Onset of centrifugal instability at a rotating cylinder in a stratified fluid

In this paper, we analyse the characteristics of the elliptic instability in a finite cylinder in the presence of both background rotation and axial stratification. A general formula for the linear growth rate of the stationary sinuous modes is derived including viscous and detuning effects in the limit of small eccentricity. This formula is discussed and compared to experimental results which are obtained in a cylinder filled with salted water for two different eccentricities by varying the stratification, the background rotation and the cylinder rotation. A good agreement with the theory concerning the domain of instability of the sinuous modes is demonstrated. Other elliptic instability modes, oscillating at the cylinder angular frequency are also evidenced together with a new type of instability mode, which could be connected to a centrifugal instability occurring during the experimental phase of spin-up. The nonlinear regime of the elliptic instability is also documented. In contrast with the homogeneous case, no cycle involving growth, breakdown and re-laminarization is observed in the presence of strong stratification. The elliptic instability in a stratified fluid seems to yield either a persistent turbulent state or a weakly nonlinear regime.

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Three-dimensionality in the wake of a rapidly rotating cylinder in uniform flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.362 ◽

2013 ◽

Vol 730 ◽

pp. 379-391 ◽

Cited By ~ 29

Author(s):

A. Rao ◽

J. S. Leontini ◽

M. C. Thompson ◽

K. Hourigan

Keyword(s):

Steady State ◽

Rotation Rate ◽

Free Stream ◽

Three Dimensional ◽

Rotating Cylinder ◽

Marginal Stability ◽

Cylinder Surface ◽

Rotation Rates ◽

Centrifugal Instability ◽

The Stability

AbstractThe flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

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Inviscid waves on a Lamb–Oseen vortex in a rotating stratified fluid: consequences for the elliptic instability

Journal of Fluid Mechanics ◽

10.1017/s0022112007009780 ◽

2008 ◽

Vol 597 ◽

pp. 283-303 ◽

Cited By ~ 16

Author(s):

STÉPHANE LE DIZÈS

Keyword(s):

Plane Waves ◽

Stratified Fluid ◽

Lagrangian Description ◽

Small Range ◽

Local Approach ◽

Coriolis Parameter ◽

Centrifugal Instability ◽

Local Plane ◽

Core Modes ◽

Oseen Vortex

The inviscid waves propagating on a Lamb–Oseen vortex in a rotating medium for an unstratified fluid and for a strongly stratified fluid are analysed using numerical and asymptotic approaches. By a local Lagrangian description, we first provide the characteristics of the local plane waves (inertia–gravity waves) as well as the local growth rate associated with the centrifugal instability when the vortex is unstable. A global WKBJ approach is then used to determine the frequencies of neutral core modes and neutral ring modes. We show that these global Kelvin modes only exist in restricted domains of the parameters. The consequences of these domain limitations for the occurrence of the elliptic instability are discussed. We argue that in an unstratified fluid the elliptic instability should be active in a small range of the Coriolis parameter which could not have been predicted from a local approach. The wavenumbers of the sinuous modes of the elliptic instability are provided as a function of the Coriolis parameter for both an unstratified fluid and a strongly stratified fluid.

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