A simple dynamical system that mimics open-flow turbulence

Abstract

The possible relevance of recent theories concerning the chaotic behavior of nonlinear dynamical systems to turbulence, especially in open flows, has frequently been questioned. Here, the issues that have led to this skepticism are investigated by studying a simple system that has been devised to include, albeit in an impressionistic way, the major mechanisms that are widely considered to operate in a broad class of turbulent flows. The variables in the system seek to represent the amplitudes of large- and small-eddy motion, respectively, and are governed by equations that allow for a Landau-Stuart nonlinear growth, a one-step Richardson cascade, and a specified time-dependent driving force. It is found that the critical value (at the onset of chaos) of the Reynolds-number-like control parameter (ν⁻¹) in the system depends on the character and magnitude of the driving force; and it is analytically demonstrated using the Melnikov technique that, with an appropriate choice of model parameters, chaos can persist at all sufficiently high values of the model Reynolds number (unlike as in most other low-dimensional models). The routes to chaos in the system when the forcing is increased at fixed ν are different from those when ν is decreased at fixed forcing, the latter being found to be more relevant to the case of streamwise-developing flows like a boundary layer. The observed routes are sensitive to the presence of even small stochastic components in the forcing. Computed spectral evolutions in the model show qualitative similarities with observations in boundary layer flow under different disturbance environments. It is concluded that many of the gross features of open-flow turbulence can be understood as dynamical chaos.