A second-order theory for wave energy converters with curved geometr

Emiliano Renzi (1251495)

Paolo Sammarco (7159373)

Simone Michele (7159403)

English

Loughborough University Institutional Repository

A second-order theory for wave energy converters with curved geometrySimone Michele1, Emiliano Renzi1, and Paolo Sammarco21Department of Mathematical Sciences, Loughborough University, Leics LE11 3TU, United Kingdom2Department of Civil Engineering and Computer Science, University of Rome ’Tor Vergata’, Via del Politecnico 1,00133 Rome, Italyemail: michele@ing.uniroma2.itIntroductionWe present a second-order nonlinear theory for a gate-type Wave Energy Converter (WEC) in a semi-infinite bi-dimensional channel. The gate model is similar to that shown in [1], except for a generalized weak displacementof the gate wetted surface about the vertical. Both free surface elevation and gate vertical displacementare assumed small if compared to the channel depth. Hence, the boundary conditions on the gate can beconveniently Taylor-expanded about the vertical. Perturbation-harmonic expansion allows us to decomposethe nonlinear governing equations in a sequence of linear boundary-value problems of order n and harmonicm. Then, we show that the effect of the gate shape forces the first-harmonic component at the second orderonly. We also show that hydrodynamic interactions between the curved gate and short waves can have eitherconstructive or disruptive effects in terms of power absorption.Mathematical modelWith reference to Figure 1(a), consider a semi-infinite channel of constant depth h′ and width w′. Define aCartesian reference system (x′, z′) with the x′ axis lying on the undisturbed free surface level and the z′ axispointing upward. Primes indicate physical variables. At x′ = 0 rests a gate-type WEC of mass M ′, allowedto oscillate horizontally along the channel without friction, under the action of incident harmonic waves. TheWEC is connected with a wall by a spring-damper system in parallel. The spring has elastic constant C ′, whilethe linear damper, which simulates a power take-off (PTO) system, has PTO coefficient ν ′pto. Hence the PTOexerts a force proportional to the gate velocity. The fluid is assumed inviscid and incompressible and the flowirrotational. As a consequence, the velocity field satisfies the Laplace equation in the fluid domain Ω (x′, z′) forthe velocity potential Φ′ (x′, z′, t′). Let us assume the following equation for the wetted gate surfacef ′(x′, z′, t′)= x′ − δ′ (z′)−X ′ (t′) = 0, (1)where δ′ denotes the deviation of the gate surface about x′ = 0, while X ′(t′) indicates the displacementdepending on time t′, positive rightward. Suppose that the mean of δ′ has zero value. Let A′ be the amplitudeof the incident waves, ω′ the incident wave frequency and g′ the acceleration due to gravity. Then introducethe following non-dimensional quantities (see also [2]):(x, z) =(x′, z′)h′, Φ =Φ′A′ω′h′, ζ =ζ ′A′, t = t′ω′, δ =δ′δ′g, X =X ′A′, G =g′ω′2h′, (2)where ζ ′ is the free surface elevation, δ′g the length scale for δ′ and G the non-dimensional frequency. Moreoverwe introduce the following two length ratios:  = A′/h′  1, µ = δ′g/h′  1. Let us assume the followingperturbation expansion for the non-dimensional unknowns up to second order O ():{Φ, ζ,X} =2∑n=1n−1 {Φn, ζn, Xn} . (3)Using the latter expansion, together with the quantities defined by (2), yields a non-dimensional form of theLaplace governing equation and a no-flux boundary condition at the bottom:∇2Φn = 0, (x, z) ∈ Ω; Φnz = 0, z = −h, (4)where the x, z, t subscripts denote differentiation with respect to the relevant variable. Taylor expansion of theboundary conditions about z = 0 gives the free-surface dynamic condition:−Gζn = Bn; B1 = Φ1t , B2 = Φ2t + ζ1Φ1tz , z = 0, (5)and the free surface mixed condition:Φntt +GΦnz = Fn; F1 = 0, F2 = −[ζ1 (Φ1ttz +GΦ1zz) + |∇Φ1|2t], z = 0. (6)Similarly, the kinematic boundary condition on the gate surface can be Taylor-expanded