Abstract
This review paper presents a unified formulation of the Kutta condition for steady and unsteady flows, implemented by removing all unbounded velocity singularities (of power‐law and logarithmic type) at the trailing edge, and including nonlinear wakes and thick swept‐back wings. A suitable boundary integral approach is adopted and the uniqueness issue is discussed for several wing configurations of interest in aerodynamics.
Sommario. Si presenta una formulazione unificata della condizione di Kutta per flussi stazionari e non stazionari, ottenuta imponendo la limitatezza della velocità al bordo d'uscita, e valida nel caso nonlineare anche per ali a freccia. Si utilizza un opportuno approccio integrale al contorno e si discute il problema dell'unicità per svariate configurazioni alari di interesse nelle applicazioni.
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Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva, R., 'A boundary integral formulation for the kinetic field in aerodynamics. I: Mathematical analysis', Eur. J. Mech. B/Fluids 10 (1991) 605–627.
Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva, R., 'A boundary integral formulation for the kinetic field in aerodynamics II: Applications to unsteady 2D flows', Eur. J. Mech. B/Fluids 11 (1992) 69–92.
Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva, R., 'On the trailing edge singularity and Kutta condition for 3D airfoils', Eur. J. Mech. B/Fluids 15 (1996) 809–830.
Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva, R., 'Uniqueness of the bounded flow solution in aerodynamics', Comput. Mech. 22 (1998) 12–18.
Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva R., 'Edge singularities and Kutta condition for 3D unsteady flows in aerodynamics', In: IABEM'98, Int. Symp. on Boundary Element Methods, Ecole Polytechnique, Palaiseau, France, May 26–29, 1998, to appear in Pitman Res. Notes in Math., 1999.
Bassanini, P., Casciola, C.M., Lancia, M.R. and Piva, R., 'A theoretical model for multiply connected wings', Euro. J. Appl. Math. 9(6) (1998) 607–634.
Bassanini, P. and Elcrat, A.R., Theory and Applications of Partial Differential Equations, Plenum Press, New York, 1997.
Cheng, H.K., 'Lifting-line theory of oblique wings', AIAA J., September 1978, pp. 1211–1213.
Chen, G. and Zhou, J., Boundary Element Methods, Academic Press, New York, 1992.
Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
Costabel, M., 'Boundary integral operators on Lipschitz domains: elementary results', SIAM J. Math. Anal. 19 (1988) 613–626.
Dziuk, G., Finite elements for the Beltrami operator on arbitrary surfaces, Lecture Notes in Math. 1357, Springer, 1998, pp. 142–155.
Friedrichs, K.O., Special Topics in Fluid Dynamics, Nelson, London, 1966.
Fichera G. and Ricci, P.E., The single layer potential approach in the theory of boundary value problems for elliptic equations, Lecture Notes in Math. 561, Springer, 1976, pp. 39-50.
Hauptman, A. and Miloh, T., 'On the exact solution of the linearized lifting-surface problem of an elliptic wing', Q. Jl. Mech. Appl. Math. 39(1) (1986) 41–66.
Henrici, P., Applied and Computational Complex Analysis, Wiley, New York, 1991.
Hinder, R. and Meister E., 'Regarding some problems of the Kutta-Joukovskii condition in lifting surface theory', Math. Nachr. 184 (1997) 191–228.
Hsiao, G.C. and McCamy, R.C., Solutions of boundary value problems by integral equations of the first kind, SIAM Rev. 15(4) (1973) 687–705.
Hsiao, G.C., 'Solution of boundary value problems by integral equations of the first kind – An update', In: Morino, L. and Piva, R. (Eds), Proc. IABEM Symposium on Boundary Integral Methods, Rome, Italy, 15–19 Oct. 1990, Springer Verlag, 1991, 231–240.
Hsiao, G.C. and Wendland, W.L., Variational Methods for Boundary Integral Equations and Mathematical Foundations of Boundary Element Methods, Springer, Berlin (in prep.)
