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Fractal dimension of particle clusters in isotropic turbulence using Kinematic Simulation Dr. F. Nicolleau, Dr. A. El-Maihy and A. Abo El-Azm Contact address: F.Nicolleau@sheffield.ac.uk, A.Aaboazm@sheffield.ac.ukF.Nicolleau@sheffield.ac.uk The University of Sheffield Department of Mechanical Engineering 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 ). Fractal Definition: In year 300 BC, In 1700, In 1904, The fractal dimension can thus be used to compare the complexity of two curves or two surfaces, and therein lies its importance. Fractal Dimension: Box-Counting Method: The fractal object is covered with a network of boxes and the number of boxes (Nε) having a side length (ε) needed to cover the surface is counted Fung, Hunt, Malik and Perkins (1992) developed a new Lagrangian model of turbulence flow, which they called Kinematic Simulation. Kinematic Simulation: For isotropic turbulence the KS velocity field is constructed by discretization of the Fourier transform of the Eulerian velocity: kNkN Injected Energy k -5/3 k1k1 Log k Log E Inertial Range Dissipation by the Effect of Viscosity Evolution of a line embedded in a turbulent flow at different times It is based on a kinematically simulated Eulerian velocity field generated as a sum of random incompressible Fourier modes of homogeneous isotropic turbulence. Energy Spectrum: Fractal Line: t/t d =0t/t d =0.1t/t d =0.3 t/t d =0t/t d =0.1t/t d =0.3 Cube advected in turbulent flow at Re=464, with initial side length 0.2 L Fractal Volume: t h is the Kolmogorov time scale Fractal Surface: Square advected in turbulent flow at Re=464, with initial side length 0.2 L Conclusions: 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. 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 cube’s initial size 1. KS is able to predict the fractal dimension of lines, surfaces and volumes and in good agreement with theoretical, experimental and LES results 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. t h is the Kolmogorov time scale

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Atmospheric turbulence Richard Perkins Laboratoire de Mécanique des Fluides et d’Acoustique Université de Lyon CNRS – EC Lyon – INSA Lyon – UCBL 36, avenue.

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