Structural stability in two-dimensional model flows: Lagrangian and Eulerian turbulence

Abstract

A two-dimensional approximation of the Navier–Stokes equations illustrates a specific instance of transition from Lagrangian to Eulerian turbulence. As the Reynolds number (Re) increases, the system describing the dynamics of the velocity field undergoes a transition from steady state to a limit cycle. At this point the flow displays chaotic advection—i.e., manifolds intersect transversely and Poincaré maps show the typical chaotic structure—but the velocity field itself is time-periodic and the power spectrum presents a fundamental frequency and its harmonics. As Re increases still further, the limit cycle bifurcates into a strange attractor producing a broadband power spectrum. The model suggests that Lagrangian turbulence (complex particle trajectories) might serve as a springboard for Eulerian turbulence (complex signal at a fixed point) and indicates a possible link between a kinematical view of flows and mixing and other viewpoints of turbulence based on strange attractors. The ideas can be generalized to three-dimensional flows; however, due to their simplicity, the flows generated are unable to mimic some key features of turbulence such as spatial uncorrelation of Eulerian signals.