Hyper-reduction and calibration techniques for projection-based reduced order models of unsteady flows

An analysis of projection-based reduced order models (ROMs) applied to unsteady flows is presented in this work. Order reduction is performed through proper orthogonal decomposition (POD) and models are tied to the governing equations using Galerkin and least-squares Petrov-Galerkin (LSPG) methods. Further cost reduction is made possible using hyper-reduction which, in this work, is performed by the accelerated greedy missing point estimation (MPE) method. Additionally, data-driven closure modeling strategies are discussed. These are based on the error minimization of the POD temporal modes and may require regularization. Also, closure coefficients can be either linear or non-linear. Methods are tested on incompressible and compressible flow problems involving a disparity of spatial and temporal scales. The incompressible flows consist of convective heat transfer in rectangular cavities and a comparison to data-driven ROMs is provided. For compressible flows, airfoils provide the necessary motivation and are studied in both subsonic and supersonic regimes. Furthermore, a more challenging periodically plunging airfoil undergoing deep dynamic stall problem is also examined. All compressible flow ROMs use a non-conservative form of the compressible Navier-Stokes equations motivated by the quadratic non-linearity of the spatial operators. Summarily, projection-based ROMs are generally capable of obtaining stable and accurate models for complex flow problems but may require closure techniques to perform adequately for long-term temporal integration.