Stability of Taylor-Dean flow in a small gap between rotating cylinders
Resource
Journal of Fluid Mechanics 243: 443-455
Journal
Journal of Fluid Mechanics
Pages
443-455
Date Issued
1992
Date
1992
Author(s)
Chang, M. H.
Abstract
A linear stability analysis has been implemented for Taylor—Dean flow, a viscous flow between rotating concentric cylinders with a pressure gradient acting in the azimuthal direction. The analysis is made under the assumption that the gap spacing between the cylinders is small compared to the mean radius (small-gap approximation). A parametric study covering wide ranges of µ, the ratio of angular velocity of the outer cylinder to that of inner cylinder, and ß, a parameter characterizing the ratio of representative pumping and rotation velocities is conducted. For —1 ≤ µ < 1, results show that non–axisymmetric instability modes prevail in a wide range of ß. The most stable state is found to occur within — 3.9 < ß —3.6 for µ < 0.3 and at —ß≈ 1.59µ + 3.5 for µ > 0.3. The most stable state is always accompanied by a shortest critical axial wavelength. Instability modes with different azimuthal wavenumber have similar stability characteristics because the basic state is either close to or at the most stable situation. This similarity is absent from either Taylor or Dean flow. We have conducted a complete analysis for the onset of secondary motion of Taylor—Dean flow in a small-gap spacing between two infinitely long rotating cylinders. We first show that previous investigations for the stability of small-gap Taylor—Dean flow were incomplete owing to the lack of consideration of the existence of non-axisymmetric oscillatory modes which in reality prevail in wide ranges of both μ and β. In particular, the peculiar neutral curves of m = 0 for near β* = —3.667 presented by Hughes & Reid (1964) are replaced by the neutral curve of m = 5, which is unimodal and most unstable. We then identify the instability mode for each (/£,/?) pair in the (μ,β)-plane which covers — 1 ≤ μ < 1 and —10 ≤ β ≤ 10. The range of β in which the non-axisymmetric modes predominate decreases with increasing /a. For a fixed μ, the mode with higher m is of greater stability and smaller axial wavelength; the most stable state with smallest axial wavelength occurs at a β at which the travelling wave changes its direction. When the instability mode is changing into the other mode, the neutral curve in terms of T and a consists of two connected branches, each of which accounts for the neutral curve of different m. For β < βmax, modes with different azimuthal wavenumber m are of similar stability characteristics in terms of Tc, ac, c, and associated eigenfunctions of disturbance velocities. © 1992, Cambridge University Press. All rights reserved.
Other Subjects
Bodies-Cylinders; Flow Stability; Rotating; Taylor-Dean Flow; Linear stability analysis; Azimuthal direction; Concentric cylinders; Inner cylinder; Instability modes; Parameter characterizing; Parametric study; Rotating cylinders; Rotation velocity; Cylinders (shapes)
Type
journal article
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