Two-dimensional potential flows

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Title:
Two-dimensional potential flows = Ebene potentialströmungen
Parallel title:
Ebene potentialströmungen
Physical Description:
24 p. : ; 28 cm.
Language:
English
Creator:
Schäfer, Manfred
Tollmien, W
United States -- National Advisory Committee for Aeronautics
Technische Hochschule Dresden
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
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Differential equations   ( lcsh )
Speed   ( lcsh )
Aeronautics   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Abstract: The characteristic differential equations are set up for two-dimensional potential flows. The second form of the characteristic differential equations, involving only the magnitude of the velocity and the deflection angle, are integrated. The methods of O. Walchner and of Prandtl and Busemann are discussed. Busemann's development of the pressure variation for small deflection angles is given and applied to the example of a circular arc profile. Tables are provided giving the relation between deflection angle, pressure, velocity, Mach number and Mach angle for isentropic changes of state for air (k kappa = 1.405).
Bibliography:
Includes bibliographic references (p. 24).
Statement of Responsibility:
by Manfred Schäfer and W. Tollmien.
General Note:
"Originally published Technische Hochschule Dresden, Archiv Nr. 44/3, Kapitel III, March 22, 1941."
General Note:
"Report no. NACA TM 1243."
General Note:
"Report date November 1949."
General Note:
"Translated by Mary L. Mahler, NACA."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 003690307
oclc - 76883341
sobekcm - AA00006235_00001
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Full Text
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NATIONAL ADVISORY COMMITTEE FOR AEr.ONAUTICS


TECHNICAL MEMORANDUM 1241


TWO-DDIEESIOIAL POTENTIAL FLOWS

By Manfred Schilfer and W. Tollmien


Outline: I. CHARACTERISTIC DIFFERENTIAL EQUATIONS INITIAL AND
BOUNDARY CONDITIONS
II. TITEGRATION OF THE SECOND CHARACTERISTIC DIFFERENTIAL
EQUATIONS
III. DIRECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGRAPH
TABLE FOR CONSTRUCTION OF TWO-DIMENSIONAL POTENTIAL
FLOWS
IV. PRANDTL-BUSEMANN METHOD
V. DEVELOPMENT OF THE PRESSURE VARIATION FOR SMALL DEFLEC-
TION ANGLES
VI. NUMERICAL TABLE: RELATION BETWEEN DEFLECTION, PRESSURE,
VELOCITY, MACH NUMBER AND MACH ANGLE FOR ISENTROPIC
CHANGES OF STATE ACCORDING TO PRANDTL-MEYER FOR AIR
(K = 1.405)
VII. REFERENCES


I. CHARACTERISTIC DIFFERENTIAL EQUATIONS -

INITIAL AND BOUNDARY CONDITIONS.


For setting up the characteristic differential equations one
starts from the differential equation for the velocity potential since
the velocity components can be expressed more simply by the velocity
potential than by the stream function. This differential equation
reads, according to I(15):


S-2 %x '2 av- ,,y -2 + 0 (1)

with

u = q v = p (2)


*IEbene Potentialstromungen. Technische Hochschule Dresden,
Archiv Nr. 44/3, Kapitel ITT, March 22, 1941.






NACA TM 1241


and the sonic velocity a determined by


a2 = ( 1 q2 (?)


in the selected non-dimensional representation.

The characteristic condition (cf. chapter 11(8), NACA TM 1242) is



a- y2 + 2xvy + 2 (4)


If one puts


u = q cos 3 v = q sin ( (5)


the characteristic condition is written


a2 q2cos2, 2 + 2q sin' cosa x (a2 q2sln2 2 = 0 (6)

Hence result two roots X' and X" for k, the slope of the charac-
dx'
teristic base curves toward the x-axis:


qsin -3 cos 3 + a. q2 a2 (a)
q2cos23 a2



q2sin cos 6 a lq2 -a2 ()
q2cos2S a2

As differential equation for the first family of the characteristic
base curves results
dy X' dx = 0 (8a)

and for the second
dy dx = 0 (8b)







