cylindrical coordinates are introduced with the vertical axis 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 with , having the same frequency, , as the incident wave.
The time harmonic complex velocity potential of the flow field around the structure can be expressed as We decompose the above velocity potential as Here, is the velocity potential of the undisturbed incident harmonic wave; 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; , , is the radiation potential resulting from the forced body motion in the th mode of motion with unit velocity amplitude with the duct open to the atmosphere; is the radiation potential resulting from the oscillating pressure head, , in the chamber for the body fixed in otherwise calm water.
The undisturbed incident wave potential, , can be expressed in cylindrical co-ordinates as  where Here, is the th order Bessel function of the first kind, is the Neumann’s symbol defined as and for and : with being its derivative at . Frequency and wave number are related by the dispersion equation: The diffracted potential, , around the restrained structure is described by the velocity potential: In accordance to (2.3) the diffracted velocity potential of the flow field around the OWC device can be written in the form: The fluid flow caused by the forced oscillation of a vertical axisymmetric body in otherwise still water is symmetric about the plane and antisymmetric about plane for surge, , and pitch, , whereas it is symmetric about both these planes for heave, . Thus the corresponding velocity potentials for these modes of motion can be expressed as As for the radiation potential , 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 . Thus, since the forcing of the internal free surface is independent of , the radiation potential will be axially symmetric and so it will include only the angular mode, that is, it holds: In the functions of (2.8)–(2.12) the first subscript denotes the respective boundary value problem each time considered, whereas the second one indicates the value of which must be taken into account for the solution of the corresponding problem. Thus the functions , , , , and remain the principal unknowns of the problem.
The velocity potentials , , have to satisfy the Laplace equation within the entire fluid domain, the linearized boundary condition at the outer and inner free sea surface (), that is, and the zero normal velocity on the sea bed, that is, Furthermore, the velocity potentials , , have to fulfill following kinematic conditions on the mean device’s wetted surface : where in (2.16) denotes the derivative in the direction of the outward unit normal vector , to the wetted surface of the device and are its generalized components defined as and , where 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 as , which has the form: Moreover, the velocity potentials, , and their derivatives, , must be continuous at the vertical boundaries of neighbouring fluid regions (Figure 1). This results in: The superscripts , , 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, , in each fluid domain (Figure 1; ) 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
Infinite Fluid Domain. Type I
One has where and and are the th order Hankel function of the first kind and the modified Bessel function of the second type, respectively, and are unknown Fourier coefficients to be determined by the solution procedure. Moreover, are orthonormal functions in defined by (2.5) for and by The eigenvalues are roots of the transcendental equation: which possesses one imaginary, , and infinite number of real roots. Substituting the value of in (3.3) and (3.4), (2.5) and (2.6) can directly be obtained.
Fluid Region of Type III
One has where and is the th order modified Bessel function of the first kind and , are Fourier coefficients to be determined by the solution procedure. The functions 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.
Interior Fluid Region of Type M
One has where and Here, the orthonormal functions are defined by (2.5) and (3.3), for and , respectively, and are given by (3.4). Moreover, are unknown Fourier coefficients in the th fluid region. The function 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 functions and ), 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 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, . 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 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 at and is expressed by (2.18) and (2.20), respectively. Multiplying both sides of equations by and integrating over their region of validity, that is, the following set of equations can be obtained: where are defined in Appendix A.
Now, the condition for the continuity of the first derivative of the potential at and 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 , integrating over the region of their validity, that is, and , respectively, and adding the resulting expressions, the following set of equations is obtained: where and , 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 , , fluid region will be truncated after , , and terms, respectively.
Substituting (4.1) into (4.5) will provide the unknown Fourier coefficients, , in the fluid domain in relation with the Fourier coefficients, , in the fluid domain, in the following matrix form, that is, where, and are both complex vectors, the elements of which are the unknown Fourier coefficients in the th and th fluid domain, respectively, is a diagonal matrix given by (4.7), is a matrix given by (4.3), is the Neumann’s symbol diagonal matrix with and for , and are diagonal matrices defined in (4.8), and is a column vector defined in (4.4).
From (4.9) and (4.1), the Fourier coefficients, and , in the th fluid domain are connected with the bellow relation: where is the unit matrix and where, is a complex vector, the elements of which are the unknown Fourier coefficients in the th region.
Substituting (4.2) into (4.6) will provide the unknown Fourier coefficients, , in the th fluid domain in relation with the Fourier coefficients, , in the fluid domain, in the following matrix form, that is, where, is a complex vector, the elements of which are the unknown Fourier coefficients in the fluid domain, is a () diagonal matrix given by (4.7), and is a 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 by where Here denotes the vertical velocity of the water surface, and 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, , into three terms associated with the diffraction, , the motion-dependent-, , and the pressure-dependent-radiation problems, , as follows:
Assuming uniform pressure distribution inside the chamber, only the pumping mode for affects the volumetric oscillations in evaluating . Moreover, by substituting (2.8) into (5.1) it can be shown that only modes with contribute to . Finally, the volume flow rate due to motion-dependent radiation potentials 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 of the device will contribute to the volume flow rate, that is, By accounting (5.2), (5.1), and (5.3) it can be obtained that Next, by substituting (3.10) into (2.8), (2.10), and (2.12), we have, respectively, The volume flow , induced by the heave mode forced oscillation, can be written as where , are the real and imaginary part of (5.7). The volume flow associated with the pressure radiation problem , (5.6), can be decomposed into real and imaginary part, , , respectively, as where is in phase with the flow acceleration and amounts to the added hydrodynamic inertia (i.e., radiation susceptance) and 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 flow is proportional to the chamber air pressure [7, 22]: representing a pneumatic complex admittance.
