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In concentrically rotating double cylinders consisting of a stationary outer cylinder a rotating inner cylinder, Taylor vortex flow appears. Taylor vortex flow occurs in journal bearings, various fluid machineries, containers for chemical reaction, and other rotating components. Therefore, the analysis of the flow structure of Taylor vortex flow is highly effective for its control. The main parameters that determine the modes of Taylor vortex flow of a finite length are the aspect ratio, Reynolds number Re. Aspect ratio is defined as the ratio of the cylinder length to the gap length between cylinders, and
*Re* is determined on the basis of the angular speed of the inner cylinder. Aspect ratio was set to be 4.0, and Re to be values in the range from 100 to 1000 at intervals of 100. Thus far, a large number of studies on Taylor vortex flow have been carried out; however, the effects of the differences in initial conditions have not yet been sufficiently clarified. In this study, we changed the acceleration time of the inner cylinder in a numerical analysis, and examined the resulting changes in the mode formation and bifurcation processes. The acceleration time was changed from 1.0 s to 10.0 s. As a result, a difference was observed in the final mode depending on the difference in the acceleration time. From this finding, non-uniqueness, which is a major characteristic of Taylor vortex flow, was confirmed. However, no regularities regarding the difference in mode formation were found and the tendency of the mode formation process was not specified. Moreover, the processes of developing the vortex resulting in different final modes were monitored over time by visual observation. Similar flow behaviors were initially observed after the start of the calculation. Then, a bifurcation point, at which the flow changed to a mode depending on the acceleration time observed, and finally the flow became steady. In addition, there was also a difference in the time taken for the well-developed flow to reach the steady state. Both EFD (Experimental Fluid Dynamics) and CFD (Computational Fluid Dynamics) results show good agreement qualitatively.

Since the classical research by G. I. Taylor in 1923 [

Let us briefly explain the Taylor vortex. When fluid flows in a narrow space between inner and outer cylinders of two coaxial rotating cylinders sharing the central axis, the fluid flow in the circumferential direction of the cylinder with the rotation of one of the cylinders or both cylinders. Thereby, if the centrifugal force working on the fluid and surrounding pressure are imbalanced, the vortex rotating alternatively in the opposite direction having an axis in the direction of flow is formed. This vortex is called “Taylor vortex”.

When the main velocity of the inner cylinder is gradually increased from zero, a Couette flow first appears in the space between two rotating cylinders. If the velocity is further increased, the flow changes to Taylor vortex flow in which a torus-like flow called a cell overlaps, wave Taylor vortex flow, and turbulent flow several times. Since the Taylor vortex flow is observed in journal bearings, various fluid machines, and chemical reaction vessels, it is technologically important to elucidate its mechanism. Especially, the prediction and control of the Taylor vortex flow is important because the changes in its mode induce the unsteady changes in torque and reaction rate [

It is known that one of the characteristics of the Taylor vortex is its non-uniqueness. Namely, even at the same aspect ratio and same Reynolds number, the final mode, which is the finally stabilized mode of flow, is sometimes different due to the changes in the acceleration time of the inner cylinder. Benjamin and Mullin [

However, the previous studies have been limited to the elucidation of the non-uniqueness of the Taylor vortex only by experiments or only by numerical calculations. There has been no research that compares the experimental results and numerical calculations. The objective of the present study is to elucidate more deeply the non-uniqueness of the Taylor vortex by comparing experimental results and calculations.

_{in} = 20 mm and r_{out} = 30 mm, respectively, and the gap between the cylinders is r_{out} − r_{in} = 10 mm. The inner cylinder is rotated at an angular velocity of ω. A pulley is attached to a belt wheel below the center of the cylinder to rotate the inner cylinder. Here, only the inner cylinder is rotated because the outer cylinder is fixed. A slide ring is attached to the upper ends of the cylinders to fix their positions and to change the aspect ratio of the gap, which is used as the observation target. The setup also includes a servo motor, a servo amplifier (MR-J2-A, Mitsubishi Electric Corporation), a servo motor control unit (Tateiwa Shoji), and aNd-YAG laser device (1.5 W, DSPP Green Laser, Japan Laser Corporation).

