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Transitions in the Small Gap Limit of

Taylor-Couette Flow

Kathleen A. Hinko, The Ohio State University, REU Summer 2003

Advisor: Dr. C. D. Andereck, The Ohio State University

Introduction

Fluid dynamics plays a powerful role in nature. Whether it is the movement of

underwater currents, the path of a hurricane, or local weather conditions, fluid motion affects

people’s daily lives. Thus, it is important to study and understand such phenomena from a

physical perspective. In order to isolate specific fluid behaviors from the plethora of variables

present in an environmental setting, certain experimental setups are utilized by physicists. Two

of these systems, the Plane Couette and Taylor-Couette, have been used to varying effects to

examine turbulent patterns in fluids.

In theory, the Plane Couette system is arguably the most straightforward fluid dynamical

modeling system; experimentally, however, it may present the greatest challenge. Figure 1 is a

diagram of the Plane Couette system. Mechanical energy is transferred from the motion of two

shearing plates to the fluid that lies in between them. The relative velocity of the two plates

determines the amount of energy transferred to the fluid, which characterizes the resulting flow

state. This type of setup can been achieved using a conveyor belt-like apparatus, but concerns

about conditions at the boundaries of the fluid, as well as the distance over which the plates

actually move, prevent the plane Couette system from producing reliable flow states.



r ir o

v2

v1

Ωi

Ωo

Figure 1. Plane Couette System Figure 2. Taylor-Couette System

In contrast, the Taylor-Couette system, due to its realizable construction, simple

geometry, and reproducible flow states, has been used by physicists for over a century. Two

rotating cylinders with differing radii, one inset in the other, transfer energy to a fluid that lies

between them. This apparatus, shown in Figure 2, was developed independently in the late

1800s by Mallock in England and Couette in France as a means to measure viscosity. In 1923

Taylor published groundbreaking observations of the flow states in this system, which we refer

to today as the Taylor-Couette system, rather unfortunately for Mallock. Research since this

time has given physicists a thorough understanding of Taylor-Couette flow under certain

commonly used parameters. Andereck et al. published in 1986 the graph shown in Figure 3,

which identifies the characteristic flow states of Taylor-Couette systems where the ratio of the

inner cylinder to the outer cylinder is from 0.7 to 0.9. The axes of the graph are the outer and

inner Reynolds numbers of the cylinders, as defined by

In contrast, the Taylor-Couette system, due to its realizable construction, simple

geometry, and reproducible flow states, has been used by physicists for over a century. Two

rotating cylinders with differing radii, one inset in the other, transfer energy to a fluid that lies

between them. This apparatus, shown in Figure 2, was developed independently in the late

1800s by Mallock in England and Couette in France as a means to measure viscosity. In 1923

Taylor published groundbreaking observations of the flow states in this system, which we refer

to today as the Taylor-Couette system, rather unfortunately for Mallock. Research since this

time has given physicists a thorough understanding of Taylor-Couette flow under certain

commonly used parameters. Andereck et al. published in 1986 the graph shown in Figure 3,

which identifies the characteristic flow states of Taylor-Couette systems where the ratio of the

inner cylinder to the outer cylinder is from 0.7 to 0.9. The axes of the graph are the outer and

inner Reynolds numbers of the cylinders, as defined by

υ

ioioio

r d R //

/

Ω⋅⋅=

where is the radius of either the outer or inner cylinder, d is the difference between the outer

and inner radii, Ω is the angular velocity of either the outer or inner cylinder, and

where is the radius of either the outer or inner cylinder, d is the difference between the outer

and inner radii, Ω is the angular velocity of either the outer or inner cylinder, and

ior /

io /

ior /

io / υ is the

viscosity of the fluid between the cylinders.



Figure 3. Diagram of Taylor-Couette flow states by Andereck et al.

It has been proposed that as the radius ratio of the cylinders approaches one, the local

topology of the flow will resemble that of a plane Couette system while still maintaining the

global topology of the Taylor-Couette system. Therefore, we believe that we will be able to

study plane Couette flow in the context of a Taylor-Couette system.



Experimental Setup

CCD cameraInner cylinder

Outer cylinder

Stepper motorsGear box

Controllers

Rubber boundary

Metal cap

Drain

Figure 4. Diagram of Taylor-Couette system and CCD camera as set up in the lab.

The inner cylinder is made of nylon and has an outer radius of 29.54 cm. The outer

cylinder is made of clear Plexiglas and has an inner radius of 29.84 cm. This gives a radius ratio

of 0.99 and a gap width of 0.30 cm. A metal cap on the top of the outer cylinder keeps the

cylinder rim from becoming elliptical. A piece of rubber sheeting is glued to the inside of the

outer cylinder at the bottom to form a boundary between the fluid underneath the inner cylinder

and the fluid on the sides of the system. With this boundary we hope to isolate the turbulence on

the sides of the system from whatever fluid motion may be occurring in-between the bases of the

cylinders. Small valves have been inserted into the base of each cylinder at various locations,

two on the inner and two on the outer, which act as air vents and drains respectively.



Figure 5. A picture of the Taylor-Couette

system with one motor attached.

The rotation of each cylinder is controlled by a Compumotor stepper motor, which is

controlled by a Compumotor 2100 Series indexer. Worm gears inside the gear box facilitate the

rotation of the vertical shafts connected to the cylinders.

