Anisotropic Coverings of Fractal sets

Harry Kennard

h.r.kennard@open.ac.uk PhD Candidate

Department of Applied Mathematics

The Open University

Supervisor: Professor Michael Wilkinson

1 Motivation

2 Method

3 Generalised Sierpinski Model

4 Inertial particles in a random flow

Outline of the paperOutline of the Talk

2~)( DkkI

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

Locally Cartesian product structure

k

k’

NI ~

2~ NI

Isotropicscattering

Coherent scattering

Require a function to characterise

fractal anisotropy

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

ε

δ

)(),( N

10

Nlog logplot vs. get straight lines

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

2)1( D

Just circles!

1

Motivation Method Sierpinski Inertial

)(11 22 ~~ DDN

)1(1)( 2 D

Upper bound

2~ DN

N is independent of ellipse orientation

in the disk

of this lie in the ellipse

explain sier generalizations

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

Motivation Method Sierpinski Inertial

Inertial particles move in an incompressible fluid (velocity field u) under a synthetic turbulent velocity field

Particles, and hence the flow and distribution of them, is characterised by η [0:1]

η=0.9

Motivation Method Sierpinski Inertial

η=0.1

Motivation Method Sierpinski Inertial

η

M. Wilkinson, B. Mehlig and K. Gustavsson,Eurohysics Lett., 89, 50002,

(2010).

η=0.4

Motivation Method Sierpinski Inertial

η

η

M. Wilkinson, B. Mehlig and K. Gustavsson,Eurohysics Lett., 89, 50002,

(2010).

Motivation Method Sierpinski Inertial

η

M. Wilkinson, B. Mehlig and K. Gustavsson,Eurohysics Lett., 89, 50002,

(2010).

Thank You – Any Questions?

Harry Kennard

h.r.kennard@open.ac.uk PhD Candidate

Department of Applied Mathematics

The Open University

Supervisor: Professor Michael Wilkinson