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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 "twosided" 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 585
coordinates (x,r,0) are such that in the yzplane the
angle ~ vanishes at the positive yaxis and increases
towards the positive zaxis. The screw has a rotational
velocity ~ > 0 (rad/sec) around the xaxis in the
negative Bdirection. There is an incoming homogeneous
flow U in the positive xdirection.
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<~
OCR for page 585
the pressure jump between the "+" and "" side of a
wing section per unit of length in the chordwise Xi
direction. The parameters 0`i = ot`(a`) and pi = p`(a`)
determine the shape of EPIC+, they are taken in the
range O < 0`i < pi < 1.
t[P,]
CPtAi~ 1h ~ ~
~ /! \
V. I\
ALE XLEto~ C XLEt4IC' NOTE
X,
Fig. 3 Prescribed chordwise pressure jump distribution
of a wing section, O < Hi < ~` < 1.
A wing section (profile) at the radial position r
has the velocity V`=V`(a`) with respect to the water
V(a) ((wr)2+(U+v )2~1/2 (8)
where p is the fluid density and where the constant
Aita`) equals
Alta`)=2/~1+pi(a`)or`(ai)~ · (10)
In agreement with the sign definition of ~ and orb, the
circulation ['i(a`) is taken positive with a "righthand
screw" in the positive direction of the length
parameter ai.
The actual pressure at pressure and suction side
of a wing (in our case a wing can be a screw blade as
well as an end plate part) also depends on the
thickness distribution of the wing. For instance,
consider a two dimensional flow around a symmetrical
profile with zero angle of attack. The fluid flow at
infinity is the uniform parallel flow Vitai). When the
thickness ratio ti~ai)/ci(ai) is sufficiently small,
the greatest drop of pressure is quadratic in Vital)
and about proportional to the thickness ratio.
Therefore, omitting terms which are of quadratic or
higher order in the thickness ratio, we can write
pi (ai)TOO= p A' (~i) V'(ai) i~ai ~
where the corrective velocity AX will be discussed
later on. At the end plate we have r = rp, hence
Vitai)=Vb(abe) for all values of Hi, (i=p,s).
To create the pressure difference over a planform
we assume that the planforms and the reference surfaces
downstream of the trailing edges of the planforms are
covered with vorticity. To represent the thickness of
the blades and end plates we assume an appropriate
distribution of sources and sinks over the planforms.
The vorticity ~ita`,Xi) has two components. The free
vorticity, denoted by /(ai'Xi), is the component in
the helicoidal xidirection, hence it does not deliver
a lift. It is reckoned positive with a "right hand
screw" in the positive direction of the length
parameter X`. The bound vorticity, denoted by
a), is the component perpendicular to the
helices in a planform Pi, hence it isb the pressure jump
generating component. The sign of yi is taken equal to
the sign of the corresponding pressure jump [Pi]. Both min ~
components ~ and yi are taken per unit of length in Pi (ai)Poo=PilAifai)~
their perpendicular directions in the reference
surfaces. Note that these perpendicular directions at
Hb are not the coordinate directions of the (ab,Xb)
system, which is not orthogonal. At the reference
surfaces Hp and Hs belonging to the end plate however
the perpendicular directions are the same as the
coordinate directions (a`,Xi), (i = p,s). We remark that
we call free vorticity that component of the vorticity
which does not give rise to a pressure jump, even when
it is situated at the planform of blade or end plate.
In our theory the picture of the chordwise
pressure jump along a wing section (Figure 3) is
proportional to the picture of the bound vorticity
yb(a`,X'). Hence the prescribed pressure jump
distribution determines how the circulation [` = lipid`)
around a profile is distributed as bound vorticity
(ai'Xi) along the chord. This means that when F`(a`)
and octal) and Midair are fixed for a specific wing
section (that is i and Hi are specified), the pressure
jump is inversely proportional to the chord length
ci(ai), hence
max ~ [P`~(ai,X`) ~ =
o
use of an adequate weight function wb=wb(ab) and we
suggest the following relation between circulation and
chord length
ci(ai)=Bwi {~ A' Vi ~T`~ +A' V' ti}
lo _ h of ol oft.` 1 {1~\
where we introduced the constant B defined by
B p/(pmin p )
If we had considered a variable static pressure and
hence a variable ambient pressure pa, then from the for
cavitation most dangerous vertical upward position of a
screw blade, we could derive that it is better to use
functions Bi=Bi(ai) instead of the constant B. However,
as remarked, for simplicity this is not considered.
