Flat plate cascades at supersonic speed

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Title:
Flat plate cascades at supersonic speed
Series Title:
NACA TM
Physical Description:
130 p. : ill. ; 28 cm.
Language:
English
Creator:
Badrawy, Rashad M. El
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Cascades (Fluid dynamics)   ( lcsh )
Aerodynamics -- Research   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: A brief review of exact two-dimensional supersonic flow theory and Ackeret's linearized theory are first presented. The lift and drag coefficients of a cascade of flat plates are calculated exactly and compared to those obtained using the linearized theory. The forces on the cascade are determined for unsteady inlet flow. The flat plate cascade theory is extended to compute the efficiency of a supersonic propeller with friction and finite blade thickness.
Bibliography:
Includes bibliographic references (p. 78).
Statement of Responsibility:
by Rashad M. El Badrawy.
General Note:
"Translation of "Ebene plattengitter bei Überschallgeschwindigkeit." from Mitteilungen aus dem Institut für Aerodynamik an der E.T.H., no. 19, 1952."
General Note:
"Report date May 1956."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003807842
oclc - 128147818
sobekcm - AA00006138_00001
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AA00006138:00001

Full Text
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1369
FLAT PLATE CASCADES AT SUPERSONIC SPEED By Rashad M. El Badrawy
Translation of "Ebene Plattengitter bei Uberschallgeschwindigkeit." Mitteilungen aus dem Institut flir Aerodynamik an der E.T.H., no. 19, 1952
32611-7011 USA
Washington May 1956




\Li n i ore
CONTENTS
Page
PREFACE.............................. iii
INTRODUCTION........................... 1
CHAPTER I. THE FLAT PLATE.................... 1+
1. General Considerations Stipulations ............ 4
2. Conditions at Expansion Around a Corner ........... 6
3. Conditions of Oblique Compression Shock ........... 8
k. Lift and Drag of an Infinitely Thin Plate (Exact Solution) 10
5. Lift and Drag at High Mach Numbers.............. 13
6. Calculation of Lift and Drag by Linearized Theory...... 15
7. Comparison of the Results of the Linearized Theory
With Those of the Exact Method............... 18
CHAPTER II. INTERSECTION, OVERTAKING AND REFLECTION OF
COMPRESSION SHOCKS AND EXPANSION WAVES ............. 19
1. Introduction......................... 19
2. Small Variations ....................... 19
3- Overtaking of Compression Shock and Expansion Wave ...... 22
h. Intersection of Compression Shock and Expansion Wave ..... 26
5- Crossing of Expansion Waves ................. 31
6. Reflection of Compression Shocks and Expansion Waves ..... 33
CHAPTER III. THE CASCADE PROBLEM ................. 3^
1. Problem...........................
2. Method of Calculation.................... 34
3- Example........................... 35
k. Calculation of Thrust, Tangential Force and Efficiency .... 37
CHAPTER IV. LINEARIZED CASCADE THEORY ............... kO
1. Assumptions......................... kO
2. Linearization of Cascade Problem.............. kO
3- Calculation of Lift and Drag................. kl
h. Numerical Example...................... kh
5. Comparison With Exact Method................. k^>
CHAPTER V. SCHLIEREN PHOTOGRAPHS OF CASCADE FLOW ......... k6
1. Cascade Geometry ....................... 46
2. Experimental Setup ...................... V7
3- Schlieren Photographs.................... k'J
CHAPTER VI. THE FLAT PLATE CASCADE AT SUDDEN ANGLE-OF-ATTACK
CHANGE............................. k-9
1. Problem........................... k9
2. The Unsteady Source..................... 50
i


Page
3< Pressure and Velocity of a Periodically Arising
Source Distribution .................... 52
k. Single Flat Plate in a Vertical Gust (Biot 19^5)...... 59
5. The Straight Cascade.................... 6l
6. Numerical Example ...................... 67
CHAPTER VII. EFFICIENCY OF A SUPERSONIC PROPELLER ........ 71
1. Introduction........................ 71
2. Effect of Friction on Cascade Efficiency .......... 71
3. Effect of Thickness..................... 73
k. Appraisal of the Efficiency of a Supersonic Propeller .... 74
SUMMARY............................. 77
REFERENCES............................ 78
TABLES.............................. 79
FIGURES............................. 91
ii


PREFACE
The work on the present report was carried out at the institute for Aerodynamics of the E.T.H., Zurich, under the direction of Prof. Dr. Ackeret, during the time from December 19^9 to June 1951-
I want to express here my deep gratitude to Prof. Ackeret for his suggestions and for the great interest he took in my work.
I am very grateful to Dipl.-Eng. B. Chaix, scientific assistant at the Institute, and to Mr. E. Hurlimann, precision mechanic, for their indispensable help in taking the schlieren pictures.
I should like to acknowledge that the "Faruk University", Alexandria (Egypt) made my studies in Zurick possible.
iii


Digitized by the Internet Archive in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation
http://www.archive.org/details/flatplatecascadeOOunit


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1369
FLAT PLATE CASCADES AT SUPERSONIC SPEED* By Rashad M. El Badrawy
INTRODUCTION
The cascade problem in the subsonic range can be analyzed under certain assumptions either by mapping or substitution of the blades by singularities sources, sinks and bound vortices where the separation of flow from the blades can cause various departures from the obtained results.
Raising the flow velocity to a given value is accompanied by sonic velocity within the cascade, which usually renders the solution of the problem even more difficult. The same complication exists on the cascade in flow at supersonic speed, in which the velocity is retarded to subsonic by shocks.
But when the cascade operates entirely in the supersonic range, the conditions become clearer. All disturbances act downstream only from the sources of disturbance, so that the pressures and velocities at the surface of a sufficiently thin airfoil in the stream can be readily determined.
The present report deals exclusively with problems of cascade flow in the supersonic range. As is known the flat infinitely thin plate is the best airfoil with respect to wave resistance in supersonic flows; hence it is logical to start with the cascade of flat plates. The last chapter deals with the case of finite thickness.
Lift and wave resistance of an isolated plate are computed first since the cascade problem can often be reduced to this special case. The well-known theories of two-dimensional supersonic flow are applied -that is, the laws of oblique compression shocks and the expansion around a corner.
The air forces are then calculated again and compared with the previously obtained exact values by means of Ackeret's formulas of linearized theory.
*"Ebene Plattengitter bei Uberschallgeschwindigkeit." Mitteilungen aus dem Institut fur Aerodynamik an der E.T.H., no. 19, 1952.


2
NACA tm 1369
The cascade problem was to be solved in such a way as to be free from the inevitable inaccuracies of the graphical method. For this reason the cases of overtaking, crossing and reflection of compression shocks and expansion waves frequently occurring on supersonic cascade flows, which usually are solved by graphical method, are analyzed in chapter II.
In chapter III the cascade problem is discussed and its solution described in the light of the results obtained in chapter II. A numerical example is also given. The same chapter gives further a definition of the efficiency of the simple supersonic cascade and an evaluation for several angles of stagger and attack.
The small angles of attack involved justified the use of a linearized cascade theory.-'- This is done in chapter IV. The numerical example of chapter III is thus linearized and the results compared with those of the exact solution. The supersonic cascade flow at various angles of attack was recorded by schlieren photographs of the flow between two parallel plates, in the high-speed wind tunnel of the Institute (chapter V).
Chapter VI deals with the specific case of unsteady flow through the cascade, caused by abrupt angle-of-attack changes.
According to Ackert's linearized theory, the lift and drag of a double-wedge profile of thickness d and chord I at angle \|j in supersonic flow m is, in the presence of friction (cf)
For the best drag-lift ratio e = put = 0. This means that
ca air
the wave resistance should be equal to the sum of friction drag and thickness effect. In that event
Assuming possible values for d/l and cf results in comparatively small optimum angles ty0pf


NACA TM 1369
3
In chapter VII the effect of friction and thickness in a special case on the cascade efficiency is analyzed. Since there might be a possible application of the supersonic cascade to the supersonic propeller, a simple evaluation of the efficiency of such a propeller is made. A parallel steady two-dimensional flow is with exception of chapter VI postulated.
The conventional notation is used unless specifically stated other wise in the text.


It
NACA TM 1369
CHAPTER I. THE FLAT PLATE 1. General Considerations Stipulations
The general equation of continuity of any compressible flow is
dp t d(pu) { d(pv) t d(pw) Q ,x dt dx dy dz
The rate of propagation of a small disturbance, that is, the sonic velocity, is, as is known
a2=|UK (2) op P
CP
where k = .
cv
In flows, in which a flow potential cp exists, the continuity equation can be written as
2 2 2
1 /dp dcp dp dcp dp dp dp\ dcp dcp dcp .
+ rr_^+r!_:+rL + z. + z. + z. = 0 (3)
pa2\dt dx dx dy dy dz dZy ^2 ^y2 ^2 The momentum theorem gives the following relations
1 dp d2cp djp_ d^ dcp_ d2cp o dcp d cp
p dx dx dt T dx dx2 dy dx dy dz dx dz
1 dp >.2 d cp + ^ d2cp + dcp d2cp + dcp d2cp
p Sy dy dt dx dy dx dz dy dz
1 x2 a cp + ^2. v2 a cp dp d2cp dcp d2cp
p dz dz dt dx dz dx dy dz dy + dz2


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5
In two-dimensional flow the potential must therefore satisfy the equation
2 2 2 dcp dcp d cp 2 dcp d cp
d2cp 1 .2 i
dx2 a2W Sy2 a2W
2 dy dx dy 2 dx dx dt
2 2 2 dcp d cp 1 d cp
a2 dy dy dt a2 dt2
(5)
The velocity of sound is then
a2 = aQ2
( 1)
where ag = velocity of sound in state of rest
+ fa)2
dt
(6)
-,2 d cp
dx'
the equation is reduced to
-,2 d cp 1 1M2 2 2 cfcp dcp d cp
Sy2 *2WI a2 dx dy dx dy
= 0
(7)
and the flow is completely identified, if the function cp(x,y), which is to satisfy the boundary conditions, is determined.
This equation is either elliptic, parabolic, or hyperbolic, depending
upon
(1 M2) ^ 0
where
M = grad cp
3. I
(8)
is the local Mach number.


6
NACA TM 13b9
The use of this equation is difficult if its type in the particular range, as in the transonic range, is changed. However, the flows analyzed here, are of identical character everywhere, that is, the flow is of the hyperbolic type.
One of the known solutions is the expansion around a corner, developed by Prandtl and Meyer (ref. l).
2. Conditions at Expansion Around a Corner
The two-dimensional flow past the wall AE at a Mach number M^_
(fig. 1) is deflected by a convex bend at E through an angle 0, through which an expansion is initiated. The disturbance proceeding from E spreads out solely in the range lying downstream of the Mach line EB^_,
where
Mx
b1ea1 = Mach angle ux = sin-1 ^- (9) and stops at the Mach line eb2, where
b2ed = m-2 = sin"1
m2
In it m2 is the Mach number of the flow after the expansion.
The streamlines in range b1eb2 are curved similarly and run parallel to the wall ED downstream of this range.
It can be proved that the Mach lines in this flow are the characteristics of the differential equation which define the potential.
When the expansion proceeds from a Mach number Mi = 1, ^u-j_ = the following relations can be proved (ref. 2):
tan U2 = A cot Acu (10)


NACA TM 1369
7
P2
k + 1
k
Po ^2 cos2?^ (Pq = stagnation pressure)
(11)
M2 ='
Obviously
(k + 1) 2 cos2A(o ( kc l)cos Atu
8 = a) + uo -
* 2
As function of M2 (ref. 3)
e =
cos-1 + A cos"1/1 -M2 2
k + 1
k 1 2 1 + -M2
(12)
(13)
(14)
This equation gives a maximum angle of expansion 0max> which corresponds to a Mach number M2 = after expansion (k = 1.400)
max = 150.45
If the Mach number before the expansion M]_ is assumed other than 1, the maximum angle of expansion becomes obviously
e
max M]_ ~ omax
= 9mav V
where v is the angle of expansion from M = 1 to M^. The values for various Mach numbers of the inflow (Mi) are
Mi 1.00 1.50 118.55 2.00 2.50 5 8 10 00
0max Mi 130.45 104.07 91-32 53-55 34.53 28.14 0


6
NACA TM I369
3. Conditions of Oblique Compression Shock
The discontinuities that may appear in supersonic flows and across which velocity, pressure, density, temperature, and entropy undergo a discontinuity, while the total energy, thermic and mechanical, remains constant, were predicted by Riemann (i860) and Rankine and Hugoniot (1887) as normal compression shocks.
In oblique shocks (Prandtl-Meyer) only the velocity component normal to the shock front is modified.
In figure 2 the supersonic flow past the wall AE is deflected at E by an angle 5. A compression shock is produced and the shock front ES is inclined at an angle 7 the shock angle toward the air flow direction.
With subscript 1 denoting the state before the shock and subscript 2 that after the shock it can be proved that (refs. 3 and k)
p1 k + 1\ 1 2k J
(15)
!2_
PO
k ~ 1
k. + 1
1
,poj
Hk .2 [k 1 -- sm^7 -
(k + l)2 U + 1
1 +
sm^7
(k 1)'
tt-1
(16)
1 K 1) + -
P2 K + 1 M^sin2/
(it)
cot 6 =
(k + 1 m12
V 2 M12sin27 1
1 tan 7
(18)
u2 cos 7 u]_ cos (7 5)
(19)


