Fig. 1 Flood on the Cuiaba river, Mato Grosso, Brazil, on January 10, 1995.
A DIMENSIONLESS CONVECTION-DIFFUSION-DISPERSION EQUATION OF FLOOD WAVES
The convection-diffusion model of flood waves [
The convection-diffusion model of flood waves is originally
due to
in which
The convective celerity, or flood wave speed, is defined as follows [
in which A = flow area.
The diffusion coefficient, or hydraulic diffusivity, for Chezy friction in hydraulically wide channels is [
in which q = unit-width discharge, _{o}S = bottom slope, and _{o}F = Froude number, defined as follows [Chow, 1959]:
in which u = mean velocity, _{o}D = hydraulic depth, and _{o}g = gravitational acceleration. y.
_{o}
The dispersion coefficient, or hydraulic dispersivity, is the following [
Ponce [1991] has expressed the convective celerity as a function of the Froude and Vedernikov numbers, as shown by Eq. 6.
The Vedernikov number is the ratio of relative kinematic and dynamic wave celerities [Craya, 1952].
Following Dooge et al. [1982], Ponce [1991] expressed the diffusion coefficient in terms of the Vedernikov number, in effect generalizing it for all friction types (turbulent Manning or Chezy, and laminar)
and cross-sectional shapes,
including
hydraulically wide, triangular, and inherently stable [Ponce and Porras, 1995]:
in which [ Dooge et al. 1982; Ponce 1991]:
with
Given u, and defining a reference channel length _{o}y_{o}L = _{o}y /_{o}S [_{o}Lighthill and Whitham, 1955], the diffusion coefficient is:
Furthermore, with Eq. 9, the dispersion coefficient is:
To nondimensionalize Eq. 1, we choose appropriate
dimensionless variables such that
t' = t (u /_{o}L).
Then, Eq. 1 converts to:
_{o}
in which
and
Therefore, the three coefficients of the dimensionless convection-diffusion-dispersion equation
(Eqs. 12 to 14) are shown to be
The online calculator D, which may be
approximated as flow depth _{o}y, _{o}S,
and (d) exponent of the rating _{o}β.
The coefficients of the dimensionless partial differential equation of convection-diffusion-dispersion of flood waves are shown to be functions
Chow, V. T. 1959.
Craya, A. 1952.
Dooge, J. C. I. 1973.
Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski. 1982.
Ferrick, M. G., J. Bilmes, and S. E. Long. 1984.
Hayami, S.,
Lightbill, M. J., and G. B. Whitman. 1955.
Ponce, V. M. 1991.
Ponce, V. M. and P. J. Porras. 1995.
Seddon, J. A. 1900. |

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