Wave-number selection at finite amplitude in rotating Couette flow

Abstract

Measurements have been made of the wavelength of Taylor vortices between rotating cylinders. It is shown that the relaxation time of such a vortex system is approximatelyL2/6v, whereLis the length of the vortex column andvis the kinematic viscosity. Previous measurements reported in the literature have not been steady-state measurements because of the long relaxation time. The present data are accurate to 1% and extend to 40 times the critical Taylor number. The variation of wavelength with Taylor number is linear and the slope is exceedingly small and negative. The non-uniqueness of wave-number observed by Coles (1965) in doubly periodic flows is here shown to occur in the rotationally symmetric case. It is argued that variational methods are inapplicable in determining the wave-number of finite-amplitude secondary flows. The experimental results show that the wave-number is determined uniquely by the initial conditions of the system. It is suggested that any method which neglects the time-dependent behaviour of the system cannot select the final state from the manifold of solutions which occur in non-linear problems.

A note on power-law scaling in a Taylor–Couette flow

“…He noted that the flow states depended not only on the initial conditions, but also on the manner in which the inner cylinder was accelerated to the final speed. This finding was supported by the subsequent studies of Snyder, 6 Burkhalter and Koschmieder, 7 Koschmieder, 8 Park et al, 9 and more recently by Lim et al 10 ͑see also the numerical studies of Antonijoan and Sanchez 11 and Rigopoulos et al 12 ͒.…”

“…Notable workers in this area include Wendt, 13 Taylor, 14 Donnelly, 15 Donnelly and Simon, 16 Nakabayashi et al, 17 Lathrop et al 18 and more recently Lewis and Swinney. 19 In 1933, Wendt 13 18 and Lewis and Swinney 19 using high precision torque measurements for Reynolds number of 800 ϽReϽ1.23ϫ10 6 show that there is no Reynolds number region which can be described by GϳRe ␣ where ␣ is a constant value. Their finding raises an inevitable question as to whether this is also true in the lower Reynolds number range.…”

On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder

“…Much of the existing analytical work and stability analyses is discussed by Chandra~ekhar.~ Recent studies of the Taylor problem have concentrated on the existence of multiple solutions, which have secondary states characterized by a unique wavelength.8-'6 The wavelength that is selected as the primary state depends upon the initial conditions of the flow. 16 Of these studies, those of Benjamin and Mullin8q9 have provided many experimental results revealing the nature of the Taylor cell evolution. Numerical studies by Cliffe" have verified these results and have helped to provide further insights in their meaning.…”

Instability of surfactant solution flow in a Taylor cell

“…Figure 3 illustrates the flow conditions of several test fluids in the gap. The time for the flow to become stable was ts determined using Snyder's equation (Snyder, 1969), as given by…”