Flocking and mesoscale turbulence in three-dimensional active fluids

Abstract

We numerically study the three-dimensional turbulence in a minimal model of an active fluid--the Toner-Tu-Swift-Hohenburg equation. For small activity, we observe bacterial turbulence, while for large activity, we uncover hitherto unexplored regime of a turbulent flock where a global order coexists with turbulence. We present a simple closure model that predicts the turbulent flock and also qualitatively explains the transition to the bacterial turbulence regime via a transcritical bifurcation.

Figures (5)

• Figure 1: (a) Mesoscale turbulence [$\mathcal{A}=0.44,\lambda=1.7$ (run R1)]: Streamlines of the velocity field, and the pseudo-color plot of the magnitude of the vorticity field in the XY and XZ plane. (b) Turbulent flock [$\mathcal{A}=1.1,\lambda=1.7$ (run R1)]: Velocity vectors showing large scale ordering, and two dimensional slice of the magnitude of the vorticity field in a plane perpendicular to the average velocity shows turbulence-like flow structures.
• Figure 2: (a) Numerically obtained phase-diagram showing regions of mesoscale turbulence and turbulent flock. The hash-region denotes the cross-over region, and the blue dots mark the simulations used to identify the phase boundary (runs R3-R4). (b) A typical ${\bm e}_{{\bm V}}$ trajectory in the turbulent flock regime, and the density map showing the time spent by the trajectory about an orientation (dark red regions indicate longer time). (c) Plot of $V^2/U^2$ vs. $\mathcal{A}$ with fixed $\lambda=1.7$ (runs R1). A transition from mesoscale to flocking turbulence appears around $\mathcal{A}=0.73$. (Inset) Plot of $E_b$ vs. $\mathcal{A}$ (open circle) and $E$ vs. $\mathcal{A}$ (cross). (d) Plot of $V^2/U^2$ versus $\lambda$ for fixed $\mathcal{A}=6.6$ showing transition from mesoscale to turbulent flock around $\lambda=0.86$ (runs R2). (Inset) Plot of $E_b$ vs. $\lambda$ (open circle) and $E$ vs. $\lambda$ (cross). The orange lines in (b,c) show that the predictions for $(E_0,E_b)$ from closure theory \ref{['eq:steady1']} are in excellent agreement with the data.
• Figure 3: Plots of $\langle {\bm u}^2 {\bm u} \rangle \cdot {\bm V}/4 E_0^2$ (filled circle) and $\langle {\bm u}^2 {\bm u} \cdot {\bm b}\rangle/4 E_0^2$ (open circle) versus $E_b/E_0$. The curves result from a least squares fitting (utilizing approximations \ref{['eq:approx1', 'eq:approx2']}) applied to data within the flocking regime, defined by $V^2/U^2 \geq 0.8$. The solid black line shows the plot $1+E_b/E_0$ (rhs of \ref{['eq:approx1']}), whereas the dashed black line corresponds to the fit $E_b/E_0 (1+ C E_b/E_0)$ (rhs of \ref{['eq:approx2']}) with $C=1.17\approx 6/5$. (Inset) Plots showing $\varepsilon$ vs. $\mathcal{A}$ (circle, fixed $\lambda=1.7$), and $\varepsilon$ vs. $\lambda$ (circle, fixed $\mathcal{A}=6.6$).
• Figure 4: Phase portrait showing the dynamics in the positive $(E_0,E_b)$ quadrant obtained using \ref{['eq:closure']} with $\mathcal{A}=6.6$, $\mathcal{B}=0.09$, and $\lambda=1.7$. The fixed points are marked with black squares and the nullclines are shown with black dashed lines. The fixed point $(0,0)$ is a unstable-node, the non-zero $(E_0,E_b)$\ref{['eq:steady1']} is a stable node, and $(0,E_b)$\ref{['eq:steady2']} is a saddle. The red trajectories, obtained from DNS for different initial conditions (circles), all terminate near a star-marked point that is located close to the stable node \ref{['eq:steady1']}.
• Figure 5: Energy spectrum for different values of $\mathcal{A}$ (see legend) while keeping fixed $\mathcal{B}=0.09$ and $\lambda=1.7$. At low $k$, we observe $E(k)\sim k^\delta$, where the exponent $\delta$ decreases from $-3.2$ to 1 as $\mathcal{A}$ increases bratanov2015sanjay2020. The inset shows components of the spectra in the flocking state ($\mathcal{A}=6.6$): $E_\parallel$ corresponds to the components of ${\bm b}$ aligned with the order direction ${\bm V}/{|\bm V|}$, and $E_\perp$ represents a direction orthogonal to it. As pointed out in the approximation for the closure model, we observe the transverse fluctuations dominate, $E_\|\sim 25 E_\perp$.