The way in which we choose the weight function
Wb  wbtab) is a subject which is discussed in [13. It is
taken into account in the optimization method, which
determines among others the functions Pi = [i(ai) and
ti = titai), (i = b,p,s), occurring in relation (13), and
its choice has an influence on the determination of
these functions. In relation (13) we also introduced
for the end plate the weight functions wp=wpfap) and
wS=wstas). We introduced the weight functions wi as a
tool which can be of use in the optimization method, to
give weight not only to the constraint related to the
cavitation danger, but also to constraints concerning
hydrodynamics, strength and other aspects, see [1~.
4. DISTRIBUTION OF VORTICITY AT THE REFERENCE
SURFACES
We assumed already that the vorticity belonging to
the lifting surfaces of blades and end plates, is lying
at the planforms P.`, (Figure 2). Because the vorticity
field is free of divergence, we can define at each
planform Pi and its corresponding reference surface Hi
a "vorticity stream function" ~` = lPi~a`,x`), (i = b, p,s).
The meaning of such a stream function is the following.
For two points (cri,X`) and (a`,X`), both situated at
Hi, we have
(~i,xi)
~ita`,Xi)  ~ikai,X`) = Jl An ds , (i = b, p,s) , (15)
( ~i,x`)
where Tn is the component of the vorticity y
perpendicular to the line element ds of an arbitrary
line connecting the two points
The leading edges, X`=Xi (a`), of the planforms Pi
are "stream lines" of~kthe vorticity field. Upstream of
the planforms, (Xi < Xi ), there is no vorticity. Hence
we can choose
~i(ai,Xi)O , Xi
line I according to a singularity which is logarithmic
in the distance to the line 1. This second consequence
implies for instance that at leading and trailing edges
of a planform Pi the induced velocity would be
infinite, when we had chosen 0`` = 0 and Hi = 1 in the
chordwise bound vorticity distribution given in
Figures. Hence from this point of view values of Hi
and pi with °`i>0 and p`<1 are to be preferred.
To illustrate some basic ideas we direct our
attention in the remainder of this section to a screw
with a twosided end plate of which both end plate
parts have equal span a ,e = as,e or kp = ks. When the
functions Job' Cb, °`b and i3b are given as in Figure
5(a), (b) and (c) respectively, we can draw pictures of
the vorticity field in the expanded screw blade
planform by using (1S)~18). In Figure 5(d), (e) and
(f) for different choices of leading edges of the screw
blade, results are drawn.
We remark that the motivation for the choice of
the shape of the functions fib, Cb, Cab and fib iS given
3.2

b
to
A/// , ol /
0 Pb (m2/sec) ~ 17 0 cat (m)
(a)
\\\\\\\\ ~ ~
LE
in later sections. By ASP`, (i = b, p,s), we denote the
difference in the stream function Hi between the
leading edge and the nearest depicted vorticity line
and between each two depicted neighbouring vorticity
lines. We must realize that the density of the
vorticity lines in Figure 5(d), (e) and (f) is not an
exact picture of the density of the vorticity lines in
space, because the surface Hb is not flat and even not
developable. However a good insight can be obtained in
this way.
For the three types of considered screw blade
planforms it is seen that at the blade tip there are
ranges of Xb in which the free vorticity strength
does not vanish
/(ab,e,Xb) ~ O ~ Xb
OCR for page 585
discontinuity arises of the type given in Figure 4. It satisfies
is observed that this type of discontinuity is less
strongly in the case of Figures(d) than in the case of
Figures 5(e) and (f).
Now we come to the question as to what kind of
twosided end plates have to be mounted to screw blades
whose planforms are for instance as depicted in
Figure 5. In order to avoid the creation of a
concentrated vortex line segment at the junction of
blade and end plate planforms, it is necessary to
choose the bound vorticity of the end plate parts such
that at the line ap_ag=O, hence at the roots of these
parts, their sum equals the bound vorticity at the tip
of the screw blade
rp( °,xp ) + As ( O. AXE ) = ~b( lab, e'Xb ) ~
Xb ((7b,e)
OCR for page 585
plate, compare Figures 14 and 17.
However a disadvantage of the above end plate is
that its chord lengths are relatively too large
compared with the chord length of the screw blade.
Since only half of the circulation of the blade tip is
conveyed to each end plate part, in relation with the
avoidance of cavitation only about half of the chord
length of the blade tip would suffice for the chord
lengths at the roots of the end plate halves, see
relation ( 13 ). Obviously, shorter chord lengths will
cause smaller viscous losses, and min~rnization of the
viscous loss especially is important for the end
plates, because they have a relatively large velocity
with respect to the water. This will be discussed now.
