The aim of this work is to develop a simple nonlinear model of a wave energy converter (WEC) for capturing power from ocean waves and converting it into electrical power. A generic axisymmetric device is considered, which consists of a vertical circular cylinder surrounded by a circular annulus. The nonlinear system of equations of motion of this generic WEC are derived; these include the nonlinear term arising from viscous drag due to boundary layer separation. The expressions for radiation damping and added mass are determined by dimensional analysis. These equations are then solved numerically and the results are displayed graphically in a number of figures. Consideration of these graphs leads to conclusions that should be taken into account by the design engineer

John J.H. Miller

Lawrence J. Crane

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MATHEMATICAL COMMUNICATIONS 303Math. Commun. 17(2012), 303–312A simple nonlinear model of a generic axisymmetric waveenergy converter∗Lawrence J.Crane1 and John J.H.Miller1,†1 Institute for Numerical Computation and Analysis (INCA), 7-9 Dame Court, Dublin 2,IrelandReceived October 31, 2011; accepted February 20, 2012Abstract. The aim of this work is to develop a simple nonlinear model of a wave energyconverter (WEC) for capturing power from ocean waves and converting it into electricalpower. A generic axisymmetric device is considered, which consists of a vertical circularcylinder surrounded by a circular annulus. The nonlinear system of equations of motion ofthis generic WEC are derived; these include the nonlinear term arising from viscous dragdue to boundary layer separation. The expressions for radiation damping and added massare determined by dimensional analysis. These equations are then solved numerically andthe results are displayed graphically in a number of figures. Consideration of these graphsleads to conclusions that should be taken into account by the design engineer.AMS subject classifications: 76B15, 76M20, 76M55Key words: ocean wave energy device, mathematical model, numerical model, nonlinearsystem, dimensional analysis, added mass, radiation damping, power take-off, viscous drag1. IntroductionThe generic axisymmetric wave energy converter (WEC) considered here consists ofa floating buoy-like device that is loosely moored to the seabed. It can freely moveup and down in response to the rising and falling of the waves. Since it is assumedthat the horizontal dimensions of the buoy are small compared to the wavelength ofthe incident wave, it follows that it operates primarily in heave mode. It also hasa power take–off device, an electrical generator, a power electronics system and acontrol system, all of which are sealed in the device. The WEC converts mechanicalenergy, due to the heaving, into electrical energy, by means of an electrical genera-tor. The power electronics system then conditions the output from the generator intouseable electric power. The heaving motion resonates with the ocean waves, whichis vitally important in a resonating energy absorber, if it is to recover useful powerfrom ocean wave energy. In the next section, the mathematical model is discussedand the corresponding nonlinear system of two equations of motion are derived.The treatment is novel, because viscous drag is taken into account. In Section 3,dimensional analysis is used to derive, in a new manner, expressions for the added∗Presented at the Seventh Conference on Applied Mathematics and Scientific Computing, Trogir,Croatia, June 2011.†Corresponding author. Email addresses: lc@incaireland.org (L. J. Crane), jm@incaireland.org(J. J. H.Miller)http://www.mathos.hr/mc c©2012 Department of Mathematics, University of Osijek304 L. J. Crane and J. J. H.Millermass and the radiation damping coefficient. The second order equations of Section2 may be reduced to first order equations in a standard manner, which is describedin Section 4. This results in a nonlinear system of four first order equations, whichcan be solved by a standard numerical solver contained in, for example, Matlab. InSection 5, graphs of the numerical results, obtained by a systematic exploration ofthe parameter space of the buoy, are presented and discussed. The goal of this ex-ploration is to determine optimal parameter values, which correspond approximatelyto those required for maximum power