Analytical Model of Two-Dimensional Dispersion in Laterally Nonuniform Axial Velocity Distributions
Publication: Journal of Hydraulic Engineering
Volume 123, Issue 10
Abstract
An analytical model of the advection-dispersion phenomenon in rivers with a laterally nonuniform axial velocity distribution is presented. The dispersion phenomenon is assumed to be governed by the two-dimensional advection-diffusion equation with constant but anisotropic turbulent diffusion coefficients. An infinitely long river with a prismatic cross section bounded laterally by parallel nondispersive banks is assumed. The velocity distribution is allowed to vary in an arbitrary functional form, but was restricted in this work to a family of power-law velocity distributions. The method of moments is used to derive the important statistical parameters of the concentration distribution. The concentration moment equations are solved analytically using the method of Greens function coupled with the method of images. The moments are then used to construct an approximate model of two-dimensional dispersion for an arbitrary velocity function, an initial distribution, and source injection scenarios. A one-dimensional simplification of the two-dimensional dispersion model with a time-dependent dispersion coefficient is also outlined. The dependence of the concentration moments on the velocity distribution and the shape of the source distribution is analyzed, and numerical results for a plane source and vertical line sources at the centerline and at the side are compared. The effect of the asymmetry of the velocity profile on the mixing length and time and on the skewness of the concentration distribution is shown to be significant, which might partly explain the persistence of the skewness observed in the field. However, the effect of the source injection scenario was not significant at large times. The analytical results can be used to model the fate and movement of pollutants and to better assess the effect of discharge siting on the dispersion of a contaminant. The model can be also used as a simple practical tool in simulating transport in a nearly prismatic river system.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Aris, R.(1956). “On the dispersion of a solute in a fluid flowing in a tube.”Proc., Royal Soc., London, England, Ser. A, 235, 67–77.
2.
Barry, D. A., and Sposito, G.(1989). “Analytical solution of a convection-dispersion model with time-dependent transport coefficients.”Water Resour. Res., 25, 2407–2416.
3.
Barton, N. G.(1978). “The initial dispersion of soluble matter in three-dimensional flow.”J. Australian Math. Soc., B, 20, 265–279.
4.
Barton, N. G.(1983). “On the method of moments for solute dispersion.”J. Fluid Mech., 126, 205–218.
5.
Basha, H. A., and El-Habel, F. S.(1993). “Analytical solution of the one-dimensional scale-dependent transport equation.”Water Resour. Res., 29, 3209–3214.
6.
Beck, J. V., Cole, K. D., Haji-Sheikh, A., and Litkouhi, B. (1992). Heat conduction using Green's functions. Hemisphere Publishing Corp., Bristol, Pa.
7.
Beltaos, S.(1980). “Longitudinal dispersion in natural streams.”J. Hydr. Div., ASCE, 106, 151–172.
8.
Chatwin, P. C.(1970). “The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe.”J. Fluid Mech., 43, 321–352.
9.
Cleary, R. W., and Adrian, D. D.(1973). “New analytical solutions for dye diffusion equations.”J. Envir. Engrg. Div., ASCE, 99, 213–227.
10.
Fischer, H. B.(1967). “The mechanics of dispersion in natural streams.”J. Hydr. Div., ASCE, 93, 187–216.
11.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H. (1979). Mixing in inland and coastal waters. Academic Press, New York, N.Y.
12.
French, R. H. (1985). Open-channel hydraulics. McGraw-Hill, Inc., New York, N.Y.
13.
Gradshteyn, I. S., and Ryzhik, I. M. (1980). Tables of integrals, series and products. Academic, Orlando, Fla.
14.
Greenberg, M. D. (1971). Application of Green's functions in science and engineering. Prentice-Hall, Inc., Engelwood Cliffs, N.J.
15.
Haberman, R. (1987). Elementary applied partial differential equations, 2nd Ed., Prentice-Hall, Inc., Engelwood Cliffs, N.J.
16.
Harvey, C. F., and Gorelick, S. M.(1995). “Temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers.”Water Resour. Res., 31, 1895–1911.
17.
Kendall, M. G., and Stuart, A. (1977). The advanced theory of statistics, Vol. 1, 4th Ed., MacMillan, New York, N.Y.
18.
Liu, H., and Cheng, H. D.(1980). “Modified Fickian model for predicting dispersion.”J. Hydr. Div., ASCE, 106, 1021–1040.
19.
Nordin, C. F., and Troutman, B. M.(1980). “Longitudinal dispersion in rivers: the persistence of skewness in observed data.”Water Resour. Res., 16, 123–128.
20.
Purnama, A.(1988). “The effect of dead zones on longitudinal dispersion in streams.”J. Fluid Mech., 186, 351–377.
21.
Reichert, P., and Wanner, O.(1991). “Enhanced one-dimensional modeling of transport in rivers.”J. Hydr. Engrg., 117, 1165–1183.
22.
Sabol, G. V., and Nordin, C. F.(1978). “Dispersion in rivers as related to storage zones.”J. Hydr. Div., ASCE, 104, 695–708.
23.
Sayre, W. W.(1968). “Discussion of `The mechanics of dispersion in natural streams' by H. B. Fischer.”J. Hydr. Div., ASCE, 94, 1549–1556.
24.
Sayre, W.(1969). “Dispersion of silt particles in open channel flow.”J. Hydr. Div., ASCE, 95, 1009–1038.
25.
Schmid, B. H.(1995). “On the transient storage equations for longitudinal solute transport in open channels: temporal moments accounting for the effects of first-order decay.”J. Hydr. Res., 33, 595–610.
26.
Smith, R.(1981). “The early stages of contaminant dispersion in shear flow.”J. Fluid Mech., 111, 107–122.
27.
Tsai, Y. H., and Holley, E. R.(1978). “Temporal moments for longitudinal dispersion.”J. Hydr. Div., ASCE, 104, 1617–1634.
28.
Valentine, E. M., and Wood, I. R.(1977). “Longitudinal dispersion with dead zones.”J. Hydr. Div., ASCE, 103, 975–990.
29.
Valentine, E. M., and Wood, I. R.(1979). “Dispersion in rough rectangular channels.”J. Hydr. Div., ASCE, 105, 1537–1553.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 123 • Issue 10 • October 1997
Pages: 853 - 862
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Oct 1, 1997
Published in print: Oct 1997
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.