A mixture of distilled water and glycerin with a mixing ratio of 8:2 is used as the fluid specimen in this study. The specific gravity of glycerin is 1.26. The advantages of using glycerin are that 1) the Reynolds number can be controlled by varying its viscosity and 2) it helps tracer particles float in the fluid specimen. The tracer particles used in this study are fine fluorescent particles with a high luminescence, which were added to the fluid to visualize flows. FLUOSTAR (EBM Corporation) particles are used in this study. When laser light is irradiated onto these particles perpendicularly to the observation direction, flows can be observed owing to the reflection of light.

The aspect ratio was determined, and the rotation speed of the servo motor required to give the desired Reynolds number and the time required to reach this value (i.e., the rising time) were input to the servo amplifier. Next, the motor was driven and photographs were taken using a high-speed video camera (VCC-H1000, Digimo Co., Ltd.) when the vortex stabilized to form the final-mode flow. The parameters that govern Taylor vortex flow are the Reynolds number and aspect ratio, as mentioned above. 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

R e = V D ν (1)

here, V is a representative velocity, given as the rotation speed of the inner cylinder (r_{in} 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. The aspect ratio Γ is defined as the ratio of the cylinder length L to the gap width r_{out} − r_{in}, and is given by

Γ = L / D (2)

Various final modes of Taylor vortex flow are formed by changing these parameters. In this study, we define a flow in which the relative time change in the torque applied to the cylinder wall is less than 10^{−6} as a steady flow. The Stokes flow function (Equation (3)) is used to visualize the flow.

u = − 1 r ∂ ψ ∂ z w = 1 r ∂ ψ ∂ r (3)

The governing equations are the axisymmetric unsteady incompressible Navier-Stokes equation with cylindrical coordinates (r, θ, z) (Equation (4)) and the equation of continuity (Equation (5)).

∂ u ∂ t + ( u ⋅ ∇ ) u = − ∇ p + 1 R e ∇ 2 u (4)

∇ ⋅ u = 0 (5)

here, u is a velocity vector with the components of (u, v, w), t is a dimensionless time based on a representative time, and p is the pressure. The discretization of the governing equations is based on the MAC method. For time integration, Euler’s explicit method is used. For space integration, the QUICK method is used for the convection terms, and the secondary central difference method is used for other terms. As an initial condition, the velocity is set to 0 in all regions. As boundary conditions, the no-slip condition at each cylinder wall is applied for the velocity, and the Neumann condition based on the equation of motion is applied for the pressure. Staggered grids are used as calculation grids and are assumed to have regular intervals in each direction.

In the present study, we observed the vortex flow that reached steady state by changing the acceleration time and Reynolds number under four conditions, i.e., the free end and fixed end at Γ = 4 and Γ = 5. Generally, it is known that an odd number of cells are produced when the boundary conditions is a free end with Non-Newtonian fluid. In the case of the fixed end, we observed that both even number and odd number of cells were produced.

The Taylor vortexes that occurred are illustrated in easy-to-understand manner in

In the case of a free end, an odd number of cells in which the flow rotates clockwise are normal cells at both lower and upper ends. In the other cases, all the cells become anomalous cells. The kinds of the cells obtained in the present analysis are shown in

In the present study, the acceleration time of the inner cylinder was changed. In this experiments, we have analyzed how the Taylor vortex flow, produced between two coaxial rotating cylinders sharing the central axis, changes with the aspect ratio, the Reynolds number and the acceleration time for the both free and fixed ends. Non-uniqueness of Taylor vortex flow is investigated both experimentally and numerically. Both results show good agreement qualitatively.

We thank Prof. Takashi Watanabe, Nagoya University who provided insight and expertise that greatly assisted the research. We would also like to show our gratitude to members in our laboratory for sharing their pearls of wisdom with us during the course of this research, and we thank “anonymous” reviewers for helpful comments that greatly improved the manuscript.

The authors declare no conflicts of interest regarding the publication of this paper.

Furukawa, H. and Suzuki, T. (2018) Study on Non-Uniqueness of Taylor Vortex Flow Changing Inner Cylinder Acceleration Time. World Journal of Mechanics, 8, 301-310. https://doi.org/10.4236/wjm.2018.88023