IJACA TM 1243 3


For explanation of these relations (somewhat difficult to survey
due to the complicated form of X' and X") one uses an artifice which
is permissible in two-dimensional flows. Since in the two-dimensional
flow, for instance, in contrast to the rotatlonally-symmetrical flow,
no direction is preferred, one may place the x-axis of a Cartesian xy
system in the direction of the flow at the location under investigation;
thus there becomes


S= 0
a

-a: (9)


X" k Sa -
2 a2



The Mach angle a is defined by


sin a = -
q

so that

tan a = a (10)
\q a2

This signifies according to (8) that the characteristic base curves
form with the stream lines the Mach angle a. The first family (8a) of
characteristic base curves forms the Mach angle toward the left (looking
in the flow direction), the second family (8b) forms the same angle
toward the right. The first family of characteristicbase curves were
thereupon denoted as left-hand, the second as
right-hand Mach waves.

The second characteristic equation of the first
q family of characteristics is, according to II(28b),
(NACA TM 1242):

q2sin cos 9 a q2 2
du + dv = 0 (lla)
q2 cos 2 a2






NACA TM 1243


and for the second family of characteristics, according to II(29b)
(TM 1242):


2 2 2
q sin 1: cos i + a /q2 a-
du + sin cos o + aq dv = 0 (11b)
q2cos 2 a2

with

du = cos 9 dq q sin 3 d8
(12)
dv = sin i dq + q cos do

If one applies the same artifice as in selecting the special coordinate
system at the considered location, one obtains, since a there becomes
0,

du = dq, dv = q dO

and one has on the first family of characteristics



dq a q de = 0 (13a)
\/q2 a2


and on the second


dq + --- q d = (13b)



The two relations (13) contain only quantities independent of the
special selection of a coordinate system. Together with the remark
made initially on the admissibility of the last used coordinate system
there results, accordingly, the general validity of the relation (13a)
for the first, of the relation (13b) for the second family of charac-
teristics.

The relations (13) are often, in an elementary manner, deduced
from the fact that the infinitesimal velocity variation is perpendicular
to the Mach wave:







NACA TM 1243


sin- a
q + dq _in_-

sin ( + a dd)


cos a

S- cos a + sin m d-a

9 = 1 tan a d3


dq -q tan a d1

a
= q d1
\q2 a2


Thus the velocity variation in crossing a left-hand Mach wave is
regulated according to this equation (of. (13b)); the crossing has to
take place along a right-hand Mach wave. Regarding this elementary
derivation the fundamental remark has to be made that here tacitly the
existence of a relation between u and v alone (and between q
and i alone, respectively) is assumed. In the rotationally-symmetrical
case where this presupposition no longer holds, one obtains accordingly
another characteristic equation although the velocity increment occurs
as before perpendicularly to the Mach wave.

A few remarks concerning the secondary conditions which supervene
the differential equation of the two-dimensional potential flow are to
be inserted at this point.

The secondary conditions may for instance be given by initial
conditions, that is, on an initial curve (for example, in a certain
cross section of a channel) a few or all flow values are prescribed.
The front of a compression shock also may serve as initial curve. The
initial distributions for the approximation method discussed here are
approximated by distributions constant over small distances. Therein
it was often used as approximation principle that the jumps in i- are
to be of a certain magnitude, for instance +10 or +20. Correspondingly,
the boundary distributions which are given by boundary conditions, the
main types of which will now be discussed, are also approximated.






NACA TM 1243


J is prescribed at a solid wall. If one denotes as compression
wave a Mach wave behind which a pressure increase and a velocity decrease
takes place, compression waves are reflected at a solid wall as compres-
sion waves. This can readily be seen in the figure.
The crossing of the right-hand Mach compression wave
occurs along a left-hand Mach wave; thus the pre-
supposed decrease of q according to (13a) is
connected with a decrease of 3. The crossing of the adjoining left-
hand wave along a right-hand wave must cause due to the boundary
condition at the wall which is assumed to be unbroken an increase
of 3 which according to (13b) produces a decrease of q, therefore a
compression.

