Retrospective Theses and Dissertations
1964
The effects of free-stream turbulence on heat
transfer from a flat plate with a pressure gradient
George Hanbury Junkhan
Iowa State University
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Retrospective Theses and Dissertations. Paper 2672.
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JUNKHAN, George Hanbury, 1929THE EFFECTS OF FREE-STREAM TURBULENCE
ON HEAT TRANSFER FROM A FLAT PLATE WITH
A PRESSURE GRADIENT.
Iowa State University of Science and Technology
Ph.D., 1964
Engineering, mechanical
University Microfilms, Inc., Ann Arbor, Michigan
THE EFFECTS OF FREE-STREAM TURBULENCE ON HEAT TRANSFER
FROM A FLAT PLATE WITH A PRESSURE GRADIENT
by-
George Hanbury Junkhan
A D i s s e r t a t i o n Submitted, t o t h e
G r a d u a t e F a c u l t y i n P a r t i a l F u l f i l l m e n t of
The R e q u i r e m e n t s f o r t h e Degree of
DOCTOR OF PHILOSOPHY
Major S u b j e c t s :
Mechanical E n g i n e e r i n g
T h e o r e t i c a l and Applied Mechanics
Approved :
Signature was redacted for privacy.
I n Charge^bf^M^Jor Work
Signature was redacted for privacy.
'or d e p a r t m e n t
Signature was redacted for privacy.
Iowa S t a t e U n i v e r s i t y
Of S c i e n c e and Technology
Air . s , Iowa
1964
il
TABLE ûïr CûiNl'tiiNïS
Pa g e
SYMBOLS
iv
S u b s c r i p t s f o r U n c e r t a i n t y Symbol w
INTRODUCTION
REVIEW OF PREVIOUS INVESTIGATIONS
vii
'
1
4
C y l i n d r i c a l , and S p h e r i c a l G e o m e t r i e s
4
F l a t P l a t e Geometries
8
ANALYSIS OF THE EFFECTS OF FREE-STREAM TURBULENCE ON
THE LAMINAR BOUNDARY LAYER EQUATIONS
14
E f f e c t on F l u i d - F l o w C h a r a c t e r i s t i c s
14
E f f e c t s on Heat T r a n s f e r
21
EQUIPMENT USED FOR EXPERIMENTAL WORK
24
A i r Flow F a c i l i t y
24
Flat Plate
2?
Plate parts
Pressure taps
Thermocouples
Power w i r i n g
Instrumentation
Thermocouples
Velocity-profile instruments
Electrical instruments
30
33
34
35
35
35
38
43
P r o c e d u r e f o r Taking Data
47
Methods of C a l c u l a t i o n
48
C a l c u l a t i o n of N u s s e l t number
C a l c u l a t i o n o f Reynolds number
B o u n d a r y - l a y e r measurements
T u r b u l e n c e i n t e n s i t y measurements
48
52
52
53
ill
rage
Uncertainties in Experimental Results
U n c e r t a i n t y i n t h e N u s s e l t number
U n c e r t a i n t i e s i n t h e Reynolds number
Uncertainties in the pressure gradient
Uncertainties in turbulence intensity
measurement
RESULTS OP THE EXPERIMENTAL INVESTIGATION
5^
55
57
57
57
59
Zero Pressure Gradient
59
Low F a v o r a b l e P r e s s u r e G r a d i e n t
71
Low f a v o r a b l e p r e s s u r e g r a d i e n t , n o g r i d
Low f a v o r a b l e p r e s s u r e g r a d i e n t , 0.090 i n c h
grid
Low f a v o r a b l e p r e s s u r e g r a d i e n t , 0 . 2 5 0 - i n c h
grid
High F a v o r a b l e P r e s s u r e G r a d i e n t
High f a v o r a b l e p r e s s u r e g r a d i e n t , no g r i d
High f a v o r a b l e p r e s s u r e g r a d i e n t , 0 . 0 9 0 i n c h
grid
High f a v o r a b l e p r e s s u r e g r a d i e n t , 0.250 i n c h
grid
D i s c u s s i o n of t h e R e s u l t s and. Conclusions
71
79
90
100
100
105
112
121
REFERENCES CITED
139
ACKNOWLEDGMENTS
1^4
APPENDIX A
145
Turbulence I n t e n s i t y
1^5
S c a l e of Turbulence
1^5
APPENDIX B
147
APPENDIX C
154
Assembly Drawing of P l a t e
APPENDIX D
15^
156
iv
SYMBOLS
The f o l l o w i n g symbols a r e used i n t h i s d i s s e r t a t i o n and a r e
defined as indicated:
A
Area of s u r f a c e
B
Constant
C
Constant
cf
F r i c t i o n f a c t o r d e f i n e d i n Equation 34
D
Constant
d
Diameter
e
Average h o t - w i r e b r i d g e v o l t a g e w i t h o u t
flow
ë
Average h o t - w i r e b r i d g e v o l t a g e w i t h f l o w
Root-mean-square v a l u e of f l u c t u a t i n g h o t ­
w i r e v o l t a g e ( s e e n o t e a t end of l i s t )
g
A c c e l e r a t i o n of g r a v i t y
H
Head
h
Convective heat transfer coefficient
i
Current
k
Thermal c o n d u c t i v i t y of a i r e v a l u a t e d a t
a t t h e mean boundary l a y e r t e m p e r a t u r e
8
k
Thermal c o n d u c t i v i t y of p l a t e m a t e r i a l
L
S c a l e of t u r b u l e n c e
m
Exponent i n Equation 33
N.ux
N u s s e l t number based on d i s t a n c e from
leading edge, x
V
Reynolds number b a s e d on d i s t a n c e from
leading edge, x
Reynolds number based on boundary l a y e r
thickness, 5
Power
Pressure
Mean p r e s s u r e
Fluctuating pressure
Heat t r a n s f e r r a t e
Net e n e r g y l o s s by c o n v e c t i o n
Energy l o s s by c o n d u c t i o n
Energy l o s s by r a d i a t i o n
Resistance; result function in uncertainty
analysis
Resistance at a reference condition a
Correlation coefficient
Turbulence intensity
Absolute ambient temperature
A b s o l u t e t e m p e r a t u r e of p l a t e s u r f a c e
Tempera t u r e
Temperature of p l a t e back
Temperature of a i r s t r e a m o r f l u i d
Temperature o f p l a t e s u r f a c e , g e n e r a l
surface in Equation 1
Free stream velocity
Mean f r e e s t r e a m v e l o c i t y
F l u c t u a t i n g component of f r e e s t r e a m
velocity
vi
R o o t - m e a n - s q u a r e v a l u e of f l u c t u a t i n g
component of f r e e s t r e a m v e l o c i t y ( s e e
n o t e a t t h e end of l i s t )
V e l o c i t y i n x - d i r e c t i o n i n boundary l a y e r
Mean v e l o c i t y i n x - d i r e c t i o n i n boundary
layer
Fluctuating velocity in x-direction in
boundary l a y e r
Defined by Equation 35
V e l o c i t y i n y - d i r e c t i o n i n boundary l a y e r
Mean v e l o c i t y i n y - d i r e c t i o n i n boundary
layer
Fluctuating velocity in y-direction in
boundary l a y e r
Variables in uncertainty analysis
U n c e r t a i n t y i n a q u a n t i t y ( f o r l i s t of
s u b s c r i p t s u s e d w i t h t h i s symbol, s e e end
of symbol l i s t )
D i s t a n c e measured p a r a l l e l t o s u r f a c e of
p l a t e , d i s t a n c e from l e a d i n g e d g e , c o o r d i ­
nate direction
Unheated. s t a r t i n g l e n g t h
Plate thickness
D i s t a n c e measured p e r p e n d i c u l a r t o s u r f a c e
of p l a t e , c o o r d i n a t e d i r e c t i o n
Defined by E q u a t i o n 36
Thermal d i f f u s i v i t y
Boundary l a y e r t h i c k n e s s
Emissivity
Mean v a l u e of Pohlhausen p a r a m e t e r d e f i n e d
i n E q u a t i o n JQ
vil
yV
Viscosity
v
Kinematic v i s c o s i t y
g-
S t e f a n - B o l tzmann c o n s t a n t
P
Density
T
Time
Tq
Shearing stress at wall
S u b s c r i p t s f o r U n c e r t a i n t y Symbol w
1, 2, 3• •
Uncertainties in variables 1, 2, 3• • •"
A
Area
E
Voltage
e'
Root-mean-square b r i d g e v o l t a g e
e
No-flow b r i d g e v o l t a g e
e
Mean f l o w v o l t a g e
T
Turbulence
T
s
Absolute s u r f a c e t e m p e r a t u r e
Tg
Absolute ambient temperature
%
(ts
t
(ts - tf)
-
tb)
Q
N
N u s s e l t number
R
Reynolds number
U
Free stream velocity
x
Distance x
x
Plate thickness
viii
v
Kinematic v i s c o s i t y
r
Calculated result
n
Resistance
MOTE:
The symbols /y e '
a n d 7 U' , a s w r i t t e n i n t h i s
t h e s i s , imply t h a t t h e t i m e - a v e r a g i n g p r o c e s s i s
performed, on t h e squared v a l u e of t h e f l u c t u a t i n g
quantity.
1
INTRODUCTION
The i n v e s t i g a t i o n of h e a t t r a n s f e r phenomena concerned
w i t h t h e c o n v e c t i v e mode of t r a n s f e r i s f r e q u e n t l y r e s o l v e d t o
t h e problem of d e t e r m i n i n g t h e c o e f f i c i e n t of h e a t t r a n s f e r h
d e f i n e d by t h e e q u a t i o n
Q = hA(ts-tf)
(l)
Common p r a c t i c e f o r d e s i g n p u r p o s e s i n e n g i n e e r i n g i s t h e
u s e of e m p i r i c a l f o r m u l a s based on e x p e r i m e n t a l d a t a taken f o r
g e o m e t r i e s and f l o w c o n d i t i o n s s i m i l a r t o t h e problem a t h a n d ,
a s , f o r example, i n McAdams ( 2 4 ) .
T h i s p r a c t i c e , however,
d o e s n o t e l i m i n a t e t h e d e s i r a b i l i t y of f i n d i n g methods f o r
c a l c u l a t i n g t h e h e a t t r a n s f e r c o e f f i c i e n t by a n a l y t i c a l means.
The a n a l y t i c a l a p p r o a c h o b v i o u s l y r e q u i r e s a knowledge of t h e
i n t e r a c t i o n of t h e v a r i a b l e s a f f e c t i n g t h e v a l u e of t h e
c o e f f i c i e n t of h e a t t r a n s f e r , and a l t h o u g h a mass of d a t a and
e x p e r i m e n t a l c o r r e l a t i o n s h a v e been c o m p i l e d , t h e problem of
a n a l y t i c a l p r e d i c t i o n of t h e h e a t t r a n s f e r c o e f f i c i e n t h a s n o t
been s o l v e d f o r more than a few r e s t r i c t e d c a s e s .
One reason
for the difficulty in obtaining analytical solutions is the
l a c k of i n f o r m a t i o n c o n c e r n i n g t h e i n t e r a c t i o n o f some v a r i ­
a b l e s w i t h o t h e r s , and t h e magnitude of t h e s e e f f e c t s .
C e r t a i n l y t h e l a c k of a g e n e r a l m a t h e m a t i c a l f l o w model f o r
e x p r e s s i n g t h e c o n s e r v a t i o n of e n e r g y and t h e e q u a t i o n of
motion f o r a f l u i d i n a r e a d i l y s o l v a b l e e q u a t i o n i s a n o t h e r .
2
A n a l y t i c a l a p p r o a c h e s a r e u s u a l l y based on t h e s o l u t i o n
of v a r i o u s forms of t h e e q u a t i o n s of motion and of c o n s e r v a ­
t i o n of e n e r g y w r i t t e n f o r t h e boundary l a y e r of f l u i d n e x t t o
a solid wall.
I f enough s i m p l i f y i n g a s s u m p t i o n s a r e u s e d , t h e
e q u a t i o n s may b e s o l v e d .
Examples a r e t h e b o u n d a r y - l a y e r
a p p r o x i m a t i o n s , o r t h e a s s u m p t i o n of f l u i d p r o p e r t i e s i n v a r i ­
a n t w i t h t e m p e r a t u r e o r p r e s s u r e and of z e r o t u r b u l e n c e
i n t e n s i t y i n many t e x t s ( 3 ^ ) ( 8 ) ( 2 2 ) .
Experimental investi­
g a t i o n s a r e f r e q u e n t l y aimed a t f i n d i n g o u t how c e r t a i n
v a r i a b l e s a f f e c t t h e boundary l a y e r when h e a t i s t r a n s f e r r e d ,
through i t .
Those v a r i a b l e s which do n o t have a l a r g e e f f e c t
may t h e n b e n e g l e c t e d i n an a n a l y t i c a l a p p r o a c h , w i t h a
r e s u l t i n g s i m p l i f i c a t i o n of e q u a t i o n s .
In general, the effect
of v a r i a b l e s s u c h a s f l u i d p r o p e r t i e s , v e l o c i t i e s and w a l l
geometry i n t h e boundary l a y e r h a v e been i n v e s t i g a t e d e x p e r i ­
mentally.
For many y e a r s i t h a s been r e c o g n i z e d t h a t c o n d i t i o n s i n
t h e f r e e s t r e a m e x t e r n a l t o t h e boundary l a y e r a f f e c t t h e h e a t
t r a n s f e r p r o p e r t i e s of t h e boundary l a y e r .
These f r e e - s t r e a m
c o n d i t i o n s can c a u s e a change from l a m i n a r t o t u r b u l e n t f l o w
i n t h e boundary l a y e r and can c a u s e t h e p o s i t i o n of t h i s
transition to change.
Moreover, c h a n g e s have been found i n
h e a t t r a n s f e r c o e f f i c i e n t s f o r c e r t a i n w a l l g e o m e t r i e s which
have a static pressure distribution along the surface.
This
d i s s e r t a t i o n i n c l u d e s t h e r e s u l t s of an e x p e r i m e n t a l i n v e s t i ­
g a t i o n and a d i s c u s s i o n of t h e i n t e r a c t i o n of a f r e e - s t r e a m
3
c o n d i t i o n , t u r b u l e n c e i n t e n s i t y , w i t h p r e s s u r e g r a d i e n t and
l o c a l Reynolds number i n d e t e r m i n i n g t h e r a t e of h e a t t r a n s f e r
from a f l a t p l a t e .
Emphasis h a s been p l a c e d on t h e c o l l e c t i o n
of d a t a which may s e r v e a s t h e b a s i s f o r f u t u r e a n a l y t i c a l
studies.
4
REVIEW OP PREVIOUS INVESTIGATIONS
F r e e - s t r e a m t u r b u l e n c e h a s been a p a r a m e t e r i n s e v e r a l
investigations concerning heat transfer.
Perhaps the f i r s t to
n o t i c e and a p p r e c i a t e i t s e f f e c t s were i n v e s t i g a t o r s working
on h e a t t r a n s f e r from c y l i n d e r s i n c r o s s f l o w .
h a v e a l s o been c a r r i e d on f o r o t h e r s h a p e s .
Investigations
A r e v i e w of t h o s e
c o n s i d e r e d most p e r t i n e n t t o t h e c u r r e n t i n v e s t i g a t i o n i s
presented in the following paragraphs.
D e f i n i t i o n s of t h e terms t u r b u l e n c e i n t e n s i t y and s c a l e
of t u r b u l e n c e used i n t h i s s e c t i o n a r e i n Appendix A.
C y l i n d r i c a l and S p h e r i c a l Geometries
G i e d t ( 1 5 ) p r e s e n t e d d a t a f o r an e x p e r i m e n t a l i n v e s t i g a ­
t i o n of t h e e f f e c t of f r e e - s t r e a m t u r b u l e n c e i n t e n s i t y on
l o c a l h e a t t r a n s f e r and s k i n f r i c t i o n f o r a 4 - i n c h d i a m e t e r
cylinder in crossflow.
The two t u r b u l e n c e i n t e n s i t i e s used
were e s t i m a t e d a t l e s s t h a n one p e r c e n t and a t a b o u t f o u r
p e r c e n t by methods s u g g e s t e d i n work done on t u r b u l e n c e damp­
i n g s c r e e n s i n wind t u n n e l s (5) •
A s i n g l e s t r i p of n i c k e l -
chromium r e s i s t a n c e a l l o y was wound i n a h e l i x around t h e
c i r c u m f e r e n c e of t h e c y l i n d e r w i t h thermocouples a t t a c h e d t o
t h e back s i d e of t h e s t r i p .
L o c a l c o e f f i c i e n t s were o b t a i n e d
from t h e c i r c u m f e r e n t i a l t e m p e r a t u r e d i s t r i b u t i o n , t h e e l e c ­
t r i c a l power i n p u t t o t h e s t r i p and c a l c u l a t e d v a l u e s of
c o n d u c t i o n and r a d i a t i o n l o s s e s .
5
The magnitude of t h e l o c a l h e a t t r a n s f e r c o e f f i c i e n t a t
t h e upstream s t a g n a t i o n p o i n t was found t o i n c r e a s e a b o u t 25
p e r c e n t w i t h t h e t u r b u l e n c e i n t e n s i t y i n c r e a s e n o t e d above
f o r a p p r o x i m a t e Reynolds numbers based on c y l i n d e r d i a m e t e r of
95» 000, 1 7 1 , 0 0 0 and 2 1 3 , 0 0 0 .
In addition, the average heat
t r a n s f e r c o e f f i c i e n t f o r t h e whole c y l i n d e r was found t o
i n c r e a s e a p p r o x i m a t e l y 20 p e r c e n t .
K e s t i n and Maeder ( 1 8 ) performed e x p e r i m e n t s on a c y l i n ­
d e r i n which b o t h t h e t u r b u l e n c e i n t e n s i t y and s c a l e of
t u r b u l e n c e were measured w i t h a h o t - w i r e anemometer.
These
t e s t s were performed i n an o p e n - c i r c u i t wind t u n n e l on a
c y l i n d e r which was heated, i n t e r n a l l y by s a t u r a t e d steam a t a
p r e s s u r e s l i g h t l y above t h a t of t h e a t m o s p h e r e .
The s u r f a c e
t e m p e r a t u r e f o r t h e c y l i n d e r was t a k e n a s t h e a v e r a g e of t h a t
r e c o r d e d by f i v e t h e r m o c o u p l e s l o c a t e d around a c i r c u m f e r e n c e
of t h e c y l i n d e r .
Measurements were made b o t h w i t h and w i t h o u t
boundary l a y e r t r i p w i r e s on t h e s u r f a c e of t h e c y l i n d e r i n an
e f f o r t t o show t h a t t h e e f f e c t of f r e e - s t r e a m t u r b u l e n c e d i d
n o t o n l y change t h e p o s i t i o n o f t r a n s i t i o n from a l a m i n a r t o a
t u r b u l e n t boundary l a y e r , b u t had an i n d e p e n d e n t e f f e c t of i t s
own.
The r e s u l t s confirmed t h a t t h e f r e e - s t r e a m t u r b u l e n c e
i n t e n s i t y had an e f f e c t of i t s own.
F o r example, a t a
Reynolds number based on c y l i n d e r d i a m e t e r of 1 8 0 , 0 0 0 , t h e
N u s s e l t number a t t h e u p s t r e a m s t a g n a t i o n p o i n t showed a 3 °
p e r c e n t i n c r e a s e f o r a t u r b u l e n c e i n t e n s i t y r i s e from 1 p e r
cent to 2.5 per cent.
The o v e r a l l mean N u s s e l t number was
6
found t o i n c r e a s e by 26 p e r c e n t f o r a r i s e i n f r e e - s t r e a m
t u r b u l e n c e i n t e n s i t y from O.75 p e r c e n t t o 2 . 6 6 p e r c e n t when
t h e t r i p w i r e s were i n p l a c e .
Without t h e t r i p w i r e s i n
p l a c e , t h e mean N u s s e l t number i n c r e a s e d 1 4 p e r c e n t under
otherwise identical conditions.
The s c a l e of t u r b u l e n c e was
measured o v e r a s m a l l r a n g e , 0 . 1 6 2 cm t o 0 . 5 7 4 cm and n o
e f f e c t s were n o t i c e d .
K e s t i n and Maeder s p e c u l a t e d t h a t f r e e -
s t r e a m o s c i l l a t i o n s may c a u s e c h a n g e s i n t h e v e l o c i t y and
t h e r m a l boundary l a y e r s which i n t u r n w i l l c a u s e d e p a r t u r e s
from t h e Reynolds a n a l o g y between s k i n f r i c t i o n and h e a t
transfer.
Van d e r Hegge Z i j n e n ( 4 o ) measured h e a t t r a n s f e r from a
c y l i n d e r i n c r o s s f l o w f o r Reynolds numbers based on c y l i n d e r
d i a m e t e r from 60 t o 2 ^ , 8 0 0 , f o r t u r b u l e n c e i n t e n s i t i e s r a n g i n g
from 2 p e r c e n t t o 1 3 p e r c e n t and f o r r a t i o s of s c a l e of
t u r b u l e n c e t o c y l i n d e r d i a m e t e r v a r y i n g from 0 . 3 1 t o 2 4 0 .
