The Aspect Ratio Dependence of the Attractor Dimension in Taylor-Couette Flow.

An investigation of the aspect ratio dependence of the attractor dimension in flow between concentric cylinders (Taylor-Couette flow) was performed with the inner cylinder rotating and the outer cylinder at rest. An increase in attractor dimension with aspect ratio might be expected since more modes should be required to describe a larger system. Phase space attractors were reconstructed from experimental velocity data by time delay methods and the attractor dimension calculated by both the Grassberger -Procaccia and Badii-Politi algorithms. The aspect ratio, the ratio of the height of the flow to the gap between the cylinders, was varied between 19.9 and 34.48 and the inner cylinder Reynolds numbers ranged between R/R _{rm c} = 11 and R/R _{rm c} = 15, where R _{rm c} is the critical Reynolds number for the primary instability. The variation of the dimension with Reynolds number was similar to that reported by Dr. A. Brandstater in an earlier study. No aspect ratio dependence of the attractor dimension was found in this study in the parameter space explored. A second area explored was the bifurcation from Couette flow to secondary flow for counter-rotating cylinders. The bifurcation boundaries were located to an accuracy of 0.3% for outer cylinder Reynolds numbers ranging from 0 to -250. The locations of the codimension -2 points and the wavespeeds and axial wavelengths measured at the transition boundaries compared well with a recent linear stability analysis of M. Golubitsky and W. Langford. No hysteresis was found in the primary bifurcation boundary, but hysteresis of at least 3% of the outer cylinder Reynolds number was found in the secondary bifurcation boundary at the first codimension-2 point. No new states arising from mode competition were observed around the codimension -2 points.