Experimental assessment of fractal scale-similarity in turbulent flows. Part 1. One-dimensional intersections

Results are presented from an assessment of the applicability of fractal scale-similarity in the spatio-temporal structure of Sc ≫ 1 conserved scalar fields ζ(x, t) and scalar energy dissipation rate fields ∇ζ·∇ζ(x, t) in turbulent flows. Over 2 million spatial and temporal intersections were analysed from fully resolved three-dimensional (256³) spatial measurements as well as fully resolved four-dimensional spatio-temporal measurements containing up to 3 million points. Statistical criteria were used to assess both deterministic and stochastic fractal scale-similarity and to differentiate between fractal and random sets. Results span the range of spatio-temporal scales from the scalar diffusion scales (λ_D, T_D,) to the viscous diffusion scales (λ_V, T_V) and to the outer scales (δ, T_δ). Over this entire range of scales, slig htly over 99.0% of all intersections with the scalar dissipation support geometry showed scale-similarity as fractal as stochastically self-similar fBm sets having the same record length. Dissipation values above the mean were found to have support dimension D = 0.66. The dissipation support dimension decreased sharply with increasing dissipation values. Virtually no intersections showed scaling as random as a random set with the same relative cover. In contrast, intersections with scalar isosurfaces showed scaling only approximately as fractal as a corresponding fBm set and only over the range of spatio-temporal scales between (λ_D, T_D) and (λ_V, T_V). On these inner scales the isosurface dimension was D = 0.48 and was largely independent of the isoscalar value. At larger scales, scalar isosurfaces showed no fractal scale-similarity. In contrast, isoscalar level crossing sets showed no fractal scale-similarity over any range of scales, even though the scalar dissipation support geometry for the same data is clearly fractal. These results were found to be unaffected by noise.