The Oscillating Water Column Wave-energy Device

J. Inst. Maths Applies (1978) 22, 423-433 The Oscillating Water Column Wave-energy Device D. V. EVANS Department of Mathematics, University of Bristol, Bristol [Received 19 September 1977 in revised form 10 April 1978] An expression is obtained for the efficiency of wave-energy absorption of a float connected to a spring-dashpot system on the top of a column offluidbounded by two closely-spaced vertical parallel plates or a narrow tube immersed under waves. The method makes extensive use of the approximate solution using matched asymptotic expansions obtained by Newman (1974) to the corresponding problem when the float-spring-dashpot system was absent. It is shown that for plates of equal length the maximum possible efficiency is 50%, that for the three-dimensional case it is theoretically possible to capture the energy in a wave whose crest length is greater than the tube diameter. 1. Introduction ONE OF THE numerous devices which has been suggested for absorbing energy from ocean waves is the oscillating water column device. In its simplest form this consists of a partially immersed vertical tube. An oscillating pressure at the open lower end due to a passing wave causes the column of water in the tube to oscillate compress air at the upper end forcing it through a valve connected to a small turbine. This idea has been successfully developed by Masuda in Japan for use as self-powered navigation buoys. In this paper the device is modelled by two closely-spaced plates in two dimensions or a narrow tube of circular cross-section in three dimensions, in each case the plates or tube being fitted at the upper end by a smoothly fitting float attached to a spring-dashpot system. The problem is solved assuming linearized water wave theory is applicable. The assumption that the plates are closely spaced or the tube narrow, allows the method of matched asymptotic expansions to be used indeed the problem reduces to a straightforward extension of work by Newman (1974) on the reflection from two closely-spaced plates. 2. The Two-dimensional Problem We consider the two-dimensional problem of regular waves incident upon a pair of thin vertical closely-spaced plates. At the water surface between the plates it is supposed that energy can be absorbed from the fluid by means of a spring dashpot system acting on a float. Cartesian coordinates are chosen with y measured vertically downwards, with y = 0 the undisturbed free surface. The plates occupy x = ±b, 0 ^ y ^ a regular 423

424 D. V. EVANS waves are incident from x = + co. Under the usual assumptions of linearized water wave theory there exists a velocity potential 0 ", (1) where co is the radian frequency of the assumed simple harmonic motion, (j)(x,y) satisfies + " (2, in the fluid. d 2 d 2 The pressure in the fluid is given from the linearized Bernoulli equation as p pgy>p = Po v-'/ dt where p 0 is atmospheric pressure. For x > b,p = p 0 on the linearized free surface, y = 0 the surface elevation rj(x,t) (measured downwards) satisfies r](x,t) =. (4) 9 dt Taking this with the linearized kinematic condition gives the usual condition = on y = 0 (5) dt dy ^n 0,, y = 0, \x\>a x >a (6) dy where K = a> 2 /g (1) has been used. For x < b, the pressure just under the float is related to its dynamics. Thus the force on the float per unit width is p.2b.1 = mij dfj kri, (7) where m is the mass of the float d,k are dashpot spring constants. Here rj(x,t) = r](t) is the displacement of the float in time, assumed independent of x. It follows from (1), (3), (4) (5) that where At large distances we assume d<j) K l <p+^ = 0, y = 0, \x\<b (8) dy 5 = (k ma> 2 )/2bpg, y = dco/2bpg. (e~' Kx + Re' Kx, x-> + c [Te~ lkx, x-> oo,

OSCILLATING COLUMN WAVE-ENERGY DEVICE 425 corresponding to an incident wave of amplitude a>/g (from (4)) from x = + oo being partially reflected partially transmitted. We also have <j>, V</> -» 0, y -» oo 36-2- = 0, x = ±b, 0<y<a. (10) dx The problem described above will be solved under the assumption that b/a = e is small while Ka,\K l a\ are 0(1). Note that if 8 = y = 0, K t = K the problem is identical to that solved by Newman (1974), using matched asymptotic expansions. Exactly the same technique is applicable to this problem, only a brief description of the method is given as the full details can be found in Newman (1974). The idea, as described by Newman, is to consider an outer region far from the plates where the solution may be regarded as a combination of the solution to the single plate problem solved explicitly by Ursell (1947) together with an oscillating source-type flow describing the flow into out of the inner region. This inner region is described by the solution for the potential flow leaving the opening between two semi-infinite flat plates is given in Lamb (1932). The free surface condition between the plates is used to determine one of the constants in the inner solution. The outer solution is precisely as in Newman (1974) is <j>(x,y) = <l> 0 (x,y) + mg(x,y,a), (11) where <j> 0 (x,y) is the Ursell potential for the single plate m is the unknown source strength. G is the source potential, also given in Newman (1974, equation (3.6)). The inner limit of this outer solution, valid for r = [x 2 + (_y a) 2 ]* -»0is (Newman, 1974, equation (3.7)) 4>(x, y ) = ~ log(r/2a) + *-*"+ - 2.TI U e- 2K Ei(2Ka)- The inner solution for the flow between the plates away from the edges (Newman, 1974, equation 4.3) gives the constant /? is then determined from (8) as so that n na (12) The outer limit of the inner solution gives 1 <t> * ( - - I log (nr/b) + P/OL + O((b/r) \og\r/b)), r/b>i;

