The instability of the flow in a suddenly blocked pipe

This paper considers the decay of Poiseuille flow within a suddenlyblocked pipe. For small to moderate times the flow is shownto consist of an inviscid core flow coupled with a boundarylayer at the pipe wall. A small-time asymptotic solution isdeveloped and it is shown that this solution is valid for timesup to the point at which the boundary layer fills the wholepipe. A small-time composite solution is used to initiate anumerical marching procedure which overcomes the small-timesingularity that arises in the flow and so allows us to describethe ultimate decay of the flow within a blocked pipe. The stabilityof this flow is then considered using both a quasi-steady approximationand a transient-growth analysis based upon marching solutionsof the linearized Navier–Stokes equations. Our transientstability analysis predicts a critical Reynolds number, fortransition to turbulence, in the range 970 < Re < 1370.