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

Abstract

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 spatio-temporal 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.

Anisotropic mesoscale turbulence and pattern formation in microswimmer suspensions induced by orienting external fields

“…It is seen that the two amplitude equations (22) correspond to two coupled Ginzburg-Landau equations [80]. This result was already obtained in [61], where the pattern formation in the reduced model equation , we introduce a small parameter ε quantifying the distance to the bifurcation. In the present case, we define ε via where we have used equation (19).…”

“…This stands in contrast to inertial turbulence observed in the Navier-Stokes equation where one finds a broad spectrum of vortex sizes [53]. For a more detailed discussion on the turbulent state without the influence of external fields, see [47,58,59,61] for the phenomenological model and [18,19] for the present model. Note that, compared to most of the listed publications, the coefficient λ 0 is rather large and, therefore, the shape of the vortices is highly irregular.…”

“…Recently, a phenomenological model [47,[55][56][57][58][59][60][61] has been proposed that reproduces the main features of mesoscale turbulence including the emergence of a characteristic length scale, i.e. vortex size [49].…”

“…Introducing an effective microswimmer velocity proportional to P then reproduces the phenomenological model briefly introduced in section 2, with the notable difference that we can calculate all the coefficients as functions of parameters of the microswimmer model. This phenomenological model, consisting of the dynamics of only one vector field, is indeed frequently used to describe mesoscale turbulence [47,[55][56][57][58][59]61]. Note that the coefficient Γ 2 has to be negative in order to obtain a mesoscale-turbulent state.…”

“…In principle, a potential for the vorticity can be constructed that favors a certain direction of rotation, which then allows for a hexagonal pattern consisting only of clockwise or counter-clockwise rotating vortices. However, for the parameter regime considered in this work, our theory does not break this symmetry and therefore we only exhibit a square lattice (see [61] for further discussion). figure 4) the bulge for P 5 r < which corresponds to a<0 is absent.…”