Flap gate farm: From Venice lagoon defense to resonating wave energy production. Part 2: Synchronous response to incident waves in open sea

We consider a ﬂap gate farm, i.e. a series of P arrays, each made of Q neighbouring ﬂap gates, in an open sea of constant depth, forced by monochromatic incident waves. The effect of the gate thickness on the dynamics of the system is taken into account. By means of Green’s theorem a system of hypersingular integral equations for the velocity potential in the ﬂuid domain is solved in terms of Legendre polynomials. We show that synchronous excitation of the natural frequencies of Sammarco et al. (2013) yields large amplitude response of gate motion. This aspect is fundamental for the optimisation of the gate farm for energy production. equation approach to solve the radiation and scattering problem for a submerged horizontal circular plate. Here we ﬁnd the solution in terms of Legendre polynomials. The Haskind–Hanaoka relation is utilised to check the accuracy and the computational cost of the semi-analytical method. We show that in the open sea there are P × ( Q − 1) out-of-phase natural modes similar


Introduction
The flap gate systems, i.e. one or more floating bodies hinged at the bottom of the sea and rolling under incoming waves, have recently proved very effective to extract energy from the sea (Whittaker et al. [1]). The mechanical behaviour of a rolling flap gate was initially investigated during the design phase of the storm barriers for protecting Venice Lagoon from flooding. For one array of gates spanning the entire width of a channel, experiments showed that the gates can be excited to oscillate at half the incident wave frequency with a very large amplitude (Mei et al. [2]). In that case, resonance occurs through a nonlinear mechanism when the frequency of the incoming wave is twice the eigenfrequency of the system (Sammarco et al. [3,4]). Li and Mei [5] found the (Q − 1) eigenfrequencies of one array made by Q identical gates spanning the full width of a channel. Later, Sammarco et al. [6] in Part 1 of this paper considered a P × Q gate farm, and showed that there exist P × (Q − 1) eigenfrequencies and associated modal forms. If the gates are not completely confined in a channel, radiation damping is always present, i.e. wave trapping is imperfect and therefore linear resonance of the eigenmodes is possible (Adamo and Mei [7]).
In this paper a linear theory is developed in order to analyse the resonant behaviour of the P × Q gate farm in an open sea of constant depth. Unlike in previous models available in the literature (Renzi et al. [8], Renzi and Dias [9][10][11][12][13], Renzi et al. [14,15], Sarkar et al. [16]), all based on the "thin-gate hypothesis" (Linton and McIver [17]), in this work the gate thickness is assumed finite, i.e. comparable with the other gate dimensions. By means of Green's theorem a system of hypersingular integral equations for the radiation and scattering potential on the boundaries of the gate farm is obtained. Achenbach and Li [18] and Martin and Rizzo [19] adopted a similar procedure to solve crack and acoustic problems, while Parsons and Martin [20][21][22] used this method to solve scattering and trapping of water waves by rigid plates. Subsequently, Martin and Farina [23] and Farina and Martin [24] used the hypersingular integral equation approach to solve the radiation and scattering problem for a submerged horizontal circular plate.
Here we find the solution in terms of Legendre polynomials. The Haskind-Hanaoka relation is utilised to check the accuracy and the computational cost of the semi-analytical method. We show that in the open sea there are P × (Q − 1) out-of-phase natural modes similar in shape to the case of the gate farm in a channel. The irregular frequencies (Linton and McIver [17]-Mei et al. [25]) are then evaluated. We also investigate the response of the gate farm to plane incident waves of varying frequency. The gate farm is designed to work in the nearshore, hence normal incidence of the waves is assumed. Large amplitude motions of the gates occur when the incident wave frequency approaches the eigenfrequencies. Hence a linear resonant mechanism of the natural modes in the open sea is effective. Finally, the P × Q gate farm and a system of P × Q isolated and independent gates are compared in terms of energy production.