He
used a 0.01-cm d i a m e t e r p l a t i n u m w i r e , a 0.08-cm o u t s i d e d i a ­
m e t e r n i c k e l t u b e and b r a s s t u b e s of e i g h t o u t s i d e d i a m e t e r s
r a n g i n g from 0 . 3 0 6 cm t o 4 . 1 9 cm.
The w i r e and t u b e s were
h e a t e d e i t h e r by p a s s i n g an e l e c t r i c a l c u r r e n t t h r o u g h them,
o r by an e l e c t r i c a l h e a t e r i n s i d e a t u b e .
The c o n c l u s i o n s
r e a c h e d were t h a t f o r a c o n s t a n t Reynolds number t h e h e a t
transfer increases continuously with the turbulence intensity
and when t h e t u r b u l e n c e i n t e n s i t y i s h e l d c o n s t a n t , t h e
N u s s e l t number i n c r e a s e s w i t h t h e Reynolds number.
Moreover,
when t h e Reynolds number and t u r b u l e n c e i n t e n s i t y a r e h e l d
7
constant, the heat transfer either increases or decreases with
i n c r e a s i n g r a t i o of s c a l e of t u r b u l e n c e t o c y l i n d e r d i a m e t e r ,
t h e maximum o c c u r r i n g when t h i s r a t i o i s around. 1 . 5 t o 1 . 6 .
L a s t l y , van d e r Hegge Z i j n e n noted, t h a t v a r i a t i o n s i n b o t h
s c a l e and t u r b u l e n c e i n t e n s i t y were more e f f e c t i v e i n c h a n g i n g
t h e h e a t t r a n s f e r than an i n c r e a s e i n Reynolds number.
Seban (35)$ i n s t u d i e s of t h e e f f e c t of f r e e - s t r e a m t u r ­
b u l e n c e on h e a t t r a n s f e r from c y l i n d e r s i n c r o s s f l o w , found
t h a t t h e h e a t t r a n s f e r c o e f f i c i e n t i n c r e a s e d , a s found i n
p r e v i o u s work, b u t h e a l s o n o t e d t h a t t h e maximum i n c r e a s e was
a t t h e p o i n t of l a r g e s t p r e s s u r e g r a d i e n t a l o n g t h e c y l i n d e r
s u r f a c e , and. a minimum i n c r e a s e a t t h e p o i n t of s m a l l e s t p r e s ­
sure gradient, thus leading to speculation that the turbulence
i n t e n s i t y h a s i t s g r e a t e s t e f f e c t when a l a r g e p r e s s u r e
gradient is present.
He a l s o found t h a t t h e t r a n s i t i o n from
l a m i n a r t o t u r b u l e n t boundary l a y e r f l o w took p l a c e a t lower
Reynolds numbers, b u t t h a t once t r a n s i t i o n had taken p l a c e
t h e r e was n o e f f e c t of t h e t u r b u l e n c e on t h e h e a t t r a n s f e r
i n t h e r e g i o n of t h e t u r b u l e n t boundary l a y e r .
H i s work was
performed on c i r c u l a r c y l i n d e r s of 1 . 2 5 - i n c h and 1 . 8 7 - i n c h
d i a m e t e r s and on a c y l i n d e r of e l l i p t i c a l c r o s s s e c t i o n w i t h
a 6 - i n c h major a x i s and a 2 - i n c h minor a x i s .
The h e a t
t r a n s f e r e l e m e n t s were i n d i v i d u a l c i r c u m f e r e n t i a l s t r i p s of
n i chrome.
The r a n g e of a i r v e l o c i t i e s used ranged, from 150
t o 350 f t p e r s e c , w i t h t e m p e r a t u r e d i f f e r e n c e s between t h e
c y l i n d e r and t h e a i r of from 15F t o 20F.
Turbulence
8
i n t e n s i t i e s were measured w i t h a h o t - w i r e anemometer.
The
s c a l e of t u r b u l e n c e was s m a l l compared t o t h e c y l i n d e r d i a ­
m e t e r , and t h u s n o n - c o n t r i b u t o r y a c c o r d i n g t o t h e p r e v i o u s l y
noted, work of van d e r Hegge Z i j n e n .
K e s t i n , Maeder and Sogin
( 1 9 ) d i d f u r t h e r e x p e r i m e n t a l work on a c y l i n d e r showing t h a t
t h e e f f e c t of f r e e - s t r e a m t u r b u l e n c e i s l a r g e s t when t h e g r a d i
e n t of t h e f r e e - s t r e a m v e l o c i t y i s g r e a t e s t and when t h e
f r e e - s t r e a m v e l o c i t y i s a t low t u r b u l e n c e i n t e n s i t y l e v e l s .
Work s i m i l a r t o t h a t on c y l i n d e r s h a s been performed on
spheres.
I n g e n e r a l , s i m i l a r c o n c l u s i o n s have been f o u n d .
An
i n c r e a s e i n t h e magnitude of t h e h e a t t r a n s f e r c o e f f i c i e n t a t
t h e upstream s t a g n a t i o n p o i n t and an i n c r e a s e i n o v e r a l l c o e f ­
ficient results as free-stream turbulence intensity increases.
The most f r u i t f u l of t h e i n v e s t i g a t i o n s i n v o l v i n g s p h e r e s
a p p e a r t o be t h o s e by S a t o and Sage ( 3 2 ) and by S h o r t and
Sage ( 3 6 ) .
F l a t P l a t e Geometries
Page and F a l k n e r ( 1 0 ) , i n e x p e r i m e n t s concerned w i t h t h e
a n a l o g y between s k i n f r i c t i o n and h e a t t r a n s f e r from a f l a t
p l a t e and from c y l i n d e r s i n c r o s s f l o w r e p o r t e d some of t h e
f i r s t d a t a on t h e e f f e c t of f r e e - s t r e a m t u r b u l e n c e on h e a t
t r a n s f e r from a h o r i z o n t a l f l a t p l a t e o r i e n t e d w i t h t h e flow
parallel to the surface.
The p l a t e c o n s i s t e d of a p l a t i n u m
f o i l 0.70-cm l o n g , 1.30-cm wide and. 0.00127-cm t h i c k .
The
f o i l was h e a t e d by p a s s i n g an e l e c t r i c c u r r e n t t h r o u g h i t ,
9
and current and r e s i s t a n c e measurements were used to compute
the heat t r a n s f e r .
Thus, the e n t i r e p l a t e was, in e f f e c t , a
hot-film anemometer.
Page and Falkner concluded t h a t changes
in f r e e stream turbulence did not a f f e c t the r a t e of heat
t r a n s f e r from the f i l m .
The conclusions f o r the cylinder were
t h a t changes in d i s t r i b u t i o n s of surface f r i c t i o n and heat
t r a n s f e r do take place, but no detailed investigation was made.
Sugawara,
a i . ( 3 7 ) performed e x p e r i m e n t s on a f l a t
p l a t e f o r Reynolds numbers based on d i s t a n c e from t h e l e a d i n g
edge from a b o u t 3*9 x l e P t o 3*5 x 10^ w i t h v a r y i n g t u r b u l e n c e
intensity in the free stream.
Heat t r a n s f e r was measured by
h e a t i n g t h e p l a t e , p l a c i n g i t i n t h e a i r s t r e a m , and m e a s u r i n g
temperatures as the plate cooled.
V a l u e s were c a l c u l a t e d
a c c o r d i n g t o an e q u a t i o n p r e v i o u s l y developed f o r n o n - t i m e steady cooling.
T u r b u l e n c e i n t e n s i t y was measured by a h o t ­
w i r e anemometer.
The r e s u l t s showed a l a r g e i n c r e a s e i n h e a t
t r a n s f e r c o e f f i c i e n t w i t h an i n c r e a s e i n f r e e - s t r e a m t u r b u ­
lence intensity.
Edwards and F u r b e r ( 9 ) i n v e s t i g a t e d t h e e f f e c t of f r e e s t r e a m t u r b u l e n c e on h e a t t r a n s f e r from a f l a t p l a t e w i t h z e r o
pressure gradient.
The p l a t e c o n s i s t e d of a p l a n e s u r f a c e
a b o u t 3 f e e t l o n g w i t h a 6 - by 4 - i n c h h e a t e d c o p p e r p l a t e
imbedded i n t h e s u r f a c e 33 i n c h e s downstream from t h e l e a d i n g
edge.
They were a b l e t o measure a v e r a g e c o e f f i c i e n t s over
t h i s a r e a , which was e l e c t r i c a l l y h e a t e d by nichrome s t r i p s
i n t h e i n t e r i o r o f t h e model.
The r e m a i n d e r of t h e model was
10
vmheated.
T h r e e f r e e - s t r e a m t u r b u l e n c e I n t e n s i t i e s were u s e d ,
t h e m a g n i t u d e s of which w e r e e s t i m a t e d from t h e g r i d s i z e s
used t o g e n e r a t e t h e t u r b u l e n c e .
The r a n g e of Reynolds num­
b e r s based on t o t a l p l a t e l e n g t h was from 200,000 t o 2 , 5 0 0 , 0 0 0 .
The c o n c l u s i o n s r e a c h e d were t h a t t u r b u l e n c e i n t e n s i t i e s up t o
an e s t i m a t e d 5 p e r c e n t d i d n o t i n f l u e n c e t h e h e a t t r a n s f e r ,
that turbulence intensity has l i t t l e effect In the laminar
boundary l a y e r f l o w r e g i o n , and. t h a t t h e p o s i t i o n of t r a n s i ­
t i o n from l a m i n a r t o t u r b u l e n t f l o w i n t h e boundary l a y e r i s
m a r k e d l y a f f e c t e d by f r e e - s t r e a m t u r b u l e n c e i n t e n s i t y .
No
e f f e c t of f r e e - s t r e a m t u r b u l e n c e was found f o r h e a t t r a n s f e r
t h r o u g h a t u r b u l e n t boundary l a y e r .
A n o t e t o t h e p a p e r of Edwards a n d F u r b e r ( 9 ) by
W h i t e f o o t s t a t e s t h a t , u s i n g t h e same g e n e r a l methods, n o
e f f e c t of f r e e - s t r e a m t u r b u l e n c e i n t e n s i t i e s up t o a b o u t 1 1
per cent could be found.
Wang ( 4 3 ) i n v e s t i g a t e d t h e e f f e c t of f r e e - s t r e a m t u r b u ­
l e n c e on t h e l o c a l h e a t t r a n s f e r r a t e s from two u n i t steam
h e a t e r s mounted w i t h t h e i r s u r f a c e s f l u s h w i t h t h e s u r f a c e of
a flat plate.
The h e a t e r s were m a i n t a i n e d a t c o n s t a n t tempera­
t u r e by e l e c t r i c h e a t i n g e l e m e n t s i n s i d e t h e u n i t s .
The r e s t
of t h e p l a t e s u r f a c e was m a i n t a i n e d a t v e r y n e a r l y t h e same
t e m p e r a t u r e a s t h e u n i t h e a t e r s by c i r c u l a t i n g steam from a
small external boiler inside the plate.
Free-stream turbu­
l e n c e i n t e n s i t y measurements were made w i t h a h o t - w i r e
anemometer, and v e l o c i t y p r o f i l e s i n t h e boundary l a y e r were
11
obtained with a total-head probe.
Wang conducted d e t a i l e d
e x p e r i m e n t s on t h e p l a t e f o r a n e g l i g i b l e p r e s s u r e g r a d i e n t ,
b u t some e x p l o r a t o r y p o i n t s f o r a f a v o r a b l e p r e s s u r e g r a d i ­
e n t were a l s o o b t a i n e d .
He found t h e l o c a l c o e f f i c i e n t of
h e a t t r a n s f e r t o b e u n a f f e c t e d by t u r b u l e n c e I n t e n s i t y l e v e l
i n t h e l a m i n a r boundary l a y e r r e g i o n i n t h e c a s e of n e g l i ­
gible pressure gradients.
I n t h e boundary l a y e r t r a n s i t i o n
r e g i o n , l o c a l c o e f f i c i e n t s i n c r e a s e d by a s much a s 220 p e r
c e n t f o r a change i n f r e e - s t r e a m t u r b u l e n c e i n t e n s i t y from
0.80 to 2.50 per cent.
I n t h e t u r b u l e n t boundary l a y e r
r e g i o n , changes i n t h e f r e e - s t r e a m t u r b u l e n c e i n t e n s i t y p r o ­
duced no s y s t e m a t i c e f f e c t s on t h e l o c a l c o e f f i c i e n t .
For t h e
c a s e of a p r e s s u r e g r a d i e n t , t h e e x p l o r a t o r y p o i n t s i n d i c a t e d
a s i z e a b l e i n c r e a s e i n l o c a l c o e f f i c i e n t f o r l a m i n a r boundary
layers.
Wang a l s o p r e s e n t e d an a n a l y s i s of t h e e f f e c t of f r e e s t r e a m o s c i l l a t i o n s on t h e l a m i n a r boundary l a y e r .
I t assumed
o n l y one f r e e s t r e a m o s c i l l a t i n g v e l o c i t y component, and shows
t h a t t h e d e g r e e t o which t h e boundary l a y e r i s a f f e c t e d i s
dependent on t h e a m p l i t u d e of t h e f r e e s t r e a m o s c i l l a t i o n s
o n l y when t h e o s c i l l a t i o n s v a r y i n t h e d i r e c t i o n of t h e mean
flow.
Wang n u m e r i c a l l y c a l c u l a t e d s o l u t i o n s f o r t h e e q u a t i o n s
of motion f o r a f l a t p l a t e i n a s t r e a m of c o n s t a n t mean
v e l o c i t y w i t h a s i n u s o i d a l l y v a r y i n g wave imposed on t h e f r e e
stream.
His s o l u t i o n shows t h a t a t low f r e q u e n c y and a m p l i ­
t u d e , t h e e f f e c t on h e a t t r a n s f e r c o e f f i c i e n t s w i l l b e s m a l l .
12
T h i s a n a l y s i s and i t s c o n c l u s i o n s w i l l be d i s c u s s e d i n g r e a t e r
detail in subsequent sections.
Reynold s , Kays and K l i n e (30) performed h e a t t r a n s f e r
and. t h e r m a l - and v e l o c i t y - b o u n d a r y l a y e r e x p e r i m e n t s on a n
i s o t h e r m a l f l a t p l a t e w i t h a t u r b u l e n t boundary l a y e r .
They
did. n o t d i r e c t l y d e t e r m i n e t h e e f f e c t of f r e e - s t r e a m t u r b u ­
l e n c e on t h e s e v a r i a b l e s , b u t t h e t u r b u l e n c e i n t e n s i t y of t h e
f r e e s t r e a m was r e p o r t e d .
The r e s u l t s showed, l i t t l e d i f f e r ­
e n c e from p r e d i c t i o n s f o r n o f r e e - s t r e a m t u r b u l e n c e .
This
work thus t e n d s t o c o n f i r m t h e work of Edwards and F u r b e r ( 9 ) •
F e l l e r and Yeager ( 1 2 ) r e p o r t e d on t h e e f f e c t of l a r g e
a m p l i t u d e o s c i l l a t i o n s on l o c a l h e a t t r a n s f e r c o e f f i c i e n t s .
The o s c i l l a t i o n s c o n s i s t e d , of f r e e - s t r e a m t u r b u l e n c e b o t h w i t h
and w i t h o u t a sound f i e l d p r e s e n t .
The sound, f i e l d was g e n e r ­
a t e d by a s i r e n i n s t a l l e d upstream of t h e f l a t p l a t e under
test.
The f r e q u e n c y of t h e sound f i e l d , was v a r i e d from 34 t o
680 o p s and t h e r o o t - m e a n - s q u a r e flow a m p l i t u d e s v a r i e d up t o
65 p e r c e n t .
The N u s s e l t number i n c r e a s e d by a s much a s 65
p e r c e n t o v e r a r e f e r e n c e f l o w a t t h e same Reynolds number.
An e m p i r i c a l c o r r e l a t i o n f o r t h e N u s s e l t number a s a f u n c t i o n
of t h e Reynolds number and t h e r a t i o of t h e r o o t - m e a n - s q u a r e
a m p l i t u d e of t h e f l u c t u a t i n g v e l o c i t y t o t h e mean f l o w v e l o c ­
i t y was d e v e l o p e d .
S c h l i e r e n p h o t o g r a p h s of t h e b o u n d a r y -
l a y e r flow r e v e a l e d a f l o w r e v e r s a l i n t h e l a y e r t h a t was
c o r r e l a t e d w i t h t h e f r e q u e n c y of o s c i l l a t i o n s .
The t i m e -
averaged. v a l u e of t h e t h e r m a l b o u n d a r y - l a y e r t h i c k n e s s was
13
found. "Co v a r y i n v e r s e l y a s t h e w u s s e l t number.
i n c r e a s e s f o r t u r b u l e n t f l o w were a l s o r e p o r t e d .
Heat t r a n s f e r
The I n c r e a s e
i n h e a t t r a n s f e r f o r l a m i n a r and. t u r b u l e n t f l o w s was a s c r i b e d
t o t h e same mechanism, a s y e t unknown, f o r b o t h f l o w s .
14
ANALYSIS OF THE EFFECTS OF FREE-STREAM TURBULENCE
ON THE LAMINAR BOUNDARY LAYER EQUATIONS
The c o n v e c t i v e h e a t t r a n s f e r from a body i s c o n t r o l l e d by
t h e r e s i s t a n c e t o h e a t t r a n s f e r t h r o u g h t h e boundary l a y e r .
The r e s i s t a n c e of t h e boundary l a y e r t o h e a t t r a n s f e r i s d e t e r ­
mined b y t h e n a t u r e of t h e f l o w i n t h e boundary l a y e r .
In
g e n e r a l , l e s s r e s i s t a n c e t o h e a t t r a n s f e r i s encountered w i t h
a t u r b u l e n t boundary l a y e r t h a n w i t h a l a m i n a r o n e .
The
f o l l o w i n g a n a l y s i s c o n s i s t s of a d i s c u s s i o n of t h e e f f e c t s of
t h e f r e e - s t r e a m t u r b u l e n c e on t h e e q u a t i o n s of m o t i o n , c o n ­
t i n u i t y and e n e r g y f o r t h e l a m i n a r boundary l a y e r , and i s
s i m i l a r t o t h e a n a l y s i s i n S c h l i c h t i n g ( 3 4 ) and Wang ( 4 3 ) .
E f f e c t on F l u i d - F l o w C h a r a c t e r i s t i c s
I n f l o w o v e r b o d i e s immersed i n a f l u i d , i t i s found t h a t
t h e boundary l a y e r b e g i n s a s a l a m i n a r t y p e and may undergo a
transition to the turbulent type as flow progresses over the
body.
The e x a c t mechanism i n v o l v e d i n t h e t r a n s i t i o n i s n o t
completely understood.
I t i s known, however, t h a t t h e t r a n s i ­
t i o n of t h e boundary l a y e r a s w e l l a s t h e c h a r a c t e r i s t i c s of
t h e d i f f e r e n t b o u n d a r y - l a y e r t y p e s can b e changed by c o n d i ­
t i o n s b o t h i n t e r n a l and. e x t e r n a l t o t h e boundary l a y e r .
This
a n a l y s i s i s c o n f i n e d t o t h e e f f e c t s of f r e e - s t r e a m t u r b u l e n c e
on t h e boundary l a y e r .
According t o S c h l i c h t i n g ( 3 4 ) , t h e f r e e - s t r e a m t u r b u l e n c e
i n t e n s i t y i s a rough measure of t h e a m p l i t u d e of t h e random
velocity fluctuations in the free stream.
These f l u c t u a t i o n s
a r e n o t c o m p a t i b l e w i t h t h e u s u a l a s s u m p t i o n s of a s t e a d y s t a t e f r e e s t r e a m v e l o c i t y a s a boundary c o n d i t i o n f o r t h e
s o l u t i o n of t h e t w o - d i m e n s i o n a l i n c o m p r e s s i b l e boundary l a y e r
e q u a t i o n s , s i n c e t h e y make t h e f r e e s t r e a m v e l o c i t y t i m e dependent.
Of c o u r s e , most s o l u t i o n s of t h e t w o - d i m e n s i o n a l
equations take the time-averaged free stream velocity as a
boundary c o n d i t i o n and t h u s d i s g u i s e t h i s problem.
However,
K e s t i n and Maeder ( 1 8 ) p o i n t o u t t h a t d u e t o t h e n o n - l i n e a r
e q u a t i o n s i n v o l v e d , i t i s n e c e s s a r y t o employ t h e t i m e dependent boundary-layer equations together with the timed e p e n d e n t boundary c o n d i t i o n s and t o perform t h e t i m e
a v e r a g i n g i n t h e p r o c e s s of s o l v i n g t h e e q u a t i o n s .
F o r t h e c a s e of a f l a t p l a t e w i t h z e r o p r e s s u r e g r a d i e n t ,
some e x p e r i m e n t s have shown l i t t l e i n f l u e n c e of f r e e - s t r e a m
t u r b u l e n c e on t h e boundary l a y e r .
