Dynamical instabilities and the transition to chaotic Taylor vortex flow

Author :P. R. Fenstermacher, Harry L. Swinney, and J. P. Gollub
Publication :Journal of Fluid Mechanics
Volume :94
Pages :103-128
Year :1979

We have used the technique of laser-Doppler velocimetry to study the transition to
turbulence in a fluid contained between concentric cylinders with the inner cylinder
rotating. The experiment was designed to test recent proposals for the number and
types of dynamical regimes exhibited by a flow before it becomes turbulent. For
different Reynolds numbers the radial component of the local velocity was recorded as
a function of time in a computer, and the records were then Fourier-transformed to
obtain velocity power spectra. The first two instabilities in the flow, to time-independent
Taylor vortex flow and then to time-dependent wavy vortex flow, are well
known, but the present experiment provides the first quantitative information on the
subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex
flow the velocity spectra contain a single sharp frequency component and its harmonics;
the flow is strictly periodic. As the Reynolds number is increased, a previously
unobserved second sharp frequency component appears at RIR, = 10.1, where R, is
the critical Reynolds number for the Taylor instability. The two frequencies appear to
be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears
in the flow a t RIR, 2: 12, where a weak broadband component is observed in addition
to the sharp components; this flow can be described as weakly turbulent. As R
is increased further, the component that appeared a t RIR, = 10.1 disappears at
RIR, = 19.3, and the remaining sharp component disappears at RIR, = 21.9, leaving
a spectrum with only the broad component and a background continuum. The observance
of only two discrete frequencies and then chaotic flow is contrary to Landau’s
picture of an infinite sequence of instabilities, each adding a new frequency to the
motion. However, recent studies of nonlinear models with a few degrees of freedom
show a behaviour similar in most respects to that observed.