On the vortex theory of screw propellers

Abstract

The vortex-theory of screw propellers develops along similar lines to aerofoil theory. There is circulation of flow round each blade; this circulation vanishes at the tip and the root. The blade may be replaced by a bound vortex system, which, for the sake of simplicity, may be taken, as a first approximation, to be a bound vortex line. The strength of the vortex at any point is equal to Γ, the circulation round the corresponding blade section. From every point of this bound vortex spring free, trailing vortices, whose strength per unit length is —∂Γ/∂r, where r is distance from the axis of the screw. When the interference flow of this vortex system is small compared with the velocity of the blades, the trailing vortices are approximately helices, and together build a helical or screw surface. Part of the work supplied by the motor is lost in producing the trailing vortex system. When the distribution of Γ along the blade is such that, for a given thrust, the energy so lost per unit time is a minimum, then the flow far behind the screw is the same as if the screw surface formed by the trailing vortices was rigid, and moved backwards in the direction of its axis with a constant velocity, the flow being that of classical hydro dynamics in an inviscid fluid, continuous, irrotational, and without circulation. The circulation round any blade section is then equal to the discontinuity in the velocity potential at the corresponding point of the screw surface. Further, for symmetrical screws, the interference flow at the blade is half that at the corresponding point of the screw surface far behind the propeller. An approximate solution for the irrotational motion of a screw surface in an inviscid fluid was given by Prandtl. The accuracy of the approximation increases with the number of blades and with the ratio of the tip speed to the velocity of advance, but for given values of these numbers we have no means of estimating the error, since the exact solution of the problem has not yet been found. It is the main object of this work to find the exact solution.