Low-dimensional models for fluid flow
Low-dimensional models for fluid flow
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Date
2004-08-24
Authors
Kalb, Virginia
Advisor
Deane, Anil
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Abstract
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.