THE APPLICATION OF THE HYDRAULIC ANALOGIES
TO PROBLEMS OF WO-DIMENSIONAL CO^PHESSIBLE GAS FLOW
A THESIS
Presented to
the Faculty of the Division of Graduate Studies
Georgia School of Technology
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Aeronautical Engineering
by
John Elmer Hatch, Jr.
March 1949
1<B&
ii
THE APPLICATION OF THE HYDRAULIC ANALOGIES
TO PROBLEMS OF TWO-DIMECTSIONAL COMPRESSIBLE GAS FLOW
Approved:
r
Date Approved by .Chairman
y^txi^g X
ItYf
T
$
•
iii
ACKNOWLEDGMENTS
I wish to express my sincerest appreciation to Professor
H. W. S. LaVier for his valuable aid and guidance in all phases
of the work required for the completion of this project.
I should
also like to thank HE*, R. S. Leonard, of the Engineering Experiment
Station Machine Shop, and Mr, J. E. Garrett, of the Engineering
Experiment Station Photographic Laboratory, for their full cooperation
in the use of their respective facilities. Constructive criticism and
suggestions of the staff of the Daniel Guggenheim School of Aeronautics
greatly helped in the prosecution of the problems involved in this work.
fcr
T&HLE 07 CONTENTS
PAGE
Approval Sheet
*
ii
Acknowledgements
•
ill
L i f t of Figures
v
Summary • « . • • . • . . • • • • » • • • • • • • • # • • • » . . . . . • . • . • . . • • . * • • . . • • • • • • • •
1
Introduction,
1
•
Theory
Equipment
•••»•«
,
••...*••
••••••.•....•....
Procedure
•••*•••••»
Conclusions
BIBLIOGRAPHY..
APPENDIX 1, Figures
'
13
18
Di scus8ion
He commendations
4-
••••••••».,....*
•
••••• 19
»»•
«
,
#.,..
..
„
•••••23
....23
..
25
2?
fi
LIST OF ILLUSTEATIONS
TITLE
PASS BO.
General VI ew of *Vater Channel
28
General View of Water Channel
29
Top View of V a r i a b l e Speed Drive Mechani sm
30
Schematic Diagram of Water Channel
and L i g h t i n g Equipment,
Shock P o l a r Diagrams For
Diamond A i r f o i l T e s t Speeds
, 3I
,
Shock P o l a r Diagrams For
C i r c u l a r Arc A i r f o i l Test Speeds
Shock P o l a r Diagrams..*
..
32
. 33
3^
Diamond A i r f o i l Model
35
C i r c u l a r Arc A i r f o i l Model
36
Model A i r f o i l Angle of A t t a c k
C a l i b r a t i o n Curve
37
V a r i a t i o n of Ware V e l o c i t y With Wave Length For
A S t a t i c Water Depth of One Q u a r t e r of an
Inch
38
A n a l y s i s of R e f r a c t i o n P a t t e r n s Formed By An
Uhdular Hydraulic Jump, and
C a p i l l a r y Waves
39
Flow About C i r c u l a r Arc A i r f o i l , M =1.56
**Q
Flow About C i r c u l a r Arc A i r f o i l . M z 2.34
41
Flow About C i r c u l a r Arc A i r f o i l , Ms 3.28
1+2
Flow About L e n t i c u l a r A i r f o i l , M =3.71
43
Flow About L e n t i c u l a r A i r f o i l , M= 4 . 2 6 . . . .
^it-
Flow About L e n t i c u l a r A i r f o i l , M = 6 . ? 8
45
Flow About Diamond A i r f o i l , M* 2 f 6 3
46
Flow About Diamond A i r f o i l , M =2.47
^7
J+l ST OF ILLUSTBATIONS
TITLE
PASS NO.
Flow About Diamond A i r f o i l , Ifc3.02
^8
Flow About Diamond A i r f o i l , 8 = 3 . 7 2 . . f .
*+9
Flow About Diamond A i r f o i l , M=4.53
50
Flow About Diamond A i r f o i l , M=5.08.
51
V a r i a t i o n of Shock Wave Angle With F r e e
Stream Mach Number For Diamond A i r f o i l
•
52
V a r i a t i o n of Shock Wave Angle With F r e e
Mach Number For Symetrical C i r c u l a r
Arc A i r f o i l
.
,.
53
Diamond A i r f o i l Local Mach Number D i s t r i b u t i o n
For Various F r e e Stream Mach Numbers.
•.
5^
C i r c u l a r Arc A i r f o i l Local Mach Number D i s t r i b u t i o n
For Various F r e e Stream Mach Numbers
P r e s s u r e C o e f f i c i e n t For riamond
A i r f o i l a t M s K5%.*.*,.,
s
P r e s s u r e C o e f f i c i e n t For C i r c u l a r
Arc A i r f o i l a t M - ^ . 2 6 . . . . . .
s
,
55
56
57
THE APPLICATION OF THE HYDRAULIC ANALOGIES
TO PROBLEMS OF TtfO-DIMJSIONAL COMPRESSIBLE GAS FLOW
SUMMARY
A water channel, with a glass bottom, four feet wide and twentyfeet long, was designed and constructed for the purpose of investigating
the application of the hydraulic analogy to problems of two-dimensional
compressible flow,
Two supersonic airfoil profiles were tested at various speeds.
The speed range covered by analogy was from M * #75 to M »•6,0. Pictures
were taken of the wave formations set up by each airfoil model. Pictures
were also taken of the local water depth distribution about the models.
From a study of the flow photographs, and by applying the hydraulic analogy,
an analysis was made of the local conditions about the models.
INTRODUCTION
There is an analogy between water flow with a free surface, and
two-dimensional compressible gas flow.
The analogy may be used for the
investigation of aerodynamic problems in high speed compressible gas flow.
Vtfhere no supersonic or high speed wind tunnels are available, the application of the hydraulic analogy is particularly useful for the study of compressible gas flow problems.
2
Riabouchinsky
f i r s t presented the mathematical "basis for the
hydraulic analogy, and applied the analogy to the i n v e s t i g a t i o n of
2
the flow in a Laval nozzle,
Binnie and Hooker further investigated
the application of the hydraulic analogy by obtaining surveys along
the center of a channel with a c o n s t r i c t i o n .
Entirely satisfactory
agreement between compressible gas flow theory and the hydraulic anal3
ogy was reached.
