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New Compressibility Correction for Two-
Dimensional Subsonic Flow 
E. V . Laitone 
Associate Professor, University of California at Berkeley 
February 26, 1951 
S U M M A R Y 
By using the local Mach Number rather than the free-stream Mach Num-
ber in the ordinary Prandtl-Glauert correction for subsonic two-dimensional 
isentropic gas flow, a simple new relation is obtained between the com-
pressible and the incompressible surface pressure coefficient. The new 
relation includes the effect of the specific heat ratio and gives a better agree-
ment with experimental data than does the von Karman-Tsien relation. 
I N T R O D U C T I O N 
TH E P R A N D T L - G L A U E R T C O M P R E S S I B I L I T Y C O R R E C T I O N for t h e local surface p re s su re coefficient is genera l ly w r i t t e n (see 
reference 1, p a g e 140) a s 
cv = cm/Vi - M ^ (i) 
E q . (1) is de r ived from t h e l inear ized p o t e n t i a l e q u a t i o n b y neg-
lec t ing all t h e s q u a r e or h igher o rde r t e r m s of t h e p e r t u r b a t i o n 
p o t e n t i a l (see reference 1, p a g e 125). T h e surface p res su re co-
efficient Cp is t h e n of t h e s a m e o rde r a s t h e first power of t h e 
p e r t u r b a t i o n p o t e n t i a l (see reference 1, p a g e 126). 
I n t h i s form, E q . (1) is r igorous ly app l i cab le on ly t o a v e r y 
s lender b o d y h a v i n g a cusped nose w i t h n o s t a g n a t i o n p o i n t . 
W h e n app l i ed t o even re l a t ive ly t h i n airfoils, i t gives a reason-
able p r ed i c t i on on ly for t h e p o r t i o n s n e a r f ree-s t ream pressu re 
a n d on ly for low va lues of t h e f ree-s t ream M a c h N u m b e r . 
B y a n a n a l y t i c a l c o n t i n u a t i o n i t is easi ly shown, however , 
t h a t a b e t t e r a p p r o x i m a t i o n for t h e local compress ib i l i ty effect 
on a n y finite b o d y is o b t a i n e d b y rep lac ing t h e f ree-s t ream M a c h 
N u m b e r Mco b y t h e local M a c h N u m b e r ML SO t h a t 
Up = CpQ/ v i ML2 (2) 
T h i s essen t ia l ly app l ies t h e P r a n d t l - G l a u e r t l inear ized p o t e n t i a l 
e q u a t i o n t o t h e local flow field cons idered a p p r o x i m a t e l y un i -
form. T h e s a m e p rocedure , us ing t h e local M a c h N u m b e r in 
t h e P r a n d t l - G l a u e r t cor rec t ion , w a s first sugges ted in reference 
2 a n d w a s l a t e r just if ied b y t h e e x p e r i m e n t a l r e su l t s p r e s e n t e d in 
reference 3 for t h e p a r t i c u l a r case of t h e subson ic flow field exist-
ing b e h i n d a d e t a c h e d shock w a v e in t r a n s o n i c flow. 
T H E N E W C O M P R E S S I B I L I T Y C O R R E C T I O N 
T h e surface p re s su re coefficient m a y be w r i t t e n (see reference 
1, p a g e 28) a s 
2 bL - pa te-) ( l /2jpco /7co2 yMa 
where for i sen t rop ic flow (see reference 1, p a g e 125) 
(3) 
PL^ 
P«, \T 
[(y - l ) / 2 ] M c o 2 ) y/(y ~ 1} y (4) + f ( T - 1)/2]ML2 
T h e n s u b s t i t u t i n g E q . (4) i n to E q . (3) a n d solving for t h e local 
M a c h N u m b e r ML, we o b t a i n t h e exac t express ion 
- ( T - D / Y 
ML2 = [Mm2 + —^-: )( 1 + - ^ - Cv ' 
-(-^)(-**4 - 1 
(5) 
N o w , before s u b s t i t u t i n g E q . (5) in to E q . (2), we m u s t be cog-
n i z a n t of t h e fact t h a t on ly t h e first-order t e r m s of Cp, co r re spond-
ing t o t h e first o rde r of t h e local p e r t u r b a t i o n p o t e n t i a l , were r e -
t a i n e d in der iv ing E q . (2) for t h e local flow field. C o n s e q u e n t l y , 
E q . (5) m u s t be r e d u c e d t o t h e first o rde r in Cv so t h a t 
1 
ML2 = Ma 1 + • Ma Mco2 C», (6) 
T h e n , s u b s t i t u t i n g E q . (6) in to E q . (2) a n d r e t a i n i n g o n l y t h e 
first-order t e r m of Cv in t h e local flow field cor rec t ion in t h e de -
n o m i n a t o r , we o b t a i n 
V l - Mo: 
Ma ^V\ 
+ 
y - 1 
(7) 
V l - Moo2 2 
Moo2 
E q . (7) is a new re la t ion for t h e two-d imens iona l compress ib le 
p res su re coefficient a n d p rov ides a b e t t e r a g r e e m e n t w i t h exper i -
m e n t a l d a t a for airfoils t h a n does t h e von K a r m a n - T s i e n rela-
t ion (see e i the r reference 4 or reference 1, p a g e 185) 
CD = 
Lsr> 
V i ~ M a Moo2 + i + Vi 
(8) 
M o 
I t is in t e re s t ing t o n o t e t h a t t h e t e r m c o n t a i n i n g t h e specific 
h e a t r a t i o y is assoc ia ted w i t h t h e t e r m Moo4, a s w o u l d b e ex-
p e c t e d from a compar i son of t h e von K a r m a n - T s i e n s t r a i g h t -
l ine p re s su re -dens i ty v a r i a t i o n (cor responding t o y = —1) w i t h 
t h e exac t i sen t rop ic v a r i a t i o n (see reference 1, p a g e 173). 
R E F E R E N C E S 
1
 Liepmann, H. W., and Puckett, A. E., