Roll waves can occur in an unstable free-surface flow regime, i.e., when the Vedernikov number exceeds 1. The flow condition can be either laminar or turbulent. In the laminar regime, an example of roll waves is the pulsating flow often observed in steep urban catchments following intense rainfall. In the turbulent flow regime, roll waves may require a specific type of boundary condition for their inception.
The Vedernikov number The objective of this note is to use Brock's (1967) laboratory data to test the validity of Ponce and Simons' (1977) theory. Experimental wavenumbers, calculated from wave periods measured by Brock, are compared with wavenumbers predicted by the theory. An approximate match of wavenumbers confirms the validity of the theory. In a practical setting, this knowledge can be used to promote or inhibit the development of roll waves.
The Vedernikov number is defined as (Chow 1959)
in which
in which
in which
The linear stability analysis of Ponce and Simons (1971) expresses the propagation characteristics of shallow water waves in terms of: (1) the Froude number
The propagation characteristics of shallow water waves are: (1) the dimensionless relative wave celerity δ. The dimensionless relative wave celerityc = (_{r}c - u_{0}) / u_{0},c = wave celerity. The logarithmic decrement δ is s a measure of the tendency of the wave disturbance to attenuate or amplify during propagation. For δ < 0, waves attenuate; conversely, for δ > 0, waves amplify. A value of δ = 0 indicates neutral stability, i.e. neither wave attenuation nor amplification. The theory states that roll waves are most likely to form in the unstable flow regime (V > 1; δ > 0), within a narrow band of dimensionless wavenumbers, at or near the peak of the functional relation between logarithmic decrement and dimensionless wavenumber [δ = f(σ)].
The theoretical expressions for dimensionless wave celerity and logarithmic decrement (Ponce and Simons 1977) are included here for completeness. The dimensionless relative wave celerity is:
in which
and
The logarithmic decrement of the downstream-propagating shallow wave is:
The Brock (1967) data were assembled as shown in Table 1, which
shows the following:
The solid curves of Figs. 1 and 2 show logarithmic decrement
Using Brock's (1967) laboratory flume data, an experimental verification of a theory of roll-wave formation (Ponce and Simons 1977) was presented. Results show that dimensionless wavenumbers corresponding to the roll waves documented by Brock's study plot near the peak of the theoretical functional relation of logarithmic decrement
Brock, R. R. 1967. "Development of roll waves in open channels."
Chow, V. T. 1959.
Craya, A. 1952. "The criterion of the possibility of roll wave formation."
lwasa, Y. 1954. "The criterion for the instability of steady uniform flows in open channels."
Liggett, J. A. 1975. "Stability -Chapter 6,"
Mayer, P. G. W. 1957. "A study of roll waves and slug flows in inclined open channels," PhD dissertation, Cornell University, Ithaca, N.Y.
Ponce, V. M., and D. B. Simons. 1977. "Shallow wave propagation in open channel flow."
Ponce, V. M. 1991. "New perspective on the Vedernikov number,"
Vedernikov, V. V. 1945. "Conditions at the front of a translation wave disturbing a steady motion of a real fluid,"
Vedernikov, V. V. 1946. "Characteristics features of a liquid flow in an open channel." |

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