For simplicity we choose in first instance chord
lengths based on relation ( 13 ) without taking into
account the thickness. Furthermore at the end plate
near the blade tip (0
x=cons t
+U+vx
Ekin = Y2p JJI Igrad~(x,y,z) l2dxdydz , (24)
x=const .
where ~ = ~(x,y,z) is the velocity potential of the
trailing vorticity lying in the twosided infinitely
long reference surfaces Hi j (i= b,p,s; j= 1,...,Z). It
is a straightforward task to reduce equation (24~) to
fib , e
E~in(ri) =v2pZ ~ J ~nbrb(ab)Vb(ab)dab +
o
or i ~ e
Vb(ab,e) ~I ~Uiri(ai)dai1 ~(25)
where ni (i=b,p,s) is the unit normal at the reference
surfaces Hi', directed from the "" to the "+" side.
When the screw blades and the end plates move
through the water, frictional forces arise. Consider
across a wing from leading to trailing edge an
elementary wing strip of width dad. The strip has the
chord length ci(ai) and maximum thickness t`(ai) and is
moving with relative velocity V2(ai) through the water.
For the resultant viscous force dFt2 ~C(a`) on the strip
we use the formula
drip (hi)= 2 p (at) Vt(ai) ci(ai) dai , (26)
where the section drag coefficient C7(ai) is defined by
C7(ai)=2C:(a`) (l+A' (a`)titai)/cifa`~) (27)
In expression (27) there occurs the skin friction drag
coefficient C:(a`), for which we take
C:(a`~=0.075 ~ logReyi(ai)2)~2 . (28)
The constant AII (Al) in a section depends on the
location of maximum thickness of the wing section. For
instance for sections with maximum thickness located at
or near 30% of the chord, AiIi can approximately be
taken as 2.0, however for sections where this location
is at 40 or 50% of the chord, the value 1.2 will be
more appropriate, see A. The local Reynolds number
Reyi(ai) is taken as
Reyi(ai) =Vi(ai) ci(ai) / v , (29)
with in the denominator the kinematic viscosity of
water ~ = 1.2 106 m2/sec. The approximative formula (26)
should not be used for Reyi below 5*105. Further we
remark that the drag, coefficient C7, (27), depends on
the lift coefficient C: of the wing section, in that at
higher lift coefficients, a greater part of the drag is
contributed by pressure or form drag resulting from
separation of the flow from the profile. See for
instance [83. One can think for example, for some types
of profiles of the "bucketshaped" drag minimum of the
function C3~=C7(C:). For simplicity we did not include
this effect in (27), however in the iteration method
that we discuss in section 7, one can account for this
influence on C: in step 5 of the iteration scheme of
Figure 7.
The work done by the frictional forces causes
energy loss. This energy loss per unit of time we
denote by EWC Denoting the contribution of the wing
strip to this energy by dE:t~C(ai), we have
dF:i8C(a`)=Vi(ai) dF:i8C(ai) , (i=b,p,s) . (30)
Using (26) and (27), choosing the chord length
ci = ci(a`) according to relation (13), and taking the
sum of the contributions of all Z blades and end
plates, we can write EWC as a function of the
circulation distributions F`, (i=b,p,s),
Ire
EWC([it)=PZ i, 2 ~ C:AtI! V3 tidal ~
_
i=b, pa, O
i, e
BJ~V2 Wi I  t Ti +Ai Vi ti} daiJ (31)
o
The dependence of EWC on the circulation
distributions is a complicated one, when we realize
that the skin friction drag coefficient C: depends via
the Reynolds number Reyi and the chord length ci on the
circulation loci, see formulae (28), (29) and (13).
The component of the frictional forces in the
direction of the screw axis counteracts the propeller
thrust. Using (26) and again relation (13) we find the
resultant thrust deduction TIC due to viscosity
bite
TWC([i)=P(U+Vx)Z ~ J HI ~Vitid~i +
i =b, pa,
lo.
',e
B i~ Vt hi { hat ri +A' Vi ti} deli . (32)
o
For the potential theoretical thrust Tpot we take
Tpot=P~Z J ~(~b)rb(~b)dab ~( )
o
whereby the thrust T becomes
T= 1 pot1 DISC
(34)
