Development of roll waves in open channels

Abstract

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope. The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity. The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel. The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow. The Froude number F (F = u_n/√gh_n, u_n = normal velocity, h_n = normal depth, g = acceleration of gravity) ranged from 3.4 to 6.0 for channel slopes S_0 of .05 and . 12 respectively. In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile. These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_(max) increased with distance. Once the shock waves began to overtake, the mean wave period T_(av) increased approximately linearly with distance. For a given Froude number and channel slope the observed quantities h_(max)/h_n, T' (T' = S_0 T_(av) √g/h_n), and the standard deviations of h_(max)/h_n and T', could be expressed as unique functions of ℓ /h_n (ℓ= distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows. A given value of h_(max)h_n occurred at smaller values of ℓ/h_n as the Froude number was increased. For a given value of h_(max) /h_n the growth rate ∂h_(max)/∂ℓ of the shock waves increased as the Froude number was increased. A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h_(max)/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h_(max)/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent. For h_(max)/h_n sufficiently large (h_(max)/h_n > approximately 1.5) it was found that the h_(max)/h_n vs. T' relation for natural roll waves was practically identical to the h_(max)/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate ∂h_(max)/∂ℓ of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments.

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