NACA TM 1369
9
tan(7 5) Pi
tan S
k + 1
M22sin2(7 5)
(20,21)
A direct relation between the Mach numbers before and after the shock can be established
M.
cos 7
M-^ cos (7 6
(22)
The relation for the change of the static pressure by the shock is the same as for the normal shock when it is applied to the velocity component perpendicular to the shock front. Consequently
K + 1 u 2 2 \ K~1 -M-, sin^ \
2 1 '
P0l
1 +
M-|_ 2sin2y
2k ..2.2 K --M-i sin^7 -
k + 1 x k + 2 j
1
1-k
(23)
7
From these equations it follows that the shock angle 7 is greater than the Mach angle, that is, the speed of propagation of a finite disturbance is greater than the sonic velocity. When the angle of deflection 5 approaches zero, 7 = u and the shock changes to a Mach wave.
Also of interest is the shock angle at which the Mach number after the minimum shock becomes equal to unity. Denoting this angle by 7g
it can be proved (the weak stable compression shock is always allowed for) (ref. 3, P- ^7) that


10
NACA TM I369
This equation is used to determine the maximum shock angle which corresponds to a Mach number before the shock M^ = and a Mach number M2' = 1 after the shock. The result is
sin27s = ^_L2 (25)
hence
2k
7S = 67.8
at k = 1.400 (air).
By equation (l8) the corresponding deflection angle 5S is
6S = 45-58
p
Table 1 and figure 3 represent the values of 7S and 6S at various Mach numbers M^.
4. Lift and Drag of an Infinitely Thin Plate (Exact Solution)
An infinitely thin plate ab in parallel flow at supersonic velocity Uj_ is placed at the angle i(r. It is assumed that the width of the
plate transverse to the flow direction is so that the problem is two dimensional.
The streamlines above oa (fig. 4) experience a deflection which is associated with an expansion. So the state at the upper side of the plate can be defined by equations (10) to (l4). But below the plate a compression shock ad occurs. The state of the flow on the lower side of the plate is accordingly determined from the formulas (16) to (21).
The force on the plate per unit area is
K =
= (p2' P2) (26)
2
The weak stable shock is always taken into account. See Richter, ZAMM, 1948 and Thomas, Proc. N.A. Sc., Nov. 1948.


NACA TM 1369 11
where p2 1 and p2 represent the pressure on the lower and upper side of the plate.
Obviously, the lift A and the drag W per unit width are
A = K cos ljrL (27) W = K sin \|rL (28)
To compute a lift coefficient, a reference dynamic pressure of the inflow
H = 2 plul
or
*l = I PlMl2 (29)
is utilized.
As function of the Mach number M^, the ratio of dynamic to airstream pressure is
ql K 2 = Mi
Pi 2
that of dynamic to stagnation pressure is
= M-, 2 = ^M^LlM!2 + K (30)
PQ2PQ1 2 1 \ 2 1 J
The results are represented in table 1 and figure 5-


12
NACA TM 1369
Lift and drag coefficients are herewith
cos \|r
sin t|(
or, if all pressures are referred to stagnation pressure Vq>
P2' P2 PO Po 1
-a cos \|i cw =--- sin \|j
p2\


po
51
po
(31)
(32)
The drag/lift ratio is
e = = tan \|/ ca
(33)
Table 2 gives the values of ca, cw, and e up to Mj_ = 10 as computed by the formulas (32) and (33)-
In the calculation of the Mach numbers up to Mj = k, the tables by Keenan and Kaye (ref. 6) as well as those by Ferri (ref. 3) were used to define p2/p0 anc^ p2/p1 (K = 1-400).
For higher Mach numbers, the formulas of sections 2 and 3 were employed. At each Mach number, the angle of attack was varied up to *S(M2' = !)
Figures 6 and 7 show the variation of cw and ca over the angle of attack figure 8 shows the polars ca plotted against cw.
The boundary curves show the maximum lift and drag coefficients that can be expected without getting in the transonic range.


NACA TM 1369
13
Other values for the boundary curve are given in table 3- Since the pressure distribution on the upper and lower side is constant, the resultant force is applied at plate center and is normal to the plate. There is no suction force as in subsonic flow.
5. Lift and Drag at High Mach Numbers
At high Mach numbers the angle of attack of the plate can exceed the maximum expansion angle Qmax (section 2) corresponding to the Mach
number of the airstream max = a^ ^1 = 6.4). Hence, when assuming
continuous flow, an empty wedge-shaped zone between plate and flow appears. This zone is largest at constant angle of attack when mq_ = in that
event, no deflection of flow is possible.
Owing to this vacuum space, the pressure at the upper side is zero. The resultant force k is obtained then from the pressure on the lower side, behind the compression shock. Hence, per unit area
k=p2'' (34) or, when referred to the dynamic pressure of the airstream,
*1 *1 KM!2?!
Introducing Pp'/Pl from equation (15) gives
k 4 .0 ft 1 2 = sir7---
1l + 1 + 1 KMX2
where the term containing l/Mj_2 can be disregarded without great error.
k 4.P = sir7
qi k + 1
(36)


Ik
NACA TM 1369
So the lift and drag coefficient are
k p sin^7 cos \|/
k + 1
k p Cy. = sin^ sin
k + 1
(37)
Both formulas are dependent on 7 and \(f only. Between these there exist the relation given in equation (l8), which can be written as follows (5 = \|r) :
cot \Jf
'k + 1
2.2 1 siry -
M]2
1 tan 7
where the term can be disregarded again. Then
M
1
k + 1 \
cot \Jr = J----1\ tan y
2 sin27
(38)
The values and curves designated with M = in table 2 and figures 6, 7 and 8 were defined by equations (37) an(i (38). For comparison the lift and drag was also computed by Newton's formula (the normal component)
ca = 2 sin2i|f cos ^
cw = 2 sin3\|
(39)
The corresponding values and curves carry the subscript N.


NACA TM 1369
15
6. Calculation of Lift and Drag by Linearized Theory According to Ackert's linearized theory, the members of higher order in can be disregarded without great error in the potential equation
-,2 0 cp
dx<
1 -
1 /dcp
ac
2\dx
-,2 o cp
5/
a2Uyy
2 dcp dcp d cp
a2 dx dy dx dy
= 0
for slender bodies at small angles of attack, because the interference flows are small compared to that of the airstream.
The equation reads accordingly
^2 o cp
dx'
i/dcp;
s2 dx,
.2
By2
= 0
Inserting
-L&f = M2 a2\dXy/
and observing that M is greater than unity, the equation reads
^P(M2 1) ^ = 0
(40)
dx'
dy
The general solution of this equation is
cp = f
M2 1
(41)
It indicates, as stated in section 2, that the lines of constant
potential are the Mach lines of flow, and their slope has the Mach angle u.


16 NACA TM I369
This solution shows further that the flow velocities
dtp dcp
u = and v =
dx dy
satisfy the condition
u =
(42)
Vm2 1
At the surface of a body in the stream
u dy_ U dx
is applicable.
The pressure variation by the momentum equation reads
^ = U AU = Uu (43) P
where U is flow velocity and u is interference flow in stream direction.
Accordingly
Ap 4 dy
2 1 [7^2 7 dx
2'
ipu2 \ju2 1
(44)


NACA TM 1369
17
The pressure difference between both sides of a flat plate is
Ap
4 dy
q i/m2 dx
(45)
The lift and drag coefficients at the angles in question are
M2 1
-w
4\|f2
M2 1
(46)
The drag /lift ratio according to this theory is
-w
e = = i|r
(47)
Instead of the expansion wave and the compression shock at the leading edge, it has simple Mach lines as interference lines (fig- 9), in contrast to the exact theory.
The values given in table 4 and plotted in figures 10 and 11 were computed by these formulas. The calculations were carried out at each Mach number up to angle of attack i|rs from the exact theory. The corresponding ca and cw values lie on the curve G1
At sonic velocity on the lower side of the plate p2vp0 = 0.5283. This value, introduced in the following directly obtainable relation
P2' Pi = ~ Ap =
2^i
(48)
- 1


18
NACA TM 1369
and
P2'/pq Pi/pq = 2^
ql/P0 ^Ml2 1
gives the angle of attack (ig^) corresponding to M2' = 1, which in general is greater than i|/s .
7. Comparison of the Results of the Linearized Theory With
Those of the Exact Method
In table 5> "the difference is (cq c^), where cq is the coefficient of the exact method and c^ is that of the linearized theory at Mach numbers M]_ = l.kO and Mq_ = ^>.00.
It follows that the linearized theory is a very good approximation for small angles (up to about 10) For greater angles the values of ca
and cw are too small.
In fi gures 6 and 7j the ca and c-^ curves by linearized theory marked A' and A are included for comparison.


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19
CHAPTER II. INTERSECTION, OVERTAKING AND REFLECTION OF COMPRESSION SHOCKS AND EXPANSION WAVES 1. Introduction
Overtaking of expansion waves and compression shocks in supersonic flows occurs when the marginal streamlines or boundary walls change their direction twice in the opposite sense (fig. 12(a)).
If expansion waves or compression shocks strike a fixed wall and their slope toward the wall does not exceed a given angle, they are reflected as expansion waves or compression shocks (fig. 12(c)). Crossings occur in flows through channels and free jets (fig. 12(b)). All these events can occur in cascade flows (fig. 12(d)).
2. Small Variations
(a) Suppose that a small expansion occurs at B in the supersonic flow Mi, pi, ai, past the wall AB (fig- 13). The angle of
expansion is AS. If A9 is sufficiently small, differential considerations are permissible.
Bernoulli's equation gives
APl = p1XJ1 AU1 (49)
where U is the magnitude of the velocity and AU its variation; Ap is the pressure variation.
Since the vectorial velocity variation is normal to the Interference line, the variation of U is
U-, A9
A|U| = U-l tan U;L A6 = f (50)
Mi2 1


20
NACA TM 1369
hence
Ap = -== (51)
y^i2 1
But, as the dynamic pressure q is given "by
q = \ piui2 = \ *pimi2
the pressure variation can be written as
Ap = 1 1 (52)
vV -1
The variation of the Mach number M follows at (ref. 3> P- 26)
The Mach line BEj_ forms with flow direction AB the Mach angle u-^, the Mach line BE2 at the end of the expansion the Mach line u2. Now
it may be assumed that this small expansion takes place on the interference line BE1, whereby
u-, + u2 A8 A'BE = --- (54)
(b) A simple differentiation gives the change of the shock angle 7 as well as the pressure change p2' after the shock, due to a small variation of the angle of deflection 5, for the compression shock AB (fig. 14).


NACA TM 1369
Between 7 and 5 the relation (eq. 18)
cot & = tan y\- 1
1M^2sin27 l
exists, and therefore
A5 = sin26
/A ,\ 2 2 2A sec y\--1 M-i siry
B 1 w2
Ay = C Ay
where
A =
n + 1 ,, 2 M-i
and
M]_2sin27 l\
C = sin25
2 lA A 2 2 2A sec^7j -11- M-j_ sin^7
Bc
The
pressure p2' after the shock is (eq. 15)
2k p ? k + 1
p2' = pi(r~r Mi sin 7 "
ik + 1
k 1
and the result for a small variation of the shock intensity i
2k
AP2 = PiM]_ sin 27 Ay
k + 1


22
NACA TM I369
3. Overtaking of Compression Shock and Expansion Wave
(a) The supersonic flow Mj_, p-^, past the wall AE (fig- 15(a))
undergoes a directional change 5 at E. The compression shock EF and the flow direction form the shock angle 7^ in zone (l). At C an expansion takes place about an angle Q, and the expansion wave FCG overtakes the compression shock at F. To simplify the calculation, the continuous expression is replaced by a given number of expansion waves of finite intensity, whereby a successive expansion through these waves is assured. If n is the number of waves, the expansion due to a wave is &/n = A3. The number n must be so chosen that A6 is sufficiently small.
Now consider the intersecting of wave CF^A) and compression shock EFi(5), figure 15(b).
From Fj_ the compression shock advances with weaker intensity in direction Gj_, that is, it deflects the flow less say by 6'. FiGj_ forms
with the flow direction the angle 7' at (l). Indicating the various zones by (l), (2), (3), and (4), the streamline through Fj_ splits the zone (k)
into the portions (kQ) and (^Hi). The flow in (2) and (3) is fully known, because the angles 6 and A0 are known.
To define the conditions in (k), the streamlines S^, S2. and S5
are examined. The directional change of S2 amounts to (5 A). But along S* the flow experiences the directional change 6'. To maintain
equilibrium in (4), the pressure as well as the velocity direction above and below the streamline S^ must be equal.
In general, the pressure change from (l) toward (3) is not the same as from (l) to (4), so that a reflected expansion wave possibly a small compression too must appear between (3) and (4), say along a line F^Hi.
Supposing that this reflected wave is an expansion wave of intensity A0*. By "intensity" of an expansion wave or a compression shock is meant the deflection, which the flow experiences in the process.
The pressure in (4u) is, (according to section 2)
p4u = p3|
AG'
(58)
Ms


NACA TM I369
23
The difference of the deflection angles amounts to (&' 6) = a5. In general, this intensity decrease is small, because the compression shock is much stronger than the expansion wave. The corresponding change in shock angle 7 is a7.
The pressure in (hQ) follows from the change in shock intensity. Hence we can say that
Pi, = P2' AP2'
(59)
This is again the equation for small variations derived for shocks from equation (57)- Accordingly
p4d = p2* piM:
2 2K
k + 1
sin 27 a7
(60)
Posting p^ = p^ gives
p3
M 2 1
3
a3'
p 2k
= P2' PiMl sin 27 a7
k + 1
(61)
For the velocity direction in zone (k) to be unequivocal, it must
a9 = a5 + a9' (62)
The relation between shock intensity variation and angle of shock (eq- (55)) together with the two previous equations gives
a91
as) r 1)
|a3FG Q c -')
(63)


2k
NACA TM 1369
a7 = a9 AB' (6
C C
with the constants
kMj2 kM22 M-^sin 2y
^Mj2 1 ^M22 1 m^sin2/ -
2k
The condition for equality of static pressures is not identical with that for equality of velocity magnitude above and below the streamline Sj_.
As the shock losses on either side of the intersection point F are unlike, the stagnation pressures in the wake above and below streamline
are different, hence there is a small vortex layer along this streamline.
Figure 16 represents the graphical solution of the problem by means of the characteristics and the shock polars. The condition for equality through equality of velocity magnitude in the entire zone (k) is approximated.
(b) The reflected wave is disregarded:
In general, the angle of deflection a01 intensity of the reflected wave is very small (compare numerical example). Thus the pressure in (3) is not much unlike that in (k), so that this reflected wave F^R^ can
be discounted.
In this event the flow directions in zones (3) and (k) are identical, or in other words
a6 = Ad (65)
With equation (55) a5 can be defined and from it the new direction 7' of the compression shock. The velocities in (3) and (k) have then obviously the same direction but not the same magnitude by reason of the small vortex layer developing between (3) and (k).


