Control mechanisms for the global structure of scalar dispersion in chaotic flows

Scalar dispersion has complex interactions between advection and diffusion that depend on the values of the scalar diffusivity and of the (possibly large) set of parameters controlling the flow. Using a spectral method which is three to four orders of magnitude faster than traditional methods, we calculate the fine-scale structure of the global solution space of the advection-diffusion equation for a physically realizable chaotic flow. The solution space is rich: spatial pattern locking, an order-disorder transition, and optima in dispersion rates that move discontinuously with Peclét number and boundary condition type are some of the discoveries. We uncover the mechanisms which control pattern locking and govern the global structure of dispersion across the parameter space and Peclét number spectrum.

Figure 1

Rotated arc mixer (RAM) flow (a) geometry and parameters; (b) streamlines for Newtonian Stokes flow [12].

Figure 2

Distribution of the fraction $f_j$ for $j=1,2,...,10$ (distributions narrow with increasing $j$) of locked orbits in the RAM flow as a function of flow parameters $\Theta$ and $\tau$ for locking angles $2\pi p/j$.

Figure 3

Map of the fraction $f(\Theta ,\tau )$ of symmetry-locked orbits in the RAM flow parameter space $\mathcal{Q}$, ranging from $f=0$ (pink) to $f=1$ (red).

Figure 4

Dominant strange eigenmodes (finite Pe) and Poincaré sections (infinite Pe) for select points in the $\{\tau ,\Theta \}$ plane.

Figure 5

Persistence of a strange eigenmode (with decaying amplitude removed) through one reorientation interval. At $t=0$ the moving boundary indicated by the arrow moves. Region (a) has been stretched and folded; diffusion heals the pattern along folds. Simultaneously advection stretches and makes new folds around the region (b). The pattern at $\tau$ persists, albeit rotated by $\Theta$.

Figure 6

Map of the asymptotic transport rate $q\equiv \text{Re}\left({\lambda }_0\right)\text{Pe}/{\alpha }_{0,1}^2$ for Dirichlet boundary conditions, $\Delta =\pi /4$ over the control parameter space $\mathcal{Q}$. Note the logarithmic scaling of $q$ contours—different for each Pe—and the $\tau$ axis.

Figure 7

Symmetry-locked intervals of dominant strange eigenmodes in the RAM flow parameter space $\mathcal{Q}$. $k$ is the wave number of the forcing and $m$ is the azimuthal wave number of the pattern. Dots show the $k$ intervals over which patterns are locked. (Inset) Locking is symmetric about $\Theta =0$.

Figure 8

Energy spectra for mixing protocol (e) at $\text{Pe}={10}^4$ (black) and $\text{Pe}={10}^3$ (gray).

Figure 9

(a) Poincaré section, (b) finite-time Lyapunov exponent distribution, (c) barriers to diffusive transport, and (d)–(h) dominant five eigenmodes at $\text{Pe}={10}^4$, $\tau =8$, $\Theta =4\pi /5$, and $\Delta =\pi /4$.