Scaling transition of active turbulence from two to three dimensions

TL;DR

This study investigates how active turbulence in bacterial flows transitions from two to three dimensions, revealing universal scaling laws and critical heights that influence flow structures and energy spectra.

Contribution

The paper provides experimental measurements and a hydrodynamic model elucidating the dimensional transition in active turbulence, highlighting universal scaling laws and critical parameters.

Findings

Identification of three turbulence regimes with increasing plate separation H.

Discovery of universal scaling laws in energy spectra for 2D and 3D regimes.

Development of a hydrodynamic model predicting scaling behaviors consistent with experiments.

Abstract

Turbulent flows are observed in low-Reynolds active fluids. They are intrinsically different from the classical inertial turbulence and behave distinctively in two- and three-dimensions. Understanding the behaviors of this new type of turbulence and their dependence on the system dimensionality is a fundamental challenge in non-equilibrium physics. We experimentally measure flow structures and energy spectra of bacterial turbulence between two large parallel plates spaced by different heights $H$. The turbulence exhibits three regimes as H increases, resulting from the competition of bacterial length, vortex size and H. This is marked by two critical heights ($H_0$ and $H_1$) and a $H^{0.5}$ scaling law of vortex size in the large-$H$ limit. Meanwhile, the spectra display distinct universal scaling laws in quasi-two-dimensional (2D) and three-dimensional (3D) regimes, independent of bacterial activity, length and $H$, whereas scaling exponents exhibit transitions in the crossover. To understand the scaling laws, we develop a hydrodynamic model using image systems to represent the effect of no-slip confining boundaries. This model predicts 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 study suggests a framework for investigating the effect of dimensionality on non-equilibrium self-organized systems.