NACA TM 1369
25
Numerical Example
Free stream:
Pl/P0 = 0.12780 Mx = 2.000
Before overtaking:
shock intensity 5=6
shock angle 7 = 35-24
hence the state in zone (2)
p2/p0 = 0.1773 M2 =1.780
intensity of the expansion wave ab = 1
state in zone (3):
P3/P0 = O.I693 M3 = 1.818
Determination of constants:
c = 1.078
F = 3.04
G = 3-015 Q = 3-175
Inserted in equation (63) and (64) gives: intensity of the reflected wave
a7 = 0.89 ag' = 0.06
By equation (62)
a5 = ab a8' = 0-94
Therefore the shock intensity after overtaking is
5 = 5-04
The new shock angle is
7' = 7 a7 = 34-35
The reflected wave disregarded, leaves
a7 = C a6 = C ab = 1.078
shock intensity 8' = 5
angle of shock 7' = 34.l6
It is readily apparent that the reflected wave is very small, hence scarcely affects the pressure in zone (3)-


26
NACA tm i369
k. Intersection of Compression Shock and Expansion Wave
Figure 18(a) represents an expansion wave AG of intensity 9 proceeding from the corner A. At F this wave crosses a compression shock of intensity 6 emanating from the corner B.
As before, the continued expansion is replaced again by n expansion waves between which the flow is straight. The deflection by each wave is
After crossing (fig. 18(b)) the compression shock has an intensity 5' and a shock angle 7'. Now the expansion wave has the intensity AS1.
The zones produced this way are numbered (l), (2), (3), and (4). The streamline Fj_S splits the zone (k) into (hQ) and (4u)-
Looked for now is the shock intensity &', shock angle yx, and expansion angle A3' after crossing, and the state of flow in (4), when the state of flow in (l) 5, 7 and AS are known.
According to chapter I the state of flow in (2) and (3) can be determined directly.
The pressure in {\xl follows through a small expansion AS1 according to the laws in chapter II at
where AB is still unknown.
The method of solution consists in first making an assumption for the shock intensity after intersecting, which is
(66)
&i = (5 AS)
(67)
The corresponding angle of shock would be 7^


NACA TM 1369
27
The state after this shock, indicated by kQ', can again be defined according to chapter II. The pressure in (4Q') is
1 / 2k o p k l\ ,so\ Pl = P*-M/sin^.-- (68)
40 ^3^k + 1 p 1 k + 1 I
Now the flows in (4Q) and (2) have identical directions, but the pressures and the magnitudes of the velocity are different.
To assure equilibrium within (4), the pressures and velocity directions in (kQ) and (4^) must be equal. And to satisfy this condition the assumed shock must be intensified by Ab.
Obviously it shall
Ac-i = a9* (69) The new pressure in (4Q) is (according to section 2b)
p40 = V + P3M32 sin 27i A?i
(70)
where Ay^_ is the change in shock angle 7^ and is computed by equation (55)
A5,
A7i = (71)
(For the calculation of C see section 2b)
Posting p^ = p^, equations (66), (68), (70), and (71) give
2
- 1
Poll--: I = Pj M_2sin27l \ +
\J^2r '
p^2 rrisin 2? ir (72)
This equation is linear in A8' .


28
NACA TM 1369
Now the quantities 5' and 71 can be defined
5' = Bi + Ao = Bi + A9' (73)
AS'
7 + A7 = y + (74)
The pressure in (4Q) can be obtained directly from equation (70) The Mach number M^o itself can be determined according to chapter i,
if S1 and 7' are known. That in (4U) is likewise directly obtainable from M2 by the isentropic expansion a9*.
The slight discrepancy between the values M^o and M^ is due,
as stated in section 3> to the fact that the condition for pressure equality, owing to the change in static pressure after both shocks, does not require equal magnitude of velocity. So a small vortex layer along streamline Fq_S is to be expected.
Before intersecting the expansion wave forms with the flow direction in zone (l) the angle (section 2)
x = l + ^3 ae
After intersecting the angle with the stream direction in zone (2)
is
Uo + uj, A1 Y =---
But there is a difference 5 between the flow directions in (l) and (2), so that the looked-for directional change is
ux + u5 AB u2 + m, AB' b = X + S- Y= --- + S---- (75)


NACA TM 1369
29
The directional change of the shock front is
c = 7 (7' + A3) (76)
A negative angle b and a positive angle c indicates that the expansion wave or shock front after crossing is in more downstream direction.
For illustrative and comparative purposes, the graphical solution in figure 19 was made with the aid of the characteristics and shock polar. Here also the condition for pressure equality was replaced by velocity equality.
The described mode of calculation is used in the following numerical example for illustration.
Numerical Example
The flow in zone (l) is:
Pl/po = 0.22905 Mi = 1.1+35 u1 = 44.18 (See cascade
example of the following chapter III, zone (3))-
Data before crossing:
intensity of expansion wave A3 = 1
intensity of compression shock 5 = 3-03
shock angle of compression shock 7 = 47-91
With it the states in zone (3) become:
P5/P0l = 0.28478 M5 =1.469 = 42.89
Therefore
P2/Pi = 1-157 P2/p0i = 0.34560 M2 =1-332 u2 = 47-75
Assumed shock intensity
Si = 6 A9= 3-03 1 = 2.03


30
NACA TM 1369
corresponding angle of shock y 45-29
pressure after shock PLq^Pq^ = 0-3153
Determination of the constants
A = l^Ll m52 = 2-592
B = (m32sin27i 1) = 0.092
C = sin26j_
J2 /A
. p 2 2A
sec^/W 11 sin^7-M
\B / B'
1 -2
0.7504
Equality of pressure in (4q) and (4^ gives
Pku = P2(l
A6'
Mo 1
. 2 2k A31
P40 = p4q + P3m3 k~TT Sin 27i
which inserted gives
A9' = A5i = 0.89c
A/i = 1 = 1.19 c
After the crossing:
shock intensity o' = b + A5j_ = 2.03 + O.89 = 2-92
shock angle 71 = y + A7i = 45-29 + 1.19 = 46.48


NACA TM I369
31
Hence
P4/P0 = 0-3303 uk = 47-08
Mk = 1.3643 Hu
Mk = 1.3640
The comparison of the two Mach numbers indicates that the difference is quite small and lies within the calculation accuracy.
Directional changes:
Expansion Wave <) b = 43-036 + 3-034 47-47 = 1.4
Shock front c = 47.91 46.48 + 1 = + 0.43
5. Crossing of Expansion Waves
Each expansion wave is again replaced by n small waves. In figure 20(a), two waves of intensity A9]_ and A32 cross each other in F.
After crossing, the intensities are A9-j_' and A92' In this case, only
one stream direction is obtained in zone (4), when
A8X + A82 = Bi + A92' (77) Application of the relations of section 2 results in
pk = P5 Apj = p2 Ap2
that is


52
NACA TM 1369
Since all other quantities are known, ABp/ and ASq1 can be computed from these two equations.
The directional change of the Mach lines is like that in the preceding section
Ii, + 11, Aa |ip + |i, AG, 1
u-, + Up A9? |i. + uk A9?' <$ c = ---- + A92 ---- (80)
Since all changes follow the same adiabatic curve, the condition for pressure equality yields equal velocity values at both sides of the streamline FS. Hence, no vortex layer will appear. Figure 20(b) represents the graphical solution.
Numerical Example
Airstream:
Pl/p0 = 0.5295 M1 = I.566 u = 47.050
The intensity of the first expansion wave is: AB-]_ = 0.99- Therefore
P2/P0 = -5154 M2 = 1-402 u2 = 45-50
The second expansion wave intensity is: ASp = 1.06 The conditions in zone (3) are then
P5/P0 = 0.31245 M^ = 1.4041 u3 = 45.433
The two equations defining A9j_' and- A}' are:
A9i + AB2 = 0.03579 = A91' + A92'
2 2
KP-zM5 KppM2
Pk = P3 7 y = P2 z==r A92'
1/m52 1 \f
M2 -. 1


NACA TM 1369
33
hence
ASq' = 0.0184 = 1.054
AOo/ = 0.0174 = O.9970 after which the conditions in zone (4) become:
Pk/P0 = 0.29722 VLk = 1.44 = 44.01
By equation (79) and (80) the directional changes of the waves are
$ b = + 0.5 and c = 0.02
6. Reflection of Compression Shocks and Expansion Waves
No difficulties occur in the determination of the conditions existing behind the reflected compression shock FB (fig. 21). Those in zone (2) can be defined according to chapter I, if the state of the airstream and the intensity & of shock AF are known. Obviously the reflected shock is of the same intensity as the impinging shock, so that the shock angle 7 of the reflected shock and the conditions in zone (3) can be defined.
The same holds true for the reflection of expansion waves, when the intensity of the expansion waves and their slope with respect to the wall are known.


34
NACA TM I369
CHAPTER III. THE CASCADE PROBLEM 1. Problem
Visualize a cascade of infinitely many and infinitely thin flat plates, of which two adjacent plates AB and A'B* are represented in figure 22. The angle of stagger is 90 P, the spacing t and the blade chord L. This cascade is exposed at angle of attack ^ to a supersonic flow M}_, pj_, p-^.
It is assumed that the flow is the same in all planes perpendicular to the plates and determines the force, that is, lift and drag as well as the pressure variation along the plate (blade).
2. Method of Calculation
To each plate there correspond interference lines (chapter I), that is, the expansion wave issuing from the leading edge and compression shock (fig. 22).
At wide spacing, the separate blades of the cascade will not affect each other and the problem reduces to the single plate.
Now if the spacing decreases for constant chord, the interference lines of one plate intersect those of the other, without, however, any force being exerted on the plates themselves for the time being. In this event, the force on each plate is the same as on the single plate, except that the wake flow is slightly disturbed.
The values of t/L, below which the interference line of a plate begins to exert an effect on the adjacent one, are called (t/l)crit "critical chord-spacing ratio."
At t/L < ^t/Lcri-t^ the interference lines are reflected on the
plates. After the crossings and reflections, new zones appear on both sides of the plate where the pressure as well as the velocities are unlike the uniform pressures and velocities to be found at either side of the plate. As a result, there is a change in the total force as well as the lift and drag on each plate.
The mode of calculation consists in defining each intersection and reflection with the laws of chapter II and from it determining the conditions in the several zones. Integration of the various pressures on both sides of the plate gives then the total force, that is, the lift and drag.