Governing equations for the P × Q gate farm
As shown in Fig. 1, consider P arrays of neighbouring flap gates. Each array, p = 1, 2, . . ., P, is composed by Q identical floating gates (q = 1, 2, . . ., Q). Let a and 2b be, respectively, the width and the thickness of each gate and let w = Qa. Consider a three dimensional Cartesian coordinate system with the x and y axes lying on the mean free surface and the z axis pointing vertically upward. The y-axis bisects the first array (p = 1), while the x-axis is orthogonal to the arrays and is centred among them. All the gates of the pth array are hinged on a common axis lying on x = (p − 1)L, z = − h, where L is the distance between the arrays and h the sea constant depth. The symbol G pq denotes the qth gate of the pth array, while pq indicates the angular displacement of G pq , positive if clockwise. Monochromatic plane normal incidence waves of amplitude A, period T and angular frequency ω = 2 /T, coming from x =+ ∞, force the gates to oscillate back and forth.
Let p (y, t) indicate the angular displacement function of the pth array: p (y, t) is a piece-wise function of y, still unknown. The analysis is performed in the framework of irrotational flow and in the limit of small-amplitude oscillations. Therefore, the velocity potential ˚(x, y, z, t) must satisfy the Laplace equation in the fluid domain ˝: On the free surface, the kinematic-dynamic boundary condition reads: while the no-flux condition on the seabed requires: On the p = 1, . . ., P arrays the kinematic boundary conditions are: Note that the no flux condition (6) is given on the finite edges of each array facing the open sea, without channel walls. The time dependence of ˚ and p can be separated by assuming a harmonic motion of given frequency ω: p (y, t) = Re{Â p (y)e −iωt }.

Semi-analytical solution
The linearity of the problem allows the following decomposition of the potential (x, y, z): where is the potential of the plane incident waves incoming from x =+ ∞, S is the potential of the scattered waves and R pq is the potential of the radiated waves due to the moving gate G pq while all the other gates are at rest. In (10), k denotes the wave number, root of the dispersion relation ω 2 = gkth kh, while i is the imaginary unit. ch, sh and th indicate shorthand notation respectively for cosh, sinh and tanh. According to the separation (7) and (8) and the decomposition (9), both R pq and S must satisfy the Laplace equation (2), the kinematic-dynamic boundary condition on the free surface (3), and the no-flux condition on the seabed (4). Let x ± p indicate the x-coordinate of the rest position of the vertical surface of the pth array: Each gate G pq spans a y-width given by: The kinematic boundary conditions on the gate-farm surfaces then become: p = 1, . . ., P, q = 1, . . ., Q.
Finally R pq and S must be outgoing when x 2 + y 2 → ∞. Separation of variables gives: where Z n (z) represents the normalized eigenfunctions: which satisfy the orthogonality property 0 −h Z n (z)Z m (z) dz = ı nm , n, m = 0, 1, . . .
with ı nm the Kronecker delta. In (15), k n are the roots of the dispersion relation: Following (14), for each of the ϕ R n,pq , ϕ S n , the Laplace equation becomes the Helmholtz equation Now define the boundary S pq of the gate G pq as and the end boundaries of the pth array of width 2b We can so refer to the entire gate farm boundary S G as: The boundary conditions (13a)-(13e) become Note that in (24) only d 0 is non-zero. We also require ϕ R n,pq and ϕ S n to be outgoing as x 2 + y 2 → ∞. The solution of the boundary value problem defined by the Helmholtz equation (18) and by the boundary conditions (22a)-(22e) can be found by using Green's theorem and Green's functions.
Consider the plane fluid domain ˙ enclosed within the boundary of the gate farm S G and a circle of large radius S ∞ surrounding the gate farm. Define the Green function G n (x, y ; , Á) as the solution of the Helmholtz equation: LG n (x, y; , with G n must be outgoing as r→ ∞, hence the solution of (25) and (26) is: In the latter, H 0 is the Hankel function of the first kind and order zero. Application of Green's theorem yields ˙ ϕ R n,pq (x, y)ϕ S n (x, y) LG n (x, y; , Á) − G n (x, y; , Á)L ϕ R n,pq (x, y)ϕ S n (x, y) d= = S G +S∞+S ϕ R n,pq (x, y)ϕ S n (x, y) ∂G n (x, y; , Á) ∂n − G n (x, y; , Á) ∂ ∂n ϕ R n,pq (x, y)ϕ S n (x, y) dS (28) where ˙ = ˙ \ ( , Á), S is a semicircle of radius → 0 centred at ( , Á) and finally ∂(·)/∂n is the derivative of (·) in the direction of the outward normal to the boundaries of ˙.
Because of the governing Eqs. (18)- (25) and the behaviour of G n for r → 0 (26) and r→ ∞, Eq. (28) simplifies to (see also Linton and McIver [17]-Mei et al. [25]) where the line integral is now evaluated in terms of ( , Á) on the boundary S G . The radiation potential ϕ R n,pq and the scattering potential ϕ S n are expressed in integral form. Define ± p and Á q as follows: Since: substitution of the boundary conditions (22a)-(22e) inside Eq. (29), yields: Note that (32) and (33) are more complex than their thin-gate counterparts of Renzi et al. [14]. Since the radiation potential ϕ R n,pq and the scattering potential ϕ S n on the boundary of the gate-farm are unknown, the first four integrals inside the expressions (32) and (33) are still unknown. The integrals inside the summations are evaluated on the boundary of each array, except for the last integral of (32) which is evaluated on the boundary of the moving gate G pq . Imposing the boundary conditions (22a)-(22e) to (32) and (33) yields a system of hypersingular integral equations for ϕ R n,pq and ϕ S n evaluated on the boundaries of the gate farm. The solution of the system is found by expanding ϕ R n,pq and ϕ S n in terms of Legendre polynomials P m of integer order m = 0, . . ., M (see Appendix for details). Finally the radiation potential R pq due to the motion of the gate G pq , on the lateral surfaces of each arrayp = 1, . . ., P, is expressed as follow: while the scattering potential on the same surfaces is given by: where x p and y are dimensionless variables defined in [−1, 1]: while ˛R ± nmp,pq , ˛S ± 0mp , ˇR ± nmp,pq and ˇS ± 0mp are complex constants determined by solving the linear systems (A.38a)-(A.38c) and (A.39a) and (A.39b) with a numerical collocation scheme (see Appendix for further details).