At a free let boundary the pressure p is prescribed as constant;
thus the velocity q also is a known constant there. The free jet
boundary must because of the cinematic boundary condition of vanishing
normal velocity coincide with a stream line which will be determined
in the course of the solution of the flow problem. A compression wave
is reflected at a free jet boundary as rarefaction wave since the drop
in velocity which occurred first must be made good again by an increase,
in order to satisfy the condition of constant velocity at the free jet
boundary.


2. INTEGRATION OF THE SECOND CHARACTERISTIC

DIFFERENTIAL EQUATIONS.


Of the characteristic equations (8a), (8b), (13a), (l?b) the last
pair can be very easily integrated; this was already done by Th. Meyer
(cf. Th. Meyer, particularly p. 38). Following, a derivation is given
which fits into our general theory.

On the left-hand family of characteristics


S- q (13a)
vq2 a2-

with a2_ 1 ( q2). Hence 3 may be determined as function
2
of q by quadrature. The execution of this elementary integration
results in







NACA TM 1243


K + 1
- are tan
K-iI


- arc tan






arc tan a*


2 +1
K I +
V( )(1 -q2) K




r1 1 1 1
are tan + 4
a2 a*2 a2 -*2 1


+ 1
or = ---
V 1


rt 1 3 v2
are tan 1 arc tan -1
K + 1\a2 a2


1 1)
is the critical velocity I---i and 81 represents
J(K + 1)


+ a7, where a*



an Integration


constant. It has to be noted that all velocities have been made non-


/ 2K Po
dimensional by Vm = 1 po
K P


(Po tank pressure, po tank density).


For this relation a table of data particularly convenient for the
practical calculation has been given by 0. Walchner for air (K = 1.405)
(cf. pp. 22-23). Aside from the velocities q referred to the sonic
velocity a and the critical sonic velocity a*, the pressure p,
referred to the tank pressure p, is given, for which the equation


L = (1 q2)K/(K-1)
Po

is valid. In the last column the Mach angle a is indicated. The
integration constant 01 is selected as O, v is used instead of b
for this special integration constant, the reason for this will be
shown in the next paragraph. The application of the Meyer-Walchner
table for the approximated construction of two-dimensional potential
flows will be discussed in the next paragraph.


/- 1 V 2 K +1

+1 V( 1)(1 q2) K -1


1
a*


(14)






NACA TM 1243


For the right-hand characteristics one obtains correspondingly by
integration of (13b)




1 1
arc tan a arc tan I- (15)
a* \ a*2 Va2 a*2 r


or

.- = +-i are tan i 1 arc tan 1 -
arc tan V 2 a2



with br being an integration constant. On the left-hand character-
istics increasing q is, according to (13a), connected with
increasing 0, on the right-hand characteristics increasing q, accord-
ing to (13b), with decreasing 0. Hence the designation left-hand and
right-hand, at first introduced through the characteristic base curves
of the Mach waves, is immediately comprehensive also for consideration
of the "characteristic hodographs" the equation of which is given in
polar coordinates (q radius vector, angular coordinate) in
differential form by (13a) and (13b), in the integrated form by (14)
and (15). Cl. Thiessen (1926) first drew attention to an interesting
geometrical interpretation of these characteristic hodographs. A
geometrical proof which, however, requires longer preparation, has
been given by A. Busemann. Following, we give our own proof, relin-
quishing the non-dimensional representation, in order to conform to
customary representation.

According to Thiessen the characteristic hodograph curves are
epicycloids originating from the rolling of a circle of diameter vm a*
on a circle of radius a*







NACA TM 1243


Since the rolling
point A, the point Q
the fixed circle 0 -
increment dq lies on


circle rolls in the denoted position about the
- which has the distance q from the center of
traces a small circular arc about A. Thus the
the line QB, since QB is perpendicular to QA.


On the other hand, dq is, according to the end of the previous
paragraph, for two-dimensional potential flows unequivocally determined
by the fact that dq is perpendicular to the Mach wave which with q
(here represented by OQ forms the Mach angle a = arc sin a/q. Thus
the proof will be given if the angle 4- OQA is found to be equal to
the Mach angle.

