Emergent scar lines in chaotic advection of passive directors

We examine the spatial field of orientations of slender fibers that are advected by a two-dimensional fluid flow. The orientation field of these passive directors are important in a wide range of industrial and geophysical flows. We introduce emergent scar lines as the dominant coherent structures in the orientation field of passive directors in chaotic flows. Previous work has identified the existence of scar lines where the orientation rotates by $\pi$ over short distances, but the lines that were identified disappeared as time progressed. As a result, earlier work focused on topological singularities in the orientation field, which we find to play a negligible role at long times. We use the standard map as a simple time-periodic two-dimensional flow that produces Lagrangian chaos. This class of flows produces persistent patterns in passive scalar advection and we find that a different kind of persistent pattern develops in the passive director orientation field. We identify the mechanism by which emergent scar lines grow to dominate these patterns at long times in complex flows. Emergent scar lines form where the recent stretching of the fluid element is perpendicular to earlier stretching. Thus these scar lines can be labeled by their age, defined as the time since their stretching reached a maximum.

Figure 1

(a) Poincaré section of the standard map showing the regular regions and chaotic regions for $K=2$. (b) Stretching experienced by the fluid over four periods at the same value of $K$, where ${\mathrm{\Lambda }}_1$ is the eigenvalue of the Cauchy-Green strain tensor defined in the Appendix.

Figure 2

(a) Angle of the advected directors ${\theta }_p$ of initially horizontal fibers. (b) Angle of the eigenvectors of the left Cauchy-Green strain tensor ${\theta }_e$. (c) and (d) Director representations of (a) and (b), respectively. All angles are measured with respect to horizontal. Animations of all four of these figures are included as Supplemental Material [46]. (Here $K=2$ and $t=4$.)

Figure 3

Gradient of the angle of the fiber orientation field after advection in the standard map for four periods. (a) Gradient of the advected director field starting with initially horizontal fibers. (b) Gradient of the eigenvector field. The large gradients lie on thin lines called scar lines [2]. (Here $K=2$ and $t=4$.)

Figure 4

The two different types of singularities. (a) and (b) Angle of ${\widehat{e}}_{L1}$ with respect to the horizontal. (c) and (d) Director representation of the eigenvector field. (a) and (c) show a singularity with a Poincaré index of $+\frac{1}{2}$ and (b) and (d) show a singularity with a Poincaré index of $-\frac{1}{2}$. To determine the Poincaré index, circle the singularity of (c) in the clockwise direction. Around the circle, the orientation of ${\theta }_e$ rotates by $\pi$ in the clockwise direction, giving a Poincaré index of $+\frac{1}{2}$. (Here $K=2$ and $t=2$.)

Figure 5

(a)–(d) Angle of advected directors ${\theta }_p$ measured from horizontal over the time interval $t=2$, 4, 6, and 8. (e)–(h) Angle of stretching eigenvectors ${\theta }_e$ over the same time intervals with singularities marked: $◯,+\frac{1}{2}$ singularities; $△,-\frac{1}{2}$ singularities. (i)–(l) Stretching experienced by the fluid for the same time intervals. Animations of these three fields are provided as Supplemental Material [46]. (Here $K=2$.)

Figure 6

Number of singularities $N$ as a function of time $t$. At later times the number of singularities increases exponentially. (Here $K=2$.)

Figure 7

Orientation fields at longer times that show the persistent pattern that appears in the chaotic region of both fields. (a) Orientation field of the advected directors ${\theta }_p$. (b) Orientation field of the stretching eigenvectors ${\theta }_e$. (Here $K=2$ and $t=10$.)

Figure 8

Locations of type 1 scar lines where the initial fiber orientation is perpendicular to the direction the fluid will be stretched. (a) Dot product of the initial orientation with the maximum eigenvector of the right CGST ${\widehat{p}}_0·{\widehat{e}}_{R1}$. (b) Locations where ${\widehat{p}}_0·{\widehat{e}}_{R1}<0.03$ superimposed on the stretching eigenvector orientation field. (Here $K=2$ and $t=4$.)

Figure 9

Sensitivity of the advected directors and type 1 scar lines to initial orientation. (a) Orientation field ${\theta }_p$ of an initially uniform grid of directors with an angle of $+{45}^{\circ }$. (b) Orientation field ${\theta }_p$ of an initially uniform grid of directors with an angle of $+{135}^{\circ }$. (Here $K=2$ and $t=4$.)

Figure 10

Locations of emergent scar lines. (a) Dot product of the maximum eigenvector of the left CGST for three periods and the maximum eigenvector of the right CGST for one period ${\widehat{e}}_{L1}(\mathrm{\Delta }t=3)·{\widehat{e}}_{R1}(\mathrm{\Delta }t=1)$. (b) Locations where ${\widehat{e}}_{L1}(\mathrm{\Delta }t=3)·{\widehat{e}}_{R1}(\mathrm{\Delta }t=1)<0.03$ superimposed on the stretching eigenvector orientation field. (Here $K=2$ and $t=4$.)

Figure 11

Simple schematic flow that generates singularities and emergent scar lines. (a) Uniform circular fluid elements. (b) Fluid elements after deformation by a spatially nonuniform flow field. (c) Fluid elements after a uniform pure strain flow with horizontal extension direction. Emergent scar lines form in the second row of fluid elements where the uniform stretching is orthogonal to the initial nonuniform stretching. (d) Director representation of the stretching eigenvector after the two stretching steps. The circle represents a $+\frac{1}{2}$ singularity and the triangle a $-\frac{1}{2}$ singularity.

Figure 12

Quantifying the age of emergent scar lines. (a) Field showing the time at which each fluid element experienced maximum stretching, which we use to define the age of emergent scar lines. Most of the field is white, indicating that maximum stretching occurs at the current time, but several generations of scar lines appear with maximum stretching before the present time. (b) Stretching as a function of time for points that lie on emergent scar lines of different ages: $*$, age 1 emergent scar line; $+$, age 2; $\diamond$, age 3; $\square$, age 4; $△,-\frac{1}{2}$ singularity on emergent scar line of age 2. (Here $K=2$ and $t=6$.)