Convective Space-Time Chaos as a Dynamical Model of Deterministic and Stochastic Turbulence

Abstract

Recently, a concept of deterministic and stochastic turbulence has been introduced based on experiments with a boundary layer. In these experiments, the flow was driven with controlled random perturbation; in addition, natural ambient noise was also present. Deterministic property manifested itself as repeatability of turbulence patterns induced by identical random perturbations at the inlet (deterministic turbulence). A stochastic non-identical component originating from natural noise grows and eventually dominates the flow further downstream (stochastic turbulence). We argue that these properties can be explained by exploring the concept of convective space-time chaos, where secondary perturbations on top of a chaotic state grow but move away in the laboratory reference frame. We illustrate this with two simple models of convective space-time chaos, one is a partial differential equation describing waves on a film flowing down a plate, and the other is a set of unidirectionally coupled ordinary differential equations. To prove convective space-time chaos, we calculate the profiles of the convective Lyapunov exponent. The repeatability of the turbulent field in different identical experimental runs corresponds to the reliability of stable dynamical systems in response to random forcing. The onset of the stochastic component is quantified with the spatial Lyapunov exponent. We demonstrate how an effective randomization of the field is observed when the driving is quasiperiodic. Furthermore, we discuss space-time duality, which links sensitivity to boundary conditions in the convective space-time chaos to the usual sensitivity to initial conditions in a standard chaotic regime.