Poiseuille flow in rough pipes: linear instability induced by vortex–wave interactions

The instability of Hagen–Poiseuille flow in a rough pipe is considered and it is shown that for arbitrarily small roughness amplitudes the flow is unstable for sufficiently large values of the Reynolds number. Various models of wall roughness are considered and, if , the height of the roughness. The instability mechanism described is closely linked to vortex–wave interaction theory and applies to both two- and three-dimensional roughness shapes and takes the form of a roll-streak-wave flow. The interaction sustaining the instability occurs in a viscous boundary layer at the pipe wall but the roll-streak flow persists throughout the pipe. The most dangerous roughness shapes are found and generic results are also given for when the roughness length scale is small compared to the pipe radius.