AP04_2web.vp


1 Introduction

One of the main tasks during the introduction stage of
aeroplane design is to determine the basic aeroplane perfor-
mance. One of the input is the thrust curve of the power
plant – available thrust versus flight velocity. The optimisation
procedure requires combinations of suitable engines and pro-
pellers offered on the market to compare different thrust
curves and, consequently, aircraft performance. Designers
of small sport aircraft very often have only the shape and
the number of propeller blades without any aerodynamic
characteristics.

It is evident that very sophisticated and precise numerical
methods (helix vortex surfaces or sophisticated solutions by
means of FEMs of the real flow around the rotating lift
surfaces) require large input data files. These conclusions
have led the author to present an easy and sufficiently precise
procedure for calculating the integral propeller aerodynamic
characteristics with minimum demands on geometric and
aerodynamic propeller input data. An inspection of various
aerodynamic propeller theories indicated that a suitable
method can be gained by enhancing Lock’s model of the

referential section connected with Bull-Bennett mean lift and
drag propeller blade curves.

2 Lock’s propeller model of the
referential section
Lock’s model [1] considers a referential section on a pro-

peller blade located at 70 % of the tip radius to be representa-
tive of the total aerodynamic forces acting on the blade (thrust
Tbl and tangential force Qbl or lift Lbl and drag Dbl) – see Fig. 1.
It is assumed that these forces are configured according to the
local relative wind W determined by the incoming flow W0 at
this section (composed of tangential sped U and flight speed
V ) and induced speed vi. The induced speed is related to
lifting line theory.

The propeller blade lift and drag expressed by means of
the lift and drag coefficient:

L W c b rbl l�
1
2

2
0 7 0 7� �( ) . . (1)

D W c b rbl d�
1
2

2
0 7 0 7� �( ) . . (2)

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 103

Acta Polytechnica Vol. 44  No. 2/2004

Preliminary Determination of Propeller
Aerodynamic Characteristics for Small

Aeroplanes
S. Slavík

This paper deals with preliminary determination of propeller thrust and power coefficients depending on the advance ratio by means of some
representative geometric parameters of the blade at a specific radius: propeller blade chord and blade angle setting at 70 % of the top radius,
airfoil thickness at the radius near the tip and the position of the maximum blade width. A rough estimation of the non-linear influence of
propeller blades number is included.
The published method is based on Lock’s model of the characteristic section and the Bull-Bennett lift and drag propeller blade curves. Lock’s
integral decomposition factors and the loss factor were modified by the evolution of the experimental propeller characteristics. The
numerical-obtained factors were smoothed and expressed in the form of analytical functions depending on the geometric propeller blade
parameters and the advance ratio.

Keywords: propeller, propeller aerodynamics, thrust coefficient, power coefficient, propeller efficiency, propeller design.

Fig. 1: Lock’s scheme of the referential blade section



enables us to write the total thrust of a propeller with z blades
as:

� �T zT z L Dbl bl bl� � �cos( ) sin( )� � (3)
in the form:

� �T W b r c cl d� �
1
2

2
0 7 0 7� � �. . cos( ) sin( ) (4)

Substitution of apparent relations from Fig. 1 for the resul-
tant velocities W:

W W V n r

n R r

i s i

s

� � � �

� �

0
2

0 7

2 2
0 7

2

2

cos( ) ( ) cos( )

( ) (

.

.

� � �

� � ) cos( ) ,2 �i

(5)

and the angle of the real incoming flow �:

� � � � � � �
�

�
��� � � � � � � �0 7 0 7 0 7

0 7
. . .

.
i i

r
arctg (6)

(� is the advanced ratio: � � V n Ds and �0.7 is the blade angle
setting) into the expression for the thrust (4) and she use of
non dimensional geometry (b b R0 7 0 7. .� , r r R0 7 0 7 0 7. . .� � )
the expression of the propeller thrust coefficient:

c
T

n D
T

s
�

�
2 4

(7)

is achieved in the final form:

c z b r r
r

T � � �
�

�
	
	




�

1
8 0 7

2
0 7 0 7

2
0 7

0 7
. . . .

