runoff diffusion in
the Pantanal of Mato Grosso, Mato Grosso do Sul, Brazil.
IS THE DYNAMIC HYDRAULIC DIFFUSIVITY ALTOGETHER
Runoff diffusion
is the property of free-surface flows to diffuse appreciably, attenuating unsteady flow stages
in Nature's apparently determined quest
to return the flow to its steady uniform condition. Diffusion
is characterized by the coefficient of the second-order term of the partial differential
equation governing the flow
(
Since the 1950s, several mathematical expressions for hydraulic diffusivity have been developed
as more information became available.
ν = ^{_______}2 S
_{o}(1) |
in which S = bottom slope.
_{o}
Since Hayami's formulation explicitly
ν = _{k}^{_______}2 S
_{o}(2) |
in which kinematic hydraulic diffusivity.
Using linear theory,
F^{2}ν = _{d}^{_______} ( 1 - ^{ ______} )2 S 4
_{o}(3) |
in which dynamic hydraulic diffusivity.
In Equation 3, F = Froude number of the equilibrium flow, defined as follows:
F = v / (g D)^{1/2},v = mean flow velocity, g = gravitational acceleration, and
D = hydraulic depth,D = A / T, in which A = flow area, and
T = channel top width.
ν
= _{d}^{_______} [ 1 - (β - 1)^{2} F^{2} ]2 S
_{o}(4) |
In fact, for Chezy friction in a hydraulically wide channel,
Ponce (
ν
= _{d}^{_______} ( 1 - V^{2} )2 S
_{o}(5) |
We note that while the kinematic
hydraulic diffusivity (Eq. 2) is independent
of the Vedernikov number, the dynamic
hydraulic diffusivity (Eq. 5)
has a clearly defined threshold for
While Equation 2 does not include the inertia terms, Equation 5 clearly does. Therefore, Eq. 5 should be a more accurate description of hydraulic diffusivity. To throw additional light on the issue, Table 1 shows the various types of waves in unsteady open-channel flow, including classical and common names.
Equation 2 was derived
by Hayami by
We have confirmed that Eq. 2 is an approximation for hydraulic diffusivity and that Eq. 5 is the complete expression. The question arises as to what is the actual difference between the two formulations when practical applications are considered. In the following section (Section 5) we evaluate the difference between the two alternative formulations for hydraulic diffusivity (Eqs. 2 and 5) using a catchment model of overland flow.
In this section, we use an overland flow
model to test the difference between the two formulations of hydraulic diffusivity,
Eqs. 2 and 5. The model simulates catchment dynamics using an open-book
geometric configuration,
i.e., two planes (left and right) and one channel in the middle (Fig. 1)
The results show negligible differences between the two diffusivities under a wide range of flow conditions. Kinematic waves (Table 1, Line 1) and diffusion waves (Table 1, Line 2) were used in the tests because these were the only ones likely to be encountered in actual practice. The Vedernikov number in the planes was quite low, around 0.03. Figure 2 shows the outflow hydrographs for drainage areas varying from 18 hectares (small) to 576 hectares (midsize). Figure 3 shows the outflow hydrographs for plane slopes (left and right) varying from 0.01 (steep) to 0.00001 (mild). No appreciable differences are observed between hydrographs generated using kinematic [(a) figures] and dynamic [(b) figures] hydraulic diffusivities.
The lack of any appreciable difference between output hydrographs when comparing
results using kinematic and dynamic diffusivities (Eqs. 2 and 5) is attributed to the
type of modeling used in this study.
ν._{k}The examples shown in Figs 2 and 3 depict kinematic waves (Table 1, Line 1) and diffusion waves (Table 1, Line 2), since these are the ones likely to be encountered in actual practice. By definition, these waves are not subject to substantial diffusion; thus, featuring relative small values of either of the two diffusivities. We conclude that when modeling catchment flows using kinematic and diffusion waves, the difference between hydraulic diffusivities is likely to be negligible.
A question to be addressed at this juncture is
the nature of mixed kinematic-dynamic waves
Further confirmation of the correctness of Lighthill and Whitham's observations regarding
the strong dissipative nature of dynamic waves was given by
The findings of Ponce and Simons (1977) confirm the very strong dissipative tendencies of mixed
kinematic-dynamic waves, commonly referred to in the hydraulic engineering literature
simply as "dynamic waves" (Fread, 1985).
The peak attenuation amount occurs at the point of inflection
in the dimensionless relative celerity
_{*}
vs dimensionless wavenumber
σ_{*}
[compare Fig. 4 (a) with Fig. 4 (b)].
Note that the amount of wave attenuation increases markedly with a decrease in equilibrium flow Froude
number. For the lowest Froude number shown in Fig. 4,
F = 0.01, the peak logarithmic decrement is δ = 180, corresponding to
σ_{*} = 90
[Fig. 4 (b)].
Table 2 shows peak logarithmic decrements (Column 3) for
the range of subcritical Froude numbers between
Having demonstrated that the mixed kinematic-dynamic wave may not be there for us,
we still have a choice of using or not the dynamic hydraulic diffusivity
(Eq. 5) in
A comparison between kinematic and dynamic hydraulic diffusivities using an actual catchment model of overland flow is accomplished. The kinematic hydraulic diffusivity is the well known Hayami diffusivity, which is independent of the Vedernikov number. On the other hand, the dynamic hydraulic diffusivity is a function of the Vedernikov number. Several examples of rainfall-runoff conversions using the catchment model show that the difference between these two formulations of hydraulic diffusivity is negligible. This may be attributed to the very low Vedernikov numbers typically featured in catchment models, or to the specification of kinematic and diffusion waves in the test cases, since these waves are the only ones likely to be encountered in actual practice. The question of the true nature of mixed kinematic-dynamic waves is examined using available analytical data. It is concluded that given their extremely strong dissipative tendencies, mixed waves may not be there for us to calculate. Nevertheless, the dynamic hydraulic diffusivity is advocated as the method of choice because it is the complete solution, it is applicable to all types of routing, including catchment and channel flow, and it does not significantly complicate the methodology.
Dooge, J. C. I. 1973.
Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkoswki. 1982.
Fread, D. L. 1985. "Channel Routing," in
Hayami, I. 1951.
Lighthill, M. J. and G. B. Whitham. 1955.
Natural Resources Conservation Service (NRCS). 1985.
Ponce, V. M. and D. B. Simons. 1977.
Ponce, V. M. 1986.
Ponce, V. M. 1991a.
Ponce, V. M. 1991b.
Ponce, V. M. 2014a.
Ponce, V. M. 2014b.
Ponce, V. M. and A. C. Scott. 2022.
Ponce, V. M. 2023a.
Ponce, V. M. 2023b.
Wylie, C. R. 1966. |

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