Vertical-Axis Propeller Efficiency 65 
Eq. (1) is quite reasonable. In fact it is found that, apart from the obvious change in sign, 
the angle of incidence for the blades on the downstream half of the propeller must be about 
1.3 to 1.8 the value for the upstream half, depending on the advance ratio and the total lift. 
On the other hand, if Eq. (12) is employed, the resulting incidence angle pattern for the 
downstream side is completely senseless as it includes abrupt changes and even infinities. 
It must further be noted that one might consider describing the mixing and decay of the 
free vortices not by Eq. (1), but by introducing the known Lamb’s law* describing the decay 
of vortices in a real liquid into Eq. (12). I have abstained from this for two reasons: First, 
Lamb’s law holds only for isolated vortices in purely laminar flow whereas propeller flow is 
turbulent, and second, the mathematical realization of this method would have been rather 
complicated without enabling a definitely better physical description of the problem. 
While this problem is discussed here to such an extent, it is scarcely considered in Dr. 
Sparenberg’s paper. Equations (5.1), (5.2), and (5.3) used therein for angle of incidence de- 
termination will render approximate solutions for the upstream half of the propeller blade cir- 
cle only. One must keep in mind that the blades on the downstream side are exposed to the 
free wake vortices induced by the upstream blades, and that on the downstream side the 
velocity field of these free vortices cannot justly be omitted from the boundary condition for 
the flow past the profile since it exerts a decisive influence. 
In closing I should like to remark that in my opinion the theory of quickly rotating pro- 
pellers evolved by Dr. Sparenberg in sections 6 and 7 with circulation (6.9) and free wake 
vortices (6.10) has the disadvantage of introducing excessive simplification. It is therefore 
probable that this theory will not render physically reasonable results. This can be seen 
already from the fact that Eq. (7.6) for incidence angle @ yields & = exactly for 9 = 0 and 
Q = 7, whereas in reality the value of @ will be relatively small or even zero for these values 
of 9. 
J. A. Sparenburg 
Equation (1) in paper II by Prof. Isay is the basis of his theoretical work. This formula 
describes the velocities induced by free vortices which are shed by a system of rotating and 
periodically varying bound vortices, placed in a homogeneous stream. The density of the 
free vorticity decreases when the velocity of the incoming stream increases. Hence it may 
be expected that the induced velocities of the free vorticity decrease in this case. However, 
in the above mentioned Eq. (1) the incoming velocity does not occur, which is surprising. 
The remark of Prof. Isay, that the increase of the incoming velocity causes a decrease in 
the free vorticity by changing the bound vorticity, describes a secondary effect. Here the 
basis formula itself and not its relation to a special boundary value problem is discussed. 
Prof. Isay further questions the validity of author’s formulae (5.1), (5.2), and (5.3) for 
the downstream side of the circle. When we look at (5.2), for instance, we see that a pole 
of the integrand crosses the line of integration when the point z, where we consider the in- 
duced velocities, crosses the vortex layer. This means a discontinuity of the value of the 
integral, which corresponds to the discontinuity of the velocities at the vortex layer. Hence 
with respect to potential theory, this formula is valid also in the downstream region. 
*H. Lamb, “Lehrbuch der Hydrodynamik,” 2nd ed., Leipzig, 1931, p. 669 
