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 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 Reλ≈ 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 36=729 discrete states. The imbalance Δn=nW−nS between rotation-promoting and strain-promoting pairs is shown to act as a robust proxy for the sign of Q: P(Q>0|Δn) increases monotonically from 0% at Δn≤−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 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.
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BibTeX
@misc{boukharfane2026discrete,
title = {{Discrete topology sensing in turbulence via signed pairwise graphlets of the velocity gradient tensor}},
author = {Radouan Boukharfane},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23799},
eprint = {2609.23799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23799},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete topology sensing in turbulence via signed pairwise graphlets of the velocity gradient tensor
AU - Radouan Boukharfane
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23799
UR - https://arxiv.org/abs/2609.23799
AB - A signed pairwise graphlet encoding of the velocity gradient tensor 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≈ 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|Δn) increases monotonically from 0% at Δn≤−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 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.
ER -