Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-10-04T03:52:55.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution and breakdown of a vortex street in two dimensions

Published online by Cambridge University Press:  20 April 2006

Hassan Aref  and
Eric D. Siggia
Hassan Aref
Affiliation:
Laboratory of Atomic and Solid State Physics, Clark Hall, Cornell University, Ithaca, NY 14853, USA Division of Engineering, Brown University, Providence, Rhode Island 02912.
Eric D. Siggia
Affiliation:
Laboratory of Atomic and Solid State Physics, Clark Hall, Cornell University, Ithaca, NY 14853, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The initial-value problem defined by two parallel vortex sheets of opposite sign is studied. Strictly two-dimensional, incompressible, nearly inviscid dynamics is assumed throughout. The roll-up of the sheets into a vortex street is simulated numerically using 4096 point vortices. Much longer runs than in previous work are performed, and it is found that only for a finite range of values of the ratio, h/λ, of sheet separation to perturbation wavelength, does a long-lived vortex street emerge. For h/λ [gsim ] 0·6 a pairing transition within each row intervenes. For h/λ [lsim ] 0·3 we find oscillatory modes.

Using up to 16384 point vortices, we also study the breakdown of the metastable street to a two-dimensional, turbulent shear flow. The vortex blobs that made up the street may merge with others of the same sign after the breakdown, but otherwise persist throughout the turbulent regime. Neither their disintegration nor amalgamation with vortices of opposite sign was observed. Using dimensional arguments we derive the relevant scaling theory, and show that it applies to a flow started from two random vortex sheets. The resulting turbulence is not self-similar. For the turbulent flow that follows from the breakdown of a regular vortex street two length scales with different power-law growth in time appear to be necessary. The important differences in the asymptotic structure of flows initialized from random and regular sheets leads us to question the idea of universality. The influence of the symmetry of the initial perturbation on the subsequent development is also considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 109 , August 1981 , pp. 435 - 463
Copyright
© 1981 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

Abernathy, F. H. & Kronauer, R. E. 1962 The formation of vortex streets. J. Fluid Mech. 13, 1–20.Google Scholar
Aref, H. & Siggia, E. D. 1980 Vortex dynamics of the two-dimensional turbulent shear layer. J. Fluid Mech. 100, 705–737.Google Scholar
Ashurst, W. T. 1979 Numerical simulation of turbulent mixing layers via vortex dynamics. In Turbulent Shear Flows I (ed. F. Durst, B. E. Launder, F. W. Schmidt & J. H. Whitelaw), pp. 402–413. Springer.
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. 12, II 233–239.Google Scholar
Bearman, P. W. & Graham, J. M. R. 1980 Vortex shedding from bluff bodies in oscillatory flow: A report on Euromech 119. J. Fluid Mech. 99, 225–245.Google Scholar
Beavers, G. S. & Wilson, T. A. 1970 Vortex growth in jets. J. Fluid Mech. 44, 97–112.Google Scholar
Birkhoff, G. & Fisher, J. 1959 Do vortex sheets roll up? Rend. del Circ. Mat. Palermo 8, 77–90.Google Scholar
Chorin, A. J. 1973 Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796.Google Scholar
Chorin, A. J. 1980 Vortex models and boundary layer instability. SIAM J. Sci. Stat. Comput. 1, 1–21.Google Scholar
Christiansen, J. P. 1973 Numerical simulation of hydrodynamics by the method of point vortices. J. Comp. Phys. 13, 363–379.Google Scholar
Christiansen, J. P. & Zabusky, N. J. 1973 Instability, coalescence and fission of finite-area vortex structures. J. Fluid Mech. 61, 219–243.Google Scholar
Clements, R. R. & Maull, D. J. 1975 The representation of sheets of vorticity by discrete vortices. Prog. Aero. Sci. 16, 129–146.Google Scholar
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 547–591.Google Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: Stationary V-states, interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859–862.Google Scholar
Delcourt, B. A. G. & Brown, G. L. 1979 The evolution and emerging structure of a vortex sheet in an inviscid and viscous fluid modelled by a point vortex method. 2nd Symp. on Turbulent Shear Flows, Imperial College, London, p. 14.35.
Durgin, W. W. & Karlsson, S. K. F. 1971 On the phenomenon of vortex street breakdown. J. Fluid Mech. 48, 507–527.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Lamb, H. 1945 Hydrodynamics, 6th edn. Dover.
Leonard, A. 1980 Vortex methods for flow simulation. J. Comp. Phys. 37, 289–335.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1 (ed. J. L. Lumley). Massachusetts Institute of Technology Press.
Papailiou, D. D. & Lykoudis, P. S. 1974 Turbulent vortex streets and the entrainment mechanism of the turbulent wake. J. Fluid Mech. 62, 11–31.Google Scholar
Roshko, A. 1954 On the development of turbulent wakes from vortex streets. N.A.C.A. Rep. 1191.Google Scholar
Roshko, A. 1961 Experiments on the flow past a circular cylinder at very high Reynolds number. J. Fluid Mech. 60, 345–356.Google Scholar
Taneda, S. 1959 Downstream development of the wakes behind cylinders. J. Phys. Soc. Japan 14, 843–848.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. Massachusetts Institute of Technology Press.
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.Google Scholar
Zabusky, N. J. 1977 Coherent structures in fluid dynamics. In The Significance of Nonlinearity in the Natural Sciences, pp. 145–205. Plenum.
Zabusky, N. J.& Deem, G. S. 1971 Dynamical evolution of two-dimensional unstable shear flows. J. Fluid Mech. 47, 353–379.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comp. Phys. 30, 96–106.Google Scholar