1.  INTRODUCTION

There are three characteristic speeds in open channel hydraulics: (1) the mean flow velocity u, (2) the relative celerity of kinematic waves v, and (3) the relative celerity of dynamic waves w [Ponce and Simons, 1977]. These three speeds give rise to two independent dimensionless ratios: (1) the dimensionless relative celerity of kinematic waves c_k = 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. The Vedernikov number is the third ratio, i.e., the ratio of relative celerity of kinematic waves to relative celerity of dynamic waves: 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 [Dooge, 1973; Dooge et al., 1982; Ponce, 1986] have prompted this review of the Vedernikov number. Together with the dimensionless relative celerities of kinematic and dynamic waves, the Vedernikov number is shown to characterize free-surface flow. In addition, the Vedernikov number is shown to have a significant role in extending the concept of hydraulic diffusivity [Hayami, 1951] to the realm of dynamic waves [Ponce and Simons, 1977; Ponce, 1990, 1991]. This new perspective on the Vedernikov number makes possible the modeling of catchment dynamics using diffusion waves and including inertial effects, under a wide range of boundary friction and cross-sectional shape specifications.

2.  THE VEDERNIKOV NUMBER

The Vedernikov number [Chow, 1959; Jolly and Yevjevich, 1971] is defined as:

V = x γ F

(1)

in which x is the exponent of hydraulic radius R in the mean velocity relation u = f(R), defined as:

1 + b x = ________          2 - b

(2)

in which b is the exponent of Reynolds number R in the frictional power law: f = α R ^-b with 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:

dP γ  = 1 - R _____                   dA

(3)

in which R = hydraulic radius, P = wetted perimeter, and A = flow area.

The shape factor γ varies in the range 0-1, with γ = 0 applicable to a hypothetical channel of constant hydraulic radius (herein defined as the inherently stable channel), and γ = 1 for the case of a hypothetical channel of constant wetted perimeter (commonly referred to as a hydraulically wide channel).

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: P = k₁A^d, with k₁ and d constants; then the derivative is dP/dA = d (P/A) = d/R. Therefore, in Eq. 3, γ = 1 - d; and the Vedernikov number is recast as follows:

V = x ( 1 - d ) F

(4)

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.

Craya [1952] has given an improved physical interpretation of the Vedernikov criterion. According to Craya, the flow will become unstable when the kinematic wave celerity (u + v) exceeds the primary dynamic wave celerity (u + w), i.e., that propagating always in the downstream direction [Ponce and Simons, 1977]. This reduces to v > w, and therefore to V > 1, confirming that the Vedernikov and Crays criteria are one and the same [Jolly and Yevjevich, 1971].

3.  SIGNIFICANCE OF V /F RATIO

Following Eq. 4, the ratio V/F is equal to x ( 1 - d ). It is noted that the product of the Vedernikov number V = v/w and the relative celerity of dynamic waves c_d = w/u leads to V c_d = V /F = v/u = c_k, i.e., to the relative celerity of kinematic waves. Furthermore, given Eq. 4, the relative celerity of kinematic waves can also be expressed as c_k = x ( 1 - d ).

The relative celerity of kinematic waves, or V/F ratio, is significant in that it is the only variable appearing in the exponent of the discharge-flow area (Q versus A) rating. In effect, assume a mean velocity formula of the following type: u = k R ^x S_o^1/2, in which k is the friction coefficient, x has the same meaning as in Eq. 2, and S_o is the bottom slope. The rating equation is Q = u A = k R ^x S_o^1/2 = k A^1+x P ^-x S_o^1/2. Since P = k₁ A^d, it follows that the rating equation is Q = α A^β, in which α = f ( k, k₁, x, S_o ); and β = 1 + x (1 - d). Furthermore, given Eq. 4, β = 1 + (V /F). Then, the kinematic wave celerity or Seddon speed [Seddon, 1990] is: u + v = dQ /dA = β (Q /A) = βu; and the dimensionless relative celerity of kinematic waves is c_k = v /u = β - 1 = V /F.

