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 c 2 i , where −αα c 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 α R 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.