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Onset and development of turbulence in two-dimensional periodic shear flows

Published online by Cambridge University Press:  21 April 2006

Roland Grappin ,
Jacques Leorat  and
Pasquale Londrillo
Roland Grappin
Affiliation:
Observatoire de Meudon, CNRS UA173, F-92190 Meudon, France
Jacques Leorat
Affiliation:
Observatoire de Meudon, CNRS UA173, F-92190 Meudon, France
Pasquale Londrillo
Affiliation:
Osservatorio di Astronomico, Universita di Bologna, Italy
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Abstract

We investigate numerically the time evolution of a two-dimensional flow submitted to a spatially periodic shear force. Initially, the flow is at equilibrium, the forcing balancing viscous stresses. At Reynolds numbers slightly above critical, a large-scale, linear instability drives the fluid towards a stable laminar state. At larger Reynolds number turbulence finally develops after several transient states. These transient states are described by measuring the divergence rate of linearized trajectories from the turbulent flow. This rate gives asymptotically a measure of the first Lyapunov exponent of the flow. We find that the first Lyapunov exponent scales as the characteristic frequency of the flow at large scale. We show here data on incompressible, isothermal and perfect gas (subsonic) two-dimensional flows with unit Prandtl number, and Reynolds number around 30.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 195 , October 1988 , pp. 239 - 256
Copyright
© 1988 Cambridge University Press

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References

Benettin L., Galgani, L. & Strelcyn J. M. 1976 Phys. Rev. A 14, 2338.
Couder Y. 1984 Two-dimensional grid turbulence in a thin liquid film. J. Phys. Lett. 45, 353–360.Google Scholar
Frisch, U. & Sulem P. L. 1984 Numerical simulation of the inverse cascade in two-dimensional turbulence. Phys. Fluids 27, 1921.Google Scholar
Grappin R., Léorat, J. & Pouquet A. 1986 Computation of the dimension of a model of fully developed turbulence. J. Phys. Paris 47, 1127.Google Scholar
Grappin, R. & Léorat J. 1987 Computation of the dimension of two-dimensional turbulence. Phys. Rev. Lett. 59, 1100.Google Scholar
Green J. S. A. 1974 Two-dimensional turbulence near the viscous limit. J. Fluid Mech. 62, 273.Google Scholar
Herring J. R. 1983 The predictability of quasi-geostrophic flows. Proc. AIP Conf. No. 106. In Predictability of Fluid Motions (ed. G. Holloway & B. J. West), pp. 321–332.
Kells, L. C. & Orszag S. A. 1978 Randomness of low-order models of two-dimensional inviscid dynamics. Phys. Fluids 21, 162.Google Scholar
Klyatskin V. I. 1966 Izv. Atmos. Ocean. Phys. 2, 474.
Kolmogorov A. N. 1960 In Seminar Notes ed. V. I. Arnold & L. D. Meshalkin. Uspekhi Mat. Nauk 15, 247.
Kraichnan R. H. 1967 Inertial ranges in two- and three-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Lafon A. 1985 Etude des attracteurs pour des écoulements bidimensionnels de fluides visqueux incompressibles. Thèse d'état, Université Paris VI.
Léorat J., Pouquet, A. & Poyet J. P. 1984 Numerical simulations of supersonic turbulent flows. In Problems of Collapse and Numerical Relativity, p. 287. Reidel.
Lilly D. K. 1969 Phys. Fluids Suppl. 12, 240.
Lorenz E. N. 1969 The predictability of a flow which possesses many scales of motion. Tellus 21, 289–307.Google Scholar
Meshalkin, L. D. & Sinai Ya, G. 1961 Z. angew. Math. Mech. 25, 1700.
Métais, O. & Lesieur M. 1986 J. Atmos. Sci. 43, 857.
Obukhov A. M. 1983 Russ. Math. Surveys 38, 113.
Passot, T. & Pouquet A. 1987 Numerical simulations of compressible homogeneous flows in the turbulent regime. J. Fluid Mech. 181, 441.Google Scholar
Pouquet A., Lesieur M., André, J. C. & Basdevant C. 1975 J. Fluid Mech. 72, 305.
Pouquet A., Meneguzzi, M. & Frisch U. 1986 Phys. Rev. A 33, 4266.
She Z. S. 1987 Large scale dynamics and transition to turbulence in two dimensional Kolmogorov flows. Preprint Observatoire de Nice, France.
Sivashinsky, G. & Yakhot V. 1985 Negative viscosity effect in large-scale flows. Phys. Fluids 28, 1040.Google Scholar
Sommeria J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139.Google Scholar
Thompson P. D. 1957 Uncertainty of initial state as a factor in the predictability of large-scale atmospheric flow patterns. Tellus 9, 275–295.Google Scholar