The Use of Approximate Inertial Manifolds for Chaotic Systems including Turbulent Flows

While much of turbulence modeling focuses on the statistical approach, a dynamical systems perspective provides useful insights that can enable more accurate representation of the small-scale features. For an ergodic system, the turbulent flow can be assumed to prescribe an inertial manifold. An IM is defined as a finite-dimensional positively invariant Lipschitz manifold which exponentially attracts all trajectories and contains the global attractor. The existence of IM has been proven for many systems described by dissipative PDE; however, the theory does not provide an explicit form for the IM of such systems. Thus, to describe dynamical systems in inertial form, an approximation is necessary. When a suitable projection operator is used to split the full state of the system into resolved and unresolved scales, it is possible to develop such an approximation, and this has been demonstrated elsewhere for canonical chaotic systems. In this work, this approximate inertial manidfold method is tested for a series of canonical flows and applied to homogeneous isotropic turbulence.