Development of an impact-pressure probe for flow vector measurements
by Travis William Chevallier, Jr
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Travis William Chevallier, Jr (1980)
Abstract:
The development and testing of a miniature probe for determining fluid flow direction and magnitude
is considered. The new probe obtains pressure data from a pair of angled tip impact tubes at a point in a
flow. This is combined with a static pressure measurement to obtain the flow vector. Basic theory and
calibration of the probe is discussed. A numerical example of how the probe is used is included. STATEMENT OF PERMISSION TO COPY
In p r e s e n tin g t h i s th e s is in p a r t i a l
f u l f i l l m e n t o f th e requirem ents
f o r an advanced degree a t Montana S t a t e U n i v e r s i t y ,
L i b r a r y s h a ll make i t
I agree t h a t the
f r e e l y a v a i l a b l e f o r in s p e c tio n .
I f u r t h e r agree
t h a t perm ission f o r e x te n s iv e copying o f t h i s th e s is f o r s c h o l a r l y pur­
poses may be granted by my m ajor p r o fe s s o r , o r , in his absence, by the
D ire c to r o f L ib ra rie s .
It
is understood t h a t any copying o r p u b l i c a t i o n
o f t h i s th e s is f o r f i n a n c i a l
p erm ission.
S ig n a tu re
Date
^
A
gain s h a ll not be allow ed w ith o u t my w r i t t e n
DEVELOPMENT OF AN IMPACT-PRESSURE PROBE
FOR FLOW VECTOR MEASUREMENTS
by
TRAVIS WILLIAM CHEVALLIER JR.
A th e s is subm itted in p a r t i a l f u l f i l l m e n t
o f th e requirem ents f o r th e degree
of
MASTER OF SCIENCE
in
Mechanical
E n g in ee rin g
Approved:
C h a irp e rs o n , Graduate Committee
Head, M ajor Department
Graduate cOean
MONTANA STATE U NIVERSITY..
Bozeman, Montana
December, 1980
ACKNOWLEDGMENTS
The a u th o r wishes to thank Dr. A-, Demetriades and Dr. T.
f o r t h e i r help and guidance in the performance o f t h i s stu d y .
C. Reihman
S incere
thanks a re a ls o due to Tim Boucher, M argaret Roukema and Charlene Townes
f o r t h e i r help in p r e p a rin g the f i n a l
copy o f the t h e s i s .
TABLE OF CONTENTS
Page
V I T A ....................... ....................................................... ■ ...............................................
ACKNOWLEDGMENTS
......................................................................................................
LIST OF T A B L E S ...........................................................................................................
LIST OF FIGURES
. ..................................................................................................
NOMENCLATURE...........................
ABSTRACT .
.
CHAPTER I .
CHAPTER I I .
iii
v
vi
v ii
. . ........................................................ .... .........................................
INTRODUCTION . .
ii
. . ■ ............................ .... ................................
DESCRIPTION OF PREVIOUS WORK
..........................................
ix
I
3
:
10
. ....................................................
22
CHAPTER V.
EXPERIMENTAL PROCEDURE .............................................................
36
CHAPTER V I .
DISCUSSION OF RESULTS .............................................................
41
CHAPTER I I I .
THEORETICAL BACKGROUND ...................................................
CHAPTER IV .
EXPERIMENTAL APPARATUS
CHAPTER V I I .
APPENDIX I .
APPENDIX I I .
APPENDIX I I I .
SUMMARY AND CONCLUSIONS
...................................................
73
DESCRIPTION OF YAW PROBE TIP GEOMETRIES USED IN
THE INVESTIGATION.............................................................................. 75
DATA REDUCTION PROGRAMS
...................................................
DESCRIPTION OF YAW PROBE TEST TO FIND FLOW SPEED
AND DIRECTION IN AN UNKNOWN F L O W ............................
LITERATURE CITED ......................................................................................................
77
82
90
V
. LIST OF TABLES
T able
Page
6.1
Location o f n e g a tiv e
6 .2
Mean value o f S a t which data s c a t t e r reached ten degrees
of a
. . . . ? ..................................... ...........................................................
70
A 3 .1
Test re s u lts f o r
<J>W = 3 0 ° ,
OD = 0 .0 8 3 " p r o b e ..................................
88
A3.2
Test re s u lts f o r
<f% = 3 0 ° ,
OD = 0 .0 3 2 " probe . . . . . . . . .
84
A 3 .3
Test re s u lts f o r
= 30°,
OD = 0 .0 2 2 " p r o b e ..................................
85 1
q
peaks in CL . Cnr p l o t s ...................
p a . ps
53
Vl
LIST OF FIGURES
Figure
Page
2.1
T ip geometry o f a t h r e e - t u b e yaw probe ...................................................
4
2 .2
T ip geometry o f a f i v e - t u b e , th re e -d im e n s io n a l yaw probe . . .
5
2 .3
T ip geometry o f a f l a t t e n e d v e rs io n o f a t h r e e - t u b e yaw probe
7
2 .4
T ip geometry o f a tw o -tu b e yaw probe .......................................................
8
3.1
I n v i s c i d flo w around a c o rn e r o f angle 3 ..............................................
12
3 .2
I n f i n i t e wedge a t a ngle o f a t t a c k , - a ,
15
3 .3
T ip o f tw o-tube yaw probe a t angle o f a t t a c k , - a .......................
4.1
Wind tunnel schematic
4 .2
Flow speed in t e s t s e c tio n
in an i n v i s c i d flo w . .
.
18
.....................................
23
.................................................................
24
4 .3
Rotary s t r u t a s s e m b l y ............................ ' ........................................................
26
4 .4
C a l i b r a t i o n o f Transducer I
..........................................................................
29
4 .5
C a l i b r a t i o n o f Transducer 2
..........................................................................
30
4 .6
Schematic o f pressure measurement system ..............................................
31
4 .7
Pressure sensors in t e s t s e c tio n
.
33
4 .8
V a r i a t i o n in impact pressure w ith d is ta n c e from t e s t s e c tio n
w a l l ................................................................................... .... .........................................
34
Convention d e f i n i n g angle o f a t t a c k , a ..................................... ....
. .
37
Cna and Cnc w ith angle o f a t t a c k .......................
pa
ps
43
5.1
6 .1 -6 .9
V a ria tio n o f
..............................................
. . .
.
................................................... ....
. . . .
55
6 .1 9 V a r i a t i o n in 6a w it h angle o f a t t a c k .......................................................
69
A 3 .I Use o f Dpy , Dpy" 1 p l o t to determ ine a c ...................................................
87
A3.2 Use o f Cp a , Cpg p l o t to determ ine qc .......................................................
89
6 .1 0 -6 .1 8
V a ria tio n o f
and Dp^"^ w it h angle o f a t t a c k
v ii
NOMENCLATURE
Symbol
D e s c r ip tio n
Dimensionless pressure c o e f f i c i e n t
Dimensionless pressure c o e f f i c i e n t ; e q u a l s
■pc—Ph '
Dimensionless pressure c o e f f i c i e n t ‘, equals — ^—
D
py
D iffe re n tia l
Pa-Pb
Ps-Pb
pressure c o e f f i c i e n t o f a wedge;equals
D iffe re n tia l
Pa-Pb
Ps-Pb
pressure c o e f f i c i e n t o f a yaw probe-,equal S
In v e rs e o f Dpy
f,
9, z
Functions
In s id e d ia m e te r
Mach number
O utside d ia m e te r
S t a t i c pressure a t a p o in t
Pressure seen by a s u rfa c e A
Pressure seen by a s u rfa c e B
Freestream impact pressure
Freestream s t a t i c pressure
I
2
Dynamic pressure equals Pq- P s= ^pU
Radial d is ta n c e from a c o rn er
Reynolds number based on o u ts id e dia m e te r
A bsolute tem perature
viii
Symbol
D e s c r ip tio n
t*
Gas r e l a x a t i o n time
U
Freestream v e l o c i t y
U
Local v e l o c i t y
Vr
ve
Radial v e l o c i t y
T a n g e n tia l v e l o c i t y
W
Wall th ic kn es s
a
Angle o f a t t a c k
3
Angle between a s u rfa c e and the flo w d i r e c t i o n
Y
R a tio o f s p e c i f i c heats ( 1 . 4 f o r a i r )
6
Angular p o s i t i o n from a c o rn e r w a ll
P
D e nsity
Wedge angle o f probe t i p s and i n f i n i t e wedges
4>
P o te n tia l
fu n c tio n o f i n v i s c i d flow
Stream fu n c tio n o f i n y i s c i d flo w
ix
ABSTRACT
The development and t e s t i n g o f a m in ia t u r e probe f o r d e te rm ining
f l u i d flo w d i r e c t i o n and magnitude is considered.
The new probe o b ta in s
pressure data from a p a i r o f angled t i p impact tubes a t a p o in t in a
flo w .
This i s combined w ith a s t a t i c pressure measurement to o b ta in th e
flo w v e c to r .
Basic th e o r y and c a l i b r a t i o n o f the probe i s discussed.
A
numerical example o f how th e probe is used is in c lu d e d .
CHAPTER I
INTRODUCTION
Mapping o f flo w f i e l d s over bodies immersed in a moving f l u i d has
re c e iv e d a g r e a t deal o f a t t e n t i o n in r e c e n t y e a r s .
In th e realm o f
f l i g h t v e h i c l e s , s tu d ie s o f the flo w over a i r f o i l s , c o n tr o l
s u r fa c e s ,
engine i n l e t s , antennaes, and radomes have re c e iv e d in te n s e s c r u t in y as
engineers seek to reduce tu r b u le n c e , d r a g , b u f f e t i n g and in c re a s e range
and performance.
This e f f o r t has taken two t h r u s t s .
One is to n u m e r ic a lly model
i n v i s c i d flo w f i e l d s by s o lv in g the equations o f c o n t i n u i t y , momentum,
and energy over v e ry f i n e g r id s about areas o f i n t e r e s t .
This has met
w ith some sudcess but the c o m p le x itie s o f th e flo w make s i m p l i f i c a t i o n
o f th e equations d i f f i c u l t .
The r e s u l t is t h a t the i n v e s t i g a t o r f r e ­
q u e n tly o b ta in s l a r g e systems o f n o n l i n e a r , coupled equations to solve
t h a t r e q u i r e la r g e amounts o f computer tim e .
The i n c lu s i o n o f v i s c o s i t y e f f e c t s can make the problem o f o b t a i n ­
ing a n a l y t i c a l
s o lu tio n s to flo w f i e l d s
insurmountable f o r a l l
but th e
s im p le s t geom etries.
The second major t h r u s t has been e x p e r im e n t a l.
Mapping o f flo w
f i e l d s by making measurements o f v e l o c i t i e s , tem peratures and pressures
r e q u i r e sensors t h a t a re s im p le , rugged and r e l i a b l e .
The purpose o f
t h i s i n v e s t i g a t i o n was. to develop such a device f o r measuring flo w mag­
n itu d e and d i r e c t i o n in a i r ,
t h a t had th e c a p a b i l i t y o f making a p r e c is e
measurement in regions o f high v e l o c i t y g r a d ie n t s .
Such a .d e v ic e must be
2
s u i t a b l e f o r m i n i a t u r i z a t i o n to m inim ize flo w d is turba nce s and to p r e ­
vent spurious readings caused by a l a r g e sensing s u rfa c e in a high
g ra d ie n t.
The sensor s tu d ie d was a p a i r o f impact tubes fa s te n e d s id e by s id e
whose t i p s were o r i e n t e d a t d i f f e r e n t angles to the flo w .
Mach numbers in t h i s study ranged over 0 .0 2 5 - 0 . 0 6 5 , flo w speeds
from 8 -2 5 meters/secorid and the Reynolds number based on probe diam eter
had a range o f .300 - 3000.
Results o b ta in e d were c a l i b r a t i o n curves o f flo w a ngle o f a t t a c k
as a fu n c tio n o f pressure re a d in g s , and dynamic pressure as a fu n c tio n
o f pressure c o e f f i c i e n t s and angle o f a t t a c k .
CHAPTER I I
DESCRIPTION OF PREVIOUS WORK
Previous a tte m pts have been made to c o n s t r u c t yaw probes based on
impact tube bundles whose t i p s a re o r i e n t e d a t d i f f e r e n t angles to the
flo w .
H a ll
2.1
[ 1 ] c o n s tru c te d t h r e e - t u b e probes o f the type shown in Figure
in 1962» using them f o r t u r b u l e n t boundary l a y e r s t u d i e s .
He
used 0 .0 2 0 inch o u ts id e d ia m e te r tu b in g and a wedge a ngle o f 6 0 ° .
s iz e o f probe was used o n ly .
i.e .