When a p r e s s u r e g r a d i e n t
i s p r e s e n t , however, changes have o c c u r r e d i n t h e boundary
layer.
Thus, an a n a l y s i s i n c o r p o r a t i n g a p r e s s u r e g r a d i e n t
is desirable.
I n o r d e r t o a n a l y z e t h e two-dimensional boundary l a y e r
w i t h n o n - t i m e - s t e a d y v e l o c i t i e s , t h e s i m p l i f y i n g assumption of
o n l y one v a r y i n g f r e e - s t r e a m component w i l l b e made.
These
v a r i a t i o n s w i l l b e assumed t o o c c u r i n t h e d i r e c t i o n of t h e
mean v e l o c i t y of t h e s t r e a m .
The p r e s s u r e and v e l o c i t y com­
p o n e n t s a r e assumed t o c o n s i s t of t i m e - a v e r a g e v a l u e s w i t h
16
f l u c t u a t i n g v a l u e s superimposed on them.
Thus,
u = u(x,y) + u*(x,y,r)
(2)
v = v(x,y) + v'(x,y,r)
(3)
u = u( x , y )
(4)
+
u* ( x , r )
p = p ( x ) + p ' ( X,T)
(5)
where u and v a r e boundary l a y e r v e l o c i t y components, r i s
t i m e , U i s t h e f r e e s t r e a m v e l o c i t y which v a r i e s o n l y i n one
d i r e c t i o n and w i t h t i m e , p i s p r e s s u r e and t h e b a r o v e r a
symbol d e n o t e s t h e t e m p o r a l mean v a l u e and t h e prime r e p r e ­
sents the fluctuating part.
Since this analysis is restricted
t o two dimensions i n s p a c e , a l l components of v e l o c i t y i n t h e
z - d i r e c t i o n a r e assumed a s z e r o .
I n a d d i t i o n , t h e assumption
of a uniform p r e s s u r e a t any s e c t i o n a c r o s s t h e boundary l a y e r
a t any i n s t a n t i s c o n s i d e r e d i n k e e p i n g w i t h t h e o r d e r of
magnitude a n a l y s i s involved i n o b t a i n i n g t h e boundary l a y e r
equations.
Using t h e t i m e - a v e r a g i n g t e c h n i q u e s of S c h l i c h t i n g ( 3 4 ) ,
i t i s found t h a t
u 1 = v 1 = U f = p* = 0
(6)
These f u n c t i o n s f o r t h e v e l o c i t i e s and t h e p r e s s u r e a r e
to be substituted in the two-dimensional non-time-steady
17
boundary l a y e r e q u a t i o n s of motion
du
du
i 7
+
u
du
+
i l
v
^
2
du
1 dp
=
" p ^
+
( 7 )
the continuity equation for incompressible flow
du
dv
^
+
i 7
= 0
( 8 )
and t h e n o n - t i m e - s t e a d y e q u a t i o n f o r t h e f r e e s t r e a m
dU
âî
+
u
dU
1 dp
aï = "7 57
(9)
A f t e r s u b s t i t u t i o n of t h e v e l o c i t y f u n c t i o n s , t h e e q u a ­
t i o n of motion i s
d(u + u')
âî
d(u + u')
+ (û + « • ) — ^
d(u + u')
+ (v + v ) — ^
1 d(p + p')
- 7 - 1 7
d2(u + u1)
—
d
y
( i = )
2
t h e e q u a t i o n of c o n t i n u i t y i s
D(Û + U*)
— —
D(V + V')
+
— — -0
(11)
and. t h e f r e e s t r e a m e q u a t i o n i s
d(U + U»)
— +
d(U + U')
— — =
1 d(p + p')
—
( 1 2
>
18
The f r e e s t r e a m e q u a t i o n i s t h e n expanded and t h e p r e s s u r e
g r a d i e n t term r e p l a c e d w i t h t h e l e f t s i d e of t h e f r e e s t r e a m
equation, giving
du
dT
du'
+
—
du
+
u
du*
+
u
dT
du
+
u ,
du'
d^+
u î
~
du
du'
+
v
~
+
v
'd7
+
dU«
du
dû»
_ dU
_ dU*
dU
T=" + T— + U -— + U —
+ U' -— + U 1
+
dr
dr
dx
dx
dx
dx
du
+ V
dy
dl
du'
dy
d2u'
(13)
V
3y^
9y"
No boundary c o n d i t i o n s can b e p l a c e d on Equation 1 3 b e c a u s e of
t h e random n a t u r e of t h e f l u c t u a t i n g p a r t s of t h e d e p e n d e n t
variables.
In order to further analyze this equation, the
t i m e - a v e r a g e w i l l b e t a k e n , k e e p i n g i n mind t h a t a l o s s of
generality is involved.
du1
du
u
dx
+
U*
dx
+
V
du
dy
The time-averaged, e q u a t i o n i s
_dU
du'
+
v'
dy
dU'
d2û
(14)
T h i s e q u a t i o n h a s been n o t e d by o t h e r s and may be f o u n d , f o r
example, i n S c h l i c h t l n g ( 3 ^ ) .
I n a s i m i l a r f a s h i o n , t h e c o n t i n u i t y e q u a t i o n , when t i m e averaged. i s
du
^
+
dv =
^
0
(15)
19
The only differences between the above equations of
motion and the usual time-averaged boundary layer equations
du 1
a r e the terms
au*
u ' rox
— ,
v ' to—
y
au*
and
U ' ox
t— .
I t must be these
terms t h a t represent the e f f e c t s of free-stream o s c i l l a t i o n s
on the boundary l a y e r .
au'
Both of t h e terms
u'r—
dx
au*
and
U—
dx
a r e dependent on x
^
due to the partial derivative portions, therefore the values
of t h e s e depend on a change on u ' a n d / o r U ' w i t h x .
du'
v'^y
The term
can be shown t o depend, on x by n o t i n g t h a t i f u ' were a
au'
f u n c t i o n of y o n l y ,
would b e z e r o .
Thus, from t h e c o n -
dv'
tinuity equation,
= 0 , a n d , due t o t h e continuum f l o w
au'
assumption of n o s l i p a t t h e w a l l , v ' = 0 and t h e term
v'^~~
au'
would v a n i s h .
Since v' is not zero, except at the wall,
v'g
x
must be d e p e n d e n t on x a s w e l l a s y o r t h e above h y p o t h e s i s i s
valid.
I n t h e e v e n t t h a t U' i s a c o n s t a n t n o t e q u a l t o z e r o , and
au»
= 0 , i t i s p o s s i b l e t h a t u ' and v ' may c o n t i n u e t o depend
on x and y , b u t i f U' = 0 , u ' and v ' w i l l n o t e x i s t .
Thus, one c o n d i t i o n n e c e s s a r y f o r t h e f r e e s t r e a m t u r b u ­
l e n c e i n t e n s i t y t o a f f e c t t h e boundary l a y e r i s t h a t t h e
20
oscillations in the x-direction in the free stream and/or the
boundary layer vary with x, i . e . , that a t least one of the
c o m p o n e n t s u ' , v ' , and. U ' v a r y w i t h x .
A second effect necessary for the free-stream turbulence
t o a f f e c t t h e b o u n d a r y l a y e r i s t h a t u ' , v* a n d / o r U' n o t b e
t o o s m a l l compared w i t h u , t h e mean v e l o c i t y i n t h e b o u n d a r y
layer.
To show t h i s , c o n s i d e r E q u a t i o n 1 4 i n a r e a r r a n g e d form
du
5â; +
du
__ d U
dU*
du*
u'^~
= /â; +
du'
Y
-
d^û
'~|+
( l 6 )
A l l of t h e t e r m s u n d e r l i n e d by t h e b r a c k e t now a p p e a r i n t h e
same f a s h i o n a s a p r e s s u r e g r a d i e n t t e r m i n t h e s t e a d y s t a t e
boundary layer equation.
Thus, if
du7
dlP"
u"
sr
•
v
'~
Su7
ana
u
' ïïT
are a l l small enough, no effect of oscillations in the free
stream will be noticeable.
Since a l l other non-pressure-
g r a d i e n t t e r m s i n t h e e q u a t i o n depend on u , i t i s n e c e s s a r y
that a t least one of the terms
du'
u ' —
du*
'
T
' ây~
dU'
o r
u
' ~
b e l a r g e enough compared t o û t o c a u s e a c h a n g e i n t h e
boundary layer.
L o o k i n g a t i t a n o t h e r way, i f t h e s e o s c i l l a ­
tions are small enough, the boundary layer velocity profile
s h o u l d remain u n a l t e r e d from t h a t o f a mean v e l o c i t y p r o f i l e .
21
E f f e c t s on H e a t T r a n s f e r
The a b o v e d i s c u s s i o n o f t h e e f f e c t s of t h e f r e e - s t r e a m
o s c i l l a t i o n s on t h e b o u n d a r y l a y e r i s b a s e d on t h e e q u a t i o n s
o f m o t i o n and c o n t i n u i t y .
To show how t h e e n e r g y e q u a t i o n
for two-dimensional flow, usually written as
dt
IT
dt
+
u
Sx
dt
+
v
3y
d^t
=
U7)
17
i s affected, i t can be seen that if a constant temperature
d i f f e r e n c e b e t w e e n t h e p l a t e and t h e f r e e s t r e a m i s m a i n t a i n e d ,
that i s , an isothermal plate, the temperature profile will
c h a n g e o n l y w i t h c h a n g e s i n u and v (when t h e s h a p e of t h e
profile changes).
I t may b e n o t e d from t h e a b o v e a n a l y s i s t h a t two c o n d i ­
tions are necessary for free-stream turbulence to affect heat
transfer.
They a r e ( l ) t h e v e l o c i t i e s u ' , v ' , a n d / o r U* m u s t
c h a n g e w i t h x , and (2) t h e m a g n i t u d e of u ' , v 1 , a n d / o r U* m u s t
n o t b e t o o s m a l l compared t o Û.
I t i s i n t e r e s t i n g t o a p p l y t h e s e two r e q u i r e m e n t s t o t h e
f l o w a b o u t a c y l i n d e r and o v e r a f l a t p l a t e .
For the cylinder,
t h e f i r s t r e q u i r e m e n t i s f u l f i l l e d , b y t h e s l o w i n g down o f t h e
stream as i t approaches the stagnation point.
I t h a s been
shown by P i e r c y and R i c h a r d s o n ( 2 7 ) t h a t n e a r t h e s t a g n a t i o n
p o i n t o f a c y l i n d e r , t h e a m p l i t u d e of t h e o s c i l l a t i n g p o r t i o n
of the flow increases.
As t h e f l u i d moves o v e r t h e f o r w a r d
part of the cylinder, the amplitude of the oscillations
22
decreases.
P i e r c y and R i c h a r d s o n ( 2 6 ) made s i m i l a r m e a s u r e ­
m e n t s n e a r t h e s t a g n a t i o n p o i n t of a s t r u t .
They found an
oscillation amplitude increase near the stagnation point (but
n o t i n t h e b o u n d a r y l a y e r ) of a b o u t 4 . 5 t i m e s t h a t o f t h e
free-stream oscillation amplitude value.
As t h e s t r e a m
a p p r o a c h e s t h e s t a g n a t i o n p o i n t , and t h e mean v e l o c i t y
decreases, the increased amplitude of the oscillations is not
s m a l l w i t h r e s p e c t t o u and t h e second, c o n d i t i o n i s t h e n f u l ­
filled.
In addition to the above, i t should be noted that a
n e g a t i v e p r e s s u r e g r a d i e n t w i l l e x i s t on t h e f o r w a r d p o r t i o n
of the cylinder (accelerating flow).
T a y l o r ( 3 8 ) h a s shown
t h a t f o r a c o n t r a c t i o n s e c t i o n ( s i m i l a r t o t h a t used i n wind
t u n n e l s ) t h e l o n g i t u d i n a l component of t h e t u r b u l e n t o s c i l l a ­
tions is reduced in the accelerating flow.
The amount of
r e d u c t i o n i s d e p e n d e n t on t h e p a r t i c u l a r form of d i s t u r b a n c e
initially.
T h e r e i s a l s o a s t r o n g d e c e l e r a t i o n ahead o f t h e
stagnation point, indicating an increase in relative magnitude
of the fluctuating components.
The f i x e d f l a t p l a t e a t z e r o p r e s s u r e g r a d i e n t d o e s n o t
satisfy the above conditions.
The t u r b u l e n c e i n t e n s i t y of t h e
free-stream approaching the leading edge does not materially
c h a n g e p r o v i d e d t h e p l a t e i s s i t u a t e d f a r enough downstream
f r o m a t u r b u l e n c e - p r o d u c i n g body f o r t h e i n t e n s i t y o f t u r b u ­
l e n c e i n t h e s t r e a m t o b e a p p r o x i m a t e l y i s o t r o p i c and. of n e a r l y
constant magnitude, as, for example, the turbulence field
23
downstream of a g r i d . ( 5 ) .
T h i s h a s been t h e c a s e f o r e x p e r i ­
m e n t a l o b s e r v a t i o n s by Wang ( 4 3 ) and Edwards a n d F u r b e r ( 9 ) •
For a fixed flat plate with negative pressure gradient
a l o n g i t s s u r f a c e , t h e r e s u l t i n g a c c e l e r a t i o n of t h e f l o w i s
consistent with the condition of change of oscillation ampli­
tude with x.
Thus, i f the i n i t i a l free-stream turbulence
i n t e n s i t y a p p r o a c h i n g t h e p l a t e i s l a r g e enough t o make t h e
oscillation amplitude not small with respect to u, an effect
should be noticed because both conditions are fulfilled.
The
e x p l o r a t o r y e x p e r i m e n t a l work of Wang ( 4 3 ) h a s shown t h a t t h i s
i s true for a large favorable pressure gradient.
Wang found
a s u r p r i s i n g l y u n i f o r m i n c r e a s e of a b o u t 65 p e r c e n t i n t h e
N u s s e l t number f o r a n i n c r e a s e of f r e e - s t r e a m t u r b u l e n c e
i n t e n s i t y from 0 . 3 6 p e r c e n t t o 1 . 7 1 p e r c e n t o v e r t h e l a m i n a r
b o u n d a r y l a y e r r a n g e o f R e y n o l d s numbers between 5 0 , 0 0 0 and
100,000.
EQUIPMENT USED FOB EXPERIMENTAL WORK
The a n a l y s i s i n c l u d e d i n t h e p r e v i o u s s e c t i o n shows t h a t
some e f f e c t o f f r e e - s t r e a m t u r b u l e n c e may b e expected, on t h e
t e m p e r a t u r e and. v e l o c i t y d i s t r i b u t i o n s i n t h e b o u n d a r y l a y e r .
I t is not possible at present to solve the equations for a
g e n e r a l s e t of b o u n d a r y c o n d i t i o n s .
T h u s , i t was deemed
a p p r o p r i a t e t o i n v e s t i g a t e t h e problem e x p e r i m e n t a l l y i n o r d e r
t o o b t a i n a c l e a r e r p h y s i c a l p i c t u r e of t h e c o n n e c t i o n between
f r e e - s t r e a m t u r b u l e n c e and t h e b o u n d a r y l a y e r .
The i n v e s t i g a ­
t i o n was c a r r i e d o u t on a f l a t p l a t e , equipped, w i t h heated,
s t r i p s , which was placed, i n a n a i r f l o w f a c i l i t y w h e r e m e a s ­
urements were taken.
The e q u i p m e n t used i s d e s c r i b e d , i n t h i s
section.
A i r Flow F a c i l i t y
The a i r - f l o w f a c i l i t y used was a n o p e n - c i r c u i t s u c t i o n type tunnel equipped with a constant-speed centifugal fan
r a t e d , a t 20 i n c h e s o f w a t e r head and 1 3 , 0 0 0 cfm c a p a c i t y .
The
t u n n e l c o n f i g u r a t i o n and some p e r t i n e n t d i m e n s i o n s a r e shown
in Figure 1.
The t e s t s e c t i o n o f t h e t u n n e l was 1 4 i n c h e s s q u a r e and.
6 6 i n c h e s l o n g , and was c o n s t r u c t e d of P l e x i g l a s p l a s t i c and
aluminum.
I t was p r o v i d e d w i t h a s i x - i n c h - l o n g r e m o v a b l e s e c ­
t i o n a t t h e u p s t r e a m end f o r i n s e r t i o n o f t u r b u l e n c e - p r o m o t i n g
grids »
V e l o c i t y p r o f i l e s a t t h e u p s t r e a m end of t h e t e s t
section were uniform within one per cent over the range of
, Figure 1.
Air flow facility configuration
K>
on
Motor
Blower
Test section
Filters
27
velocities involved in this work.
Turbulence intensities for the tunnel were measured using
t h e h o t - w i r e anemometer e q u i p m e n t d e s c r i b e d u n d e r t h e i n s t r u ­
mentation heading in this chapter.
The f r e e t u n n e l had a
measured minimum t u r b u l e n c e i n t e n s i t y o f 0 . 4 p e r c e n t f o r t h e
higher tunnel speeds, the turbulence intensity increasing to
0 . 8 p e r c e n t a t t h e low t u n n e l s p e e d s .
Turbulence intensities
h i g h e r t h a n t h a t f o r t h e f r e e t u n n e l w e r e o b t a i n e d by u s i n g
g r i d s p l a c e d a t t h e u p s t r e a m end. of t h e t e s t s e c t i o n .
Two
g r i d s were u s e d , o n e of 0 . 0 9 0 - i n c h d i a m e t e r w i r e on o n e - i n c h
c e n t e r s , and t h e o t h e r o f 0 . 2 5 0 - i n c h d i a m e t e r rod on o n e - i n c h
centers.
The mesh o f b o t h g r i d s was s q u a r e .
The 0 . 0 9 0 - i n c h
g r i d p r o d u c e d t u r b u l e n c e l e v e l s from 1 . 8 p e r c e n t t o 3 . 2 p e r
cent during tests with the plate, while the 0.250-inch grid
produced, t u r b u l e n c e i n t e n s i t i e s of from 2 . 0 t o 8 . 3 p e r c e n t
during these tests.
The 0 . 2 5 0 - i n c h g r i d was used t o c h e c k t h e
t e s t s e c t i o n b e h a v i o r f o r t u r b u l e n c e downstream o f a g r i d .
R e s u l t s of t h e s e d a t a a r e shown i n F i g u r e 2 .
The recommended
e q u a t i o n of B a i n e s and P e t e r s o n ( l ) i s r e p r e s e n t e d by t h e l i n e
on t h e f i g u r e .
Flat Plate
The f l a t p l a t e used was o f t h e t y p e used by Drake ( 4 ) ,
F e i l e r and Yeager ( 1 2 ) and S c e s a and S a u e r ( 3 3 ) among o t h e r s .
The a s s e m b l e d p l a t e was 1 4 i n c h e s w i d e , 42 i n c h e s l o n g and
about 5/8-inch thick.
I t was composed, o f f i v e m a j o r p a r t s —
Figure 2.
T u r b u l e n c e I n t e n s i t y downstream o f a t u r b u l e n c e — p r o m o t i n g g r i d
30-0-
Grid
Rod diameters downstream of grid, —
0.250-inch rods
on 1-inch square
mesh
30
two s i d e r a i l s , a n o s e p i e c e , a p l a t e b a c k and a h e a t t r a n s f e r
surface.
The a r r a n g e m e n t o f t h e s e p a r t s i s shown i n F i g u r e 3
i n expanded form a n d i n Appendix C a s a n a s s e m b l y d r a w i n g .
Plate parts
The n o s e p i e c e was c o n s t r u c t e d o f wood and p r o v i d e d w i t h
s t a t i c p r e s s u r e t a p s and a t h e r m o c o u p l e f o r t e m p e r a t u r e m e a s ­
urement.
The s t a t i c p r e s s u r e t a p s w e r e used t o h e l p l o c a t e
t h e s t a g n a t i o n p o i n t f o r t h e oncoming a i r f l o w s o t h a t a v e r y
s l i g h t l y f a v o r a b l e a n g l e of a t t a c k was m a i n t a i n e d =
The l e a d ­
i n g e d g e of t h e n o s e p i e c e was rounded w i t h a s m a l l r a d i u s t o
aid in maintaining a stable stagnation point.
This arrange­
ment resulted in a negligible pressure gradient along the
p l a t e s u r f a c e a f t e r some e x p e r i m e n t i n g i n p o s i t i o n i n g t h e
plate in the test section.
Pressure gradients along the plate
s u r f a c e w e r e o b t a i n e d b y u s e of a f a l s e t u n n e l w a l l .
The s i d e r a i l s and p l a t e b a c k w e r e p r i m a r i l y t o add
s t r u c t u r a l s t r e n g t h and i n s u l a t i o n a r o u n d t h e h e a t t r a n s f e r
surface.
The p l a t e b a c k had g r o o v e s c u t l e n g t h w i s e i n i t s
i n t e r i o r s u r f a c e t o c a r r y e l e c t r i c a l w i r e s and p r e s s u r e t u b i n g .
