- Email: [email protected]
Final reply by Michael McCormick
The authors wish 1o thank Drs Couni and Jefferys for lBrcing us to review our results.
REFERENCES 1
2
McCormick, M. E. An analysis of optimal wave energy conversion by two single-degree-of-freedom devices, Journal of the Waterway, Port, Coastal and Ocean Div., ASCE in review, 1982 McCormick, M., Johnson, D., Hebron, R. and Hoyt, J. Wave energy conversion in restricted waters by a heaving cylinder/ linear-inductance system, Proceedings, Oceans 81, MTS/IEEE, Boston, 1981 July, 1982
M. E.
McCormick
Department of the Navy, US Naval Academy, Annapolis, Maryland 21402, USA
Further Comments by Richard Jefferys In response to Professor McCormick's reply to the criticism of his paper, I would like to make the following points. 1. Remarks about symmetry refer to fore and aft symmetry. The waves radiated up and downstream by a symmetrical device will be equal and in phase, independent of non-linearities and shallow water effects. High efficiency is achieved when the radiated wave cancels the transmitted and reflected waves to the greatest possible extent (i.e. see Evansl). It is difficult to see how non-linearities or shallow water effects might alter this basic physical result and lead to efficiencies of greater than 50%. 2. The expression used by McCormick et al. to evaluate the absorbed-power is wrong and the cause of the high calculated efficiencies. Their value ffz is the (averaged) total work done by all the forces acting on the device. This is simply the (averaged) change in kinetic energy and depends only on the amplitude of the response. T
ffz= T1 f
These will be unable to cancel any diffracted wave and will therefore carry away energy and decrease efficiency. 4. It is stated that the radiation damping is approximately one tenth of the dissipative viscous damping and bearing friction. If this is the case, linear theory shows (adapting Evans, 1976) that the peak efficiency is approximately 16.5%. The result is unchanged by shallow water effects and it is hard to see how non-linearities, unlikely to be helpful in any case, could triple the efficiency predicted by linear theory. 5. The log decrement method is entirely adequate for measurement of damping where viscous, frequency independent terms dominate the radiation damping. But radiation and dissipative damping must be approximately equal in wave energy experiments where high efficiencies are desired. 6. Damping measurements in calm water only yield a lower limit to the friction which acts in waves; once the roller bearings are loaded by the surge forces which they are designed to resist, their friction increases substantially. To summarise, there is no need to invoke 'blockage' non-linearities and shallow water effects to explain the high calculated efficiency. If the correct expression is used to calculate the power dissipated, the efficiency will be less than the 50% predicted as the maximum for a symmetrical device in a channel.
REFERENCE 1
Evans, D. V. A theory for wave power absorption by oscillating bodies, J. FluidMech. 1976, 77, 1
August, 1982 E. R i c h a r d J e f f e r y s
Final reply by Michael McCormick
T
mk(t)~(t)dt=
T f me(t)d(e(t))=rt 1 [m~Z]T 2 J0
o
o
If T is one q_uarter of a cycle and the device is at rest at t = 0, then Pz is positive (McCormick's result). However, in the next quarter cycle, Pz is negative since all the hydrostatic spring work is returned and the work done (over the half cycle) by the wave exciting force acting on the body exactly balances the work done by the body on the power absorbing mechanism. Obviously, Pz is zero over a whole cycle since no net work can be done on a body in steady oscillation. McCormick et al. evaluated Pz over one quarter of a cycle and multiplied it by four, thereby counting the returned hydrostatic spring energy as absorbed power (twice over). Only the dissipative work done by the body on the power absorbing mechanisnr is not returned. The correct expression for absorbed power is
1 shall attempt to answer the criticisms expressed in your previous letter in their own order. First, I agree that the phase of the fore and aft radiant waves is unaffected by nonlinearities in the body motion. The waves themselves, however, will be nonlinear in nature if the excitation motions are nonlinear. Concerning the power calculations: tire power available to an electro-mechanical energy conversion device, such as a linear-inductance device, is simply that which it 'sees' in the motion of the body, be it a heaving solid or an oscillating water/air column. Although the power is a vector quantity and, taken in a vector sense, will be zero when calculated over one cycle, the practical energy conversion device is only 'interested' in the scalar power. This is why my limits in integration were over a quarter cycle and not over a complete cycle, as is your choice. Your argument is the same as saying that the area under a sine curve is zero since T
T
/~a =
T
T 0
f sin(c~t) dt = 0
sin 2 cot dt -
b~ dt-
2
o
0
where b is the total dissipative damping excluding radiation damping. 3. Linear theory is expected to place an upper limit on efficiency since a non-harmonic response to sinusoidal forces generates harmonics of the incident wave frequency.
114 Applied Ocean Research, 1982, Vol. 5, No. 2
whereas, we know that the area is not zero. If the scalar power of the body motions is zero, then you and I should find a more fruitful technology to study. Again, I cannot argue with the results of the linear theory, as obtained by Evans and others, when applied to linear systems. My argument, simply stated, is that the
heaving motions which we observed were nonlinear, and the mechanical power exhibited by the motions was more than 50% of the wave power. I stated in my reply to the previous comments by both you and Dr Count, that measured radiation damping was negligible. Linear theory predicts that the radiation damping is proportional to exp(--2kd) where k is the wave number and d is the draft. For our experimental conditions
this exponential term is about 0.10. That is, the radiation damping is only 10% of the value it would be if the oscillations were on the free surface. Alas, it appears that we are at an impasse.
September, 1982 Michael E. M c C o r m i c k This correspondence is now closed. The Editor.
