A conditional space-Time POD formalism for intermittent and rare events: Example of acoustic bursts in turbulent jets

We present a conditional space-Time proper orthogonal decomposition (POD) formulation that is tailored to the eduction of the average, rare or intermittent events from an ensemble of realizations of a fluid process. By construction, the resulting spatio-Temporal modes are coherent in space and over a predefined finite time horizon, and optimally capture the variance, or energy of the ensemble. For the example of intermittent acoustic radiation from a turbulent jet, we introduce a conditional expectation operator that focuses on the loudest events, as measured by a pressure probe in the far field and contained in the tail of the pressure signal's probability distribution. Applied to high-fidelity simulation data, the method identifies a statistically significant 'prototype', or average acoustic burst event that is tracked over time. Most notably, the burst event can be traced back to its precursor, which opens up the possibility of prediction of an imminent burst. We furthermore investigate the mechanism underlying the prototypical burst event using linear stability theory and find that its structure and evolution are accurately predicted by optimal transient growth theory. The jet-noise problem demonstrates that the conditional space-Time POD formulation applies even for systems with probability distributions that are not heavy-Tailed, i.e.A for systems in which events overlap and occur in rapid succession.