8/18/2019 Design of Optimum Propeller
1/7
J O U R N A L O F P R O P U L S I O N A N D
P O W E R
Vol. 10, No. 5, Sept.-Oct. 1994
Design of Optimum Propellers
Charles
N .
A d k in s *
Falls
Church,
Virginia
22042
and
Robert H. Liebeckt
Douglas Aircraft Company,
Long
Beach,
California
90846
Improvements have been made
in the
equa tions
and
computational procedures
for
design
of
propellers
and
wind turbines of maximum efficiency. These eliminate the small angle approximation and some of the light
loading
approximations prevalent
in the
classical design theory.
An
iterative scheme
is
introduced
for
accurate
calculation
of the
vortex displacement velocity
and the
flow angle distribution. Momentum losses
due to
radial
o w
can be estimated b y either th e Prandtl or Goldstein momentum loss function. The methods presented here
bring
into exact agreement
the
procedure
for
design
a nd
analysis. F urthermore
the
exactness
of
this agreement
makes
possible an empirical verification of the Betz condition that a constant-displacement velocity across the
wake provides
a
design
of
maximum propeller efficiency.
A
comparison w ith experimental results
is
also
presented.
Nomenclature
a =
axial interference factor
a ' = rotational interference factor
B = num ber of blades
b =
axial slipstream factor
C
d
= blade section drag coefficient
C, =
blade section lift coefficient
C
p
-
power
coefficient, P/pn
3
D
5
C
T
= thrust coefficient,
T/pn
2
D
4
C
x
=
torque force coefficient
C
y
=
thrust force coefficient
c
= blade section chord
D
=
propeller diameter,
2R
D' = drag force per unit radius
F
= Prandtl momentum loss factor
G = circulation function
/
= advance ratio, VlnD
K
= Goldstein momentum loss factor
L' = lift force per unit radius
n
= propeller
rp s
P
= power into propeller
P
c
= power coefficient,
2P/pV
3
7rR
2
Q
=
torque
R = propeller ti p radius
r
= radial coordinate
T =
thrust
T
c
=
thrust coefficient, 2T/pV
2
7rR
2
V =
freestream velocity
v'
= vortex displacement velocity
W = local total velocity
w
n
=
velocity norm al to the vortex sheet
w
t
= tangential (swirl) velocity
x = nond imen sional distance,
£lrlV
a
= angle of attack
j8
=
blade twist angle
F =
circulation
s = drag-to-lift ratio
Presented
as
P aper
83-0190
at the
A I A A
21st
A erospace
Sciences
M eeting,
Reno, N V, J a n .
10-13,
1983; received July 23 , 1992; re-
vision
received July
6 , 1993;
accepted
for
publication
D e c. 15 , 1993.
£ = displacement velocity ratio, v'lV
17 =
propeller
efficiency
A = speed
ratio,
V/flR
g =
nondimensional radius,
r/R = A J C
\
e
= nondimensional P randtl radius
£
0
=
nondimensional
hu b
radius
p =
fluid density
a =
local solidity,
Bc/2
/
nr
f>
— flow angle
< f)
t
=
flow angle
at the tip
f l
=
propeller
angular velocity
Superscript
' =
derivative with
respect
to
r
or
£,
unless otherwise
noted
Introduction
I
N
1936,
a
classic treatise
on
propeller theory
w as
authored
by H . Glauert.
1
In this work, a combination of mo men tum
theory and blade element theory, when
corrected
for mo-
mentum loss due to radial flow, provides a good method for
analysis of arbitrary designs even though contraction of the
propeller wake is neglected. A lthoug h the theory is developed
for
low disc loading (small thrust or pow er per unit disc area ,
it works quite well fo r moderate loading, and in light of its
simplicity, is
adequate
fo r
est imating performance even
fo r
high
disc loadings. The conditions und er which a design would
have minimum energy loss were stated
by A. Betz
2
as
early
as 1919; ho wever, no organized procedure for produ cing such
a design is evident in
Glauert's
w o r k .
