Minimal model of quasi-cyclic behaviour in turbulence driven by Taylor-Green forcing

We attempt to formulate the simplest possible model mimicking turbulent dynamics, such as quasi-cyclic behaviour (QCB), using only three variables. To this end, we first conduct direct numerical simulations of three-dimensional flow driven by the steady Taylor-Green forcing to find a similarity between a stable periodic orbit (SPO) at a small Reynolds number (Re) and turbulent QCB at higher Re. A close examination of the SPO allows the heuristic formulation of a three-equation model, representing the evolution of Fourier modes in three distinct scales. The model reproduces the continuous bifurcation from SPO to turbulence with QCB when Re is varied. We also demonstrate that, by changing model parameters, the proposed model exhibits a discontinuous transition from steady to chaotic solutions without going through an SPO.