Estimating the local mean oscillation of the vorticity direction in turbulence simulations: sampling measure, stratified designs and limits of scale-response classification

Conditions on the local mean oscillation of the vorticity direction appear in localized regularity results for the Navier–Stokes equations. Estimating such oscillations from direct numerical simulation requires averages over balls computed from interpolated point samples. We study three aspects of this estimation with data from the Johns Hopkins Turbulence Databases. First, the spatial measure matters: in 80 intense-vorticity events of an 81923 isotropic snapshot (Reλ ≈ 613), points drawn from a mixture of two concentric balls give a median outer-scale oscillation of 0.770, whereas inverse-density weighting to uniform volume gives 0.832 (median paired difference 0.049, 95% interval 0.045–0.060), in agreement with a second uniform-volume estimator. Second, a stratified design with ten thousand points in each of five shells and volume weights reduces the thinning variability at the two smallest scales by a factor of about three relative to uniform sampling, at some cost at the outer scale; it met fixed precision criteria in all 12 events of a 40963 dataset and all 10 events of the 81923 dataset. Third, finite-window scale responses do not discriminate Hölder-type from logarithmic templates: no synthetic generator reproduced both the amplitude and the shape of the measured responses, and the nearest template changed with the window and with the set of scales fitted. The work provides estimators and precision diagnostics for this observable, not a classification of regularity.