Those
equations which
ar e
given
by
Betz make extensive
use of
small-angle approx-
imations and relations applicable only to light loading con-
ditions.
Theodorsen
3
showed that the Betz condition for min-
imum energy loss can be applied to heavy loading as well.
I n
1979, E.
Larrabee
4
resurrected the design equations and
presented
a
straightforward procedure
fo r
optimum design.
However ,
there
are still some problems:
first,
small angle
approximations are used; second, the solution for the dis-
placement velocity is accurate only fo r vanishin gly small val-
ues (light loading), although an approximate correction is
suggested for moderate loading; and third,
there
are viscous
terms missing in the expressions for the induced velocities.
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ADKINS
AND LIEBECK: DESIGN OF OPTIMUM PROPELLERS
67 7
Th e purpose of this article is to
correct these
difficulties
and bring the design method into exact agreement with th e
analysis.
It is
then possible
to
verify empirically
t he
optimality
of
the design.
This
work was
initiated
at
M c D o n n e ll D o u g la s
in 1980
in
response
to a
requirement
for simple
estimates
of
propeller performance.
In-house
m ethods, i f they
existed,
h ad
been irretrievably archived. A n early version was presented
as
AIAA Paper
83-0190. Continuous
requests
fo r copies of
th e paper plus some
ref inements
to the method have moti-
vated
its
publication
in the
Journal.
M omentum Equations
D etailed axial and general momentum theory is
described
by Glauert ,
1
and only a
brief summary
is
given
here
to em-
phasize
several important features. Consider
a
fluid e l e m e n t
of mass dm, far upstream
moving
toward th e propeller disc
in
a
thin, annular stream tube
with velocity V . It arrives at
the disc with
increased
velocity, V (l +
0 ), where
a is the
axial interference
factor. A t the disc, dm exists in the an nulus
27rr dr , and the mass
rate
per unit radius passing through th e
disc is 2irrpV(l + a ), neglecting radial
flow.
The
e lement
dm
moves downstream into the far wake, increasing
speed
to the
value
V (l
4 -
& ),
where
b
is the axial
slipstream
factor.
A x ia l
m o m e n t u m theory determines b to be
exactly
20 , whereas the
general theory
(which includes rotation
of the
flow) deter-
mines b to be
approximately
2a .
U s in g
th e
axial
approxi-
mation, which is generally
accepted,
the overall
change
in
m o m e n t u m
of the
e lement
is 2VaF
dm
w h e r e F, th e
m o m e n -
tu m
loss factor,
accounts
fo r
radial
flow of the
fluid.
The
thrust
p er unit radius
T',
acting on the
annulus
can now be
expressed as
rr
T = — = 27rrpV(l
+ a)(2VaF)
(la)
By
similar arguments, the
torque
pe r unit
radius
Q ' is
given
by
Q'lr = 2irrpV(l + a)(2Slra'F)
(Ib)
Flow geometry
about
a
blade
element at the disc is shown in
Fig.
1,
where W
acts on the blade element
with
a, and acts
on the disc at < j > . F
goes
from
about
1 at the hub (where th e
radial flow
is
typically negligible),
to 0 at the
tip,
and is not
unlike
th e
spanwise
loading of a
wing.
The
functional form
of this factor was first estimated by P r a n d t l
1
-
2
and a more
accurate, though
more complex, form
w as
determined
by
Goldstein
5
an d Lock.
6
8
Circulation
Equations
A t each radial position along the blade,
infinitesimal
vor-
tices are shed and
move
aft as a
helicoidal vortex sheet. Since
these vortices
follow
th e
direction
of local
flow,
the helix angle
of
the spiral surface is > , shown in
F i g . 1.
Th e
Betz condition
for
minimum energy
loss,
neglecting
contraction
of the wake,
requires
the
vortex sheet
to be a
regular screw surface;
i.e.,
r t a n
< t>
must b e a constant
i n d e p e n d e n t
o f radius. Theodorsen
3
AXIS
VORTEX FILAMENT
AFTER
TIME
INCREMENT,
A t
VORTEX FILAMENT (t = 0)
Fig.
