Quantitative assessment of low-dimensional POD-ODE models of wall-bounded flows.

We examine low-dimensional ODE models of coherent structures in wall-bounded flows derived from proper orthogonal decomposition and Galerkin projection. We show that POD-ODE models of periodized boundary-layer flow are linearly stable about the mean flow for some parameter values, despite the lack of upper-surface velocity boundary conditions. However, such models are predictively and statistically inaccurate. We compare POD-ODE models of plane Couette flow to direct numerical simulations and find that the convergence rate of the modeled dynamics is very slow --much slower than the convergence of POD expansions to instantaneous velocity fields. Eddy viscosity models do not improve the convergence of POD-ODE dynamics. However, numerical results suggest that Galerkin projection is sub-optimal, and that more accurate models of the low-dimensional dynamics can be derived through empirical modeling.