A parametric and feasibility study for data sampling of the dynamic mode decomposition: range, resolution, and universal convergence states

Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The dynamic mode decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. It aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject—the turbulent free-shear flow over a prism—for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent.

Modal Decomposition Analysis of Combustion Instability Due to External Perturbation in a Mesoscale Burner Array

In this work, the growth regime of combustion instability was studied by analyzing 10 kHz OH planar laser induced fluorescence (PLIF) images through a combination of dynamic mode decomposition (DMD) and spectral proper orthogonal decomposition (SPOD) methods. Combustion instabilities were induced in a mesoscale burner array through an external speaker at an imposed perturbation frequency of 210 Hz. During the transient onset of combustion instabilities, 10 kHz OH PLIF imaging was employed to capture spatially and temporally resolved flame images. Increased acoustic perturbations prevented flame reignition in the central recirculation zone and eventually led to the flame being extinguished inwards from the outer burner array elements. Coherent modes and their growth rates were obtained from DMD spectral analyses of high-speed OH PLIF images. Positive growth rates were observed at the forcing frequency during the growth regime. Coherent structures, closely associated with thermoacoustic instability, were extracted using an appropriate SPOD filter operation to identify mode structures that correlate to physical phenomena such as shear layer instability and flame response to longitudinal acoustic forcing. Overall, a combination of DMD and SPOD was shown to be effective at analyzing the onset and propagation of combustion instabilities, particularly under transient burner operations.