an experimental study of taylor-goertler vortices in a...

AN EXPERIMENTAL STUDY OF TAYLOR-GOERTLERVORTICES IN A CURVED RECTANGULAR CHANNEL

Robert Joe McKee

luate School

fornia 93940

Miun

onterey, California

AN EXPERIMENTAL STUDY OF TAYLOR-GOERTLER

VORTICES IN A CURVED RECTANGULAR CHANNEL

by

Robe rt Joe McKee

Thesis Advisor: M. D . Kelleher

June 19 7 3

Approved Ion pubtic n.dlexL6e,; du&iibutlon antuuXzd.

-

An Experimental Study of Taylor-Goertler

Vortices in a Curved Rectangular Channel

by

Robert Joe McKee

Lieutenant, United States Naval Reserve

B.S., University of California, Santa Barbara, 1968

Submitted in partial fulfillment of the

requirements of the degrees of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

and

MECHANICAL ENGINEER

from the

NAVAL POSTGRADUATE SCHOOL

June 1973

Library

,UCf t-y, Lal/forni

ABSTRACT

A rectangular cross section channel, with both a straight

and a curved portion, was designed and built to create the

laminar secondary Taylor-Goertler vortex flow. An aerosol

was used to visualize these flow patterns, and photographs

of the results are presented. A hot wire anemometer inves-

tigation of one location in the curved section was conducted

to obtain the mean velocity profiles and turbulence levels.

The pressure drop along both the straight and the curved

sections was measured and compared. A constant temperature

wall heater was also designed and installed in the straight

section of the channel. Approximate values of the heating

losses were obtained and the overall and local heat transfer

coefficients were calculated for the straight portion of the

channel. It was concluded that the vortices develop with

both velocity increases and distance along the channel, and

that the increase in pressure drop due to the vortices

varied from near zero to approximately 20% with increasing

Reynolds numbers

.

TABLE OF CONTENTS

I. INTRODUCTION 11

A. DESCRIPTION OF VORTICES 11

B. BRIEF HISTORY 13

II. NATURE OF THE PROBLEM 17

A. INTENT OF THIS STUDY 17

B. DESIGN REQUIREMENTS 18

C. DESCRIPTION OF APPARATUS 24

III. FLOW VISUALIZATION 39

A. EXPERIMENTAL CONSIDERATIONS 39

B. PATTERNS AND PICTURES OF THF FLOW 41

IV. VELOCITY AND TURBULENCE MEASUREMENTS 51

A. EXPERIMENTAL PROCEDURES 51

B. VELOCITY PROFILES 53

C. TURBULENCE LEVELS _ 62

V. PRESSURE DROP MEASUREMENTS 72

A. EXPFRIMENTAL PROCEDURES 72

B. PRESSURE DROP 74

VI. HEAT TRANSFER MFASUREMFNTS 85

' A. EXPERIMENTAL PROCEDURES 85

B. RESULTS 89

VII. CONCLUSIONS 102

VIII. RECOMMENDATIONS 104

APPENDIX A: ERROR ANALYSIS 106

APPENDIX B: COMPUTER RESULTS FOR VELOCITYAND TURBULENCE 108

APPENDIX C: CALCULATIONS AND COMPUTER RESULTSFOR PRESSURE DROP 119

APPENDIX D: COMPUTER RESULTS FOR HEAT TRANSFER 129

LIST OF REFERENCES 158

INITIAL DISTRIBUTION LIST 161

FORM DD 1473 162

LIST OF TABLES

Table

I. PRESSURE DROP PER FOOT FOR TWO FOOTINTERVALS CALCULATED BY LEAST SQUARES 78

II

.

PRESSURE DROP PER FOOT FOR ONE FOOTINTERVALS CALCULATED BY LEAST SQUARES 80

III. PRESSURE DROP PER FOOT FOR TWO FOOTINTERVALS MEASURED VALUES 81

LIST OF FIGURES

Figure

1. Sketch of Taylor-Goertler Vortices 12

2. Reynolds Number vs. Dean Number 21

3. Cross Section of Channel 25

4. Arrangement of Apparatus 27

5. Aerosol Generating Equipment 29

6. Lighting and Camera Arrangement 31

7. Hot Wire Anemometer Arrangement 32

8. Pressure Drop Measuring Equipment 34

9. Heating Apparatus 37

10. Photographs of Vortices Re = 609 45


12. Photographs of Vortices Re = various 47




16. Velocity Profiles Re = 609 55







23. Typical Position of Hot Wire 63

24. Turbulence Intensity Re = 609 64



27. Turbulence Intensity Re = 1298 £7




31. Pressure Below Atmospheric vs.Pressure Taps 75

32. Percentage Increase in Pressure Dropvs. Reynolds Number 82

33. Pressure Drop Coefficient vs.Pressure Taps 84

34. Heat Transfer Coefficient Tw = 125 95





TABLE OF SYMBOLS

A area

Ac cross sectional area

Aw area of the wall heater

As area of a single heater strip

a half the height of the channel

b half the width of the channel

Cp pressure drop coefficient

cp specific heat

De Dean number

Dh hydraulic diameter

d spacing of channel wall

Go Goertler number

Gr Grashoff number

H local heat transfer coefficient

Hm mean heat transfer coefficient

Ho overall heat transfer coefficient

k thermal conductivity

L a distance along the channel in the flow direction

Nu Nusselt number

Pr Prandtl number

p pressure, a relative measured value

Qc heat energy transferred by conduction and convection

Qr heat energy transferred by radiation

Re Reynolds number based on hydraulic diameter

Ri radius of inner wall or cylinder

Ro radius of concave wall

T temperature

Ta Taylor number

Tb bulk temperature of a fluid

T£m log mean temperature difference

Tw average temperature of the wall heater

U mean velocity of flow

•

V volumetric flow rate

6 boundary layer thickness

z emissivity

p mass density

a Stefan-Boltzmann Constant

y viscosity

v kinematic viscosity

ACKNOWLEDGEMENT

The author wishes to express his appreciation to all

those who made this work possible. It is impossible to give

a complete list of people to whom the author is indebted but

it is hoped that by naming those who deserve a special note

of thanks, all will know that their efforts were appreciated.

The guidance and advice of the author's thesis advisor,

Professor M. Kelleher, and his efforts made this undertaking

possible. The personnel of the Mechanical Engineering Shop

and particularly Messrs. K. Mothersell, G. Bixler, and

T. Christian deserve a special thanks for their timely and

careful assistance in preparation of the experimental appa-

ratus. The author is very grateful for the assistance of

Mrs. V. Culley of the Mechanical Engineering Office in obtaining

material and completing administrative requirements. Final-

ly, to my wife Catherine, for her patience through repeated

manuscript typings and for her support throughout the study, a

heartfelt thank you.

10

I. INTRODUCTION

A. DESCRIPTION OF VORTICES

The phenomenon best known as Taylor-Goertler vortices is

caused by a body force instability in fluid flows. Speci-

fically, Taylor-Goertler vortices are a secondary flow

induced by centrifugal forces. In a channel, such as used

in this investigation, the fluid near the center of the

channel is subjected to larger centrifugal forces than the

slower moving fluid near the concave wall. Thus, the ten-

dency is for the fluid in the center of the channel to move

outward toward the wall. The fluid near the wall is unable

to resist this action and must move in the spanwise direction

and then radially inward replacing the central fluid. Once

in the center of the channel, this fluid obtains the higher

flow velocity, and the tendency to move toward the concave

wall. This cyclic motion forms the rotating Taylor-Goertler

vortices

.

Such vortices, caused by centrifugal forces, occur in

many cases of curved flow. Taylor-Goertler vortices are

laminar vortices with their axes in the direction of the

main flow, and with secondary velocities in both perpendic-

ular directions. Such spiral vortices occur in counter

rotating pairs. There is frequently a regularly spaced

cellular structure associated with these vortices. A sketch

of this type of flow pattern is shown in Figure 1.

11

Figure 1. Sketch of Taylor-Goertler Vortices

12

B. BRIEF HISTORY

The first person to consider the instability of curved

flows was Lord Rayleigh who found a stability criterion for

inviscid fluid rotating symmetrically about an axis [Ref . 1]

.

His criterion for a stable flow required that the product of

the local circumferential velocity and the local radius of

curvature either increase or at least remain constant as the

radius increases. An example of an unstable flow described

by Lord Rayleigh is the flow between a fixed outer cylinder

and a concentrically rotating inner cylinder. In 1923,

G. I. Taylor published an extensive analytic and experimental

study of such unstable Couette flows of viscous fluids

between rotating cylinders [Ref. 2]. The Taylor number,

defined as

Ta = um nrv v Ri

was shown to be a characteristic parameter of such flows with

a critical value for the onset of the instability. The

secondary flow observed between the cylinders when the outer

was fixed and the inner rotated has been shown to be vortices

whose axes were in the circumferential direction. These

vortices were named as Taylor vortices. In recent years,

there have been a large number of investigations' of cylindri-

cal Couette flow. The torque required to rotate a cylinder

in the presence of Taylor vortices was measured by P. Castle,

et al. [Ref. 3]. The same type of instability in the boundary

layer along concave walls was first investigated by

13

H. Goertler in 1940 [Ref. 4]. The Goertler number, which

is similar to the Taylor number, is a parameter characteris

tic of curved boundary layer flows and is defined as

GO = ^V R

where 6 is the boundary layer thickness. When the Goertler

number exceeds a critical value, the presence of Goertler

vortices is observed. H. W. Liepmann [Ref. 5] was among the

investigators to continue and verify Goertler 's experimental

work. The approximate analytic results obtained by Goertler

have been verified with an exact solution by the German

investigator G. Hammer lin as reported by H. Schlichting

[Ref. 6]. These results were also verified with an extensive

numerical solution completed by A. M. O. Smith [Ref. 7]

.

The problem of instability of viscous fluids flowing in

a curved channel was first considered by W. R. Dean in 1928

[Ref. 8]. Dean found a parameter, similar to the Taylor and

Goertler numbers , which governs the onset of Taylor-Goertler

vortices in curved channels. The analytic work done by Dean

has been verified by W. H. Reid [Ref. 9] among others. The

experimental work on secondary flows in channels seems very

limited. Until just recently, the possible effects of

Taylor-Goertler vortices on heat transfer had not been

studied. In 1965, L. Persen [Ref. 10] considered the effect

of Goertler type vortices on the heat transfer from a wall

for the special cases of both a very high and a very low

Prandtl number. The first experimental work with the effect

14

of Taylor-Goertler vortices on heat transfer in a boundary

layer was reported by P. McCormack et al. [Ref. 11] in their

1970 study. With some limiting assumptions, a numerical

solution for the heat transfer in curved rectangular chan-

nels was reported by K. Cheng and M. Akiyama in 1969

[Ref. 12]. Analytic and experimental results for a fully

developed, constant wall heat flux, square cross section

curved channel were obtained in 19 70 by Y. Mori, et al.

[Ref. 13]

.

Although Taylor-Goertler vortices are a laminar flow,

they have secondary velocity components and are more complex

than one would expect. Furthermore, as indicated by H. W.

Liepmenn [Ref. 5], these vortex flows are potentially a key

to understanding transition to turbulent flow. Taylor-

Goertler vortices have similarities to other vortex flows

such as wing tip vortices and the vortex rolls in forced

convection heating of fluid layers described in the paper by

M. Akiyama et al . [Ref. 14]. The striations seen at stag-

nation points on bluff bodies and the cross hatching on

reentry bodies have also been attributed, at least in part,

to the presence of Taylor-Goertler vortices, as reported by

L. Persen [Ref. 15]

.

There are many possible applications that could result

from investigation and understanding of these vortex phenom-

enon. Potential application include improved cooling of

turbine blades and other external surfaces, reduction of

pressure losses in bends, and improved mixing in laminar

15

flows. Heat exchangers which could utilize the low pressure

drop in laminar flow as well as the enhanced heat transfer

in the presence of vortices would be a significant advance-

ment in heat transfer.

16

II. NATURE OF THE PROBLEM

A. INTENT OF THIS STUDY

Because future application of the Taylor-Goertler vortex

phenomenon may take place in closed channels, it was intended

to study the effect of these secondary flows in a curved

channel of rectangular cross sectional area. To investigate

Taylor-Goertler vortices in a curved rectangular channel, the

presence of the vortices had to be verified. It was intended

to visualize the flow in order to confirm the presence of

the vortices. How rapidly the vortices would form and to

what extent they would fill the channel was not known. It

was intended to use the flow visualization technique to

study, as far as possible, the size, the form, and the

development of the vortices.

Other planned measurements included a hot wire anemometer

check of the turbulence level to insure that the flow was

laminar. The velocity profiles in the channel were also to

be obtained to determine what effect the secondary flow would

have on the mean velocity. The additional velocity com-

ponents in Taylor-Goertler vortices produce Reynolds stress

not present in other laminar flows. It was planned to

measure the pressure drop along both the straight and

curved portions of the channel used in this investigation

to determine the effect, if any, of these Reynolds stresses

on the pressure drop. An increase in the pressure drop

would be anticipated because it is known that additional

17

energy is required to maintain the secondary flow. This

energy is supplied to the Taylor-Goertler vortices by an

interaction of the mean and secondary velocity components.

The works of G. I. Taylor [Ref. 2] and P. Castle et al.

[Ref. 3] have shown the increase in torque required to

rotate a cylinder in the presence of Taylor vortices.

The Taylor-Goertler vortices would also be expected to

effect the heat transfer from the concave wall. This effect

would result from the secondary velocity components trans-

porting heated fluid from the wall towards the center of the

channel, and carrying cooler fluid from the center to the

wall. It was intended to study the requirements for measur-

ing the effects of this action on the heat transfer from

the concave wall.

B. DESIGN REQUIREMENTS

In order to study Taylor-Goertler vortices, an apparatus

had to be built which would give the values of the non-

dimensional parameters required to produce the secondary

flow. The limit of stability of the basic laminar flow in

a curved channel against the formation of Taylor-Goertler

vortices is well known, and is presented as a critical Dean

number by W. Reid [Ref. 9]. If the Dean number, defined as

-W if

exceeds a value of 36, it is presumed that Taylor-Goertler

vortices will be present. Furthermore, it is known that the

18

rate of development of the vortices will be somewhat depen-

dent on how much greater than 36 the Dean number becomes.

It is known that the size of the vortices will depend in

some manner on the dimensions of the channel. Investigators

of Taylor-Goertler vortices on concave walls have reported

vortices from the size of the boundary layer to ten times

larger, dependent on geometric factors [Ref . 7] . The Dean

number had to be greater than the critical value of 36 and

be adjusted within a workable range. One hundred fifty was

taken as the upper limit of this range, as it was assumed

the vortices would be fully developed by at least this value.

The minimum critical Reynolds number, based on hydraulic

diameter, at which turbulent flow is observed in a parallel-

plate channel is given by G. Beavers, et al. [Ref. 16] as

Re = 2200. Several investigators including H. Liepmann

[Ref. 5] and R. Nunge [Ref. 17] have reported that the

decrease in the critical Reynolds number due to curvature is

small, especially for a large radius of curvature. There-

fore, the maximum Reynolds number, based on hydraulic dia-

meter, selected for this investigation was Re = 2000. The

Reynolds number had to be high enough for the flow velocities

to be measured accurately. A minimum of Re = 100 was used

for the design of the experimental apparatus.

The design procedure involved checking the effects of

several factors on the range of the nondimensional variables.

