Considered below is the plane, high-Reynolds-number (Re), flow of an incompressible fluid through a channel suffering a severe non-symmetric constriction, ‘severe’ meaning ‘of typical dimensions comparable with the channel width a*’. The (mainly inviscid) flow description is governed by free streamline theory, to be consistent with the viscous incompressible separation from the constriction surface and with a relatively slow eddy motion beyond. Separation also occurs far ahead of the constriction, at a distance $O(Re^{\frac{1}{7}} a^{*})$ upstream. Attention is given primarily to the flow features for a particular class of slowly varying severe constrictions, from which however the features for the probably more useful classes of slender (i.e. of large length but O(a*) height) and moderately severe (i.e. of O(a*) length but small height) constriction follow, as do those for curved or cornered channel flows. In all these cases a long-scale flow response is induced upstream and downstream and in many cases remarkably simple universal formulae for the separation and reattachment positions result. The corresponding O(a*) corrections to the upstream separation distance above are also derived.