NACA TM I369
35
The resultant force still is perpendicular to the plate, but no longer through the plate center, hence produces a moment with respect to the center. The position of the force is defined by statistical methods.
This method is illustrated in the following example.
3. Example
The cascade ABA'B1 (fig. 23) with 30 angle of stagger, that is, P = 6o, and at angle of attack of \|r = 3 is placed in a stream with M-l = 1.4004 (corresponding to v = 9)
The' blade spacing was assumed at the beginning, while the plate chord was so chosen after completion of the calculation that the expansion wave was reflected exactly once on the bottom side of the upper plate. It was found that t/L = 0.547-
The flow experiences a compression shock starting at the leading edge A1. The shock angle y = 49-57 is read from the shock tables and the shock front A*a can be plotted.
Proceeding from the leading edge A, an expansion wave spreads out between the Mach lines Ax and Ay. The first forms with the airstream direction the angle u-^ = 45.56. The characteristics tables give
My = I.503, that is, the Mach number which is obtained at an expansion
by 3 from the Mach number 1.4004. The corresponding Mach angle, that
is, the angle which direction Ay forms with the plate, would be
Uy = 4l.70o. Instead of the continuous expansion, assume an expansion
in three stages, each corresponding to a 1 deflection. The conditions in zones 1, 2, 3> and 4 are obtained from the characteristics tables, after which the directions Aa, Ab, and Ac can be defined.
By applying the methods of chapter II to the calculation of the crossings a, b, c, e, f, g, I, m, n, p, q, s and the reflections d, h, i, k, o, r, u, the static pressures, the Mach numbers (table 6), and the intensities of the expansion waves and compression shocks, as well as their directional changes (see table 7 an The static pressures were referred to the standard stagnation pressure pQ .
The stagnation pressure changes were disregarded in the determination of the Mach number. This change is rather small according to table 6,


36
NACA TM 1369
so that no appreciable advantage was to be gained by including it. calculation of Po/Pq-[_' c^P3-1"6 eQ.- (23)0
(For
The pressure distribution past the plate is obtained immediately and represented in figure 25. There the passage of compression shocks and expansion waves is accompanied by a sudden pressure variation. Since the actual expansion is continuous, the serrated line is replaced by a smooth curve, such that the areas decisive for the force calculation are identica
Note that the pressures on both sides of the plate cancel out over a large portion of the chord. The resultant force can be determined by integration of the various pressures; the various spacings I are read directly from figure 23-
The plate width was assumed at b
The result is
= o.2963 upper side = o.2896 lower side
Downward resultant force
K
P0nL
o.oo67
lift coefficient
P0iL 1 drag coefficient
K %
cos \|( = 0.01532
K P0i
c,. =--sin i|r = 0.00082
P0lL *


NACA TN 1369
37
k. Calculation of Thrust, Tangential Force and Efficiency
(a) The resultant force on the blade is resolved into two components. One the thrust S is normal to the plane of the cascade, the other -the tangential force T parallel to it (fig. 26).
If
K = resultant force per unit of area
(90 B) = cascade stagger angle
then
S = K cos B T = K sin B
(81)
As functions of lift and drag
S = A cos(p t) W sin(B \|r)
T = A sin(B i|f) + W cos(p i|<)
Referring the force to the dynamic pressure of the airstream,
;ives the coefficients
c = cos B
ql
(82)
similar to the lift and drag coefficients, which can be obtained directly from ca and cw.


38
NACA TN I369
At fixed blade chord and fixed angle of attack the resultant force reaches its maximum value when adjacent blades do not affect each other, that is, when t/L > (t/L)cri^.. In this event
K = p2' P2
where p2' is pressure at lower side (behind compression shock) and p2 is pressure at upper side (behind expansion wave).
For a given Mach number of flow and angle of attack the thrust and tangential force is maximum at B = 0 and B = 90, respectively.
At a given angle p and a given Mach number, T and S increase with increasing \|r. Owing to our assumptions \|r may not exceed \|/ in
order to prevent subsonic flows on the bottom side of the plate, (b) Definition of efficiency (no friction):
It is supposed that the air enters normal to the plane of the cascade at a speed v (fig- 26). The cascade moves with the tangential velocity \ and finds itself accordingly in a relative flow with an angle of attack ty, whereby tan(p \|/) = v/u. As a result of this flow, the two forces S and T normal and parallel to the plane of the cascade act on the plate; S and T are defined according to previous considerations. An efficiency is defined as on a propeller, by visualizing the blade being driven at speed u with respect to force T and so producing a force S in axial direction on the flowing air. Then the power input is T x u, the power output S X v and the efficiency is
i = ^ (83)
Tu
or
l tan \|rj = tan(p jQ = V tan p/
tan p 1 + tan ijf tan p
The efficiency is seen to be dependent on and p only. At constant p it decreases with increasing At \|r = constant, r) has
a maximum, if
51= 0 (84) 5p


NACA TM I369
39
that is, when
tan B = tan i|/ + \/(tan \|/) + 1
which approximately gives
3 = I15O + 1 ^ 2
(85)
The maximum efficiency is then
^max
\|/ = Constant
/1 tan |V
V
1 + tan
(86)
At small values of \|r, tan \|r = \|f and \|/2 is negligibly small, hence
at p = 45 + i ^
2
Vax= W 1 + if
The efficiency for various p and \|r is represented in table 8 and figure 27.


4o
NACA TM I369
CHAPTER IV. LINEARIZED CASCADE THEORY 1. Assumptions
The theory is based upon the following:
(a) All disturbances are small in the sense that all interference lines may be regarded as Mach lines. The expansions are simply concentrated in a Mach line and the compression shocks replaced by Mach compression waves.
(b) Intensity and direction of waves are not changed by intersection of expansion and compression waves. The justification of this assumption is indicated in the preceding numerical example, where it was shown that the directional changes of the wave fronts are small, as a rule.
On these premises, the interference lines AA1 and AAM parallel to BB1 and BB1' start from the small disturbances A and B (fig. 28(a)). At the intersection in a the directions of the waves AA1 and KB1 as well as their intensities remain unchanged. The pressure and the velocities in the zones (2), (3), and (4) are defined by the laws of chapter II. In the hodograph these assumptions imply that the characteristics network in the applied zone is replaced by a parallelogram (fig. 28(b)).
2. Linearization of Cascade Problem
The application of these simplifications to the solution of the cascade problem produces parallel Mach lines within the cascade, which remain parallel after crossings or reflections (fig. 29(a)). (L = plate chord, t = spacing and \|r = angle of attack.) The Mach lines Aa and A'a emanate from the leading edges A and A1; the angles aA'X1 and aAX are Mach angles and both equal to u-^. On passing through A*a,
the flow experiences a compression and a directional change along Aa an expansion with the same directional change.
The pressure in (2) and (3) can be defined by the laws of isentropic expansion and compression (chapter I); that of zone (4) is computed the same way from the pressure in (2) and is obviously equal to p-^, as seen
in the hodograph (fig. 29(b)). But the flow direction in (4) differs from that in (l) by an angle 2\|r.
The Mach line aC* intersects the plate at C' and is reflected along C'E, whereby C'E is parallel to A'a. The pressure in zone (5) is again equal to that in (3) and the flow is obviously parallel again to the plate.


NACA TM I369
On passing through DE' the flow from (k) and (6) is compressed -the reflected wave DE1 so that in (7) the direction and the velocity of flow are the same as in (l); the same applies to the flows in (6) and (2).
Thus it is seen that the corresponding zones repeat themselves, hence that the further conditions are completely known without new calculations.
The pressure variation on either side of the plate can be plotted (figs. 29(c) and 29(d)). The pressure remains constant over the lengths AC, CD, DE, EF and FB and over A'C', C'D', D'E1, E'F' and F'B1 -where the interference lines strike the plate.
Along CD the pressures on both sides are equal and cancel out, whereas a downward pressure difference p^ p2, obviously perpendicular
to the plate, acts on AC and EF, and an identical upward pressure difference on DE.
The pressure pattern in figure 29(e) repeats itself in length direction of the plate over the period Lq. If the plate chord is chosen
exactly like or a multiple of it, there is no resultant force, that
is, a plate of this length has neither lift nor wave resistance.
For the values of L, which satisfy the inequality
whereby n can be = 0, 1, 2, ., the resultant force reaches its maximum value, and then
Hence it serves no useful purpose to make the plate longer than Lq, because there is no more lift increase anyhow. On the other hand, a moment occurs and, in the presence of friction, the drag would increase unnecessarily. The boundary Lq(= ac) is the plate length not touched by interference lines of the other plate and can be defined geometrically in terms of cascade spacing t and angles B, v> and-
3- Calculation of Lift and Drag
L0 < (L
nLq) < (Lq Lq)
(88)


k2
NACA TM 1369
L0 = t
sin[p (Ul + i|f)J sin[p + (ux + t|t)] L^ 2Lq + "t -
sin(nq + ^)
sin(uq -
(89)
Accordingly the best ratio of spacing/chord is
t t
sind^ + i|/)
L Lq sin[p (ux + i|rj]
(90)
Now ca and c^. can be determined when
1. L = nLq then ca = cw = 0
2. Lq < (L nLi) < (Li Lq), the boundary values are
K
ca = cos i|r =
C^L
P3 P2
F cos \|/
K p3 p2 Cy- = sin \|/ = F sin \|r
qxL
(91)
where
sin
[p (ux + i|r)]
L sin( u-j_ + i|i)
3- L0 > (L nLq)
-w
p5 p2 qxL
P3 p2 qxL
(L nLi)cos >|f
(L nLq)sin \|f
(92)


NACA TM 1369
43
4. (L nLi) > (Li L0)
(L nil) (Li Lq)
Co =
P3 P2
qxL L
COS \|f
cw =
p3 P2
^1
L L
(L nLi) (Li Lq)
sin \|f
(93)
The linearization can be extended to the pressures p2 and p^;
admittedly then only when the angle of attack is sufficiently small.3
The pressures can be defined by the laws of small variations (chapter II). Thus
p2(3) = Pi
kM-i 2ib \ 1 +
Ml "
(94)
Inserting these values in the above formulas for ca and cw, while expressing the dynamic pressure with
11 = I KPiMi2
and the values 1 and \|r for cos \|f and sin gives as for the isolated plate,
3ln the following table the pressures after expansion of Pq = 0.3l4o4 (corresponding to Mj_ = 1.4004) are represented in terms of the expansion angle:
p2 = pressure according to isentropic law of expansion
PpT = pressure according to the laws of small variations (chapter II)
p2/p0 0.29906 0.25478 0.27114 0.25809 0.23363
vzl/vo 0.29865 0.28335 0.26718 0.25266 0.22196
(p2 p21j/p2 Percent 0-1 0.5 1.5 2.1 5.0


44 NACA TM 1369
ca =
cw =
M2 1
H2
>
(95)
The factor F approaches 1 when t/L = t/LQ. The theory is now illustrated on the following numerical example.
I
4. Numerical Example
The cascade of the numerical example in chapter III is applied again with the same airstream as by linearized theory, figure 50-
It was
t/L = 0.5k7
0 = 60
Mq = l.kOOk i|r = 5
pi/p0 = 0.5iko4 m = 45.56
The Mach lines within the cascade can now be plotted. By equation (i
Lq = 0.144L
Geometrically defined are
(L Lq) = 0.778l
so that
(L Li) = O.O78L


NACA TM I369
45
The tables of characteristics give
p2/po1 = 0-27H (expansion by 3 starting from P-^/pq-^)
Pj/Pq = 0.3640 (isentropic compression by 3) Assuming the plate width at one cm, gives:
K (p3 Pp)
resultant force -= (L Li) --= 0.3^36
POl p0i
resultant force per unit length = O.oo67 lift coefficient ca = O.oi56 drag coefficient c^ = 0.0008
The pressure distribution on both sides and the resultant pressure are shown in figure 31-
5- Comparison With Exact Method
Instead of the lengthy calculations of all crossings and reflections, the linearized theory affords a quick and simple solution of the cascade problem. At small angles the results are reliable and the errors small, as seen from the comparison with the numerical examples in section 3> chapter III and the preceding section.
ca(exact) ca(linearized) _
------- = -2 percent
ca(exact)
The interference lines of the linearized solution within the cascade the Mach lines are included in figure 23 for comparison. It is seen that the zones governing the resultant pressure are smaller by linearized theory.
The pressure distribution of the linearized example is also shown in figure 25-


NACA TM 1369
CHAPTER V. SCHLIEREN PHOTOGRAPHS OF CASCADE FLOW 1. Cascade Geometry
A disturbance in supersonic flow is known to spread out only downstream of the source of disturbance. So the pressures and velocities on one of the sides of a profile, stipulated by the form of the surface, are not influenced by the other side.
This property is used to represent the flow through a cascade consisting of a number of infinitely thin plates. Two profiles with a flat surface on one side are so assembled that their flat sides face each other and are parallel. The flow between the parallel sides is exactly the same as that between two adjacent plates of the cascades.
The two profiles can be moved apart or shifted relative to one another, so that any desired ratio t/L and any stagger angle can be obtained.
The experimental cascade was patterned after the cascade in the numerical example of chapter III, which had the same angle of stagger of 30 The Mach number of flow was as in previous calculations -M = l.kO; the spacing ratio was t/L = 0.517- Tne angle of attack \|r ranged from 0, 1-5, 3 to 4-5.
The maximum profile thickness was so chosen that no blocking of the tunnel (section 2) was produced at the selected Mach number and that the deflection of the profiles at maximum angle of attack is small.
Now at M = l.kO the deflection due to compression shock, which exactly leads to sonic velocity, is 5S = 9- As there is to be no subsonic flow in the test section and since the angle of attack was assumed at h.5, the leading edge of the profile may at most form an angle of about k, which corresponds to the constructed profile.
The compression shock is not separated at the leading edge of an infinitely thin plate or an infinitely sharp wedge of sufficiently small included angle. Therefore the leading edge shall be as sharp as possible. It succeeded in attaining a thickness of 0.05 0.07 mnr so that the
distance of the separated shock from the edge is scarcely visible.
The profile chord L was 118 mm, so that the cascade lies within the tunnel window. Since the tunnel itself was kOQ mm wide, the width of the profile was limited to 398 mm, figure 32.
Hurit Co., Affoltern, Zurich.


