There are three characteristic speeds in open channel hydraulics: (1) the mean flow velocity v/u; and (2) the dimensionless relative celerity of dynamic waves c = _{d}w/u. The reciprocal of c, is the Froude number F = u/w. V = v/w. A Vedernikov number V = 1 describes the condition of neutral stability, separating stable flow (V < 1) from unstable flow (V > 1). During propagation, surface perturbations attenuate in stable flow and amplify in unstable flow [Vedernikov, 1945, 1946; Chow, 1959; Jolly and Yevjevich, 1971].
Recent developments in the linear theory of surface runoff [
The Vedernikov number [
in which x is the exponent of hydraulic radius R in the mean velocity relation u = f(R), defined as:
in which b is the exponent of Reynolds number in the frictional power law: Rf = α with R^{ -b}f the Darcy-Weisbach friction factor. The parameter b varies in the range 0-1, with b = 0 applicable to turbulent Chezy friction, and b = 1 for laminar flow. In Eq. 1, γ is a cross-sectional shape factor defined as follows:
in which R = hydraulic radius, P =
wetted perimeter, and A = flow area.
The shape factor
Equation 3 can be simplified by noting that it accounts for a channel of arbitrary shape, in which the wetted perimeter is a function of flow area. Assuming the validity of a power function of the following type: k_{1} and d constants; then the derivative is P/dA = d (P/A) = d/Rγ = 1 - d; and the Vedernikov number is recast as follows:
The inherently stable channel is a hypothetical channel in which the hydraulic radius is constant, i.e., independent of the flow area. Therefore, d = 1, and γ = 0. Given Eqs. 1 and 4, it follows that V = 0, regardless of boundary friction specification or Froude number; consequently, the flow is inherently stable. Furthermore, for constant boundary friction and bottom slope, the mean velocity of the inherently stable channel is a constant, regardless of discharge, flow area, or stage. The shape of the inherently stable channel has been documented by Liggett [1975], among others.
Following Eq. 4, the ratio w/u leads to V c = _{d}V /F = v/u = c, i.e., to the relative celerity of kinematic waves. Furthermore, given Eq. 4, the relative celerity of kinematic waves can also be expressed as _{k}c = _{k}x ( 1 - d ).
The relative celerity of kinematic waves, or ^{1/2}, in which k is the friction coefficient, x has the same meaning as in Eq. 2, and S is the bottom slope. The rating equation is _{o}Q = u A = k R^{ x} S_{o}^{1/2} = k A^{1+x} P^{ -x} S_{o}^{1/2}P = k_{1} A^{d}, it follows that the rating equation is Q = α A,^{β}α = f ( k, k_{1}, x, S ); and _{o}β = 1 + x (1 - d). Furthermore, given Eq. 4, β = 1 + (V /F).Seddon, 1990] is: u + v = dQ /dA = β (Q /A) = βuc = _{k}v /u = β - 1 = V /F
To isolate the effect of boundary friction specification, a hydraulically wide channel is considered first. This condition is modeled by setting
To study the effect of cross-sectional shape, three exact cases are considered: (1) a hydraulically wide channel, with
It is further noted that the Froude number corresponding
to neutral stability (that for which V = 0, then F = ∞. This confirms the absence of the neutral stability condition in the inherently stable channel (i.e., the flow disturbances attenuate for all Froude numbers).
_{n}
Recently, the Vedernikov number has been shown to play a significant role in modeling catchment dynamics using diffusion waves and including inertial effects [
in which q is the unit-width discharge, and S is the bottom slope. In contrast to (5), the dynamic hydraulic diffusivity [_{o}Dooge, 1973; Dooge et al., 1982; Ponce, 1991] is defined as follows:
It is seen that unlike its kinematic counterpart, the dynamic hydraulic diffusivity is also a function of the Vedernikov number. This allows the simulation to be responsive to the dynamic effect. In fact, in Eq. 6, for V = 1, ν = 0, verifying the absence of wave attenuation or amplification at the condition of neutral stability, a characteristic of dynamic waves [_{d}Ponce and Simons, 1977] which cannot be simulated using the kinematic hydraulic diffusivity (Eq. 5). On the other hand, for small Froude number flows in hydraulically wide channels: F → 0. Then F ^{2} → 0, and given Eq. 1, V ^{2} → 0, leading through Eq. 6 to ν → _{d}ν. This confirms the applicability of the diffusion wave model for flows well in the subcritical regime [_{k}Ponce et al., 1978].
The case of the inherently stable channel further illustrates the concept of dynamic hydraulic diffusivity. When ν. _{k}V = 0 is that of the inherently stable channel, in which wave disturbances attenuate due to the intrinsically stable nature of the cross-sectional shape, regardless of Froude number, discharge, or stage. This confirms the observation that a wave propagating in a channel of rapidly expanding top width (with increasing stage, such that u and R are nearly constant and, therefore, V → 0, and ν → _{d}ν)_{k}Ponce and Windingland, 1985].
The Vedernikov number is reviewed in the light of a new perspective of its role in the modeling of catchment dynamics using diffusion waves and incorporating inertial effects. Together with the dimensionless relative celerities of kinematic and dynamic waves, the Vedernikov number is shown to properly characterize unsteady free-surface flows. Echoing
The Vedernikov number is also shown to have a significant role in extending the concept of hydraulic diffusivity [
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