Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-29T00:58:08.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex dynamics of the two-dimensional turbulent shear layer

Published online by Cambridge University Press:  19 April 2006

Hassan Aref  and
Eric D. Siggia
Hassan Aref
Affiliation:
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
Eric D. Siggia
Affiliation:
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
Get access Rights & Permissions [Opens in a new window]

Abstract

The role of large vortex structures in the evolution of a two-dimensional shear layer is studied numerically. The motion of up to 4096 vortices is followed on a 256 × 256 grid using the cloud-in-cell algorithm. The scaling predictions of self-preservation theory are confirmed for low-order velocity correlations, although the existence of vortex structures produces large fluctuations even in a simulation of this size. The simple picture of the shear layer as a line of vortex blobs, that merge pairwise thus thickening the layer, is not seen. On the contrary, the layer seems to thicken by the scattering of vortex structures of roughly fixed size about the midline. The size of the vortex structures does not scale with the layer thickness. A study of the entrainment of a passive marker shows that flow visualization experiments may have overestimated the size of the vortex structures. It appears that the finite area vortices have time to equilibrate between mergings, and the consequences of applying equilibrium statistical mechanics to their internal structure are explored. A simple model is presented which demonstrates how the size and separation of vortex structures may lock into a fixed ratio. This is precisely the type of mechanism that is needed to produce simple scaling in a flow that has initially several distinct length scales. A number of consistency checks on the numerical results are performed. In particular, the evolution of the same vortex configuration on two grids of different size is compared. This test showed that, although errors on subgrid scales do propagate to small wavenumbers, the dominant wavenumber of vorticity cascades back ahead of the peak in the error spectrum.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 100 , Issue 4 , 29 October 1980 , pp. 705 - 737
Copyright
© 1980 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

Acton, E. 1976 The modelling of large eddies in a two-dimensional shear layer. J. Fluid Mech. 76, 561–592.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
Batt, R. G. 1977 Turbulent mixing of passive and chemically reacting species in a low-speed shear layer. J. Fluid Mech. 82, 53–95.Google Scholar
Browand, F. K. & Latigo, B. O. 1979 Growth of the two-dimensional mixing layer from a turbulent and nonturbulent boundary layer. Phys. Fluids 22, 1011–1019.Google Scholar
Browand, F. K. & Troutt, T. 1980 A note on spanwise structure in the two-dimensional mixing layer. J. Fluid Mech. 97, 771–781.Google Scholar
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76, 127–145.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Buneman, O. 1973 Subgrid resolution of flow and force fields. J. Comp. Phys. 11, 250–268.Google Scholar
Chandrsuda, C., Mehta, R. D., Weir, A. D. & Bradshaw, P. 1978 Effect of free-stream turbulence on large structure in turbulent mixing layers. J. Fluid Mech. 85, 693–704.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
Damms, S. M. & Küchemann, D. 1974 On a vortex-sheet model for the mixing between parallel streams. Proc. Roy. Soc. A 339, 451–461.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.
Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78, 535–560.Google Scholar
Fiedler, H. 1978 Structure and Mechanisms of Turbulence, I, II. Lecture Notes in Physics, volumes 75, 76. Springer.
Frisch, U., Sulem, P.-L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719–736.Google Scholar
Hockney, R. W., Goel, S. P. & Eastwood, J. W. 1974 Quiet high-resolution computer models of a plasma. J. Comp. Phys. 14, 148–158.Google Scholar
Kida, S. 1975 Statistics of the system of line vortices. J. Phys. Soc. Japan. 39, 1395–1404.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. U.S.S.R. 30, 301–305.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.Google Scholar
Kraichnan, T.-H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1969 Statistical Physics, section 75. Addison-Wesley.
Leonard, A. 1980 Vortex methods for flow simulation. J. Comp. Phys. (to appear).Google Scholar
Lundgren, T. S. & Pointin, Y. B. 1977 Statistical mechanics of two-dimensional vortices. J. Stat. Phys. 17, 323–355.Google Scholar
Michalke, A. 1964 Zur Instabilität und nichtlinearen Entwicklung einer gestörten Scherschicht. Ing. Arch. 33, 264–276.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1 (ed. J. L. Lumley). Massachusetts Institute of Technology Press.
Moore, D. W. & Saffman, P. G. 1975 The density of organized vortices in a turbulent mixing layer. J. Fluid Mech. 69, 465–473.Google Scholar
Onsager, L. 1949 Statistical Hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.Google Scholar
Roshko, A. 1976 Structure of turbulent shear flows: A new look. A.I.A.A. J. 14, 1349–1357.Google Scholar
Rossow, V. J. 1977 Convective merging of vortex cores in lift-generated wakes. J. Aircraft 14, 283–290.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95–122.Google Scholar
Shampine, L. F. & Gordon, M. K. 1975 Computer Solution of Ordinary Differential Equations. W. H. Freeman.
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Takaki, R. & Kovasznay, L. S. G. 1978 Statistical theory of vortex merger in the two-dimensional mixing layer. Phys. Fluids 21, 153–156.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
Wygnanski, I. & Fiedler, H. E. 1970 The two-dimensional mixing region. J. Fluid Mech. 41, 327–361.Google Scholar
Wygnanski, I., Oster, D., Fiedler, H. & Dziomba, B. 1979 On the perseverance of a quasi-two-dimensional eddy-structure in a turbulent mixing layer. J. Fluid Mech. 93, 325–335.Google Scholar
Zabusky, N. J. 1977 Coherent structures in fluid dynamics. In The Significance of Nonlinearity in the Natural Sciences, pp. 145–205. Plenum Press.
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