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

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.

Figure 1

Geometry for the rotor-oscillator flow. The walls are at $x=\pm h$, with the rotor oscillating in $y$ at constant $x=c$.

Figure 2

(a) Poincaré map for the model rotor-oscillator flow [19, 77] with $c=0.54, \epsilon =0.125, \lambda =2\pi /5$ simulated for 300 forcing cycles. The blue and red colors depend on the average value of the $y$ coordinate along simulated trajectories. The Poincaré map is sampled at the rotor's oscillation period $\tau$. (b) Detailed view of the green dashed frame in (a).

Figure 3

Schematics of the experimental setup for the Hackborn-Weldon rotor-oscillator flow [19, 77], with still walls and an oscillating rotor. (a) Experimental apparatus setup. (b) Fluid system to set up a two-dimensional flow through density layers for the PT experiment. In the PIV experiment, the tracer particles were mixed to a single layer of glycerol, and a laser sheet was projected horizontally.

Figure 4

Snapshot of instantaneous streamlines of (a) the model velocity field [19, 77] with $c=0.54, \epsilon =0.125, \lambda =2\pi /5$ and (b) the PIV-recorded experimental velocity field at the same time instance.

Figure 5

A physical braid and a corresponding topological braid generated from five trajectories. In all diagrams the time flows from bottom to top. The sequence of generators (ordered from left to right in increasing time) is $B={\sigma }_2^{-1}{\sigma }_4{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_2^{-1}{\sigma }_4^{-1}{\sigma }_2{\sigma }_4{\sigma }_3{\sigma }_4^{-1}$.

Figure 6

Two topological loops before (left column) and after (right column) action of the braid from Fig. 5. The strands of the braid are shown in black as cross sections. The Dynnikov vectors before and after the braid action in (a) are $\left[111000\right]\stackrel{B}{\to }[0-230-10]$ and in (b) $\left[000001\right]\stackrel{B}{\to }[0-110-10]$.

Figure 7

Coordinate system based on intersection numbers ${\alpha }_i,{\beta }_i$ used to calculate Dynnikov coordinates (6). In this example of $N=5$ strands, ${\alpha }_{1,3,5}=0, {\alpha }_{2,4,6}=2, {\beta }_{1,2,3,4}=2$, resulting in the Dynnikov vector $\left[111000\right]$. Precise location of strands in the $x,y$ space is not important, only their order along the projection coordinate.

Figure 8

(a) Particles (dots) and loops in physical space. Particles in the identified structure (blue) are connected by straight lines (solid) and surrounded by the corresponding straight pair loop (dashed and dotted lines). The coherent structure loop (blue line) links the two straight pair loops. (b) The particles and loops in standard form, with points aligned on the real axis. The Dynnikov vectors are $[010-10-10001]$ for the blue curve, $[011-10-10001]$ for the red curve, and $[01000-10100]$ for the green curve.

Figure 9

(a) The initial positions of $N=300$ analytic model trajectories (dots), colored based on their coherent structure assignment with unassigned particles in black. The lines connecting particles indicate straight pair loops that have a growth ratio less than the threshold, with less opaque lines representing loops that grow more. (b) and (c) Zoomed-in regions corresponding to the gray dashed boxes in (a). The coherent structures corresponding to slowly growing loops for each set of points are drawn in the corresponding color.

Figure 10

(a) The initial positions of $N=33$ experimental trajectories (dots) and identified coherent loops (black lines). Three coherent structures are identified (red, blue, and green). The straight pair loops that have a growth rate of less than the threshold are represented as straight lines connecting the particles. The less opaque the line, the greater the growth rate. (b) The final position of the trajectories and the deformed coherent loops.

Figure 11

Top: Loop ${\ell }_0$ used to compute the FTBE, with Dynnikov vector ${\ell }_0=[0000-1-1-1-1]$. Bottom: The loop after action of the braid from Fig. 5, with Dynnikov vector $B{\ell }_0=[-10810-6-37]$. Black dots: strands belonging to the braid; rightmost red dot: the auxiliary “anchor” strand. The length of a loop is the number of intersections with the dashed axis ($|{\ell }_0|=4,|B{\ell }_0|=36$).

Figure 12

Poincaré plots of trajectories in braids used to compare FTBE and braid growth rate between analytic model and PIV velocity fields. Trajectories were initialized uniformly inside the highlighted rectangles (a) $(x,y)\in [-0.25,-0.5]\times [-0.5,0.5]$ for model velocities; (b) $(x,y)\in [-0.1,-0.5]\times [-0.5,0.5]$ for PIV velocities. The Poincaré map is taken at the rotor's oscillation period $\tau$.

Figure 13

Standardized distributions of FTBEs and growth rates with $S=500$ samples of $n=300$-stranded subbraids (out of $N=600$ trajectories, $T=30\tau =150$); nonstandardized histograms are inset into each graph.

Figure 14

Dependence of the maximum FTBE and mean growth rate on trajectory duration $T$. The comparison is between the maximum FTBE over a sample of $n=300$-stranded braids and a single braid of $n=600$ strands, all from analytic model and PIV flows.

Figure 15

Effect of increase in braid size (number of strands) on FTBE and growth rate of braids of trajectories from model and PIV flows. Filled-in marks correspond to maximum of a sample of $n$-stranded braids. Hollow marks correspond to the single 600-stranded braid.