The convection-diffusion model of flood waves [
The convection-diffusion model of flood waves is due to
in which Q is discharge, c is convective celerity, v is the diffusion coefficient, and η is the dispersion coefficient.
The convective celerity, or flood wave speed, is defined as [
in which A is flow area.
The diffusion coefficient, or hydraulic diffusivity, for Chezy friction in hydraulically wide channels is [
in which q is the unit-width discharge, _{o}S is the bottom slope, and _{o}F is the Froude number, defined as follows [Chow, 1959]:
in which u is the mean velocity, _{o}g is gravitational acceleration, and y is flow depth.
_{o}
The dispersion coefficient, or hydraulic dispersivity, is [
Ponce [1991] has expressed the convective celerity as a function of the Froude and Vedernikov numbers. The latter 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 (Manning or Chezy) and cross-sectional shapes (e.g., hydraulically wide, triangular, inherently stable):
in which [ Dooge et al. 1982; Ponce 1991]:
with
With u, and defining a reach 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:
Assume the dimensionless variables t' = t (u / _{o}L ). Then Eq. 1 converts to:
_{o}
in which
and
Thus the coefficients of the dimensionless convection-diffusion-dispersion equation are shown to be functions of the Froude and Vedernikov numbers only.
The coefficients of the dimensionless partial differential equation of convection-diffusion-dispersion of flood waves are shown to be functions of the Froude and Vedernikov numbers only. The Froude number is the ratio of mean velocity to relative dynamic wave celerity. The Vedernikov number is the ratio of relative kinematic wave celerity to relative dynamic wave celerity. The third-order convection-diffusion-dispersion equation can be used to analyze flood propagation problems where both diffusion and dispersion are deemed to be significant.
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