Improving analysis of near-periodic flows through alignment in phase space

Flows with prominent but imperfect periodic content are pervasive in wakes, jets, rotary flows, and several other important engineering scenarios. These near-periodic flows, with strong cyclical content but cycle-to-cycle variability, are difficult to analyze with traditional methods since the key dynamics are not aligned in time, occurring earlier in some cycles and later in others. In this work we present a local alignment scheme that relates each instance in a given cycle to the most dynamically similar point along a nominal cycle, representative of the typical evolution of the near-periodic dynamics. The alignment amounts to locally dilating and compressing time along a cycle and is carried out within a low-dimensional phase space that captures and isolates the near-periodic dynamical behavior. This approach makes it straightforward and physically meaningful to compare the flow dynamics across cycles. To demonstrate the utility of this approach, we show that its use in aligning cycles reveals the underlying cross-cycle similarity in the high-dimensional data, that would be obscured by traditional approaches. We also present an ensemble expectation operator that respects this alignment process, and demonstrate that low order statistics (mean, standard deviation) are much more illustrative of physical processes than those based on a time-average or on an ensemble average using raw time that does not respect the (near-) periodic evolution process.