The working fluid, the channel height, and the flow rate

selected, each have an effect on both the Reynolds number

19

and the Dean number. The radius of curvature also effects

the Dean number. Trial and error was used to select the

height and width of the channel's cross section, and the

following inequality in the Reynolds number was checked to

give the range of velocities

100 < Re < 2000

The inequality in the Dean number given below was then

checked to give the range of the radius of curvature.

36 £ De <_ 150

When the velocity range was practical to obtain and easy to

measure, and when the radius of curvature was reasonable for

construction, the design variables were considered feasible.

The resulting relationship between the Reynolds number,

based on hydraulic diameter, and the Dean number with the

ratio of radius of curvature to plate spacing as a parameter

is shown in Figure 2. The selected design is represented by

the line R/d = 48.

The design of the apparatus for this investigation was

also constrained by time and cost limitations. The material

and equipment to be used had to be workable, available with-

out long delay, and inexpensive. Water and air were the

only working fluids considered as practical. The flow

velocities at which water satisfied the Reynolds number and

the Dean number requirements were too low to measure by

ordinary means. Furthermore, water requires more elaborate

construction to prevent leakage from the channel. For these

20

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21

reasons, air was selected as the working fluid. For design

purposes, the value of the fluid properties at estimated

operating temperatures were taken from reference tables.

The length of the entrance region in the design channel was

checked to see that it would not be excessive. To insure

smooth entering flow, the entrance contraction section was

designed in accordance with considerations given by M. Cohen

and N. Ritchie [Ref. 18]. The intent to visualize the flow

required the use of a workable, transparent, inexpensive

material for the walls of the channel. Plexiglas was

selected for this purpose.

It was hoped that the results obtained for both pressure

drop and heat transfer would be comparable to infinite

parallel plate solutions, so a large aspect ratio was chosen.

The pressure drop for flow between parallel plates is known

from an exact analytic solution and is not of a large magni-

tude. This magnitude was compared with the sensitivity of

the laboratory micromanometer available. It was decided

that the pressure drop could be measured, but only if a

sufficiently long test section was built. No exact solution

exists for the pressure drop in a curved channel in the

presence of Taylor-Goertler vortices. However, the studies

by R. Nunge [Ref. 17] and by K. Cheng and M. Akiyama [Ref. 12]

indicates a rather small increase in the friction factor in

curved channels of large aspect ratio. It was assumed that

the curved test section would be sufficiently long to allow

measurement of the pressure drop.

22

To measure the effects of Taylor-Goertler vortices on

the heat transfer from a constant temperature concave wall

of a curved channel requires comparative data for one wall

of a flat plate channel. Heat transfer data for a flat

plate channel with one constant temperature wall is very

limited. Limiting values of the Nusselt number and other

comparative values can be obtained from R. Shah and A. Lon-

don's report [Ref. 19]. The design of a heater plate to

measure this type of heat transfer presented special require-

ments. As with any flat plate heater, more power would be

convected away from the upstream edge than from points

farther downstream. Therefore, to obtain an uniform wall

temperature, by electrical heating, separate resistance

elements with individual voltage controls were required. A

multiple channel regulated power controller was not immedi-

ately available, but it was determined that such an instru-

ment could be constructed. Grade A nichrome was selected

for the heating elements because of this metal's large

specific resistance, small temperature coefficient of resis-

tance, and relative low cost. The many power leads and

temperature recording leads would have to reach the heating

element through the channel wall. For this reason a castable

plastic resin was selected for constructing the heater sec-

tion. Although only a flat heater plate was to be constructed

for the first tests, design considerations were given to

adapting this construction method to a heater plate for the

curved section. Other aspects of the heated section con-

struction will be discussed later. To insure that the heat

23

transfer data v/ould be for forced convection, and would not

contain the effects of free convection, the value of the

governing dimensionless group was checked against the

requirement

Gr * Pr * — < 1000L -

given by J. Holman [Ref. 20]. It was found that this

requirement imposed a significant temperature difference

limitation for water as the working fluid, but allowed a

large temperature range for the selected working fluid, air.

The effects of the many design requirements discussed

above was to make the design problem somewhat difficult.

However, the working fluid, the geometry-/ and the flow

velocities to be studied are of values frequently encountered

in practice.

C. DESCRIPTION OF APPARATUS

The cross section of the channel was 0.250 inches high

and 10.0 inches wide. It had an aspect ratio of 40 and a

cross sectional area of 0.017361 square feet. The hydraulic

diameter of this channel was 0.040650 feet. The channel was

made of two one-quarter inch thick sheets of plexiglas

separated by rolled metal spacers which also formed the

sides of the channel as shown in Figure 3. Steel was used

for the spacers in the straight section of the channel and

for added flexibility, aluminum was used in the curved por-

tion. A straight portion of this channel, consisting of a

24

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25

two foot long entrance section and a two foot long test

section, preceded the curved section. The radius of curva-

ture of the interior concave wall of the curved section was

one foot. The curved section was formed by heating the

plexiglas and bending it over a carefully constructed wood

frame. This wood frame was also used to support the entire

channel. The curved portion of the channel completed a

180 degree turn and was followed by a short straight section.

The flow of air entered the channel through a contrac-

tion section attached to the beginning of the entrance region

and covered by a cheesecloth screen for most of the experi-

mental work. The flow left the channel through an exhaust

nozzel which was connected to flexible tubing of one inch

inside diameter . The tubing led the flow through a Fischer

and Porter Company variable area flow meter, model number

10 A3565. This rotometer had a 100% full scale flow rate

of 11.1 standard cubic feet of air per minute and an accuracy

of ±0.5% of full scale. The flow was drawn through the flow

meter, the flexible tubing, and the channel by an electri-

cally driven Cadillac, model G12, centrifugal blower. The

blower speed was controlled by varying motor voltage with a

variac. The voltage was supplied by a Sorenson, R 1050,

A.C. voltage regulator. The general arrangement of this

apparatus is shown in Figure 4.

The idea of using an aerosol for visualizing the flow

was taken from a paper by 0. Griffin and C. Votaw [Ref. 21].

The advantages and ease of using an aerosol are derived from

26

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the non-toxic and non-corrosive properties of the DOP

,

di (2-ethylhexyl) -phthalate , chemical used. The equipment

used to make the aerosol was built in accordance with a

United States Naval Research Laboratory report by W. Echols

and J. Young [Ref. 22] and personal communications with

0. M. Griffin. Air from a regulated and filtered supply was

piped to an atomizer nozzle submerged in a jar of DOP. In

addition to the pressure regulator, there was a throttling

valve on the air line and the depth of submergence of the

nozzle could be adjusted. The aerosol from the top of this

jar was collected and led to a jet impactor in a second jar.

The effect of the jet impactor was to remove excessively

large particles and to make a smaller, more uniform aerosol

particle size. The aerosol was then led to a settling

chamber where the pressure was reduced to nearly atmospheric

level by bleeding off the excess aerosol. Flexible injec-

tion tubing of 0.250 inch inside diameter connected the

settling chamber to the contraction section where the

aerosol entered the channel. The aerosol generating equip-

ment is pictured in Figure 5. The character of the aerosol

used was such that visualization of the flow was facilitated

by illuminating slits along the channel and viewing them at

approximately 90° to the light. A single 650 watt Colortran

light, model 100-071, was placed on the side of the channel

opposite the viewer. The light was collimated with a paper

cone lined with aluminum foil. Flat black paper was taped

to the outside of the channel walls to make four narrow

28

Figure 5. Aerosol Generating Equipment

29

slits. The light beam was directed through one of these

slits so that a small area of the flow could be visualized.

A Nikomat 35 mm camera and a Nikon 55 mm f3.5 Macro-Nikkor

lens was used to obtain photographs of the flow patterns.

A sketch of the lighting and camera arrangement is shown in

Figure 6. The visualization techniques and procedures will

be discussed more fully when the photographs are presented.

In addition to visualizing the flow it was planned to

measure the velocity profile. Due to the size of the chan-

nel, a sub miniature hot wire anemometer probe, Thermo-

Systems Inc. model 1279, which had a 90° bend in the supports

was used to measure the turbulence levels. This probe was

located three inches from the center line of the channel and

20 inches of arc length downstream from the start of the

curved portion of the channel. The probe was placed in a

plug and inserted through the convex wall of the channel.

The inside surface of this plug fit flush with -the wall of

the channel, and had the same radius of curvature. The

0.035 inch diameter body of the hot wire anemometer protruded

from the plug and was fastened to a micrometer barrel which

was mounted on a machined block attached to the outside of

the channel. In this way, the movement of the wire from

very close to the concave wall of the channel to near the

convex wall could be controlled and measured. This arrange-

ment is shown in Figure 7. The leads from this probe were

connected to a Thermo-System Inc. hot wire anemometer

system, model 1050, including a Hewlett-Packard RMS Meter.

30

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micrometerbarrel

machinedsupportblock

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probe

Figure 7. Hot Wire Anemometer Arrangement,

32

A Thermo-Systems Inc. calibrator, model 1125, was used to

calibrate the hot wire anemometer.

In order to measure the pressure drop along the length

of the channel, pressure taps and a micromanometer were

required. Pressure taps were located every three inches

along the entire length of the channel, starting two inches

from the beginning of the entrance section and alternately

set 0.2 50 inches to the right or left of the center line.

Along the curved section, the three inch spacing was

measured on an arc of one foot radius, resulting in a total

of 28 pressure taps with number 17 being two inches into

the curved section. These pressure taps consisted of a

0.040 inch diameter hole drilled into the channel with a

short plexiglas tube glued into a 0.250 inch hole counter-

bored over the smaller holes. Flexible tubing with pressure

fittings was connected to these short tubes to facilitate

connecting and disconnecting the micromanometer.. The instru-

ment used in this investigation was an E. Vernon Hill and

Company Type C micromanometer with a range up to two inches

of water and a sensitivity of 0.001 inches of water. This

micromanometer is of the re-zeroing type where the reservoir

is raised to return the meniscus to the same level it had

before the pressure was applied, and the amount the reser-

voir is raised indicates the pressure. A photograph of the

channel and the pressure measuring apparatus is shown in

Figure 8

.

33

The bulk temperature of the air was measured six inches

after the entrance of the channel , and three inches after

the heating plate, by four thermocouples spaced across the

channel at each location. Copper-constantan thremocouples

were used. They were located at different heights in the

channel and connected in parallel to give the average bulk

temperature. An ice bath reference junction was provided,

and these thermocouples and the others used in this experi-

ment were connected through a switch to a Dymec digital

voltmeter, model 2401B. These thermocouples and a sample

of the others used in this study were calibrated in a

Rosemont Laboratory Calibration System. In this constant

temperature bath, the temperature was determined by a

platinum resistance thermometer.

A wall heater was constructed for the straight test

section of the channel. This heater was formed of forty

0.004 inch thick, 0.125 inch wide strips of nichrome run

across its 11.5 inch width. These metal ribbons were

placed in a mold by hand and spaced approximately 0.010

inches apart. The streamwise length of the plate was

5.50 inches. A copper-constantan thermocouple was welded

to the center of each of the nichrome strips. Power leads

were connected to one side of each strip four inches from

the center of the plate. On the other side of the center-

line, the strips were connected together in pairs to make

20 electric circuits. The 20 heaters were individually

supplied by a specially constructed, regulated

35

high current D.C. power supply and a 20 channel power con-

troller. The D.C. power supply had a 200 amp capacity and

a maximum RMS ripple of one millivolt. The power controller

was a 20 channel potentiometer biased series regulated

voltage source with a temperature compensated reference

voltage. The components of this power controller were

mounted on a common heat sink in order to improve the

stability of operation. The power supply and controller

are shown in operation in Figure 9 . When all the power

leads and thermocouple leads were in place, APCO 210 epoxy

resin with APCO 180 hardener was poured into the mold. This

thermal set plastic was cured at room temperature for

approximately 2 4 hours. When the heater plate was removed

from the mold, it was found to be warped. A metal frame was

constructed, securely attached to the outer edges of the

0.250 inch thick casting and adjusted to produce a completely

flat plate. A further problem experienced with the casting

involved some of the nichrome strips near the trailing edge

of the plate that had pulled away from or sunk into the

plastic during drying. This resulted in rough spots on the

surface of the plate. The loose strips were cut and removed

from the heater, and the low spots, from the 3 3rd strip to

the end of the plate, were filled with epoxy. The plate was

then lightly sanded, resulting in a smooth surface. The

first 16 heating circuits remained useable, which resulted

in a heated area 8.125 inches wide by 4.384 inches in the

flow direction.

36

Hea t i n j Mpps ra tus

Fi jur?; 9

37

After the other experimental work was completed, a

5.50 inch piece of the upper wall of the channel, between

the two foot ten inch station and the three foot 3.50 inch

station, was removed. The heater was placed into this cut,

becoming part of the straight wall of the channel with the

nichrome ribbons exposed to the flow of the air. The top

of the heater and the surrounding channel walls were then

covered with a 0.125 inch thick layer of Johns-Manville

Company Min-K insulation. The Min-K was covered with a one

inch layer of glass wool. Each piece of the apparatus was

cleaned and carefully prepared before the experimental work

began.

38

III. FLOW VISUALIZATION

A. EXPERIMENTAL CONSIDERATIONS

The aerosol generating equipment was found to be very

versatile as to the amount and quality of the aerosol pro-

duced. After much trial and error, good quality aerosol was

found to be that of relatively small particle size and low

exhaust pressure. This aerosol appeared much like cigarette

smoke when discharged into the room. A supply air pressure

of 30 pounds per square inch, or slightly less, an atomizer

nozzle submergence of approximately 0.50 inches, and a low

settling chamber pressure were the conditions which produced

this aerosol. Producing this quality aerosol and getting

the right amount of it into the channel proved to be impor-

tant in visualizing the flow. Viewing the aerosol at the

proper angle relative to the incident light was also impor-

tant. For lighting, the best arrangement was to place a

single, strong light near the center of curvature of the

channel and to shine it radially outward. The collimated

beam of light was directed through narrow slits formed from

black paper taped to the channel. If this small area of

the flow was viewed from an angle tangent to the channel,

the vortex patterns could be readily seen. This viewing

location was the same as the camera location shown in

Figure 6. Four such narrow slits were made along the curved

section of the channel. The first slit was located 7.50

39

inches from the beginning of the curved portion of the

channel. The second was 12.00 inches downstream from the

first, and the third was 11.50 inches farther downstream.

The last slit was located at the end of the curved section,

8.0 inches from the third slit. At the location of this

fourth slit, the channel enters the final straight section.

To obtain photographs of the flow patterns in each of

these slits, the camera with the 55mm macro lens was mounted

on a tripod and moved as close to the channel as possible.

The camera was aligned with the axes of the vortices by

placing the camera on a tangent to the channel. In the case

of the first and second slits, the flow was coming toward

the camera, and in the case of the third and fourth slits,

the flow was away from the camera. Due to the many reflec-

tions observed from the surfaces of the plexiglas, the room

was darkened while the photographs were taken. A Gossen

"Luna-Pro" exposure meter with a variable angle spot meter

attachment was used to assist in obtaining the proper

exposure settings. Under -these difficult conditions, several

attempts were required before satisfactory photographs were

obtained.

Other arrangements for viewing the flow were also tried.

