Anomalous dispersion of Couette-Taylor spirals

Competition of spirals in Couette – Taylor system between counter rotating cylinders leads, for small supercriticality to the formation of localized source. Measuring the group velocity as function of the amplitude, we have determined that waves forming such source have anomalous dispersion, in the sense that phase and linear group velocity of each have opposite signs. Our Couette Taylor system has following characteristics: the inner cylinder with the radius 459 . 4 = a cm was made of black Delrin, the outer cylinder was made of transparent Plexiglas and its internal radius was 050 . 5 = b so the gap between cylinders is 591 . 0 = − = a b d cm and the working length of the system is 5 . 27 = L cm. So, the experiment was performed in a spatially extended system with an aspect ratio Γ = L/d = 46.5, and radius ratio 883 . 0 / = = b a η . Cylinders were rotated independently by two DC motors. Thus in this Couette Taylor system the flow control parameters are Reynolds numbers defined for the inner and outer cylinder respectively : ν / bd R o o Ω = and ν / ad R i i Ω = where i o Ω Ω , are the angular frequencies of outer and inner cylinders respectively, v is for kinematic viscosity of fluid. We have used distilled water ( s cm v / 10 2 2 − = at C T 0 21 = ) with 2% volume of flakes of Kalliroscope AQ1000 for the visualization of flow structures. To obtain information on space – time behavior of flow, we have used a linear 1024– pixel charge coupled device (CCD) array oriented along the cylinders axis to record instantaneous intensity distribution ) (X I of light reflected by Kalliroscope. The recorded length was from 20 to 25 cm in the central part of the system, corresponding to the a spatial resolution from 50 to 40 pixel/cm. The intensity is sampled in 256 values, displayed in gray level time interval along time axis to produce space – time diagrams ) , ( T X I of the pattern. To extract the spatial and temporal properties we have performed the complex demodulation technique of the signal [1] which allows in particulary, to separate the right traveling and left traveling waves and to represent them as )] ( exp[ ) , ( t x k i T X A A A ω − and )] ( [ exp ) , ( t x k i T X B B B ω − . The phase and group velocity of waves are calculated from the relation vph = ω/k and vg = dω/dκ. We may choose the length scale as d, the time scale as τ = d/ν, leading to dimensionless wavenumber and frequency q = kd and Ω = ωG/ν. The velocities are scaled by the characteristic diffusion velocity ν/d. We investigated regimes of spirals which we have observed as a first supercritical instability mode from the base flow in a case of counter-rotating cylinders [2] for 500 180 < < i R and 155 1155 − < < − o R . A typical space-time diagram shown in the Figure 1 exhibits very clearly the position of phase fronts of wave disturbances corresponding to right and left traveling spirals.