Predictions of the hydrodynamic performance of the wave rotor wave energy device

E L S E V I E R

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Predictions of the hydrodynamic performance of

the wave

rotor

wave energy device

J . R. Chaplin & C . H . Retzler

Department of Civil Engineering, City Umversity, London EClV OHB, UK

(Revised 11 March 1996)

This paper provides a linear solution for the Wave Rotor, a wave energy device that comprises two parallel counter rotating cylinders m orbital motion. Theoretical results are obtained for the radiated waves generated by the device, and for its efficiency. Comparisons with earlier measurements of radiated waves show very promising agreement. Copyright © 1996 Elsevier Science Limited.

Keywords: Wave energy, wave generation, vortex, free surface, horizontal cylmder

1 INTRODUCTION

A novel wave energy device (the 'Wave Rotor') is driven by the action of the Magnus effect on two parallel counter-rotating cylinders.^ The cylinders span between arms that turn about a central axis parallel with the wave crests as shown in Fig. 1. Circulation in opposite senses around the two cylinders generates lift forces which are many times greater than the inertia loading, and which both contribute to the moment about the central axis. In orbital flow the device rotates continuously at the wave frequency.

Another system which provides an output of a similarly non-reversing nature was described by Hermans et al? In this case the lift force generated by circulation (determined by the Kutta condition) drives a single f o i l around a circular path. By comparison, the Wave Rotor has the advantages that it is mechanically balanced, that l i f t forces on two bodies make contributions to the driving moment, and that the moment may be controlled by adjusting the cylinders' rotational speed.

This paper provides some theoretical predictions of the hydrodynamic characteristics of the Wave Rotor. It uses linearised boundary conditions at the free surface and represents the device as a simple system of rotating singu-larities. From the solution, which follows the approach adopted by Kochin et al? for the case of a vortex in a current beneath a free surface, we obtain explicit results for the radiated waves and for the available power output. Com-parisons between this model and initial measurements by Retzler* show a promising measure of agreement.

2 A LINEAR MODEL OF THE WAVE ROTOR

The problem is defined with reference to the definition sketch in Fig. 2. The rotor is represented by two cylinders of radius c which progress anticlockwise with angular velocity w around a circular path of radius a centred at a distance h beneath the mean elevation of the free surface. There is circulation F around cylinder A, and - F around cylinder B. The fluid is assumed to be of infinite depth, and apart from the concept that circulation can be generated by rotation of the cylinders, all effects of viscosity are neglected. The origin of Cartesian coordinates x,y is at the mean surface elevation, directly above the axis of the rotor, so that at time t cylinders A and B are at z = -ih-ae"^' and z =

-ih+ae'"', respectively, where z = x+iy. Incident waves of

amplitude A and frequency w travel from right to left, and have a complex potential

k

where k = o?lg, and y is the phase of the waves with respect to that of the rotor.

When the Wave Rotor is submerged at a sufficient depth for its effect at the free surface to be neglected Qi » a), and when its dimensions are small in comparison with the wave-length {a « 27r/^), fiow around it may be approximated as that around two counter rotating cylinders in an infinite fluid in which the undisturbed flow is given by

drv; , -kh-iiut-y) n \ Ui-iVi=-T-=Aoie e ' { I )

Incident wave direction

Direction of rotor rotation Fig. 1. The Wave Rotor.

Tliis motion may be resolved into two components, one perpendicular to the line joining the cylinders' axes, and one (of speed Acoe"*''cos7) parallel to the same line. Magnus lift forces due to the perpendicular velocity component act in the radial direction, and so make no contribution to the moment about the rotor's axis.

The complex potential for flow around two parallel cylinders moving through an infinite fluid is given by Basset (p.113).^ Also, it is shown (p.223) that the effect of circula-tion (of opposite senses) around the two moving cylinders may be obtained exactly by placing vortices at their common inverse points. These lie on the line joining the centres of the cylinders, at a radius of a' = {a^-c^f^ from the central axis.