Gate dynamics
Consider each gate G pq coupled with an energy generator at the hinge. Assume that the generator exerts a torque proportional to the angular velocity of the gate G pq , pto˙ pq , where pto is the power take-off coefficient. Conservation of angular momentum requires: where I is the moment of inertia of the gate about the hinge and C is the net restoring torque: with: where S A denotes the cross sectional area of the gate at the water line and V the water volume displaced by the gate in its rest vertical position. M g and z g are respectively the mass and the vertical coordinate of the center of mass of the gate. For the geometry of Fig. 1, I A xx and I V z are: Using (7)- (9) and the expressions of the potentials (10), (34) and (35), the momentum equation (37) gives where is the exciting torque due to the incident and scattered waves, while: represent, respectively, the added inertia and the radiation damping of the gate G pq due to the unit rotation of the gate G pq . Eq. (41) can be written in matrix form: where Â is a column vector of length s = P × Q that contains all the angular displacements of the gates: . .
I is the identity matrix of size s × s, M and N are respectively the added inertia matrix and the radiation damping matrix also of size s × s: where both M m m and N p p are symmetrical square matrices of size Q × Q: Finally, once the angular displacements of the gates are known, the average power absorbed over a wave cycle by the gate farm, is equal to:

Eigenfrequencies and eigenvectors
The momentum equations given by (45) are equivalent to a system of P × Q linear damped harmonic oscillators with given mass, stiffness and damping. In order to find the eigenfrequencies of the system, the exciting torque and the damping terms are set equal to zero. System (45) becomes homogeneous: To find non-trivial solutions the following implicit non linear eigenvalue condition must then be solved: Once the eigenfrequencies are known, the respective modal forms can be obtained by setting the displacement of the gate G 11 = 1 and then solving system (50).

