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World Journal of Mechanics, 2019, 9, 177-190

http://www.scirp.org/journal/wjm

ISSN Online: 2160-0503

ISSN Print: 2160-049X

DOI:

10.4236/wjm.2019.97012 Jul. 30, 2019 177 World Journal of Mechanics

Numerical Analysis of Critical Reynolds

Number of Wavy Taylor Vortex Flow with

Changing Aspect Ratio

Hiroyuki Furukawa, Takashi Watanabe, Yorinobu Toya

Department of Mechanical Engineering, Meijo University, Nagoya, Japan

Abstract

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 a

s 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 calcu-

lated 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 Rey-

nolds number, the distribution of the Reynolds number is qualitatively simi-

lar between the calculated and experimental values.

Keywords

Rotating Cylinders, Flow Instability, Attractor

1. Introduction

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 [1]. State of the flow between

inner and outer cylinders of a rotating dual cylinder transits from Couette flow

to Taylor vortex flow and to wavy Taylor vortex flow as the Reynolds number

changes and as an aspect ratio increases [2] [3] [4]. The aspect ratio is defined to

be the ratio of the gap between inner and outer cylinders to cylinder height. The

Taylor vortex flow is the one where torus flows, called cells, are stacked. The

How to cite this paper:

Furukawa, H.,

Watanabe, T. and Toya, Y.

(2019) Numeri-

cal Analysis of Critical Reynolds Number

of Wavy Taylor Vortex Flow with Chan

ing Aspect Ratio

World Journal of M

anics

, 177-190.

https://doi.org/10.4236/wjm.2019.97012

Received:

June 10, 2019

Accepted:

July 27, 2019

Published:

July 30, 2019

9 by author(s) and

Scientific

Research Publishing Inc.

This work

is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

H. Furukawa et al.

DOI:

10.4236/wjm.2019.97012 178 World Journal of Mechanics

wavy Taylor vortex flow is the one where every vortex of the Taylor vortex flow

is subjected to the axial cyclic oscillation resulting in wavy and unstable flow

over time [5] [6].

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 an unstable change in the physical quantities that character-

ize 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 expe-

riment 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 identi-

fied 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 expe-

riment 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.

2. Numerical Method

The Reynolds number

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

Re VD

(1)

Here,

is a representative velocity, given as the rotation speed of the inner

cylinder (

is the width of the gap between the inner and outer cylinders,

given by the difference between their representative radii as

out

−

, and

the kinematic viscosity of the fluid.

out

= 30 cm and

, = 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 repre-

sentative velocity. The aspect ratio

is defined as the ratio of the cylinder length

to the gap width D and is given by

LDΓ=

(2)

H. Furukawa et al.

DOI:

10.4236/wjm.2019.97012 179 World Journal of Mechanics

The governing equations are the unsteady three-dimensional incompressible

Navier-Stokes equation with cylindrical coordinates (

) and the continuity

equation.

We use both SOR and ILUCGS methods to solve Poisson’s equation for pres-

sure. 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 ini-

tial 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 ×

–6

/s. For the discretization method, we apply the QUICK method for con-

vection terms, the second-order central difference method for the other space

integration, and Euler’s method for time integration. Grids are staggered and

equidistant in each direction. The number of grid points is 41 in the radial direc-

tion, and the number of grid points in the axial direction is proportionally ad-

justed so that it becomes 41 for the aspect ratio of 1.0. The number of grid points

in the circumferential direction is 74. In order to examine the validity of the

number of grid points, we analyzed Taylor vortex flow using several types of gr-

ids under various numerical conditions, and concluded that there are no differ-

ences among the modes that are finally formed, the formation of modes up to

the final mode, and the manner of decay of the vortexes.

3. Mode Analysis Method

In the preceding studies on the numerical analysis, the changes in kinetic energy

of axial velocity in every cell have been used to identify whether the flow state is

the Taylor vortex flow or wavy Taylor vortex flow. This identification is possible

because the Taylor vortex flow is defined to be the wavy Taylor vortex flow when

the Taylor vortex flow becomes non-linear and unstable due to the cyclic axial

oscillation. The criterion for this identification is the amplitude of the kinetic

energy changes, and the flow state is identified to be the wavy Taylor vortex flow

when the amplitude in the neighborhood of the estimated critical Reynolds

number reaches or exceeds this criterion. When the amplitude is less than (does

not reach) this criterion, the flow state is identified to be the Taylor vortex flow.

The mode is decided based on the amplitude when TS is 2,000,000 (at TS =

2,000,000). The critical Reynolds number when the flow transits from the Taylor

vortex flow to wavy Taylor vortex flow are decided. The computation for a

relatively short period of time when TS is 2,000,000 (at TS = 2,000,000) reveals

that the flow between inner and outer cylinders at around the critical Reynolds

number is unsteady, and the computation is not extended to the point where the

kinetic energy changes converge. Therefore, the flow state may have been de-

cided to be the wavy Taylor vortex flow based on the kinetic energy changes

even when the Taylor vortex flow is unsteady. The criterion of the energy am-

plitude is decided by the trend of energy amplitude changes in the neighborhood

of the estimated critical Reynolds number, and thus the wavy Taylor vortex flow

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