Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Turbulent dispersion from a plane surface

Authors: Mark J. Beran and Alan M. Whitman
Journal: Quart. Appl. Math. 33 (1975), 97-113
DOI: https://doi.org/10.1090/qam/99671
MathSciNet review: QAM99671
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Abstract | References | Additional Information

Abstract: We consider here the turbulent dispersion of a neutral concentrant from a plane surface. Very near the surface the dispersion is by molecular diffusion since the turbulent velocity is zero at the surface. Within a layer near the surface the mode of transport changes to turbulent exchange. Our principal concern in this paper is to determine the concentration distribution near the surface and the flux of concentrant that is transported from or to the surface. We use as our governing equation an integro-differential equation form which was derived in an earlier paper. In this equation we are required to assume a form for the integral kernel which is compatible with a turbulent state. A kernel was chosen using simple arguments and the resulting concentration distribution is presented. It is shown that under suitable conditions the concentration decays at large distances as an inverse power of the distance from the surface. We also present the dependence of flux on the molecular diffusion coefficient. For some physical situations the concentration will decay at a rate inversely proportional to the square of the distance, and the flux will be proportional to $ {D^{2/3}}$. ($ D$ is the molecular diffusion coefficient.) For two particular physical applications the validity of the above solution is discussed. We also compare the results obtained with those found using an eddy diffusivity. It is shown that in the presence of sources the two results may differ substantially.

References [Enhancements On Off] (What's this?)

  • [1] M. Beran and G. Assaf, J. Geophys. Res. 75, 5279 (1971)
  • [2] J. McCoy and M. Beran, Application of an iteration technique to the problem of turbulent diffusion, in Proceedings of 8th Annual Meeting--Soc. of Eng. Sci., Env. Eng., Washington, D. C., November 1970, p. 284, 1972, NASA, Langley Research Center, Hampton, Va.
  • [3] A. Monin and A. Yaglom, Statistical fluid mechanics, Mechanics of Turbulence 1, The MIT Press, Cambridge, Mass., 1971
  • [4] M. Lavrentiev, Some improperly posed problems of mathematical physics, Springer-Verlag, New York, 1967
  • [5] L. Scriven, Flow and transfer at fluid interfaces, Parts I, II, and III, 1968 Award Lecture ASEE, Chem. Eng. Ed., Fall 1968--p. 150, Winter 1969--p. 26, Spring 1969--p. 94.
  • [6] G. Fortescue and J. Pearson, Chemi Eng. Sci. 22, 1163, (1967)
  • [7] W. Broecker and T. H. Peng, Earth Planet. Sci. Let. 11, 99 (1971)

Additional Information

DOI: https://doi.org/10.1090/qam/99671
Article copyright: © Copyright 1975 American Mathematical Society

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