The Vedernikov Number: Concept and Practice

Aleksandr Gostomelskiy


Open-channel flow can be steady or unsteady. Flow in an open channel is said to be steady if the depth of flow does not change or if it can be assumed to be constant during the time interval under consideration. On the other hand, in unsteady flow, the flow depth changes with time. The equations which govern the steady flow are well known (Chezy, Manning, and the conservation of energy equation). For unsteady flow, there are a number of equations which are much more complex: the continuity and momentum equations. Examples of unsteady are found in rivers and streams during flood, storm runoff in sewers, tidal flows in estuaries, and in the formation of roll waves. The case of roll waves, a special case of unsteady flow, is the focus of this article.

Usually, uniform flow in an open channel becomes unstable when the velocity is very high or the slope is very steep. Under these conditions, the instability of the free surface is characterized by the formation of a train of roll waves. The earliest theoretical description of this instability was given in 1945 by the Soviet scientist V. V. Vedernikov. The criterion, presented to the international scientific community in 1946, is now referred to as the Vedernikov number V.

In its original form, the Vedernikov number was expressed as:

V= xγv/(vw - v)

in which x= is exponent of hydraulic radius in the frictional power law (x= 2 for laminar flow, x= 0.5 for turbulent flow if the Chezy formula is used, x= 2/3 for turbulent flow if the Manning equation is used); v= mean velocity; vw= absolute velocity of the roll waves in the channel; and γ= cross-sectional shape factor (γ varies in range between 0 and 1).

To provide a physical interpretation of the Vedernikov number, a few definitions are in order. In unsteady flow, a wave is a temporal variation of flow depth or rate of discharge. Another important concept is the celerity, which represents the wave velocity taken relative to the flow it is traveling in. In other words, celerity is a measure of rapidity of surface waves, which could travel in the upstream or downstream direction. In unsteady flow, there are two main wave types: kinematic and dynamic. These concepts were developed in order to calculate overland flow. The kinematic wave technique simulates runoff concentration in the absence of runoff diffusion. The dynamic wave technique takes into account runoff concentration, diffusion, and dispersion processes. Flow in a channel will become unstable if the value of kinematic wave celerity is larger than the value if dynamic wave celerity. Thus, the Vedernikov number is best characterized as the ratio of the relative celerity of kinematic waves to the relative celerity of dynamic waves.

Essentially, the Vedernikov number is a measure of flow stability. The flow regime is stable if the V<1, neutral if V=1, and unstable if V>1. When the flow is stable (V<1) the energy-transporting waves (dynamic) travel faster than mass-transporting waves (kinematic); when the flow is neutral (V=1) these waves travel at the same speed; when the flow is unstable (V>1) the mass-transporting waves (kinematic) travel faster than energy-transporting waves (dynamic). Roll waves form when V ≥ 1. This is of practical interest, particularly in the area of steep-channel design, where minimization of cost is crucial.

Hydraulic engineers are able to design channels to accomodate roll waves. As Dr. V. M. Ponce mentioned at one of his Open-Channel Hydraulics lectures at San Diego State University, the La Joya Canal in Arequipa, Peru, serves as a good example for roll wave design. Initially engineers did not take into consideration the Vedernikov number, and roll waves would form; the resulting channel was inadequate and water was lost. To fix this problem, engineers had to design another channel with special drop structures to eliminate the surface instability. It follows that the Vedernikov number is of great importance, both in theory and practice.


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