Energy balance of reduced order models for unsteady flows using sparse coding

Galerkin projection is a commonly used reduced order modeling approach; however, stability and accuracy of the resulting models are open issues for unsteady flow fields. Balance between production and dissipation of energy is crucial for stability. Moreover, the rates of energy production and dissipation are functions of large- and small-scale information captured in the chosen modes. Due to the highly nonlinear nature of the Navier–Stokes equations, the process of choosing an ‘appropriate’ set of modes from the simulation or experimental data is non-trivial. Recent work indicates that modal decompositions computed using a sparse coding approach yield multi-scale modes that provide improved low-order models compared to the commonly used proper orthogonal decomposition. This study seeks to use energy components analysis to develop a deeper understanding of the improved model performance with sparse modes. In addition, a greedy search-based sparse coding algorithm is developed for basis extraction. The analysis is performed on two canonical problems of incompressible flows inside a lid-driven cavity and past a stationary cylinder. Results indicate that there is a direct link between the presence of multi-scale features in the reduced set of modes, balance between production and dissipation of energy, and reduced order model performance.