T h e s e p i e c e s w e r e made o f a p a p e r - l a m i n a t e d p h e n o l i c i n s u l a t ­
i n g m a t e r i a l known c o m m e r c i a l l y a s I n s u r o k T-64-0.
The h e a t t r a n s f e r s u r f a c e was composed of 37 t r a n s v e r s e
s t r i p s of n i c k e l - c h r o m i u m r e s i s t a n c e a l l o y , c o m m e r c i a l l y known
a s Nichrome V , e a c h o n e i n c h w i d e , 0 . 0 0 2 i n c h e s t h i c k and 1 2
i n c h e s l o n g on t h e w o r k i n g s u r f a c e .
T h e s e s t r i p s w e r e mounted
Figure 3.
Figure 4.
Expanded view of m a j o r p l a t e p a r t s
D e t a i l s k e t c h of p r e s s u r e t a p i n s t a l l a t i o n
32
Plate back
Side rail
Nosepiece
Heat transfer
surface —
Side rail
Heating elements
Heating elements
Dekhotinsky
•^cement
Phenolic
backing
-Epoxy adhesive
Stainless steel tubing
Copper tubing
Plastic tubing
on a p a p e r - l a m i n a t e d p h e n o l i c m a t e r i a l i d e n t i c a l t o t h a t used
f o r t h e p l a t e b a c k and s i d e r a i l s by u s e of a n e p o x y - r e s i n
adhesive.
The e n d s o f t h e s t r i p s w e r e b e n t 90 d e g r e e s o v e r
the edges of the phenolic material.
Two h o l e s were d r i l l e d
through the resistance s t r i p into the edge of the phenolic
b a s e and a s m a l l m a c h i n e s c r e w t h r e a d was t a p p e d i n t o t h i s
hole.
A c o p p e r b u s b a r was h e l d i n p o s i t i o n w i t h s c r e w s i n
t h e s e h o l e s a t t h e end o f t h e s t r i p and e l e c t r i c a l c o n n e c t i o n s
f o r power i n p u t and v o l t a g e measurement was made t o t h e b u s
bar.
The s t r i p s w e r e s p a c e d 1 / 1 6 - i n c h a p a r t on t h e h e a t
transfer surface to allow stainless steel tubing static pres­
sure taps to be installed between s t r i p s .
The r e s u l t i n g
1 / 1 6 - i n c h b y 0 . 0 0 2 - i n c h g r o o v e was f i l l e d w i t h a h i g h t e m p e r a t u r e D e k h o t i n s k y c e m e n t and e a c h s p a c e was t h e n
smoothed b y hand t o a s s u r e a smooth w o r k i n g s u r f a c e .
A sketch
o f t h i s p o r t i o n o f t h e p l a t e i s shown i n F i g u r e 4 .
Pressure taps
S t a t i c p r e s s u r e s a t t h e p l a t e s u r f a c e w e r e measured b y
0.020-inch inside diameter stainless steel tubing inserted
between a d j a c e n t r e s i s t a n c e s t r i p s .
C a r e was t a k e n t o make
sure the tubing did not cause an electrical short circuit
between strips.
The t u b i n g was i n s e r t e d t h r o u g h a h o l e
d r i l l e d i n t h e p h e n o l i c b a s e and h e l d i n p l a c e w i t h a s p o t o f
epoxy adhesive.
E a c h t u b e was a b o u t 3 / 4 - i n c h l o n g and ended
i n a 1 / 8 - i n c h o u t s i d e d i a m e t e r c o p p e r t u b e t o which a p l a s t i c
34
t u b e l e a d i n g t o a manometer was a t t a c h e d .
A detail of this
p o r t i o n o f t h e p l a t e i s a l s o shown i n F i g u r e 4 .
Thermocouples
The t e m p e r a t u r e o f e a c h s t r i p was measured w i t h i r o n con s t a n t a n t h e r m o c o u p l e s o f 3 0 - g a g e ( 0 . 0 1 0 - i n c h d i a m e t e r ) w i r e
s p o t - w e l d e d , t o t h e b a c k s i d e o f t h e n i chrome s t r i p .
The
t h e r m o c o u p l e s were made on a Weldmatic s p o t - w e l d i n g machine
w i t h a b u t t - w e l d i n g a c c e s s o r y d e v i c e and. t h e n s p o t - w e l d e d t o
t h e b a c k s i d e of t h e 0 . 0 0 2 - i n c h r e s i s t a n c e s t r i p .
C a r e was
t a k e n t o b e s u r e t h e s p o t - w e l d i n g o p e r a t i o n did. n o t l e a v e a
r o u g h s p o t on t h e o p p o s i t e s u r f a c e o f t h e s t r i p o v e r which t h e
a i r f l o w would t a k e p l a c e .
In order to check the lengthwise temperature distribution
i n t h e s t r i p s , s e v e r a l s t r i p s had. a n a d d i t i o n a l t h e r m o c o u p l e
a t t a c h e d , o n e i n c h i n from e a c h e n d .
The a s s e m b l e d h e a t t r a n s ­
f e r s u r f a c e u n i t was checked, f o r a c c u r a c y of t e m p e r a t u r e
m e a s u r e m e n t o v e r t h e r a n g e of t e m p e r a t u r e s f o r which i t was
t o b e u s e d , by p l a c i n g i t , t o g e t h e r w i t h t h e n e c e s s a r y w i r i n g
and. em f-m ea s u r i n g equipment i n a room w h e r e t h e t e m p e r a t u r e
was v a r i e d , a n d a l l o w e d t o come t o e q u i l i b r i u m o v e r a period,
of several hours.
The t e m p e r a t u r e s of t h e b a c k s i d e of t h e w o r k i n g s u r f a c e
w e r e r e q u i r e d when c a l c u l a t i n g t h e c o n d u c t i o n l o s s e s from t h e
resistance strips.
T h e s e t e m p e r a t u r e s w e r e o b t a i n e d , from
m e a s u r e m e n t s of emf on s i x t h e r m o c o u p l e s mounted on t h e b a c k
35
side of the phenolic backing of the working surface.
Place­
m e n t o f t h e s e t h e r m o c o u p l e s i s shown i n Appendix C .
A f t e r t h e c e n t r a l s e c t i o n of t h e p l a t e was c o m p l e t e d , t h e
e l e c t r i c a l power w i r i n g was a t t a c h e d . .
t h e s t r i p s i s shown i n F i g u r e 5»
The w i r i n g d i a g r a m o f
The power i n p u t t o e a c h s t r i p
was v a r i e d b y c h a n g i n g t h e v e r n i e r r e s i s t a n c e a c r o s s e a c h
strip.
The measurement o f power i n p u t t o e a c h s t r i p was accom­
p l i s h e d by o b t a i n i n g t h e s t r i p r e s i s t a n c e a n d v o l t a g e d r o p
across each strip.
In order to obtain the voltage drop of the
resistance s t r i p alone, separate potential wires were attached
to each bus bar.
The c o p p e r b u s b a r s , e a c h a b o u t two i n c h e s
l o n g , were of n e g l i g i b l e r e s i s t a n c e .
Instrumentation
Thermocouples
Thermocouple p o t e n t i a l s were r e a d w i t h a L e e d s and
N o r t h r u p Model 8686 m i l l i v o l t p o t e n t i o m e t e r .
The m a n u f a c ­
t u r e r ' s s t a t e d l i m i t of e r r o r f o r t h i s i n s t r u m e n t i s + 0 . 0 5
p e r c e n t + 3juy
tion.
when used w i t h o u t r e f e r e n c e J u n c t i o n compensa­
A l l p l a t e s u r f a c e t e m p e r a t u r e measurements were made
b y two m e t h o d s .
The f i r s t was a d i f f e r e n t i a l r e a d i n g between
t h e f r e e s t r e a m and t h e p l a t e s u r f a c e .
The second was a
p o t e n t i a l measurement a g a i n s t a n i c e b a t h .
Thermocouple
p o t e n t i a l s w e r e checked b y c o m p a r i n g t h e d i f f e r e n t i a l m e a s u r e ­
ment a g a i n s t t h e d i f f e r e n c e found between t h e p l a t e s u r f a c e
Figure 5.
W i r i n g d i a g r a m of p l a t e h e a t i n g e l e m e n t s
37
Plate surface resistance strips
Potential.measurement leads
Autotransformer
To power source
38
p o t e n t i a l measured a g a i n s t an i c e b a t h .
I t was found t h a t
t h e s e m e a s u r e m e n t s checked, w i t h i n 0 . 5 F .
The l a r g e number o f t h e r m o c o u p l e s i n t h e p l a t e were c o n ­
n e c t e d t o L e e d s and N o r t h r u p t h e r m o c o u p l e s w i t c h e s which were
used to connect the potentiometer to the desired thermocouple.
A c i r c u i t was d e s i g n e d s o t h a t e a c h t h e r m o c o u p l e i n t h e e n t i r e
s y s t e m c o u l d b e measured e i t h e r d i f f e r e n t i a l l y w i t h t h e f r e e s t r e a m t e m p e r a t u r e o r a g a i n s t an i c e b a t h .
Free stream
t e m p e r a t u r e m e a s u r e m e n t s were- o b t a i n e d w i t h a s h i e l d e d i r o n constantan thermocouple located Just upstream of the leading
e d g e of t h e p l a t e and a b o u t f o u r i n c h e s away from t h e p l a t e
surface.
Velocity-profile instruments
The b o u n d a r y - l a y e r v e l o c i t y - p r o f i l e measurements were
made w i t h a t o t a l - h e a d , p r o b e c o n s t r u c t e d from s t a i n l e s s s t e e l
hypodermic t u b i n g w i t h a f l a t t e n e d end s e c t i o n t o r e d u c e t h e
velocity gradient across the opening facing into the flow.
s k e t c h of t h i s t u b e i s shown i n F i g u r e 6 .
A
The o p e n i n g of t h e
t u b e was l a r g e enough t o g i v e a t i m e c o n s t a n t o f t h e m e a s u r ­
i n g s y s t e m o f t h e o r d e r o f two m i n u t e s when t h e p r e s s u r e
m e a s u r e m e n t s were made w i t h a Meriam Model 3^FB2 micromanometer.
The p o s i t i o n o f t h e b o u n d a r y - l a y e r p r o b e i n r e l a t i o n
t o t h e p l a t e s u r f a c e was found b y u s e of t h e m i c r o m e t e r a d j u s t m e n t p r o b e p o s i t i o n e r shown i n F i g u r e 7 .
The z e r o
a d j u s t m e n t o f t h e p r o b e a g a i n s t t h e p l a t e s u r f a c e was made by
Figure 6.
S k e t c h o f p r o b e used f o r b o u n d a r y l a y e r p r o f i l e s
40
0.03-
zC
0.01
I
0.02
-<0.02>
Enlarged view of
probe tip
Figure 7•
Micrometer probe positioner
42
MlCtLOMblEK HE/ xd
i
SP12.1MÛ
4 P20B&-
a d v a n c i n g t h e p r o b e from a p o s i t i o n some d i s t a n c e away from
t h e p l a t e u n t i l t h e t i p of t h e p r o b e and i t s i m a g e , r e f l e c t e d
in the plate surface, Just touched.
I t was found t h a t r e p e a t
a b i l i t y of t h e z e r o p o s i t i o n was w i t h i n one p a r t 'in one
t h o u s a n d by t h i s m e t h o d .
A s t a n d a r d p i t o t t u b e was used t o d e t e r m i n e t h e mean
free-stream velocity of the tunnel.
Electrical instruments
A c o n s t a n t - t e m p e r a t u r e h o t - w i r e anemometer s y s t e m was
used f o r d e t e r m i n a t i o n of t h e t u r b u l e n c e i n t e n s i t i e s .
The
w i r e was 0 . 0 0 0 1 5 - i n c h d i a m e t e r , a p p r o x i m a t e l y 0 . 0 5 0 - i n c h l o n g
and made of a p l a t i n u m - i r i d i u m a l l o y .
The h o t - w i r e a m p l i f i e r
used was an improved, model of t h a t d e s c r i b e d b y L a u r e n c e and
L a n d e s ( 2 3 ) which h a s a f r e q u e n c y r e s p o n s e of t h e p r o b e ,
b r i d g e , a m p l i f i e r and c a b l e s t h a t i s e s s e n t i a l l y f l a t between
5 and 2 0 , 0 0 0 c y c l e s p e r second..
The v a l u e s of t h e m a g n i t u d e of t h e f l u c t u a t i n g b r i d g e
v o l t a g e were d e t e r m i n e d w i t h a m o d i f i e d a v e r a g e - s q u a r e com­
p u t e r s i m i l a r t o t h a t d e s c r i b e d by L a u r e n c e and L a n d e s ( 2 3 ) •
A w i r i n g d i a g r a m of t h e c o m p u t e r used i s shown i n F i g u r e 8 .
V a l u e s of ê , t h e a v e r a g e b r i d g e v o l t a g e when t h e w i r e i
i n t h e a i r s t r e a m , a n d e Q , t h e a v e r a g e v o l t a g e when t h e w i r e
i s i n s t i l l a i r , were made u s i n g a T e k t r o n i x Model 502
o s c i l l o s c o p e t o m e a s u r e t h e D.C. l e v e l .
T h e s e v o l t a g e s were
o f t h e o r d e r of 6 v o l t s and t h e d i f f e r e n c e between ë w and e o
^
Figure 8.
Wiring diagram of average-square computer
470K
input
( 12AX7)
i(12Ax7)
OP.B.
X
50mf
4.7K
200
33
v\
1
9
W
6K
o
©-W
o
to line
4-40/450
46
was l e s s t h a n 1 v o l t .
In order to measure this difference
a c c u r a t e l y , a c o n s t a n t b u c k i n g v o l t a g e c o n s i s t i n g of c a l i ­
b r a t e d m e r c u r y c e l l s was u s e d t o r e d u c e t h e v o l t a g e i n p u t t o
t h e o s c i l l o s c o p e t o v a l u e s which w e r e r e a d i l y m e a s u r a b l e on
the higher ranges of vertical amplifier gain, thus affording
a l a r g e r movement o f t h e t r a c e f o r a s m a l l v o l t a g e c h a n g e .
V a l u e s of e
were measured w i t h t h e h o t w i r e i n s t i l l a i r a t
t h e same t e m p e r a t u r e a s t h e t u n n e l a i r and v a l u e s o f ë
were
m e a s u r e d w i t h t h e h o t w i r e i n p o s i t i o n i n t h e wind, t u n n e l .
The h o t - w i r e b r i d g e was o p e r a t e d , w i t h a c o n s t a n t s l i g h t u n b a l ­
a n c e o f 2 microamperes t o p r e v e n t t h e a m p l i f i e r from o s c i l ­
lating.
The s m a l l s y s t e m a t i c e r r o r t h u s i n t r o d u c e d , i s
considered negligible (25).
Power i n p u t t o t h e r e s i s t a n c e h e a t e r s was measured by
o b t a i n i n g t h e r e s i s t a n c e o f t h e h e a t i n g e l e m e n t s and d e t e r m i n ­
i n g t h e v o l t a g e d r o p a c r o s s t h e i n d i v i d u a l h e a t e r s t r i p s by
means of t h e p o t e n t i a l - m e a s u r e m e n t w i r e s connected, t o e a c h b u s
bar inside the plate.
The r e s i s t a n c e of each h e a t e r was m e a s ­
u r e d on a n E l e c t r o - S c i e n t i f i c I n s t r u m e n t s impedance b r i d g e
a f t e r i n s t a l l a t i o n i n t h e w o r k i n g s u r f a c e of t h e p l a t e .
C a p a c i t i v e and i n d u c t i v e e f f e c t s of t h e h e a t e r s y s t e m were
found to be negligible.
L i n e v o l t a g e t o t h e h e a t e r s was
r e d u c e d t o a s u i t a b l e v a l u e by u s e of a n a u t o t r a n s f o r m e r , and.
t h e o u t p u t of t h e a u t o t r a n s f o r m e r t o t h e p l a t e was c o n t i n u ­
ously monitored during the testing time to maintain a constant
voltage supply.
4?
P r o c e d u r e f o r T a k i n g Data
The c h r o n o l o g y f o r a r u n began when t h e t u n n e l f a n was
s t a r t e d and t h e dampers and. v a n e - i n l e t c o n t r o l a d j u s t e d , f o r
the velocity required.
The p l a t e - h e a t e r v o l t a g e was a d j u s t e d u n t i l t h e d e s i r e d
d i f f e r e n c e between t h e f r e e s t r e a m t e m p e r a t u r e and t h e p l a t e
s u r f a c e was o b t a i n e d .
The v e r n i e r r e s i s t o r s a c r o s s e a c h
h e a t e r w e r e a d j u s t e d u n t i l t h e p l a t e s u r f a c e was a t a u n i f o r m
temperature.
M o n i t o r i n g of t h e t e m p e r a t u r e s and r e a d j u s t m e n t
o f t h e v o l t a g e s was r e q u i r e d u n t i l s t e a d y - s t a t e c o n d i t i o n s
were reached.
After steady-state conditions prevailed, the
h e a t t r a n s f e r i n f o r m a t i o n was r e c o r d e d s i x t i m e s o v e r a p e r i o d
of about fifteen minutes.
On some r u n s , most f r e q u e n t l y when
o p e r a t i n g when t h e b o u n d a r y l a y e r was i n t h e t r a n s i t i o n r a n g e
and somewhat u n s t a b l e , a d d i t i o n a l h e a t t r a n s f e r d a t a were
taken in order to be sure a true steady-state average could be
obta ined .
I m m e d i a t e l y a f t e r t h e h e a t t r a n s f e r i n f o r m a t i o n was
o b t a i n e d , a b o u n d a r y l a y e r s e a r c h was made w i t h t h e t o t a l head
tube, using the plate surface static pressure to obtain the
velocity head.
I t was assumed t h a t t h e s t a t i c p r e s s u r e a c r o s s
t h e b o u n d a r y l a y e r was c o n s t a n t .
T h i s was v e r i f i e d w i t h i n t h e
l i m i t s of measurement of t h e manometer s y s t e m by c h e c k i n g t h e
s t a t i c p r e s s u r e d i s t r i b u t i o n s a l o n g t h e p l a t e s u r f a c e and i n
the free stream Just above the plate surface outside the
b o u n d a r y l a y e r a t a d i s t a n c e s u f f i c i e n t t o a s s u r e t h e r e was n o
48
error in the reading due to the proximity of the wall.
After the boundary layer information had been obtained,
t h e p r o b e was removed and a h o t - w i r e a n e m o m e t e r p r o b e i n s e r t e d
i n t h e t e s t s e c t i o n s o t h a t t h e w i r e p o r t i o n was p a r a l l e l t o
the plate surface and perpendicular to the flow direction.
The w i r e was k e p t a t a d i s t a n c e o f a p p r o x i m a t e l y o n e i n c h f r o m
the plate surface.
H o t - w i r e d a t a was t h e n r e c o r d e d .
Methods of Calculation
The m e t h o d s u s e d t o c a l c u l a t e t h e e x p e r i m e n t a l r e s u l t s
are discussed in this section.
A set of sample calculations
f o r o n e r u n i s i n A p p e n d i x D.
GaACWAetlçm
Nussslt number
In the presentation of the results of this investigation
the Nusselt number,
N NU
x
"
hx
—
a
LL7)
was f o u n d f r o m v a l u e s o f h o b t a i n e d f r o m p o w e r and t e m p e r a t u r e
m e a s u r e m e n t s on t h e p l a t e , t h e d i s t a n c e d o w n s t r e a m f r o m t h e
leading edge, x, and the thermal conductivity of a i r taken
from reference (16).
From t h e d e f i n i t i o n o f t h e c o n v e c t i v e h e a t t r a n s f e r
coefficient h,
S>
h
-
A(+-
\
(18)
49
where A i s the s t r i p surface area, Q
i s the net rate of
e n e r g y l o s s f r o m t h e p l a t e s u r f a c e b y c o n v e c t i o n , and ( t
- t„)
i s the temperature difference between the plate surface and
the free stream.
The n e t r a t e o f e n e r g y l o s s Q.
was found, by m e a s u r i n g t h e
electrical power input to the resistance s t r i p and deducting
the losses by radiation and conduction.
0%
=
P
"
qr
"
Thus,
qc
(19)
where P i s the total power input to the s t r i p , q^ i s the
radiation loss and qQ i s the convection l o s s .
The t o t a l p o w e r i n p u t was o b t a i n e d f r o m v o l t a g e a n d
resistance measurements of each s t r i p .
The t e m p e r a t u r e d i f f e r e n c e b e t w e e n t h e p l a t e s u r f a c e a n d
t h e f r e e s t r e a m was t a k e n a s t h e a v e r a g e o f t h e r e a d i n g s m e a s ­
ured by both differential measurements and against an ice bath.
I n o r d e r t o o b t a i n t h e c o n d u c t i o n l o s s , i t was n e c e s s a r y
t o f i n d t h e t e m p e r a t u r e d i s t r i b u t i o n on t h e b a c k s i d e o f t h e
w o r k i n g s u r f a c e , t o know t h e t h e r m a l c o n d u c t i v i t y o f t h e
p h e n o l i c b a c k i n g m a t e r i a l and t o e s t i m a t e a n y end e f f e c t s d u e
to the finite length of the resistance s t r i p s .