The moment acting on the rotor for the deeply submerged case is therefore

Mo = 2prAcoe~*''a'cosY (3) where p is the fluid density. The unsteadiness of the motion

may be ignored in this respect, because the acceleration of the flow due to the ambient motion is the same for each cylinder, while that due to the motion of the cylinders is in the radial direction. Neither can generate forces that contribute to the moment.

We turn now to the problem of finding the complex potential when the motion at the free surface cannot be neglected. Here the Wave Rotor's cylinders are represented by doublets. The validity of this approach relies on the assumption that the diameter of the cylinders 2c is small

Direction or

in comparison to their separation 2a. (It also requires that c is much smaller than the submergence h and the wave-length lirlk, but these conditions are likely to be less restric-tive.) In fact, the concept of the Wave Rotor is not necessarily limited to small values of c/a, but we are con-cerned here only with a first approximation to the motion. The variation of the incident flow over the region occupied by the rotor \z+ih\ < a is again neglected, so that the ambient motion at the position of each cylinder is given by eqn (2). The doublets are placed at the positions of the centres of the cylinders and the vortices at their common inverse points, in accordance with the result cited above. On this basis, the complex potential associated with the two moving and spinning cylinders, i f their effect at the free surface were neglected, would be

z-ae z + ae" -lawe

^ r ^Jz+ih + a'é"'

27ri \z + ih-a'é"' (4) The first two terms in eqn (4) represent the potentials of the cylinders associated, respectively, with the ambient flow and with their motion, and the third term represents the effect of the vortices. It is convenient to express eqn (4) as an expansion about the point z = -ih in the form

2B„

\z+ih\ > a ₍₅₎

^ „ 2Acoc2e-*"e-'> r f a ,

where B„ + — ; - for n odd,

a nm \ a ,

and B„ = - 2ioic^ for n even.

The problem is now to derive a complex potential Ws{z) that takes the form WQ in the vicinity of the point z = -ih, satisfies the linearised free surface boundary condition

and has no other singularities in the lower half-plane y <Q. These requirements will be met by the solution of the equation

i d^Ws

(7)

Fig. 2. Definition sketch of the Wave Rotor.

providing that the unlcnown function G(z,t) has no singula-rities in the lower half-plane, and that

3m{f{z,t)} = 0ony = 0. (8)

However, eqn (8) is the condition from which it follows, by Schwarz's reflection theorem, that f{z,t) = f{z,t). Therefore, the singularity in

i B^WQ g'dF dwo = lna"B„e" dz ,1 = 1 ink (z+ih)" (z + ih) H + l (9) at z = -ih must be mirrored by one at z = ih in G(z,t). From tliis requirement, eqn (7) can be constructed as

- Ina" i d Ws dWs ~~dz g dt' « = 1 ink -B„e" (z+ih)" (z+ih)"^^ ink 1 [{z-ihy-{z-ihr*'\<

where the overbar represents the complex conjugate. The solution of eqn (10) is obtained by expressing the nth har-monic component of w, as w*"'=wf\z)e''™'+ w<"'(z)e and extracting the corresponding parts of the right hand side to give dw' .(>>). _na"B., ink {z +

ih)"

ih)

•2na"B„e - i{imt + n kz) (inh)"

J'I ks

•ih (13)

in which the first two terms represent evanescent waves whose effect does not extend far from x = 0, and the third term repre-sents the outgoing waves which propagate only in the negative X direction, having an orbital motion in the same sense as that of the rotor. The limits of the integral in eqn (13) reflect the fact that, as X - > 00, the amplitude of the radiated waves diminishes to zero. It is easily shown that Wj satisfies all the requirements stated above. And the proof of uniqueness given by Kochin

et al? applies here also.

