Comparison of Methods for Computation of Ideal Circulation

Comparison of Methods for Computation of Ideal Circulation Distribution on the Propeller

Ing. Jan Klesa

vedoucí práce: doc. Ing. Daniel Hanus, CSc.

Abstrakt Ideální rozložení cirkulace na vrtuli je vypočteno za předpokladu platnosti Betzovy podmínky pro rozložení indukované rychlosti (podmínka pro práci vrtule s maximální účinností odvozená A. Betzem). Pro výpočet bylo použito několik metod pro návrh vrtule. Je uveden jejich popis a porovnání jimi vypočtených průběhů ideálního rozložení cirkulace. V závěru je zhodnocena použitelnost jednotlivých metod pro návrh vrtulí. Klíčová slova Aerodynamika vrtulí, rozložení cirkulace, návrh vrtule, Betzova podmínka, Biot-Savartův zákon . Abstract Ideal circulation distribution on the propeller is computed assuming validity of Betz condition for induced velocity distribution (condition for propeller working with maximum possible efficiency derived by A. Betz ). Several methods for design of propellers were used for computing of ideal circulation distribution. These methods are described and computed ideal circulation distributions are compared. Possibilities of application to the propeller design are discussed. Keywords Propeller aerodynamics, circulation distribution, propeller design, Biot-Savart law.

1. Introduction Propellers are the most important mean of propulsion of airplanes flying at low speeds. It converts engine power to thrust. So it is impossible to reach high efficiency of the propulsion system without propeller working with high efficiency. Circulation distribution along propeller blades has great influence on

2. Propeller Working with Maximum Possible Efficiency The condition for propeller working with maximum possible efficiency was derived by A. Betz (see [1]). This condition says that if the propeller should work with maximum possible efficiency than the vortex system generated by the propeller should move downwards like a rigid body. Physically this condition represents the request for accelerating of the airflow with minimum energy consumption (i.e. propeller working with minimum induced loss). The solution of this problem leads to the simple result mentioned above.

3. Larrabee’s Method

E. E. Larrabee developed in 1970s quite simple method for design of propellers with maximum possible efficiency (see [2]). This method is based on the work of A. Betz [1] and uses Prandtl’s loss function described by L. Prandtl in the appendix to [1]. Small angles assumption is used in his solution. The ideal distribution of the dimensionless circulation can be expressed by (1).

2

22

2

2

ππ I

r

r

B

IF

w +⋅=Γ

(1)

Where F is Prandtl’s loss function defined by (2) and (3).

( )feF −⋅= arccos2

π (2)

( )rI

IBf −+⋅= 1

2

22 π (3)

The method itself is very simple and very fast. Larrabee himself programmed it on the programmable pocket calculator in 1970s. It is intended to be used only for the propellers with light loading. It was used for the design of the propellers for human-powered airplanes Gossamer Albatross and Gossamer Condor (see fig. 1 to 3).

Fig. 1. Gossamer Albatross during in flight (NASA photo collection).

Fig. 2. Gossamer Albatross on ramp with crew (NASA photo collection).

Fig. 3. Gossamer Albatross on lakebed, propeller shape is clearly visible on this photo

(NASA photo collection).

4. Method Developed by Adkins and Liebeck The Larrabee’s computational method was further developed by C. N. Adkins and R. H. Liebeck (see [3]). The limitation for slightly loaded propellers was removed and no small angles assumption is used. This method could be used for the propellers with high loading. But it is more complicated than the original one. Iteration scheme has to be used in this case (Larrabee’s method uses straight computation without any iteration). The ideal circulation distribution computed by this method can be expressed by (4).

Φ⋅Φ⋅=Γsincos

2B

Fr

w (4)

In this case the inflow angle Φ is the function of velocity of the translation of the vortex system w . Comparison of the results of both above mentioned methods are compared in fig.

4. Different computation of Prandtl’s loss function was used in [3] – modified formula can be seen in (5). However it gives practically the same results as (3).

( )

t

rBf

Φ−⋅=

sin

1

2 (5)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

r [1]

Γ [1

]w=0w=0.1w=0.2w=0.3w=0.5

Fig. 4. Comparison of the ideal circulation distribution on the propeller for I = π/5

computed by the method of Adkins/Liebeck for different values of the parameterw (for 0=w the results are identical to Larrabee).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

r [1]

Γ [1

]

w=0w=0.1w=0.2w=0.3w=0.5

Fig. 5. Comparison of the ideal circulation distribution on the propeller for I = π computed

by the method of Adkins/Liebeck for different values of the parameterw (for 0=w the results are identical to Larrabee).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

r [1]

Γ [1

]

w=0w=0.1w=0.2w=0.3w=0.5

Fig. 6. Comparison of the ideal circulation distribution on the propeller for I = 4π computed

by the method of Adkins/Liebeck for different values of the parameterw (for 0=w the results are identical to Larrabee).

5. Goldstein’s Method S. Goldstein computed ideal circulation distribution according to the Betz’s condition (see [4]). His solution modified to the dimensionless form that is comparable with other ones can be seen in (6). For further understanding to this formula and its nomenclature see [4] – the solution itself and its structure is quite complicated.

