Reduced-Order Modeling for Complex Flow Emulation by Common Kernel-Smoothed Proper Orthogonal Decomposition

Abstract

In the present study, we propose a new surrogate model [common kernel-smoothed proper orthogonal decomposition (CKSPOD)] to emulate spatiotemporally evolving flows. The model integrates and extends recent developments in Gaussian process learning, high-fidelity simulations, projection-based model reduction, uncertainty quantification, and experimental design, rendering a systematic, multidisciplinary framework. The novelty lies in the construction of a common Gram matrix: the Hadamard product of Gram matrices of all observed design settings. The common Gram matrix synthesizes the temporal dynamics by transferring proper orthogonal decomposition (POD) modes into spatial functions at each observed design setting, which remedies the phase-difference issue encountered in the kernel-smoothed POD (KSPOD) emulation. The CKSPOD methodology is demonstrated through a case study of flow dynamics of swirl injectors with three design parameters. A total of 30 training design settings and eight validation design settings are included. The CKSPOD emulation outperforms the KSPOD counterpart, and it is capable of capturing small-scale flow structures faithfully. The CKSPOD prediction of turbulent kinetic energy reveals lower uncertainty than KSPOD. The turnaround time of the CKSPOD emulation is about five orders of magnitude faster than the corresponding high-fidelity simulation, which enables an efficient and scalable framework for design exploration and optimization.