Srnst Preiswerk, at the suggestion of Dr. J ,
Ackeret, conclusively proved that the methods of gas dynamics can be
applied to water flow with a free surface.
In 19^0 the National
Advisory Committee for Aeronautics constructed a water channel for
the i n v e s t i g a t i o n of the hydraulic analogy to the flow through nozzles
and about c i r c u l a r cylinders a t subsonic v e l o c i t i e s and extending into
the s u p e r c r i t i c a l velocity range.
Reasonable satisfactory agreement
was found between the water flow and a i r flow about similar bodies,
although i t was concluded from t h i s work that accurate q u a n t i t a t i v e
Riabouchinsky, D., Mecanique des f l u l d e s , Conrptes Rendus. t .
195, No, 22, November 28, 1932, pp. 993-999.
2
Binnie, A.M., and S. G. Hooker, "The Plow Under Gravity of an
Incompressible and Inviscid Fluid Through a Constriction in a Horizont a l Channel", Proceedings of the Royal Society. Vol, 159 (London,
England: 1937), pp". 592-608, "
-^Preiswerk, E r n s t . , "Application of the Methods of Gas Itynamics
to Water Hows With Free Surface,"
P a r t I . "Flows with No Energy Dissipation," NAOA TM. No. 93^,
19^0.
Part I I , "Flows with Momentum D i s c o n t i n u i t i e s . " NACA TM. No.9?5.
19^0.
James Orlin, Norman J . Linder, and Jack G. B i t t e r l y , "Applicat i o n of the Analogy Between Water Flow With a Free Surface and TwoDimensional Compressible Gas Flow." NACA TM. No. 1185, 19^?.
3
Results would require additional i n v e s t i g a t i o n s , "both theoretical and experimental.
North American Aviation Incorporated recognized the posslbil-«
i t i e s of extending experimental compressibility research by means of the
hydraulic analogy, and in 19^+6 constructed a water channel i n which the
water remained stationary and the t e s t models moved through the water. I t
was indicated by North American Aviation Incorporated that with accurate
equipment, q u a n t i t a t i v e as well as q u a l i t a t i v e r e s u l t s could "be expected
from water channel experiments.
The hydraulic analogy offers a convenient and inexpensive way of
i n v e s t i g a t i n g two-dimensional compressible gas flow of phenomena occurring
in a i r a t speeds too high for visual observation.
By means of the water
channel, the two-dimensional flow in a i r at high Mach numbers can be simulated to low speeds, and the r e s u l t s may be easily photographed,
The advantages of using the hydraulic analogy for i n v e s t i g a t i n g
problems of high speed aerodynamics may be summarized as follows:
(1) Cost i s low compared with wind-tunnel or free f l i g h t t e s t s .
(2) Visual observation of flow in the transonic region i s p o s s i ble since the t e s t model can pass through the sonic range with r e l a t i v e
slowness and s t i l l avoid choking phenomena.
(3) Various features of the flow such as shock wave formation,
v o r t i c e s , and turbulence, may be observed and photographed.
(^) High supersonic Mach numbers are simulated at speeds of a
few feet per second.
Bruman, J..E., "Application of the Water Channel Compressible Gas
Analogy." (North American Aviation Incorporated Engineering Report. No.
ttA-^'M-8, 19^7). ~
THEORY
1
The t h e o r y of t h e h y d r a u l i c analogy a s g i v e n "by P r e i s w e r k
w i l l be
reviewed h e r e .
The "basic assumptions made i n the mathematical development of t h e
h y d r a u l i c analogy a r e :
(1) The flow i s i r r o t a t i o n a l .
(2) The v e r t i c a l accelerations a t the free surface are negligible
compared to the acceleration of g r a v i t y .
By t h i s assumption the pressures
a t any point i n the f l u i d depend on only the height of the free surface
above that p o i n t .
(3) The flow i s without viscous l o s s , and i s uniform along any vertical line.
Throughout the following discussion of the development of the hydraul i c analogy the symbol dQ represents the t o t a l head (water depth for V = 0 ) ,
while the symbol d represents any water depth in the flow f i e l d .
discussion of the compressible gas equations the symbols T a o
represent values in the undisturbed gas, and T, P
in the disturbed flow f i e l d .
In the
, and p
and p represent values
Other symbols used are as follows:
L i s t of Symbols
c
Specific heat a t constant pressure
"O
cy
Specific heat a t constant volume
0
Adiabatic gas constant, r a t i o of c to c of gas
P
v
Preiswerk, E r n s t . , "Application of the Methods of Gas Dynamics to
Water Hows *fith Free Surface."
Part I . "Plows With No Energy Dissipation," KACA TM Ho. 93^.
19^0.
5
T
Absolute temperature of gas
^
Mass density of gas
p
Pressure of gas
h
Enthalpy or t o t a l heat content
q
Dynamic pressure of gas
(j*
Surface tension of l i q u i d
P
Pressure coefficient
X
Viscosity of l i q u i d
a
Speed of sound in gas
a*
C r i t i c a l v e l o c i t y . (Velocity a t which M s i ) .
V
Velocity of flow
d
Water depth
M
g
Mach number, stream value unless otherwise indicated
(V for gas; V for water)
a
gd
Acceleration of gravity
P
Compressibility factor
x,y
Rectangular coordinate axes
u,v
Components of velocity in x and y d i r e c t i o n s , respectively
$
Velocity potential i n two-dimensional flow
\
Wave length of surface waves i n f l u i d
U
Velocity of propagation of surface waves i n fluid
R
Reynolds number
p>-p
(L£?)
{M. )
u v Nondimensional velocity components i n x and y d i r e c t i o n s , r e s p e c t i v e l y . (Reference velocity a*; in hydraulic jump a j , the c r i t i c a l
velocity "before the jump, u" u, T v)
a*
a*
Subscripts
No subscript
any value of variable
o
value at stagnation (V= 0)
I
0
l o c a l v a l u e of v a r i a b l e ; v a l u e a t s u r f a c e of model,
a t channel w a l l s , o r I n f i e l d s of flow
s
value i n undisturbed
max
maximum v a l u e of v a r i a b l e
x
p a r t i a l d e r i v a t i v e w i t h r e s p e c t to x
streams
i e (L*$£ ' $ -d£
y
p a r t i a l d e r i v a t i v e with r e s p e c t to y
1
Conditions b e f o r e t h e h y d r a u l i c jump
2
C o n d i t i o n s a f t e r t h e h y d r a u l i c jump
The development of the analogy between t h e flow of water w i t h
a f r e e s u r f a c e and t h e flow of a compressible g a s may be o b t a i n e d by
s e t t i n g up the energy e q u a t i o n s f o r the two f l o w s .
