EXPERIMENTAL INVESTIGATION
OF
SHEAR FLOW IN
RECTANGULAR
BENDS
by
Warren
Jby
B.S.,Massachusetts institute of Technology(1949)
Submitted in partial fulfillment of requirements
for the Degree of Master of Science
at the
Massachusetts Institute of
Technology
(1950)
Signature of the Author
Department of Aeronautical
tngineering,May 19,1950
Certified by
Thesis
.p e rv is oY
Uhkairman, .oepartmental committee on Graduate students
Massachusetts Institute of Technology
CambridgeMassachusetts
May 19,1950
Professor Joseph S. Bewell
Secretary of the Faculty
Massachusetts institute of Technology
Cambridge ,Massachusetts
Dear Sir:
Submitted herewithin partial fulfillment of the
requirements for the degree of Master of Science,is a
thesis entitled "Experimental Investigation of shear
Flow in Rectangular Bends."
Respectfully submitted
ACKNOWLEDGEMENTS
Grateful acknowledgement is made to the following
persons and organizations:
a.) to Professor Hawthorne for guidance in
this project and for contributions to the theory presented herein,
b.) to Jim Hands and the staff of the Gas Turt.
bine Laboratory for the use of tools,eguipmentand
apace,
c.) to F. Lustwerk for help in obtaining
required materials,
d.) to the Westinghouse Electric Corporation for financial assistance,
e.) to M. Demayo for assistance in preliminary investigations,
f.) and to ZisfeinFelbeck,Valentine,and
others for the use of their thesis equipment and
apparatus.
ABSTRACT
Experiments on Secondary Circulation induced in
rectangular bends are discussed in this thesis.All
experiments were performed in the 4as Turbine Laboratory at M.I.T. in 1950. a velocity gradient was
pruduced upstream of the bends testedand an analysis of the flow in the bend and downstream of the
bend was ma&e.Experimental results are presented in
the form of velocity distribution "maps". The bends
tested had five inch by ten inch cross-sections
(the shorter side was parallel to the plane of bend).
The radii of turn tested were seven inches and fifteen inches.Angles of turn up to 180 degrees were
produced.
An approximate theory was derived from the results of the work of Squire and Winter and of
vawthorne and is presented in this thesis.This
approximate theory corrolated fairly well with
the experimental results.byurther work of a similar nature wiil be required to check the accuracy of the theory presented.
The results of this thesis indicate that
this approximate theory can be applied to design
I'
problems in an effort to minimize losses due to
secondary circulation.
Figures S-1,S-2,and S-3 are included here to
give an indication of the type of flow patterns
encountered in this project.
9Q0 RECTANGULAR
STA.
R=15
BEND
STA.
iNS.
2C = 5 INS.
2U
= JOiNS.
87
x
50
(1
0
STA.
VELOCITY
IN
F.P.S.
2.C]
2b
_L
0
STA
G)
1600 RECTANGULAR
STA.
STA.
R = 15iN5
STA.
®
50
aC r 5 INS.
zb
BEND
6o0
o=10
INS.
808
8f0
mT
50
0
STA.
bJ
hu- lE
VELOCITY
IN F. P S.
STA. @
STA.
o(f)
ZI
900 RECTANGULAR BEND
EFFECT OF BEND RADIUS
71iN).
ca-5Ns.
b -t1NS.
3-
,
7.
AVERAGE
PROFI LE
2-
T
R
STA
STA. ID
STA
t -
-
0*
0,
v (fps.)-
14
q+
t
-- a
x
2b
L2
6
15
INS.
I
AVERAGE
PROFIL E
-5ms
b -low.5
~1
vi
TABLE OF CONTENTS
Abstract
Table of contents
PO. i
p.
Vi
p.
Table of Symbols
vii
Sections:
I. Introduction and Background
II. Description of Apparatus
III. Test Procedure
Discussion of Errors
IV. Results
V. Theoretical Considerations
VI. Conclusions
VII. Recommendations
VIII. Bibliography
P. 2
p.
10
p.
25
p.
33
p.
48
p.
94
p.
113
p.
117
P. 121
Appendices:
A. Raw Run Data
p.
125
B. Data for C.F. derivations
p. 133
C. Data for vorticity caleulations
p. 141
Figures :
1-1 to 1-3
pp. 6-8
2-1 to 2-6
pp. 17-23
3-1 to 3-3
pp. 43-46
4-1 to 4-37
pp. 55-92
5-1 to 5-6
pp. 105-111
Y11I
TABLE OF SYMBOLS
2a
width of bend cross-section
Ax
cross-sectional area
AR
Aspect Ratio ( Lb/2a )
2b
height of bend cross-section
CF1
correction factor for correcting velocity
for variations in Q, and P,.9
GFz
correction factor for correcting velocity
for vakiations in upstream velocities.
CF3
correction factor correcting for incorrect manometer calibration.
D
outside diameter of pitot tube
d
diameter of cylindrical tube
g
acceleration of gravity
gradA- total pressure gradient
A h
height of liquid head
A1
axial distance a particle moves in time
interval &t .
Pamb
ambient air pressure
Pb
stagnation pressure inside settling
chamber
Pbd
b""am )measure& value of stagnation
pressure lnside settling chamber.
Pbd
ref
Phead
Pind
an arbitrary value of Pbd chosen as a
standard
pressure under a head of liquid
measured ambient pressure of outside air
Vill
P,0
stagnation pressure in flow
Pstat
static pressure in flow
P
air pressure at standard temperature and
pressure conditions (29.92" Hg )
stp
Ptt
stagnation air pressure in flow
AP E (Pstat"'
pressure
q
b) measured value of the static
unction
dynamic pressure ( i-47
2
)
dynamie pressure gradient
R
radius of turn
RE
Reynolds bumber
Ref.
Reference in sect. 8
a a.
linear distance along the manometer tube
a s
are moved by rotating flow in time interval At
(As)
ave
average value of .As
T
measured ambient air temperature
T
standard ambient air temperature (59 deg.F)
A
axial velocity
116
Amn
minimum velocity in a velocity gradient
maximum velocity in a velocity gradient
V
velocity
VII
velocity inside settling chamber
Vind
Vindoorr
velocity indicated by the manometer
ind corrected for &p error
Vmin
minimum velocity at a section
Vmax
maximum velocity at a section
Vref
indcorr corrected for errors in
and Pbd
Vrefcorr
Vref corrected for error due to
changing upstream velocities
Vref 3 .7
Vref where Pbd = 3.7" H2 0
ref 340
Vref for run number 340
Vref 4 6 1
1ref for run number 461
Istd
Vindoorr corrected for errors in
only
AV
correction on velocity due to Ap
error
(d)
velocity gradient
V
tangential velocity induced by the
seconctary circulation
x
horizontal coordinate at cross-section
y
vertical coordinate at cross-section
c
angle of flow rotation
r,
circulation
B
a deviation in coordinates due to the
effect of the total pressure gradient
on the pitot tube reading
e
angle of turn
e
angle of turn when
OC
-
vorticity
maximum value of vorticity tat
e
)
radius from center of circulation
air density
density of liquid
measured air density
?i
air density at standard temperature and
pressure conditione (.002378 slugs/ft 3 )
stagnation densit y of air
angle between the normal to the stream
surface and the direction of acceleration
£2. initial velocity gradient (
IT
3.1416
)
-7
SECTION
1
Introduction and Background
Section I
The existence of secondary flows has been known
for some time.These flows were first observedin
connection with the flow of riversby J.Thompson
in 1876.Yen and aowe state: "The lateral currents
induced in bends were first observed by J. Thompson in 1876 in a series of experiments conducted
in an open channel having a 180 degree bendthe
spiral motion being observed by seeds and drops of
aniline dyes.I 2
A study was made by BlueHerbertand Lancefield
concerning the flow around a bend in the Iowa
River.They state: "Current velocities at the bed
of the stream are very much less than those near
the surface.....&ear the outside bank the water
moves from top to bottom...near the inside it
moves from bottom to top....(this)combination...
results in a helicoidal flow."4 Figure 1-1 is
taken from their work and shows definitely the
"scouring" action of the helicoidal currents.
l.Ref. 31,p.5
2.Ref. 30,p. 714
3.Ref. 3 2,p. 259
4.Ibid
3.
Some experimental work has been done by Shukry
on the flow of water in channels. Figure 1-2 shows
flow patterns taken from his work. In this case,
since there are two velocity gradients upstream, we
get two distinct spirals downstream.
The secondary flows produced by the turning of
a fluid are not confined to the field of hydraulics
alone. They are found in the field of aerodynamics
also. The flow of air in ducts, pipes, and in cascades of airfoils may contain secondary circulation
due to initial velocity gradients. it has been estimated that secondary flow may acoount for 50% of
the losses in compressor cascades.
Most of the analysis of air flow around bends
done hertofore has been confined to flow with no
initial velocity gradient 3 4 5
.
Squire and Win-
ter 6 7 were the first to work out the theory for
simple flows containing initial velocity gradients.
kawthorne has modified this theory to apply to the
flow in cylindrical pipes 8 .6ection 5 in this
thesis contains the results of Squire and Winter
1.
2.
3.
4.
5.
6.
7.
8.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
29
33
13
14
21
25
11
12
4%.
and the Hawthorne modification cited above.
A variety of experimental work has been done on
fluid flow in bends. 12 3 4 5 6 7 8 9 10
mere again,
the flows were uniform upstream and contained no velocity gradients.A bibliography of this work is presented in reference 19
.
Squire and winter II were
the first to experiment with a flow containing an
initial velocity gradient and to corrolate the experimental results with their theory.They observed
the flow in the boundary layer of a wind tunnel and
its action as it moved through the turning vanes at
the corners. Some work has been done also on the
same type of flow by uemayo 12 at vi.I.T. in 19491950 with an apparatus similar to that used by the
author.
Because the results of Squire and winter and
those of Demayo indicated that more work needed to
be done on this problem, Professor Rawthorne 13
instituted two simultaneous proects. One project
was
concerned
with
ninety-degree
bends
1. Ref. 1
7. Ref. 19
2.
8. Aef. 20
Ref. 3
3. Ref. 15
9. fief. 21
4. Ref. 16
10. Ref. 22
5. Ref. 17
11. Ref. 25
6. Ref. 18
12. Ref. 4
13. Westinghouse Professor in Mechanical
Engineering at m.I.T.
5.
of circular cross-section to be carried out by
Eichenberg
.The other was concerned with bends
of rectangular cross-section and was to be carried
out by the author. Fig. 1-3 shows a general picture of the type of flow to be investigated. The
two projects were proposed in order to obtain
quantitative data on the vorticity in ninetydegree bends , and to ascertain the effect of
changing the design variables of the bends
(such
as aspect ratio, radius ratio). Work done by
Eichenberg 2 on 180 degree bends of circular
cross-section led to similar work by the author
on rectangular 180 degree bends.
It was hoped that the results of these propects would either corroborate the present
theories or serve as a foundation for modified
theories. When the theory connected with this
problem is made more adequate and when more is
discovered about the problem
,
attempts can be
made to minimize losses due to secondary circulation by more efficient design of ducts and
cascade passages.
1. Ref. 26
2. Ibid
6.
SECONDARY
FIG. I RIVERS
FLOW IN
DEPOSITED
URING"
1925
1933
BOUNDARY
BOUNDARY
SECT10N
TOP
AA
FIGURE FROM
REF.
32
FIG. I TAKEN
FLOW
2
FROM
VELOCITY
LOOKING
I
REF.
PATTERNS
29
MAPS
DOWN
STREAM
5' UPSTREAM
0'
FLOW
PATTERNS
LOOKING DOWNSTREAM
600
90
150
VELOCITY
IN
CM/SEC.
F
FIG. I -3 SHEAR FLOW
TH ROUGH BENDS
INITIAL
VELOGITY GRADIENT.,..
r
Low v.
HIGH V.
HI GH
V.
9.
SEOTION II
Description of Apparatus
10.
Section II
A schematic diagram of the apparatus set-up is
shown in fig. 2-1 . The apparatus can be described
as a combination of five units. These units are ;
the blower-nozzle, the extension containing the
velocity gradient screen, the "exploration chamber,
the bend, the extension downstream of the bend. A
description of each follows below.
The blower-nozzle unit was designed and constructed by Ziefein.A detailed description of it is
available in his thesis 1.
its function was to pro-
vide the required air flow for the experimental
work. The rate of flow produced was approximately
1200 cu. ft./min. through an exit cross-section of
five by ten inches. The unit contained provisions 2
for measuring the stagnation pressure inside the
settling chamber (Pb )*
The extension containing the screen served to
change the flow from one with a uniform velocity
distribution to one with a velocity gradient.Fig.
2-la
shows qualitatively how this was accomplished.
This unit was designed and constructed by ielbeck
and is more fully described in his thesis 3.
1. Ref. 1 ,pp. 10-15,21
2. Ref. 3 ,p.16
3. Ref. 2 ,p.lll8
The
extension had a cross-section of five by ten inches
throughout.The screen used was of the 60 mesh size
and of a double layer. The exact shape is given on
page 18 in ref. 2 . Typical velocity profiles are
given on figures 4-2,4-12, and 4-24 in section 4 of
this thesis. it should be noted that the performance
of this screen configuration can be sharply impaired
by any accumulation of dust that may gather upon it.
The so-called "exploration chamber"
was designed
and constructed by Valentine. A detailed description
of it is given in his thesis
. it was used by this
author as part of the apparatus so that the velocity
distribution upstream of the bend could be determined
accurately. Its cross-section was five by ten inches
throughout.
The above three components of the apparatus remained unchanged as a single unit during all tests.
it was dis-assembled only when it was necessary to
remove dirt from the gradient.producing screen.
The relative size and shape of the three bends
used is shown in fig. 2-2 . The 15 inch bends were
designed by the author and were constructed with
the cooperation of the carpenter shop and the Gas
Turbine Laboratory
.
The non-turning sides (i.e,the
1. Ref. 3 ,pp.19,114,119-122
12.
top and bottom ) were made of plywood. The turning
sides were made of sheet brassproducing a smooth
surface as well as a rigid structure. The seven inch
bend was used by Zisfein and Valentine and is des12
cribed in their theses
.The non-turning sides were
made of plexiglass *The turning sides were made of
laminated hardwood.
Investigations by Demayo on a similar apparatus,
but one without the extension downstream of the bend,
demonstrated that both mixing and separation phenomena distorted the flow downstream of the bend.This
indicated the need for an extension to cut down these
effects. Such an extension was incorporated by the
author and served several purposes:
1.) It eliminated mixing effects.
2.) It out down separation effects by applying a slight back-pressure. t3,4)
3.) It more nearly simulated the flow of
air in a duet.
4.) It served as a channel for investigation
of the flow downstream of the bend.
5.) It served to stabilize the flow somewhat
by its damping effect.
l.hef.
2.ref.
3.nef.
4.aef.
5.Aef.
1 ,p.20
3 ,p.14
18
21
4
t3.
This extension was designed and constructed by the
author with a five by ten inch cross-section throughout.it is 6.5 feet long. Its top and bottom (the five
inch sides) were made of plywood. The first four feet
of the sides was constructed of 1/8 inch plexiglass.
The remaining part of the sides was constructed of
t
inch plywood.,with provisions for the installation of
plexiglass there also,if required. The plexiglass served
three purposes:
1.) It provided a smooth
surface.
2.J It made orientation of the pitot and
static tubes easy and accurate.
3.) It made visual observation of the flow
possible in future experimentation,if
desired.
The traversing mechanism used was a milling
machine similar to that used by Valentine 1
This
machine provided motion in three mutually perpendicular directions which could be readily calibrated. An angle iron was bolted to the bed of the
milling machine and was used as a
pitot tube.
support for the
By varying the length and position of
the angle iron, and bend-extension configuration
could be investigated without moving the milling
machine.
1. Ref. 3 ,p.ll,12
'4.
Built into the bottom of the fifteen izch bends
and the extension were slots with moveable slides similar
to that used in the "exploration chamber" by Valentine l.
The slides were designed to be flush with the
inside of the ductand each contained a hole.This hole
had a diameter approximately equal to the pitot tube
barrel diameterin order to insure a tight fit. This
design enabled the traversing of the flow with a minimum of disturbance to the flow. Figures 2-3 and 2-4
show the slot and slide configuration in a fifteen
inch bend. Slots were cut into the bottom of the
seven inch bendbut clearance limits prevented the
incorporation of slides.These slots were kept taped
over when not in use.
The position of each slot in the bend and extension was given a Station Number. The system of numbering varied from bend to bend. The system is given in
the results section before the velocity distribution
maps of the applicable bends .The figure numbers are
fig. 4-1,4-ll,and 4-23 .
Velocities were measured with the pitot tube
shown in figures 2-4,and 2-5 * it was constructed
by the author out of thin-walled 0.057 inch bypodermic tubing.The nose was slightly rounded off.
l.
hef 3.,p.
19,122
15.
This tubing was wedged into a barrel of .410 inches
diameter with a wooden wedge of very small size. The
nose of the pitot tube was approximately two inches
upstream and above the barrel ,lessening interference effects. To the bottom of the hypodermic tubing
was soldered an adapter,to which was attached the
tubing leading to the manometer. uuring all tests,
the nose of the pitot was placed perpendicular to the
plane of the cross-section and was pointed upstream.
The manometer used was the one used by Valentinel
with a slight modification in the reservoir arrangement. The arrangement is shown schematically in fig.
3-1 in the next section of this thesis.
Static pressures were measured by a static probe
designed by Valentine and described in his thesis 2,
it consisted of a bent piece of .045 inch tubing with
two .020 inch holes drilled in opposite sides.Figure
2-6 shows the shape and the location of the holes.
This piece of tubing was soldered to a brass barrel
to facilitate traversing. In all runs this static
probe was traversed manually by removing the slides
from their slots. The manometer configuration is
shown
in fig. 3-1 . The static probe was set up so
that it measured the pressure differential Ap.
1. Ref.3 pp. 30,33 (called #3)
2. ibid pp.15,17
16.
The temperature in the flow was measured with a
mercury thermometer placed through the hole in the
slide at each station.
Ambient pressure was measured with a mercury barometer located in the room next to the cell where the
experiments were carried out.
The stagnation pressure in the settling chamber
was measured by a simple vertical manometer constructed by the author. The arrangement was somewhat like
that shown in fig. 3-1 .
All apparatus mentioned in this thesis was located in the center combustion cell on the second
floor of the Gas Turbine Laboratory (unless specified elsewhere).
17.
FIG.
2 -1
A PPARAT US
ARRANGEMENT
TOP
VIEW
-
EXTENSION
CONTAINING SCREEN
SETTLING
CHAMBER
"EXPLORATIO N
\
CHAMBER
vs
BEND
BLOWER-NOZZLE
UNIT
EXTENSION
OF BEND
t8.
F G. 2-IA
SCREEN
CONFIGURATION
CON FIGURAT ION
OF
SCREEN
FLOW
DIREC TION
SCREEN
VELOCITY DISTRIBUTION
DOWNSTR EAM OF SCREEN
'9.
FIG.2-2 COMPARISON
OF BEND DIMENSIONS
15" x 180" BEND
RI1"
NOTE: ALL CROSS-SECTIONS
THROUGHOUT
ARE 5"x 10"
I5 " x 90' BEND
7"1- 90' BEND
R= 15"
R a7
20.
FIG. 2-3
SLOT
MECHANISM
21.
FIG. 2-4-
PITOT
POSITION
22.
FIG. 2-s
PITOT
TUBE
23.
FIG.
2-6
0.35'
STATIC
PROBE
PRESSURE
jA
A
.o-.045"
.020,
DRI LLED
HOLE
HOLE
SECT.
AA
24.
SECTION
Test
III
Procedure
and.
Discussion of
Errors
25.
Section III
The manometer configurations are shown in fig.
3-1.The manometer used for velocity measurements
is the one labeled g3 by Valentinel. It
was first
set up to read impact pressure in inches of water
by the author as follows:
Ah x (Boa.)
sin 14.5
where &s is the linear movement of the
liquid along the tube,
where s.g. is the specific gravity of the
liquid used,
and where a h
water.
is the head in inches of
The author used kerosene in the reservoir and
the value of s.g. was taken as 0.82
Thus: As'= 2.92(Ah)
(eq. 3-1)
That is,for a pressure of one inch of water,the
movement along the tube of the column of kerosene (As' ) will be 2.92 inches.
From aerodynamic theory we have:
Ptot
!t at
where: P
l.±ef. 3 p.30,03
z2
is total pressure,
26.
the density of air,
QA
is
V
is the velocity,
Pstat is the static pressure.
From bydrodynamic theory we have:
head
where: P
0
a
is the total pressure exerted
ti9e liquid in the reservoir,
Pamb is the ambient pressure in the
room,
qL is the density of the liquid in
the manometer,
Combining the above two relationsand assuming
Pstaf
amb
, we get:
tggtV2
and:
=tge
2(eq.
3-2)
This function is plotted on figure 3-la and was
obtained from reference 5.1t is based on a manometer using water at 24 deg. 0 in the reservoir.
By combining eq. 3-1 and eq. 3-2 ,the author
calibrated the manometer to read velocity direetly. Since this calibration depended on the
assumption that
P stat - .amb ,a correction was
l.for simplification in reduction of data
27.
Pamb
derived to allow for cases where P
amb
stat
This correction chart is plotted as fig. 3-2,and
was derived from the chart on fig. 3-la. The author
defined
V
indcorr
as follows:
Vind corr
where:
ind+A V
(Eq. 3-2a)
Vin&corr is the corrected indicated
velocity,
is the reading of the velocity
rind
scale on the manometer,
&V is a function of &p and V and is
plotted on fig. 3-2,
aP
stat ~Pamb
is the true
Thus V
indoorr
velocity at the point
being investigated.
in order to eliminate
the effects of variation
in outside atmospheric conditions and the effects
of conditions in the blower-nozzle unit, it was
decided to standardize all velocities ( Vindcorr
in the manner described below. in the analysis that
follows the flow is assumed to be incompressible.
1. subject to the errors discussed later in
this section
28.
To correct for variations in R. due to temperature and pressure
Vstd
,
we note that :
(from eq. 3-2)
-
stp
incorr
thus:
rind
Vstd
Vind
indcorr
stp
---------
stp
stp
(eg.
3-3 )
Tind
where:
st is the velocitystandardized to
standard temperature and pressure
conditions,
, and Tind are actual,
ind IP
measured conditions in air,
ot
=.002378
T
=.. 9 deg.F ,
stp
Ps
slugs/cu. ft.
= 29.92 in Hg.
To correct for errors in velocity due to errors
in the total pressure in the settling chamberthe
author defined a quantity
Pbd
Pbd 2 Pb *amb
where:
?bd was measured directly with a
manometer,
is the total pressure inside
Pb
the settling chamber
29.
Pam is ambient air pressure outside the settling chamber.
To standardize the results,the author chose an
arbitrary value of Pbd for each bend and called
it Pbdref . in most eases Pbdref was the value of
Pbd found when investigating station 2 or 3
arom aerodynamic theory we have:
Pb =Ptt+
thus:
pbd+
Now if we assume
then:
amb
ab
Vindcorr2
i
stat
stat
Pbd
iA(
Vindorr
)
*(ie.
Ap*
Vindoorr
thus:
indoorrO"b
The author now defined Vref by the'following
relation:
ref
Vstd
=
lbdref
Pbd
(Eq.
3
-4)
1. see section 4 for station locations
2. all such superscripts are exponents
3. tests showed this assumption to be valid.
30.
where:
V
Pbd
is defined in eq. 3-3,
is the measured value of Pbdo
Pbdref is the arbitrary value of
Pb chosen.
Combining equations 3-3 and 3-4
ref
indoorr
Ebdref
Tstd
Tin
bd
,
we get:
Pind
std
The author further defined a correction factor:
bdref
Pbd
std
Pid
Tind
2std
therefore:
Yref
indcorr
X(cF1 )
(eq. 3-5)
Therefore,if we were to take readings at the
same location at different times ( allowing the
pressure,temperature and Pbd to vary ) , we would
expect Vind
and Vindoorf to differ in each case.
kowever, we would expect
Vref to be the same in
each case. bow it was found that Vref
did not
31.
remain the same. This was not the fault of the theory
proposed above,but was due to the accumulation of
dirt on the velocity gradient-producing screen.
Because of the above factthe author made duplii.
cate runs: at various stations (usually sta. 2 ) at
different times to note the change in velocities
(i.e. Vref ) and to derive a second correction
factor ( C.F.2 ) to account for this effect.Appendix
B gives calculations involved. The author defined
C.F.2 as follows:
C.F*2
=Vrefideal
Vrefactual
where:
Vref is defined in eq. 3-5.
By experiment C.F. 2
was found to be a function of
either the run number or Pbd. Whichever was more
convenient was used in the calculations.The author
further defined
Vrefcorr as follows:
Trefcorr =Vrefx"G.F2
(eq. 3-6)
In runs 100 to 200 an incorrect manometer scale
was used( it assumed water to be in the reservoir).
