Turbulence and turbulent pattern formation in a minimal model for active fluids

Active matter systems display a fascinating range of dynamical states, including stationary patterns and turbulent phases. While the former can be tackled with methods from the field of pattern formation, the spatiotemporal disorder of the active turbulence phase calls for a statistical description. Borrowing techniques from turbulence theory, we here establish a quantitative description of correlation functions and spectra of a minimal continuum model for active turbulence. Further exploring the parameter space, we also report on a surprising type of turbulence-driven pattern formation far beyond linear onset: the emergence of a dynamic hexagonal vortex lattice state after an extended turbulent transient, which can only be explained taking into account turbulent energy transfer across scales.

Figure 1

The continuum model, Eq. (1), displays a range of dynamical phases of the vorticity field depending on the nonlinear advection: (a) classical pattern formation ($\lambda =0$, simulation 1 in Table 1), (b) active turbulence ($\lambda =3.5$, simulation 2 in Table 1), and (c) turbulent pattern formation ($\lambda =7$, simulation 3 in Table 1). Notably, the dispersion relation shown in (d) along with the nonlinear damping is kept fixed for all examples. The dashed green line corresponds to the most unstable wave number, given by $k=k_c$, which sets the wave number of the pattern in (a). The horizontal orange lines in (a) and (c) correspond to five times the length scale of the patterns, i.e., $10\pi /k_c$ and $10\pi /k_0$, respectively, exemplifying that the wave-number selection in the turbulent pattern forming phase (c) differs from the classical pattern forming phase (a).

Figure 2

(a) Energy budget of active turbulence: direct numerical simulation (DNS) results (dashed lines, simulation 2 in Table 1) vs EDQNM closure theory. The black, green, and blue curves correspond to the energy spectrum, the transfer term, and the effective linear term, respectively. (b) Spectra from DNS of active turbulence compared to EDQNM closure theory. (c) Longitudinal velocity autocorrelation of active turbulence: DNS vs EDQNM closure theory. The blue, black, and green curves in (b) and (c) correspond to the simulations 2, 5, and 6, respectively, as listed in Table 1.

Figure 3

Emergence of hexagonal vortex lattice after a turbulent transient (simulation 4 in Table 1). (a)–(c) Vorticity field after $t=20,150,850$. The insets show the two-dimensional vorticity spectra with the wave vectors corresponding to the most unstable wave number indicated by an orange circle. The inset (c) clearly shows six isolated peaks at $k_0\approx 0.57$ which characterize the vortex lattice. For visualization purposes, these figures were obtained through a simulation on a smaller domain with half the domain length compared to Fig. 1. Note that the final vortex crystal state selects a sign of vorticity different from that of Fig. 1, exemplifying spontaneous symmetry breaking in this system. Panel (d) shows the evolution of the enstrophy, as well as the maximum and the minimum vorticity through the transient to the final quasistationary state.