A method used by Tsien to derive similarity rules for hypersonic flows is utilized to derive Von Karman's similarity rules for transonic flows. At the lower limit of the transonic region of flow the theory yields a formula for the critical stream Mach numbers of a given family of symmetrical profiles. It is further shown that this formula can also be obtained by means of the Prandtl-Glauert small-perturbation method. Investigation of the behavior of the similarity parameter in the region where the thickness coefficient approaches zero and the critical stream Mach number approaches unity shows that it possesses a limiting value characteristic of the prescribed family of shapes

Kaplan, Carl

English

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ON
REPORT No. 894
SIMILARITY RULES FOR TRANSONIC FLOWS
BY CARL KAPLAN
SUMMARY
xl method used by Tsien to derire m“milan”tyrules for hyper-
sonic $OWS ia utilized to derice J-on K4rmdn’s similam”iy rules
for transonic $ow8. A 8[ight generalization i8 introduced by
the inclution of y, the ratio of 8pecijic heai%, a8 a parameter.
At the lower limit of the tran80nic region of jlow the theory
yie[d8 a formula for the critical 8tream Mach numbers of a
@“t’enfamdy of aym?netn”cal profi[e8. It i8 jurth.er 8hown. that
thi8 formula can also b8 obtained by mean8 of the PrandtL
Qlauert small-perturbation method. ke8tigation of the be-
harior of the similarity parameter in the region where the
thickness eoejitient approaches zero and the cm”tical stream
Mach number approaehe8 unity 8hows that it po8sesse8 a
limiting value’ characteristic of the prescribed family of shape~.
INTRODUCTION
The rigorous solution of the subsonic flow of a compressible
fluid past a prescribed closed body thus far has proved to
be of insurmountab~e dif%culty. As a consequence of b
difficulty the emphasis has been pIaced on the estabhhraent
of a correspondence between the flow past a given body in an
incompressible fluid and the same body in a compressible
fluid. Among the best known results of this mode of attack
are the PrandtLGIauert rule ~d the Von Kdrm&n-Tsien
velocity or pressure correction factor-both based on some
form of Linearizationof the fundamental nordinear flow equa-
tions. None of tie methods based on the linearization of the
flow equations, however, can yield correct results in the
t.ransonicrange where the flow is partly subsonic and partly
supersonic. For this case a certain amount of the feature of
nonlinearity of the flow equations must be retained in order
to obtain useful and nontrivial results. In the present paper
a detailed derivation is given of the transonic similarity
ruks recently given by Von K6rm&n (reference I).
FUNDABiENTAL EQUATIONS
In plane steady flow the equation governing the flow of a
nonviscous compressible fluid can be written in the form
(c’– d) ~ -U’(%+a+(c’+-%=’“)
In the derivation of this equation the pressure is assumed to
be rLfunction of the density onIy. If, further, the motion is
irrotational, then
btdl)o——— =
dy ax (2)
Here u and o are, respectively, the component velocities along
thq Cartesitm x- and y-axes and c is the local velocity of
sound given by
‘y-l ~&=@2-- 2 (?.4+tl’) (3)
where COis the velocity of sound at a stagnation point
u=O, a=O and -y is the ratio of specfic heats at constant
pressure and constant vokne.
Equation (1) is far too complicated to afford an insight into
the properties of potential flow in the neighborhood of
Ifach number unity. The discussion is, therefore, restricted
to the flow past a thin proEIe. Thus, at first Dis assumed
to be small in comparison with the sound velocity CO. Equa-
tion (1) is then simplified to
By the introduction of the sound velocity for which the
local fluid and sound velocities are equal
d–2C*= _-y+ 1 co (5)
and of the maximum possible fluid velocity
r!zna.=~& f% (6) ‘“-
equation (4) becomes
(7)
It is desirable to_simplify thk equation still further but yet
to retain those features which yield nontrivial and useful
results. Thus, if it is assumed that u is of the order of the
critical velocity c* and if onIy terms up to the order 1—~
are retained, then equation (7) can be written as
–(7+1) (u–c”) ~+c” *=O (8)
Thus far the irrotationality condition, equation (2), has .
