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We have numerically and experimentally investigated the flow modes of Taylor-Couette system consisting of coaxial two cylinders with vertical axes. The inner cylinder rotates and the outer cylinder and the bottom end of the cylinders remain stationary. The upper top boundary is the free surface of the working liquid between the inner and outer cylinders and it contacts with the air. While this flow appears in fluid machinery and chemical reactors and includes industrial interests, it also contains problems of fluid mechanics, which is about the behavior of the free surface in the rotating field. In this paper, we concretely show the developments of the one cell mode flow and the three cell mode flow at a small aspect ratio. We also represent the bifurcation diagram of the flow at the moderate aspect ratio about 5.5. In the numerical simulation, the flow is rest in the initial state, and the inner cylinder is linearly or suddenly accelerated to attain a flow with a prescribed Reynolds number. When the acceleration of the inner cylinder is high, an imperfect bifurcation occurs and the flows of the secondary modes emerge. At high Reynolds numbers, the flow first has many vortices and then some of the vortices collapse and the final stable flow arises. The loci of the normal five cell mode, the anomalous six cell mode and the secondary seven cell mode are determined.

Taylor-Couette system has been investigated as a typical model of bioreactors, filtration devices and fluid machinery. In addition, this system presents a basic problem about rotating flows and its dynamical characteristics have been examined. In this paper, the mode transition of the flow between two coaxial cylinders with vertical axes is investigated. The inner cylinder rotates and the outer cylinder is stationary. The bottom end wall is fixed and the glycerin solution is filled in the annulus between the two cylinders. At the top, the solution contacts with the air and the surface is free.

One of the reasons why the Taylor-Couette system attracts many interests is the finiteness of the cylinder lengths and the bifurcation of the flow phenomena. While an analytical approach can be used for the flow between two cylinders with infinite lengths or periodical conditions, the finite lengths of the cylinders introduce an additional effect of the end wall(s) and the straightforward approach cannot be applied. This additional effect makes the flow bifurcation imperfect, and the flow patterns different from those found in the perfect condition appear [

The studies of the flow which develops between two vertical cylinders and has a free surface at the top are classified in three groups. The first group assumed the infinite length of cylinders even though they are finite. The second group examined the stratified flow, and the third group investigated an asymmetric system with the free surface at the top and the stationary end wall at the bottom.

In the first group, Cole [

The stratified flow was regarded as a model of the flows in atmosphere and ocean. Caton et al. [

When the height of the working fluid is lower than that of the cylinders, the free surface appears between cylinder walls. The finite flow field with the stationary bottom end wall and the top free surface is called an asymmetric system in the vertical direction. When the convection effect is large, the entrainment of the air into the working fluid occurs at the free surface. Djéridi et al. [

On the other hand, the characteristics of the asymmetric flow without the entrainment of the air have also been investigated. Similarly in the symmetric system with both stationary end walls [

In this paper, we first show the flow development processes at small aspect ratio. Then, we numerically confirm the experimental results [

Taylor-Couette system consists of the rotating inner cylinder and the stationary outer cylinder. The cylinders are coaxial and their axes are parallel to the direction of the gravitational force. The radii of the inner and outer cylinders are r_{in} and r_{out}, respectively. The gap width d of the cylinders is given by r_{out} – r_{in}. The bottom end of the cylinders is a stationary wall and no wall appears at the top. The height of the working fluid is L and it is lower than the height of the cylinders. This means that the free surface appears between the inner and outer cylinders and the flow field is asymmetric in the vertical direction. The reference velocity is the circumferential velocity component of the inner cylinder and the reference length is the gap width d, and the Reynolds number is defined by these reference values and the kinematic viscosity of the working fluid. The aspect ratio Γ is the ratio of the cylinder length L to the gap width d, and the radius ratio η is given by r_{in}/r_{out}.

The flow pattern, that is the mode, is determined from the number of Taylor vortices and the rotation direction of the vortices. In the asymmetric system, when the flow near the bottom end wall is radially inward and the flow at the top is outward, the flow is a normal mode. Otherwise, the flow is an anomalous mode, and an anomalous vortex appears near the bottom end wall or the free surface, whose rotating direction is opposite to the direction of the normal mode flow. Depending on whether the anomalous vortex appears near the bottom or near the top, the flows of the anomalous mode are classified into the lower anomalous mode and the upper anomalous mode. The primary mode flow is the Taylor vortex flow that first appears when the inner cylinder is gradually accelerated from rest. This flow mode coincides with the flow mode that appears just before the appearance of the annular Couette flow when the flow at higher rotation speed is gradually decelerated [

The experimental result shown in this paper has been obtained by one of the authors, who have been carrying out a series of experiences [^{−5} m^{2}/s and the surface tension is 6.59 × 10^{−2} N/m. The working fluid includes a small amount of aluminum flakes and the flow is visualized. Toya et al. [

