1. Three characteristic velocities in openchannel flow There are three characteristic velocities in openchannel flow (Ponce 1991):
The Manning equation is (Chow 1959):
in which u = the mean velocity of the steady uniform flow, R = hydraulic radius, S = friction slope, and n = Manning friction coefficient. The Chezy equation is (Chow 1959):
in which C = Chezy friction coefficient. The Seddon formula expresses the kinematic wave celerity c_{k} as follows (Seddon 1900; Chow 1959):
in which T = channel top width, Q = discharge, and y = flow depth.
The Seddon formula is alternatively expressed as (Ponce 1989):
in which β = exponent of the dischargeflow area rating:
The relative kinematic wave celerity v, or Seddon celerity relative to the flow, is:
The relative dynamic wave celerity w, or Lagrange celerity relative to the flow, is (Chow 1959):
in which g = gravitational acceleration, and y = flow depth. 2. Two dimensionless numbers in openchannel flow The three velocities u, v, and w, expressed by Eqs. 1 or 2, and 6 and 7, respectively, give rise to two independent, dimensionless numbers, the Froude and Vedernikov numbers (Ponce 1991). The third number, or third possible combination, is expressed in terms of the other two. The Froude number is the ratio of u and w :
The Vedernikov number is the ratio of v and w :
The third number is the ratio of v and u :
Note that the quantity
The Froude number describes the condition of (1) subcritical, (2) critical, or (3) supercritical flow (Chow 1959).
Under subcritical flow, dynamic waves can travel upstream because
The Vedernikov number describes the condition of (1) stable, (2) neutral, or (3) unstable flow (Chow 1959).
Under stable flow, kinematic waves travel slower than dynamic waves because
The parameter β characterizes the type of friction and crosssectional shape; for instance,
Table 1 shows values of F_{ns} for selected
values of β corresponding to typical combinations of types of friction and shape of cross section,
from a high of
Table 1 shows that the value
From Eq. 5, for
3. Ven Te Chow: A synopsis Ven Te Chow was born in Hangchow, China, on August 14, 1919. He received his B.S. in civil engineering from the National Chiao Tung University in 1940, his M.S. in engineering mechanics from Pennsylvania State University in 1948, and his Ph.D. in hydraulic engineering from the University of Illinois in 1950 (Ackermann 1984). He became a naturalized U.S. citizen in 1962, and was on the civil engineering faculty at the University of Illinois from 1948 until his death in 1981. Chow is the author of the following widely acclaimed books: (1) Openchannel hydraulics, published in 1959, from which Fig. 3 was extracted; (2) Handbook of Applied Hydrology, published in 1964; and (3) Applied Hydrology, with coauthors David Maidment and Larry Mays, published in 1989.
4. The Froude number The Froude number is attributed to William Froude, who was born in Dartington, Devon, England in 1810, and died from a stroke on a cruise to South Africa in 1879, at age 69 (Fig. 3). In 1861, Froude wrote a paper on the design of ship stability in a seaway (http://froude.sdsu.edu). Later, between 1863 and 1867, working with physical models of ships, he showed that the frictional resistance in the model (at reduced scale) and prototype (the actual ship) were equal when the speed V was proportional to the length L to the power 1/2:
in which k is a constant that applies to both model and prototype. Froude called this physical law the "Law of Comparison." He was the first to identify the most efficient shape for the hull of ships, as well as to predict ship stability based on studies using reducedscale models. In openchannel hydraulics, Froude's Law is embodied in the Froude number, defined as follows (Chow 1959; Brater and King 1976):
in which D is the hydraulic depth, defined as the flow area divided by the wetted perimeter. Note that for application to openchannel flow, the horizontal length L in Froude's Law (Eq. 12) has been replaced by the hydraulic depth D (Eq. 13). This equation is basically the same as Eq. 8, wherein the hydraulic depth D has been approximated as the flow depth y.
5. The Vedernikov number The concept of Vedernikov number was first published in a Soviet journal (Vedernikov 1945; 1946). Craya wrote about the same concept in a paper published in 1952 (Craya 1952). The VedernikovCraya criterion states that roll waves will form when the Seddon celerity equals or exceeds the Lagrange celerity, that is, when the kinematic wave celerity, governed by gravity and friction, exceeds the dynamic wave celerity, governed by the pressure gradient and inertia. This is the condition that the Vedernikov number V ≥ 1. Note the unfortunate confusion in Craya's seminal paper, where the Lagrange [dynamic wave] celerity is described as governed by gravity [sic] and inertia. The role of the several forces acting in unsteady flow in open channels (gravity, friction, pressure gradient, and inertia) has been clarified by Ponce and Simons (1977), who calculated the dimensionless relative wave celerity throughout the dimensionless wavenumber spectrum. Under Chezy friction, for Froude number F = 2, that is, V = 1, all waves propagate at the same celerity, regardless of size. To reiterate, the VedernikovCraya criterion states that roll waves will form in an open channel under the following condition, in terms of absolute celerities:
In terms of relative celerities, roll waves will form when the relative Seddon celerity is greater than or equal to the relative Lagrange celerity:
Figure 4 shows roll waves forming on the spillway of Turner reservoir, in San Diego County, California,
following spillage after heavy rains, on February 24, 2005. Note that the noslip condition on the vertical walls
makes the waves appear as if the flow were threedimensional.