32.
Rence all values of velocity read were too high,
by a factor
1/s.g. . Therefore the author defined
a third correction factor to be used on runs 100 to
200 only.
C.F.3
g.
= .905
(eq. 3-6)
In all other runs, C.F' 3.E 1.00
Finally,combining equations 3-2a,3-5,and 3-6,
we get:
Vrefcorr
OF1 x CF2
F3 ( Vindcorr
or:
Trefcorr
OF1 x 0F2 t OF3 x( Vind+AV ).
The procedure for a set of runs is outlined
below.
The apparatus was turned onand the pitot tube
was placed and oriented at the proper station.
before any run data was taken ti.e. Vind ),readings of ambient pressure ( Pa
(Tind
j
,
and 2bd
jtemperature
were taken.During this time
the flow had had time to become stabilized.
Each traverse was considered a "run" ,for
the relation between the coordinates (y,x )
33.
during any traverse was well determined (due to
the rigidity of the milling machine arrangement).In
most runs,readings were taken at every
i"
in each
directionas determined by preliminary investigations 1.
After data was taken for a few runs , another
reading of Pbd was taken. At the end of the runs for
each station final readings of ±ebd and Pamb were
taken. Then readings of A p were taken with the
static probe. During runsfrequent visual measurements of pitot tube position were made and orientation noted.
Each of the following paragraphs discusses a
possible error in the veloc.ity plotted on the velocity distribution maps.These errors were either
due to inaccuracies in measurements of the velocities themselvesor due to inaccuracies in the
measurements of the coordinates x and y .in each
case the author has underlined either the reason
for the error)or the assumption made which leads
to the error. The percentage error is estimated
whenever it
is appreciable.
1. Ref. 4
3+4
Specific Gravity of Kerosene is 0.82. in the
apparatus used the actual specific gravity was never
determined by measurement. The figure of 0.82 was
taken as a weighted mean of values found in the
international uritical Tables'. These tables give
an absolute variation of from .792 to .869,with
the bulk of values listed falling between 0.80 and
0.84 approximately. Pavian 2
gives a variation of
.789 to .878. The variation of specific gravity due
to temperature variations was neglected because it
is small when compared to the uncertainties listed
above. Since the velocity depends on N s.g. (eq.
3-2 )
the error in velocity is approximately one-
,
half the error in the specific gravity (on a percentage basis). The error in specific gravity is
3% ,therefore the estimated error in velocity is
S1.15%
.
The Effect of Yaw on Pitot Tube Readins is
veglected . We know that our pitot tube was in yaw,
because the flow had a secondary circulation and
therefore a component of velocity perpendicular $o
1. ±ef. 6
2. Ref. 7
35.
the direction of velocity.The author has calculated this secondary circulation roughly in Appendix C
*
A measure of the angle of yaw is the quan-
tity as/Al
(see fig. 4-34a )
where: As is the tangential displacement
in the time At
Al is the axial displacement in the
time &t.
Thus the angle of yaw is tan" 1
As/ Al . Reference
to Appendix 0 gives the following values of As/ Al :
bend
(
as/Al)max
(As/ Al )ave
15"-90 deg.
.36
.20
15"-180 deg.
.39
.15
1.32
.30
7"1-90 deg.
The corresponding angles of yaw are:
bend
max.yaw
1511-90 deg.
20 deg.
15"-180 deg.
23 deg.
V"-90 deg.
53 deg.
Figure 3-3
ave.yaw
11.5 deg.
9
17
deg.
deg.
gives the results of some testsl run on
pitot tubes at 55 ft/see. Oonsidering the relative
1. Ref. 8
36.
shapes of the pitot tubesthe author estimated the
error to be approximately that of the top one pictured (to be conservative). Thus the average error
in the dynamic pressure(q ) is about +6% for the
15" bends and about + 10% for the 7" bend. The
error in velocity will be one-half that in the
dynamic pressure. Therefore, the estimated error
in velocity is +3%
on the 15" bends and +5% on
the 7" bend,.
Error in C.F* 2 - A look at the spread of values
used in derivation of C.F*2 (in Appendix
.8
)shows a
possible error of ±5* in velocity.
Blocking Error due to Pitot Barrel. The flow
velocity is inversely proportional to the crosssectional area, if we assume the flow to be essentially incompressible.
Area of Cross-section is 50 in. 2
maximum height of barrel is 8 in.
(at y 9.9")
Diameter of pitot barrel is 0.4. in.
Thus:
Maximum blocking area is 3.2 in2
Blocking Area Ratio is .062
37.
therefore:
Maximum error is 4 6%
Estimated average error is + 3%
Error in Velocity due to Errors in Measurement
of Ap
Appendix A gives measured values ofap.The
*
maximum value recorded was ± .08 in. ki
20
fig. 3-2 shows the average
A look at
.
V to be ± 2 to ± 3
ft./sec.. The average recorded velocity was 60 ft./sec.
(Vind )
.
If &p were in error by t 20% ,the error in
V would be approximately i 0.5 ft./sec.. Therefore,
the estimated error in the velocity is ± 2%.
Effects of Uompressibility are Beglected
.
The
maximum velocity encountered (Vrefcorr ) was 87
ft./sec.. Therefore, the maximum Mach Number was
~.08 . ieference 28 gives a zero error at this low
Mach Number. aeference 10 gives
=.996 ,which
means that the error in Q is about + 0.4%
the error in the velocity is negligible
.
Thus,
.
Apparent Displacement of Lines of Constant
Velocity due to the velocity Gradient's Effect on
Pitot
Tube
Readings . Young and Maas 1 have per-
formed experiments on the performance of a pitot
tube in a velocity gradient. Their results give
1. Ref. 9
38.
the function 1
d
as a function of
D
q
g
where:
& is the apparent shift in a
line of constant total pressure,
D is the outside dia. of pitot,
q is the dynamic pressure,
(AI)is the dynamie pressure
gradient.
We know that:
2
Investigation of plots given in section 4 shows:
minimum
V= 40 ft./see.
maximum local
,
-960
=d) see-1
7 in the seven inch bend )
diameter (D)
(at sta.
,
.057 in.
thus:
[24
V
Cx
max.
xoung and Maas 1 state that
V)
from 0.1 to 1.2
,
,
= .17
in the range of
the maximum value
of D is 0.30 . lence, the maximum value of 8
in .0012 inches. This error is negligible when
compared with other errors.
1. Ref. 9
I
39.
Calibration of Manometer Assumes Tube to be
Straight . Valentine had previously calibrated the
manometer used by the author and a plot given in
his thesis 1
shows a linear relation between the
pressure and the reading. This demonstrates that
the tube was indeed straight.Thereforethere
is
no error incurred.
Errors in Vind
, Vind was read to the nearest
i0.5 ft./sec. in the range 50 to 9U ft./see. and to
the nearest21 ft./sec. in the range 20 to 50 ft./sec.
In aduition to the reading error above,an error
might have occured due to dynamic lag in the manometer system. However, sufficient time was allowed
between readings to let the fluid come to within
the reading error before another value was read
on the manometer scale.Thusthe lag error was incorporated into the reading error. TXhe estimated
error in Vmd is therefore t2o .
The Flow in the Apparatus is Assumed to be
Steady . Any unsteadiness would probably have been
damped out by the manometer fluid-air system.Thus
the author assumes no error in readings occurred
due to this cause.
1. Ref. 3,p. 33
40.
Error due to Uncertainty in Values of x and y.
Since there were velocity gradients in the flow,
any error in x or y could result in an error in the
velocity.(That is,
the velocity reading would be
correctbut its place on the velocity map would be
in error, giving an erroneous flow picture.).The
maximum velocity gradient was
960 sea-1 . The
maximum estimated error in x or y is 11/8 in.
Thusthe potential maximum "error" is±10 ft./see
(i.e. error in YVef
). This error seems large,
but an analysis of the flow patterns shows that
such an error would actually be insignificant in
the high velocity gradient producing it. Therefore, such errors were not considered.ihus, the
flow patterns can be assumed to be free of this
error.
The analysis contained in the ten previous
paragraphs points to a potential error of 416%
and -11% in the velocity. The purpose of this
thesis was to obtain data on secondary circulation.The theory used in the calculation of the
circulation was approximate anyway,
so this
error in velocity is not really too high when
compared to the errors involved in the theory.
4-I.
A look at figures 4-35,4-36,and 4-37 shows a very
large spread in the calculated values of
.
srrors in other measured quantities are discussed below.
The largest potential error inAp was due to
the angle of yaw between the streamlines and the
static probe. Ower 1 states that a 20 degree yaw
may give -8v error in static pressure reading.
Merriam and Spaulding 2 give -11o error for 20
degrees of yaw.kioweverthese experiments were
static tubes in "symmetrical'
carried out on
yaw (that is,
the tube was yawed in a uniform
stream). in our case we had a helical flow across
the two static holes, The error is probably
larger that that predicted abovebut a numerical
estimate is difficult to make.kiowever,during
measurements of
ap ,the probe was yawed from
calIy
side to side slightly and a very small change
in the indicated value of
p was noted at mod-
erate angles of yaw.
There was another error in Ap possibledue
to the fact that the sections investigated were
not completely isolated from the ambient air.
1. uef. 27
z. Aef. 24
42.
This came about because the slides 1 were removed from the slots for the measurements of Ap.
The error in Ap estimated from the above considerations is ±30% . we note that an error as
large as this would still have only a small effect
on the accuracy of the velocities.
Pbd was measured with a total pressure probe
designed by Valentine
ometer to the nearest
.Lt
±
was read on a man-
0.05 in. H20 .he
mean
value of Pbd read throughout the runs was 4.0
in. H2 0
.
The
bdd
estimated error in Pbd is therefore
There was a small error possible in Pamb
because the ambient pressure reading was taken
on a mercury barometer in an adjoining room.
Ihis error affects the value of k.F.1 ,but is
small when considered on a percentage basis.
±hereforeit was neglected.
There
was a small error possible in the temp-
erature measurement, due to conduction and convection effects. xhis,also,was neglected because
of its small size when considered on a percentage
basis (an error of 5 degrees i is 1
absolute scale of temperature
,F
error in the
abs. ).
1. see section 2 for a description
2.
nef. 3 , p.13,*16
4+3.
FIG. 3MANOMETER
CONFl GURATIONS
VELOCITY
PROBE
(VIND.)
PITOT
TUBE
AMB.
KEROSENE
RESERVOIR
STATIC
V
PROBE
STATIC
PRESSURE
PROBE
.AMB.
WAT
+4-.
I.
I
45.
4-6.
FIG. 3-3
PITOT
CON FI GURATIO N
TO YAW
AND
TUBE
ERROR
DUE
% ERROR IN
- +1t0
...
*0.
-
I.-
-200
20
- -10
YAW
PITOTS FOUND
IN REF. 8
X ERROR IN
V
A.,
-20*
V
41ZK2zzzzz
PITOT
USED
AUTHOR
BY
a -a
Ita
fta
20'
YAW
NOTE'. FIGURES
ARE NOT TO
SCALE RELATIVE
TO EACH OTHER
4-7.
SECTION
Results
IV
48,
Section
IV
iigures 4-1 to 4-34 give the results of the
velocity measurements taken during this investigation. The results are given in the form of
"maps" of lines of constant velocity at the vara
ious stations investigated. Figures 4-1, 4-11, and
4-23 give the location of these stations relative
to each other. rigures 4-2, 4-12, and 4-24 give
typical upstream velocity profiles. ivor purposes
of comparison some of these maps are presented
together on one pageon fig. 3-1,
S-2, and B-3.
because the static pressure did not vary
appreciablythese lines shown on the maps can be
considered lines of constant total pressure.
4e note that frictional effects are appreciable, The minimum velocity changes from 40
ft./sec. to 50 ft./sec
. Also, the maximum
velocity changes from 87 to 65-70 ft./seo.
Thus,a
particle with an initial velocity of
70 ft./sec. may end up with a velocity of only
50 ft./seo.
herefore , in order to follow
streamlines in the flow we must follow the
high-velocity and low-velocity contours.
1
1. a more accurate method involves the
use of tuftssmoke,etc...
4-9.
if we do follow the high-velocity and lowvelocity particles,as suggested, we see a definite tangential (rotational) motion in the flows
in all the bends investigated. This denotes the
presence of a secondary circulation, We note
also that the flow in the seven inch bend is
distorted by separation and back-flow phenomena.
it should be noted that the initial boundary
layer
,
originally at the edge of the flow, ends
up finally in the center of the flow! If this
were to happen in an inlet duct
,
an undesireable
flow would be present, in addition to pressure
losses due to vorticity.
eigures 4-32 ,4-34 , and 4-33 show the variation of the static pressure function
where:
&p
Pstat*
P
ap/q
amb
In these plots the value of ap used is the average Ap
over the cross-sectional area.Actual
measured values of
&p can be found in appendix
A ,and vary widely ,especially in the bends. A
50.
sample variation is as follows:
151-90 deg. bend - sta.6
x (in.)
ap (in.H2Q
i.
-.04
1
-.02
lt.
0
2
21
3
31
4
4j
0
4.01
I.02
4 .03
+.03
+.03
thus +.007 average
ihe value of q taken is the mean value of the
dynamic pressure at station 2 (i.e. upstream
of the bend). In the fifteen inch bends the
velocity varied from 50 to 85 ft./sec. across
the gradient. i:his gives a mean V9 of 5000
and a qmean of 1.15 in.
2O
* -in the seven
inch bend the velocity varied from 45 to 65
ft./sec.,giving a
qmean of 0.75 in. ki
20 .the
author realizes that the mean value of q changes
throughout the duct system because of frictional
effects.
doweverthe error inbp (see section 3)
is sufficiently large so that this change in
qman can be neglected.
Data on &p/q is too unreliable to make any
valid conclusions.We appear to get oscillations
,
5.
perhaps due to the oscillations in the vorticity.
indeed, the period seems to be roughly equal to
that of the vorticity (see figs. 4-35,4-36,4-37).
This data ,inaccurate as it is
,
is included in
this thesis simply because no other similar data
is available.
The calculations of vorticity given in Appene
dix u are plotted in figures 4-,5, 4-36, and 4-37.
The vorticity plotted was obtained by an approx-
imation, based on the following assumptions:
1.) Viscous effects are neglected and
we assume that the velocity of the
particles does not change much from
station to station.Thus, the rate
of movement of lines of constant
velocity approximates the rate of
movement of the streamlines.
2.) We assume the circulation is centered at(x =2.5 ' y = 2.5) and at
(z a 2.5 ; y
=7.5
3
3.) We neglect the effect of the cor-
ners and assume the flow to be concentrated in two adjacent tubes,
five inches in diameter.
4.) We assume
area.
is constant over the
5.) We neglect errors in the axial
velocity (ItL) due to yaw.
in actual calculations the author used the formula derived in section 5
52.
(
r
)
(see also fig. 4-34a)
where:
is the vorticity,
L is the velocity of the particles,
Q is the radial distance of the partical from the center of the circulation,
As is the motion in the tangential
direction in the time At,
al is the motion in the axial direction in the time &t.
The sign of the vorticity was arbitrarily chosen
by the author to be positive when:
a.) it causes flow to circulate in a
counter-clockwise direction in the
upper half ,when looking upstream,
or
b.) it causes the flow to circulate in
a clockwise direction in the lower
half,when looking upstream.
The large spread in the values of
shows that
our approximation is indeed only an approximation.
viscous effects partially account for this large
spread. The curves plotted were faired through the
center of the spread of the calculated values.The
peaks were placed at the value of E where the normal to the lines of constant velocity (or constant
53.
total pressure) was in approximately the same direction as the acceleration.This configuration is
denoted by vertical lines of constant velocity on
the velocity distribution maps.
it is interesting to note that there seems to
be no marked difference between values of
cal-
culated at Q2.5 inches and those calculated at
= 1.75 inches. £his means that
9
may indeed be
approximately constant over the cross-sectional
area.
iie see a definite change in the vorticity as
the flow goes through the bend. in the 180 degree
bend the vorticity builds up,
decays ,and(going
through zero)builds up again in the opposite direction. All this happens inside the bend. Txhis is
of interest because it means that,with this type
of velocity gradient, similar bends can be designed so that no secondary circulation would be present at the exit. ience, no energy would be stored
in the form of rotation and no losses would occur
due to energy unavailability.
.ownstream
of the bends the vorticity falls
off due to viscous effects.
ihe results of the tests run with the seven
54-
inch bend are probably in error due to the following reasons:
1.) Separation and backflow distorts the
flow so that the assumption that the
flow is confined in a tube is not
valid,
2.) Errors in velocity measurements,due
mainly to yawing
,
are excessive.
55.
FIG. 4-1
STATION
15 "x
90'
LOCATIONS
BEND
30
R=15"
NOTE: STATION 11
IS 12" DOWNSTREAM OF
STATION 10.
PLANE OF BEND
12"
12"
12
SECTION
AA
Ii:
56.
FIG.
AXIAL
15
4-2
VELOCITY
90' BEND
STATION
X= I"
YIN.
PRO FILES
2
10-
IN.
10-
8-
8-
X =2"'
642 -
0
20
40 60
I
80
0
I
2p040
6
60 80
VFP.s
(DATA FROM
RUNS
VF,.RS.
200-209)
X=3"
Y IN.
mmmmp
80
X-4"
IN.
10-
108 -
8--
6-
6-
4,-
4-
0
20 40
60
80
VFPS.
0
20 40
60
80
V F.P,S,
57.
FIG.4-3
VELOCITY DISTRIBUTION
BEND
15"x90'
STATION
2 -LOOKING
UPSTREAM
450
50
60
65
0
75
80
85
8580
775
65
60
55
'4 5
VELOCITY IN F. P. S.
5"x O" CROSS-SECTION
DATA FROM RUNS 200 -209
MAP
C-
FIG.4-* VELOCITY
15 "x90'
STATION
DISTRIBUTION
BEND
4 - LOOKING
UPSTREAM
VELOCITY IN
FP. S.
5"'i 10 " CROSS- SECT IO N
DATA
FROM
RUNS
210 -217
MAP
59,
FIG. 4-5
VELOCITY
15"x90
STATION
DISTRIBUTION
BEND
6 - LOOKING
VELOCITY
IN
UPSTREAM
F. P.S.
5"x10'" CROSS-SECTION
218-226
FROM RUNS
DATA
MAP
60.
DISTRIBUTION
15"x90'
BEND
FIG. 4-6 VELOCITY
STATION
7 -
LOOKING
UPSTREAM
VELOCITY IN
F.P.S.
5"x 10" CROSS -SECTION
227-245
DATA FROM RUNS
MAP
61.
FIG. 4-7
VELOCITY DISTRIBUTION MAP
15 "x 90'
STATION
8 -
VELOCITY
5"
x
DATA
BEND
LOOKING
IN
UPSTREAM
F. P. S .
10'" CROSS - SECTION
FROM
RUNS
246 -265
62.
FIG.4 -8
VELOCITY DISTRIBUTION
15" x90' BEND
STATION
9- LOOKING
UPSTREAM
VELOCITY
IN F. P, S.
5"x 10" CROSS - SECTION
DATA
FROM
RUNS
266 - 285
MAP
'3.
FIG. 4 -9
VELOCITY DISTRIBUTION
15"x 90'
BEND
STATION
10 - LOOKING
UPSTREAM
VELOCITY IN F. P. S.
5"x 10" CROSS -SECTION
DATA
FROM
RUNS
286 -296
MAP
64.
FIG. 4-io VELOCITY DISTRIBUTION MAP
BEND
15 //-9o
STATION It
-
LOOKING
UPSTREAM
55
55
4.
VELOCITY
5"x10"
IN
F.P.S.
CROSS -SECTION
DATA
FRO M RUNS
307-315
5.
FIG. 4 -u
STATION
15 " x I80'
LOCATIONS
BEND
R=is"
PLANE OF BEND
K 5q
SECTION
AA
66.
FIG.
AXIAL VELOCITY
15" x 180' BEND
4-12
STATION
PROFILES
2.
X =2"
YIN.
YIN
.10-----
642-
0
20 40
60
80
V
DATA
0
60
RUNS
80
VF.P.S.
F.P S.
FROM
461-468
x =3"
YIN.
20 40
X 4"
10-
86-
42-
I
2
I
0 20 4060
I
80
v--
V EPS.
0
20 40
60 80
VF.P.S.
67.
FIG.
4-13
VELOCITY DISTRIBUTION MAP
15 " x 180 '
BEN D
STATION 2-LOOKING
UPSTREAM
40
50
~-
-53"
50
60
70
80
(=86
82
80
70
60
50
40
F. P. S.
IN
VELOCITY
CROSS-SECTION
5"x 10"
DATA FROM RUNS 461-468
68.
FIG. 4 -14
VELOCITY
DISTRIBUTION
15 " x 180'
BEND *
STATION
DATA
4 -LOO KING
UPSTREAM
VELOCITY
IN F. P. S.
5"x 10" CROSS-SECTION
TAKEN WITH
15"x90* BEND
(R U NS 210 - 217)
MAP
69.
FIG. 4 -15 VELOCITY
DISTRIBUTION
15"x 180'
BEND *
STATION
DATA
6-
LOOKING
UPSTREAM
IN F.P.S.
VELOCITY
5" x 10" CROSS-SECTIO N
15ix90* BEND
WITH
TAKEN
(RUNS 218-226)
MAP
70
)ISTRI BUT ION MAP
15"x180'
BEND
FIG. 4-16 VELOCITY
STATION
7-
LOOKING
UPSTREAM
VELOCITY
IN
F. P. S.
5"x 10" CROSS -SECTION
DATA
FROM
RUNS 469-481
7/.
FIG.4
-17
VELOCITY
15"
x
STATION
DISTRI BUTION
Ix8O'
BEND
8-LOOKING
UPSTREAM
IN
F.P.S.
VELOCITY
CROSS-SECTION
5" x 10'I
488-497
DATA FROM RUNS
MAP
7'~.
DISTRIBUTION
15" x 180'
BEN D
FIG.4 -18 VELOCITY
STATION
9-LOOKING
VELOCITY
511
DATA
X 10"
IN
CROSS
FROM
UPSTREAM
F. P. S.
-
SECTION
RUNS
498-517
MAP
73.
FIG. 4-19 VELOCITY
DISTRIBUTION
15" x 180
BEND
STATION
10-LOOKING
UPSTREAM
F. P.S.
VELOCITY IN
CROSS-SECTION
5" x 10"
RUNS 518-536
FROM
DATA
MAP
74.
FIG.
4-20
DISTRIBUT ION MAP
15"x [80'
BEND
VELOCITY
STAT10N
12 -LOO KING
UPSTREAM
IN F. P. S.
VELOCITY
5"x 10" CROSS-SECTION
DATA
FROM
RUNS 537-555
C
-
-
75.
FIG.4 -21
VELOCITY
15"
STATION
x
1800
DATA
BEND
13 -LOOKING
VELOCITY
5"x
DISTRIBUTION MAP
o'
IN
UPSTREAM
F.
P.S.
CROSS-SECTION
FROM
RUNS
566-574
76.
FIG.
4-22
VELOCITY DISTRIBUTION
15"x 180'
STATION
BEND
14-LOOKING
UPSTREAM
VELOCITY IN F. P.S.
5"x 10 " CROSS-SECTION
DATA
FROM
RUNS
580-597
MAP
7T.
FIG.
4-23
STAT ION
7"x 90
PLA
LOCATIONS
BEND
78.
FIG.
4-24.
VELOCITY PROF I LES
7"/ x 90
BEND
AXIAL
STATION
YIN.
3
YIN.
x= I/
10-
108-
8 -
642-
2 m
I
I
i
20
40 60
80
0
20 40
60 80
V F.PS.
DATA FROM
VF.P.S.
RUNS
343-351
YIN.
YIN.
10-
10-
8-
8-
6-
6-
4-
4-
21
1
I
20 40 60
I
1
I
80
VF.PS.,
0
-
I
20 40
I
I
60
80
VF.P.S.
79.
FIG. 4
-25
VELOCITY
7"x
STATION
90*
DISTRIBUTION
BEND
3 - LOOKING
UPSTREAM
IN F. P. S.
VELOCITY
5"x 10" CROSS- SECTION
343 -351
RUNS
FROM
DATA
MAP
80.
FIG. 4
-26
VELOCITY
7"x
STATION
DISTRIBUTION
90'
BEND
5 - LOO KING
UPSTREAM
VELOCITY
IN
F. P. S.
5 "x 10"' CROSS-SECTION
352-360
DATA
FROM RUNS
MAP
FIG. 4
-27
VELOCITY DISTRIBUTION
7"x 90'
BEND
STATION
7-
LOOKING
UPSTREAM
VELOCITY IN F. P.S.
CROSS -SECTION
5"x 10"
361 -379
FROM RUNS
DATA
MAP
82.
FIG. 4
-28
DISTRIBUTION
VELOCITY
7"x 90 *
STATION
7A-
BEND
LOO KING
UPSTREAM
VELOCITY IN F. P. S.