For water* t h e energy e q u a t i o n g i v e s
V2
=
2g(d r t -d)
f
max
&
and
o
I n a g a s t h e energy e q u a t i o n g i v e s
^2
Z
Vmax =
2
&% ( T 0 - T )
an d
/2gcA
3y e q u a t i n g the r a t i o of J
f o r w a t e r to t h e r a t i o of X-
T
max
g a s i t i s seen t h a t
dQ
- d
TQ
f°r
- T
or
O
0
i- - JL
d
o
It is seen, therefore,
*o
that the ratio of the water depth corresponds
to the ratio of the gas temperatures.
7
By a comparison of the equations of continuity for the gas and the
liquid a further condition for the analogy may he derived.
The continuity equation for water i s
3jM)+
&x
£l2&
d y
z 0
For the two dimensional gas flow the equation of continuity i s
$bU4*dkJL
d x
=o
tfy
I t i s seen that the continuity equations of the two flows have the
same form, and from the two equations a further condition for the analogy may he derived
JL* t
a0 "TT
This r e l a t i o n s h i p shows that the r a t i o of the l o c a l density (* to
stagnation d e n s i t y ^ i s analogous to the r a t i o of the local depth d to
the stagnation depth d «
dimensional.
I t i s assumed that the flow i s steady and two
For the liquid flow i t i s assumed that the flow i e steady
and "bounded by a free surface on the top and a f l a t surface on the hot torn,
and that the l i q u i d i s incompressible^
I t has he en shown that the water depth i s analogous both to the
mass density of the gas and to the temperature of the g a s .
These relation-
ships can only e x i s t i f an assumption i s made i n regards to the nature of
the g a s .
The analogy requires that for the compressible gas, T - 2 .
r e l a t i o n s h i p i s shown as follows:
For adiahatic gas flow
/_
£.= (w_Y"
This
8
Since t h e h y d r a u l i c analogy r e q u i r e s t h a t * » jfc and T - d_ i t
X
obvious t h a t d o
d
5
T
o
o
is
d
o
( & )*w and t h e equation i s s a t i s f i e d only when f o
'
From t h e a d i a b a t i c r e l a t i o n
» = <£)!r
Po
?o
i t i s seen t h a t , s i n c e 0 = 2 ,
£ p *o
(i )
The d i f f e r e n t i a l
d
2
o
equation f o r the v e l o c i t y p o t e n t i a l of t h e w a t e r
lZ>-fi)+l,<>-g>^&-<>
The d i f f e r e n t i a l
e q u a t i o n for t h e v e l o c i t y p o t e n t i a l of a two
dimensional compressible gas flow i s
6YKX (i-4?)+d>
a? ' iyr (l-£)-24
<xz/
c
V
The two e q u a t i o n s become i d e n t i c a l when
£d
wI^
=o
gd 2ga0_
a^
2ghQ
That t h e v e l o c i t y J gd i s t h e v e l o c i t y of p r o p a g a t i o n of
surface
waves (small d i s t u r b a n c e s ) i n shallow water may be shown a s f o l l o w s :
Page
2
g i v e s t h e f o l l o w i n g e q u a t i o n f o r t h e v e l o c i t y of w a t e r
waves under t h e a c t i o n of g r a v i t y and s u r f a c e
V.I? g X
X
2][<ntajih
tlzjr
T
fW
tension:
2 7T d- 7
k
AJ
Inspection of the equation shows that as wave length becomes
great in comparison with the water depth ( i.e. d _^.0, tanh 2 Tt d
X
X
27*d)
A
2
Page, Leigh, I n t r o d u c t i o n to T h e o r e t i c a l P h y s i c s . New York:
D. Van No s t r a n d Co. I n c . , 1935, p.250
9
and i f t h e s u r f a c e t e n s i o n i s equal to z e r o , t h e wave v e l o c i t y 1)6001068
Vr/fJ
Since t h e v e l o c i t y of sound, a, i s the v e l o c i t y a t which small
d i s t u r b a n c e s a r e propagated i n ges flow, t h e r a t i o V f o r
t h e shallow
m
water flow i s analogous to the Mach number V i n the gag flow.
a
The local Mach numbers about the model are determined from
(d
d }
v [; °: * 1*
which i s o b t a i n e d by s u b s t i t u t i n g the equation f o r t h e l o c a l v e l o c i t y
i n the e x p r e s s i o n f o r Mach number*
The p r e s s u r e c o e f f i c i e n t f o r t h e w a t e r flow may be determined
as follows:
For a compressible g a s ^
P
*
- 1
s
^o
- 1
Ps"
When t h e analogous r e l a t i o n s h i p s f o r water a r e s u b s t i t u t e d i n t h e
equation for the pressure coefficient the r e s u l t i n g equation i s
P -F
" c
2* - l
is
d 2 o
1
3
O r l i n , J a m e s , , Norman J . L i n d e r and J a c k G. B i t t e r l y , " A p p l i c a t i o n
of t h e Analogy Between Water Flow w i t h a Free Surface and Two-Dimensional
Compressible Gas Flow," HACA TN, No* 1185, 1 W . p . 8
10
k
For a compressible gas the compressibility factor
1
=
c
is
rT^[ ft*V--.,)* -1]
Since the hydraulic analogy requires that
# - 2 the compressibility
factor for the water flow i s
*« =
i+iM
C
I
2
s
I t may "be noticed that no theory has been presented concerning
shock waves or supersonic flow.
In the hydraulic analogy a compressible shock i n the gas i s
analogous to a hydraulic jump in the water flow.
The hydraulic jump
may he defined as an unsteady motion in which the velocity of flow may
strongly decrease for short distances, and the water depth suddenly
increase.
Hydraulic jumps occur in water whose velocity of flow
i s greater than the wave propagation v e l o c i t y .
Preiswerk developed
the r e l a t i o n s h i p s between conditions in front of the shock wave and
behind the shock wave for the hydraulic analogy.
In developing the
theory the assumption i s made that the water motion i s e n t i r e l y unsteady
which i n d i c a t e s that the water jumps suddenly along a l i n e from the lower
water l e v e l to the l e v e l a f t e r the jump.
The assumptions o r i g i n a l l y
made are also carried through the discussion of the shock wave analogy.