Looking radially inward through the channel toward a less

intense light located near the center of curvature showed a

different type of pattern. The patterns seen were streaks

or lines in the direction of the flow. These patterns were

similar to those seen by other investigators including

40

L. Persen [Ref . 15] . Several methods of injecting the aero-

sol into the flow were also tried. The methods involving

small tubes or fittings did not work because sufficient

aerosol would net pass through these passages. The results

obtained from the other methods were for the most part the

same as for the method chosen. In a few cases, the early

development patterns appeared differently, depending on how

the aerosol entered the channel. The fully developed vortex

pattern, however, always appeared the same.

B. PATTERNS AND PICTURES OF THE FLOW

The pictures of the vortex patterns are presented in

Figures ten throvigh 15. Below each picture is a slit number,

a Reynolds number and a Dean number indicating the location

of the slits, the flow rate and the value of the character-

istic parameter for secondary flow respectively. For the

channel used in this investigation, the Dean number is

related to the Reynolds number by

De = 0.073973 * Re.

The line below the vortex patterns in these enlarged pictures

is the concave wall of the channel. The line of reflected

light above the vortices is the inner wall of the channel

which is 0.25 inches from the concave wall. The photographs

taken at slit four appear slightly different than the others

because the lighting angle and therefore the reflections

from the plexiglas were slightly different. The pictures in

each figure are arranged with slit one at the top of the page

41

and the downstream slits in order below the first. The

Reynolds numbers of the pictures in each figure are intended

to be of the same magnitude. There is not complete repre-

sentation of the same Reynolds numbers or an even sampling

of the slit location because of the moderately high failure

rate of obtaining pictures. In spite of this difficulty,

the photographs presented are indicative of the Taylor-

Goertler vortices as observed. The figures are in order of

increasing average Reynolds number.

A number of distinct patterns appear repeatedly in these

pictures. At the lowest velocities, the aerosol remains

near the concave wall in a pattern that is not clearly a

fully developed vortex flow. Examination of these patterns

in Figures 10a, 10b, 11a, and 12a show that the ends are

curled up. This same pattern has been observed farther

downstream in slit three at flow rates lower than pictured

here. This pattern occurs predominately near the beginning

of the curved section. From extended observation of both

this pattern and the motion at the curled end, it is felt

that they represent the beginning of a developing weak vortex,

The next form considered looks much like a "mushroom."

This pattern is exemplified by Figures 12b, lib and the right

side of 10c. The appearance of two different patterns under

the same conditions, as in Figure 10c, was observed occasion-

ally. Such occurrances are possibly explained in terms of

the actual finite width of the channel which produces a

stronger velocity gradient near the side walls than in the

42

middle. A difference in the placement of the aerosol injec-

tion tubes could also contribute to the appearance of dif-

ferent patterns at the same flow rate. Furthermore, as the

vortices develop and occupy more space in the channel, some

will be distorted to make room for the others. At higher

Reynolds numbers, the top of the curl pattern, as in

Figures 13a, 14a, and 15a, could be considered a "mushroom"

on a small scale. The "mushroom" shape occurs farther down-

stream or at higher velocities than the first pattern

discussed. This shape always proceeds in distance or flow

rate, the first clear vortex patterns. The "mushroom" has

been observed to lead directly into distinct vortices and

therefore should be considered as a developing vortex.

The next pattern is the round form shown in Figures lie,

13b, 14b, and 15b. These forms represent a counter rotating

pair of vortices with flow leaving the concave wall in the

center and curling back toward the wall on either side.

These round vortex patterns do not appear to occupy all the

space in the channel. In Figure 13b there appears to be

space above the vortices, and in Figure 15b the space appears

below the vortices. However, the aerosol which is what was

photographed does not necessarily show the entire vortex

pattern. The aerosol shows a streakline line pattern which

may be somewhat dependent on how the aerosol enters the

developing vortices. As the flow rate and the downstream

distance increases, the fully developed vortex patterns

appear. These vortices, as shown in Figures lOd, lid, and

43

12c are the full height of the channel and occupy a nearly

cellular area. Such patterns could be considered as fully

developed Taylor-Goertler vortices.

As the velocities continued to increase/ the vortices

grew in width until they began to crowd one another, as seen

in Figures 13c, 14c, 14d and 15d. If the distance or the

Reynolds number were increased farther, an unsteady oscil-

latory motion developed in the spanwise direction. This

action which could be observed by eye, was difficult to show

in a still photograph. Figure 13d is the result of such an

attempt. The observations indicated that the sideways oscil-

lations of the vortices may involve the absorption of one

vortex by an adjacent growing one. It was apparent that an

increase of oscillatory motion would have led to turbulence.

It was evident from the photographic study that the

vortices did require some distance to develop. Several

different flow patterns were observed during the development

At high-flow rates, less distance was required for the

vortices to become full size. There were several complex-

ities observed in the flow patterns which could lead to

further studies.

44

(a). Slit 1 Re = 609 De = 45.0

(b) . Slit 3 Re = 609 De = 45.0

(c) . Slit 4 Re = 662 De = 49.0

(d) . Slit 4 Re = 742 De = 54.9

Figure 10

45

(a). Slit 1 Re = 874 De = 64.7

(b) . Slit 3 Re = 742 De = 54.9

c) . Slit 3 Re = 874 De = 64.7

(d) . Slit 4 Re = 874 De = 64.7

Figure 11

46

(b) Slit 2 Re = 954 De = 70.6

c) . Slit 3 Re = 927 De = 68.6

(d) . Slit 4 Re = 874 De = 64.7

Figure 12

47

(a). Slit 1 Re = 1033 De = 76.4

(b) Slit 2 Re = 1033 De = 76.4

(c) Slit 3 Re = 1033 De = 76.4

(d) Slit 4 Re = 1033 De = 76.4

Figure 13

48

(a) Slit 1 Re = 1166 De = 86.3

(b) Slit 2 Re = 1166 De = 86.3

c) Slit 3 Re = 1033 De = 76.4

(d) Slit 3 Re = 1166 De = 86.3

Figure 14

49

a) Slit 1 Re = 1457 De = 107

(b) Slit 2 Re = 1245 De = 92.1

c) Slit 2 Re = 1457 De = 107

(d) Slit 3 Re = 1298 De = 96.0

Figure 15

50

IV. VELOCITY AND TURBULENCE MEASUREMENTS

A. EXPERIMENTAL PROCEDURES

The sub miniature hot wire anemometer probe to be used

to obtain the mean velocity profiles was connected to the

anemometer. The cold resistance of the wire was measured

as 7.90 ohms. Based on an overheat ratio of 1.5, the

anemometer's operating resistance was set at a value of

11.85 ohms. The anemometer's signal was fed into an oscil-

liscope so that the patterns could be observed. Stability

and trim controls were adjusted for maximum frequency

response. The probe was centered in the chamber of the

calibrator. A filtered air supply was connected to the

calibrator, and the pressure drop across the calibrator's

flow nozzle was measured by an incline manometer. With the

anemometer switched to run, the flow rate through the

calibrator was adjusted to different levels. The pressure

drop in inches of water, the bridge output voltage, and the

RMS voltage were recorded. This data was reduced by the

calibration program shown in Appendix B and the square of

the output voltage was plotted as a straight line against

the square root of the flow velocity in feet per second.

The slope and intercept of that straight line were the

values required to reduce further data, and were found to

be 0.710 and 2.020 respectively.

51

The hot wire anemometer was removed from the calibrator

and inserted as far into the plexiglas mounting plug as

possible. The mounting block for the micrometer barrel was

attached to the stem of the probe, and the distance from

the base of the block to both the top of the plug and to

the hot wire were measured. The difference in these machine

vernier readings was 0.050 inches. This was as close as

the hot wire anemometer could come to the convex wall of the

channel. The micrometer barrel and a straight piece of

steel wire had been used to measure the distance across the

channel at the location of the plug. This distance was

found to be 0.240 inches, indicating that the probe would

have a travel of 0.190 inches across the channel. When the

plug was inserted into the channel and the micrometer was

attached to the mounting block, the micrometer reading was

0.460 inches. The hot wire anemometer was carefully moved

until the supports for the wire contacted the concave wall

of the channel. The micrometer reading at this point was

0.270 inches, verifying the 0.190 inches of travel. The

wire itself was very close to, but not touching, the wall.

The data was taken between these micrometer settings of

0.270 inches and 0.460 inches.

Before the data was taken, the channel, the blower, and

the electronic instruments were allowed a warm up period of

about one hour. After the warm up, the flow rate was

adjusted and a minimum of 30 minutes was allowed for steady

state to be achieved. At each flow rate, the temperature of

52

the incoming fluid was measured and the micrometer was moved

0.010 inch between data positions. At each position, the

flow rate was checked to see that it was stable, and the

micrometer setting, the output voltage, and the RMS voltage

were recorded. This data was reduced by a computer program

which plotted the mean velocity and the turbulence intensity

against the distance from the concave wall. This program

and the printed results are contained in Appendix B.

B. VELOCITY PROFILES

The velocity recorded and shown in the figures which

follow was not just the mean velocity component down the

channel, but the vector sum of the streamwise and the radial

velocity components. The portion of the response that was

due to the radially inward or outward flow could not be

determined by this one set of measurements. However, at the

flow rates of this experiment, a nearly parabolic velocity

profile would be expected if the secondary flow was not

present [Ref . 17] . This has also been shown for curved

channels of square cross section [Ref. 13] . The position

of the hot wire relative to the vortices was not determined

during the tests. Considering the nature of vortex motion,

a shift in the position of the vortex relative to the probe

could have made a difference in the velocity indicated. The

location of the hot wire anemometer was very close to the

second slit used in flow visualization. Thus, for each flow

rate, the vortices may be considered to have developed to a

corresponding level.

53

Figures 16 through 22 show the mean velocity in feet per

second versus the distance from the concave wall in hun-

dredths of an inch. There was a maximum uncertainty of

approximately 0.010 inches on the location of the wall. The

uncertainty in the velocity measurement using the T.S.I,

system was at most 0.05 feet per second. Figure 16 shows

a parabolic shape with a small increase in velocity indicated

near 0.060 inches. This point may be associated with an

active point of a developing vortex. The step in the velo-

city shown near the wall could be associated with the probe

actually contacting the wall, or with the error in position-

ing. In this case, and in all other graphs, the curve should

be extended to zero at the wall. Although a vortex was most

probably present, Figure 17 shows no irregularities in the

velocity profile. Perhaps the probe was located near the

center of a vortex at this time. The same comment could

be true of Figure 18. The magnitude of the velocity is

increasing as would be expected.

Figure 19 shows a variation from the parabolic profile.

As indicated by Figure 15b, the vortices were growing at

this Reynolds number and a different portion of the vortex

may have been acting on the wire. In Figure 20 the velocity

profile appears much different. In this case, the radial

velocity component was larger near the wall than in the

center of the channel. Figure 21 is much the same as

Figure 20. A different profile appears in Figure 22. This

last profile could be explained in terms of the hot wire

54

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moving through the edge of a vortex. Near the wall the

outward radial velocity would increase the mean velocity

measured. In the center of the vortex there may be little

effect on the hot wire anemometer. Near the top of the

vortex the radial component of the velocity would again

affect the probe. At the other edge of this vortex, the

inward movement of the fluid would produce the same velocity

profile. The length of the hot wire, 0.06 3 inches, would be

sufficiently small to traverse the vortices in this manner.

A schematic drawing of the probe in a typical position

relative to the vortices is shown in Figure 23.

C. TURBULENCE LEVELS

The turbulence intensity was defined as the root-mean-

square of the velocity fluctuation divided by the mean

velocity. These values were calculated by the data reduc-

tion program from the RMS meter readings and the bridge

output voltages. The results were plotted against the

distance from the • concave wall in Figures 24 through 30.

The turbulence level was quite low in the center of the

channel and higher near the walls as a result of the low

mean velocity near the walls.

The change in the turbulence level near the wall in

Figure 24 was due to the step change in the velocity profile

for that location and Reynolds number. Otherwise, Figures

24, 25, and 26 are self explanatory. In Figure 27, the

curve representing the turbulence intensity is not as smooth

as the other curves. Large increases in the fluctuations of

62

convex wall

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Figure 23. Typical Position of Hot Wire

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the RMS meter started at this flow rate, resulting in an

increased uncertainty in the level of turbulence. With the

RMS meter reading at the 0.00350 volt level, the value would

suddenly increase as much as 0.010 and drop back again.

The frequency of this random increase was of the order of

30 cycles per minute. One possible cause of these large

jumps in the local velocity would be the sideways oscillatory

motion of the vortices. During such sideways motion, dif-

ferent portions of the vortices could cross the hot wire

anemometer causing the fluctuations. These fluctuations

were present during the measurements that led to Figure 28.

This figure represents a uniform turbulence level across the

channel. In Figures 29 and 30, the level of the turbulence

was larger than in the previous cases. The magnitude of

the fluctuations was also doubled or larger. The occasion-

ally high value of the turbulence intensity would indicate

turbulent flow. In other words, there was bursting to

turbulent flow at the higher flow rates.

The growth and motion of the vortices within the channel

has been indicated by the mean velocity profiles and turbu-

lence levels presented here. Evidence for the transition

to turbulent flow has also appeared.

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V. PRESSURE DROP MEASUREMFNTS


To measure the pressure drop along the channel required

a great deal of care. Each of the 28 pressure taps were

cleaned and checked for tightness with high pressure air.

The flow rate in the channel was adjusted and allowed to

stabilize for approximately one hour before data was taken.

The micromanometer was positioned on the table, leveled,

and filled with the special fluid provided by the manufac-

turer.

The meniscus of this instrument was viewed under a

magnifying glass and carefully adjusted so that the curve of

the meniscus just touched the hair line on the glass tube.

This was established as the zero position. It was found

that the movement of even a small weight on the laboratory

table would effect the level and therefore the zero of this

micromanometer. Once the micromanometer was zeroed, no

weight movement was permitted and the investigator did not

touch the table while measurements were taken.

The pressure fittings for each of the pressure taps were

numbered and taped to the edge of the channel. The pressure

drop between pairs of pressure taps at one foot intervals

was measurable, but contained too much uncertainty to be of

use. Typical values of 0.007 inches of water were measured

as the pressure drop per foot. However, values of 0.001

72

inches higher or lower could be obtained in the same posi-

tion of the channel at the same flow rate. It is possible

that part of these variations were due to small fluctuations

in the flow rate caused by the centrifical blower. Other

causes were considered, but such fluctuations could not be

positively identified or eliminated.

The difference between the local pressure at each tap

and atmospheric pressure proved to be more useful. At

successive locations down the channel the magnitude of this

pressure difference became larger and hence the relative

error was less. In taking each pressure reading, the zero

of the manometer was checked and then the flexible tubing

from the micromanometer was connected to the pressure fit-"

ting. The pressure tap number was checked and the inlet

temperature reading was taken and both were recorded. The

manometer reservoir level was adjusted with the fine screw

dial until the meniscus of the fluid had returned to its

zero location. The rotometer was checked to insure that

the flow rate was correct and stable. The meniscus of the

fluid was then rechecked and if it had remained in the zero

position, the pressure indicated by the dial was recorded

along with that flow rate. The micromanometer was discon-

nected from the pressure fitting and then adjusted to

return the meniscus to its zero position. If the dial

indication was again zero, the pressure difference obtained

was considered to be correct. Occasionally the micromano-

meter would not re-zero and the pressure measurement at

that location was retaken after the instrument was adjusted.

73

The pressure drop over two foot portions of both the

straight section and the curved section were taken to verify

the other results. For measurement over these intervals

the upstream pressure tap was connected to the top of the

micromanometer . The downstream tap was connected to the

reservoir of the micromanometer as in the previous setup.

