Finite-amplitude stability of pipe flow

Abstract

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c²_i, where −ααc_i is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R⁻², and the wave-number α is of order R^1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R^½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.