2 Definition of
displacement
velocity v in the
propel ler wake.
develops
th e
Betz condition
fo r
heavy loading
by
including
th e
contraction
of the wake. He shows
that sufficiently
fa r
downstream
in the contracted
w a k e ,
th e vortex
sheet
must
be the
same
regular screw
surface
for a
propeller
of m i n i m u m
induced energy
loss. This
opt imum
vor tex sheet acts
as an
A rchimedean screw, pumping
fluid
af t
between
rigid spiral
surfaces.
A t th e
blade station,
r, the
total lift
per uni t radius is
given
by
L >
=
f=
BpWT
(2)
and in the wake, the circulation in the
corresponding
a n n u l u s
is
BY
= 27rrFw
f
3
Setting
th e
circulation F
in Eq. (2) equal to that in Eq. (3)
will ult imately determine that
circulation distribution
F(r)
that
minimizes
the induced p o w e r of the
propeller.
I n order to obtain F(r), it is necessary to
relate
v v
r
to a
more
measurable quant i ty.
Figure
2
shows
the wake vortex f i lament
at station r and the definition of the var ious
velocity
com-
ponents there. T he motion of the fluid must be normal to the
local
vortex sheet, and this
normal velocity
is
w
n
.
T h e r e f o r e ,
the tangential velocity is given by
W
t
= w
n
sin < >
However, for a
coordinate
system
fixed to the propeller disc,
th e
axial
velocity of the vortex
filament would
be
v '
= H ^ / C O S />
where the
increase
in magnitude of
v' over w
n
is due to ro-
tation of the f i lament.
This
is
analogous
to a
b a r b e r pole
w h e r e
it appears
that the
stripes
are translating in
spite
of the
fact
that only
a
rotat ional
velocity exists. It
will become
clear that
it
is convenient to use v' , and the
corresponding
displacement
velocity rat io, £ =
v'lV.
Th e
tangential velocity
is
then
W
t
= V£
sin
c f>
cos
4 >
and the circulation of Eq. (3) can be expressed as
F
27rV
2
£G/(Bty
(4)
G = F x cos
/>
sin
< >
(5 )
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678
ADKINS AND LIEBECK:
DESIGN
OF OPTIMUM PROPELLERS
DISC
P L A N E -
Fig.
3 Force diagram for a
b l ade
element.
T he circulation
equations
for thrust
T
, and torque Q', per
unit radius
can be
written
by inspection of Fig. 3 as
T
=
L'
cos
0
-
D'
sin 4 >
= L'
cos
0(1
-
e
tan
< £ )
(6a)
Q'lr = L' sin < £ + D' cos
0
= L' sin
< / > (
+ e/tan
(6b)
where
e is the drag-to-lift-ratio of the
blade element. N e x t,
using
Eq.
(2), L'
can be
replaced
by
F(r) which,
in turn, is
related
to conditions in the w a k e b y E q . (3).
Based
on the
flow
in the
w a k e ,
F(r) is
given
by
Eqs.
(4) and
(5),
an d T
an d
Q' lr
are
reduced
to
being functions
of
f>
and the
dis-
placement velocity,
£ =
v'lV.
Th e
local flow angle
f> will
clearly be a function of the
radius;
however , at this stage of
th e
analysis,
th e
opt imum distribution £(r)
is not yet
deter-
mined.
Several diagrams and an excellent
photograph
of the
vortex
sheet can be found in a 1980 work by Larrabee.
9
Condition for M inimum Energy
Loss
A t
this
point , a departure
from
Larrabee's
4
design
proce-
dure is made, and the momentum equations, Eqs. (1), an d
the circulating
equations,
Eqs.
(6), are required to be
equiv-
alent.
