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Development of mixing and isotropy in inviscid homogeneous turbulence

Published online by Cambridge University Press:  20 April 2006

Jon Lee
Jon Lee
Affiliation:
Flight Dynamics Laboratory, Wright-Patterson AFB, Ohio 43322, U.S.A.
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Abstract

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 120 , July 1982 , pp. 155 - 183
Copyright
© 1982 Cambridge University Press

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References

Arnold, V. I. & Avez, A. 1968 Ergodic Problems of Classical Mechanics. Benjamin.
Bellman, R. 1953 Stability Theory of Differential Equations. McGraw-Hill.
Benettin, G., Galgani, L. & Strelcyn, J.-M. 1976 Phys. Rev. A 14, 2338.
Benettin, G., Casartelli, M., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1978 Nuovo Cim. 44B, 183.
Birkhoff, G. & Fisher, J. 1959 Rend. Circ. Mat. Palermo 8, 77.
Birkhoff, G. D. 1927 Dynamical Systems. Am. Math. Soc. Coll. Publ. vol. 9.
Brissaud, A., Frisch, U., Léorat, J., Lesieur, M. & Mazure, A. 1973 Phys. Fluids 16, 1366.
Glaz, H. M. 1977 Statistical study of approximations to two dimensional inviscid turbulence. Lawrence Berkeley Lab. Rep. LBL-6708.Google Scholar
Glaz, H. M. 1981 SIAM J. Appl. Math. 41, 459.
Hemmer, P. C. 1959 Dynamic and stochastic types of motion in the linear chain. Thesis, Trondheim, Norway.
Hénon, M. & Heiles, C. 1964 Astron. J. 69, 73.
Kells, L. C. & Orszag, S. A. 1978 Phys. Fluids 21, 162.
Khinchin, A. I. 1949 Mathematical Foundations of Statistical Mechanics. Dover.
Kraichnan, R. H. 1973 J. Fluid Mech. 59, 745.
Lanford, O. E. 1976 Qualitative and Statistical Theory of Dissipative Systems. Lecture notes of 1976 CIME School of Statistical Mechanics.
Lebowitz, J. L. 1972 In Statistical Mechanics: New Concepts, New Problems, New Applications (ed. S. A. Rice, K. F. Freed & J. C. Light), p. 41. University of Chicago Press.
Lee, J. 1975 J. Math. Phys. 16, 1367.
Lee, J. 1979 Phys. Fluids 22, 40.
Lee, J. 1980 Emergence of periodic and nonperiodic motions in a Burgers’ channel flow model. In Proc. Int. Conf. on Nonlinear Phenomena in Mathematical Sciences, University of Texas at Arlington, June 1980.
Lee, J. 1982 Phys. Fluids (to appear).
Lee, T. D. 1952 Quart. Appl. Math. 10, 69.
Leith, C. E. 1978 Ann. Rev. Fluid Mech. 10, 107.
Liepmann, H. W. 1979 Am. Scientist 67, 221.
Lorenz, E. N. 1969 Tellus 21, 289.
Mclaughlin, J. 1976 J. Stat. Phys. 15, 307.
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT Press.
Nemytskii, V. V. & Stepanov, V. V. 1960 Qualitative Theory of Differential Equations. Princeton University Press.
Orszag, S. A. & Patterson, G. S. 1972 Phys. Rev. Lett. 28, 76.
Prigogine, I. 1979 Astrophys. Space Sci. 65, 371.
Robinson, G. D. 1967 Quart. J. R. Met. Soc. 43, 409.
Ruelle, D. 1979 Ann. N.Y. Acad. Sci. 316, 408.
Shampine, L. F. & Gordon, M. K. 1975 Computer Solution of Ordinary Differential Equations. Freeman.
Sinai, Ya. G. 1973 In The Boltzmann Equation — Theory and Applications (Acta Phys. Austriaca, Suppl. X) (ed. E. G. D. Cohen & W. Thirring), p. 575. Springer.