Figure Captions

Figure 1. The growth rates of the two most dangerous odd modes at a position 100cm from the leading edge in the experiment of Swearingen and Blackwelder (1987). Figure 2. The contours of the streamwise and normal velocity components for the most dangerous odd mode 100 cm from the leading edge in the experiment of Swearingen and Blackwelder (1987). Figure 3. The various regions of the three{dimensional boundary{layer ow: Ia,b The outer ow; IIa,b The diiusion layers of relative thickness O(Re ?1=4 ?1=2); III Critical layer of relative thickness O(Re ?1=6); IV Outer wall layer of relative thickness O(Re ?1=6); V Inner wall layer of relative thickness O(Re ?1=4). Note that (i), these thicknesses are given relative to the boundary{ layer thickness O(Re ?1=2), and (ii), the critical and diiusion layers are distinct from the wall being centered around the level Y = f(x; Z). 20 Hultgren, L. S. (1992). Nonlinear spatial equilibration of an externally excited instability wave in free shear layer. (1987) The growth and breakdown of streamwise vortices in the presence of a wall. On the weakly nonlinear three{dimensional instability of shear layers to pairs of oblique waves | the stokes layer as a paradigm. (1994) On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of GG ortler rolls. (1988), Nonlinear spatial evolution of an externally excited instability wave in a shear layer. Taylor-GG ortler instabilities of Tollmien-Schlichting waves and other ows governed by the interactive boundary layer equations. On the onset of three-dimensionality and time-dependence in the GG ortler vortex problem. 18 In this paper we have described the evolution of the inviscid modes found by Hall and Horseman (1991); this has done using viscous{critical{layer and diiusion{layer theories in the context of a weakly nonlinear instability theory. In particular, we have considered the evolution of a mode near the critical streamwise location where the vortex structure has developed suuciently to ((rst) render the now three{dimensional boundary{layer ow unstable to inviscid modes. At such a location, the ow is marginally unstable and we can consider the evolution of the most dangerous (important) mode. We note that our theory is not directly applicable to modes excited at streamwise locations where the ow supports a band of unstable modes (i.e. at an O(1) distance downstream from the critical x{location); in such cases the most dangerous mode has too large a growth rate and the wavenumber …