[Paper Review] Route to turbulence via oscillatory states in polar active fluid under confinement

Sora Shiratani, Kazumasa A. Takeuchi+1 more|arXiv (Cornell University)|Apr 6, 2023

Micro and Nano Robotics4 citations

TL;DR

This study reveals a novel route to active turbulence in polar active fluids under confinement, using GPU-accelerated simulations of the Toner-Tu-Swift-Hohenberg model. As confinement size increases, the system undergoes a hysteretic transition from a stationary vortex to oscillatory states, followed by chaotic and quasiperiodic dynamics, ultimately reaching active turbulence—distinct from conventional turbulence routes.

ABSTRACT

We report a novel route to active turbulence, observed in numerical simulations of a polar active fluid model under confinement. To deal with large-scale computations with arbitrary geometries, we developed a GPU-based scheme that can be used for any boundary shape in a unified manner. For the circular confinement, as the radius was increased, we found a series of transitions first from a single stationary vortex to an oscillating pair of vortices, then through reentrant transitions between oscillatory and chaotic dynamics before finally reaching the active turbulence. The first transition turned out to be hysteretic, with the emergence of the oscillatory state consistent with the subcritical Hopf bifurcation. In dumbbell-shaped boundaries composed of two overlapping circles, we observed a transition comparable to the ferromagnetic-antiferromagnetic vortex-order transition reported in previous experiments, but the transition point turned out to show a qualitatively different geometry dependence.

Motivation & Objective

• To investigate the dynamical transitions leading to active turbulence in confined polar active fluids, a regime where such routes remain poorly understood.
• To overcome computational challenges in simulating complex geometries and large-scale hydrodynamic systems with arbitrary boundary shapes.
• To identify and characterize intermediate states—oscillatory, chaotic, and quasiperiodic—between ordered vortices and fully developed turbulence.
• To develop and validate a GPU-based numerical scheme capable of efficiently solving the Toner-Tu-Swift-Hohenberg equation for arbitrary confinement geometries with high resolution.

Proposed method

• A GPU-accelerated spectral solver was developed to efficiently integrate the Toner-Tu-Swift-Hohenberg (TTSH) equation for polar active fluids under arbitrary boundary conditions.
• A novel mask processing algorithm was implemented to prevent aliasing in spectral methods, using a low-pass filtered DFT of a binary mask followed by inverse DFT and squaring to ensure non-negativity.
• The method employs a unified framework for arbitrary geometries, including circular and dumbbell-shaped confinements, by defining spatial masks that distinguish fluid regions from boundaries.
• High-resolution simulations were performed on GPU clusters (NVIDIA A6000, Kugui, nekoya/ai), achieving ~60× speedup over CPU for large lattices.
• Lyapunov exponents were computed to characterize chaotic dynamics, requiring extensive computation enabled by GPU parallelization.
• Boundary conditions were informed by experimental observations of bacterial turbulence, ensuring physical relevance.

Experimental results

Research questions

• RQ1 What dynamical transitions occur as the size of a confining domain is increased in a polar active fluid?
• RQ2 How does the route to active turbulence in confined polar fluids differ from known routes in Navier-Stokes or active nematic systems?
• RQ3 What role do oscillatory and chaotic states play in the transition from ordered vortices to fully developed turbulence?
• RQ4 How does the geometry of confinement—specifically circular vs. dumbbell-shaped—alter the nature and location of transitions?
• RQ5 Can a unified GPU-based numerical framework accurately simulate large-scale active fluid dynamics with arbitrary boundary shapes?

Key findings

• The first transition from a single stationary vortex to an oscillating vortex pair is hysteretic and consistent with a subcritical Hopf bifurcation.
• As the confinement radius increases, the system undergoes a sequence of transitions: periodic oscillations → chaotic dynamics → quasiperiodic states (with irrational frequency ratios) → active turbulence.
• The transition to active turbulence occurs via a complex, non-monotonic path involving reentrant transitions between oscillatory and chaotic states.
• In dumbbell-shaped confinements, a vortex-order transition resembling ferromagnetic-antiferromagnetic transitions was observed, but its dependence on geometry differs qualitatively from previous experimental reports.
• The GPU-based solver achieved a ~60× speedup over CPU for large-scale simulations, enabling high-resolution analysis of transient and chaotic dynamics.
• Lyapunov exponent analysis confirmed the emergence of chaos, with positive exponents indicating sensitive dependence on initial conditions in the turbulent regime.