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Subsonic flow around the leading edge of a thin aerofoil with a parabolic nose

Published online by Cambridge University Press:  26 September 2008

Z. Rusak
Z. Rusak
Affiliation:
Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
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Abstract

Subsonic potential flow around the leading edge of a thin two-dimensional general aerofoil with a parabolic nose at small angles of attack is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed Mach number in terms of the thickness ratio of the aerofoil in an outer region around the aerofoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of linearized aerofoil theory and its second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary-value problem is formulated in the inner region for the solution of a compressible uniform flow with the same Mach number, past a semi-infinite two-dimensional parabola. The analysis shows that the drag of any general aerofoil with a round nose is zero for every subsonic Mach number below the critical Mach number, where the drag component that is calculated by the integration of the outer pressure distribution over the aerofoil is cancelled exactly by the drag of the parabolic nose. The numerical solution of the flow field in the inner region results in the pressure distribution on the parabolic nose. The stagnation point tends to shift toward the leading edge due to the compressibility effects as the Mach number is increased at a fixed angle of attack and camber of the aerofoil. Good agreement is found in the leading edge region between the present asymptotic solution and numerical solutions of the Euler equations. Using the outer linearized solution and the nose solution a uniformly valid pressure distribution on the entire aerofoil surface in a subsonic flow is derived.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 3 , September 1994 , pp. 283 - 311
Copyright
Copyright © Cambridge University Press 1994

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References

Abbott, I. H. & von Doenhoff, A. E. 1959 Theory of Wing Sections. Dover, New York, pp. 111–118.Google Scholar
Finn, R. & Gilbarg, D. 1957 Asymptotic behavior and uniqueness of plane subsonic flows. Comm. Pure Appl. Math. X, pp. 23–63.Google Scholar
Hughes, T. J. R., Franca, L. P. & Hulbert, G. M. 1989 A new finite element formulation for computational fluid mechanics: Viii The Galerkin least squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng. 73, pp. 173–189.Google Scholar
Imai, I. 1951 Application of the M 2-expansion method to the subsonic flow of a compressible fluid past a parabolic cylinder. Proceedings of the 1st Japan Congress for Applied Mechanics, pp. 349–352.Google Scholar
Jones, R. T. 1950 Leading-edge singularities in thin-aerofoil theory. J. Aeronautical Sci. 17, pp. 307–310.CrossRefGoogle Scholar
Kusunose, K. 1979 Two-dimensional flow past convex and concave corners at transonic speed and two-dimensional flow around a parabolic nose at subsonic and transonic speeds. PhD Thesis, University of California at Los Angeles, pp. 56–92.Google Scholar
Lighthill, M. J. 1951 A new approach to thin aerofoil theory. Aeronautical Quart. 3, pp. 193–210.CrossRefGoogle Scholar
Prandtl, L. & Tietjens, O. G. 1934 Applied Hydro and Aeromechanics. McGraw-Hill, New York, pp. 158–184.Google Scholar
Rusak, Z. 1990 Subsonic Flow Around a Thin aerofoil With a Parabolic Nose. Department of Mathematical Sciences Report No. 186, Rensselaer Polytechnic Institute, Troy, NY.Google Scholar
Van Dyke, M. D. 1954 Subsonic edges in thin-wing and slender-body theory. NACA TN 3343, November.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford, CA, pp. 54–56, 59–66.Google Scholar
von KármáN, T. & Burgers, J. M. 1935 General aerodynamic theory–Perfect fluids. In: Div. E. ‘Aerodynamic Theory’ (Ed. Durand, W. F.), Vol 2. Springer, Berlin, pp. 51–52.Google Scholar
Webster, B. E., Shephard, M. S., Rusak, Z. & Flaherty, J. E. 1992 An adaptive finite element method for rotorcraft aerodynamics. 7th IMACS Conference on Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky D. Knight & G. Richter (eds.), pp. 799–805.Google Scholar