A dynamical-systems approach to understanding turbulence in plane Couette flow
Dynamical systems theory is used to understand the dynamics of low-dimensional spatio-temporal chaos. Our research aimed to apply the theory to understanding turbulent fluid flows, which could be thought of as spatio-temporal chaos in a very-high dimensional space. The theory explains a system's dynamics in terms of the local dynamics of its periodic solutions; these are the periodic orbits in state space. We considered the development of a model for the dynamics of plane Couette flow based on the theory. The proposed model is essentially a set of low-dimensional models for the local dynamics of the periodic orbits of the Navier-Stokes equations with plane Couette boundary conditions. We considered various aspects of the proposed model, including the possibility that the local very-high dimensional dynamics about a periodic orbit could be approximated with a low-dimensional model, and the possibility of building the set of local models on a certain Poincare section. The Poincare section is associated with the constraint that the rate of kinetic energy in the flow is zero. Our research suggests that the dynamics of plane Couette flow can, in fact, be organized in terms of the system's periodic orbits, and that, for at least one periodic orbit, the local dynamics could be approximated in 16 dimensions. Ultimately, we conclude that building the proposed model is impractical for various reasons. More likely, a better approach is to attempt a model based on a set of judiciously chosen linear Poincare sections.
direct numerical simulations, dynamical systems theory, low-dimensional models, periodic orbits, plane Couette flow, spatio-temporal chaos, Mathematics, Physics
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