5''x 10" CROSS-SECTION
380-391
RUNS
FROM
DATA
MAP
83.
FIG. 4
-29
VELOCITY
7"
STATION
90
x
8 - LO DKING
VELOCITY
5"
DATA
x
DISTRIBUTION
'
BEND
10"
IN
UPSTREAM
F.P.S.
C ROSS - SECTION
FROM
RUNS
394 -405
MAP
8+.
FIG. 4
-30
VELOCITY
DISTRIBUTION
7"x 90'
STATION
9 -
BEND
LOOKING
UPSTREAM
IN
F. P.S.
VELOCITY
5"1 x 101" CROSS -SECTION
DATA
FROM
RUNS
406 -
418
MAP
85.
FIG.4-31 VELOCITY DISTRIBUTION
BEND
7"x 90'
STATION
10 -
LOOKING
UPSTREAM
50
45
UNEXPLORED
VELOCITY
5"
x
DATA
10"
MAP
IN
FPS.
CROSS-SECTION
FROM
RUNS
419 -430
86.
717fl7F VV
14
Ll
I
-,
-~
"
iI r
-
lit
I
-41
I I
I
-1
--
~-
.
I
.4.
L~
1
At
}
I t
I
I-
___If ___
1K
jil
I 11 '1 I I ~
-A
TI
t
ifII
-~-4~-
-- - -- --
I
-
1
1
1
1
-~-
-
~
- If
-
t
--
--
1T
4
T
-
4
01
-
--
-
-
4-4.-
4-
4;
-
.--
3---~;--r-j
Kt
---
-
-
-
.,..-
4--.. .....
4
-
i
- n
[--
--
I
-
---
i4_T7K
~I+II
I ~ I 1 ~o~Ido~ IAt4tI
J..~ Li_ ~__T;711
~1
i
-
I
LL4
1i i
! i
-
11
-
- - I
I
:
-I
-
I
|
___
flT
-
ii
4
I
-4,
1
---
.4I
I 41
ii1~
~ I 1KJ Jv
TT
1
-
-
-I
*
-
-
-
4
s
-1
I
.
-
__
47-7
iT
~
j
-
4
-
.
[
.
.
i
.
I
.
.
0i i
1
|
t
"~
Ji j
I
~2f
87.
t7+-1
~
1
-+
T-
4-T
-T
Z1
1J
4
T
4-4-
- -
-
.
-
-
--
-~
i
4-
--
-
-
ftti
t
t
-
_-
4
T
-
i
--
I
1
f
4
_
_
IT~~
'
---
~ ~.1
1. ' I
-t
jI
t
4
--
-
-
-
-
4-
4-P
_
-
Y
'7
-
TTT
4-41
'
I
u
t
-
-------
-
--8+
+-4~~~,-
4j
-
+-
-
4
TIT
- -I4
'-
4
4- _
1
1
-4j
d
41
4-4---~
4
4
~ j-4
T:1
4-I
-j
-j.
L
.-
1
_
-
1
88.
i
I
K
i
!
-
F
.I
L
!
_
]
44
JU
1
-
~1
-
-
~
- -
--.
-
~
--
'V.
111
I
I
I
4
'
--
'_
Ii
4
-
t KK
-
1'
I-
-
~
-
~
---
4
III
II~ItItl
I
IiItlf'II!tI
'
-
-4+
I I I I I-~ r1j~
Ii
if-i
II
I
I
I
I
I
-~
-I-I
II
I
I .
IIi
-
4
-
-
1
-
-
-
T
t
-- T -
4.
--
iL
41 ~
--
4
{4
7 ~J7i7J 7717177177
-
{f
-
4
i
af
+-T'j-4
--
-
j,4~f44K
iLL
-7-J-
11
<r
i 1-4
I II .
-
4-
1Lj-
L
j ~
14
I
f1
89.
FIG. 4-34A METH OD OF
CALCULATION OF VORTICITY
ALSO
(SEE
m
-
C)
APPENDIX
AAt
I
10.
I
STA.
STREAMLINE
OF CONSTANT
VELOCITY (Vz'l.,)
ENLARGEMENT, LOOKING
UPSTREAM
90.
'f6
r
92.
III
"
Ii
I
I
- '
---
_.
I-,
-
-
L -,
I''
,'_
f- --
4- -
-
,
t
-! -1 -
1 -,
f
-
. 4I'll
-
-
--
i-i i
I
I-
- 4-t
I
I-
;
I,
11
I~
i
1 -. 1- -
-i
-Ir-1-7
II -II li
I
, I, '_
J1
;1
-I
-i-
, 7-,
-
I
I
.
i! i. ,Ii -II
.I
-- I-'
., I
'LI I I
I I II
_*: iI
;
I
II I ] r
I
:I
I
,
; -1 !
. ; -,
.
- 7
ii
4,
i )
,
I
.
Ij
I,1 t'
t'jt ;; :'
.
I
I
I II I
I
-,
I
: ! j I
I
'
;
I
I .- :
,
II
I
i
_'
.
_T -
.I
;
11
Ill.. _
,
: f
.
I
I . ..
-4
I ,
I1-_
. T
I
!
,
; I , I
I
,
4 ), 1
I rr
I
I
I A ,
1,
... I
I-
,
,
I
11; .
"
1,;
t
- r
;
, ,1--'--'
- -1 I
,
_
T
;
.
._i
-
I
,
:
I
,
--
1f
r
I.'
I, -
- -.
!
, .
.
; i- L_
,: r
i.
-1i r.i
I
-
i
Ii
'I 4. ;I
144
I
!
- I.
II
!i I i
__1 t; .
_ I..;
:
i
. 1II
-i
1,
I
I ! 1 1t
-
' -
:- I
. i
I
I
, 4 I
;. 1, I!:
:
i
-
I
I , I I
I
.
i
,
! -I
I
I
-
4
;
.
I
I
1. I
,
I
7I .
: 1
1
1
I
I
11
,
:
,, I
I
!
I 1 1
i
1 I! !
I
I il
i ,
fI
I
I
t_ _ , ,
I
,
-
.
I1 1
4
1
!
I
;
,i
_,
:
, ,
4._i
-,
, 'i
A-1
;-
'.
:,
t
i; 1, " I
I
-1
I
, ".
i .
.
I
: I
: ,
1
,
I 1 - . .i;I_I
,I -'I
1,1,
, r
: "
.
-4
.1 I I.I .
f I I
II
__
t
II
.
I
I
- .
.
.
I "
1 IL I.,
,,
-
-i I I I
, I
I
.
I
II
i
1, !-
. _.
Ii
- "I
II
F
.
.
.I
__
. .I . . . ; , I
".II,I 1I; :I I
1 ,
j ;
T'
i- I
I
I
I
I
I I
I I
t
I
-
I
;
1 ! ;_ _ '
-1
___ I
-
; '
:-
,I
'- .
I' *
I
I
I
.
i
:
! I
II
II I, :, f
I
I !
I
I
.
i
,
.__
I
, . : ; :1I
II I:_I
I iI : .
_ 4 , _ j - LI
_;
! I.
I I 1 4,_ - , , 1 I, I,
I I , II I ; ,
I
I I I
I
I
,,,
II
i
; ; ,
-
, ,
iI
I
I
i1 I
I,
I
I
I
!
1I
I- 4
I -
-; r
I 7,
t I- -1I-- I
I
.-
I
- --
I
t r
II_
,1-I4
-
1-
i
4 1-
1
-
i
f
-4 ,
- ,.11
-i
11+'- 11 ft
i-
i_
_! -1
+
: I I
.11.
+
A _i
i
-
.
- -
-
4H __
-1
I
-
-
-,-'- i 4-__--i - I- t-i _ 11
_11
- 4-r - - __ -- r rr I - -I- -1-1_
-_
II
I_ - I. , -t:!:r
- d - - 4 T, ' I--f - i - _ _ __
i-1
1--;.
r14 1-I
IFIHI-l
_:- r- - -1 - 1,
I fT
i I
11
I
I i -l
I.
1
I
i! I-
I
-
_
:-I
IJ_
"I I.jI i III,I
t
--
.- ,
-
.
... t
'
10
- .4
I - J- - -1 _. I I'it,
1.I -1 .
__.
I
1. ___
' U '_ __ I'I-i,1, ' _t--,-
; --1-.I
I . .' I , _i
'_
T_L71
17, 5 I: iI -KI -tt.j _4-t1L .rjy
__
-, , + .- L4-f j: 41- L _.
t
,I
1t I ';
'j.
,I 1Ii , ''.I.I
,
::
,
[
I
:
I
. I -
I
I I
,I
I
I I.,
I
-
-
4
,-,
I
11
I .
; 1
ri
,
I
,I I
1:
!- 4
I
- ,I -
1,
- _+
- i _1
. j I I I-
I
- I
I-
-
.
'_L
" I,
, I I
_p
.I I I 1 I f
;
-_
II .IL!i
'1:
I I i.I .1, j ," f_i-1 _l , i: I
' : T, I i i I I I ;_A
.I ,: ;, . . , .'
I- -_
4 I
:
I IT _ ;
I
II 1 I , . I
I:i I I I
.
.. i
I II I
t
11
I
4
4
t
j
I
4
T
I
FiAt
4
;
- t-,
!-4
II- !
1J . i I,
-IV
I : t r t _-, i- q -11-4--f . 14 1 -I
-Ii _l
47
_.
I '-Il-
, .
I. .
1 ,
,
I
I
'
4 I
, ;[-1 14. I-L
I III'II--t
t I1I111
Ii-I
-
Ii 1:11
-_. I.
14- -44
I
-; '7
.
11
I
p ,1
t
.1 . I ,
-
" ; . ! I I !
-
:
I2 .
;I,"
II I
.
I ,
I
: I I- II
I
I ..
I ,
_0
,
1,
'I -
I_
, II f
;
I
!,
1-1 I
t
I
"
I
I
I
r
.
-i I_".L
1I . - i
I, , .1 -
4
I -- .
4t - 1
4I I 1I .1
: 1 t,
-1
,
4 ''_ 3 _.
,
II
7
I.
I
I
1 ,
I,
'
I
I
I
,
I ,
I
%J
-
.
:I ; :
I.II
.
I
I
' l
I
I I
I
; ,
I I,
I
,,. I
I I, I,t
I
,
.
:
II
I
I
* I
__l.__
-
71-1-
_ __-1
II
I
I
I
'
i -
I
.
I
I
.
.
i! I
.
I
I
;I
I ij ;, ,.
;' - , -
. !
.
I
, I ,I-
mi
.I
I ';msi - - I -;
I i 4a
I
I II .
. -
I_.,
- v
'L t ;i I : I
,j , 11 .
1 I, ,I jI Ii ;
I
I
III
I ' II
I
i
I i A
I
I
I
_F'-7
I
!
I .
_
!
I. I
i
i-Ii
I 4I - ;
- ,
,
-4
-11I
-
,
Ij-
"II
_
:
I
-11
-1,i
-
i I i fI-f K
_
. -,
4 . !- II - .I
i _.I
tl
I -1 f
!
-
-t-ri-
t
,
- i .
, ,
...
f, _ _
-
i- -,- ,
__
_ II I
I
I
---
.'+ _4 1 1.
. I
i" - ___
..
I
_H
-fI1I -_f#17
I-,--- _ I ,-_'_
,,-t-; --1..- 11II -.-I
I
I
I
I
I
I
. I II -
I
" ,
4
: I
.1.
- I
:
" ,
__ _
,
zi ,.
-
I
,
-1 " ; I
.
i- f
.I
, ,
-I
11 i
i
7 I
!_I
7i
,4 1
_
7
. .
! . i,
- r-
I
1 ,
I
I
:10
.
__ _!II -
I I-
I I I1.
.
. I,
I
I
, I,
. I - I - I -1
I
-. _
II
,
I
II
.
.I
i p,
-
-
II .
I
I
I
j
7I '
-I
i
I.
.
I LI I
I
,
, II
,
.
I
I
-
, ,
I, ,
-: I :
II
jIT 4 II1 I - _l , ! T I;
: m
i
i
I
,
..I I
I.
_1
- _;
1 I I.
, I
'i - -
_
: -,
' iI
--- -'I- JJ
7
-
:
II
1-t 14
I
I "I I: f 1 1 I;;
I
I
, ,I
.
__
.
,
. -,
's
,
I
: ,
i
; .
i "
-
!
!
-, v t-
Ii I- I-
-
II
.
'
1 - -1 tlI - I-., -,
; -I'I ,I , I * -i 1 , II f --i -11_
.f
- i , ; 1 4 t -1,1
j ,
I ,, I
I
I- t
__:
'. ; _ j
,
,
, ,
I
,
TI.
- .-
,
!
-I1
It
-
-11
I I
!!I
I
'
. -"
- .I "
.
- I 1 - I i- I _ _
I
J-t': I-
, _L_!_
- - ..
- - 1 I- -I- .
I I
'
- ! J.-;- # --'j
1.
4_
* _ ---!:''--, i T
-- _.J_
-+ , -1-? - -: -1111
F T It !
-t 1 -++
I'
, -
Ti7 .1I
-f ..r' '_'
4
I '__L._
L -
_ _L4 - -
I--1 __
I kL' - -il_J I _ _f
-t
.
I ii
I 4I
-
I
-r -
-f
- . ---
f-
+
__ -
i!'. !_
I, "1 Ljj L
._t'I 11.i - f
[
l
t
--
I
-
i
-
_r
I
[ 1
11
- _
-_-
__
I .
'
: I
-j
-
A
I
-, ;-- - I-
f,
, . t -t-t . '- I
f I i ;;;I
11I
4 1 _ -" - - _ _4 ! - , 4
.. -11
_.It' . f: -, , 4,
.
- 44- - . .l
I T_.
- _. -II- I - I - i-T
-_
-1
t;- -_F
__
!-1 --, .
I
, ! 1 -I 7 ' j, IT L L
I I 11i I I , I-, ,I - I -T 4 -,'-T-f-...- fl'
..T
--iII
id
I
l. -
i I r.
-_ -
I I I
I
I
i tI I
, t 1,
I . , ,
!
.
.
-1 1.
I'
: -4- 1I
. __
1
I
1
4 '
: ':4d-, _' , I I I Ii f 1, . . I
j I+ '.- ,-I
I
i I
- I-I
1_ ,.I- I . . - -i 1--i,- _1__I
4 T,I
_ : _: I
- ,- -4H j - .'_--i -_ - ; ,. - I... -4_t - 4 :"
-1 _'4 - !_ -I_
' _i
-!-_, i
I 1- 1. Ll- , ,
,T
-! 1-1
I .
I
I , , , I :,
I ;
; I
1 _-L'i-
,_ 1
:: I I
I
.
t
-
ttt
i
I
,
-I1.
- ,
-
-
_!I
! I f- -r-T
i
1
1
'
_ -I'
', -I-r, -t7
__ l-1 - I
IT
-i I
I
I
Et
I
I
I
.
-
,
- *
I
--
I J-II
_1*1
I. _.. I 11
I
I.
I
.
, i ! I ;:. :
I
,
T -
I
I 1
-1 .
I I
;
'
.
.
. -
- , 7 7 7 t-'
i1
I .II . , ,
1
_
.
f T
i
--
. I
VA
11__
I i i I
_t 1.
__I .
.
!- -I- II:
.
I
I I:
-
I
I II I .
7I -
.
I
I
:
, I
-. II -i+
1:
- - i - -T
-, I
1 _-
'
I !
-j
t . !
_ _
-,
.
-
rrrrr
I
1
1 ,I
" I ;;
,
I I
'
.
I I
I
-1
,
'' ' ,
,
I
,I
_
-"
I-F
I .. I.
I
-1
,
I i j4-1
1 ;- ;1 ,
I
I
,,
i
-
t
-
; " 4 ii 1_
_, _-L
: : -,I ,
I I "II I..I-- I.- I, . 4- - .I
E
4
i : , : 1:
. !_ _. I , 1:
I :
+ ,
, - - It 1-t. , t --t- ; iI
t
! . T
._ -. -L
_
I
__4
,j
__ .
, '
I.,
.I1.'. .I ; .
.
;I "
,
;
,
,
1
I
1
,!
1I i, . ', -,I
I tI- -: .;_
.'f-. i 1_
*,,
: -H - -1- - _L - -1
II
I- - I 1I ;_I_
, , I..
I
.
;
_
f
,
f
.
t- -,t.I.I I 1 , - - _ _1t-T"
1. , t,_I_11 't; H
.L , , i'
I "
'
1 , ,-I- 1 1I -1-1 -'
II : 1 I I
LJ
- t I
I !I
I
II ; - I
.
I t : ._
i I.1, . I I. I _l_,;;' _, ,,_ '-_ -_. "i ,dLI I11__T
- I 1,
_l -. __.1, - ( ."I ' '
-I I - , , I,I : ; I, .!
I-I
1
I'
I
,Ii
i -1, , i - ,
T; ! !
IOI
)1 "I'll
- - -.
. - '
12
.
.
I
I
I I
I-
-
i - __
I
I Ii
. __
I : .I . t I
^
I I i Nd V
"I . .
.
-
I
-
II I : I ;
. I,
i ;
I
:
I,
:
I
.1 .
.
II- II
:
I , ,
I
.._,f-
i
-1I
I
11
. I
I
I :. .
.,
I
I
-I
I
1
i I
..
! ,
I
I
I
-4- -1
[
4
I
i
,
,
,: I , .,I;
I
, :I )
I
-- t- 7---
- -
I
"I
I II I
..- I-_ _ _ -_
I
I
-
1
I
.4- .t- ,j 1. I -iI - iI I , I ;, [ , ,r , i" :i
I I,
1-1 . I 11 - 1 _41
II
I ,
- 4, i1
-i
I , I,
. _ A I- I, !
: I
t 1
i
[
I
I
I,
_j
. I
1III ,: t.- 7 ' T_'
!.. I-7I , t-__
4,
"
'
1
L
, ,
,
.
i
I I ;I .I
. . :
-
__
% II
L1 ^
11
t-H
.
f
11It, If I
! II I
I
. 7
i ; I
II
'
.' 11
' I t ' ' i , I- i
' ; ; I
II
I
-
- 11
T
4 L A ,L -I
;
- -
:
:N A
-1
-
i
I
-
II: Ii,II1r , I-f--J__"
I
-f- I 1'1_
' I I -, - I+'LI_ _f
i_ T ,I,
,
t
.
-
I
I I I- I
; I
1 !
I I
I! 1 1
,
,
I I
f
--
01%
rkI
-
L
__4-
I
i
'I I f ;
.
9
i I
, .
I-
.
1
I
1,
:I
I :!
.
1
I
II I
.
I; I I
II
. . I
I
i-
I
C'
---
. I
I _
-1 -1 --.- '1- _t-__ 1
I
I - ; ; -I i - .- ,
,_
I.
- I
I
i, j 11!-
, I
I I -77'"
I -,I I'I II, I , , ,
I1 I' ., I
,
I . L
, , ,
I
I ; I
i
- 7- _-7-7 77
11
I
- -_; T - t
L
f-
,
i
I
t +
41I i
I III .. P
, ,
1 11
'
- ;_.'
*1i
I I
+
I
TI
.
I
_1 II 1
,
. 1I ;!;
-I
I-,
1-i
' I
+ L'
I _
..4-i41 1,
__ - - _4
1-1..............
I
II
'.
-1
i
I
II
, 1 ; ii
_';
; 1,I II
11
r , +i
i, I,.1 I -1 ,I i
;
;
I
II
,.
_L_ 1 I
I !,
" I
A
I
1 t. 1
"I
-i
I
:
r ; . f r
1
t
-
:-1
"I ''
I I-1 It
I
7 _
i i ; Ii
I I
! 1
;
i i
-"! I f jI r, I '
- ; I It
1 4 I Ii
I
I
I.
I I ,'T
f
.
s' I
,
f-
-, _ ; ,_;
i,I i
.. I ,
II
i![t I lj;
I
-
, i , :
+
.
;
!
7_!
- , -, :11, -I-I-,-,- -_ _ 1. -
I
I ;
f I
f.
- t
I :
- i-
.
l
I
,
,
T -1 - ,
,-1
- __I
+ -f
-'t i 'r LT
f - k
i
I
;
v. I t
i , 4%
" 1111r
I 11
1-f '
IL ,
I I.. '
I
T:
1- ' '
,:
. "T
:, I I
. II
I
I
'
1,
f 1! -i
-I'414 - , I i 1 ; I
j ".'
II ,
!
I
I -
I __
_
_;.I . _I .
;
_iI."
_ - - 4., -i ..
- _'
I:
I
i
! I
!
I
,
7 1 iT
-
I'll-,I'll
.-,II ....II7_f-._I_1-I
. I -.-I .' -1
; . J - ' ,
.
.I
, 3.
I
!*
T
i
II
I
-
T
[ + 'i I
1-11
- ---
--- -+*-4 1-1r V --,± - ,- ' t
t- - 4_
,I !
It I-I"I-T_
-, I_ I'll
_J
4__'_
_ i - , I _'+ tT
I ''- _I_;_
, -1 1 --,_ , __t, 1I,f , ., *I I _ _ ,-!-.
-1 -, F
- --1 -t l' , I I
_ i._
.
. -
!
I
"-I
I.
'11
IT
,
-
-
-Ir
I -,
I
r: ,
;
+--I
-
1 I : I
:
.
I
.
-
I
I:
1 , , I _. II
;
, t i
-
I
'I I,
. I,
ij
,
I
I 1
II , d
-,
.
I
I
I I
T
4+ II
iII
] , ;:iI! I j
4:
I *1
I,
,,
y
-
__ - ---
I
I
I
, ,
-
- -__Li_
- __
-
-11, I
I_ _ ' II1+ -f -* f- I
I~ E'__
_
- _ i
4 1H ,- ___;_4, t
i I
I +_t 1
r - .1
-tt-i
I H i ,_, Il'
f._[I f L,
. II1..I
_ 1_ i t! _T
. _: -'-f ....-1
I - -t-i-TP-.
" _ -! i,I - ;_.L-,f I, , _
i ,;_
,I I
I
i.,
- -, _ _ - _l
'L - _'
__
.1i I _. -I- 1.4-7
:1 1
I L , _- - - - - f -I -1, -.
! ,I
fitT
, : ! i t f . . i !, -- . I-_.
I
I- -r
Iq Jr -I-,
T
HI+ t- 11i 11
I I,
,
_;
.-
I, I
i- , -;-I
I
II i -1I
. ,
,
:: :
, 1
. ;
ft
_.
.
-
I
__ _!
, , i 7:
I;.Ii-'. , 7
,, "
I
: . i I , - 11II ;11
I
1
I
: I.II
,
II I
I
. :
I
,
i
"
I
I
I
I
i
IV
. . .. I . I I t , 1
;
I , I ;I
,
I
__
I
I
j [ 4 f
I
,-
-
.
.
I
7
, :
''
-
I__ , - I
I
'
-
1
_
I
,
i
7.
I,
-
--.--.-
I
-
_!._r - --iII L
*
4- t , 77, i
,,I
II .I ,. .I
I
j
I
I
-+.
4 1I , , .
t I'
i I
I
- I+ I
7
:
. ;1
I
!
4I
- ,
iI-! 1I I - I I
+-
' ' 'I
, I
I I r
1t f
#WJ" ^
-
.
I
ii
j
,L
,
iI
I
_-J
,I I r .,
I
1
.
I
;
I I; ;;I :,:.,..jI...,
'
I:
. I
I.
I
LI
.
r
I i ,
i
% I -!I
'
, 1,
_;
,; !.i
1 j
. I
I - ' III :.I III rI II-- -- , - I:i
i -H tIi - -__
4 1f..11I I,
I , I. I
IifTT.. + I.,f - _1_.1
.- ..."III I-.-i-1
, fIII ,_, , I, , t-;i ,IT,I 2,I I"I1,T..I ,I "I . i1, I Ii-.I-[,I-I-,
-1
I -1
,
i
I
j
'i
I
i
1-1
,
4
!
,
1
-1
-I
1A
.
,.iI
,
. _
1- -4-i
I
'i
,- I
t
I
11
- I fI 4 ,I
;
! ;
i _
; ,I
tI
I II I1: __
I I,, I I'll
:L-ji ,
-,I', i f. 11_1
I I I- I:'
'" , if -1
.
,,
i ,
I
I.
. ,! :
.
I
.
A
i -1 I, -
I
,
I
,
_'+
" '
11 1 ,
4 r -[ '11
m [-ii ,I
'M 4 +'I
.II, ,'A .! "I_l J. rt%t t 1
j
, 11 1 ,
ii
I tr I ,
II
I
- i
.1I I
I i, L 1
- 1 11..
,
_j
1
i- 1-1 , I I
I
.
I
- I L 1, 1 ! I , I
, I j7-+- ' 4
I[1- I f f; ,I 1
I
-1 :
i
' 1 44 i 4
I
I,
-,
- I !-
H
j . ",
, , ' ;I , t
;
-
i -
-f
-1
r
-1-.i I .
