On Zhukovsky's Vortical Theory for Describing Flow Past a Propeller

Abstract

The blade-based theory of a propeller is developed by providing exact solutions which do not require numerical integration of velocities from semi-infinite spiral vortices. The induced speed on a lifting line is shown to be describable in terms of coefficients of speed harmonics being multiples of the total number of blades. Speed harmonics of a k-blade propeller are shown to be equivalent to the respective harmonics of a single-blade propeller which are multiplied by the number of blades; this simplifies computation drastically. The first term in the series is the zero-order harmonic of an instantaneous speed; it is the Zhukovsky solution for the induced speed of a propeller with infinitely numerous blades. The second term in the series is the coefficient for the k-th speed harmonic, the third one for the 2k-th one, etc. This growth of the order of harmonics is revealed to drastically reduce amplitudes of the harmonics and accelerate convergence, especially for propellers with large numbers of blades. It is shown that kernels of these coefficients may be integrated analytically over the interval {0,∞} and transformed into a modified first-order Bessel function whose argument is a multiple of the harmonic order. The solutions are implemented in a “rapid” program for aerodynamic calculation of a propeller. An example analysis of circulation, instantaneous speeds, and total propeller characteristics is provided. Also, the article includes a brief review of fundamental studies by Professor Zhukovsky which have been the basis for the development of propeller aerodynamics research and helicopter technologies.