Water is used as the fluid in the system and has a viscosity of 0.100. One percent

Kalliroscope is added to the water for visualization purposes, and 1% Kalliroscope Stabilizer is

added as well to keep the Kalliroscope flakes suspended. Kalliroscope is made of flat polymers

that face outward and reflect light in the direction of flow. The viscosity of the solution is

increased by just 1% due to the presence of these substances.

Data Acquisition

We control our experimental variables automatically using LabView software. This

software, written by Christopher Carey at The Ohio State University, sets the speed for each

cylinder, allows for a period of rotation at this speed, collects data, and then increments the

Reynolds number to repeat the process. It is necessary for the system to run for a time after the

rotation rate has changed in order for residual turbulence effects from the cylinder acceleration to

settle out. For the data discussed here, this period of time was 5 minutes, followed by a 5 minute

data acquisition. A CCD camera takes an image of the side of the system at a rate of 10 frames



per second. A row of pixels from each image is projected in sequence onto a space-time

diagram, an example of which is shown in Figure 6. For our experiment, space-time diagrams

were created for every fifth Reynolds number between 450 and 1600; thus with this method, we

were able to look at many flow patterns effectively and efficiently.

Space

Time

Figure 6. Example of the projection of still frame onto space-time diagram

Observations

Data was taken with the outer cylinder stationary, while the Reynolds number for the

inner cylinder was changed. For Reynolds numbers R i = 0 to 350, we observe Couette flow, a

featureless fluid state. Starting at approximately R i = 350, we see the onset of Taylor Vortex

Flow (TVF). This flow state is characterized by horizontal vortices that are about as wide and as

tall as the gap and form slowly until they span the space from the bottom to top of the cylinders.

As shown in Figure 7 the TVF has stabilized at Reynolds number R i = 450.



Figure 7. Still photo and space-

time diagram at Reynolds number

R i = 450. We see the TVF flow

state.

Still picture at R i = 450 Space-time diagram

At slightly higher Reynolds numbers, R i = 455, we observe the secondary phase

transition, Wavy Vortex Flow (WVF), which is TVF with a vertically oscillating wave

superimposed over the vortices (Figure 8). The formation of TVF followed by WVF is seen in

Taylor-Couette systems with smaller radius ratios as well.

Figure 8. Still photo and space-

time diagram at Reynolds number

R i = 455. We see the WVF flow

state.

Still picture at R i = 455 Space-time diagram



At Reynolds number R i = 460, however, we observe a new type of transition in the flow,

in the form of isolated patches of high frequency turbulence; a still photo of such a patch is

shown in Figure 9. These patches, which we have named Very Short Wavelength Bursts

(VSWB), increase in number as the Reynolds number of the inner cylinder gets larger, until the

flow state is saturated with VSWB. Figure 10 shows the progression of Reynolds numbers and

the corresponding space-time diagrams. Although the fluid is in a very turbulent state, the flow

retains a laminar background of horizontal vortices which we see as dark bands in spatial

direction. Theoretical predications for Plane Couette flow indicate that the flow states will

contain localized areas of weak turbulence. Thus, we believe that the flow states we observe at

Reynolds numbers greater than 500 may be a combination of both Plane Couette and Taylor-

Couette flow.

Figure 9. Still image of VSWB.



R i = 510 R i = 565 R i = 1200

Figure 10. Space-time diagrams of VSWB at various Reynolds numbers. For R i = 1200, the flow

is clearly saturated with VSWB; however, the dark bands in the spatial direction indicate that there

is still a back round of laminar flow.

Conclusions

At Reynolds numbers greater than 500, we observe high frequency patches of weak

turbulence; this instability is a bifurcation of WVF. These chaotic formations, which we call

Very Short Wavelength Bursts, saturate the flow at high Reynolds numbers. The VSWB may be

a superposition of flow states common to both Taylor-Couette and Plane Couette systems.

Future Work

Over the next year, I plan to explore the full range of both inner and outer Reynolds

numbers and calculate the turbulence percentages as a function of the Reynolds number in order

to gain a more quantitative understanding of the flow in our system. I will also design and build

additional inner cylinders with shorter radii than the current inner cylinder in order to determine

in total the parameters of our particular Taylor-Couette system.



Acknowledgments

I would like to thank my advisor, Dr. C. David Andereck, for his guidance throughout the

summer. I also must recognize Christopher Carey, who designed and built our Taylor-Couette

system, wrote the appropriate operational and analytical programs, and graduated from Ohio

State with his B.S. in June 2003. I have deeply appreciated his invaluable advice and assistance.

Finally, thank you to The Ohio State University Department of Physics for giving me this

research opportunity.

This research is supported by the National Science Foundation.

Bibliography

Andereck, C. David, et. al. Flow Regimes in a circular Couette system with independently

rotating cylinders. J. Fluid Mech. Vol. 164, pp 155-183: 1986.

Carey, Christopher. Undergraduate Senior Thesis: Critical Transitions and Pattern Formation

In the Large Radii, Small Gap Limit of the Taylor-Couette System. The Ohio State

University: 2003.

Colovas, Peter William. PhD Dissertation: The Formation of Time Dependent Patterns in Non-

Equilibrium Fluid Dynamical Systems. The Ohio State University: 1996.

Degen, Michael Merle. PhD Dissertation: Time-Dependent Pattern Formation in Fluid

Dynamical Systems. The Ohio State University: 1997.

Donnelly, Russell. Taylor-Couette Flow: The Early Days. Physics Today: November 1991.

Faisst, Holger. PhD Dissertation: The transition from the Taylor-Couette system to the plane

Couette system. Philipps-Universitat Marburg: 1999.

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