Correlation and decomposition framework for identifying and disentangling flow structures: canonical examples and application to isotropic turbulence

Turbulence organization, long conceptualized in terms of spatial coherent-structures, has resisted clear description. A major limitation has been the lack of tools to identify instantaneous spatial organization, while unravelling the superposition of structures. To address this, we present a generalized correlation framework, using: (i) correlation measures to identify instantaneous vector-field patterns, and (ii) a Helmholtz-decomposition based structure-disentanglement paradigm. After examples using canonical flows, we apply these methods to homogeneous isotropic turbulence fields. We show that high kinetic energy ($E_k$) regions manifest as interspersed, localized, velocity-jets, contrary to the prevalent view of high $E_k$ regions as large swirling structures (eddies). High enstrophy ($\omega^2$) regions form small vorticity-jets, invariably surrounded by swirling-velocity. The jet-like and swirling-velocity structures are spatially exclusive. Decomposing the Biot-Savart contributions from different levels and regions of the vorticity-field reveals the organization of velocity-field structures. High $E_k$ jets are neither self-induced (due to their low vorticity contents), nor induced by strong vorticity, being almost entirely induced, non-locally, by the permeating intermediate range (rms level) vorticity. High $\omega^2$ swirls, instead, are a superposition of self-induced swirling-velocity along with a background-induced flow. Moreover, intermediate vorticity dominantly induces the velocity-field everywhere. This suggests that turbulence organization could emerge from non-local and non-linear field interactions, dominated by permeating intermediate vorticity, leading to an alternative description of turbulence, contrary to the notion of a strict structural hierarchy. The tools presented can be readily applied to generic vector and scalar fields associated with diverse phenomena.