Low-dimensional models for fluid flow

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.