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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The concept of boundary-layer decambering seems to have been developed by M. J. Lighthill [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.
All boundary-layer thicknesses depend on the inverse of some power of the Reynolds number, Appendix A.5.4.
- 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.
A large trailing-edge angle \(\theta \) may even lead to a negative \(dC_{L}/d\alpha \) at small angles of attack [14].
- 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.
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.
In [3] a detailed discussion of the fluid mechanical aspects of this case can be found.
- 9.
The prime indicates that the coefficient is related to the original airfoil length of 20 cm.
- 10.
We gratefully acknowledge that the results shown in the following figures were provided by C. Weiland [23].
References
Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014)
Lighthill, J.: On displacement thickness. J. Fluid Mech. 4, 383–392 (1958)
Anderson Jr., J.D.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2011)
Hirschel, E.H., Lucchi, C.W.: On the Kutta Condition for Transonic Airfoils. MBB-UFE122-AERO-MT-651, Ottobrunn, Germany (1983)
Zierep, J.: Der senkrechte Verdichtungsstoß am gekrümmten Profil. ZAMP, vol. IXb, pp. 764–776 (1958)
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)
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)
Klopfer, G.H., Nixon, D.: Non-Isentropic Potential Formulation for Transonic Flows. AIAA-Paper 83–0375 (1983)
Hirschel, E.H.: Basics of Aerothermodynamics, 2nd, revised edition. Springer, Cham (2015)
Küchemann, D.: The Aerodynamic Design of Aircraft. Pergamon Press, Oxford: also AIAA Education Series, p. 2012. Va, AIAA, Reston (1978)
Vos, R., Farokhi, S.: Introduction to Transonic Aerodynamics. Springer, Dordrecht (2015)
Obert, E.: Aerodynamic Design of Transport Aircraft. IOS Press, Delft (2009)
Abbott, I.H., Von Doenhoff, A.E., Stivers, Jr., L.S.: Summary of Airfoil Data. NACA Report No. 824 (1945)
Hoerner, S.F., Borst, H.V.: Fluid-Dynamic Lift. Hoerner Fluid Dynamics, Bricktown (1975)
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)
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)
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)
Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)
Herr, M.: Trailing-Edge Noise—Reduction Concepts and Scaling Laws. Doctoral Thesis, Technical University Braunschweig, Germany, also DLR-FB 2013-32 (2013)
Berg, D.E., Zayas, J.R.: Aerodynamic and Aeroacustic Properties of Flatback Airfoils. AIAA-Paper 2008–1455 (2008)
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)
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)
Weiland, C.: Personal Communication (2018)
Newsome, R.W.: A Comparison of Euler and Navier-Stokes Solutions for Supersonic Flow Over a Conical Delta Wing. AIAA-Paper 85–0111 (1985)
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)
Deister, F., Hirschel, E.H.: Self-Organizing Hybrid-Cartesian Grid/Solution System for Arbitrary Geometries. AIAA-Paper 2000–4406 (2000)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2021 Springer-Verlag GmbH Germany, part of Springer Nature
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-61328-3_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-61326-9
Online ISBN: 978-3-662-61328-3
eBook Packages: EngineeringEngineering (R0)