:
-
.
I
;
1-
I
.I
I
..I .I.
_7 - -
I, ,
I
-7
;
I.-,,
I
,
I III
I
,j I .I
I I
_
I ,
I
: '
i
, !
- 4 +t
--1 I, I
:,, I,
I
" ,
. I
j
,
1
__.
I
:, .
I
,
11I
r1l"
, ,
. I '! .'
I-, ,,
It I, ,
-
4
, I I.
-i
--,-
_'
t
T-r-
; t
"OI, I
!
I
, ,, ;-
I
II I
f
..
! I
;,
I - - : 1I I! :
I'
-1
4
I
1.I III,
4- 4I ! "
; , -
-
r
,
I I I f
4-4-
,
!
I, !
; 1 1
I. I ,
, I I,
I
; , ,
1 , ,
I
I
;
i i
1 1
11
11I
I
; t i II _1
. i
-
I 1--
I
.1 j i
.
Ip
I
I ,
1 1,"
' lT
j
P
I
I
- I,.. , I 1
-I
_1II - ; I
I .
-
_.
-I f 1 ,
,
, -
I-i -, '
i.
I
I
I,
I- 7
i
I
.
1
I
;
,
- l"I,4 _
'_
1i I I. .
_
. l-
.
1
1
...
ii . ; I , ,
,
I Itj -1: -I+ -I i I: , , ,., I, : IIi -t-Ii;
t.1
' . -. 1j
j-, - , _;_f i - :;.11
I- L
-.
.
I
I
.
i
.
I
I
IT
i-_
11_i_ __
1. -'.-I
; T
i
1,
t'.
,I
,I ..."I.P.'01.- ..
,; , I,
F r I k" .#%14
I,
I i-I
; Ii
!,. , -%,
I- Its-I-14-1
t- -I
r _-- f
tr F
I 4
1 111
1t ; I m
1
1 1:
-7-, 1 1
I i I - !
'
I
f
! , r ,.'
I i I !I I-! ; , I'll
I ,-(
;!
4 -.i1,
;
, , ! !r
.;
- -1 t
I .Ii
I
It II
I
:
1'.,: :
I ,
, ,
::
-
:
,
I
'
...
I
t i t __ t
!
7il!-I- I1 :
1
I
-
I
1-I 4 1
1:
I iII,. 11
.
It
I
"t !
;I !"
I 1L
t
i. t
1-1 I
i 1. I
I
.
1f, , _
, I
11
I1 I;
1j
'
I
! 1I I. I -
.
:
-
_
, I I_"
-- i . II II I -,
. , ''
- - I I, i I ' 1
-
li - I I
It
Ii 1I I
t i I
I '
j
I - i,i -,I: ! i I
II
I
! II'll
I
!-]I . -
_+
-
,
I I
.;_
I,
14
_
I
.1
I
I I ii ;I I I
-1
___
I - ";A
,
iL--_ I' --- ,- _.'-.",I- 1, .- -1I '- .".-
-4
T
_
i
i-T
I - '_ _'_
I
- -
,
I
- ..., I,-71
!_'__ _1- , 17
IIIi _rt_-,
!,-3, IT , '11I '( )--1
1I-1i_L__'
, :T'i- i,-L.[)
11t,4-, L
I-
I
i 7 Ii ,1 -1
I
_ff 1.r 4. e-1 II
I- I , I :,I ;,
1-- t -, , i.. FI 1
i I
++ __ - I _j
;
ri1
! i 7 , :; !1 :
1' II i
.-
-
-1
--i-
1-f-
_' 4 f- I
. ii
_- -
-
-
4- ,
r,
", I,t'I"I t-t-f
-_'
I-1 i t I '4
_"'.. _"
-, .
-1
__
t
11 -i-f--
-t
_ 1
I
4__
!
1
fH 1- -'- ] ; i ,-t- -T
L_
i j 1i- ' I II., - - I -1,Ii-1,
__
I-
kI
II'
-.1
-
, I T
;
T_
L
fl___
'
I
t -.- JH__
- Ii .I-,I1-1
f rI-1-1
t-" _1
-4 1 - -1
'
_. -
-
.
-
11-1
I -It1 - !', __.- -I-7 -t-
I
I
4.
-
1-, i- ..- -
- rr
, - '__._
I i 44-
i - I -- -, - F _T: -1 !- f_11.' _ -, __ - - .. - -1-, -. .. :7-' Tt
"
_TI
- -4 - T.' T I I 1-1 I I -P'- I
I' ' i I.I t - I i iii
I - I--ii- II_-I- HrI I1 - 4-t_i___.
ii
_;
-1
1 . .
IIIIIIIIII
93.
SECTION
V
Theoretical Consiaerations
94.
section V
The theory presented here is by no means complete.
it is carried out only as far as is necessary for application to the experimental results given in section
4 .
Squire and winter obtained, the expression 1 a:
grad L
-2
sin d6
teq.5-1)
where:
is the secondary vorticity,
V is the velocity
grad_,
is the total pressure gradient,
is the angle between the normal to
the stream surface and the acceleration,
G is the angle of turn.
£Wow,by assuming :
grad
is constantand
sink
is unity,
we obtain:
(S-S.)=- 2. ..
1
(2)
(eq. 5.2)
where:
2 is the initial velocity gradient.
1.jief 11
2..bid
95.
hquation 5-2 appears to predict that
will build
up indefinitely as G increases. hovveverit doesn't,
because this relation becomes invalid at the point in
=
tne flow where
.In actual flows
can and does
become zero. The experimental results plotted on figs.
4-b and. 4-;66 show that
3
increases and then starts
to decrease at a rate similar to its rate of increase.
it remains,therefore, to obtain some analytical reand (
lation between
,in order to determine the
value ofG when # does become zero.
nawthorne considers the flow of a fluid in a cylindrical tube 1 and depicts it rotating to an angle
om
under the influence of the circulation r .iee
figure 5-1 for a picture of the flow. lawthorne
assumes that the velocity gradient stays approximately constant and that it does not distort.He ob2. :
tains the expression
-
2
fcoscCO
(eg. 5-3
0
where:
Um is the maximum velocity in the
gradient,
%Ab is the minimum velocity in the
gradient,
1.Ref. 12
2.Ibid
96.
d
is the diameter of the tube,
oc
is the angle of rotation,
e is the angle of turn in the bend,
r
is the circulation,
is the vorticity.
If we assume
to be constant over the area
F'
then
(eq. 5-3a)
If we def ine v,the tangential velcoity of the flow
due to the secondary circulation,by the following:
V = d/2
then
dct
p = Tr d v
and
v=
Nqow we note that:
&,
./4
(eq. 5-4 )
1
= ( R + d/2 sin-
) d
hence:
d/2
v
dc..
-U(R + d/2 s inec.)d g
or or
v
lkvw
..
-
.1
/2
40MMW
---d.
(R +d/2 sinC) &
l.Ref. 12
(eq.
5-5)
97.
5-3, 5-4, and 5-5 ,we get:
Combining equations
dcc'
-U6
(R d/2 sinoc
cosc&e
(eq. 5-6 )
can be shown to be equal
Hawthorne says that
to the following
dc
lv-14
:
(R+- d2 sin*c)
dn(i i<
(eq. 5-7
or:
Oc
dec
(2R/d+s inoc)
It
in
d
(
sin ac)
looks like the above equation could be
d<c
solved graphically . kowever , d9
when oc.=O
goes to
00
and,hence, any graphical solution
would not be accurate . in addition to this fact,
the experimental evidence shows that the gradient
distortsso the exact theory
lf we go back to formula
(oes
not apply.
5-2 , we see that
we should like~nowto find the value of
where
9 or
reaches its maximum. A combination of
1. itef. 12
98.
equations 5-4 and 5-5 gives:
2 Vol
(eq. 5-8 )
R+d/2 sin .c)
dw.
d9
By combining equation 5-7 with 5-8 we get:
"in
.1
-4
d.
(i+-sin.c)
2R
(eq. 5-9 )
is a maximim when o(=
and we note that
define
and
0
the point where
-4
d
-
.
if we
0
as the values of
and
is a maximum (ct:
) ,then :
F!
- -
at
&
2R
V'in (
and combining this with equation 5-2
9
-U
-m
n
in
(
(
d
M
but:
ICU VWk
-U6
by definition,
so:
2
F in
+
99.
You will note that equation 5-9 does not predict a
linear rise in the value of
S. however,
for the
purpose of this thesis the author assumes the value
of
to be approximately correct and uses equation
5-2 as a first approximation.
Thus
from
0:0 to eO
*
:
- 2
and from 9z9to i:3Q :
These functions are plotted in figures 5-la
,
5-2
,
and 5-3 for different values of U1 and same values
of 'UL for the applicable values of
on the plots the sign of
d/2R . Note that
has been arbitrarily cho-
sen initially positive by the author for convenience,
Jote also that a change inlh. changes values of
and 94 slightly
,
but changes the value of I leav-
ing the bend considerably.
The above theory was derived for the case of a
flow in a circular pipe. This author tried to derive a comparable theory for the flow in a rectangular duct and found that the problem required the
100.
use of superimposed image vortices around the outside of the channel to satisfy the boundary condit-ions at the four walls. A derivation of this kind
is extremely involved and is not correct because the
actual flow distorts, thereby invalidating one of
the original assumptions. Therefore
,
in order to
corrolate the experimental results with a theory,
the author has chosen the approximation cited in
section 4 . This approximation is based on the fol'
lowing assumptions:
1.) We assume the rotation of the flow
to center about the points x=2.5
and y= 2.5 ; and x= 2.5 and y--7.7.
(these coordinates are in inches)
2.) We neglect the effects of corners
and assume the flow in the rectangle
to be approximated by two adjacent
circles of diameter = 5 in. ii.e.
d=2a ).
3.) We assume j to be constant over the
cross-section.
4.) We neglect the small change in the
velocity of particles from one station to anotherhence the rate of
turning of lines of constant velocity approximates the rate of turning of streamlines.
5.) We assume the flow in the upper half
to be unaffected by the flow in the
lower half.
1.NOTE:this type of approximation works best
for rectangles of AR= 2
101.
T1he above theory is designated "pipe theory" by the
author.
The calculation of the vorticity (
from the velo-
city distribution maps is carried out by the formula
derived below :
Al
at
where:
'L. is the velocity of the particles,
A 1 is the axial distance
moved,
A t is the time interval considered.
how we know:
as
V =
V
thus,
a=s(
=
1
(
where:
v is the tangential velocity induced by the circulation,
As is the are distance the particle rotates in time &to.
i02.
fOW
2 Tr(
where:
rl
is the circulation,
is the radius from the center of
the circulation.
But ,by definition :
thus:
2
The values of vorticity calculated in Appendix
Q are plotted on figures 4-35, 4-36, and 4-37
is interesting to note that the value of
to be independent of the radius
Z
*
it
appears
,and hence
appears to be constant over the area of the flow.
The author realizes that the limited results of this
investigation d.o not prove that
the area *iowever
,
is constant over
we now have some grounds for
assuming it to be constant(as a first approximation).
103.
In calculations on the fifteen inch bends the
lower half of the flow was investigated because the
velocity gradient upstream of the bend seemed to be
more nearly linear. see figures 4-3 and 4-13. On
the seven inch bend the top half of the flow was
used because the lower half was not investigated
by the author (due to poor gradient shape upstream).
A comparison of the values of
calculated from
the tests and the values predicted by the approx"
imate ("pipe") theory is given in figures 5-4,5-5,
and 5-6. T!here is remarkable agreement,especially
on the fifteen inch bends, viscous effects tend to
damp down the amplitudes of oscillation ,as would
be expected.Howeverthe period of oscillation
seems to check with the theory. A look at the velocity maps shows that
the value of
e
oc
reaches
r at approximately
predicted by the theory.
more experiments on 180 degree bends of different radius ratios should be made before this
approximate theory is accepted for engineering
purposes.
ihe results of the seven inch bend do not
corrolate weli. This is due mainly to the fact
that the flow separates from the inner wall and
J0+.
so distorts that it no longer can be considered to
be flowing in a cylindrical tube. In addition to
this separation phenomenon ,the results of section
3 point out that errors in velocity measurements
were probably excessive.
105.
FIG. 5-i SHEAR FLOW
TH ROUGH BENDS
INIT IAL
VELOCITY GRADIEN T, -.
r
LCN V.
HIGH V.
F
HIGH
V.
106.
I
!
4
I,
I
I- - I
,
:
----
--
i
-,
:- ,
I
I 1, I
I
I I
I i-f- I-
[
IA
1,
i
i
Ii
..
-
.i4 1-1
- ,
f -r
-
I-I
-I
I-
-
- ;-. -1
- --
--I
i 1
-,--r - 1-1- I
t
-- - I ;--,'t, .
,
--
,- i iI. .
I- 1111- -i
1-1 ,
. f,
I
:- -
j -
I
-
I
I 1,
I
I
i
Iil i
,,
!
r -1
I
"I11 1
, i 1 -r
, iI
I I- , . ,
, t ,
I ,,,
-
,
, -1,
1-L.- i -I- 1-1
,
,
-T-4
,
-
- II---.1
I- t 'T
I
i ,-I
.
r4.
i ,
r , ,
-
-;
.
,
i
I
,,
;
; , ,, -
- I ,-
-
1
-
,
T,-I 1
4 1
;
-I- I I
i ,
i , 1
I
jI I... -I . -,, , ,
.,4 , - i- .,, j ,--,
",
- tI-,i -i , -, i I !i r. .- L
- '-T -. -I- -t- ;
,
I -4 ,.
11:1 - , - -140, 1111i
1
lifr 111 ,Ltf
-W
I
-
I-- ,
-
A
, I ,'.
I
'I
, ,
I
T
i
-t I
4
I
I
, !
I
,
I I
I
I -- I 1-1
I
'-
I -1-, T
f - -I-j-,
i-
-
4 1T
4
1,
I
,
I-i , I
;
-1
L
.
I 11
I I
tj
,
-+
-
; iIt
-
4
i 4
t F
I
i I
i
:i
I
,
I
I
I
1-
Ij
-
-
, ,
1-I- H
i
-1
1I,- I;
i I.
I-i
-
;
1,
I
i
:
-
,
-
.
I- 1,.
i I
!- 1
I 1
,-,
i - -
I
, : , ,
- -"
i
, I
f
:11 ; i
i
1! I r
r
1 i ;
L
;
--
I !
I ,i TI
;
;I
- --
I
t
,
!
411 ,
! 4 ;
7' -
i
I
-
-
-
T
I II
I
,
I
t-f
lp, f ;
i- - i I
, -- - i,- ,
I
I ,
t
- i i- i -t
I
:
, .I III
I
.
,
I
I II
- - ,
-
,
i
-
,
I I
, ,
;
!
t '
I
I
-
,
i ,
1
I
I
, I4
f
1
,
i ; I,
I
,
,
I
! I ,
I
LIL,
I
I
I
i
,
I
: !
I
j ,"
, :
: ,
,
II
; ,
.I 1
- i1- .
:
i ,
I
, -
1,
:-i i
I
I I
.
I
1t
1
; 1
--- _ - , , :
4
1 . .
,
, ,
4 1 .
;
-
I
i
-
'I i I ,
I
t ,
i
I
-
,
:1
,-
,
,, ,i I
: 1
,
-
!. !
7 '
T
1
Ft -, ,
1
4,
-,
I
T7
1
- 1.
-
I
--
I
. tI I 4I ! 4 , [ F -
. .
-
,
T" ,
I
f
-- ..
f
1
--
,
-i
I t " .. -
-
I- i. I, -. ,,, ::i
,
-1 i .
I
-,
i-
.
-1.
I
I
:I ,
I
II
I
-1 I_ ,
1
T.
j
11
1:1
1, -t-
i
!
-
;
I-
,
.
I . L
,- t
-i
j 1 , 1
f
j
.,
I
!
,
Li
, ,
II
! 1
li
!
1 t
i
IIi
11I -
,
,11
I ,
i
;
11.1
. :
1
I .
i
I I
-
i
-i i
I
I
11
I1
,
-
I
!
I t
,
iI
I
;
-
,
,
I
Li ,
T--, T
-
i
-
,
; ,
" 1 t, tI
I:
i
, .
1
,
I
,
r
-I t I.
11 T
I: I
t 1
I t!
! ;
i
I
, , + ,I
t
.
I
t I
I !
- i- t
-I
4 -,
II
-
-
.
- - -. -- -T------
-
H- I I
-,
-
-
- r
i
' FF I
"I
r
i I I
I
.
f
I-11
+
,
I
I
i i
-- - ,
I,
-i
I
t
I
I
-
-
L,
I
4+
,2
. !
.ho
:
1
,-
i-
; I
.. ;
;
I
.
i
i ;
, I; I
I
!
: .
r
.I
I
, ,:I
,
+
11
-
i
f
-- -
I I
:
i
I I
I ,i I
+, .
i
;
i
:
I
,
j
-
.. F tF
.-
i
, i i
.-
-
1
r
t
N
! , ,*
,
f
I
4-t
-
, ,
.
11i ;
I
,
I:
I-
t 1:
I
11
fI
: 4 j
-
!
I
i
I
.
,
,
.
,
,
i.
: I
,
L
I
I
,
-
.
i . ,
I
I
1
,
-
;
! ,
I
. .
;
:
I1 i !
; I, 4.1
1
-
I
I
i
;
,
4 I.
I
.
*
i t
-1
II
I 't
,
I
4 1
- I
.
t
-
I
--
- -
-
I
.
-I
I
i.
-
+
,
II
I
1-1
...
.11- -, ..
t
-
-
-!.
f-
-
t
-
II
; : TI-
.
-
,
-.
i .
.
-
-
r 11
I
!
-
,
i i ,
i
i
I
1
I-
1
I
I
I i-
-
-
I
.
"I
f-
-
L - ,
;
j t
- ,..
-,I
4
-,-L
I
1 .
T- [ -1 4
,I
I
1I F
-,I- , - ,-
+
,-4
.- --- :1 -, 4 _r._t
f . 1 , 4,,,4
: t-j
,
-1 "11,
-
1,
4i
I 11
I ,..
I-I
..
-4 -
-h
i
,
I
[ "
i-
--
i I1
-
! f
1
- I -!
-1
I
f
I
I,
Ii-I
, t , It
i 3 - -.-I,
II,
L , - --i I+Ir
I
t - :
I
-1
I
j
v', ! I
i j
i , --I-!
L.
I 'I.
j i ,,
I
i "I 4
I It
; '- '
+ I
; i "
- 5, ,
I
T T
-'- r
-
i 4-11- f- ill
tt-f
,
r
!
- t i - t : i, I iI
,11
L
I
! l
I
i ++i
I
I
I
j
-
':
i
I .
I
I i
I
:
.
1r I
1
.
-A. + I
IL, , - I, . I"! .1
i'
,
.
,
I
-
! -11
i
-
kI
11
I, I -
d
'
.
I
..
vI
T,
.,
I
I
L
. ti. ;
-
4
t
q-
- f I f I -T
i , - F.
i-t- 4 1
! 11
I5
- ,
I
- ;
.
-. ;
- ,
! i1
.1.11
1
, 1, I
, - ,- ,
I .
i
I
-i
,I I I-1
-
,
I! j I
ii
;
I I
,
I
-!
--
i
I t I 11I I t -- -- -.1
I
f -4
4-
- -,
- 4-
I
-
f t1,
.i- * " I ,I 4,
il,
I
!
I
!1
"
!
-
-T .
-1
-i
I
,
,r
I , --r F -1
T
- f. -,+ 4" -, - It f
I
,
,-
--
T
IT
-
- -
I i"
1 ;
.
;
- , -*-
---
: ,
11
; I
I
:;
I ,
I
1,
-
Ft
I
-
;-
i I:
:
.
; '
I
q
-:-'
-
,- t, - -
11
t- i ]
- 11
"
i-i I- I-
4 -"
1;
-I
j .
I
- - "I
-I
-
- f- -- -t 1-I - I11
- -. -- -t
.11,- -,-:
-1
-.L- - 1 4-i
,-- i
-,,- I - .
- 11
-I
TI I-,,- I
4
I, - !I [" i, I I -i _-7I - - -C
- . :-.
I
- +
t
-TI!i "I
- 11- tr- F ....
I
+- F I I 4-1- I ,I-t
- ,4
f I
I
A
, I , ,,h II ! I
, i
I
,
,
' i -4 1 , L -I
7
-1 ,-
I -, I ,
, 1
,
; i
I
U , III4
- Ii
j
11
t
I
I I- L
I
I
I
1
, ,
j , iI I 1
I I
4
--;
" , , ,,
i
t
-i
-
, -
-
-
,
i ; ,
oi ;
,
ITI
. . !
i : ,
.
.
':
, -i- , "
11
, - -, , I ! - . . .
..I
J.;
I
V
I 11-- t 1,I +.- - ,
-
, : ,
II I
I, I
,
.1I!
I
I: I i11 i
t ..
1;
! i
I
I
! t
;
I1
.t - -
I
; I
,
I
f
! ;
-
,
I
f
,
11
I
f
I
vI
I
, , ,
:
I
-,
4 -
i I
I
, ;
I
,
, i : .i
:
I I ;
. ,
i
;
i
f t Ii
,
1L
1
i ,!
11It I I :
I I: I
-
, i I
1
t -,
I
,
4 ,
I
I
-it
i
!
-
I;
t II I L
14
I
I i
-,
-
,
11
I
,
I
-
I ,
i
r
1
I! 1 1
T
I II, i . i
:
I
i
f- - j t- - I I
t
; II
; ,
I It
I 4- ;
,f 11
- ,
t
t
-
T, -I
,
-
I I
,-
I
;
I
!
I1
11.11
I
-
- ; ,
;
I
I i I
r
i , I :
1 1
I,
,
,
:i-"
.- ......
IJ- .
, ,--. ..
I.. -:
1,.i- 4 I-1TI
,
-F -;-,
LT
,
-1 t
t R
, 7i
-1
f
f .
11 11
r
I1 , I
i
4i
-
j-
- -
-.
I
-, ,
-I'
- I
- -I I,
, T,
i
1,
- 7-.- -
-
I j ,
,: .I I t
;-
,
-
-I--
-
I r
- - 1 -!
,
,I ,,,,
, I .
I
.. :
L
..
I-
; ; j ;I ,
I
i
1
I
I i:,
I-
i
r ",
-
I i
,; ,1 ; !, t A
,
r
,
-
T
I
1 4 1 ,
1 4 , i
I
"I
.
i
,
I
.
, ,
14,
) I! .; -.I : .
I
II
I;
,, r
;,
,'
!-17 ,
:
, *
I
, i
,
I
II
4 1
.
"I'll
1
;
+
I- I
T
,," ,-- .
T,
- -
I
II
7, 4
,
: ,
I , .
-, I
I
I
I
i i
. . 1
I- , i! i+
+
14
I I
-
, !
1
;
i 1,
1,
;
t
I .
;
,
I , , . II ,,
,
,
1 1
I
-
-
;
1 i
,
I
, .
j i
. ! i
1
1
I
II
: ,
. I
;
..
.
1,
-
I
; 1 . ,
i ;
. Ii t
I
,
T
I
I :
I I-;,
1'
I
-
I
(
!
.
.
,
4
i
.
;
f
I
,
I
f
I
I j , j ;
-t I-
r
-
i
,I- ,
I
11
4
i I I
ii
- -H- -I I
I 4
I- j
1
,I I
t ;
I i
k" 4
I r
1 ,
i
1;
f
-I
4
I I
,
II I
I ,1
: ,- IJ T
i
i
:
.
I
i II
I
- "I ... .,."*
:
- -
1. [tH! i --tT I I
! , , +"
,--I1" II r I
11Ati l''_41 't '_1t t 1,
11I!!T-, , 11 I 1
I
-,-
, ,I i
, ,4
.
II
I
I - ,,: ,
14
- I
I - - I , , , T-4- f r
f - H+
-- I
I I I . ,"I , , -i, 1
'i , - I 4--'I
4-1
I--
I - f -
.,'I ,".
I
;
I
-
i
I
11
II
I -
i
i
Ii
t
I -, I I ,
f i
, L - i
Li I
.
-I
- T - -- I " 1 ! i- I , ! JI I- Ii " -11
-I i - 1 i!, :if T, ;- Iil I Ij 1i-I r,1-1, ,L
11
I t- r ,
-'
1-1 j i
1
,
- -
.,
.
.
I- I
1, t I
.-"
: , II
1
I . .!
-
[I,i
,
11 I
I
.,
' 1 -1:
I --4
- 4
,;
j
-,
I I
I
, t 1!
, ,
- I
,
,_
-
-
-f
i
I
Ii
, - . ,
I- -4- [ "I .. 7-.
t ,
-
I
;
I i Ii
.
-
,
-i- I
I- -
,
------
4-
I --'
.
t
[
I
--*
-
i-,
I
, - -! i1
- I
: , i
!,
-4 .- . iI -, ...
, I1 If - 1 1 I ,i I t- -A -I- -
: t
I1
1.
-
Ii I
: I
-
1
t ;
. 11
,
I
TI j !
" , -
!
1- I
,
I1 ,
! ; ,r!- t
-
,
I
I
I
,
.. -
II-
I
,
.
-t
i
1 11
,: -,, I
i
i
-
- - ! I--H
.
1
.-
rr
- -
I.- , -
.
I ,
-
- -- ..
: -
L
1
--,,
-1;- --
I,
-
,- ,
, ,:
... ..
,4
.-
.
t
-t -
t. 1, .11 - i
-, ..,
I
I
t
- .
",-;
, F,
IVT
,,
-,
-
I i- -4'-;0 ,-tr-
-
1 1.
- - -I
-.1 -p
,
t 7rj
- -
11
-L- LL
.. 11-1
+. i
Ii- -- -
. -
.
--I
-, - -
--t-+-
-1-
4-,- -4- .
-"-
1
-
.
-
I
I
i
-1- T.r
t + . - .-I-. L. I'll,
. I'll : 1_,1
i,
-
I
.
;
I
_- -
--
I ...
'K
---
I
,,-, ,
;
I
,
I
4 .
--
,
. II
:
,
J-.r
.,
:
,
!
,
I
; i
;
,
:
- -"7
I
fI
I I
TI
--
iI,
;-;
I
I I .
1, /
1
4
T
r
.
7 -
,
,
,
-,
,
,
-.;..;-
r -
-
r I
-L.
1, , , 1 .,
I.1 I
I;
;
r
, ,
. I
I i , j
I
A
I
I
,
I
,
1
1
. ;
II
1i . .
, ,
_
1
I
IN
;
7
,
-
;
. ;
I
,
I
N
,
I
,+-4 - , t7
I
ls
I)
t
!
,
,
I-
. .
.
,
- - .+- 1
_N
,
.:
1 ; I
i
1 i :
, 4
:
,,^
,0, -,
;;
I
I
,
,
.
;
It
4T
1
,
.
I
:
I
I
I1 ,
I
--
,
t !v
. I
,
-.4-i
;
I
.
I
,
,
V
I
I
I
.I r .
.
I
I + ,
.I
1-1-
:
I,
!
,
, ,-, , .
I..
. ,
.
. I
,
I,* . I
. I
'. I I I
/
;
VF
,
. ,
:
I
; ,t
!
I
.
,
':
I: I-, -.I
/
,
I I
,
I
,
f
-
I :
.
!
:
I r II
.
-
.
,
77
--
: I
,
,
:
I
I
-
I
I I! I .
.
II
1
,
i
,
.
,
,
I.
.
,
,
I
1
I
1: ,
I
7%
! N,,,
,
I , ,
, 11 ,
L,
; !
;
,
,
,
,
-,
!
I
I
*j V
-tt
I
:
,'
,
Ik,
4
I
N
I I
I
II.
1
.
I
,I
.
.
--.,
I
1
, , ,
I:
.
---,
I -
.
I -
--
.
I L
-
I
:
-
I
I
,
i
I
. I
I .
,
1 I .I
. -
,
.
I
-
, I
.- I...
--
.I)
, ,
.1I Ir I . .:
1,
I
I
..I
:
. ,
,
., -I
.
.
, r
t
.
i
;
.
,
I I iI
a
I
I
I
I I., !
..
I
i
I
,I ,
I
I
;!
: ii :
.1--I I
,
I
- -,
,
i,
i,
;
i
I
i ,
-1-1
ji 1 .
,
-4-+
- '-I 7
-, 'J, _
i
I
1-1
;
i 1
-t ! ,
.
11
!
,
.
I
,
-
+
-
! !. !: .
.
.
.,
.
, .
i !
11
1 III;III
i , --
I
i
I
,
.
I
I
,
I I,
4
.
1
I . 7
I, .
. .1
,
I
II
.
i
:
I i
. I
I.
,
-
1
,
,
:-
i
; 1
;
b
iI ., .
.
.
I
"
1:
-.
.
,
! i I I
. .
I.
+
11, 11
1 .
,
-
'A
-I
: I,
,
II
--
;
!
-
,
.; I:
I
.
-; .
I
1
I
I : ,i
,
t
I
, I
.
I
1I
II
,
, 11 ;
. . ,
., I ; i
,
, " ,
I
,
,, ,.
I
j
I
.. -I
I I
.
:
A
,
:
I
1.
I
I
.
. i p
D- I
,
: j ,
II
-
..... ,
I.
I.
.
,
, ;--.t . , : 1: iL .I I 1 : ,
I
,
I I
7
.
,
.
!
,
I
;
4
: I:
., II I
, I
,
,
; , r .
:
-- I
II
:
i
Ii ,
I
:
-
-,
,, , ,
. II ;
7
I
:
7 , ;
-r
. . e .
. : ..
I
, . i i
.
,
11
I
r
,
I
,, - ,1 I,
j
I"
F%
--
I
'
' r
I
; !
,
I
I.
.
,
.
!
I
-
:
. I .
II I I
. 41
,
i -i I
;
,-.11,.
1
I !
.
I
I1 ,
II I
I
.
.I.
-
,
. :-
,
,
,
' 7
4
I :
I
I
;
-
-
L-
I I
- z
: -4
I
;
_-I
,
-
t i i
I
-
I
I ,- !
I
;
, i,
14
-
-
f I
!
-1
11
;
I I
-
I
4
11I
1
, ,
I
; " ,
1 !
1 :
i
-
: 1
!- I- i I
-i
,
-7-f
I
, -,
I
I
,:
f : ! 1
i
1 ,
I 1.1:,
I
'.
I
: ,
-1
T-i
J
-I 1 ! I ,
-
-
TI
I
; - , i .-i I
1
! I I
, ,
-, , - -t
,
I
_
, ,
i I
t ,
,
_I
-I
4 ,1
",- ,
L4
,
I
"-
-
,
i
i f i
t .,
, 1 ,-
-
-
t
ft
,-,,-'
,
, I
,
f
, I, t -I
;
.
I-iT- , 11- I- I
- , 4- - 1
,- I
,
I-4 ,1 I
, :
,
I
I 'I f
I tI
.- I . . , ,
: Ii
I Ii I..
.
) - I,
I- - t-14 1
J-1i.
.11
-
,
i
-
-
-I
1-I
1
-,
+-I - .
,-
-
. -..
- . -j -- -I
-1I I
14-f-
-T+
, I
Ij
-
-
- -..
--T
!
4-1I I
i-
,I
-1I -1
-
-- .. J.-----
, --,
4
1- +
- , L
--, f
'.
i -
f
-
t
-1-i
- -
" -1
i
-
--- 1 -.-
+
: I
i
-,- I ;_.
I I I
!! I
1..
T-
-
T
L
-
-,-
.,
I -- (-.
ii,
L
I
-.I , ,--
- - IT
-, '.. - TI
, i I
i - I
, --- i
-
T
-
-. ,j , -,
I Ii I
- - 4-t-I
- , I.[-H -i
- -
I
,
: - 1-
I11
i i
;L
-I
-- -
I-I..-.- 1
,
- - - - - , f1- T ,
I I
-
.
-
- i
--.--
t
7 I-41--
- II
..
I--
+
-i --I- i
I f 1, , i I I I t
I -1. i .. . - - .
i
I 1 -1
I I
I
,
t, -
IL
, -jt
"
-1
.
-
I
:
I
- ,,-r.i
TI -1
, ITI
-I
I
- ! I- . ,
1i - -
I+
L . -fl
,
I
-
,
,
1,[ 4 i-i--1 1
i
.
. i
i ; , 4- ,
,- 1 I- I
-
, j
H-
-i-, , ,.
t-LI . ;
-
i
I--,, ,k -L
- t -I . . ; ! LT
1
,
1 ,1
-
;
I;
I- - T-;
, I'll
:
-
II
1 T f T - '- 1
- ,!
I
i-,,
I
I
.
t - Ii ,
i _j ,-.
- I - 11
1 i , f- 7 : -. I
-i
i I_
- , L- i
-1-I I . 1. I .
-1
I.- ij
I
: r
, I
--I
i -
I I
. .
+
--
I-' -,I
1, I
- 4, ,, -I"
11-j..
.
I,-, -
I-- I
,
; I
I: I,
I
- , -1"
;.
L" -L
:
I t I
.
-,j :
i -I ,
: I I I, I I ;
, 4 I j Ii I
I I , , I i1 1;,!4, , L 4"
L
I
.
--
-1 I ..I
'I
1
i
i " ' -1
I.
I
I
- . I I
I
I
; ,, j I
,: ,,!
!; . -11
I
I
1,
;
i
1 1
,
1!
'. -1
J !,
L
.
,
, I-!-! -,:
,,
.
I
, !
-1 ,
,
-
I
:
1*-
!
- 'I,-- - 4.
I
- t1 L Ij -
,
-1.
-- -- ,- - -.
-1,
, .., I
4-t- - -- I ,Z , -- I I 1,
14
,' 1I 1I
II L' I , I
I
i I I1-
I
_ '
.
i -
I i. 4 I
1
I
I
---- T-, L
.1 I I
!
I
1
:
11 "1
I
) 1.
.
,
I
I
I
.
. .. .
, ,
"I,,,
I
.00
,
:
1I
- '
I.
-11
Ir
I
. I
a
; L:
,
-,
, -, ,
I
., ., : .4
- -1
--,,+ T
-- H I
-- -- '
t
1 I !-1,
r
-
.
;
I II
.
I
,
I
77-. "
j-
^
-A
S
I 11
;
. 11
i I
I
, ;
i .I
[I
I
I
-
-f
- t --
I
i-
1
;
I
I I;
L I,
i
i
.
,
i
,' t
I
I
I
-
I
-
;
- -
I
I
1
,
I
I
,, ; ,
-
I
: I
. i I,
11
. ,I
:
:
I; i
1
,
1-i
: , .
I
.
.
.
11 ,
-
-
:1
:
,
.
tI .
:
!
1 1, , ;
. .
, ,1
I4
I -.
-
I
,
!
1
.
-
I
I
t
t I
; .
j
I
I
.
-"-I i
:
I
, 1
,
,
.
. I ,
-
i
-i I -
,
Ij
:-
-j :, I II I
i
41, 1.
.
i
,
I"
I ,
!
,- L !
4 --. 4I
I1
r Tt
A
f
- i, It
, : I
,
4
I
j-
II
---
;
,
II
I .
J 1," .
.
, 1 ,
, ,
;
i I , I
, , ,
;1
1 : ,
-; ,
I
j
,
.
L i
I I-i ,
1'.1i
:11-
I-
, -, , , ,I
,
, , I
!-
,
III I
.
iI
I
I.
i
I
I
I
I
k
:
i I
I
-1 , ,
I i" I
.
. .
I: I
1- ,
: ,
I
:
I
I
; ,
i ,
I
-,
-:
.
-
.
-
,-
,
,.
1
-
.
I
.
;;
,
;
,
I, 1 I
t
.
.
,
-4 I
I
,
! .
i.
I.
I
I-;-- I-
- -i ;
I
I
I I
;
-'-
"I-1
,
!
,
,
,
: ,
I . i.
-
"
4:
* ,
i
I
,, A- , i -i
, "
, .
s-,
r
,
,_; ,
I
i ,
I .
-U !
I- I
: I - ,1 -. ,
!.i
, 1
+ i i
I II i
;
I i
i
I
, j ; ,
I
- !
: *
. i
I . I . ; I :
,
, ,
. II
.
, _
-
N
.
,
-
i , ,
I
.,
. I
.
-77 , I
I
II -
,
I 'I
:
I
-
, ,
-I
, , ,
, ,, , ,
:
I
-
: I
T
.
-11
!
1,
: I
I
, ,
.
f
'
-,--
I
! I I
,
--
-+---
i
,
+
41
1
I
I
. I 1
I
!
1 1 1,
i
II
1
! -
7 f -,
I- --.
I
11
-
t-1
II
--,
.
-
i
-! - -
-
-r-,-;
!-
-
-f
-
,
-1
,
-T
---
f i
1
i
I'll - - 11- - -.11,--
.-
I
--
!!-
. L ,-
-
I4,
I .
- -I-
107.
JOB.
109.
(10.
T
_
Ill.
I. .
I.
I
:- - -
,i l -
-1. _ - -1
;_
!
-I-H - L.
I
--
I
.
I I,
-
-t tj __1
I4- - , I 1
-F
'- L
, - -I 1, --;-t,-
i-
I
!
t
-_
.;
-,
- _
- II i,ir - I-.- 7
-
I
I
-r , - t.
I
T_11
-
!
I I 1. I
-1,
I
i
:
1 ! ,
I
I
i
t i1 i1 iII t -I
I I- I
,
-, f t ,
, !
, 1 .
t t , ,.
I
I
!,
t -" ; i
. I j , -
i i
-
1-
,
TI
, , -
.- -
....
-
- - 4
-_ I--f-Ift
II
-, --
I F-- !-
,
3,
,1 ;
I ;
-
, i
,
- i-t 1,
,
,,
i
I I.; I-
L , ,_
-11
,
t1
- tI
,1,
I
. I
, .
t -
-I
I
t
,
i - I---,
I
I
,
,
,I,
:
,
I
: -, ,
I- -
,
,
I
,,
1 !
:
-
-4- ,
7
I
,
.I
.
-.
I
,
.
I_,
I
r
1
1,.
III , ,
-t
I-
j I- I
-i _ 1
,
,
I
,
-
I,
,
I ,
I - _i
-- - I I
-
- -i
i
.
,
- , I
t
j-1
R
I
rl; i 1
t
i , I
j
!
f I I
, I I I
-7-t-
11
1_1 I , I i
;
-_ T II
i ii .
- ,
i t _t I i :
, ;
.- .I r
,, : - j
1--'
-4 t -_t I
_ I i ,
i ,
T_
I I
-
.
11
,
. i_
I
, . , .
1I - - ,;i,
-,1 11 I
-,j ;. t. 1[ ,
11
.1 t,
, i
4
f -- f :
I ,
I
-
i
, ,
.
- I, LI-. I.
1:I
I
L i, t -1 - - i I- L-1
!i IIri I
-,, L ' '1,1 i' It
4 - I- - I t t- I
,
: 1 I , 1:
t , ,
I
I I
I ; 1I
._I=
1
i:
TI
I i
I I
11
:
,
;
,
: L
!0
1
-
I
*-
f
,_ l
-P
- ;-
-
i , i- ,_ .. .
I ,t ,I; -, 1I'!ff
L
-
.L _
,
. ,
,
7
;,t-i: ,II
_
.
; ,
77
: ,
I .
,
, I
.
I- : t
,
t
, .I
;
: t
:. 1
,
U I
I
I
,
II
F
'-
I. I
tt r
.*r
f
I
1
Ne
41v'
1: 1
,
. -
I
I . .
I , 1
t i=t;
.
-T .
I
i ,
I
7 7 -,
__
, ,
,
, ,
-" - ,
i
,1
;
11 I II
i
t
_ _
I,
,
,+ "! _j
1 I- _.
,_:
, ,I
.
i :
1-1 t
I
-4 4
-
I
I, t I
I . ,
I
4
-
Ii
,
I
1 ,i
1
.
i
,
-
; .
7 . I I _j
.
I-
,
I
,
II
-
,
i
:
,
1.
I.
t
t
I,
i 1
1
-
;
.
j
.
,
I
I
I
I
I
,
I
. I
I
,
,I
: 4-
II
, I
I
I.,il.
- -,- -
-
.
-
t .
.
I
I
r
\
'.)
,
L
,
.I
I
,
I
I
, i :
t .
.L, _1 : I ' ,
- 1 1*
:
'-
I
1
,I .
:
I
I
I
-
r
.
I
,
I
I
.
,-.1 11 ,.1 1 ,.
I
771- I !,,
'I .
I
;
I
; 1
-
1 -
I
i-+ I
I-1I
. i ,
,
I
I
; ,
1
t
iI .1
:
,
1 ,
!-I . , i
1
i I t
!
t
11
,
.
t I
I
-
t
4.
t I,'T
'
!
I
.
I
.
I
-t-
I
I
! 4
.
1
T!
, ,
: I
...
,
, I, ,- I, t: i,
-j
1
I, I I
t
I
-7-7
i4
I
t i
,
I
, :, tL
44
i ,
.
, t
,
7
-
_ _L
-_
1 - -i,. ! Ii , ;- -
-T+
I
L I
'Ity"
I -
,
,
: I -ir .I: I
... L. ...
t i
I
It
I
I
t
7
-
I I
,
-
1.
;I
1 It
-1
,
-f
-
I-
,
--
I
_.
,,
I
I
'
I
:
,
:
If
II
,
:,
i I I ,
I
:
-
:
;
, ,
I : ,
, , .!* ,
,
_I_]
Al
I
- I r
T_ j
I ;_
- . _
,i t__
"I - - - ( I -
I-,I -]
i
l
-j-,
,
"I
* ;;
I -
!
,
.
4-
t I l;I -1t
_ _1
-T,
I-
I .
_
1 J-4-1,
ii
I, ' I
:i
_ I
1- i r
-I -f
T
I __
-
j
__. -
I
1
- '1
I-,
-
r
'
i
:- I
l
I
il
. I
- 11
6- ,_
414
11-4I__-
-
.
F - -
i 1.
i
-
1
-4- .
: .
1
,
- 11
- -
-
_- I I
I 1,_ f-f-I
- -:f,
I ill 1-1
0 - ,,
.
- __I
,.- ,
, ill;--,I
_;# _i
,;I
f4
t-
I 4
_l
, _FI
I , I
1 . I
1
i i -7I1 11
i I -
to
;
.
: "
- -
1. 1
i-
...
t.
I
_
- - --i-r:-i-__ 4i__
-1 'L ,L _4
-4
41;_11
! r, L
4I
I, t. _:_
j ,
L
I
'_
I 1.
I
I
I
- -1t
,
I- , I
"_r
.,
411 - '-41
I,
_
.
-: , ,-i
-
r Hlr -
-1
T 4
_1
f ,
_
II- Ij
:
I I
-
_;
I
r-
-
I - 4- -
_!
11
I
J - it
-1 t
, If --
I
. .1 I
.
::::: iI
r,
- 'It-il
4, I L- i , i -_
i_ 11
I
I tt
, 1L
I
,
-_-
1
, .
..
!
I -f-1-- - I - , ,4, ! , I , 1:
, f , 14-j
1-
_ _ - _lt
I
I
i
_. -- I--- - - T i
I - __-,
,
- .,I _j_ T_ I ,--+
,
H i- _j- i.,I
A"
T
i4 _t
,
I -, ., - -1 4-, -1 i
i t i
".
,
11
.
I
i i .j 1,
I
1
'
:
_,_.
Ti7i
_, 1 ,__jI 717!
, ,
,
;
I
I
,
, . :
1-f 1 4
,
,
, . ,
;
It
I FF
...
f T;- ! j I , ,
[ -, -41
- I I
, I TI
- I_
I
I t4
, 1,i ,,I 11
I
,
it t
I
I ,
: "I
I11 - i,
i t-I ,
;
TIf I
:I
I
I
, t-
- - I
- - - - -,
1I
,.
, 1I ! - 4i 14
.*it i ,
I :, II i , ; ,
, 11
_I t
4I
f
II , t
I , 41 t
;_
, , -4-,4
I
. I I
I+1.
II -ti t- , iI. ; iI , 1 _1
- -, I
I
I1 ;
I_
- , ,
,
.i f 1
.
I
,
tI
,
I I I
, -, i
- . - -- I _ -1 L
..
-_ ,- . - , 1
_ __
- 4, _ -t f ;
i I 1:I
1 -1
I
I i
,
t t
,
'.
_ f
,
11 --i
I
-j
, T I 1 i
I4,
,
-4T
I
I ,
t
I 111,
II t
:1 I
.
4 1
,
I
I
1
!
-
I
: ,
. 7
. ,
,
,
,
,,
--
---t
%,".
,
.%', _j,
i, ,1
4
II j
I
I:
: ;
i -
;
4.1-
-
,: :
- A- I
::
I:
I
I i i
1
-
:
,
:
L .: ,
,
.
f
i
. .
,
.40
I
I
I
I:
i
...,
,,, I
;, I, -t,_, _
-
,
.
.
,
;
,
.
4
,
II
^
,
I
; ,1
, .1 4-,
,
I . 4 1
,
:
L.
,I ,
:
,
L
I
:
, , :
II
:
i
, !
*
I
I_ :
j
1 1 :'
I
!
1
:
-
I
t
,I
.11
.
;
I
, , I
I I
:
11
1
I ;_
;
,
. t,
1,
. t-j
'o
I
,
.
,I
I
: I
,
.:.
-
I ',
II I
i ,
4 .
I
i,- ,
,
7
.
, i
I ,,, i
? ! i
,t
t
t
,
It
. I
r
.
i+
j. ;
I I
!
I
1- I- .
, , ; -I
,
i ;
I ,: I I,
I
ii ,
_
'!.,
,
-.
, i _jt :
I
"I'll,
- ".,
_.,
_- I-
!_tJ
-T
,t jj
i
I
;
11
I
- - f
-
, #
- - -
-
I _1
7-,
-I i-l- 4-
; I L
: -,
_4 :
, , 1
4 7'. - I
I
I
J_
II I
i1 i-f
I , It
; I
,
I
. Ii
.
,
_
I
,-+
_
,
j ,_t I
,
-
!I
-
; I - _.-1- i- - I
I
H
t ,I :
I.,
I
i
ilI '
;
I
-j
, 1
I II
,
jI :i I I
f- I.: --,i : I1 1 -jL
I , . -1
I
I
_.
_L
- - 1. t , It-i 11,
1
:
__
1
.
7+
1,
I
I;
-, :
, . - _. I "
-1 ,_ L :
4
. 1+ ;
I , 1r
! , ,I,
I I ! : I
.. 1.I,
, , , .
:
L.I j I
ii
r : I
I
,_
I
+1
I-
- I I
1 -,
-
t I_
4- [, 1-i ! - 1_1,1
-,
r
1,
_11.,
-I i
'
I
-1
If
I I1 I
.
:
;I
.
I
-11
-,_.
,
,
,
i .. __
-
-
,
1
.
_
.
I-
I IF
I
, - 4-i
- i- - _i [ T4- -
-
I
I
_
,- -
-
- -
-1
,
!I,
I
I
- I
:
- __Lr 1 I
-1
- !-- 1 -, ,
- 11. ..
,I
I
4-i t i .- t
1 L
I, .__I__
:...
t-
:
I4
t.
I.
,! __
- . I
__
,_ ,t _
!_,
I I+,- 1,
.. l
I
.
i
I
I - I
: j -q
I 't
t
I-
-
I __
-1
1, I .
I
I
,
_ _!
i"I
-- t-
' I
I ,
,
- -t- -
_
17j
;
.
I
r
_
-
ill
-
4
t r T I I "II
;:J i - - 1 T-,-
-1
_j
,
,
__
-;_
-4
I
__
...1-1 I ,, I
. _
I
-
. ,I-11-T
- , .''..
.
i
I, I
I .;-. !t-. ,,_ I
-i-
i I
I
-7.
-iI - 4- 1
4+ ,
i -111
+
;_ t -f t
, _
j.-
__
-
-T- I'l-I
-
- I-I - ..
...
I~
.
-
-4
.._
,:
- -1 .
I. _t
-
-
..
-
, I_ - - __
- __
-, r t
I
+
- - - I..
, - - ____
- I
i
_. .-
1.
..
..
......
-1.
,
- -!-- tl 4--t
L
-, T
I T -rjill
- - ll-_
I
1-T . -1
1_111. -1.
.
--,
_r
- : ,-:lf- ' _T
1__
t 6 - - A-_t
I- F
, t, ; -_
-
-
t- II iI- .- t ,
.
i1 , : -I
I. I I
T ol-1-1,
T - T
-- _j_
1
-
11
i ,
t
f 11
, --1 :g
,
-, - F -i
t+-f+
1
-I
-1 "
,
I
.
.
It
_
.1 f
.
.
--_-
,
I
I
t ,
, - I
_jj
Ll _ __A __
.. -
, ,
-: ,
-_,'J,-4
, _-- L-1Ii- ,
'..
I
-
- _
t
::
I
4- F
-
.
i
I
4 T,
f
__
, !
I
I 1, i
I
I-
Lttt
,
,
-
-i- I I
1,T
-1.A _i .1,
I
I "
+!_
_ i_- - 66 - I -
I- I
:
i
Fi :
i
,
I- , I :
Ii Ii I
t
L;1; " 4
-, ... : -i
- -
I
9
-
,I,
L'. - t,.
-! -
, - _L
:
t 14I ;- II- - - -_ .- ,- I
-1
Ii-I- , , I
_l, it- .I , :
I
I
.
