Practical Ship Hydrodynamics

Propellers 45
This formula can be interpreted as follows. The smaller increase in velocity
due to the propeller, the better is the efficiency. If the velocity downstream
is the same as the velocity upstream, the efficiency would be an ideal i D 1.
(But no thrust would be produced.) The ideal efficiency can also be expressed
in terms of the thrust loading coefficient cTh as:
2
i D
1 C � 1 C cTh
Thus a large thrust loading coefficient decreases the efficiency.
The conclusion for practical propeller design is that usually the propeller
diameter should be chosen as large as possible to increase the efficiency.
2.3.3 Lifting-line methods
Lifting-line methods still form a vital part of practical propeller design. They
find the radial distribution of loading optimum with respect to efficiency as
a first step to determine the corresponding blade geometry. Alternatively, the
radial distribution of loading may be specified to determine the corresponding
blade geometry (Lerbs (1952, 1954)). Of course, this approach works only
within limits. If unrealistic or too demanding pressure distributions are specified,
either no solution is found or the error in framework of the theory is so
large that the solution does not reflect reality.
Lifting-line methods for propellers were adapted from lifting-line theory
for straight foils. We shall therefore briefly review the lifting-line theory for
straight foils.
A straight line of vorticity creates lift orthogonal to the direction of the
vortex line and the direction of the inflow, (Fig. 2.6). Helmholtz’s first and
second laws state:
1. The strength of a vortex line is constant along its length.
2. A vortex line must be closed, it cannot end in the fluid.
As a consequence, the vortex lines on a foil are bent downstream at the end
of the foil. Far downstream these vortex lines are closed again, but often ‘far
Horseshoe
vortex Foil
Γ
Lift
Inflow
Figure 2.6 Lifting-line theory is based on representing the foil by bound vortex and trailing
vortices