Experiments on Wave Number Selection in Rotating Couette-Taylor Flow

Abstract

Experimental results are reported for the selection of axial wavelengths of Taylor vortices in a system where the Reynolds number, R 9 is "ramped" spatially from above to below the critical value, R c , for the onset of vortex flow. It is found that a sufficiently slow ramp connecting the supercritical region (containing vortices) to a subcritical region (of purely azimuthal flow) results in the selection of a unique wavelength. A more rapid ramp results in a small R-dependent band of allowed wavelengths which grows in width as R-^R C from above.PACS numbers: 47.20. + m, 05.70.LnIn nonlinear dissipative systems subjected to an external stress, R, a transition frequently occurs to a state of reduced symmetry having a spatial structure with a characteristic wavelength. Examples are Rayleigh-B6nard convection, 1 Taylor vortex flow, 2 " 4 certain chemical reactions, 5 flame-front propagation, 6 and crystal growth. 7 The equations governing systems of infinite spatial extent usually possess a continuum of linearly stable solutions , 8 corresponding to a band of wave numbers having a width which varies as e 1/2 (e = R/R C -1, and R c is the value of the stress parameter at the transition). Imposition of nonperiodic boundary conditions may reduce the width of the allowed band to order e, 9 * 10 but for the investigated cases a finite band remains. Such finite systems generally select discrete states from the band of stable solutions, with the selected wave numbers dependent upon the particular boundary conditions imposed.For Taylor vortex flow (TVF) between concentric rotating cylinders, the multiplicity of stable states was first explored in detail by Coles 3 and Snyder. 11 They observed TVF to be stable for a range of axial wavelengths when the Reynolds number, R, was above R c , the value for vortex formation, but below the value where azimuthal waves appear. The observable range of wavelengths 11 * 12 is roughly equal to that expected theoretically, 13 but the factors determining the wavelength selected in a particular experiment are not understood. It has been suggested quite recently, 14 » 15 however, that a gradual axial variation of e from positive to negative values (a spatial ramp) could permit a continuous wavelength adjustment mechanism to become effective, which might result in the selection of a unique although possibly e -dependent state. The question of the existence and nature of such selection processes is largely unexplored. One might expect that their nature would have a strong influence on the development of temporally nonperiodic (or turbulent) behavior in large real systems. 16 More specifically, we would expect a "weak" selection mechanism to render the primary modes of the system relatively sensitive to the influence of noise and other degrees of freedom of the system.To investigate wavelength selection experimentally, we used apparatus in which the gap between the cylinders tapered linearly from supercritical to subcritical values as shown in Fig. 1. The working fluid was a 30% solution of glyc...

Circulation in a Short Cylindrical Couette System

“…One idea for reducing the effect of endcaps is to use a tapered section [24]. Another idea is to divide the cap into two parts: the inner one fixed to the inner cylinder, and the outer one fixed to the outer cylinder.…”

Boundary limitation of wave numbers in Taylor-vortex flow

“…One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii. [4,7,9,10,14] Theoretically, phase pinning (and thus the Eckhaus instability) at small ǫ is expected when the boundary conditions at the system ends z = 0, L correspond to a large amplitude A(0) = A(L) = A 0 of the velocity field, say A 0 = O(1), while in the system interior the amplitude A(z) is small, say O(ǫ 1/2 ). This situation closely corresponds to the TVF system with rigid ends where the influence of the Ekman vortex can be approximated by this boundary condition.…”

“…The (more narrow) stable band of states is limited by the long-wavelength Eckhaus instability. [3,4,5,6,7,8,9,10,11,12,13,14] The Eckhaus instability is a bulk instability which manifests itself in the system interior where one pattern wavelength (one pair of Taylor vortices) is either gained or lost, depending on whether k is larger or smaller than the stable band limited by k E (ǫ). One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii.…”

Over two decades of pattern formation, a personal perspective

“…5. Figure 6 shows again the Eckhaus boundary, and in addition as plusses the data obtained by Marco [99,100] for the state selected by a ramp in the outer-cylinder radius. ε Although the investigation of wavenumber selection by ramps was at least in part motivated initially by a hope of finding something equivalent to an extremum principle for these nonequilibrium systems, it soon became clear from the theoretical work [98,106,107] that the unique wavenumber had a different origin.…”

“…The experiments soon revealed a unique wavenumber when the outer cylinder of the TVF apparatus had a gentle ramp in its diameter. [99] Here I might say that it is a great tribute to our machine shop and to the skills of Rudy Stuber (the head of our shop) that the required precise geometries [100] could be produced. After the initial excitement, the issue of stability ranges and selection processes became a serious program for us.…”