The mass flow rate of air though the turbine can be written as  Here is the air chamber volume, is the air density, which can be related to the atmospheric density , by assuming isentropic density-pressure relations, small air density variation and linearization with respect to the pressure , that is, whith being the adiabatic constant and the atmospheric pressure. By inserting (5.11), (5.12) into (5.10), the air volume flow rate through the turbine is  where is the value of in undisturbed conditions. The turbine aerodynamic performance can be written in dimensionless form as [23, 24] where is the rotational speed (radians per unit time) of turbine blades, the turbine power output, and the turbine rotor diameter. It was found  that for the Wells turbine: where is constant for a given turbine geometry (independent of turbine size or rotational speed). Substitution of (5.14), (5.15) into (5.13) gives  From (5.10) and (5.16): where .
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: where is the velocity potential in each fluid domain , , .
The horizontal wave force, , on the device is given by where is the horizontal force on the device’s external wall, and 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 the 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 .
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, , is equal to the sum of the vertical wave exciting force, , on the device when the duct is open to the atmosphere (diffraction load) and the vertical force, , due to the internal pressure head when the duct is closed: Substituting (2.8) and (2.12) into (6.3) we extract where and , , , and are the diffraction and pressure radiation Fourier coefficients for the th type of ring element and , , , and are defined at (4.8).
The vertical force induced by the oscillating pressure head is related [8, 22] to the volume flow , induced by the heave mode forced oscillation, (see (5.8)). The above force, , can be written as where are the real and imaginary part of (6.5)
The moment on the device about a horizontal axis lying at an arbitrary distance from the sea bed is the real part of , where is made up of and arising from the pressure distribution on the device’s vertical walls and on its bottom, respectively, :
7. Added Mass and Damping Coefficients
The hydrodynamic reaction forces and pitching moment acting on the oscillating water column device in the th direction due to its sinusoidal motion with frequency and unit amplitude in the th direction can be obtained as the real part of , from (6.1): Herein is defined at (2.16) and is obtained from (3.1), (3.5), and (3.10) in each fluid domain , , .
The complex force may be written in the form : where and are the added mass and damping coefficients, respectively, both real and dependent on frequency . Substituting the appropriate expressions for in (7.1) and using (7.2), the relations for nondimensionalized hydrodynamic coefficients can be found .
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 where is the 6-degree displacement vector, is the mass matrix, is the frequency-dependent added mass matrix, is the frequency-dependent damping matrix, is the stiffness matrix, and 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 (: surge, : heave) and one rotation (: pitch). So the (8.1) can be reduced to where Also, is the water plane area, is the metacentric height, is the vertical distance of the centre of gravity from the reference point of motions, and 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: with being the force on the chamber’s wall due to inner pressure; it is equal to Combining (8.7), (8.6) we obtain From (5.16) and (5.2) it is obtained that which in turn by considering (5.3) is reduced to Solving the system of (8.3), (8.4), (8.8), and (8.10), we calculate the unknown motion components of the device and the internal chamber’s pressure . When the body is considered restrained in the waves, the air chamber pressure is deduced from (8.10) for .
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: 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 : where is the device’s mass, is the transformation matrix, containing only first-order angular displacement (see, Appendix B), is the acceleration matrix of the body’s centre of gravity (see, Appendix B), is the water density, is the acceleration due to gravity, is the relative wave height (see, Appendix B), is the time harmonic complex velocity potential of the flow field around the structure, and 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, , is obtained from  where is the turbine’s pneumatic admittance, (5.17). By inserting (5.2), (5.16), and (5.17), into (11.1), we obtain Alternatively, (11.1) can be expressed as: where , are defined in (5.9). Using (11.3), the maximum absorbed power can be obtained as corresponding to an optimum inner pressure head , equal to The capture width [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, where is the group velocity of the incident wave, given by and 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 speed and the turbines diameter . The optimum value of , (5.17), is obtained from and from (11.6) it is equal to When takes the above optimum value, the maximum value of the absorbed power can be written in the bellow form:
12. Numerical Results
The calculation of the Fourier coefficients , , , , , and 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 th ring element, , terms were used, while for the third ring element , 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  for a device open to the atmosphere. So, at first for the sake of computer code’s verification, a geometric configuration investigated in  has been chosen. It is a body with immersion and water depth for three cases of wall thickness , , and . In Figure 2 the horizontal and vertical exciting force on the OWC device is compared with the results by Mavrakos  for the moon pool body when the inner pressure is equal to the atmospheric.
Then, a three-dimensional oscillating water column device is investigated, which is floating in a water depth 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.
Next, we examine how the results’ accuracy is affected by the number of terms considered in the potentials’ series expansions. In Figures 4 and 5 the values of and are plotted for various numbers of series terms . Here, and 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 m5/(N·s).
In Tables 1 and 2, the module of the air pressure trapped in the OWC chamber is presented for the various values of and . 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 , the device’s chamber is approximating the open condition, so the inner pressure head tends to zero. On the contrary, by letting the turbine at the top of the chamber is approximating the closed condition, thus the air chamber pressure is increased.