6. The Vedernikov number in Chow's Openchannel Hydraulics In Section 88, Instability of Uniform Flow, of Openchannel hydraulics, Ven Te Chow describes a criterion "which may be called the Vedernikov number" as follows:
in which x = exponent of hydraulic radius R in the general velocity formula
in which b = exponent of Reynolds number R
in the frictional power law The parameter γ is a crosssectional shape factor defined as follows:
in which R = hydraulic radius; P = wetted perimeter; and A = flow area.
The shape factor γ varies in the range 01. The value
The derivative is: dP/dA = d_{1} (P/A) = d_{1}/R.
In Eq. 18, for With Eq. 8, Eq. 16 reduces to (Chow 1959):
which implies that the Vedernikov number is a function of the Froude number, a statement which is strictly not correct. The examination of Eqs. 8 to 10 reveals that the Froude and Vedernikov numbers are totally independent of each other. The confusion arises only circumstantially because the relative kinematic wave celerity v is expressed in terms of the mean velocity u (Eq. 6). Given Eq. 10, it follows that:
which states that β contains information on both friction (x) and crosssectional shape (γ). Table 2 summarized the relations between b, x, γ, d_{1} and β for a wide range of flow conditions.
The question remains as to why Chow placed the Vedernikov number in Chapter 8 of his book, as the
last section [Section 88] of the chapter entitled "Theoretical Concepts ..." instead of placing it in Chapter 1,
together with the Froude number and other fundamental concepts.
This fact may have contributed to the relative obscurity of the Vedernikov number, which persists to this date despite the passing of more than half a century.
Many practicing engineers, while they acknowledge having consulted the book many times, have yet to discover the Vedernikov number
(Ponce 2003).
7. Concluding remarks The concepts of Froude and Vedernikov numbers are reviewed on the occasion of the 50th anniversary of the publication of Ven Te Chow's Handbook of Hydrology. While the Froude number (F) is standard fare in hydraulic engineering practice, the Vedernikov number (V) remains to be recognized by many practicing engineers. It is surmised here that this may be due in part to the fact that Chow placed the Vedernikov number in Chapter 8 of his book, instead of placing it in Chapter 1, together with the Froude number. A comprehensive description of the variation of β, the altogether important
exponent of the dischargeflow area rating References
Ackermann, W. C. 1984. Ven Te Chow, 19191981. Memorial Tributes: National Academy of Engineering, Vol. 2. Brater, E. F., and H. W. King. 1976. Handbook of Hydraulics. 6th Edition, McGrawHill, New York.
Chow, V. T. 1959.
Openchannel hydraulics. McGrawHill, New York.
Craya, A. 1952. The criterion for the possibility of rollwave formation. Gravity Waves,
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Liggett, J. A. 1975. Stability. Chapter 6 in Unsteady flow in open channels, Vol. 1, K. Mahmood and V. Yevjevich, editors, Water Resources Publications, pages 259281. Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves: I. Flood movement in long rivers. Proceedings of the Royal Society, Vol. 29, A, No. HY12, pages 281316. Ponce, V. M., and D. B. Simons. 1977. Shallow wave propagation in open channel flow. Journal of Hydraulic Engineering, ASCE, Vol. 103, No. HY12, pages 14611476, December. Ponce, V. M. 1989. Engineering Hydrology: Principles and Practices. Prentice Hall, Englewood Cliffs, New Jersey. Ponce, V. M. 1991. New perspective on the Vedernikov number. Water Resources Research, Vol. 27, No. 7, pages 17771779, July. Ponce, V. M. 1992. Kinematic wave modelling: Where do we go from here? International Symposium on Hydrology of Mountainous Areas, Shimla, India, May 2830, pages 485495. Ponce, V. M., and P. J. Porras. 1993. Effect of crosssectional shape on freesurface instability. Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 4, pages 376380, April. Ponce, V. M. 2003. That's the one we skip! Legacy Tales, link 970 in ponce.sdsu.edu.
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