Theoretical Advances on Generalized Fractals with Applications to Turbulence

Theoretical advances toward a purely meshless framework for analysis of the generalized fractal dimension, with applications to turbulence, are considered. The key basic theoretical idea is the formulation of the probability density function of the minimum distance to the nearest part of the flow feature of interest, e.g. a turbulent interface, from any location randomly chosen within a reference flow region that contains the feature of interest. The probability density function of the minimum-distance scales provides a means to define and evaluate the generalized fractal dimension as a function of scale. This approach produces the generalized fractal dimension in a purely meshless manner, in contrast to box-counting or other box-based approaches that require meshes. This enables the choice of a physical reference region whose shape can be based on physical considerations, for example the region of fluid enclosed by the turbulent interface, in contrast to box-like boundaries necessitated by box-counting approaches. The purely meshless method is demonstrated on spiral interfaces as well as high-resolution experimental turbulent jet interfaces. Examination of the generalized fractal dimension as a function of scale indicates strong scale dependence, at the large energy-containing scales, that can be described theoretically using exponential Poisson analytical relations.