| Summary: | The separation of a boundary layer from an aeroplane wing can have severe effects on aeroplane safety and efficiency, as its occurrence directly results in decreases in lift and increases in drag. Similar considerations apply to other technologies that rely on airfoils, such as drones, helicopters, propellers and wind turbines. Hence recent experimental and numerical work on dynamic roughness elements---small bumps that are made to oscillate up and down at a given frequency---is exciting, as it suggests that these elements are able to delay separation or increase the angle of attack at which it occurs, provided that the Reynolds number is such that the flow remains laminar (Grager et al. (2012), Huebsch (2006), Huebsch et al. (2012), Rothmayer & Huebsch (2011)). Our aims are to gain further insight into whether this is indeed the case; to determine the possible impact of the roughness parameters on the separation of a boundary layer from a surface; and to attempt to understand the physical mechanisms that may be involved, with our focus very much on the pressure gradient. To this end, we will make use of a mathematical approach and exploit asymptotic methods throughout. Three scenarios will be considered, and we will study both dynamic and static roughnesses. The first consists of small roughness elements, which are able to modify the mean flow pressure gradient, on a flat plate. The second will revolve around flow over a hump within a condensed boundary layer, first described by Smith & Daniels (1981), but with the addition of roughness elements on its lee side, in the region in which local separation occurs and the advent of full breakaway separation is seen. The final scenario is set near the leading edge of an airfoil, inclined to the oncoming flow at or near the critical angle of attack, where marginally separated flow exists and a small separation bubble is possible (Ruban (1982), Stewartson, Smith & Kaups (1982)).
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