B. PRESSURE DROP

The pressures measured were plotted downward on Figure

31. The top of this graph represents atmospheric pressure,

and the amount that the static pressure in the channel was

below the atmospheric level can be seen. The pressures are

plotted against pressure tap number. The 28 pressure taps

were three inches apart along the channel, and numbers one

through 16 were on the straight portion of the channel.

The pressure for six different Reynolds numbers were plotted,

and the same patterns can be seen in each. The irregulari-

ties in the curves around pressure taps 14 and 15 and around

taps 18 and 19 were of a smaller magnitude than the uncer-

tainty for these measurements. Pressure taps 14 and 15 were

in the straight portion of the channel and no cause for the

irregularities were apparent. However, burs or other dis-

turbances which were undetected during construction of the

channel may have caused these irregularities. Pressure taps

18 and 19 were in the curved portion of the channel. In

this case, the cause for the pressure irregularities may

have been a difference in the position of the pressure taps

relative to the vortices. A pressure tap that was being

74

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75

affected by the stagnation of outward moving fluid and one

under the influence of moving fluid would probably give dif-

ferent pressure indications. Further study would be required

to confirm that this was the cause of the irregularities.

The slope of the pressure curve in the straight section

should compare with the exact analytic solution for laminar

flow between parallel flat plates. The results of such an

analysis [Ref. 6] show that the pressure drop is

d* a2

where a is half the plate spacing and y is the viscosity of

the fluid. U is the mean velocity of the parabolic velocity

profile between the parallel flat plates. However, the

experimentally determined U was the volumetric flow rate

through the rectangular channel divided by the cross sec-

tional area. For a rectangular channel of aspect ratio 40,

the difference in these two values of U would be small.

The analytic value of U for a rectangular channel could be

obtained from an infinite series given in the literature

[Ref. 19]. The first term of this series would reduce U

for the rectangular channel to 98.5315 percent of the value

for an infinite parallel plate construction. The remaining

terms of this series would rapidly become smaller. The

experimental uncertainty was larger than this analytic

difference, so that no correction was applied to the theo-

retical pressure drop.

76

The experimental pressure drop was calculated by a com-

puter program for both the straight and the curved portions.

This program fit a least square straight line to nine con-

secutive pressures representing a two foot section of the

channel. The straight section of the channel was represented

by pressure taps three through eleven inclusive. This

avoided both the entrance region and the irregularities that

were noted before. The curved portion was represented by

pressure taps 19 through 27, where the vortices were most

developed. The slope of each line was considered as the

pressure drop in that portion of the channel. This program

and its results are shown in Appendix C. These pressure

drops in inches of water per foot of length, along with the

analytic pressure drop calculated with the experimental

values of U are shown in Table I. The percentage increase

in the pressure drop for the curved section was calculated

with respect to both the theoretical value and .the measured

value of pressure in the straight section. These results

were also tabulated in Table I. The negative values shown

were probably due to experimental uncertainties.

Because more significance might be obtained from a com-

parison of pressure drops in the straight section with

pressure drops in the curved section/ where the vortices

were fully developed, the same computer program was used to

consider one foot section of the channel. The straight

portion was represented by pressure taps six through ten,

and the curved section by the last one foot, pressure taps

77

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24 through 28. The computer results for these calculations

are shown in Appendix C. The resulting pressure drops and

percentage increases are shown in Table II. These results

showed no negative values for the increase, and had the same

magnitude as the other results.

To verify and compare the other results, direct measure-

ments of the pressure drop in the two foot intervals used

above were made. The error in this type of measurement could

have a magnitude as high as 0.002 inches of water. Never-

theless, the results, along with the resulting percentage

increases in the pressure drop, are presented in Table III.

The values of the pressure drop measured in the straight

section of the channel compared moderately well with the

theoretical values. This comparison further supported the

methods used to obtain the data.

The percentage increase in the pressure drop from the

straight section to the curved section was arrived at with

three different measurements, and calculated with both the

theoretical and the measured values in each case. The six

different sets of percentage increases were plotted against

the Reynolds numbers in Figure 32. The average increase for

each Reynolds number is also identified in Figure 32. These

results have considerable scatter due to the effects of

experimental uncertainty. With an increase in the Reynolds

number, there was an increased influence of the Taylor-

Goertler vortices on the pressure drop. Even though the

secondary flow was present, the increase in pressure drop in

the curved section was small at the lower flow rates. At

79

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Figure 32. Percentage Increase in Pressure Dropvs. Reynolds Number.

82

higher Reynolds numbers, where the vortices became more

active, the pressure drop in the curved section increased

more than in the straight section.

This data was nondimensionalized as the pressure drop

coefficient, which was defined as

Cp = -*Hp u

2

where p was the pressure measured at each pressure tap.

There was a larger error bound on these values because the

uncertainty in U was added to that in the pressure drop.

The calculation of the pressure drop coefficient was done by

hand on a desk calculator. A sample calculation for the

lowest Reynolds number is given in Appendix C. The results

were plotted in Figure 33.

83

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8 4.

VI. HEAT TRANSFER


In order to determine the effect of Taylor-Goertler

vortices on heat transfer it was necessary to measure the

heat transferred from a uniform temperature surface in one

wall of the straight portion of the rectangular channel.

The heat transfer coefficients of interest were defined by

the equation

Q = H * A*AT

where Q was the heat convected, H was the heat transfer

coefficient and A was the area of the heated surface. The

temperature difference/ AT, used depended on the particular

heat transfer coefficient to be found. For the overall

heat transfer coefficient, the log mean temperature differ-

ence was used. The mean heat transfer coefficient was

defined with the difference equal to the wall temperature

minus the inlet bulk temperature of the fluid. The temper-

ature difference used for the local heat transfer coefficient

was the difference between the temperature of the individual

heating element and the bulk temperature of the fluid enter-

ing the control volume below that element. The differences

in these definitions and the areas used will become clear

during the discussion of the results.

A heater plate with 16 separate resistance heating

elements placed along the downstream length of the heater

85

was designed and built as described earlier. To measure the

heat transferred from a uniform temperature surface it was

necessary to control the voltage to each of the 16 heating

elements. The resistance and voltage at each heating strip

was to be measured to determine the power generated. The

temperatures were to be measured at two locations on each

heating elements

.

After the heater was in place in the channel wall, the

resistance of each heating elements was measured with a

Rosemont Commutating Bridge, Model 920A. The high current,

regulated D.C. power supply fed the power controller which

governed the voltage across each heating element. This

specially constructed equipment had never been previously

used or tested. The output level of each of the 20 circuits

was set as low as possible. The input voltage was adjusted

until the power controller's reference voltage was between

the maximum recommended value of 8.800 volts and 8.700 volts.

To insure that the heater plate did not become too hot, flow

was established in the channel. The voltage in each of the

first 16 power circuits was slowly increased, and the tem-

perature along the plate was checked frequently. At any

given power setting, four to five hours were required for

the heated to come to equilibrium. As expected, in order

to maintain the same temperature for each strip, the circuits

near the leading edge of the plate required more power than

the other circuits. If the variation of the power settings

was not correct, the temperature distribution approached by

86

the plate was not uniform. When it was determined that the

temperature would not be uniform, the power to the low

temperature heating elements was increased and another

settling period was allowed. Considerable time was required

to obtain the first uniform temperature distribution. There-

after, to obtain higher temperatures across the plate, each

potentiometer's adjustment screw was turned equal amounts.

As equilibrium was approached, small adjustments were made

to remove temperature irregularities along the plate. In

this manner, a uniform temperature could be achieved in a

minimum time of about six hours.

During continued testing of the equipment, it was found

that the heater plate could sustain temperatures as high as

200° F without damage. Higher operating temperatures were not

attempted. During the early testing of the power controller

the power circuit voltages were found to fluctuate even

when the reference voltage was steady. Examination of the

power circuit voltage showed a 60 cycle A.C. voltage super-

imposed on the controlled D.C. voltage. The source of this

A.C. voltage was the grounded side of the regulated power

supply. Consequently, downstream from the protective

diodes, a 0.05 uf capacitive connection was made across the

inputs to the power controller. This connection removed the

A.C. voltage and the fluctuation in the power levels. When

the power circuits were loaded, the stability was found to

be approximately ±.004 volts over a 30 minute period. The

power available was actually more than required for these

tests

.

87

After the equipment was tested, data collecting began.

At the beginning of each data run, the flow rate was estab-

lished and power was supplied to the heater. After two

hours, an ice bath was made and the temperatures along the

plate were periodically checked. During the third hour,

adjustments were made in the temperature and flow rate. The

reference voltage was also checked at this time so that its

variation could be determined later.

The temperatures measured and recorded included the bulk

temperatures of the entering and exiting air, the tempera-

tures at the 32 thermocouple locations along the heater, and

the temperatures of the four locations on the outside of

the heater plate. The thermocouples were connected to a

digital voltmeter by a set of switches. The switching and

recording of the thermocouple outputs were accomplished by

hand. As equilibrium was approached, the temperatures were

recorded at 30 minute intervals. When the temperatures had

stopped changing within the limits of accuracy of this

experiment, the values were recorded as data. The reference

voltage and flow rate were rechecked to insure no change had

taken place. Then the digital voltmeter was connected to

the power controller and the voltage across each heating

element was recorded. The digital voltmeter was reconnected

to the switches. The temperatures were recorded one addi-

tional time, and the flow rate was again checked. The data

was considered valid if no changes had taken place. The

process of taking data took approximately 30 minutes.

88

B. RESULTS

In order to calculate heat transfer coefficients from

the data collected, it was necessary to account for the heat

losses. In the geometry of this experiment, there were

several potential losses. The largest was expected to be

the conduction through the back side of the heater plate.

For this reason, the insulation, described earlier was placed

over the heated area. To obtain an estimate of the conduc-

tion losses through the plate, a finite difference model of

the experimental situation was used.

A computer program was used to obtain a numerical solu-

tion of the transient and steady state temperature distribu-

tions and the heat flows of the three dimensional model.

The program used is entitled TRUMP and was developed at the

Lawrence Radiation Laboratory [Ref. 23]. This program was

implemented for the IBM 360 computer at the Naval Postgrad-

uate School by C. Erbayram [Ref. 24]. Because .of the length

of this program and its output, only the steady state results

of one run are included in Appendix D.

In the computer program, one half of the symmetric

arrangement of the heater plate, the insulation, and the

surroundings were modeled by a total of 197 node points and

606 thermal connections. The initial temperature of all

the nodes, the heat generated at the nodes representing the

heating elements, and the heat transfer coefficients repre-

senting convection to the surroundings, were input data.

The results were the steady state temperature at, and the

89

heat flow from each of the nodes. The time to reach steady

state was also given by the TRUMP program, and compared well

with the six hours required in the actual experiment.

After the TRUMP program was tested, the heat generation

rates actually measured were input to the program. The heat

transfer coefficients representing the convection to the air

in the channel were initially estimated and then iterated

until the wall temperature arrived at by the computer program

was the same as that actually measured. In this way, the

TRUMP program was used to model each experimental run. The

heat flows from the nodes representing the heating elements

were used to calculate losses in the reduction of the exper-

imental data. This process will be described next.

Temperatures and several heat transfer coefficients were

computed from the data collected by the computer program in

Appendix D. A table of the thermocouple calibration data

was included in the computer program. All of the tempera-

tures were calculated by interpolating the thermocouple

readings in this table. The local temperatures along the

wall and the average wall temperatures are shown in the

computer results contained in Appendix D.

The log mean temperature difference was calculated from

the equation

Tbo - TbiT£m Tw - Tbi

Ln Tw - Tbo

where Tw was the average wall temperature, and Tbi and Tbo

were the bulk temperatures of the air into and out of the

90

test section respectively. Using a table of fluid prop-

erties vs. temperatures, the fluid properties were evaluated

at the average bulk temperature. The overall heat transfer

coefficient was then calculated from the equation

Ho = P * CP * V * (Tbo - Tbi)Aw * TZm

where V is the volumetric flow rate, and Aw is the area of

the heated surface. The mean heat transfer coefficient was

also calculated in the program. With the mean temperature

difference defined as

Tm = Tw - Tbi

the mean heat transfer coefficient was calculated from the

equation

_ p * cp * V (Tbo - Tbi)rim — ——

—

.

Aw * Tm

The symbols are the same as used for the overall heat trans-

fer coefficient. These two values appear in the computer

output.

To calculate the local heat transfer coefficients, the

power generated at each heating element was calculated from

the resistance of the strip and the voltage across it. Two

values of heat flow obtained from the TRUMP program were

also read into the computer for each heating element. Since

one node in the TRUMP program represented two adjacent heat-

ing elements, the heat flow values from the TRUMP program

were divided in half to represent the losses from the indi-

vidual heating elements. At the end elements, the conduction

•91

losses along the channel wall were attributed entirely to

the first and last heating element. The two heat flows for

each heating element represented the conduction loss and

the heat convection into the channel. A loss factor was

calculated by the computer program by dividing the conduc-

tion loss by the sum of the heat flows. The power generated

in each strip was multiplied by one minus the loss factor

to give the useful heat at each element.

The radiation loss was accounted for by the equation

Qr = e * a * As * (Tw4

- Tbi4

)

where e was the emissivity of the nichrome taken as 0.7 and

where As was the surface area of each heating element,

0.014783 square feet. The temperatures used were converted

to degrees Rankine, and the Stefan-Boltzmann Constant, a,

was equal to 0.199967 x 10~ 10 BTU/min ft2

°R. The radiation

loss was subtracted from the heat generated at each element

less the heat conducted out of the plate to give the heat

convected into the channel.

The last loss accounted for was the convection from the

heated air in the channel to the surroundings. Two thermal

resistances were calculated for this type of loss. The

first included terms for the forced convection in the chan-

nel, the conduction through the plexiglas lower wall, and

the free convection from the horizontal outside surface of

the plexiglas. The second thermal resistance included terms

for the forced convection in the channel, the conduction

92

through the aluminum side walls, and the free convection

from the vertical outside surfaces. The forced convection

was represented by the mean heat transfer coefficient

calculated by the program. The free convection was repre-

sented by correlation formulas given in the text, Heat

Transfer , by J. Holman [Ref. 20]. The convection loss from

the air through the walls of the channel was calculated from

n _ A * (Tb - 70.0)gCR

where Tb was the bulk temperature of air entering the con-

trol volume below each heating element, R was the thermal

resistance for each case, and A was the corresponding area.

The total loss of this type was the sum of the two Qc's

calculated with each of the two thermal resistances. In

summary the heat convected into the channel from each heat-

ing element, designated as p(j), was the power generated

at each element minus the conduction losses through the

plate and the radiation losses. The heat which raised the

bulk temperature of the air, was this convected heat, p(j)

,

minus the convection losses from the air.

The new bulk temperature of the fluid after passing a

heating element was therefore calculated by the computer

according to

Tb(j+D = Tb(j) + [p(j>-Q;(jil.J J p* cp* v

The local heat transfer coefficient was then easily calcu-

lated from the equation

93

HM) = P* cp* V * [Tb(j + 1) - Tb(j) ]KJ ' As * [Tw - Tb(j) ]

where As was the area of one heating element. The distance

of each heating element from the leading edge of the plate

was also calculated and printed out along with the power

convected, the bulk temperature of the fluid, and the local

heat transfer coefficient. The average of these local heat

transfer coefficients was also displayed on the computer

outputs found in Appendix D.

