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A Markovian random coupling model for turbulence

Published online by Cambridge University Press:  29 March 2006

U. Frisch ,
M. Lesieur  and
A. Brissaud
U. Frisch
Affiliation:
Centre National de la Recherche Scientifique, Observatoire de Nice, France
M. Lesieur
Affiliation:
Centre National de la Recherche Scientifique, Observatoire de Nice, France
A. Brissaud
Affiliation:
Ecole Nationale Supérieure de l'Aéronautique et de l'Espace, Toulouse, France
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Abstract

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.

The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 1 , 12 August 1974 , pp. 145 - 152
Copyright
© 1974 Cambridge University Press

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References

Brauner, C. M., Penel, P. & Teman, R. 1974 Sur un problème intervenant en théorie de la turbulence. Dept. Maths., Orsay Preprint.
Brissaud, A. & Frisch, U. 1974 J. Math. Phys. 15, 527.
Brissaud, A., Frisch, U., Leorat, J., Lesieur, M. & Mazure, A. 1973 Phys. Fluids, 16, 1366.
Brissaud, A., Frisch, U., Leorat, J., Lesieur, M., Mazure, A., Pouquet, A., Sadourny, R. & Sulem, P. L. 1973 Ann. Géophys. 29, 539.
Burgers, J. M. 1950 Proc. Acad. Sci. Amst. 53, 247.
Edwards, S. F. 1964 J. Fluid Mech. 18, 239.
Edwards, S. F. & McComb, W. D. 1969 J. Phys. A2, 157.
Frisch, U. 1973 High Reynolds number fluid turbulence and energetic catastrophes. In Proc. Symp. Turbulence in Fluids & Plasmas, Culham. To appear.
Frisch, U. & Bourret, R. 1970 J. Math. Phys. 11, 364.
Frisch, U., Leorat, J., Mazure, A. & Pouquet, A. 1973 MHD helical turbulence. Observatoire de Nice Preprint.
Herring, J. 1965 Phys. Fluids, 8, 2219.
Herring, J. 1966 Phys. Fluids, 9, 2106.
Herring, J. & Kraichnan, R. H. 1972 In Statistical Models and Turbulence, p. 148. Springer.
Hopf, E. 1962 Proc. Symp. Appl. Math. 13, 157. Am. Math. Soc.
Kraichnan, R. H. 1958 Second Symp. of Naval Hydrodyn. (ed. R. Cooper). Washington: Office of Naval Research, Publ. ACR-38.
Kraichnan, R. H. 1959 J. Fluid Mech. 5, 497.
Kraichnan, R. H. 1961 J. Math. Phys. 2, 124.
Kraichnan, R. H. 1967 Phys. Fluids, 10, 1417.
Kraichnan, R. H. 1971 J. Fluid Mech. 47, 513.
Kraichnan, R. H. 1973 J. Fluid Mech. 59, 745.
Kubo, R. 1963 J. Math. Phys. 4, 174.
Lee, T. D. 1952 Quart. Appl. Math. 10, 69.
Leibowitz, M. A. 1963 J. Math. Phys. 4, 852.
Leith, C. E. 1971 J. Atmos. Sci. 28, 145.
Lesieur, M. 1973 Ph.D. thesis, University of Nice.
Ogura, Y. 1963 J. Fluid Mech. 16, 33.
Orszag, S. A. 1969 J. Fluid Mech. 41, 363.
Pouquet, A., Lesieur, M. & André, J. C. 1973 Evolution of high Reynolds number two-dimensional turbulence. Observatoire de Nice Preprint.
Prigogine, I. 1962 Nonequilibrium Statistical Mechanics. Interscience.
Tatsumi, T. 1957 Proc. Roy. Soc. A, 239, 16.