Kondrat'ev V.A. and Oleinik, O.A., 'Boundary value problems for partial differential equations in nonsmooth domains', Russian Math. Surveys 38 (1983) 1–86.
Legras, J., 'Influence de l' attaque oblique et de la non permanence du mouvement sur une aile de grand allongement', Les Cahiers de l' aerodynamique 7 (1947) 9–28.
Lewis, R.I., Vortex Element Methods for Fluid Dynamics Analysis of Engineering Systems, Cambridge Univ. Press, Cambridge, 1991.
Lighthill, M.J., An Informal Introduction to Theoretical Fluid Mechanics, Clarendon Press, Oxford, 1986.
Mangler, K.W. and Smith J.H.B., 'Behaviour of the vortex sheet at the trailing edge of a lifting wing', Aero. J. Royal Aeron. Soc. 74 (1970) 906–908.
Marchioro, C. and Pulvirenti, M., Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, 1994.
Martensen, E., Potentialtheorie, Teubner, Stuttgart, 1968.
Maz'ya, V.G., Boundary Integral Equations, Encyclopedia of Math. Sci. 27, Springer-Verlag, (1991), pp. 127–222.
Maz'ya, V.G., Plamenevskii, B.A., 'Problems with oblique derivatives in regions with piecewise smooth boundaries', Funct. Anal. Appl. 5 (1971) 256–257.
Milne-Thomson, L.M., Theoretical Aerodynamics, McMillan, London, 1966.
Milne-Thomson, L.M., Theoretical Hydrodynamics, McMillan, London, 1968.
Moran, J., An Introduction to Theoretical and Computational Aerodynamics, Wiley, New York, 1984.
Morino, L., 'Helmholtz decomposition revisited: vorticity generation and trailing edge condition', Comput. Mech. 1 (1986) 65–90.
Nazarov, S.A., 'Singularities at angular points of a trailing edge under the Joukowskii-Kutta condition', in: Pitman Res. Notes in Mathematics Series 379, 1998, Longman, 153–157.
Nedelec, J.C., 'Integral equations with non-integrable kernels', Int. Eq. Oper. Theor. 5 (1982) 563–572.
Nedelec, J.C., Approximation des équations intégrales en méchanique et en physique, Lect. Notes, Ecole Polytechnique, Palaiseau, France, 1977.
Reissner, E., 'Boundary value problems in aerodynamics of lifting surfaces in non-uniform motion', Bull. Amer. Math. Soc. 55 (1949) 825–850.
Sedov, L.I., Two-Dimensional Problems in Hydrodynamics and Aerodynamics, Wiley, New York, 1965.
Serrin, J., Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik VIII/1, Springer, Berlin, 1959.
Spalart, P.R., 'Airplane trailing vortices', Ann. Rev. Fluid Mech. 30 (1998) 107–138.
Thurber, J.K., 'An asymptotic method for determining the lift distribution of a swept-back wing of finite span', Comm. Pure Appl. Math. XVIII (1965) 733–756.
Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
Weissinger, J., 'Theorie des Tragfluegels bei stationaerer Bewegung in reibungslosen, inkompressiblen Medien', in: Handbuch der Physik Bd. VIII/2, Springer, 1963, pp. 385–437.
Wendland, W.L., 'Variational methods for BEM', In: Morino, L. and Piva, R. (Eds), Proc. IABEM Symposium on Boundary IntegralMethods, Rome, Italy, 15–19, Oct. 1990, Springer-Verlag, 1991, pp. 15–34.
Wolkowich, J., The joined wing: an overview. AIAA J. Aircraft, March 1986, 161–178.
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Bassanini, P., Casciola, C., Lancia, M. et al. Edge Singularities and Kutta Condition in 3D Aerodynamics. Meccanica 34, 199–229 (1999). https://doi.org/10.1023/A:1004571915758
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DOI: https://doi.org/10.1023/A:1004571915758