Global stability of spiral flow. Part 2

Abstract

The stability of spiral flow between rotating and sliding cylinders is considered. In the limit of narrow gap, a’ modified’ energy theory is constructed. This theory exploits the consequences of assuming the existence of a preferred spiral direction along which disturbances do not vary. The flow is also analyzed from the viewpoint of linearized theory. Both problems depend strongly on the sign of Rayleigh's discriminant, – 2Ωζ. Here Ω is the component of angular velocity, and ζ is the component of total vorticity of the basic flow in the direction perpendicular to the spiral ribbons on which the disturbance is constant. When the discriminant is negative, there is evidently no instability to infinitesimal disturbances, and the spiral disturbance whose energy increases at the smallest R is a roll whose axis is perpendicular to the stream. This restores and generalizes Orr's non-linear results for disturbances having a preferred spiral direction. When the discriminant is positive, the critical disturbances of linear theory and the modified energy theory are spiral vortices. The differences between the energy and linear limits can be made smaller in the restricted class of disturbances with coincidence achieved for axisymmetric disturbances in the rotating cylinder problem in the limit of narrow gap. For the sliding-rotating case, the critical disturbance of the linear theory appears as a periodic wave in a co-ordinate system fixed on the outer cylinder. This wave has a dimensionless frequency equal to - ½ a sin (χ-ψ), where a is the wave-number, χ is the angle between the pipe axis and the direction of motion of the inner cylinder relative to the outer one, and ψ is the disturbance spiral angle.

Instability limits, frequencies and wave-numbers are computed numerically when the cylinder gap is not narrow. These are in even closer agreement with Ludwieg's experimental results than the approximate results which were given in part 1.