Discrete topology sensing in turbulence via signed pairwise graphlets of the velocity gradient tensor

Abstract

A signed pairwise graphlet encoding of the velocity gradient tensor $\bm{A}$ is introduced as a minimal discrete representation of its off-diagonal structure, and its statistical relationship with the PQR topology of homogeneous isotropic turbulence is investigated using direct numerical simulation data at $Re_λ\approx 433$. Each off-diagonal component is assigned a ternary label based on a local threshold, mapping the thresholded off-diagonal sign pattern to one of $3^6=729$ discrete states. The imbalance $Δn=n_W-n_S$ between rotation-promoting and strain-promoting pairs is shown to act as a robust proxy for the sign of $Q$: $P(Q>0\midΔn)$ increases monotonically from $0\%$ at $Δn\leq-2$ to $100\%$ at $Δn=+3$, and graphlets with a single rotation-promoting pair $(Δn=+1)$ predict the stable-focus-stretching region with probability approximately $65\%$ -nearly double the global base rate of $36.5\%$- despite discarding all magnitude information. This predictive power can be largely attributed to the off-diagonal contribution of $\bm{A}$, which dominates the sign of $Q$ in a majority of flow states. The graphlet--PQR association remains statistically correlated over several Kolmogorov time scales and exhibits spatial coherence comparable to that of the $Q$ field at dissipative scales, providing a consistency check that the discrete thresholding preserves the short-range organisation inherited from the velocity-gradient field. An out-of-sample predictive log-loss test further shows that $Δn$ captures most, but not all, of the predictive content of the complete $729$-state graphlet code: the full code retains a small, reproducible gain of order $10^{-3}$ bits per observation in predicting future PQR topology beyond $Δn$ and the continuous invariants, present at finite lag and absent at zero lag.