Irregular dependence on Stokes number, and nonergodic transport, of heavy inertial particles in steady laminar flows

Small heavy particles in a fluid flow respond to the flow on a timescale proportional to their inertia or Stokes number $\text{St}$. Their behavior has been thought to be gradually modified as $\text{St}$ increases. We show, on the other hand, in the steady spatially periodic laminar Taylor-Green vortex flow, that particle dynamics, and their effective diffusivity, actually change in an irregular, nonmonotonic, and sometimes discontinuous manner with increasing $\text{St}$. At $\text{St}\sim 1$, we show chaotic particle motion, contrasting with earlier conclusions for heavy particles in the same flow [Wang et al., Phys. Fluids 4, 1789 (1992)]. Particles may display trapped orbits, or unbounded diffusive or ballistic dispersion, with the vortices behaving like scatterers in a soft Lorentz gas [Klages et al., Phys. Rev. Lett. 122, 064102 (2019)]. The dynamics is nonergodic.

Figure 1

Typical particle trajectories exhibiting ballistic, diffusive, and trapped dynamics. The thin gray lines are streamlines of the TG vortex flow. (a) Particles initialized at (i) (0.12,2.6) with $\text{St}=1.12$ (green, diffusive), (ii) (2.6,2.8) with $\text{St}=0.95$ (orange, diagonal “square ballistic”), (iii) (2.48,1.07) with $\text{St}=1.83$ (red, diagonal smooth ballistic), and (iv) (1.5,1.5) with $\text{St}=1.133$ (magenta, horizontal ballistic). The (left) zoomed inset shows the initial positions by filled circles. The other insets (right) show the ballistic (smooth) and diffusive trajectories in compactified physical space. (b) Particles initialized at (i) (1.0,0.42) with $\text{St}=1.12$ (black, limit cycle), (ii) (2.33,0.81) with $\text{St}=1.37$ (blue, chaotic attractor), (iii) (0.664,0.11) with $\text{St}=0.89$ (brown, trapped in square limit cycle), and (iv) (1.2,0.6) with $\text{St}=0.5$ (violet, trapped at the SP shown by the filled square). All trajectories plotted conserve the quantity $\mathcal{Q}$ to better than $0.1\%$, as shown in Fig. 12 in Appendix pp1.

Figure 2

Trapped, diffusive, and ballistic particle trajectories at the same Stokes number $\text{St}=1.18$ have distinct signatures. Histograms of the number of trajectories vs (a) $\alpha$ and (b) the mean characteristic speed ${log}_{10}\left(v_m\right)$ defined in Secs. 3a and 3b are shown respectively. A total of $100\times 100$ trajectories are integrated until $T={10}^4$ and grouped using a bin size of ${10}^{-2}$. The standard color scheme of blue: trapped, green: diffusive, and red: ballistic is used.

Figure 3

The time evolution of Lyapunov exponents ${\lambda }_i\left(t\right)$ for (a) three $\text{St}=1.18$ particles showing trapped (regular), diffusive (chaotic), and ballistic (regular) dispersion at the large time respectively starting at positions (1.5,0.5), (0.55,0.15), and (0.5,1.0) and (b) $\text{St}=0.5$ particle starting at initial position (1.5,0.5) and getting trapped (in a SP) at the large time. The initial velocity of the particles in all the cases is fixed to be zero. Observe the symmetry of the Lyapunov spectrum in each case.

Figure 4

(a) Distribution of large-time Lyapunov exponents of $316\times 316$ trajectories with $\text{St}=1.18$ (starting from basic vortex cell at uniformly distributed locations, with zero initial velocity) after $T={10}^4$ plotted in the ${\lambda }_1\text{−}{\lambda }_2$ plane. (b) Histograms, generated with 1000 bins, of the Lyapunov exponents ${\lambda }_1$ and $-{\lambda }_2$ of $316\times 316\text{St}=1.18$ particles, simulated up to ${10}^5$ nondimensional time units.