If one puts preliminarily

4OQA =
and
an -QBO =
one has
4. O-B = n/2 + e

4- QAO = n/2 + 0


According to the sine
one obtains


theorem applied to the triangles OQB and OQA


sin( +
sin




sin E

sin( + +


cos C
sin P




sin e
cos B


Vm
q



a*

Q


From these two equations the angle c which is of interest to us
may be calculated by elimination of 0. This brief calculation may be
performed about as follows. One has directly


sin 0 = -' cos
Vm


cos = sin F
a*






NACA TM 1243


Hence follows by squaring and adding


or


q2 cosB2 E +

\ m



S-2 sin2, +
v2
m


a n 1
an2)




K + sln2 E 1
K -


Next, there results


sin2 = K -
2


2 2
Vm q2
q2


a2
q2


This, however, is just the equation for the Mach angle; thus E equals
the Mach angle. Therewith the fundamental statement dealing with the
behavior of dq with regard to the characteristics has been obtained
again. The directional field produced by rolling of the circle coin-
cides, therefore, with the directional field of the characteristic hodo-
graph curves. The characteristic hodograph curves are epicycloids.


I. DIRECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGRAPH

TABLE FOR CONSTRUCTION OF TWO-DIMENSIONAL

POTENTIAL FLOWS.


Meyer's solution for the characteristic hodographs indicated
in the previous paragraph has been used directly for the construction
of two-dimensional potential flows by J. Ackeret and, recently, by
0. Walchner. We shall explain the method in one of the examples cal-
culated by 0. Walchner which for the first time showed the general
validity of the method.

First it is to be noted that the one table given actually is
sufficient for all characteristic hodographs, since, according to (14),
for the left-hand characteristic hodographs


.a = v + (a6


(16)







NACA TM 1243


according to (15) for the right-hand characteristic hodographs


b = -V+ + r
r


(17)


It has to be pointed out that the crossing of a right-hand Mach wave
takes place along a left-hand characteristic, so that (16) is valid.
The crossing of a left-hand Mach wave occurs along a right-hand charac-
teristic with (17) being valid.


The transition from one field with the index
the index i + 1 is regulated at the crossing of
wave, thus according to the equations:


i to the next with
a right-hand Mach


whence follows, with


S= Vi +b


=+ vi+1 + 19

-8 eliminated:


v+1 = v + )


For crossing of a left-hand Mach wave

di = -Vi +


1+1" i -+1 + r


(18)






NACA TM 1243


must be valid or, after elimination of Or:


1 v+l 1 (i+1 (19)


According to the two equations (18) and (19) Meyer's table is used for
the approximated calculation of two-dimensional potential flows.

The selected example of Walchner (fig. 3 of Walchner's report Lufo
Bd. 14 (Aviation Research, vol. 14) p. 55, 1937) deals with the flow
about a biplane at an angle of attack, with relatively rough approxi-
mation since large Jumps in direction from one field to the next are
admitted. Furthermore compression shocks are approximated by Mach
waves which in view of the weakness of the occurring shocks is
still justified (cf. chapter V, par. 5).

The field numbers are put as indices to the pertaining flow
ql pi
values. In the initial field = 1.64, = 0.221, 91 = 0 is
al Po

prescribed. According to the table the value V1 = 160 pertains to
this value of q/a or p/po. In field 2 02 = -10, due to the
geometrical boundary condition. Since in the transition from field 1
to field 2 a right-hand Mach wave is crossed, v2 = 60 according to (18);
ql
according to the table, the pertinent values are -= 1.293,
aP
Pi
S= 0.363. For field 3 one has, for geometrical reasons, 33 = 40; the
Po
transition from field 1 to field 2 leads, according to (19), to v3 = 120
3 P3
with = 1.504, = 0.270. The calculation of the q- and b-values
a3 p0
in the next field 4 represents the general case of the method. From
field 3 one arrives at field 4 by the crossing of a right-hand Mach
wave, thus according to (18): vq = v3 + 4 3 = 80 + V4. From
field 2 one arrives at field 4 by the crossing of a left-hand Mach wave,
thus according to (19): V4 = V2 -4 2) = -40 04. From the two

equations for -4 and V4 set up just now follows 4 = -60, v4 = 2o
q4 P4
with = 1.132, = 0.449. In all remaining fields 0 is prescribed
a4 Po







