Intrinsic flow structure and multifractality in two-dimensional bacterial turbulence

The active interaction between the bacteria and fluid generates turbulent structures even at zero Reynolds number. The velocity of such a flow obtained experimentally has been quantitatively investigated based on streamline segment analysis. There is a clear transition at about 16 times the organism body length separating two different scale regimes, which may be attributed to the different influence of the viscous effect. Surprisingly the scaling extracted from the streamline segment indicates the existence of scale similarity even at the zero Reynolds number limit. Moreover, the multifractal feature can be quantitatively described via a lognormal formula with the Hurst number $H=0.76$ and the intermittency parameter $\mu =0.20$, which is coincidentally in agreement with the three-dimensional hydrodynamic turbulence result. The direction of cascade is measured via the filter-space technique. An inverse energy cascade is confirmed. For the enstrophy, a forward cascade is observed when $r/R\le 3$, and an inverse one is observed when $r/R>3$, where $r$ and $R$ are the separation distance and the bacteria body size, respectively. Additionally, the lognormal statistics is verified for the coarse-grained energy dissipation and enstrophy, which supports the lognormal formula to fit the measured scaling exponent.

Figure 1

Two snapshots of the instantaneous streamline colored with the velocity magnitude. Visually the flow field is smooth with a typical large-scale structure size of $\sim 50 \mu \mathrm{m}$.

Figure 2

Experimental dissipation spectrum $E\left(k\right)k^2$, where a peak is found to be around $k_{p}R=0.17$, corresponding to a spatial scale ${\ell }_p\simeq 6R$. The inset shows the Fourier power spectrum $E\left(k\right)$, where a power-law fitting $-8/3$ on the range $0.2\le kR\le 0.4$ is illustrated as a solid line.

Figure 3

(a) Illustration of a streamline with respect to a specified grid point (denoted by $+$). Along the streamline the local extremal points, either maximum (in red) or minimum (in blue), can be identified according to the velocity magnitude $u$. The given streamline is then divided into streamline segments, which are defined as the part confined by two adjacent local extremal points. According to the sign of $\mathrm{\Delta }u$ along the velocity direction, the streamline segment can be positive (p) if $\mathrm{\Delta }u>0$ or negative (n) if $\mathrm{\Delta }u<0$. (b) With respect to selected grid points (denoted by $+$), the corresponding streamline segment structure is extracted from a snapshot of experimental data with local maximal points and local minimal points.

Figure 4

(a) Experimental joint PDF $p(\ell ,\mathrm{\Delta }u)$. For display convenience, $p(\ell ,\mathrm{\Delta }u)$ is measured at the logarithmic scale. For a better display, the scale is in the range $0\le \ell /R\le 15$. (b) The corresponding marginal PDF $p\left(\ell \right)$. For comparison, normal and exponential distributions are illustrated as dashed and solid lines, respectively.

Figure 5

(a) High-order intrinsic structure function $M_q\left(\ell \right)$ for $q$ from 1 to 4. The power-law behavior is observed in the range $2\le \ell /R\le 10$, corresponding to a wave-number range $0.1\le kR\le 0.5$. The solid line indicates a power-law fitting in this scaling range via the least-squares fitting algorithm. (b) The corresponding compensated curve using the fitted parameters to emphasize the observed power-law behavior.

Figure 6

(a) Scaling exponent $\zeta \left(q\right)$ in the range $2\le \ell /R\le 10$ for $0\le q\le 4$. A lognormal formula with a Hurst number $H=0.76\pm 0.01$ and an intermittency parameter $\mu =0.20\pm 0.01$ is shown as a solid line. (b) The corresponding singularity spectrum $f\left(\alpha \right)$. The error bar indicates a $95\%$ confidence interval.

Figure 7

Scale-to-scale flux for (a) energy ${\mathrm{\Pi }}^E\left(r\right)$ and (b) enstrophy ${\mathrm{\Pi }}^{\mathrm{\Omega }}\left(r\right)$. A negative value indicates an inverse flux transferring from small to large scales.

Figure 8

PDF of (a) the coarse-grained energy dissipation ${\epsilon }_r$ and (b) the coarse-grained enstrophy ${\mathrm{\Omega }}_r$. The lognormal formula is also shown (dashed line) for comparison.