not been used. If, now-, the undisturbed stream past a
slender body is of velocity U slightly dif%erent from the
83
https://ntrs.nasa.gov/search.jsp?R=19930091963 2020-06-17T03:26:35+00:00Z
S4 REPORTZYQ.89 4—NATIONALAIWISORYICOMMITTEEFOR AERON~UTIC~
velocity of sound and in the direction of tho positive z-axis;
then, according to equation (2) and the awunptions leadiig
to equation (8), a velocity potential@ can be introduced with
@=”c*2!+(l-M*)p (9)
where M*=: and (1—M*) q is the disturbance-velocity
potential.
Then
ancl equation (/3) becomes ~:: . ... . . -:: ‘_:_
– (7+1)(1–M*) * g.+c” ~po (11)
Equation (11) is a nonlinear simplified forni of the fundamen-
tal differential equation (1) and has been treated recently
by Von Kdrmfi in commction with similarity rules in two-
dimensional trammnic flow (reference 1). Equation .(11),
when expressed in hodograph variablw, is of the type treatccl
by the Italian mathematician F.. .~c.omi somQ years ago
(referencti). .-.
DERIVATION OF SIMILARITY CONDITIONS
Recently, Tsien (reference’3) derived similarity rules for
hypersonic flows where the fluid velocity is much larger than
the velocity of sound, In””thepresent paper the same proce-
dure is employed to derive similarity rules for transonic flows
where the fluid velocity is very nearly that of sound.
According to the assumptions leading to equation (11), it
is implied that the solid body is thin and possessa no stag-
nation points smd that the velocity of the fluid is everywhere
in the neighborhood of the local velocity of .scmnd. NTOW,
suppose the. profile of the obstacle ti. be symmetrical with
respect to both the x- and y-axes and to possess cusps at both
the leading and the trailing edges. Such profiles with uni-
form flow in the direction of the long axis-of-symmetry x
fulfdl the assumptions leading to equation (11). The flows
past these profiles are said to be similar if the equation of
motion (11) and the boundary conditions can be expressed
in nondimensional variabhs in such a way that only a single
constant factor is involved. Thus, if 2a is the chord and 2b
is the maximum thickness of the body, then the following
nondimemional variables are introduced:
(12)
where t=: and m and n are exponents yet to be determined.
It is clear that the nondhnensiomd quantity invokccl k the
thicknepscoefficient t aimx.this quantity dctw’mines the mag-
nitude of the disturbance mlocitiw. The exponents m and n
are to be determined in such a way that the same constant
factor appears in both the equation of motion and the
boundary conditions.
The appropriate nondhncnsiona.1 form for the velocity
potentifd P is
~=ac~(t,~) (13)
By substitution from equations (12) and (13), k cquation
of motion (1I) becomes
–(.y+l)’”+’(l-ilf”) tfi &y+:$=o (14)
The hounclary conditions at infinity require thut lhr flow
vglocity be Z% Hence, from the first of cquaf ions (1O),
*_ C*
3X
}
(at CO)
.. k... . ~–
The bcnmdary condition at the surfaco of tlw sicnder body is
()“/l”-hJ*)$=c*g : t (at y=O, —aSzSa)
where g
()
~ describes the distribution of ~sIopc along tho
surfacg of the body.
By rncarik of equ~tions (12) and (]3), tho boundary c{
ditio~_can be written as:,
}
QJ=-l , ,
--(at co)
< ~.f_ (-J
. ?rq–
(1;)
A comparison of the differential equation (14) and the bound-
ary conditions, equations (15), shows that a single parnrnctm
is invoIved if
2n =–(n+l) or 1“n=-– 3
2m+l=-7n or m=-; --- -“
that is,
(7+ I)’m(l –ilf*)t-’fi=2K
ON SIMILARITY RUZES FOR TR.4W301WCFLOWS 85
The undisturbed+tream llach number Mm=! can be
.
introduced in the folIowing way:
The general reIation between M* and Mm is
or
{.
