A study of mixing in coherent vortices using braiding factors

This paper studies the use of braiding fluid particles to quantify the amount of mixing within a fluid flow. We analyze the pros and cons of braid methods by considering the motion of three or more fluid particles in a coherent vortex structure. The relative motions of the particles, as seen in a space–time diagram, produces a braid pattern, which is correlated with mixing and measured by the braiding factor. The flow we consider is a Gaussian vortex within a rotating strain field which generates cat’s eyes in the vortex. We also consider a modified version of this strain field which contains a resonance frequency effect that produces multiple sets of cat’s eyes at different radii. As the thickness of the cat’s eyes increase they interact with one another and produce complex Lagrangian motion in the flow which increases the braiding of particles, hence implying more mixing within the vortex. It is found that calculating the braiding factor using only three fluid particles gives useful information about the flow, but only if all three particles lie in the same region of the flow, i.e. this gives good local information. We find that we only require one of the three particles to trace a chaotic path to give an exponentially growing braiding factor. i.e. a non–zero ‘braiding exponent’. A modified braiding exponent is also introduced which removes the spurious effects caused by the rotation of the fluid. This analysis is extended to a more global approach by using multiple fluid particles that span larger regions of the fluid. Using these global results we compare the braiding within a viscously spreading Gaussian vortex in the above strain fields, where the flow is determined both kinematically and dynamically. We show that the dynamic feedback of the strain field onto the flow field reduces the overall amount of braiding of the fluid particles. 1 Current address: Sir Harry Ricardo Laboratories, School of Computing, Engineering and Mathematics, Cockcroft building, University of Brighton, Lewes Road, Brighton. BN2 4GJ. UK 2 Corresponding author: [email protected]