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In the present study, we propose a new surrogate model, called commonkernel-smoothed proper orthogonal decomposition (CKSPOD), to efficientlyemulate the spatiotemporal evolution of fluid flow dynamics. The proposedsurrogate model integrates and extends recent developments in Gaussian processlearning, high-fidelity simulations, projection-based model reduction,uncertainty quantification, and experimental design, rendering a systematic,multidisciplinary framework. The novelty of the CKSPOD emulation lies in theconstruction of a common Gram matrix, which results from the Hadamard productof Gram matrices of all observed design settings. The Gram matrix is aspatially averaged temporal correlation matrix and contains the temporaldynamics of the corresponding sampling point. The common Gram matrixsynthesizes the temporal dynamics by transferring POD modes into spatialfunctions at each observed design setting, which remedies the phase-differenceissue encountered in the kernel-smoothed POD (KSPOD) emulation, a recent fluidflow emulator proposed in Chang et al. (2020). The CKSPOD methodology isdemonstrated through a model study of flow dynamics of swirl injectors withthree design parameters. A total of 30 training design settings and 8validation design settings are included. Both qualitative and quantitativeresults show that the CKSPOD emulation outperforms the KSPOD emulation for allvalidation cases, and is capable of capturing small-scale wave structures onthe liquid-film surface faithfully. The turbulent kinetic energy predictionusing CKSPOD reveals lower predictive uncertainty than KSPOD, thereby allowingfor more accurate and precise flow predictions. The turnaround time of theCKSPOD emulation is about 5 orders of magnitude faster than the correspondinghigh-fidelity simulation, which enables an efficient and scalable framework fordesign exploration and optimization.

Common kernel-smoothed proper orthogonal decomposition (CKSPOD): An  efficient reduced-order model for emulation of spatiotemporally evolving flow  dynamics