Unsteady turbulence in expansion waves in rivers and estuaries: an experimental study

Abstract

A sudden decrease in water depth, called a negative surge or expansion wave, is characterised by a gentle change in free-surface elevation. Some geophysical applications include the ebb tide flow in macro-tidal estuaries, the rundown of swash waters and the retreating waters after maximum tsunami runup in a river channel. The upstream propagation of expansion waves against an initially steady flow was investigated in laboratory under controlled flow conditions including detailed free-surface velocity and Reynolds stress measurements. Both non-intrusive free-surface measurements and intrusive velocity measurements were conducted for relatively large Reynolds numbers with two types of bed roughness. The data showed that the propagation of expansion waves appeared to be a relatively smooth lowering to the water surface. The wave leading edge celerity data showed a characteristic trend, with a rapid acceleration immediately following the surge generation, followed by a deceleration of the leading edge surge towards an asymptotical value: $(\mathrm{U}+\mathrm{V}_\mathrm{o})/(\mathrm{g}\times \mathrm{d}_\mathrm{o})^{1/2}=1$ for both smooth and rough bed experiments. The results indicated that the bed roughness had little to no effect, within the experimental flow conditions. Relatively large fluctuations in free-surface elevation, velocity and turbulent shear stress were recorded beneath the leading edge of the negative surge for all flow conditions. The instantaneous turbulent shear stress levels were significantly larger than the critical shear stress for sediment erosion. The present results implied a substantial bed erosion during an expansion wave motion.

Appendix: on ensemble-averaging

A fundamental challenge was the flow unsteadiness associated with the very-rapidly-varied unsteady motion during the surge propagation. While phase-averaging can be easily performed in periodic flows (e.g., [4, 23]), the technique is not suitable to very-rapidly-varied flows including the present study. Instead the experiments must be repeated in a carefully controlled manner and the results must be ensemble-averaged [1, 15]. Herein each experiment was conducted 25 times, although it is acknowledged that the number of repeated runs was relatively limited. A sensitivity analysis was performed on the effects of experiment number in terms of the free-surface properties, longitudinal velocity component and tangential stress $\overline{\mathrm{v}_\mathrm{x} \times \mathrm{v}_\mathrm{z}}$ for the rough bed configuration. The results showed that, during the propagation of the negative surge, including the leading edge passage, the time-variations of the free-surface elevation and free-surface fluctuations $(\mathrm{d}_{90}-\mathrm{d}_{10})$ were basically independent of the number of experiments for a minimum of 15 runs. And the time-variations of the longitudinal velocity component and velocity fluctuations $(\mathrm{V}_{90}-\mathrm{V}_{10})$ were basically independent of the number of experiments for a minimum of 15 runs. Similarly the time variations of the median tangential stress $\overline{\mathrm{v}_\mathrm{x} \times \mathrm{v}_\mathrm{z}}$ were little affected by the number of runs for a minimum of 20 runs. Altogether it is believed that the selection of 25 repeats was a reasonable compromise and may be compared with the results of Perry et al. [24] who needed 10 samples for convergence of the phase-averaged data. Herein the data were presented in terms of the median and decile difference values. This approach is commonly used in statistics with small to medium size data sets for which the mean and standard deviation values may be biased by outliers.