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

Abstract

Results are presented from an assessment of the applicability of fractal scale-similarity in the spatio–temporal structure of Sc [Gt ] 1 conserved scalar fields ζ(x, t) and scalar energy dissipation rate fields ∇(x, t) in turbulent flows. Over 2 million spatial and temporal intersections were analysed from fully resolved three-dimensional (256) 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 (ΛD, TD) to the viscous diffusion scales (Λv, Tv) and to the outer scales (δ, Tδ). Over this entire range of scales, slightly 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 (ΛD, TD) and (Λv, Tv). 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.