output. Since the equations are nonlinear, ananalytic solution is not available. However, it should be emphasised that, even if ananalytic solution were available, the dimension of the parameter space is such thatit would not be feasible to determine analytically these optimal parameter values.The present approach is made feasible by the careful formulation of the math-ematical model and the availability of standard packages containing fast nonlinearsolvers. Because of the nonlinearity of the mathematical model, its solution is be-yond the scope of standard approaches based on frequency domain models. Theapproach described here enables the WEC designer to derive approximate optimalvalues of the design parameters in a few minutes using a desktop computer. Eachpoint in the graphs presented in Section 5 requires the solution of the nonlinearsystem. Nevertheless, the three-dimensional graphs can all be obtained in a fewminutes. This low cost rapid determination of approximate optimal values of theparameters is a useful supplement, at the early stages of a WEC design, to the highcost slow determination of accurate values by methods using a three-dimensionalboundary element or CFD techniques.2. Mathematical modelConsider first the heaving motion of the cylinder in isolation. Let L be its totallength, d its draft, a its cross-sectional area, z the distance measured verticallyupwards from the draft level to the mean level of the water surface, so that z ispositive when the draft level is below the mean level of the water surface. In addition,let x be the height of the wave above the mean water level and y the distance fromthe wave to the top of the cylinder. It follows thatx + y + z + d = L,and thatẍ + ÿ + z̈ = 0.Now the vertical acceleration of the top of the cylinder, referred to the constantmean water level, isẍ + ÿ,so that the vertical acceleration of the cylinder is−z̈.Wave energy converter 305Neglecting damping, the vertical forces on the cylinder consist of the upward pressureforce on the bottom of the cylinder, namelyρga(x + z + d),and the downward weight of the dead mass, m, of the cylinder,mg,where ρ is the density of the water and g is the gravitational acceleration. Now, thedead weight of the cylinder is equal to the weight of the water displaced when thecylinder is at rest, so thatmg = ρgad.It follows that, neglecting damping, the net vertical force on the cylinder isρga(x + z).Damping forces are of two types. First, the linear radiation damping force, which isgiven by−b(vertical velocity) = bż,where b is a constant. The determination of its value is discussed in the next section.Secondly, there is a nonlinear viscous drag force, which is caused by boundary layerseparation and is proportional to the square of the velocity. This force is−cż|ż|,where c is the constant of proportionality. The determination of its value is alsodiscussed in the next section. To summarise: taking damping into account, the netvertical force on the cylinder isρga(x + z) + bż + cż|ż|.Now, as the cylinder moves through the water, it drags a volume of water behind itof mass µ, say, so that the total inertial mass of the cylinder is m + µ. The quantityµ is called the added mass and its determination is discussed in the next section. Itfollows that the equation of motion of the cylinder is(m + µ)(−z̈) = ρga(x + z) + bż + cż|ż|,or, rearranging,(m + µ)z̈ + bż + cż|ż|+ ρgaz = −ρgax. (1)Assuming that the water wave is monochromatic, x can be written in the formx = αsin(ωt),306 L. J. Crane and J. J. H.Millerwhere t is the time, ω the angular frequency and α the amplitude. The use of sinrather than cos here permits the convenient choice in Section 4 of homogeneousinitial conditions for the numerical solutions. In deep water, where the depth ismuch larger than the wavelength λ,ω =√2πgλ.See [1] for an introduction to the classical theory of monochromatic waves.The equation of motion of the annulus in isolation is similar to (1). In isolation,therefore, the two bodies of the WEC satisfy the following uncoupled system of twosecond order nonlinear ordinary differential equations(m1 + µ1)z̈1 + b1ż1 + c1ż1|ż1|+ ρga1z1 = −ρga1x, (2)(m2 + µ2)z̈2 + b2ż2 + c2ż2|ż2|+ ρga2z2 = −ρga2x, (3)where the subscript 1 refers to the cylinder and the subscript 2 to the annulus. Theundamped natural angular frequencies of the two separate bodies are thenf1 = 2π√m1 + µ1ρga1, f2 = 2π√m2 + µ2ρga2. (4)However, to generate power, both cylinder and annulus must interact. For ex-ample, electric power may be generated by constructing the cylinder out of a solidmagnet and the annulus containing a cylindrical coil of electrical conductor. In thepresent case, the instantaneous force between the two parts of the device is propor-tional to their relative velocityż1 − ż2,the constant of proportionality being denoted by pto, an abbreviation for ”powertake–off”. It follows that the WEC satisfies the following coupled system of twosecond order nonlinear ordinary differential equations(m1 + µ1)z̈1 + b1ż1 + c1ż1|ż1|+ pto(ż1 − ż2) + ρga1z1 = −ρga1x, (5)(m2 + µ2)z̈2 + b2ż2 + c2ż2|ż2| − pto(ż1 − ż2) + ρga2z2 = −ρga2x. (6)Note that the instantaneous power output is equal to the product of instanta-neous force and velocity.pto(ż1 − ż2)2.3. Added mass, radiation damping and dragIn this section expressions for the added mass and the radiation damping coefficient,up to an unknown multiplicative constant, are derived using standard techniques ofdimensional analysis, see, for example, [5].Wave energy converter 307Consider first the added mass µ; this is a function of ρ, g, a, λ and may be ex-pressed in the formf1(µ, ρ, g, a, λ) = 0. (7)In the Table, the dimensions of these quantities are given in terms of the fundamentaldimensions mass M̃, length L̃ and time T̃ .Quantity µ ρ g a λDimensions M̃ M̃L̃3L̃T̃ 2L̃2 L̃Thus, there emerge two dimensionless variables, namely µρa3/2and a1/2λ . Conse-quently, the functional relationship (7) reduces tof2(µρa3/2,a1/2λ) = 0.Solving for µρa3/2then givesµρa3/2= f3(a1/2λ).In the limiting case of small acceleration, which corresponds to low frequency andtherefore large wavelength, the added mass remains finite; see, for example, [6].Therefore, if a1/2λ << 1,µρa3/2≈ K,where K is a dimensionless constant, which depends on the shape of the WEC. Fora cylinder, K can be uniquely determined by either analytic methods or experiment.In the case of an annulus, the value of K depends on the ratio of the inner and outerradii, and can be similarly determined.On the other hand, the radiation damping coefficient b has dimensions of mass/time,and there are again five dimensional quantities in the functional relationship, namelyb, ρ, g, a, λ.Again, out of these five dimensional quantities emerge two dimensionless quantities,which are related bybρa3/2√λg= F (a1/2λ).The above formulae are in agreement with the work of [2] for the case of a spherehalf submerged in water. Havelock shows that, when a1/2λ is small, then the latterrelationship is linear, that isbρa3/2√λg= βa1/2λ,308 L. J. Crane and J. J. H.Millerwhere β is a constant which depends on whether the shape is a solid cylinder or anannulus. The value of β can be determined by analogous methods to those used todetermine K.Next, the coefficient c is considered. It can be shown (see [6]) that, in developedturbulent flows for which the Reynolds number is greater than 106, c = 12ρaCD,where CD is the non-dimensional drag coefficient. CD depends on the body shapeand for a sphere it is of order 10−1.4. Numerical solutionObserve that the nonlinear system of two second order differential equations (5), (6)can be written in the formMZ̈ + BŻ + PTO.Ż + C.diag(|Ż|).Ż + DZ = F, (8)where Z and F are the column 2-vectorsZ =(z1z2), F =(−ρga1x−ρga2x)and M,B,C, D, PTO, diag|Ż| are the 2× 2 matricesM =(m1 + µ1, 00, m2 + µ2), B =(b1, 00, b2), C =(c1, 00, c2),D =(ρga1, 0 00, ρga2), PTO = pto(1, −1−1, 1), diag|Ż| =( |ż1|, 00, |ż2|).Introducing the derivatives Ż of the displacements Z as new dependent variables, thesecond order system (8) can be rewritten as a first order system of four first orderequations in a standard way. Specifying homogeneous initial values the resultinginitial-value problem is then solved numerically with the ODE45 code in Matlab.In the following section some results of extensive numerical experiments are de-scribed. Values of the parameters are chosen, which reflect typical values arisingfrom typical WEC.5. Discussion of the resultsIn the graphs of this section all quantities are made dimensionless by using thephysical parameters of the WEC device. The four components of the numericalsolution, as functions of the dimensionless time T , are shown in Fig.1. Y (1), Y (2) arethe dimensionless displacements and Y (3), Y (4) are