This
condition results in the interference factors being
related
to
£
by the
equations
a = (£/2)cos
2
(£(l - £ tan < / > ) (7a)
a' '= (£/2*)cos /> sin
) (7b)
where Eqs. (4) and (5)
have
been used to express L' in terms
of £, and the terms in
epsilon correctly describe
th e viscous
contribution. E q u a t i o n s
(7),
together
with
th e geometry of
Fig. 1,
lead
to the important simple relation
tan < £ =
£/2)/x
(1 + £/2)X/(
(8)
Here,
A i s a
constant ,
an d
£ varies from
£
0
at the hub to unity
at
th e
edge
of the disc. The relation between the two non-
dimensional
distances
and the
constant
speed
ratio
is
=
=
(r/R)/\
= f/A
R ecalling
th e Betz
2
condit ion,
r
ta n < > = const, Eq. (8)
proves that for the vortex sheet to be a regular screw surface,
Constraint
Eq uations
For design, it is necessary to
specify either
7, delivered by
the propeller or the power P, delivered to the propeller. The
nondimensional thrust and power coefficients
used
for design
are
T
c
=
2T/(pV
2
irR
2
)
(9a)
P
c
=
2P/(pV
3
7rR
2
)
=
2Q(l/(pV
3
>7TR
2
)
(9b)
an d
using
these
definitions,
Eq. (6) can be written as
T'
c
=
I (£
- Itf ( lOa)
P'
C
=J[{
+ J ( l O b )
where the pr imes denote derivatives with respect to f, and
/; =
4£G(1
-
e
tan
0) (lla)
1
2
=
A ( / ;/ 2 £ ) ( l + e/tan < / > ) s i n < j> cos < > (lib)
/;
4fG(l
+e/tan 0) (lie)
J '
2
=
(/;/2)(l
-
s
tan
< / > ) co s
2
0
(lid)
Since £ is
constant
for an opt imum design, a specified thrust
gives
the constraint equations
£
=
( A /2/
2
) - [(A/2/,)
2
-
T
C
/I
2
]»
2
(12)
P C = J^
+
/
2
f
2
(13)
S imilar ly, if power is specified, the constraint relations ar e
£ = ~(JJ2J
2
)
+
[(V2/
2
)
2
+ PJJ '̂
2
(14)
T
c
=
U
-
I
2
£
2
(15)
where the integration has been
carried
out over the region
f =
& t o f
= 1.
Blad e G eometry
For the element
dr
of a
single blade
at radial station r, let
c be the chord an d C
l
th e local
lift coefficient.
Then, th e lift
pe r
unit
radius of one
blade
is
=
P
w r
(16)
w h e r e F is given by Eq.
(4).
I t follows directly
that
We
= 47rXGVR£/(C
l
B)
A ssume for the
m o m e n t that
£ is know n;
then
th e
local
value
of
c j ) is known
from
E q.
(8),
and the
above relation
is a
function only
of the
local lift
coefficient.
Since
the
local Rey-
nolds number is
We divided
by the
kinematic viscosity,
E q.
(16) plus
a choice fo r C/
will
determine th e
R eynolds
n u m b e r
an d £, from th e airfoil section data. The total velocity is
then
determined by
Fig.
1 as
W = V(l + fl)/sin
(17)
w h e r e a is given by E q.
(7),
and the
chord
is
then
k n o w n
from
E q. (16). If the
choice
fo r C/ causes £ to be a m i n i m u m , then
viscous as well as m o m e n t u m losses will in most cases be
minimized, and overall propeller efficiency will be the highest
possible
value. For
preliminary
considerations,
it is usually
sufficient
to
choose
one C
h
the design C
h
for
determining
blade geometry. ( A n y
C
l
specification is
permissible
as long
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ADKINS
AND
LIEBECK:
DESIGN OF
OPTIMUM PROPELLERS
679
at
th e
edge
of the
disc,
and the tip
chord
is therefore
always
zero for a finite lift coefficient.
Design
Procedure
Either F or K, relation for the momentum loss function can
be selected. For the sake of simplicity,
only
th e P r a n d t l re -
lation
is
described
as
where
F = (2/77)arc
f=
(18)
(19)
an d < / > , is the flow angle at the tip.
