A Lagrangian perspective towards studying entrainment

Abstract

This study presents a method for characterizing entrainment using Lagrangian data. Specifically, the method exploits the temporal evolution of enstrophy along pathlines to determine if pathlines undergo entrainment. The method only recently became feasible with the development of four-dimensional particle tracking velocimetry [4D-PTV; see Schanz et al. (Exp Fluids 57(5):7, 2016)], which produces pathlines of temporal length and spatial density that was previously unattainable. The proposed entrainment method was tested on experimentally acquired 4D-PTV data of a turbulent starting vortex forming behind a linearly accelerating circular plate. The method is shown to be insensitive to its control parameters. The first control parameter is an enstrophy threshold used to identify pathlines that always exhibit low enstrophy. The second control parameter is the size of an “enstrophy-source region”, which is placed within the measurement domain to identify pathlines that gain enstrophy through interactions with an enstrophy source. In the case of the starting vortex, the method reveals topological features significant to the entrainment process, such as the roll-up of irrotational fluid between the shear layer and vortex core, and pockets of entrainment that exist within undulations on the shear layer’s outboard side. Whereas the method proposed here measures the entrainment ratio directly, the use of Lagrangian coherent structures (LCSs) to quantify entrainment is shown to be infeasible for turbulent flows due to the presence of complex material manifolds for such flows. Furthermore, unlike results from Rosi and Rival (J Fluid Mech 811:3750, 2017), which analyzed the spreading of enstrophy isocontours to characterize entrainment into a starting vortex, the proposed method produces an entrainment rate that is insensitive to the enstrophy threshold, and clearly reveals mechanisms of entrainment. Using the starting vortex data, the proposed technique measured an entrainment ratio range similar to that for laminar vortex rings. The result suggests that, for this particular class of flows, turbulence does not play a significant role with respect to entrainment. The proposed method is seen as an effective tool, given its ability to quantify the entrainment, reveal its salient topological features, and potentially be easily adapted to other classes of flows.