426 D. V. EVANS matching with (12) in the overlap region b ^ r ^ a gives m = -2n/<x matching the 0(1) terms to eliminate a gives m _ ~ -r. l-i So from the behaviour of G as x -> ± oo. Now we may write Thus where So where K l b\_l-2e- 2Ka Ei+2nie- 2K "-\l. (15) 2nie- 2Ka -2e- 2Ka Ei+l+\og - + - 0 0 A^ Kb 27ie- 2K A 1 = 1 +log ^ + ^ -2 e " 2Ka i- -^- (1 + 5). (17) Now the efficiency of power absorption is just since Hence E=1-\R\ 2 -\T\ 2 = 2ReC-2 C (20) I«ol 2 + I^ol 2 = l R 0 +T 0 =l. (21) Af + (1+S) 2 Aj + V + B) 2 4 2 + (l + B) 2 ' V ;

OSCILLATING COLUMN WAVE-ENERGY DEVICE 427 As a check on this result we can compute the mean power absorbed by the device which is just P = \ df] 2 dt= 2n Jo 271 J o \dyj y = o = \d from (8). Now, from (13) (14), So But So P = 2 2 I JLI2,\ 2 \4>\ 2 n m </>(0,0) = txkib 2Kx b' p_dh 2 8b 2 ' Imp = e 2K "\Q 2 = e 2Ka - d e 2Ka copb l But the power in the incident wave is just since the amplitude is co/g. So E = agreeing with (12). The maximum efficiency occurs when A x = 0 B, = 1, whence max = \ C =. Then G)-G)-» since = i IB 1 pco ~4~ 0 Similarly, Tp = \ in this optimum case. So the efficiency is expressed in terms of A l B,, where

428 D - v - EVANS For high efficiency, we require A {to be small. Since bja <$, 1, this can only be achieved if FLU (23) Ka =l+o lc when A x =^-[e 2lCa (l+o(kb))-2ei(2ka)-]. 2% The maximum efficiency if A t # 0 is 1 achieved by choosing B x = ^J\+A\. The magnification factor M, defined as the ratio of the maximum displacement of the float to the incident wave amplitude is just If <5 = y = 0 then WMAl + il+btfy M = e Ktt /2Kb\A 1 +i\ which is identical to Newman (1974, equation (6.1)) for the case when the float is absent. With 5 = 0 y chosen for resonance so that B x = 1 A x = 0, 3. Extension to Three Dimensions (24) M ~ e Ka /8Kb. (25) The method can be extended to consider waves incident upon a narrow vertical open-ended pipe immersed to a depth a of radius b. It is assumed that far from the pipe the solution is <f) = e- ikx - Kz + mg(x,y,z), Kr = 0(1), Kb < 1, where z is measured vertically downwards x,y are horizontal coordinates. Thus the outer solution reduces to the incident wave plus an oscillatory point source of unknown strength describing the flow into out of the bottom of the pipe. Here r = ( x 2 + 3,2 + ( z _ a )2)i. Now, from Thorne (1953), 1 f KA-k G(x,y,z) = 1 ^^-e- k^ r Jo K k = -+G 0 (x,y,z), +

OSCILLATING COLUMN WAVE-ENERGY DEVICE 429 where so the inner limit of the outer solution, valid for r/a <? 1, is G o (0,0,fl) = - -+mg o (0,0, (27) 2a The inner solution describes potential flow out of a semi-infinite tube the required expansions are given in Noble (1958). Thus the inner solution has the expansions -+B, r r/b>l where / = 0-61336 is the "end correction" for an open pipe. Matching in the overlap region b 4 r <^ a gives But so, from (29), It follows that m = B +^ ^oz = 0, r<a,,-ka Am -2Ka (26) (28) (29) (30) (31) (32) We can now define an absorption length n, being the ratio of the power absorbed by the device to the incident power per unit length. The power absorbed is still z = O