.
( ) cos� � �

�

�
arctg

� 

�
� �

� � � �c cl dcos( ) sin( ) .. .� � � �0 7 0 7

(8)

Calculation of the thrust coefficient at given advance ra-
tios requires not only the lift cl(�) and drag cd(�) blade curves
but also the relationship to determine angle of attack �. Lock
[1] developed the induced equation as the dependence of the
lift coefficient on the tip loss factor 	 (function of advance ra-
tio � and angle � of the real incoming flow):

s cl i0 7 4. sin( ) ( )� 	 � �tg (9)
where s0.7 is the propeller solidity factor related to the refer-
ential section:

s
z b

r
r0 7

0 7

0 7
0 72

0 7.
.

.
., ( . ) .� �

�
(10)

The relations among thrust Tbl and tangential force Qbl
acting on the propeller blade and the equivalent blade lift
Lbl and drag Dbl forces were designed by Lock [1] in a de-

composition form based on the angle of the real incoming
flow � at the referential section. This resolution is corrected
by integration factors E and F:

c s Ec Fcl T M0 7. cos( ) sin( )� �� � (11)

c s Fc Ecd M T0 7. cos( ) sin( )� �� � . (12)

Torque coefficient cM represents the tangential force due
to

c
M
n D

z Q r

n D
M

k

s s
� �

�




�
2 5

0 7
2 5

.
. (13)

The introducing of a propeller power P and power coef-
ficient cP:

c
P

n D
P

s
�

�
3 5

(14)

provides a constant relation between the power and torque
coefficient: c cP M� 2� . Integration factors E and F were de-
veloped by Lock in dependence on the advance ratio:

E �
�

3 276
4 336 2

.
. �

(15)

F
E

r
r� �

2
0 7

0 7
0 7

.
., ( . ) . (16)

Decomposition equations (11) and (12) can be used for an
inverse procedure to calculate the thrust and power coeffi-
cient by means of the integration factors and blade lift and
drag curves:

c
s

E
c

c c
T l

d l� �
�

�

�

�

	
	




�

�
�

0 7

1
.

cos( )
( )

( )

( )�
�

�

�
tg

tg

tg2
(17)

� �
� �

c c
s c c

F
P M

d l� �
�

�
2 2

1
0 7� �

�

� �

. ( )

cos( ) ( )

tg

tg2
. (18)

3 Lift and drag of the propeller blade
Bull and Bennett [2] published the lift and drag propeller

blade curve gained by an applying Lock’s scheme (11) and
(12) to a set of experimental propeller aerodynamic charac-
teristics. To calculate the induced values the set of Lock’s
induced equation (9) and the decomposition relation (11) for
the lift was used. The results of the calculations were pre-
sented in [2] and are shown in Fig. 2. These curves represent
different RAF-6 section propellers covering a wide range of

104 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 2/2004

Fig. 2: Propeller blade lift and drag curve



setting angles and advance ratios. The tip Mach number
never reached 0.7.

The mean values of the lift and drag blade curve are de-
scribed by simple linear and quadratic forms – [2]:

c

c

l

l

� � � �

� � �

0 4996 01096 4 98

0 9867 0 0001 0 0024

. . .

. . .

� �

� �

�

2 4 98� � � �.
(19)

cd � � �0 0258 0 00318 0 00173
2. . .� � (20)

4 Modification of Lock’s method
There are at least three reasons for improving Lock’s pro-

cedure to obtain a more effective and accurate method for
quick preliminary calculations of integral propeller aerody-
namic characteristics.
1) Lock’s loss factor 	(�, �) is given only in tabular form, and

requires interpolation procedures.
2) The blade geometry represented by the referential section

(b0.7 and �0.7) is too reduced to affect the entire propeller
acceptably.

3) Lock’s method involves a number of blades in linear form.

Use was made of experimental thrust and power coeffi-
cients and a presumption of the Bull-Bennett lift and drag
blade curve independence of the geometry and flight regime
(fixed curves in Fig. 2 for all types of propellers) to meet the
above outlined requirements. Eleven two-blade propellers
with RAF-6 sections were involved in the calculations. Two
other geometric parameters were added: blade thickness at
90 % of the propeller tip radius – t0.9 and the position (radius)
of the maximum blade chord – rmax. All of the geometric pa-
rameters and the tip Mach numbers M are presented in Ta-
ble 1, where t t b0 9 0 9 0 9 100. . .( )� and r r Rmax max� .