The radiation potential in the far field
Consider the polar coordinates r and defined by (x, y) = r(cos , sin ).
Following a similar procedure as in Renzi and Dias [10], the radiation potential in the far field (i.e. for r→ ∞), for unit rotational velocity of the gate G pq , can be approximated as R pq (r, , z) where represents the angular variation of the radially spreading wave (Mei et al. [25]). The latter can be used to derive some useful formulas that relate the hydrodynamic parameters.

The Haskind-Hanaoka relation for the gate farm
Consider the 3D Haskind-Hanaoka relation (Mei et al. [25])  where F pq is the exciting torque given by expression (42) while A R pq (0) represents the wave amplitude in the direction opposite to the incident waves Expression (55) has been used to check the numerical computation via the relative error

One gate in the open sea: the effects of the gate thickness
In order to evaluate the effects of the finite gate thickness 2b, the simplest case of P = Q = 1, i.e. the case of one gate in the open sea is considered. Inertia, buoyancy and width of the gate, and water depth, are listed in Table 1. Different values of the thickness 2b have been chosen, i.e. 2b ∈ [0.1; 1.5] m. The limit value of 2b = 0.1 m corresponds to the case where the "thin-gate" hypothesis can be applied (b/a 1 -Renzi and Dias [9]). Fig. 2 shows the values of the added inertia , the radiation damping and the magnitude of the exciting torque |F| versus the frequency of the incident waves for different values of b. The effects of the gate thickness on the added inertia and radiation damping are significant for ω ∈ [1, 3.5] rad s −1 . In particular, the larger the gate thickness the larger the added mass and radiation damping. As a consequence the eigenfrequency of the system decreases if the gate thickness increases. The eigenfrequency ω 1 of the single gate for five different values of 2b is listed in Table 2.

The gate farm in the open sea
With reference to Fig. 1, we consider P = 3 arrays each with Q = 5 gates. The input parameters are defined in Table 1.

Eigenfrequencies and eigenvectors
The eigenvalue condition (51) has been solved in order to find the eigenfrequencies of the system within a range of ω from 0 to 1.2 rad s −1 . The frequency range includes the P × (Q − 1) = 12 eigenfrequencies of the out-of-phase motion and the first two eigenfrequencies of the in-phase motion, where the pth array moves at unison. The numerical values of the eigenfrequencies are listed in Table 3 for the out-of-phase motion and in Table 4 for the in-phase motion. Solution of the momentum equations (50) gives the corresponding modal forms. Note that the generic out-of-phase natural mode N ij follows the same definition of Sammarco et al. [6], that is: for modes N 11 , N 21 , N 31 , and N 41 , each array has the same modal shape, but for the central array (p = 2); modes N 12 , N 22 , N 32 , and N 42 , are characterized by having the middle array (p = 2) with null angular displacement, while the last array (p = 3) is in opposite phase with respect to the first (p = 1); for the remaining modes N 13 , N 23 , N 33 , and N 43 , modal deformation is the same, but for the middle array (p = 2), which is in opposition of phase with the other two. N(ω 1 ) represents the in-phase natural mode characterized by the middle array in opposite phase with respect to the first and the last array. Similarly N(ω 2 ) represents the in-phase natural mode characterized by the middle array (p = 2) with null angular displacements while the arrays p = 1 and p = 3 are in opposition of phase. Let K be   the number of the gates per modal wavelength of the first array, p = 1; the eigenfrequencies of the out-of-phase modes decrease as K increases.

Irregular frequencies
Because of the geometry of the gate farm, the integral equations (32) and (33) possess the so-called irregular frequencies when n = 0 (Linton and McIver [17]-Mei et al. [25]).
Define the boundaries of the pth array as and let ˙ p be the interior of S p . We can so define ϕ p as the interior potential that satisfy the Helmholtz equation in ˙ p with boundary conditions The eigensolutions of the homogeneous Dirichlet problem (59) and (60) are found by separation of variables: where A nm is an arbitrary constant and n, m = 0, 1, . . .. The corresponding eigenvalues are while the related eigenfrequencies ω nm can be found via the dispersion relation These eigenfrequencies are the so-called irregular frequencies (Linton and McIver [17]-Mei et al. [25]). The lowest value of ω nm corresponds to the case of n = 0 and m = 1 and it is equal to ∼2 rad s −1 , i.e. higher than the range of our interest. For this reason we do not need to exclude them from the analysis.