NACA TM 1243


by geometrical boundary conditions so that the calculation is easy and
takes place very similarly to that for field 2 and field 3. One has


5 = 4, V = 120 = 1.504, = 0.270;


q6 P6
6 =-100, v = 60, = 1.293, 6 = 0.363;
6 O


-7 = "30, 7 = 190, = 1.743, = 0.190;
a7 Po

08 P8
g = -30, vg = 130 = 1.538, = 0.257
8a8 p0

A drawing machine is desirable for plotting the Mach waves which
close any newly calculated field.

With a finer subdivision of the angular variations, a greater
accuracy seems attainable by means of this method than with the aid
of the Prandtl-Busemann method described below.


4. PRANDTL-BUSEMANN METHOD:


For approximated construction of two-dimensional potential flows
mostly the Prandtl-Busemann method is used, the main expedient of which
is a diagram with the characteristic hodograph curves (14) and (15),
respectively.

The Prandtl-Busemann method is, according to our terminology
introduced in chapter II, (NACA TM 1242), a field method, that is, a
pair of values q, d is coordinated to each field formed by the
characteristic base curves or Mach waves. For the sake of a simple
representation of the method we assume this pair of values q, j to
be valid precisely for the field center, the definition of which was
given in chapter II, paragraph 7, NACA TM 1242.1 We now visualize the
field centers as connected with each other; these connecting curves
1 The field centers are very useful for explanation of the method;
they are, however, in case of two-dimensional potential flows, in contrast
to rotationally-symmetrical ones, not required for the construction so
that the exact definition of the field centers would here not yet be
necessary.






NACA TM 1243


give, as was shown, again characteristic base curves, thus here the
Mach waves. To these Mach waves, not perhaps to the Mach waves of the
field boundaries, the characteristic hodograph curves were coordinated.

The net of the characteristic hodographs may be drawn once and for
all according to the expositions in section 2 for a given K. According
to former representations the crossing of a left-hand Mach wave in the
flow plane is connected with a progressing along a right-hand charac-
teristic hodograph in the velocity plane. Correspondingly, crossing of
a right-hand Mach wave is coupled with progressing along a left-hand
characteristic hodograph. Busemann and Preiswerk gave a net of charac-
teristic hodographs for K = 1.405. This diagram is customarily denoted
simply as "characteristics diagram".

In order to calculate from the known
/ pairs of values q, B in field I and II
m the unknown pair of values q, B in the
field III adjoining downstream, one pro-
iT gresses from the point in the character-
istics diagram corresponding to field I
along a right-hand epicycloid, since one
has crossed a left-hand Mach wave in the
transition from I to III; correspondingly
one progresses from the point of the characteristics diagram corres-
ponding to field II along a left-hand epicycloid. The point of
intersection gives the pair of values q, 6 for the field III. The
field that had been open so far is then closed by two Mach waves the
direction of which is determined from the values of q and 9 found
just now. The modifications of the method for fields at the boundary
of the region are obvious.

In order to facilitate the reading of the q- and S-values from
the characteristics diagram, one may take the net of the characteristic
hodographs as net of coordinates. According to (14) and (16), respec-
tively, one has for the left-hand epicycloids:

q)J =' (20)

according to (15) and (17), respectively, for the right-hand epicycloids

v(q = --r (21)

Thus 3 and through the tabulated function v(q) q as well may be
very easily expressed by the parameters O1 and 9r of the epicycloids.







