Mandelbrot extended the mathematical theory of fractal dimensions to applications in Engineering and Science (Kac, 2003; Falconer, 2014). The frameworks that have been briefly laid out in this report are based on Mandelbrot's works, but also encapsulate some details on abstract mathematical objects, primarily measures. Fractals were first derived as strict mathematical objects and have been used in some manner since the 17 th century, known previously as self-similar sets. It was not until later that these mathematical objects began to be used in science, and in particular, in the 20th century, their use in the field of Fluid Mechanics became significant (Kac, 2003; Falconer, 2014; Turcotte, 1988). It was investigated whether or not some element of the motion of a fluid behaved like a fractal (Turcotte, 1988; Falconer, 2014), specifically if the dissipation of fluid flow was dependent on a fractal dimension (Falconer, 2014). Fractals are now being used for geometries of bodies moving through fluids, such as aircraft in flight, or fluids moving over bodies, such as cities in the atmosphere. Nedic and Vassilicos (2015) conducted a study where a NACA0012 wing's trailing edge was modified with fractal and multiscale geometries, where the aerodynamic performance was improved. The improvements were in the reduction of the wing's vortex shedding energies and the wing's lift-to-drag ratio. Finite fractals naturally occur in nature, an example being canopies of forests. There have been many studies on canopy flows. Bai et al. (2015) conducted an experimental investigation to understand the mixing of fluid through canopies and to understand the dispersive fluxes these 'tree-like' fractals generate. These are just a few examples where naturally occurring near-fractals are of interest to the field of Fluid Mechanics.

This document deals with the basic theory of fractal objects and provides a brief introduction into their applications in Fluid Mechanics. The main interest is to then apply fractal geometries to urban environmental flows. The document provide a detailed account of how to use the minkcomp.m MATLAB script. This uses the idea of roughness elements (Grimmond and Oke, 1999; Cheng and Castro, 2002b,a; Cheng et al., 2007) to represent an urban environment. The urban flow environment can be considered a rough wall system (Grimmond and Oke, 1999; Castro, 2007; Kanda et al., 2013; Yang et al., 2016). The script then calculates the number of elements needed to represent a rough wall with certain user inputted geometric parameters. This rough wall can scale to that of the atmospheric boundary layer, and aerodynamic properties can be derived from wind tunnel testing across the roughness elements designed by the minkcomp.m script. The outputs of the minkcomp.m script will provide all the geometric details in output .txt files and .py, which can be used to generate CAD models for visualisation. The CAD files can be converted into the necessary formats for manufacturing.