Double-Averaged Open-Channel Flows with Small Relative Submergence

Abstract

We investigate the turbulent structure of shallow open channel flows where the flow depth is too small ͑compared with the roughness height͒ to form a logarithmic layer but large enough to develop an outer layer where the flow is not directly influenced by the roughness elements. Since the log layer is not present, the displacement height d, which defines the position of the zero plane, and the shear velocity u * cannot be found by fitting the velocity data to the log law. However, these parameters are still very important because they are used for scaling flow statistics for the outer and roughness layers. In this paper we propose an alternative procedure for evaluating d in laboratory conditions, where d is found from additional experiments with the fully developed log layer. We also point out the appropriate procedure for evaluating the shear velocity u * for flows with low submergence. These procedures are applied to our own laboratory flume experiments with uniform sphere roughness, where velocities were measured using Particle Image Velocimetry. Results were interpreted within the framework of the double-averaged Navier-Stokes equations and include mean velocities, turbulence intensities, Reynolds stresses, and form-induced normal and shear stresses. The data collapse well and show that in flows without a developed log layer the structure of turbulence in the outer layer remains similar to that of flows with a log layer. This means that even though the roughness layer in the experiments reported herein was sufficiently high to prevent the development of the log layer, influence of the bed roughness did not spread further up into the outer layer. Furthermore, the results show that flow statistics do not depend on relative submergence except for the form-induced stresses which increase when relative submergence decreases.

Turbulence structure of open channel flows over permeable and impermeable beds: A comparative study

“…This methodology relies on an averaging procedure which follows two steps: ͑i͒ time averaging is applied in order to compute flow statistics for each measurement point contained within the flow regions captured by PIV; ͑ii͒ flow statistics resulting from time averaging are spatially averaged along thin boxes parallel to the bed. 18 The streamwise extent of such boxes is one bead diameter and, in the lateral direction, they include measurement points located in the top and valley section. By virtue of the PIV methodology the thickness of the averaging boxes is equal to the height of the interrogation areas.…”

“…This allowed to have enough measurement detail to estimate spatially averaged flow statistics. 18 For each experiment, 2200 images were recorded at a sampling frequency of roughly 13 Hz. The random error of a PIV measurement was estimated to be ⌬v = 0.0025 m / s. It was computed as ⌬v = ⌬s / ⌬t, where ⌬s corresponds to 1/10 of 1 pixel physical dimension, and ⌬t is the time delay between the subsequent images.…”

Free surface flow over square bars at intermediate relative submergence

“…Indeed, Gimenez-Curto and Corniero (1996) and Gimenez-Curto and Corniero Lera (2003) suggested that at very low submergences the form-induced stress could become the dominant component in the streamwise momentum balance. Interestingly Manes et al (2007) found that the relative contribution of form-induced stress increased as submergence decreased in their experiments on open channel flow over closely-packed spheres.…”

“…Nikora (2009) extended the use of the double-averaging methodology to develop a theoretical expression that explicitly showed that the friction factor can be accounted for by six additive components, of which only three are present in two-dimensional uniform spatially-averaged flow without secondary currents: viscous stress, turbulent stress and forminduced stress. The contribution of the form-induced stress, ũw , has recently attracted substantial attention as it has been shown to play an important role in the overall streamwise momentum balance in the near-bed region (Aberle, Koll, & Dittrich, 2008;Dey & Das, 2012;Ferreira et al, 2010;GimenezCurto and Corniero, 1996;Gimenez-Curto and Corniero Lera, 2003;Manes, Pokrajac, Coceal, & McEwan, 2008;Manes, Pokrajac, & McEwan, 2007;Manes, Pokrajac, McEwan, & Nikora, 2009;Mignot, Barthelemy, & Hurther, 2009a,b;Nikora et al, 2007b;Sarkar & Dey, 2010). Indeed, Gimenez-Curto and Corniero (1996) and Gimenez-Curto and Corniero Lera (2003) suggested that at very low submergences the form-induced stress could become the dominant component in the streamwise momentum balance.…”

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