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Convergents to turbulence functions

Published online by Cambridge University Press:  29 March 2006

Robert H. Kraichnan
Robert H. Kraichnan
Affiliation:
Dublin, New Hampshire, U.S.A.
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Abstract

A method is described for constructing approximations, to statistical functions, that are uniformly convergent in time, starting with the expansion of the functions as Taylor series in time. The principal tool is a technique for expanding the Fourier transform of the unknown function by use of a set of orthonormal functions. Application to the Lagrangian velocity correlation and the eddy diffusivity for marked particles in a three-dimensional random velocity field yields results that agree excellently with computer simulations. The approximation procedure is extended to expansions in strength parameters (e.g. Reynolds number expansion) and to an expansion about the direct-interaction approximation. The latter is based on a new model representation of the direct-interaction approximation. An implication of the work is that the usual diagram expansions, obtained through term-by-term averaging over a Gaussian distribution, may not uniquely determine the functions they represent; it may be that truly meaningful expansions are possible, in general, only for distributions which bound the amplitudes in the individual realizations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 41 , Issue 1 , 26 March 1970 , pp. 189 - 217
Copyright
© 1970 Cambridge University Press

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References

Batchelor, G. K. 1953 Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. 1959 J. Fluid Mech. 5, 113.
Corrsin, S. 1952 J. Appl. Phys. 23, 113.
Edwards, S. F. 1964 J. Fluid Mech. 18, 239.
Edwards, S. F. & McComb, W. D. 1969 J. Phys. (2) A 2, 157.
Herring, J. 1965 Phys. Fluids, 8, 2219.
Herring, J. 1966 Phys. Fluids, 9, 2106.
Kadomtsev, B. B. 1965 Plasma Turbulence. London: Academic Press.
Kraichnan, R. H. 1958 Phys. Rev. 109, 1407.
Kraichnan, R. H. 1961 J. Math. Phys. 3, 496.
Kraichnan, R. H. 1964 Phys. Fluids, 7, 1030.
Kraichnan, R. H. 1966 Dynamics of Fluids and Plasmas (ed. S. I. Pai), pp. 239–255. New York: Academic Press.
Kraichnan, R. H. 1968 Phys. Rev. 174, 240.
Kraichnan, R. H. 1970a The Padé Approximant in Theoretical Physics (ed. G. Baker & J. Gammel). New York: Academic Press.
Kraichnan, R. H. 1970b Phys. Fluids, 13, 22.
Lee, L. L. 1965 Ann. Phys. 32, 292.
Lighthill, M. J. 1958 Fourier Analysis aud Generalised Functions. Cambridge University Press.
Lumley, J. L. 1962 La Mécanique de la Turbulence. Paris: C.N.R.S.
Novikov, E. A. 1964 Zh. Eksper. Teor. Fiz. 47, 1919.
Orszag, S. A. 1966 Dynamics of Fluid Turbulence. Princeton University, Dept. of Astro-physical Sciences Report PPLAF-13.
Orszag, S. A. 1970a J. Fluid Mech. 41, 385.
Orszag, S. A. 1970b The Statistical Theory of Turbulence. Cambridge University Press.
Phythian, R. 1969 J. Phys. (2) A 2, 181.
Roberts, G. E. & Kaufman, H. 1966 Laplace Transforms. Philadelphia: Saunders.
Shohat, J. A. & Tamarkin, J. D. 1943 The Problem of Moments. New York: American Mathematical Society.
Taylor, G. I. 1921 Proc. London Math. Soc. (2) 20, 196.
Wall, H. S. 1948 Analytic Theory of Continued Fractions. New York: Van Nostrand.
Wyld, H. W. 1961 Ann. Phys. 14, 143.