NACA TM 1243


Instead of the parameters 3 and 3, which probably first seemed
obvious, Busemann selected others with only the starting points of the
count shifted from and V. The angles 5 and v are measured
in degrees. The degree sign (0) is omitted below. One may then express
Busemann's epicycloid numbering so that as equation of the left-hand
epicycloids


v(q) = 2(X 400) (22)

as equation of the right-hand epicycloids


-9 v(q) = 2(i 600) (23)

is written with the new parameters X and p. Hence there results


200 8 = p X (24)


1000 -V(q) = p + X (25)

Thus the difference of the new parameters' X and p gives the angle b
except for an insignificant shifting of the initial point and reversal
of the sense in which one is counting. The center line of Busemann's
characteristics diagram (9 = 0) obtains the "direction number"
p X equal to 200, whereas the sum of X and p yields the function
v(q) and therewith also q and the pressure p. The numbering of the
epicycloids according to (22) and (23) is carried out very easily if
one considers additionally that v(q) just vanishes for q = a*
(critical velocity). Eusemann writes the parameters X and p as
field numbers" into the fields of the flow plane; a table for the
connection of the "pressure number" p + X with q and p must be
given as supplement.

If one approximates the initial and boundary conditions in such a
manner that one replaces the prescribed angles by sectionally constant
distributions with jumps of +10 or +20, one may assure by a suitably
fine-meshed characteristics diagram that one gets by without interpola-
tion. The customary characteristics diagrams are in their main part
arranged for angular jumps of 10.






NACA TM 1243


Additionally developed graphical expedients for facilitated
plotting of the Mach waves will not be discussed, since one can dispense
with them when a drawing machine is used.

We will be content with these observations regarding the Prandtl-
Busemann method and will cmit the carrying out of a standard example
since the method has been represented in detail by Heybey (HVP Archiv
Nr. 66/31 and 66/32).


5. DEVELOPMENT OF THE PRESSURE VARIATION

FOR SMALL DEFLECTION ANGLES.


In a flow unilaterally bounded by a wall the flow variations en-
forced by the boundary conditions are propagated from the wall along
one family of Mach waves; in the figure it is the left-hand family.

60>








This property of two-dimensional potential flows follows immediately
from the characteristic differential equations (13) which connect
velocity and directional variation. This property is by no means
transferable to other than two-dimensional potential flows, for instance
potential flows with rotational symmetry.

Since for the conditions assumed in the figure the flow variations
occur at the crossing of left-hand Mach waves, they may be calculated
by progressing along a right-hand characteristic, thus according to
(13b) from
aq
dq = da (13b)
2 a2

A development of the pressure difference for deflection of the
flow from the angle 3= 0 to the angle 3 = 5 is to be given; the
deflection angle 5 is to be small and the third powers of 5 are
still to be included in the development. Busemann has for the first
time set up such a development, using a method totally different from






NACA TM 1243 17

ours. In order to develop the pressure, first the velocity q2 which
is to pertain to = 8 is developed according to equation (13b) with
respect to 6. For b = 0, q is to equal ql. We set up:


2 = q1 + c + 02 +2 +3 3 (26)


and determine the unknown coefficients c, c2, c3 fran (13b) by com-

parison of the coefficients. One has only to equate = Cl+ 2c2 +3c3

aq
with the expression originating from when q is replaced
q2 a2

by q2 according to (26). Therein is a2 = 1 (Vm2 q2). It is

most convenient to square the two sides of the relation mentioned before
comparing the coefficients. There results


1 __ = -2 q 1)ql + 2aI
a12 4 (q2 a 12)2 L


3 = a1q, 1)(2K 3)q6 (27)
12 2 a12 7/2 U


+ 9( )a12q14 + 6a1 q2 + 2a16

In order to obtain frcm the development for q2 that of the pressure
P2, one starts from the isentropic pressure equation

21 i 2 2 A-
P2 m .2 -2 21 q1
2- = =m K- (28)
p1 2 q i2 2 q
m m






NACA TM 1243


This expression is binomially expanded, making use of (26) and (27).
For transformation of same of the terms one applies the formula

2K 91
2 Pl
K 1 2 q12 01


which follows from the energy theorem (1(6)), and


al2 -1 m2 12)



with the Mach number M = ql/al one then finally obtains


S P 2 52(M2 2)2 + KM
2 -1 -1 4(M2 1)2


&3 (29)
+ (a + 1)M8
12(M2 1)7/ (/2



+ (292 7' 5)M6 + I0(K 1)M 12M2+8 j


The first term of this development may be found already in Th. Meyer's
report; Busemann gives the development up to 53 (Volta congress, p. ?37).
It is true that the coefficient of 53 as calculated by us, completely
differs from Busemann's. The comparison of the pressure difference
occurring for isentropic deflection with the one in case of a compres-
sion shock is made in chapter V, paragraph 5.

