Two-dimensional unsteady flows in complex geometries that are characterized by simple (low-dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the proper orthogonal decomposition or POD (also called method of empirical eigenfunctions) is applied to the resulting data for two examples: the flow in a periodically grooved channel and the wake of an isolated circular cylinder. Low-dimensional dynamical models for these systems are obtained using the empirically derived global eigenfunctions in the spectrally discretized Navier–Stokes equations. The short- and long-term accuracy of the models is studied through simulation, continuation, and bifurcation analysis. Their ability to mimic the full simulations for Reynolds numbers (Re) beyond the values used for eigenfunction extraction is evaluated. In the case of the grooved channel, where the primary horizontal wave number of the flow is imposed from the channel periodicity and so remains unchanged with Re, the models extrapolate reasonably well over a range of Re values. In the case of the cylinder wake, however, due to the significant spatial wave number changes of the flow with the Re, the models are only valid in a small neighborhood of the decompositional Reynolds number.

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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 ͑DNS͒ followed by proper orthogonal decomposition ͑POD͒ 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 ͑Re͒ 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.

A parametric study is made of chaotic Rayleigh-Be'nard convection over moderate Rayleigh numbers. As a basis for comparison over the Rayleigh number (Ra) range we consider mean quantities, r.m.s. fluctuations, Reynolds number, probability distributions and power spectra. As a further means of investigating the flow we use the Karhunen-Loe've procedure (empirical eigenfunctions, proper orthogonal decomposition). Thus, we also examine the variation in eigenfunctions with Ra. This in turn provides an analytical basis for describing the manner in which the chaos is enriched both temporarily and spatially as Ra increases. As Ra decreases, the significant mode count decreases but, in addition, the eigenfunctions tend more nearly to the eigenfunctions of linearized theory. As part of this parametric study a variety of scaling properties are investigated. For example it is found that the empirical eigenfunctions themselves show a simple scaling in Ra.