Intrinsic flow structure and multifractality in two-dimensional bacterial turbulence

Abstract

The active interaction between the bacteria and fluid generates turbulent structures even at zero Reynolds number. 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 of 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 $r/R>3$. 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.