Vertical-Axis Propeller Efficiency 63 
an t+ 12) RS aE 
D9) = = (t# 4 RU LA on Me ly (7.5) 
The constant bound vorticity [’,* is still arbitrary and can be used also in this case to 
determine the activity of the blades in the front or in the back position. 
With respect to the tangent at the cycloid we obtain for the angle of incidence 4(9) 
i Bla 
Vip) =7 V9) 4 
ap) = a, - (7.6) 
where ©, is the angle of zero lift, V(@) follows from (2.2), and 4 is the length of the chord 
of the profile. The angles 0(() and @, are positive when they open to the left with respect 
to the direction of V(9). 
Because we have not taken into account the velocities induced by the oscillating motion 
of the blades we have to place the turning point T of the profiles at one quarter of the chord 
length from the trailing edge, as follows from the end of section 5. 
REFERENCES 
[1] Mueller, H.F., “Recent Developments in the Design and Application of the Vertical Axis 
Propeller,” Trans. S.N.A.M.E., 1955 
[2] Isay, W.H., “Zur Berechnung der Stromung durch Voith-Schneider-Propeller,” Ing. Arch. 
XXIV Band, 1956 
[3] Kotschin, N. J., Kibel, I. A., and Rose, N.W., “Theoretische Hydromechanik,” Band I, 
Akademie Verlag, Berlin, 1954 
[4] Muskhelishvily, N.J., “Singular Integral Equations,” 2nd ed., transl. P. Noordhoff, N.V., 
Groningen, 1954 
DISCUSSION 
G. Weinblum (Institut fiir Schiffbau, University of Hamburg) 
I wish to raise the question, who should decide when mathematicians disagree? I have 
not understood what you have done. Have you calculated some numerical values so that we 
can compare what comes out? 
J. A. Sparenburg 
This is a case in which mathematicians do not agree upon the interpretation of the 
basic formulae needed for the theory. In my opinion the taxation of the various ways of ap- 
proach has to be made by executing experiments, not only of the over-all results but also of 
the basic formulae. We have made some calculations based on the simple theory of sections 
6 and 7. Before publishing results, however, we shall compare them with results based on 
the more refined theory of sections 2-5 and on experiments. 