4.  EFFECT OF BOUNDARY FRICTION AND CROSS-SECTIONAL SHAPE

To isolate the effect of boundary friction specification, a hydraulically wide channel is considered first. This condition is modeled by setting d = 0, which leads to γ = 1. Within this context, four typical cases are examined: (1) laminar, b = 1; (2) mixed laminar-turbulent, b = 1/2; (3) turbulent Manning, b = 1/5; and (4) turbulent Chezy, b = 0. With d = 0, and b as defined, Eqs. 2 and 4 lead to the V /F ratios shown in Table 1.

To study the effect of cross-sectional shape, three exact cases are considered: (1) a hydraulically wide channel, with d = 0 and γ = 1; (2) a triangular channel, with d = 1/2 and γ = 1/2: and (3) an inherently stable channel, with d = 1 and γ = 0. It is noted that for most trapezoidal channels, d varies in the range 0-0.5, i.e., between the hydraulically wide and triangular channel shapes. For some natural channels, d may be in the range 0.5-1, between the triangular and inherently stable channel shapes. With b corresponding to the turbulent Chezy and Manning friction specifications, and d as defined in this paragraph, Eqs. 2 and 4 lead to the V /F ratios shown in Table 2.

It is further noted that the Froude number corresponding to neutral stability (that for which V = 1) is the reciprocal of the ratio V /F. In effect, for V = 1, F /V = F_n, i.e., the neutral stability Froude number. In the case of the inherently stable channel, since V = 0, then F_n = ∞. This confirms the absence of the neutral stability condition in the inherently stable channel (i.e., the flow disturbances attenuate for all Froude numbers).

5.  ROLE OF VEDERNIKOV NUMBER IN MODELING CATCHMENT DYNAMICS

Recently, the Vedernikov number has been shown to play a significant role in modeling catchment dynamics using diffusion waves and including inertial effects [Ponce, 1986, 1991]. For diffusion waves, the kinematic hydraulic diffusivity [Hayami, 1951: Lighthill and Whitham, 1955] is defined as follows:

q vk = ______          2So

(5)

in which q is the unit-width discharge, and S_o is the bottom slope. In contrast to (5), the dynamic hydraulic diffusivity [Dooge, 1973; Dooge et al., 1982; Ponce, 1991] is defined as follows:

q vd = ______ ( 1 - V 2 )          2So

(6)

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_d = 0, verifying the absence of wave attenuation or amplification at the condition of neutral stability, a characteristic of dynamic waves [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_k. This confirms the applicability of the diffusion wave model for flows well in the subcritical regime [Ponce et al., 1978].

The case of the inherently stable channel further illustrates the concept of dynamic hydraulic diffusivity. When V → 0, regardless of Froude number, then V ² → 0; and therefore ν_d → ν_k. The case 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) will attenuate governed by the kinematic hydraulic diffusivity (Eq. 5), regardless of discharge, flow area, or stage [Ponce and Windingland, 1985].

5.  SUMMARY

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 Craya [1952], the Vedernikov number is defined as the ratio of the relative celerity of kinematic waves to the relative celerity of dynamic waves. Furthermore, the ratio of Vedernikov to Froude numbers V/F is shown to properly characterize the discharge-flow area rating.

The Vedernikov number is also shown to have a significant role in extending the concept of hydraulic diffusivity [Hayami, 1951] to the realm of dynamic waves [Dooge, 1973; Dooge et al., 1982]. This new perspective makes possible the modeling of catchment dynamics using diffusion waves and including inertial effects [Ponce, 1986, 1990, 1991], under a wide range of boundary friction and cross-sectional shape specifications.

REFERENCES

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Craya, A. 1952. The criterion for the possibility of roll ways formation, Gravity Waves, Circ. 521, 141-151, Natl. Inst. of Stand. and Technol., Gaithersburg, MD.

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

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Vedernikov, V. V. 1946. Characteristic features of a liquid flow in an open channel. Dokl. Akad. Nauk SSSR, 52(3), 207-210.