( )

( )

+

+

+−

+−

+⋅=Γ

∑ ∑∞

=

∞

=

+

++

0 0

0

2

1

2

1

20

20

2

12,

22

2

2

1

2

1

1

2

12

8

12 m m

mB

mB

mmB

BmI

BmI

Am

F

B

I

wµ

µ

µµ

πµ

πµµ

π (6)

The tables of Goldstein solution from [5] are used for comparison with other methods.

6. Numerical Solution of the Vortex Model of the Propeller The numerical solution of the velocity induced according to the Biot-Savart law by the system of helical vortices was done. General formulation of the Biot-Savart law for fluid mechanics can be seen in (7).

34 r

rsdvd i

×⋅Γ=π

(7)

Modification of this solution for axial component is in (8) for velocity induced at radial coordinate r from helicoidal vortex filament at radial coordinate 1r . Numerical integration of this expression was done in order to compute induced velocity

zd

zI

zrrrr

I

zrrr

Ivd iz ⋅

+

+−+

+−⋅Γ=

2

3

21

221

11

cos2

cos

σπ

σπ

π

(8)

where

B

i 12

−= πσ (9)

Axial component of induced velocity viz is related to the velocity of the translation of vortex system w by (10).

Φ

=2cosizv

w (10)

If matrixes are used for the computation then the relation between circulation and dimensionless velocity of the translation of the vortex system can be expressed by (11). kjkj aw Γ⋅= (11)

If the matrix inversion is made, than following relation is obtained. kjkj wa ⋅=Γ −1 (12)

Ideal circulation distribution is computed like the distribution of ΓΓΓΓ for constant w along the blade length.

7. Comparison Computed ideal circulation distributions for propellers with two blades are compared in the following figures for various advance ratio I . Values of the advance ratio I go from π/5 to 4π. Results of Larrabee's, Goldstein's and numerical solutions are compared in fig. 7 to 11. Comparison of Larrabee's solution and the results of the method developed by Adkins and Liebeck can be seen in chapter 4 in fig. 4 to 6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

r [1]

Γ [1

]

Larrabeenumerical vortex methodGoldstein

Fig. 7. Comparison of ideal circulation distribution for I = π/5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

r [1]

Γ [1

]



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

r [1]

Γ [1

]



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

r [1]

Γ [1

]


Fig. 10. Comparison of ideal circulation distribution for I = π.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

r [1]

Γ [1

]


Fig. 11. Comparison of ideal circulation distribution for I = 4π.

8. Conclusion Ideal circulation distribution for propeller with two blades are compared. It is clearly visible that the distribution of circulation on the propeller blade from Goldstein's solution is identical to the distribution computed by the numerical solution of Biot-Savart law. These both solutions can be considered correct solution of the problem (if we assume inviscid incompressible flow). With increasing advance ratio I the difference between Larrabee's and Goldstein's solution increases. So Larrabee's method can be successfully used for design of propellers with low advance ratio when it gives almost identical results as more complicated method (like in the text mentioned case of design propellers for man-powered aircraft, when you can profit from the simplicity of the calculation). However Larrabee's method (and also method developed by Adkins and Liebeck) gives quite inaccurate results for higher values of advance ratio and its use is not advisable. Goldstein's method should be used instead in these cases. Nomenclature

jka components of matrix A [1] 1−

jka components of inverted matrix A [1]

D propeller diameter [m] ds element of vortex filament [m] B number of blades of the propeller [1] F Prandtl’s loss function [1] f argument of the Prandtl’s loss function [1]

I advance ratio Dn

VI

s

= [1]

ns propeller revolution per second [s-1] R propeller radius [m] r position vector [m] r dimensionless radius [1] vi induced velocity [m·s-1] viz induced velocity in the direction of the propeller axis [m·s-1] w translation velocity of the propeller vortex system [m·s-1] w dimensionless translation velocity of the propeller vortex system [1] Γ circulation [m2·s-1]

Γ dimensionless circulation Ω

Γ=Γ24 Rπ

[1]

Φ flow angle [rad] Φt flow angle at the blade tip [rad] Ω propeller angular velocity [s-1] References [1] Betz, A. with Appendix by L. Prandtl, Schraubenpropeller mit Geringstem

Energieverlust, Göttinger Nachrichten, Göttingen, 1919, pp 193–217. [2] Larrabee, E. E., Practical design of minimum induced loss propellers, SAE Technical

Paper 790585, 1979. [3] Adkins, C. N., Liebeck, R. H., Design of optimum propellers, Journal of Propulsion

and Power, vol. 10 (1994), no. 5, pp 676–682. [4] Goldstein, S., On the vortex theory of screw propellers, Proc R Soc London A, 1929,

123, pp. 440–465.

[5] Wald, Q. R., The aerodynamics of propellers, Progress in Aerospace Sciences 42 (2006), pp 85–128, doi:10.1016/j.paerosci.2006.04.001

[6] Alexandrov, V. L., Letecké vrtule, SNTL, Praha, 1954 [7] www.nasa.gov

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Comparison of Methods for Computation of Ideal Circulation