Hydrodynamics of Dense Active Fluids: Turbulence-Like States and the Role of Advected Activity

Abstract

Dense suspensions of self-propelled bacteria and related active fluids exhibit spontaneous flow generation, vortex formation, and spatiotemporally chaotic dynamics despite operating at vanishingly small Reynolds numbers. These phenomena, commonly referred to as active turbulence, display striking visual and statistical similarities to classical inertial turbulence while arising from fundamentally different nonequilibrium mechanisms. In this article, we present a combined review and theoretical study of hydrodynamic models for dense active fluids, with particular emphasis on bacterial suspensions described by the Toner--Tu--Swift--Hohenberg (TTSH) framework. We review key experimental and theoretical developments underlying the analogy between active and inertial turbulence, highlighting the emergence of multiple dynamical regimes and the conditions under which universal spectral and intermittent behavior arises in homogeneous systems. Moving beyond the conventional assumption of spatially uniform activity, we introduce a minimal model in which the activity field is heterogeneous and dynamically advected by the flow it generates. Thus treating activity as a spatiotemporally evolving field coupled to the TTSH dynamics, we investigate how advection and diffusion lead to sharp activity fronts, confinement of turbulent motion, and complex interfacial morphologies. Our numerical results demonstrate that spatial variations in activity can induce transient coexistence of distinct spectral regimes and that universality in active turbulence is inherently local and time-dependent in heterogeneous systems. These findings underscore the importance of treating activity as a dynamical field in its own right and provide a framework for studying active turbulence in more realistic, spatially structured biological and synthetic active matter systems.

Figures (3)

• Figure 1: Representative plots of the activity field $\alpha$ at time $t=15$ for Schmidt numbers (a) $\mathrm{Sc}=1.8$ and (b) $\mathrm{Sc}=0.3$, with the white curve denoting an iso-contour of the activity field defining the evolving activity front. The Schmidt number controls the morphology of this front, which becomes increasingly convoluted as diffusion weakens. This is best seen in a time evolution of the scalar field for a large range of Schmidt numbers as shown in Ref. scalar_contours. Panel (c) shows the temporal evolution of the fractal dimension $D_F$ of the activity front for different Schmidt numbers, illustrating a rapid growth from the initially smooth interface followed by saturation to a statistically steady value. Panel (d) shows the corresponding steady-state values of $\langle D_F \rangle$, with error bars characterizing statistical fluctuations, as a function of the Schmidt number. The activity fronts become more fractal with increasing $\mathrm{Sc}$.
• Figure 2: Representative plots of the (normalized) vorticity field (left panels) and activity field (right panels) for Schmidt number $\mathrm{Sc}=1.8$ at (a) $t=1.0$ and (b) $t=20.0$. At early times, the turbulent flow is largely confined within the region of high activity, indicated by the white broken line in panel (a), while the surrounding region remains comparatively quiescent. At later times, as the activity spreads and becomes more homogeneous, this confinement weakens and vorticity structures extend more uniformly across the domain. Panel (c) shows the magnitude of the activity gradient field, $|\nabla \alpha|$, at $t=20$, revealing intermittent, worm-like structures associated with sharp activity fronts. We refer the reader to Ref. fields for a complete time evolution of the vorticity, activity and gradient of activity fields.
• Figure 3: (a) Kinetic energy spectra $E(k)$ at four representative times for a system initialized with a strongly active patch of activity $\alpha_{\mathrm{in}}=-8$ inside a central disc of radius $r=10$ and $\alpha_{\mathrm{out}}=4$ outside. This extreme value is chosen to clearly expose the mechanism underlying dual scaling. At early times, the spectrum exhibits two distinct regimes: a universal $k^{-3/2}$ scaling at larger wavenumbers, characteristic of strongly active turbulence ($\alpha<\alpha_c\approx -5$), and a non-universal scaling at smaller wavenumbers associated with weakly active regions. The crossover between these regimes occurs at a time-dependent wavenumber $k_R(t)\sim R^{-1}(t)$, where $R(t)$ is the effective linear size of the region with $\alpha<\alpha_c$. As shown in panel (b), $R(t)$ initially grows due to advection, causing $k_R(t)$ to shift to smaller values, and later decreases as diffusion and mixing erode the strongly active core, shifting $k_R(t)$ back to larger values and eventually eliminating the $k^{-3/2}$ regime. (b) Snapshots of the activity field $\alpha(\mathbf{x},t)$ at the same four times, with regions satisfying $\alpha<\alpha_c$ highlighted in blue. These blue regions define $R(t)$ and thus directly set the crossover scale $k_R(t)$ observed in panel (a): their growth supports the coexistence of dual spectral scaling, while their eventual shrinkage explains the disappearance of the universal regime. (c) The $\alpha$-spectrum shown at times as in panels (a) and (b) show distinct scaling regimes $k^{-2}$ (dotted line), $k^{-4}$ (dashed line) and $k^{-1}$ (dot-dashed line). (Inset) Probability density functions (PDFs) of $\alpha$ at these times, showing an initially near-bimodal distribution reflecting the coexistence of strongly and weakly active regions, which progressively flattens and eventually disappears for $\alpha \lesssim \alpha_c$, as the activity field becomes well mixed.