NACA TM I369
47
2. Experimental Setup
The previously described profiles were mounted in the test section of the supersonic tunnel of the Institute5 on four supports (fig. 33)-
The compression shock issuing from the leading edge of the top profile could not be reflected at the upper tunnel wall at maximum \[f, because the deflection to be made retrogressive at the wall was too great for the Mach number prevailing behind the shock. To avoid blocking in this region, a bend had to be made in the upper nozzle wall (fig. 5*0 The position of the bend was so chosen that the fan of expansions emanating from it hits the cascade downstream from the entering edge. This adjusts the wall to the flow direction after the shock to some extent as well as raises the Mach number between the upper plate and the nozzle wall.
The Mach number in the test section before the cascade was determined by pressure measurements at the upper, lateral, and lower walls. The investigation was carried out at a moisture content of air of about O.p g water/kg air.
3. Schlieren Photographs
The schlieren photographs illustrating the flow through the plate cascade at \|f = 0, 1.5 an The equality of Mach's angle in figure 57 (i = 0) i-s' indicative of an unchanged Mach number in the cascade. The visible disturbances within the cascade may be due to the fact that the plate surfaces do not exactly agree with the flow direction, or to thickening of the leading edges by a boundary layer.
In figure 58 (\|( = 1.5) "the interference lines inside the cascade are almost parallel, as stipulated by the linearized theory.
5See Report No. 8 of the Institute for Aerodynamics, at the E.T.H, ref. 1.
Vor description of schlieren apparatus see Report No. 8 of E.T.H. Institute.


1+8
NACA TM 1369
At i|( = 3 (fig- 39) the deflection of the shock front at crossing
of the expansion wave emanating from the top leading edge is plainly-visible. Figure 1+0 represents an enlargement of the crossing to illustrate the numerical example in chapter III. The interference lines inside the cascade for this example are again shown in figure 1+1 at smaller scale (compare also fig. 23), whereby the perspective effect is indicated.
In the majority of photographs the retardation of the flow near the tunnel wall leads to separation of the head waves.
The flow in all photographs is from left to right.


NACA TM 1369
49
CHAPTER VI. THE FIAT PLATE CASCADE AT SUDDEN ANGLE-OF-ATTACK CHANGE 1. Problem
Visualize a cascade of flat plates in a flow with relative velocity W at an angle of attack \|r. A supersonic flow which may be regarded as two-dimensional prevails throughout the cascade. At a given moment the angle of attack of the airstream changes from \|r to i|r' within an infinitely short time interval. The transition to the new state, which is to last for a period, is analyzed.
Such a change in the angle of attack takes place when the cascade moves in an absolute flow which has not the same speed at every point, or when one of the velocity components of the flow, normal or parallel to the plane of the cascade, varies with respect to time.
Resolving the velocity W in two components V and U (fig. h-2) normal and parallel to the plates, the change of the angle of attack, small in itself, can be regarded as a change of component V. This change in V is obtained by superposition of a velocity vq, which has the same
direction as V and is obviously small compared to V and consequently smaller than sonic velocity. From the assumption of a small angle of attack, it follows that velocity component U remains greater than sonic velocity. Besides, an eventual variation of this component U is disregarded.
The problem therefore reduces to the study of the new forces on the cascade, resulting from a gust vq 7 which, together with the velocity U
enters perpendicular to the plates.
Biot (ref. 5) solved the problem of an isolated plate by means of "unsteady sources." This method is applied to the cascade problem. But first the unsteady source is described in more detail. Since the plates are to be partly replaced by such sources, the pressures and velocities originating from a source distribution are analyzed. Then Biot's results for the isolated plate are correlated and extended to the cascade. The special case of straight cascade (nonstaggered) is examined.
^By "gust" is meant a continued, uniform vertical velocity distribution Vq.


50
NACA TM I569
2. The Unsteady Source
According to linearized theory, the general potential equation (5) for two-dimensional unsteady flow can be simplified to
_ m2) + ^-31 2 5 9 -IOL=o dx2 dy2 aSxat *2dt2
(96)
cp = flow potential.
For a system of coordinates moving with velocity U(u/a = M), that is, air at rest at infinity, this equation gives the acoustic wave equation for two-dimensional motion
2>2 ^2 ;,2
d cp | d cp 1 d cp Q
^2 -.2 2 -^2
dx dy a dt
(97)
One solution for a linear sound source is
, .1 at cp = k cosh x T r
(98)
where r = yx2 + y2 and K = constant with dimensional length times velocity.
This solution is rewritten in the form
cp = kiogel^+ J^i2-
= k loge
-k lot
Mat
*4
a2t2
i/a2t2 r2
(99)


NACA TM 1369
51
It represents a cylindrical wave varying in time rate. At t = r/a, 9=0, that is, if such a singularity appears in the zero point of the coordinate system, its effect is diffused inside a circle of radius r = a-t.
If such a source appears at the point (x,0) on the x-axis at period tj, the potential in a point P(x,y) of the surroundings of this
source at a given period (fig. 43) is
cp = k loge
i(t t-,) + 1/ a2(t t
,2 2
(100)
In this case
(x X!)2 + y2
and the following velocity components are obtained by simple differentiation
cfcp Sr"
v-p = = k r
a(t ti
^a2(t ti)
2 r2
(101a)
cto (x xl) Vx = = k -
dx r^
i(t t!)
^a2(t t!)2 r2
(101b)
vy = *t = k X a(t *1> (10lc) Sy r2 L?(^ t 2 r2
When y is small compared to a(t tj_) near the source the formula (101a) becomes
vr = k/r


52
NACA TM 1569'
the same as that of a steady wave in incompressible flow, hence with Q denoting the strength of the source (dimensional length x speed)
K
2a
The pressure in the same point is computed by
Ap = _p <*P = paQ
dt 2rt \Ja2(t tx)2 r2
:i02)
It will be noted that Vy always equals zero for y = 0, except at r =
in the source itself. It means that such a source delivers at no other place on the x-axis a velocity component parallel to the y-axis.
3. Pressure and Velocity of a Periodically
Arising Source Distribution
Consider a continuous distribution of infinitely small sources over the length OA (fig- 44) along the negative x-axis. The distance OA increases linearly with the time: OA = Ut, where U is a constant velocity and the sources on the x-axis appear momentarily at the point where A arrives at the moment. The strength of this source distribution per unit length of OA is assumed equal to q (dimension of a velocity) and remains constant in time.
(a) Pressure
At point P(x,y) (fig- 44) the pressure p of the source distribution at time T is, by equation (102)
P =
paq
,Xq=0
dxn
Xl="Ut \/a2(t tl)2 (x xq)2 y2
(105)
t-i the time of origin of the source in point x^_.


NACA TM 1369
53
With the following variable transformation
=
at
at
(104)
5l -
*1_
atn
= sm u U
1 M
we get
- k / dC (io5>
The boundaries should be defined before the integral is evaluated. For the function in the denominator is real only in the zone affected by the source distribution; this is bound by the Mach line AM and the circle with center 0 and radius a-t. Hence the integration must be made between the zero places of the function where it is real.
Posting
(1 + i sin p.)2 (5 ^)2 -^2=0 (106)
the new boundaries are found at
5
(1)
(2)
({; + sin u)
4
(l + sin u)2 -n2cos2u
(107)
-cos2p.
To get an idea of the integrating process as function of the position of point P, l("^ and ]_(2) are plotted in terms of It results in two curves of the second degree, which cross in point Q(5,5i) (fig- ^5(a)) whereby


5k NACA TM 1369
5 = -
T| COS U
sin u I
cos u
.sin u cos u.
(10
The shaded area represents the ru.nge in which the integration should be made. At small values up to = y|l t\2, integrate between
and q^2^ and then between ^3 and the axis. In figure 45(b)
the integral limits are shown p ;ted in the x,y-plane for explanation. The reason for not integrating over positive q values is the absence of sources in the right-hand half plane.
Two integration cases are differentiated
1 T] COS U
sin
that is
j and
T)^ < t, < +
ri ? CS % < x < J>tg yg
sin u / V
and
a2t2 y2 <
x < + \j
a2t2 y2
In the first instance the pressure integral is
_p_= i_ r^i
paq 2k J (l
(2)
1 ^(1 + ^ sin p.)2 ( ^)2 T!2
>(10
(ik


NACA TM 1369
55
But as and ^(2) are the solutions of the expression below
the root, it can be rewritten as
p (2) A
paq 2rt j f (l) si
- C1)(?1(2) tx)
With the substitution
this integral gives
pc
paq 2 cos u
a formula that is independent of t\ = y/at.
In the second case, if point P is so situated that
it results in
(111)
fc- ^q(l) + ^q(2) ^^1^0 (112)
(115)
paq 2n J ft1) / a o
&1 \/(l + 5i sin p)2 ( 5i) T
o
As long as the function in the denominator can be brought, with the aid of the integral limits, into the form of equation (ill), the integral gives the same value.


56
NACA TM 1369
which after evaluation gives
paq jt cos u
cos'
(t; + sin (i)
2 2 2 (l + sin u) T| cos u
(115)
The ensuing pressure pattern along a line y = Constant is represented in figure k6. For each y the pattern consists of two pieces. In the first piece the pressure is constant and equal to pc, that is,
along the length EF between the points where the Mach line emanating from A and the circle with center 0 and radius a-t intersects the line y = Constant. The second piece is composed of length
FO' = -\/a2t2 y2 and O'G = +\/a2t2 y2, where the pressure is variable; at G the pressure is zero. At y = yc the constant portion
disappears and wherever y = at, the pressure becomes zero.
At y = 0 it represents Biot's case with the integral limits
sin u
< 5 < 1
and
that is
at
sin u
< x < at and
1 < 5 < + 1
-at < x < + at
(116)
The pressure p0 has the same value as before
paq. 2 cos u
(117)'
but the second piece of the pressure distribution becomes
py=o
paq 2rt cos u
cos'
+ sin u
at
\1 + sin
at
(118)
9q corresponds to Biot's 2vf


NACA TM 1369
57
It should be noted that a pressure effect appears also outside the area in which the sources are distributed, because the source in 0 affects the area inside the circle a-t as mentioned before.
(b) Velocity Calculation
In general, the velocity component is defined by the integration of the portions stemming from a single source (eq. (101)). In our problem the velocity Vy is of particular interest.
It becomes
1
^Xq=0
Xq=
a(t tq)y dxq
r / ---
[U xx)2 + y2j l/a2(t tx)2 f(x XjJ2 + y2
If r = \/(x xq)2 + y2 is small compared to a(t tq), that is, for the places close to the x-axis, this equation simplifies to
^x1=0 y dXn
2*
xn = -Ut
[(x xq)2 + y2]
2jt
tan + x| tan"1
(119)
Letting y approach 0, positive y, the results for negative values of x are
which may be designated by Vq (as in Biot's report). For positive x values, vy = 0.
It indicates that such a source distribution gives a uniform vertical velocity vQ over the distance of the x-axis where the sources are. (For negative y, inverse velocities result.)


58
NACA TM 1369
Biot mentioned this fact in his report and used it to calculate the pressure distribution over a plate in a vertical gust (compare next section) .
The same variable change as in the pressure calculation gives
vv = > it
5i
(2) (1)
t](1 + 5-l sin u)d^
U ~ q)2 + I2 1/(1 + ?! sin u)
U ?i)2 + ti2
(121)
The arguments for the integral limit are the same as for the pressure integral and (jq^^j q^2^ is given by equation (107)- Integrating
between q^"^ and !^2^, that is, when
'1 T) cos u sin u
<
1 t)<
it is seen that the integral gives the value so that vv has the constant v0 for this range of .
For the second case
between f^"^ and 0 indicates that1^
^_ r)2 < < +\J 1 t|2 j the integration
v = ^< J it
jt tan"
M/i a2 + n2)
sin |i(^2 -- r)2) ^
(122)
This identifies the velocity distribution on the lines y = Constant (fig- k7)-
10,
See note on p. 66.


