Abstract

When two nearly parallel streams of different velocities join together at a river confluence, a mixing layer develops along the joining interface. The mixing layer is termed a shallow mixing layer (SML) when the flow has a small depth and thus is significantly influenced by riverbed friction. SML has impacts on the riverine environment, river ecosystems, and hydraulic engineering design. The effects of the velocity ratio (Vr) between the two incoming streams on SML characteristics have not been addressed adequately from both Lagrangian and Eulerian perspectives. This study employs the Lagrangian and Eulerian approaches to explore the SML. Specifically, this study aims to answer these main questions: How does SML behave under different ratios of incoming flow velocities? How does the shear caused by the velocity gradient influence the fluid particles in SML? To what extend previous relations proposed to determine mixing layer width are practical? What factors control the pairing of adjacent eddies and the growth of large-scale coherent structures? How are the fluid mass and momentum exchanges affected by Vr?
The dye visualisation and particle tracking velocimetry (PTV) techniques were used in the laboratory experiments of SML. In the dye visualisation technique, a tracer’s motion is recorded using a single camera. PTV is considered an optical flow measurement technique in which neutrally buoyant particles are tracked in consecutive image frames. PTV provides the essential means for Lagrangian studies of SML. For numerical simulations of SML, the smoothed particle hydrodynamics (SPH) model was used. SPH is considered a Lagrangian CFD method in which a continuum is discretised using a set of material points or particles. Laboratory experiments of SML were conducted at three velocity ratios: Vr = 1, 1.14, and 1.5, and PTV measurements were made from the region between the joining location (x = 0 m) and 0.3 m downstream (x = 0.3 m) in the confluence. The dye visualisation experiments consisted of the same three velocity ratios, and the visualisation covered the region of x = 0–1.2 m. The SPH simulations included three velocity ratios: Vr = 1.14, 1.5, and 3; the domain covered x = 0–1 m.
The PTV measurements show that the boundary layers, which develop on the sidewalls of a splitter plate used to separate the two incoming streams before joining, and the wake effect cause a velocity deficit in the confluence and limit the mixing layer growth. The SPH results reveal that a smaller velocity ratio results in a more visible velocity deficit in streamwise velocity profiles due to the relative importance of the wake versus velocity gradient in the SML. PTV application for SML investigation has been found to require special technical considerations, some of which were introduced in this study. The technical measures in the data acquisition and the Python script developed for particle trajectory analysis improved the PTV technique in studying SML. Finite-time Lyapunov exponents (FTLE) results indicate that for particle trajectories located inside the mixing layer, a divergence was evident with a positive value of FTLE. However, for the particle trajectories out of the mixing layer, both positive and negative FTLE can be observed. The dye visualisation results show that turbulent instabilities still form in the absence of velocity gradient (Vr = 1). When the velocity gradient exists (Vr > 1), the instabilities persist, and a pairing of eddies is observed. The intermittency of SML is observed with a lack of a temporally fixed pattern in the vortex arrangement at a fixed location. The dye visualisation results show a linear relation between eddy spacing and downstream distance, with the most frequent eddy spacing being 0.42x.
A new approach to the determination of mixing layer width was proposed based on the boundary layer definition. Results show that for smaller velocity ratios, the mixing layer width determined by the boundary layer method is smaller than those from existing empirical relations. A pairing process of vortices occurs less often when Vr is as small as 1.14, compared to that for Vr = 1.5 and 3. The results also show that pairing activities in SML are affected mainly by the average vorticity magnitude of two neighbouring eddies rather than their relative distance. The Okubo-Weiss parameter of SML indicates that the general form of an eddy in SML consists of an inner vorticity-dominated region at the core and an outer region, which is strain-dominated and surrounding the inner region. The strain-dominated boundary of the eddies performs such a barrier for the particles inside the eddy until the eddies decay or are paired with other eddies. Smaller velocity ratios result in lower mass transfer with a constant rate from the tip of the splitter plate to the downstream whilst for Vr = 3, the mass transfer rate increases moving downstream due to the larger eddies and more profound pairing process. In the SML, the intermittent crests and troughs in momentum transfer indicate the evolution of eddies.