For the water flow without a jump the energy equation holds
V2=
where d
the total head ie a constant.
o
place the t o t a l head d
°2
4
2g(d 0 " d)
LoC. Glt»
After a hydraulic jump has taken
i s smaller than d
°1
, since p a r t of the k i n e t i c
energy of the xvater i s converted into heat, and i s treated as l o s t energy,
For the flow a f t e r the jump the energy equation again applies and now "becomes
2
f9
=
2
23(4
- d)
o2
To solve t h i s equation for d
, the new t o t a l head, use i s made of the
°2
shock polar diagram to determine V , the water velocity behind the shock
wave.
The equation for the shock polar for the hydraulic analogy is
vA/a.-uJfc-u, «/(3,;:^3^~]
By varying the parameter ^r "
and
family of curves will r e s u l t .
solving for v 2
in
terms of
~
a
The curves are similar to the shock
p o l a r s of an i d e a l gas, "but they show one c h a r a c t e r i s t i c difference. For
the maximum velocity "both cases "become c i r c l e s .
However, for water
the c i r c l e passes through the origin while for a perfect gas the c i r c l e
does not pass through the o r i g i n .
The shock p o l a r s for water were con-
structed for the speeds investigated in t h i s t h e s i s find are presented
in Figures 5 anu 6.
In Figure 7 shock polars are drawn for a i r and for water for the
same i n i t i a l flow conditions.
I t may "be seen that for continually de-
creasing shocks the polars approach each other.
The energy l o s s in the flow with hydraulic jumps i s developed
to he
4e =
I-K
~*n
This i s the r e l a t i v e energy converted into h e a t .
However,
for gas the
r i s e i n heat during the shock i s not l o s t energy since the t o t a l heat
12
content remains the game before and after the shock.
energy for water flow are presented by Preiswerk.
Curves of constant
From the carves it
may "be seen that the energy loss is small over a large region. For the
velocities investigated in this thesis the relative loss is less than
one per cent. Because of this small shock loss the analogy between the
two types of flow is still satisfied,
By employing the shock polar diagrams and the equations given
by Preiswerk conditions across a shock wave may be determined*
The various quantities which may be obtained from a shock polar
diagram are illustrated in Figure 7, No further explanation will be
given, since the theory and development of shock polar diagrams is well
covered in appropriate textbooks*
A summary of the analogous relationships is given below.
Two-Dimensional
compressible gas flow
Incompressible liquid flow
(water)
r=2
Temperature Eatio, T,
*o
Water depth r a t i o ,
Density r a t i o ,
Water depth r a t i o ,
£.
d
o
d
d
?o
Pressure r a t i o ,
O
p
r
Square of w a t e r - d e p t h r a d i o / d }
o
V e l o c i t y of sound, a = / p V
fa
Mach number,
Shock wave
V
a
Wave: v e l o c i t y ,
Mach number
Jgd
V
/fr
H y d r a u l i c jump
Liepmann, Hans Wolfgang, and A l l e n E. P u c k e t t , I n t r o d u c t i o n to
Aerodynamics of a Compressible Fluid.(New York: John Wiley and Sons, I n c . ,
1W
p. 53) '
13
EQUIPMENT
In designing the equipment for conducting experiments with the hydraul i c analogy, two methods of approach were a v a i l a b l e .
These methods were:
(1) The model was to remain stationary with the water flowing p a s t
the model,
(2) The water was to remain stationary with the model moving through
the water.
The f i r s t type of channel would necessarily "be the more elaborate,
and therefore, the more expensive.
I t would "be d i f f i c u l t to change Mach
number or to have decelerating flow or accelerating flox*,
The presence of
a "boundary layer on the bottom and sides of the channel would further comp l i c a t e a channel of the f i r s t type.
However, with the model remaining
stationary i t would be easy to measure the water depth ahout the model,
which i s very important i n the hydraulic analogy.
A channel of the second type r e s u l t s i n an easy measurement of
speed and acceleration, and therefore, Mach number.
The big disadvant-
age of t h i s type of channel i s the d i f f i c u l t y of measuring the water depth
about the model, since the model moves r e l a t i v e to the channel.
Taking al] f a c t o r s into consideration, i t was decided to construct
a channel of the second type, Figure 1.
The water channel frame work was made of s t r u c t u r a l steel bolted
together.
The channel i s four
feet wide and twenty feet long.
The
channel bottom i s p l a t e g l a s s one quarter of an inch thick i n two five
foot sections and one ten foot section.
i s located
The ten foot section of g l a s s
a t the " t e s t " end of the channel.
The g l a s s bottom i s sup-
ported by transverse members at t h i r t y inch i n t e r v a l s .
The purpose of
Ik
the glass "bottom i s to provide a smooth surface for the models to s l i d e upon,
and to provide for a method of photographing the flow of water about the
models.
The channel "bottom was maintained to a l e v e l of ,01 inch "by a
screw-jack type device on each of the channel' 8 supporting l e g s .
The ac-
curacy of the channel "bottom l e v e l was determined "by using a surveyor' s
transit.
A drain i s provided at one end of the channel,
The model carriage i s made of steel tubing welded together, and
t r a v e l s the e n t i r e length of the channel on a track which i s located along
each side of the channel, Figure 2„
t i r e d , "ball "bearing wheels.
in a horizontal plane.
The carriage t r a v e l s on four rubber-
Pour additional wheels r e s t r a i n the carriage
Safety stops are provided on the carriage track
to prevent the carriage and model from running off e i t h e r end of the channel.
The carriage i s driven by a r e v e r s i b l e , d i r e c t current motor through
a set of reduction gears and a continuous cable.
19.5 amperes, 2k v o l t s ,
The motor i s rated a t
A d i r e c t current motor i s used to provide a high
s t a r t i n g torque i n order to quickly accelerate the carriage to the desired
speed.
The driving cable i s 3/32 inch diameter conventional a i r c r a f t
s t e e l cable.
The cable goes one end one-half times around the driving
pulley, and i s rigged with about twenty pounds tension.
The speed of the
carriage drive motor i s controlled by a 6.7 amperes, 18 ohm r h e o s t a t .
The
rheostat has a calibrated dial with s e t t i n g s to give carriage speeds from
one to ten feet per second.
By proper operation of the rheostat the car-
riage may be run a t increasing, decreasing, or constant speeds.
After
15
<experimenting with the water channel i t was observed that the d i r e c t current motor did not r e s u l t in constant carriage speeds "below a speed of
one foot per second.