F r o m
Eq. (8)
tan < t >
t
= A(l +
£ / 2 ) (20)
so that a choice for £ determines the
function
F as well as
f>
by
ta n
=
(tan
< £ ,
(21)
which i s simply t he
condition that
th e
vortex sheet
in the
w a k e
is a rigid screw surface (r tan
< £
= const). For an
initial
value,
£ =
0
will
suffice.
The design is initiated with the
specified
conditions of power
(o r
thrust), hub and tip radius, rotational rate, freestream
velocity,
number of blades, and a
finite n u m b e r
of stations
at which
blade
geometry is to be
determined.
Also, the design
lift
coefficient—one
fo r
each station
if it is not
constant—
must be specified. The design then proceeds in the following
steps:
1)
Select
an
initial estimate
for
£
(£ = 0 will
work) .
2) D e t er m i ne the
values
for
F
an d < j) at each blade station
by Eqs. (18-21).
3 )
Determine
th e
product
W e,
a n d R e y n o ld s n u m b e r
from
Eq. (16).
4)
Determine e
and
a from
airfoil
section
data.
5 ) If e is to be
minimized,
change C, and repeat
Steps
3
an d 4 until this is accomplished at each station.
6 )
D e t e rm i n e
a a nd
a ' from
E q. (7), an d
W f ro m
E q.
(17).
7) Compute the chord from step 3, and the blade
twist
)3 = a + f > .
8 ) Determine the four derivatives in / and
from Eq.
(11)
and numerically
integrate
these from £ =
£
0
to
£
= 1.
9)
D e t er m i n e £
an d
P
c
from
Eqs.
(12) an d (13), or
£
an d
T
c
from Eqs. (14) an d (15).
10) If this new value fo r
£
is not sufficiently
close
to the
ol d one
(e.g., within 0.1%) start
over at step 2 using the
ne w
£.
11 ) D e t er m i n e propeller
efficiency
as TJ P
C
, an d other fea-
tures such
as
solidity.
The above steps converge rapidly, seldom taking m o r e than
three or four cycles. A n
accurate
description of viscous
losses
ca n
be obtained by creating another design
with
e equal to
zero
and
noting
the
difference
in
propeller efficiency.
Analysis of Arbitrary Designs
The analysis method is
outlined here
in
order
to
discuss
problems
of convergence for off design and for square-tipped
propellers in general, and to point out two minor errors in
Glauert's w o r k . Figure 4, which is
simply
an alternate version
of Fig. 3,
shows
th e relation between t he propeller
force coef-
ficients, C
y
an d C
x
, and the
airfoil coefficients,
C, an d C
d
. Th e
equations are
C
y
= C, co s f> - C
d
si n < £ = C,(cos < > - e sin < / > )
DISC P L A N E -
Fig. 4 Force
coefficients
for propeller blade element analysis.
and the relations for the thrust
7"
an d torque
Q '
per unit
radius are then
T
=
( )pW
2
BcC
y
Q'lr =
( )
P
W
2
Bc C
x
(22a)
(22b)
Again,
it is required
that
the
loading Eqs.
(22) be
exactly
equal to the
momentum
result Eqs.
(1).
With the use of the
flow
geometry in Fig. 1,
this requires
the interference factors
to be
where
and c r is given by
a
= o-KI(F -
< r K )
a '
=
)
K '
= C
x
/(4 co s 0sin < / > )
e r = Bc/(2m)
(23a)
(23b)
(24a)
(24b)
Equations (23)
correct
th e placement of the factor
F
used by
Glauert in his equations (5.5) of Chapter V I I a s identified by
Larrabee.
4
The
relation
for the flow angle is
obtained
from
Fig.
1 and
Eqs.
(23)
as
ta n < t>
=
[V(l +
a)]/[nr(l
- a')]
(25)
F or
determining the
function,
F, in
Eq. (18), Glauert suggests
the relation sin $, = f sin 4 > be used in E q. (19). It is rec-
ommended that Eq.
(21)
be used instead, i.e.,
tan < / > , = £ ta n < >
which
is
exact
for the
analysis
of an
optimally designed pro-
peller at the design point.
The analysis
procedure requires
an
iterative solution
for
the
flow angle < >
at each radial position,
£.
