Shape Statistics of Particle Clusters in a Turbulent Flow

Grant, John (2019). Shape Statistics of Particle Clusters in a Turbulent Flow. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.00010b2b

Abstract

The attractor of a chaotic dynamical system may have a multi-fractal measure which can be described by a spectrum of fractal dimensions. These dimensions do not characterize local geometrical structures that may exist within the attractor and in physical situations these may be important. It is therefore of interest to have a more effective means of characterizing the local structure of fractal sets and it is this problem that is addressed in this thesis. The problem is approached by considering the statistical distributions of the size and shape of very small triangular constellations of points sampling the fractal measure. The approach is illustrated, and validated, using fractal clusters of particles formed by advection and diffusion in a two-dimensional compressible random flow, which models turbulence. Our numerical simulations show that as the compressibility parameter of the fluid passes through a critical value the distribution of the flatness of constellations undergoes a phase transition. We develop a theoretical model for this phenomenon which correctly predicts the critical value of the compressibility. Also, by representing the effects of the flow as a stochastic matrix process, we show that for a range of values of compressibility the probability density of the size of constellations is a modified power law. For a fractal cluster generated by the random flow we derive an expression for the Renyi dimension of order three, D₃, in terms of the probability density of the size of constellations and find it is in agreement with the results of other authors, obtained using other methods.

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