Experimental and numerical investigation of the competition between axisymmetric time-periodic modes in an enclosed swirling flow

Abstract

Time-periodic vortex flows in an enclosed circular cylinder driven by the rotation of one endwall are investigated experimentally and numerically. This work is motivated partly by the linear stability analysis of Gelfgat et al. ͓J. Fluid Mech. 438, 363 ͑2001͔͒, which showed the existence of an axisymmetric double Hopf bifurcation, and the purpose of the experiment is to see if the nonlinear dynamics associated with this double Hopf bifurcation can be captured under laboratory conditions. A glycerin/water mixture was used in a cylinder with variable height-to-radius ratios between ⌫ = 1.67 and 1.81, and Reynolds numbers between Re = 2600 and 2800 ͑i.e., in the neighborhood of the double Hopf͒. Hot-film measurements provide, for the first time, experimental evidence of the existence of an axisymmetric double Hopf bifurcation, involving the competition between two stable coexisting axisymmetric limit cycles with periods ͑nondimensionalized by the rotation rate of the endwall͒ of approximately 31 and 22. The dynamics is also captured in our nonlinear computations, which clearly identify the double Hopf bifurcation as "type I simple," with the characteristic signatures that the two Hopf bifurcations are supercritical and that there is a wedge-shaped region in ͑⌫ ,Re͒ parameter space where both limit cycles are stable, delimited by Neimark-Sacker bifurcation curves.

Quenching of vortex breakdown oscillations via harmonic modulation

“…The current experiments were performed using the same apparatus as in Lopez et al (2006), but with the experimental procedures modified for this particular study. The apparatus is shown schematically in figure 1.…”

“…It is inverted from the actual experimental setup for ease of comparison with the experimental results of others which have the rotating disk at the bottom of the cylinder (flow visualization photos are also inverted). A detailed description of the apparatus can be found in Lopez et al (2006), and only the essential features are described here.…”

“…The investigation is both numerical and experimental. For the unforced flow, it is well known that for cylinders of height-to-radius aspect ratio between about 1.6 and 2.8, the onset of unsteadiness as the rate of rotation of the endwall (measured non-dimensionally by the Reynolds number) increases is via a supercritical axisymmetric Hopf bifurcation (Gelfgat, Bar-Yoseph & Solan 2001), and that the resultant time-periodic axisymmetric flow is stable to three-dimensional perturbations for a considerable range of Reynolds numbers beyond onset (Blackburn & Lopez 2000Blackburn 2002;Lopez, Cui & Lim 2006;Lopez 2006). Here, we report in detail on the response to variations in the forcing amplitude and forcing frequency for a time-periodic axisymmetric state in a cylinder of aspect ratio 2.5 at a Reynolds number of 2800, which is characterized by a large double vortex breakdown bubble undergoing largeamplitude pulsations along the axis.…”

Harmonically forced enclosed swirling flow

“…It is inverted from the actual experimental setup for ease of comparison with the experimental and numerical results of others which have the rotating disk at the bottom of the cylinder ͑flow visualization photos are also inverted͒. A detailed description of the apparatus has been previously presented 7,29 and only the essential features are described here. The apparatus consists of a Plexiglas cylinder, with a matching rotating disk at the bottom and a stationary disk at the top of the cylinder.…”

“…Nevertheless, calibration is not an issue for concern when measuring the temporal frequencies in a flow, and this is confirmed by the previous experimental results and detailed comparisons with numerical simulation. 7,29 Given that the frequencies of interest in the present investigation are below 1 Hz, the output signal of the hot film from the constant temperature anemometer ͑CTA, Dantec 55 M01͒ was conditioned by a low-pass filter with a cutoff frequency of 10…”