Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations

TL;DR

This paper introduces a novel minimal subspace rotation method on the Stiefel manifold to stabilize and improve projection-based reduced order models for the compressible Navier-Stokes equations without empirical turbulence modeling.

Contribution

It proposes a trace minimization approach on the Stiefel manifold to stabilize ROMs by accounting for truncated modes a priori, maintaining consistency with the full model.

Findings

Effective stabilization of ROMs for compressible flows.

Enhanced predictive accuracy in flow simulations.

No need for empirical turbulence models.

Abstract

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier-Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the full order model from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.