The t e m p e r a ­
t u r e d i s t r i b u t i o n on t h e b a c k s i d e o f t h e w o r k i n g s u r f a c e was
o b t a i n e d f r o m t h e t h e r m o c o u p l e s m o u n t e d on t h e p h e n o l i c m a t e ­
rial.
The t e m p e r a t u r e d i f f e r e n c e was r e a d b y a d i f f e r e n t i a l
reading with the plate working surface thermocouples.
Since
t h e w o r k i n g s u r f a c e o f t h e p l a t e was l a r g e w i t h r e l a t i o n t o
50
the thickness of the phenolic material, a one-dimensional flow
o f e n e r g y b y c o n d u c t i o n was a s s u m e d .
The l o s s was c a l c u l a t e d
from
(ts - tb)
%
=
V-ç—
(20)
where xp i s the thickness of the phenolic backing, ( t g - t^)
i s t h e d i f f e r e n c e b e t w e e n t h e p l a t e s u r f a c e t e m p e r a t u r e and
the back side of the working surface, k^ i s the thermal con­
d u c t i v i t y o f t h e m a t e r i a l and A i s t h e a r e a o f t h e r e s i s t a n c e
strip.
The t h e r m a l c o n d u c t i v i t y was found from v a l u e s g i v e n
in reference (13).
The t e m p e r a t u r e v a r i a t i o n a l o n g t h e l e n g t h
o f a r e s i s t a n c e s t r i p ( i . e . s p a n w i s e on t h e p l a t e s u r f a c e ) f o r
t h i s t y p e o f c o n s t r u c t i o n was shown t o b e n e g l i g i b l e b y
Drake (4).
However, s e v e r a l s t r i p s had a d d i t i o n a l t h e r m o ­
c o u p l e s a t t a c h e d o n e i n c h i n from e a c h end o f t h e s t r i p .
T h e s e w e r e c h e c k e d and f o u n d t o h a v e l e s s t h a n o n e d e g r e e P
d i f f e r e n c e f r o m t h e c e n t e r l i n e t e m p e r a t u r e o f t h e same s t r i p
f o r m o s t v a l u e s o f f r e e s t r e a m t e m p e r a t u r e and v e l o c i t y ,
a l t h o u g h some of t h e h i g h e r v e l o c i t y r u n s had v a r i a t i o n s o f
about 1.5 degrees P.
I t was n o t f e l t t h a t c o n d u c t i o n a l o n g
t h e s t r i p was a s i g n i f i c a n t f a c t o r s i n c e t h e t e m p e r a t u r e d i f ­
ference between the working surface and the back side of the
working surface remained constant.
calculated from
Radiation losses were
51
qr
where
= 6 <5-A(Ts4 - Tg4)
(21)
i s the emissivity of the s t r i p material, A i s the
s t r i p a r e a , cr i s t h e S t e f a n - B o l t z m a n n c o n s t a n t , T g i s t h e
absolute temperature of the s t r i p surface and Tg i s the
temperature of the surroundings.
The v a l u e o f e m m i s s i v i t y was
t a k e n from S c e s a a n d S a u e r ( 3 3 ) f o r N i c h r o m e V r e s i s t a n c e
a l l o y . . The t e m p e r a t u r e o f t h e s u r r o u n d i n g s was t a k e n a s t h e
room w a l l s u r f a c e t e m p e r a t u r e .
Corrections for absorption in
t h e room a t m o s p h e r e a n d i n t h e p l a s t i c t u n n e l w a l l w e r e
assumed t o b e n e g l i g i b l e .
The r a d i a t i o n g e o m e t r i c " v i e w
factor" was assumed to be unity as implied in Equation 21.
The N u s s e l t n u m b e r s p r e s e n t e d i n t h e r e s u l t s s e c t i o n h a v e
been corrected for the unheated starting length of the nosepiece according to the equation of Eckert (?) for the laminar
range,
X0\ 3 / 4
N.
Nu
=
N
x, corrected
Nu
1
x
- 1/3
(22)
-\T
and the Nusselt numbers in the turbulent range have been
corrected according to the equation of Rubesin (31)
\ 39/40
N
Nu
=
x, corrected
N
Nu
1 -
- 7/39
(23)
52
Calculation of Reynolds number
The R e y n o l d s number f o r a n e x p e r i m e n t a l p o i n t f o r f l o w s
w h e r e t h e p r e s s u r e g r a d i e n t a l o n g t h e p l a t e s u r f a c e was
n e g l i g i b l e was formed, f r o m
N He
=
Ux
~
(24)
X
w h e r e U i s t h e mean f r e e s t r e a m v e l o c i t y , x i s t h e d i s t a n c e
from the leading edge of the plate and v i s the kinematic
v i s c o s i t y o f a i r a t low p r e s s u r e s t a k e n f r o m r e f e r e n c e ( 1 6 ) .
The f r e e s t r e a m v e l o c i t y was d e t e r m i n e d f r o m t h e v e l o c i t y
head, density and temperature of the free stream according to
the expression
U
=
where H i s the velocity head,
yiiH
(25)
The d e n s i t y w a s c a l c u l a t e d from
the perfect gas law using the static pressure in the test sec­
t i o n a n d t h e a v e r a g e o f t h e p l a t e s u r f a c e and f r e e s t r e a m
temperatures.
Boundarv-laver measurements
The b o u n d a r y l a y e r d a t a a s t a k e n d u r i n g t h e e x p e r i m e n t a l
runs involved an inherent error due to the height of the total
head probe.
A correction to the measured distance above the
p l a t e s u r f a c e was made a c c o r d i n g t o t h e f i n d i n g s o f Young and
Maas ( 4 4 ) .
This correction involved adding a correction
f a c t o r b a s e d on t h e r a t i o o f t h e p r o b e h e i g h t t o t h e m e a s u r e d
distance of the probe from the surface.
When t h e b o u n d a r y l a y e r t h i c k n e s s w a s r e q u i r e d i n c a l ­
c u l a t i o n s , i t was t a k e n a s t h e d i s t a n c e a b o v e t h e s u r f a c e
w h e r e t h e b o u n d a r y l a y e r v e l o c i t y was 0 . 9 9 3 o f t h e f r e e s t r e a m
velocity.
S i n c e some i n v e s t i g a t o r s a s s u m e t h e p o s i t i o n o f t h e
boundary layer thickness to be a t 0.99 of the free stream
v e l o c i t y and o t h e r s u s e 0 . 9 9 5 » a n i n t e r m e d i a t e v a l u e was
chosen.
The s e l e c t i o n o f t h e v e l o c i t y w h e r e t h e f r e e s t r e a m
and t h e b o u n d a r y l a y e r a r e c o n s i d e r e d t o m e r g e i s a n a r b i t r a r y
choice.
Tttrftttleqge i n t e n s i t y m e a s u r e m e n t s
Data taken with the hot-wire anemometer were reduced
u s i n g e q u a t i o n s b a s e d on t h e r e l a t i o n
(26)
R - Ra
King (20) developed the original form of Equation 24 for
incompressible flow, assuming that heat transfer from the wire
did not change the flow field.
The v i s c o u s e f f e c t s o f t h e
flow about the wire were not taken into account.
These limi­
tations have not been proved to be serious for low-speed
c o n t i n u u m f l o w s , and a l a r g e a m o u n t o f d a t a i n t h e t e c h n i c a l
l i t e r a t u r e i n d i c a t e s t h e e q u a t i o n may b e u s e d w i t h l i t t l e
error for applications such as the present study.
54
E q u a t i o n 26 g i v e s a r e l a t i o n s h i p b e t w e e n f l u i d v e l o c i t y
and. c u r r e n t t h r o u g h t h e w i r e .
For a wire of constant resist­
a n c e ( i . e . , c o n s t a n t t e m p e r a t u r e ) , R i s c o n s t a n t and t h e
t u r b u l e n c e i n t e n s i t y may b e f o u n d f r o m t h e r e l a t i o n
U
ew
-(?)
w h i c h was o r i g i n a l l y d e v e l o p e d by L a u r e n c e and L a n d e s ( 2 3 ) and
in which ë
i s t h e a v e r a g e b r i d g e v o l t a g e m e a s u r e d when t h e
wire i s in the a i r stream, e
is the average bridge voltage
when t h e w i r e i s i n s t i l l a i r and / y ë ' ^
is the root-mean-
square of the fluctuating voltage about the average bridge
v o l t a g e when t h e w i r e i s i n t h e a i r s t r e a m .
Uncertainties in Experimental Results
I t would b e o f v a l u e t o h a v e s u f f i c i e n t r e p l i c a t i o n o f
data for each flow condition to treat i t statistically.
In
engineering experiments, however, i t i s difficult to obtain
m o r e t h a n o n e o r two d a t a p o i n t s f o r a g i v e n r u n o r o p e r a t i n g
condition.
This fact precludes statistical treatment of data.
N e v e r t h e l e s s , some e s t i m a t e o f e r r o r i s d e s i r a b l e .
The e s t i ­
m a t e s o f e r r o r f o r t h i s e x p e r i m e n t a r e b a s e d on t h e " s i n g l e
s a m p l e " p r o c e d u r e s o f K l i n e and M c C I i n t o o k ( 2 1 ) and t h e
examples of Thrasher and Binder (39)•
55
The method o f K l i n e and M c C l i n t o c k ( 2 1 ) r e q u i r e s t h a t
e a c h v a r i a b l e used, i n t h e c o m p u t a t i o n o f a q u a n t i t y b e
a s s i g n e d a n u n c e r t a i n t y i n t e r v a l t o g e t h e r w i t h odd s t h a t t h e
value of a given variable lies within this interval.
Were i t
possible to treat the data for any variable statistically, the
u n c e r t a i n t y i n t e r v a l would c o r r e s p o n d t o some p r e c i s i o n i n d e x ,
such as a standard deviation.
Lacking enough data for such
treatment, the investigator must use his familiarity with the
e x p e r i m e n t a l e q u i p m e n t , h i s k n o w l e d g e o f t h e phenomena
involved, and his assessment of the care used in gathering
data to state an interval of uncertainty for the values of
t h e s e d a t a a n d g i v e odd s t h a t t h e v a l u e s a r e w i t h i n t h a t
interval.
Uncertaintv
j & £ NnSSS3.t number
Prom K l i n e and M c C l i n t o c k ( 2 1 ) , t h e u n c e r t a i n t y i n a c a l ­
c u l a t e d r e s u l t w h i c h i s f o u n d from a l i n e a r f u n c t i o n o f
variables is
1 1/2
w.
r
(28)
w h e r e w^ i s t h e u n c e r t a i n t y i n t e r v a l i n t h e c a l c u l a t e d r e s u l t ,
R i s t h e f u n c t i o n , v ^ , v 2 , . . . v^ a r e v a r i a b l e s i n t h e f u n c ­
t i o n R a n d t h e w ^ , w^, . » « w^ a r e t h e u n c e r t a i n t y l i m i t s
p l a c e d on t h e s e v e r a l v a r i a b l e s b y t h e e x p e r i m e n t e r .
56
Applying the above to the calculation for the Nusselt
n u m b e r , f r o m E q u a t i o n s 1 8 a n d 1 9 , t h e N u s s e l t number i s
«n X
N
Nu
Alca(ts
"
(29)
V
T h u s E q u a t i o n 2 8 b e c o m e s , f o r t h e N u s s e l t number
BN Nu
»NKu
wN
%
WQ
5x
BN
Nu
BN Nu
x
BA •
"x
Wa
+
1/2
x
<Hts - t f )
(30)
wtj
I t should be noted that the uncertainty in
in Equation
30 i s d e p e n d e n t on t h e u n c e r t a i n t i e s o f s e v e r a l o t h e r v a r i ­
a b l e s.
These have been treated in a manner similar to the
N u s s e l t n u m b e r t o o b t a i n t h e u n c e r t a i n t y i n Q^.
Values of the
uncertainties used in Equation 30 are tabulated in Appendix B.
When u n c e r t a i n t y i n t e r v a l s w e r e c a l c u l a t e d , i t w a s
r e a d i l y a p p a r e n t t h a t two e x p e r i m e n t a l d a t a r e a d i n g s w e r e
responsible for the major portion of the uncertainty in the
Nusselt number.
T h e s e w e r e t h e v o l t a g e r e a d i n g f o r t h e power
i n p u t and t h e t e m p e r a t u r e d i f f e r e n c e u s e d f o r c a l c u l a t i n g t h e
conduction loss through the plate back.
These readings were
known t o be c r i t i c a l when t e s t i n g s t a r t e d and e v e r y e f f o r t was
57
made t o s e c u r e a c c u r a t e d a t a .
The maximum p r e d i c t e d u n c e r ­
t a i n t y i n t h e N u s s e l t n u m b e r f o u n d f o r t h e e x p e r i m e n t was 7 . 5 1
per cent.
A l i s t i n g o f t h e maximum u n c e r t a i n t i e s f o r v a r i o u s
runs i s in Appendix B.
UflgerfraiUltiss l a
Rf r yflo;i<3fi n u m b e r
The e q u a t i o n f o r t h e R e y n o l d s n u m b e r u n c e r t a i n t y i s
1/2
(3D
wR
The u n c e r t a i n t i e s f o r t h e i n d i v i d u a l v a r i a b l e s may b e
found in Appendix B.
The maximum u n c e r t a i n t y i n t h e R e y n o l d s
n u m b e r f o r a l l r u n s was f o u n d t o b e 2 . 5 3 p e r c e n t .
I t should
b e n o t e d t h a t t h e m e a s u r e m e n t s h e r e w e r e much l e s s c r i t i c a l
t h a n t h o s e f o r t h e N u s s e l t n u m b e r , and l i t t l e v a r i a t i o n f r o m
r u n t o r u n was e n c o u n t e r e d .
U n c e r t a i n t l e s j j l t h e PfSSRUrS KMdiWnt
T h e o n l y m e a s u r e m e n t made f o r t h i s p a r a m e t e r w a s t h e
+
s t a t i c pressure, which has an estimated uncertainty of - 0.001
inch of water (odds of 20 to l ) .
Uncerta in ties in turbulence intensity measurement
The t u r b u l e n c e i n t e n s i t y was m e a s u r e d w i t h a h o t - w i r e
anemometer of the constant temperature type, which, for lower
levels of turbulence, say less than 2 per cent, i s not as pre­
cise as other types.
The u n c e r t a i n t y i n t e r v a l f o r t h e
58
turbulence intensity i s expressed as
and values of the calculated uncertainty for a l l runs are in
Appendix B.
The maximum v a l u e o f u n c e r t a i n t y i n a l l r u n s was
1 7 =5 p e r c e n t o f t h e t u r b u l e n c e i n t e n s i t y r e p o r t e d .
It is
i m p o r t a n t t o r e c o g n i z e t h a t i n t h i s work t h e r e l a t i v e v a l u e s
o f t u r b u l e n c e i n t e n s i t y a r e more o f v a l u e t h a n t h e a b s o l u t e
magnitude.
In addition, only a very low turbulence inten­
s i t i e s , w h e r e t h e f l u c t u a t i n g c o m p o n e n t s a r e s m a l l and
t h e r e f o r e h a r d t o m e a s u r e , was t h e u n c e r t a i n t y t h i s l a r g e .
59
RESULTS OF THE EXPERIMENTAL INVESTIGATION
The r e s u l t s o f t h e e x p e r i m e n t a l i n v e s t i g a t i o n a r e p r e ­
s e n t e d and d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n c l a s s i f i e d
according to the variables of primary interest in each series
of tests.
C o m p a r i s o n s t o p u b l i s h e d w o r k s o f o t h e r s a r e made
where possible.
Zero Pressure Gradient
The f i r s t s e r i e s o f t e s t s was c a r r i e d o u t w i t h n o p r e s ­
sure gradient along the plate surface.
The o b j e c t i v e o f t h e s e
t e s t s was t o c h e c k t h e wind t u n n e l and p l a t e e q u i p m e n t a g a i n s t
e a r l i e r a n a l y t i c a l and e x p e r i m e n t a l work t h a t h a d b e e n d o n e on
t h i s c o n f i g u r a t i o n , and t o o b s e r v e t h e e f f e c t s o f a n i n c r e a s e d
free-stream turbulence, intensity to check the results of other
investigators.
The pressure distribution for this series of
t e s t s i s shown i n F i g u r e 9 a s t h e r a t i o o f t h e s t a t i c p r e s s u r e
to the stagnation pressure of the free stream measured oppo­
s i t e the f i r s t heat transfer measuring station back from the
leading edge.
All other pressure distributions are presented
i n t h e same w a y .
T h e h e a t t r a n s f e r r e s u l t s a r e shown i n F i g u r e 1 0 , w h e r e
the two sets of points correspond to different ranges of free
stream turbulence intensities, one for the free tunnel with no
g r i d , w h e r e t h e r a n g e was f r o m 0 . 4 0 t o 0 . 8 0 p e r c e n t , and t h e
other for the range from 1.30 to 1.80 per cent.
The h i g h e r
Figure 9.
Static pressure distribution for zero pressure
gradient
61
0.70
0.60 -
0.50-
Q]0
o o Do
O
•
o
Don
on
o
o
0.40O Plate surface taps
• In stream above plate surface
0.30-
~~î
10
i
i
i
i
15
20
25
30
Distance from leading edge, inches
35
40
Figure 10.
Heat transfer results for zero pressure gradient
63
ioJ
von Karman (42)
. 34
J
OO
Prandtl (29),
eq. 35
2-i
Gw
M
O/
O no grid, 0.4%< T< 0.8%
• no grid, check points
^-0
0.5 - i
A 0.090-inch grid,
1.3%<T<1.8%
Pohlhausen (28),
eq. 33
o.:
"i—r—i—r
"i
10
Reynolds number, N_
26
, x 10
1
1
i
r
60
64
range i s due to the insertion of the 0.090-inch grid upstream
of the plate.
The r e s u l t s f o r t h e l a m i n a r r a n g e o f R e y n o l d s n u m b e r s
with no grid in place agree within - 5 per cent of the line
representing the equation of Pohlhausen (28) for the laminar
boundary layer.
For the higher Reynolds numbers, the points
are above the line and indicate that they are deviating
further from the line as the Reynolds number increases.
This
i s to be expected as the transition from laminar to turbulent
flow in the boundary layer occurs near these Reynolds numbers.
The r e s u l t s o b t a i n e d w i t h t h e 0 . 0 9 0 - i n c h g r i d i n p l a c e
i n d i c a t e t h a t t h e p o s i t i o n o f t r a n s i t i o n moved t o w a r d t h e
l e a d i n g e d g e a n d t h a t t h e f l o w o v e r t h e p l a t e s u r f a c e was p r i ­
marily turbulent.
T h e p o i n t s i n t h e t u r b u l e n t R e y n o l d s number
r a n g e a g r e e w i t h t h e e q u a t i o n s f r o m t h e l i t e r a t u r e a s shown i n
Figure 10.
The e q u a t i o n s r e p r e s e n t e d b y l i n e s on t h e f i g u r e ,
a l l calculated for a Prandtl number of 0.7> are that of
Pohlhausen (28) for a laminar boundary layer,
[,Nu
x
"
° - 2 9 5 (NRex ) ° ' 5
(33)
t h e e q u a t i o n o f von ICarman ( 4 2 ) f o r a t u r b u l e n t b o u n d a r y l a y e r
X
•
°-241(X)°'8
(34)
65
and the equation originally due to Prandtl (29) for a turbu­
lent boundary layer
ux
kN
\ °-8
=
(35)
In order to check the behavior of the boundary layer as
indicated by the heat transfer results, velocity profiles of
the boundary layer were taken a t several positions along the
plate surface.
Typical profiles for Reynolds numbers in the
l a m i n a r r a n g e a r e shown i n F i g u r e 1 1 w h e r e t h e y a g r e e w e l l
with the line representing the Blasius solution to the twodimensional boundary layer equations.
Note that points are
p r e s e n t e d f o r f l o w w i t h and w i t h o u t h e a t t r a n s f e r .
Velocity
p r o f i l e s f o r t h e t u r b u l e n t b o u n d a r y l a y e r a r e shown i n
Figure 12 where the agreement with the universal turbulent
velocity profile i s good.
T h e s e p r o f i l e s w e r e p l o t t e d on t h e u n i v e r s a l t u r b u l e n t
b o u n d a r y l a y e r c o o r d i n a t e s s u g g e s t e d , by C l a u s e r (3)•
Plotting
the data to these coordinates involves knowledge of the
coefficient of friction, c ^, defined as
2TO
cr
1
=
(36)
ptr
D e t e r m i n a t i o n o f v a l u e s o f c ^ f o r e a c h r u n was made a c c o r d i n g
to a procedure proposed by Clauser (2).
were then used to compute values of u
4*
These values of c^
+
and y
which are
Figure 11.
Laminar boundary layer profiles for zero pressure gradient
1.0
0.8-
0.6
48,000 (cold)
48,000 (hot)
0.4-
117,900 (hot)
187,900 (hot)
0.2Blasius profile (34)
Figure 12.
