Abstract
We examine the critical viscous mode of the Taylor–Couette strato-rotational instability, concentrating on cases where the buoyancy frequency and the inner cylinder rotation rate are comparable, giving a detailed account for . The ratio of the outer to the inner cylinder rotation rates and the ratio of the inner to the outer cylinder radius satisfy and . We find considerable variation in the structure of the mode, and the critical Reynolds number at which the flow becomes unstable. For , we classify different regions of the -plane by the critical viscous mode of each region. We find that there is a triple point in the -plane where three different viscous modes all onset at the same Reynolds number. We also find a discontinuous change in along a curve in the -plane, on one side of which exist closed unstable domains where the flow can restabilise when the Reynolds number is increased. A new form of viscous instability occurring for wide gaps has been detected. We show for the first time that there is a region of the parameter space for which the critical viscous mode at the onset of instability corresponds to the inviscid radiative instability of Le Dizes & Riedinger (J. Fluid Mech., vol. 660, 2010, pp. 147–161). Focusing on small-to-moderate wavenumbers, we demonstrate that the viscous and inviscid systems are not always correlated. We explore which viscous modes relate to inviscid modes and which do not. For asymptotically large vertical wavenumbers, we have extended the inviscid analysis of Park & Billant (J. Fluid Mech., vol. 725, 2013, pp. 262–280) to cover the cases where and are comparable.
Highlights
We consider the linear instabilities of the vertically stratified Boussinesq Taylor– Couette system, unbounded in the vertical direction, under gravity g and with angular velocity Ω(r) at radius r
We extend the Wentzel–Kramers–Brillouin– Jeffreys (WKBJ) inviscid analysis performed by Park & Billant (2013) to derive the corresponding (η, μ)-stability curve for any given Fr
centrifugal instability (CI) are stabilised by the presence of stratification, increasing the critical Reynolds number compared to unstratified flow
Summary
We consider the linear instabilities of the vertically stratified Boussinesq Taylor– Couette system, unbounded in the vertical direction, under gravity g and with angular velocity Ω(r) at radius r. The centrifugal approximation, implicit in the foregoing work, was discussed explicitly by Shalybkov & Rüdiger (2005) who extended the search for viscous instabilities beyond narrow gaps They considered a range of rotation ratios for both η = 0.78 and η = 0.3 in the viscous problem, with the Froude number fixed at Fr = 0.5. Rüdiger et al (2017) performed numerical simulations of Taylor–Couette flows for η = 0.52, and at finite Re 3000 found instabilities only for 0.3 < Fr < 5.5 They discussed the dependence of critical wavenumbers on Fr and Re. Laboratory experimentation with a heat stratification was used to check their numerical results, with which they saw a good correlation.
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