A n
initial estimate
for
< >
can be
obtained
from Eq. (8) by
setting £
equal to zero.
Since j3 is k n o w n , the value for a in Fig. 3 is /3 — < £ , and the
airfoil
coefficients
are known from th e
section
data. The
Reynolds n u m b e r
is determined
from
the
known
chord and
W,
which is obtained
from
Fig. 1 and Eq.
(23a),
and the new
estimate
for
< £
is then
found
from E q. (25). A direct
substi-
tution of the new f> for the old value will cause adequate
8/18/2019 Design of Optimum Propeller
5/7
680
ADKINS
AND LIEBECK:
DESIGN
OF
OPTIMUM
PROPELLERS
nonoptimum
designs, some recursive combination of the old
and new
values
fo r
f>
is required to cause adequate conver-
gence.
U n d e r some
conditions (usually
near
th e tip),
con-
vergence
may not be
possible
at all due to large values fo r
the interference factors,
a and a' , in Eq. (23). Since Fis zero
at the tip and
a
is not for a
square
tip
propeller,
th e value
for
a
is 1 and a ' is +1. S uch values are physically
impossible
since
the slipstream
factors
are approxim ately twice the values
at
th e
rotor
plane.
W ilson
an d Lissaman
10
suggest empirical
relations for resolving this
problem, whereas Viterna
an d
Janetzke
11
give empirical arguments fo r
clipping
the magni-
tude of a an d a ' at the value 0 .7 (a/F at the tip is finite at the
design
point
for an
optimum propeller).
F or
analysis,
the conventional thrust and
power coefficients
ar e
C
T
= TI(pn
2
D
4
)
C
p
=
P/(pn
3
D
5
)
Using
Eqs. (22) and (24), th e differential forms with respect
to f ar e given by
C
T
=
8/18/2019 Design of Optimum Propeller
6/7
ADKINS
AND LIEBECK: DESIGN OF OPTIMUM
PROPELLERS 68 1
Tab le 1
Propeller
design
solution
5
8958
1 2917
1 6875
2 833
2 4792
2 875
3424
46 5
4269
3569
2796
1913
58 3125
41 8645
32 2669
22 2978
18 7971
15 9619
13 8552
54 8118
38 3637
28 7661
22 7927
18 7971
15 9619
13 3552
4449
81 4
9834
1 295
974
783
348
644
8 4
89
938
968
a
633
365
219
142
98
72
Input : brake horsepower = 70, 2
blades;
hub diam = 1 ft, tip
diam
=
5.75
f t ; b l ade sect io n : N A CA
4415, C,
=
0.7,
velocity
= 110
mph, rpm
=
24001.
O utput: thrust = 207.61 I b, 77 = 0.86996.
N o t e : a an d
a'
have been se t eq u al to
zero
at the tip.
T a b le 2
Propeller
analysis solution
5
8958
1 2917
1 6875
2 833
2 4792
2 875
54 8116
38 3638
28 7661
22 7927
18 7971
15 9619
12 5862
c
7
7
7
7
7
7
7
4449
81 4
9834
1 295
974
783
348
644
8 4
89
938
968
a
633
365
219
142
98
72
Input : propeller geometry
from Table
1;
r, C, and /3 at the
same radial locations; velocity =110 m p h ,
rpm = 2400.
O u t pu t :
brake horsepower
= 70, thrust =
207.61 Ib ,
17 =
0.86996.
DISC PLANE
Fig. 6
Force
coefficients for w indmill bla de element.
0.08
0 06
0.04
0 02
0.8
0.6
0.30
0 40 0 50 0 60 0 70 0 80 0 90
Fig. 7 Example of
propeller
performance.
1 00
require an additional layer of
iteration
to
achieve
a specified
design
thrust or power. I n
light
of the favorable agreement
between
the present theory and the exp erimental results
given
later in
this
article, it is argued that such an increase in com-
plexity
is not
justified.