Consequently, a "Speed-Ranger" constant speed
device powered by one-quarter horse power, three phase, a l t e r n a t i n g
current, motor was i n s t a l l e d .
The "Speed-Ranger" device resulted i n
carriage speeds over the range of from .50 feet per second up to one foot
per second.
The "Speed-Ranger" speed control serves as an a l t e r n a t e
drive device since the d i r e c t current motor was found to "be more convenient
for high speed runs, and for accelerating or decelerating runs.
A photo-
graph of the drive mechanises i s shown ir. Figure 3.
An o b s e r v a t i o n p l a t f o r m i s l o c a t e d a t t h e channel t e s t
section.
Mounted on the platform i s a control panel which Includes a l l switches
and controls for operating the water channel.
The control panel and
observation platform are so set up that only one person i s required to
operate the water channel during a t e s t i n g program.
Included on the
platform are mounts for the cameras used to photograph the water flow
about the models.
The arrangement of the equipment for photographing the flow about
the models i s shown schematically in Figure 4.
obtain the flow photographs.
Two cameras are used to
One camera, mounted on the platform floor,
photographs the l o c a l water depth about the models, while the second camera,
mounted on a "boom extending out over the channel, photographs the wave
formation caused "by the models.
As seen in Figure ht the camera used to photograph the flow about
the models i n the horizontal plane p r o j e c t s through a diffusion screen
located approximately four f e e t above the water surface.
A second diffu-
sion screen i s located immediately under the g l a s s i n the area to be
16
photographed.
The l i g h t i n g arrangement consists of a boom spotlight
directed on the model from above and to the rear of the model, and a
large spotlight directed on the diffusion screen above the t e s t section,
Four flood l i g h t s illuminate a cellotex screen located on the floor beneath the channel.
The four flood l i g h t s r e s u l t i n an even l i g h t i n g
effect over the t e s t area,
The spotlights r e s u l t i n the wave formations casting shadows on
the screen d i r e c t l y under the g l a s s , and these shadows are then photographed,
An Eastman Medalist Camera with a s e t t i n g of one four-hundredth
of a second at f: 7,5 lens opening i s used to photograph the wave formations.
The camera shutter was tripped e l e c t r i c a l l y a t the correct time
by a cam device, mounted on the model carriage, a c t i v a t i n g a micro switch
located on the channel track.
The r e s u l t s of using the photographic tech-
nique here described are e n t i r e l y s a t i s f a c t o r y ,
From the development of the theory of the hydraulic analogy i t may
be seen that the water depth d i s t r i b u t i o n about the model i s very important i f a q u a n t i t a t i v e analysis of the flow i s to be made.
To determine
the water depth about the model, a Speed Graphic camera i s used to photograph the water depth d i s t r i b u t i o n i n the v e r t i c a l plane.
The camera i s
mounted on the platform above the water l e v e l to insure no i n t e r f e r r e n c e
from waves between the model and the camera.
by a 500 watt slide p r o j e c t o r .
The model i s illuminated
The projector i s so masked that at the
time the camera i s tripped only the model i s illuminated.
This feature
was found necessary because any extensive side l i g h t i n g i n t e r f e r r e d with
the horizontal flow photographs.
The camera i s tripped by the same switch
and cam arrangement that operates the overhead camera.
The best camera
17
petting was found to be one four-hundredths of a second a t fj 3»? lens
opening*
The a i r f o i l s chosen for t e s t i n g i n the water channel are 5, A, C. A.
a i r f o i l s developed by the National Advisory Comitdttee for Aeronautics, hav1
ing the following designations:
(1) NACA 2S-(50) (03)-(50) (03)
(2) NACA lS-(30)
c i r c u l a r arc a i r f o i l
(03M30) (03) diamond a i r f o i l ,
The designation of the c i r c u l a r arc a i r f o i l i n d i c a t e s that the maximum
thickness of both the upper and lower surface i s located at the f i f t y percent chord point, and that the maximum thickness i s six percent of the
chord length.
The designation of the diamond shaped a i r f o i l shows that
the maximum thickness of the upper and lower surface i s located a t the
t h i r t y percent chord point, and that the t o t a l thickness i s six percent
of the chord.
These p a r t i c u l a r a i r f o i l s were chosen because they are
basic supersonic a i r f o i l s ,
The a i r f o i l models are made of wood.
chord.
Each model has a twelve inch
The models are painted a f l a t white on the bottom and sides while
the top i s painted a f l a t black.
This paint scheme ifas selected to pro-
vide good photographic p r o p e r t i e s for the flow p i c t u r e s .
A single black
l i n e was ruled on each side of the models from the leading edge to the
t r a i l i n g edge.
The l i n e i s used as a reference l i n e in the determination
of the water depth about the models, since the water depth i s scaled from
photographs of the flow in the v e r t i c a l plane.
Figure 8 and Figure 9 show
the model dimensions.
Lindsey, W.F., Bernard N. Daley, smd Milton D, Humphreys, "The
Flow and Force Characteristics of Supersonic A i r f o i l s at High Subsonics Speeds."
U. S. National Advisory Committee for Aeronautics Teclinical Note.No. 1211.
19^7. P . 3 .
18
Since t h e h y d r a u l i c analogy a p p l i e s only to two dimensional flow
,the a i r f o i l models a c t u a l l y s l i d e on t h e g l a s s "bottom of t h e channel to
cause t h e r e q u i r e d flow,
The models a r e a t t a c h e d to t h e c a r r i a g e by
a l i n k a g e a s shown i n F i g u r e 2 .
The a n g l e of a t t a c k of t h e models i s determined by p o s i t i o n i n g
the l e a d i n g edge and t h e t r a i l i n g edge i n r e l a t i o n to t h e channel s i d e s ,
To f a c i l i t a t e the s e t t i n g of each a n g l e of a t t a c k a c a l i b r a t i o n curve
was drawn
g i v i n g t h e d i s t a n c e s of t h e l e a d i n g edge and t r a i l i n g edge
from the channel s i d e s f o r each d e s i r e d a n g l e .
This curve i s p r e s e n t e d
i n H g u r e 10.
^ROCEDUBE
The p r o c e d u r e for model t e s t i n g i n the water channel c o n s i s t e d of
running the two-dimensional a i r f o i l models through t h e water i n t h e chann e l a t predetermined s p e e d s .