Turbulent boundary layer profiles for zero pressure gradient
25
Profile from Clauser (3)
Reynolds number, N
Re x
O 145,500
•
10-
O
o
M3
• 250,400
0.090-inch grid installed
O
1
1
r—1 1—I—r
10
1
r—i—|
100
r
600
70
defined as
+
u
y
+
u
(37)
U
Uy
(38)
v
I t should be noted that the temperature difference between
t h e p l a t e s u r f a c e a n d t h e f r e e s t r e a m was m a i n t a i n e d , a t
a p p r o x i m a t e l y 3OP f o r t h e d a t a shown, w i t h t h e e x c e p t i o n o f t h e
check points which were a t temperature differences of 15 and
20 F .
L i t t l e e f f e c t o f h e a t t r a n s f e r on t h e b o u n d a r y l a y e r
could be expected, a t these temperature differences since the
variation in fluid properties across the boundary layer i s
small.
The r e s u l t s o f t h i s s e r i e s show t h a t , f o r t h e R e y n o l d s
number r a n g e i n v e s t i g a t e d , t h e a n a l y t i c a l s o l u t i o n o f
P o h l h a u s e n and t h e s e m i - e m p i r i c a l e q u a t i o n s o f von Karman a n d
Prandtl agree with the experimental data.
The p o i n t s a l s o
a g r e e w i t h t h e e x p e r i m e n t a l d a t a o f E d w a r d s and. F u r b e r ( 9 ) e n d
Wang ( 4 3 ) , who showed t h a t t h e e f f e c t o f i n c r e a s e d f r e e s t r e a m
turbulence intensity only serves to advance the position of
t r a n s i t i o n o f t h e b o u n d a r y l a y e r and h a s n o e f f e c t on t h e h e a t
transfer in the laminar range.
Since the free stream turbulence generated by the .090i n c h w i r e g r i d was s u f f i c i e n t t o c a u s e t r a n s i t i o n t o b e
c o m p l e t e a t a R e y n o l d s number o f a b o u t 1 0 0 , 0 0 0 , a n d , s i n c e t h e
71
results indicate only a further change in the position of
t r a n s i t i o n , a h i g h e r f r e e s t r e a m t u r b u l e n c e i n t e n s i t y would
show o n l y t u r b u l e n t b o u n d a r y l a y e r f l o w o v e r t h e R e y n o l d s
number range.
For this reason, no higher free stream turbu­
lence intensities were run.
A further comparison of the data with published results
c a n b e made b y u s i n g t h e h e a t t r a n s f e r d a t a t o d e t e r m i n e t h e
R e y n o l d s n u m b e r s o f t r a n s i t i o n a n d c o m p a r i n g them w i t h t h e
w o r k o f G a z l e y ( 1 4 ) on t r a n s i t i o n o f t h e l a m i n a r b o u n d a r y
layer.
This i s done in Figure 13 where i t appears that the
e x p e r i m e n t a l p l a t e u s e d had somewhat l o w e r t r a n s i t i o n R e y n o l d s
numbers than Gazley predicts.
Low F a v o r a b l e P r e s s u r e G r a d i e n t
T h i s s e r i e s o f t e s t s was r u n w i t h t h e l o w e r o f two p r e s ­
sure gradients used.
The p r e s s u r e d i s t r i b u t i o n f o r t h i s
s e r i e s i s shown i n F i g u r e 1 4 .
Low f a v o r a b l e p r e s s u r e g r a d i e n t . £ £ e r i d
The h e a t t r a n s f e r r e s u l t s f o r t h e l o w p r e s s u r e g r a d i e n t
w i t h o u t a g r i d a r c shown i n F i g u r e 1 5 i n t h e form o f t h e l o c a l
Nusselt number
as a function of the local Reynolds
x
n u m b e r , N^ e .
x
The N u s s e l t n u m b e r s shown h a v e b e e n c o r r e c t e d
f o r t h e u n h e a t e d s t a r t i n g l e n g t h o f t h e n o s e p i e c e b y t h e same
procedure used for the zero pressure gradient data.
Figure 13.
Transition Reynolds numbers for zero pressure
gra dien t
73
2-
1-
0.5-
0. 2 Gazley (14)
O
0-05-
Transition Reynolds
numbers from this
work, values obtained
from heat transfer data
•H
0.02-
0.01
0.5
1
2
5
Reynolds number, N
10
Re
20
x 10 ^
x
50
Figure 14.
S t a t i c p r e s s u r e d i s t r i b u t i o n f o r low f a v o r a b l e
pressure gradient
75
0.40
0.30
U
•H
Cu
CO
CO
c
0.20
eu
CL
0 . 1 0 -i
_i
,
I
i
I
I
5
10
15
20
25
30
Distance from leading edge, inches
35
Figure 15»
H e a t t r a n s f e r r e s u l t s f o r low f a v o r a b l e p r e s s u r e
gradient with no grid
77
10
O no grid, 0.42%<T<0.74%
0-5-
0.
Pohlhausen (28),
eq. 33
i
2
|
|
5
|
|
:
i
i
|
10
i
20
Reynolds number, N_
, x 10
Re%
-4
i
r
60
The p o i n t s o b t a i n e d b y e x p e r i m e n t l i e o n l y s l i g h t l y a b o v e
t h e P o h l h a u s e n r e l a t i o n f o r t h e l a m i n a r b o u n d a r y l a y e r and n o
significant increase in heat transfer is apparent.
No i n d i c a ­
tion of boundary layer transition i s evident from these data.
The a b s e n c e o f a t r a n s i t i o n r e g i o n i n t h e h e a t t r a n s f e r
data i s confirmed, by boundary layer profiles taken a t several
stations along the plate surface.
S i n c e t h e r e was a p r e s s u r e
gradient along the plate surface, the free stream velocity
could be expected to vary with distance back from the leading
e d g e o f t h e p l a t e , and t h e b o u n d a r y l a y e r p r o f i l e s c o u l d n o
longer be expected to conform to the shape of the Blasius
profile.
However, the fact that the heat transfer data
c o r r e l a t e i n a s t r a i g h t l i n e on t h e c o o r d i n a t e s u s e d s u g g e s t s
a similarity in boundary layer profile shape.
This can be
e x p l a i n e d b y n o t i n g t h a t t h e r e a r e some s o l u t i o n s o f t h e
laminar boundary layer equations which admit similar profiles
and have a variation in free stream velocity.
One s o l u t i o n o f
this type applies to the free-stream velocity variation
U(x)
w h e r e C and m a r e c o n s t a n t s .
=
Cxm
(39)
This corresponds to a solution
o f P a l k n e r and S k a n ' s ( 1 1 ) e q u a t i o n s o f t h e t w o - d i m e n s i o n a l
b o u n d a r y l a y e r f o u n d by H a r t r e e ( 1 ? ) •
The p r e s e n t t e s t c o n d i t i o n s a p p r o x i m a t e d t h i s t y p e o f
solution.
The f r e e s t r e a m v e l o c i t y was p l o t t e d a s a f u n c t i o n
o f l e n g t h on l o g a r i t h m i c c o o r d i n a t e s a n d t h e s l o p e f o u n d .
79
T h e e x p o n e n t m a s f o u n d b y t h i s m e t h o d was 0 . 1 4 9 f o r t h e l o w
pressure gradient, independent of the magnitude of the free
stream velocity.
The b o u n d a r y l a y e r s w e r e t h e n p l o t t e d u s i n g
H a r t r e e ' s c o o r d i n a t e s a s shown i n F i g u r e 1 6 w h e r e c o o r d i n a t e s
for m = 0.149 were interpolated between the tabulated values
for m = 0.10 and m = 0.20 given in Hartree's paper (17).
It
i s s e e n t h a t t h e d a t a c o r r e l a t e w e l l on t h e s e c o o r d i n a t e s a n d
that the analytical solution by Hartree i s closely followed
for the range of Reynolds numbers investigated.
The d i f f e r ­
e n c e b e t w e e n H a r t r e e ' s d e f i n i t i o n o f t h e p a r a m e t e r r\ a n d
m + 1
Blasius' definition is the factor
2
The t u r b u l e n c e i n t e n s i t y r a n g e o v e r w h i c h t h e d a t a w e r e
o b t a i n e d , was f r o m 0 . 4 2 t o 0 . 7 4 p e r c e n t .
No e f f e c t o f t h i s
low level of turbulence i s noticeable from either the heat
transfer data or the boundary layer data.
The t e m p e r a t u r e d i f f e r e n c e b e t w e e n t h e p l a t e and t h e f r e e
s t r e a m was a n o m i n a l 3 °
The actual temperature difference
varied by less than 1 F from s t r i p to s t r i p along the plate
surface while the deviation from the nominal terperaturc
d i f f e r e n c e f o r t h e w h o l e p l a t e was a b o u t 1 F .
Because of the
small temperature difference between the plate surface and the
f r e e s t r e a m , l i t t l e e f f e c t o f h e a t t r a n s f e r on t h e p o s i t i o n o f
the transition point of the boundary layer could be expected.
Low f a v o r a b l e p r e s s u r e
P*P9Q
The a d d i t i o n o f t h e 0.090 i n c h g r i d u p s t r e a m o f t h e p l a t e
Figure 16.
Boundary layer profiles for low pressure gradient with no grid
1.0
Reynolds number, N p
3
O
39,900
n
59,400
o 104,700
7 157,300
A
222,000
^ 275,300
N 329,600
t\ 391.900
Ilartree (17),
m = 0.149
i1 + m
"!\ = V
V^x
82
r e s u l t e d i n a n i n c r e a s e i n N u s s e l t number f r o m t h e v a l u e s
found for data taken with no grid, the plate being undisturbed
from the position i t occupied for the runs with no grid.
The
variation in local Nusselt number corrected for unheated.
s t a r t i n g l e n g t h w i t h R e y n o l d s number i s shown i n F i g u r e 1 7 •
The h e a t t r a n s f e r d a t a p o i n t s l i e f r o m 15 t o 35 p e r c e n t
a b o v e t h e P o h l h a u s e n r e l a t i o n l i n e a t t h e low R e y n o l d s n u m b e r s .
A transition from laminar to turbulent flow in the boundary
layer i s indicated, in the Reynolds number range from 200,000
to 250,000.
The d a t a p o i n t s f o r f l o w s w i t h R e y n o l d s n u m b e r s
higher than 250,000 l i e close to the lines representing
E q u a t i o n s 3^ a n d 35 f o r t u r b u l e n t f l o w .
Boundary layer profiles for the data are plotted in
Figures 18 and 19•
The p r o f i l e s i n F i g u r e 1 8 a r e shown com­
p a r e d w i t h H a r t r e e ' s s o l u t i o n a s was d o n e f o r t h e b o u n d a r y
l a y e r p r o f i l e s t a k e n when n o g r i d , was p r e s e n t .
The experi­
mental points in this figure are for Reynolds numbers up to
244,900, which i s in the region of transition indicated by the
heat transfer data.
I t i s clear from this figure tnat the
velocity distributions for different Reynolds numbers are more
scattered than those taken with no grid.
Since Hartree's
solution results in profiles which are similar in profile for
a g i v e n v a l u e o f m, b u t d i f f e r e n t R e y n o l d s n u m b e r s , a n d , s i n c e
the data obtained with no grid behaved in this manner, the
increase in free-stream turbulence intensity seems to have a t
least partially destroyed the similarity of the profiles.
gure 1 7 .
H e a t t r a n s f e r r e s u l t s f o r low f a v o r a b l e p r e s s u r e
gradient with 0.090-inch grid
84
von Karman, (34),
CM
o
x
Prandtl (29),'
eq. 35
x
• 0.090-inch grid,
1.8%<T<3.2%
Pohlhausen (28),
eq. 33
0
0
?
5
10
20
Reynolds number, N^ e , x 10
60
gure 18.
Boundary layer profiles for low favorable pressure gradient with
0.090-inch grid
0.8
Reynolds number, N
0.6
O
n
45,700
55,100
O
v
110,200
A
145,500
80,000
208,300
0.4
21.0,800
244,900
0.2 -
Hartree (17),
m = 0.149
1 + m
i\ " \
x
Re,
Figure 19.
Turbulent boundary layer profiles for low favorable gradient with
0.090-inch grid
20-
15 u+
Profile from Clauser (3)
V
Reynolds number, M
O
10-
428,100
372,200
293,100
244,900
0-
™r"T~r"'
100
600
89
I t i s a l s o i n t e r e s t i n g t o n o t e t h a t t h e p r o f i l e d a va
taken with the 0.090-inch grid present are generally to the
l e f t of the analytical line, whereas the profile data taken
w i t h no. g r i d p r e s e n t l i e on t h e l i n e .
Thus, i t also appears
that the shape of the profile has been changed slightly with
the addition of the grid.
A further examination of the boundary layer profiles
indicates that the distributions for Reynolds numbers of
2 0 8 , 3 0 0 , 2 1 0 , 8 0 0 a n d 2 4 4 , 9 0 0 d e v i a t e m o s t from t h e r e s t o f t h e
data.
These profiles are a l l in the transition region as
determined by the heat transfer data and might therefore
exhibit erratic behavior.
The b o u n d a r y l a y e r p r o f i l e s f o r R e y n o l d s n u m b e r s o f
2 4 4 , 9 0 0 a n d h i g h e r w e r e a p p a r e n t l y t u r b u l e n t on t h e h e a t
transfer plot.
To c o n f i r m t h i s , t h e s e p r o f i l e s a r e p l o t t e d , on
the universal turbulent boundary layer coordinates in Figure
19, where the profile for the transition Reynolds number of
244,900 deviates most from the other profile data.
I t may
thus be concluded that the transition indicated by the
b o u n d a r y l a y e r p r o f i l e s and t h a t shown i n t h e h e a t t r a n s f e r
c o r r e l a t i o n a r e i n a g r e e m e n t , and t h a t t h i s p a r t i c u l a r p r o ­
f i l e was i n d e e d , i n t h e t r a n s i t i o n r e g i o n .
The t u r b u l e n c e i n t e n s i t i e s f o r t h e s e d a t a r a n g e d f r o m
1.84 to 3.20 per cent.
The e f f e c t s o f r a i s i n g t h e f r e e s t r e a m
t u r b u l e n c e seem t o h a v e b e e n a n i n c r e a s e i n t h e N u s s e l t number
for the laminar boundary layer range and a lowering of the
90
transition Reynolds number range to approximately 200,000 to
250,000.
In addition, similarity of the boundary layer pro­
f i l e s h a s b e e n somewhat d e s t r o y e d a n d t h e g e n e r a l s h a p e o f
these profiles has been affected.
lpw
favorable pressure gredieat,
grW
The t u r b u l e n c e p r o d u c e d b y t h e 0 . 2 5 0 i n c h g r i d , r a i s e d t h e
Nusselt numbers for low Reynolds numbers approximately the
same a m o u n t f o u n d f o r t h e 0 . 0 9 0 i n c h g r i d
e
A value for the
increase a t a higher Reynolds number i s hard to estimate since
i t i s not possible to t e l l where the transition from laminar
to turbulent flow occurs.
The h e a t t r a n s f e r d a t a shown i n F i g u r e 20 d o n o t i n d i ­
c a t e a n y t r a n s i t i o n o f t h e b o u n d a r y l a y e r , and i t a p p e a r s t h a t
t h e p o i n t s f a l l on a l i n e o f t h e t u r b u l e n t h e a t t r a n s f e r
correlation equations extended to the lower Reynolds numbers.
Investigation of the boundary layer profiles tends to support
this idea.
The b o u n d a r y l a y e r p r o f i l e f o r t h e l o w e s t R e y n o l d s
number in Figure 21 nearly matches the line for Hartree's
solution for the plate without free stream turbulence.
As t h e
Reynolds numbers increase, the points fan out until poor
agreement with the solution i s apparent.
The l i n e on F i g u r e
21 that deviates most from the Hartree solution i s for a
R e y n o l d s number o f 1 4 1 , 9 0 0 .
T h i s same p r o f i l e , when p l o t t e d
on t h e u n i v e r s a l t u r b u l e n t b o u n d a r y l a y e r c o o r d i n a t e s o f
F i g u r e 22 a g r e e s f a i r l y w e l l w i t h t h e u n i v e r s a l p r o f i l e , a n d
Figure 20.
Heat transfer results for low favorable pressure
with 0.250-inch grid
92
von Karman (42)
eq. 34
CN
o
Prandtl (29)7
eq. 35
%
z:%
2 -
1-
A
0.250-inch grid,
2.90%<T<5.07%
0
Pohlhausen (28),
eq. 33
0
9
5
20
10
Reynolds number, N
,
x
-4
10
60
gure 21.
Boundary layer profiles for low favorable pressure gradient with
0.250-inch grid
1.0
o
b o O
v
0.8
Reynolds number, N.
Re
O
45,900
0.6
•
62,700
O
87,600
v
115,400
0.4-
0.2
Ilartree (17),
m = 0.149
n
1 + m
U
V x
Figure 22.
Turbulent boundary layer profiles for low favorable pressure gradient
with 0.250-inch grid
20-
15-
Profile from Clauser (3)
u+
Reynolds number, N„
Re
O 369,700
10-
•
280,900
O
230,200
V
141,900
!
100
1" —|
600
the points of profiles for higher Reynolds numbers agree even
better.
Thus the change from laminar to turbulent flow i s not
characterized by à distinct transition in this case, but in a
g r a d u a l c h a n g e w h i c h i s a p p a r e n t l y b r o u g h t a b o u t by t h e
increased, free stream turbulence.
The region of transition
has thus been "stretched out" from a relatively small and
easily identifiable location to one in which the "point" of
t r a n s i t i o n i s l a r g e and. n o l o n g e r d i s c e r n a b l e from t h e h e a t
transfer data.
A comparison of the boundary layer profiles for the data
t a k e n a t t h e l o w p r e s s u r e g r a d i e n t shows t h a t t h e g e n e r a l p r o ­
f i l e s h a p e h a s c h a n g e d e v e n more f r o m t h e c h a n g e s n o t e d , f o r
the 0.090 inch grid.
T h i s c h a n g e i s e v i d e n t from F i g u r e 2 3
w h e r e t h e open p o i n t s r e p r e s e n t t h e i n t e r m e d i a t e f r e e s t r e a m
turbulence level of the 0.090 inch grid and the flagged, points
represent the turbulence level of the 0.250 inch grid.
In
g e n e r a l , t h e t h i c k n e s s o f t h e b o u n d a r y l a y e r s r e p r e s e n t e d , by
the flagged points i s greater than the thickness of those for
t h e open p o i n t s .
I t might then be hypothesized that the
boundary layer i s thickening with increasing free stream tur­
bulence level, under the influence of a pressure gradient.
I t i s recalled, that the thickening of the boundary layer i s a
characteristic of transition from laminar to turbulent flow.
Thus, one might expect that the heat transfer results for the
higher free-stream turbulence intensities are justified in
appearing as a continuation of the correlation for turbulent
Figure 23•
Comparison of boundary layer profiles for low favorable pressure
gradient with 0.090- and 0.250-inch grids
1.0
% o%n^ b
Vv
" £V*
0.8 -
%
?o
|3lfc>
•V-
0.6
->J
u
•U
cd
u
u>>
•H
U
O
<D
>
9^
0.4
S»
43
O
A
0.2 -
Open points from 0.090-inch g r i d
Flagged points from 0.250-inch g r i d
vo
XO
100
flow since the boundary layer i s no longer purely laminar
under these conditions.
High Favorable Pressure Gradient
The r e s u l t s o f t h i s s e c t i o n a r e f o r t e s t s c a r r i e d o u t
with the highest of the pressure gradients used.
The p r e s s u r e
d i s t r i b u t i o n f o r t h i s s e r i e s i s shown i n F i g u r e 2 4 .
H i g h f a v o r a b l e p r e s s u r e g r a d i e n t , jdo g r i d
The h e a t t r a n s f e r d a t a f o r a n i n c r e a s e d p r e s s u r e g r a d i e n t
w i t h n o t u r b u l e n c e - p r o d u c i n g g r i d shown i n F i g u r e 2 5 a r e s i m i ­
l a r to those obtained, with the lower pressure gradient in that
most of the points l i e only slightly above the line for the
Pohlhausen relation for a laminar boundary layer.
In this
case, however, the change of position of the false tunnel wall
required an increased pressure drop through the test section,
which in turn necessitated operation of the tunnel a t pressure
r a t i o s f o r w h i c h t h e f r e e s t r e a m t u r b u l e n c e i n t e n s i t y was
somewhat h i g h e r t h a n t h e r a n g e o f t u r b u l e n c e i n t e n s i t i e s u s e d
with the lower pressure gradient.
This operation resulted in
a t u r b u l e n c e i n t e n s i t y r a n g e o f from 0 . ? 4 t o 0 . 9 2 p e r c e n t .
The increased level of turbulence advanced the position of the
b o u n d a r y l a y e r t r a n s i t i o n s o t h a t t h e da t a t a k e n a t h i g h e r
Reynolds numbers entered, the transition range.