the corresponding dimensionlessvelocities of the cylinder and annulus respectively. Similarly the dimensionless strokeand dimensionless power versus T are seen in Fig.2. In Fig.3, it is seen that the WECresonates with the waves at a particular frequency, which is close, but not equal,to the frequency of the waves. It is observed that the dimensionless mean powergenerated by the WEC falls off rapidly on both sides of the resonant frequency.Wave energy converter 309In Fig.4, the dimensionless mean power is plotted against the frequency and thepto. It is easy to see that mean power at the resonant frequency increases withincreasing pto. In Fig.5, the dimensionless mean power is plotted against the fre-quency and the dimensionless mass of the cylinder. It can be seen that the meanpower does not change significantly with varying mass. Fig.6 shows the dependenceof the dimensionless mean power on the frequency and the dimensionless cross sec-tional area of the annulus. This demonstrates that there are significant variationsin the power output with changing cross section. In Fig.7, the dimensionless meanpower is plotted against the frequency and the dimensionless mass of the cylinder. Itis clear from this graph that the resonant frequency depends strongly on this mass.Finally, Fig.8 shows clearly that the mean power output is highly dependenton the position of the WEC in its parameter space. The power output changesrapidly due to small variations in the parameter values. Indeed it can be seen that acatastrophic fall-off in power occurs in certain regions of the parameter space. Thedesigner of a WEC must take such properties into account in order to ensure thatthe device is not constructed in such a way that it lies within or near such a regionof parameter space.The simple mathematical model discussed here provides qualitative guidance tothe design engineer, because it identifies finite regions of the parameter space inwhich the design parameters should lie. This is important at the early stages of thedesign process, since it gives rapid, inexpensive guidance, which protects the engineerfrom gross design errors. Later, if more accurate values of the design parametersare required, more complex, mathematical models may be employed, which involveboundary element techniques or full three-dimensional CFD methods. These are,of course, orders of magnitude slower and more expensive than the simple modelsemployed in the present paper.0 100 200 300−1−0.500.51TZ(1)0 100 200 300−1−0.500.51TZ(2)0 100 200 300−1−0.500.51TZ(3)0 100 200 300−1−0.500.51TZ(4)Figure 1: Numerical solution of the initial value problem310 L. J. Crane and J. J. H.Miller0 50 100 150 200 250 300−1−0.500.51TStroke0 50 100 150 200 250 300−1−0.500.51TPowerFigure 2: Stroke and power versus time0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20246810121416x 105FrequencyMean powerFigure 3: Mean power versus frequency0.511.522.533.5x 1060.050.10.15051015x 105PTOFrequencyMean powerFigure 4: Mean power versus frequency and ptoWave energy converter 31122.22.42.62.83x 1050.050.10.15051015x 105Annulus massFrequencyMean powerFigure 5: Mean power versus frequency and annulus mass204060801001201400.050.10.15051015x 105Cylinder cross sectionFrequencyMean powerFigure 6: Mean power versus frequency and cylinder cross section246810x 1060.050.10.15051015x 105FrequencyCylinder massMean powerFigure 7: Mean power versus frequency and cylinder mass312 L. J. Crane and J. J. H.Miller246810x 10620406080100120140051015x 105Cylinder massCylinder cross sectionMean powerFigure 8: Mean power versus annulus cross section and cylinder massAcknowledgementThe authors are grateful to the unknown referees whose comments led to significantimprovements in the exposition.References[1] C.A.Coulson, Waves, Oliver and Boyd, Edinburgh, 1955.[2] J. Falnes, Ocean Waves and Oscillating Systems, Cambridge University Press, Cam-bridge, 2002.[3] T.H.Havelock, Waves due to a floating sphere making periodic heaving oscillations,Procs. Roy. Soc. Lond. 231(1955), 1-7.[4] J.N.Newman, Marine Hydrodynamics, MIT Press, Cambridge, 1977.[5] J.H. Spurk, Dimensionsanalyse in der Strömungslehre, Springer, Berlin, 1992.[6] J.H. Spurk, N.Aksel, Strömungslehre, Springer, Berlin, 2006.

A simple nonlinear model of a generic axisymmetric wave energy converter

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