Figure 5

Trajectories in the stationary state [i.e., $t\in ({10}^3,{10}^4)]$ showing typical dynamics, for particles initialized (a) at (1.0,0.42) with $\text{St}=1.12$ (trapped-regular), (b) at (2.48,1.07) with $\text{St}=1.83$ (smooth ballistic-regular, diagonal), (c) at (1.5,1.5) with $\text{St}=1.133$ (ballistic-regular, horizontal, and square type), (d) at (2.33,0.81) with $\text{St}=1.37$ (trapped-chaotic), (e) at (0.11,2.59) with $\text{St}=1.12$ (diffusive-chaotic), and (f) at (0.25,0.5) with $\text{St}=1.06$ (ballistic-chaotic). The insets show zoomed-in views near SP regions. A vertical ballistic trajectory similar to (c) also exists in rare parameter combinations, which is not shown here.

Figure 6

The autocorrelation function $C\left(\tau \right)$ for representative trajectories of the different types mentioned in the text (cf. Fig. 5). The horizontal velocity $v_x\left(t\right)$ is used to calculate $C\left(\tau \right)$ [see Eq. (14)]. The same color scheme as Fig. 5 is used here.

Figure 7

Frequency spectrum for the representative trajectories in Figs. 5 or 6. The DFT is performed with a sampling frequency of 0.1. We note that square-type trajectories display a larger multiplicity of frequencies than smooth trajectories.

Figure 8

(a) Colormaps, with $316\times 316$ particles and $\text{St}=1.18$, of (bottom) $\alpha$ such that ${\sigma }^2\left(t\right)\sim t^{\alpha }$ at large times, and (top) ${\lambda }_1$, the largest of the “large-time Lyapunov exponents.” (b) ${\sigma }^2\left(t\right)$ (solid lines) for particles starting at (1.5,0.5), (0.53,0.15), and (0.5,1.0) and displaying typical ballistic, diffusive, and trapped large-time dispersive behavior, respectively; dashed lines show the corresponding scalings.

Figure 9

(a) Fraction, $f$, of trapped, diffusive. and ballistic particles. The inset shows effective diffusivity $D_{\text{eff}}$ (black) and the average $\alpha$ (green) of diffusive particles. ${\alpha }_d\approx 1$ indicates normal diffusion. (b) The fraction ($f$) of particles with regular and chaotic Lyapunov spectrum are plotted for various Stokes numbers. The fraction of (trapped $\cup$ ballistic) particles and diffusive particles are shown in the background using light colors. The one-to-one mapping between the two kinds of classification is evident (excluding certain windows of Stokes number).

Figure 10

Colormaps of (a) $\alpha$, similar to Fig. 8, bottom, and (b) the large-time Lyapunov exponent ${\lambda }_1$ similar to Fig. 8, top, with the same parameters as in those figures except with the initial particle velocity set equal to the local fluid velocity.

Figure 11

(a) Streamlines of the triangular flow field. Observe the qualitative similarity of the flowfield with the TG vortex flow field shown in the background of Fig. 1. (b) Color map of $\alpha$, indicative of the large-time dynamics, of $316\times 316$ inertial particles with $\text{St}=1.15$ initialized uniformly in the basic vortex cell $(0,\pi )\times (0,\pi )$ of triangular flow; trapped (blue), diffusive (green), and ballistic (red) trajectories are possible at this Stokes number.

Figure 12

(a) Evolution of constant of dynamics $\mathcal{Q}$ with time for the cases in Fig. 1 with the same color codes. (b) The percentage relative error in the numerically evaluated value of $\mathcal{Q}$ with the theoretically expected value ${\mathcal{Q}}_0$ is plotted with time and shown to be negligible and bounded for all the cases until $t={10}^5$.

Figure 13

Bifurcation diagram plotted using local maxima of $v_x\left(t\right)$ vs $\text{St}$ for a particle starting at the initial location (2.33,0.81) with zero initial velocity. The values in the $\text{St}$ axis ranges from 0.75 to 2. The insets zoom in on certain regions where the system transition from regular to chaotic dynamics as $\text{St}$ is changed.