Least-order model for the transient dynamics of the fluidic pinball

We propose a least-order mean-field model for a flow system undergoing two successive supercritical bifurcations. The fluidic pinball, an incompressible two-dimensional flow crossing three equidistantly spaced cylinders, is numerically investigated using Direct Numerical Simulation. Two generic bifurcations in fluid mechanics are observed: the primary Hopf bifurcation leads to a statistically symmetric vortex shedding and the following pitchfork bifurcation breaks the symmetry at higher Reynolds number. Interestingly, this symmetry-breaking instability works on the steady solution simultaneously, illustrated by the global stability analysis and Floquet analysis. The elementary degrees of freedom are identified under mean-field considerations exploiting the symmetry/asymmetry of the base flow and the fluctuation. An easily interpretable five-dimensional Galerkin model compatible with the quadratic non-linearities of the Navier-Stokes equations is derived, which can reproduce the main features of bifurcating dynamics and the transient behavior to the asymptotic regime. This generalized mean-field Galerkin methodology is considered to be applicable to other transition scenarios and nonlinear model-based control.