The propagation of dam-break waves on different rough beds was observed to be quasi-steady in the range < [CDATA[11.3 < x/h dam, where 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 becomes greater than, in the range <[CDATA[0.001< k_s/h_{dam}, where is the roughness and 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, in analogy with the series in terms of, derived for Stoker's model, being the tailwater depth. Compared with the retarding effect of the tailwater, 1 mm of roughness is found to be equivalent to 13 mm of tailwater, and 1 m of viscous length (, where is the kinematic viscosity and g the acceleration due to gravity) is equivalent to 1700 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 is an increasing function of for <[CDATA[h 2.

Shear plates have previously been used to measure bed shear stresses under swash and dam-break waves. The present study has been focused on the large bed shear stresses near the tip and even at distances less than one plate length from the tip. In order to resolve this rapid stress variation, the shear plate was calibrated with respect to its step response as well as statically. Step response calibration enables the effects of the time lag and the natural frequency of the shear plate system to be removed. Thus, the initial variation of the bed shear stress during and just after the rise of the water level is resolved and opens the opportunity to investigate the boundary layer development close to the tip where other measurements have not previously succeeded. The bed shear stress τ_bed(t) initially increases almost linearly with time to a maximum, which occurs about 0.2 s after the passage of the contact point, while it takes about 2.0 s for the water surface level h(t) to rise to a quasi-steady level under the present experimental conditions. Thus, with h(t) and τ_bed(t) varying with different timescales, the simple assumption of τbed=τbed(roughnessdepth) is not supported. For a given dam-break, the peak bed shear stress depends on the tailwater level. In our tests with initial dam depth 400 mm and bed roughness 84 mm, τ_bed peaks at 210 Pa ± 20 Pa on dry beds decreases to peak values of the order 40 Pa with 98 mm tailwater depths. Quasi-steady τ_bed-values reached while the water surface is still quasi-steady, which are of the order 5–10 Pa. For dry beds, τ_bed shows a single peak followed by a smooth monotonical decay, while tailwater depths above 40 mm may lead to two almost equal, successive τ_bed-peaks and an oscillating decay. Graphic abstract: [Figure not available: see fulltext.]

This paper focuses on the safety analysis of a nonlinear hydro-generating unit (HGU) running under different loads. For this purpose, a dynamic balance experiment implemented on an existing hydropower station in China is considered, to qualitatively investigate the stability of the system and to obtain the necessary indices for safety assessment. The experimental data are collected from four on-load units operating at different working heads including 431 m, 434 m, 437 m, and 440 m. A quantitative analysis on the safety performance of the four units was carried out by employing an integration of entropy weights method with grey correlation analysis. This assisted in obtaining the safety degree of each unit, providing the risk prompt to the operation of nonlinear hydro-generating units. The results confirm that unit 4 has the highest level of safety while unit 3 operates with the lowest safety condition. This provides the optimal operational schedule of HGUs to cope with the fluctuations of electricity demand in the studied station. The proposed methodology in this paper is not only applicable to the HGUs in the studied station but could also be adopted to assess the safety degree of any hydropower facility.