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Double-Averaged Momentum and Energy Balances in Open-Channel Flows over Streamwise Ridges

Author(s): Stuart Cameron; Andrea Zampiron; Vladimir Nikora

Linked Author(s): Stuart Cameron, Andrea Zampiron, Vladimir Nikora

Keywords: Double-averaging methodology; Secondary Currents; Energy Budget; Momentum Balance; Open-channel flows

Abstract: Flows over beds covered with streamwise ridges exhibit counter-rotating cells in the time-averaged velocity field with associated streamwise vorticity. These secondary current cells scale with the inter-ridge spacing, interact with large-scale turbulent structures, and contribute to momentum and energy transfer within the flow with implications for mixing, bed friction, and sediment transport. A better understanding of the mechanisms that control the formation of secondary currents, their size and their strength may lead to improvements in the modelling, design, and management of open channels. The double-averaging (averaging in time and in space) methodology provides a useful framework to study flows over ridge covered beds, as with careful selection of the spanwise size of the averaging domain to a multiple of the ridge spacing, the equations are greatly simplified and terms in the momentum and energy balance equations are neatly partitioned into turbulence and secondary current contributions. In this study, we explore the distributions of these terms and their dependence on the interridge spacing for the case of smooth triangular shaped ridges overlying a hydraulically rough bed. Our data, obtained with stereoscopic PIV, indicate that the normalised momentum and energy fluxes associated with secondary currents collapse when plotted as functions of (z-d)/s, where z is the vertical coordinate, s is the transverse ridge spacing, and d is a constant that aligns the elevation of secondary current cell centres. Contributions of the secondary currents to the total transport of momentum and energy were similar in magnitude to the contributions of turbulence. We also observed the data collapse for the vertical gradient of the double-averaged streamwise velocity when normalised with u*/s, where u* is the shear velocity, suggesting that the shape of the mean velocity profile was dominated by the secondary currents. Analysis of the double averaged equations for the Reynolds stresses indicates that energy flows initially to the double-averaged streamwise velocity through a gravity source term where it is then distributed to streamwise velocity spatial and turbulent fluctuations. Energy is then distributed from the streamwise to the vertical and transverse turbulent fluctuations via a pressure-strain term. Finally, energy of the secondary currents does not come directly from the double-averaged streamwise velocity, but it is instead supplied by harvesting the energy of the transverse and vertical turbulent velocity fluctuations, confirming a strong association between turbulence and secondary currents generation.

DOI: https://doi.org/10.3850/IAHR-39WC252171192022419

Year: 2022

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