An experimental study of fractal and multifractal scale similarity in turbulent flows.

Results are presented from an assessment of the applicability of fractal and multifractal scale similarity to the spatio-temporal structure of $Sc\ \gg$ 1 conserved scalar fields $\zeta$(x,t) and scalar energy dissipation rate fields $\nabla\zeta\cdot\nabla\zeta$(x,t) in turbulent flows. Over 4 million spatial and temporal intersections are analyzed from fully-resolved three-dimensional (256$\sp3$) spatial measurements as well as fully-resolved four-dimensional spatio-temporal measurements containing up to 3 billion points. Statistical criteria are used to assess fractal and multifractal scale similarity and to discriminate between scale similar and random sets. Over the range of scales ($\lambda\sb{\nu},\ T\sb{\nu}$) to the outer scales ($\delta, T\sb{\delta}$), slightly over 99.0% of one-dimensional 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 are found to have support dimension D = 0.66. With increasing intersection dimension n, the dissipation support data show a decrease in the fraction of intersections displaying fractal scale similarity, consistent with the presence of localized nonfractal inclusions. Local scale similarity analyses on three-dimensional (64$\sp3$) intersections with the dissipation support directly show such intermittent nonfractal inclusions with characteristic length scale comparable to $\lambda\sb{\nu}$. These inclusions lead to a failure of the relation among codimensions $D\sb{n}\equiv D-(3 - n)$ which has formed the basis for most previous assessments of fractal scale similarity. Examinations of the scalar isosurface geometries exhibited no fractal scaling over any range of scales. Further analyses show the scalar energy dissipation field to exhibit multifractal scaling at scales larger than 1.4$\lambda\sb{D}$, and evidence is also found for multifractal scale similarity in the conserved scalar field but only over a range of scales larger than 0.5$\lambda\sb{\nu}$. Analyses show that the random multiplicative cascade process responsible for the multifractal structure of the scalar energy dissipation rate field may be accurately characterized by a bilinear P(M) distribution.Ph.D.Aerospace engineeringApplied SciencesMechanical engineeringPlasma physicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/129850/2/9635519.pd