Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

Abstract

We consider the upper-branch neutral stability of flow in pipes of large aspect ratio, basically extending the work of F. T. Smith to the nonlinear regime. The inclusion of weak nonlinearity leads to an eigenproblem whose solution depends on the properties of three-dimensional nonlinear critical layers. Two special cases are considered. The first is for very small (≪O(R - 14 33 )) amplitude perturbations, where R is a Reynolds number based on the height of the tube and which is assumed large. Then a fully analytical solution of the three-dimensional critical layers is possible, from which the linear results of Smith may be deduced. The second case studied is that of flow in a rectangular pipe, where a solution of the nonlinear critical layer problem can be obtained. Further analysis of neutral modes in this latter case suggests the possible existence, inter alia, of neutral modes for finite aspect ratio tubes. These modes depend on the scaled amplitude and have O(1) wavespeeds.