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The evolution of a localized vortex disturbance in external shear flows. Part 1. Theoretical considerations and preliminary experimental results

Published online by Cambridge University Press:  26 April 2006

Vladimir Levinski  and
Jacob Cohen
Vladimir Levinski
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
Jacob Cohen
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
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Abstract

The evolution of a finite-amplitude three-dimensional localized disturbance embedded in external shear flows is addressed. Using the fluid impulse integral as a characteristic of such a disturbance, the Euler vorticity equation is integrated analytically, and a system of linear equations describing the temporal evolution of the three components of the fluid impulse is obtained. Analysis of this system of equations shows that inviscid plane parallel flows as well as high Reynolds number two-dimensional boundary layers are always unstable to small localized disturbances, a typical dimension of which is much smaller than a dimensional length scale corresponding to an O(1) change of the external velocity. Since the integral character of the fluid impulse is insensitive to the details of the flow, universal properties are obtained. The analysis predicts that the growing vortex disturbance will be inclined at 45° to the external flow direction, in a plane normal to the transverse axis. This prediction agrees with previous experimental observations concerning the growth of hairpin vortices in laminar and turbulent boundary layers. In order to demonstrate the potential of this approach, it is applied to Taylor-Couette flow, which has additional dynamical effects owing to rotation. Accordingly, a new instability criterion associated with three-dimensional localized disturbances is found. The validity of this criterion is supported by our experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 289 , 25 April 1995 , pp. 159 - 177
Copyright
© 1995 Cambridge University Press

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References

Acarlar, M. S. & Smith, C. R. 1987a A study of hairpin vortices in a laminar boundary layer. Part 1. Hairpin vortices generated by a hemisphere protuberance. J. Fluid Mech. 175, 1.Google Scholar
Acarlar, M. S. & Smith, C. R. 1987b A study of hairpin vortices in a laminar boundary layer. Part 2. Hairpin vortices generated by fluid injection. J. Fluid Mech. 175, 43.Google Scholar
Aref, H. & Flinchem, E. P. 1984 Dynamics of a vortex filament in a shear flow. J. Fluid Mech. 148, 477.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics, pp. 517–520. Cambridge University Press.
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160.Google Scholar
Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359.Google Scholar
Breuer, K. S. & Haritonidis, J. H. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 1. Weak disturbances. J. Fluid Mech. 220, 569.Google Scholar
Breuer, K. S. & Landahl, M. T. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances. J. Fluid Mech. 220, 595.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390.Google Scholar
Grigoriev, Yu. N., Levinski, V. B., Yanenko, N. N. 1982 Hamiltonian vortex models for turbulence theory. Chisl. Met. Mech. Spl. Sred. 13, No. 3, 13 (in Russian).Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241.Google Scholar
Hagen, J. P. & Kurosaka, M. 1993 Corewise cross-flow transport in hairpin vortices – the ‘tornado effect’. Phys. Fluids A 5, 3167.Google Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layers structure. J. Fluid Mech. 107, 297.Google Scholar
Henningson, D. S. 1988 The inviscid initial value problem for a piecewise linear mean flow. Stud. Appl. Maths 78, 31.Google Scholar
Henningson, D. S., Lundbladh, A. & Johansson, A. V. 1993 A mechanism for bypass transition from localized disturbances in wall bounded shear flows. J. Fluid Mech. 250, 169.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28, 735.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, vol. 6, 2nd Edn., pp. 99–100. Pergamon.
Moin, P. & Kim, J. 1985 The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical conditions. J. Fluid Mech. 155, 441.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347.Google Scholar
Roberts, P. H. 1972 A Hamiltonian theory for weakly interacting vortices. Mathematika. 19, 169.Google Scholar
Rott, N. & Cantwell, B. 1993 Vortex drift. I: Dynamic interpretation. Phys. Fluids A 5, 1443.Google Scholar
Russell, J. M. & Landahl, M. T. 1984 The evolution of a flat eddy near a wall in an inviscid shear flow. Phys. Fluids 27, 557.Google Scholar
Theodorsen, T. 1952 Mechanism of turbulence. Proc. 2nd Midwestern Conf. on Fluid Mech., Ohio State University.