The w a t e r depth for a l l t e s t s was one
q u a r t e r of an i n c h w h i l e the a n g l e of a t t a c k of the models i n each case
was zero d e g r e e s .
The time for the c a r r i a g e and model to move a c r o s s
the t e s t s e c t i o n was recorded by an e l e c t r i c timing clock which was
s t a r t e d by a cam d e v i c e on t h e model c a r r i a g e a c t u a t i n g a m i c r o - s w i t c h
l o c a t e d on t h e c a r r i a g e t r a c k .
At the end of the t h r e e f o o t r u n t h e cam
r e l e a s e d t h e m i c r o - s w i t c h and stopped the c l o c k .
The model v e l o c i t y
could then be c a l c u l a t e d .
Simultaneous p i c t u r e s were taken of the wave f o r m a t i o n s set up
by each model and of the l o c a l water d e p t h d i s t r i b u t i o n about the models.
A complete d i s c u s s i o n of p h o t o g r a p h i c t e c h n i q u e i n v o l v e d i s i n c l u d e d i n
t h e p r e v i o u s c h a p t e r on equipment.
19
DISCUSSION
Before t h e h y d r a u l i c analogy may be a p p l i e d to t h e i n v e s t i g a t i o n
of problems of compressible flow, the flow p i c t u r e s must be p r o p e r l y
interpreted.
From an i n s p e c t i o n of t h e flow photographs i n the h o r i z o n t a l
p l a n e a s e r i e s of a l t e r n a t i n g l y dark and l i g h t bands of d e c r e a s i n g i n t e n s i t y may be seen s t a r t i n g a t t h e l e a d i n g e&^e and t r a i l i n g edge of
each model,
A s i m i l a r c o n d i t i o n a l s o e x i s t s a t the s e c t i o n of maxi-
mum t h i c k n e s s of t h e diamond shaped a i r f o i l ,
The problem immediately
a r i s e s a s to which p a r t of the p a t t e r n d i r e c t l y a p p l i e s to t h e h y d r a u l i c
analogy ?
Of major importance i s t h e d e t e r m i n a t i o n of t h a t p a r t of t h e
flow p a t t e r n which r e p r e s e n t s a h y d r a u l i c jump o r shock wave.
Waves on
t h e water s u r f a c e a c t a s l e n s e s i n t h a t they d i v e r t t h e l i g h t r a y s i n
t h e d i r e c t i o n of i n c r e a s i n g w a t e r d e p t h .
Therefore,
the sharp b r i g h t
l i n e s r e p r e s e n t t h a t p a r t of each wave or r i p p l e which h a s t h e maximum
convex c u r v a t u r e .
The d a r k a r e a p r e c e d i n g t h e sharp w h i t e l i n e s r e p -
r e s e n t s t h e remainder of each wave.
I t may be then concluded t h a t a shock
wave i s recorded on t h e photographs a s a white l i n e preceded by a dark
area,
In a d d i t i o n t o , and i n f r o n t of,
g r a p h s show r i p p l e s o r c a o i l l a r y waves.
t h e shock waves t h e flow p h o t o The c a p i l l a r y waves a r e n o t
p a r t of the h y d r a u l i c analogy s i n c e they a r e p u r e l y w a t e r waves and depend p r i m a r i l y on t h e s u r f a c e t e n s i o n of the l i q u i d .
t h e r e l a t i o n of wave p r o p a g a t i o n v e l o c i t y to
depth of one q u a r t e r of an i n c h .
F i g u r e 11
shows
wave l e n g t h f o r a w a t e r
D i s t u r b a n c e s of wave l e n g t h s l e s s than
t h a t corresponding to the minimum w a v e ' v e l o c i t y a r e c a p i l l a r y waves.
The refraction p a t t e r n formed by a hydraulic jump and c a p i l l a r y
waves are shown i n Figure 12.
From these p a t t e r n s a q u a l i t a t i v e study
of the photographs may he made.
I t was found "by North American Aviation Incorporated
t h a t expan-
sion waves in a water channel do not agree with the theory of the hydraul i c analogy, but rather tend to travel a t a constant speed not dependent
on the s t a t i c water depth, or the model speed.
This fact i s "borne out
by an examination of the diamond a i r f o i l flow photographs in the horizont a l plane.
At the point of maximum thickness where an expansion takes
place there i s no d e f i n i t e or consistent p a t t e r n to indicate an expansion
wave.
However,
from the photographs showing the water depth d i s t r i b u -
t i o n about the diamond a i r f o i l a d e f i n i t e l e v e l drop may be seen a t the
change i n section,
Prom the side photographs, then, an expansion wave
may be detected.
A comparison was made between the shock wave angles obtained by
measurement from the flow p i c t u r e s and the shock wave angle as obtained
from the shock p o l a r s for each speed investigated,
A further comparison
was made with the r e s u l t s obtained by using the shock wave charts p r e 2
sented by Lai tone.
The r e s u l t s are given in Figures 25 and 26. I t
may be seen that the three methods of obtaining the shock wave angles
for a given free stream Mach number and semi-wedge angle agree very
closely.
This close agreement i n d i c a t e s that the application of the
Bruman, J . R . , "Application of the Water Channel Compressible Gas
Analogy." North American Aviation Incorporated Engineering Reoort, Ho.
H&-47-87, 1957. '
Lai tone, E.V., "Exact and Approximate Solutions of Two-Dimensional
Oblique Shock Flow." Journal of the Aeronautical Sciences, Vol. 14, Bb.l,
January, 194?, pp. 25-^1.
21
hydraulic analogy i s a useful tool i n the study of
«
shock wave formation.
To insure no d i s t o r t i o n as a r e s u l t of enlarging the flow p i c t u r e s ,
a comparison was made "between contact p r i n t s of each wave and enlargements
of the same wave formation.
No d i s t o r t i o n , as a r e s u l t of enlargement,
was noted.
Curves showing the v a r i a t i o n of l o c a l Mach number about the a i r f o i l
models are presented i n Figure 27 and Figure 28.
The l o c a l Mach number
v a r i a t i o n for each model was calculated from the M* equation^.
I t may be
observed that the local Mach numDer i s a function of the t o t a l head and
the l o c a l water depth.
Since the t o t a l head consists of a velocity
head and a s t a t i c head the shock polar diagrams were employed to find the
water velocity "behind the shock wave, and the s t a t i c head was determined
from the side photographs.
The s t a t i c head was taken as that water depth
where the water l i n e on the model was horizontal.
