Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T11:23:42.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Final equilibrium state of a two-dimensional shear layer

Published online by Cambridge University Press:  26 April 2006

J. Sommeria ,
C. Staquet  and
R. Robert
J. Sommeria
Affiliation:
Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46 al.d'Italie, 69 364 Lyon, France
C. Staquet
Affiliation:
Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46 al.d'Italie, 69 364 Lyon, France
R. Robert
Affiliation:
21, Avenue Plaine Fleurie 38240 Meylan, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 233 , December 1991 , pp. 661 - 689
Copyright
© 1991 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.)

Article purchase

Temporarily unavailable

References

Antipov, S. V., Nezlin, M. V., Snezhkin, E. N. & Trubnikov, A. S. 1986 Rossby autosolution and stationary model of the Great Red Spot. Nature 323, 238–240.Google Scholar
Corcos, G. M. & Sherman, F. S. 1984 The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid Mech. 139, 29–65.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Gidas, B., Ni, W. M. & Nirenberg, L. 1979 Commun. Math. Phys. 68, 203–243.
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.Google Scholar
Liouville, J. 1853 Sur l’équation aux différences partielles 2logLD/u LD/2a2 = 0. J. Maths 18, 71–72.Google Scholar
Montgomery, D. & Joyce, G. 1974 Statistical mechanics of negative temperature states. Phys. Fluids 17, 1139–1145.Google Scholar
Nguyen, Duc, J. M. & Sommeria, J. 1988 Experimental characterization of steady two-dimensional vortex couples. J. Fluid Mech. 192, 175–192.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento suppl. 6, 279.Google Scholar
Patterson, G. S. & Orszag, S. A. 1971 Spectral calculations of isotropic turbulence: efficient removal of aliasing interactions. Phys. Fluids 14, 2538–2541.Google Scholar
Rabaud, M. & Couder, Y. 1983 A shear flow instability in a circular geometry. J. Fluid Mech. 136, 291–319.Google Scholar
Robert, R. 1989 Concentration et entropie pour les mesures d'Young. C. R. Acad. Sci. Paris I 309, 757–760.Google Scholar
Robert, R. 1990 Etat d’équilibre statistique pour l’écoulement bidimensionnel d'un fluide parfait. C. R. Acad. Sci. Paris I 311, 575–578.Google Scholar
Robert, R. 1991 Maximum entropy principle for two-dimensional Euler equations. J. Statist. Phys. (to appear).Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.Google Scholar
Sommeria, J., Meyers, S. D. & Swinney, H. L. 1988 Laboratory simulation of Jupiter's Great Red Spot. Nature 331, 689–693.Google Scholar
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar