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
11. Photographs of Vortices Re = 875 46
12. Photographs of Vortices Re = various 47
13. Photographs of Vortices Re = 1033 48
14. Photographs of Vortices Re = 1166 49
15. Photographs of Vortices Re = 1457 50
16. Velocity Profiles Re = 609 55
17. Velocity Profiles Re = 875 56
18. Velocity Profiles Re = 1166 57
19. Velocity Profiles Re = 1298 58
20. Velocity Profiles Re = 1457 59
21. Velocity Profiles Re = 1722 60
22. Velocity Profiles Re = 1934 61
23. Typical Position of Hot Wire 63
24. Turbulence Intensity Re = 609 64
25. Turbulence Intensity Re = 875 65
26. Turbulence Intensity Re = 1166 66
27. Turbulence Intensity Re = 1298 £7
28. Turbulence Intensity Re = 1457 69
29. Turbulence Intensity Re = 1722 70
30. Turbulence Intensity Re = 1934 71
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
35. Heat Transfer Coefficient Tw = 133 96
36. Heat Transfer Coefficient Tw = 143 97
37. Heat Transfer Coefficient Tw = 139 98
38. Heat Transfer Coefficient Tw = 131 99
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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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
plexiglaswall
flow
electricallead
mountingblock
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
34
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
>— concave wall
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
A. EXPERIMENTAL PROCEDURES
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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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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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
A. EXPERIMENTAL PROCEDURES
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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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
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112
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INITICAL PCSITION= C.270VC2 = 2.C20000 BETA = C. 7LOOOO
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4.663301
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4.283268
4.026724
3.719356
3.369696
0.03C997
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0.023128
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0.014C43
0.013969
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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
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0.015C16
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0.012514
0.012398
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0.0127C4
0.014266
0.014929
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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
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C.C6C000
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C.14CC0C
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2.084258
3.600484
4.825512
5.407611
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6.C577C6
6.111675
6.220504
6.3C2920
6.335982
6.497612
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0.03C549
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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
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6.3C2920
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6.247911
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. 1 7 6 £ 1
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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
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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
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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
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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
PCSITION (FT) PRESSURE (IN H20)
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
THE PRESSURE DROP IN THE STRIGHT SECTICN IS 0.016133
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
THE PRESSURE DROP IN THE STRIGHT SECTICN IS 0.004000
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
PCSITION (FT) PRESSURE (IN H20)
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 LOCAL FEAT TRANSFER COEFFICIENTS AND
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 AVERAGE LCCAL HEAT TRANSFER COEFFICIENT IS 0.0630
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 LOCAL FEAT TRANSFER COEFFICIENTS AND
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 AVERAGE LCCAL HEAT TRANSFER COEFFICIENT IS 0.0615
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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