5 Induced velocity
Lock’s expression for the thrust coefficient (8) with

Bull-Bennett lift (19) and drag (20) blade curves was equal to
the experimental thrust and numerically solved the unknown

induced angle �iexp for the corresponding advance ratio and
blade geometry:

� �c c z b r c c cT T y x T� � �
�

2 0 7 0 7 0 7

0

, , , , , ( ), ( ),. . . exp

.

� � � � �

� � 7
0 7

� �arctg
�

�
�

r
i

.
.

(21)

The calculated induced angles �iexp were then used to
determine of the loss factor directly from Lock’s induced
equation (9):

	
�

� �
�

s cl
i

0 7
4

. ( )
sin( ) ( )tg

. (22)

A three-step procedure was used to simulate the influence
of the blade geometric parameters on the induced values.
The first step smoothed the loss factor only as a function of
the advance ratio and induced angle. Subsequently this ana-
lytical expression of the loss factor was used to calculate in-
duced angles �i by solving the induced equation (9) for all the
experimental propellers. Finally, these induced angles �i were
correlated with the experimental set �iexp.

The function of the smooth loss factor that approximates
the numerical results was stated in the form:

	
� �	 	

�
�

1
a bi i

(23)

with coefficients:

a 	 � �� � �0 3254 0 3529 0 4449
2. . . (23a)

b	 � �� � �0 8213 0 0854 0 0628
2. . . . (23b)

A comparison of experimental set of the induced angles
�iexp with induced angles �i calculated by means the smooth
loss factor showed differences that were evaluated by regres-
sion analysis into the final linear correction function:

� � �i i iA Bexp � � �cor (24)

A s r� � � �1088 0 0149 174 0 4620 7 0 7. . . .. . max� (24a)

B t� �1 286 0 113 0 9. . . . (24b)

This linear expression gives a good approximation in the
region of higher angles. In order to keep the simple linear
correlation through all the angles, a slightly different form
based on coefficients A (24a) and B (24b) is used for a range of
small induced angles:

� � � �i i i i A B� � � � � �0 5 1 3 0 5 0 65. : . . .exp cor (25)

6 Integral factors
The calculated experimental induced angles �iexp can also

be used to express more precisely the integral factors E and F
of Lock’s decomposition equations (11) and (12). Such modi-
fied factors are necessary for more accurate calculations of
the thrust and power (torque) coefficients directly with use of
Lock’s decomposition equations (17) and (18). The integral
factors were explicitly derived from decomposition equations
(11) and (12):

E
s c F c

c
l P

T
�

�2
2

0 7� �

� �
. sin( )

cos( )
(26)

� �
� �

F
s c c

c
d l

P
�

�

�

2 0 7� �
� � �
. ( )

cos( ) sin( ) ( )
tg

tg
. (27)

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 105

Acta Polytechnica Vol. 44  No. 2/2004

s0.7
[1]

t0 9.
[%]

rmax
[1]

�0.7
[°]

M Ref.

M 337 0.0620 10.5 0.313 14.60 0.45 0.5 0.55 [3]

M60-180 0.0568 13.4 0.355 18.77 0.5 0.6 0.7 [3]

M60-130 0.0606 13.5 0.320 15.74 0.5 0.6 0.7 [3]

M30-011 0.0576 11.7 0.365 12.66 0.5 0.6 0.7 [3]

R503-2V 0.0502 13.8 0.300 10.59 0.5 0.6 0.7 [3]

M30-04A 0.0605 12.0 0.335 11.53 0.5 0.6 0.7 [3]

N5868-15 0.0596 8.3 0.500 16.06 0.46 [4]

N5868-25 0.0596 8.3 0.500 21.06 0.46 [4]

N3647-15 0.0888 8.3 0.500 16.06 0.46 [4]

