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The effect of aspect ratio on longitudinal diffusivity in rectangular channels

Published online by Cambridge University Press:  20 April 2006

P. C. Chatwin  and
Paul J. Sullivan
P. C. Chatwin
Affiliation:
Department of Applied Mathematics and Theoretical Physics, The University of Liverpool, U.K.
Paul J. Sullivan
Affiliation:
Department of Applied Mathematics, The University of Western Ontario, Canada
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Abstract

In a recent paper Doshi, Daiya & Gill (1978) showed that the value of Taylor's longitudinal diffusivity D for laminar flow in a channel of rectangular cross-section of breadth u and height b is about 8D0, for large values of the aspect ratio a/b, where Do is the value of the longitudinal diffusivity obtained by ignoring all variation across the channel. This superficially surprising result is confirmed by an independent method, and is shown to be caused by the boundary layers on the side walls of the channel. The primary purpose of the paper, however, is to consider the value of D in turbulent flow in a flat-bottomed channel of large aspect ratio, for which arguments based on physics are adduced in support of the formula D≈[1 + B][1 - λ(b/u)], where B and λ are positive constants independent of b. It is shown that this result is consistent with laboratory experiments by Fischer (1966). The paper concludes with a discussion of the practical effects of aspect ratio on longitudinal dispersion in channels whose cross-section is approximately rectangular.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 120 , July 1982 , pp. 347 - 358
Copyright
© 1982 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Dover.
Allen, C. M. 1982 Numerical simulation of contaminant dispersion in estuary flows. Proc. R. Soc. Lond. A (to appear).
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67.Google Scholar
Chatwin, P. C. 1970 The approach to normality of the concentration distribution of solute in solvent flowing along a straight pipe. J. Fluid Mech. 43, 321.Google Scholar
Chatwin, P. C. 1971 On the interpretation of some longitudinal dispersion experiments. J. Fluid Mech. 48, 689.Google Scholar
Chatwin, P. C. 1972 The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube. J. Fluid Mech. 51, 63.Google Scholar
Chatwin, P. C. 1973 A calculation illustrating effects of the viscous sub-layer on longitudinal dispersion. Quart. J. Mech. Appl. Math. 26, 427.Google Scholar
Chatwin, P. C. 1977 The initial development of longitudinal dispersion in straight tubes. J. Fluid Mech. 80, 33.Google Scholar
Chatwin, P. C. 1980 Presentation of longitudinal dispersion data. J. Hydraul. Div. A.S.C.E. 106, 71.Google Scholar
Dewey, R. & Sullivan, P. J. 1977 The asymptotic stage of longitudinal turbulent dispersion within a tube. J. Fluid Mech. 80, 293.Google Scholar
Dewey, R. & Sullivan, P. J. 1979 Longitudinal dispersion in flows that are homogeneous in the streamwise direction. Z. angew. Math. Phys. 30, 601.Google Scholar
Doshi, M. R., Daiya, P. M. & Gill, W. N. 1978 Three dimensional laminar dispersion in open and closed rectangular conduits. Chem. Engng Sci. 33, 795.Google Scholar
Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544.Google Scholar
Fischer, H. B. 1966 Longitudinal dispersion in laboratory and natural streams. Caltech Rep. KH-R-12.Google Scholar
Fischer, H. B. 1973 Longitudinal dispersion and turbulent mixing in open-channel flow. Ann. Rev. Fluid Mech. 5, 59.Google Scholar
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic.
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1. M.I.T. Press.
Saffman, P. G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8, 273.Google Scholar
Smith, R. 1980 Buoyancy effects upon longitudinal dispersion in wide well-mixed estuaries. Phil. Trans. R. Soc. Lond. A 296, 467.Google Scholar
Smith, R. 1981a A delay-diffusion description for contaminant dispersion. J. Fluid Mech. 105, 469.Google Scholar
Smith, R. 1981b The importance of discharge siting upon contaminant dispersion in narrow rivers and estuaries. J. Fluid Mech. 108, 43.Google Scholar
Sullivan, P. J. 1971 Longitudinal dispersion within a two-dimensional turbulent shear flow. J. Fluid Mech. 49, 551.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. 20, 196.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing through a tube. Proc. R. Soc. Lond. A 219, 186.Google Scholar
Taylor, G. I. 1954 The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A 223, 446.Google Scholar