Is dispersion important in flood hydrology? Victor M. Ponce

WaterArchives.org

Teton dam failure, June 5, 1976, Teton Canyon, Idaho.

IS DISPERSION IMPORTANT IN FLOOD ROUTING?

Victor M. Ponce

Professor Emeritus of Civil and Environmental Engineering

San Diego State University, San Diego, California

14 November 2023

ABSTRACT. A third-order convection-diffusion-dispersion differential equation of flood flows (Ferrick and others, 1984) is used as the basis for the development of a dimensionless convection-diffusion-dispersion equation. This equation shows that its three coefficients, of convection, diffusion, and dispersion, are functions only of the Froude and Vedernikov numbers, the two conceptual pillars of unsteady open-channel hydraulics. The computer program ONLINEDISPERSIVITY is used to establish the order of magnitude of the diffusion and dispersion coefficients.

1. INTRODUCTION

Flood routing is the calculation of the movement of a flood wave in space and time along a stream or channel. The governing equations are the equations of water continuity and motion of open-channel flow, referred to as the Saint-Venant (1871) equations (Chow, 1959; Ponce, 2014a). The solution of these equations leads to a mixed kinematic-dynamic wave. In hydraulic engineering practice, this wave is commonly referred to as the "dynamic wave" (Fread, 1985).

The realization that the contribution of inertia is very often minimal led Hayami (1951) to simplify the flood routing problem by combining the set of two governing equations into one equation, with space x and time t as independent variables and discharge Q as the dependent variable. This equation is referred to as the convection-diffusion equation. It is a partial differential equation of second order, describing convection, of first order, and diffusion, of second order. This approach to flood routing has been referred to as Hayami's diffusion analogy (Ponce, 2014b).

Ferrick and others (1984) added another term to the convection-diffusion equation, effectively creating a third term, which they referred to as dispersion term. Thus arose the third-order convection-diffusion-dispersion equation of flood routing. Ferrick's work was further enhanced by Ponce (2020), by expressing the convection-diffusion-dispersion equation in dimensionless form. In addition, Ponce expressed the coefficients of this latter equation in terms of only the Froude and Vedernikov numbers, confirming the strong theoretical foundation of the entire subject of one-dimensional unsteady open-channel flow (Ponce, 2023).

This article expands on the significance of diffusion and dispersion in the theory and practice of hydraulic engineering. The online calculator ONLINEDISPERSIVITY is used to calculate the relative magnitude of the routing coefficients.

A word of explanation.
The term dispersion was used by Ferrick et al. (1984) to denote the third-order term in the governing differential equation of one-dimensional unsteady open-channel flow. Actually, the term "dispersion" may mean different things in different fields. For instance, in fluid mechanics, dispersion generally refers to the spreading of mass. In this article, we use the term dispersion in the mode of Ferrick et al., to refer to the third-order spreading of momentum in open-channel flow.

2. CONVECTION-DIFFUSION-DISPERSION EQUATION

Table 1, Equation 1, shows the convection-diffusion-dispersion equation. The convection coefficient is the Seddon celerity, or kinematic wave celerity (Seddon, 1900; Ponce, 2014b). The diffusion coefficient is the Hayami diffusivity (Hayami, 1951; Ponce, 2014b). The dispersion coerficient is the Ferrick dispersivity (Ferrick and others, 1984; Ponce, 2020).

Table 1. Elements of the convection-diffusion-dispersion equation.
Equation	Q_t + c Q_x = ν Q_xx + η Q_xxx	(1)
Convection coefficient	V c = ( 1 + ^____ ) u_o F	(2)
Diffusion coefficient	L_o ν = ^____ u_o ( 1 - V² ) 2	(3)
Dispersion coefficient	L_o η = ( ^____ ) ² u_o ( 1 - V² ) F² 2	(4)
Symbol definition. Q = discharge; A = flow area; x = space; t = time; g = gravitational acceleration; α = coefficient of the discharge-area rating Q = α A^β; β = exponent of the discharge-area rating Q = α A^β; u_o = mean velocity; y_o = flow depth; S_o = channel bottom slope; F = Froude number = u_o / (g y_o)^1/2; V = Vedernikov number = (β - 1) F; L_o = reference channel length = y_o /S_o.

3. DIMENSIONLESS CONVECTION-DIFFUSION-DISPERSION EQUATION

Table 2, Equation 5, shows the dimensionless convection-diffusion-dispersion of flood waves (Ponce, 2020). To accomplish the nondimensionalization, we used the reference channel length L_o, which is the distance along the channel in which the channel drops a head equal to its flow depth (Table 1, bottom line) (Lighthill and Whitham, 1955; Ponce and Simons, 1977). All three dimensionless coefficients are shown to be functions only of the Froude and Vedernikov numbers. Therefore, we conclude that these two numbers effectively constitute the pillars of unsteady open-channel flow. Together, they describe and characterize wave motion.

It is observed that the dimensionless convection coefficient c' (Eq. 6), which may also be referred to as dimensionless kinematic wave celerity, is in fact the exponent of the discharge-flow area rating: β = 1 + (V/F). Thus, β may be regarded as perhaps the most significant parameter in the entire field of unsteady open-channel flow (Ponce, 2023).