__+11_
F
i
__
*
Ii_
.____ li "
_ -1, -!-]
t
: Ij , ii :,- -, . . 11I - 1! .1.1
j
-"!
I I I -;, -,
-I-1i--; I , t , -- , 1,
I, ; , -, 1I I - t4 i I 14 )II 1!- -11I-I-i ."
I, 4
iI
-
I . t : :I, -
I , tT
.
,
-4
LL
;--,-
11
t Ii
l
, , I ,
_- , " i _,_
_ 1I
L.
i
i :
1,
ti_
I __
-:I I I 1- I
- - rl_1 :-_._ tI r 1
- -f .
1
I :1
! II
.
-
-
i, 1_t
-, I
_
,
11
,7! r .,7 ..
I
. I . -
.
11
; 4 i ,
. . . I-L
I It-
I
I
I!
I . - , t _. __ i i
L : I .
!
, - I. _4_
; 'i -
-
I:
I
I
"I
; .
i,
, , ,
-
L
I
I ;
, ,,,
, 1:
4
I -
I
I ,I .
I I_;-
I ,
:
- . I
*,
I
I
I .
---
I
I
I
. , It
t-!! . I
,
_
,.
t '!
I
:T _ 4
i- II
., I_
I
I . I I
I
.
I
;, t
-
i I-
- - ; 11
t I
I
, ot ;
4_
1
1
iI
.1
I t !4
I
,L
- t- t
!
; 7
1 ",
,i r
I
I
,
,
i
-, 4 !I ,
-
I
:
I
I
; I
,
1,
- I . .i ,i__, Ili
i '_
I I r II , .
I- I
i- . ,
I
1 7 i ,_ I
, i
;-I __
_.
"
i j , i,
i I , I
i,
I
i
: I II
,
I
I .1 I
I I
. I It
:_
t
: I
.1
i
.
I
j
if-1!_',
j 1I 1,
I -1- 1 1 4 ",.
I, It ,i
I
1L
i, _
.1 t f
. :1
II , ,
I
, ,
14
:
;I ;I I. . ;I. :!
I
I
I
L
I
'
.
i , 9 -,
: -I .
. - , I
I
, .
,
I
i
. II
I
,
; :
I
,
____
. .
I
I
I
.
tI I - II
1111!06-16A.
I ,v
.I
I
,
I-
---
j
II
.
,
I
!
I ..__
,
,
I I II . I.
;
-+_
7
,
.
I
I I.
I
;
_ _
:
. I , ,
I
I
:
.
I
I
.
It
I
T
,
.
,
"
, -i
,
I"
1
I
.
,
I
-
, :
-
_L -
1 41
-4:.
,
- .I
! -i
I :
t
T-- '-
,.
. I
I
.I
,
!
1 1I
; ! t ;
-, 0 : ,
i .
1 4I-
-
-T
;
I
II
,I
L.
_ -tI ,.- 1
,.
I
.
,:
,
I
;
; i
.
.
!
.
I
.1 !
I
1 ,
I
-4 ,
.,
, I ;._
I
- ,
I
i
I
--
I I
,
I
1*
4 . 5 -I ,!tb1r, 4 i .II! I
.
:.1
, --
i ; 1 I1 ,I
:
,
.
I .
-, I
;
- , ,
! I
: !
!
,
; i
, 1
I
- -.
J- - -
t , I ,.
i 11 ; H7-,
I - I-11III: , A .,I- .I I, ,,
. ,
:
I .
.
: I
i
.,-6
I..I
-
,
I
,
:
.
I
I
; !
4 ,
,
, -- I
I
,
:
I
.
I
I . -
.
I
"I .:
4 . ., , ,, . ' O I I
,
-
-I - l
_
.- I...
-i
t 1,
_F_I i. H .w
--- f
-1
,
i !
: ;
! ,
V
, '.7-, E
I ;: I :,, : , - ; t,I Ii III- ijJ__II
,
,
, ;
i
I
1
1.
1 41 1
1
I
i
-
I
..-. 4"'+.
t
L
1
I
1
i
f
:
.
.
,
!
I
, , -
tI if "t 1-j
,
,-L7
I . , L . ; , ,,
- , ; i ,i
, I, :
It I ,
,
r
-i
i
; !
4
,
:
-1
I
I
I
!
-
-
,
_0 ! !
-I
,
DI
I
I
i
!
fI
j
;
,
.
,
-
,
II
-
i,
:-1 i - i,
- -
;
t
;! ,
I I 1,
I.
,
.
t
. t
!
,
L
: I
__
,I
z
1
1
,
f-
1 1, 4
;
-- _I.- -- - t.,I. ,
I- - - --f-fI I I, I
- tt- I11T I -1
!
[
- I- - I
i-I.1 Ift It1
I-1I t
- - fl-;"-, -_ _
.[i II 1-1
Ij __,_ , , -1
1
t
I, I I
I
1 '4 I
H
4 1 f r 11
t, i
,
.1.. I , _,l__I,_
- ,-11- - 1. _I
t.
,
I
?
1
;
: : " ; t.I-,t-,,
I
--"t:
i--t
-1
,
I
,
T
+ - - i,
i,
.!
_ , . I ,,tt T ,,
- . 1_1
- -_t- i I, - +4
Tr
r r -I
i _t
-
1, - :
i -.- I ,I
7-,
; ,
I :
i .
! '. I111
I :
Ii -I ,
1
: .
,
i
,
i _- r_- _
!
I
4
, t
1
:
,
.
1
I
7-!- -
,
_
+
-L
-
,-
i
!
. I
-T , +---
_.
i . t
- II . , . , , ,
-, 1 . _,
I I4
I . I
I-
-
i I
A
,;,
I
.,
__
__l
___ -t-
I-
,
-, : t t! i t - I
i , i I t 1! -- f I :I It A.,- ,4-, I ,__
. , ___L
I:_
i - I iL_ _;_
, ; I_ __ . , I - t i _i_ t- - -f-
.1 I
. .
i
:1, 1
i
4
t , I
it i - "
I
:
%o
i
,, I,
i
I ,i
t I
;
:
,
;:
Iii ,, ; . t . 1- , - - . I , , . i , : I : : -.I ,; , ,. L,
t I I iI II
t
I
- :-,
:I -,I . :, , , . I .
i
, i 11 ,
1,
, I : 1, , I 11I I, Iit 1 I I f -, I ! !
i
I 4 1 77- .1 ,
I
I
I
,
f-
I !
. ..
- . - , , -.
.:: , ;, I
.1 : I ,
'\j . 4 ,; .I Ii . , *I
, ,,
-1-1
I I t.. -_. +,. ,I. ' -: -,. ,, I, I I, : I;- I t-..II, .L:, , T--, , 4 i-, - t-.
. I k' . I II I I . . .
, jI,
_t
_
_ _T,
, ,
- - _ !t
I
- "-;__,
,_I I - -- 4
i -1,4--j ; I , . .l.'I..
177
. ii
. t l-1
-Il I-.1
-11 IItL
- t
i
,,,,,,,,,,, I I
I . I . I :i I, . i
. - ..I,_: . I ; ,_
:I
I
.
I
I t .1
I
,
,Jtf
I
; ! ,
-,i
:
:
I'll,
;
ii
i
.
_,
i f 1 . ,,- ,
,
;,
I
; i
. I ; ,
I1,: ;I _ ti_ , 1
; I _
.
, , ,
11
, 1
!
!. ,
,
,
I
, 7:
.
i
i .
,
,
:
II f
,
i .
, . i I
,
I
I
I
1
, _.
_ _ a
I
i ,;
,
i
-44_
_ _;
I I I I i
I
I :
I . I
I
4
t!
-i ! t
, I
- 1 1, t,
i 1,
,:
, I I
"I"I iI
iI t
i
I .
i, ,i' , 4,
,i ,
-
I __,
,
,
,
t
I
1
,
,
- ,
;
I
I ll,
-
I
;
J
,I ,
i . . i
1 , ;
I .
-
t , -
, ;
:
,
i , 4,
,
Iz
Ii
;
.
I
t
j ,
I !
, j
.
I .;
III
1
I
i
,_
, .
. I
,, , , ,
t
- " , . , ;; , , . ,
-1 , 1 t f i
I I ;
i- i -1;
i _I i- I , i ,I ; : : ,
1, 61
-
.
.
,
_i
-1r Ii
t-1
I
I, ,
1 ,-N ' ,1' - 7-7
11
f
;% I
I-i I. -
I
!
1
,; ,
i
i'i T
j "i,
4 I
I I
,
-
_:._L,_
,--;
; :;v; j
i
,
-- I
I
II
. .
,
I
,I _
,
,
I
t
,
I
.
.
1
I I It :
If
. If ! I :
.
:
-*I ,
,
i
: -,
,
.
.
'I !
I
,
:I
i f 441 I
"
-H
. t
I ! I ,
I
t
1
T
i!-
I
I t
I _ _! I.:, t
Ii
. .
I :
- ,
I :
,
1
t
t
I
!
II, "
,, ;,
, ,
I
,
- _ ;
, II I 1
!
i i I ,
It: ,I.I I ,t .,iI
I
1-11I
: :
I ;
:
I
I I
.
1
..
7
-j
, ;I- I11
I I !'I
;, ,
1, iI-
tI I" if i
I t I 'I
i
,
,
1
,
I , i1,J!-.
, _ , ,
4
i, - !i,17
$ :-, I1
- I.
? i ...
4, i. ! I, -.1,
I_1
",
,L,
I
t. I ,,
7. :
I
T,
I
I
.
I I
, ,
___
I
-,
"-
i Ii
-f
; I !t
I I
-, I
.
-
11I I
It,
--
I
, ,
- i I
I
I ,
1t 1
- -,
_1
,
I
i
I-
I q; i .
II , ,
.
, .
..
1 ,
I
I ^
%J,
I
I
, ,
I
.
I
!
I
1
1-t t i
i-
I
i
1
i
I .
;
I
,
,
i
I1 ,
: ;
t
1
I ;
i
r 1 t r- , I; i 1 - ! -1- .
I
t, I
;-
I I
i ,
;
I itI ; I.
I
I
I
i
;; jI :.
-
,
.
I
;;
: t
L
I-
;
I
I:
I
-. I j I
,, : -H- 4 , , , .
. t I
t
,i ,
I
t
II I .
,
_
_
-
II I
-r-r-
4 75r I i_:
I I I I ,!, I--
,
I.. , i - : I, I
I !I I, I1 t .; ,1 I, it ,
1 -,
; ,
,V
.
:
,
. .
,
I
.
.
I
I
, , I
-,
I
I
I I
i ;
,I i
t
_
+ , rll,
ly
,
j., I- -1, t i
t f
-I
-1 1, I
11
i
j-1; 11
,- t- I
;
1
,
I
I
, ,
-
;
,
I
_
i
I I I!
, ,
,.
1
i
-
I
: ,
41
,
.
_
..:
I .
I
t
I I
I
-1 t iI tl
_
: I I f 1!
?%
_!!k.f
i
I i
-I ,
,
. ::;..
.. , ,,
-
.
! I
,
I
i :
I .
II I;
,
- *- ,
-,
.
. i
I
-
:
- -
, , +
I
I
: 1
t t. , ; 1
_
I
,i- ,I . I . 41,,
i
.
.
_
t
"I II
, ,_1IT;: -i11
i
witi,04 A
, . i ,
;. .
____ _
I .
.
I
.
II .
I t 1,
;j
"-
,
IL_I II I 1-7 I,.I
.
1
I
i _ _1-ii,
I,
,
, , .
it
i
II i
I
+_7 - I
I r- - I ; II - i 4 . - : II ,- 7. : I I I: ; I I I , , , +! : I , !
-I-I- I I t
- .
.lI "I 1
'I, I
I
i ;
1
,
1 - 1
II .
II
I
,
:
tl IIt i,- ti I
.
-1,
! I
, I
Yi ,
I
. 31
I
1_ I
;
Im
N
1
I
t , !
I
I j
,
. ! , 111[
I I I 11
, ,
I
.
i t fII
iIi ,I A 1, r 1 -
-
.
t
"I
,
': ! t I -,
,
I_4_- 1
,, : __
1. ,
I i Ii
, 1
, .
,
1t i 1
I
I
i ,. , , 1
I
.
I
1
:
16,
T
11
" , ! 1:__j , , ,
1_
*
-
!
t
I
_ -
,
- t-
tt
f
t ri
i1__t I t 4 .i I-iA I: -, ,11I!,I- .. : I iII,i ,I;-,
I , II Ii ;i t,I I , - .ot . . : ii "i ! - II_ I , ,TT .e
-,- - I'll,- ...
-1
_t f, L,_I 1t ,I: t !4 ! ;, i 1I7I 11i - I
, 1,T7 i -,LI II : I
11 I y
:
II
I
I
,_ ,
i
; , 11
I1 I -6 "I I
I
;-t
;
:
,
t
i
I
1
4
4
!
:
I, t , I , . , - 4-, - I- t 11 ,;-- I! I.
, 1Ii, , , , I , I
i Ii ,-it I 1,-1
i I "I...Ili - Ii ! ,- I I - I I
+ L
; , ,_
'I,
,
_ ., .,II !
"" I I . ,
! I
:t 1
Ii i
II
,
1 , ! I I 4- li j 1 ,
r
I 11
'I i. i- ,
,"
1
I
t 1
, ,
,
I, (
_!- , -
4
::
I
. -
11
1-f-'r !!,I --''- r-,1j
-
-
II
"
-.- ,.
-i-t-
t r :
- I
-i
1
.:L,,i
__, . ,- i
f,
t--,-,
I
,
r
"I
I I Ti
! 1Ij-i
I, I i-:
i
I
I,
I
!
,
,
I-
I- ,1! Ii i
;
;
i-
I
I
,
, t i I
I I- i[ , -1
I
,!_, II- ! , ;I . 1 .1
I
j
1iIr1
1;
,'
14 ! f
I . _ , it-
t-1
t,
I
i
tI I 4 i
I, .I
_11- , ,i. i I _:
I I 1,i
i
_
[i
;
i
I
t,
I ! tI I1 ,
I:
I !I 11
I I
i :
i
--it
.
I
t
f
I I- I ;111 i - 4. 1 !
11 ; ,
, ,:_ j- 1 iI I !
I ! i _
_
I , -L I ,
- I
- - F1 i
I i_
. _l
-I ! :1I -777
., - . 1,I -1.- I-4
.
1 1 I ,i
t
. L
L
I
I!
;i II I I
L -- L
I
, , I , P I,
i ,
. , ,i I; -
-
I ,
, .- I , I - f
I-i
I
t I II
4I
I; i I - I "i ,, I,
:-,
I i i .
L .
_
t-I ; , il.o iti
1I
.
-. I
, T I"II,
I I
LI ,I
I I
0
,, t
I
IIY
I
i
I
: ;_
i
;
,
-, , I
I
,
t ,
,
-
44
T I- ,
,
I I
-
L_, ;
444
I
I4
I f -,0 , F 1OR 1.0l ,Ir
, H 11
I
-I
i ii _i
.j,
:
I
1-1
'i
-; I-:1i l- -I1i!J, It--I -i,-_.
,l
_
-44-- -i ,! "I -- - ,- -- ,,11
iI
I
I
I
"
"
:
t
-H -__l
6 - ,I I I1,--- f
-I1- II , I ( i t ' tti ! i. AI. I
i- -... i f- I If..1-I Ilk-1.r tp i ,I 1 -----i- 11-k" I tI -,,! i! _,. ;_tI!, "I
I
I I-11 I I ,,.
I
111 f f -i i I- '
I -I r -_ I It_I 11
! -11
I- ;
! T; t I
I L -i_ t i "
-F
I
.. F t T
I
i
-
I
j
I'll :
II
i
_i
-
!,
1 11
_ , 4
:
; I,
!
I
t_ I
:
t , .. . ,( i_ W,- * 1 ."."
i, 17- It
, I
11
,rlt''
I 'l
.
!
_
!. -i- i-TI
L !- I 4 -:
'_i I I'll
I-
,
I
1
- 71
,
7 ,
i_
I
i I
,
,
!-
-
I
I
I
i - . , -, I .
.-i _1 I- ,I _ ;.
!--
I
.
:
_+ 11i !_
11_-,
!! --i 'T 5 ,- ''I
, 14
-
I Ii ,
_
it I I
, ,
II
,I I
I
_ I
. -
,
-
-I
1 4
i ,
I
!
'
I
Rt
- T,
I - , I
1
.:, I t
I I
I L
;
i I
"
1
1
.11i
.
.1
f-
I -
I
, ,- ,T
__- -, I- 1
1 ll -, .
-1. I
. , .__
I 1r- - I- i _4
4 - . T_I ,- - : -1-- I- - I, A
I I, ,
11. II. , - , - - 1:LI-;- -. - i34.- i T_
,-, -1 , , -4,11
t . r ,! _r
-, t -- I't
- I !- i 1 i , T
-; -*,-I
I .
r- 1 f 4 - !J
I :;
-
I-
__- --I __-
I
--
112.
SECTION
VI
Conslusions
"13.
;ection VI
It is clear that secondary circulation is
undesireable.The results of this experiment
indicate that it may be possible in the future
to reduce secondary circulation(or eliminate
it) by careful design of the bend configuration
and/or by careful control of upstream flow
conditions.Therefore,there exists an optimum
radius of turn for any given cross-section and
angle of turn.Uonverslythere exists an optimum angle of turn for any given cross-section and radius of turn.
The approximate theory presented by the
author( called "pipe theory" ) seems to corrolate with the experimental evidence fairly
wellconsidering the possible errors in the
measured quantities. roecause of this there
seems to be no present need for the derivation of a more exact theory. If further
experiments show this approximation to be
greatly in errorthen,of course, a new,modified theory will be necessary.
The results of the calculations show that
is approximately constant over the crosssectional area.
114.
The data presented in this thesis applies
only to bends of R/dZ 3 and AR *2 and having
similar initial velocity gradients (the data
on the 7 inch bend is not reliable). The effects
of changing AR,RN, and gradient configuration are
not predictable by the theory presented.because
of this,more work should be done with rectangular
bends containing similar gradients upstream.
!he effect of AR, RN, k/d should then be eval-.
uated and the theory modified,if necessary.
Bo attempt was made to follow the path of
any particle in order to determine the position
of the streamlines.T±he results presented in the
form of velocity maps cannot be used to indicate the positions of any streamlines.The use
of tufts,sparklers,
or smoke 'will be required.
for accurate studies of the movement of the
streamlines.
The use of screensto produce velocity
gradients, seems to be satisfactory.±t is
also noted that screens are relatively simple
to design and install. 1
Velocity measurements were sufficiently
accurate to be used in corrolating any
1. rief. 2
15.
approximate theory. in the future,however, more
careful velocity measurements should be made,
with special care being made to correct for the
blocking effect and to eliminate the yaw error
bee section 3 for a discussion of the sources
of error.
ihe static pressure data presented here is
not reliable. it may be used only as a qualitative guide.
ihe results of the seven inch ninety degree
bend are to be used for comparison purposes
only.±he errors present in measurements and in
the application of the theory are too large.
1. nef. 23
1(6.
SECTION VII
Recommendat ions
117.
bection VII
The following recommencaations are made to
persons continuing the investigation of secondary
circulation on the same apparatus( or on similar
apparatus).It is recommened that:
1.)
other radii be investigated,keeping the
same cross-section (5"1 x 10")
2.) tests be run with different radii,using
a cross-section of 4"
x 10" *This can be accom-
plished easily on the present apparatus.
3.) in future tests,readings be made at 15
degree intervals in the curved sections and at
6" intervals in the straight sections.
4.) tuftssparklers,or smokel 2 be used to
get more accurate information on streamlines.
5.) yaw meters be used to check caiculated
values of circulation (i.e.-
).
6.) a pitot tube be used which is relatively
insensitive to yaw3,
7.) a pitot-static tube be used to obtain
simultaneous total pressure and static pressure
readings
1. Ref. 7
2. Ref. 1
3. def. 23
118,
8.) provisions be made to prevent the clogging of the velocity gradient screen.
The following recommendations apply to
experiments using new apparatus.it is recommended that:
1.) different velocity gradients be investigated.
2.) larger Aspect Aatios be investigated.
3.) different methods of producing velocity
gradients be developedl2
The following recommendations apply to
theoretical considerations.It is recommended
that:
1.) pressure losses due to seconiary
circulation alone be calculated.This data
should then be corrolated with that presented
in ±eference 19.
2.) effects of viscosity on vortex decay3be experimentally determined by using the
bend extension or one similar.very little
experimental work has been done on this prublem.
buch data would be helpful in modifying the
1. Ref. 4
2. hef. 15
3. Ibid
II9.
theory to account for the effects of viscosity.
3.) the existing theory be modified if experimental results shov it to be greatly in error.
I20.
SECTION VIII
Bibliography
121.
Section VIII
iiumbers appearing in footnotes throughout
this thesis refer to the sources listed below.
1.) M.B. Zisfein , M.S. thesis entitled,
"An Investigation of vorticity and Losses in
albows." - 1948
2.) D.F. .kelbeek , iv.S. thesis entitled,
"An investigation of litot Tube aeasurements
in a Steep xransverse Total Pressure uradient."1949
3.) P.J. Valentine ,M.S. thesis entitled,
"Experimental investigation of Air Flow in
±ectangular iinety-Degree bend's."1-1949
4.) M.J. Demayo ,.. thesis entitled, "An
Experimental investigation of ;hear .dlows in
Elbows. "1950
5.) Instrument sectionAeronautical Engineering epartment, MAassachusetts institute of
Technology - inotes on 16.33 instrumentation
iaboratory, Experiment iio. 4. -1949
6.) International Uritical Tables ,Vol. 2,
section on petroleum sources edited by B.H.
Leslie, and '.0. Geniesse -1927
7.) H.C. Pavian ,"Experimental Aerodynamics" - 1940
8.) H. Kumbruoh , "Pitot->tatie wubes for
±etermining the velocity of Air -1925 (NACA
Tech. iviemo. 303 )
9.) A.D. Young and ..N. Maas, "The Behavior
of a Pitot Tube in a Transverse Total kressure
Gradient." (H and M No. 1770 ) -1937
10.) H.W. Emnons ,"Gas iynamics wables for
Air." -ciro. 1948
12.
11.) vi.R. Hawthorne ,unpublished paper entitled,
"A Preliminary "ote of shear flow in ends.,-1949
12.) W.R. hawthorne ,unpublished paper entitled,
"Shear elow in a Bent uircular .ipe : an Approx.
imate solution for a inear Initial velocity uradient." - 1950
13.) M.H. Vavra , "Steady rlow of Bon-viscous,
iWlastio rluids in Axially 6ymmetrical .,hannels." ,
Jour, of Aero. so.,oivarch 1950
14.) W.R. Dean ,"Fluid iotion in a Uurved
uhannel." Proo. ioyal 6oo. of London ,J6l 1211,928
15.) J.R. Weske ,"Experimental investigation
of the velocity oistribution iownstream of 6ingle
Duct zends." -1948 (NACA Tech. liote 1471 )
16.) J.R. Weake ,"Pressure Losses in uucts
with compound Blbows." -1943 (NACA War Aept.
W-39
17.) Stuart,varnerand aoberts ,article in
Heating,Piping and air Qonditioning ,iept. 1942
18.) L. vvirt ,"New Data on Design of Elbows
in Duct Systems." ,General Electric ieview
vol. 30 ,1927
19.) A.G. Anderson ,"Fluid Flow Diversion ; A
Summary and Bibliography of Literature.",Project
Report bo. 1 ,St. Anthony Falls ydraulic Laboratory at the University of Minnesota -1947
20.) J.R. Zwickel ,"Flow of Air and Its Dist-
ribution Through Ducts."
, Heatin
entilation,
February and miarch 1939
21.) R.D. Madison ,article in Aeating and
ventilating ,July 1936
22.)
J.R. Henry
,
"Design of kower Plant
Installations;Pressure loss Characteristics of
Duct Components." -1944 ( NACA War xiept. L-208)
23.) G. Kiel, "Total ead Meter with small
sensitivity to Yaw."-1936 j NACA Tech. Memo. 775)
1z3.
24.) K.G. Merriam and B.R. Spaulding, "Comparative Tests of kitot-6tatic Tubes."-1935 ( NACA
Tech. iiote 546
25.) Squire and winter ,"Secondary Flow"
Report #Aero 2317 )
,
(RAE
26.) H. Eichenberg ,uncompleted thesis on shear
flows in circular pipes (at MIT)
27.) E. Ower ,"Design of Pitot-Statie Tubes.",
-1925 (R and M 981 )
28.) 0. Walchner ,"Effect of Qompressibility
on Pressure neadings of a Prandtl Pitot Tube at
Subsonic Plow Velocities." -1935 (NACA Tech. hote
917
29.) A. Shukry ,"Plow around bends in an upen
Flume." , Proc. of American Society of uivil Engineers , vol. 7b, -1949
30.) C.H. Yen and 4.W. Howe ,"The Effects of
Qhannel Shape on Losses in a vanal Bend.",Civil
Engineering ,vol 12 1942
31.) J. Thompson ,"On the Origin of Winding of
±ivers in alluvial Plainswith memarks on Flow
Around bends in .ipes." ,Proc. Aoyal Soc. of London,
Vol.-- ,karch 1876
32.) F.L. Blue,.j.K. kierbertand R.L.Lancefield,
"Flow Around a River Bend investigated." ,Civil
Engineering ,Vol. 4-1934
33.) W.R. Hawthorne ,remarks given at a seminar
of the ivechanical Engineering Department of m.I.T.
in imay 1950.