N3647-25 0.0888 8.3 0.500 21.06 0.46 [4]

VR 411 0.0947 6.4 0.680 9.50 0.6 [5]

Table 1: Set of experimental propellers



The numerical dependences of the two factors on the
blade geometry and flight regime were obtained by using the
set of experimental induced angles �iexp in the expressions (6)
for angles � and � and introducing these angles with the cor-
responding experimental values of thrust and power coeffi-
cients into equations (26) and (27). The numerical results
were smoothed by the following functions:

E a b cE E E� � �� �
2 (28)

F
a b cF F F

�
� �

1
� �( )

(29)

with parameters describing the influence of the blade geome-
try and also partly the flight regime:

a b c

a b
E E E

F

� � � � �

� �

0 565 0 0825 0 0375

0 639 18189 0 7

. , . , .

. . ,. b F b a

F b
F m F r

m

� �

� � � �

0 965

0 393 0 9731 0 027
0 7

0 7 0 7

. ,

. . .
.

. .

�

� 0 414

0182 0 0234

0 0475 10777
0 9

0 7

.

. .

. .
max .

.

�

� �

m

m

r t

b

�

� �

� � 0 7

0 7

01

0165

0

9 8676 2

.

.

. ,

.

:

: ( . .

�

� �

� �

� � �

� �

� �

� �

r m

r F

r F

c

c b 542

1 9144 1

2

0 7
3

)( )

( . )( ).

� �

� �

� �

� � �

r

rb

(29a)

The analysis confirmed the independence of the E factor
from the propeller geometry in acceptance with Lock’s origi-
nal model.

7 Number of blades
Lock’s scheme considers the linear dependence on the

number of blades with the use of the solidity factor (10) both
in the tip loss factor (22) and in the relations for thrust (17)
and power (18) coefficients derived from the decomposition
equations. The linear model gives thrust and power coeffi-
cients that are higher than they are in reality, and the propel-
ler propulsive efficiency does not depend on the number of
blades.

To preserve the simplicity of the developed procedure for
two-blade propellers, an initial correction of the linear model
was designed on the basis of the evolution of experimental
thrust and power coefficients. By comparing different blade
propellernumber having the same blade geometry [4], it was
found that the mean value of the rate between the thrust
(power) per blade of a two-blade propeller and a z-blade pro-
peller systematically increases from 1 (z � 2) to higher values
( z > 2). The analytical expressions of the mean thrust KT and
power KP ratio are as follows:

K
c z

c z z

z z

T
T

T
�

�
� �

� � � ��

( )
( )

.

. . .

2 2
0 837

0 08583 1 5 10 3 3333 2 33 10 4 3� � z

(30)

K
c z

c z z

z z

P
P

P
�

�
� �

� � � ��

( )
( )

.

. .

2 2
0 764

016533 27 10 016663 2 6 10 4 3� � z .

(31)

The ratio KT(z) and KP(z) can therefore be used as conver-
sion factors between two-blade and z-blade propeller thrust
and power coefficients:

c z
c z z

K
T

T

T
( )

( )
�

� 	2
2

(32)

c z
c z z

K
P

P

P
( )

( )
�

� 	2
2

. (33)

In order to ensure the correct internal calculation of a
two-blade propeller even if the right solidity factor (10) of the
z-blade propeller is given, an effective solidity factor must be
considered during the calculation:

s s
z0 7 0 7
2

. .ef � (34)

8 Calculation procedure
1) Geometric input data:

referential section
(chord, setting angle) – b R0 7. [1], �0.7 [°]

relative thickness – ( ). .t b0 9 0 9 100 [%]
position of the max. blade chord – r Rmax [1]
number of blades – z

2) Flight regime input:
advance ratio – � [1]

3) Calculation of the effective solidity factor – (34)
4) Solution of the induced angle – the root of the transcen-

dent induced equation (22) with the modified loss factor
(23), (blade lift curve (19) is required)

5) Linear correction of the induced angle – (24) and (25)
6) Calculation of the modified integral factors E and F – (28)

and (29)
7) Calculation of the thrust and power coefficient with the ef-

fective solidity factor – (17) and (18), (blade lift and drag
curves (19) and (20) are required)

