Communities for the Lagrangian dynamics of the turbulent velocity gradient tensor: A network participation approach

In this paper, we present a network-based framework for analyzing the Lagrangian dynamics of the velocity gradient tensor (VGT). Each node represents a flow state, and link weights correspond to the transition probabilities between states, derived from Direct Numerical Simulation (DNS) of statistically steady, isotropic turbulence. The network provides a compact representation of the VGT’s continuum dynamics by discretizing it into a finite set of states. We investigate the optimal variables for this discretization, classifying VGT states into groups that best capture the flow’s physics. To this end, we test several classifications based on topology and various properties of the background flow coherent structures. We do this using the notion of “community” or “module”, namely clusters of nodes that are optimally distinct while also containing diverse nodal functions. The most effective classification, informed by VGT invariants frequently used in the literature, combines the signs of the principal invariants $Q$, $R$, and the discriminant $\Delta$, distinguishing regions of real and complex eigenvalues. We refine this further by incorporating the relative magnitude of the non-normal contributions to enstrophy and straining, derived from the Schur decomposition of the VGT. Accounting for non-normality significantly enhances classification fidelity and underscores the critical role of the unclosed and complex contributions to VGT dynamics from the pressure Hessian and viscous terms. A comparison between the DNS data and an enhanced Gaussian closure model reveals the challenges for conventional modeling approaches in accurately capturing the non-normal contributions to the VGT dynamics.