Application of proper orthogonal decomposition to flow fields around various geometries and reduced-order modeling

This study is focused on a reduced-order model (ROM) based on proper orthogonal decomposition (POD) for unsteady flow around a stationary object, which allows prediction with different object geometry as a parameter. The conventional POD method is applicable only to data with the same computational grid for all snapshots. This study proposed a novel POD methodology that performs on flow snapshots, including some time-series data of flow fields around objects of different shapes and numerically computed by different computational grids. The concept of the proposed POD involved mapping the flow fields computed on different grids in computational space. Consequently, the optimal POD basis for minimizing reconstruction errors in physical space was obtained in the computational space. The proposed POD was applied to the flow around ellipses and airfoils generated via conformal mapping to a cylinder. The ROM formulated using the proposed POD bases reconstructed the flow fields around the ellipses with different aspect ratios and airfoils with varying shapes. Using the modes obtained by the proposed POD, the ROM was demonstrated to stably predict the time evolution of the flow around objects, which is not included in the snapshots. In the ROM, the difference between the frequency of the flow field in the POD snapshot and that of the reconstructed flow field resulted in a phase error owing to the time evolution. The mean squared error between the flow fields obtained via the ROM and the directly solved Navier–Stokes equations was under $10^{-7}$ when the reconstructed flow and the flow included in the snapshot had the same frequency as that of Kármán vorticities behind the objects. Based on these observations, the proposed POD is suitable for constructing an ROM to reconstruct the flow around various geometries.