Interaction of POD modes in canonical pipe flows quantified by transfer entropy

We investigate the interactions of coherent structures in turbulent pipe flows using Proper Orthogonal Decomposition (POD) and Transfer Entropy (TE) analysis. Comprehensive Direct Numerical Simulation datasets from OpenPipeFlow and Neko codes span Reynolds numbers of Re_τ = 180 - 1000, across three flow configurations: canonical smooth pipes, rotating pipes (rotational speed/bulk velocity = 0.5-4), and rough pipes (roughness height in wall unit = 10-40). The temporal coefficients of the POD modes are analysed using TE to quantify directional information transfer between mode families.

A critical discovery is the strong sensitivity of TE results to relative phase shifts between POD mode families. For canonical pipe flows, systematic phase validation reveals minimal TE dependence (~10% variation), due to the statistical homogeneity of the flow where mode interactions are robust to azimuthal phase relationships.

Validation using stepped cylinder data reveals TE variations up to 100% with phase shifts, while directional dominance remains preserved. These findings show fundamental limitations in the state-of-the-art causality metrics to turbulent flows and establish the critical need for phase-invariant validation protocols when interpreting energy transfer mechanisms through POD temporal coefficient analysis.