## Abstract

High Reynolds number $(Re)$ flows through large aspect ratio $(\lambda\mu)$ tubes of rectangular cross section are studied. One wall of the tube is slightly deformed to produce a two-dimensional distortion of length $\lambda$. We determine conditions for the flow at the centre of the tube and near the distortion to approximate the appropriate two-dimensional solution: namely, $\lambda\mu \gg (\lambda^{-1} Re)^\frac{1}{6}$ if $Re^\frac{1}{7} \precsim \lambda$ and $\mu \gg 1$ if $Re^\frac{1}{7} \gtrsim \lambda$. However, the latter condition needs to be strengthened to $\lambda\mu \gg Re^\frac{1}{7}$ if the flow is additionally to be approximately two-dimensional far up- and down-stream. The method of solution includes a numerical calculation for the flow in the sharp corners of the tube. We deduce that for sufficiently short distortions ($\lambda \ll Re^\frac{1}{9}$(ln $Re$)$^-\frac{11}{9}$), the sharp corners can effectively isolate disturbances in the wall boundary layers from each other. However, for larger distortions the disturbances in the boundary layers are all of comparable magnitude owing to interactions at the corners. Our examination of the corner regions also enables us to confirm a hypothesis, due to Hocking (1977) and others, that to leading order the pressure is constant in approximately square regions at the sides of the tube.