Dynamics in Turbulence

Kennard, Harry Robert (2013). Dynamics in Turbulence. MPhil thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000f188

Abstract

This MPhil thesis explores two themes associated with the dynamics of inertial particles in turbulent fluids. Firstly, we consider an optimal partial covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the semi-minor axis is ϵ and the semi-major axis is δ, we set δ = ϵα, where 0 < α < 1 is an exponent characterizing the anisotropy of the ellipses. The optimization involves varying the angle of the principal axis to maximize the measure covered by each ellipse. For point set fractals, in most cases we find that the number of points N which can be covered by an ellipse centred on any given point has expectation value (N) ~ ϵβ, where β is a generalized of the fractal dimension. β(α) is investigated numerically for various sets, showing that it may be different for sets which have the same fractal dimension. Secondly, we examine an analytically solvable limit of a model for the alignment of microscopic rods in a random velocity field with isotropic statistics. The vorticity varies very slowly and the isotropic random flow is equivalent to a pure strain with statistics which are axisymmetric about the direction of the vorticity. We analyse the alignment in a weakly fluctuating uniaxial strain field, as a function of the product of the strain relaxation time τ_s and the angular velocity ω about the vorticity axis. We find that when τ_sω >> 1, the rods are predominantly either perpendicular or parallel to the vorticity. We also review the current literature on the dynamics of inertial particles in turbulent flows.