1– (1–.M*)[2- (1–3f*)]
}
1/2
M.=
1+% (1–M”)[2-(1 –M”}]
Then, if powers of 1–M” higher than the first are neglected,
1—AI*= ~+ (1–M.}+ . . .
Therefore
(1–MJ[(7+l)t]-’~=K (17)
The results obtained thus far are such that by means of the
substitution equations
y=a[(7+l)t]-u3q
t
(18)
P= ac*’(& T)
K= (1 –illa,)[(T+ l)t]-’@J
the differential equation for f(& q) and the boundfi~ con-
(19)
(20)
The meaning of the similarity rule irnpIied in the defi-
nition of the parameter K is the foUowing:
If a series of bodies having the same distribution function
g(~) for the slope but dfierent thickness ratios t are placed
in flows of different. undisturbed-stream Mach numbers
Me and different vtdues of 7, such that the parameter
K=[(;;:;mremains constant, then the flow pat t ems are
similar in the sense that the same function f (,$,q) describes
the flows.
RESULTS DERIVED FROM THE SIMILARITY RULE
PRESSURECOEFFICIENT
In the case of a uniform flow past rLfixecl boundary, the
pressure coe%kient is defined as
-. ——
where pm and pmare, respectively, the pressure and density
in the undisturbed stream and the static pressure p in the
fluid is given by
p=pm
[ +’=’(%+lfi ’21).~_7—1
Then
{ [ -+’.2(%913} ’22)cpjM.=7&3–1+ 1
By means of equations (10) and (18), if powem of 1–.llm
higher than the first are neglected, the foUowing result is
obtained:
c,,..=–+(l–w) (l+%)
or
p!a
cPJJf. = (7+ ~)1/3m ~ ;m (23)
where ~(& q; K) depends on the form of the solution f (~, q)
for the particular family of profiles treated.
LIFT COEFFICIENT .
The lift 1 of the body is given by
$1= (p)q.(dz
By a similar procedure, as in the derivation of equation (23),
the lift coefficient is given by
(24)
where
L(K)=J:lP(E,o;K)4
and where for an extremely thin straight-line profile the
thickness coefficient t has been replaced by the mgle of
attack a.
DRAGCOEFFICIENT
The pressure drag d of the body is givm by the following
expression:
~=2J:.w14Yx
.=-.
-..
86 . REPORTNO. 89 4-NATIONAL ADVISORYCO_.TTEE FO.RAERONAUTICS —
Hence the drag coefficient is given by
(25)
where
Um =2J”1 !I(:)W, 0; m @
ADDITIONAL CONSIDERATIONS
The results derived in the present paper apply to two-
dimensional near-sonic flows past thin shapes. .Such flo~vs
have been calculated for. a family of symmetrical shapes
with cusped leading and traiLingedgm (r~erence 4) and for
a family of elliptic cylinders (referenm 5). These calcula-
tions are valid at least up to the critical stream Mach rmm-
ber h!.,. The critical Mach number may be considered
from two points of view. First, it may be considered. to
denote the lower limit of a mixed subsonic-supersonic flow,
that is, where. the..imbedded supersoriic re~on is simply the
point of muinmm fluid velocity at the surfwe of the solid.
From this point of view, according to equation (17), a
critical value of the similarity parameter K can be defied.