H a ll o p erated his probe in th e n u l l mode,
th e probe was r o t a t e d u n t i l
was e q u a l .
One
the p ressure sensed by the o u te r tubes
Using a yaw probe in t h i s manner does n o t n e c e s s i t a t e making
c a l i b r a t i o n curves o f flo w a ngle o f a t t a c k versus p ressure re a d in g s .
It
does r e q u i r e , however, t h a t th e aerodynamic c e n t e r o f th e probe be found.
In 1969, Dudzinski and Krause [ 2 ] developec probes based on the
same t i p
geometry as H a l l ' s , but th e y c a l i b r a t e d t h e i r probes so t h a t
th e y could be o p erated in a n o n - r o t a r y , f i x e d p o s i t i o n .
probes have s e v e ra l advantages over r o t a t i n g ones.
F ix e d -p o s itio n
They a r e les s com­
p le x to f a b r i c a t e and use,, th e y r e q u i r e les s space, and s in c e they, do
not have to be r o t a t e d to a n u ll p o s i t i o n t h e i r response i s much q u ic k e r .
Also developed and c a l i b r a t e d by Dudzinski and Krause were t h r e e dim ensional probes c o n s is tin g o f f i v e tubes as in F ig u re
2 .2 .
T h e ir
tw o-dim ensional probes were c o n s tru c te d o f 0 .0 6 4 inch OD tu b in g and
t h e i r th re e -d im e n s io n a l probes were made o f 0 .0 3 2 inch OD tu b in g .
4
I
F ig u re
2 .1
T ip
g e o m e t r y o f a t h r e e - t u b e ya w p r o b e
Fig u re 2 . 2 T ip geometry o f a f i v e - t u b e ,
yaw probe
th re e dimensional
In 1975, S p a id , H u r le y , and Heilman [ 3 ] designed m in ia t u r e probes
as in Figure
2 .3
f o r tr a n s o n ic boundary l a y e r mapping.
T h e i r probe is
a f l a t t e n e d v e rs io n o f th e standard t h r e e - t u b e arrangem ent, shown in
Fig u re
2.1 .
The probe was manufactured from 0 .0 1 0 inch OD tu b in g t h a t
was f l a t t e n e d to a h e ig h t o f 0 .0 0 6 in c h .
dimensions o f 0 .0 4 0 inch x 0 .0 0 6 in c h .
The f i n i s h e d probe had t i p
The wedge angle o f th e beveled
faces was 4 5 ° .
The Spaid group c a l i b r a t e d t h e i r probe so t h a t i t could be used in
a f i x e d mode, i . e .
once th e probe was placed in an area o f i n t e r e s t i t
was moved in a t r a n s l a t i o n a l manner o n ly .
The probes mentioned b e fo re were a l l
o f th e t h r e e - t u b e ty p e .
Two
beveled tubes a re used to o b ta in flo w d i r e c t i o n and an unbeveled c e n t e r
tube is used to o b ta in impact p re s s u re .
From these th r e e measurements
th e flo w speed and d i r e c t i o n can be deduced.
Two-tube yaw probes, having no c e n t e r tu b e , can be used to d e t e r ­
mine flo w d i r e c t i o n ,
(see Fig u re 2 . 4 ) .
According to Roberson and Crowe
[ 4 ] , such devices a re always used in th e n u l l mode.
T h is n e c e s s ita te s
mounting th e probes so they r o t a t e and as mentioned p r e v i o u s l y , t h i s has
some dis adv a nta ges .
A ls o , no flo w speed i s obtained from th e probe.
The development o f a f i x e d yaw probe c o n s is tin g o f two tubes o n ly
would p ro v id e some advantages to th e e x p e rim e n te r.
F i r s t , since only
two tubes a re used, the probe would be e a s i e r and le s s expensive to
fa b ric a te .
Secondly, i f th e probe t i p c r o s s -s e c t io n a l area must be
7
Upward fa c in g
F ig u re
2 .3
T ip
g e o m e try o f a f l a t t e n e d
th re e -tu b e
yaw p ro b e
v e rs io n
of a
JL
F ig u re
2 .4
T ip
g e o m e t r y o f a t w o - t u b e ya w p r o b e
9
l i m i t e d in s i z e , as in a boundary l a y e r probe, two tubes w i l l
s m a lle r c r o s s -s e c t io n than th r e e tubes o f th e same s i z e .
have a
S m a lle r OD
tu b in g could be used in th e t h r e e - t u b e probe to decrease i t s
c ro ss -
s e c t i o n , but f o r small OD1s u n d e s ira b le low Reynolds number e f f e c t s
begin t o ta k e
e ffe c t,
(Re^ <_ 2 0 0 ) .
Thus f o r a tw o-tube probe and a
t h r e e - t u b e probe having th e same c r o s s - s e c t io n , the t h r e e - t u b e probe is
more l i k e l y to e x p e rie n c e d e l e t e r i o u s v i s c o s i t y e f f e c t s due to lower
Reynolds numbers.
The disadvantage to a tw o-tube probe i s t h a t o n ly two pressure
measurements a re o b ta in e d .
These are inadequate by themselves i f the
e x p e rim e n te r wishes to determ ine both speed and d i r e c t i o n o f th e flo w .
T h e r e f o r e , an a d d i t i o n a l
pressure measurement i s r e q u ir e d .
pressure to measure w ith o u t a d d i t i o n a l
This can be done w ith a w a ll
The e a s i e s t
probes is the s t a t i c p ressure.
s t a t i c pressure ta p .
The d e te r m in a tio n o f
th e flo w v e c to r w ith two yaw probe pressures and the s t a t i c pressure is
d e t a i l e d in the f o l l o w i n g c h a p te rs .
CHAPTER I I I
THEORETICAL BACKGROUND
For an id e a l
flu id
in s te a d y , i r r o t a t i o n a l
flo w th e r a t i o o f s ta g ­
n a tio n to s t a t i c pressure i s expressed by th e i s e n t r o p ic flo w e q u a t io n :
^
= (I +
H2Jt ' 1
(3.1)
S
Where Pg i s the s t a t i c p re s s u re , Pq th e s ta g n a tio n p re s s u re , y is
th e r a t i o o f s p e c i f i c heats and. M i s th e Mach number.
numbers encountered in t h i s i n v e s t i g a t i o n
For th e low Mach
(M ^ 0 . 0 6 5 ) , Equation ( 3 . 1 )
can be approximated by:
= ( I + J M2 )
(3 .2 )
s
which by a p p ly in g th e e quation o f s t a t e can be r e w r i t t e n as:
(3 .3 )
Po - i ^
where U i s fr e e s tr e a m v e l o c i t y and p i s th e fr e e s tr e a m d e n s i t y .
tio n
Equa­
( 3 . 3 ) i s commonly known as th e inc om pres s ible B e r n o u lli e q u a tio n .
In th e case o f a r e a l
f l u i d Equation ( 3 . 3 ) is m o d ifie d t o :
(3 .4 )
where C i s a c o n s ta n t t h a t is a p p ro x im a te ly u n i t y except f o r Reynolds
numbers based on d ia m e te r o f le s s than 200 [ 5 ] .
In t h i s i n v e s t i g a t i o n
Tl
th e range was 300 < Re^ < 3000.
Thus f o r s te a d y , i r r o t a t i o n a l
flo w o f low Mach number and moderate
Reynolds number, Equation ( 3 . 3 ) i s t o t a l l y adequate f o r d e s c r ib in g the
flo w .
This has been r e p e a te d ly v e r i f i e d
[5 ],
When a s o l i d body is immersed in a moving f l u i d , th e o n ly p o in ts
t h a t e xp e rie n c e th e s ta g n a tio n p ressure a re the ones a t which th e flo w
i s brought i s e n t r o p i c a l l y to r e s t .
Al I o th e r s u rfa c e p o in ts experience
a pressure low er than Pq .
An i n v i s c i d flo w a n a ly s is o f flo w around a c orner i l l u s t r a t e s t h i s
v a r i a t i o n in pressure (see Figure 3 . 1 ) .
As S t r e e t e r and W y lie [ 6 ] show,
th e stream fu n c tio n f o r c o rn er flo w i s :
u
ip = - U r 6 s in y
(3 .5 )
where ip is th e stream f u n c t i o n , U i s the f r e e stream v e l o c i t y , r is the
ra d ia l
d is ta n c e from th e c o r n e r , 6 i s th e angle o f the c o r n e r , and 6 is
th e angle between the upstream c o rn e r w a ll and the p o in t o f i n t e r e s t .
The p o t e n t i a l
f u n c t i o n , <(>,. f o r c o rn e r flo w i s :
TT
(J) = - U r 6 cos
(3 .6 )
The v e l o c i t y components in c y l i n d r i c a l
coord in ates a r e d e fin e d as
fo llo w s :
r
= M = I M .
3r
r 96 ’
v
= 1 M = _ M
9 . r 36'
3r
12
Figure 3.1
I n v i s c i d flo w around a c o rn er o f angle B
13
S u b s t i t u t i n g Equations ( 3 . 5 ) and ( 3 . 6 )
i n t o th e v e l o c i t y expressions
y ie ld s :
TT-6
(3 .7 )
(3 .8 )
For 0=0, Equations ( 3 . 7 ) and ( 3 . 8 ) reduce t o :
I I I
(3 .9 )
(3 .1 0 )
In th e case o f B =
tt,
( f lo w along a f l a t w a l l ) . Equation ( 3 . 9 ) reduces
to :
Vr = -U
'
(3 .1 1 )
which is th e expression f o r uniform flo w in the - x d i r e c t i o n .
For 6 = 8 ,
Equations ( 3 . 7 ) and ( 3 . 8 ) reduce t o :
T T -8
V
r
M
8
rT e
V 0
(3 .1 2 )
(3 .1 3 )
V a le n tin e [ 7 ] s t a t e s t h a t the pressure c o e f f i c i e n t f o r any p o in t in a
flo w could be s ta te d as:
C
P
(3 .1 4 )
14
where P is the l o c a l pressure and u i s th e lo c a l v e l o c i t y .
s t a t e s t h a t Equation ( 3 . 1 4 )
He also
is independent o f the e x t e n t o f th e f lo w ,
and independent o f the magnitudes o f th e v e l o c i t i e s and p r e s s u re s , w i t h ­
in th e low Mach number app ro xim a tio n .
In flo w around a c o r n e r , f o r 9 = B, U = V
r
sin ce V0 = 0.
b
Thus
Equation ( 3 . 1 4 ) can be r e s t a t e d as:
P-Ps
V
2
Cp ' r ^ 2 ' 1 ' fT 1
(3.14b)
2 PU
S u b s t i t u t i n g Equation ( 3 . 1 2 )
in to
( 3 .1 4 b ) y i e l d s :
Z(TT-B)
3
‘
(3 .1 5 )
' I PU2
T h e re fo re th e pressure c o e f f i c i e n t on the corner w a ll a t 9 = B i s a
fu n c tio n o f . B and r o n ly .
This argument can be extended to th e case o f an i n f i n i t e wedge a t
an a ngle o f a t t a c k as f o l l o w s .
The s tr e a m lin e t|;=0 in F ig u re
re p re s e n t th e w a ll a t 9=0 in the pre vious case.
3 .2
can
The angles Bg and Bj5
a r e given by th e exp re s sio n s :
where
6a = *K + f - a
(3.16)
6b = *>« + £ + “
(3 .1 7 )
i s the wedge a ngle and a is th e angle o f a t t a c k o f th e wedge
15
Surface B
Wedge
c e n te rp la n e
Surface A _ S
Figure 3 .2
I n f i n i t e wedge a t angle o f a t t a c k , -a , in an i n v i s c i d flo w
16
c e n t e r l i n e to the flo w d i r e c t i o n .
By Equation ( 3 . 1 5 ) th e pressure c o e f­
f i c i e n t s on surfa ce s A and B become:
2 (Ti--Ba)
2
TT
Ip u 2
B=
(3 .1 8 a )
’
Z(TT-Bb )
Pb- P s
Ip U 2
TT2
1
(3 .1 9 a )
" 2U2
The expressions on th e r i g h t sides o f Equations ( 3 . 1 8 a ) and ( 3 . 1 9 a )
can be expressed as Z ( r , 8 ) thus:
V!S2
= z (r,3 a)
(3 .1 8 b )
2 = z ( r , B b)
(3 .1 9 b )
Ip U
h r!S
y PU
R e w ritin g Equations ( 3 . 1 8 b ) and (3 .1 9 b )
pa
1
V
(3 . 1 8 c )
Ps + ^pU2 [ z ( r $3a ) ]
Ps + ]p U 2 [ z ( r , B b) ]
.
(3 .1 9 c )
S u b tr a c t in g ( 3 . 1 9 c ) from ( 3 . 1 8 c ) y i e l d s :
Pb = Y pU2 [ z ( r , B a ) - z ( r , B b) ]
(3 .2 0 )
17
S u b tr a c t in g ( 3 . 1 9 c ) from Pgl the s t a t i c p re s s u re , y i e l d s :
Ps - Pb = f pU2[ z ( r , 3 b) ]
(3.21)
D iv id in g Equation ( 3 . 2 0 ) by ( 3 . 2 1 ) y i e l d s th e wedge d i f f e r e n t i a l
p re s ­
sure c o e f f i c i e n t (Dpw) .
Pa - P b _ Z ( r -Bb> - z ( r , y
PW " Ps-Pb
(3 .2 2 a )
z ( r , 3 b)
Z(r ,S a)
'
(3 .2 2 b )
1 "
2(it-3a)
2
I - ^
a -
r
^
Ba
Dpw " 1
_2
T h e r e f o r e , th e d i f f e r e n t i a l
f u n c tio n o f r ,
(3 .2 2 c )
2 ( Tr- 3b )
■ ■-a - - —
pressure c o e f f i c i e n t on th e wedge (Dpw) i s a
3a > and 3b o n ly .
Since 3a and 3b are both fu n c tio n s o f
a and <j)w, as in Equations ( 3 . 1 6 ) and ( 3 . 1 7 ) , the d i f f e r e n t i a l
pressure
c o e f f i c i e n t can be expressed as:
V
■ f (0 ' r -+w>
. (3 - 23)
The t i p o f a yaw probe (see Fig u re 3 . 3 )
aspects to an i n f i n i t e wedge.
is very s i m i l a r in some
The m ajor d i f f e r e n c e s a re two.
F irs t,
th e pressure faces o f th e yaw probe do not extend i n f i n i t e l y as the
\
18
Fig u re 3 .3
T ip o f tw o-tube yaw probe a t angle o f a t t a c k , - a
19
wedge's do.
Second, the yaw probe t i p
i s th r e e dimensional whereas the
wedge was considered i n f i n i t e normal to th e plane o f F ig u re
3 .2 .
Both f a c t o r s suggest t h a t ( 3 . 2 2 c ) cannot be used to express the
d iffe re n tia l
pressure c o e f f i c i e n t o f a yaw probe (Dpy) .
However, they
do not r u l e out th e p o s s i b i l i t y t h a t a s i m i l a r r e l a t i o n s h i p e x i s t s such
th a t:
Dpy = g i (a,r,(j,w)
where Dpy i s the d i f f e r e n t i a l
(3.24)
pressure c o e f f i c i e n t o f a yaw probe, and
g i s some fu n c tio n o th e r than f .
An example o f an analogous r e l a t i o n s h i p between a th re e -d im e n s io n a l
body and i t s
two-dim ensional model i s t h a t o f a sphere and a c y l i n d e r .
A c y l i n d e r in c ro s s flo w can be considered to be a two-dim ensional model
o f a sphere, as the tw o-dim ensional wedge was used as a model o f a yaw
probe t i p .
The pressure c o e f f i c i e n t d i s t r i b u t i o n on a c y l i n d e r i s given by
V a l e n t i n e [ 7 ] t o be:
p - p s
Cn = T-------- 1 = 1 - 4
P
sin^Q
(3 .2 5 )
£ P r
Cp o f th e model i s a f u n c tio n o f one v a r i a b l e , 6 , thus one m ight expect
th e Cp o f th e sphere to be a d i f f e r e n t fu n c tio n o f the same v a r i a b l e .
The Cp o f th e sphere i s such a fu n c tio n and i t
W y lie [ 6 ] to be:
is given by S t r e e t e r and
f
C
P
Since the d i f f e r e n t i a l
s in 26
(3 .2 6 )
pressure d i s t r i b u t i o n o f the wedge is not. a
simple fu n c tio n o f a s in g le v a r i a b l e , as th e c y l i n d e r ’ s i s , th e drawing
o f p a r a l l e l s between th e c y lin d e r - s p h e r e analogy and th e wedge-probe t i p
analogy cannot be considered a rig o ro u s p r o o f t h a t
e x is ts .
= g ^ (r,^ ,a )
However, th e p r o b a b i l i t y t h a t such a w e d g e -to -p ro b e - t i p analogy
does occur is high enough to w a rra n t f u r t h e r i n v e s t i g a t i o n .
The d is ta n c e r in Equation ( 3 . 2 4 ) , a t which the pressure is sensed,
i s not a s in g le p o in t f o r th e case o f a probe t i p .
Rather i t
is a
nominal v a lu e sin ce the pressure i s sensed over the area o f an e l l i p s e ,
not a t a p o i n t .
There does e x i s t a v a l u e , r ,
a t which th e pressure is
equal to th e average pressure over the e l l i p t i c a l
opening.
i s a. fu n c tio n o f the probe tu b in g OD and I D , <b,, and a .
I t s value
Thus r can be
s ta te d as:
F = r(0D,ID,<j>w, a )
(3 .2 7 )
S u b s t i t u t i n g th e fu n c tio n s governing r in pla ce o f r in Equation
(3 .2 4 ) y ie ld s :
Dpy = Q2 ( M wsO DJD)
(3 .2 8 )
The purpose o f th e e xpe rim ental p o r t i o n o f t h i s study was to d e t e r ­
mine i f fu n c tio n s such as Equation ( 3 . 2 8 ) e x i s t s f o r yaw probes and to
21
determ ine t h e i r form.
This was done by measuring Pg-Pj3 and Pg Pb fo r
d i f f e r e n t values o f OD, I D ,
, and a in e xperim ental t e s t s .
CHAPTER I V
EXPERIMENTAL APPARATUS
A.
The MindtunneI
The experim ent was conducted in a small subsonic w indtunnel manu­
fa c t u r e d by Aerolab Supply Co.
The tunnel
is a sim ple v e n t u r i w ith an
e l e c t r i c a l l y d r iv e n fan p r o v id in g s u c tio n to d r i v e th e a i r flo w .
The
t e s t s e c tio n i s c i r c u l a r in c r o s s -s e c t io n w it h a 508 cm (12 in c h )
d ia m e te r .
The t e s t s e c tio n w a ll
is c l e a r p l e x i g l a s a llo w in g the e x p e r i ­
menter to observe probes w ith the tunnel
f o u r t h inch t h i c k and i t
running.
The p l e x i g l a s is one-
is e a s i l y d r i l l e d and tapped f o r mounting ex­
pe rim e n ta l equipment.
The tunnel
f o r e - s e c t i o n , which includes the i n t a k e , n o z z l e , and
t e s t s e c t i o n , is on a s l i d i n g t r a c k .
T h is a llo w s opening the tunnel
between the t e s t s e c tio n and the d i f f u s e r . (see Figure 4 . 1 ) .
speed o f th e tunnel
The flo w
is c o n t r o l l e d by changing the w id th o f t h i s opening,
which was c a l l e d th e shroud gap, th e w id e r th e opening, th e slower the
a i r f l o w . ( s e e F ig u re 4 . 2 ) .
The tunnel has a maximum flo w speed o f a p p ro x im a te ly 22 meters per
second, which a t th e tem peratures encountered in the t e s t s e c tio n (1 7 °
to 29°C) meant t h a t the maximum Mach number a t t a i n a b l e was. about 0 .0 6 5 .
The t e s t s e c tio n te m p e ratu re was assumed to be equal to ambient.
T h is in tro d u c e s l i t t l e
a p p r e c ia b le e r r o r because t h e . r a t i o o f t e s t sec­
t i o n te m p e ratu re to am bie n t, a ccording to Liepman and Roshko [ 8 ] , is
expressed by:
D iffu s e r
—
122
Test
s e c tio n
------
Nozzl e _
—T '
I
2 7 .9 - ^ 9 .5 ^
Shroud gap-
F ig u re 4.1
Windtunnel
schematic (Dimensions in cm)
2 7 .9
In ta k e
7.8>
Flow speed (m /s ec )
24
_____________i_____________i_____________ i-----------------6
5
1
0
15
Shroud gap (cm)
F ig u re 4 .2
Flow speed in t e s t s e c tio n
25
T
= (1 +
M2 ) - l
(4.1)
O
which f o r M=O.065 is 0 .9 9 9 1 6 , o r n e a rly u n ity .
B.
F a b r ic a tio n o f Probes
The. yaw probes were m anufactured from s ta in le s s s te e l
hypodermic
tu b in g o f o u ts id e diam eters ran g in g from 0 .0 8 3 inch to 0 .0 2 2 in c h .
In te rm e d ia te s iz e s were used to te le s c o p e th e s m a lle r s iz e s up to more
manageable d ia m e te rs .
To c o n s tru c t each p ro b e , two tubes o f equal d ia m e te r were s i l v e r
s o ld ere d s id e by s id e .
They were then p laced in a j i g
beveled to th e d e s ire d wedge a n g le , <j>
and th e t ip s were
(see F ig u re 2 . 4 ) .
The tubes were then s i l v e r s o ld ere d to th e edge o f 3 /4 inch x 3 /4
inch x 1 /4 inch brass p la t e s .
The brass p la te s were d r i l l e d
and notched
so t h a t th e y could be r i g i d l y fix e d to th e r o ta r y s t r u t assembly (see
F ig u re
4 .3 ).
The fin is h e d yaw probes could be r a p id ly in te rch a n g e d by
removing a s e t screw t h a t fa s te n e d th e brass p la te to th e s t r u t .
o f th e brass p la te s was stamped w ith th e probe 00 and (J)w.
Each
A to ta l o f
n in e probes were c o n s tru c te d and u s ed ,h avin g th re e d i f f e r e n t diam eters
w ith th re e wedge angles ((Jjw) f o r each d ia m e te r (see Appendix I ) .
C.
R otary S t r u t Assembly
The r o ta r y s t r u t assembly c o n s is te d o f th re e main p ie c e s ;
s t r u t , th e s tre a m lin e d c ro s s p ie c e , and th e r o t a t in g c y l i n d r i c a l
the
base
26
S tre a m !ined
s tru t
— Crosspiece
R otary
c y lin d r ic a l
post
R o ta tio n
handle '
A d ju s ta b le
needle -
P r o tr a c to r
F ig u re 4 .3
R otary s t u t assembly
(1 /2 s c a le )
27
(see F ig u re 4 . 3 ) .
The s t r u t , to which th e probes were a tta c h e d , is a
3 /4 inch x 1 /4 inch s ta in le s s s te e l
bar t h a t is fo u r inches lo n g .
It
is
p o lis h e d and rounded to m inim ize d is tu rb a n c e s o f the flo w f i e l d . (see
S e c tio n 4 D ).
It
is notched and d r i l l e d a t th e upper end to f i t
brass p la te a f f ix e d to each probe,
the
th e base o f the s t r u t is a tta c h e d to
a th re e inch aluminum c ro s s p ie c e th a t is s tre a m lin e d .
The upstream end o f th e c ro ss p ie ce is a tta c h e d to a r o t a t in g c y l i n ­
d r ic a l p o s t.
fram e.
The c y l i n d r i c a l post is a tta c h e d to th e w indtunnel support
The probes and th e s t r u t assembly were designed so t h a t the
probe t i p
remained fix e d a t midstream and th e a x is o f r o t a t io n passed
through th e t i p
and the c e n t e r lin e o f th e c y lin d r ic a l p o s t.
At the base
o f th e c y l i n d r i c a l post an a d ju s ta b le n eedle is a tta c h e d t h a t r e g is te r s
th e probe angle o f a tta c k on a p r o t r a c t o r .
The p r o tr a c to r is mounted on
th e w indtunnel support fra m e .
D.
P ressure Measuring System
P ressure measurements were made w ith two d i f f e r e n t i a l
tra n s d u ce rs o f th e v a r ia b le - r e lu c ta n c e diaphragm ty p e .
pressure
In t h is ty p e ,
th e p ressu re diaphragm fle x e s in response to pressure d iffe r e n c e s across
i t ; moving m agnetic cores th e re b y in d u c in g a v o lta g e .
The v o lta g e is
a m p lifie d and f i l t e r e d
The o u tp u t o f th e
by th e tra n s d u c e r c o n d itio n e r .
c o n d itio n e r is a D.C. v o lta g e which can be read w ith an o s c illo s c o p e , an
X-Y r e c o rd e r, o r a v o lta g e m eter.
28
The pressure tra n s d u ce rs were c a lib r a t e d using a pressure measuring
panel
c o n s is tin g o f a m a n ifo ld whose pressu re can be measured w ith
Bourdon tube a b s o lu te gages.
An atm ospheric ven t and a vacuum pump
a llo w e d th e pressure to be v a rie d from atm ospheric to 0.1 mmHg.
D i f f e r e n t i a l pressures were measured o v er a range o f 0 .1 0 to 4 .0 0
mmHg.
T h is corresponded to th e range o f d i f f e r e n t i a l
pressures encoun- .
te re d in p re lim in a r y probe te s ts w ith a w a te r manometer.
The c a l i b r a ­
t io n d a ta was p lo tte d and then c u r v e - f it t e d w ith the use o f a le a s t
squares com putational scheme [ 9 ]
E x c e lle n t l i n e a r f i t s
(see program CAL
in Appendix l l ) .
were o b ta in e d f o r both tran sd u cers w ith regards to
th e slo p e o f th e c a lib r a t io n
lin e
(see F igures 4 .4 and 4 . 5 ) .
D r ift o f
th e zero p o in t occurred f o r both tra n s d u ce rs making p e r io d ic re z e ro in g
necessary.
Rechecks o f th e c a lib r a t io n s showed th a t l i t t l e
a p p re c ia b le
slo pe change had o c cu rre d .
The p ressu re m easuring system was assembled as shown in F ig u re
(4 .6 ).
Both th e d i g i t a l
v o ltm e te r and th e X-Y re c o rd e r could be used
f o r d e te rm in in g tra n s d u c e r o u tp u t.
T y p ic a lly , the X-Y re c o rd e r was used
to d eterm ine when steady s ta te had been reached and th e a b s o lu te v o lta g e
v alu e was then read o f f th e d i g i t a l v o ltm e te r.
The d i g i t a l v o ltm e te r had a tim e a ve ra g in g fe a tu r e t h a t could be
s e t to 0 . 1 , 1 .0 , 1 0 , o r 100 seconds.
Transducer response was n e a rly
in s tan tan eo u s to any change in p re s s u re ; thus i t was necessary to tim e
average th e s ig n a l o u tp u t o f th e c o n d itio n e r over 10 seconds to o b ta in
Pressure (mm o f Hg)
P= .002185331 x (mV) - .023961067
o r ig in a l
c a lib r a t io n
recheck
1200
1400
Transdu cer o u tp u t (mV)
F igu re
4.4
C a lib ra tio n
o f Transducer
I.
p = . OOl63497 x (mV) -
.008392692
& - o r ig in a l
o - recheck
-1 6 0 0
c a lib r a t io n
-1200
Transducer o u tp u t (mV)
F igure
4.5
C a lib ra tio n
of
T r a n s d u c e r 2.
31
F ig u re 4 .6
Schematic o f pressure measurement system
_
..
IR I
P
I
Yaw probe
pressure B
Yaw probe
pressure A
II —
S t a t ic
pressure
L ----- ----- ,--------
Impact
p ressure
pTt
p
Valve
L*
P_ or P
------- IR 2
1-4-
p -p
L u. -S . ->
(mV)
T ra n sd u ce r
v
c o n d itio n in g
c irc u it
(mV)
V
I
D igi ta l
voltm e te r
X-Y
re c o rd e r
v
v
(mV)
’t
:
32
a s ta b le re a d in g .
This tim e averaged s ig n a l was tra c e d by th e X-Y r e ­
c o rd e r to d eterm ine stead y s t a t e .
T ransducer I was used f o r measuring th e pressure d iffe r e n c e between
im pact and s t a t i c p re s s u re .
Transducer 2 was used to measure th e p re s ­
sure d iffe r e n c e between Pg and
fitte d
o r Pg and P^.
The tra n s d u c e rs were
to th e pressure sensors w ith c le a r p la s t ic tu b in g .
The j o i n t s
were p ressure te s te d and le a k y j o i n t s were sealed w ith T e flo n ta p e .
The
p ressu re sensors used were an impact tu b e , a w all-m ounted s t a t i c p re s ­
sure ta p , and th e tw o -tu b e yaw p ro b e .(s e e F ig u re 4 . 7 ) .
Freestream im pact pressure was measured w ith the im pact tu b e .
im pact p ressure v a rie d o ver the c ro s s -s e c tio n as in F ig u re
The
4 .8 .
In te r fe r e n c e e f f e c t s w ith yaw probes became measurable when th e impact
tube was w ith in I c e n tim e te r l a t e r a l l y o f a yaw probe.
Hence, th e im­
p act tube was placed in a re g io n where n e ith e r in te r fe r e n c e w ith th e yaw
probe o r th e t e s t s e c tio n w a ll was m easurable.
A ty p e o f yaw probe to im pact tube in te r fe r e n c e t h a t d id occur was
a drop in dynamic pressure a t h ig h e r yaw probe angles o f a t t a c k .
This
was caused by th e s tre a m lin e d s t r u t being a t an angle o f a tta c k where i t
began to s t a l l
s e c tio n .
and cause s e rio u s in te r fe r e n c e w ith th e flo w in the t e s t
For angles o f a tta c k up to ±35° th e drop in dynamic pressure
was le s s t h a t I p e rc e n t.
At h ig h e r a n g le s , q decreased more r a p id ly
re ac h in g a v alu e 7 p e rc e n t low f o r a = ± 6 0 °.
be used in t e r p r e t in g pressure data f o r
For t h is reason c a u tio n must
|a |> 3 5 ° .
■
33
Shroud gap
S t a t ic tap
Impact probe
—
Yaw probe
R otary s t r u t assembly
Test s e c tio n w a ll
F ig u re 4 .7
P ressure sensors in t e s t s e c tio n ( 1 /2 s c a le )
1.00
0 .9 5 -
C
O
0 .9 0
0
F ig u re 4 .8
5 .0
V a r ia t io n
P_. = lo c a l
1 0 .0
1 5 .0
D is ta n ce from w a ll
(cm)
in im pact p re ss u re w ith d is ta n c e from t e s t s e c tio n w a ll:
im pact p re s s u re , Pq = fre e s tre a m im pact p ressure
35
A w all-m ounted s t a t i c
pressure ta p was the s t a t i c source used.
It
was lo c a te d 8 .9 cm ( 3 .5 in c h es ) upstream o f the im pact tube (see F ig u re
4 .8 ) and in a check f o r in te r fe r e n c e w ith th e im pact tube none could be
measured.
Even w it h .t h e im pact tube removed from th e t e s t s e c tio n
e n t i r e l y no change in s t a t i c
pressure was o b s e rv a b le .
CHAPTER V
EXPERIMENTAL PROCEDURE
A.
Data C o lle c tio n
The f i r s t step in c a lib r a t in g each o f th e yaw probes o f a c e r t a in
OD, ID , and ^
was to f in d th e p ro b e 's n u ll
w indtunnel speed th e probe was r o ta te d u n t i l
were e q u a liz e d .
p o s itio n .
At th e maximum
th e pressures Pg and Pb
The p r o tr a c to r n eedle was then, s e t to 0 ° .
At th e beginnin g o f a c a lib r a t io n run th e am bient p re ss u re and
te m p e ratu re were re co rd e d .
The w indtunnel was s ta r te d and when Pq -P s
reached a n e a rly c o n s ta n t v a lu e , i t was recorded (PQ-P s= q ) .
The probe
was then yawed u n t i l a was a p p ro x im a te ly a t n e g a tiv e 70° as in F ig u re
5 .1:
Pg-P b was recorded a t t h is a , a was then changed in increm ents
o f f i v e degrees and Fg-P b was recorded f o r each a .
re p ea te d u n t i l a reached p o s itiv e 7 0 °.
This process was.
The yaw probe was then s e t to
0 0 to check f o r tra n s d u c e r z e r o -p o in t d r i f t .
I f a d r i f t o f more than
1 p e rc e n t had occurred th e data run was re p e a te d .
Also checked a t ct=0° was th e dynamic pressure q.
change q.
s h i f t in q.
Two fa c to r s would
F i r s t , tra n s d u c e r z e r o -p o in t d r i f t would cause an apparent
Second, an a c tu a l change in th e dynamic p ressu re would
occur because o f flo w flu c tu a tio n s o f th e tunnel i t s e l f .
To preven t
sp urious q read in g s due to tra n s d u c e r d r i f t th e tra n s d u c e r was rezeroed
b e fo re each check o f q.
A ctual q changes o f g re a te r than 3 p e rce n t were
con sid ered cause f o r re p e a tin g th e data ru n .
37
<<:•\
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V
F ig u re 5.1
Convention d e fin in g angle o f a t t a c k , a
•<-
38
I f no a p p re c ia b le change had o ccurred in q and th e ze ro p o in ts o f
th e tra n s d u c e rs , th e data ta k in g process was rep eated f o r th e same
an g les o f a tta c k re c o rd in g P -P ^ .
At th e end o f re c o rd in g Pg- P ^ f o r
each a ,t h e checks f o r q changes and tra n s d u c e r zero p o in t d r i f t were
re p e a te d .
The a m b ie n t p r e s s u r e a n d te m p e ra tu re were a ls o rechecked and
re co rd e d .
At th e end o f a c a lib r a t io n run a t a c e r t a in dynamic pressure
th e shroud gap was changed to change th e flo w speed thus th e dynamic
p re s s u re , and a new c a lib r a t io n run was begun.
For each probe P -P^j and P5 -Pj 3 were recorded as fu n c tio n s o f a f o r
dynamic pressures o f a p p ro x im a te ly 1 . 8 , 1 . 0 , 0 . 6 , and 0 .3 mmHg.
Thus
th e d ata was o b ta in e d such th a t the fu n c tio n D p y = O g ( O D ) I D , a ) could be
d eterm in ed .
B.__ Sources o f E xperim en tal E rro r
P robable sources o f measurement e r r o r in th e e xp e rim en t were the
a n g le measurements and th e pressure measurements.
The m a jo r ity o f e r r o r
p ro b a b ly o ccurred in th e pressure measurements because th e an g le measure
ments were q u ite sim ple where as th e p re ss u re measurements were r a th e r
complex.
The n eed le and p r o tr a c to r system a llo w e d the a n g le o f a tta c k to be
measured w ith in ± 0 .2 5 ° ,
As long as th e a n g le o f i n t e r e s t was approached
from th e same d ir e c t i o n , n e g a tiv e to p o s i t i v e , no problems w ith mechani­
c a l h y s te re s is o f th e r o ta r y s t r u t o c c u rre d .
39
The p ressu re measurements were much more i n d i r e c t .
E rro r in the
tra n s d u c e rs , th e c o n d itio n in g c i r c u i t , th e d i g i t a l v o ltm e te r , th e X-Y
re c o r d e r, and in th e tra n s d u c e r c a lib r a t io n s could a l l
in a c c u ra c ie s in pressure measurements.
c o n tr ib u te to
For t h is reason none o f the
e le c t r o n ic equipment was sw itched o f f between yaw probe c a lib r a t io n
ru n s .
C o n s is te n t e r r o r s o f th e p ressu re system should have been accoun­
te d f o r in th e tra n s d u c e r c a l i b r a t i o n .
I f so random e r r o r s due to. each
component and m istakes o f th e e x p e rim e n te r would account f o r th e rem ain­
d e r o f th e e r r o r s .
For th e tra n s d u c e r-c o n d itio n e r. p a rt. o f th e system , d r i f t o f the
• .
.■
■
v
z e ro p o in t was th e p rim ary e r r o r encou ntered.
This was c o n t r o lla b le by.
fre q u e n t checks o f t h e z e r o p o in ts .
D e v ia tio n from the tra n s d u c e r c a lib r a t io n slo pe was th e second type
o f e r r o r encou ntered.
d iffe r e n tia l
p re ss u re s.
This n o n - l i n e a r i t y was most s e rio u s a t , t h e low er
The diaphragm in th e tra n s d u c e r was ra te d a t
I psi which is a range o f about 52 mmHg.
tia l
The maximum p re ss u re d i f f e r e n ­
encountered in th e e xperim ent was a p p ro x im a te ly 3 mmHg o r o n ly
about 6 p e rc e n t o f t h a t range.
At the lo w est tunnel speed (8 m /sec) th e
change in P^-P^ p e r degree o f a was about .01 mmHg w hich is le s s t h a n
0.02% o f th e tra n s d u c e r's range.
In th e c a lib r a t io n o f th e tra n s d u c e rs ,
d e v ia tio n s o f 10 p e rc e n t from th e l i n e a r f i t were encountered a t d i f f e r ­
e n t i a l pressures o f 0 .1 0 mmHg, o r in o th e r words, d e v ia tio n s from the
c a lib r a t io n l i n e o f 0.01 mmHg were seen.
T h is tr a n s la t e s to tra n s d u c e r
p re c is io n o f 0.02% o f f u l l
range.
As s ta te d b e fo re , such a 0.01 mmHg
p ressu re d e v ia tio n would mean an a n g u la r d e v ia tio n o f ±1° a t th e low est
tu n n el speed.
The d i g i t a l
v o ltm e te r 's zero had d r i f t e d two m i l l i v o l t s when
checked w ith a v o lta g e standard a t th e end o f th e e xp e rim e n t.
a tio n was c o n s is te n t through out th e m e te r's te n v o l t ran g e.
The d e v i­
A 2 mV
e r r o r in v o ltm e te r in d ic a tio n corresponds to an a n g u la r d e v ia tio n e r r o r
of ±
a t th e lo w es t tunnel speed.
The X-Y re c o rd e r was used o n ly to judge when th e tim e averaged
pressures had reached steady s t a t e .
M isalig n m en t o f th e t r a c in g paper
could cause f l a t steady s ta te tra c e s to appear to r is e o r f a l l .
re v e rs e a ls o could happen.
The
This was an e xp e rim e n te r m istak e t h a t was
c o r r e c ta b le by checking th e tr a c in g paper a lig n m e n t.
M is alig n m e n t was
n o tic e d when th e X-Y re c o rd e r in d ic a te d steady s ta te but th e d i g i t a l
v o ltm e te r d id not o r v ic e v e rs a .
The p ressu re measurement e r r o r s were most serio u s a t th e low est
tu n n el speed becuase o f th e small pressu re change f o r each an g le o f
a tta c k change.
Assuming a w orst case o f low speed (8 m /sec) and maximum
e r r o r in each p ressu re system component plus maximum p r o tr a c to r e r r o r a
p o s s ib le e r r o r o f ± 1 .5 8 ° could r e s u l t .
CHAPTER V I
DISCUSSION OF RESULTS
The raw pressu re d a ta o b ta in e d f o r each yaw probe were v o lta g e s
re p re s e n tin g P^-P^, Pg-P b . and Pq-P s and th e tra n s d u c e r c a lib r a t io n s
were used to c o n v e rt th e d a ta to mm o f Hg, a usable form .
Pa~Pb and Ps~Pb was o b ta in e d f o r each a n g le o f a tta c k .
A v alu e o f
Pq -P s . which is
th e dynamic pressu re q , was c o n stan t d u rin g each c a lib r a t io n run a t a
fix e d flo w speed.
The d e s ire d r e s u lt o f th e in v e s tig a tio n was to c o n s tru c t a data
base from which th e fu n c tio n
could be o b ta in e d .
The fu n c tio n , r e ­
s ta te d here f o r convenience, is :
Dpy = g2 (a,0D,ID,<f>w)
(3 .2 8 )
To o b ta in Dp^'s fu n c tio n a l form v a rio u s p lo ts o f the c o n verted pressure
d ata were made.
R ecognizing t h a t graphing D
as a fu n c tio n o f fo u r v a ria b le s would
be im p o s sib le on a s in g le p l o t , s e p a ra te p lo ts were made, f o r each yaw
probe thus f i x i n g OD, ID , arid <J> .
The ID is d i r e c t l y r e la t e d to th e OD
by th e e x p re s s io n :
ID=OD-
2w
where w is th e probe tu b in g w a ll th ic k n e s s .
(6 .1 )
For the s te e l tu b in g used
in th e in v e s tig a tio n to c o n s tru c t yaw p ro b e s, the w a ll th ic k n e s s was
c o n s ta n t f o r a given OD o r ID , thus f i x i n g OD a ls o f ix e s ID (see Appen­
.
42
d ix I ) .
T h is s im p lif ie s Equation ( 3 .2 8 ) to th e form :
(6. 2)
Dpy = g3 (a,0D,<i>w)
Hence, f o r a given probe o f fix e d OD and wedge a n g le , th e angle o f
a t t a c k , a , becomes th e s in g le independent v a r ia b le .
The f i r s t group o f p lo ts o f t h i s ty p e t h a t were c o n s tru c te d were o f
q
and —s- r-— as fu n c tio n s o f a .
q
Cpa and Cps r e s p e c tiv e ly .
shown in F ig u re s
6 .1
These p lo ts w i l l
be r e fe r r e d to as
The Cpa and Cps p lo ts f o r each probe are
through
6 .9 .
The combined Cpg, Cps p lo ts were c o n s tru c te d f o r two reasons.
F i r s t , i t a llo w e d s e p a ra te , but n o n -d im e n s io n a l, exa m in atio n o f th e com­
ponents o f th e q u o tie n t
This was d e s ir a b le because a phenomenon
t h a t appear in Pg-P^ may not have appeared in P5 -Pj3 to th e same degree
and v ic e v e rs a .
th e combined form
T h is d is t i n c t io n disap p eared when th e y were p lo tte d in
Pa-Pb
Ps-Pb-
An example o f a phenomenon th a t is much more a p p are n t f o r Cps p lo ts
than i t
is f o r Cpg p lo ts is th e v alu e o f th e n e g a tiv e peaks.
The nega­
t i v e peaks in th e Cpg p lo ts show no p a r t i c u l a r p a tte r n f o r th e a valu e
a t which th e peak o c cu rs .
The a v alu e s range over -4 6 ° to -6 0 ° w ith no
ap p are n t r e la t io n s h ip to <j)w o r GD.
However, in th e case o f th e Cps
p lo t s , th e peaks occur a t a values t h a t appear to be <f>w dependent as is
shown in T a b le ( 6 . 1 ) .
a = 0 °.
The h ig h e r <f>w th e f u r t h e r th e peaks occur from
The s ig n if ic a n c e o f th e lo c a tio n o f th e n e g a tiv e Cps peaks is
. 43
EXPLANATION OF SYMBOLS FOR FIGURES 6 . 1 - 6 . 9
- open p o in ts a re Cpa=- ^ pb
- s o lid p o in ts a re Cps=p^~Pb
O o in d ic a t e q ^ 1 .8 ironHg
^
A in d ic a t e q = 1 .0 mmHg
D D in d ic a t e q = 0 .6 mmHg
v
v in d i c a t e q = 0 .3 mmHg
Also see Appendix I
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-T_L
probe 5.
49
II*
I ■« . . -t- — f
------k
■*------- k—— 4 ------<C--------r—— g
ecf -/c 0' -I'J -‘■t'-scr-tcr -.7
- -1 - — I ------- ♦---------I
"
io
-i;Tr z.v o* -o' ta
'4 - — + —
ic*
45*
0 0 -0.08 3
F igure
6.6
V a ria tio n
of
Cp a , Cps w i t h
a
; p ro b e 6.
SC
r:f ri
F ig u re 6 .7
V a r ia tio n o f C
51
Vr *
F ig u re 6 .8
V a r ia tio n o f C1
i
'V
52
: <2.0
i r -•:"-^V-SCi --O'-/a -e'cT-i'c“
■*■■! 1-!
i'c"
:o /o' Vo■CO' Co* io"
i
kvu = 3 0 *
0 0 -0 .0 0 3 "
..I ■ j . I . .
I
F ig u re 6 . 9
V a r ia tio n o f C
i '
Table
L o c a tio n o f n e g a tiv e Ps~P^
U
OD
= 30°
6.1
peaks in C
Dq
* * = 45°
, Cnc p lo ts
US
= 60°
0 .0 2 2 "
. -2 4 °
-3 5 °
-5 3 °
0 .0 3 2 "
-2 8 °
-3 7 °
-5 5 °
0 .0 8 3 "
-2 5 °
-4 0 °
-5 0 °
54
t h a t th e y re p re s e n t th e maximum-value o f P^.
The second reason f o r making th e Cpg and Cps p lo ts is t h a t they can
be used to d eterm ine th e magnitude o f th e dynamic p ressu re i f a and
Pg-Pj3 o r P5 -Pj 3 a re found in some o th e r manner.
The method o f t h is q
d e te rm in a tio n is covered in Appendix I I I .
The second group o f p l o t s . Figures
o f p-s-_-p-^ and i t s
6 .1 0
in v e rs e as fu n c tio n s o f a .
through
6 .1 8
a re p lo ts
Once again s e p a ra te p lo ts
a re made f o r each probe so th a t a is th e independent v a r ia b le and OD and
<{>w become param eters c h a r a c te r iz in g each yaw probe.
ps~p^ and pg~p^ f a l l
in g ra p h ic a l form .
The d a ta p o in ts
in bands along curves th a t re p re s e n t Dp^ and Dp^
The f a c t th a t th e d a ta p o in ts do f a l l
in f a i r l y
narrow bands on each p lo t is some c o n firm a tio n th a t th e fu n c tio n Dp^
does e x is t and i t
is m ostly a fu n c tio n o f a o n ly f o r a g iven probe.
However, th e s c a t t e r o f Figures
6.10
through
what was p re d ic te d in th e e r r o r a n a ly s is
d ata p o in ts to f a l l
6 .1 8
a re w id e r than
( ± 1 . 5 8 ° ) and th e f a i l u r e o f
in t h is p re d ic te d bandwidth along a s in g le curve
in d ic a te s th a t fa c to r s o th e r than a , GD, and cp may be re s p o n s ib le .
Vv
In C hapter I I
in v is c id flo w th e o ry was used to p r e d ic t th e e x i s t ­
ence o f a d i f f e r e n t i a l
pressure fu n c tio n f o r tw o -tu b e yaw probes o f the
form :
Dpy = g2 (a ,0 D ,ID ,(J > w)
Which, through th e use o f Equation ( 6 . 1 ) was reduced to :
(3.28)
55
EXPLANATION OF SYMBOLS FOR FIGURES 6 .1 0 - 6 .1 8
- open p o in ts a re Dpy =
- s o lid p o in ts a re Dpy' 1 =
O O in d ic a te q s 1 .8 mmHg
a
a
in d ic a te q s ' 1 . 0 mmHg
Cl □ in d ic a te
v
q s 0 .6 mmHg .
v i n d i c a t e q s 0 .3 mmHg
Also see Appendix I
56
I .!. '
z BCT - Yc -CCT -55* - A C - - I T ■ 20" -IS*
-I ' -I
v JCj To" Z 'S t C
cZ a 6 0 “ 70"
80*
• I
J__ J ____
•i—
r
i -3.0
Figure 6 .1 0
V a ria tio n o f D
57
H
- H
T r n ' T I
I
i" I " i
T
i
i SOj- I
I
I
!
I
!
'
: I
! i ■' !
I
-I-
...
I
I
’
I .....
'
j
I
1
*t
2 . 0 \ • ........... i-
* I
=■ ■ •
i ! '
. I .
• . I
; ! :
. \
T
O.3
....J.
%
> 4%
, *
J VV V VV \ . '
.............
i
!.. i . - J . - 1 . . ! .
*
;: I
■■ • ^ IlOr- ; - ■| **"j
I
»
,
- , I i—r
I
"
•
|
—. j . ~ j —j- ^ - —j- ... j.. ^ -i —
I
,'*•-*+ •
„
—; -jI '. I
Yr z o -- f——
1 : ■j
.2
^ . , • - i—
-1.0 t
I
I——- I—i
•
r
Ijl
v]:r
;
j -2 .0
[..I
I
....... i — ! -
I
i
- r
■ iiii.n -I-Ti
I I I -3.0 -
I T T
I
1:
I ‘ 1
IV
j-L.
Fig u re 6.11
I
-
«.- •
!
\w"GO
1
B
; I -
!I
;
i
.
|
- _
HT
-i-
.. j..
T
.
:I
» i j .I...
.L
. i
T . -
I
...
..!
it
!....I . 4- -
.
I
GD-0.052''
... I ■ i
V l-
u li.
I .
I
i - • t-—
—••• ... *. .. i
- .
i..
I::
"I :t r
T"l I" I ' —
-1
:
-
— I
■ ..
. - I
V - T - V -I
—-
,
—- •■
-f—■
! .. I
I
-j
:
v
vj*
•—
I .. j .-f-
!
11:
I
:
j . . .j.
■ - — --f —
•
I ! !
1 ■'
'L
i .
n i T
I
!
I
I' I .i
I I
'
o ?
- i
I : rri
.
I
i i
p
I
I ■;
!
' ! '" ! ' '
I •I
'
: i:i *
I ;
:I
i
I
V a r i a t i o n o f Dpyj Dpy' 1 w ith a ; probe 2.
58
tc T O-I'
L
Figure 6 .1 2
;j .
V a r i a t i o n o f Dp y , Dp^
w ith a ; probe 3
59
rI ' | !
—
----4 —
—
■ ♦'----*- -- -- I — —* -
—• - •■—
»—
n
■
1-
Figure 6 .1 3
V a ria tio n o f D
«
t—-I - I—
•
-♦
V-C1
53' / , '/ : /
dor -? c r-6 y - L y ^ c --X I" --.T1 - 3'
1-
•;* =Cr
60
i" Vc--J:’’7:'7, --I:'
2.0
Figure 6 .1 4
V a ria tio n o f D
*
61
* r 1.0•
I
I- ' "!— I'
!su,=4s*: :
OU=O C83"
!
Figure 6 .1 5
V a ria tio n of D
i
i
62
■
1
v tI
— I • ••
I v T o^ d- W mV- So5^ ovTF'* ;~7Ficv..V- V6"sV"w"ro“
I--• • — —-i L *—
I
I '
i .I . .
1I ; i
Figure 6 .1 6
V a ria tio n of D
I" ^
I
63
4 1
—
— ♦ - — t-------- — -
•
•EJ’ -TO' C* =-517 -W -JL i'
0 0 = 0 .0 3 2 "
Fig u re 6 .1 7
V a r i a t i o n of D
64
Figure 6 .1 8
V a ria tio n o f D
65
Dpy " S3 (Ot5OD5^w)
: (6 .2 )
The t h e o r e t i c a l a n a ly s is used to o b ta in t h i s fo r m u la tio n was f o r s te a d y ,
in v is c id ,
irro ta tio n a l
flo w o f an id e a l
flu id .
The im p o s itio n o f these
l i m i t a t i o n s n e g le c ts many phenomena a s s o c ia te d w ith the flo w o f a re a l
gas.
According to Chambre and Schaaf [ 5 ] th e pressure c o e f f i c i e n t e x p e r­
ienced by any impact tube such as a yaw probe can be expressed as the
fu n c tio n o f th e Reynolds, Mach, P r a n d t l , and Knudsen numbers, th e tu r b u ­
lence i n t e n s i t y , the r a t i o s o f s p e c i f i c h e a t , gas r e l a x a t i o n tim e to
OD/U, and ID/OD, and the angle between flo w d i r e c t i o n and the probe
opening.
dered.
In the i n v i s c i d a n a ly s is o n ly the l a s t two f a c t o r s a re c o n s i­
The o th e r f a c t o r s w i l l
now be c onsidered.
Mach number dependence can be s a f e l y n e g le c te d , as was e x p la in e d in
th e beginning o f Chapter I I ,
because o f the low speed o f th e flo w .
P ran d tl number dependence can be s a f e l y ignored s in ce n e g l i g i b l e
heat t r a n s f e r occurred in th e i n v e s t i g a t i o n .
This was due to the lack
o f any s i g n i f i c a n t te m peratu re g r a d ie n ts anywhere in th e t e s t s e c tio n .
ChambrA and Schaaf [ 5 ] s ta te d t h a t a decrease in measured impact
pressure due to s l i p begins to occur f o r Knudsen numbers g r e a t e r than
_O
10" . The h ig h e s t Knudsen number encountered in t h i s i n v e s t i g a t i o n was
_4
on the o r d e r o f 10
thus s l i p was not a f a c t o r and Knudsen number e f ­
f e c t s can be n e g le c te d .
66
The r a t i o o f s p e c i f i c h e a ts , y ,
i s c onstant f o r a i r over the narrow
range o f tem peratures encountered in the i n v e s t i g a t i o n .
Hence v a r i a t i o n
o f Y can a ls o be s a f e l y ig n o re d .
The r a t i o o f gas r e l a x a t i o n tim e , t * ,
to OD/U c h a r a c t e r iz e s the de­
crease in impact pressure due to the n o n is e n tr o p ic d e c e l e r a t i o n o f the
flo w a t an impact probe t i p .
According to ChambrS and Schaaf [ 5 ] t h i s
e f f e c t becomes s i g n i f i c a n t f o r values o f t* U /0 D g r e a t e r than u n i t y .
At
the tem peratures and a b s o lu te pressures encountered in the i n v e s t i g a t i o n
t * i s a p p ro x im a te ly IC f^ as s ta te d by Liepman and Roshko [ 8 ] ,
i n v e s t i g a t i o n t*U/OD had a maximum v a lu e o f 4 . 5 x 10
-5
In t h i s
, t h u s , th e re was
no decrease in impact pressure due to t h i s e f f e c t and i t can be ne glec ­
te d .
Reynolds number and tu rb u le n c e i n t e n s i t y are two f a c t o r s t h a t are
not inclu d e d in the i n v i s c i d flo w p r e d i c t i o n and which cannot be r e n ­
dered i n s i g n i f i c a n t in a manner s i m i l a r to the f a c t o r s j u s t considered.
Thus v a r i a t i o n s i n Reynolds number and tu rb u le n c e i n t e n s i t y may be the
mechanism o f data s c a t t e r in the -pfqsjj- and
In s p e c tio n o f Figures ( 6 . 1 0 )
g r e a t e r data s c a t t e r in
w itt
p lo ts .
through ( 6 . 1 8 ) r e v e a ls a tre n d toward
in c r e a s in g magnitude o f the angle o f
I
a tta c k .
F u r th e r in s p e c tio n r e v e a ls t h a t t h i s data bandwidth in c re a se
occurs sooner f o r the g r e a t e r wedge a n g le s .
Another n o ta b le f a c t is
t h a t f o r a given dynamic p re s s u re , q , the data f a l l s v ery c lo se to a
s in g le c u rv e .
The e xc e ss ive bandwidth s iz e is due to the f a c t t h a t the
67
curves f o r each q do not c o in c id e .
Al I o f these trends were c o n s is te n t
f o r the th r e e diam eters s tu d ie d .
I f the s c a t t e r f o r a f i x e d
i s examined, a llo w in g the OD to v a r y ,
a n o th e r les s obvious tre n d becomes a p p a re n t.
g r e a t e r the d a ta s c a t t e r .
g le s .
The s m a lle r the OD the
This is c o n s is te n t f o r each o f the th r e e an­
The group o f p o in ts having the low est q value i s r e s p o n s ib le f o r
most o f the s c a t t e r .
As s t a t e d b e f o r e , p o s s ib le mechanisms o f data s c a t t e r in excess o f
e r r o r p r e d i c t i o n s , could be Reynolds number a n d /o r tu rb u le n c e i n t e n s i t y
e ffe c ts .
Both Reynolds number and tu r b u le n c e i n t e n s i t y have a f u n c t i o n ­
al dependence on the fr e e s tr e a m v e l o c i t y .
In t h i s i n v e s t i g a t i o n p was
n e a r l y c o n s ta n t thus q is d i r e c t l y p r o p o r tio n a l
to the square o f the
fr e e s tr e a m v e l o c i t y and the veloc t y dependence o f Reynolds number and
tu rb u le n c e i n t e n s i t y becomes in te r c h a n g e a b le w ith a
dependence.
Finding t h a t Reynolds number is dependent on a c o n s ta n t times / q
and knowing i t a ls o depends on OD o f f e r s an e x p la n a tio n to th e band­
w id th v a r i a t i o n noted as the OD v a r i e s .
small OD both reduce the Reynolds number.
The combination o f low q and
In the case o f the 0 .0 2 2 inch
OD probes being t e s te d in the flo w having a q o f ~ 0 .3 mmHg, the Reynolds
number was around 300.
f o r each <|>w.
This corresponded to the case o f w id e s t s c a t t e r
T h e r e fo r e i t can be s a id t h a t exc e ss ive s c a t t e r has an
ap p are n t dependence on Reynolds number such t h a t s c a t t e r in c re a se s w ith
de cre a sing Reynolds number.
68
Changes in cf>w and a have no e f f e c t on the Reynolds number based on
probe d ia m e te r , but th e y can e f f e c t the tu rb u le n c e i n t e n s i t y in the f o l ­
lowing manner.
The i n t e n s i t y o f tu rb u le n c e on the leeward sid e o f the
f l a t p l a t e is a fu n c tio n o f " t h e p l a t e angle o f a t t a c k .
The pressure
sensing surfa ce s A and B o f a yaw probe a re n o t.le e w a rd sid es o f f l a t
p l a t e s but th e y a re s i m i l a r in t h a t th e y move i n t o wakes behind le a d in g
surfaces w ith
changes in the angle o f a t t a c k .
yaw probe is the apex o f
the t i p and the
The
angles
le a d in g s u rfa ce f o r a
o f a tta c k th a t
the s u r ­
faces A and B encounter are c h a r a c te r iz e d by Equations ( 2 . 1 6 ) and ( 2 . 1 7 )
as:
= 4>w + -rr/2 - a
(2 .1 6 )
Ba = <t>w + 7tZ? + a
(2 .1 7 )
As i t can be seen from Figure
6 .1 9
the g r e a t e r the wedge a n g le ,
<|>w, the sooner one o f the sensing surfaces f a l l s
angle o f a t t a c k in c r e a s e s .
i n t o a wake region as
Thus the onset o f a d e l e t e r i o u s tu rb u le n c e
e f f e c t s a t s u rfa ce A i s p robably e x p r e s s ib le as a fu n c tio n o f some
c ritic a l
v a lu e o f Bg .
Table ( 6 . 2 )
in c lu d e s s o - c a l l e d c r i t i c a l
f o r which the s c a t t e r o f data reached about 10 degrees o f a .
seen from Table ( 6 . 2 ) the c r i t i c a l
10 p e rce n t o f each o t h e r .
Bg values f o r a l l
In a d d i t i o n , the c r i t i c a l
Bg values
As can be
probes f a l l w i t h i n
Bg values f o r the
0 .0 8 3 inch probes, f o r which d e l e t e r i o u s Reynolds e f f e c t s a re a minimum,
a ll
fa ll
w i t h i n th r e e p e rc e n t o f each o t h e r .
This is s t r o n g l y suggestive
69
+w = 6 0°
8
d
=
200°
Surface A
Surface A
Surface A
Figure 6.19
Variation in 6 with angle of attack
d
T a b le
6 .2
Mean v a lu e o f Bq a t which Dp^ data s c a t t e r reached ten degrees o f a
0.022
0 .0 3 2
0 .0 8 3
O
71
t h a t the mechanism o f s c a t t e r i s a fu n c tio n o f 3 and tu r b u le n c e i n t e n ­
s i t y is probably t h a t mechanism.
data as a fu n c tio n of'<f>
U n f o r tu n a te ly tu rb u le n c e i n t e n s i t y
was not o b ta in a b le w it h the expe rim ental e q u ip ­
ment used in the i n v e s t i g a t i o n .
Continuing on the premise t h a t the data s c a t t e r in the
i s e x p la in a b le in the terms o f the re a l
p lo ts
gas e f f e c t s o f Reynolds number
and tu rb u le n c e i n t e n s i t y , the accuracy o f yaw probes can be maximized by
t r e a t i n g these f a c t o r s .
In examining f ig u r e s
6 .1 0
becomes obvious t h a t s c a t t e r as a fu n c tio n o f <j>
through
6 .1 8
it
(presumably tu rb u le n c e
r e l a t e d ) i s much more severe than s c a t t e r as a fu n c tio n o f OD (presumably
Reynolds number r e l a t e d ) .
Thus to o p tim iz e the accuracy o f a yaw probe,
tu rb u le n c e re d u c tio n should be the f i r s t p r i o r i t y .
M in im iz in g
fu l
to 30° seemed to reduce tu rb u le n c e measurably.
shaping o f the probe t i p i s a ls o c r u c i a l
Care­
to p re ve n t burrs on cut.
tu b in g edges, bent o r p a r t i a l l y c o lla p s e d tubes and improper a li g n i n g o f
the two tubes w it h each o t h e r .
C a refu l
im p o rta n t to p re ve n t damage to the t i p s .
handling o f the tubes is a ls o
Any f a c t o r t h a t a l t e r s th e t i p
geometry a d v e r s e ly can induce undue tu r b u le n c e .
Avoidance o f adverse low Reynolds number e f f e c t s can be accomplished
by r a i s i n g probe OD o r . a v o i d i n g very slow flo w speeds.
Examining the
30° probes, where s c a t t e r due to tu rb u le n c e was a minimum and Reynolds
number s c a t t e r was most n o t i c e a b l e , the g r e a t e s t s c a t t e r due to Reynolds
number e f f e c t s occurred f o r Reynolds numbers less t h a t 350.
When the
72
Reynolds number was r a is e d to 450 the s c a t t e r dim inished to w i t h i n the
bandwidth p r e d ic te d in the e r r o r a n a l y s i s .
