Meso-scale turbulence in living fluids

Abstract

Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.

Fig. 1.

(A) Schematic non-equilibrium phase diagram of the 2D SPR model and snapshots of six distinct phases from simulations: D-dilute state, J-jamming, S-swarming, B-bionematic phase, T-turbulence, L-laning (see also SI Appendix, Fig. S2 and Movies S1–S6). Our analysis focuses on the turbulent regime T. (B) Enstrophy per unit area Ω in units (V/λ)² for different aspect ratios a = ℓ/λ, obtained from SPR simulations with N ∼ 10⁴ to 10⁵ particles. The maxima of the enstrophy indicate the optimal filling fraction for active turbulence and mixing at a given value of the aspect ratio a. Note that values ϕ > 1 are possible due to the softness of the repulsive force (see SI Appendix for simulation parameters).

Fig. 2.

Experimental snapshot (A) of a highly concentrated, homogeneous quasi-2D bacterial suspension (see also Movie S7 and SI Appendix, Fig. S8). Flow streamlines v(t,r) and vorticity fields ω(t,r) in the turbulent regime, as obtained from (B) quasi-2D bacteria experiments, (C) simulations of the deterministic SPR model (a = 5, ϕ = 0.84), and (D) continuum theory. The range of the simulation data in D was adapted to the experimental field of view (217 μm × 217 μm) by matching the typical vortex size. (Scale bars, 50 μm.) Simulation parameters are summarized in SI Appendix.

Fig. 3.

Velocity statistics of self-sustained turbulent phases in active suspensions. (A) The marginal distributions of the normalized Cartesian velocity components are approximately Gaussian (thin gray line) for experiments, SPR model, and continuum theory. (B) The distributions of the longitudinal and transverse velocity increments δv_‖,⊥, normalized by their first and second moments are shown for three different separations R. (C) Longitudinal and transverse velocity structure functions normalized by . The maxima of the even transverse structure functions reflect the typical vortex size R_v , which is significantly larger in the 3D experiments. Experimental and theoretical data points are spatio-temporal averages over two orthogonal directions in A and B and all directions in C, yielding a typical sample size > 10⁶ per plotted data point in C. Histograms and structure functions for quasi-2D (3D) curves were obtained by combining PIV data from two (15) movies, respectively, representing an average over 2 × 1,000 (15 × 300) frames. Simulation parameters are identical to those in Fig. 2 and summarized in SI Appendix. Error bars are smaller than symbols.

Fig. 4.

Equal-time velocity correlation functions (VCFs), normalized to unity at R = ℓ, and flow spectra for the 2D SPR model (a = 5,ϕ = 0.84), B. subtilis experiments, and 2D continuum theory based on the same data as in Fig. 3. (A) The minima of the VCFs reflect the characteristic vortex size R_v (48). Data points present averages over all directions and time steps to maximize sample size. (B) For bulk turbulence (red squares) the 3D spectrum E₃(k) is plotted (k_ℓ = 2π/ℓ), the other curves show 2D spectra E₂(k). Spectra for the 2D continuum theory and quasi-2D experimental data are in good agreement; those of the 2D SPR model and the 3D bacterial data show similar asymptotic scaling but exhibit an intermediate plateau region (spectra multiplied by constants for better visibility and comparison).