On dynamical systems obtained via Galerkin projections onto low-dimensional bases of eigenfunctions

Abstract

In this paper, some implications of the technique of projecting the Navier-Stokes equations onto low-dimensional bases of special eigenfunctions are explored. Such low-dimensional bases are typically obtained by truncating a particularly well-suited complete set of eigenfunctions at very low orders, arguing that a small number of such eigenmodes already captures a large part of the dynamics of the system. In addition, in the treatment of inho-mogeneous spatial directions of a flow, often eigenfunctions are used that do not satisfy the boundary conditions, and in the Galerkin projection the corresponding boundary conditions are ignored. We will show that both the severe truncation as well as an improper treatment of boundary conditions can severely restrict the validity of these models. As a particular example of an eigenfunction basis, systems of Karhunen-Loève eigenfunctions will be discussed in more detail, although most of the results presented are valid for any basis.