The growth of Taylor vortices in flow between rotating cylinders

In flow between concentric rotating circular cylinders, it was shown by Taylor (1923) that instability may occur in the form of toroidal vortices spaced regularly along the axis. When the vortex motion occurs additional torque is required to keep the cylinders in motion at given speeds. Stuart (1958) used an energy-balance method, in the case when the annular gap is small compared with the radius, to estimate the additional torque and the associated finite amplitude attained by the vortices. He included the effect of distortion of the mean motion, but ignored the generation of harmonics of the fundamental mode and the distortion of the velocity associated with the fundamental mode. It is now known that these are not valid mathematical approximations and a rigorous perturbation expansion is developed here to remedy the deficiency. The analysis is valid for any gap width and any angular speeds of the containing cylinders, but requires the amplification rate of the disturbance to be small. Numerical results using a digital computer are obtained for the shape and amplitude of the vortices in three cases: (i) when the outer cylinder has twice the radius of the inner one and is kept at rest, (ii) when the gap is small and the cylinders rotate with nearly the same speeds, and (iii) when the gap is small and the outer cylinder is kept at rest. The equilibrium amplitude obtained in the last case is substantially the same as that found by Stuart. The results for cases (i) and (iii) give close agreement with the experimental values obtained by Taylor (1936) and Donnelly (1958) for the torque required to keep the inner cylinder rotating with constant speed while the outer one is at rest, for a certain range of speeds. In the small-gap problem it is shown that the equilibrium amplitude is almost proportional to 1 − m , where m is the ratio of the angular speeds of the outer and inner cylinders.