Analysis Problem

Consider a wing with elliptic planform, of span b and chord c(y) = c0A/1 – (2 y/b)2. The wing has constant relative camber d(y)/c(y) = d/c = const and has washout given by t(y) = -4tx(2y/b)2, where -4tx is the twist at the wing tips.

Calculate the coefficients A1, A2,… ,An, n = 1, to. Hint: Write Prandtl integro – differential equation, where the circulation in the l. h.s. is represented by the Fourier series, and the wing geometry is introduced in the r. h.s. Use the change of variables y = -§ cos t, 0 < t < n and the identity 4cos21 sin t = sin t + sin 3t.

From the equation for A1 give the equation of the lift curve CL(a). What is the incidence of zero lift, a0?

Calculate the induced drag if a = 3tx – 2d/c.

6.14.1 Design Problem

Considering the lifting problem, design a rectangular wing of span b and chord c, (AR = b/c), with zero twist, and equipped with parabolic thin airfoils, such that the circulation is elliptic at the given lift coefficient CL, des = 0.5. Furthermore, at the design CL, des the wing has an adapted leading edge.

Find the relative camber distribution d(y)/c that will satisfy all the requirements. Where is the maximum camber located along the wing span? Calculate the value of the maximum camber.

Give the value of the geometric incidence at design, a, in terms of CL, des and AR. Show that at a different incidence, в = a, the circulation is no longer elliptic.

If viscous effects limit the maximum local lift coefficient to a value Ci, moa, which is the same for all profiles, where will stall occur first on the wing?


1. Viviand, H.: Ailes et Corps Elances en Theorie des petites Perturbations, Ecole Nationale Superieure de L’Aeronautique et de l’Espace: Premiere Partie, Class Notes (1972)

2. Prandtl, L.: Applications of Modern Hydrodynamics to Aeronautics. NACA, report 116 (1922)

3. Glauert, H.: The Elements of Aerofoil and Airscrew Theory. Cambridge University Press, Cambridge (1926)

4. Munk, M. M.: The Minimum Induced Drag of Aerofoils. NACA, Report 121 (1921)

5. Chattot, J.-J.: Low speed design and analysis of wing/winglet combinations including viscous effects. J. Aircr. 43(2), 386-389 (2006)

6. Chattot, J.-J.: Analysis and design of wings and wing/winglet combinations at low speeds. Comput. Fluid Dyn. J. 13(3), 597-604 (2006)

7. Ashley, H., Landhal, H.: Aerodynamics of Wings and Bodies. Addison Wesley Longman, Reading (1965)

8. Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge University Press, New York (1967)

9. Moran, J.: An Introduction to Theoretical and Computational Aerodynamics. Wiley, New York (1984)

10. Weissinger, J.: NASA TM 1120 (1947)

11. Liepmann, H. W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)

12. Nixon, D., Hancock, G. J.: Integral equation methods—a reappraisal. Symposium Transson – icum II. Springer, New York (1976)

13. Tseng, K., Morino, L.: Nonlinear Green’s function method for unsteady transonic flows. Tran­sonic Aerodynamics. Progress in Aeronautics and Astronautics, vol. 81. AIAA, New York (1982)

14. Garrick, I. E.: Non steady wing characteristics. Aerodynamic Components of Aircraft at High Speed. Princeton University Press, Princeton (1957)

15. Evvard, J. C.: Use of Source Distributions for Evaluating Theoretical Aerodynamics of Thin Finite Wings at Supersonic Speeds. NACA, Report 951 (1950)

16. Puckett, A. E., Stewart, H. J.: Aerodynamic performance of delta wings at supersonic speeds. J. Aerosp. Sci. 14, 567-578 (1947)

17. Klunker, E. B.: Contribution to Methods for Calculating the Flow About Thin Wings at Tran­sonic Speeds—Analytic Expression for the Far Field. NACA TN D-6530 (1971)

18. Cole, J. D., Cook, P.: Transonic Aerodynamics. North Holland, Amsterdam (1986)

19. Cheng, H. K., Meng, S. Y.: The oblique wing as a lifting-line problem in transonic flow. J. Fluid Mech. 97, 531-556 (1980)

20. Hafez, M.: Perturbation of transonic flow with shocks. Numerical and Physical Aspects of Aerodynamic Flows. Springer, New York (1982)

21. Jameson, A.: Transonic potential flow calculations using conservative form. In: Proceedings of 2nd AIAA CFD Conference (1975)

22. Caughey, D., Jameson, A.: Basic advances in the finite volume method for transonic potential flow calculations. Numerical and Physical Aspects of Aerodynamic Flows. Springer, New York (1982)

23. Holst, T., Thomas, S. D.: Numerical solution of transonic wing flow field. AIAA paper 82-0105 (1982)

24. Hafez, M., South, J., Murman, E.: Artificial compressibility methods forthe numerical solution of the full potential equation. AIAA J. 16, 573 (1978)

25. Hafez, M., Osher, S., Whitlow, W.: Improved finite difference schemes for transonic potential calculations. AIAA paper 84-0092 (1984)