Finally we remark that in the approximation of the
viscous energy loss ERIC and of the thrust deduction
due to viscosity TIC we have neglected the positive
or negative interference drag produced in the comers
of the bladeend plate junctions. The interference drag
is caused by the interaction of the boundary layers of
blades and end plates and it depends among others on
the thickness ratios ti/ci alla on the lift coefficient
CL. Of the wing sections near the junction.
6. FORMtL\TION AND SOLUTION OF A VARIATIONAL
PROBLEM
In our screw propeller model the energybalance
equation reads
Q(ri) ~ = (T',ot(ri)Tvisc(ri)) U+Ekin(ri)+Evisc(ri), (35)
where Q(ri) is the torque about the propeller axis and
Tpot' Tvisc' Ekin and Evisc' as functions of the
593
OCR for page 585
circulation distributions pi, are given in (33), (32),
(25) and (31) respectively. Evidently it is demanded
that Tpo`(Ti) > T~,C(ri). For the propeller efficiency
we have
useful work (Tpot([ i )  T2,, ~c(ri) ) U
total work Q( r i )~
In this section we assume rh/rp, U. ox, Z. rp, I),
kp and k,, to be given, that is the geometry of the
reference surfaces is given. We minimize the sum of the
kinetic and viscous losses, (Ekin+Evgac)' under a
number of constraints. One constraint is that a
prescribed thrust T has to be delivered
Tpo~(Ti) TV) = T . ( )
Notice that when we place a bar on top of a symbol, we
want to stress the fact that the variable in question
is prescribed.
Another constraint is that the distribution of
circulation and of the chord length along the span of
blades and end plates has to be such that the danger of
cavitation is about the same for blades and end plates.
It can be demanded that along the spans of blades and
end plates the chordwise magnum pressure at the
suction side of blade or end Slate equals a prescribed
minimum pressure level porn = p Ant or equivalently B = B.
see (14~. Instead of this last condition, another one
can be required, namely that the screw has a specific
blade area ratio Ae/Ao, (4~. Thus we formulate two
problems.
Problem I: We assume T=T and B=B to be prescribed.
Then we want to find the optimum circulation
distributions PiPt (i = b, p,s), belonging to the given
quantities, from which follow automatically the optimum
chord length distributions coins by relation (13~. Hence
for PiPt we require the efficiency A, (36), to be
maximum and therefore (E~(Pi, )+E~ec(Pi )) to be
minimum, under the constraint that 15 satisfies (37).
Then the blade area ratio Ae/Ao follows from the
optimum chord length distributions C.Pt by formula (4).
Problem II: We assume the thrust T= T and the blade
area ratio Ae/Ao = Ae/Ao to be prescribed. Then it is
possible to calculate the corresponding minimum
pressure level porn in the optimum case. We remark that
problemII can be important for the purpose of tuning
the constant B in our cavitation criterion on the basis
of existing screw propeller designs, for which the
blade areas are known.
For both problems we consider the functional
J=J(Il`) given by
J(I`i) =E`c`.,([i) +Ev='c(l~i) A (Tpo`(~i) Tv~c([i)  T) ,
(i = b, p,s) , (38)
where ~ is a Lagrange multiplier.
For a functional J=J(I,i), (i=b,p,s), we denote the
Gateauxvariation at [i with respect to a set of
perturbational functions hi, (i=b,p,s), by [J([i; hi)
The Gateauxvariation, also called the first variation,
is given by
~J([i; hi)=Em (J(IIi+~hi)J(~i))/6 ,
provided that the limit exists.
Here we remark that in our model rift vanishes
automatically at the root and the tip of the blade when
the screw propeller has no end plates (ratio of
covering k=0~. Otherwise a concentrated hub vortex or
tip vortex would arise which theoretically gives an
infinite energy loss. For the same reason, when the
screw propeller does have end plates, I:Pt has to
vanish at the end plate tips when 0
if=) (~1+B (P2)+B dp3
To satisfy the condition of vanishing total circulation
around the free vortex sheets, we require each of the
three potentials to be uniquely valued in the whole
space, their boundary conditions are
Hb ~ 1 = _~) , Hi : aft =0 , (45)
.=p,8
(ib p a) Mini W(U+vx)C:ViwiA' , (46)
Hi : [~33473 = hi C Vi3 wi A,
( i=b,p, s)
, (47)
where the potential Al is the optimum potential
belonging to the optimization in which only kinetic
energy loss is minimized.
For the boundary value problems (45), (46) and
(47) some properties of the corresponding free
vorticity strength in the reference surfaces if`,
(i = b, p,s), infinitely far behind the propeller are
discussed in [1~. These properties are necessary to
formulate wellposed systems of singular integral
equations which are equivalent to the boundary value
problems and which can be solved numerically by a
collocation method analogous to [3].
In the beginning of this section we formulated two
problems, problem I and problem II, to be solved.
Assuming we have solved numerically the circulation
distributions [!~]_(a`), [~2]_(ai) and [~33_(ai),
(i= b, p,s), we now explain how the resulting optimum
circulation distributions
Pi (of)= { [~l]_(ai)+B Eqi2](~i) }+B [~33_(ai), (48)
are obtained for each of the two problems.
