Turbulence and Turbulent Length and Time Scales in Skimming Flows on a Stepped Spillway. Dynamic Similarity, Physical Modelling and Scale Effects

Stepped spillways are designed to increase the rate of energy dissipation on the chute and to reduce the size of the downstream energy dissipator. It is essential therefore to predict accurately the turbulent dissipation above the steps for large discharges per unit width corresponding to the skimming flow regime. Skimming flows are highly turbulent and aerated. Through the air-water interface, air is continuously trapped and released, and the resulting two-phase mixture interacts with the flow turbulence yielding a complicated air-water structure. Herein the two-phase turbulent flow measurements were conducted in a large facility with a channel slope of 21.8º and a step height of 0.05 m. Several types of phase-detection intrusive probes were used, and detailed turbulence data were collected including turbulence levels and integral turbulent time and length scales. The experiments were performed with dimensionless discharge dc/h between 1.17 and 3.16, and flow Reynolds numbers ranging from 1.7 E+5 to 7.2 E+5. The skimming flow characteristics presented some air-water flow properties that were qualitatively and quantitatively similar to previous studies. Very strong free-surface aeration was observed downstream of the inception point of air entrainment. The void fraction distributions exhibited a S-shape that followed closely an analytical solution of the advective diffusion equation for air bubbles. The distributions of both bubble count rate and turbulence intensity presented a marked maximum on the intermediate region (0.3 < C < 0.7). However, the relationship between turbulence level and dimensionless bubble count rate showed some alternate trend between adjacent step edges. This pattern was consistent with the longitudinal distributions of maximum bubble count rate and maximum turbulence intensity which exhibited a distinct see-saw pattern. It is believed that these oscillations are caused by the vortex shedding in the developing shear layers downstream of each step edge, their interactions between successive adjacent step cavities and their interference with the free-surface. The distributions of integral turbulent length and time scales presented some distinctive features. All the measurements highlighted maximum integral turbulent scales in the intermediate region. This intermediate region (0.3 < C < 0.7) between the bubbly and spray regions played a major role in the development the large eddies and turbulent dissipation. Turbulence level maxima were observed consistently for 0.4 < C < 0.5, while maximum integral turbulent scales were seen for 0.5 < C < 0.7. The findings suggested that, while the stepped cavities contributed to intense turbulence production, some turbulent energy was dissipated in the form of large-scale vortices in the bulk of the free-stream flow. The present measurements (
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=21.8º, h = 0.05 m) were compared systematically with the study of CHANSON and CAROSI (2007a) (
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=21.8º, h = 0.10 m). The comparative results were used to test independently the validity of the Froude and Reynolds similarities. The results showed consistently that the Froude similitude was not satisfied using a 2:1 geometric scaling ratio. Lesser number of entrained bubbles and comparatively greater bubble sizes were observed at the smaller Reynolds numbers, as well as lower turbulence levels and larger turbulent length and time scales. The present results with a 21.8° chute slope confirmed the earlier findings of CHANSON and GONZALEZ (2005) with 3.4° and 15.9° chute slopes. In practical terms, the small-size models did underestimate the rate of energy dissipation and the aeration efficiency of prototype stepped spillways for similar flow conditions. Similarly a Reynolds similitude was tested herein, and the results showed also some significant scale effects. The present experimental results showed however a number of self-similar relationships that remained invariant under changes of scale. The findings were significant and suggested that self-similarity may provide a picture general enough to be used to characterise the air-water flow field in large prototype spillways.