The separating flow through a severely constricted symmetric tube

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Abstract
The axisymmetric flow of an incompressible fluid through a pipe (of radius a) suffering a severe constriction is studied for large Reynolds numbers R, the features of symmetric channel flows being virtually the same. Here ‘severe’ refers to a constriction whose typical dimensions are finite, and the oncoming velocity profile is taken to be of a realistic type, i.e. with no slip at the wall. The study adopts (Kirchhoff) free-streamline theory, which, for the mostly inviscid description, affords a rational basis consistent with viscous separation. The major (triple-deck) separation takes place on the constriction surface and is followed by a downstream eddy of length O(aR). Another, less familiar, separation is predicted to occur at a distance 0.087a In R + O(a) ahead of the finite obstacle. Free-streamline solutions are found in the two main extremes of moderately severe and very severe constriction. In both extremes, and in any slowly varying constriction, the major separation is sited near the maximum constriction point. The upstream separation point is also derived, to O(a) accuracy in each case. The upstream separation can be suppressed, however, if the constriction has no definite starting point and decaysslowly upstream, but then the upstream flow response extends over a much increased distance. Comparisons with Navier-Stokes solutions and with experiments tend to favour the predictions of the free-streamline theory.

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