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The turbulent trailing vortex during roll-up

Published online by Cambridge University Press:  20 April 2006

W. R. C. Phillips
W. R. C. Phillips
Affiliation:
Department of Engineering, University of Cambridge[dagger] Presently, National University of Singapore, Kent Ridge, Singapore 0511
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Abstract

The turbulent trailing vortex forming from a rolling-up vortex sheet is considered. The inviscid, asymptotic roll-up of a vortex sheet is briefly reviewed, as are the effects to the sheet of merging by viscous and turbulent diffusion. The merged region is found to rapidly attain a state of equilibrium and similarity variables are used to describe it. The detailed distributions of circulation and Reynolds stress are seen to depend to some extent upon the initial spanwise distribution of circulation on the wing. However, a tiny region which is independent of the wing circulation distribution is found to exist near the point of peak tangential velocity. It is suggested that this region is described by Hoffmann & Joubert's logarithmic relationship. Assuming this to be the limiting form for the distribution of circulation near r1, the radius where the tangential velocity takes its peak value v1, an approximate form for the distribution of circulation is found and this is used to determine the form of the Reynolds-stress distribution. It is found that two modes for the decay of v1 with time are possible: one when r1 is much less than ½s, the wing semi-span, and v1 decays like t−½n; and the other when r1 = O(½s) and v1 may decay like t½(n-2); 0 < n < 1, for elliptic wing loading n ≃ ½.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 105 , April 1981 , pp. 451 - 467
Copyright
© 1981 Cambridge University Press

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References

Batchelor, G. K. 1964 J. Fluid Mech. 20, 645–658.
Bisgood, P. L., Maltby, R. L. & Dee, F. W. 1971 Aircraft Wake Turbulence and its Detection, pp. 171–206. Plenum.
Fage, A. & Simmons, L. F. G. 1925 Phil. Trans. Roy. Soc A 225, pp. 303–326.
Fink, P. T. & Soh, W. K. 1978 Proc. Roy. Soc. A 362, 195–209.
Govindaraju, S. P. & Saffman, P. G. 1971 Phys. Fluids 14, 2074–2080.
Graham, J. A. H., Newman, B. G. & Phillips, W. R. C. 1974 Proc. 9th Cong. Int. Counc. Aero. Sci. no. 74–40.
Graham, J. A. H. & Phillips, W. R. C. 1975 McGill Univ. MERL TN 75–1.
Hilton, W. F. 1938 Aero. Res. Counc. R. & M. 1858.
Hoffmann, E. R. & Joubert, P. N. 1963 J. Fluid Mech. 16, 395–411.
Kaden, H. 1931 Ing. Arch. 2, 140–168.
Leibovich, S. 1978 Ann. Rev. Fluid Mech. 10, 221–246.
Maskell, E. C. 1962 Proc. 3rd Cong. Int. Counc. Aero. Sci., pp. 737–750.
Mason, W. H. & Marchman, J. F. 1972 N.A.S.A. CR 62078.
McCormick, B. W., Tangler, J. L. & Sherrieb, H. E. 1968 J. Aircraft 5, 260–267.
Moore, D. W. 1974 J. Fluid Mech. 63, 225–235.
Moore, D. W. & Saffman, P. G. 1973 Proc. Roy. Soc. A 333, 491–508.
Nielsen, J. N. & Schwind, R. G. 1971 Aircraft Wake Turbulence and its Detection, pp. 413–454. Plenum.
Owen, P. R. 1970 Aeronaut. Q. 21, 69–78.
Pullin, D. I. & Phillips, W. R. C. 1981 J. Fluid Mech. 104, 45–53.
Saffman, P. G. 1973 Phys. Fluids 16, 1181–1188.
Slater, L. J. 1960 Confluent Hypergeometric Functions. Cambridge University Press.
Squire, H. B. 1965 Aeronaut. Q. 16, 302–306.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
Widnall, S. E. 1975 Ann. Rev. Fluid Mech. 7, 141–166.
Wygnanski, I. & Fiedler, H. E. 1969 J. Fluid Mech. 38, 577–612.