Thus, an
i n c r e a s e i n N u s s e l t number o v e r t h e P o h l h a u s e n r e l a t i o n i s
found, a t R e y n o l d s n u m b e r s a b o v e 2 5 0 , 0 0 0 i n F i g u r e 2 5 .
The
h e a t t r a n s f e r d a t a t a k e n a t a R e y n o l d s number o f 4-56,300
Figure 24.
Static pressure distribution for high favorable
pressure gradient
102
0.40 -
0.30 -
to
eu
O.
I
0.20-1
Cu
0.10
r~
15
25
30
20
Distance from leading edge, inches
10
35
F i g u r e 25.
Heat transfer results for high favorable pressure
gradient with ho grid
ÎO
lo
Ln
1
Nussel t number, N., , x 10 ^
N"x
h-1
K>
__L_J
1
!
I
Vl
I
L
1
fi> hj
•O» O3"
M
CO 3"
co m
c
CO
fDy
t*
m
3
•
o
M
CL
W
S
c
3 o
h-*
cr
fi)
M
N5
co
[\
b.
ro
o
ix
bv
tx
o>
O
t
!.
105
a p p e a r t o be i n t h e t u r b u l e n t r a n g e .
The v a l u e of t h e exponent m i n t h e f r e e s t r e a m was found
t o b e 0=160, i n d e p e n d e n t of t h e magnitude of t h e f r e e s t r e a m
velocity.
The boundary l a y e r p r o f i l e s i n F i g u r e 26 a r e com­
pared. w i t h t h e l i n e r e p r e s e n t i n g H a r t r e e ' s s o l u t i o n f o r t h i s
v a l u e of m.
These p r o f i l e s d i s p l a y t h e same s i m i l a r i t y of
p r o f i l e shape found, f o r t h e low p r e s s u r e p r o f i l e s t a k e n w i t h ­
o u t a grid, f o r Reynolds numbers up t o 3 2 2 , 0 0 0 .
One f u r t h e r
p r o f i l e was t a k e n f o r t h e h e a t t r a n s f e r p o i n t a t a Reynolds
number of 4 5 6 , 3 0 0 .
T h i s p r o f i l e was p l o t t e d on t h e u n i v e r s a l
t u r b u l e n t boundary l a y e r c o o r d i n a t e s i n F i g u r e 27, where t h e
agreement with the universal profile i s only fair.
I t was
concluded, t h a t t h i s p o i n t was i n t h e t r a n s i t i o n r a n g e .
High f a v o r a b l e p r e s s u r e g r a d i e n t . 0 . 0 9 0 i n c h g r i d
The h e a t t r a n s f e r d a t a f o r t h e h i g h e r p r e s s u r e g r a d i e n t
w i t h t h e 0 . 0 9 0 i n c h grid, i n p l a c e behaved s i m i l a r t o t h e d a t a
f o u n d f o r t h e c o r r e s p o n d i n g t u r b u l e n c e l e v e l w i t h t h e lower
pressure gradient.
The low Reynolds number p o i n t s a r e from
1 5 t o 35 p e r c e n t h i g h e r than t h e recommended e q u a t i o n , a s
shown i n F i g u r e 2 8 .
A t r a n s i t i o n from l a m i n a r t o t u r b u l e n t
f l o w i s found i n t h e Reynolds number r a n g e from 200,000 t o
250,000 a s was found f o r t h e lower p r e s s u r e g r a d i e n t .
The
p o i n t s i n t h e t u r b u l e n t Reynolds number r a n g e a r e i n agreement
w i t h e q u a t i o n s a n d t h e recommended e q u a t i o n s of P r a n d t l and
von Karman r e s p e c t i v e l y .
Figure 26.
Boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e g r a d i e n t w i t h n o
grid
1.0-
0.8-
Reynolds number, N
0.6-
o
u
u
u
•r-i
•fH
U
O
0)
>
0.4-
i—I
0.2-
O
46,600
n
65,500
O
1.26,000
v
178,300
A
252,900
ÎS
322,200
Hartree (17),
ra = 0.160
1 + m
A
2
U
v x
Re x
H
o
-x}
Figure 27.
T u r b u l e n t boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e
gradient with no grid
25
20-
Oo
O O
o_
o
Profile from CLnuser (3)
o
o
Reynolds number, N.
Re,
O
I —f-• |
+
100
456,300
Figure 28.
Heat transfer results for high favorable pressure
gradient with 0.090-inch grid
Ill
von Karman (42),
eq. 34
Prandtl (29),
eq. 35
0.090-inch grid,
2.06%<T<3.20%
Pohlhausen (28),
eq. 33
2
5
10
Reynolds number,
20
, x 10
60
112
The boundary l a y e r p r o f i l e s i n F i g u r e s 29 and 3 ° w i t h t h e
h e a t t r a n s f e r d a t a show t h a t t h e p o s i t i o n of t r a n s i t i o n and
t h e c h a r a c t e r of t h e boundary l a y e r i s much t h e same a s p r e ­
d i c t e d above from t h e h e a t t r a n s f e r d a t a and. might b e
p r e d i c t e d , from t h e b e h a v i o r a t t h e l o w e r p r e s s u r e g r a d i e n t .
The p r o f i l e s tend toward, n o n - s i m i l a r i t y f o r t h e l a m i n a r r a n g e
and. d i f f e r from t h e H a r t r e e s o l u t i o n , e s p e c i a l l y f o r t h e
Reynolds numbers a p p r o a c h i n g t h e t r a n s i t i o n r e g i o n .
The p r o ­
files for the turbulent region agree with the universal
profile.
The r a n g e of t u r b u l e n c e i n t e n s i t y f o r t h e s e d a t a was from
2 . 0 6 t o 3•20 p e r c e n t .
High f a v o r a b l e p r e s s u r e g r a d i e n t . 0.2S0 i n c h g r i d
An e f f e c t of i n c r e a s e of N u s s e l t number w i t h f r e e s t r e a m
t u r b u l e n c e i n t e n s i t y i s found i n F i g u r e 3 1 f o r t h e 0 . 2 5 0 i n c h
g r i d which i s s i m i l a r t o t h a t found f o r t h e 0.090 i n c h g r i d .
At t h e low Reynolds numbers, an i n c r e a s e of a b o u t 1 4 p e r c e n t
was f o u n d , b u t e s t i m a t i o n of f u r t h e r i n c r e a s e s f o r t h e l a m i n a r
boundary l a y e r a t h i g h e r Reynolds numbers was n o t p o s s i b l e d u e
t o t h e d i f f i c u l t y i n e s t a b l i s h i n g a Reynolds number r e g i o n f o r
transition.
The d a t a a g a i n a p p e a r t o b e an e x t e n s i o n of t h e
P r a n d t l and von Karman r e l a t i o n s f o r t u r b u l e n t f l o w .
The boundary l a y e r p r o f i l e s shown i n F i g u r e 32 e x h i b i t
t e n d e n c i e s s i m i l a r t o t h o s e found, f o r t h e low p r e s s u r e g r a d i ­
ent.
The p r o f i l e s s u g g e s t t h a t a t a Reynolds number of a b o u t
Figure 29.
Boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e g r a d i e n t w i t h
0.090-inch grid
1.0
""C7"
17-
0.8 -
Reynolds number, N
0.6
o
•H
-U
et)
U
4J>>
O
37, 700
•
55, 900
O
71, 800
V
1.02, 700
A
L50, 200
196, 200
•H
U
O
r—<
>m
Re,
0.4 -
0.2
A
225, 400
V
235, 100
H
H
•Çr
Ilartree (17),
m = 0.160
r
4
1 + m
"'X
5
U_
^x
Figure 30.
T u r b u l e n t boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e g r a d i e n t
with 0.090-inch grid
i
25
°o o o
Profile from Clauser (3)
1
1
1— i I I
10
Reynolds number,
O
430,300
•
407,800
O
235,100
"T"
100
r
600
Figure 31.
Heat t r a n s f e r r e s u l t s f o r h i g h f a v o r a b l e p r e s s u r e
gradient with 0.250-inch grid
118
5-
von Karman (42)
eq. 34
CNJ
o
Prandtl (29)
x
2-
1-
L
0.250-inch grid,
3.6%<T<8.3%
Pohlhausen (28),
eq. 33
0
0
2
5
10
Reynolds number, N
20
, x 10
K.ex
60
Figure 32.
Boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e g r a d i e n t w i t h
0.250-inch grid
0.8-
Reynolds number,
O
44,800
n
80,000
O
121,300
•H
0.4-
Ilartree (17)
m = 0.160
01 + m
U
V x
121
121,300 the transition region i s reached, but that the region
i s much l a r g e r than was found, f o r l o w e r f r e e stream t u r b u l e n c e
l e v e l s , and n o t s o e a s i l y d e f i n e d .
The t u r b u l e n t boundary
l a y e r s i n F i g u r e 33 a g r e e w i t h t h e u n i v e r s a l p r o f i l e a s h a s
been t h e c a s e f o r a l l t u r b u l e n t p r o f i l e s included i n t h e work.
The changes of p r o f i l e shape and l a c k of s i m i l a r i t y a r e
again evident in the curves.
i s made i n F i g u r e
A comparison of t h e s e changes
where t h e f l a g g e d p o i n t s r e p r e s e n t t h e
t u r b u l e n c e l e v e l g e n e r a t e d by t h e 0 . 2 5 0 i n c h g r i d .
The f r e e
s t r e a m t u r b u l e n c e l e v e l s found f o r t h e d a t a presented, h e r e
a r e from J , 6 k t o 8 . 2 0 p e r c e n t .
D i s c u s s i o n of t h e R e s u l t s and. Conclusions
The p a p e r s of Wang ( 4 3 ) and Edwards and F u r b e r ( 9 ) and.
t h e r e s u l t s of t h i s i n v e s t i g a t i o n a g r e e on t h e e f f e c t of f r e e
s t r e a m t u r b u l e n c e i n t e n s i t y on t h e h e a t t r a n s f e r from a f l a t
plate with zero pressure gradient.
I n a l l of t h e s e r e s e a r c h e s ,
o n l y a change of t h e p o s i t i o n of t r a n s i t i o n from l a m i n a r t o
t u r b u l e n t f l o w i n t h e boundary l a y e r t a k e s p l a c e when t h e f r e e
stream turbulence intensity is increased..
The d a t a obtained, i n t h i s work d o n o t a g r e e w i t h t h e
r e s u l t s of Sugawara, £ £ £ i . ( 3 7 ) .
greement a r e n o t f u l l y a p p a r e n t .
between t h e d a t a of Sugawara,
The r e a s o n s f o r t h e d i s a ­
P e r h a p s some d i f f e r e n c e
and. t h a t of t h i s p a p e r
i s d u e t o t h e n o n - t i m e - s t e a d y method of energy t r a n s f e r meas­
urement used i n t h e f o r m e r .
I f s o , f u r t h e r i n v e s t i g a t i o n of
F i g u r e 33•
T u r b u l e n t boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e g r a d i e n t
with 0.250-inch grid
25
20-
15Reynolds number, N
Profile 1'• mi Clauser (3)
O
220,300
D
217,000
O
121,300
Re
10-
5-
1
i
r
i
10
y4"
i
T "-|—n
100
600
Figure 3^.
Comparison of boundary l a y e r p r o f i l e s f o r h i g h f a v o r a b l e p r e s s u r e
g r a d i e n t w i t h 0 . 0 9 0 - and. 0 . 2 5 0 - i n c h g r i d s
1.0-
0.8
&•
CAO
z•
o
•H
4J
U0
0.6
i
%
4J
«H
U
O
y
>
r—i
A
aD
0.4
Open points from 0.090-inch grid
Flagged points from 0.250-inch grid
H
tv
Ux
%
A
V
0.2
'1 " i
/H3 /
7/ X
126
n o n - t i m e - s t e a d y c o o l i n g on t h e l a m i n a r boundary l a y e r i s i n
order.
The d a t a of F e l l e r and Yeager ( 1 2 ) were t a k e n w i t h a
minimum f r e e s t r e a m t u r b u l e n c e i n t e n s i t y which was much l a r g e r
t h a n t h e maximum t u r b u l e n c e i n t e n s i t y used i n t h e z e r o p r e s ­
s u r e g r a d i e n t p o r t i o n of t h i s work.
In addition, the static
pressure distribution along the plate surface is not given.
F o r t h e s e r e a s o n s , no d i r e c t comparison can b e made.
It is
i n t e r e s t i n g t o n o t e , however, t h a t t h e s l o p e s of t h e r e l a t i o n s
p r e s e n t e d by F e i l e r and. Yeager a r e d i f f e r e n t f o r h e a t t r a n s f e r
w i t h o u t a sound f i e l d p r e s e n t and f o r h e a t t r a n s f e r w i t h a
sound f i e l d .
Surprisingly, the slopes for data taken with a
sound f i e l d , a r e n e a r l y t h e same a s t h o s e found i n t h i s work,
w h i l e t h e s l o p e s of d a t a t a k e n w i t h o u t a sound f i e l d d o n o t
c o r r e s p o n d w i t h o t h e r r e s u l t s found i n t h e l i t e r a t u r e .
Comparison of t h e p r e s e n t d a t a w i t h t h a t t a k e n on a f l a t
p l a t e w i t h a p r e s s u r e g r a d i e n t by Wang ( 4 3 ) i s a l s o a p p r o p r i ­
ate.
Wang's p o i n t s a r e p l o t t e d i n F i g u r e 3 5 a l o n g w i t h l i n e s
r e p r e s e n t i n g d a t a from t h i s i n v e s t i g a t i o n .
The p r e s s u r e
g r a d i e n t used i n Wang's work was n o t t h e same a s e i t h e r of
t h o s e used i n t h i s work, based on a comparison of t h e d a t a of
t h i s work w i t h c u r v e s g i v e n by Wang.
I t i s also probable
t h a t t h e p r e s s u r e g r a d i e n t u s e d by Wang was one f o r which t h e
expression for the free-stream velocity as a function of the
d i s t a n c e from t h e l e a d i n g edge was d i f f e r e n t t h a n t h e f u n c t i o n
used. h e r e .
N e v e r t h e l e s s , F i g u r e 35 shows t h a t Wang's d a t a
F i g u r e 35»
Comparison of p r e s e n t d a t a w i t h t h a t of Wang ( 4 3 )
Numbers n e x t t o p o i n t s r e p r e s e n t p e r c e n t t u r ­
b u l e n c e i n t e n s i t y found by Wang. L i n e s r e p r e s e n t
t r e n d s of d a t a from b o t h p r e s s u r e g r a d i e n t s f o r
t h i s work.
128
10 -,
Ol.71
5 -
O Data of Wang (43)
-
O 1.58
CN
I
o
O 1.44
X
0.250-inch grid
0.62
X
0.090-inch
3
z
55 2 -
O 0.56
grid-^/
u
0.46C
no grid
.42
0.38
CJto
M 1
1.22
C0.52
O0.62
0.5
_ii
0.3-
2
!
5
1
1
] -- -;
10
Reynolds number, N
1
20
-4
, x 10
1
|
i
60
129
g e n e r a l l y f o l l o w t h e t r e n d s of t h e d a t a from t h i s work.
Wang used, t u r b u l e n c e i n t e n s i t i e s i n t h e r a n g e s of 0 . 3 8 t o
0 . 6 2 p e r c e n t and 1 . 1 2 t o 1 . 7 1 p e r c e n t .
In the higher turbu­
l e n c e r a n g e , f o r Reynolds numbers g r e a t e r than 1 5 0 , 0 0 0 , t h e
p o i n t s a g r e e w e l l i n s l o p e , b u t Wang's N u s s e l t numbers a r e
a b o u t 20 p e r c e n t lower t h a n t h e l i n e r e p r e s e n t i n g t h i s work.
The p o i n t s f o r Reynolds numbers below 150,000 a r e s c a t t e r e d ,
and d o n o t a g r e e .
S i m i l a r l y , t h e p o i n t s above a Reynolds number of 125,000
i n t h e lower t u r b u l e n c e i n t e n s i t y r a n g e g e n e r a l l y a g r e e i n
s l o p e , b u t h a v e N u s s e l t numbers from 1 0 t o 3 ° p e r c e n t l o w e r .
The p o i n t s below a Reynolds number of 1 2 5 , 0 0 0 f o l l o w a l i n e of
lower s l o p e than o t h e r p o i n t s i n t h i s t u r b u l e n c e i n t e n s i t y
range.
The above d i s c u s s i o n shows t h a t t h e work of Wang and t h e
present data agree at least qualitatively.
The g r e a t e s t
d i f f e r e n c e s a r e i n t h e magnitude of t h e N u s s e l t number
increase.
One f u r t h e r comparison w i t h p u b l i s h e d d a t a can b e made.
A proposed t h e o r y on t h e t r a n s i t i o n of t h e l a m i n a r boundary
l a y e r i n t h e p r e s e n c e of a p r e s s u r e g r a d i e n t h a s been pub­
l i s h e d by van D r i e s t and Blumer ( 4 l ) .
From t h i s t h e o r y , an
e q u a t i o n can b e d e r i v e d which r e l a t e s t h e f r e e s t r e a m t u r b u ­
l e n c e i n t e n s i t y , t h e Pohlhausen p r e s s u r e g r a d i e n t p a r a m e t e r A,
t h e Reynolds number based on t h e l o c a l boundary l a y e r t h i c k ­
n e s s and t h e l o c a l f r e e s t r e a m v e l o c i t y .
Thus, f o r a f l a t
130
plate,
3 e 3 6 ( N R e ) 2 ( T 2 ) + ( 1 - 0,0485A)N R e - 9860
5
S
=
0
(40)
where Be^ i s t h e Reynolds number, A i s t h e Pohlhausen p a r a ­
meter
A
™
6% d p
> a dx
(4l)
and T i s t h e f r e e s t r e a m t u r b u l e n c e i n t e n s i t y .
The e q u a t i o n
i s p l o t t e d i n F i g u r e s 36 and 3 7 f o r t h e h i g h and low p r e s s u r e
gradients respectively.
The a r e a s t o t h e l e f t and below t h e
lines represent stable or laminar-flow conditions, while the
a r e a s above and t o t h e r i g h t of t h e l i n e s r e p r e s e n t u n s t a b l e
or turbulent conditions.
The two v a l u e s of A shown b r a c k e t
t h e r a n g e of e x p e r i m e n t a l v a l u e s of A found f o r a l l d a t a i n
t h i s work.
I t i s i n t e r e s t i n g t h a t t h e magnitude of A h a s
l i t t l e e f f e c t on t h e t r a n s i t i o n boundary l a y e r Reynolds number
e x c e p t a t v e r y low t u r b u l e n c e i n t e n s i t i e s .
The e x p e r i m e n t a l
p o i n t s n e a r t h e l i n e r e p r e s e n t i n g t r a n s i t i o n a l boundary l a y e r
Reynolds numbers shown on F i g u r e s 36 and 37 a r e l a b e l l e d w i t h
t h e i r r e s p e c t i v e Reynolds numbers based, on d i s t a n c e from t h e
l e a d i n g edge o f t h e p l a t e .
Both f i g u r e s show n o a p p r o a c h t o t h e l i n e r e p r e s e n t i n g
t h e t r a n s i t i o n Reynolds number e x c e p t f o r t h e p o i n t a t a
Reynolds number of 456,300 based on d i s t a n c e from t h e l e a d i n g
Figure 36.
Low p r e s s u r e g r a d i e n t d a t a compared w i t h t h e b o u n d a r y - l a y e r t r a n s i t i o n
c r i t e r i o n of van D r i e s t and Blumer ( 4 1 )
I
75
10-
00i
o
X
Ono grid
"A= 0
A
f 6
pd
to
T3
O 0.090-inch grid
<0
A
0)N
!
A
VY= 5.0
r—J
0.250-inch grid
H
V)
N>
o
A
o
4-
Turbulent region
n
Q)
P5
•
• •
2 -
n
•
B
•
O244,900
210,80?
A
oo
Oo o
A
O
Laminar region
i
3
Turbulence i n t e n s i t y , T, %
45,900
Figure 37.
High pressure gradient data compared with the boundary-layer transition
of van Driest and Bluraer (4l)
10-
A= 5.0
8 -
O no grid
co
0 0.090-inch grid
o
A 0,250-inch grid
6 —
Turbulent region
456,300
4-
^ 225,400
2 -
O
Laminar region
0
1
2
80,000
O
3
4
Turbulence intensity, T, %
5
6
7
135
edge for the high pressure gradient with no grid.
This point
i s on the l i n e representing t r a n s i t i o n , and, comparing with
the boundary layer profiles for that series of data, appears
as the f i r s t turbulent point a t the end of the transition.
The intermediate turbulence levels for both pressure
gradients agree with the van Driest and Blumer equation also,
with t r a n s i t i o n Reynolds numbers based on distance from the
leading edge bracketing the line in both cases.
Again, these
correspond to the transition Reynolds numbers found from the
boundary layer profile analysis.
The high turbulence level data in both cases have points
which approach the recommended l i n e from the turbulent side,
thus indicating that the points were above the transition
Reynold s number based on boundary .layer thickness.
This
suggests that these points were a t least in transition, i f not
turbulent.
