Low-dimensional models for fluid flow
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Despite the temporal and spatial complexity of fluid flow, model dimensionality can often be greatly reduced while both capturing and illuminating the nonlinear dynamics of the flow. This dissertation follows the methodology of direct numerical simulation (DNS) followed by Proper Orthogonal Decomposition of temporally sampled DNS data to derive temporal and spatial eigenfunctions. The DNS calculations use Chorin's projection scheme; 2-d 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. Truncated to retain only the most energetic modes produces a low-dimensional model for the flow at the decomposition Re. This dissertation demonstrates that these low-dimensional models reproduce the flow dynamics, but with small errors in amplitude, phase, and particularly long term dynamics. A new stabilization algorithm is presented that projects the error onto the derived temporal eigenfunctions, then modifies the dynamical system coefficients to significantly reduce these errors. Its effectiveness is demonstrated with low-dimensional dynamical systems for driven cavity flow in the periodic regime, quasi-periodic flow at Re 10000, and the wake flow. This dissertation also addresses the task of obtaining more useful models that are valid over a range of Reynolds numbers. Straightforward Re-based parameter continuation applied to extrapolate the model proves inadequate for successful flow prediction. A new concept of parameterizing the dynamical system coefficients is introduced that utilizes the kinetic energy transfer between modes as a function of Re to predict the flow dynamics correctly. Results for the driven cavity flow include a minimal four-mode dynamical system that captures the flow dynamics for Re up to 10000. A four-mode dynamical system for the square cylinder wake flow demonstrates accurate amplitude predictions for Re up to 100. The most robust low-dimensional models are obtained by further including a model for the frequency variation with Re. Low-dimensional models that incorporate spatial mode changes with Re are developed and quantitatively assessed for both test flows.