The local water depth
was obtained "by measurement from the side photographs.
Airfoil theory p r e d i c t s that for a diamond a i r f o i l i n a supersonic
flow a t zero degrees angle of attack the local Mach number remains constant over the forward p a r t of the a i r f o i l up to the point of maximum
thickness, at which time i t increases abruptly to a new value.
From
the curves of l o c a l Mach number d i s t r i b u t i o n for the diamond a i r f o i l
it
may be seen that the l o c a l Mach number i s not constant over the forward
thirty
percent of the a i r f o i l chord, but goes to r e l a t i v e high values
a t the leading edge.
The curve does not change abruptly a t the point of
maximum thickness, as theory p r e d i c t s , but there i s a smooth t r a n s i t i o n
3
Cf Ante,p.10
22
=£rom the lower value to the high value as the expansion takes place.
The l o c a l Mach number then "becomes constant over the remainder of the
airfoil.
The high values of the Mach numbers at the leading edge may "be
explained since in the development of the hydraulic analogy i t was a s sumed that the hydraulic jump took place along a single l i n e , and the
water depth changed abruptly along that l i n e .
Actually, these conditions
did not r e a l i z e since the hydraulic jump required a f i n i t e distance
r e l a t i v e to t i e a i r f o i l i n which to form.
I t therefore appears that
the hydraulic analogy w i l l not give the correct local Mach number d i s t r i b u t i o n immediately following the leading edge for the diamond a i r f o i l .
For the c i r c u l a r arc a i r f o i l an e n t i r e l y different trend of the
l o c a l Mach number curve may be observed.
As for the diamond a i r f o i l ,
the local Mach number goes to a r e l a t i v e high value a t the leading edge.
After the minimum value i s attained the local Mach number increases
over the e n t i r e chord length.
This d i s t r i b u t i o n of local Mach number
i s i n agreement with theory.
An attempt was made to p l o t the l o c a l Mach number d i s t r i b u t i o n
about the models over the range of free stream Mach numbers from .75
to .90.
At these slow speeds the meniscus effect resulted in an e r r a t i c
water depth d i s t r i b u t i o n about the models, and a s a r e s u l t no dependable
data was obtained from the side photographs.
After t h i s condition was
noticed the models were waxed and polished to reduce the surface roughness.
No marked improvement in the water depth d i s t r i b u t i o n was
noted.
I t would seem, then, th&t for slow speed t e s t s deeper water
should be used to reduce the r e l a t i v e magnitude of the meniscus
effect.
23
*COHCLUSIOHS
(1)
The hydraulic analogy o f f e r s a r e l a t i v e l y inexpensive meth-
od of conducting research i n supersonics,
(2)
The water channel i s especially suited for c l a s s room dem-
o n s t r a t i o n of shock wave phenomena.
The r e f l e c t i o n of shock waves
from the channel w a l l s , and the i n t e r s e c t i o n of shock waves may "be
observed as well as the shock waves formed by various shapes.
(3)
With a more a c c u r a t e method of depth measurement more quan-
t i t a t i v e d a t a may be o b t a i n e d ,
(k)
The d e t e r m i n a t i o n of shock wave a n g l e s from t h e flow p h o t o -
graphs for t h e a i r f o i l s t e s t e d agrees very closely with r e s u l t s obtained by a p p l i c a t i o n of t h e o r e t i c a l methods*
(5)
The r e l a t i v e v a l u e of t h e l o c a l Mach number d i s t r i b u t i o n
about the a i r f o i l s t e s t e d a g r e e w i t h t h e o r y .
(b)
A c c u r a t e q u a n t i t a t i v e r e s u l t s w i l l not be o b t a i n e d when
t h e h y d r a u l i c analogy i s a p p l i e d t o t h e study of a i r flow because t h e
w a t e r flow i s analogous to a g a s flow w i t h
.
- 2 , whereas f o r a i r
S Ijfc,
RECOMMESTMIONS
(l)
With t h e p r e s e n t e x c e l l e n t d r i v e mechanism i n s t a l l e d ,
a c c u r a t e model speed i s a s s u r e d .
very
However, t h e d r i v e d e v i c e t r a n s m i t s
a c e r t a i n amount of v i b r a t i o n to the model which c o m p l i c a t e s the flow
photographs by s e t t i n g up e x c e s s i v e secondary r i p p l e s .
To o f f s e t
this
o b j e c t i o n a b l e v i b r a t i o n the model support l i n k a g e should be s t i f f e n e d .
The model i t s e l f
should be made h e a v i e r to h e l p reduce v i b r a t i o n .
(2)
U!he use of fresh water i n the water channel r e s u l t s in
the hydraulic Jump inducing a great number of r i p p l e s ahead of i t self which has no p a r t i n the hydraulic analogy.
By allowing the
water to remain i n the channel over a period of twenty four to forty
eight hours r e s u l t s in clearer wave photographs,
(3)
By further experimentation with the photographic technique
the present large amount of equipment involved could be eliminated
except for the underwater l i g h t s , and the screen r e s t i n g on the floor.
(*4-)
The sides of the model should he wet "before each run
to r e s u l t i n constant meniscus effect,
(5)
For t e s t speeds of l e s s than Mach numher
a water depth
somewhat greater than one quarter of an inch should he used i n the
channel,
25
BIBLIOGRAPHY
i
B i n d e r , R. 0 , ,
F l u i d Mechanics. Hew York: P r e n t i c e - H a l l I n c . , 1 9 ^ 7 . 307 p p .
B i n n i e , A. M., and S.G. Hooker, "The Flow Under G r a v i t y of an Incompressi b l e and I n v i s c i d F l u i d Through a C o n s t r i c t i o n i n a Horizont a l Channel." P r o c e e d i n g s of the Royal S o c i e t y . Vol. 159,
(London, England: 1937), PP. 592-608.
Bruman, J . R., " A p p l i c a t i o n of t h e Water Channel Compressible Gas Analogy."
North American A v i a t i o n I n c o r p o r a t e d Engineering R e p o r t .
No. NA-47-87, W .
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Surface Waves i n Water and Shock Waves i n a Compressible
G a s . " C a l i f o r n i a I n s t i t u t e of Technology H y d r a u l i c s
L a b o r a t o r y P r o g r e s s Report. No. 5^,19^6.