Five data runs were taken and reduced in the manner

described above. The five runs cover four different Reynolds

numbers and a moderate range of wall temperatures. The

resulting local heat transfer coefficients were plotted

against the heating element numbers in Figures 34 through 38.

The center of the first heating element was 0.137 inches

from the leading edge of the plate, and each following ele-

ment was 0.274 inches farther downstream.

The local heat transfer coefficients tend to be large

at the beginning of the plate, decreasing more slowly with

increasing distance along the heater plate. There is con-

siderable scatter on these graphs due to the experimental

uncertainty

.

The heat transfer coefficient at the second heating

element appeared consistantly low. This was due primarily

to the method of calculating loss factors, which did not

charge this heating element for any of the end losses. How-

ever, insufficient information about the heat flow was

available to improve the technique. Other factors affecting

94

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CN

LO<N

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99

the scatter included the variation of temperature along the

wall. This variation was as high as ±2.0° F and was probably

due to small irregularities in the surface of the plate.

The first nichrome strip on the fifth heating element was

known to be indented, and its temperature was usually 1.5

to 2.0 degrees higher than the surrounding nichrome ribbons.

Typical error bounds are included in each graph. Dis-

pite the scatter, the decrease in the local heat transfer

coefficient is very much like that for a flat plate. How-

ever, the heat transfer coefficients are consistantly higher

than those for a flat plate.

A brief attempt was made to find some result in the

literature with which to compare the heat transfer coef-

ficients. The majority of the literature for developing

heat transfer in rectangular channels considers both walls

being heated. The Nusselt number of thermally fully

developed flow between parallel flat plates with one wall

at constant temperature was found to be 4.0 [Ref. 16]. For

this experiment, a Nusselt number of 4.0, based on hydraulic

diameter, corresponds to a heat transfer coefficient of

0.0244 BTU/min ft2

°F. For the lowest Reynolds number

tested, the local heat transfer approaches this value. For

the other Reynolds numbers, the local heat transfer coeffi-

cient is higher than the fully developed value.

This data was not nondimensionalized because it was not

compared to any other results. The intended use of this

data was for comparison with similar heated plate data from

100

the curved section of the channel. The procedures used to

obtain these results were considered adequate for comparison

to later work. However, several possible improvements were

identified by this testing. Such improvements would include

designing a system with fewer heating losses and developing

a construction technique with would result in a smoother

heated surface.

101

VII. CONCLUSIONS

The observations made during this investigation have led

to several conclusions about the nature of Taylor-Goertler

vortices in a curved rectangular channel. The vortices were

observed to develop with distance along the channel. The

importance of this observation was that an analytic descrip-

tion of the flow would have to account for this growth. The

vortices were observed to increase in amplitude and complex-

ity with increasing flow rate. Thus at a higher Reynolds

number, less distance along the channel was required to

obtain the same degree of development. Several identifiable

patterns of the developing flow were observed. The patterns

were useful in determining the degree of development of the

vortices

.

The study of the velocity profiles in the channel indi-

cated that Taylor-Goertler vortices are a complex laminar

flow. Furthermore, at flow rates below that necessary to

sustain turbulent flow, momentary bursting to turbulent

flow was observed.

The effect of Taylor-Goertler vortices on the pressure

drop in the curved rectangular channel was observed to vary

from no effect to approximately a 20 percent increase. The

amount of the increase in pressure drop due to the Taylor-

Goertler vortices increased with increasing Reynolds numbers,

This seems reasonable considering that as the secondary flow

102

increases in activity the interaction of the Reynolds

stresses with the mean flow intensifies.

The investigation of heat transfer from a constant

temperature plate in one wall of the channel resulted in a

procedure that would be adequate for determining the effects

of Taylor-Goertler vortices on heat transfer. The heat

transfer coefficients obtained for the straight section of

the channel showed the trends expected and compared with

the limiting data available. Improvements in the heat

transfer measuring procedures could be made.

103

VIII. RECOMMENDATIONS

At the end of this investigation, it was clear that

several recommendations could be made regarding continued

study of Taylor-Goertler vortices in a curved rectangular

channel. More flow visualization should be done to clarify

the patterns in the vortex flow. Different aerosol injec-

tion techniques, including a wall slot, should be tried.

More intense lighting and perhaps narrower slits should be

used to obtain clearer pictures of the flow. An improved

flow visualization technique should then be used to examine

more locations along the curved portion of the channel. Due

to the fact that there were motions in the flow patterns,

still photography could not indicate all of the observations

A motion picture study of flow patterns would be ideal.

More could be learned from a hot wire anemometer study

where the probe could be moved sideways across the channel

as well as radially in and out. In this manner, the perio-

dic variation in the spanwise direction could be determined.

The size of the vortices and their locations could then be

determined. In addition, by knowing where the probe was

in a vortex, the magnitude of the other velocity components

could be determined.

With respect to pressure drop, the placement of more

pressure taps along the last portion of the straight section

and along the entire curved section would improve the

10 4

reliability of the average pressure drop obtained. An

investigation should be made to determine if the pressure

variation in the direction perpendicular to the flow could

be measured.

It was clear that work should continue towards deter-

mining the effect of Taylor-Goertler vortices on heat trans-

fer. A heater plate for the curved section could be built

in the same manner as the flat heater plate used in this

experiment. However, it is recommended that a casting

material with less shrinkage and better thermal insulation

properties be used. If this is done, a smoother heated

surface and fewer thermal losses would result. It is

recommended that the development of a good heat transfer

measuring technique be given considerable thought.

For any analytic solution attempted, it was felt that

it should account for the development of the vortices with

distance and flow rate. If the magnitudes of the velocity

components are determined, as suggested above, they should

compare with any analytic solution obtained. The inter-

action of the Reynolds stresses with the mean flow would

have to be accounted for to predict the increase in pres-

sure drop. Only a non-linear analysis such as J. Stuart's

work [Ref. 25] could be expected to give such results.

Other analytic techniques including numerical methods ought

to be tried.

105

APPENDIX A

ERROR ANALYSIS

A calculation of the uncertainty for each of the major

variables of this experiment was made in accordance with

the methods of S. Kline and F. McClintoch [Ref. 26]. The

estimates of the uncertainty in the measured quantities were

made quite conservative so that there was considerable con-

fidence in the uncertainties arrived at. As an example of

the calculations the uncertainty in Reynolds number is

calculated below. The Reynolds number is given by

U * DhRe =

v

and the uncertainty is taken as

dRe.

/ dU2

dDh2

dvRe ' V U Dh v

2

The uncertainty in' the flow velocity is obtained from other

calculations like this one based on estimates of the uncer-

tainty in reading the volumetric flow rate through the roto-

meter and the uncertainty in the cross sectional area of the

channel. The results of those calculations were — = 0.02805

The uncertainty in hydraulic diameter and kinematic visco-

sity are obtained from estimates as 0.00426 and 0.00136

respectively. Then the uncertainty in the Reynolds number

is

106

dRe / ,_ ^ , n ,„,-".,» ^.^r-x"Re" "V u * !68 + 0.1815 + 0.0185) x 10 = 0.02840

Other quantities and their uncertainties are given below

The error bound for any quantity can be obtained by multi-

plying its magnitude by its uncertainty.

Quantity Uncertainty

Ac 0.00412

Aw 0.00320

Cp 0.07515

cp 0.00010

De 0.02 838

dH 0.00426

H 0.27150

Ho 0.03517

p (pressure) 0.05000

Ph (heat generated) 0.00502

p(j) (heat convected) 0.25502

Re 0.02840

T 0.00750

T £m 0.02000

Tw 0.01667

U 0.02805

V 0.02775

0.00080

107

APPENDIX B

COMPUTER RESULTS FOR VELOCITY AND TURBULENCE

The computer program for reducing the hot wire anemometer

calibration data is shown below. This program was adopted

from a program written by Mr. Loyd Smith while he was study-

ing at the Naval Postgraduate School. The resulting cali-

bration curve is also shown.

The hot wire anemometer data reduction program was also

adapted from a program by Mr. Smith and is shown below. The

reduced data for the seven velocity and turbulence profiles

is then included.

108

.0,4.0,5.0/SUMXY/O.O.C.C, 0.0,0.0/

ICO

200

200

IOC

200

DIMENSION PD{20) ,DATE(20)DIMENSION X(16) ,Y(16)DIMENSION XG(6),VG2(6)REAL*8 LABEL1/8H /REAL LAbEL2/4H /REAL*8 ITITLEU2)DATA XG/0.0,1 .0,2.0,3DATA SUMX,SUMX2, SUMY,DATA A/3.55447/READS IN PROBE DATA AND CALIBRATION DATAREAD( 5, 10COPDREAD(5, 10C0)DATEFCRMAT(20A4)WRITE(6,2000)PDFORMAT ( »1» , « PROBE DATkRITE(6,2001)DATEFCRMATl • , 'DATE: ' ,2READ(5, 1001 )NFORMAT ( I1C

)

WRITE(6,2002)A,NFORMAT* ' » , ' FLOW

A: ',20A4)

0A4)

110CCNS//,61TA POINTS:

2'Y' ,//

)

DC 10 1=1,

N

C READS IN EXPERIMENTALC ANC THE DC VOLTAGE OUC STANT A IS SET FOR H

READ(5, 10C2)H,V1002 F0PMAT(2F10.3)

0=A*SQRT(H)X( I ) = SQRT(U)Yd )=V**2X2 = X( I )**2XY=X( I )*Y( I)SLMX=SUMX+X( I

)

SLMX2=SIMX2+X2SLMY=SUMY+Y( I

)

SLMXY=SUMXY+XYKRITE(6,20C3)H,V,U,X(I),Y(I)

3 FCPMAT(5F10.3,/)C CONTINUE

COMPUTE CONSTANTS BYD=N*SUMX2-(SUMX**2)BETA=(N*SUMXY-SUMX*SUVC2=(SUMY*SUMX2-SUMXYkRITECb ,2004)BETA, VC2

TANT: ' ,3X, F10.3,/, ' NUMBER OF DAX, , H , ,9X, , V , ,9X,«U',9X, , X',9X,

VALUES OF TFE MANGMETER READINGTPUT OF THE ANEMOMETER. THE CCN-IN IN. OF F2G ON A STANDARD CAY

2001

LEAST SQUARES ANALYSIS

MY)/D*SUMX)/D

200

IOC

4 FORMAT*' S'BETA • , F 10. 3 , 10X , • VC2 = ',F1C3)) ,1=1,6)) ,1 = 7, 12J

READ(5, 1003) ( ITITLE( I

READJ5, 10C3X ITITLEUFCRMAT( 6A3)PCINT PLOTS DATA

, ,,,--,CALL DRAW(N,X,Y,1,2,LABEL1,ITITLE,1.0,1.0,1,1,0,0,4,

35, C, LAST)C TFE LEAST SQUARES EQU,C PCINTS FOR A STRICHT

DC 20 1=1,6VG2(I )=VC2+BETA*XG( I)

^ CCALL

I

DrL(6,XG,VG2,3,0,LABEL2,ITITLE,1.0,1.C,1,1,0,0,44 ,5,0, LAST)STOPEND

IATION IS USEC TO COMPUTE SIXLINE

109

4.0

3.0

0)

u(d

D1

CO

(1)

tr>

(d

-PrHO>

2.0 <•

1.0

0.0

0.0 1.0

/ v

2.0

sec

3.0

Hot wire anemometer calibration curve

110

30sec

100C

2000

20C1

1CC1

2002

DIMENSION CAT A It 20) , CAT A2 ( 20 ) , H t 25 ) ,U ( 25 ) ,7(25)REAL*8 IT1TL1 (12) , ITITL2Q2)RE£L LAEEL/4h /READ(5, 90ONPR0BFCRMATt UC)IF(NPROB.LT.O) STOPREAC(5 T 10CO0ATA1REAC( 5, 100ODATA2FCRMAT (20A4)hPITE(6, 2.000)FCRMATt '1' ,//////////)hFITE(6,20Ql )CATA1WRITE(o,2C0i)DATA2FCRMAT( • ' ,21X,20A4)REAC(5,1001) RG,VC2,6ETAFCRMATt 2F10.6)hPITE(to f 2002) RG,VC2,BETAFCPMAT(23X,' INITICAL PCS I T I CN= ' , F7 .3 , / , 24X , ' V02 =

dETA =»f F10.6,// f 30X, »H» f 19X 19X2F1C.6,9X,

NCF=01C PEAC(5, 10C2)R,V,E

10C2 FCRMAT(2F10.fc)IF(R.LT .0.0) GO TC 20NCP=NDP+1I=NCPh( I )=R-RGU(I)=( ( (V**2)-VC2)/EETA)**2T(I)=(4.C*V*E)/((V**2)-y02)VvRITE(6,2003)P( I ) ,U(I >,T( I )

20 03 FCRMAT(21X,F10.6,1CX,F10.6,10X,F10.6,/)GC TO LO

2C kRITE(6 ,2G04)2CC4 FCRMATt //////////)

PEAC(5, loC3)(ITITLKI),I=i,6)RE AD (5, 1003) ( IT I TL1 (I) ,1 = 7, 12)FCRMATt £A8)kPITE(6,2CG5 )

FCRMATt ///////////////)CALL DR/JMNDP , U , H , C , , LABEL , ITITLi,1.0,0.C5,C, 0,0,0,

88 ,5,0, LAST),1=1,6),1 = 7, 12)

//)

1CC3

20C5

1CC4

READ(5, 10C4)( ITITL2( I)REAC15, IOCh) t IT I TL2( I

)

FCRMAT16A8)GALL DRAk(NDP,T,H,0,0,LABEL, ITITL2.0

66 ,5,0, LAST)GC TO 3CEND

010 , .05,0,0,0,0,

111

CGNCAVE TO CONVEX WALL DISTANCE, VELCCITY AND TLRBULENCEFRCBE NC. TSI1279 DATE 5 FEB 73 22? FLOW RATE RE= 6C9

INITICAL PCSITICN= 0.27CV02 = 2. £20000 BETA = C. 710000

H U

c. CICCOO

c

.

