Stretching fields and mixing near the transition to nonperiodic two-dimensional flow

Although time-periodic fluid flows sometimes produce mixing via Lagrangian chaos, the additional contribution to mixing caused by nonperiodicity has not been quantified experimentally. Here, we do so for a quasi-two-dimensional flow generated by electromagnetic forcing. Several distinct measures of mixing are found to vary continuously with the Reynolds number, with no evident change in magnitude or slope at the onset of nonperiodicity. Furthermore, the scaled probability distributions of the mean Lyapunov exponent have the same form in the periodic and nonperiodic flow states.

Figure 1

(Color online) Stretching fields (see the text) at a constant phase of the forcing (negative going zero crossing) with red and blue indicating forward and backward in time fields, respectively: (a) periodic flow $\left(\text{Re}=7.8\right)$, with a map interval $\Delta t=10\text{s}$; (b) weakly turbulent or nonperiodic flow $\left(\text{Re}=108\right)$, with $\Delta t=1\text{s}$; (c) same as in (b) but one forcing period later (10 s); (d) reverse stretching field for the case shown in (b) with virtual dye placed in the flow and advected by the velocity fields to demonstrate that passive impurities do not cross lines of large past stretching.

Figure 2

(a) Logarithm of stretching $⟨\text{ln}S⟩$ vs time increment $\Delta t$ for periodic ($\text{Re}=7.8$, $○$) and turbulent ($\text{Re}=64$, $\square$ and $\text{Re}=108$, $▲$) flows; the growth is exponential with dashed line fits to the curves yielding the Lyapunov exponent. (b) Areal particle density $n\left(\Delta t\right)$ vs $\Delta t$ showing dispersion of particles seeded uniformly over a box of size $L=6\text{cm}$ for different Re: 7.8 $(○)$, 32 $(\square )$, and 108 $(▲)$. The decay, averaged over space and over several phases of the forcing, is exponential and yields a normalized decay constant $\alpha$ indicated by the dashed line fits to each curve.

Figure 3

(Color online) Probability distribution of the normalized stretching $\text{ln}S/⟨\text{ln}S⟩$ for Re: 7.5 (black continuous line), 32 (red long dashed line), and 108 (blue short dashed line). The inset shows the PDFs of $⟨\lambda ⟩=\text{ln}S/\Delta t$ for the same Re as in (a). The statistics are not significantly affected by the transition to nonperiodic flow.

Figure 4

The average Lyapunov exponent $⟨\lambda ⟩$ $(▲)$, the normalized (for comparison) rate of strain $\sigma ={\sigma }_{rms}/3$ $(○)$, and the areal particle density decay rate $\alpha$ $(■)$, as functions of Re. The dashed line is a linear plus quadratic fit to the Lyapunov data. The uncertainties in the measured quantities are estimated to be about 10%. The transition from periodic to weakly turbulent flow occurs around $\text{Re}=35$, where the velocity fields separated by a forcing period begin to differ. All of these measures of mixing vary smoothly across the transition from periodic to weakly turbulent flow.