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OCR for page 585
On the Optimization, Including Viscosity Effects,
of Ship Screw Propellers with Optional End Plates
K. de long (University of Groningen, The NetherIands)
ABSTRACT
For ship screw propellers an optimization theory
is discussed, that can be applied to propellers with
and without end plates. The method differs from the
classical optimization in that for the derivation of
the optimum circulation distributions the sum of
kinetic and viscous energy loss of the propeller is
minimized instead of kinetic energy loss only. A
criterion for the risk of cavitation, a simple strength
model and an optional nonlinear correction are
incorporated in the optimization method. Hydrodynami-
cally favourable planform shapes of screw blades and
end plates are discussed. Numerical results of optimum
screws are given, including a comparison between the
application of different types of end plates. It is
shown under what circumstances and to what extent the
theory predicts a higher optimum efficiency for
propellers with certain types of end plates than for
propellers that have no end plates.
This paper is a summary of [1i, in which some
topics are treated at greater length and in which more
numerical results are given.
1. INTRODUCTION
In [2] the design of a model screw propeller with
end plates is discussed, and results of experiments are
given. These results are promising for the end plate
concept. In this paper we want to obtain a better
insight into the circumstances under which optimum
propellers with end plates can have a higher efficiency
than optimum propellers without end plates. In the
classical optimization theory for ship screw
propellers, as used for instance in [2], [3], [4] and
[5], when the propeller reference surfaces are given
the optimum circulation distributions are determined by
solving a variational problem such that the kinetic
energy loss of the propeller is a minimum. A major
subject in this paper will be an optimization theory in
which in the variational problem the sum of kinetic and
viscous energy loss is minimized. In this paper we will
speak of the latter optimization theory as
"optimization including viscosity".
Bearing in mind the results of [3] we confine
ourselves to propellers with zero rake and with end
plate planforms lying in a circular cylinder.
Furthermore, for reasons of simplicity, we restrict
ourselves to propellers with homogeneous inflow.
K. de Jong, University of Groningen, Department of Mathematics,
P.O.Box 800, 9700 AV Groningen, The Netherlands.
585
Viscous energy loss depends mainly on the
distributions of chord length and of maximum profile
thickness along the spans of screw blades and, if it
does apply, of the end plates. When we want the risk of
cavitation to be approximately equal along all spanwise
stages of blades and end plates of a screw propeller,
the distributions of chord length and maximum profile
thickness can not be chosen arbitrarily. Therefore in
sections we will introduce a criterion for the risk of
cavitation, being a relation between circulation, chord
length and maximum profile thickness, depending on some
parameters. That the cavitation criterion can be
satisfied for a "two-sided" end plate, is discussed in
section 4 by explaining the concept of an end plate
consisting of two "shifted" parts. The required
strength of the propeller also puts constraints upon
the distributions of chord length and maximum profile
thickness. Therefore an approximative strength
calculation is embedded in the optimization method, as
is explained in [1~.
The relation associated with our cavitation
criterion, makes the viscous energy loss of a screw
propeller depend in essence only on the circulation
distributions of the screw, see section 5. Because it
is evident that also the kinetic energy loss depends on
the circulation distributions, we are able to formulate
in sections a variational problem for the minimization
of the sum of kinetic and viscous energy loss, which
both can now be considered as functionals of the
circulation distributions. The optimization including
viscosity seems to be important especially when
propellers with end plates are optimized. This we
expect because end plates have the effect of relatively
increasing the circulation and chord length at large
radii with respect to a propeller without end plates,
while at large radii the viscous energy loss becomes
important. Also for lightly loaded propellers which
have a relatively large viscous energy loss the
optimization including viscosity seems useful.
For more heavily loaded propellers the kinetic
loss becomes predominant, while the prediction of the
kinetic loss for higher loadings is more uncertain
because of the nonlinear character of the problem.
Therefore we made a rough attempt to correct the linear
theory by iteratively adapting the advance ratio of our
vortex wake.
Some design requirements, hydrodynamical aspects,
strength aspects, aspects of the viscous energy loss,
our cavitation criterion and our nonlinear correction
OCR for page 586
coordinates (x,r,0) are such that in the yz-plane the
angle ~ vanishes at the positive y-axis and increases
towards the positive z-axis. The screw has a rotational
velocity ~ > 0 (rad/sec) around the x-axis in the
negative B-direction. There is an incoming homogeneous
flow U in the positive x-direction.
In our model we neglect the influence of the ship
and of the hub, hence we consider freely moving screw
blades. The reason for the neglect of the hub is that
then in the optimum case the circulation vanishes at
the root of the blade, and it seems reasonable that
when such a blade is mounted to a hub of finite length,
no strong hub vortex will occur. A more fundamental
approach for the hub of finite length is given in [6~.
The blade and end plate are situated in a close
neighbourhood of their planforms, which are parts of
the corresponding reference surfaces. The reference
surface Hb of the blade is
Hb: 0=~x/(u+vx), rh<~
