Reply by the Authors to J.G. Leishman

This model can be extended to finite span wings, i.e., three dimensions. For a helicopter, the relationship between u and A/7 cannot be derived directly from Eq. (9) in Ref. 1. One then must question if the leading-edge suction model is valid for helicopters as an extension of two-dimensional theory to three dimensions. Eqs. (10-20) in Ref. 1 show that this extension is possible. The second purpose was to show the difference between the induced drag per unit length predicted by the two models, i.e., the induced angle model and the leading-edge suction model. As mentioned in our paper, this difference must exist for helicopter rotor blades. It is emphasized that either lifting-line theory or lifting-surface theory could be used in both models. Our purpose was to compare the two theories and not to calculate the induced power exactly. The latter needs further correlation. In Ref. 1 we stated that "lifting-line theory is not adequate in describing helicopters in forward flight." Leishman considers this to be rather "misleading and largely untrue." His reason is that the lifting-line theory, even the classical momentum theory, will do a reasonable job in predicting the induced power. But the sense of the words "describing helicopters" can include several meanings, such as lift distribution, drag distribution, blade vibration, noise, ambient flowfield, etc. The rotor performance is only one item; it is not the unique item of these areas. There are many codes that use the lifting-line theory to predict the lifting distribution, including the very complicated free wake analysis. But we cannot be satisfied with their results, especially at higher harmonic loading (see Ref. 2, for example). In fact, it is Professor Leishman himself who pointed out many of the deficiencies of lifting-line theory in his comments. We consider that if Professor Leishman agreed on the aforemention sense of the phrase "not adequate in describing helicopters," his comment about this problem would not be presented. It appears that Professor Leishman does not believe Table 1 in Ref. 1, because of the lower values of/. His reason is that lifting-line theory can predict induced drag reasonably. However, this is a misunderstanding. Both factors, yW/U\ and (ya-S), come from the same lifting-line theory. The only modification is the two-dimensional Theodorsen function. Since in Ref. 3, which was used for Table 1 in Ref. 1, it was assumed that the number of blades is infinite, then the theory used is a steady theory. There are three factors for the case when/ 5* 1.0. The first is the inherent difference between the two models, which we wanted to show by using Table 1 in our paper. For example, if the unsteady modifications make the slope of the lift curve decrease by 3%, or the angle of attack increase by 3%, then/will decrease by 6% (with S/ya == 0.5). The second is the lower than usual slope of the lift curve, 5.73, used in Ref. 3 as usual. The third perhaps is the problem in the theory used to calculate the lift distribution. Our purpose for presenting Table 1 in Ref. 1 was to show the first factor, but indeed/showed the combined effect of the three factors already mentioned. When we use the leadingedge suction model to predict induced power, practically, we must choose the theoretical value 2ir in using lifting-line theory or lifting-surface theory for the lift distribution and angle of attack, or an alternative method. In this sense, we thank Professor Leishman for his comments. It is emphasized that since the values of/are far from 1.0, then perhaps this means there may be some problems in the lifting-line theory used. It certainly does not mean that the induced power is too low. However, we do not consider that the two-dimensional Theodorsen function and Table 1 in Ref. 1 were used and presented arbitrarily. We now present Table la which shows the difference between the two models. It is the rotor of a CH-34, at /* = 0.0873, \l/ = 90 deg (y, W, and a are calculated). The calculation is a free wake analysis, using lifting-surface theory, completely unsteady, and similar to that used in Ref. 4. Although the lift distribution does not show good agreement with experiment, the calculated result still can be used. Since the values for / are not far from 1.0, they may reveal the difference between the two models because the slope of the lift curve is lower than 2-jr. In fact, the comparison between the two models (i.e., the values of/) is so simple that those who doubt the results can do the calculation themselves with whatever data is at hand. We acknowledge Leishman's point that in describing helicopters the leading-edge sweep angle should be included in the leading-edge suction. When the blade is at azimuths not at 90 or 270 deg, the sweep angle exists due to the pcosil/ term. Usually, in existing lifting-line or lifting-surface theory applications for predicting the lift distribution, or in the liftingline/blade element analysis, the term ^cosi/' is omitted. We do not have an available method to include /ucosi/'. It is unreasonable to use the lift in which the j-icosi/' is not included, or the leading-edge suction in which the JKCOS^ is included for Table 1 in Ref. 1. Our purposes were to demonstrate that the leading-edge suction model could be used for helicopters and to show the difference between the two models; a complicated planform (e.g., tip sweep) was beyond the scope of Ref. 1. Coincidentally, in Ref. 5, for incompressible unsteady flow, we obtained theoretically