Kaplan, Carl

English

NASA Technical Reports Server

NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 
TECHNICAL NOTE 
No. 1527 
ON SIMILARITY RULES FOR TRANSONIC FLOWS 
By Carl  Kaplan 
Langley Memorial Aeronautical Laboratory 
Langley Field, Va. 
Washing ton 
January 1948 
FEB 2 1948 BUSINESS, SCIENCE & TECHNOLOGY uEPT. 
https://ntrs.nasa.gov/search.jsp?R=19930082226 2020-06-17T20:49:48+00:00Z
By C a r l  &plan 
A method used by Tsien t o  derive s i m i w t y  rules f o r  hypersonic 
flaws is ut i l ized  t o  derive von ~ & ' s  similarity rules f o r  transonic 
flows. A s l ight  generalization is introduced by the inclusion of y ,  the 
r a t i o  of specific heats, as a parameter. A t  the lower l i m i t  of the 
transonic region of flow the theory yields a fonrmla f o r  the c r i t i c a l  
stream Mach nwnbers of a given family of symmstrlcal profiles.  It is 
f'urther shown tha t  this formula can a lso  be obtained by means of the 
Prandtl-Glauert mall-perturbation method. Investigation of the behavior 
of the s imilari ty parameter in the region where the thickness coefficient 
approaches eem and the c r i t i c a l  stream Mach number approaches unity 
shows tha t  it possesses a limiting value characteristio of the prescribed 
family of shape.  
The rigorous solution of the subsonic f l a w  of a compressible f lu id  
past a prescribed closed body thus far has proved t o  be of insurmountable 
diff icul ty,  As a consequence of t h i s  diff icul ty the emphasis has been 
placed on the establishment of a, correspandence between the flow past a 
given body i n  an incompressible f lu id  and the sams body i n  a compressible 
fluid. Among the beat known results  of this node of attack are the 
Prandt1-Glauer-t rule and the von Id rdk-~s ien  velocity o r  pressure correc- 
t ion factor  - both based on same fonn of l inearization of the fundamental 
nonlinear flow equations. None of the methods based on the linearization 
of the flow eqyations, however, can yield correct resul ts  i n  the transonic 
w e  where the flow is partly subsonic and partly supersonic. For this 
case a certain amount of the feature of nonlinearity of the f l a w  equations 
must be retained i n  order t o  obtain usef'ul and nontrivial results.  I n  
the present paper a detailed derivation is given of the transonic similarity 
rules recently given by von J S & d  (refemnce 1). 
mTNDrnAL EQUATIONS 
I n  plane steady flow the equation governing the flow of a nonviscous 
compressible f lu id  can be written i n  the form 
NACA TN No. 1527 
I n  the derivation of this equation the pressure is  assumed t o  be a 
function of the density only. If  further,  t he  motion is  i r rotat ional ;  
then 
Here u and v are,  respectively, the camponent veloci t ies  along the 
Cartesian x- and y-axes a,nd c is  the loca l  velocity of sound given by 
where co is  the velocity of sound at a stagnation point u = 0, v = 0 
and y i s  the r a t i o  of specif ic  heats at constant pressure and constant 
volume. 
Equation (1) is  f a r  too c q l i c a t e d  t o  afford an insight  in to  the 
properties of potent ia l  fluw i n  the neighborhood of Mach nmber unity, 
The discussion is, therefore, r e s t r i c t ed  t o  the flaw past a th in  profi le .  
Thus, at first v is  asswned t o  be s m a l l  i n  camparison with the sound 
velocity co. Equation (1) is then simplified t o  
By the introduction of the sound v e l m i t y  f o r  which the loca l  f lu id  
and sound veloci t ies  are equal 
and of the maximum poasible f l u i d  velocity 
NACA TN No.  1527 
equation ( 4) bec ones 
It is desirable t o  simplify t h i s  equation s t i l l  further  but yet  t o  re ta in  
those features which yield qontr ivial  and useful results.  Thus, 
i f  it is assumed that  u is of the order of the c r i t i c a l  velocity c* 
and i f  only t e r m  up t o  the order 1- am retained, then equation (7) 
c* 
can be written 8,s 
Thus far the irro.tationality oondi tion, equation ( 2), has not been 
used, I f ,  now, the undisturbed stream past a slender body is of velocity 
U s l ight ly  diTferent from the velocity of sound and i n  the direction of 
the positive x-axis, then, accardfng t o  equation (2) and the assumptions 
leading t o  equation (8), a velocity potential. 4 can be introduced with 
and ( 1  - W)p i e  the distwbance-velocity potential.  where W = - 
c* 
Then 
m d  equation (8) beccanss 
Equation (ll) is a nonlinear simplified f o m  of the fundamental d i f ferent ia l  
equation (1) and has been treated recently by von K&&n i n  connection with 
4 NACA TN No. 1527 
similarity rules i n  twodimensional t m o n i c  flaw (reference 1). 
