The critical layer during transition of the spiral Poiseuille flow in an annular gap

Abstract

Subject of this paper falls under the broad heading of the propagation of disturbances causing transition in a fully developed spiral annular flow. The problem is approached through reformulation of the linearised equations of motion governing small-amplitude disturbances in a parameter space that differs from that conventionally employed for this purpose. The alternative parameter space comprises a suitably defined Reynolds number that is formed with a resultant characteristic velocity which is a vector sum of the axial and azimuthal characteristic velocities, a swirl parameter that is the ratio of the azimuthal to the axial characteristic velocity, and the ratio of the annular gap to the mean diameter of the cylinders. In the limits of the swirl parameter assuming very small or very large values, the newly derived generalised Orr-Sommerfeld and Squire equations for disturbance propagation in the revised parameter space reduce to the corresponding equations that are known and well established for the limiting cases of disturbance propagation as the swirl parameter goes to zero or infinity respectively. Furthermore, they also lead naturally to the criterion that determines from the equations themselves the analytical expression for the location of the critical layer in spiral annular flow, the scaling behaviour of the thickness of the critical layer with the Reynolds number and the swirl parameter, and the equation governing the dynamics of the flow in the critical layer. They thus explicitly reveal the effect of swirl on the critical layer.