Induced Velocity by Helical Vortices

A I A H E application of the vortex theory of airfoils has •*• contributed much to the development of propeller theory. However, owing to the difficulty in calculating the induced velocity by a finite number of helical vortices, the theory is generally confined to the hypothetical case where the number of blades is infinite. Consequently, important problems such as the fall of circulation near the tip of the blades were placed outside of the theoretical consideration. Goldstein treated the problem of finding the most efficient distribution of circulation for a propeller with any number of blades and succeeded in elucidating the nature of tip effect. In the present report the author treats the problem of finding the induced velocity with uniform distribution of circulation along the radius and intends to pave the way for the solution of this problem in general. The method adopted by the author follows closely the lines indicated by Goldstein. Take a p-bladed propeller and let co = angular velocity of the propeller, v = translational velocity, r, 6, z •==. cylindrical coordinates. On the assumption of the smallness of the induced velocity, the equation to the surfaces described by the blades of the propeller becomes: n coZ _ n 2 IT 4w 2(p—1)7T v p p V