Conditional space-time POD extensions for stability and prediction analysis

Abstract

The correlation and extraction of coherent structures from a turbulent flow is a principal objective of data-driven modal decomposition techniques. The Conditional space-time Proper Orthogonal Decomposition (CST-POD) offers insight into transient dynamics, revealing the causes of specific flow phenomena, or events, in a customizable manner. This work targets the temporal evolution of CST-POD modes in a reduced subspace, resulting in new extensions and adaptations that are distinct from capabilities of other decomposition methods. First, the properties of CST-POD and its relation to the ensemble average are instantiated with a simple Lorenz system. Other CST-POD properties are examined with applications to problems displaying intermittent properties that test data-driven methods, including transitional supersonic boundary layers and schlieren video processing of unstarted inlet buzz. These examples also explore the pragmatic concerns of oversampling and offer theoretical connections to the sub-sampling of a larger Hankel space through time-delay embedding. Through these insights, it is demonstrated that the subsequent application of dynamic mode decomposition (DMD) to CST-POD modes, provides a flexible tool to investigate targeted flow instabilities, both absolute (tonal) and convective in nature. It is shown, using a resonating jet as an example, that Spectral POD modes associated with tones can be recovered if the time-horizon of CST-POD is suitably extended. Regarding convective instabilities, a multi-resolution framework (CST-mrDMD) yields a refined “cause and effect” stability analysis, capable of diagnosing the natural forcing mechanisms within the jet, and the resulting unstable shear-layer mode. In the final interpretation, the potential for real-time flow prediction of extreme events is derived from an active sensor correlated to a CST-POD mode using the example of bluff-body wake structures impinging on a channel wall.