New perspective on the convection-diffusion-dispersion equation


Victor M. Ponce and P. T. Huston


Online version 2016

[Original version 1994]



ABSTRACT

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.


1.  INTRODUCTION

The convection-diffusion model of flood waves [Hayami, 1951; Dooge, 1973] is improved with the addition of the third-order dispersion term [Ferrick et al., 1984], in effect constituting a convection-diffusion-dispersion model. With the appropriate dimensional technique, the coefficients of the resulting third-order partial differential equation can be expressed in terms of the Froude and Vedernikov numbers only. This underscores the role of these two dimensionless numbers in describing the dynamics of flood wave propagation.


2.  BACKGROUND

The convection-diffusion model of flood waves is due to Hayami [1951]. Dooge [1973] improved the Hayami model by including inertia in the formulation of the diffusion coefficient, resulting in a convection-diffusion model with inertia. Dooge et al. [1982] generalized the convection-diffusion model with inertia for any friction type and cross-sectional shape. Ponce [1991] expressed the diffusion coefficient in terms of the Vedernikov number. Ferrick et al. [1984] derived the most complete third-order linear equation to date, which includes convection, diffusion, and dispersion processes.

Ferrick et al.'s equation is:


Qt + c Qx = v Qxx + η Qxxx

(1)

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 [Seddon, 1900; Chow, 1959]:


        dQ
c = _____
        dA

(2)

in which A is flow area.

The diffusion coefficient, or hydraulic diffusivity, for Chezy friction in hydraulically wide channels is [Dooge, 1973]:


         qo                  F 2
v = _______ [ 1 - ( ____ )]
        2 So                4

(3)

in which qo is the unit-width discharge, So is the bottom slope, and F is the Froude number, defined as follows [Chow, 1959]:


           uo
F = _________
       (g yo)1/2

(4)

in which uo is the mean velocity, g is gravitational acceleration, and yo is flow depth.

The dispersion coefficient, or hydraulic dispersivity, is [Ferrick et al., 1984]:


                  yo
η = F 2 ( ______ ) v
                 2So

(5)

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]:


                 V
c = ( 1 + ____ ) uo
                 F

(6)

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):


         qo
v = ______ ( 1 - V 2 )
        2 So

(7)

in which [Dooge et al. 1982; Ponce 1991]:


V = ( β - 1 ) F

(8)

with β equal to the exponent of the discharge-flow area rating Q = α Aβ.

With qo = uo yo, and defining a reach length Lo = yo / So [Lighthill and Whitham, 1955], the diffusion coefficient is


           Lo
v = ( _____ ) uo ( 1 - V 2 )
           2

(9)

Furthermore, with Eq. 9, the dispersion coefficient is:


           Lo
η = ( _____ ) 2 uo ( 1 - V 2 ) F 2
            2

(10)


3.  DIMENSIONLESS CONVECTION-DIFFUSION-DISPERSION EQUATION

Assume the dimensionless variables x' = x / Lo, and t' = t (uo / Lo ). Then Eq. 1 converts to:


Qt'  + c' Qx'  = v' Qx'x'  + η' Qx'x'x'

(11)

in which c'  is a dimensionless celerity


               V
c' = 1 + _____
               F

(12)

v'  is a dimensionless diffusivity


         1
v' = ____ ( 1 - V 2 )
         2

(13)

and η'  is a dimensionless dispersivity


         1
η' = ____ ( 1 - V 2 ) F 2
         4

(14)

Thus the coefficients of the dimensionless convection-diffusion-dispersion equation are shown to be functions of the Froude and Vedernikov numbers only.


4.  SUMMARY

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.


REFERENCES

Chow, V. T., Open-channel Hydraulics, McGraw-Hill. New York. 1959.

Craya, A., The criterion for te possibility of roll wave formation, in Gravity Waves, Circ. 521, pp. 141-151, Natl. Inst. of Stand. and Technol., Gaithersburg, Md., 1952.

Dooge, J. C. I., Linear theory of hydrologic systems, Tech. Bull 1468, 327 pp., U.S. Dep. of Agric., Washington. D. C., 1973.

Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski, Hydrodynamic derivation of storage parameters of the Muskingum model, J. Hydrol., 54, 371-387, 1982.

Ferrick, M. G., J. Bilmes, and S. E. Long, Modeling rapidly varied flow in tailwaters, Water Resour. Res., 20(2), 271-289, 1984.

Hayami, S., On the propagation of flood waves, Bull. Disaster Prev. Res. Inst. Kyoto Univ. Jpn, 1, 1-16, 1951.

Lightbill, M. J., and G. B. Whitman, On kinematic waves, I, Flood movement in long rivers, Proc. R. Soc. London A, 229, 281-316, 1955.

Seddon, J. A., River hydraulics, Trans. Am. Soc. Civ. Eng., 43, 179-243, 1900.

Ponce, V. M., New perspective on the Vedernikov number, Water Resour. Res., 27(7), 1777-1779, 1991.


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