ON THE EFFICIENCY OF A VERTICAL-AXIS PROPELLER 
arenberg 
J. A. 
Netherlands Ship Model Basin 
In order to minimize the kinetic energy left behind in the wake of a 
vertical-axis propeller the oscillating motion of the propeller blades is 
discussed. The blades are assumed to be infinitely thin and infinitely 
long. The chord of the profiles is small with respect to the curvature of 
the cycloidal orbit of the blades. Also the case of a propeller with many 
blades or with a high rotational velocity is considered. 
1. INTRODUCTION 
The oscillatory motion of the blades of a vertical-axis propeller is governed by a system 
of rods and hinges [1]. This system prescribes the angle of incidence of the blades when 
they perform their rotational motion. The forces which act on the blades have been calcu- 
lated by Isay [2], making approximations which give rise to some doubt and which we dis- 
cuss at the end of section 6 of this paper. 
One of the difficulties in a theoretical investigation is the complicated wake which is 
crowded with vortex layers. Another difficulty, which is discussed in section 2 is that for 
rigid blades with a finite chord no rigorous linearized theory for the perturbation velocities 
can be given. This is caused by the varying radius of curvature of the cycloidal orbits de- 
scribed by the blades. 
In this paper we do not assume a prescribed oscillatory motion of the blades. We de- 
scribe a method to construct the angle of incidence of the profile, as a function of position, 
in such a way that the kinetic energy which remains in the wake becomes a minimum. The 
following simplifying assumptions are made. The blades are infinitely thin, tip effects are 
ignored, and the chord of the blades is assumed to be small with respect to the smallest 
radius of curvature of the orbits of the blades. 
The kinetic energy of the wake is expressed in terms of the bound vortices. Then by 
the Ritz-method this energy is minimized. The following property (section 4) of the bound 
vorticity of the blades, under the conditions just stated, will be proved: In order to obtain 
the highest efficiency, it is necessary that the sum of the bound vorticities of a blade in its 
two positions on a straight line parallel to the direction of translation of the propeller, be 
Note: The results presented in this paper belong to research sponsored by the Office of Naval Re- 
search under Contract N 62558-2630 with Ned. Scheepsbouwkundig Proefstation. 
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