NACA TM 1369
59
From the calculation of the pressure and velocity distribution over the lines y = Constant, it is apparent that the flow outside the circle of radius a-t is steady. This is true from the physical standpoint too, since the gust front does not affect this area.
4. Single Flat Plate in a Vertical Gust (Biot 19k5)
The flat plate AB of length I at supersonic velocity U' enters a gust with the transverse velocity Vq (fig- 48(a)). Since the transverse velocity on the plate must be zero (no flow through plate) an equal and opposite velocity is superposed on the gust velocity vq in place
of the plate. This velocity can be visualized as reflection of the gust on the plate (fig. 48(b)).
Since velocity U is greater than the sonic velocity, the sides of the plates are not affected by one another, so that one side of the plate can be analyzed separately. The pressure acting on one side is exactly the same as on the other, except with inverse prefix. As the interference velocity vq is much smaller than velocity U, the linearized potential
equation can be applied to the stream potential. Selecting a system of coordinates that moves with the velocity U, (eq. (97)) according to which the disturbances are diffused with sonic velocity, can be applied to the flow potential.
Biot's method replaces the part AO of the plate struck by the gust, by unsteady sources. This source distribution, which increases in time, yields a uniform velocity Vq normal to the plate, hence satisfies the boundary condition on the bottom side of the plate.
If the plate enters the gust at time t = 0, the distance at time t is AO = -Ut, the origin of the coordinate system being located in the gust front.
The results of section 5 can be applied directly, and the pressure variation along the plate defined (fig. 49(a)). As it is dependent solely on x/at the patterns are like those for the different Mach numbers. The total force on the plate the lift is obtained by integration of the pressure pattern. Three phases are involved here (fig. 49):
I (U + a)t = 1
that is, the trailing edge is outside the effective range of the gust front;


6o
NACA TM 1369
II (U + a)t > I > (U a)t
that is, the trailing edge is inside the effective range of the gust front}
III (U a)t 2 I
that is, the entire plate is outside the effective range of the gust front, hence is no longer exposed to any unsteady effect.
The integration gives the following lift values of the three phases:
A-,
Ut
2pav02 I sin u
(123)
*II
2pav0Z it cos u
cos
1
sin u
Ut p 1
I
Ut _i + sin -1-
Jtl
sin u\Ut
l! + *
(124)
i:iThe integral
I =
u A)\/i $
appears in the calculation of Aj and Ajj. With no boundary, the solution is
I = sin-1^
A i/l + Ag sm-1--
\/A2 1 \A + J
which gives
I = it
between the limits -1 and +1.
A
A2 1,


NACA TM 1369
61
The sin 1 to be taken between it and + i
i 2 2
JHH- = _L_ (125)
2pav0i cos u
In phase I and II the lift increases continuously with the time and reaches a maximum in phase III, where it becomes independent of the time. In the last phase the lift is the same as on a plate at angle vq/U In steady flow.
5- The Straight Cascade
The cascade problem is unlike that of the plate to the extent that the plates mutually interfere. The sources replacing the portion of the plate struck by the gust create a pressure on the adjacent plates. They also produce a velocity vu, which in order to satisfy the boundary condition of no through flow of the plate, makes a change in that source distribution necessary.
Since the disturbances are small the solution of the single plate can be superposed in the sense of the linearized theory of the adjacent plate effect.
As shown in sections 2 and 3, the unsteady source and the source distribution which lies on the x-axis, produces no vertical velocity component along this axis, outside the distance, where it is.
This characteristic enables the velocity component vv to be
replaced by an additive source distribution along the particular parts of the plate, which gives the velocity at each point. The new sources create a further pressure on the plate itself and in general react on the adjacent plates.
The total force the lift on each plate consists then of the lift of the undisturbed plate (Au), the lift from the pressure pv of
the sources of the adjacent plates (A^) and the lift (Av) of the new source distribution due to velocity Vy.
Suppose that h is the plate spacing and L the plate chord of the straight cascade ff (fig. 50).. The lines AM represent the


62
NACA TM I369
Mach lines emanating from the leading edge, where angle MAB is the Mach angle u = sin-"'" u/a.
Then the following approximation is made: the relative flow and
the plate form in reality the angle \|/ = tan_1(vo/u). But as v0 is
small compared to U, this \|r is negligibly small with respect to u, and it can be assumed that the Mach line itself rather than the relative flow direction forms the angle u. In this event, the Mach lines form the same angle with both sides of the plate. The Mach lines emerging from the leading edge strike both sides of the plate at the same distance AE from the leading edge. At (h/Z) > tan u the points are not located on the plates, and the plates do not influence each other. Consequently the cases where (h/l) < tan u are examined.
At time t = 0 the cascade is directly in front of the gust; the origin of the coordinates is placed in the gust front. In the first time intervals of the phenomenon the disturbances have not spread out enough to be able to influence the adjacent plates. As in figure 51* the distance is a-t < h, so that the circles with center 0 and radius a-t do not touch the plates. Lift and pressure distribution are the same as on the single plate.
As soon as t > h/a, the plate AB comes within the effective range of its adjacent plates. On EFG (fig. 52) the source distributions A'O1 and A"o" create an additional pressure which can be computed according to section 5- The points F and G are then the points of intersection of both circles with center 0' and 0" and radius a-t with plate AB. It is readily apparent that the additive pressure on EF is constant and, according to equation (113)> has the value
Pc
pavQ cos u
(126)
If tj = h/at is inserted (y = h) in equation (115), the pressure on FG follows at
Ph
cos"
pavQ it cos u
+ sin u
x v 2 h.2 p
1 + sin u\ cos u
at
a2t2
The ensuing additive pressure is represented in figure 52-
(127)


NACA TM 1369
63
In addition, the following condition must be satisfied: The normal velocities vy = h created by the source distribution A'O1 and A"o"
are reflected on EO, so that at that point the gust is partly compensated. The source distribution to be applied is to compensate the velocity (vq vv) The velocity Vy is computed as in section J>> and the
pressure pv along the particular plate is obtained by integration of the pressure contribution of each source. Assuming the local velocity Vy on a small distance dx-^ to be constant, the yield of the source
distribution per unit length on this small distance is then q = 2Vy.
Along this area of the plate the source distribution produces the pressure (compare section 2)
pavv dx-,
APy=Q = 1 (128)
t tn)2 r2
hence
y=o
xi
Xl(2) 1 pa
(1) ~
Vy dxq
a2(t tq)2 r2
(129)
with Vy periodically and locally variable. It is best to solve the
integral graphically for each particular case. The arguments for the integral limit are the same as before. The lift contribution Av at
any instant is obtained by integration of the ensuing pressure plot.
To obtain the resultant pressure, this pressure is superimposed on the two previous pressure distributions.
In the following, the pressure contribution due to the additive pressure p^ is calculated. Three phases, depending on time and ratio h/l, are involved:


6k
NACA TM 1369
I. When I ^ ^Ut + \ja^t2 b2 j (fig. 52), the lift is
r -Wa2t2-h2 P + \/a2t2-h2
Vr = 2 // \ Pc ^ + 2 / 1- Ph dx
V- Jl .ut + -J!) j_l/a2t2-h2 h
tan (ii 1/
with the previously employed variable change and with y = h
A^-r = 2at / /. Pc d.? + 2at / .- p d
J1 J Ut+y/tan u c J v/1_TJ2
at
-V7!^2 r + \/W /dPh\
2atp(0 tt^ + 2at(pO ,- at / .-- U M
Ut+y/tan u ^-V/l^2 V/ at
(130)
By equation (127)
dPh pav0 (1 + ^ sin (i) T]2
hence the integral
(1+5 sin M-)2 T)2cos2|i \^1 2 tj2
(13D
P+ \/l^2 (1 + ^ sin (i T)2K
Vl-T2 \jl ^ t)2 |(i + 5 sin n)2 t^cos2!!
45


NACA TM I369
65
Its evaluation gives
it
sin u cos u
(cos u 1)
1 +
T|(CQS U + 1) T) + COS U
(132)
consequently
2pa2uq
sin u cos u
(l t) cos u) + (cos u 1)
1 + T1(CQS U + 1) (COS U + Tj)
II. But if (fig. 53)
|ut + v/a2t2 + y2^ > j > (ut ^a2t2 y2J
the evaluation of the integral gives the formula
(133)
I Ut 2pav0 /,
at / cos u
2pav0
it sin u cos u
sin
is
4
A -1 1 + AS it =- sin"1 + 1
A
A^ 1
A + S 2
(t] cos u)
sin_1S
B
. _q 1 + BS L It /, sin 1 + II -
B + S 2|
B2 1,
(134)


66
NACA TM 1369
where
s = 1 ut
V
a2t2 y2
1 + Tl cos U. 1 T) cos U
a = -!- and B = --
sin |i \j 1 T]2
sin p\/ 1 T]2
The pressure pe is obtained from equation (127), when x = (Z Ut is inserted, at
Pe
cos
-1
pavQ it cos u
Z Ut at
+ sin u
1 +
1 Ut
at
sin u\
_h2_
i2t2
cos2u
(135)
III. If I (U at), py = pc along the entire distance EB, so
that the additive lift
Ayili = 1 L h
2pavQ cos ul tan u
(136)
reaches a value that is independent of the time.
Note on the Velocity Integral
By a simple transformation the integral can be rewritten in the following form:
(2) (a + BCi)!^
I = (1) (?!2 + a)^^ + B^ + 7


NACA TM I369
67
This integral can be solved by means of tables (Integral table, Part I, by Grobner and Hofreiter, Springer Co., Wien). Although the general solution is quite complicated, the result is found to be independent of once the limits have been inserted.
Bearing in mind that the integration limits are the solutions of the function below the root, the integral is rewritten as function of the limits
fPN (l 1 sin rfdt,!
fSl1 J -
I =
(1 E1 sin n)2 cos2n(^ EjCD) (g^2) ^
The
The following substitutions are made consecutively:
1. X = 1 5l sin \i limits are thus Xj_ and X2 (X2 > Xq)
1 2X-iXo
2. Y = -- 1
X Xq + X2
3. Z = Y(X2 + Xq)/(X2 Xq) k. t = Z2
-, 4XnXp cos2M.(X-1 + Xp)2 5. s = A' = x *---
A' + t cos2u(X1 X2)2
6. Numerical Example Dimensions of cascade h/l = 0.55
Mach number of flow M = 1.4l4 (= \/ 2) corresponding to a
Mach angle u = 1+5


68
NACA TM I369
The period te up to the end of the phenomenon is determined by
(U a)te = I
whence
0.klka
All time intervals are referred to y/a in order to obtain a dimen-sionless ratio (= i/t]). Then
^_=k.38 (y=h)
The three pressure contributions pu, p^, and pv are defined by
the formulas of the preceding section at various time intervals indicated by the digits 0, 1, 2, 3> 10.12 The time intervals were chosen as I follows: The time interval denoted by 3 represents the end of the first phase of the undisturbed plate (compare section k) At r\ = 1 (period: < the influence of the adjacent plates begins and ends at r\ = 0.2kk (period: 9)- At r\ = 0-707 (period: 6), the Mach line emerging from the! leading edge of the plate strikes the adjacent plate.
Figure represents the position of the gust front and the area disturbed by it at the different time intervals.
Figure 55 illustrates the pressure of the undisturbed plate pu-
Figure 56 illustrates the pressure contribution p^.
The pressure contribution pv is computed graphically, the velocity distributions vv/vq required for it are obtained by equation (122) for the time intervals 5> 6, 7> 8 and reproduced in figure 57-
-L^The corresponding curves in figures 55> 56, 57> 60, and 6l are denoted by the same digits.


NACA TM 1369
69
Since the integrand
f = _vy/vo_
i/^-tJ-^-w2
becomes infinite at the two limits ^t^"^ and 2(2) which are, as known, the solution of the function below the root, the graphical solution is continued to f?l^^ ~ ) 3111 (j>l^^ ~ e'j wnere e> e' are small real values in comparison to f^*
Figure 58 represents several of the functions f for different time intervals.
The integration over e and e' is made analytically, by putting vv/vq = constant mean value.
The relation for tj_/t at i\ < O.707, that is, when the additional source distribution is bound by a Mach line, is
tj_/t = t] cos u + t sin u
If the source distribution is limited by the circle of radius R = at, a similar relation
t-j_/t = T) cos p + ^ sin p
is applicable (fig. 59)-
Figure 60 represents the pressure contribution pv, figure 6l the resultant pressure distributions.
The lift of the plate (fig. 62) is obtained by graphical integration over the resultant pressures. At the appearance of the adjacent plate effect the lift decreases with the time interval; A1 represents the steady lift of the undisturbed plate.


TO
NACA TM I369
Figure 63 shows the moment distribution M plotted against plate center; Mst represents its steady value. Here the moment increases with
the time because of the built-up negative pressure from t = h/a.


NACA TM 1369
71
CHAPTER VII. EFFICIENCY OF A SUPERSONIC PROPELLER
1. Introduction
The cascade efficiency defined from thrust and tangential force is suitable also for the propeller. But in the preceding arguments the flow was assumed parallel and the blades as infinitely thin plates, which now must be modified. The friction at the plates must be allowed for and the infinitely thin plates replaced by profiles of finite thickness. Then the results are used to calculate the efficiency of a real propeller in order to obtain an approximate picture of the efficiency to be expected.
When the friction at the plate surfaces is taken into account, the resultant force K without friction defined in chapters III and IV, is supplemented by an additional resistance F, so that K1 is the total force acting on the plate (fig. 6k).
The frictional force is parallel to the plate. But since its component normal to the airstream direction is small at the angles of attack in question, the total frictional force can be assumed to be in the flow direction.
The drag coefficient is expressed by
2. Effect of Friction on Cascade Efficiency
c
H2
(137)
w
The lift coefficient remains
(138)
\/m2
- 1
as for parallel flow.


72
NACA TM 1369
According to the definition introduced in chapter III, the cascade
efficiency is
n = (139)
T'u
where S1 and T' are the thrust and tangential force corresponding to the new force K'.
In terms of angles B, y and a (fig. 64) the efficiency is
T) = tan a tan(B \|/)
where
a = tan"1 (l40) T'
Introducing the drag/lift ratio
, = WJ:= c^= I + cfXJM2 1
A ca \ 2^ the efficiency r\ becomes
t, 1 tan(P *> (142) 1 + e'/tan((3 1)/)
Hence it is apparent that, contrary to the earlier results, the efficiency is now dependent on the Mach number.