124-.
APPE1DIL
A
125
APPENDIX
A
DATA -
VI ND.
RAW
15"x 90
MAR. 15, 1950
I5"x 90* 13END
MAR. 15, 1950
STA. 8
P=.0Q3 N.H,0
Pa=
3.75 IN.
8
6 7
5
60
-7S
-
I±5'4
(6 5
7b
-
s
4
46
-
431
'7
4
48Fo48
17
7
70r
15
719
8
7x
7592.5'7
S
AP
10
29
F9P86
7
2.513.0
2.
1.011.5
.5
T=81'
-
RU N.s
F
t.
CF718 9 3
*3
-
-
-
Sr
.
t0
S-9
551
175
7/29
(s
4.0
3.5
-7
7
' 5
4.5 '74
&')B.
;
73
*.s
3
8
54 5 &JO . l
r1 7457.37(. 4S
8.
574 732- S4 S
SS
.70.77.
5.o 2.5
'2.
-
I
2
-
-
'.-
64
51
' .
49
_
___
_
51
-
(05
7+
67 8~
1
'f
64
53
77
&8.;5
-
1
4
_
1.3
3
8(
171
7S
so
?3
i
- 5
4(.
S56
So
90 'I s 7~ 7 ~9
5751
84.5
o.5
M
9 -7M
71.5
L5
54
t.49479
5to
SA09
-.5(4
9 5l +4 5 1 .C 171 71-S
_S
4'f
S3
+
4
4y5
47
0
T
7
0
2.
.5
2
7 49
0FP s-L
45
AR27/0ST.
45
.5
'9
18. S
SS
7
72-1.
( 8
47
to0
SIS
-4
E- - 70 21. L
57"
57
-S $5
1
7
8i
/
5-+
77
_-S_5+
'
1
431St.
74
17 581047 70 0
TDi8
819
.8
70
87.5-
1g44eS(85.5
,2-7-4+1,
8
4
S7
7
7
5 2
222 223 72
419 22022
78/ 38
04
03
847-1
43
0-t04-&060Bs
3+.5 40 845
952.027530
5 .60
3
96
-
_
1
.5
4.0
3.0.3.
CS7
1
-7rS
157
3
.
.
+.01+.02
1
49
f-/F77.( T7C2
1
1
1
43
(C
3r2
42,S
47
40to
62
1
(---f
.3.583
5
1
146
P
5(,-7 3
8
7F 2 t0f-75 -7C77F41
3 I7
47
-,9
2
-
43
4
(0
0
l.0
47
213 214 25 21
0
8+.5.2.
0
00
8896
1.82.5i57.5815
212
02
-.
04
5.6
(.3t E isM5
1B55
2
-.
= 98
STA47Is
f5+
o4-0206
.9
s
5 7.
"9 t-43 9. 490
4
3i4F'779;t9 b'1+3
8.
423
'40
(o
59
3
3,
3
10
183. 5
5
4.5
7I1&
-o.
"7
55
8
18
46
-
3.0
3
-
L
0
0
0
0
2.5
18
1.3
0
4 829 F Go 4(/
2324
1122
65,
s
G
4 t
r.o0
3757 t7.0
t
5+t 02 70
20910 2s
SS
Z
.9
02
-.
-
-
-
43
(64
O 91 54 s 14
-
2.5
-
-.
15
7
4
-
-
-4
8
71
4
525a
1
91
45
00F79
.0
-
2a--MAR531422+75,
89-8-90671
571.01.5o2.0
'7
-
+1.
If7
425208017208
RUNNB32087202720
53
9.5
172 17
-
1S1
57
5
5.
-
-
4.5
4.0
B -9
79 7
7 5CF.=-.0-70
G'£.
r7
"7
(2.
5-7
-S
177187
-
N1
oF1140C
M
Sq 5r
3
7t-L 47S5418047
5S772.s
CF,=.
5s__4
"56
5- SD S,
f85
6
59
£9
-
05.
%
ll
(lI
-
7+
9
5
~r e 0D
IN4 HG.zT(2
42A /.
051
720.
62o0
32504
t-to
(7
45
-3 2. 7 5
6 &2
1 41 9.520
51
(1
0
53
737-2222
3.57
90 7.is 69.8
SZ.
59
-.
MARS.290STA.52
58
2 1
SD
99
--
1 > 75
7+54.7
73
79
G 5
.9
t.5 -7
82
s*9
53
-
170 171
20
8 0 .x 49
.- 57
9*3
75
-7-4-2.
207
206
205
-
q3S5+
s-
g4C
-
8 795 0 6 6 8
R4.A2 7
q7
(.'
55
r.0
67
.
-5-
-
4
3
S7 4 y .s 59
s -o'
70-0K68T(.
44.3-T
_-
_
-
RUNNO.2037204
M
-
75-4.
34
5%
58
67 4.7 (0
C
70
S6
9
8
-
-
-
-
168 169
167
7
6
5
1
-
-
-
-
-
-
-
CFa= 1.06
152 164 165 166
6.5 2
STA. 9
-3.78 EST.
CFg.905
pa
CF,= .970
CF 3-. 90 5
8
7
6
go
AP=.00
(61
S-
'75
-
I' P0-
-. 2-5-
CFL=
3
4
5
-
1
-.
-
-
--
r5
61
5.0 8
2
3
4-
9.5
9
74
-
-
-
1.0
1.0O4
134 135 136 137138 13 140 141 142 143 144 145 146 147 148 149 1501 51
RUN NC 133
47-
5
PabV3.7
CF, =.975
CF 3=.90 5
C.F2 =l.04
C.F =.975
STA, 10
AP= .00
H0
15"x
MAR. 18, 1950
2
38
j-
47
4
S+
.~iz
7
0-'
7S
£7 &&
0L
4 7
47J 5
4C
-+.524 24 0 '(1.. E
38
17
4o
5S+s7
47
(7
-
7.
1Fl
77f(.4
9 .
47 1
E
70
7)
7
7L &
7
-
A -3
126
APPENDIX
RAW
A
VI ND.
D ATA-
5'x 90*
MAR.27)
PA:= 30.10
1950
T= 810 F
F,=.970
C
RUN NO. 227
22B 229
9.5
9.0
8,5
59
(67
2
2.5_
233
234
235
236
237
238
239
240
241
242
8.0
7,5
5
7.0
6.5
6.0
5.5
5.0
4.5
3.5
3.0
2.5
2.0
S'2
50
33
47
'7
35
42
4.0
45
46
19
f9
5D
_(__
3
45.
((70
79
S.
~71.5
57 5?
59
5-/
59
55,
57
7j
-49
(.
(70,49-4.59
it
7
79
&
4I9
40
-Z
73
~73.5
5
59
. 7._71.5-D71
1-7
o -7 57
44
57.5
o7
7-
-2
.7
-79
70
37
47.
1
4
1
4
S
T=81*
249
250
251
252
253
4.0
3.5
3,0
2.5
2.0
1.5
C5 -77
(.7
(-7
gu
50
46
4-7
4 6.
%
45
5L
Z.5
1
46
__
3.14
4.s(
4_L
5. -0
5L(
4-7
_____
(.7
r1.
4.
7+
-
j
13
.
-1
254
255
256
1.0
0.5
5.5
70.5
7
L
71
75
54
7,74
-/l.4
54
73
77,5
j74 76
5D
1
i6
46
1
(.9
7
,/
71--
5_
C F
272
273
274
275
276
40
35
30
25
20
1 5
10
0.5
77
7f-
73
4f
(o
z(.9
5.5
(,3
45
57
42.
(1.5
42.5
o
53
S'7
_
S5
4.s
5
4.7=
5--7
5
-3
5s
.5
r9
76- 3 & 44 (64z2 1 5 S 39 41
47
5
19
5"
S*O
3
47
ho
53
5o
.L
i5 <u -4L -45
48
49
7
7T
46. s
49
46
5fo5
s 55
--------
569
6.0
6.5
7.0
(,4
(0.. .2
4-&
54
-74
7f
78
260
261
262
7.5
8.0
8.5
(6.5
7/
71
77
6. 4 . . 7 .74
41
(4.
579'o
"
54.5
55
.
.
''/. IL
53
I
s
4i
s
5z+
55
5259.5
2
s7
17
45
'14.5
99
44
I
4s.s
$5
+7
4L
It.5
7.
263
264
265
9.5
9.9
7
---...
j
-
.49g
<
70
L '2
-L
7/
76
-+97
4
.3
(.-
7'
40
---
-
277
278
7
77
6.0
(, C)
280
282
6.5
7.0
7.5
8.5
59
sLi5
.o
53
So
57
51
±.48
40
55o 7 5
5
7
46
51,5
57
/
g4
_
. 067 57
53
945I
11 3
So_
s?
4L
4-1
95 34
41.
40
4.
53
sI
47
281
9.0
9.5
9.7
8.0
(1_____
j
3
.5
57
4s
_-
.71
24
5f
5-?
52
_
1
285
49.5
9_ 49
_s
284
40
5T ,0f
52 SS7
55
s5
6
283
-46
7
'5
64-
_1
_
_1
60.
(9_7
6.e
it
s
.5
98
_.5"_
G,-
LFIG.
.
-
,7
-
279
45
99.s
%5
V3
5.9
259
(..3
(.-3
4. 'Li
4
5 1'.
515
is555I
5-.
44 4 .43
49
T.
1 .07
271
2
4
=-
270
2.5
2.5
*6 44c
269
(7
258
57
' -0
268
70_7o7
257
4-.
267
"7
_0
-2.
STA. 9
L P= 0 Pet= 3.89
CF,= .945
50
-8 /
77
174
'T
70
15"x 90*
MAR.28, 1950
T=80 0 F
P -= 29.57
RUN NQ 266
G3
a2 .
42.
.
7/) 7176
Pe,3.8 8
.4-
(.4
7
Lv
S3
C7
06
-) tA712-744L
4o
71
7
LP--O
F
-S157
SM
(.7
1e
.00.5
S.
5~/.
2
4.
7/
1.5
5o
51
ggrG
4
24245
STA. 8
0
L..
o
-7
- 10,5
1;
53
- (.i .6 72. 7 3 57
C3
50, So f-7 LF5s 55
15' iL
4Lm
st
iq. 5.
4(0
1947 4-7 48 49 52
S'/
70
(
51eS7
5+
3
(.o
CF.=L
248
63
.40
452 - 5
s
1--7
5 3o
5 773
5+ +18 51r
243
90 *
CF=955
5.04.5
T 4) 5-1
47
tS
L
If_'t
1 5"x
247
4
1-
_f-S_
-7Z.
Pwr~30.05
4v
4
9
MAR.27) 1950
RUN NO. 246
0.0
1.06
232
73-7
S'
C Fa=
231
(of
3
7
AP=
230
- 75
(.3
BEND
STA.
PaD=3.87
A-
127
A
APPENDIX
DATA -VIND
RAW
90*
10
15"x 90* o15"x
MAR.
28,1950
STA.
P4m=
2 9. 5 8
P:= 3.95
AP
0.0
CF=.9 32
CF,= 1.09
T=82*F
RUN NO. 286 28T 2e8
MAR,29,1950
T481*
STA.
F
PO=3.
251+ 295 254
2 9
PA,=
98
151 x900
MAR.29, 1950
T=82 0 F
.7 4
302
297 298 299 300 301
309 3O(
-10S330
56
19
5
19 195
~
1V
50
_5.3
1 0
-
4L..s
-46
-
-m
--
-
-
-
____So
-ME
2
STA.
PA -29.96
AP
CF, =z.9 35
CF,=
RUN NO.336 337 338
A P .03 .02 .02
__-=
41
9~-
44
.~
r
1.9
7
--5 E
344 345 346
347 348349350
.02
.02
.02
.02 -.05
-. 040.0
+.02
.02
.02 .02
2.0
2.5
3.03.54.0
15
E1E1.
9.,s
342 343
4.0
3.5
0
5-1
49
467
_
O.so
0.2.5~
1
20
1 +0
2-7
-
45
-
-
39
v
t.( 145.
58 TF 61
45
8
51.45
5 ss
5 '57
4.7
-IF(
44
4+7
.
L
49
b
(-1
( .2'
5
(499
-11
-7 . 4 75..5
son
-
49.5
45
g5
5/
9-5 +
-
-
+6
o50
49
to 5 0
49
46
5d
495
E3F9E53
7.E55E57.E
E E9 5 9
6 7.S
9
1 5 f 64 74 + .9 6 159
10
2
4
57
/
50
4-f
4
4,.
-
-
- -
f.
4
1
-
c9
-
-
4+9
-
~
1E
1-9
+9
1.01
E
*.5A
4
--
4 -
n9
358
359 360
.08
.08
.08
,08
-
4.
4.5
49
*?
53
EE3
60
32.
-
-
SI
-
5, it
47
EE
si -t%5.o
1
5
57
'?
57
6
97 44.. (1zE07
S8.I F?
69.5.
(.7 4 4*
4.
59
46
o
+ 44
55
s1-59
4
5 ,50
5.
4
70
tl l.
47
_S
S7+ST. 5*
53
4.577 3
6o
47'(T
M &
(1-
43
s ;-+ 5
J
i
6
f8
ME
5
r
5
49
n
3
46
9
4
-_..._
-__-_11__
-
357
-
0
2
(9 .95to 9s
66
49
--
+
-
CF 2 :
4-
4
4.
-
,
-
E
5'9
53
E0 1 E61.5E 5
E6.8 4 4 9.
56
8
'73 - ( s6 o
1+
(.
63
47
70
(.954573
44
1f
54
'.457-71 4
f-t
43
4(?
1- 4G.5
7T CL L 5 -7 4- 73 41.5 57
7z.
-8
.
to
(9
4
" 49
.
714 <.+ 5
S+.s 5 j.51 . ST
41
T
9 G5
51577~
1.4.s*
+9-5,Fl
f
(0 40
(0 D
555
513
so
1
g
IMEM~(,EE E
,E
5,4
5,1
MM
F Z.1;3 M T 1
52
I2
47
4L
2
2. 13
5;7fF2..
st' V3
5.2
*51+9 4 41 44V.
-L
49
73
PAhE
STA, 5
30.33
EQ..
2.5
-
-
o57
50
C
+
EE
.07
.03
2.0
1.5
1.0
0.5
11- 7 , -4
/ 71 5
(-3
6.5
C3
4.
c4 0
'(.45.4.s5.1j51
+
so
4-2
5T3gggg44gg
43
7
7+.5 ?.5 ?.1.572
75
/4
7+
69 9 4.
.1
1 14
0.0 +.03
.04 - 05
EEEE1ME'L51E49-S
.
49.5
50
-
49ETE
ES
51 .
73
E
-
F775EsFEb5E,9
/-40
G5
~ S~C. 2.5 5
c
72
63 5.9
s
4
7.5
7 3 65.5 '?7.5
6.
71 . ,5 ,5
5.E - ,1
5 ~o~~ g 7 , - 70. i7
45- -7
4.5~ - (.c 48
-~1n 4 4. - 56 69
s 45
SSo
41
6C
3.I
40
51+
49
60
3
55
55
4; 4 ~,51 51
5+If
5"
5SF'I0
2.s
.
S/
53
-5 5o 53 2.5
5/ IIF1.3
1.0~~ +7.6
0.19
+4
3-
46.
0
50
43
o
5EM
_-EMMEMM
-
1
355 356
352 353 354
+4~
-
-
-
-
38
.03
351
-
4S
Y7
3+
3o
47
51
+4.';
9 1
5 +5
.z
55g7Sq. 5r5 3 54 51.5-+9
9
91
s;i
9 .9
.
195
S3 5o
5 2 E E-7. 5ES ,a
4+
48
rM
31
9+
1.5
1.0
05
4. 5
.
7
.149. 5 7
5.s1.t r2
5*5
5-
EE
E
47S
+9
CF,=.91 5
CFz: 1.005
_,
340 341
3.0
+9
6
PBD r 4.36
339
2 2.5
I
5-5-
T=78*F
PAs = 30.30
CF,=.910
1.20
MAR.
3
STA.
31)1950
0
T=78 F
Pa)= 4-.20
MAR.
5
7'"it90*
31)1950
7"90*
30) 1950
Tm78 0 F
Po= 4.07
-E
-ME
-
-EEEEEMEM -EE
-
-M
E-
S!454.1 5
.5+9. s S
5
F 57
57
S"77
-
5
5
_ _
_5
-E
15"x 90*
MAR.
1
3 0 Y3SSo
27-60
s
E
5C.S. EM
1_9 41-.15 05,4
6 5 93
1 1 5/
57.5 5
So 4r+ 5(55.
60D 57.5 57
51
<60 5.7
5 /. '
51
sLI 521 54-5
50
50
53
5-7 -5-157
53
G3
52.
s3
sl -I 5.0. 5 '1 1.I 54.S 5o
57 54
T.3 154
5
si15+
513
S4SM5E.E
E1EEE5MEE
-741__s_-3K1 5-1TF1
49
1
-EE
51
5D 54
- 54,T 5
3
9
8
-
-
--
4
7
6
5
53.5
3
312.
u
4.
54
4 o50 5 . > 554.
S.4& 159 f
4(t
57
54
E+7 E4 M1 52 sfz.11s5 5E1 EE5E4.E5 5E9.5E
55 140
1+9 5s[ 52- 153
+-7
.54
3
_ o3 r 44 48 50 52.5
3
3
53
1.5
45 5-9
'93
50
G73
47 44.5.49
J. 5 53 -49
s-Ez.1 EE M5E3EME53EEE51
151. 1
45~~
1_
53
5 50
.5,jj
5-7 55.51 5 555* S
45
z s
310
3
2
1
5.5 6. G65 7.0 7.5 8.0 8.5 9.09.5 9.8 5.0 454.03.73.22.82.41.51.00.7
53.5
-3
.
5
57.547.9
515
S'9
(0
.5
(.0
5-3
5j
59o(47s3.5
5" 53 59 GO Go 53 r/ I (Z.
' 7 4/ 4/ 7 QI(3 . / SS*75354 S1 52.
51 '? 62 7o z 6o6.5
54 5G-5 51 5Zs 5/ 53
1-125
CF2
30 3D9
'307
=5.0
.s
= 0.0
CF; -. 921
1.= 2
CFa:
STA.I1
PAM= 29.75
PBD=4.0 O0P
AP =0.0
CF;=.925
233
292
233 290 291
10
S9
44
49
44
-
-
'-~~
-
-
--
-
-FIG.
A-3
128
A
APPENDIX
RAW
DATA-VIND.
x 9 0 * BEND
7"
MAR, 31.)1950
STA.
T=75 0 F
P,:
362
RUN N. 361
5.0
4.5
4.0
-30
-7
-3
- -s -3
/
-25-5
0
2.5
25
2
443
41
(.0
.z
47
(4
31 ,
.
___
9,5
3
46
s1 50
49
3.+ 92..
i
.4
L
51
C,
43
4.s-
5-
+9
$9.5
5"+
.5.
52
+
386 387
4-2
+8
41
31
49
38
49
4845
49
5 i
7.7
57
(r
15~
/0
/
3o
39 .9
5I 7 1
3
4&
(.0
(7
7
C7
4G
43
?
(z
(__6
(15
STA
4.5
A P%=
S.
io
5
o10
10
-.01
-- os
2.1
1
271
3
1
c
-. o
= 9.5
0
0
0
31
0
4C G 6
~
-
+T557.±14
s
9.0
8.5
8.0
45
7.5
+6
4
f.-x
-+/1
+
42.
o
40
39
39
T-'5 5+5553 19
59
I
5s09
2
59
2.5
5-9
3
-
4
--
5z
47
44
+.5
47
44
-
T8
5
S___W
413
0
7.0 6.5
4
+C.
+/ 41
~
5t
+X5
2.5".
5-+1
_-j
v
to
5
9.5
/P
6/
41.5
43
_
3.o
=1.5
=9.5
9.0
55
-7
72
5
57
1
0
59
5&
8.5
S+L
i
f9
7.5
S
52
4C
55'2.
50
vs
-
_s.
I
4.
.. - 'C+s £ .
49)
41
414 41 5 416 417 418
0
0
0
0
5.5
4?
5.0 4.5 4.0
47
+8
51
3.5
45
44
47
+1-8
1
SO
45
4
% S
4$45
4' 44 j4+4-++ 4
43
43
.+2.5
'1
9 I
4.5
f-+
4L
4
±
47
+Z+2
47
97,8
420421
-
-
4-03 404
6.0
5.5
5.0
4.0
41
3 41
3/
3/
35
2.9)
3123
zj
32.
4
34
31
30
+- o 3I-a35 3(.
44
41
42.
31
3+
e
7
3
42.
-
7
_
4.0 4.5
5.0
5.5
6.0
16
49
So
6.5
49
53
o
49
47
-7
44
4C
45
43
43.5
43
4+
4.
+-
-q47
9
42.
So 4
4, -72
51
'17
14
+1
47
4L
4543
4-2.
4L.5
45
45
+31
+3
43
+14-3
-1
45
45
4,71 +5
4+
4-3
44
-
I.
42.
5K43.5L43
7.0
P. 0 =4,30
C-F=r.0S5
24-30
C=9
426 427 42 8 429
7.5
-
8.0
8.5
42
1-5
L 42
42 4-z.
47
4,
j.
45
45
4
42
41.5
44'
4
43
4+
45
14
143 1
-
9.0
9.5
49
47
47
.
So
53
2
47.5
+7
-7
44-+144
430
-
5o /.
o il S3
%
4.-1
43
4
STA3
4-
(-
42
o (-
7"x90
-
-
43
____i
-7
e0
AP=0
-
-
3__1
STA.10
422 423 42442
-
32.
70
=- 29.97
I.5
__5f.
41
4
419
~
+,o1
900
x
PA,
-
5
59
.03
F2=
7 1 T7
T
i
402
3633
§
z.
1
-
STA. 8
Pa: 4.26
&P= OC
43
39
4r 71
+.8
44
-
-- 03
x 90*
7.0 6.5
45
+2
5
2.
(4
63
-
_
__
55
43
+413934
3-f
4--L 39
T= 81* F
CF; =.900
6.0
C.
4
50
'S
4(1
142.9
3_T
2. LC3
4 .2 7
0
+f
47
49
4
5C+.s 57
-4-
0
+0
4-
4-7
4,l950
PAma =30. 00
8.0
.45 C 5949
f9
+C
5.5/--55C.
3
S0
5/ 17
S 5s l 48
57- s4CC3
_
50
394 395 396 397 398 399400 401
0
_
+,
-. 0
T V1- 4.5
?31 3 s.+
4-6
5o
So
57
__/J
_
-. 08
4 9 +5
CF,=.910
393
-9/49
75+e
-s
_ _
+
o
o
4o.5
4f11F 0-5 41
4- LI+ I +2.
12.
±S'2.1151+94
9.0
7"
+4.41 39.5
53 53
3.S
_
19
ri
8.5
%4
APR.4,1950
PD=
0
8.0
7"
CF, r1.0 37
+.02
7.5
z7
APR.
T=79"F
STA. 9
RUNNl 406 407 408 409 410 411 412
.02
379A
5f
56
(og_
3
s
-
75-i153
CF; =.910
6P
379
3
49
5,4.5
CF.925
5.0
49
+
4-+
4)1950
Pa-= 29.92
T= 77 0 F
378
53
3 55
53
7" x 90*
APR.
377
5
13 ;
55
"x 90*
MAR.31
5.5
5.
3
+0
-s
.±.+.1
4
(o
49
45
376
7.0
-2o
36
-30
___3f
388 389 390 391 392
6.0
6.5
'2 1 15
'27/S
22.
33
3
6.5
375
7
7A
Pero= 4.32
7.0
-
55
9
90*
7.5
8.0
841
50
5
3 F9
I
(2-
383 384 38
4
51
54
59
57
6.0
so
49
5V
91.-
Go 6 69 0o
5.5
374
373
-30
-30
7
25 -20
1? 27
4f
47
_
CFzr .025
T4
4274538
t+8 - +5
49s 4
2.5
85
_
372
C-5,
_
51
STA.
j3 40
350
5 t
5
_$7r
So
il
£3
-9
f5-
_
___
8.5
9.0
49 47
49
50
I6 50
4
5i
I.02
F:-
371
0.5
3
9 47
5
PeS30. 2 5
RUN N. 380 381 382
2
31
1950
C0F; =.9 20
=
1.0
31
C3
43
T =75* 0 F
X=.5
1.5
to
Ft
7"1x
MAR
2.0
31
C
370
369
2.5
_5+
+
(3
4
368
367
=-.917
F,
3.0
_1
_
Co
C
366
5Z-5
_5
34
41
4.31
.9 to
_7+:
(.5
PAma= 30.30
365
-20
Vz 57
8
7
-2
z7.