Table 2. Elements of the dimensionless convection-diffusion-dispersion equation.
Equation	Q_t' + c' Q_x' = ν' Q_x'x' + η' Q_x'x'x'	(5)
Dimensionless convection coefficient	V c' = 1 + ^____ F	(6)
Dimensionless diffusion coefficient	1 ν' = ^____ ( 1 - V² ) 2	(7)
Dimensionless dispersion coefficient	1 η' = ^___ ( 1 - V² ) F² 4	(8)
Symbol definition. x' = dimensionless space = x /L_o; t' = dimensionless time = t (u_o /L_o);

4. ANALYSIS

Table 3 shows the results of script ONLINEDISPERSIVITY. We varied mean velocity u_o (Col. 2), flow depth y_o (Col. 3), and channel bottom slope S_o (Col. 4) as shown. The focus was on unit-width discharge (q_o = u_oy_o) and channel bottom slope S_o (Col. 4), since these two variables are strongly related to the diffusion (Eq. 3) and dispersion (Eq. 4) coefficients. The value of β, the exponent of the discharge-area rating, was fixed at β = 1.5 (Col. 5), which constitutes a central value in the normal variability (1.33-1.67).

Specific observations regarding the coefficients of diffusion and dispersion are the following:

Diffusion (Col. 9) increases with an increase in unit-width discharge, i.e., with an increase in u_o and/or y_o (Cols. 2 and 3).

Diffusion (Col. 9) increases with a decrease in channel slope (Col. 4).

Dispersion (Col. 10) increases with an increase in unit-width discharge, i.e., with an increase in u_o and/or y_o (Cols. 2 and 3).

Dispersion (Col. 10) increases with a decrease in channel slope (Col. 4).

The dimensionless coefficients (Cols. 11, 12, and 13) are independent of channel slope.

Table 3. Results of script ONLINEDISPERSIVITY.
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Line	u_o (m/s)	y_o (m)	S_o (m/m)	β	F	V	c (m/s)	ν (m²/s)	η (m³/s)	c'	ν'	η'
1	1	1	0.01	1.5	0.32	0.16	1.5	4.97	248.34	1.5	0.49	0.025
2	1	1	0.001	1.5	0.32	0.16	1.5	49.7	24834.	1.5	0.49	0.025
3	1	1	0.0001	1.5	0.32	0.16	1.5	497.	2483475.	1.5	0.49	0.025
4	2	2	0.01	1.5	0.45	0.225	3.0	38.69	3870.	1.5	0.47	0.048
5	2	2	0.001	1.5	0.45	0.225	3.0	386.9	386964.	1.5	0.47	0.048
6	2	2	0.0001	1.5	0.45	0.225	3.0	3869.	38696497.	1.5	0.47	0.048
7	4	4	0.01	1.5	0.64	0.32	6.0	292.94	58585.	1.5	0.45	0.092
8	4	4	0.001	1.5	0.64	0.32	6.0	2929.4	5858924.	1.5	0.45	0.092
9	4	4	0.0001	1.5	0.64	0.32	6.0	29294.	585892404.	1.5	0.45	0.092

5. SUMMARY

A convection-diffusion-dispersion equation of flood flows (Ferrick and others, 1984) is used as the basis for the development of a dimensionless convection-diffusion-dispersion equation. This equation shows that its three coefficients are functions only of the Froude and Vedernikov numbers, recognized as the two conceptual pillars of unsteady open-channel hydraulics (Ponce, 2024). The computer program ONLINEDISPERSIVITY (https://ponce.sdsu.edu/onlinedispersivity.php) establishes the order of magnitude of diffusion (second-order), and dispersion (third-order) in one-dimensional unsteady open-channel flow.

REFERENCES

Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, Inc, New York, NY.

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

Fread, D. L. 1985. "Channel Routing," in Hydrological Forecasting, M. G. Anderson and T. P. Burt, eds. New York: John Wiley.

Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention Research Institute, No. 1, December.

Lighthill, M. J. and G. B. Whitham. 1955. On kinematic waves. I. Flood movement in long rivers. Proceedings, Royal Society of London, Series A, 229, 281-316.

Ponce, V. M. and D. B. Simons. 1977. Shallow wave propagation in open channel flow. Journal of Hydraulic Engineering, ASCE, 103(12), December, 1461-1476.

Ponce, V. M. 2014a. Fundamentals of Open-channel Hydraulics. Online textbook.
ponce.sdsu.edu/openchannel/index.html

Ponce, V. M. 2014b. Engineering Hydrology: Principles and Practices. Online textbook.
ponce.sdsu.edu/enghydro/index.html

Ponce, V. M. 2020. A dimensionless convection-diffusion-dispersion equation of flood waves. Online article. ponce.sdsu.edu/dimensionless_convection_diffusion_dispersion_equation.html

Ponce, V. M. 2023. The states of flow. Online article. ponce.sdsu.edu/the_states_of_flow.html

Ponce, V. M. 2024. Froude and Vedernikov: Pillars of open-channel hydraulics. Online article.
ponce.sdsu.edu/froude_and_vedernikov_pillars_of_open_channel_hydraulics.html

Saint-Venant, B. de. 1871. Theorie du mouvement non-permanent des eaux avec application aux crues des rivieres et l' introduction des varees dans leur lit, Comptes Rendus Hebdomadaires des Seances de l'Academie des Science, Paris, France, Vol. 73, 1871, 148-154.

Seddon, J. A. 1900. River hydraulics. Transactions, ASCE, Vol. XLIII, 179-243, June.

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