Thus
(26)
This equation can also be written in a form that yields the
critical stream Mach number for a given family of shapes;
that is,
Me,= l–K,, [(7+ l)t]~a (27)
Second, the critical-stream hfa.ch number may be consid-
ered to denoti the upper limit of the purely subsonic range
of speeds. This point of view suggestsa derivation of equa-
tion (27) by means of the Prandtl-Glauert smalI-perturbation
method (reference 6). Th~ procedure is as follows:
The relation between the local and the undisturbed-stream
Mach number, within the approximation of the small-
disturbance theory, is given by
‘=’’-’[1+2$(1+=’’ 2)1’28)
where u’ is tho disturbance velocity and U is the undisturbcd-
stream velocity,
By cle6iition, M= 1 for M. =M,, and, $nce 0,, ~~ = -2$-
approximately, equation (28) becomw
The relation between CP,~,, and the pressure coefficient CP,~
of the incompressible fluid is given by the Prandtl-Glauert
rule
c,, ,
c’”‘~=dl–lwc:” ““““ x. --
Hence equation (29) becomes
(1–M,?)”2
‘ =R=’rC”o (30)
As seen from equation (29), 1—M,? is of the first order in the
small perturbation u’/ U and, accordingly, equation (30}
includes”terms .of higher order. This fact can be seen by
rewriting equation (30) in the following form:
(1–mf,?)’i’
=– c,,,
7+1 ~_T—1
~ [1–(1–M,;)] ~ (1–M.’)
Then, to within the lowest order in the small pwturbation,
( ‘+1Cp,o 2/8~—Jf*r2= —— 2 )
or, approximately,
(31)
It is qui~ easy to show the connection between equation (31)
and equation (27). Thus, for an incompressible fluidl tie
pressure coefficient is given by
0,,0=1–$ - (32)
where Qis the magnitude of the fluid velocity at any point in
the field of flow.
.In thecase of the family of shapes of refcrcncc 4, the maxi-
mum v?locity at the surface is given by
+=l+; t+ . . .
Hence --
.-
c,,o=–3t+ . . . -
ant{ equation (31) becomes
—.
()ill.,= 1– & ‘“ [(-y+ 1) ~]zfl (33)
In thti case of the family of elliptic cylindcm of refer-
ence 5, the maximum velocity at. the surface is given by
Hence
and equation (31)
—.
cp,o=–2t+ , . . -
becomes
M.r= 1–; [(7+ wlgm (34)
An emwninationof equations (33) and (34) stiggest.sthat
K., possesses a limiting value; that is,
(35)
ON S~IHJARITY RULES FOR TRANSO~C FLOWS 87-
lt is noteworthy that the value of (Km) ,k depends on the
family of shapes, although in the limit t~ the profile in
every case ia a straight-line segment. The numerical VdUH
of (ZGJ~i~ahown in equations (33) and (34) repremut, how-
ever, only the effect of the PrandtI-G1auert or first-order
term in the power-series development in t of the maximum
velocity at the solid surface. It is rather surprising that the
higher-order terms imrohing the higher powers of t also
contribute to the value of (Km) ~f~. The procedure is simply
to replace the maximum velocity ’by the critical apeed; that is,
(3)..=(5).=(-.:~
2 cr
or
where al, az, aa, . . . depend on the given family of shapes and
involve only 7 and 31,,.
Then as suggested by equation (26), vvhen 1—Mmzis re-
placed by 2&[(7+ l)t]V8, equation (36) is identically satis-
fied for t~O, with each additional higher-order term cort-
tributing to tie value of Km. In this fashion there are
obtained, successively, linear, quadratic, cubic, and higher-
order equations for the determination of (Kti) ~,m.
The foregoing consideratio.m have been applied to the
family of symmetrical shapes of reference 4 and to the famiIy
of elliptic cylinders of reference 5. The values of KW for
TABLE I
VALUES OF ~H FOR THE FAMILY OF S“YMMETRICAL
SHAPESOF REFERENCE4
[Far ak, -pl.q far Frem-1~ T-1.12Sl
t MC* I K.
t I AI)wxfmath)n I Ap,rd,na,,on
1
I
Fkst Ssrmd l’hfrd Fkst Sacond Thkd
AIr
I I I I 1 I
o
:K
.Cao
.020
.040
.Im
.W
. m
L~
.$3s
.92s
.961
.924
.Ssl
.s4.s
.SIo
.7e4
Frsml-12
l.al&I
.CJ2a
:%
.Stt7
.s0
:E
0..
