In this article, the multipolar vortex instability of the flow in a finite cylinder is
addressed. The experimental study uses a rotating elastic deformable tube filled
with water which is elliptically or triangularly deformed by two or three rollers.
The experimental control parameters are the cylinder aspect ratio and the Reynolds
number based on the angular frequency.
For Reynolds numbers close to threshold, different instability modes are visualized
using anisotropic particles, according to the value of the aspect ratio. These modes
are compared with those predicted by an asymptotic stability theory in the limit
of small deformations and large Reynolds numbers. A very good agreement is
obtained which confirms the instability mechanism; for both elliptic and triangular
configurations, the instability is due to the resonance of two normal modes (Kelvin
modes) of the underlying rotating flow with the deformation field. At least four
different elliptic instability modes, including combinations of Kelvin modes with
azimuthal wavenumbers m = 0 and m = 2 and Kelvin modes m = 1 and m = 3 are
visualized. Two different triangular instability modes which are a combination of
Kelvin modes m = −1 and m = 2 and a combination of Kelvin modes m = 0 and
m = 3 are also evidenced.
The nonlinear dynamics of a particular elliptic instability mode, which corresponds
to the combination of two stationary Kelvin modes m = −1 and m = 1, is examined
in more detail using particle image velocimetry (PIV). The dynamics of the phase and
amplitude of the instability mode is shown to be predicted well by the weakly nonlinear
analysis for moderate Reynolds numbers. For larger Reynolds number, a secondary
instability is observed. Below a Reynolds number threshold, the amplitude of this
instability mode saturates and its frequency is shown to agree with the predictions
of Kerswell (1999). Above this threshold, a more complex dynamic develops which is
only sustained during a finite time. Eventually, the two-dimensional stationary elliptic
flow is reestablished and the destabilization process starts again.