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Analysis of non-symmetrical flapping airfoils

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Abstract

Simulations have been done to assess the lift, thrust and propulsive efficiency of different types of non-symmetrical airfoils under different flapping configurations. The variables involved are reduced frequency, Strouhal number, pitch amplitude and phase angle. In order to analyze the variables more efficiently, the design of experiments using the response surface methodology is applied. Results show that both the variables and shape of the airfoil have a profound effect on the lift, thrust, and efficiency. By using non-symmetrical airfoils, average lift coefficient as high as 2.23 can be obtained. The average thrust coefficient and efficiency also reach high values of 2.53 and 0.61, respectively. The lift production is highly dependent on the airfoil’s shape while thrust production is influenced more heavily by the variables. Efficiency falls somewhere in between. Two-factor interactions are found to exist among the variables. This shows that it is not sufficient to analyze each variable individually. Vorticity diagrams are analyzed to explain the results obtained. Overall, the S1020 airfoil is able to provide relatively good efficiency and at the same time generate high thrust and lift force. These results aid in the design of a better ornithopter’s wing.

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Abbreviations

c :

airfoil chord

C d :

drag coefficient

C l :

lift coefficient

\({\bar {C}_l }\) :

average lift coefficient

C p :

pressure coefficient

C t :

thrust coefficient

\({\bar {C}_t }\) :

average thrust coefficient

f :

frequency, Hz

h :

instantaneous heaving position

\({h'_0 }\) :

heaving amplitude

h 0 :

heaving amplitude, non-dimensionalized by airfoil chord

k :

reduced frequency f c/U ∞

L :

lift force

M :

moment created by the lift and drag forces at the pitching axis

p :

pressure

P :

power input

\({\bar {P}}\) :

average power input

p s :

statistical value calculated by Minitab to determine if the variable or interaction is significant

Re :

Reynolds number

St :

Strouhal number \({fh'_0/U_\infty }\)

t :

non-dimensionalized time

t 0 :

time when flapping starts

T :

thrust force

u b :

grid velocity

u i :

Cartesian velocity components

U ∞ :

freestream velocity

x i :

Cartesian coordinates

θ :

instantaneous pitch angle, in degrees

\({\phi }\) :

phase difference between pitching and heaving, in degrees

θ 0 :

pitch amplitude, in degrees

η :

propulsive efficiency

References

  1. Shyy W., Lian Y., Tang J., Liu H., Trizila P., Stanford B., Bernal L., Cesnik C., Friedmann P., Ifju P.: Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Acta Mech. Sinica 24, 351–373 (2008). doi:10.1007/s10409-008-0164-z

    Article  Google Scholar 

  2. Koochesfahani M.M.: Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27, 1200–1205 (1989)

    Article  Google Scholar 

  3. Triantafyllou G.S., Triantafyllou M.S., Grosenbaugh M.A.: Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205–224 (1993)

    Article  Google Scholar 

  4. Taylor G.K., Nudds R.L., Thomas A.L.R.: Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425, 707–711 (2003). doi:10.1038/nature02000

    Article  Google Scholar 

  5. Read D.A., Hover F.S., Triantafyllou M.S.: Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17, 163–183 (2003)

    Article  Google Scholar 

  6. Hover, F.S., Haugsdal, O., Triantafyllou, M.S.: Effect of angle of attack profiles in flapping foil propulsion. J. Fluids Struct. 19, 37–47 (2004). doi: 10.1016/j.jfluidstructs.2003.10.003

    Article  Google Scholar 

  7. Ramamurti, R., Sandberg, W.: Simulation of flow about flapping airfoils using finite element incompressible flow solver. In: AIAA 37th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, pp. 253–260 (1999)

  8. Wu J.H., Sun M.: The influence of the wake of a flapping wing on the production of aerodynamic forces. Acta Mech. Sinica 21, 411–418 (2005). doi:10.1007/s10409-005-0064-4

    Article  MathSciNet  Google Scholar 

  9. Aono H., Shyy W., Liu H.: Near wake vortex dynamics of a hovering hawkmoth. Acta Mech. Sinica 25(1), 25–36 (2009)

    Article  Google Scholar 

  10. Pedro G., Suleman A., Djilali N.: A numerical study of the propulsive efficiency of a flapping hydrofoil. Int. J. Numer. Methods Fluids 42, 493–526 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tuncer, I.H., Kaya, M.: Optimization of flapping airfoils for maximum thrust and propulsive efficiency. In: AIAA 41st Aerospace Sciences Meeting, Reno, NV, pp. 2329–2336 (2003)

  12. Mathews, P.: Design of Experiments with MINITAB (2005)

  13. Hirt C.W., Amsden A.A., Cook J.L.: An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 135, 203–216 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kim D., Choi H.: A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids. J. Comput. Phys. 162, 411–428 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tay, W.B.: Analysis of non-symmetrical flapping airfoils and their configurations. Ph.D. thesis, National University of Singapore, Singapore (2009)

  16. Balay, S., Buschelman, K., Eijkhout, V., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc 2.0 Users Manual: Revision 2.1.5. United States (2004)

  17. Falgout, R.D., Jones, J.E., Yang, U.M.: The design and implementation of hypre, a library of parallel high performance preconditioners. In: Numerical Solution of Partial Differential Equations on Parallel Computers, vol. 51, pp. 267–294. Springer, Berlin (2006)

  18. Pauley L.L., Moin P., Reynolds W.C.: The structure of 2-dimensional separation. J. Fluid Mech. 220, 397–411 (1990)

    Article  Google Scholar 

  19. Young, J.: Numerical simulation of the unsteady aerodynamics of flapping airfoils. Ph.D. thesis, New South Wales University, Canberra (2005)

  20. Tuncer I.H., Kaya M.: Thrust generation caused by flapping airfoils in a biplane configuration. J. Aircr. 40, 509–515 (2003)

    Article  Google Scholar 

  21. Lai J.C.S., Platzer M.F.: Jet characteristics of a plunging airfoil. AIAA J. 37, 1529–1537 (1999)

    Article  Google Scholar 

  22. Mueller T.J., DeLaurier J.D.: Aerodynamics of small vehicles. Ann. Rev. Fluid Mech. 35, 89–111 (2003). doi:10.1146/annurev.fluid.35.101101.161102

    Article  Google Scholar 

  23. Videler J.J., Stamhuis E.J., Povel G.D.E.: Leading-edge vortex lifts swifts. Science 306, 1960–1962 (2004). doi:10.1126/science.1104682

    Article  Google Scholar 

  24. Box G., Behnken D.: Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475 (1960)

    Article  MathSciNet  Google Scholar 

  25. Jones, K.D., Castro, B.M., Mahmoud, O., Pollard, S.J., Platzer, M.F., Neef, M., Gonet, K., Hummel, D.: A collaborative numerical and experimental investigation of flapping-wing propulsion. In: AIAA 40th Aerospace Sciences Meeting, Reno, Nevada (2002)

  26. Dallal, G.E.: The Little Handbook of Statistical Practice. http://www.tufts.edu/~gdallal/LHSP.HTM. Accessed 20 March 2007

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  1. Department of Mechanical Engineering, National University of Singapore, 21 Lower Kent Ridge Road, Singapore, 119077, Singapore

    W. B. Tay & K. B. Lim

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  1. W. B. Tay
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Tay, W.B., Lim, K.B. Analysis of non-symmetrical flapping airfoils. Acta Mech Sin 25, 433–450 (2009). https://doi.org/10.1007/s10409-009-0259-1

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