Fractal dimension of particle clusters in isotropic turbulence using Kinematic Simulation

1
Fractal dimension of particle clusters in
isotropic turbulence using Kinematic Simulation
The University of Sheffield Department of
Mechanical Engineering
Fractal Definition
Kinematic Simulation
The word fractal was coined in 1975 by
mathematician Mandelbrot to describe shapes or
objects too irregular to include in traditional
geometry and of which are detailed at all scales,
( fractus in Latin means broken ).
Fung, Hunt, Malik and Perkins (1992) developed a
new Lagrangian model of turbulence flow, which
they called Kinematic Simulation.
It is based on a kinematically simulated Eulerian
velocity field generated as a sum of random
incompressible Fourier modes of homogeneous
isotropic turbulence.
In year 300 BC,
In 1700,
In 1904,
For isotropic turbulence the KS velocity field is
constructed by discretization of the Fourier
transform of the Eulerian velocity
Fractal Dimension
Energy Spectrum
The fractal dimension can thus be used to compare
the complexity of two curves or two surfaces, and
therein lies its importance.
Box-Counting Method
The fractal object is covered with a network of
boxes and the number of boxes (Ne) having a side
length (e) needed to cover the surface is counted
kN
Fractal Line
Fractal Surface
Evolution of a line embedded in a turbulent flow
at different times
t/td0
t/td0.1
t/td0.3
Square advected in turbulent flow at Re464, with
initial side length 0.2 L
th is the Kolmogorov time scale
th is the Kolmogorov time scale
Fractal Volume
Conclusions
1. KS is able to predict the fractal dimension
of lines, surfaces and volumes and in good
agreement with theoretical, experimental and LES
results
2. The fractal dimension of the line increases
linearly with time, up to its maximum value. The
time required to reach its maximum is a function
of the Reynolds number.
3. The fractal dimension of a square increases
linearly with time, up to its maximum value. The
time required to reach its maximum is a function
of the Reynolds number.
t/td0
t/td0.1
t/td0.3
Cube advected in turbulent flow at Re464, with
initial side length 0.2 L
4. The fractal dimension of a cube is found to
decrease regularly towards 2, reflecting that
fact that the volume is progressively converted
into an elongated sheet. Also, the fractal
dimension is found to be independent of the
Reynolds number and a function of the cubes
initial size
Future work

1. Use the kinematic simulation technique to study
   heavy particles as a cloud (multi-particle
   dispersion) and investigate the different
   parameters of turbulence.
2. Use the KS results to investigate the value of
   the fractal dimension of a surface and volume of
   clouds.
3. Use this study to control turbulence and noise in
   pipes and piping systems by introducing fractal
   objects in these systems.

Dr. F. Nicolleau, Dr. A. El-Maihy and A. Abo
El-Azm
Contact address F.Nicolleau_at_sheffield.ac.uk,
A.Aaboazm_at_sheffield.ac.uk