  In the present study, we propose a new surrogate model, called common
kernel-smoothed proper orthogonal decomposition (CKSPOD), to efficiently
emulate the spatiotemporal evolution of fluid flow dynamics. The proposed
surrogate 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 of the CKSPOD emulation lies in the
construction of a common Gram matrix, which results from the Hadamard product
of Gram matrices of all observed design settings. The Gram matrix is a
spatially averaged temporal correlation matrix and contains the temporal
dynamics of the corresponding sampling point. The common Gram matrix
synthesizes the temporal dynamics by transferring POD modes into spatial
functions at each observed design setting, which remedies the phase-difference
issue encountered in the kernel-smoothed POD (KSPOD) emulation, a recent fluid
flow emulator proposed in Chang et al. (2020). The CKSPOD methodology is
demonstrated through a model study of flow dynamics of swirl injectors with
three design parameters. A total of 30 training design settings and 8
validation design settings are included. Both qualitative and quantitative
results show that the CKSPOD emulation outperforms the KSPOD emulation for all
validation cases, and is capable of capturing small-scale wave structures on
the liquid-film surface faithfully. The turbulent kinetic energy prediction
using CKSPOD reveals lower predictive uncertainty than KSPOD, thereby allowing
for more accurate and precise flow predictions. The turnaround time of the
CKSPOD emulation is about 5 orders of magnitude faster than the corresponding
high-fidelity simulation, which enables an efficient and scalable framework for
design exploration and optimization.