Applied Ocean Research, 1983, Vol. 5, No. 2
115
REFERENCES 1
2
McCormick, M. E. An analysis of optimal wave energy conversion by two single-degree-of-freedom devices, Journal of the Waterway, Port, Coastal and Ocean Div., ASCE in review, 1982 McCormick, M., Johnson, D., Hebron, R. and Hoyt, J. Wave energy conversion in restricted waters by a heaving cylinder/ linear-inductance system, Proceedings, Oceans 81, MTS/IEEE, Boston, 1981 July, 1982
M. E.
McCormick
Department of the Navy, US Naval Academy, Annapolis, Maryland 21402, USA
Further Comments by Richard Jefferys In response to Professor McCormick's reply to the criticism of his paper, I would like to make the following points. 1. Remarks about symmetry refer to fore and aft symmetry. The waves radiated up and downstream by a symmetrical device will be equal and in phase, independent of non-linearities and shallow water effects. High efficiency is achieved when the radiated wave cancels the transmitted and reflected waves to the greatest possible extent (i.e. see Evansl). It is difficult to see how non-linearities or shallow water effects might alter this basic physical result and lead to efficiencies of greater than 50%. 2. The expression used by McCormick et al. to evaluate the absorbed-power is wrong and the cause of the high calculated efficiencies. Their value ffz is the (averaged) total work done by all the forces acting on the device. This is simply the (averaged) change in kinetic energy and depends only on the amplitude of the response. T
ffz= T1 f
These will be unable to cancel any diffracted wave and will therefore carry away energy and decrease efficiency. 4. It is stated that the radiation damping is approximately one tenth of the dissipative viscous damping and bearing friction. If this is the case, linear theory shows (adapting Evans, 1976) that the peak efficiency is approximately 16.5%. The result is unchanged by shallow water effects and it is hard to see how non-linearities, unlikely to be helpful in any case, could triple the efficiency predicted by linear theory. 5. The log decrement method is entirely adequate for measurement of damping where viscous, frequency independent terms dominate the radiation damping. But radiation and dissipative damping must be approximately equal in wave energy experiments where high efficiencies are desired. 6. Damping measurements in calm water only yield a lower limit to the friction which acts in waves; once the roller bearings are loaded by the surge forces which they are designed to resist, their friction increases substantially. To summarise, there is no need to invoke 'blockage' non-linearities and shallow water effects to explain the high calculated efficiency. If the correct expression is used to calculate the power dissipated, the efficiency will be less than the 50% predicted as the maximum for a symmetrical device in a channel.
REFERENCE 1
Evans, D. V. A theory for wave power absorption by oscillating bodies, J. FluidMech. 1976, 77, 1
August, 1982 E. R i c h a r d J e f f e r y s
Final reply by Michael McCormick
T
mk(t)~(t)dt=
T f me(t)d(e(t))=rt 1 [m~Z]T 2 J0
o
o
If T is one q_uarter of a cycle and the device is at rest at t = 0, then Pz is positive (McCormick's result). However, in the next quarter cycle, Pz is negative since all the hydrostatic spring work is returned and the work done (over the half cycle) by the wave exciting force acting on the body exactly balances the work done by the body on the power absorbing mechanism. Obviously, Pz is zero over a whole cycle since no net work can be done on a body in steady oscillation. McCormick et al. evaluated Pz over one quarter of a cycle and multiplied it by four, thereby counting the returned hydrostatic spring energy as absorbed power (twice over). Only the dissipative work done by the body on the power absorbing mechanisnr is not returned. The correct expression for absorbed power is
1 shall attempt to answer the criticisms expressed in your previous letter in their own order. First, I agree that the phase of the fore and aft radiant waves is unaffected by nonlinearities in the body motion. The waves themselves, however, will be nonlinear in nature if the excitation motions are nonlinear. Concerning the power calculations: tire power available to an electro-mechanical energy conversion device, such as a linear-inductance device, is simply that which it 'sees' in the motion of the body, be it a heaving solid or an oscillating water/air column. Although the power is a vector quantity and, taken in a vector sense, will be zero when calculated over one cycle, the practical energy conversion device is only 'interested' in the scalar power. This is why my limits in integration were over a quarter cycle and not over a complete cycle, as is your choice. Your argument is the same as saying that the area under a sine curve is zero since T
T
/~a =
T
T 0
f sin(c~t) dt = 0
sin 2 cot dt -
b~ dt-
2
o
0
where b is the total dissipative damping excluding radiation damping. 3. Linear theory is expected to place an upper limit on efficiency since a non-harmonic response to sinusoidal forces generates harmonics of the incident wave frequency.
114 Applied Ocean Research, 1982, Vol. 5, No. 2
whereas, we know that the area is not zero. If the scalar power of the body motions is zero, then you and I should find a more fruitful technology to study. Again, I cannot argue with the results of the linear theory, as obtained by Evans and others, when applied to linear systems. My argument, simply stated, is that the
heaving motions which we observed were nonlinear, and the mechanical power exhibited by the motions was more than 50% of the wave power. I stated in my reply to the previous comments by both you and Dr Count, that measured radiation damping was negligible. Linear theory predicts that the radiation damping is proportional to exp(--2kd) where k is the wave number and d is the draft. For our experimental conditions
this exponential term is about 0.10. That is, the radiation damping is only 10% of the value it would be if the oscillations were on the free surface. Alas, it appears that we are at an impasse.
September, 1982 Michael E. M c C o r m i c k This correspondence is now closed. The Editor.
Applied Ocean Research, 1983, Vol. 5, No. 2
115