The water surface elevation of the radiated waves y]s{x,t) can be found from the linearised dynamic boundary condition

, dw >.=o

(14) by performing the integration in eqn (13) along the real axis. As A: -oo, -q, tends to

„ 2 1 , 5 /,• 2 i

,, " 47rK kB„{in ka)"^_„^u,^-i(no,t+,?kx)

(15)

to which may be added the displacement due to the incident waves TJi = ??Me" '•{^+"'-^>. The amplitudes A„ of the trans-mitted waves at frequency nco are therefore given by

A , = CO , . 2 r , D ('.•„21 4Trn^kB„{in'-ka)"^_„2u, CO n = 2,3... (16)

The mean moment acting about z = -ih is found by Blasius's theorem to be M = 2pFAcoe 00 X

2

The first term in eqn (17) (identical to the moment for the deeply submerged case eqn (3)) represents that applied to the rotor by the incident waves, and is maximum or mini-mum when the rotor is aligned with the direction of the incident flow. The second term represents the moment asso-ciated with the production of the radiated waves. It may be seen that the mean power Mco is equal to the difference between the power of the incident waves pgA^co/4/f and that of the transmitted waves ^pgAf,o}/4nk.

3 RESULTS

To check the implications of using doublets in place of the cylinders, it is of interest to make some comparisons between the present approximate theory and a complete linear solution. For this purpose we use the linear solution obtained by Ogilvie'* for the problem of a stationary circular cylinder without circulation beneath waves, and compare computed results for the phase shift experienced by the waves as they pass over the cylinder. Within the framework

« no 5-30-a. 20-4 IS \ \ \ \ V \ \ \ \ \ Ogilvie (1963) Approximation \ ^ \ \ ^ \ \ ^ \ \ o . 2 \ \ fcc=0.5

\

Fig. 3. Phase shift in waves passing over a submerged horizontal cylinder; comparison of the present approximation with results

0.02

Fig. 4. Tlie driving moment for cases when the wave amplitude at the rotor's elevation is equal to its radius a. The broken line represents the deeply submerged case, eqn (3); ka = 0.2, a = 3. of the present approach, a cylinder of radius c placed at submergence h and represented by a doublet, results in a radiated potential of the form given in eqn (13), with a replaced by c, and

5i=Acoce-*^', 5„ = 0 , « > 1 . (18) The phase shift can be obtained from the form of in

eqn (16) with 7 = 0, and is plotted in Fig. 3 for kc = 0.1, 0.2 and 0.5 as a function of relative submergence h/c. Errors in the phase shift (which reflect those in the forces) are about 6% at h/c = 3 and about 15% at h/c = 2. This is a reasonable basis for the first approximation required for present purposes.

The effect of the submergence of the Wave Rotor on the mean moment that it experiences is shown in Fig. 4. This refers to conditions in which the amplitude of the motion induced by the incident wave at the elevation of the rotor is

Fig. 5. Amplitudes of the first four frequency components of waves produced by the rotor when driven in initially still water;

ka = 0.2, a = 3.

0.0 0.5 1.0 1.5 U / 2 J T (Hz)

2.0

Fig. 6. Predictions of waves generated by the Wave Rotor com-pared with measurements by Retzler.*" The line is given by eqn

(16) with a factor 0.20 on the circulation.

equal to the rotor's radius (Ae" = a), and the ratio of the cylinders' circumferential speed to the local wave induced flow velocity a = VI2-KC03Ae~ is 3. The phase angle 7 is zero, corresponding to a maximum in the applied moment, and the results are plotted here as functions of c/a, the ratio of the cylinder radius to that of the rotor. The results of the linear model converge towards eqn (3) as hia is increased, and the conditions for the maximum moment approach the corresponding Umit of c/a = I/V2. Results presented below are for this geometry.

