Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration)

The incompressible boundary layer in the corner formed by two intersecting, semiinfinite planes is investigated, when the free-stream flow, aligned wi th the corner, is taken to be of the form UinfinityF(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duc k (1997) similarity solutions for F(x) = x(n) were considered, and it was f ound that solutions exist for only a range of values of n, whilst for infin ity > n > -0.018, approximately, two solutions exist. In this paper, we ext end the work of Dhanak & Duck to the case of non-90 degrees corner angles a nd allow for streamwise development of solutions. In addition, the effect o f transpiration at the walls of the corner is investigated. The governing e quations are of boundary-layer type and as such are parabolic in nature. Cr ucially, although the leading-order pressure term is known n priori, the th ird-order pressure term is not, but this is nonetheless present in the lead ing-order governing equations, together with the transverse and crossflow v iscous terms. Particular attention is paid to flows which develop spatially from similari ty solutions. It turns out that two scenarios are possible. In some cases t he problem may be treated in the usual parabolic sense, with standard numer ical marching procedures being entirely appropriate. In other cases standar d marching procedures lead to numerically inconsistent solutions. The sourc e of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the f irst scenario), analogous to those found in the computation of some two-dim ensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; B rown & Stewartson 1975). In order to circumvent this difficulty, a differen t numerical solution strategy is adopted, based on a global Newton iteratio n procedure. A number of numerical solutions for the entire corner flow region are prese nted.