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The asymptotic study of dissipation and breakdown of a wing-tip vortex

Published online by Cambridge University Press:  26 April 2006

V. N. Trigub ,
A. B. Blokhin  and
I. N. Simakin
V. N. Trigub
Affiliation:
INTECO srl., Via Mola Vecchia 2A, I-03100 Frosinone, Italy
A. B. Blokhin
Affiliation:
INTECO srl., Via Mola Vecchia 2A, I-03100 Frosinone, Italy
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Abstract

The steady axisymmetrical wing-tip vortex is studied in this paper by means of asymptotic methods within the limit of high Reynolds numbers. The smooth regrouping of the vortex under the action of viscous forces is described by a quasicylindrical approximation. The solutions of the quasi-cylindrical approximation are thoroughly analysed numerically and it is shown that a saddle-point bifurcation appears at certain critical values of circulation. At these values the solution may be continued in two ways: as a supercritical branch which approaches the Batchelor limit far downstream; and a subcritical one, which passes the second, nodal-point bifurcation. The parabolic quasi-cylindrical equations past this point allow the downstream disturbances to propagate upstream, like for example, boundary-layer equations in the regime of strong hypersonic interaction. The flow past the second bifurcation point was studied numerically and it was shown that solutions of the quasicylindrical approximation with large reversed-flow regions exist. An asymptotic expansion of such solutions far downstream was constructed, and it turned out that the reversed-flow region expands exponentially. This process is halted by elliptical effects in the external flow. An asymptotic theory of large reversed-flow regions is suggested including viscosity and elliptical effects. Numerical solutions for unbounded vortex breakdown parabolically expanding far downstream are presented. Then the general asymptotic problem statement which describes the flow near the bifurcation points is used to study the asymptotic solutions near the first bifurcation point. The problem is investigated numerically and two kinds of solution, which may be treated as transcritical jumps and marginal vortex breakdown, are found and discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 274 , 10 September 1994 , pp. 293 - 337
Copyright
© 1994 Cambridge University Press

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