Bridging Large Eddy Simulation and Reduced Order Modeling of Convection-Dominated Flows through Spatial Filtering: Review and Perspectives

TL;DR

This review surveys a bridging framework between Large Eddy Simulation and Reduced Order Modeling for convection-dominated flows via spatial filtering, centering on the Evolve-Filter-Relax (EFR) strategy. It details how NL spatial filtering and approximate deconvolution form the core of LES closures and how these ideas translate into LES-ROMs (e.g., EFR-ROM, Leray ROM, AD-Leray ROM, time-relaxation ROM), with attention to consistency between FOM and ROM and to preliminary numerical analysis. The paper highlights successful incompressible and compressible flow applications, including cardiovascular hemodynamics and atmospheric simulations, and discusses open problems in parameter selection, boundary treatment, and data assimilation. It argues that LES-ROMs offer an easily implementable, highly efficient path to accurate predictions in convection-dominated regimes, enabling multi-query tasks and potential integration with machine learning and physics-informed surrogates for enhanced predictive power.

Abstract

Reduced order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, we focus on one particular type of ROM closures and stabilizations that are inspired by Large Eddy Simulation (LES). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in many applications. LES-ROM are constructed by leveraging spatial filtering, i.e., the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also ``bridges'' two distinct research fields (LES and ROMs), disconnected until now. This paper is a review of LES-ROMs. It starts with a description of a versatile LES strategy called evolve-filter-relax (EFR) that has been successfully used as a full order method. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs. Several applications of LES-ROMs are presented. Finally, we draw conclusions and outline several research directions and open questions in the LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which has a lot of potential in practical numerical simulation of convection-dominated flows.

Figures (19)

• Figure 1: Overall view of the energy cascade, from injection to dissipation of energy, and associated types of modeling.
• Figure 2: LES schematic showing the input flow variable, $\mathbf u$, that cannot be represented on a given coarse mesh, and the filtered flow variable, $\overline{\boldsymbol u}$, that can be accurately represented on the coarse mesh.
• Figure 3: Schematic of the concept proposed in maulik2020spatiotemporally.
• Figure 4: Images of a patient-specific AoD showing the true lumen and the false lumen.
• Figure 5: Simulation in a patient-specific AoD. Top left: pressure. Top right: velocity (in $cm/s$) in the descending aorta and at the entrance of the false lumen. The two bottom panels outline the complexity of the flow induced by the entry tear for the velocity (left) and the Wall Shear Stress (right).

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2
• Theorem 1