8) Conversion of the gained thrust and power coefficients by
means of the KT and KP factors with respect to the number
of blades – (32) and (33)

9) Calculation of the propulsive efficiency: � �� ( )c cT P [1]
(or � � 0 8 3 2. ( )c cT P in case of � � 0)

9 Validity
It was proved by systematic reversal calculations of the

experimental propeller set, Table 1, that the maximum rela-
tive error of both thrust and power coefficients is less than
10 % and the mean error is about 5 %. These differences
are valid from the start regime up to flight regimes with
maximum propeller efficiency. The range of geometric pa-
rameters that ensures a relative error limit of 10 % can
therefore be directly estimated from the data in Table 1:
blade width b R0 7 0 09 0 22. ( . . )� � ,
blade angle setting j0 7 9 23. ( )� � 
,
maximum blade width r Rmax ( . . )� �0 3 0 7 and
airfoil thickness ( ) ( ). .t b0 9 0 9 100 6 14� � %.

The tip Mach number should not exceed 0.75. All analy-
ses were performed with RAF-6 blade airfoil propellers.

106 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 2/2004



10 Examples
The first example presents calculations of the two-blade

propeller VLU 001 [6] with upper geometric limits of the
blade chord and angle setting. The tip Mach number
M � 0.45. Numerical results are compared with experimental
values. The input geometric parameters are as follows:
� blade angle setting at 70 % of the propeller diameter –

�0.7 � 24.4 [°],
� relative chord of the blade at 70 % of the propeller diame-

ter – b0.7/R � 0.227 [1],

� relative position of the maximum blade width –
rmax/R � 0.535 [1],

� relative airfoil thickness at 90 % of the propeller diameter –
(t0.9/b0.9) 100 � 8.5 [%].

The thrust and power coefficients are shown in Fig. 3. The
propeller efficiency is presented in Fig. 4. The relative error of
the power coefficient cP reached about 10 %. The thrust coef-
ficient gives better results.

The second example shows the possibilities of a para-
metric study. Propeller efficiency in a static regime ( � � 0,
non-forward movable propeller V � 0 – see Fig. 1) defined as
� � 0 8 3 2. ( )c cT P is calculated for the case a two-blade propel-

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 107

Acta Polytechnica Vol. 44  No. 2/2004

Fig. 4: Efficiency of two-blade propeller VLU 001

Fig. 6: Efficiency of a two-blade propeller at � � 0.2 with different
geometric parameters

Fig. 3: Thrust and power coefficients of two-blade propeller
VLU 001

Fig. 5: Efficiency of a two-blade propeller at � � 0 with different
geometric parameters



ler with fixed r Rmax , two thickness parameters ( ). .t b0 9 0 9
and four b R0 7. . The results are plotted in Fig. 5 as a function
�0.7. Fig. 6 depicts the propulsive efficiency of the same pro-
peller at the advance ratio � � 0.2.

11 Conclusion
The published method presents a simple and quick calcu-

lation procedure for thrust and power propeller coefficients
based on Lock’s 2D scheme of the referential section. The nu-
merical demands are restricted to the solution of a non-linear
algebraic equation to obtain the induced angle. The thrust
and power coefficients are consequently calculated directly by
explicit analytical algebraic formulae.

The aerodynamics characteristics of the propeller are
obtained with an acceptable error for preliminary aircraft
performance analyses: the maximum relative deviation of
both the thrust and the power coefficient does not exceed
10 % from the start regime up to flight regimes with maxi-
mum propeller efficiency. The mean error is about 5 %. The
range of blade geometric parameters was set to keep the
calculations within this error limit.

In addition to applications in the small aeroplane industry
the presented method is also suitable for student study
projects at technical universities with aerospace study
programmes. Parametrical input data of the propeller blade
geometry and the number of blades enables easy studies of
the influence of propeller geometry on the aerodynamic
characteristics.

This procedure can be further enhanced by considering
the tip Mach number effect, the aerodynamics of a blade air-
foils and by a more detailed analysis of the influence of blade
numbers. Systematic use of FEMs (e.g. FLUENT) can supply
the experimental basis.

