Minimal modelling of turbulence driven by steady 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.