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Abstract

The Kutta condition in all physical and mathematical flow models defined in Sect. 1.5 is generally understood as appearing at acute edges of aerodynamic surfaces. That can be the “zero thickness” trailing edge of a large aspect-ratio wing or the “sharp” leading edge of a delta wing. Usually it is assumed that the flow leaves the trailing edge in bisector direction.

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Notes

  1. 1.

    The concept of boundary-layer decambering seems to have been developed by M. J. Lighthill [2].

  2. 2.

    Flow-off separation, like ordinary separation, locally is characterized by strong interaction of the external inviscid and the boundary-layer flow, Sect. 2.2. Nevertheless we can speak here simply of the boundary layers at the trailing edge.

  3. 3.

    All boundary-layer thicknesses depend on the inverse of some power of the Reynolds number, Appendix A.5.4.

  4. 4.

    Although D. Küchemann with the term “aerodynamically sharp edge”, [10], to some degree circumvented the topic of geometrical properties of trailing edges, we believe that a closer look is justified.

  5. 5.

    A large trailing-edge angle \(\theta \) may even lead to a negative \(dC_{L}/d\alpha \) at small angles of attack [14].

  6. 6.

    The authors are grateful to the Deutsches Museum in München, where it was possible to investigate the wings of a large number of aircraft. Thanks go also to colleagues who made data available.

  7. 7.

    The actual shapes of blunt trailing edges vary from sharply cut off to well rounded. Although the detailed shapes locally may play a role, we do not take them into account.

  8. 8.

    In [3] a detailed discussion of the fluid mechanical aspects of this case can be found.

  9. 9.

    The prime indicates that the coefficient is related to the original airfoil length of 20 cm.

  10. 10.

    We gratefully acknowledge that the results shown in the following figures were provided by C. Weiland [23].

References

  1. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014)

    Book  Google Scholar 

  2. Lighthill, J.: On displacement thickness. J. Fluid Mech. 4, 383–392 (1958)

    Article  MathSciNet  Google Scholar 

  3. Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011)

    Google Scholar 

  4. Hirschel, E.H., Lucchi, C.W.: On the Kutta Condition for Transonic Airfoils. MBB-UFE122-AERO-MT-651, Ottobrunn, Germany (1983)

    Google Scholar 

  5. Zierep, J.: Der senkrechte Verdichtungsstoß am gekrümmten Profil. ZAMP, vol. IXb, pp. 764–776 (1958)

    Google Scholar 

  6. Lucchi, C.W.: Shock correction and trailing edge pressure jump in two-dimensional transonic potential flows at subsonic uniform Mach numbers. In: 6th Computational Fluid Dynamics Conference, Danvers, MA. Collection of Technical Papers (A83-39351 18-02), AIAA, pp. 23–29 (1983)

    Google Scholar 

  7. Lucchi, C.W.: Ein Subdomain-Finite-Element-Verfahren zur Lösung der Konservativen vollen Potentialgleichung für Transsonische Profilströmungen (A Sub-Domian Finite-Element Method for the Solution of the Conservative Full Potential Equation for Transonic Airfoil Flow). Doctoral Thesis, Technical University München, Germany (1984)

    Google Scholar 

  8. Klopfer, G.H., Nixon, D.: Non-Isentropic Potential Formulation for Transonic Flows. AIAA-Paper 83–0375 (1983)

    Google Scholar 

  9. Hirschel, E.H.: Basics of Aerothermodynamics, 2nd, revised edition. Springer, Cham (2015)

    Google Scholar 

  10. Küchemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford: also AIAA Education Series, p. 2012. Va, AIAA, Reston (1978)

    Google Scholar 

  11. Vos, R., Farokhi, S.: Introduction to Transonic Aerodynamics. Springer, Dordrecht (2015)

    Book  Google Scholar 

  12. Obert, E.: Aerodynamic Design of Transport Aircraft. IOS Press, Delft (2009)

    Google Scholar 

  13. Abbott, I.H., Von Doenhoff, A.E., Stivers, Jr., L.S.: Summary of Airfoil Data. NACA Report No. 824 (1945)

    Google Scholar 

  14. Hoerner, S.F., Borst, H.V.: Fluid-Dynamic Lift. Hoerner Fluid Dynamics, Bricktown (1975)

    Google Scholar 

  15. Ackeret, J.: Versuche an Profilen mit abgeschnittener Hinterkante. Vorläufige Mitteilungen der Aerodynamischen Versuchsanstalt zu Göttingen, Heft 2, 18ff. (1924), also NACA Technical Meomorandum No. 431 (1927)

    Google Scholar 

  16. Henne, P.A.: Innovation with computational aerodynamics: the divergent trailing-edge airfoil. In: Henne, P.A. (ed.), Applied Computational Aerodynamics. AIAA Educational Series. pp. 221–262. AIAA, Washington, D.C. (1990)

    Google Scholar 

  17. Lawaczeck, O., Bütefisch, K.A.: Geplante Untersuchungen über v. Kármánsche Wirbelstraßen als eine mögliche Ursache für Buffet-Onset. In: Probleme der experimentellen transsonischen Aerodynamik, W. Lorenz-Meyer (ed.). DFVLR Bericht 251-77 A45, Göttingen, Germany, 6-1–6-8 (1977)

    Google Scholar 

  18. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)

    Book  Google Scholar 

  19. Herr, M.: Trailing-Edge Noise—Reduction Concepts and Scaling Laws. Doctoral Thesis, Technical University Braunschweig, Germany, also DLR-FB 2013-32 (2013)

    Google Scholar 

  20. Berg, D.E., Zayas, J.R.: Aerodynamic and Aeroacustic Properties of Flatback Airfoils. AIAA-Paper 2008–1455 (2008)

    Google Scholar 

  21. Bangga, G.S.T.A., Lutz, Th., Krämer, E.: Numerical investigation of unsteady aerodynamic effects on thick flatback airfoils. In: Proceedings of the 12th German Wind Energy Conference DEWEK, May 2015, Bremen (2015)

    Google Scholar 

  22. Schlichting, H., Truckenbrodt, E.: Aerodynamik des Flugzeuges, Vol. 1 and 2, Springer, Berlin/ (1959). Also: Aerodynamics of the Aeroplane, 2nd edition (revised). McGraw Hill Higher Education, New York (1979)

    Google Scholar 

  23. Weiland, C.: Personal Communication (2018)

    Google Scholar 

  24. Newsome, R.W.: A Comparison of Euler and Navier-Stokes Solutions for Supersonic Flow Over a Conical Delta Wing. AIAA-Paper 85–0111 (1985)

    Google Scholar 

  25. Eberle, A., Rizzi, A., Hirschel, E.H.: Numerical Solutions of the Euler Equations for Steady Flow Problems. Notes on Numerical Fluid Mechanics, vol. 34. Vieweg, Braunschweig Wiesbaden (1992)

    Google Scholar 

  26. Deister, F., Hirschel, E.H.: Self-Organizing Hybrid-Cartesian Grid/Solution System for Arbitrary Geometries. AIAA-Paper 2000–4406 (2000)

    Google Scholar 

  27. Deister, F.: Selbstorganisierendes hybrid-kartesisches Netzverfahren zur Berechnung von Strömungen um komplexe Konfigurationen (Self-Organizing Hybrid-Cartesian Grid System for the Computation of Flows Past Complex Configurations). Doctoral Thesis, University Stuttgart, Germany, Fortschrittsberichte VDI, Reihe 7, Strömungstechnik, Nr. 430 (2002)

    Google Scholar 

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Hirschel, E.H., Rizzi, A., Breitsamter, C., Staudacher, W. (2021). About the Kutta Condition. In: Separated and Vortical Flow in Aircraft Wing Aerodynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-61328-3_6

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