Forced response
Extensive computations have been carried out for the range of interest of the incident wave frequencies ω = 0.1 − 1.2 rad s −1 without the PTO. The amplitude of the incident wave is A = 1 m. Resonance occurs at eight frequencies whose values are near the natural frequencies of the homogeneous system previously calculated. Because of the direction of the incident wave, orthogonal to the axes of the arrays, only the symmetric natural modes with respect to the x-axis can be excited; i.e, P × (Q − 1)/2 =6 out-of-phase and 2 in-phase natural modes are resonated. Let ω ij be the eigenfrequency of the out-of-phase mode N ij . In Fig. 3 we show the amplitude of the angular displacements versus the incident wave frequency and indicate the eigenfrequencies of the resonating natural modes. Note that the high and unrealistic values of the peaks are related to the weakness of the radiation damping corresponding to the resonance frequencies. In this case the gate-farm is almost undamped and radiates low energy at infinity. Fig. 4 and Fig. 5 the shapes of the gate-farm forced at the resonance frequencies ω ij are shown. Note that the number near each gate G pq represents Re{Â pq } normalized with respect to Re{Â 11 }. The values of Re{Â 11 } at the resonance frequencies are listed in Table 5.

The influence of the power take-off on the capture width
A parametric analysis is performed to investigate the effect of the power take-off coefficient pto on the generated power P over a wave cycle (see (49)). Define the capture width ratio C F as the ratio of the generated power P per unit gate-farm width to the incident power per unit width of the crest (see Renzi et al. [15]): where C g is the group velocity: Waves of amplitude A = 1 m are normally incident on the flaps. Different values of the PTO coefficient have been chosen, i.e. pto ∈ 10 4 ; 10 8 kg m 2 s −1 . Fig. 6 shows the behaviour of the capture width ratio C F versus the incident wave frequency for three different values of the PTO coefficient. When pto = 10 6 kg m 2 s −1 and ω > 0.6 rad s −1 , the capture width ratio is equal to ∼0.5 for a wide range of frequencies. Consider the case of pto = 10 8 kg m 2 s −1 and the behaviour of the magnitude of the exciting torque |F p3 | on each gate G p3 shown in Fig. 7: the behaviour of C F is quite similar to |F p3 |. In other words, the dynamics is dominated by the exciting torque due to diffracted waves (see Renzi and Dias [10]). Differently, the behaviour of the capture width ratio for pto = 10 4 kg m 2 s −1 , resembles that of the amplitude of the angular displacements shown in Fig. 3, hence in this case the dynamics is dominated by the resonance effects.

Wave power generation and efficiency: (P × Q) gate farm versus (P × Q) isolated gates
In this section the (P × Q) gate farm and a system of (P × Q) isolated and independent gates are compared in terms of energy production. The single flap gate has the same characteristics for both systems (see Table 1 for the values).
Consider the PTO coefficient that maximize the power output for incident wave frequency ω = 0.9 rad s −1 , i.e. a typical value in the Mediterranean Sea. The optimal PTO coefficient for a system of isolated gates pto,IG can be designed such that ( where and represent respectively the added inertia and the radiation damping of a single isolated gate at ω = 0.9 rad s −1 (see Fig. 2 for the values). The optimal PTO coefficient for the gate farm pto,GF is found numerically by maximizing the function (49) for a fixed ω. For ω = 0.9 rad s −1 , pto,GF = 7 ×10 6 kg m 2 s −1 . The difference between pto,IG and pto,GF is related to the behaviour of the exciting torque. Inspection of the different relations between radiation damping and exciting torque (Renzi and Dias [11]-Mei et al. [25]) shows that when ω is far from resonance the larger the exciting torque the larger the optimal PTO coefficient. In the present case the value ω = 0.9 rad s −1 is very close to the peaks of the exciting torque for the gate farm (see Fig. 7), while is distant from the peak of the exciting torque for a single isolated gate (see Fig. 2). As a consequence, pto,GF is larger than pto,IG . Hereafter, both pto,GF and pto,IG are fixed. Now define the capture width ratio of the gate farm C GF and the capture width ratio of (P × Q) isolated gates C IG as where P GF and P IG represent respectively the averaged power generated by the gate farm and by the single isolated flap gate. Fig. 8 shows the capture width ratio curves of both systems. The gate farm captures significantly more energy than a system of isolated gates. Also the bandwidth of the gate farm curve is larger than the other. Note that C GF behaves as the exciting torque magnitude shown in Fig. 7, hence the performance is dominated by diffracted waves. In Renzi et al. [15] have been obtained similar results. Now consider the amplitude of the angular displacements Â 33 of the gate G 33 and the amplitude of the angular displacements Â IG of the isolated gate shown in Fig. 9. The maximum value for |Â 33 | is ∼0.2 rad, hence the influence of the PTO coefficient decreases significantly the unrealistic amplitudes of the gates without PTO damping (see Fig. 2 for the gate farm). This fact justifies the hypothesis of small-amplitude oscillations and the applicability of the linear theory.