Amplitudes of radiated waves generated by the rotor when driven in initially still water as a wave-maker are shown in Fig. 5. For this motion it is appropriate to relate the circumferential speed of the cylinders about their own axes to their tangential velocity about the centre of the rotor. Accordingly, defining a' as r/27rcwa, the figure shows for

ka = 0.2 and a' = 3 the amplitudes of the first 4 harmonic 100 60 60 20 H 2 \ ^ i \ . . . , . ^ ^ 4 5 h/a ^1 7

Fig. 7. Efficiency of the Wave Rotor at various cylinder spin speeds as a function of relative submergence; ka = 0.2.

wave components [eqn (16)], normalised with respect to a, and plotted as functions of hIa. High frequency components diminish rapidly as the rotor is submerged more deeply beneath the free surface, and at a given elevation the dominance of the fundamental wave component increases with the speed ratio a'. At hIa = 3 the fundamental compo-nent accounts for 98.1% of the outgoing energy for a' = 3, and 99.5% at a' = 6.

Predictions of the amplitude of the fundamental wave component generated by the Wave Rotor in initially still water are compared in Fig. 6 with measurements by Retzler.* In the experiments, a = 35mm, c = 17.5mm, h = 87.5mm and the angular velocity of the cylinders on their own axes 0 was 67 rads/s. In this case the circulation used in eqn (16) was F = 0.20(27rc^0), reflecting typical ratios between the theoretical Uft force and that obtained from measurements by load cells built into the supporting rig.

When operating as a wave power device there is an optimum elevation for the rotor, between the region near the surface where significant power is lost in the generation of radiated waves, and that at greater depths where, for the same efficiency, very high rotational speeds would be required. This is illustrated in Fig. 7 where the efficiency of the device as a wave absorber is plotted against relative submergence h/a for various speed ratios a. In the near surface region, maximum theoretical efficiency falls well below 100% at any speed ratio because of the presence of radiated waves. The effect of the attenuation of the motion of the incident waves at greater depths can however be off-set by a corresponding increase in the speed ratio. Thus, at a speed ratio a of 8, a theoretical efficiency of 98% is achieved at a depth of hIa = 3.55.

4 CONCLUSIONS

The purpose of this paper was to provide to a first approxi-mation a theoretical framework for the evaluation of the

Wave Rotor wave energy device. In the process we have obtained a more general form of the linear theory by Kochin

et al? and shown that except for the near surface region it is

in reasonable agreement with an exact solution of the linearised problem, of a horizontal cylinder beneath the waves. Predictions of radiated waves generated by the Wave Rotor are found to be in promising agreement with measurements, providing allowance is made for the actual production of circulation.

However, there are many other factors to be considered in studying the performance of the Wave Rotor as a practical device. The most important is probably the effect of separa-tion in generating form drag, and in limiting the producsepara-tion of circulation around a spinning cylinder in a cross flow. Other considerations that have been neglected here include the fact that the phase position 7 = 0 which corresponds to maximum power (used in computing the above examples) would not be achieved in practice, since it represents a point of unstable equilibrium with respect to small angular dis-placements. These factors are the subject of continuing experimental investigations.

ACKNOWLEDGEMENTS

This work was supported by EPSRC grant GR/J 17906.

REFERENCES

1. Basset, A. B., A Treatise on Hydrodynamics, vol.1, Deighton Hall, London, 1888. Reprinted by Dover, New York, 1961.

2. Hermans, A.J., van Sabben, E. & Pinkster, J.A., A device to extract energy from water waves. Appl. Ocean Res., 12 (1990) 175-179. 3. Kocliin, N. E., Kibel, I . A. & Roze, N. V., Tiieoretical

Hydro-dynamics, Interscience Publishers, New York, 1964.

4. Ogilvie, T.F., First- and second-order forces on a cylinder submerged under a free surface. J. Fluid Mecli., 16 (1963) 451-472.

5. Retzler, C. H., Wave energy converter, Patent GB2262572, 1991. 6. Retzler, C. H., The wave rotor. In Proceedmgs of tiie second