In transferring formula (29) to the case of the flow variation
taking place at the crossing of right-hand Mach waves, one has to
establish the connection with the characteristic differential equation
(13a). The simple rule results that one has to select the sign of 5 in
(29) as it corresponds to a reflection on the freestream direction of the
conditions assumed above (cf. the figures on p. 16).







NACA TM 1243 1i


As application of this formula (29) we use the calculation of ca
and ow for circular segment profiles in second approximation which
was already performed by A. Busemann.


yr


According to the figure one has
the profile 8ob = -0 Bob and for
If one replaces cos 6 by 1, sin 6


c--
t
Ca 2- -q12f
t--0o
2




a 2_
cw 0
t2


for
the
by


the upper side (index ob) of
lower side (index u) 8u = 0.
8, one has


Pu Pob) dl






(Pup Pob PObPob) dl


with the integration to be extended over the entire wing chord. If one
uses formula (29) to the second term in 6 for the expression of pu
and pob:





NACA TM 1243


S1 2 2 M2 ( 2 2)2 M4
P Pl = -I + -
2 -1 (M2 )2 J


for which one writes abbreviatedly:


2
51 q 1 + C252)
P P1 (18 + C2


2 (M2 2)2+ +M4
S M 1M s C2 4(M2 )2

one obtains

Ca = 2Ci Pob2 d


t
Therein it is already taken into account that f Bob dl vanishes

(for the expression for yo we shall also use the vanishing of

O ob dl for circular segment profiles). In the foregoing deriva-

tion ca is then determined to include terms which are quadratic in the
angles P and. ob. For ow one obtains analogously the expression
including terms of the third order in the angles 0 and Pob:


cw = 2Cji2 tot+ f -d P T o2 dZ
0 ob d -- ob
ft 0






NACA TM 1243


Since one has for approximation of the circular arc by a parabolic arc

81d 4d
Pob = t


one obtains


ob2 16 t
0 ob 3 t


Therewith becomes


ca 2C1 -


c = 2C1,2


d2
\t/


C1 4d\2 4d2
+ 3M C2 t)


It is noteworthy that for the angle of attack P = 0 a negative lift of
the circular segment profiles exists which was also determined experi-
mentally.





NACA TM 1243


TABLE

RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND

MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO

PRANDTL-MEYER FOR AIR (K = 1.405).


p q m a a
P-- '- a, = arc sin a
Po a* a q

00 0.527 1.000 1.000 90000'
1 .478 1.067 1.081 67 45
2 .449 1.106 1.132 62 05
3 .424 1.140 1.177 58 10
4 .402 1.170 1.217 55 20
5 .382 1.199 1.256 52 50
6 .363 1.225 1.293 50 40
7 .346 1.250 1.330 48 45
8 .329 1.275 1-366 47 05
9 .314 1.299 1.401 45 35
10 .299 1.322 1.436 44 10
11 .284 1.345 1.470 42 50
12 .270 1.367 1.504 41 40
13 -257 1.388 1.538 4o 30
14 .244 1.408 1.572 39-30
15 .232 1.428 1.606 38 30
16 .221 1.447 1.640 37 35
17 .210 1.466 1.674 36 40
18 .200 1.485 1.708 35 50
19 .190 1.504 1.743 35 00
20 .18o 1.522 1.778 34 15
21 .170 1.540 1.812 33 30
22 .161 1.557 1.848 32 45
23 .153 1.575 1.883 32 05
24 .145 1.592 1.918 31 25
25 .137 1.609 1.954 30 50
26 .130 1.625 1.990 30 10
27 .123 1.642 2.027 29 35
28 .116 1.658 2.083 29 00
29 .109 1.674 2.100 28 25
30 .103 1.689 2.138 27 55







NACA TM 1243


RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND

MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO

PRANDTL-MEYER FOR AIR (K = 1.405)- Continued.