C2C000

c. C3CCC0

c,-C4C000

c .C5CCCC

c,.C6CC0O

c .07C000

c..ceccoo

c .csccoo

c .ICCCC0

c .11CCC0

c .12C00O

c .13CCC0

c .14CCC0

c .15CC0O

c .16CCC0

c .L7C000

c .16C0C0

c .19C000

0.613295

C. 620402

0.9516C6

1.536921

1.958789

2.504096

2.665120

2.90C582

3.111880

3.239269

3.332104

3.407528

3.426528

3.407528

3.2321C4

3.2C2550

3.C4C466

2.815131

2.504096

0.031175

0. 032164

0.028536

0.022217

0.02C265

0.017990

0.017536

0.016941

0.016465

0.0159 17

0.015727

0.0 155 96

0.015561

0.015596

0.015727

0.015990

0.016621

0.017149

0.017990

112

CCNCAVE TO CONVEX WALL DISTANCE, VELOCITY AND TURBULENCEFRCEE NC. TSI1279 DATE 5 FEB 73 332 FLOW RATE RE= 875

INITICAL PCSITION= C.270VC2 = 2.C20000 BETA = C. 7LOOOO

H U

C.CiCOOO

C.C2CCC0

C.C3CCC0

C.C4CC00

C.C5CC0C

C.06COGO

C.C7CC00

C.C8CCC0

C.C9C00O

C.1CC000

C.11C000

C.12C00Q

C.13CC00

0.14COOO

C.15CG0C

0.16C000

C.17CC00

c.ieccco

C.19CC0O

0.701554

1.2987C2

1.945134

2.665120

3.111880

3.580913

3.943310

4.153802

4.370939

4.549487

4.640411

4.686278

4.663301

4.572118

4.459676

4.283268

4.026724

3.719356

3.369696

0.03C997

0.027440

0.023128

. C 1 9 6 9 C

0.017332

0.016380

0.015237

0.014926

0.014629

0.014152

0.014C43

0.013969

0.014C16

0.013377

0.014514

0.014746

0.015111

0.015865

0.0165C5

113

CONCAVE TO CONVEX WALL DISTANCE, VELOCITY AND TLBEULENCEPPC6E NC. TSIL279 DATE 5 FEB 72 442 FLOW RATE RE=1166

INITICAL PCSITICN= 0.270VC2 = 2.C20000 BETA = 0.710000

H U

C.OLCOOO

C.C2CCC0

C.C3CC00

C.C4C0CG

C.C5CC0C

C .060000

C.C7CCC0

C8C000

C9CC0O

1CCCC0

110000

12CCC0

13CC00

14CCCC

15CC00

,160000

17CCC0

,180000

c,

c

c

C

c

C

c

c

C

c

c

C.19CCC0

0.942583

1.958789

2.815131

3.659631

4.153802

4.594827

5.015061

5.258296

5.483244

5.688062

5.844732

5.977282

6.003999

5.95C615

5.844732

5.636421

5.282988

4.919745

4.459676

0.032475

0.025157

0.021C60

0.018945

O.O175C0

0.016618

0.015774

0.015C16

0.014213

0.014111

0.012514

0.012398

0.012152

0.012421

0.012514

0.0127C4

0.014266

0.014929

0.015515

114

CONCAVE TO CONVEX WALL DISTANCE, VELCCITY AND TUREULENCEFPCEE NC. TSIL279 DATE 7 FEB 73 49£ FLOW RATE RE=L298

IMTICAL PCSITICN= 0.270VC2 = 2.G20000 BETA = 0.7LOOOO

H U

0.010000

C.C2CCC0

C.C3C000

C .C4CG0C

C.C5CC00

C.C6C000

C.C7CCC0

C.C8CC00

C.C9CCC0

C.1CCCC0

C.L100CO

C.12CC00

C.13CC00

C.14CC0C

C.15CC0C

C.16C000

C.17CC00

C.18CC00

C.L9CCC0

1.320405

2.084258

3.600484

4.825512

5.407611

5.871078

6.C577C6

6.111675

6.220504

6.3C2920

6.335982

6.497612

6.582447

6.667787

6.639285

6.554168

6.3C2920

5.977282

5.457953

0.03C549

0.027238

0. 0245 17

0. 021835

0.02C888

. C 1 7 9 8 7

0.017772

0.017269

0.0167 12

0.016191

0.015239

0.015138

0.014633

0.014133

0.014156

0.014226

0.014440

0.015184

0.015726

115,

CCNCAVE TO CONVEX WALL DISTANCE, VELOCITY ANC TLRBULENCEPRCBE NC. TSI1279 DATE 6 FEB 73

INITICAL PCSITIGN= CVC2 = 2.C2C000 BETA

H U

552 FLOW R/HE RE=145727C= C. 710000

C.C1CCC0

C.C2CC00

C.C2CC00

C.C4CCC0

C.05C00C

C.C6CC0O

C.C7CC0O

C.C8C000

C.C9CCC0

C.1CCC00

11CCC0

12CCC0

130000

14CCCC

15C000

leccco

17CCC0

18C00O

C.19CCC0

2.348951

4.872481

6.138762

6.811521

7.045468

6.927876

6.869542

6.725045

6.497812

6.441764

6.330523

6.3C2920

6.275369

6.247911

6.247911

6. 193172

6.030813

5.766069

5.357558

0.023978

0.019336

. 1 7 6 £ 1

0.014869

0.014679

0.014773

0.014821

0.014941

0.015138

0.015188

0.015290

0.015316

0.C149C3

0.014928

0.014928

0.014528

0.014667

0.014943

0.0163C5

116

CONCAVE TO CONVEX WALL CISTANCE, VELOCITY ANC TUREULENCEPROBE NO. TSU279 DATE 6 FEB 73 65? FLOW RATE RE=1722

INITICAL PCS1TICN= C.270V02 = 2.C20000 BETA = 0.710000

H U

C.C10000

C.C2CCC0

C.C3CC00

C.C4CC00

C.C5CCC0

C .C6CC00

C.C7CCC0

C .C8C000

C .C9CCC0

C.1CCCC0

C.UCOOO

C.12CCC0

C.13C000

14CCCC

15CCCC

L6C000

17CCC0

18CC0O

C.19CCC0

2.488311

5.C87305

6.358225

7.25421C

7.497396

7.466729

7.405624

7.284319

7. 164276

7.104720

7.C15953

6.957165

6.927876

6.698683

6.811521

6.696287

6.554168

6.330523

5.924045

0.05C629

0.047523

0.042613

0.041479

0.032772

0.022822

0.024694

0.02C7C7

0.02C828

0.02C9C4

0.C169C3

0.016964

0.016664

0.016911

0.016993

0.C171C3

0.019399

0.024C27

0.024647

117

CCNCAVE TO CONVEX WALL DISTANCE, VELCCITY AND TLRBULENCEPRCBE NC. TSI1279 DATE 6 FEB 73 72? FLOW RATE RE=1934

INITICAL PCSITION= C.27CVC2 = 2.C2CG00 BETA = C. 710000

H U

c. C1C000

c. C2C0C0

c.,020000

c, C4CCC0

c..05C00C

c,,C6CCC0

c..C7CCC0

c .C80000

c ,C9ccco

C .1CC000

c .11CCC0

c .12CCG0

c .120000

c .14CCCC

c .15CC0C

c .16CCC0

c .17CCC0

c .18C000

c .19CCC0

2. 425794

4. 778824

6,,020813

7. 164276

7,,466739

7,.62C902

7 ,589893

7 .528167

7 ,284319

7 .254210

7 .405624

7 .405624

7 .466729

7 .466739

7 .558975

7 .558975

7 .466739

7 .214532

7 .C15953

0.095928

0.097417

. C 8 9 C 1

1

0.075C16

0.061543

0.052928

0.053CC9

0.C4CC62

0.041414

0.041479

0.037C41

0.0288C9

0.C2672C

0.024617

0.02C420

O.C2C420

0.02C514

0.02C675

0.0252C4

H8

APPENDIX C

CALCULATIONS AND COMPUTER RESULTS FOR PRESSURE DROP

The computer program written to find the slope of the

pressure curve is shown below. The results for six dif-

ferent Reynolds numbers are shown. The first set of results

represent nine pressure readings or two foot intervals. The

second complete set represents selected one foot intervals.

The last item shown in this appendix is a tabular calcula-

tion of the pressure drop coefficient for the lowest Rey-

nolds number tested.

119

cccccccc

CALCULATION OF THE LEAST SQUARES SLOPE FOR PRESSUREDRCP DATA. DATA READ IN IS NUMBER OF DATA POINTS,REYNOLDS NUMBER, POSITION OF PRESSURE TAP IN INCHESANC PRESSURE IN INCHES OF WATER.

11

2CC1C

D

FL

RFIc;

SSSLI

GHLWF

1*2'CDRF

X

I

xXs

ssCcAAhF

3«G

3CC S

E

21C

21C20

22

23

ICC

24

I MENS I

PITE(6ORMAT(=EAD(5,CRMAT(F(N.LTLNX=0.UMY=0.UNX2=0LMXY--0= L + 1F(L.GTC TO 2PITE(6= 1

RITE( 6CRMATtREYNOLPRESSU= NC 100EAD(5,CPNAT (

=T/12.RITE(6CPMAT(2=X**2> = X*YLNX=SULMY=SULMX2=SLNXY=SONTINU=(SUMXC=(SUM1=(SUMPITE(6CPMAT(SECT 10C TO 2TCPND

ON TYPE(2),11)1« ,////)

10) N,RE,TYFE(1),TYPE(2)16, FS. 1,2A4).1) GO TO 3C0CC.0.0

.3 ) GO TO 3101C,11)

,20) RE1 ,31X, 'SLOPE OF PRESSURE DROP CLPVE FCR

DS NUMBER =• , F7 . 1, // / , 35X

,

'POSITION (FT)',20XRE (IN H2G) ' ,//)

1 = 1 ,N22) T,Y2F10.5)C,23) X,Y39X,F7 .5,25X,F7.5)

MX +MV +UMXLMXE) **X*Sx*s,245X,N IS' ,F12.6,////)00

XV

2+X2;y+xy

2^C-SUMX2UMXY-SUMY*SUMX2)/CUMY-C*SIMXY) /D) TYPEil), TYPE(2), Al

TrE PRESSURE CRCF IN TFE ',2A4, IX,

120

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUNEER = 609.0

POSITION (FT) PRESSURE (IN F20)

0.00.25CCC0.500CC0.75000l.OOOCG1.250001 .500001.7500G2.00000

0.011C00.01200C. 013000.014CC

01450C15C00165C0180C019C0

ThE PRESSURE DROP IN THE STRIGHT SECTION IS 0.003667

SLCPE OF PRESSURE CROP CURVE FOR REYNCLCS NUMEER = 609.0

PCSITION (FT) PRESSURE (IN H20)

0.00.250000.500000.750001.000001.250001.500001. 75C0C2.00000

C. 030000.03 200.03400C.C35CC0.03650C.038C0C.C39CC0.040000.04150

THE PRESSURE DROP IN THE CURVED SECTION IS 0.005533

SLCPE OF PRESSURE DROP CURVE FOR REYNOLDS NUMBER = 875.0

POSITION (FT) PRESSURE (IN H20)

0.00.250000. 50GCC0.750001 .000001.250001.500001.750C02.00CCC

C.014C0C.C16CC0.013000.C1950C.021CC0.02300C.C250C0.O27C00.02900

THE PRESSURE DROP IN THE STRIGHT SECTION IS 0.007367

12 3

SLCPE OF PRESSUFE DROP CURVE FOR REYNOLCS NUMBER = 875. C


0.,00.,250000. 500CC0,,750CO1

,

,000001.,250CC1.•500GC1

,

,750002.,OOOCC

C.042CC0.044C00.04550C.Q47CC0.C49CC

05000C52CCC540C05600

THE PRESSURE DROP IN THE CURVED SECTION IS 0.0068C0

SLOPE OF PRESSURE DROP CURVE FOR REYNOLDS NUMEER = 1166.0

POSITION (FT) FFESSURE (IN F2C)

0.00.250C00.500000. 75CCC

0000025000500CC750CC00000

0.02400O.C260CC.028CC0.030500.033000.035C00.03700C.C39000.04200

THE PRESSURE DROP IN THE STRIGHT SECTION IS 0.CC89C0

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMEER = 1166.

C

PCSITION (FT)- PRESSURE (IN h20)

0.0C.2500C0.500CC0.75000

00000250005000075000

2.000CO

.C59C0

.06200

.06300

.066CC

.06900

.071C00.073C00.07600C.079C0

THE PRESSURE DROP IN THE CURVED SECTICN IS 0.0098C0

122

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMEER = 1484.0

PCSITION (FT) PRESSURE (IN H20J

C.0320C0.03500C. 03800C.C4CCC0.04300C.C46CCG.C49CG0.052000.05500

0,,00.•25CC00..500CC0,,750001.,000001,.250001.,500001.,750002,.OOOCO

THE PRESSURE DROP IN THE STRIGHT SECTICN IS 0.0L1400

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUFBER = 1484.0

POSITION (FT)

0.00.250000.500000.75CCC1.000001.250001.500CC1.750002.00000

PRESSURE (IN H2C)

0.0780CC.C82CC

085C0C88CCC93C009f3CCC980C

THE PRESSURE CROP IN THE

C. 102CCC.IOdCC

CURVED SECTION IS 0.0134CC

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUNEER = 1748 .C

POSITION (FT) PRESSURE (IN F20

)

0.00.250C00.500CC0.75000l.OOOCC1.250001.500001.750CC2. OOOCO

0.0390C0.04300C.047C00.050CC0.05300C.057CC0.061CC0.06400C.067CC

THE PRESSURE DROP IN THE STRIGHT SECTICN IS 0.014000

123

SLCPE OF PRESSURE DRGP CURVE FCR REYNCLCS NUNEER = 1748.0

POSITION (FT) PRESSURE (IN F20)

n'oznno 0.C93CC0.250CC 0.090000.5000C C.1CCCC0.75000 0.103CCl.OOOCC C.1LC001.25000 C.112CC1.50000 0.116CC1.75CC0 C.121CC2.00000 C.1260C

THE PRESSURE CROP IN THE CURVED SECTICN IS 0.016133

SLCPE CF PRESSURE CROP CURVE FOR REYNCLCS NUMBER = 1987.

C

POSITION (FT) PRESSURE (IN h20)

0.0 C.048CO0.25000 0.C52CC0.500CC 0.0560C0.75C0C C.C6CCC1.00000 0.G64CC1.25CCC C. 068001.500CC C.C73CC1.7500C C.076CC2.000CC 0.08000


SLOPE OF PRESSURE DROP CURVE FOR REYNCLCS NUNEER = 1987.0

POSITION (FT)' PRESSURE (IN F2C)

0.0 0.108000.2500C C.114CC0.50000 CU7CC0.75CCC 0.121001.00000 0.129CC1.25000 0.132CC1.500CC 0.137001.75000 0.142C02.00000 0.147CC

THE PRESSURE DROP IN THE CURVED SECTION IS C.C194CC

124

SLOPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMBER = 609.0

POSITION (FT) PRESSURE (IN H20J

0.00.250CC0.500000.75CCC1.00000

0.01400C. 01450C.C15GC0.01650O.C18C0


SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMBER = 609.0

POSITION (FT)

0.00.25000C. 500CC0.750C01.00000

PRESSURE (IN H2C)

C.C380CC.C39CCC. 04000C. 041500.043CC

THE PRESSURE DROP IN THE CURVEO SECTION IS 0.CC5CCC

SLCPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMEER = 875.0


0.0C.250C00.500CC0.750001.00000

C. 019500.02100C.C23CC0.025CC0.02700

THE PRESSURE DROP IN THE STRIGHT SECTION IS 0.007600

SLOPE OF PRESSURE DROP CURVE FOR REYNCLCS NUNEER = 875.0

POSITION (FT) PRESSURE (IN F2C)

0.00.250000.500000.7500C1.00000

0.05000C.C52CCC.C54CC0.05600C.058C0

THE PRESSURE DROP IN THE CURVED SECTICN IS 0.C080C0

125

SLCPE OF PRESSURE DROP CURVE FOR REYKCLCS NUMEER = 1166.0

PCSITION (FT) PRESSURE (IN h20)

0.00.250000.500CC0.750001. ooocc

C.C3C500.033000.03500C.037CC0.03900

THE PRESSURE DROP IN THE STRIGHT SECTION IS C.CC84CC

SLOPE OF PRESSURE DROP CURVE FOR REYNCLCS NUMEER = 1166.0

POSITION (FT)

0.00.250000.500000.750001.00000

PRESSURE (IN F20)

0.07100C.073C0C.C76CC0.079000.C82C0

THE PRESSURE DROP IN THE CURVED SECTICN IS 0.0U2C0

SLCPE OF PRESSURE DROP CURVE FOR REYfvCLCS NUN.EER = 1484. C

POSITION (FT)

