# Effects of the airfoil section, the chord and pitch distributions on the aerodynamic performance of the propeller

Technical Paper

## Keywords

Small propeller Momentum theory Blade element theory Panel method Blade aerodynamics Airfoil section

## List of symbols

$$a$$

Inflow factor

a0

Lift curve slope at zero Mach number (i.e., in incompressible flow) (radians−1)

aM

Lift curve slope at zero Mach number (radians−1)

$$b$$

Swirl factor

$$B$$

Number of blades of the propeller

$$c$$

Local blade chord (m)

$$C_{\text{d}}$$

Two-dimensional drag coefficient of the local blade chord

$$C_{\text{l}}$$

Two-dimensional lift coefficient of the local blade chord

$$D$$

Diameter of the propeller (m)

$$f_{\text{tip}}$$

Tip loss correction used to calculate Prandtl loss factor F

$$f_{\text{hub}}$$

Hub loss correction used to calculate Prandtl loss factor F

$$F$$

Prandtl loss factor for combined tip and hub losses which arise due to the finite number of the propeller blades

$$J$$

Advance ratio of the propeller $$J = {V \mathord{\left/ {\vphantom {V {(n{\kern 1pt} D)}}} \right. \kern-0pt} {(n{\kern 1pt} D)}}$$

$$k_{\text{P}}$$

Power coefficient of the propeller $$k_{\text{P}} = {P \mathord{\left/ {\vphantom {P {(\rho {\kern 1pt} n^{3} D^{5} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{3} D^{5} )}}$$

$$k_{\text{Q}}$$

Torque coefficient of the propeller $$k_{\text{Q}} = {Q \mathord{\left/ {\vphantom {Q {(\rho {\kern 1pt} n^{2} D^{5} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{2} D^{5} )}}$$

$$k_{\text{T}}$$

Thrust coefficient of the propeller $$k_{\text{T}} = {T \mathord{\left/ {\vphantom {T {(\rho {\kern 1pt} n^{2} D^{4} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{2} D^{4} )}}$$

M

Local Mach number of the relative flow

$$n$$

Rotational speed of the propeller (rps)

$$N$$

Rotational speed of the propeller (rpm)

$$p$$

Geometric pitch of the blade section (m)

$$P$$

Power supplied at the propeller axis (Nm/s)

$$Q$$

Torque applied on the propeller (Nm)

$$r$$

Radius of the transversal section of the blade of the propeller (m)

$$R$$

Radius of the blade tip of the propeller (m)

$$Re_{75}$$

Reynolds number of the propeller based on the local chord and resultant velocity at a radial distance of 0.75 of the tip radius

$$T$$

Thrust force of the propeller (N)

$$V$$

Advance velocity of the propeller (m/s)

$$V_{0}$$

Axial component of the flow velocity relative to the blade (m/s)

$$V_{\text{R}}$$

Resultant flow velocity relative to the blade (m/s)

$$V_{\text{S}}$$

Axial component of the flow velocity relative to the propeller at exit of the slipstream (m/s)

$$V_{\text{w}}$$

Rotational component of the flow velocity relative to blade (m/s)

$$\alpha$$

Angle of attack is the angle between the resultant velocity vector $$V_{\text{R}}$$ and the zero lift line of the blade airfoil (radians)

$$\alpha_{\text{c}}$$

Angle between the resultant velocity vector $$V_{\text{R}}$$ and the chord line of the blade airfoil (radians)

$$\delta {\kern 1pt} k_{\text{Q}}$$

$$\delta {\kern 1pt} k_{\text{T}}$$

$$\eta$$

Efficiency of the propeller

$$\theta_{\text{c}}$$

Pitch angle of the blade section (radians)

$$\lambda$$

Taper ratio of the propeller blade

$$\rho$$

Specific mass of the fluid (air) (kg/m3)

$$\sigma$$

Solidity of the rotor

$$\phi$$

Angle of the resultant velocity $$V_{\text{R}}$$ with the plane of rotation of the propeller (radians)

