An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models

Abstract

Despite the temporal and spatial complexity of common fluid flows, model dimensionality can often be greatly reduced while both capturing and illuminating the nonlinear dynamics of the flow. This work follows the methodology of direct numerical simulation (DNS) followed by proper orthogonal decomposition (POD) of temporally sampled DNS data to derive temporal and spatial eigenfunctions. The DNS calculations use Chorin’s projection scheme; two-dimensional validation and results are presented for driven cavity and square cylinder wake flows. The flow velocity is expressed as a linear combination of the spatial eigenfunctions with time-dependent coefficients. Galerkin projection of these modes onto the Navier-Stokes equations obtains a dynamical system with quadratic nonlinearity and explicit Reynolds number (Re) dependence. Truncation to retain only the most energetic modes produces a low-dimensional model for the flow at the decomposition Re. We demonstrate that although these low-dimensional models reproduce the flow dynamics, they do so with small errors in amplitude and phase, particularly in their long term dynamics. This is a generic problem with the POD dynamical system procedure and we discuss the schemes that have so far been proposed to alleviate it. We present a new stabilization algorithm, which we term intrinsic stabilization, that projects the error onto the POD temporal eigenfunctions, then modifies the dynamical system coefficients to significantly reduce these errors. It requires no additional information other than the POD. The premise that this method can correct the amplitude and phase errors by fine-tuning the dynamical system coefficients is verified. Its effectiveness is demonstrated with low-dimensional dynamical systems for driven cavity flow in the periodic regime, quasiperiodic flow at Re=10000, and the wake flow. While derived in a POD context, the algorithm has broader applicability, as demonstrated with the Lorenz system.