; :6
.S12
.Km
.Ina
. 6i6
0:;l
.ma
.704
:%
:%!
.647
O.iw
.?36
.782
.72s
.716
.m
%!
.aw
values of t different from zero -were obtained from equa-
tion (26) with the required values of Mmdetermined by means
of the theoretical results of references 4 and 5. The limiting
values of& were obtained from equations corresponding to
equation (36). Tables I and II show the results of these
calculations for air and for Freon-12. hTote the approach
to the limiting vahes of K. as t approach= zero and Mm
approaches unity. The &ratcolumn of values of K. shows
the effect of the first-order or Prandtl-Glauert term; the
second and third columns show, respectively, the effech of
the second-order and third-order terms in the thickness
coefficient t. The successive vahes of Km for the various
values of the thickness coefficient tindicate good convergence.
LANGLEY MEMORML AERONAUTICAL LABORATORY,
lVATIONAL ADVCISORYCOMMITTEE FOR AERONAUTICS,
LANGLEY FIELD, VA., December 9, 1947.
REFERENCES
1. Von K6rm6n, Theodore: Supersonic Aerodynamics-Principles and
Applications. Jour. Aero. Sci., vol. 14, no. 7, JuIy 1947, p. 396.
2. Tricorr& F.: SuUe equazioni lineari alle derivate piraiali di 2°
ordine di tipo misto. Atti dells R. Accad. Naz. dei Lincei, ~fem.
dells CL Sci. Fis., Mat. e Nat., ser. 5, voi. 14, 19%, p. 13-I. “””““
3. Tsien, Hsue-shen: Similarity Laws of Hype~onia Flows. Jour.
Math and Phys., VO1.XXV, no. 3, Oct. 1946, pp. 247-251.
4. Kaplan, Carl: The Flow of a Compressilie Fluid psst a Curwd
Surface. NACA Rep. No. 768, 1943.
5. Kaplan, Carl: Effect of (20mprassibiIity at High Subsonic Veloclties
on the Moment Acting on an Elliptic Cylinder. NACA TN No.
1218, 1947.
6. Liepmann, Hans Wolfgang, and Pucket\, Allen E.: Introduction to -
Aerodynamics of a Compressiblsi Fluid. John ‘Wiley & Sons, Inc.;
1947, pp. 142-148.
TABLE II
VALUES OF K., FOR THE FAMILY OF ELLIPTIC CYLINDERS -
OF REFERENCE 5
[Frr air, ~_L4 for Frcix-~ pl.13Sl
=
Atr
o L~ L @lo 1,m O.ml? 0.627 O.m
.(02 .flss .986 .494 .bm .650
.Q74 .972 :%?
:E
.420 .M4
.Om
.95s .4s7
:%
:%
:% .020
.b27
.4s1
.Oio .fml .s94
.616 .s26
.470 .Mi5
:%? :g .4!3’
.516
.W .s73 .M3
.Om .s4s .E37 .s22
.m7
.4’W .mo
.m .s?7 .S13 .Sw .4*7 .4s4 .404
Ra.m-12
I.og
.
. $i6
.002
.94
.0(8
.Sm
$%
1:&o
.974
.969
:%!
.s72
.S47
.Sm
l..
.07a
.9Ea
:E%
.S71
.845
.s22
Uxc
.407
.40s
.491
.4s4
.474
.4&3
:%
0.656
.6s2
:U
%
.613
.Eo6
.4m
, 1
,.


On similarity rules for transonic flows

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