Friction effects on quasi-steady dam-break wave propagation on horizontal beds

Abstract

The propagation of dam-break waves on different rough beds was observed to be quasi-steady in the range $11.3 < x/h_{dam} < 18.8$ , where $x$ is measured from the dam position. These quasi-steady propagation speeds converge with the steady ideal fluids model of Stoker (Water Waves, 1957, Interscience) when the tailwater depth $h_2$ becomes greater than ${\sim }0.5k_s$ , in the range $0.001< k_s/h_{dam}< 0.2$ , where $k_s$ is the roughness and $h_{dam}$ the depth behind the dam. Hence, this convergence encourages the use of Stoker's steady, ideal fluid solution to develop more general models, including friction effects due to bed roughness and/or viscosity. The new experimental data support a MacLaurin series for the celerity $c$ , in analogy with the series in terms of $\sqrt [4]{{h_2}/{h_{dam}}}$ , derived for Stoker's model, $h_2$ being the tailwater depth. Compared with the retarding effect of the tailwater, 1 mm of roughness is found to be equivalent to $\sim$ 13 mm of tailwater, and 1  $\mathrm {\mu }$ m of viscous length ( ${\nu }/{\sqrt {gh_{dam}}}$ , where ${\nu}$ is the kinematic viscosity and g the acceleration due to gravity) is equivalent to $\sim$ 1700  $\mathrm {\mu }$ m of tailwater. While the MacLaurin series quantifies the similar effects of small roughness and small tailwater depths acting separately, the new data illustrate for the first time the complex interplay between tailwater and roughness on ‘wet beds’ with many details yet to be investigated. In particular, it was shown that a small amount of tailwater on a rough bed acts as a lubricant, so that $c(h_2, h_{dam}, k_s)$ is an increasing function of $h_2$ for $h_2 < 0.5k_s$ .