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The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow

Published online by Cambridge University Press:  19 April 2006

R. Phythian  and
W. D. Curtis
R. Phythian
Affiliation:
Department of Physics, University College of Swansea, Wales
W. D. Curtis
Affiliation:
Department of Physics, University College of Swansea, Wales
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Abstract

The problem considered is the diffusion of a passive scalar in a ‘fluid’ in random motion when the fluid velocity field is Gaussian and statistically homogeneous, isotropic and stationary. A self-consistent expansion for the effective long-time diffusivity is obtained and the approximations derived from this series by retaining up to three terms are explicitly calculated for simple idealized forms of the velocity correlation function for which numerical simulations are available for comparison for zero molecular diffusivity. The dependence of the effective diffusivity on the molecular diffusivity is determined within this idealization. The results support Saffman's contention that the molecular and turbulent diffusion processes interfere destructively, in the sense that the total effective diffusivity about a fixed point is less than that which would be obtained if the two diffusion processes acted independently.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 89 , Issue 2 , 28 November 1978 , pp. 241 - 250
Copyright
© 1978 Cambridge University Press

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References

Batchelor, G. K., Howells, I. D. & Townsend, A. A. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. J. Fluid Mech. 5, 134–139.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1956 Turbulent diffusion. In Surveys in Mechanics, pp. 352–399. Cambridge University Press.
Kraichnan, R. H. 1964 Kolmogorov's hypothesis and Eulerian turbulence theory. Phys. Fluids 7, 1723–1734.Google Scholar
Kraichnan, R. H. 1965 Lagrangian history closure approximation for turbulence. Phys. Fluids 8, 575–596.Google Scholar
Kbaichnan, R. H. 1970a Diffusion by a random velocity field. Phys. Fluids 13, 22–31.Google Scholar
Kraichnan, R. H. 1970b Convergents to turbulence functions. J. Fluid Mech. 41, 189–217.Google Scholar
Kraichnan, R. H. 1975 Remarks on turbulence theory. Adv. in Math. 16, 305–331.Google Scholar
Kraichnan, R. H. 1976a Diffusion of weak magnetic fields by isotropic turbulence. J. Fluid Mech. 75, 657–676.Google Scholar
Kraichnan, R. H. 1976b Diffusion of passive scalar and magnetic fields by helical turbulence. J. Fluid Mech. 77, 753–768.Google Scholar
Lundgren, T. S. & Pointin, Y. B. 1976 Turbulent self-diffusion. Phys. Fluids 19, 355–358.Google Scholar
Martin, P. C., Siggia, E. D. & Rose, H. A. 1973 Statistical dynamics of classical systems. Phys. Rev. A 8, 423–437.Google Scholar
Morton, J. B. & Corrsin, S. 1970 Consolidated expansions for estimating the response of a randomly driven nonlinear oscillatior. J. Stat. Phys. 2, 153–194.Google Scholar
Phythian, R. 1972 Some variational methods in the theory of turbulent diffusion. J. Fluid Mech. 53, 469–480.Google Scholar
Phythian, R. 1975 Dispersion by random velocity fields. J. Fluid Mech. 67, 145–153.Google Scholar
Saffman, P. G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8, 273–283.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. Ser. 2, 20, 196–212.Google Scholar