Explicit analytic formulas for Newtonian Taylor-Couette primary instabilities

In this study, existing primary stability boundary data for flow between concentric cylinders, for the broad range of radius and rotation ratios examined, were found to be self-similar in a properly chosen parameter space. The experimental results for the primary transitions to both Taylor vortex flow and spiral vortex flow collapsed onto a single curve using a combination of variables technique, for both counter-rotating and co-rotating cylinders. The curves were then empirically fit, yielding explicit analytic formulas for the critical Reynolds number for any radius ratio (η) and rotation ratio (μ). For counter-rotating flows, the primary critical Reynolds number is determined by a single variable: the ratio of the nodal gap fraction to a known function of the radius ratio. The existence and influence of a nodal surface is shown experimentally for μ≅-1.7. For co-rotating flows, the important scaled variable was found to be the radius ratio divided by the nodal radius ratio. Comparisons of the resulting explicit stability formulas were made to existing analytic stability expressions and experimental data. Excellent quantitative agreement was found with data across the entire parameter space.