In the case of problem I where we have a
prescribed minimum pressure level pmin= pmin we know by
formula (14) the value of B= B in (44). Then the
Lagrange multiplier ~ can be solved from the prescribed
thrust T= T by substituting (44) in (34) with the use of
(32), (33) and relation (13). We find
~ _ ~
T/(pZ) + K  B J(4P3)
~ _
J(4P1 + B 4~2)
where K and Jo) are given by
~(49)
al e
K = (U+vx) >, J C`FVit`(A``Ii + BVi220iA2I ) dai , (so)
i=b, O
P,8
Ja e
J ( ( / ~ ) = { ( ~ ) r  ~ 2 B ( U + v x ) c F ~ v i 2 o i A ~ } [ ~ ] + d a b
o
a.
~B(U+vx) ~ jC,FVi22eiA~[~]+ dai . (51)
i=p,8 0
For Problem II we have to solve ~ and B from the
prescribed thrust T= T and the prescribed blade area
ratio Ae/Ao=Ae/Ao. This gives us two equations for the
unknowns ~ and B. one by substituting (44) in (34), the
other by substituting (44) in (4), using for the chord
length relation (13). The more complicated formulae
that arise for the solution of ~ and B are skipped
here, but are given in [1~.
7. II~RAlIVE DEI=MINA1ION OF SOME: DESIGN
REQU~
In this section we discuss how some design
parameters and functions, occurring in the solution of
the variational problem discussed in section 6, are
iteratively determined. _ _
When rh/rp, U. Z. rp, a, Up, k`,, T= T and (B_ B or
A~/Ao = Ae/Ao ) are given, an iteration method can be
carried out according to the scheme given in
Figure 7.
Prescribed quantities:
rh/rp, a, z, rp, is, kp, k`,, T=T, l
Problem I: B = B ,
Problem II: Ae/Ao = AJAo
Iteration scheme:

1 initialize :vx = 0,
:'D`(a`)_1
:Reyi(a`)_l*lO6,
:ti(a`)_lower bounds,
: a`` ( Hi )O. OS, pi ( al ) _0. 8
2 calculate ~ :[0l]+(ai) from conditions (45) .
1
3 calculate :[~2]+(a`) from (46),
+(a) from (47) .
~ ,
4 determine :A (ProblemI)
:B alla ~ (Problem II)
5 correct :,u`(a`),
:Reyi(ai), C:(a`), C7(ai),
:ci(ai), ti(ai)
:c~i(a`), fitai), A,(~i)
if the corrections are still significant, then
else
correct Vx=~. 
if vx still changes to some extent, then
1
1
Fig. 7 Iteration scheme for the optimization including
viscosity when the given quantities are prescribed.
In the initialization slept of the iteration we
introduced lower bounds for the maximum thickness
distributions ti = t`(cri). As the lower bound at the
screw blade we take 2 rp f th, where we introduced the
595
OCR for page 585
thickness factor [th. At an end plate part we choose as
lower bound for the maximum profile thickness
distribution, a tapered distribution from the) =
2 rp Ah at the end plate root to ti(ai,e) = rp fth at
the end plate tip, (i=p,s). Thus the use of the lower
bound in particular results in a nonhero thickness at
the free ending tips of blades and end plates and
according to relation (13) also in a nonzero chord
length there. In the calculations used in this paper we
have used the value fth=0.0035, see (52).
For the calculations in step 2 and step 3 of the
iteration we use the collocation method described in
[33. When we want to apply only the linearized theory,
then in the iteration we keep ox equal to zero, and
hence leave the outer loop undone.
Step4 is the determination of the Lagrange
multiplier ~ from (49) (Problem I), or of ~ and the
constant B belonging to our cavitation criterion from
more complicated algebraic formulas (Problem II ), see
[1~.
In steps the occurring functions are adapted
according to some design aspects, which are discussed
more closely in t13. For instance a strength
calculation based upon simple beam theory is
incorporated in this step. In the strength calculation
the considered force fields acting on the beams
representing the screw blades and end plates, are the
potential theoretically induced lift forces, the
centrifugal forces and the viscous forces. Another
adaptation concerns appropriate choices from a
hydrodyn~mical point of view of the functions ~xi=cxi(oi)
and ,B' = pi(ai). For some different types of screw 
propellers values of the occurring functions which are
eventually found using our iteration method, are given
in Figures 1417.
Finally in step6 we want the velocity ox to be
such that it equals the Lagrange multiplier A,
multiplied by a factor ~ with 6:~. The reason that we
take ox = 6A, is that in the linearized lifting line
theory where only kinetic energy loss is nnnm~ized, the
induced backwards translational velocity of the vortex
sheets infinitely far behind a propeller, equals the
Lagrange multiplier A. Then in the neighbourhood of the
blades and end plates of the propeller the induced
velocity is about ~ in the positive xdirection. For
higher thrusts the optimization including viscosity,
resembles the kinetic optimization more, because in
that case the kinetic loss becomes relatively more
important than the viscous loss. Therefore we expect
that the corrective velocity vx=~A, which has a greater
influence for higher thrusts, will be useful of the
optimization including viscosity also.