about x = 0:Φnx = Xnt + Gn; G1 = 0, G2 = −Φ1xx[µδ +X1]+ Φ1zµδz, x = 0. (7)Finally, the equation of motion of the gate can be written as:MXntt +GCXn + νptoXnt =∫ 0−1dzΦnt +Dn, (8)D1 = 0, D2 =∫ 0−1dz{Φ1tx[µδ +X1]+12|∇Φ1|2}+G2ζ21 + Φ1tζ1. (9)in which M = M ′/(ρ′h′2w′) is the non-dimensional mass, C = C ′/(ρ′g′h′w′) the non-dimensional elasticconstant, νpto = ν′pto/(ρ′ω′h′2w′) the non-dimensional PTO-coefficient and ρ′ the water density. Because ofharmonic motion, higher-order solutions imply higher harmonics. Hence, return in physical variables, omit theprimes for convenience and assume the following harmonic expansion [2]:{Φn, ζn, Xn} =n∑m=0{φnm, ηnm, Xnm} e−imωt + ∗, (10)where the symbol ∗ indicates the complex conjugate. Substitution of the latter expansion into the governingequation and boundary conditions gives a sequence of linear boundary problems of order n and harmonic m.Leading order problem O (1)Performing harmonic expansion (10) of the governing equations, we obtain that the zeroth harmonic problemat the leading order is unforced with homogeneous Neumann boundary conditions, hence φ10 = 0. The incidentwave field is assumed to be at the leading order O (1). As a result, this is a well-known diffraction problemforced by the incident wave field. The resulting velocity potential reads:φ11 = − igAωcosh k0 (h+ z)cosh k0hcos k0x−∞∑n=0ωX11DnknCncosh kn (h+ z) eiknx, (11)where the first term on the right represents the diffraction (incident + reflected) potential, while the secondterm the radiation potential due to body motion. In (11), the kn’s denote the roots of the dispersion relationω2 = gk0 tanh k0h, ω2 = −gkn tan knh, kn = ikn, n = 1, 2, . . . (12)while the coefficients Cn, Dn correspond to:Cn =∫ 0−hdz cosh2 kn (h+ z) =12(h+gω2sinh2 knh), Dn =∫ 0−hdz cosh kn (h+ z) =sinh knhkn. (13)The gate response is finally given byX11 = − AρgwD0/ cosh k0h−ω2M + C − iωνpto − iω2ρw∑∞n=0D2nknCn. (14)Second order problem O ()At this order we obtain a zeroth-harmonic drift, plus two harmonics ω and 2ω forced by quadratic products ofthe leading order solution O (1). The bound wave and the static displacement have expressions:η20 =1g(2ω2 |η11|2 − |∇φ11|2), X20 = −ρwC(ω2h |X11|2 + g |η11|2). (15)The related velocity potential φ20 differs from zero and is forced both on the free surface and on the gate.However, it does not affect energy extraction and will be evaluated in future work. Effects of the gate shapeon the total wave field influence the first harmonic solution only. In particular, we have:φ21 =∞∑n=0(−X21DnωCnkn+Ψn)eiknx cosh kn (z + h) , Ψn =iknCn∫ 0−hdz cosh kn (z + h) {φ11xxδ − φ11zδz} .(16)The corresponding displacement is given by:X21 = −iωρw[∑∞n=0 ΨnDn +∫ 0−h dz φ11xδ](−Mω2 + C − iωνpto − iω2ρw∑∞n=0D2nCnkn) . (17)The numerator of (17) is the complex exciting torque, and represents the effect of the vertical profile of the gate.Note that, for a flat gate with δ = δz = 0, the displacement X21 becomes null. The denominator of expression(17) is made by terms that do not depend on the gate shape. Hence the displacement X21 is maximum if theexciting torque function is maximized accordingly.The second harmonic boundary-value problem presents the following inhomogeneous forcing terms, respec-tively, on the free surface and on the gate surface:φ22z =4ω2φ22g− iωg[φ11g(−ω2φ11z + gφ11zz)− 2 |∇φ11|2] , φ22x = −2iωX22 + 1 φ11xxX11. (18)Linearity allows decomposition of the velocity potential, i.e. φ22 = φS22 + φF22, in which φS22 represents thesolution with homogenized condition on the gate, while φF22 is the velocity potential solution with homogenizedcondition on the free surface. Solution can be found by the eigenfunction expansion method (see also [2]).After some lengthy but straightforward algebra we obtain:φS22 =∞∑l=0coshκl (z + h)coshκlhgκlCl{Γ1(− 12κl− κl cos 2k0x−8k20 + 2κ2l)+ Γ2eiκlx (km + kn)− ei(km+kn)xκlκ2l − (km + kn)2+Γ3eiknxκl(k20 − κ2l + k2n)cos k0x+ kn[eiκlx(k20 + κ2l − k2n)− 2ieiknxk0κl sin k0x][k20 − (κl + kn)2] [k20 − (κl − kn)2] + Γ4(− 12κl+κl cos 2k0x−8k20 + 2κ2l)+Γ5eiknxκl(k20 − κ2l + k2n)sin k0x+ k0[−ieiκlx (−k20 + κ2l + k2n)+ 2ieiknxknκl cos k0x][k20 − (κl + kn)2] [k20 − (κl − kn)2] , (19)where κl are the real roots of the dispersion relation4ω2 = gκ0 tanhκ0h, 4ω2 = −gκl tanκnh, κl = iκl, l = 1, . . . ,∞, (20)while the Γ-terms have expressionsΓ1 =(Agω)2(iωk0 − 3iω5g2), Γ2 = ω2X211∞∑n=0∞∑m=0DnDm cosh knh cosh kmhCnCmknkm(3iω5g2− 2iωknkm − iωk2n),Γ3 = −AgX11ω∞∑n=0Dn cosh knhCnkn(6ω4g2− k20 − k2n),Γ4 = −2iA2g2k20ω,Γ5 = 4iAgX11ωk0∞∑n=0Dn cosh knhCn.