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The effect of axial stretching on the three-dimensional stability of a vortex pair

Published online by Cambridge University Press:  26 April 2006

J. S. Marshall
J. S. Marshall
Affiliation:
Department of Ocean Engineering, Florida Atlantic University, Boca Raton. FL 33431. USA
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Abstract

The stability of a pair of counter-rotating vortices to three-dimensional disturbances in the presence of a stretching flow is studied for vortices of small circular cross-section. The problem is reduced to a system of two first-order, linear ordinary differential equations, which can be integrated numerically to obtain the change in the perturbation of the vortex pair with time. The stability of the vortex pair depends upon four dimensionless constants, two of which characterize the stretching flow. Computations indicate that stretching usually exerts a stabilizing influence on the vortex pair, although in many cases the perturbation amplitude may initially increase and then decrease at some later time due to the effects of stretching. The results of the study are applied to investigate stability of hairpin vortices that are typically observed in turbulent shear flows. An estimate of the percentage increase in perturbation amplitude of a hairpin vortex in a homogeneous turbulent shear flow is given as a function of the stretch of the hairpin for different values of the dimensionless perturbation wavenumber and the microscale Reynolds number Reλ = λq/ν (based on the Taylor microscale λ and the turbulent kinetic energy ½q2). The maximum percentage growth of a perturbation of the legs of a hairpin vortex in a turbulent shear flow is found to decrease with increase in Reλ.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 403 - 419
Copyright
© 1992 Cambridge University Press

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References

Aref, H. & Flinchem E. P. 1984 Dynamics of a vortex filament in a shear flow. J. Fluid Mech. 148, 477–497.Google Scholar
Arms, R. J. & Hama F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Burgers J. M. 1948 A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics (ed. R. von Mises & T. von KaArmaAn), p. 171. Academic.
Crow S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Head, M. R. & Bandyopadhyay P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.Google Scholar
Kida S. 1981 Motion of an elliptical vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Lundgren, T. S. & Ashurst W. T. 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307.Google Scholar
Marshall J. S. 1991 A general theory of curved vortices with circular cross-section and variable core area. J. Fluid Mech. 229, 311–338.Google Scholar
Moin, P. & Kim J. 1985 The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations. J. Fluid Mech. 155, 441–464.Google Scholar
Moore D. W. 1972 Finite amplitude waves on aircraft trailing vortices. Aeronaut. Q. 23, 307–314.Google Scholar
Moore D. W. 1985 The interaction of a diffusing line vortex and an aligned shear flow Proc. R. Soc. Lond. A 399, 367–375.Google Scholar
Moore, D. W. & Saffman P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 339–353. Plenum.
Moore, D. W. & Saffman P. G. 1972 The motion of a vortex filament with axial flow Phil. Trans. R. Soc. Lond. A 272, 403–429.Google Scholar
Moore, D. W. & Saffman P. G. 1975a The density of organized vortices in a turbulent mixing layer. J. Fluid Mech. 69, 465–473.Google Scholar
Moore, D. W. & Saffman P. G. 1975b The instability of a straight vortex filament in a strain field Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Neu J. C. 1984 The dynamics of a columnar vortex in an imposed strain. Phys. Fluids 27, 2397–2402.Google Scholar
Pearson, C. F. & Abernathy F. H. 1984 Evolution of the flow field associated with a streamwise diffusing vortex. J. Fluid Mech. 146, 271–283.Google Scholar
Pierrehumbert, R. T. & Widnall S. E. 1981 The structure of organized vortices in a free shear layer. J. Fluid Mech. 102, 301–313.Google Scholar
Robinson, A. C. & Saffman P. G. 1982 Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411–427.Google Scholar
Robinson, A. C. & Saffman P. G. 1984 Three-dimensional stability of an elliptical vortex in a straining flow. J. Fluid Mech. 142, 451–466.Google Scholar
Rogers, M. M. & Moin P. 1987 The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 176, 33–66 (referred to herein as RM).Google Scholar
Rosenhead L. 1930 The spread of vorticity in the wake behind a cylinder Proc. R. Soc. Lond. A 127, 590–612.Google Scholar
Widnall, S. F. & Bliss D. B. 1971 Slender-body analysis of the motion and stability of a vortex filament containing an axial flow. J. Fluid Mech. 50, 335–353.Google Scholar