Here we emphasize that it is not claimed that the
use of the corrective velocity ox gives an exact result
for the kinetic energy loss for higher thrusts. We did
not take into account the vortex sheet deformation, and
the induced velocity by the vorticity and the
sourcesink distributions representing blades and end
plates. However the influence of the trailing
vorticity, which delivers a substantial part of the
induced velocities, is possibly treated more realistic
than in the pure linearized theory.
8. SOME ASI~IS OF l  ; OPTIMIZATION MEIlIOD
Preliminary choices
In this section we first make choices of some
design parameters and functions that are kept the same
for the calculations in this paper. Of course these
particular choices fire not essential for our
optimization method. \Ve take
P = Pseawa~er = 1023 kg/m ~ Pacrew = 7650 kg/m3
rh/rp = 0.2 , ~ = 6 m/see ~ (tlC)h = 0~2
°{b( 3(Jb,e) = 0~05 , Ab( 3~b,e) = 0.8 ~
fth = 0.0035 , SPer = 5.6*107 N/m2
for Problems: B=0.024 sec2/m2, (52)
where SPer is the prescribed permissible stress level
for use in the strength calculation. The density of the
material of the screw P'cr~' and the prescribed
permissible stress level SPer are chosen corresponding
to the material cunial bronze. SPer is based on load
variations of 50 %.
We choose for blades and end plates chordwise
thickness distributions belonging to the NACA16series
sections (see for instance [83), which for ship screw
blades are commonly used.
The location of maximum thickness of the section
is at 50% of the chord length from the leading edge, so
that for the constant A`l`(ai), introduced in the
section drag coefficient C7(ai), (27), we will choose
the value 1.2. The symmetrical section at zero lift has
its minimum pressure located at 60% of the chord from
the leading edge, and from the "basic thickness formsn
tables given in [8] it can be derived that the constant
A'I(a`), introduced in relation (13) can approximately
be taken as 1.25.
When, in our calculations, propellers with end
plates are considered, we assume the twosided shifted
end plate with the anterior end plate part located at
the suction side of the screw blade, unless we
explicitly state that another type of end plate is
considered.
Comparison with the classical optimization
In the classical screw propeller optimization, as
used for instance in [2], [3], [4] and [5], circulation
distributions are derived from a minimization of only
the kinetic energy loss E,Cin of a screw propeller under
the constraint that the potential theoretical thrust
T. equals a prescribed thrust T. Hence in the
classical theory instead of our functional J = J(l~i),
(38), the functional J~,u,=J~.~,(Ti) given by
]~i)=Etin(~i)  11 (Tpo`(Ti)T), (i=b,p,s), (53)
is considered, where ~ is the Lagrange multiplier. By
demanding the first variation of JC`~(li) to vanish we
obtain the conditions
Job Vb ~ Hi Toni = 0 ~ (i = p,S) ~ (54)
in analogy with the derivation of (41) and (42). We put
~=~1 ,
{55)
so that if, is the potential already introduced in
relation (44), and which was solved from condition
(45).
We want to obtain a comparison between the
596
OCR for page 585
optimization including viscosity and the classical
optimization. Therefore we use for the classical theory
the relation between circulation and chord length which
is explained in sections and we derive our design
requirements by using the iteration scheme discussed in
section7. For the classical optimization as well as
for our optimization including viscosity we solve
problernI, hence wit_ a prescribed thrust T=T and a
prescribed value B=B. Then we derive the value of the
Lagrange multiplier ~ from the condition
Tq' ~
pot1 vise = 1 7
which, for a fair comparison, contains the thrust
deduction T2,~,C due to viscosity. Analogous to the
derivation in section 6 we find from (56) that ~ can be
solved from
T/(pZ) +K
~ ,
J ( ~1 )
with K and Jo) given in (50) and (51) respectively.
Now we give some numerical results of the
comparison between the optimization including viscosity
and the classical optimization theory. Note that,
strictly speaking, we do not make a comparison with the
exact classical theory, because we make the comparison
with the classical theory embedded in our numerical
iteration method, in which we incorporated various
nonclassical aspects. In both theories we use the
linearized versions (ox = 0), and we solve Problem I
introduced in section 6. We consider propellers
satisfying
Z=3,
rp = 4 m , ~ = 6 ,~ad/sec , (58)
for a case without end plates (k=0) and a case with end
plates (k=2kp=2k`,=0.5). Results are given in Tablel
for three different values of the prescribed thrust T=T
for which the dimensionless thrust coefficient CT
defined by
CT= T / (I* p U2 or rp)
attains the values 1, 2 and 3. We remark that, for the
optirr~ation including viscosity as well as for the
classical optimization, for the efficiency ~ we used
relation (36).
