_{1}

^{*}

Taylor vortex flow between two concentric rotating cylinders with finite axial length includes various patterns of laminar and turbulent flows, and its behavior has attracted great interests. When mode bifurcation occurs, quantitative parameters such as the volume-averaged energy change rapidly. It is important to visualize the behaviors of vortices. In this study, a three-dimensional visualization system with respect to time is devised. This system can change the viewpoint of flow visualization, and we can observe the track of a vortex from any point. The volume-averaged energy is projected to the track of the center of a vortex. The proposed system can help to investigate the relationship between the mode bifurcation process and the volume-averaged energy.

Taylor vortex flow has been studied as an important vortex flow since it was first reported by Taylor in 1923 [

Since Taylor’s study, Taylor vortex flow has been studied by many researchers, and the complexity of the flow has been clarified. Unsteady flow (e.g. Taylor vortex flow) causes unstable change in the physical quantities that characterize the flow. Pacheco et al. [

In this study, the flow structure of Taylor vortex flow is investigated numerically, where the inner cylinder is rotating, and the outer cylinder and both the upper and lower end walls are stationary. The main parameters in this study are the aspect ratio which is the ratio of the cylinder length to the gap between the cylinders, and the Reynolds number, which is estimated from the velocity of the inner cylinder. Changes in these parameters lead to the generation of various flow structures. In this study, the mode formation process and the bifurcation of Taylor vortex flow are analyzed.

The most common method of identifying vortices is to use the velocity vector (

(N4). The center figure and right-hand figure respecttively show the velocity vector and the contours of Stokes’ stream function. The contours of Stokes’ stream function can be clearly used to identify the centers of vortices. Moreover, using Stokes’ stream function it is possible to distinguish the borders of vortices.

The governing equations are the axisymmetric unsteady incompressible Navier-Stokes equation with cylindrical coordinates (r, θ, z) and the continuity equation.

We use both SOR and ILUCGS methods to solve Poisson’s equation for pressure. The stress-free boundary

condition was used for the upper end wall and the stationary (non-slip) condition is used for the lower end wall. We applied Neumann conditions based on the momentum equation for pressure. As the initial condition, all velocity components are zero. Mixed solution of water and glycerin is assumed to be the working fluid, and its dynamic viscosity is 6.0 × 10^{–6 m2}/s. For the discretization method, we apply the QUICK method for convection terms, the second-order central difference method for the other space integration, and Euler’s method for the time integration. Grids are staggered and equidistant in each direction. The number of grid points is 41 in the radial direction, and the number of grid points in the axial direction is proportionally adjusted so that it becomes approximately 42 for the aspect ratio of 1.0. In order to examine the validity of the number of grid points, we analyzed Taylor vortex flow using several types of grids under various numerical conditions, and concluded that there are no differences among the modes that are finally formed, the formation of modes up to the final mode, and the manner of decay of the vortexes.

(1)

Here, u and w are the velocity components in the radial and circumferential direction, respectively, and r denotes the coordinate in the radial direction. In

There are extreme in the contours of the stream function. The positions of which the extreme values of the stream function appear are defined as the centers of vortices. The black lines in

The modes of Taylor vortex flow are roughly divided into two: normal modes and anomalous modes. The normal and anomalous modes are defined as below and depend on the end-wall boundary condition of the cylinder and the flow direction in the vicinity of the end walls. When the cylinder end wall is stationary, the normal mode has a flow from the outer cylinder to the inner cylinder (inward flow) in the vicinity of the end wall, while the anomalous mode has a flow from the inner cylinder to the outer cylinder (outward flow). In the previous study, it was confirmed that the anomalous mode has extra vortices [

The following variables are defined: the radii of the inner and outer cylinders are r_{i} and r_{o}, respectively, and the radius ratio η ( r_{i}/r_{o}) is set to 0.667. The aspect ratio Γ is the ratio of the cylinder length, L, to the radial difference between the cylinders, D (= r_{o} – r_{i}). The angular velocity of the inner cylinder is ω, and the Reynolds number, Re, is estimated using r_{i}, ω and D. All physical parameters are made dimensionless using the characteristic length L and characteristic velocity r_{i}ω.

It is necessary to confirm that the history of each center completely corresponds. When the past and current centers of vortices are identical, it is called complete correspondence. When they are different, it is called noncorrespondence. Incomplete correspondence is defined as a flow in which the past and current vortices belong to same vortex but do not appear to be identical. The conditions used to classify the correspondence of vortices are as follows:

1) The distance between the past and current centers of a vortex is less than 5 lattices.

2) The past and current centers of a vortex are included in the same vortex.

Complete correspondence satisfies both 1) and 2), noncorrespondence does not satisfy both 1) and 2), and incomplete correspondence is defined as other than these cases.

lower ends remain, and where the four central vortices are absorbed and eventually disappear. The two remaining vortices become stable, and the normal two-cell mode develops. The figure clearly shows that the visualization system in this study successfully captures the tracks of the centers of the vortices.

In the previous analysis, we presented the calculated results in two dimensions. However, to investigate the time dependence of the mode formation process, it is necessary to present the position of each vortex with respect to time. To clarify the complex process of vortex development with respect to time, we represent the behaviors of vortices in three dimensions using the Java3D library.

system, the viewpoint of flow visualization can be changed. (

Such vortices play an important role in mode formation processes.

Three-Dimensional Display of Normal ModeThe centers of the two vortices move symmetrically with respect to the midplane in the axial direction.

When the Reynolds number is changed after a stable mode appears, a mode bifurcation occurs and the flow becomes another mode.

time of 600. After the Reynolds number begins to decrease, the pair of vortices near the midplane in the axial direction become smaller. The two center vortices weaken and disappear, and finally the flow becomes the normal two-cell mode.

During the mode bifurcation, the characteristic parameters such as the volume-averaged energy oscillate and affect the flow structure. In this study, we develop an interactive visualization system that can project the quantitative parameter of each vortex to the track line in three dimensions.

The tracks of the centers of the vortices numbered by 1 - 4 are shown in

To display the tracking results in three-dimensions, we followed the centers of the vortices using the Java3D library. Using this method, we can observe the mode formation processes in three-dimensional coordinates. Tracks showing the movement of vortices before their full development and vortices with complicated behavior that are difficult to observe by a two-dimensional representation can easily be observed. Observation of the development process from various viewpoints is an effective means of tracking the development of vortices.

In the early stage of mode formation, vortices develop on the midplane in the axial direction and at the upper and lower ends of the inner cylinder. During the mode bifurcation, vortices whose volume-averaged energies are weak disappear. After the bifurcation, the volume-averaged energies of the remaining vortices become large for a while. The volume-averaged energies then decreases and the flow becomes stable. The mean energy is closely related to the bifurcation process.

The development of Taylor vortex flow generated in finite-length rotating dual cylinders whose upper and lower ends were fixed was studied. The behavior of the vortices in three dimensions are presented by the Java3D library. This visualization system can be used to analyze the fusion and disappearance of vortices. The formation process for each final mode is nonunique, multiple and complicated. Tracks were colored to analyze the changes in the values of physical quantities. Using this method, changes in the values of physical quantities were measured in detail. During the calculation, the Reynolds number was changed to investigate the behavior of the volumeaveraged energy. When vortices disappear, the volumeaveraged energies of the disappearing vortices are lower than those of the other vortices. When mode bifurcation occurs, the volume-averaged energy becomes large for a while, then decreases gradually, and the flow becomes stable.