NACA TM 1369
73
With the assumption of a turbulent boundary layer
0.1+55 log10 Re
2.58
(143)
according to Schlichting, where Re is the Reynolds number based upon chord I and relative velocity w.
The values plotted in table 9 and figure 65 as functions of the angle of stagger were calculated with.the Mach numbers M = 1.40 and M = 2.50, then at an angle ^ = 3 and the optimum anglesl3 i|/ = 2.65 (M = 1.40) and \^ = 1+.11 (M = 2.50). The Reynolds number assumed at Re = 106 corresponds to cf = 0.0045-
The effect of friction is illustrated in figure 65, along with the efficiency curve for \|/ = 3 with friction discounted.
3. Effect of Thickness
To assure minimum wave resistance the contour of a supersonic profile must consist of straight lines and its maximum thickness lie in the center, hence a double-wedge profile is recommended (fig. 66).
By linearized airfoil theory (ref. 1) the thickness causes a drag which increases quadratically with the thickness ratio d/l, and which can be directly superposed on the lift coefficient and the frictional drag of the plate.
Hence the drag coefficient of a profile of finite thickness ratio with friction is
'w
,2 /d
+ Cf
N/m2
(144)
But the lift coefficient remains unchanged
Co =
4\|
M2 1
13
As stated in the introduction, iDpt the plate, obviously d/l = 0.
(d/Z)2 + Cf^u22- 1, for


74
NACA TM 1369
and the drag/lift ratio to be inserted in equation (l4l) in place of e' is
4r + =
(145)
Table 10 shows the efficiencies of two cascades of double-wedge profiles and the relative maximum thickness ratio
d/l = 0.05 and 0.10 with friction at M = 1.40
J = 3 cf = 0.0045
These values are also shown in figure 67 together with those for d/l = 0 (the plate) for comparison.
The angle of stagger S with friction and finite thickness -for maximum efficiency at fixed angle of attack and fixed Mach number is found by simple differentiation at
(146)
4. Appraisal of the Efficiency of a Supersonic Propeller
On a supersonic propeller the blades are struck at a relative speed which at every point of the blade is greater than the sonic velocity. Two types of propellers are differentiated. The one moves forward at supersonic speed, so that supersonic speed occurs at every rpm and every On the other the supersonic speed is reached without it having to move for ward with supersonic speed. The efficiency of the first type propeller is calculated.


NACA TM 1369
75
The propeller has a forward speed v of about ko6 m/sec (M = 1.20 -sonic velocity a = 338 m/sec); it has four blades of 2m outside diameter and lm inside diameter and 0-5 hub ratio. The cross section of the blade is a double-wedge profile, with maximum thickness ratio of d/l = 0.07 at the hub, and tapering to d/l = 0.05 at the tip.
The maximum efficiency of a profile is reached with (3 = 45, according to chapter III. At the blade tip where the thrust is highest, this condition gives a tip speed of kOO m/sec; that corresponds to 3,820 rpm. For reasons of strength the blade chord tapers from L]\j = hO cm
at the hub to Lg = 30 cm at the tip.
In each coaxial cylindrical section with respect to the propeller axis the angle between the relative flow direction and the profile axis -the angle of attack was assumed at iQpt (compare introduction). To
satisfy this condition, the angle of stagger in each section, that is, the angle between profile axis and direction of peripheral speed U must be varied. The relation
tan(B if) = v/u
must be satisfied.
With reference to a system of coordinates fixed in space, each point of the propeller moves on a helical line. Disturbances issue from each point which at the assumed pressure conditions and angles of attack can be regarded only as sound disturbances. The zone disturbed by each blade is then limited by the enveloping curve of all spheres whose centers lie on the various helical lines and whose radii at the same time are equal to sonic velocity x time. Figure 68 represents the disturbed zone of an edge OA, which, for example, moves at a forward speed of 1.2 x a and whose maximum tip speed equals the sonic velocity; O'A' represents the position of the same edge after a time interval At, which corresponds to a fourth of a revolution.
Considering that the blades are twisted, that the disturbances of different sections can influence one another and be reflected on propeller hub, it is readily apparent that an exact calculation of the forces on each blade represents a difficult problem. When each blade is outside the zone of disturbance of the other blade, the blade can be examined separately. Assuming homogeneous flow and coaxial cylindrical areas, that is, radial equilibrium, the blade forces can be determined from a two-dimensional consideration of the developed.blade (fig. 69), by computing the lift and drag and from it the thrust and tangential force in each cross section by linearized theory.


76
NACA TM I369
At the velocities selected the boundary effect is confined to a moderately large zone compared to blade area, so that its effect within the framework of the intended appraisal on the total forces can be disregarded.
The thrust of the whole blade is then
The integration is made by graphical method (fig. Jl(a)) with
as, H pw l cosl> *> + 7] (m fu2 1 2 cos 7
computed for five sections (fig. 69).
The corresponding Reynolds number for all sections was assumed at Cf = 0.004 (turbulent boundary layer).
The torque D of a blade is defined the same way as the thrust by integration (fig. 71(b)). The following relation applies:
J hub clr J Jrf> ~ 2 cos 7
(149)
The characteristics for the five sections are correlated in table 2. The integration gives
Sblade = ^25 kg
Dblade = 600 Wm


NACA TM I369
77
hence for the propeller
S =
1700 kg
D = 2400 kgm
The efficiency of the propeller is given by
Sv
(150)
*1 =
2nhD
with the values inserted gives
t] = 71-8 percent
A quick and close estimate of the efficiency is obtainable directly
from the calculation of and f where these values are appli-
1. Lift and drag coefficients for the flat plate at various Mach numbers, ranging from 1.20 to 10, and for different angles of incidence are calculated, account being taken of the exact flow over both sides of the plate. These values are tabulated and also given in the form of charts. The same coefficients are also calculated under the assumptions of linearized flow over the plate, according to the Ackeret theory. A comparison of both methods shows reasonable agreement between the linearized theory and the exact method within the usual range of angles of incidence (max 10) and for the usual Mach numbers. Special formulas for calculating the lift and drag coefficients for very high Mach numbers are derived.
2. An analytical solution of the problem of the interaction between shock waves and expansion waves has been established.
3- A method for calculating the lift and drag coefficients for a cascade of flat plates is described and applied to an example, with the aid of the formulas derived in the foregoing item. A definition for the
cable to the whole blade.
SUMMARY


78
NACA TM I369
efficiency of the cascade without friction is introduced and the efficiency is evaluated for two Mach numbers and different angles of blading.
4. A linearized theory for supersonic flow through a cascade of flat plates is established and applied to the example already treated. Comparison of the lift coefficients shows reasonable agreement.
5. For demonstration purposes, schlieren photographs were made showing the flow between two flat surfaces. They serve to confirm the established linearized theory for small angles of incidence and show clearly the interaction between shock and expansion waves.
6. Under the assumption that the flow through the cascade of flat plates undergoes a small sudden change of direction, that is, a small change in the angle of incidence, the nonstationary flow in the cascade is discussed to show the kind of forces which act on the plates during the transition period. An example has been calculated in detail.
7- The definition of the efficiency mentioned in 3> is especially suitable for application to a supersonic propeller. The effect of friction and blade thickness on that efficiency is shown. A rough estimation of the efficiency of a supersonic propeller is then made.
Translated by J. Vanier National Advisory Committee for Aeronautics
REFERENCES
1. Ackeret, J.: Gasdynamik, Handbuch der Physik, Bd. 8, S: 289-342, 1925.
Gasdynamik, Vorlesungen an der E.T.H., u.a. Mitteilung Nr. 8 des Institutes fur Aerodynamik.
2. Sauer: Einfilhrung in die technische Gasdynamik. Springer-Verlag,
Berlin, 1943-
3- Ferri: Elements of Aerodynamics of Supersonic Flows. The Macmillan Company, New York, 1949.
4. Schubert: Zur Theorie der stationaren Verdichtungsstobe. Z.A.M.M.,
Heft 3, Juni, 1943.
5. Biot: Loads on a Supersonic Wing Striking a Sharp-Edged Gust. Journal
of Aeronautical Sciences, May, 1949-
6. Keenan and Kaye: Gas Tables. John Wiley & Sons, New York, 1945-


NACA TM 1369
79
TABLE 1
Mj_ c O s y 0 Pl/PO q/p0 q/pi
1.00 0 90 0.52850 O.5698I 0.700
1.10 1.40 75-68 .46855 .39704 .847
1.20 3.70 68.08 .41258 .41567 1.008
1.30 6.32 65.12 .56092 .42689 1.185
l.kO 9-03 63.33 .51424 .43114 1.572
1.50 11.67 62.25 .27240 .43050 1-575
1.60 14.21+ 61.65 .25527 .42182 1-792
1.70 16.63 61.37 .20259 .40995 2.025
1.80 18.84 61.28 .17404 .59472 2.268
1.90 20.87 61.55 .14924 .57714 2.527
2.00 22.71 61.48 .12780 55750 2.800
2.20 25.90 61.90 09552 .51684 5.588
2.50 29.67 62.4o .05855 .25610 4-575
3-00 34.01 65.77 .02722 .17145 6.300
4.00 38.75 65.25 .00658 07375 11.20
5.00 4l.ll 66.20 .00189 .05506 17-50
6.00 42.44 66.75 .0006^ .01588 25.20
8.00 45.79 67.00 .00010 .00448 44.80
10.00 44.45 67.12 .00002 .00165 70.00
00 45.58 67.70 0 0 00


NACA TM 1569
TABLE 2 LIFT AND DRAG COEFFICIENTS
Ml P2/P0 70 P2'/Pi P2'/p0 ca cw e
1.20 1 0.39145 58.75 1.056 0.43527 0.1054 0.0018 0.0175
2 .37210 61.10 1.120 .46187 .2158 .0075 0349
3 35403 64.37 1.199 .49444 3373 0177 .0524
3-7 35952 68.08 1.277 .52681 .4011 0259 .0641
1.40 1 .29910 46.87 1.051 .33027 0723 .0013 .0175
2 .26480 48.19 1.104 .34692 .1440 .0050 .0349
3 .27114 49-57 1.159 .36420 .2156 .0113 .0524
4 .25824 51.15 1.219 .38306 .2888 .0202 .0699
6 23376 54.62 1-353 42517 .4415 .0464 .1052
8 .21130 59.36 1.527 .47984 .6168 .0867 .1406
9-03 .20050 63.17 1.655 .52007 7321 1159 .1584
1.60 1 .22374 39.67 1.060 .24939 .0608 .0011 0175
2 .21269 40.73 1.104 25978 .1116 .0039 .0349
3 .20207 41.82 I.161 .27300 .1679 .0088 .0524
4 19191 42.93 1.219 .28679 .2244 0157 .0699
6 .17280 45.36 1.345 .31637 .3385 .0356 .1051
8 15525 48.04 1.484 .34938 4557 .0641 .l4o6
10 13914 51.14 1.644 .38685 5783 .1019 .1762
12 .12438 54.89 1.832 .43101 .7110 .1511 .2125
14.21+ .11022 61.65 2.143 .50418 .9052 .2532 .2532
1.80 1 .16503 34.64 1.054 .18352 .0468 .0008 0175
2 .15640 35-53 1.110 .19318 0931 0033 0349
3 .14813 36.48 1.170 .20363 .1404 .0074 .0524
4 .14019 37-44 1.230 .21407 .1867 .0131 .0699
6 .12534 39-49 1.362 .23704 .2814 .0296 .1051
8 11175 41.69 1-505 .26193 .3768 0530 .1406
10 09935 44.06 1.661 .28908 .4734 .0834 .1763
12 .08806 46.70 1-835 .31936 5732 .1218 .2126
15 07303 51-35 2.139 37227 7323 .1962 2679
18 .06011 58.OO 2.551 .44398 .9249 3005 3249
18.81+ .05788 61.28 2.740 .47687 1.0045 3427 .3412
2.00 1 .12076 30.82 1.058 .13521 04o4 .0007 .0175
2 .11401 31.65 1.118 .14288 .0807 .0028 -0349
3 10757 32.58 1.181 1-15093 .1211 .063 0523
4 .10141 33-40 1.247 15937 .1618 .0113 .0699
6 .08994 35-24 1-377 .17726 .2429 0255 .1051
8 .07949 37-22 1-539 .19668 .3246 .0456 .11+06
10 .07005 39-32 1.707 .21815 .408 0719 .1763
12 .06149 41.59 I.889 .24141 .4923 .1046 .2125
15 .05024 45.34 2.195 .28052 .6222 .1667 .2679
18 .04070 49.78 2.555 .32653 .7604 .2471 3249
21 .03265 55.67 3-014 .38519 .9207 3534 -3838
22.71 .02886 61.48 3.460 .44219 I.0659 .4462 .4187


