On the stabilization and enhancement of the reduced-order models for compressible flows

Abstract

Projection-based model reduction offers a physically informed, and mathematically rigorous framework to bypass the prohibitive amount of computational resources required by the direct numerical simulations in fluid dynamics, and enable the recurrent computations that dominate many-queries applications. Projection of the governing equations onto a low-dimensional space, however, does not guarantee to naturally inherit the stability properties of the high-fidelity model. Symmetrization of the Reduced-Order Model (ROM) through a least squares Petrov-Galerkin projection, or by Galerkin projection using the symmetry inner product, provides theoretical error bounds, and generates more stable ROMs. This study shows that besides being more stable, the symmetrized ROMs are more controllable and robust. The stability guarantees by symmetrization or energy-based inner products, assume that the subspace constructed for projection, accurately captures the coherent structures that are the main ingredients in the dynamics of the flow. However, when the high-fidelity simulations contain nonlinear phenomena (e.g. unsteady shock waves, and turbulence), truncation of the high-frequency modes through dimensionality reduction with a linear approach like Proper Orthogonal Decomposition (POD), that is biased towards the most energetic modes, may result in losing structures with critical contributions in the dynamical evolution of the system. As a result, especially when the governing equations lack any intrinsic dissipative mechanisms to contain the generated errors (e.g. the Euler equations), symmetrization alone is not sufficient to preserve stability. Therefore, a complete framework is proposed in this study for the enhancement of ROMs for compressible flows, through ROM symmetrization, and post-ROM stabilization. Two optimization-based non-intrusive stabilization methods are developed here: a Hybrid method for the stabilization of ROMs as Linear Time-Invariant (LTI) systems, and an eigenvalue reassignment method for stabilization of nonlinear ROMs (ERN algorithm). The Hybrid method is a two-step approach: in step one (efficiency-oriented), the left reduced order basis of the ROM is minimally modified in a convex optimization problem; in step two (accuracy-oriented), an eigenvalue reassignment method is used to stabilize the most energetic eigen-modes. The ERN algorithm, on the other hand, confines the nonlinear ROM to maintain a negative total power for stability; and the distance between the nonlinear ROM and Full-Order Model (FOM) attractors is directly minimized as the eigenvalues of the linear dynamics matrix (control parameters) are reassigned in the complex plane. A computational bottleneck occurs in strongly nonlinear systems (e.g. advection-dominated flows), where the slow decay of the projection error requires more base functions to accurately span the high-fidelity solutions with a linear subspace. Hence to sufficiently describe a strongly nonlinear system, ROMs have higher dimensions intrinsically. Nevertheless, the truncation of such ROMs may still bring in instability, and their relatively higher dimension (i.e. large coefficient matrices) leads to a large number of control parameters which may potentially prevent the stabilization algorithm being feasible in computation. As a remedy, this study introduces a multi-stage layout for robust stabilization of nonlinear ROMs with the ERN algorithm, in strongly nonlinear systems, where a linear ROM typically fails to capture the true dynamics. The proposed methods are applied on POD-Galerkin ROMs based on the snapshots of two supersonic flow applications. The high-fidelity simulations are performed with a Weighted Essentially Non-Oscillatory (WENO) shock capturing scheme, integrated with the immersed boundary method. The two applications involve strong shock-wake interactions in the downstream, where the unsteady shock oscillations as a result of the interaction of shock waves with vortices, exhibit strong nonlinearities that are not completely resolved in the leading POD modes. Thus, the missing high-frequency contributions of this phenomenon trigger strong instabilities in the linear and nonlinear ROMs, and enable a thorough investigation of the ideas that are developed in this research for the stabilization and enhancement of ROMs.