Review of the Application of Hodograph Theory to Transonic Aerofoil Design and Theoretical and Experimental Analysis of Shock-free Aerofoils

  • J. W. Boerstoel
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Summary

After an introductory sketch of hodograph theory the mathematical concepts, on which a few modern hodograph methods for computational aerofoil design are based, are summarized. The used mathematical methods lead to two different types of computation procedures; these are discussed.

Typical computation results of a few advanced aerofoils are shown. As far as available, typical experimental results are also presented. An attempt is made to summarize what maximum thickness ratio’s of shock-free aerofoils can be realized at a given design point (M , C ) according to present-day knowledge.

An analysis of the leading edge shape of an interesting type of aerofoil shows, that the applicability of finite difference methods for the design of such aerofoils should be considered a more difficult matter than perhaps is generally appreciated.

It is concluded, that a computational design method for shock-free aerofoils based on hodograph theory is nowadays a valuable tool for transonic aerofoil and wing designers.

Keywords

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Literature

  1. 1.
    Bers, L.: Mathematical aspects of subsonic and transonic gas dynamics, John Wiley and Sons Inc., New York (1958).MATHGoogle Scholar
  2. 2.
    V. Mises, R., and H. Geiringer and G.S.S. Ludford: Mathematical theory of compressible fluid flow, Academic Press Inc., New York (1958).MATHGoogle Scholar
  3. 3.
    Manwell, A.R.: The hodograph equations (An introduction to the mathematical theory of plane transonic flow), Oliver and Boyd, Edingburgh (1971).MATHGoogle Scholar
  4. 4.
    Nieuwland, G.Y.: Transonic potential flow around a family of quasi-elliptical sections, NLR TR T 172 (1967).Google Scholar
  5. 5.
    Chaplygin, A.A.: On gas jets, NACA Techn. Mem. 1063 (1944).Google Scholar
  6. 6.
    Lighthill, M.J.: The hodograph transformation in transonic flow. II. Auxiliary theorems on the hypergeometric functions, Proc. R. Soc. A 191, London (1947) 341-351.Google Scholar
  7. 7.
    Cherry, M.T.: Uniform asymptotic formulae for functions with transition points, Trans. Am. Math. Soc., vol. 68 (1950) 224–257.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Sobieczky, H.: Analog-analytische Konstruktion überkritischer Profil Strömungen, DGLR-Symposium “Tragflügel-Aerodynamik bei schallnahen Strömungen”, Göttingen, 1972, rep. DGLR 72–129 (1972), DLR-Mitt. 73–04 (1973) 224-240.Google Scholar
  9. 9.
    Boerstoel, J.W.: A transonic hodograph theory for aerofoil design, IMA Conf. Comp. Methods and Problems in Aeron. Fl. Dyn., Manchester (1974), NLR rep. MP 74024 U (1974).Google Scholar
  10. 10.
    Boerstoel, J.W. and G.H. Huizing: Transonic shock-free aerofoil design by an analytic hodograph method, AIAA 7th Fl. Plasma Dyn. Conf., Palo Alto, AIAA paper 74-539 (1974), NLR MP 74025 U (1974).Google Scholar
  11. 11.
    Morawetz, C.S.: Non-existence of transonic flow past a profile, Comm. Pure and Appl. Math., vol. 17, (1964) 357–367.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lighthill, M.J.: The hodograph transformation in transonic flow. III. Plow round a body, Proc. R. Soc. A. 191, London (1947) 352-369.Google Scholar
  13. 13.
    Goldstein, S., Lighthill, M.J., Craggs, J.W.: On the hodograph transformation for high speed flow. I. A flow without circulation, Quart. J. Mech. Appl. Math., 1 (1948) 344–357.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Takanashi, S.: A method of obtaining transonic shock-free flow around lifting aerofoils, Trans. Japan Soc. Aero. Space Sci., 16, 34 (1973).Google Scholar
  15. 15.
    Bauer, P., Garabedian, P.R., Korn, D.: Supercritical wing sections II, Lecture Notes in Economics and Mathematical Systems no 108, Springer, New York (1975).Google Scholar
  16. 16.
    Swenson, E.V.: Geometry of the complex characteristics in transonic flow, Comm. Pure Appl. Math., 21 (1968) 175–185.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Sobieczky, H.: Entwurf überkritischer Profile mit Hilfe der rheo-elektrischen Analogie, rep. DLR-PB 75-43 (1975).Google Scholar
  18. 18.
    Eberle, A.: Transsonischer Profilentwurf, MBB-Bericht UPE 1108 (Ö) (1974).Google Scholar
  19. 19.
    Pearcy, H.H.: The aerodynamic design of section shapes for swept wings, Advances in Aeron. Sc, 3, Pergamon Press (1962), 277-323.Google Scholar
  20. 20.
    Nieuwland, G.Y., Spee, B.M.: Transonic shock-free flow, fact or fiction?, AGARD Conf. Proc. no. 35 (1968), NLR MP 68004 U (1968).Google Scholar
  21. 21.
    Boerstoel, J.W., Uijlenhoet, R.: Lifting aerofoils with supercritical shock-free flow, ICAS paper 70-15 (1970), NLR MP 70015 U (1970).Google Scholar
  22. 22.
    Loeve, W.: Appraisal of wing aerodynamic design methods for subsonic flight speed, AGARD Lecture Series no. 67 (1974); NLR MP 74021.Google Scholar
  23. 23.
    Van Egmond, J.A., Boerstoel, J.W.: unpublished computation results (1975).Google Scholar
  24. 24.
    Shigemi, T.: Research on the transonic aerofoil sections at the National Aerospace Laboratory, Japan, ICAS paper, Haifa (1974).Google Scholar
  25. 25.
    Kacprzynski, J.J., Ohman, L.H., Garabedian, P.R., Korn, D.G.: Analysis of the flow past a shockless lifting airfoil in design and off-design conditions, Nat. Res. Counc. Van., Aer Rep. LR-554 (1971).Google Scholar
  26. 26.
    Zwaaneveld, J., Rohne, P.B.: unpublished NLR Pilot-tunnel results (1974).Google Scholar
  27. 27.
    Zwaaneveld, J.: Semi-empirical methods for predicting the transonic off-design characteristics of airfoils, rep. NLR TR 75045 C (classified), 1975.Google Scholar
  28. 28.
    Boerstoel, J.W.: A survey of symmetrical transonic potential flows around quasi-elliptical aerofoil sections, NLR TR-T. 136 (1967).Google Scholar
  29. 29.
    Nieuwland, G.Y., Spee, B.M.: Transonic airfoils: recent developments in theory, experiment and design, Ann. Rev. Fl. Mech., 5 (1973) 119–150.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag, Berlin/Heidelberg 1976

Authors and Affiliations

  • J. W. Boerstoel
    • 1
  1. 1.National Aerospace LaboratoryAmsterdamThe Netherlands

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