In unsteady open-channel flow, several types of waves may be construed for purposes
of analysis.
Some waves attenuate, i.e., diffuse, decay, or dissipate; others do not.
The question is: What condition causes waves to decay? Is it friction? Or is it
the
pressure gradient? Both of these terms have been regarded as responsible for the
wave attenuation observed in practice (
In this article, we review the work of
The presence of absence of wave attenuation in unsteady open-channel flow
is explained in Table 1.
This table has been prepared based on the findings of
The first approach (A) consists of the
exclusion of one or more terms in the governing equation of motion, leading to the formulation
of several wave types, or wave models: (1) kinematic wave;
The kinematic wave is formulated by neglecting the local acceleration, advective acceleration, and pressure
gradient terms of the
equation of motion ( = a = 0). The only term that remains is the
kinematic term (bottom friction and gravity) (p = 1). This wave model does not attenuate kat all.
Ponce, 2024a). In practice, the kinematic wave model is that
of Seddon (1900),
The diffusion wave is formulated by neglecting the local and advective acceleration terms in the
equation of motion ( = 0).
The remaining terms are the
pressure term (a = 1) and the
kinematic term (p = 1).
This wave model does attenuate, albeit a ksmall amount.
It is used in the modeling of the large majority
of flood waves, characterized by a small dimensionless wave number
(Hayami, 1951;
Ponce, 2024a).
The unnamed wave is formulated by neglecting the local acceleration and pressure-gradient
terms in the
equation of motion ( = 0).
The remaining terms are the
advective acceleration (p = 1) and
the kinematic term (a = 1).
This wave model attenuates
(kPonce, 1982);
however, it is not generally used in unsteady flow modeling
because the neglect of only the local acceleration term ( = 0),
while at the same time keeping the advective acceleration term (e = 1),
is generally not warranted.
a
The mixed wave is formulated by keeping = a = p = 1).
kPonce, 2024b).
Its used is recommended particularly in the case of a dam-breach flood wave, where the suddenness of the
phenomenon may justify the use of this unusual type of wave.
The dynamic wave is formulated by neglecting the kinematic term (friction plus gravity) ( = e = a = 1).
This wave model does not attenuate pat all. It is used in the
modeling of relative small waves,
characterized by a very large dimensionless wave number (Ponce, 2024a).
A plot of dimensionless relative wave celerity vs. dimensionless wavenumber across the spectrum of shallow-water waves is showm in Fig. 2. Kinematic waves lie to the left of the wavenumber domain; dynamic waves to the right. The avowed constancy of wave celerity depicts the absence of wave attenuation. Therefore, neither kinematic nor dynamic waves attenuate!
Diffusion waves lie immediately to the right of kinematic waves (Fig. 2).
They are subject to a small but appreciable attenuation. Since flood waves usually attenuate very little,
diffusion waves are seen to be very
applicable to the modeling of flood waves
(
Mixed kinematic-dynamic waves, for short, mixed waves, lie towards the middle of the dimensionless wavenumber spectrum.
The sharpness of the variation in dimensionless relative
wave celerity with dimensionless wavenumber depicts
strong to very strong wave attenuation. [Note that peak attenuation occurs at the point of inflexion, i.e., where the second derivative
is equal to zero].
Figure 2 confirms the unsuitability of the mixed wave as a general basis for
flood wave computations. Indeed, the mixed waves are so dissipative
that they are not there for us to calculate them!
The unnamed wave ( = 0; and p = a = 1),
which does attenuate (Table 1, Line A3),
is not used in practice. We conclude that, kunless all terms are
included in the equation of motion = 1)p = 1)
is the konly one responsible for wave attenuation (Lines A2, A4, and B6;
note that the indicated columns are shown with light yellow background color). We also conclude that the kinematic term
= 1)k
A second approach to study open-channel
flow wave attenuation has nothing to do with which terms are missing, or negligible, in the
governing equations (this approach is contained in Table 1, Part A). Instead, in this approach
all terms are present (Part B1), so that wave attenuation is not due to the presence or absence of
certain terms of the equation of motion. Indeed, in unsteady open-channel flow, waves
will attenuate or amplify depending on the value of the Vedernikov number -
The (primary) dynamic waves, which transport energy, travel**V**< 1:*faster*than the kinematic waves, which transport mass; consequently, the flow is*stable*(**Craya, 1952****;**Stability implies that roll waves do not occur. Roll waves are periodic surface perturbations which travel downstream.**Ponce, 1992**). -
The (primary) dynamic waves travel at the same speed as the kinematic waves; consequently, the flow is neutrally stable, i.e., at the onset of surface-flow instability.**V**= 1: -
The kinematic waves, which transport mass, travel**V**> 1:*faster*than the (primary) dynamic waves, which transport energy; consequently, the flow is*unstable*. Instability implies that roll waves may occur (Fig. 3).
In conclusion, open-channel flow waves will be subject to attenuation
when the Vedernikov number is
The present article answers the question of what is the cause of wave attenuation in unsteady
In the modeling approach, four distinct models are recognized:
Chow, V. T. 1959.
Craya, A. 1952.
Hayami, I. 1951.
Lagrange, J. L. de. 1788.
Ponce, V. M. and D. B. Simons. 1977.
Ponce, V. M. 1982.
Ponce, V. M. 1991a.
Ponce, V. M. 1992.
Ponce, V. M. 2023a.
Ponce, V. M. 2023b.
Ponce, V. M. 2023c.
Ponce, V. M. 2024a.
Ponce, V. M. 2024b.
Seddon, J. A. 1900. |

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