Numerical Analysis of Critical Reynolds Number of Wavy Taylor Vortex Flow with Changing Aspect Ratio

Taylor vortex flow is one of the important vortex flows that have been studied since its classic study made by G. I. Taylor in 1923. State of the flow between inner and outer cylinders of a rotating co-axial cylinder transits from Couette flow to Taylor vortex flow and to wavy Taylor vortex flow as the increase of Reynolds number. This study has identified the critical Reynolds number when the flow changes from Taylor vortex flow to wavy Taylor vortex flow. The numerical analysis making use of the attractor in the chaos theory has been used in this identification of the critical Reynolds number. The calculated critical Reynolds numbers of each flow mode are almost identical to the values obtained by the visualization experiment at small aspect ratios. In the region where the aspect ratio is larger than the ratio at the peak critical Reynolds number, the distribution of the Reynolds number is qualitatively similar between the calculated and experimental values.

vortex flow) causes an unstable change in the physical quantities that characterize the flow. Pacheco et al. [7] showed experimentally that in small aspect-ratio, Taylor-Couette flows have a band in the parameter space where rotating waves become steady non-axisymmetric solutions via infinite-period bifurcations. Martin et al. [8] showed that imposing axial flow in the annulus and radial flow through the cylindrical walls in a Taylor Couette system alters the stability of the flow. To analyze these unsteady flows, authors focused on quantitative values such as mean energy [9]. The kinetic energy and enstrophy for flows with different final modes are compared.
The critical Reynolds number when the flow state transits from the Taylor vortex flow to wavy Taylor vortex flow has been found by a visualization experiment of the flow when the aspect ratio is small. In the numerical analysis of the transition to the wavy Taylor vortex flow, the critical value, at which the wavy Taylor vortexes are generated by the changes of the aspect ratio, has been identified by obtaining changes in the kinetic energy due to axial velocity in every cell.
However, the analysis has not yet extended until the changes in the kinetic energy converge. It should be noted that comparison between result of the experiment and of the numerical analysis is insufficient. In this study, the Taylor vortex flow in the normal mode, when the cylinder ends are fixed, is numerically analyzed based on chaos theory methodology. The purpose of the numerical analysis is to identify the critical Reynolds number when the flow state transits from the Taylor vortex flow to wavy Taylor vortex flow at various aspect ratios and at various number of cells.

Numerical Method
The Reynolds number Re is a dimensionless number, defined as the ratio of the inertial force in the equation of motion to the viscous force, and is given by Here, V is a representative velocity, given as the rotation speed of the inner cylinder ( in r ω ), D is the width of the gap between the inner and outer cylinders, given by the difference between their representative radii as r out −r in , and ν is the kinematic viscosity of the fluid. r out = 30 cm and r in , = 20 cm, and the radius ratio is 0.666. The physical parameters are made dimensionless by using the gap as the representative length and the velocity of the inner cylinder as the representative velocity. The aspect ratio Γ is defined as the ratio of the cylinder length L to the gap width D and is given by

Mode Analysis Method
In the preceding studies on the numerical analysis, the changes in kinetic energy In this study, axial velocity changes in the Taylor vortex flow is expressed with an attractor in the chaos theory by taking the above into considerations. The mode of finally stabled flow and convergence of energy changes are decided respectively based on the changes and convergence of attractor orbit.

Mode Identification
The flow state of the Taylor

Creation of Attractor
The attractor means an "orbit on which the physical quantity in the phase space converges", and when the attractor shows signs of remaining the constant orbit,  When the Reynolds number exceeds Re2, the flow becomes to have one additional frequency and the attractor takes the form of T3 torus. When the Reynolds number exceeds Rec, the flow becomes turbulent, and the attractor becomes the chaotic attractor. In this state, the attractor has a wide spectrum.
In this study, the functions, W (w (t), w (t + τ), and w (t + 2τ)) that can draw the attractor in the three-dimensional coordinate system are embed as shown in Figure 3 by incorporating the delay time (τ) in the axial velocity changes (w (t)).
The attractor based on the axial velocity changes is drawn as shown in Figure 4 by plotting the function (W ) on the phase space of the three-dimensional coordinate system. In drawing the attractor, the delay time (τ) is set to be "τ = f/4" which is equal to one cycle (f ) of the autocorrelation function of the axial velocity changes (w (t)) as shown in Figure 5.

Mode Determination with Attractor
The mode that can finally and stably exist under the condition(s) is decided based on the created attractor. In the case where the vortex mode is the Taylor

Determination of Critical Reynolds Number
It is assumed that the accuracy of the Reynolds number is +/−10 at a certain aspect ratio (Γ), and the critical Reynolds number is defined to be the number when the state of vortexes becomes completely wavy (i.e., the wavy Taylor Figure 9 indicates the changes of the attractor at Re = 1100 for the number of time steps from 7 million to 10 million. It is shown that the attractor draws the same orbit and that it converges into the limit cycle. Therefore, the flow state is decided to be wavy (i.e., the wavy Taylor vortex flow).

Result of Numerical Analysis
Based on the above observation, the critical Reynolds number (Rec) at the aspect ratio of 7.3 is determined to be 1100 (i.e., Rec = 1100).     Figure 12 indicates the attractor at Re = 1040 for the number of time steps from 7 million to 10 million. The attractor orbit keeps diminishing and its change keeps decreasing as the time steps increase. That is, the flow state is decided to be the Taylor vortex flow because the flow seldom changes with time. Figure 13 indicates the attractor at Re = 1060 for the number of time steps from 7 million to 10 million. The flow state is decided to be wavy (i.e., the wavy Taylor    values. But, the experimental stability limit of the Taylor vortex flow is lower than the calculated stability limit. The difference between the experimental and calculated stability limit is significant at around the peak of the critical Reynolds number, and the aspect ratio at which the calculated value hits its peak is larger than the aspect ratio at which the experimental value hits its peak. It is also shown that the difference between the calculated and experimental values increases as the number of cells increases. This difference will be attributable to the Reynolds number in order to identify the factor(s) contributing these differences.

Conclusions
This study has identified the critical Reynolds number when the state of the flow whirling between the inner and outer cylinders of the rotating dual cylinder transits from the Taylor vortex flow to wavy Taylor vortex flow. The numerical analysis making use of the attractor in the chaos theory has been used in this identification of the critical Reynolds number.
The calculated critical Reynolds numbers at various number of cells are almost identical to the values obtained by the visualization experiment in the region between the Reynolds number at small aspect ratio and the vicinity of the peak of Reynolds number. In the region where the aspect ratio is larger than the ratio at the peak critical Reynolds number, the distribution of the Reynolds number is qualitatively similar between the calculated and experimental values. But, the experimental stability limit of the Taylor vortex flow is lower than the calculated stability limit. The difference between the experimental and calculated stability limit is significant at around the peak of the critical Reynolds number, and the aspect ratio at which the calculated value hits its peak is larger than the aspect ratio at which the experimental value hits its peak. This difference is assumed to be attributable to the pattern of changes in cell boundary plane of the wavy Taylor vortex flow at the aspect ratio after the peak and to the relaxation time of the flow. It is necessary to review the experiment method and conditions of the numerical analysis in order to verify this assumption.