Reduced Order Modeling of Turbulent Flows using Sparse Coding

Abstract

Basis identification is a critical step in the construction of accurate reduced order models using Galerkin projection. This is particularly challenging for turbulent flow fields due to the presence of multi-scale phenomena that cannot be ignored when building a reduced order model. The ubiquitous proper orthogonal decomposition approach seeks to truncate the basis spanning an observed data set into a small set of dominant modes, leading to loss of small scale information in turbulent flow fields. Ignoring the small scale information results in under-resolved rate of dissipation of energy, and consequentially, over-prediction of kinetic energy by constructed reduced order models. This study focuses on this issue by exploring an approach known as sparse coding for the basis identification problem. The sparse coding approach seeks the best compact basis to span the entire data set, and capture the multi-scale features present in the turbulent flows. These approaches are demonstrated for a canonical problem of an incompressible flow inside a 2-D lid-driven cavity. Results indicate that Galerkin reduction of the governing equations using only a few sparse modes produces reasonably accurate predictions of second order statistics of the fluid flow. Additionally, the sparse modes based models are found to maintain balance between the production and dissipation of energy. Whereas, models constructed using the same numbers of proper orthogonal decomposition modes are found to perform poorly.