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Twenty-First Symposium on NAVAL HYDRODYNAMICS
For tip-fin propellers designed according to the guidelines outlined above, experimental verification of predicted improvements in efficiency was reported in . Although the propellers of the small series tested were designed for open water conditions, experiments with a propeller in behind condition and cavitation experiments gave promising results in form of predicted power savings of 5 to 6 per cent for a 225 m, 22 knot container ship.
The examination reported in the present paper was conducted to see, whether the promises would be fulfilled for a tip-fin propeller designed for a given ship. The idea was to make a complete tip-fin propeller design computation and model tests consisting of resistance, open-water, self-propulsion and cavitation tests and compare and analyse those results with similar results for the conventional propeller, designed for the same ship. Although such model tests with the conventional propeller had been conducted earlier, all tests were repeated to exclude as many errors and inaccuracies as possible. This comparative study with analyses of results will be presented in the following sections after an outline of the tip-fin propeller design method which is given next.
In the first publications on tip-fin propellers in which the present author was involved , , a simple criterion was used to find the optimum distribution of loading over the span of the blade and tip fin. Since then improvements have been implemented and now a numerical method is used based on a variational approach. It is essentially the same procedure as that published by Kerwin and his colleagues  but modified to suit the special geometry. It will be outlined below.
The mid-chord line is given and is described in a cylindrical coordinate system by its coordinates (x(s), r(s), θ(s)), all functions of the arc length s, cf. Figure 1. The velocity in a point on this line is
where Va and Vr are the axial and radial components of the circumferentially constant but radially varying wake velocit y, ωr is the tangential component due to rotation and ua, ur and ut are the velocity components induced by the propeller.
If all bound circulation over the blade were concentrated in the mid-chord line and had the strength Γ(s) the thrust and torque without friction could be written as
Figure 1 Geometry of propeller with lattice of free, trailing vortices.
where Z is the number of blades. Note that if θ≡0, x≡0, then s=r, ∂r/∂s=1 and (2) would be the force expressions used in normal lifting-line theory.
To proceed as in  the mid-chord line is subdivided into segments as indicated in Figure 1. The integrations in (2) then become sums and, moreover, the induced velocities are given as
where ua(i) is the axial velocity induced in the mid-chord line segment no. i. This gives the following expressions for the thrust and torque