Conclusions
A semi-analytical model has been developed in order to solve the dynamic behaviour of the P × Q gate farm when excited by planar incident waves. By means of the Green theorem, a system of hypersingular integral equations for the radiation and scattering potential on the wet surfaces of the gate farm is obtained. The system is solved in terms of Legendre polynomials of integer order. Then the expressions of the added inertia, the radiation damping and the exciting torque are derived. The theory takes into account the thickness of each gate without resorting to the "thin-gate" hypothesis.
A parametric analysis of one gate in the open sea reveals the effect of the gate thickness on the eigenfrequency and on the gate response to incident waves. We have shown that the larger the thickness the larger the added inertia and the lower the eigenfrequency. Moreover, the radiation damping increases as the thickness increases, while the exciting torque shows negligible variations.
The solution of the eigenvalue condition for the P × Q gate farm, gives P × (Q − 1) out-of-phase natural modes similar in shape to those of the P × Q gate farm in a channel of Sammarco et al. [6]. The system response is then evaluated for a wide range of incident wave frequencies.
Numerical results show that the resonant peaks are close to the natural frequencies of the system. In particular, the narrow resonant peaks indicate that the radiation damping is small, hence synchronous excitation of the natural modes is significant. An asymptotic expression of the radiation potential is obtained in order to apply the Haskind-Hanaoka relation to the gate farm. The (P × Q) gate farm and a system of (P × Q) isolated gates are compared in terms of energy production. The results show that the gate farm capture more energy than a system of isolated gates.
The amplitude response at the resonance frequencies is large and non-realistic, hence the hypothesis of small-amplitude oscillation at the basis of this linear theory, is not satisfied. However, the amplitude response is significantly reduced when the gates are coupled with a PTO device at the hinge. Also fluid viscosity and vortex shedding should be considered in order to better evaluate dissipation effects (see Wei et al. [27]). For this reason, the development of a non-linear theory is necessary. This will also allow the evaluation of the gate response when the natural modes are excited sub-harmonically by incident waves.  Expressions (A.5a)-(A.6b) form two systems of 4 × P integro-differential equations whose unknowns are respectively ϕ R n,pq and ϕ S n evaluated on the boundary of the gate farm. Consider the case where the index of the summation p * is equal top. The integrals inside (A.5a)-(A.6b), given by ∂ ∂x are hypersingular when Á = ± y and = ± x. In this case, the inversion between the outer derivative and the integral sign is possible by means of the Hadamard finite-part integral H .
Recalling the expression of the Hankel function H Expressions (A.38a)-(A.38c) and (A.39a)-(A.39b) define two systems of linear equations whose unknowns are respectively ˛R ± nmp * ,pq and ˇR ± nmp * ,pq for the radiation problem, ˛S ± nmp * and ˇS ± nmp * for the scattering problem. Each system has 4 × P × M + 1 unknowns, hence M + 1 evaluation points must be chosen for each side of the single array. A good choice for the collocation points (x p,j , y j ) is given by the roots of Chebyshev polynomials of the first kind ( R pq x ± p , y, z