Table 1 Comparison between optimization including
viscosity and classical optimization for
propellers without end plates (k=0) and propellers
with end plates (k=2kp=2kS=0.5~; linear theory
(Vx=o).
(56)
k Ekin ( 106 Nm/sec)
.
0 2.369 9.445 121.23
2.368 9.444 21.23
_
0. ~ 1. 932 7. 644 17. 16
1 2 3
(57)
Evil (10 Nm/sec)
2.871 4.552 6.114
2.895 4.569 6. 129
4.640 7.447 10.02
4.888 7.630 10. 17
1 2  3
C.~.
. '0(0 .
. 67.65 152.87 143.27
67.64152~87 143.21
69.86 56.97 47.85
69.79 56.95 47.84
.
2 3
Optimization 
incl. vise. 
classical _ 
incl. vise.
classical
From Table 1 it is seen that in the classical
theory the kinetic energy loss is slightly smaller than
in our theory, however the sum of kinetic and viscous
loss is larger, resulting in a lower efficiency.
To obtain some insight into how the difference
between optimization including viscosity and classical
optimization affects in our theory the corresponding
 screw propellers, we give in Figure 8 for the
propellers of Tablet with k=0.5 the corresponding
circulation distributions.
It is seen from these figures that in the
linearized theory for the optimization including
viscosity there occur somewhat smaller values of
circulation at large radii than for the classical
optimization. Smaller values of circulation in our
theory imply smaller values of chord length, see
relation (13). Obviously the cause of the occurrence at
large radii of smaller chord length for the
optimization including viscosity, is that at large
radii the relative velocity of the wing sections is
59) large and therefore the viscous energy loss is
25 25 .
~ at= _ ~
O
0 ab (m) 3.2 0.51ap (m) O
(a) (b)
rs
(m2/sec )
. hi
as (m)> 051
Fig. 8 Distributions of circulation Pi (ibyp,s) of screw propellers with end plates
(k = 2kp = 2kS = 0.5); linearized theory (AX = 0)
classical optimization
optimization including viscosity
597
OCR for page 585
relatively important there, while the classical
optimization does not worry about large chord lengths.
This also explains why the difference between classical
optimization and optimization including viscosity is
greater for propellers with end plates than for
propellers without end plates, see Table 1.
Since kinetic energy loss becomes relatively more
important for propellers with higher loadings, the
difference in the results obtained from the two
optimization methods is relatively smaller for larger
values of the thrust coefficient CT than for smaller
values of CT, as can be seen from Table 1 and Figure 8.
Summarizing we can state that the results of both
optimization theories differ only slightly.
When no solution is found
In section 6 in the paragraph following relation
(42) we have assumed that our circulation distributions
are positive along all spans and therefore we replaced
the quotients l~ifai)/~l~i(ai)~, (i=b,p,s) by the value
one. When the circulation distributions are numerically
calculated we can verify whether this assumption is
correct.
For pTOblet?! I and problem II there exists a region
in the (w,rp,k)space, for which our formulation of the
variational problem does not lead to a solution of the
optimization including viscosity. This appears for
large values of a, rp and k. For these cases we
numerically find only circulation distributions which
change sign along the span of a screw blade or end
plate, and which therefore are not solutions of our
problem. For instance for a screw propeller without end
plates (k = 0) the numerically calculated circulation
distributions along the screw blade can be as in
Figure 9. It is understandable that this type of
circulation distribution is found, because since the
quotient l~i(ai)/~Ti(a`)~ has unjustly been taken equal
to one, at the spanwise stages of negative circulation
Pb there can occur viscous energy gain EWC and thrust
production TVUIC due to viscosity, instead of viscous
energy loss and thrust deductum due to viscosity.
rblL
t
ab c
Fig. 9 Example of circulation distribution of a screw
propeller without end plates, that is found when
the assumption ri(ai)/lri(ai)l=1 is incorrect.
It seems that, to solve this problem, another
variational approach for the optimization, including
viscosity effects, has to be undertaken. In this paper
this will not be carried out and in sections we will
designate the regions in which our opti~r~ation method
does not give a solution. Fortunately it seems that the
relevant optima are found in those regions where our
method does give a solution.
