Turbulence and turbulent pattern formation in a minimal model for active fluids (1711.10865v2)

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.

• The paper introduces a minimal active fluid model that quantitatively captures turbulence using hydrodynamic statistical techniques.
• It demonstrates that nonlinear advective energy transfer triggers an upscale energy cascade, leading to distinct spectral peaks.
• The study reveals that turbulent transients can evolve into hexagonal vortex lattices, offering new insights into active matter dynamics.

Minimal Model for Active Fluid Turbulence and Pattern Formation

The paper explores the dynamics of active matter systems, focusing on turbulence and pattern formation in a minimal model of active fluids. These systems, encompassing both biological and synthetic agents that exhibit self-organized behavior, manifest in various dynamic states. The research presented investigates a two-dimensional flow field representing a dense suspension of bacteria, using a continuum model described by non-dimensionalized equations. The key aim is to elucidate statistical descriptions of correlation functions and spectra associated with active turbulence and the surprising emergence of regular vortex lattices after turbulent transients.

Key Contributions

This research delivers several significant contributions to the understanding of active fluid dynamics:

1. Statistical Description of Turbulence: Techniques from hydrodynamic turbulence theory are adapted to provide a quantitative statistical framework. This includes derivations for correlation functions and energy spectra applicable to active turbulence.
2. Advective Nonlinearity and Energy Transfer: The model combines advective nonlinearity—characteristic of Navier-Stokes dynamics—with pattern-forming elements, enabling exploration across a parameter space. A noteworthy discovery is the role of nonlinear advective energy transfer in forming dynamic hexagonal vortex lattices.
3. Emergence of Vortex Lattices: Contrary to classical pattern formation mechanisms, the paper finds that a turbulent transient can evolve into a hexagonal vortex lattice. This pattern emerges due to the large-scale energy distribution, driven by nonlinear advective effects. The transition to this ordered structure highlights the importance of nonlinear mean advection and energy transfer mechanisms not accounted for in traditional linear analysis.

Numerical Analysis and Findings

The paper extensively utilizes numerical simulations to validate theoretical insights. A pseudo-spectral code is employed to explore different dynamic phases under varying parameters. For instance:

• Active Turbulence: Simulations reveal that when nonlinear advection is moderate (e.g., with specific values of advection strength parameter $\lambda$), the energy transfer occurs upscale, generating structures larger than the bacterium scale. The spectrum peaks are located at wavelengths longer than the typical inter-bacterial distance, illustrative of an inverse energy cascade.
• Pattern Forming Transients: Upon increasing the advection strength dramatically, the model undergoes a transient turbulent phase before stabilizing into a hexagonal vortex lattice. This ordered state is a product of spontaneously broken symmetry endorsed by interactions among emergent vortices.

Theoretical and Practical Implications

Theoretically, this work bridges concepts between classical turbulence and active matter, generating a statistical mechanical approach to active fluid systems. The elaborate use of EDQNM (Eddy-Damped Quasi-Normal Markovian) closure further provides clarity on spectrum and correlation functions, offering predictions that resonate well with simulation data.

Practically, the insights offer potential avenues to leverage these collective behaviors in designing materials or systems mimicking biological activities at micro and macro scales. The understanding of turbulence-driven pattern formation might guide technological applications where ordered patterns with defect control are desired.

Future Prospects

The paper’s insights suggest various expansions for future research. Generalizing the framework to three-dimensional systems or incorporating more detailed descriptions of agent interactions could yield deeper understanding. Further, exploring active nematic systems gives another avenue to observe if similar pattern-forming phenomena occur, which can contribute to the broader comprehension of active matter systems' dynamic complexity.

This paper enriches the discourse on active turbulence in complex fluids, presenting a solid foundation for continued investigation into the equilibrium and non-equilibrium dynamics of a broad spectrum of self-organized systems.