Equation (ll), when expressed i n  hodograph variables, is of the type 
treated by the Italianniathematician F. Triccsni some years ago (reference 2). 
Recently, Tsien (reference 3) derived slmllaritg rules for  hypersonic 
flows where the f lu id  velocity is much larger than the velocity of sound. 
In the present paper the same procedure i s  employed t o  derive similarity 
rules for  transonic flows where the f lu id  velocity is very nearly that of 
sound. 
According t o  the assumptions leading t o  equation ( l l ) ,  it is implied 
that  the solid body is thin, possesses no stacgnation points, and the 
velocity of the f lu id  is everythere i n  the neighborhood of the local  
velocity of sound. Now, suppose the profile of the obstacle t o  be 
symmetrical with respect t o  both the x- and y-euces and t o  possess cusps 
a t  both the leading and the t ra i l ing  edges. Such profiles with uniform 
flow i n  the direction of the long ax i s -of - -~3mtry  x f u l f i l l  the 
assumptions leading to  equation (11). The flows past these profiles are 
said t o  be similar i f  the equation of motion (11) a d  the boundam 
conditions can be expressed in nondimensional variables i n  such a way 
that  only a single constant factor is  involved. Thus, i f  2a is  the chord 
and To is  the maximum thickness of the body,. then the following 
nondimensi onal variables are introduced : 
where t = and m and n are exponents yet to  be detemlned. It ~ E J  
a 
clear  that the nmdimensional quantity involved is the thickness coeffi- 
cient t since th i s  quantity determine8 the magnitude of the disturbance 
velocities. The exponents m 83Ld n are t o  be determined in  such a way 
that  the same constant factor appears i n  both the equation of motion and 
the boundary conditions. 
The appropriate nondimensional form for  the velocity potential cp is 
By substitution from equations (12) and (13), the equation of motion (11) 
becames 
NACA TN No .  1527 5 
The bontihqy conditions at  in f in i ty  require tha t  the flow velocity 
be U. Hence, fram the f i r s t  of equations ( lo )>  
!L'he boundary condition a t  the surface of the slender body is 
(1 - w)& = c*@t ( a t  y = 0, 4 L r s a) 
where g(t) describes the  d is t r ibut ion  of slope along the surface of the 
body. 
By means of equations (12) and (13), the boundp.ry conditions can be 
wr i t ten  as: 
-7 
( a t  
"la = g ( ~ )  ( a t  7 = 0, (y + l ) - ( l -  W)t-  -1 5 S 5 1)) 
all 
A comparison of the d i f f e ren t i a l  equation (14) and the boundary conditions, 
equations ( ~ 5 ) ~  shows tha t  a s ingle  pallameter is  involved i f  
t h a t  is, 
6 NACA TN No. 1527 
The undisturbed--stream Mach number = can be introduced i n  
C, 
the following way: 
The general relation between Mw and M, is 
& =  I 1- 1 - 2  - ( l ) ]  1112 
Then, i f  powers of ( 1  - W) higher than the f i r a t  are neglected, 
Theref ore 
The results obtained thus far are such t h a t  bx m e a n s  of the substitution 
e quat i om 
the di f ferent ia l  equation for  f ( ~ ,  T) md the boundary conditions become: 
NACA TN NO. 1527 
and 
b (a t  OD) 
The meaning of the similarity rule implied in tb definition of the 
parameter K is the following: 
If a series of bodies having the same distr3.bu.tion function g(f, ) 
f o r  the slope but different thickness rat ios t are placed in fluws of 
different undisturbed-stream Mach rruniberrs & and different values of y ,  
such that  the pssamster K = 1-k renuxlm c m t a n t ,  then the 
[ ( T  + l)tI2'3 
flow patterns are stmilax i n  the amse that the s- f b c t i o n  f(5, q) 
describes the flare. 
H E S m  DHiNED FROM THE sIMI:LRHTn RUI;E 
Pressure Coefficient 
I n  the case of a uniform f l a w  past a fixed b 
coefficient ia defined aa 
where Pa Poo are, respectively, the pressure and density i n  the 
undisturbed stream and the s t a t i c  pressure p i n  the f lu id  is  given by 
8 
Then 
NACA TN N o .  1527 
By means of equations (10) and (18), i f  powers of (1 - M,,,,) higher than 
the first are neglected, the following result  is obtained 
where P((, q; K) depends on the form of the so lu t im f(5, q) fo r  
the particular family of profiles treated. 
