Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-07T21:55:27.452Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I. 1971 Ordinary Differential Equations. Moscow: Nauka.
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.Google Scholar
Blokhin, A. B., Grubin, S. E., Simakin, I. N. & Trigub, V. N. 1992 TURLEN Library INTECO srl, Frosinone, Italy.
Brown, S. N., Stewartson, K. & Williams, P. G. 1975 Hypersonic self-induced separation. Phys. Fluids 18, 633–639.Google Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400.Google Scholar
Gartshore, I. S. 1963 Some numerical solutions for the viscous core of an irrotational vortex. NRC Can. Aero. Rep. LR-378.
Goldstein, S. 1960 Lectures on Fluid Mechanics Interscience Publ.
Hafez, M., Ahmad, J., Kuruvila, G. & Salas, J. D. 1987 Vortex breakdown simulation. AIAA Paper 87-1343.
Hall, M. G. 1967 A new approach to vortex breakdown. Proc. 1967 Heat Transfer Fluid Mech. Inst. pp. 319–340. Stanford University Press.
Hall, M. G. 1972 Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–218.Google Scholar
Harvey, J. K. 1962 Some observation of the vortex breakdown phenomenon. J. Fluid Mech. 14, 585–592.Google Scholar
Leibovich, S. 1970 Weakly nonlinear waves in rotating fluids. J. Fluid Mech. 42, 803–822.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. & Kribus, A. 1990 Large-amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459–504.Google Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.Google Scholar
Lessen, M., Singh, P. I. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Mager, A. 1971 Incompressible, viscous, swirling flow through a nozzle. AIAA J. 9, 649–655.Google Scholar
Mager, A. 1972 Dissipation and breakdown of a wing-tip vortex. J. Fluid Mech. 55, 609–628.Google Scholar
Morton, B. R. 1969 The strength of vortex and swirling core flows. J. Fluid Mech. 38, 315–333.Google Scholar
Neiland, V. Ya., 1970 Upstream propagation of the disturbances in interacting hypersonic boundary layer. Izv. Akad. Nauk, SSSR, Mekh. Zhid. i Gaza, No. 4, 40–49.Google Scholar
Neiland, V. Ya., 1971 The flow past the separation point at supersonic speed. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza, No. 3, 19–25.Google Scholar
Peckham, D. H. & Atkinson, S. A. 1957 Preliminary results of low speed wind tunnel tests on a Gothic wing of aspect ratio 1.0. Aeronaut. Res. Counc., CP 508.Google Scholar
Powell, M. J. D. 1970a A hybrid method for nonlinear equations. In Numerical Methods on Nonlinear Algebraic Equations, pp. 87–114. Gordon and Breach.
Powell, M. J. D. 1970b A Fortran subroutine for solving systems of nonlinear algebraic equations. In Numerical Methods on Nonlinear Algebraic Equations, pp. 115–161. Gordon and Breach.
Randall, J. D. & Leibovich, S. 1973 The critical state: a trapped wave model of vortex breakdown. J. Fluid Mech. 53, 495–515.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdown. J. Fluid Mech. 45, 545–559.Google Scholar
Sychev, V. V. 1972 On laminar separation. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza, No. 3, 47–59.Google Scholar
Sychev, Vic. V. 1992 Asymptotic theory of the vortex breakdown. Izv. Akad. Nauk Russ., Mekh. Zhid, i Gaza (to appear).Google Scholar
Trigub, V. N. 1985a The problem of the vortex breakdown. Prikl. Matem. i Mekh., No. 2, 220–226.Google Scholar
Trigub, V. N. 1985b The problem of the vortex breakdown in inviscid fluid. Uchenye Zap. TsAGI, No. 3, 100–104.Google Scholar
Trigub, V. N. 1986 An analysis of the flow near the stagnation point in an axisymmetrical wake. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza, No. 2, 53–59.Google Scholar
Trigub, V. N. 1987 Asymptotic theory of the origination of recirculation zones in an axisymmetrical wake. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza, No. 5, 54–60.Google Scholar