9. SOME RESULTS OF OPTIMIZATION
Efficiency ~ as a function of T. Z. is, up and k in some
theories
To obtain some feeling for the dependency of the
efficiency ,7 on the various parameters we have drawn in
this section for some cases pictures with level lines
of the optimum efficiency q. As for all calculations in
this paper, we made choices from (52).
The equiefficiency lines are constructed by
calculating on equidistant 21*21 grids of (w,rp), (sA),k)
or (rp,k) values, propellers which all are optimized by
one of the methods discussed in this paper. Although in
reality the level lines of the propeller efficiency are
smooth lines, they are sometimes drawn less smoothly.
This is due to the discretization on the equidistant
grids. The efficiency '7 as a function of two considered
variables in each picture has not more than one local
extreme. The location at which the absolute maxims of
the efficiency is attained is designated by an asterisk
in the corresponding pictures. In the pictures on the
(w,k) and (rp,k) grids we have designated by the little
ball ~ the location of the maximum efficiency ~ of an
optimum screw propeller without end plates (k = 0).
Notice that the propellers ~ and ~ are in a sense
optima of optimum propellers, because all the
propellers for which the equiefficiency lines are drawn
have optimum distributions of circulation, chord length
and the other relevant functions occurring in our
model. By ~7 we denote the difference in the efficiency
7' between propeller ~ and its nearest level line and
between each two neighbouring level lines, so that from
the value of ~ at the point ~ the efficiency it at each
depicted level line can be derived.
~ the pictures the ratio of covering k is chosen
in the range 0
a)
11
· 
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4=
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599
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OCR for page 585
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OCR for page 585
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=: °
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as, Cal ~
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OCR for page 585
From Figures 1012 we conclude that, in the
regions where the optimization including viscosity
gives an answer, the efficiency obtained by the
optimization including viscosity is quite the same as
the efficiency obtained by the classical optimization.
The nonlinear vdcorrection of Figure lO(c) gives
values which differ not much from the values of the
linear theories of Figures lO(a) and (b). The reason is
that the propellers have very large propeller radii rp
(25m
OCR for page 585
I. al e
: :
. .
: :
: ~:
Ma ~TE
~a,
I, .... al = Glib
: :
~ cb(a~e) ..
Fig. 13 Leading and trailing edge and corresponding
chord lengths of our twosided symmetrical end
plate planform; kpkS=YJc; i=p,s.
plate. Assuming the span of an end plate part to be
smaller than the chord length of the screw blade tip,
the chord lengths of the twosided symmetrical end
plate are
Ci(ai) = Cb(ab,e) + (a2,e  at )  (Ti,e , 0 < Hi < hi e ~
(ai,e < Cb(ab,e); i = pus) (61)
To account for the difference in chord length
distribution we make use of our weight functions wi
belonging to the end plate, (i = p,s). In our
optimization method each time we iteratively correct
these weight functions such that the desired chord
lengths (61) result.
Table 4 Influence of the type of end plate planform for
screws satisfying (60); screw ~ of Figure Lila)
with twosided shifted end plates; screw with
onesided end plates at suction sides of screw
blades (k = ks; kp = O); screw with twosided
symmetrical end plates of Figure 13 (k = 2kp = 2kS).
.
screw w
.
~ 6.10
.
8.20
6.45
8.20
6.10
8.20
Had/
see
k
0.90
0.00 .
~0.70
0.00 .
0.55
0.00
AJA
0.789
0.815
0.802
0.815
0.780

0.815
Ehn
8.437
9.459
8.443
9.459
8.857
9.459
o6
see
1=
1.011
1.156
1.137
. 156
. 126
. 156
106Nm/
see
~ y(~)~(.J
56.0 2.9
55.6 2.5
54.6 1.5
_ 96 % ,
end plate

twosided,
shifted
onesided
twosided,
symmetric
The results of Figure ll(a) and Tabled show some
clear and understandable tendencies. Under the
considered circumstances the best optimum propellers
ordered with respect to decreasing efficiency ~ are as
follows. First the propeller with the twosided shifted t5]
end plate, ~ of Figure llta), second the propeller with
the onesided end plate ~ of Tabled, third the
propeller with the twosided symmetrical end plate ~ Of t6]
Table 4, and finally the propeller without end plates
of Figure ll(a). That the application of a onesided
end plate appears to be somewhat less favourable than
the application of a twosided shifted end plate, was
already predicted from some approximate considerations
given in [33.
To obtain some insight into the underlying
propellers we have drawn in Figures 1417 some results
of the optimum propellers occurring in the Figurell(a)
and Table 4, at the optimum values of ~ and k. In these
figures we have chosen the leading and trailing edges
of the screw blades symmetrically with respect to the
604
blade generator line, which is the line segment rh<~
OCR for page 585