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A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence

Published online by Cambridge University Press:  19 April 2006

P. A. Durbin
P. A. Durbin
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

A new definition of concentration fluctuations in turbulent flows is proposed. The definition implicitly incorporates smearing effects of molecular diffusion and instrumental averaging. A stochastic model of two-particle dispersion, consistent with this definition, is formulated. The stochastic model is an extension of Taylor's (1921) model and is consistent with Richardson's $\frac{4}{3}$ law. Its predictions of concentration fluctuations are contrasted with predictions based on a more usual one-particle model.

The present model is used to predict fluctuations in three case studies. For example (case (i) of § 6), downstream of a linear concentration gradient we find $\overline{c^{\prime 2}}=\frac{1}{2}m^2(\overline{Z^2}-\overline{\Delta^2})$. Here m is the linear gradient, $\overline{Z^2}$ is related to centre-of-mass dispersion and $\overline{\Delta^2}$ is related to relative dispersion (see equation (3.1)). The term $\frac{1}{2}m^2\overline{Z^2} $ represents net production of fluctuations by random centre-of-mass dispersion, whereas $\frac{1}{2}m^2\overline{\Delta^2} $ represents net destruction of fluctuations by relative dispersion. Only the first term is included in the usual one-particle model (Corrsin 1952).

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 100 , Issue 2 , 25 September 1980 , pp. 279 - 302
Copyright
© 1980 Cambridge University Press

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References

Arnold, L. 1974 Stochastic Differential Equations: Theory and Applications. Wiley.
Batchelor, G. K. 1952 Proc. Camb. Phil. Soc. 48, 345.
Chatwin, P. C. & Sullivan, P. J. 1979 J. Fluid Mech. 91, 337.
Corrsin, S. 1952 J. Appl. Phys. 23, 113.
Csanady, G. T. 1967 J. Atmos. Sci. 24, 21.
Csanady, G. T. 1973 Turbulent Diffusion in the Environment. Dodrecht, Holland: Reidel.
Durbin, P. 1980 A random flight model of inhomogeneous turbulent dispersion. Phys. FluidsGoogle Scholar
Gad-el-hak, M. & Morton, J. B. 1979 A.I.A.A. J. 17, 558.
Lamb, R. G. 1976 3rd Symp. on Atmos. Turbulent Diffusion and Air Pollution, Rayleigh, N.C.
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, volumes 1 and 2. Massachusetts Institute of Technology.
Murthy, C. R. & Csanady, G. T. 1971 J. Phys. Oceanog. 1, 17.
Taylor, G. I. 1921 Proc. London Math. Soc. 20, 196.
Thiebaux, M. L. 1975 J. Atmos. Sci. 32, 92.
Townsend, A. A. 1976 Structure of Turbulent Shear Flow. Cambridge University Press.
Wax, N. 1954 Selected Papers on Noise and Stochastic Processes. Dover.