References
[1] Lock C. N. H.: A Graphical Method of Calculating the Perfor-

mance of an Airscrew. British A. R .C. Report and Memo-
randa 1675, 1935.

[2] Bull G., Bennett G.: Propulsive Efficiency and Aircraft Drag
Determined from Steady State Flight Test Data. Mississippi
State University, Dep. of Aerospace Engineering. Society
of Automotive Engineers, 1985.

[3] Slavík S., Theiner R.: Měření aerodynamických charakteristik
vrtulí pro motor M-60 a M-30 v tunelu VZLÚ � 3 m. Czech
Technical University in Prague - Faculty of Mechanical
Engineering, Department of Aerospace Engineering,
Prague 1989.

[4] Hartman E. P., Biermann D.: The Aerodynamic Character-
istics of Full Scale Propellers Having 2, 3, and 4 Blades of
Clark Y and R.A.F. Airfoil Sections. Technical report
No. 640, N. A. C. A., 1938.

[5] Hacura E. P.: Charakteristiky vrtule VR 411 - 4003. Aero-
nautical Research and Test Institute, Prague, 1952.

[6] Hacura E. P., Biermann D.: Zkoušky rodiny vrtulí R & M
829 v aerodynamickém tunelu � 3 m. Aeronautical Re-
search and Test Institute, Prague, 1949.

Doc. Ing. Svatomír Slavík, CSc.
phone: +420 224 357 227, +420 224 359 216
e-mail: svatomir.slavik@fs.cvut.cz

Department of Automotive and Aerospace Engineering

Czech Technical University in Prague
Faculty of Mechanical Engineering
Karlovo náměstí 13
121 35 Prague 2, Czech Republic

108 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 2/2004


	Table of Contents
	Biological Systems Thinking for Control Engineering Design 3
	D. J. Murray-Smith

	Computational Fluid Dynamic Simulation (CFD) and Experimental Study on Wing-external Store Aerodynamic Interference of a Subsonic Fighter Aircraft 9
	Tholudin Mat Lazim, Shabudin Mat, Huong Yu Saint

	Dynamics of Micro-Air-Vehicle with Flapping Wings 15
	K. Sibilski
	The Role of CAD in Enterprise Integration Process 22
	M. Ota, I. Jelínek


	Development of a Technique and Method of Testing Aircraft Models with Turboprop Engine Simulators in a Small-scale Wind Tunnel – Results of Tests 27
	A. V. Petrov, Y. G. Stepanov, M. V. Shmakov

	Developing a Conceptual Design Engineering Toolbox and its Tools 32
	R. W. Vroom, E. J. J. van Breemen, W. F. van der Vegte
	Knowledge Support of Simulation Model Reuse 39
	M. Valášek, P. Steinbauer, Z. Šika, Z. Zdráhal

	The Effect of Pedestrian Traffic on the Dynamic Behavior of Footbridges 47
	M. Studnièková


	Control of Systems of Reservoirs with the Use of Risk Analysis 52
	P. Fošumpaur, L. Satrapa

	A coding and On-Line Transmitting System 56
	V. Zagursky, I. Zarumba, A. Riekstinsh
	Speech Signal Recovery in Communication Networks 59
	V. Zagursky, A. Riekstinsh


	Simulation of Scoliosis Treatment Using a Brace 62
	J. Èulík

	Image Analysis of Eccentric Photorefraction 68
	J. Dušek, M. Dostálek

	A Novel Approach to Power Circuit Breaker Design for Replacement of SF6 72
	D. J. Telfer, J. W. Spencer, G. R. Jones, J. E. Humphries
	Numerical Analysis of the Temperature Field in Luminaires 77
	J. Murín, M. Kropáè, R. Fric

	Computer Aided Design of Transformer Station Grounding System Using CDEGS Software 83
	S. Nikolovski, T. Bariæ


	Recycling and Networking 90
	T. Bányai
	Response of a Light Aircraft Under Gust Loads 97
	P. Chudý

	Preliminary Determination of Propeller Aerodynamic Characteristics for Small Aeroplanes 103
	S. Slavík