0.00.250000.500000.75C0C1.00000

PRESSURE (IN H2C)

0.04000C.043CG0.C460C0.049C0C.C52CC

THE PRESSURE DROP IN THE STRIGHT SECTICN IS C.0120C0

SLCPE CF PRESSURE DROP CURVE FOR REYNCLCS NUMBER = 1484.0

POSITION (FT) PRESSURE (IN h20)

0.00.25000C.500CC0.750001.00000

C.095CCC.098CC0.10200C.105CC0.10900

THE PRESSURE CROP IN THE CURVED SECTION IS 0.014CC0

126

SLCPE OF PRESSURE DROP CURVE FOR REVNCLCS MMEER = 1748. C

POSITION (FT) FFESSURE (IN F2C)

0.0C.250CC0.500CG0. 75C0C1.00000

C. 05000C.052CCC.057C00.06100C.C64CC

THE PRESSURE DROP IN THE STRIGHT SECTION IS 0.0144C0

SLCPE OF PRESSURE DROP CURVE FOR REVNCLCS NUMBER = 1748.0

POSITION (FT)

0.00.250000. 500CCC. 750001.00000

PRESSURE (IN H20)

C. 112C00.116CC0.12100C. 126CCC.13C0C

THE PRESSURE CROP IN THE CURVED SECTION IS C.C184CC

SLCPE OF PRESSURE DRCP CURVE FOR REVNCLCS NUNEER = 1987.0

POSITION (FT)

0.00.250CC0.5000C0.75000l.OOCCC

PRESSURE (IN F20)

C.06CCCC.C6400C.068CC0.072CC0.07600

THE PRESSURE DRCP IN THE STRIGHT SECTIGN IS C.C164CC

SLOPE OF PRESSURE DROP CURVE FOR REVNCLCS NUNEER = 1987.0

POSITION (FT) FPESSURE (IN F20)

0.00.250000.500000.750001.00000

0.13200C.137CCC.142C00.147C0C.152CC

THE PRESSURE DROP IN THE CURVED SECTICN IS 0.0200CO

127

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128

APPENDIX D

COMPUTER RESULTS FOR HEAT TRANSFER

The final results for one run of the TRUMP program are

shown on pages 130 through 144. The computer program for

the calculation of the heat transfer coefficients follows

the TRUMP program. The five sets of heating data results

are presented with the temperature distribution shown on

the first page and the heat transfer coefficients on the

second page of each of the results.

129

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cccccccccccccc

cccc

DIMENSION TRC( 1.4,4) ,R( 18) ,V( 18 J , TM(40) , TD(4C) ,TB(2C)DIMENSION h(16),U(lS),X(18),P(18),GT(18)DC 100 1=1,14

1) ,TRC(1,2),TRC( 1,3) ,TRC( I ,4)5,F10.3)

TRC( I,1.2F10

REACC5, 10)1C FCPMAT(F10

ICC CONTINUECC 200 J=l,18PEAC(5,12) MJ),V(J)

12 FCRMAK 2F10.4)2CC CCNTINUE

DC 340 K=l,40READ(5, 14) TM(K)

34C TC(K)=0.014 FCRMAT(F10.3)

CC 320 1=1 f 2032C TE(L)=0.0

DC 318 M=l,18F (M ) = 0.0

318 P (P ) =0 . CDC 322 L=l,18

322 REAC(5, 20) U(L) ,GT(L)2C FCFMAT(2F8.3)

READ(5, 11) XO,DX,AW,AS11 FCPMAT(4F10.6)

PEAC(5,13) w,T0,S,Al,A213 FCRMAT(3F1C.4,2F 10.8)

TFE DATA HAS BEEN READ, NOWTORES AND WRITE THE LABELS.

INTRUPULATE TFE TEMPERA-

DI

LDI

L41C C99C T

GDT

Ch

r

GO TO 990

4)) GC TO 4 12

412

4CC

15

17

16

42C

2,3'RW

i FWF

11TCwFT

C

C 400 1=1,40F(TM( 1 ) .LT. .834)= 1

C 410 N=2,14F(TM( I ) ,LE.TRC(N= L + 1

CNTINUEC(I)=TC+S*TM(I)

= (TM(t)-TPC(L,4) )/(TRClL+l,4)-JPC(L,4) )

CCIi=TBC(L f l)+B*(TRC(L*ii l)-TRC(Lf II J

CNTINUERITE (6,15) QCRMAT( • 1' ,////, 41X,/,39X, 'WALL HEATERTHE FLOW RATE IS « ,

E=238. 7024-QPITE(b,25) RECRMAT141X, 'THE REYNOLDS

CRMAT?47x!»THE WALL TEMPE RATUR E ' ,// ,41 X , • ST AT ION •

IX, 'TEMPERATURE ' ,/)

•HEAT TRANSFER CALCULATIONS FCP A*

IN A RECTANGULAR CHANN E L ',///, 35X

,

«,F5.2,' CUBIC FEET PER NINLTE',///)

NUMBER IS" ,F7.0, /////)

SUM=0.0C 420 J=l,32PITE(6, 16) J,TD( J)CRMAT(43X,I2 ,16X,F6.2)SLM=TSOM*TD( J)CNTINUE

145

18

Cc

cc

TAVG=TSUM/32.0TB( 1)=TD(39)TE(20I=TD{40)hFITE(6,16) TAVG,TB(1 ) ,TB(20)FCRMAT(///,33X, 'THE AVERAGE WALL TEMPERATLRE IS *,

2F6.2,* DEGREES F ,/// t 35X ,• ThE ENTERING AND EXITING

4*BULK TEMPERATURES ARE « ,/ ,49X , F6 .2 , AND «,F6.2,///)

CALCULATE THE LOG MEAN TEMPERATURE ANDHEAT TRANSFER COEFFICIENT.

ThE CVERALL

Ccc

D=ATLM1F(N = l

DCIF(N = NCCNGCE=(RC =

C=THAVWRIFCF

3'EN5' C7' STA =FN =

hRIIt FCR6F6.8F8.

LOG( (TAVG-Tb( 1) ) / ( T AVG-TB ( 20 J ) )

= ( TB(20)-TB(1) )/DTAVG.LT.70.0) GO TO 999

51C

52C

19

510 K =

TAVG .L+ 1

TINUETO 999TAVG-TTRC(N,RC(N,3G=(RC*TE(6, 1

MAT( • 1

CE ISOEFFICCUARE(TAVG-HAVG*(TE(6,2MATC352,//,

3

4,////

2,14E.TRC(K, 1)) GO TO 520

RC(N,1) )/(TRC(N+l,l)-TRC(N,l))2)+E*( TRC(N+i,2)-TRC(N , 2) )

i+E*(TRC(N +l,3)-TRC(N»3) )

C*Q=MTb(2C)-T£Q)

)

)/(Ak*TLM)9) TLMtHAVG*,////, 35X, 'ThE LOG MEAN TEMPERATLRE

F* ,/////)

• ,F7.2,//IENT IS' ,Ffc.4,/,33X, • INFEET DEGREESTb( 1))TLM/TA)6) TA,HMX,*THE fcALL FLLID5X f 'THE MEAN HEAT)

CIFFER'HEAT

67U PERTRANSFER*MM7E» ,

TEMPERATLRE DIFFERENCETRANSFER COEFFICIENT IS

IS*',

CALCULATE THE POWER AND LOCAL FEAT TRANSFER COEFFICIENT

61C

62C

6CC

<:<:

TTTRDwLFPRRCCCI

NDI

NCGGRCT

KLT

FC

F2«3/

h=TAVGV=TB(1TR=.COA=.000C 600= ( .056T=U(J)=L(J)/(J)=w*1=(2092=175.Cl=Al*C2=A2-CN=CC1F(TB(J= 1

C 610F(TB(

J

= 1^ + 1

CNTINUC TO 9= (TB( JC=TRC(=TRC(MB( J+l)= 2*J= K-1X=(TD((J)= (RCNTINURITEC6CPMATiAND* ,

//,28X

+ 45)+4CGO019J = l

884+ GTUT(1..15596(TB(TB+C0) .L

9.7259 .721*( TW**4-TV**4)99667*AS*TTR,16*V< J)**2)/R( J)(J)

C-FJ-RA18+1 .O/HM)852(JJ-70.O/R1(J)-70.0)/R2

70.0) GO TO 999

GO TO 6201=2,14) .LE.TRC( 1,1))

E99)-TRC(M,l ))/(TRC(M+l,l)-TRC(M,l))M,.2) + G*( TRC(M+l,2)-TRC(M,2)i,3)+G*(TRC(M+l,3)-TRC(M,3))=TB(J)+(P( J)-CCN)/(RC-C*C)

K)+TD(L) ) /2.CC*C*C*(TB(J+1)-TE( J) ))/(AS*(TX-TB( J) ))

E,22)//,37X, 'THE// ,39X,'THE, 'STATION* ,

LOCAL HEAT TRANSFER CC EF F I C I EN TS

* ,

BULK TEMPERATURES Ak E SFCwN BELCW'5 X, 'DISTANCE' ,5X,« POWER' ,5X,

146

4 , TEMPERATURE',5X, I H',/)hS=C.OCC 62C N=l,16F;S = HS + H(N)SS=FLOAT(N)X(N)=-XG+SS*DXWRITE (6,23) N,X(N),P(N),TB(N)

21 FCRKAT(31X,I2,7X,F8.6,4X,F8.o63C CCNTINUE

FA=HS/16.0WRITE(6,27) HA

27 FCRMAT(///,3IX,2' COEFFICIENT I

WRITE(6,24) TB(24 FCPMAT(//7 ,41X,

347X,' AS CALCULATEC4F6. 2, /////)GC TO 9C0

9 9S V, RITE (6, 29)29 FCRMAT(//,47X, 'SOMETHING

9CC S1CFEND

,H(N),5X,F6.2,5X,F8 .6)

•THE AVERAGES' .Fd.4, ///)17) ,Tb(2C)•THE FINAL BULK

LOCAL FEAT TRANSFER*

TEMPERATURE WAS',//,F6.2,//,47X, 'AS MEASURED

WENT WRONG' ,////)

147

HEAT TRANSFtR CALCILATICNS FCR 4WALL HEATER IN A RECTANGULAR CHANNEL

THE FLOW RATE IS 2.55 CLBIC FEET FER MINLTE

THE REYNOLDS NUMBER IS 6C9.

STATION

1

23456789

1Cil

i!141516171819202l2223242526272829303132

THE UALL TEI^FERATURE

TEMPERATURE

124125125124125125125124

83tl297155344629

125.97124.58125124125125125124125125125

13624217714529295C

124. 92125125124,125125124125124

OC468 7

2138960445

125. C6124.41124.45125. 13

THE AVERAGE kALL TEMPERATLRE IS 125. C8 DEGREES F

THE ENTERING AND EXITING BULK TE f^ F ERATUR ES /IRE80.13 AND 90.87

148

THE LOG MEAN TEMPERATURE DIFFERENCE IS 39.34

THE OVERALL HEAT TRANSFER COEFFICIENT IS 0.C460IN 6TU PER MINUTE SQUARE FEET CEGREES F

THE WALL FLUID TEMPERATURE DIFFERENCE IS 44.95

THE MEAN HEAT TRANSFER COEFFICIENT IS 0.04C3

THE LOCAL FEAT TRANSFER COEFFICIENTS ANC

THE BULK TEMPERATURES ARE SHChN BELGW

STATION DISTANCE POWER TEMPERATLRE H

1 0.011416 0.088965 8C.13 0.1321042 0.034249 0.038459 £2.C8 C.C59C253 0.057082 0.043506 82 .92 C.C674864 0. 079915 0.C33487 83 .£6 0.0533175 0. 1C2748 0.028635 84.58 0.0455726 0.125581 0.026795 85.20 C.C435117 0. 143414 0.027666 £5.77 C.C454278 0.171247 0.021361 86.36 C.C349219 0. 194030 0.018317 86. £1 0.029719

10 0.216913 0.020849 £7. 19 C. 034529li 0.239746 .0 ic224 8 7.62 C.C2652612 0.262579 0.020555 87 .95 .03474713 0.285412 0.015C31 ££.38 C.C246C714 0.30S245 0.013437 88 .69 C.C2226815 0.331C78 0.011493 88.95 0.01871816 0.353911 0.019532 £9. 18 0.C34C48

THE AVERAGE LCCAL HEAT TRANSFER COEFFICIENT IS 0.C442

THE FINAL BULK TEMPERATURE MSAS CALCULATEC 89.58

AS MEASURED 90.87

149

HEAT TRANSFER CALCULATIONS FCR AWALL HEATER IN A RECTANGULAR CHANNEL

THE FLOW RATE IS 3.66 CUEIC FEET FEB MINUTE

THE REYNCLDS NUMBER IS 874.

THE WALL TEMPERATURE

STATION TEMPERATURE

12a

456789

1011121314151617ie9o

21222324252627

\

293C'31

32

131,,99133.,24132.,86132,,07133,,32132.,82133..94131,,45134.,52131,,99133,,24132, , 74133,,82133 ,.15133,,82132,,37132,.99133,,28133,.24132,.45132,,03132,,53132,.41132,.74133,.49132,.78133,.15132,,41133,.36132,,24132,,61133,.03

THE AVERAGE WALL TEMPERATURE IS 132.88 CEGREES F

THE ENTERING AND EXITING BULK TEMPERATURES ARE76.99 ANL 88.51

150

THE LOG MEAN TEMPERATURE DIFFERENCE IS 49.91

THE OVERALL HEAT TRANSFER COEFFICIENT IS 0.0551IN BTU PER MINUTE SQUARE FEET CEGREES F

THE WALL FLUID TEMPERATURE DIFFERENCE IS 55.69

THE MEAN FEAT TRANSFER CCEFFICIENT IS 0.C492

THE LOCAL FEAT TRANSFER COEFFICIENTS AND

THE BULK TEMPERATURES ARE SFCkN EELGW

STATICN DISTANCE POWER TEMPERATURE H

1 0.0114160.034249

0. 123293 76. cs 0. 1492152 .063262 78.88 0.C788923 0.057C82 0.064357 79.84 0.0807154 0.0799J.5 0.056232 8C.E2 C.C721195 0.102748 0.045533 81.67 C.C584C36 0. 12556 1 0.046C22 82.26 C.060C997 0. 14 8414 .0460 4 83. C6 C.C602678 0. 171247 0.C36775 83.76 0.0438159 0. 194080 0.C316C5 84.21 C.C42C99

10 0.216913 0.036077 84 .79 0.C49CC411 0.239746 0.025993 85.33 0.03556112 0.262579 0.C36541 85.71 C. 05081313 0.285412 .023612 86 .26 0.C3928914 0.308245 0.027 137 86.69 0.03773415 0.331078 0.022355 8 7 . C 9 C.C3092216 0.353911 0.029529 87.42 0.C41776

THE AVERAGE LCCAL HEAT TRANSFER COEFFICIENT IS 0.0585

THE FINAL BULK T E ,* F ERATU RE hAS

AS CALCULATED 87.85

AS MEASURED 88.51

151

HEAT TRANSFER CALCULATIONS FCR AWALL HEATER IN A RECTANGULAR CHANNEL

THE FLOW RATE IS 4.88 CLBIC FEET FER MINUTE

THE REYNOLDS NUMEEP IS 1166.

