Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem

The remarkable results obtained by Benjamin (1978), who investigated experimentally the nature of Taylor vortex flows in short cylinders have stimulated much theoretical work on the role of end effects in stability theory. Most of this work has been in connection with the Rayleigh-Bénard problem in finite rectangular or circular containers where the side-wall conditions are simple enough for perturbation methods to be used. The experiments of Benjamin were performed in cylinders of variable length having end walls held fixed to the stationary outer cylinder. In this case the basic state set up when the inner cylinder rotates is never close to the purely circumferential flow of the corresponding infinite problem. Thus perturbation methods cannot be used directly to investigate Benjamin’s problem but Schaeffer (1980) has proposed a model problem having simpler end-wall conditions. The conditions used by Schaeffer in fact correspond to porous end walls. Some predictions about Benjamin’s problem can be made by perturbing the end-wall conditions of the model problem towards those of Benjamin’s apparatus. Schaeffer’s analysis is not valid for most of the available experimental results, which correspond to cylinders so short that only two or four Taylor cells occur. However, the bifurcation picture in the length-Reynolds number plane obtained by Benjamin for this configuration is probably similar to those appropriate to longer cylinders. Using qualitative methods and assuming certain numerical constants have the required behaviour, Schaeffer argues that Benjamin’s results are plausible. In this paper we investigate quantitatively the model problem proposed by Schaeffer using perturbation methods and determine explicit values for the numerical constants, which are so crucial in determining the possible equilibrium configurations. Moreover, the present formulation allows us to calculate the stability properties of these equilibrium configurations. Results are obtained for a wide range of possible ratios of cylinder diameters. We also investigate the possibility of equilibrium configurations having an odd number of cells. The results obtained are compared with the observations of Benjamin.