For example, in a i r flows
a t about 25°C, i f using a probe o f . 0 .0 1 0 inch diam eter i s d e s ir e d , c a l i
b r a tto n flo w speeds and a c tu a l flo w speeds less than 20 m/sec should be
avoided.
CHAPTER V I I
S L W R Y AND CONCLUSIONS
I t was demonstrated t h a t impact pressure probes using two t i p beveled tubes can be used to o b ta in th e flo w v e l o c i t y v e c t o r .
Previous
work using t h r e e - t u b e probes was dis cus s ed, c i t i n g advantages o f f i x e d
versus r o t a r y probes.
An i n v i s c i d a n a ly s is was used to examine the
p o s s i b i l i t y o f using tw o-tube probes in th e f i x e d mode.
t i o n was made t h a t th e r e could be fu n c tio n s Dn , D
py
py
The d e term ina­
and C
pa
, C
ps
th a t
would a llo w the d i r e c t i o n and dynamic pressure o f a flow to be found
by measuring the two impact pressures and the s t a t i c p re s s u re .
Experim ental t e s t s were conducted to o b ta in a data base t h a t would
prove o r dis prove th e e x is ta n c e o f Dp^.
t h a t a fu n c tio n D
rJ
The data was reduced to show
e x is te d but the s c a t t e r o f the data f o r many o f the
probe geom etries was g r e a t e r than p r e d ic te d in the e r r o r a n a l y s i s .
Real
gas e f f e c t s were in tro d u c e d to e x p la in s c a t t e r and methods were d is c u s - ■
•
sed to m inim ize the s c a t t e r by t r e a t i n g the r e a l gas e f f e c t s .
A p r a c t i c a l example o f how D
py
is used to o b ta in the flo w v e c to r
i s inc lu d e d in Appendix I I I . . The r e s u l t s show t h a t in th e Reynolds
number range o f 400 - 300 0 , proper choice o f probe wedge angle
(<j>w < 3 0 °) w i l l
y i e l d flo w v e c to r p r e d i c t i o n s t h a t a re w i t h i n t
d ire c tio n a lly .
Using t h i s p re d ic te d angle o f a tta c k and p r e v io u s ly
1 .0 °
o b ta in e d pressure data the f l o w 's dynamic pressure can be p r e d ic te d
w i t h i n t 2.4%.
Two-tube yawprobes, used in the f i x e d mode, t h a t a llo w flo w
74
v e c to rs to be determ ined to t h i s accuracy should be a v a lu a b le
new to o l f o r the f l u i d d y n a m ic is t.
APPENDIX
I
DESCRIPTION OF YAW PROBE TIP GEOMETRIES
USED IN THE INVESTIGATION
Nomenclature:
OD = O utside d ia m e te r
■
ID = In s id e diameter*
w = Wall th ic kn es s
= Wedge angle
A l l probes were 2 inches in le n g th measured from th e le a d in g edge
o f th e s t r u t .
Probe 3:
Probe I :
OD = 0 .0 2 2 "
V .
"
OD = 0 .0 8 3 "
. I'D = 0 ,012"
ID = 0 .0 6 3 "
w = 0 .0 0 5 "
. w = 0 .0 1 0 "
d> = 60°
w
4 * = .6 0 °
I
- "
Probe 4:
Probe 2': .
OD = 0 .0 3 2 "
OD = 0.Q22"
■ ID. = 0 .0 2 0 "
■ID = 0.012"-
w. = 0 .0 0 6 "
■ w = 0 .0 0 5 "
■ 60°
'
76
Probe 9:
Probe 5:
QD =? 0 .0 3 2 "
OD = 0 .0 8 3 "
ID = 0 .0 2 0 "
ID = 0 .0 6 3 "
w = 0 .0 0 6 "
w = 0 .0 1 0 "
'f’w = 4 5 °
Probe 6:
OD = 0 .0 8 3 "
ID = 0 .0 6 3 "
w = 0 .0 1 0 "
Ow = 450
Probe 7:
OD = 0 .0 2 2 "
ID = 0 .0 1 2 "
w = 0 .0 0 6 "
* „ = 30°
Probe 8:
OD = 0 .0 3 2 "
ID = 0 .0 2 0 "
w = 0 .0 0 6 "
tf5W =
30°
* „ = 30 ”
APPENDIX I I
DATA REDUCTION PROGRAMS
CAL
PLOT
TEST
78
COMPUTER PROGRAM ' CAL1
O OOO O O
MME = O l
oirnkArrcN
NOS I S THE
VARIA^LtS•
or
NU4GER
J=
ru
p /u s s u r e
transducers
0 6 1 A P O I N T S , NV
IS
THE
N j N QER
OF
v 0 ?v * 2 ‘ j ‘ l l l 0 1 i c t l v j I S t 1^ , 1 ^ > » ANP Cl v ) , V ( I G S ) 1TClOO )
11
PO I ! I = I , N U J
R E A D ( Z ) J , I T ) V ( I ) 1T ( I )
DO ZZ 1 = 1 , NJJ
X ( N V t l l I ) =I ( I )
Xll , D =I . )
X IZ 1D = V ( I )
ZZ C ON T I N U E
CAL L C J R c T ( N V ( N J P t Xl C )
CALL ERROR ( N V 1NJ = , X , C , S S Q )
WR I T = I U . R , ! ' . ) V l D ( U I )
I C F ORt AT C Z = I v . 3 )
END
SUS R I O T l N E C J R F T C N V ( N O P 1 X 1 C )
O I mEn SION X C D , ! . ) ) , CCI
C
C
C
) , 3 ( 1 2 , 1 2 ) , ANP(D)
L t A S T SOUARt S - NV I N ) ' Pu NDt NT VAP I t E L E S
Y = X ( N V t l t I ) - h o 3 » f . J . JF U S S t R V A T I O N S
C ( D = S ) E F F I C I i h T Or X ( I )
M=NVt;
MP=Vt;
I
PO
OP
S d
DC
30
DO
I
I
(J
Z
Z
I
I = I 1N
J=l,vp
) =) . ) )
I=I(N )Q
J = Id
R = I1R
Z S ( J , O = S ( J , K ) t x ( J , D = X ( K 1I )
Svl , '») = ! . O
IF I '.V-I )
R N ? , 9 ) 1 , VRJ
S97 S ( 1 , > ) « S ( 1 , Z ) / S ( 1 . 1 )
99*
H
IZ
Z3
CJ TO
UU I v
IF I 3(
V-RITt
FJR«;t
999
L * = l , NV
I , . j ) I j,lZ ,13
(1 D ,> -)
( '
t JJ A T I JNS I N S1J S R J U T I NE
ANALYSIS')
CU'FT
ARE
Ct P = NCiNT
I LOW: .G F R R J .
3(
IJ
IR
OP ;
I I I = I 1NV
C dID =:-. •
CO T l 31
OO I t J = I f W
S ( V 1J ) = S C , J t D X S ( I d )
CU I S I = Z , , V
O =S (Id)
PU I S J = I , M
15 S d - I ( J ) = S C , J t D - O * S ( M1J )
DO I j J = I , 1
16 S( NV 1J) = S(W1J )
399 DU ) 1 = 1 , NV
C d O= S ( I d )
J WRI T!
(D d d )
A FORviT ( '
!(C d )
*i)X,'COEFFICIENT
JF X ' I Z , ' » 'E I 5 . A)
-
I GNORE
FCL
79
31
RETlPN
ENO
SU=C )UT I NE
E U lP C N V . N O e .X , C.SSO)
< c i u . I N I i C U ■ > , S d 2 . : 2 1 , ANP C l 03
OCUEL= pc= : ; s i j n y : , r s . te
E k R O l A N A L Y S I S - NOL O e S E R V A T I O N S
V = X C N V e l , I ) - S S J * STANDARO D E V I A T I O N
OI = CNSI O n
C
C
c
C
cci ) -
c o n s i ants
N=NVtI
TS=C.0)
T E = O . 03
ENX = - I .
00 5 1=1,5
5 ANP( I 3 * C
W H T t ( 1 C A, I )
1 EORv AT < ' , * 5 X , ' r : - X P E R I v t N T A L ' d X , ' V CAL CULATED "5 X, " N Ov E R I C A L ERRO
I R " E X , " P E R CENT E R R O R " )
0 3 2 I = I 1NJP
YE=XC',IJ
AOO EQU A T I O N RuR VC AT T H I S P O I N T
TC=!.0 )
do 2 :
i j = i , .v
20 T C = Y C * C C I J ) * X < I J , I )
E=TC-T =
LP = I D J . t c / T E
EPA=ANS(EP)
IR C - = A - E v X) 6 , 4 , 7
7 E v X = E PA
6 OO e J= 1 . 5
ACO=J=S
I F C- = A - A C J ) 3 , 3 , 8
« AN=C J J = ANP*. JJ
6 CONTINJt
TS=TS=O=E
Tc=TL=ARSCr P)
2 J t T T ' - C l - 3 , 0 ) V : , V C , E , EP
3 FCRPAT ( A c O , . 8 )
A = N Ol
SSJ=(TSZA)=,.5
Tt=TlZA
WRI TE U - S 1 AJ S S J 1TE
a F O = v AT C "J " I ) X , " S T A N D A R D D E V I AT 1 1JN = - E l S . a, S X , * A VE = T ' E
P R CENT DE V
I I A T I ' N= = E I S . 8 )
W - I T E C l ' - , I . * ) ENX
V
F D = v AT ( " : " . U X , " MAXI MUM P c W CENT Of V l A T I ON= " E l 5 . 3 3
Oj J ! 1 = 1 , 3
XN==AN=(I)
X N P = I ) 3 . =X i p Z A
A C J * ! »3
11 K R t T t C V 8 , 1 2 ) X N P 1 ACD
1 2 FU = v T
C S X , ' I 5 . 3 , 2 X , " P : P CENT OF DATA WI TH I N " 2X ,E I E . 3 , 2 X , * P ER CENT
I OF E Q U A T I O N " )
RETURN
ENO
30
name
C
c
= P L OT
plo ttin g
program
fur
cpa
>cps p l o t s
C
D I M E N S I O N A O U l O O > , PT C I O * ) , D P 2 C l U G )
I N P U T ,NP1 DP I
D P l = . G G 2 1 3 5 3 3 l * D P l - . r' 23961067
DO 1 1 1 = 1 , NP
C
C
C P R E S S U R E S ™ £ a N 1 L E S 0 'S A T T A C K
C
INPUT A O A ( I ) , O P Z ( I )
11
AN3 THE
DP 2 ( I ) = . ; : J I 6 3 4 9 7 * I * q p 2 ( 1 ) - . W0 8 3 9 Z 6 9 1 6
y T C D = 9 P ? 11 ; / D P I
W R I T E d C3 , 1 0 )
DO 2 2 I = I , N P
C
C WRITING
OUT THE R E S U L T S
C
22
IfI
20
DIFFERENTIAL
V 0 ITE ( 1 ( 3 , 2 3 ) A J A ( I ) 1P T ( I )
FO R M ATC //,'
AOA
DP2/0P1 ' , / / )
FJRMATCF6.2,2X,F7.4)
ENO
81
O O O O
NAME = TEST
TEST PR3GRA1 FQR YAW PR03ES
O O O O
REAL IDPY
INPUT N
DO 11 1 = 1 , N
INPUT ING THE PAW PRESSURE DATA AND APPLYING THE
TRANSDUCER CALIBRATIONS
O O O O
IN P U T DP A,DPS ,OPO
DPA=OPA..
16 3t>y 7 0 1 3 * 3 3 3 2 6 9 1 6
OPS=DPS* . - 11 6 3 5 9 7 0 1 v >63926916
QM=DPO* . : * 2 1 3 5 3 3 1 - . 3 2 3 9 6 1 J t l
DPY=DPAZDPS
ICPY=L. ZDP Y
OUTPUTINS THE VALJSS FOR PA - PP, PS-PD, AND Qw IN MMHG
and OUTPJTING
the v a l u e s for JPY and IZDPY
WRITECl C3 , I D DP A, DPS, DP Y, IOPYt QM
10 FORMoTCZZ,' PA-P E-= ' , S *7 ,
PA-P S= ' , F.. 7 , 'D PY=
>1E .7 , ' IZOPY= ' , 6 . 7 , ' QM= ' , 6 . 7 , ZZZj
11 CONTINUE
END
APPENDIX I I I
DESCRIPTION OF YAVJ PROBE TESTS TO FIND FLOW
SPEED AND DIRECTION IN AN UNKNOWN FLOW
The t e s t s were performed using the th r e e probes having wedge angles
o f 30°.