The heat transfer data for these points f a l l along
the l i n e s recommended f o r turbulent boundary layers and thus
these results are to be expected, according to van Driest and
Blumer.
The transition criterion used above has an interesting
further point that should, be considered.
In deriving their
equation, van Driest and Blumer used, an analysis which assumed
that the scale of turbulence in the free stream was of the
same order of magnitude a s the boundary layer thickness.
To
check their theory against the present data, an estimation of
the scale of turbulence used in t h i s work i s necessary.
Since
136
no measurements of scale were made due t o lack of equipment,
t h e a p p r o x i m a t e e d d y s i z e may b e e s t i m a t e d f r o m t h e w o r k o f
D r y d e n £ £ iLL= ( 6 ) w h e r e d a t a o n t h e s c a l e o f t u r b u l e n c e d o w n ­
stream of grids i s published.
For a l l tests performed, the
scale of turbulence varies from 0.125 inches up to 0.288
inches.
Since these eddy sizes are easily of the order of
magnitude of the boundary layer thicknesses involved, the
comparison with van Driest and Blumer's theory i s valid.
The previous analysis of the experimental r e s u l t s leads
to several conclusions:
1.
The new data presented in t h i s thesis support the
proponents of the theory that there i s no effect of the freestream turbulence intensity on heat t r a n s f e r through a laminar
boundary layer with zero pressure gradient.
There i s nothing
in the present results that suggests changes other than moving
the position of transition of the boundary layer for a change
in free-stream turbulence intensity.
2.
T h i s w o r k p r o v i d e s e v i d e n c e t h a t when a p r e s s u r e
gradient i s imposed on the laminar boundary layer, and the
free-stream turbulence intensity i s raised, an increase in the
Nusselt number will r e s u l t , assuming the Reynolds number of
the flow remains unchanged.
Whether the boundary layer
r e m a i n s t r u l y l a m i n a r d u r i n g t h i s p r o c e s s i s shown t o b e
doubtful.
The boundary layer profiles indicate a change in
shape due to the increase in free-stream turbulence, and they
a l s o show a l o s s of the s i m i l a r i t y t h a t existed p r i o r t o the
137
i n c r e a s e of t u r b u l e n c e i n t e n s i t y .
The r e v i e w of t h e t u r b u l e n t
b o u n d a r y l a y e r by C l a u s e r (3) p o i n t s o u t a s i m i l a r p r o c e s s
u n d e r g o n e by a l a m i n a r b o u n d a r y l a y e r i n t r a n s i t i o n .
Because
t h e boundary l a y e r e q u a t i o n s f o r f l o w w i t h a p r e s s u r e g r a d i e n t ,
i n c l u d i n g t h e f l u c t u a t i n g components o f t h e v e l o c i t y , h a v e
terms that resemble the Reynolds stresses of turbulent flow,
i t i s r e a s o n a b l e t o c o n c l u d e t h a t t h e f l o w i s qua s i - l a m i n a r ,
or in transition.
3.
The d a t a o b t a i n e d i n t h i s s t u d y i n t h e t u r b u l e n t
boundary layer range do not exhibit an increase in Nusselt
number f o r a n y of t h e c o n d i t i o n s t e s t e d . .
Although the range
o f t e s t i n g i n t h i s r e g i o n was l i m i t e d , i t a p p e a r s t h a t n o
increase i s to be expected.
T h i s p o i n t o f view i s i n c o n ­
currence with the general ideas of the wall layers of turbu­
lent flow.
Again r e f e r r i n g t o C l a u s e r (3), i t i s f o u n d t h a t
t h e t u r b u l e n t b o u n d a r y l a y e r shows l i t t l e c h a n g e i n t h e
v e l o c i t y p r o f i l e of t h e l a m i n a r s u b l a y e r and t h e i n n e r w a l l
l a y e r w i t h t h e i m p o s i t i o n o f a. p r e s s u r e g r a d i e n t a l o n g t h e
plate surface.
Since the velocity distribution in this region
i s all-important in changing the temperature distribution,
l i t t l e change in the l a t t e r could be expected.
4.
The v e l o c i t y p r o f i l e s f o r t h e b o u n d a r y l a y e r s o f f e r
s u b s t a n t i a l s u p p o r t f o r t h e t r a n s i t i o n t h e o r y of van D r i e s t
a n d Blumer ( 3 9 ) .
A l t h o u g h t h e y h a v e compared t h e i r t h e o r y
a g a i n s t t h e d a t a of o t h e r s t a k e n on a body of r e v o l u t i o n , t h e
d a t a of t h i s work o f f e r t h e f i r s t d i r e c t comparison f o r t h e
138
flat plate,
i n o r d e r t o f u l l y s u b s t a n t i a t e t h e t h e o r y of
van D r i e s t a n d B l u m e r , more work i s n e e d e d i n t h e v e r y low
turbulence area.
The c o n c l u s i o n s s t a t e d a b o v e s u g g e s t f u r t h e r a n a l y t i c a l
a n d e x p e r i m e n t a l i n v e s t i g a t i o n s t h a t may b e b a s e d on t h e
findings of this work.
1.
Some p o s s i b l e p r o j e c t s a r e :
An a n a l y t i c a l a n d / o r e x p e r i m e n t a l s t u d y o f t h e i n t e r ­
a c t i o n o f f r e e - s t r e a m t u r b u l e n c e and. a f a v o r a b l e p r e s s u r e
g r a d i e n t t o d e t e r m i n e how t h e s h a p e o f t h e b o u n d a r y l a y e r p r o ­
f i l e changes with changes in free-stream turbulence and
pressure gradient.
2.
An a n a l y s i s o f h e a t t r a n s f e r i n a p r e s s u r e g r a d i e n t
t h a t i n c l u d e s t h e e f f e c t o f s c a l e o r eddy s i z e on t h e h e a t
t r a n s f e r c o e f f i c i e n t from a f l a t p l a t e .
3«
An e x p e r i m e n t a l i n v e s t i g a t i o n a l o n g t h e l i n e s of
paragraph 2, but including measurements of the Reynolds
stresses in the boundary layer as the pressure gradient and
the free-stream turbulence intensity are varied.
These could
t h e n b e compared, w i t h R e y n o l d s s t r e s s t e r m s found i n t h e e q u a ­
tions discussed in the analysis section of this thesis.
139
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t i o n o f u n c e r t a i n t y c a l c u l a t i o n s t o measured d a t a .
American S o c i e t y o f M e c h a n i c a l E n g i n e e r s T r a n s a c t i o n s .
79: 373-376. 1957.
40.
van d e r Hegge Z i j n e n , B . G. H e a t t r a n s f e r from h o r i ­
zontal cylinders in a turbulent airflow. Applied Science
R e s e a r c h e s . S e c t i o n A, 8 : 2 0 5 - 2 2 3 • 1 9 5 8 •
41.
Van D r i e s t , E . R . and B l u m e r , C. B . Boundary l a y e r
t r a n s i t i o n : f r e e - s t r e a m t u r b u l e n c e and p r e s s u r e g r a d i e n t
e f f e c t s . American I n s t i t u t e of A e r o n a u t i c s and
Astronautics Journal. 1: 1303-1306. 1963•
42.
von Karman, T . The a n a l o g y between f l u i d f r i c t i o n and
h e a t t r a n s f e r . American S o c i e t y o f M e c h a n i c a l E n g i n e e r s
Transactions. 61: 705-710• 1939.
43.
Wang, H. E . The i n f l u e n c e o f f r e e - s t r e a m t u r b u l e n c e on
t h e l o c a l c o e f f i c i e n t o f h e a t t r a n s f e r from a f l a t p l a t e .
Xerox c o p y . U n p u b l i s h e d PhD. t h e s i s . P r o v i d e n c e , Rhode
I s l a n d , L i b r a r y , Brown U n i v e r s i t y . 1 9 5 9 .
143
44.
Young, A. D. and. Maas, J . N. The b e h a v i o r of a P i t o t
tube in a transverse total-pressure gradient. Great
B r i t a i n A e r o n a u t i c a l R e s e a r c h C o u n c i l . R e p o r t s and.
Memorandum 1 7 7 ° • 1 9 3 6 .
144
ACKNOWLEDGMENTS
The a u t h o r would l i k e t o acknowledge t h e p a t i e n c e and
u n d e r s t a n d i n g o f t h e members o f h i s g r a d u a t e s t u d y c o m m i t t e e ,
composed o f P r o f e s s o r H. M. B l a c k , D r . Glenn Murphy,
P r o f e s s o r S . J . C h a m b e r l i n , D r . D. S . M a r t i n , and D r . G. K .
Serovy.
T h i s t h e s i s was p r e p a r e d u n d e r t h e d i r e c t i o n o f
D r . G . K. S e r o v y whose p a t i e n c e , u n d e r s t a n d i n g and sound
counsel are profoundly appreciated.
The e x p e r i m e n t a l work performed was d o n e i n t h e Iowa
Engineering Experiment Station flow facility using equipment
p r o v i d e d t h r o u g h a g r a n t from t h e P r e s i d e n t ' s P e r m a n e n t
O b j e c t i v e Committee of t h e Alumni Achievement F u n d .
The
author gratefully acknowledges the help of these organizations.
145
APPENDIX A
The f o l l o w i n g d e f i n i t i o n s of t h e t e r m s t u r b u l e n c e i n t e n ­
s i t y , s c a l e o f t u r b u l e n c e and s p e c t r u m o f t u r b u l e n c e h a v e been
used, i n t h e t e x t m a t e r i a l .
Turbulence Intensity
The t u r b u l e n c e i n t e n s i t y f o r t h i s work i s d e f i n e d a s
(42)
T
U
where U 1 i s t h e f l u c t u a t i n g component of t h e f r e e - s t r e a m
v e l o c i t y and U i s t h e t i m e - a v e r a g e v a l u e of t h e f r e e - s t r e a m
velocity.
I n g e n e r a l , t h e d e f i n i t i o n of t u r b u l e n c e i n t e n s i t y
should include the fluctuations in a l l three coordinate
d i r e c t i o n s , b u t d u e t o t h e i m p l i c i t l y assumed c o n d i t i o n o f
isotropy of the free-stream fluctuations, the definition i s
t h a t used a b o v e .
Scale of Turbulence
The s c a l e o f t u r b u l e n c e i s d e f i n e d f o r t h e r e f e r e n c e s
made t o i t i n t h i s work a s
r<*>
(43)
L
where the correlation coefficient
i s defined as
14-6
U'AU'B
Rxu'
~
r~z~2 f - g
V U'A Z\j u'b
The s c a l e o f t u r b u l e n c e d e f i n e d , a s a b o v e i s sometimes known
a s t h e m a c r o - s c a l e o r i n t e g r a l s c a l e o f t u r b u l e n c e a n d may
b e t h o u g h t o f a s a m e a s u r e o f t h e eddy s i z e i n t h e f l o w .
The
v e l o c i t y f l u c t u a t i o n s U ' ^ and U ' g a r e measured a t two p o i n t s
l o c a t e d on t h e x - a x i s a n d s p a c e d a d i s t a n c e x a p a r t .
14?
APPENDIX B
The uncertainty in the Nusselt number i s calculated from
the final form of Equation 26, given below
xw Q
W'N
Akalts
s ."x
"
tf)/
+
\Aka(ts - tf) '
V
XW,
x
2
+
1/2
xwt
(45)
_ tf)
Aka ( t s
"
tf)
I n u s i n g t h i s e q u a t i o n t o c a l c u l a t e t h e u n c e r t a i n t y , i t was
assumed that the value of thermal conductivity of the a i r was
precise enough that any error in i t s magnitude had negligible
e f f e c t on t h e r e s u l t a n t u n c e r t a i n t y i n t h e N u s s e l t n u m b e r .
Each of the uncertainty intervals in Equation 45 must be
evaluated giving odds that the values for each variable will
l i e in the interval specified.
The uncertainty intervals for
the quantities used for computing the Nusselt number are given
in Table 1.
Table 1.
Variable
(t
x
A
- t )
Values of uncertainty used for calculating the
uncertainty in the Nusselt number
Uncertainty interval
t 0.02 in.
± 0.001 sq. f t .
± 0.5 F
varies - see text
Odds
20 t o 1
20 t o 1
20 t o 1
20 t o 1
148
The u n c e r t a i n t y i n t e r v a l f o r ^ i s a f u n c t i o n o f t h e
v a r i a b l e s used t o c a l c u l a t e
and t h u s must b e t r e a t e d , i n a
.
manner s i m i l a r t o t h a t f o r f i n d i n g
The r e l a t i o n used.
x
to obtain
was
%
~
P
qr
"
™
(19)
qc
S u b s t i t u t i n g from E q u a t i o n s 20 and 2 1 f o r P , q^ and q ^ ,
R
- €<3"A[T
The u n c e r t a i n t y i n
7 dS:
wQ
+l2-5rMA
- T
4.
] -
(46)
x_
i s then
\%
\ W VW'E" J
4
/
+
j
+
\
+
a
\
u( t
(
dTs
%)
1
- ow t ,3
+
+
(
axp
8Ia"Ta)
™x-
2 n 1/2
(47)
i n which i t h a s been assumed, t h a t k ^ , d", and 6 a l l h a v e s m a l l
enough v a r i a t i o n s i n t h e i r t r u e v a l u e s t h a t t h e c o n t r i b u t i o n
o f c a c h and t h e t o t a l c o n t r i b u t i o n of t h e i r a g g r e g a t e u n c e r ­
t a i n t y w i l l bo n e g l i g i b l e .
I t was n o t p o s s i b l e t o o b t a i n e x p e r i m e n t a l v a l u e s of t h e
e m i s s i v i t y s o v a l u e s were t a k e n from t h e t e c h n i c a l l i t e r a t u r e .
U n f o r t u n a t e l y , e x p e r i m e n t e r s d o n o t a g r e e on a s i n g l e v a l u e of
149
emissivlty, probably because of surface conditions or the like.
As a r e s u l t , values found in the l i t e r a t u r e varied from 0.05
to 0.I5.
Because this variation i s not random, i t cannot be
included, in the uncertainty analysis.
I t was a s s u m e d f o r t h e
analysis that the value of emissivlty was 0.10 and that the
random variation in t h i s assumption was small.
tion means that errors in the value of %
Calculations of
This assump­
may r e s u l t .
w i t h t h e maximum a n d minimum v a l u e s o f
e m i s s i v i t y n o t e d a b o v e show t h a t u n d e r t h e w o r s t p o s s i b l e c o n ­
ditions for this work,
cent.
c o u l d v a r y b y a s much a s 2 . 3 p e r
Interestingly enough, of a l l the experimenters quoted
i n r e l a t i o n t o t h i s w o r k , o n l y o n e (Wang, ( 4 1 ) ) h a s u s e d a
correction for the radiant loss.
as unimportant.
The u n c e r t a i n t y i n t e r v a l s f o r t h e v a r i a b l e s
used to calculate G
Table 2.
Variable
E
B
T
s
T
(t
A
- t, )
s
b
x
The others have neglected i t
are given in Table 2.
Values of uncertainty interval for variables used in
calculating the net heat loss %
Uncertainty interval
Odd s
± 0.005 V
± 0.003 0
± 0.50 F
20 to 1
20 t o 1
20 t o 1
± 0.50 F
20 t o 1
± 0.001 sq. f t .
± 0.25 F
20 t o 1
20 t o 1
± 0.01 in.
20 to 1
150
When t h e u n c e r t a i n t y i n t e r v a l s i n T a b i c 2 a r e used, t o
calculate uncertainties in
, i t q u i c k l y becomes a p p a r e n t
t h a t o n l y two o f t h e v a r i a b l e s were o f i m p o r t a n c e , t h e v o l t a g e
r e a d i n g E and t h e t e m p e r a t u r e d i f f e r e n c e ( t
- t, ).
more , when t h e r e s u l t i n g v a l u e o f u n c e r t a i n t y i n
Further­
i s used t o
c a l c u l a t e t h e u n c e r t a i n t y i n t h e N u s s e l t number, i t i s found
t h a t t h i s q u a n t i t y i s d e p e n d e n t t o a l a r g e d e g r e e on t h e
u n c e r t a i n t y i n Q. .
Thus,, m e a s u r e m e n t s o f
point for meaningful data.
are the critical
B e c a u s e of t h i s d e p e n d e n c e on t h e
u n c e r t a i n t y i n Q^, u n c e r t a i n t i e s i n o t h e r q u a n t i t i e s which
were n o t s i g n i f i c a n t w i t h r e s p e c t t o t h e u n c e r t a i n t y i n
were neglected.
R e s u l t s showing t h e maximum u n c e r t a i n t y f o r
each set of data are given in Table 3•
The u n c e r t a i n t y i n t h e R e y n o l d s number was c a l c u l a t e d
from
1/2
(48)
wR
using the uncertainties listed, in Table 4.
The a n a l y s i s
showed, t h a t t h e u n c e r t a i n t y i n t h e R e y n o l d s number was d e p e n d ­
e n t a l m o s t e x c l u s i v e l y on t h e u n c e r t a i n t y i n t h e v e l o c i t y
measurement.
B e c a u s e o f t h e s m a l l d e p e n d e n c e on t h e d i s t a n c e
x and the kinematic viscosity v, these variables were neg­
lected in the final computations.
The maximum u n c e r t a i n t y i n
The R e y n o l d s number i s t a b u l a t e d i n T a b l e 3 •
Table 3.
Maximum u n c e r t a i n t i e s i n N u s s e l t n u m b e r , R e y n o l d s number a n d t u r b u l e n c e
intensity
d
uri
Maximum
u n c e r ttaa1i n t y i n
N u s s eIl t number
Maximum
uncertainty in
R e y n o l d s number
Maximum
uncertainty in
turbulence intensity
%
%
Zero pressure gradient,
no grid
+
Zero pressure gradient,
0.090 in. grid
+
Low p r e s s u r e g r a d i e n t ,
no grid
+
Low p r e s s u r e g r a d i e n t ,
0.090 in. grid
+
4.68
+
+
6.42
+
2.19
+
+
4.80
+
2.16
+
4.99
+
2.66
+
6.19
+
2.22
+
Low p r e s s u r e g r a d i e n t ,
0.250 in. grid
High pressure gradient,
no grid
High pressure gradient,
0.090 in. grid
+
High pressure gradient,
0.250 in. grid
+
4.96
5.23
7.51
+
+
+
2.41
2.56
2.53
2.23
+
+
+
+
16.8
8.2
17.1
9.5
9.6
17.5
11.8
9.7
152
Table 4.
Variable
V a l u e s o f u n c e r t a i n t y used, i n t h e c o m p u t a t i o n of
u n c e r t a i n t y i n t h e R e y n o l d s number
Uncertainty interval
Odds
Ù
- 0.5 f t . p e r s e c .
20 t o 1
X
- 0.02 i n .
20 t o 1
The u n c e r t a i n t y i n t h e t u r b u l e n c e i n t e n s i t y was found
from
The u n c e r t a i n t i e s f o r t h e v a r i o u s q u a n t i t i e s a r e l i s t e d i n
T a b l e 5*
Values of the uncertainty in the turbulence
i n t e n s i t y a r e g i v e n i n T a b l e 3*
153
T a b l e 5»
Variable
e"2
Uncertainties for turbulence intensity calculation
Uncertainty interval
.Odds
- 0.002 v
20 t o 1
eo
t
0.05 v
20 t o 1
e~~
w
- 0.05 v
20 t o 1
154
APPENDIX G
Assembly Drs win g of P l a t e
155
+ <> 4-
FIGURE 38. ASSEMBLY DRAWI!
4-
f
+
+
f<»4- 4~ <> +
-h
+
-f- o -f* < » -H> < >
O
-j-
f
f
4-
z.
MOSEPIECE
ML.
NTS
8«Pg RAIL
W
w
IBLY DRAWING OF PLATE
PRESSURE
TAP
THERMOCOUPLE
1^6
APPENDIX D
A s e t o f s a m p l e h e a t t r a n s f e r c a l c u l a t i o n s i s worked, o u t
below.
The d a t a a r e t a k e n from t h e low p r e s s u r e g r a d i e n t
series with the 0.090-inch grid installed.:
Power i n p u t
P
E2
g (conversion factor)
=
1.21
0.308 (3*^13)
13.43 Btu/hr
Radiation loss
'V - V'
q.
a
r
O.I36 Btu/hr
Conduction loss
q
c
v
— (t. - V
(0.143)(0.O833)(12)(1.48)
(0.1405)
1.51 Btu/hr
157
N e t l o s s by c o n v e c t i o n
%
=
P
=
13.43 - 0.136 - 1.51
=
II.78 Btu/hr
- % -
qo
NuSSS^fr number
S ,x
N
Nu
A(ts
- tf^a.
11.78(0.891)
0.0833(14.4)(0.0154)
568
Correction for unheated starting length (turbulent region)
39/40-1-7/39
%Nu ' c o r r e c t e d
x
=
N
-
x
=
568(0.969)
=
550
(k)
Bomber
pUx
'k r~
(0.07013)( 5 8 . 9 ) (0.891)
125.5
=
293,100
X
10"7