Lift Coeff icient  
The l i f t  2 of the body is given by 
By a similar prwedure, as i n  the derivation of equation (23), the l i f t  
coefficient is given by 
where 
NICA TN No. 1527 
and where f o r  an extremely t h i n  straighb-Une prof i le  the thickness 
coeff ic ient  t has been replaced by the tmgle of a t tack a, 
D r a g  Coefficient 
The pressure drag d of the body is given by the following expression: 
Hence the drag coeff icient  i s  given by 
where 
ADDITIONAL C O E I D ~ o m  
The resu l t s  derived i n  the present paper apply t o  two-dimensional 
n s w a 9 l f c  flm past t h i n  shapes, Such flows hagre been calculated 
f o r  a famlly of symmetrical shapes with cusped Peeding and t r a i l i n @ ;  
edges (reference 4) and f o r  a family of e l l i p t i c  cylinders (reference 5 ) ,  
These ca lcuh t ions  a re  valid. at  l e a s t  up t o  the  c r i t i c a l  stream WB1 
number &re The c r i t i c a l  Mach number mtq be considered fram two points 
of view. F i r s t ,  it m y  be coxlsidered t o  denote the lowe r l imi t  of 
a mixed subsonic-supersonic flaw, tha t  is, where the imbedded supersonic 
region i s  simply the point of imum f lu id  velocity at the surface of 
the solid.  Frau t h i s  point of view, according t o  equation (17), a 
c r i t i c a l  value of the s imilar i ty  parameter K can be defined. Thus 
This  equation can a lso  be wri t ten i n  a form tha t  yields  the c r i t i c a l  stream 
Mach nmber f o r  a given family of shapes; t ha t  is, 
NACA TN N o .  1527 
Second, the critical-stream Mach nwrber m q y  be considered t o  denote the 
upper llmlt of the purely subsonic range of speeds, This point of view 
suggests a derivation of equation (27) by means of the Prandt1-Glauer-t 
small-perturbation method (reference 6). The procedure is as follows; 
The relation between the local  and the undisturbed--stream Mach 
number, within the approximation of the 8m&ll-disturbance theory, is  
given by 
where u V s  the disturbance velocity and U is the undisturbed-stream 
velocity. 
1 
By definition M = 1 for  M, = & and, tiime C p 9 ~ -  = -& 
U 
approximately, equation (28) becosnes 
The relation between C p , ~ c r  and the pressure coefficient Cp,o 
of the incampressible f lu id  is given by the P~and.tl-Clauert rule 
Hence equatim (29) beccanes 
As  seen ern equation (29 ) ,  1 - i s  of the f i r s t  order i n  the small 
perturbation and, accordingly, equation (30) includes tenus of higher 
order, This fact  can be seen by rewriting eq~a t ion  (30) in  the fo l luwix  
form: 
NACA TN No. 1527 11 
Then, t o  within the lowest order i n  the amabl perturbatian, 
or, approximately, 
It is quite easy t o  show the connection between equation (31) and 
equation (27). Thus, fo r  an inca~n.pressible fluid, the pressure coefficient 
i a  given by 
where q i s  the magnitude of the f lu id  velocity a t  an;g point i n  the 
f i e ld  of flow. 
I n  the case of the family of shapes of reference 4, the 
velocity at the surface is given by 
Hence 
Cp,O = -3t + * e * 
and equation ( 31) bec ames 
12 NACA TN No. 1527 
I n  the case of the family of e l l ip t i c  cyUndem of refemme 5,  
the velocity a t  t h e  surface is  given by 
cp,o = -2t + * * * 
and equation (31) becomes 
An examination of equations (33) and (34) suggests tha t  ;TEr 
possesses a limiting value; that is, 
It is  noteworthy that  the value of depends on tb famAl;y of 
shapes, although i n  the ULmi -sO W p r o f i l e  i a z e v e ~  cage i e  a 
s tmight-line segaent , The l l d a l  values of sh- i n  
tiorsfl  (33) an& (34) represent, hawever, only the sffeot  of t'km 
dtl-Glauert or  firs+-order tern  i n  the p m ~ e r i e e  d e v e l o p n t  i n  t 
of ths  loci ty at the solid surface, It is rather surprfsinl?; 
that t rder t e r m  nvo em of t also 
contribute t o  the value of i s  simply t o  m g b e  
the velocity by the c r i t i c a l  speed; that  is, 
NACA TN No. 1527 13 
where al, a2, a3, . . . depend on the @ven I"dQ of shapes and 
involve only y and 
m e n  as suggested by equation (26), when 1 - &,2 i s  replaced by 
2&, [(7 + l ) t ]  2/3, equation (36) is  ident ical ly  sa t i s f i ed  f o r  t + 0, 
wTth each addi t ional  highe-rder t e r n  contributing t o  the value of Kc r e  
I n  t h i s  fashion there are obtained, successively, l inear ,  quadratic, 
cubic, and h i g h e ~ r d e r  equations f o r  the determination of ("o~.)lim* 
The foregoing considerations have been applied t o  the family of 
symmetrical shapes of reference 4 and t o  the family of e l l i p t i c  cylinders 
of reference 5. The values of Gr f o r  values of t d i f fe rent  fram 
zero were obtained fram equation (26) with the required values of kr 
detexm3ned by means of the theore t ica l  r e su l t s  of references 4 and 5.  