V o lta g e values f o r P^-P^, Pg-P^ and q were recorded along w ith
a f o r v a rio u s dynamic pressures and angles o f a t t a c k .
be th e unknowns in a c tu a l
The a and q would
p r a c t i c e , th e y were recorded in the te s t s as a
means o f checking the p r e d ic te d v a lu e s .
recorded in Tables A 3 . 1 , A 3 .2 , and A 3 .J-.
A p o r tio n o1 the r e s u l t s are
The measured value s use sub­
s c r i p t s m, the c a l c u l a t e d values use s u b s c r ip ts c.
The procedure f o r c a l c u l a t i n g
m unknown flo w d i r e c t i o n and speed
was as fo llo w s :
1.
At an "unknown" a and q the v o lta g e values f o r Pa - P b and P5 -P f3
a re d e te rm in e d , along w ith am and q .
2.
Program TEST was a p p lie d to o b ta in values o f P3 -P f3 and P5 -P f3 in
mmHg and the q u o tie n ts
3.
H o r iz o n ta l
and
and qm in rrniHg.
l i n e s were drawn on the Dpy , Dp^~^ p l o t s from the
v e r t i c a l a x is from the values o f
and p ~ t o
the data
bands.
4.
From the i n t e r s e c t i o n o f the h o r iz o n t a l
o f the a p p r o p r ia t e data bands v e r t i c a l
a a x is .
I f the v e r t i c a l
l i n e s and the centers
l i n e s were drawn to the
l i n e s were not c o - 1 i n e a r , an average
o f t h e i r a v a lu e was taken to be 'a .
T a b le
Test re s u lts
am
ac
Degree
Degree
0. .
I
I
Error
fo r V
=
A 3 .I
3 0 ° , OD = 0 . 083" probe
% E rror
E rror
%
Degree
mmHg
nrnHg
mmHg
% Error
0
0
0
1 .710
1.637
+ .073
+ 4 .5
r4 5 .0
-4 4 .5
+ 0 .5
-1 .1
1.661
1 .6 3 7
+ .024
+ 1 .5
-3 0 .0
-3 0 .0
0
0
1 .644
1.637
+ .007
+ 0 .4
-1 5 .0
-1 4 .5
+ 0 .5
-3 .3
1 .7 2 2
1 .6 3 7 '
+ .085
+ 5 .2
+ 1 7 .5
+ 1 8 .0
+ 0 .5
+ 2 .9
1 .7 6 0
1.6 3 7
+ .123
+ 7 .5
+ 3 2 .5
+ 3 2 .0
-0 .5
-1 .6
1 .6 3 9
1.6 3 7 '
+ .002
+0.1
-3 9 .5
-3 8 .7 5
+0. / o
-1 .9
.9408
.9660
- 9 .5
- 9 .5
0
0
.9316
+ 6 .0
+ 6 .0
0
0
+ 2 8 .5
+ 2 8 .5
0
+ 4 2 .5
+ 4 2 .5
0
•
CO
CO
I
i
I
-.0 2 5
-2 .6
I
.9660
-.0 3 4
-3 .6
I
.9800
.9660
+ .014
+ 1 .4
!
I
0
.993
.9660
+ .027
+ 2 .8
I
0
.92 4
.9660
-.0 4 2
-4 .3
I
.
.
T a b le A3.2
Test re s u lts
I
f o r 4>w = 3 0 ° , OD = 0 032" probe
.
Error
% Error
cxC
irenHg
mmHg
mmHg
% E rror
-2 .7
1 .416
1 .3 8 6
-2 .1
-2 1 .0
-2 0 .5
+ 0 .5
-2 .4
1 .4 1 6
1 .429
+ .013
+ 0 .9
0
0
0
1 .4 1 6
1.4 4 6
+ .030
■ +2.1
+ 6 .0
+ 6 .0
0
0
I .416
1.4 6 0
+ .044
+3.1
+ 3 7 .0
+ 3 6 .0
-1 .0
-2 .7
1 .4 1 6
T.3 98
-.0 1 8
-1 .3
-4 2 .0
-4 1 .5
+ 0 .5
. -1 .2
.89 7 3
.8862
-.0 1 1
-1 .2
-1 7 .0
-1 7 .0
0
0
.8973
.9015
+ .004
+ 0 .5
- 4.0 .
- 4 .0
0
0
.8973
.8898
-0.8
+ 1 1 .0
+ 1 0 .5
-0 .5
-4 .5
.8973
.9202
+ .023
+ 2 .6
+ 3 1 .0
+ 3 0 .0
-1 .0
-3 .2
.8973
.8737
-2 .6
0
.
•
I
+1.0
I
CM
-3 6 .0
<5*
-3 7 .0
O
S
Degree
E rro r
C
.O
I
Degree
q
m
O
O
CO
I------------------ Degree
q
2 . ■
T a b le
Test re s u lts fo r
.
E rror
am
ac
Degree
Degree
-3 9 .0
-3 5 .0
-4 .0
-1 6 .0
-1 4 .0
+ 2 .6
A 3 .3
= 3 0 ° , OD = 0 .0 2 2 " probe
% Error
Error
% E rror
qm
qc
mmHg
mmHg
mmHg
+ 1 0 .2
1 .7 0 2
1 .756
+ .054
+ 3 .2
-2 .0
+ 1 2 .5
1 .7 0 2
1 .6 9 7
-.0 0 5 . '
-0 .3
+ 3 .0
+ 0 .4
+ 1 5 .4
1 .702
1 .5 8 9
-.1 1 3
-6 .6
+ 2 3 .4
+ 2 2 .5
-0 .9
- 3 .8
1 .702
1 .689
-.0 1 3
-0 .8
+ 4 1 .0
+ 3 9 .5
-1 .5
- 3 .6
1 .702
1 .6 0 2
-.1 0 0
-5 .9
-4 9 .0
-4 7 .5
+1 .5
- 3 .0
.7201
.6929
-.0 2 6
-3 .6
-2 6 .0
-2 4 .0
+ 2 .0 .
- 7 .7
. 7201
.7 4 3 1 •
+ .023
+ 3 .2
. -1 0 .0
- 9 .0
+ 1 .0
-1 0 .0
.7201
.7221
+ .002
+ 0 .3
+ 9 .0
+ 9 .5
+ 0 .5
+ 5 .5
. 7201
.7427
+ .023
+ 3 .2
+ 2 8 .5
+28
-0 .5
- 1 .8
.7201
.7035
-.0 1 7
-2 .3
Degree
86
5.
Using the C , C
p lo ts , a v e rtic a l
Pd
Pi
l i n e is drawn from a
c
through the data bands.
6.
From the i n t e r s e c t i o n o f the v e r t i c a l
l i n e and the c en ters o f
the Cpa and Cps data bands h o r i z o n t a l
l i n e s a re drawn to the
v e rtic a l a x is .
These values on the v e r t i c a l
a x is re p re s e n t
Cpac and CpSc r e s p e c t i v e l y .
7.
q
L
is determ ined by knowing t h a t C =^ a-- !3 and C
pa
q
ps
s o lv in g f o r q
~
q
and
such t h a t qr 4 - ( Pca ~Pb + ^s-~Pb) .
cP=C
cP=C
A numerical example o f the above procedure f o r the < ^ = 3 0 °, 00=0.083
probe i s as fo llo w s :
1.
Measuring raw d a ta :
Pa - Pb = -450 mV
Ps - Pb = -941 mV
%
= -1 5 °
qm = 760 mV
2.
TEST y i e l d s :
Pg - Pb = -0 .7 5 4 5 9 mmHg
Ps - Pb = -1 .5 4 7 9 4 mmHg
qm = I .6369 mmHg
3.
See Figure A 3 . I
4.
See Figure A 3 . I ; y i e l d s a c = - 1 4 . 5 °
5.
Using a c = - 1 4 . 5 ° , a v e r t i c a l
l i n e is drawn, see. Fig u re A 3 .2 ,
87
I
I
TTt T T
i
!
.
i
.
T
I
T T
'i
'
I
I
!
! ■I I
I ' i
|
I
I
i
;
i :j
I
T
I
i :I T
11
Z I!
■'!
i
I
I
- N i ;
o
ie I I
I
I
h
: • I
s V- *
-
Tb*' tV rV
of.,* V
I
M
,
. . S «» ♦-
IC
1
,
.
!
j
I
.Ti
.
f-! Tl
-■
: :
Ti
..
...
-
I
]
j
..
••
.
r
•
I
•*
I
#
j
I
•
-ZO
i
i
I
:
IT
i"
!
!
":
' II
I ■
’ II ''
i
1
I
I
I I
i
-i '
F ig u re A 3 . I
:
■
!
I
.
•
: I
: I
i I
I
I
i
- 1 —«
*
I
I"
•• ■ ••'
I
I
I
!
I
,
•-
•] "
...
•
j.
L'
.
.
1
..
I
I
'
I * -
i
I •'i ;
h i
• •
- *• ■
•
.
I •
I .
•
—»
v." : s bu' rj1
"1
l
«'
!
* r
i Tl
I
i
.
I
I
#
•
'i'
I
i
i
♦
- ' . C1
1
T
I
o
I -
I
I !I !
I
,
!
I
I
I i
I
-:
I'
yf
I.
I
1
I . I
V
i
. i
:
’.0 - :
I
i
!
I
I
I° I
'
i
:
I*:
V
•
! •
0D-O.0SJ'f
:
i
M
I
i :
T
: I
I ;
ii
,
I
-3 0
i
M T
:
L
IL- -'-----
■
~
I 11 . I ,
I I
I :
. i- *- * ■ ■■ i n
1
i
i
:
I
.
L
Use o f Dp y , Dpy' 1 p l o t to de te rm in e a c
.
I
88
6.
See Figure A 3 .2 ; y i e l d s Cpac - - 0 . 4 3 7 5 , CpSc ~ - 0 . 9 0 0
' '
qC
„
_ I /-0 .7 5 4 5 9
~ 2 -0 ,4 3 7 5
, -1 .5 4 7 9 4 ,
-0 .9 0 0 }
qc = I . 722 mmHg
E r r o r in a c is 0 . 5 ° high which is a 3 . 3 p e rce n t e r r o r .
E r r o r in qc i s 0 .0 8 5 mmHg high which i s a 5 .2 p e rcent e r r o r .
89
1 -I —
-T 1"1 — J— T -—f—-- f—T
I
I
I
!
;
I
!
- T - T -,
i I !
I • ,
I
I I I
r-T
I'!
1 !
!
I
I
I I
i
! M
I I
'f
> I I ,,
I
!
.
i
T
... I
i
.* . I
I
.
:
!
1 I • I ! ■ ii ! J
I ;
1
.
I
r-!
I
! ! ■!
!
i
r
. ; 1
j*
!
I
.
1.0
.
i
I
i
: -
o '. i
r
i
i
i o ' <3
a ?
» .
'
! i :
! ' I
? '
M
. j
-
10
A.
.*
-------------#
*-
I C 0 ; C 4 ■'
4/
. . . .
IC-
.
•
CO® 7 0*
-Al
I Z
I.-
.- f - ■
f
;
. I
..
;
.I. E .
I
' I
; •
:i
! I!
Fig u re A3.2
’ • * • • • 1.0
. .L O
- . I
X
.I
L •■
i
I
Use o f Cna
pa
:i l
j •:
*■ »
I
..
.1
i.
U
k, ^ JO*
0 0 : 0 005 *
I i
•I
I I'
I l ''i
I
i
.
!.-J -J -
!
:
e'
y,
*e
O4■v h , •
’ •
!
.! I 11 I
! : i
-i.0
I :
I hi ' I!
"I I
I
1. .!..J..:ti
V -W tI
i' ■
..
i. i . i .
•y r
%
J.
I
:
s
. 4'
I
I '
j
I :]'I-;.,.
If
ill'!
J - L
Cnc p l o t to determ ine q
ps
'c
I
l
I:
LITERATURE CITED
H a l l , M. G ., "Experim ental Measurements in a Three-Dim ensional T u r­
b u le n t Boundary Layer in Supersonic Flow ."
Royal A i r c r a f t Estab­
lis h m e n t T e chn ical R e p o rt, p. 8 3 5 -5 3 , 1963.
D u d z in s k i, Thomas J . , and Lloyd N. Krause, " F lo u - D ir e c t i o n Measure­
ment w ith F ix e d - P o s it io n P ro b e s ."
U . S . , N a tio n a l Aeronautics and
Space A d m i n i s t r a t io n , NASA Techn ical Memorandum X -1904.
S p a id, F. W., F. X. H u r le y , and T. H. Heilm an, " M in ia tu r e Probe f o r
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Chevallier, Travis W
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for flow vector measure­
ments
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