The Boundary Layer Due to Rectilinear Vortex

J. D. A. Walker
1978 Proceedings of the Royal Society A  
The boundary layer created by th e motion of a single rectilinear vortex filam ent above an infinite plane wall is considered. In a fram e of reference which moves uniform ly w ith th e vortex the inviscid m otion is steady; however, th e possibility of a corresponding steady boundary-layer solution can be ruled out and it is concluded th a t th e boundary-layer flow is in herently unsteady for all tim e. To investigate th e n atu re of the unsteady boundary-layer flow, a tim e-dependent
more » ... , corresponding to the sudden insertion of th e plane wall a t tim e 0, is considered; separation in th e boundary layer is found to ta k e place in a short period of tim e and th e solution shows possibly explosive features as t increases. I t is conjectured th a t an eventual eruption of th e boundary-layer flow is to be expected along w ith a m ajor modification of the inviscid flow. The theory com pares favourably w ith experim ents on th e flow induced near th e ground by trailing aircraft vortices. I n t r o d u c t io n The problem of inviscid flow w ith v orticity and how this interacts w ith a solid boundary is an im p o rtan t subject ab o u t which relatively little is known th eo reti cally. There are a num ber of im p o rtan t areas such as tu rb u len t boundary layers where a good understanding of such flows should prove useful. A nother practical area of interest is th e n atu re of the flow induced near th e ground by aircraft trailing vortices. In an experim ental investigation, H arvey & P erry (1971) observed flow separation near th e ground which was induced by the trailing vortex and which ultim ately resulted in th e ejection of a secondary vortex from the boundary layer. The separation had profound effects on the m otion of the trailing vortex in the crossflow plane which was observed to slow down, rise and in some configurations actually reverse direction. H arvey & P erry (1971) also note a sim ilarity between the observed separation and the observations of the flow over slender delta wings. In the experim ents of H arvey Ornberg (1964) th e vortex sheet shed from th e leading edge of the wing rolled up into a vortex above the upper surface of the wing, which subsequently induced a secondary separation dow nstream . I t is of interest to understand how this ty p e of separation can occur. Inviscid flows w ith vorticity are often modelled w ith distributions of rectilinear vortices (see for example, Dee & Nichols 1968). However, in a flow configuration w ith solid walls, boundary layers will exist in any real fluid; consequently before [ 167 ] 6-2
doi:10.1098/rspa.1978.0038 fatcat:yvkih7s55jey7o5h4ve2i2cx5m