The l imit ing values of were obtained fram equations corresponding 
t o  equation (36). Tables I and I1 ahm the re su l t s  of these c a l c u h t i o m  
f o r  air  and f o r  Freon-12. Note the approach t o  the l imit ing values of 
hr as t approaches zero and B&r approaches unity. The f i m t  column 
of values of qr shows the e f f ec t  of the f i r s t -order  o r  Prandtl-Glauert 
term; the second and third colunms show, mspectively, the e f fec ts  of the 
second-order and thfrd-order terms i n  the thickness coeff ic ient  t, The 
successive values of qr f o r  the various values of the thickness 
coeff ic ient  t indicate good convergence, 
-ley Mc:morid $Le i c a l  Laboratory 
National advfsory CcBmnittge f o r  Aeronautics 
Ungley Field, Va,, December 9, 1947 
14' NACA TN No . 1527 
1, von ~&&n; Theodore : Supersonic Aerodynarmics - Binciples  and 
Applications. Jm, Aero. Sci., vol. 14, no. 7, July 1947, p. 396. 
2, TricaePI, F, : Sulle equazioni l inear i  alle derivate parzial i  d l  2' 
ordine d i  t ip0  misto. A t t i  del la  Re Accad. N a z -  dei Lincei, Mem. 
deUa C1,  Sci. Fie, Mat. e Hat., s e re  5, vole 14, 1923, p. 134. 
3. Tsien, Hsue-shen: S M l a r i t y  Laws of Eypersonic Flows. Jour. Math. 
and Phys., vole XXV, no. 3, Oct. 1946, pp. 247-251. 
4. Kaplan, Carl: The F l m  of a Compressible Fluid past a Curved Surface. 
NACA Rep. No. 768, 1943. 
3 .  Kaplan, C a r l :  Effect of Crsnpressibility at High Subsonic Velocities 
on the Moment Acting on an Ell ip t ic  Cylinder, NACA TN No. 1218, 
19 '~7. 
6, Lie~pnann, Hans Wolfgang, and Puckett, Allen E.: Introduction t o  
Aerodynamics of a Compressible Fluid, John Wiley & Sons, Inc., 
1947, pp. 142-143. 
NACA TN No .  1527 15 
TABU I 
VAUUES OF Kc, FOR THE FAMILY OF S=CAL SRAPES OF HEFEXE3CE 4 
[For air, y = 1,4; for F"reon-!, y = 10136] 
ICIATIONU; msom 
C 0 1 . I M T m  FOR AEROPIAUTICS 
. 005 
.010 
.020 
,040 
.060 
. ot20 
. 100 
967 
* 947 
,918 
.872 
837 
.807 
.781 
k.eoD-12 
.963 
,942 
.go9 
.859 
,818 
.784 
.753 
0.738 
.735 
-730 
.725 
.716 
.706 
.698 
.691 
-682 
.962 
.940 
.go6 
.854 
.812 
.774 
8743 
o 
. 002 
.005 
.o10 
.020 
-040 
.060 
.080 
.loo 
1.000 
. 983 
.969 
951 
.924 
.881 
.848 
.819 
*794 
.638 
.632 
.624 
. 609 
. 595 
.581 
.568 
1.000 
.981 
.966 
.946 
.915 
-867 
.830 
0797 
0769 
0.655 
.6k7 
-642 
.636 
a 627 
.612 
.600 
.588 
0 576 
1.000 
.980 
.964 
.944 
.912 
.863 
.823 
.787 
0756 
.To5 
.700 
0689 
.675 
.664 
.650 
.639 
0,717 
.713 
.709 
.704 
.694 
2 
.670 
.658 
.647 
.726 
,719 
.709 
-696 
.685 
.678 
.665 
TABIE I1 
V A I E B  OF qr FOR TRE IFMU OF EIJ3PTIC CYUMIERS OF HE-CE 5 
or air, 7 = 1,4; for Ere 
ONAL MWlSOII11 
FOR AEROMAUTICS 


On Similarity Rules for Transonic Flows

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server