Shear flow instabilities in pipes and channels

Abstract

Two broad problems are considered in this thesis. The first investigation focuses on the spatial stability of pressure driven flow in a pipe, while the second problem is concerned with temporal stability of Couette flow in a parallel wall channel. A similar approach is taken with both halves of this thesis: an asymptotic analytical model is developed and this is then solved with numerical methods.

The prominent problem in the first chapter is Hagen-Poiseuille flow with suction and injection on the pipe wall boundary. The flow is fully developed in the axial direction and the suction acts radially at the wall while the other boundary conditions are no-slip. The flow is perturbed by a small non-axisymmetric disturbance and this configuration is solved in the radial-azimuthal plane. The results are provided for 2pi-periodic suction and it is found that the non-parallel base flow is unstable in certain conditions. The suction coefficient is varied but the critical Reynolds number is found to be the same. A weakly nonlinear stability analysis reveals that there is a finite amplitude solution in the supercritical region.

The second chapter presents the vortex-wave interaction equations and a special case of the model is created to seek stationary, equilibrium solutions of the sinuous wave disturbance in Couette flow. The flow is initiated with an artificial forcing which has the sinusoidal symmetries embedded. From this initial condition the `roll' problem is solved and the `streak' can be found from this solution. The wave on this streak has Reynolds stresses which now force the `roll'. The amplitude of the wave is varied and the system is iterated until the wave is neutral. This equilibrium configuration is then marched forward in time and studied. The solutions agree with numerical calculations and experiments.