Analytical models of mean secondary velocities and stream functions under different bed-roughness configurations in wide open-channel turbulent flows

Abstract

Turbulence-induced secondary currents are commonly present in straight natural as well as artificial open channels without bed forms. Different structures of cellular secondary currents can be seen in open-channel flows due to various bed configurations. In our study, mathematical models of turbulence-induced secondary currents in the vertical and transverse directions within a straight open rectangular channel with alternate rough and smooth longitudinal bed strips are proposed. The proposed models are derived using appropriate theoretical and mathematical analysis. Most of the previous models of secondary currents in the literature are proposed empirically and without proper mathematical derivations. The effects of fluid viscosity and eddy diffusivity are included in the present study to make it more practical. Initially, the governing equation for vertical secondary flow velocity is derived from continuity and the Reynolds-Averaged Navier Stokes equations. Then, the proposed problem is divided into two sub-considerations, corresponding to the base flow and perturbed flow. Finally, these sub-problems are analytically solved using method of variables separation with suitable boundary conditions. Different models to consider two different types of bed-roughness configurations (i.e. equal and unequal lengths of smooth and rough longitudinal bed strips) are obtained. Apart from velocity formulations, models of the stream function are proposed for these two types of bed configurations. All proposed models are validated using existing experimental data for the various bed configurations in open-channel flows and satisfactory results have been obtained. These present models are also compared with empirical models from the literature and they are found to be more effective in representing both types of bed-roughness configurations. The effects of bed configuration on the streamlines of settling velocity are also investigated. Results show that laterally-skewed secondary cells (which occurs due to unequal smooth and rough bed strips), have significant effects on the closed ω-streamlines in terms of shape and location of the centre of these streamlines. More precisely, it is found that the area of the downflow zone proportionally increases with the length of rough-bed strips.