| A rigid wall constrains and simplifies turbulence in its vicinity. In the near-wall region, turbulent flow is dominated by a few energetic structures. Aubry, Holmes, Lumley, and Stone [1] developed low-dimensional dynamical systems models of the turbulent boundary layer, using the proper orthogonal decomposition (POD) to identify the dominant structures and Galerkin projection of the Navier-Stokes equations to derive their equations of motion. Their models reproduced important qualitative features of boundary-layer dynamics, most importantly, the growth, bursting, and reformation of counter-rotating streamwise vortices. We reexamine Aubry et al.'s models, in an effort to improve their quantitative accuracy, in both predictive and statistical senses. We find that POD models are stable in some parameter regions, despite the lack of upper-surface velocity boundary conditions. For predictive accuracy, however, enforcement of velocity boundary conditions and accurate specification of the upper-surface pressure signal are necessary, for the cases considered. We propose that plane Couette flow is a closely related but better-posed test case than the boundary-layer for dynamical-systems modeling of turbulence. For the simplest plane Couette systems, we find that very large truncation dimensions (0(1000)) are necessary to reproduce the qualitative behavior and statistical measures of the dominant structures. We find that eddy-viscosity is an ineffective model for the unresolved modes of moderate truncation dimensions (O(100)), but numerical results suggest that unresolved-mode modeling is possible in principle. |