THE WALL TEMPERATURE

STATION TEMPERATLRE

1

234c

6789

101112131415161718192C2122222425262728293C3132

143. 16144. 8C144.06143.52144.2.2143.61144142,144142143143143143143141142

39346238810773164C8038

142.8 7

143.07142.3814214314214^143142143,142143

54284642527C2866It

142.05142.42142.87

THE AVERAGE WALL TEMPERATLRE IS 143.13 DEGREES F

THE ENTERING AND EXITING EULK TENFERATURES ARE79. E2 ANC 91.96

152

THE LOG MEAN TEMPERATURE CIFFERENCE IS 57.02

THE OVERALL HEAT TRANSFER CCEFFICIENT IS 0.C666TN BTU PER MINUTE SCUARE FEET CEGREES F

THE WALL FLUID TEMFERATLPE DIFFERENCE IS 63.31

THE MEAN FEAT TRANSFER CCEFFICIENT IS 0.C6C0


THE BULK TEMPERATURES ARE SHCM BELCU

STATICN DISTANCE PCWEP TEMPERATURE H

1 0.011416 .14793C 79.62 0.155C862 0.034249 0.C81882 81.53 0.C878703 0.057C82 0.080664 82.47 C.C876084 0.079915 0.070126 83.39 C.C777925 0. 1C2748 0.054745 64.19 0.0610346 0. 125581 0.059199 84. £2 C.C666257 0.148414 0.C55696 85.50 C.C634488 0. 171247 0.044451 66. 13 C.C515719 0. 194080 0.03 7 6 5C 86.64 C.C4374510 0.2169L3 0.C444GC 87 .C6 0.05215711 0.239746 0.04023 1 87.56 0.C4730912 0.262579 0.045171 88 .C2 C.C542C413 0.285412 0.034C27 88.53 0.04017614 0.30S245 0.038087 88.91 C.C456C415 0.331078 0.026458 89.34 C. 03147316 0.353911 0.C35C7C 69.63 0.C42578


THE FINAL BULK TEFFERATURE bflS

AS CALCULATEC 90.03

AS MEASURED 91.96

153

HEAT TRANSFER CALCULATIONS FCR AWALL HEATER IN A. RECTANGULAR CHANNEL

THE FLOW RATE IS 6.10 CLBIC FEET FER MINUTE

THE REYNOLDS NUMEEP IS 1457.

THE WALL TENFERATURE

STATION TEMPERATURE

1

2

3456789

IS12131415161718192C212223242526272829303132

13914013913813913914-013714013713913513913813^13713813813913713813813813813913813913813913713b139

.71• --* —.71.80.42.25.41.51.61.84.05.26.13.17.21.47.09.71.05.88. 13.88.22. 22.54.51.34.42.25. S£

,*Q9

ThE AVERAGE WALL TEMPERATURE IS 138.84 DEGREES F

THE ENTERING AND EXITING EULK TEfFEFATURES ARE81.23 ANC 91.. 57

154

THE LOG MEAN TEMPERATURE CIFFERENCE IS 52.27

THE OVERALL HEAT TRANSFER CCEFFICIEM IS 0.C779IN BTU PER MINUTE SQUARE FEET CEGREES F

JHE fcALL FLUID TEMPERATLPE DIFFERENCE IS 57. 6L

ThE MEAN FEAT TRANSFER CCEFFICUNT IS O.C7C7


THE BULK TEMPERATURES ARE SHCkN eELCh

STATICN DISTANCE POWER TEMFEFATLRE H

1 0.011416 0.148667 81 .23 0.1696292 0.034249 0.C62567 82 .60 .C972663 Q.C57G82 0.082961 63.36 C. 0986604 0.079915 .073819 84 .12 0.C895285 0. 102748 0.059164 84.60 0.0719156 0.125581 0.060823 85.34 C.C754667 0.148414 .0^8258 85.90 C.C729128 0. 171247 0.C496C2 66.43 C.C627569 0. 194080 0.042977 86 ,es C.C5448C10 0.216913 0.049149 87 .27 0.062939u 0.239746 0.043632 6 7 . (2

/-> r c /_ -i n a

12 0.2 625 79 0.049fc71 68 .11 C.C6491713 0.285412 0.03926C 86.56 0.05044714 0.3C6245 0.044C19 86.92 C.C5735315 0.331078 0. 031205 89 .32 C.C4053116 0.353911 0.0395(6 6 9 . t C 0.052026

THE AVERAGE LOCAL HEAT TRANSFER COEFFICIENT IS 0.C736

THE FINAL BULK TEfPERATURE k^S

AS CALCULATEC 89.95

AS MEASURED 91.57

155

HEAT TRANSFER CALCULATIONS FCP fi

WALL HEATER IN A RECTANGULAR CHANNEL

THE FLOW RATE IS 6.10 CUBIC FEET FER MINLTE

THE REYNCLDS NUMEER IS 1457.

STATION

1

234567e9

1Cu121314151617181920212221242526272829303132

THE *ALL TENFERATURE

TEMPERATURE

132.133.132.131.132.131.132.129.132.130.131.130.131.130.131.129.130.130.13i.130.130.131.130.130.131.131.131.131.131.130.131.

tt533729125449796604376237544 1

714195244 65C16415483CE83OC875612

131. 7C

THE AVERAGE WALL TEMPERATLRE IS 131.25 DEGREES F

THE ENTERING AND EXITING SULK TENFER^TURES ARE78.32 AND 88.60

156

ThE LOG MEAN TEMPERATURE DIFFERENCE IS 47.61

THE OVERALL HEAT TRANSFER COEFFICIENT IS 0.C661IN BTU PER MINUTE SCUARE FEET CEGREES F

THE WALL FLUID TEMPERATLRE DIFFERENCE IS 52.94

THE MEAN HEAT TRANSFER COEFFICIENT IS 0.0775

THE LOCAL HEAT TRANSFER COEFFICIENTS ANC

THE BULK TEMPERATURES ARE SHCkN BELCW

AT I ON DISTANCE POWER TEMPERATLRE H

1 0.011416 0.148165 78.32 0.1820802 0.034249 O.C8175C 79.68 C. 1049423 0.057082 0.082228 8C .43 C. 1070134 0. C79915 0.C73501 61.18 0.C981915 0. 102748 0.060461 61.66 0.0812356 0.125581 .061153 82 .41 0.C836177 0. 146414 0.C58306 82.97 0.0805803 0. 171247 0.050CGC 62. 5C C.C701639 0.194080

0.216912.044486 82.96 .C62616

10 0.C51C15 84.26 0.C7238411 0.239746 .044598 64 .62 Li t lu JV-> J12 0.262579 0.0D0545 65.23 0.C7355513 0.285412 0.C40C99 85.69 C.C5722614 0.308245 0.0451C8 86 ,C5 0.C6516C15 0.331078 0.C34332 86.46 0.C4968616 0.353911 0.035976 66.77 C. 052272


THE FINAL BULK TEMPERATURE feAS

AS CALCULATEC 87.09

AS MEASURED 88. 6C

157

LIST OF REFERENCES

1. Lord Rayleigh, "On the Dynamics of Revolving Fluids,"Proceedings of the Royal Society of London , seriesA v. 93, pp 148-154, 1916. Reprints in ScientificPapers V6 , pp. 447-453.

2. Taylor, G. I., "Distribution of Velocity and TemperatureBetween Rotating Cylinders ,

" Proceedings of the RoyalSociety of London , series A, v. 135, p. 494-511, 1935

3. Castle, P., Mobbs , F. , and Markho, P. "Visual Observa-tions and Torque Measurements in the Taylor VortexRegime Between Eccentric Rotating Cylinders,"Journal of Lubrication Technology v. 93, pp. 121-129,January 1971.

4. National Advisorary Committee for Aeronautics TechnicalMemorandum 1375, On the Three Dimensional Instabilityof Laminar Boundary Layers on Concave Walls , byH. Goertler, p. 32, 1942.

5. National Advisorary Committee for Aeronautics WartimeReport W-8 7, Investigation of Boundary Layer Transi -tion on Concave Walls , By H. W. Liepmann, p. 22, 1945

6. Schlichting, H., Boundary-Layer Theory pp. 500-522,McGraw-Hill, 196 8.

7. Smith, A.M.O. , "On the Growth of Taylor-Goertler VorticesAlong Highly Concaved Walls," Quarterly of AppliedMathematics , v. 8, p. 233-362, November 1955.

8. Dean, W. R. , "Fluid Motion in a Curved Channel," Proceed-

ings of the Royal Society of London , series A,

v. 121, pp. 402-420, 1928.

9. Reid, W. H. , "On the Stability of Viscous Flow in a

Curved Channel," Proceedings of the Royal Society ofLondon , series A v. 244, pp. 186-198, 1958.

10. Aerospace Research Laboratories Report ARL 65-6 8, A Sim-

plified Approach to the Influence of Goertler TypeVortices on the Heat Transfer from a Wall , by Lief N.

Persen, p. 62, May 19 65.

11. McCormack, P. D., Welker, H., and Kelleher, M. , "Taylor-Goertler Vortices and Their Effect on Heat Transfer,"Journal of Heat Transfer , v. 92, pp. 101-112, Feb-ruary 1970.

158

12. Cheng. K. C. , and Ahiyama, M. , "Laminar Forced Convec-tion Heat Transfer in Curved Rectangular Channels/'International Journal of Heat and Mass Transfer ,

v. 13, p. 471-490, 1970.

13. Mori, Y. , Uchida, Y., and Ukon , T. , "Forced ConvectionHeat Transfer in a Curved Channel of Square CrossSection," International Journal of Heat and MassTransfer , v. 14, p. 1786-1805, November 1971.

14. Akiyama, M. , Hwang, G. J., and Cheng, K. G. , "Experimentson the Onset of Longitudinal Vortices in LaminarForced Convection Between Horizontal Plates,"Journal of Heat Transfer , v. 93, p. 335-341,November 19 71.

15. Aerospace Research Laboratories Report ARL-69-0160,Exploratory Experiments in Water on Stream WiseVortices and Crosshatching of the Surface of ReentryBodies , by Lief N. Persen, p. 55, September 1969.

16. Beavers, G. S., Sparrow, E. M. , and Magnuson, R. A.,"Experiements on the Breakdown of Laminar Flow in aParallel-Plate Channel," International Journal ofHeat and Mass Transfer , v. 13, pp. 809-815, May 1970.

17. Nunge , R. J., "Time Dependent Laminar Flow in CurvedChannels," Journal of Applied Mechanics , v. 93,pp. 171-176, March 1970.

18. Cohen, M. J. and Ritchie, N., "Low-Speed Three Dimen-sional Contraction Design," Journal of the RoyalAeronautical Society , v. 66, p. 231-236, April 1962.

19. Office of Naval Research Technical Report Number 75,Laminar Flow Forced Convection Heat Transfer andFlow Friction in Straight and Curved Ducts - A~Summary of Analytic Solutions , by R. K~. Shah andA. L. London, p. 196, November 1971.

20. Holman, J. P., Heat Transfer , pp. 111-272, McGraw-Hill,1968.

21. Griffin, 0. M. , and Votaw, C. W. , "The Use of Aerosolsfor the Visualization of Flow Phenomena," Inter-

national Journal of Heat and Mass Transfer , v. 16,

pp. 217-219, February 1973.

22. United States Naval Research Laboratory NRL Report 5929,Studies of Portable Air-Operated Aerosol Generators ,

by W. H. Echols and J. A. Young, p. 16, July 1963.

'159

23. Lawrence Radiation Laboratory, University of California,Report UCRL-14754, TRUMP: A Computer Program forTransient and Steady-State Temperature Distributionsin Multidimensional Systems , by Arthur L. Edwards,p. 60, July 1969.

24. Erbayram, Coskun, A Computer Program for Solving Tran -

sient Heat Conduction Problems , M.S. Thesis, NavalPostgraduate School, Monterey, Dec. 1971.

25. Stuart, J. T. "On the Non-linear Mechanics of Hydrody-namic Stability," Proceedings of the !Qth Inter -

national Congress of Applied Mechanics , pp. 63-97,1962.

26. Kline, S. J. and McClintock, F. A. "Describing Uncer-tainties in Single-Sample Experiments," MechanicalEngineering, v. 75, pp. 3-8, January 1953.

-160

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation CenterCameron StationAlexandria, Virginia 22314

2. Library, Code 0212Naval Postgraduate SchoolMonterey, California 93940

3. Associate Professor M. Kelleher, Code 59kkDepartment of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

4. LT Robert J. McKee, USNR5118 Angeles Crest Hwy.La Canada, California 91011

5. Mr. George Bixler, Code 59Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

6. Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

161

UNCLASSIFIEDSecuntv Classification

DOCUMENT CONTROL DATA • R & D

(Security classification o/ title, body o( abstract and indexing annotation must be entered when the overall report Is clas silled)

in A TING ACTIVITY (Corporate author)

Naval Postgraduate SchoolMonterey, California 93940

la. REPORT SECURITY CLASSIFICATION

Unclassified26. CROUP

EPORT TITLE

AN EXPERIMENTAL STUDY OF TAYLOR-GOERTLERVORTICES IN A CURVED RECTANGULAR CHANNEL

DESCRIPTIVE NOTES (Type of report and,lnclueive da tea)

Master of Science & Mechanical Engineer's Thesis; June 1973,j tmORISI (First name, middle Initial, lr.it name)

Robert J. McKee ; Lieutenant, United States Naval Reserve

lEPOR T D A TE

June 19 7 3

CONTRACT OR GRANT NO.

PROJEC T NO.

7«. TOTAL NO. OF PAGES

163

17b. NO. OF REFS

26

»a. ORIGINATOR'* REPORT NUMBER!*)

96. OTHER REPORT NOIS) (Any other numbera that mmy be aetl&iedthtt report)

DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited.

SUPPLEMENTARY NOTES

ABSTR AC

T

12. SPONSORING MILITARY ACTIVITY

Naval Postgraduate SchoolMonterey, California 93940

A rectangular cross section channel, with both a

straight and a curved portion, was designed and built to

create the laminar secondary Taylor-Goertler vortex flow.

An aerosol was used to visualize these flow patterns, and

photographs of the results are presented._A hot wire

anemometer investigation of one location in the curved

section was conducted to obtain the mean velocity Profiles

and turbulence levels. The pressure drop along both the

straight and the curved sections was measured and compared

A constant temperature wall heater was also designed and

installed in the straight section of the channel. Approx

imate values of the heating losses were obtained and the

overall and local heat transfer coefficients were calcu-

lated for the straight portion of the channel. It was

concluded that the vortices develop with both velocity

increases and distance along the channel, and that the

increase in pressure drop due to the vortices varied

from near zero to approximately 20% with increasing

Reynolds numbers.

)D ,

F°oRvM..1473

/N 01 01 -807-681 1

(PAGE 1)

162UNCLASSIFIED

Security Cl«»«ific«tion i-31408

UNCLASSIFIEDSf-'iiri! v Cl.issifii .it ion

KEY WO RDS

TAYLOR-GOERTLER VORTICES

CURVED CHANNEL FLOW

SECONDARY LAMINAR FLOW

),F

Nr 651473 <BACK)

310) -907-6821

LINK A

« 1 >- O L E

163UNCLASSIFIEDSecurity Classification

Thesis 145054M2154 McKeec.l An experimental study of

Taylor-Goertler vorticesin a curved rectangularchannel

.

3 i WAR 77 2 3 7 9 5

" 7

Thesis 145054M2154 McKeec.l An experimental study of

Taylor-Goertler vorticesin a curved rectangularchannel

.

thesM2154

An exper mental study of Tavlor-Goertler

3 2768 001 88211 1

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