Windmills
A ll
of the analyses described in this
article
ar e
directly
1.6
1.4
1 .2
1 .0
I
0 8
0 6
0 4
0 2
7 DESIGN
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r
/
R
T.P
Fig.
8 C
distributions
for
example propeller.
0 20
C P C T
0.15
0.05
REF:
NACA
TN 1834,
PROP
MODEL 5
1.0
0.8
Fig. 9 Comparison of theory and experiment .
geometry for a windmill is shown in Fig. 6 , w h e r e the primary
distinction is that the blade section is inverted (a s compared
with a propeller), and the local angle of attack is measured
from
below
the local velocity
vector. Corresponding
relations
fo r
th e
angles
ar e
8/18/2019 Design of Optimum Propeller
7/7
68 2
ADKINS AND
LIEBECK:
DESIGN OF OPTIMUM
PROPELLERS
0. 3
0.1
J = 0.914
THEORY WITH PRANDTL F
- — — THEORY WITH GOLDSTEIN K
0 NACATEST
REF: NACA
TN
1834,
PROP MODEL 5
0 0.2 0.4 0.6 0.8 1.0
r/R np
Fig.
10
Comparison
of
propeller analyses thrust coefficient.
as shown in
Figs.
6 and 1,
respectively.
I n
these figures, C,
fo r
th e windmill is negative
with
respect to that for the pro-
peller, an d this
sign
change
together
with
the angle definition
will
convert
t he propeller metho ds to the windmill application.
For the design
case,
th e
input P
c
value should be negative,
and the resulting values of v ' (and th e interference factors a
an d
a ')
an d T
c
will
also be negative. (Thrust is of
less interest
fo r
a windmill since it typically represents th e tower
load
an d
is not a main performance parameter.) Similarly, the analysis
results for a
windmill
rotor
will
yield
negative
values fo r both
P
c
an d T
c
.
Examples
A s a sample calculation, the design of a propeller for a light
airplane is
considered.
T he
design conditions
and the
resulting
design are described in
Table
1 which
gives
fo r each radial
station: blade chord, blade pitch angle, local
flow
angle, local
R eynolds
n u m b e r ,
and the interference coefficients a an d a'.
This
propeller geometry has,
in
turn, been analyzed
at the
design con dition
and the
result
is
given
in
Table
2 .
A g r ee m e n t
is virtually
exact.
A nalysis over a
range
of
values
of the ad-
vance ratio, J = V/(nD),
provides
the typical propeller per-
formance
plots
which
are shown in
Fig.
7, and Fig. 8 gives
the blade lift coefficient distribution over a range of /s where
the design condition is the C, — 0.7 and is a
constant
line at
J
=
0.7.
A calibration of the method is given by
comparing
its
the-
oretical prediction with experimental results. Reid
12
has eval-
uated several
conventional propellers
extensively by experi-
ment, and one of
these
ha s
been
chosen fo r
comparison.
Figure 9 gives C
p
,
C
T
, a nd
77
vs
/
fo r
both Reid's
experiments
an d
the
corresponding
theoretical
prediction. The
agreement
here is quite good, with
most
of the disparity occurring after
the
blade
is
stalled. This
propeller uses
N A C A
16-series
air-
foils, and no poststall data were
available.
Figure 10 gives the comparison of the blade
thrust
coeffi-
cient distribution
as
measured
by
R e i d
an d
calculated
by the
method.
Tw o
theoretical
results are shown: one
using F ,
an d
th e
other using
th e
more complex
(from
a
calculation point
of view) K .
I n
principle,
th e
accuracy
of the
method should
be better with th e Goldstein factor for a propeller with few
blades—this
example had three blades—and the two
factors
should give similar results as the n u m b e r of
blades
is in-
creased.
The results of Fig. 10 confirm
this
t r en d , and the
overall
comparison for
both
factors is regarded as quite good.
Conclusions
and
Recommendations
The propeller theory of Glauert has been extended to im-
prove the design of optimal propellers an d
refine
the calcu-
lation of the performance of arbitrary propellers.
