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Turbulent flows are observed in low-Reynolds active fluids. They areintrinsically different from the classical inertial turbulence and behavedistinctively in two- and three-dimensions. Understanding the behaviors of thisnew type of turbulence and their dependence on the system dimensionality is afundamental challenge in non-equilibrium physics. We experimentally measureflow structures and energy spectra of bacterial turbulence between two largeparallel plates spaced by different heights $H$. The turbulence exhibits threeregimes as H increases, resulting from the competition of bacterial length,vortex size and H. This is marked by two critical heights ($H_0$ and $H_1$) anda $H^{0.5}$ scaling law of vortex size in the large-$H$ limit. Meanwhile, thespectra display distinct universal scaling laws in quasi-two-dimensional (2D)and three-dimensional (3D) regimes, independent of bacterial activity, lengthand $H$, whereas scaling exponents exhibit transitions in the crossover. Tounderstand the scaling laws, we develop a hydrodynamic model using imagesystems to represent the effect of no-slip confining boundaries. This modelpredicts universal 1 and -4 scaling on large and small length scales,respectively, and -2 and -1 on intermediate length scales in 2D and 3D,respectively, which are consistent with the experimental results. Our studysuggests a framework for investigating the effect of dimensionality onnon-equilibrium self-organized systems.

Scaling transition of active turbulence from two to three dimensions