In the numerical simulation, the governing equations are the unsteady axisymmetric Navier-Stokes equations and the equation of continuity. The free surface model is the VOF (volume of fluid) and the gravitational force in the axial direction and the surface tension based on the free surface curvature are considered. Physical quantities are made dimensionless by the gap with d and the maximum circumferential velocity of the inner cylinder attained during the simulation. The equations are formulated in the cylindrical system (r, θ, z) of the radial, azimuthal and axial coordinates, and the velocity components in each direction are u, v and w. The discretization method is the finite difference method, and the time integration is the fractional step method. Initially, the velocity components are zero and the pressure is the hydrostatic pressure. The boundary condition on the solid wall is no slip condition. In order to visualize the numerical flow information, the Stokes’ stream function φ is introduced. The details of the numerical method can be found in Watanabe et al. [

The bifurcation locus between the normal three cell mode (N3) and the normal one cell mode (N1) was numerically determined at the aspect ratio from 1.44 to 1.96 [

The time variations of the bulk averaged energy of the axial velocity component E_{w} at Re = 800, 810, 820, 830 and 840 are shown in _{w} appear at Re = 810, 820 and 830, although the onset time delays as the Reynolds number becomes large. When the Reynolds number is 840, however, the increase of E_{w} was not observed even though the simulation was continued well beyond the time range shown in

The development of flow at Γ = 5.6 is shown in

direction. Therefore, the flow is inward on the bottom end wall and outward at the free surface, and the flow mode is N11. The pair of the second and third vortices from the bottom and the pair of the fourth and fifth vortices have radially inward flow at the boundaries of paring vortices. These pairs of vortices are not stable, and they disappear and N7 emerges by t = 100. This N7 flow is not stable neither. The third and fourth vortices from the free surface, which have a radially inward flow at their boundary, collapse and N5 finally appears by t = 200. The top free surface has radially outward and its profile is higher near the outer cylinder than near the inner cylinder. During this developing process, the disappearing vortices makes a pair in which the radial flow is inward at the boundary of the vortices. Similar disappearing pair of vortices was found in the symmetric Taylor-Couette system with stationary walls at both ends of the cylinders [

The development of the flow at Re = 100 is shown in

When the aspect ratio is not more than 4.2, the bifurcation between the normal three cell mode and the normal one cell mode and the bifurcation between the normal five cell mode and the normal three cell mode were numerically determined [

The experimental result by Toya et al. shows that the primary mode at Γ from 4.79 to 6.32 is N5. The numerical and experimental bifurcation diagrams are shown in Figures 5 and 6, respectively. In both cases, the bifurca

tions from N7 to Au6, from Au6 to N5 and from N7 to N5 are found. The experimental and numerical results do not give good agreement about the ranges of the aspect ratio, in which the bifurcations appear, nor the critical Reynolds numbers. However, the fact that the critical Reynolds number of the transition from N7 increases as the aspect ratio decreases is well predicted. When the aspect ratio is about 6.0, the numerical and experimental result of the critical Reynolds number for the bifurcation from N7 to Au6 almost coincide with each other. While the range of the aspect ratio from 5.4 to 5.7, at which the transition from Au6 appears, is found in the numerical and experimental results, the critical Reynolds number obtained by the experiment is larger than that predicted numerically. The flow of Au6 is an anomalous flow and it is physically unstable. Therefore, it is suggested that imperfections of the experimental apparatus make the bifurcation from the unstable Au6 to the more stable N5 occur at the higher Reynolds number.

Numerical investigation has been carried out on the flow developing between a rotating inner cylinder and a stationary outer cylinder with vertical axes. The bottom end of the cylinders is stationary fixed wall and the top is the free surface.

Depending on the Reynolds numbers, the final flow state bifurcates into the one cell mode flow and the three cell mode flow at the aspect ratio 1.6. This shows the bifurcation mechanics of the flow even in a given geometrical configuration.

When the aspect ratio is from 5.0 to 6.0, the high Reynolds number flows developing from rest have many vortices in their transient states. It is numerically shown that, as the flow grows, a pair of vortices which have a radially inward flow between the vortices disappears and the flow state with smaller number of vortices is established. When the Reynolds number is low, the Ekman vortex emerging on the bottom stationary wall induces

interior vortices and makes the flow develop into a unique mode. The experimentally estimated critical Reynolds number for the transition from the secondary normal seven cell mode is obtained by the present numerical simulation. Especially, the transition from the normal seven cell mode to the anomalous six cell mode at the aspect ratio around 6.0 is predicted well.

This work was partly supported by JSPS KAKENHI Grant Number 23560193.