Using braids to quantify interface growth and coherence in a rotor-oscillator flow

Abstract

The growth rate of material interfaces is an important proxy for mixing and reaction rates in fluid dynamics and can also be used to identify regions of coherence. Estimating such growth rates can be difficult, since they depend on detailed properties of the velocity field, such as its derivatives, that are hard to measure directly. When an experiment gives only sparse trajectory data, it is natural to encode planar trajectories as mathematical braids, which are topological objects that contain information on the mixing characteristics of the flow, in particular through their action on topological loops. We test such braid methods on an experimental system, the rotor-oscillator flow, which is well described by a theoretical model. We conduct a series of laboratory experiments to collect particle tracking and particle image velocimetry data, and we use the particle tracks to identify regions of coherence within the flow that match the results obtained from the model velocity field. We then use the data to estimate growth rates of material interface, using both the braid approach and numerical simulations. The interface growth rates follow similar qualitative trends in both the experiment and model, but have significant quantitative differences, suggesting that the two are not as similar as first seems. Our results shows that there are challenges in using the braid approach to analyze data, in particular the need for long trajectories, but that these are not insurmountable.

Braids in the heart: global measures of mixing for cardiovascular flows

“…An important point that can be seen in figure 3( a ) is that, even though no particles are being ejected by the left ventricle during the advection period, some particles do still leave the field of view. This issue is present in many braid applications, such as mixing flows (Filippi et al., 2020) and granular flows (Puckett et al., 2012). Recall that the model datasets are originally obtained from our particle image velocimetry measurements in an in vitro experiment (Di Labbio & Kadem, 2018; Di Labbio et al., 2018).…”

“…in a third dimension), their ‘world lines’ become entangled, forming a braid (Artin, 1947). Likewise, for more general two-dimensional flows, a braid can be formed by advecting or tracking particles directly and using their trajectories in two-dimensional space and time as the strands of the braid (Allshouse & Thiffeault, 2012; Boyland, Stremler, & Aref, 2003; Filippi et al., 2020; Francois, Xia, Punzmann, Faber & Shats, 2016; Gouillart, Finn, & Thiffeault, 2006; Puckett, Lechenault, Daniels, & Thiffeault, 2012; Smith & Warrier, 2016; Taylor & Llewellyn Smith, 2016; Thiffeault, 2005, 2010; Thiffeault & Finn, 2006; Thiffeault, Finn, Gouillart, & Hall, 2008; Yeung, Cohen-Steiner, & Desbrun, 2020). We illustrate how particle trajectories can be used to produce a braid in figure 2.…”

Braiding Dynamics in Active Nematics

“…The use of braids to analyze the dynamics of fluid systems is well established [11][12][13][14][15]. This topological perspective has been used to help find coherent structures in flows [16,17], to characterize point vortex motion [18,19], to investigate mixing in lid-driven cavity flow [20] and channel flow [21], and to design industrial mixing protocols [22]. Recently, braids were used to help understand the ANMT turbulent state [6].…”

Topological Entropy of Surface Braids and Maximally Efficient Mixing

“…In this context, the mixing of a 2D fluid (a fluid surface, or a bulk fluid where disparate characteristic time-scales result in effective 2D motion) is encoded in the braid formed from the space-time trajectories of stirring rods or passively advected particles. For example, the trajectories of data-collection buoys can help reveal regions of the ocean in which the flow enhances mixing or help identify coherent structures in which mixing is minimized 6,7 . These structures 8 play a central role in the transport of nutrients, oxygenation, temperature, and pollutants.…”