S- q -~ arc sin
Po a* a q

310 0.097 1-704 2.177 27020'
32 .091 1.720 2.215 26 50
33 .086 1.735 2.255 26 20
34 .080 1.750 2.295 25 50
35 .076 1.765 2.335 25 20
36 .071 1.780 2.376 24 55
37 .067 1.794 2.418 24 25
38 .062 1.808 2.460 24 00
39 .058 1.822 2.502 23 35
40 .055 .836 2.545 23 10
41 .051 1.850 2.590 22 45
42 .047 1.864 2.635 22 20
43 .044 1.877 2.661 21 55
44 .041 1.890 2.728 21 30
45 .038 1.903 2.778 21 10
46 .035 1.915 2.823 20 45
47 .033 1.928 2.872 20 20
48 .030 1.940 2.922 20 00
49 .028 1.952 2.974 19 40
50 .026 1.984 3.027 19 20
51 .024 .976 .081 18 55
52 .023 1.988 3.136 18.35
53 .021 1.999 3.191 18.15
54 .019 2.011 3.247 17-53
55 .018 2.022 3-304 17 35
56 .016 2.033 3-363 17 20
57 .015 2.044 3.424 17 00
58 .013 2.055 3.487 16 40

SNACA





NACA TM 1243


REFERENCES


1. Meyer, Th.: Uber zweidimensionale Bevegungsvorgange in einem Gas,
das mit Uberschallgeschwindigkeit stromt. Forsch.-Arb. Ing.-
Wesen Nr. 622, 1908.

2. Ackeret, J.: Luftkrafte auf Fligel, die mit grSsserer Geschwindlg-
keit als Schallgeschwindigkeit bewegt werden. ZFM Bd. 16,
pp. 72-74, 1925.

3. Walchner, O.: Zur Frage der Viderstandsverringerung von Tragflu-
geln bei Uberschallgeschwindigkeit durch Doppeldeckeranordnung.
Lufo Bd. 14, pp. 55-62, 1937.

4. Prandtl, L., and Busemann, A.: Naherungsverfahren zur zeichneri-
schen Ermittelung von ebenen Stramungen mit Uberschallgeschwindig-
keit. Stodolafeetschrift, Zurich 1929.

5. Busemann, A.: Article from. "Gasdynamik", Handbuch der Experimental-
physik Bd. 4, Teilband 1, pp. 423-459, 1931.

6. Preiswerk, E.: Anwendung gasdynamischer Methoden auf WaBserstrb'-
mungen mit freler Oberflache (Mittellungen aus dem Institut fir
Aerodynamik, Eidgenossische Technische Hochschule, herausgegeben
von Prof. Dr. J. Ackeret, Nr. 7, Zirich 1938.).

7. Heybey, W., Das Prandtl-Busemannsche Niherungsverfahren zur zeich-
nerischen Verfolgung ebener Uberschalletr&mungen. HVP Archiv Nr.
66/31 and Nr. 66/32, 1940.

8. Reale Accademia d' Italia, Fondazione Allessandro Volta Atti del
Convegni 5, Convegno di Science Fieiche, Matematiche e Naturali
30 Sett. 6 Ott. 1935. Tema: Le alte velocity in aviazione
Rcma 1936: Busemann, A.: Aerodynamischer Auftrieb bel
Uberschallgeschwindigkeiten pp. 328-368 (abgedruckt in Lufo Bd.
12, pp. 210-220, 1935.)-

9. Busemann, A., and Walchner, 0.: Profileigenschaften bel Uberschall-
geschwindigkeit. Forschg. Ing.-Wes. Bd. 4, pp. 87-92, 1933-





Translation by Mary L. Mahler
National Advisory Committee
for Aeronautics.


NACA-Langley 11-28-49 900























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