_5(_
?3
9
4$
4.5
3.5
-3
S_
L
(
3.5
364
363
7
431
8 58
I5*52
_
_
St.
5/
SI
4-9
5r/4
4'/.
46
<5 L
./
+
FIG. A - 4
129
APPENDIX
A
RAWDATA-VIND.
15''
APR. 13,, 1950
T=75*F
x
180* BEND
2
STA
PA,s=29.8I
PaI = 3. 30
0.5
3.5
9.
44
8.,5
5(63
8
1.0
464
465
3.0
1.5
2.0
:
35
+1 53
s.+
303
So 5'o
55
.2-
48.5
:1.00 0CF;
,
,.9 80
466
467
4.0
4.6
4C
-7
7o
7(.9
80
S/
81
18 I.E
e62-
(.I
7
.
S- £3 3
s
(.Z
71
-71
77 7'77
4.
7.9
+
&5
2.-8.9
'6
190s-7
/.t
79
&8
5.9
5-
3.0
81
(8.
.&L
-. 01
S-78t3
7)
.5
8
-75
-7(4 -77
9S
78
3
__
4.
, 9
5~7
f
U7
S
so
5 2
t
7Z9
_
_
73
_-__
15,
_-9
APR.
14)1950
T=72 0
Pm
9.5
9.0
(.5
45
5c4
,L
67.5- 41
36
(3
2.
73.5
8.5
472
473
8.0
7.5
145/
V8
5
S~5
5-t
15
5-8
34 Z,5
s-
4
4(..
(9
5-6
63
-7
79,5-
80
-76
'7z.-S
RN N 9S-
40
63
47.5
13.5
4
-L.j
o
1-5
9
1--.
68
473
9,5 9.077 8.5
474
475476
7.0
6.5
5/41--0
s 52.13
4-9
CS
s
(o
,
42
I/
4 44P/--
8.0
7.5
40
39
,9 Cl
475
47
7.0
6.5
-71(
f
41
(ao
53
-T-9
43
+5
____t
_
43
45
4-41-
7
41
45'
44
47.53
+8
~7-5
+4.5
o45 s
44
46
4544
.44
41-
C+.9
413
+7
+39
il
6.__5 _ co
?
4.r
53
,
57
75
/
57 3 '8-o7 4-2 3j 35
52,5
S'7.5
44
7
4651
56
45
7~42 +
(
4. (9,
_
7
_+
.5_
+,of5+
4.o+,48
53
.-
_
+,1
5
__:
.7.5
_s
.5 4
1ar,
43
+.
-o1
o.o
_
57,5
_r_
(
+3
4-5
+
98
479
480
481
482
483
484
485
486
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
3- 9
5
/5
_ 50(__
(_5_
3
'-4
4z
-0
___
754
5.5
+-7 1
1
-,6
6.0
35
-
OF -Z1.010
47747
49
-
7
CF -. 990
RUN NO. 469 470 471
0.5
-
.9
+
4C
F
lB D =.3.2
I'D -7/
-
.47
79 75.
1.0
497
I80~
STA. 7
15 1 x
.5
51
44
1S
(7
.S
.
+9
496
71
4C.5
+5
4-3
7!
_
7
49
4-6 t5
9
1.5
-DS
C.4-5
50
7.+3
5'5' 5-0
3
4-95
4 Z
S 5 4-7
3
--
,53
5______
8t_
4.5)
4
8~7.75'+
4 -44,5
__70
_
-77
5 (7.5
-7(.465
-73
2.0
-7'5_
jj
7. 62
2..5
49 4
5's -7
7Z9
85FR3 6+5 8 t s1- 7D7-1
493
45-+Tq'I5-
L-----7 IT9
4,S
73 Y9
.I
492
1-
-14
-
CF: 1.02 5
f
( 5(A
f,
5(,
(A44+94347.9..5(+._40_o57
55T
g -0 5
5.
3.5
so
15
44-7A±
82
1.5 57
I.0
.()
70.5
-71
491
'6392.s
6
-~79
1 3 5
'7
84678
9
81
7
-
Z7.5
4a
7
756-79
490
7=s45 4.0
. 77
- 7 5(.Z .1.o
2.s
10
( .55
70
'7/
AP
-
'3 .,o S +.o .
5 $o
4.02.
153
-
35o
F '3
4,5
468 488 489
54f.7
55
(5
7.5
'7
CF
462 463
STA. 8
Pem=29.82
Peg 3.30
CF =.9 85
RUN NO. 461
180"
15' x
APR. t4, 1950
T=72 0 F3
6.-
It
-4o
40
(
47
7
45--
+0 54''-.010o
40'4 0
1-Z.
3-9
3 (5' 39
140
5-C~
53 / 34,394-40I4,9
4 Go f 33
4/
38
9750.-STA.8.7.8
35
3:
+1 1+ .5-2.
1
4
f8
4S 4 9
so
5___
(7
48
47
43
48
701
76.
7
54.
75
50 52. s
-
5
4.9
59
535.9
43
5
o
&45'
482
3.0
483
2.
52, 9
53
'.0
54(
57. 5
48
9,g
64e-
_4o
sr.F.G.
-
700
-7o
7+±
484
485
46
2.0
1. 5
77.0
__
0
487
o4-0
4A
0-7,
-
AP PENDI X
130
A
DATA - VIND.
RAW
15''x 1804
APR. 14, 1950
STA.9
T= - F
P 9b:::3.31
P
- 2 9.83
CF, =.980
C
= 1.04
501
498 499 500
RUN
e'6
45-5
4
5
2.0
4
2.5
68
o
58
7.0
6.5
6.0
-47
42.5
47
4~7
+1
13
56
43
52.
5/
41
49
4-7
44
8.0
AL..._5-7 55
- 6"
.
503 504 50 5 506
7.5
50
-4
8.5
9.0
9.5
-
502
0)
119
3
.5 7 4.
78
79
4.0
75
7
7
43
7 /
42.
9
,f
-72-
_355
6c5
72-..
'71.
C F,
51
519
520
521
M 9.5
-
9.0
s- -
8.5
50
r.3
57
Z+915'
8.0
13
-9,
-49
-+9
RU N
-5
1.0
1,
S3
5 'j-
(5
('+..
2.
2--
5-1
5
Go
.3
3.0-
~
i/
2.5
4(
45
17
7.5
47
7.0
e4
6.0
-
-f 7
.- (.
6.5
455
. 5
4g
+ J3 1+<.5
51
50o
55.5
5-C,
41
(t
515
3.0
2.5
1.0
5'+
4+9
2.0
5f 9
1.5
-
6'Z.3
43
+5
41
.- (
45
'4-7
49)
47.5
4-7.5
52.53
5
-
-5
(.0
1-5
45
1+*2.
524
2.
3.0
5
59
149.
+.01
66
70
(a7
re8.5
41
7oK
#-.,0,
7/
4104
0735
SS5
55
72.75?.354
_
530
$31
4.0
31
4c
3.5
4Y4
3.0
1-5,
47
JK 45.
526
527
528
529
5.5
+
i3-4
5.0
+'?
43
4.5
1-3
-f3.5
419
'1-5
+-7
48
-452S9'47.5
53
59
51L.....
r90
5
43LE
6 7.3
47
(o
&I43(at
533
532
te__0_7
17
.
5'1
515<,
533
534
53
5
2.0
1.5
0
47
44
4-9
47
47.15
-100
53
4.
(1
.
6
C44.5
E
4
5
4.0...418.
50
7 4 ~71a5 C
.
'1+4!
.o.To
4
(044
56
45
15
18
50
5,0
53
(.06.7
570.5-' - f
4-'
G.
7 I4
CS
73
-
,os
5475
47
STA. 12
PAMS= 30.0
15 x 1800
PBD= 3.36
APR. 15, 1950
T=74 *F
.OL
.03
10
PAms = 30.0 0
5 51
4.0
-7o
0
4.0__
(l1
7C)
49
(7
73
-70
5-7
4G1.
5(0
63. .1
A
________
(3
jjM
0.5
4(05
1
43
51
4.I
5{
52.
516
C.7
CF.
35
44.5
-7 %S
7
514
STA.
AP10
oL
525
4- . 4.5 6- (.0,5 (1375 1 46 71.5
7o2.
48
15,x518.0
Pav = 3.3 6
524
40
-7
j
$13
+1,5
45
18
55
3
(2. 43 4 &1
67
75 47
7 53
_
55L Iz2
48 5
9 9
_
7_
47
50
522 '523
5,(
1-7 5
5 2 .L5 55
53
2.75
4.5
45
54
.U
2
~ -
4-,-7
-
5'0
'f9. 5
45(.s
G-s5
49
4(09
2,5*
k
+7
47.5
47
47
(.3
7 15
46
48
'1 I
q
APR.215.1950
T=73 OF
-3,6-
47
81
14-5
5
.948
73
4.0
45
3._5_-_+_(_2- 53
3.5
4.5
41
4
43
_5'
3.5
5.0
512
5H1
510
4
45
42 14547
o
5-9
5.5
47
+3
13
1
43
44
507 508 509
&P-= 0.0
_
_____
RUN
538 539
537
9.0
9.5
. 6
.5
2.0
2.5
50
5o.5
5.57 5'
595(.5
5(,5s
3.
81-6~ 5
172
542'54-3
544
6.5
6.0
541
8.0
1I -
7.0
7.5
8 49.5~-7
4.50
52
53
4
2.5
52.56
55
57
S7
(f
7+ 4
_
4.5
5.75
59
5Z.
57.5.
'
43
_
F
_C
545
546
5.0
5.5
+75 ' -4.57
52 -
5.
9.59o3
2.5
5C3
27
M
5
3 (
57
5
753 C 55
3 34
il 6 (549
I 50515_
73
57_5_
5_
545
4.5
IF.5
4.0
537
+3
549
550
551
3.5
3.0
47-
2.5
2.0
4z
4z.
45
47
50
I51
47
53
4+
(.1(4.- -7 f (
456 251
49
(4.
14
is
~
4-9.5
.55.
(o il,5,
.59.
45
46.5
59
51
_
__
__
1, 4 4T
43
G7 ( 1 G5
(5.5
.70,5
CA
GI
.
_
547
2A-
~70
59
=1.07
495
5s
.5
5.7
553.5 5.5 2 (5 &747
. 5 5A-
52. 55
.3 C9 5t(.5
C5
_
____CF_.980
540
.5 9
57
53
(S5-
3.0
4-o
8.5
47
__
s
553
552
45
14
_
_
_
_
5
4
0.5
1.0
1.5
9
+-
4
o
3
64
4~4,-
l-s
5.1
41
4
45 7-
i7 5 3050
5 8A66
S
571F+9G.53
SC.03W 60t.
-4
(5.5-
585
5
.
5
53 I
C
.
4
.
7
131
A
APPENDIX
RAW
DATA-
VIND.
180*
15 " x
APR. 14,
T=730 F
CF; =.975
557
558
559
560
9.0
8.5
8.0
7.5
49
so
51
5-
57
59
51.5
53
51
55
52
54
SO.- 2
51
53
RUN NO. 556
9.5
+ '1-+
_
Z=.5
51(
1-7
/-1.
1.5 -8.s 5
/.o
2.0
49.5
So
2.1
3.o
3.5-
51
4.0
5 548
59
59
57
5'.5
Co
'I
(;.iS
565
7.0
6.5
6.0
5.5
5.0
4.5
57.5
57.5
59
5 8
C.5
C
(.3
65.5
4 0
(03
,2
.
CA.+
40
41
sC,
5 C 54
57
.s
564
2-
;5
55
55.5
55
_9
57
,CV5
67
C+
45
6,2.5
17
C7,s
CS.5
4Z
4,7
45
4/
57
45.5
44.5
-T 5.LS-2.5
s-
1
ss-
STA.2APR.
Pet -= 30.0
RUN NO. 575576 577
-.02
.02
.02
578 579
1.57 69
2.o
2.5
3.
~7+
-70
2
-77
~-0
o
7-7
70.S ;8
'1Z.s5 ?&,
70.1;
~-7b
9.o
*7o
77
582
583 584 58 5 586
-
3 2
53
9.5
-3
-7.5
i
2
61-
.5
9.0
tS 5o5,t5
-
8.0 7.5
5584-2.
5
57
C
(4
CS
4'
53
5
r.
t.3
4L
49
5l
5C
.
(.5
44
Ct.5
59
4/
43
o
6?
55
51
55.!g5 e
f5
5-
74&(.S
5$
+4
5t
5Z
53
53.5
.- +. . C 9 5 1 2.5 5f
5 -.
5.0
5771
4
5
5.5
45
57
5.5
53.5
6.0
.5-G
5~C
4257D
5 1f9
-
57
55
,S4 9.S ~71 ~79' C9
-
40
49
5
51 , Si/
5
2.5
2.0
1.5
40
5/
5!
5s
54
3
__
-7
49
5
52.
'5
-L.5
4?
5z.
49.
52
J7
1.0
0.5
47-. t-1
+27
43
44
+1+
5
1*+
43
45
4.
___-
so1-
(o___2
se
7
T2.
41
59
is
-
591
592 593
594 59 5 596
3.0
58
2.5
55
4" 2.5
(0
54
5
C1
57
52.
5/
4/
57.5
51
ro
46,5 is.1
59
5C
51.
9
47
/
+-5
47
19
47.5
42. 5
53
f
7
57.5
6..5
53
5'f
(-
'7
i
f.
5'3.
5'2.
49.5- +
52.5
53
5-.
+7
.53.5 Ti.575.3
5t
5
2.0
55
3.5
161
4.0
4.5
&(4.5 41 42.5
1/
5' 0
597
-
-
-
-
1
s/49
574
+9
S' 2 51.57
5'S
57
_5-8
573
47
-5
4
588 589 590
.0 4C
5$15
2I 2C4 %C..s ;3
5
44
9-9
15
6.5
7.0
S .3
587
572
STA.
14
P*:mg29.9 5
AP-0.0
_CF= 1.11 EST.
-
-
-
5
'73 757C
3-5
580 581
571
180*
18) 1950
T=76* F
.02,02
6 5 4
-. 5 75 174.5 (C5
-7o 8.5 7
o
( 0CT
55
_CF=.965
=
-x.
41
=3.41
CF;=.978
P
(o Z.
5"*
Pao= 34 1Pa2
A
59
4C
3.0
570
2-$57
44 6
47
15"x 180*
APR.l5,l950
T= 74*F
5
6CL
(.5
5-7
569
3.5
C.
56.5
95
56
6! 63
+
4.0
l.5
.3
5.5
55
51
568
566 567
563
az
55
5
CF 2 z=.105
562
561
(2.5
STA. 13
P 6 =30.00
1950
PD-3.41
P O=0
1.5
5755
z. s
1-9
+9
-
9
17
S'
4
3-
-.
4-
45+.
4.4
45
7
40*
4-
1.0
4
44
4fC
+.
FIG. A -7
132.
APPENDIX
B
-
-7
+,-+
rt-t
-4
-
-
+-
-7L
-El--LL
-
r4-
-T
-j
-
-+
.L
--.
- -+--- -
-
- -
- +
-
-
4-
I
_.i
12H--
-F-
- r
-l
-,
--
1
-4
-
-
-
--...
--
*
T
-e---
4~
--' - '- -.- -
{~
-
- .
-
+
-
i+
+
+---- _L
-
-
-T-
rr
1
-
-
1
T
-I
- t -
1
T
-t--t-
.. -- 4
.-
-7
-
--.-
-
i
-+
-L--t
-
K
--
-
---
-
-
-
7 *.
L-
--
-
I
-1.7-7-
-
+r
-
-
-
-
t+
r
-
-i
-7
-
-
-
-
-4-1-1
71-7
-
--
..
-T
-
--.
-
4
-
1
--
-
-
-~~j
.
t-
-
-
-
-
.
r -
-7
--
-
'T
-T
F4.B.
1-
134-.
B
APPENDIX
WORK
-- 5-
x
SHEET
180' BENDY=3
Yx=2
RUNS
4 +CI-4L6
.0
4t-4C8
g
69
7s4~
; 58
-7-7
65
5s
-7&.
168.5
53.5
(.9
(
~79
~74
5(.
5.s
1
44.
77
-i
59
+9. q
o70
(~
76
73
5-7
4-D.5
X=2.0
s9
so.s1
X= 3.0
s9
X=4.0
X=4,5
'*f61-+r8
96
57S-579
Xc1.s
*I-446
77
26.6
76
7.5
2-
X= 1.
Z.
46
Y=6
Y= 5
X=0.5
51 S-579
+CI-In
(-37.s
S--
X=1.5
RUNS
5S-S-r
55C.5
57
X-0O.5
X=
-S-573
Y=4
-78.5
80 so
575-579
44.s
ALL
FIGURES
FAGR ES
4*
______ARE
X= 2.0
82.
X=3.0
873
X-4.0
9-7(.
X=4.5
-7
*7'
78
VREF.
71.5
(IN
F.P.S.)
27
49
73
47. 5
Fl G. 8-2A
135.
FI G. B.- 2.
136.
v
4 --
I I
-
I~
44-
1 11
I-il
2
-
1
14
- 2
vY62&
~T
i t
I,
47-4
4t4-
4
t
T
L,4
4 L
I-
1
t
,i
I-v
~I i~.i I~I Ff7V777
rT]77
i
t
I~if7ti1l
11 -~---~-1-~-t
t
-~
j4j
71-
-
-
- -~~
j
i
i Ifi T
.
Ir-
-all
-t-Y
I-I-i
TT
'lit
IIT~i-'I
-
-T
~{j1
I,'
4A T-: L
I
A
I ~4L 1~
FIG. B- 3
1-37,
FIG.
B-4
13.
FIG.
B-5
139.
FIG.
B-6
140.
APPEBDIX C
?4'.
APPENDIX
CALCULATIONS
VORTICITY
15"x 90* BEND
( BOTTOM)
(i-)
O (F7:)A
.
Pos
5o
0
(60
TO
9,
(a-5
1.00
9.8
12.5
/.t0
12.Q
.60
lo.6
76
.5o
3.42
S.5
.40
+.S0
72.5
2.9
23.
1-74
.11
.35
1.15
8(0
S T7
1
2.25
so
STA
(se-
(A
0
.75
70
.50
s0
4
80
1.75
(.0
.75
60.2
8"
70
.
70
-79
2.5
50
3-3
go
40 1--7
1. 5
'2o
STA,
4
870
1.6
go
j. i
To
0
6
2 1.
2.-7
(0
2
2s.4
2.5
_
__3
__3_
210
so
A t = 7.5 "
0 3.5
54
' .5
.1i44
t
70
o .-71.&5
-
--
.
_
---_
'.
o~
/__
?S/
'2
__
9'5
12.0
FI-G.C-I
142.
APPENDIX
C
CALCULATIONS
VORTICITY
15 "x 90 * BEND
(rr.
Pos.
'LA.(pps.)
ta>
a
3.25
Go_ _
- 51.7 5
7-2.9
18.4
13F
02. o
2.2
21. 2
5-
to
70
-L--
-70
ST A,o
.
.
2.0
1. 4
f.so
T
AV'e (se Ce'
(A s),tv
o
1. -7
)~.1
4
70
.
.
.92.0
J4..o
21~0
2.8
19.9
4
10
-
So
STA.
1.9
To
To
1_
__
So
18
5o
_
0_
-60__ 1.-o
_
._
_
Ill
12.I
_
__
_
7-5
_
_
5 .
/.5
2.0
1.75
4
Z.
4.1
100
11.9
79
7010.0
ISO
T.o
STrA.
0
s~
4s
50
To
9
-- -
A- s 12
. ,144
159
teo
1158
----
------
3.75
220ao
. 15
'1-l3
4-7
/.0
01+.1
15
F1 G.C -2
143.
APPENDIX
VORTICITY
CALCULATIONS
15"x 180* BEND
BOTTOM
(-r.)
PC S.
STA.
.2.o8
'IA. (mps.) A s (I.)
80
l,/
70
1.o 4 t.
0
6.7o7
50
_7
.7s
._
1.1
so
so
5
19.
I1o5
(_
-_7_
2.0
'7
/0
^o1.0
7o
.a
(to
4.V5
-47
4
S'0
'-
o
.5
.45
-7 t
70
4-1.7
-
1.9
-_4._s
-_-_
47
3.'2
.
-. 2-2.
__
2.3Z.
.
-.
-Is,
SS
-
.4
50
70.208
4+5
.201B
0
-.
50
S TO9
1Ist
-O
*7o
S1rA
'8.3
6.o_
o
1.
~3
70
'7To 8
/Us
12.3
40
STA
/86t--s
3.07
50.o
2, 0
--
1.85
2.2
50
Luc')
--
2.1
STAZo-70
3
IP'(Pr:)
(a s)..
_
_
-02
_
-2.3
1
-0
to
f60
-14.L
-__
0--4
5o
-S
-- s
3.
_
-45
FIG. C-3
I.9
24%__x
144.
C
APPENDIX
CALCULATIONS
VORTICITY
15x 180 BEND
BOTTOM
5)
f 0s.
70
-70
-(.70
-/.zs
(.S
-
(W4
-2.0
~
-9
'.48
-
-
-
-7e
I(bee')
&.
(.g
.zo6
~*7
-
-25--
5TA.
'.5
-'.0
9 To 10
~
-0
4-.1
-.
-/.75
8"
S 81
(D0
.14(.
g55
so
-
.7o
55-
.75
-
.0
-
I.z.
83
-12
---
~-
-%
sl
5
-4.
.
- 4.68
So
(5
-
1l4o
0.5
1.5
- -. 0
so
-/06
-5.5
o.
.-
9.0
-4
.208
S 'A.
I0OTo 12.
5o
-
.75
-
/./
-.
-2.o
-14
75
55v
'At,
1
z2"
-4.9
/-15
55
-
I. /
-
2.55
~ 1'
5
40
too
-
2.
5
A
- 1. o -9.9
-
FIG. C-4
9
145.
APPENDIX
C
VORTICITY CALCULATIONS
15"xl80* BEND
BOTTOM
Po
s,.
(FT.)
A (r. p. s.)
A 6 (m.)
2.
50
(rs),
(FT
9.1
s-
~i
.Z08
-2-0,
(00
S TA.
TO
-(
9.0
-(.
-c
(0o
-
s
-
45
-
I
e) ,-')
_
I.2
+
.-
--
-0.
(.0
-.
1
(-0
-Il
/
15
-
.72
"x 9
4
1/0
-
-7 -3
-- (
.4__4_
-
7
.-
2.5
So
.0
-
_
__
_
_34.
_
-1.2
5.
-6 3
-1.2
-5.3
-89
BEND
TOP
PO S.
"Q
_
.208
I'
(& s)
A S
-_
_5_o
__5
(.2S7
STA.
G0
3
5O
_
.__5
39
2.30
17
4.os
30
0
. V;
To
5o
50
'5
S.35
5_5___._3___4._3
5
0
7
15
2.2
33
4.3555.
FIG. C - 5
_
146.
C
APPENDIX
CALCULATIONS
VORTICITY
7"x 90" BEND
TOP
I
po S.
I (Fr.)
1
11
1
r(;-Ec).,
It (.
.45
AS (iW.)
7.0
.0
(sece-')
570
3.0
. zos
4.2
3.4f.
50
3.0
55
5 2,
18~
STA.
5 70 7
1
1.8
3.0
2.15
'0
4 la
1___
___
4o
50
390
50
Z.o
. Jgg
24.5
2.
2..25
37b
[4
Z-0
19
.Z08
STA.
7
To
200
2.0
4o
.3
7A
A t--
.~7 S
.4
49
40
-------
L. 0
'#
I .o
- .11s
. 14 .
Its
5Jo
2. 5
so)
-40
I
1--J
I
FIG. C-6
'4-7.
APPENDIX
VORTICI TY
CALCULATI ONS
7 "x 90'
BEND
TOP
5.
_____
Cvr.)
U (rpr.)
As 0mw)
+o
ST A.
7A
.. g
3C~
r____
-b 5
o
f
(As
?-.-t 5 1% 12.1
90
It
A5
-(.5
45
Z.
.AIS.5
5o
.o
1.0
+o
,I
2oS
To
10
B-
L..
+0
.144
If
99
s.0
i..
,15
+0
144
2.4 30
5
_
_
_
- 1.6
4o
7.9
S
45.(
05o
Zo
sTA.
50.9
8To
9
S~o
/0-9
.47
043
_
_
-1.0
_
_
_
.%Q0
g4.0
-d
5
4-o
,7g
2,3
/02.
l'3.4
.
J.'5.
45.
50.i
.10
-35
5-3
S7.s
FIG.
C-7
_