Vorticity-direction coherence at intensity-adapted scales in isotropic turbulence: scale normalization, angular saturation and clustered inference

Geometric criteria for the regularity of the three-dimensional Navier–Stokes equations constrain the coherence of the vorticity direction in regions of intense vorticity, and the length √(ν/|ω|) appears naturally in such estimates. We measure a sampled one-half Hölder coherence coefficient κ̂ of the vorticity direction at the intensity-adapted radius r = c√(ν/q₀) around intense-vorticity events in one snapshot of the 8192³ forced isotropic turbulence simulation of the Johns Hopkins Turbulence Databases (Re_λ ≈ 613). If κ ∝ Ωᵖ and r ∝ Ω^(−1/2), the dimensionless coefficient χ = r^(1/2) κ is independent of intensity when p = 1/4. In a confirmation sample of 78 events in 15 spatial blocks (c = 8), the slope of log χ̂ against the logarithm of local intensity is −0.05 (95% interval −0.18 to 0.07), and none of the seven inference procedures examined rejects zero; at c = 4√2 the slope is −0.06. The raw coefficient increases with intensity (slope 0.19), as the kinematic reference predicts, but its statistical significance depends on the procedure. Decisive pairs are frequently near the angular ceiling (median saturated fraction 0.39 at c = 8), and the estimator is sensitive to the number of candidate pairs and to the search radius. A smaller development sample yields a negative normalized slope (−0.27) whose significance also depends on the procedure. The results characterize a finite-sample estimator in a single snapshot; they do not test a regularity criterion.