A priori analysis of reduced description of dynamical systems using approximate inertial manifolds

The treatment of turbulent flows as finite-dimensional dynamical systems opens new paths for modeling and development of reduced-order descriptions of such systems. For certain types of dynamical systems, a property known as the inertial manifold (IM) exists, allowing for the dynamics to be represented in a sub-space smaller than the entire state-space. While the existence of an IM has not been shown for the three-dimensional Navier-Stokes equations, it has been investigated for variations of the two-dimensional version and for similar canonical systems such as the Kuramoto-Sivashinsky equation (KSE). Based on this concept, a computational analysis of the use of IMs for modeling turbulent flows is conducted. In particular, an approximate IM (AIM) is used where the flow is decomposed into resolved and unresolved dynamics, similar to conventional large eddy simulation (LES). Instead of the traditional approach to subfilter modeling, a dynamical systems approach is used to obtain the closure terms. In the a priori estimation of the AIM approach for the Kuramoto-Sivashinsky equation, it is shown that the small-scale dynamics are accurately reconstructed even when using only a small number of resolved modes. Further, it is demonstrated that the number of resolved variables needed for this reconstruction is dependent on the dimension of the attractor. (C) 2020 Elsevier Inc. All rights reserved.