Extensions
of
th e theory include 1) elimination of the small angle as-
sumptions in the optimal design theory; 2)
accurate calcula-
tion of the vortex displacement velocity which properly ac -
counts
for the blade
section drag;
and 3)
elimination
of the
small
angle assumptions in the P r a n d t l m o m e n t u m l o s s
func-
tion for
both
design and analysis. These
extensions
bring th e
design
an d
analysis procedures
to exact numerical agreement
within th e precision of
c o m p u t e r
analysis.
The pr imary approximation remaining in both
procedures
is the use of the axial m o m e n t u m equations which require th e
increase in w a k e velocities to be twice those at the
disc.
Under
certain
conditions
this approximation is not good an d gives
rise to the u n n a t u r a l
conditions
and
convergence
problems
described in the
analysis
section. Improvements might be made
by
replacing th e
axial m o m e n t u m equations with
relations
more closely aligned with th e general theory, particularly in
those differential
stream
tubes in
which
heavy
loading
ex -
ists. Such conditions appear to be more prevalent in the anal-
yses
at off-design
conditions
than in the
design
itself,
an d ,
when combined with poststall misknowledge, can lead to large
errors
in
analysis. However,
fo r design and analysis
within
the conventional operating regime, both procedures ar e sim-
ple,
accurate, and
reliable. This
method has
been
extended
by
P ag e
an d L ieb eck
13
to the design and analysis of dual-
rotation propellers.
A favorable
comparison between theory
an d experiment w as also observed.
References
l
G la u e rt ,
H.,
Airplane
Propellers,
Aerodynamic
Theory, edited
by W . D u r an d ,
D i v.
L ,
Vol.
5 , P e t e r Smith , Gloucester, M A ,
1976,
pp . 169-269.
2
Betz, A., with appendix by Pra n d t l ,
L.,
Screw
Propellers
with
M i n im u m E n e rg y Loss, Gottingen Reports, 1919, pp . 193-213.
3
Theodorsen, T ., Theory of Propel lers, M c G ra w -H il l, N ew York,
1948.
4
Lar r abee, E., Practical D esign of M in imu m Induced Loss Pro-
pellers,"
Society
of
A utomotive En gineers, Business
A ircraft
M e et -
ing
an d
Exposition,
W ic h ita , K S, A pril 1979.
5
Goldstein,
S.,
"O n the Vortex Theory of Screw
Propellers,
Pro-
ceedings of the Royal Society of London, Series
A,
Vol. 123, 1929,
pp .
440-465.
6
L o c k ,
C.,
The
A pplication
of Goldstein's A irscrew Theory to
D es ign, " British A er onautical
R esearch
Com m ittee , RM
1377, N o v.
1930.
7
L o c k , C.,
"An A pplication of P randtl 's Theory to an A irscrew,"
British
Aeronautical R es ear ch
Com m ittee , RM
1521,
A u g.
1932.
8
L o c k ,
C., Tables
for U se in an
E m pirical M ethod
of
A irscrew
Strip Theory C alculations,"
British A er onautical
R es ear ch
Commit-
tee, RM 1674, Oct.
1934.
9
Larrabee,
E.,
The Screw Propeller, Scientific American,
Vol.
243, No. 1, 1980, pp. 134-148.
10
W ilson,
R., and Lissaman, P., Applied A e ro d yn a mic s of
W i n d
P o wer M achines ," O r egon
S tate
U n i v ., N S F / R A / N - 7 4 - 1 1 3 ,
P B-
2318595/3,
Corvallis,
O R , July 1974.
H
Viterna, A., and J anetz ke, D., Theoretical an d
Experimental
Power from L arge Horizontal-A xis Wi n d
Turbines,
Proceedings from
the Large Horizontal-Axis Wind Turbine Conference, DOE/NASA-
L e R C ,
July 1981.
12
R e id , E. G., The Influence of Blade-W idth Distribution on
Propeller Char acter is t ics , " N A CA TN 1834,
M a rc h 1949.
13
Page, G . S., and Liebeck, R.
H.,
Analysis of D ual-R otation
Propellers,"
AIAA P a p e r 89-2216, A u g . 1989.