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Hydrodynamics. 6 t h E d i t i o n , London: Cambridge U n i v e r s i t y
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C h a r a c t e r i s t i c s of Supersonic A i r f o i l s a t High Subsonic
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Between Water Plow With a F r e e Surface and Two-Dimensional
Compressible Gas Flow." U.S. Advisory Committee f o r Aeronaut i c s Technical Note.
No.1185, 19^7, 20 -pv.
Page, Leigh,
I n t r o d u c t i o n t o T h e o r e t i c a l P h y s i c s . New York: D. Van
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P r e i s w e r k , E r n s t . , " A p p l i c a t i o n of the Methods of Gas Dynamics to Water
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P a r t I . , "Flows with No Energy D i s s i p a t i o n . "
U.S. N a t i o n a l
Advisory Committee f o r A e r o n a u t i c s Technical Memorandum.
No. 9 3 ^ . 19^0, 69 p p .
P a r t I I . "Flows With Momentum D i s c o n t i n u i t i e s (Hydraulic
Jumps)." U.S. Advisory Committee f o r A e r o n a u t i c s Technical
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No. 935, 19^0, 56 jiV,
Riabouchinsky, D , , Mecaniques des F l u i d e s , Comptes Rendu, t . 195 No.22,
P a r i s , November 28, 1932, p p . 998-999.
Boufee, Hunter, Fluid Mechanics for Hydraullc Engineers. New York and
London: McGraw-Hill Book Company, I n c , , 1933, 420 vv»
Vennard, John K., Elementary Fluid Mechanics. New York: John Wiley
and Sons, I n c . 19^7, 339 pp.
2?
APPENDIX I
FIGURES
03
FIGURE 1
GENERAL VIEW OF WATER CHANNEL
FIGURE 2
GENERAL VIEW OF WATER CHANNEL
30
iE MECHANISM
&OOM SPOT
L/GHT
GLASS
SUN SPOT
L/GHT
^0
*Q
PROJECTOR
SZOOD L/GHTS-.
SCREEN
FIGURE 4
SCHEMATIC DIAGRAM OF WATER CHANNEL AND LIGHTING EQUIPTMENT
BOTTOM
FIGURE 5
SHOCK POLAR DIAGRAMS FOR
DIAMOND AIRFOIL TEST SPEEDS
V<
FIGURE 6
SHOCK POLAR DIAGRAMS FOR
CIRCULAR ARC AIRFOIL TEST SPEEDS
'2 -6
FIGURE 7
SHOCK POLAR DIAGRAM
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FIGURE 9
CIRCULAR ARC AIRFOIL MODEL
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FIGURE 10
MODEL AIRFOIL ANGLE OF ATTACK CALIFRATION CURVE
:
26
26.1
26.2
26.3
26.4
26.5
26.6
DISTANCE FROM CHANNEL SIDE TO LEADING EDGE AND TRAILING
EDGE OF MODEL AIRFOIL WITH 12 INCH CHORD, INCHES
26.7
-sj
FIGURE 11
VARIATION OF WAVE VELOCITY WITH WAVE LENGTH FOR
A STATIC WATER DEPTH OF ONE QUARTER OF AN INCH
WAVE
8
10
L E N G T H , X , INCHES
14
SCREEN-'
GLASS BOTTOM
(a)
SCREEN
UN'DULAR HYDRAULIC JUMP
GLASS BOTTOM
(b) CAPILLARY WAVES
FIGURE 12
ANALYSIS OF REFRACTION PATTERNS FORMED h\ W
UNDULAR HYDRAULIC JUMP, AND CAPILLARY WAVES
1*0
^rf
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FIGURE 13
FLOW ABOUT CIRCULAR ARC AIRFOIL, M=1.56
Ul
(a)
FIGURE 14
FLOW ABOUT CIRCULAR ARC AIRFOIL, M=2.34
k2
(a)
(b)
FIGURE 15
PLOW ABOUT CIRCULAR ARC AIRFOIL,
1=3.28
hi
(a)
(b)
FIGURE 16
FLOW ABOUT LENTICULAR AIRFOIL, M=3.71
kk
(a)
(b)
FIGURE 17
PLOW ABOUT LENTICULAR AIRFOIL, M=4.26
45
(a)
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1
d ^ ^ M H | j | ^
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FIGURE 18
PLOW ABOUT LENTICULAR AIRFOIL, M=6.78
he
(a)
(b)
FIGURE 19
FLOW ABOUT DIAMOND AIRFQIL, M=2.63
47
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(b)
FIGURE 20
PLOW ABOUT DIAMOND AIRFOIL, M=2.47
48
(a)
(b)
FIGURE 21
FLOW ABOUT DIAMOND AIRFOIL, M=3.02
49
(a)
(b)
FIGURE 22
FLOW ABOUT DIAMOND AIRFOIL, M=3.72
50
(a)
(b)
FIGURE 23
PLOW ABOUT DIAMOND AIRFOIL, M=4.53
51
(a)
FIGURE 24
FLOW ABOUT DIAMOND AIRFOIL, M-5,08
FIGURE 2 5
VARIATION OF SHOCK WAVE ANGLE WITH FREE
STREAM MACH NUMBER FOR DIAMOND AIRFOIL
m
w
LxJ
cc
a
a
fcj
w
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3
4
FREE STREAM MACH NUMBER, M
FIGURL 26
VARIATION OF SHOCK WAVE ANGLE WITH FREE STREAM
MACH NUMBER FOR SYMMETRICAL CIRCULAR ARC AIRFOIL
3
4
FREE STREAM MACH NUMBER, M c
5
5*4FIGURE 27
DIAMOND AIRFOIL LOCAL MACM NUMBER DISTRII UTION
FOR VARIOUS FREE STREAM MACH NUMI ERS
20
40
DISTANCE FROM LEADING
60
80
EDGE, PERCENT CHORD
FIGURE 28
55
CIRCULAR ARC AIRFOIL LOCAL MACK NUMI ER DISTRIBUTION
FOR VARIOUS FREE STREAM MACI! NUMBERS
20
40
60
80
DISTANCE FROM LEADING EDGE, PERCENT CHORD
100
56
FIGURE 29
PRESSURE COEFFICIENT FOR
DIAMOND AIRFIIL AT Ms = 4.53
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u
UJ
o
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UJ
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UJ
cc
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G207
57
FIGURE 30
PRESSURE COEFFICIENT FOR
CIRCULAR ARC AIRFOIL AT M =4.26
<3 |
FW
i—i
CJ
w
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U
cc
wi
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cc
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PERCENT CHORD