## References

1. 1.
Theodorsen T (1948) Theory of propellers. McGraw-Hill, New York, pp 6–15Google Scholar
2. 2.
Dumitrescu H, Cardos V (1998) Wind turbine aerodynamic performance by lifting line method. Int J Rot Mach 4(3):141–149.
3. 3.
Palmiter SM, Katz J (2010) Evaluation of a potential flow model for propeller and wind turbine design. J Aircr 47(5):1739–1746
4. 4.
Slavík S (2004) Preliminary determination of propeller aerodynamic characteristics for small aeroplanes. Acta Polytech 44(2):103–108Google Scholar
5. 5.
Gur O, Rosen A (2008) Comparison between blade-element models of propellers. Aeronaut J 112(1138):689–704
6. 6.
Uhlig DV, Selig MS (2008) Post stall propeller behavior at low Reynolds numbers. In: 46th AIAA Aerospace Sciences Meeting and Exhibit. 2008-0407Google Scholar
7. 7.
Bohorquez F, Pines D, Samuel PD (2010) Small rotor design optimization using blade element momentum theory and hover tests. J Aircr 47(1):268–283.
8. 8.
Khan W, Nahon M (2015) Development and validation of a propeller slipstream model for unmanned aerial vehicles. J Aircr. (AIAA Early Edition)
9. 9.
Drela M (2013) XFOIL Subsonic airfoil development system, XFOIL 6.99. Massachusetts Institute of Technology, Cambridge, MA, USA. http://web.mit.edu/drela/Public/web/xfoil/. Accessed 12 April 2016
10. 10.
Morgado J (2016) Development of an open source software tool for propeller design in the MAATProject. University of Beira Interior, PhD Thesis, MarchGoogle Scholar
11. 11.
Silvestre MAR, Morgado J, Páscoa JC (2013) JBLADE: a propeller design and analysis code. In: 2013 International powered lift conference. American Institute of Aeronautics and Astronautics, Los Angeles, CA, USA.
12. 12.
Morgado J, Abdollahzadeh M, Silvestre MAR, Páscoa JC (2015) High altitude propeller design and analysis. Aerosp Sci Technol 45:398–407.
13. 13.
MacNeill R, Verstraete D (2017) Blade element momentum theory extended to model low Reynolds number propeller performance. Aeronaut J 121(1240):835–857.
14. 14.
Wald QR (2006) The aerodynamics of propeller. Prog Aerosp Sci 42(2):85–128
15. 15.
Glauert H (1926) The elements of aerofoil and airscrew theory. Cambridge University Press, Cambridge, pp 199–221
16. 16.
Houghton EL, Carpenter PW, Collicott SH, Valentine DT (2013) Aerodynamics for engineering students, 6th edn. Elsevier, Amsterdam, pp 643–687Google Scholar
17. 17.
Wald QR (1964) The distribution of circulation on propellers with finite hubs. ASME Paper 64WA/UNT-4, Winter Annual Meeting, New YorkGoogle Scholar
18. 18.
Glauert H (1935) Airplane propellers, Vol IV, Div L, Chap VII. In: Durand WF (ed) Aerodynamic theory. Julius Springer, Berlin [Reprinted 1963 by Dover Publications, Inc., New York], pp 251–269Google Scholar
19. 19.
Hartman EP, Biermann D (1938) The aerodynamic characteristics full-scale propellers having 2, 3 and 4 blades of Clark Y and RAF 6 airfoil sections. NACA Report No. 640. Langley Memorial Aeronautical Laboratory, National Advisory Committee for Aeronautics, Langley Field, VA, USAGoogle Scholar
20. 20.
Lyon CA, Broeren AP, Giguère P, Gopalarathnam A, Selig MS (1998) Summary of low-speed airfoil data, vol 3. Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, SoarTech Publications, Virginia Beach, VA, USAGoogle Scholar
21. 21.
Glauert H (1924) A generalised type of Joukowski aerofoil. ARC RM No. 911Google Scholar

© The Brazilian Society of Mechanical Sciences and Engineering 2019

## Authors and Affiliations

• Kamal A. R. Ismail
• 1
Email author
• Célia V. A. G. Rosolen
• 1
1. 1.Department of Energy, School of Mechanical EngineeringUniversity of CampinasCampinasBrazil