# Low-order model for successive bifurcations of the fluidic pinball

## TL;DR

This paper introduces a minimal 5-dimensional Galerkin model that captures two successive supercritical bifurcations in a complex wake flow, advancing understanding of fluid flow transitions.

## Contribution

It presents the first low-order Galerkin model incorporating symmetry considerations to describe successive bifurcations in fluid dynamics.

## Key findings

- Model accurately reproduces bifurcation sequence in fluidic pinball
- Successfully describes transition to chaos through bifurcations
- Generalized mean-field approach applicable to other flow scenarios

## Abstract

We propose the first least-order Galerkin model of an incompressible flow undergoing two successive supercritical bifurcations of Hopf and pitchfork type. A key enabler is a mean-field consideration exploiting the symmetry of the mean flow and the asymmetry of the fluctuation. These symmetries generalize mean-field theory, e.g. no assumption of slow growth-rate is needed. The resulting 5-dimensional Galerkin model successfully describes the phenomenogram of the fluidic pinball, a two-dimensional wake flow around a cluster of three equidistantly spaced cylinders. The corresponding transition scenario is shown to undergo two successive supercritical bifurcations, namely a Hopf and a pitchfork bifurcations on the way to chaos. The generalized mean-field Galerkin methodology may be employed to describe other transition scenarios.