NACA TM 1369
81
TABLE 2.- Concluded LIFT AND DRAG COEFFICIENTS
Ml P2/P0 P2VP0 ca cw e
2.50 1 0.05296 21+-35 1.068 0.06251 0.0373 0.0006 0.0175
2 .05113 25.05 1.11+1 .06678 .0611 .0021 031+9
3 .01+772 25.82 1.216 .07117 .0911+ .001+8 .0521+
h .01+1+50 26.62 1.296 .07585 .1221 .0085 .0699
6 03857 28.27 1.1+52 .081+98 .1826 .0189 .1051
8 031+55 30.00 1.658 .09701+ .21+16 .031+0 .11+06
10 .02859 31.86 1.865 .10916 3098 .0536 1763
12 .021+1+5 33-81 2.091 .12232 3738 0795 .2126
15 .01917 36.95 2.1+67 .11+1+39 1+723 .1266 .2679
18 .011+81+ 1+0.1+0 2.895 1691+9 571+2 .I865 .321+9
22 .01035 1+5.62 2.557 .20816 .7162 2893 .1+01+0
28 00575 56.35 1+.885 .28592 9659 .5136 5317
29.67 .001+82 62.65 5.602 .1+1+220 1.0960 .621+0 .5695
5.00 3.60 .00118 11+ 1.51+1 .00291 .0523 0033 .0627
6.17 .00081+ 16 2.051 .00388 .0911+ .0098 .1076
10.68 .oooi+3 20 3.21+7 .00611+ .1698 0319 .1879
16.60 .00018 26 5.1+36 .01027 .2926 .0869 .2972
20.21 .00009 30 7.129 .0131+7 .3800 1393 .3666
26.78 .00002 38 10.893 .02059 5557 .2792 .5021+
31.21 .000009 1+1+ 13.913 .02629 .6805 .1+088 .6000
36.29 .000003 52 17-953 03392 .8281+ .601+8 7301
4i.ii .000001 66 21+. 206 01+575 1.01+27 .9098 .8726
10.00 3.21 8 2.091 .0228 .0013 .0560
7.65 12 1+.877 .0611+ .0082 .131*0
13.31 18 10.981 .151+9 .0365 2356
18.53 21+ 19.135 .2613 .0877 3351
23 A7 30 29.018 .3828 .1662 .1+31+1
28.17 36 1+0.161+ 5079 .2726 .5366
32.59 1+2 52.080 .6288 .1+017 6393
36.64 1+8 61+. 263 .71+88 5565 .71+37
1+0.18 51+ 76.213 .861+1 72^2 .81+1+1
1+2.90 60 87.361 9529 .8862 9293
1+4.4-3 67.12 98.713 1.0125 .9925 9803
00 8-37 10 .01+97 0073 .11+71
16.57 20 .1873 .0558 2973
21+.50 30 3795 .1731 1+557
28.31 35 .1+71+1 2559 5386
32.05 1+0 .581+3 .3662 .6261
35-55 1+5 .6812 .1+81+6 .711+6
38.81 50 .781+0 .6121 .801+2
1+1.6 55 8735 7754 .8876
1+3-9 60 .91+52 9123 .9623
1+5.58 67.8 1.0001 1.0201+ 1.0200
N 5 .0152 .0013 .O875
10 .0591+ .0105 .1763
15 .1291+ .031+7 .2679
20 .2198 .0801 361+0
25 .3238 .1509 .I+663
30 1+330 .2500 5771+
35 .5388 3772 .7002
1+0 6331 5311 .8391
1+5 .7068 .7068 1.0000
50 751+7 .8992 1.1918


82
NACA TM I369
TABLE 3
LIFT AND DRAG COEFFICIENTS OF THE BOUNDARY CURVE
Mi *s P2/P0 7S 0 Pa'/Pi p2'/p0
1.10 l.k 0.43110 73 14 1.150 0.52924 0.2471 0.0104 0.0248
1.20 3-70 55952 68 5 1.277 .52681 .4011 0259 .0641
1.30 6.32 .26668 65 7 1.457 .52586 6035 .0668 .1107
1.40 9.03 .20050 65 10 1.656 .52009 .7521 .1159 .1584
1.50 11.67 .14392 62 15 1.894 .51595 8559 .1724 .2068
I.60 14.24 .11022 61 39 2.143 .50418 .9052 .2297 2552
1.70 16.63 .07918 61 22 2.439 .49412 .9697 .2897 .2988
1.80 18.84 .05788 61 17 2.741 .47687 1.0045 3427 .5412
1.90 20.87 .04047 61 21 3.097 .46220 1.0449 .3983 .3812
2.00 22.71 .02886 61 29 3-46o .44219 1.0665 .4465 .4187
2.20 25.90 .01417 61 54 4.250 .58746 I.0883 .5284 .4855
2.50 29.67 .00482 62 39 5.602 .44221 I.0960 .6242 5695
3-00 34.01 .00075 63 46 8.385 .22824 1-0735 7243 6747
4.00 38.75 .000014 65 15 15.25 .IOO60 1.0650 8551 .8025
5.00 4l.ll .000001 66 12 24.40 .04612 1.0515 .9107 .8652
6.00 42.44 0 66 45 35-25 .02231 1.0362 9432 .9156
8.00 45-79 0 67 0 63.11 .00630 1.0171 .9692 .9601
10.00 44.43 0 67 7 98.90 .00233 1.0121 .9895 .9805
00 45.58 0 67 41 1.0050 1.0185 1.0200


NACA TM 1369
83
TABLE It
LIFT AND DRAG COEFFICIENTS ACCORDING TO THE LINEARIZED THEORY
1 + ca =w e
1.20 1 0.10553 0.00018 0.0175
2 .21046 .0073* 0349
3 .31600 .01655 .0524
3-7 38957 .02516 .0646
l.ltO 1 .07144 .00125 0175
2 .14248 .00497 .0349
3 .21392 .01121 .0524
1* .28496 .01989 .0698
6 .42744 .04480 .1047
8 56992 07950 1396
9-03 .64259 .10112 1574
1.60 1 .05604 .OOO89 0175
2 .11177 .00390 0349
3 .16781 .00879 .0524
1* 22353 .01560 .0698
6 33530 .03458 .1047
8 .44707 .06240 1396
10 .55884 09752 1745
12 .67060 .14040 .2094
14.24 79582 .19772 .2485
1.80 1 .04676 .00082 0175
2 09325 .00326 0349
3 .14001 .00734 .0524
.18650 .01302 .0698
6 .27976 .02932 .1047
8 37301 .05205 .1396
10 .46626 .08135 .1745
12 55952 .11714 .2094
15 69953 .18314 .2618
18.84 .87855 .28884 .3288
2.00 1 .04042 .00070 0175
2 .08060 .00281 0349
3 .12102 .00634 .0524
.16120 .01125 .0698
6 .24180 .02532 .1047
8 32240 04995 .1396
10 .40301 .07030 .1745
12 .48361 .10125 .2094
15 .60463 .15829 .2618
18 .72564 .22802 .3142
21 .84643 .31040 3665
22.71 .91502 36259 .3962
2.50 1 .03054 .00053 0175
2 .06091 .00213 0349
3 .09145 .00479 .0524
It .12181 .00851 .0698
6 .18272 01915 .1047
8 .24363 .03400 1396
10 .30454 .05318 1745
12 .36544 .07650 .2094
15 .45689 .11962 .2618
18 .54834 .17230 .3142
22 .67016 .25742 .3840
28 .85288 .41675 .4887
29.67 .90366 .46789 5178
5.00 It 05699 .00398 .0698
6 .08549 .00896 .1047
12 17097 03579 .2094
18 .25654 .08061 3142
2lt .34203 .14321 .4189
30 .42752 22385 .4712
36 .47030 .32232 5760
4l.ll .58584 42033 7175
10.00 It .02806 .00196 .0698
6 .04209 .00441 .1047
12 .08418 .01762 .2094
18 .12651 .03969 3142
2lt .16840 .07051 .4189
30 .21049 .11022 .4712
36 .25258 .I58OO .5760
42 .29467 .21600 7330
44.43 .31172 .24177 7755


84
NACA TM 1369
TABLE 5
Ml ^ (caG caLJ caG (cwG cwl) cwG
1.1*0 1 0.00084 0.07228 0 0.0013
2 .00151 .14399 0 .0050
3 .ooi63 .21555 .0001 .0113
4 .OO585 .28881 .0002 .0202
6 .01408 .44152 .0010 .0464
8 .04690 .61682 .0072 .o867
9-03 .08950 73209 .0148 .1159
5.00 4 .0005 .0523 .0009 .0033
6 .0050 .0914 .0010 .0098
12 0255 .1698 .oo63 0319
18 .0696 .2996 .0246 .o869
24 1591 .3800 .0341 1393
30 .1846 5557 .1455 .2792
36 5972 .6805 .2698 .4088
41.11 .4669 .8284 .4895 .6048


NACA TM 1369
TABLE 6
Region P/POl M Po/POi
1 0.3l4o4 1.4004 45.5 1
2 .3645 1.293 50.6 .9996
3 2991 1.435 44.2 l
4 3455 1.352 48.8 9997
5 .2848 1.469 42.9 l
6 3304 1.564 47.I 9997
7 .2711 1.503 41.7 l
8 3139 i.4oi 45.7 9998
9 3285 1.368 47.0 9997
10 3139 1.401 45.6 9997
11 .2985 1.436 44.1 9997
12 2999 1.455 44.2 9997
13 .2850 1.468 42.9 9997
14 .2708 1.504 41.7 9997
15 .3624 1.154 50.1 .9982
16 .3445 1.534 48.6 9997
17 .3295 1.566 47.0 9997
18 .3124 1.404 45.4 9997
19 .3276 1.570 46.9 .9998
20 3134 1.402 45.5 9997
21 .2972 1.439 44.0 9997
22 .2996 1.434 44.8 9997
23 .2842 1.483 42.8 9997
24 .2696 1.507 41.6 9997
25 .3610


86
NACA TM I369
TABLE 7
(a) Angle of deflection and shock angle of the shocks between the zones
Regions Angle of deflection 6 Shock angle 7
1-2 3-00 49.57
3-4 3-03 47.91
5-6 2.93 46.48
7-8 2.93 45.15
8-15 2.93 49.43
11-16 2.92 47.78
13-17 2.90 46.40
14-18 2.93 45.05
18-25 2-93 49-30
(b) Intensity of expansion waves between the zones
Regions Intensity A9 Regions Intensity A9
1-3 1.00 13-14 1.02
2-4 1.03 15-16 1.00
3-5 1.00 16-17 .88
4-6 .89 16-19 1.00
4-9 1.03 17-18 1.06
5-7 1.00 17-20 99
6-8 1.00 18-21 1.00
8-11 1.01 19-20 .89
9-10 .92 20-21 1.05
10-11 1.00 21-23 .89
10-12 .92 22-23 1.05
11-15 .91 23-24 1.05
12-13 1.02


NACA TM 1369
87
TABLE 8
CASCADE EFFICIENCY tj PERCENT = f(p,\|r) (NO FRICTION)
1 3 5 7 10

10 89.8 69.6 49.6 29.72
20 94.7 83-9 73-6 63.4 48.4
30 96.O 88.3 80.8 73-5 63.I
4o 96.5 89.8 83.4 77-3 68.8
50 96.6 90.00 83.8 78.2 70.4
60 96.0 88.8 82.4 76.5 68.8
70 94. 8 85.7 78.2 71.4 63.1
80 90.6 76.4 65.8 57.65 48.4
0 for max. t) 46.5 47.5 48.5 50.0
^max* Percent 96.3 90.2 83-7 78.2 70.4


88
NACA TM I369
TABLE 9
CASCADE EFFICIENCY WITH FRICTION ALLOWED FOR
M = l.4o M = 2.50
= 5 = 2.65 = 30 + = 4.11
10 55 8 57-0 44.1 4l.l
20 74 2 74.6 63-9 63.7
30 80 5 80.4 71.2 71.8
4o 82 5 82.8 73-9 74.8
50 82 6 82.7 73-5 74.7
60 80 5 80.6 69.9 71.8
70 74 7 74.6 60.6 63.8
80 57- 8 57.2 55-5 41.4


NACA TM I369
89
TABLE 10
CASCADE EFFICIENCY WITH DOUBLE-WEDGE PROFILE AND FRICTION
M = 1.1+0 \|r = 3
d/l = 0.05 d/l =0.10
10 45.53 29.01+
20 65.28 l+T-22
30 T2-52 T4.T8
1+0 T5-11 56.93
50 T4.83 54.82
60 71-50 47.31
TO 62. Tl 29.25
80 37-18


90
NACA TM 1369
TABLE 11
1 3 5
Section hub N 2 center M 4 tip S
0.500 0.625 0.750 O.875 1.000
0.1+50 0.1+25 0.1+00 0.375 0.550
0.070 O.O65 0.06 0.055 0.050
0.0315 0.0282 0.021+ 0.0206 0.0175
200 250 500 350 400
Relative velocity, wm/sec .... 1+52.7 1+76.8 501+.6 556.O 570.0
1-339 1.1+11 1.493 I.586 1.687
Angle of attack, \|/0pt- \ ^ t 0 0.0818 O.O787 0.0761 0.0742 0.0721
I+.67 1+.52 4.38 4.25 4.13
O.3672 0.3161+ 0.2745 0.2407 0.2120
O.1637 0.1572 0.1525 0.1484 0.1442
9-30 8.95 8.66 8.43 8.21
Stagger, (3........... dS/dr, kg/m ........... 68.8 62.9 57-9 53-5 49.58
612 7I+8 825 878 904
dn/dr, kg ............ 1026 1118 1173 1212 1224


NACA TM 1369
91
Figure 1.- Expansion around a corner.
s
Figure 2.- Oblique compression shock.


92
NACA TM i369




b /
i \ \



1 23456789 10
(a) Dynamic pressure/stagnation pressure
(b) Dynamic pressure/ inflow pressure
Figure 5.


9h
NACA TM I369