430 D. V. EVANS so n = 32d\m\ 2 /pcob*, where (29) has been used. After some algebra, we obtain, from (27) (32), where H eka = 2^K (33) (34) Now, in three dimensions, y = do)/nb 2 pg, so that 4y /i 2 B 2 2b Kb A 2 2 + (\ + B 2 ) 2 ' where n/2b is the length of wave crest whose energy is absorbed compared to the pipe diameter. Generally, since Kb < 1, \ij2b = 0{Kb) unless B 2 a: 1 A 2 = 0(1), in which case fi/2b = O(l/Kb). If we choose 8 = 0 so that k = m = 0, then we have no control over A 2 then, as before, the optimum damping d, or y, is achieved by choosing B 2 = (l+^l^)* when Again, the magnification factor M is just M = M0,0) \K X e n(kb) 2 for /4 2 = 0, B 2 = 1, the resonant case. 4. An Alternative Approach An alternative approach to the problem in both two three dimensions is to use the theory developed for the efficiency of wave-evergy devices described previously (Evans, 1976a). In that work the efficiency of wave-energy absorption was calculated for various bodies including a two-dimensional cylindrical cross-section, a threedimensional body, each being symmetric about a vertical axis. In each case the expression for the efficiency required the computation of the added mass damping coefficients for the body. In this work the combination of float plates in two dimensions or piston tube in three dimensions constitute the bodies their added mass damping coefficients can be determined approximately in the same way as for the general

OSCILLATING COLUMN WAVE-ENERGY DEVICE 431 problem described here. All that is required is to solve a radiation problem with condition (8), its equivalent in the three-dimensional problem, replaced by =1, y = 0,.x < b. - This approach has the advantage of avoiding the scattering problem entirely although it is no longer possible to determine the reflection transmission coefficients in the two-dimensional problem. It does, however, provide physical insight into the various * coefficients A h B t (i = 1,2). Thus, in the notation of Evans (1976a), it is found that ^ B, = d/b 22, where a 22,b 22 are the added mass damping, respectively, of the float between the plates M = 2bap is the mass of water between the plates, each per unit width. Thus Mco In three dimensions, B 2 = d/b 22, e 2K where now M = nb 2 ap is the mass of water in the tube a 22,b 22 refer to the added mass damping, respectively, for the float the tube. Thus 5. Results t> 22 _ n Mco 2 \a Figure 1 shows the maximum possible efficiency as a function of Ka for different values of 2b/a for the closely-spaced plates, with 3 = 0. This corresponds to a light float with no spring restoring force. It can be seen that the bwidth widens as 2b/a increases. The values of Ka for which A x = 0are0-935,0-88 082 when 2b/a = 01,0-25 0-5, respectively. The corresponding values of y are00144,0038 00795 with these fixed values for the damping constant y the efficiency may be computed from (22) (19) for varying Ka.

432 D. V. EVANS 0-50 - 0-40 0-30 ft l V 0-20 0-10 / / ^ ^ 01 ^ ^--\ i 1 0-2 0-4 0-6 0-8 1-0 1-2 1-4 1-6 1-8 Ko FIG. I. Variation of maximum efficiency ma, with non-dimensional wavenumber Ka for different values of plate separation 2b/a, with float mass m spring stiffness k both zero. 4-0 - 3-5 3-0 1-5 1-0 0-5 _ =0-25 - II 1 i ^ a ^ = i i 0 0-2 0-4 0-6 0-8 1-0 1-2 1-4 1-6 1-8 Ko FIG. 2. Variation of maximum absorption length to pipe diameter, (/i/2/>) mak with non-dimensional wavenumber Ka for different values of plate separation 2b/a, with float mass m spring stiffness k both zero. Figure 2 shows the maximum absorption length ratio in the three-dimensional case of the narrow pipe, where the bwidth is extremely narrow even for 2b/a = 05. 6. Conclusion A simple approximate analytical solution has been derived for the efficiency of wave energy absorption of an oscillating water column wave energy device. The solution to the problem of the oscillation of a water column contained between two

OSCILLATING COLUMN WAVE-ENERGY DEVICE 433 totally submerged plates or, in three dimensions, in a totally submerged tube, can also be treated by the present method with no added difficulty. Such a solution would provide a simple model for the Vickers wave-energy device. A possible extension to the method in two dimensions would be to consider two plates of unequal length. It is anticipated that in this case the maximum theoretical efficiency possible would be greater than 50%. However the solution requires the potential due to an oscillating source on one side of a plate whereas the far field behaviour of the solution to this problem is given by Evans (1976b) the full solution is not available. For more realistic geometries which allow for the thickness of the plates a full numerical procedure is more appropriate in the two-dimensional case, progress in this direction has already been made (Revill, 1977, pers. comm.). The author would like to thank Mr S. Revill of the Energy Division of the National Engineering Laboratory for bringing the problem to his attention for suggesting the alternative approach described in Section 4. REFERENCES EVANS, D. V. 1976a J. Fluid Mech. 77, 1-25. EVANS, D. V. 19766 J. Inst. Maths Applies 17, 135-140. LAMB, H. 1932 Hydrodynamics, 6th ed. Cambridge: Cambridge University Press. NEWMAN, J. N. 1974 J. Fluid Mech. 66, 97-106. NOBLE, B. 1958 Methods Based on the Wiener-Hopf Technique. Oxford: Pergamon Press. THORNE, R. C. 1953 Proc. Camb. Phil. Soc. 43, 374-382. URSELL, F. 1947 Proc. Camb. Phil. Soc. 43, 347-382.