(21)The velocity potential due to forcing on the gate surface readsφF22 = −∞∑l=0(2ωX22DlκlCl+∆l)eiκlx coshκl (h+ z) , ∆n = − iκlCl∫ 0−hdz coshκl (z + h)φ11xxX1, (22)in which Cl and Dl have the form (13) with wavenumber κl. Finally the gate response X22 isX22 =ρw{−2iω[∫ 0−h dz φS22∣∣x=0−∑∞l=0 ∆lDl]+ ∫ 0−h dz [−iωφ11xX11 + ∇φ11·∇φ∗112 ]− gη2112 }(−4Mω2 + C − 2iωνpto − 4iω2ρw∑∞l=0D2lClκl) . (23)(a)0.5 0.75 1 1.25 1.5 1.75 20.60.811.21.41.6(b)Figure 1: (a) Geometry of the system in physical variables; (b) Ratio between the capture factor CF of eachconfiguration and the capture factor of a vertical flat-gate CF0Results and discussionsSince the gate response is X = (X11 + X21) e−iωt + X22e−2iωt + ∗, the average extracted power becomesP = limτ→∞1τ∫ τ0dt(dXdt)2νpto = 2ω2νpto(|X11 + X21|2 + 4 |X22|2). (24)The efficiency of the system is finally assessed by considering the capture width ratio CF defined as the ratiobetween the extracted power P and the incident wave energy flux per channel width w [1]. Now, let us comparethe flat-gate (δ = 0) with four different gate configurations, respectivelyδ1 = Ag sinpi (z + h)h, δ2 = −Ag sin pi (z + h)h, δ3 = Ag2z + hh, δ4 = −Ag 2z + hh, (25)where the water depth is h = 10 m and the maximum gate displacement is Ag = h/10. Let us assume theeigenfrequency of the system equal to ω = 1 rad s−1 and the amplitude of the incident waves A = h/10.The power generated at O (1) is maximized by assuming the PTO coefficient equal to the radiation dampingin resonance conditions, i.e νpto = 7.9 · 104 kg m2 s−1. Figure 1(b) shows the ratio between CF of eachgate configuration and the capture factor of the flat-gate CF0. Note that the curves cross each other incorrespondence of the resonance frequency ω = 1 rad s−1. At low frequencies the effect of a vertical displacementis small. Constructive or destructive effects due to the vertical displacement δ arise mainly at large frequencies.Configurations 2 and 3 are more efficient than the flat gate, especially at large frequencies ω > 1.5 rad s−1.The least beneficial effects occur in Configuration 1. It should be noted that the optimization process dependson a large number of parameters even though the analytical model is 2-D. Work is currently in progress to usegenetic algorithms able to search a family of vertical shapes which maximizes power output. Finally, we remarkthat this analytical model constitutes the basis for investigating nonlinear resonance phenomena occurring inmore complex systems ([2, 3, 4]).The work of S.M. is supported by a Royal Society - CNR International Fellowship.References[1] C. C. Mei, M. Stiassnie, and D. K.-P. Yue. Theory and application of ocean surface waves. World Scientific,Singapore, 2005.[2] S. Michele, P. Sammarco, and M. d’Errico. Weakly nonlinear theory for oscillating wave surge convertersin a channel. J. Fluid Mech., 834:55–91, 2017.[3] S. Michele, P. Sammarco, and M. d’Errico. Theory of the synchronous motion of an array of floating flapgates oscillating wave susrge converter. Proc. R. Soc. Lond.A, 472:20160174, 2016.[4] E. Renzi and F. Dias. Resonant behaviour of an oscillating wave energy converter in a channel. J. FluidMech., 701:482–510, 2012.

A second-order theory for wave energy converters with curved geometry

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A second-order theory for wave energy converters with curved geometry

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