Multiscale model reduction for incompressible flows

TL;DR

This paper develops a multiscale reduced-order modeling approach with cubic closure terms to better capture complex unsteady incompressible flow dynamics, improving stability and accuracy over traditional Galerkin models.

Contribution

It introduces a novel multiscale closure method incorporating cubic nonlinearities derived via stochastic Koopman operator averaging, enhancing flow model fidelity.

Findings

Improved stability in reduced-order flow models.

Enhanced accuracy in chaotic flow simulations.

Captures energy cascade and mean-flow effects.

Abstract

Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis approximate the flow as a low-dimensional dynamical system with linear and quadratic terms. However, these Galerkin models often fail to reproduce the true dynamics, in part because they ignore important nonlinear interactions with unresolved flow scales. Here, we use a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart-Landau equation. The leading order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modeling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows