The instability of the flow in a suddenly blocked pipe

This paper considers the decay of Poiseuille flow within a suddenly blocked pipe. For small to moderate times the flow is shown to consist of an inviscid core flow coupled with a boundary layer at the pipe wall. A small-time asymptotic solution is developed and it is shown that this solution is valid for times up to the point at which the boundary layer fills the whole pipe. A small-time composite solution is used to initiate a numerical marching procedure which overcomes the small-time singularity that arises in the flow and so allows us to describe the ultimate decay of the flow within a blocked pipe. The stability of this flow is then considered using both a quasi-steady approximation and a transient-growth analysis based upon marching solutions of the linearized Navier-Stokes equations. Our transient stability analysis predicts a critical Reynolds number, for transition to turbulence, in the range 970 < Re < 1370.