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The boundary layer on a paraboloid of revolution

Published online by Cambridge University Press:  24 October 2008

D. R. Miller
D. R. Miller
Affiliation:
Department of Applied Mathematics, University of Western Ontario
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Abstract

An analytical study is made of the boundary-layer approximation for the steady laminar flow of a viscous incompressible fluid past a paraboloid of revolution. The functional form of the vorticity distribution is found and shown to involve exponential decay with distance from the surface of the body, and inequalities are established concerning the displacement thickness.

The exact results are compared with various linearizations. The Oseen linearization (12) turns out to be incorrect except in the limit of vanishing Reynolds number; this result is connected with the infinite drag of any paraboloid of non-zero thickness. An alternate linearization, related to a method of Burgers (3) and allowing for the displacement effect of the body and of its boundary layer, is shown to give the correct functional form for these quantities. Numerical results are included which bear out these conclusions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 285 - 299
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Berker, R.Intégration des équations du mouvement d'un fluide visqueux incompressible, pp. 1–384. (Handbuch der Physik, vol. VIII/2.) (Springer-Verlag, Berlin, 1963).Google Scholar
(2)Boussinesq, M. J.Calcul du pouvoir refroidissant des courants fluides. Journal de Mathématiques (de Liouville, J.) Ser. 6, 1 (1905), 285–332.Google Scholar
(3)Burgers, J. M.Stationary streaming caused by a body in a fluid with friction. Proc. Acad. Sci. Amst. 23 (1921), 1082–1107.Google Scholar
(4)Chang, I-Dee. Navier–Stokes solutions at large distances from a finite body. J. Math. Mech. 10 (1961), 811–872.Google Scholar
(5)Coppel, W. A.On a differential equation of boundary-layer theory. Philos. Trans. Roy. Soc. London, Ser. A 253 (1960), 101–136.Google Scholar
(6)Erdelyi, A. et al. Higher transcendental functions, 3 volumes. (McGraw-Hill, New York, 1953).Google Scholar
(7)Goldstein, S.Modern developments in fluid dynamics. (Clarendon Press, Oxford, 1938).Google Scholar
(8)Kaplun, S.Low Reynolds number flow past a circular cylinder. J. Math. Mech. 6 (1957), 595–606.Google Scholar
(9)Lagerstrom, P. A. and Cole, J. D.Examples illustrating expansion procedures for the Navier–Stokes equations. J. Sat. Mech. Anal. 4 (1955), 817–882.Google Scholar
(10)Mark, R. M.Laminar boundary layers on slender bodies of revolution in axial flow. GALCIT Hypersonic Wind Tunnel Memo. No. 21, July 30, 1954.Google Scholar
(11)Milne-Thomson, L. M.Theoretical hydrodynamics, fourth edition (Macmillan and Co., London, 1960).Google Scholar
(12)Oseen, C. W.Über die Stokessche Formel and Über eine verwandte Aufgabe in der Hydrodynamik. Ark. Mat. Astronom. Fys. 6 (1910), no. 29.Google Scholar
(13)Pillow, A. F.Diffusion of heat and circulation in potential flow convection. J. Math. Mech. 13 (1964), 521–545.Google Scholar
(14)Probstein, R. F. and Elliott, D.The transverse curvature effect in compressible axially symmetric laminar boundary-layer flow. J. Aero. Sci. 23 (1956), 208–224.CrossRefGoogle Scholar
(15)Thwaites, B.Incompressible aerodynamics (Clarendon Press, Oxford, 1960).CrossRefGoogle Scholar
(16)Van Dyke, M.Higher approximations in boundary-layer theory. I. General analysis. J. Fluid Mech. 14 (1960), 161–177. II. Application to leading edges. J. Fluid Mech. 14 (1962), 481–495. III. Parabola in uniform stream. J. Fluid Mech. 19 (1964), 145–159.CrossRefGoogle Scholar
(17)Wilkinson, J.A note on the Oseen approximation for a paraboloid in a uniform stream parallel to its axis. Quart. J. Mech. Appl. Math. 8 (1955), 415–421.CrossRefGoogle Scholar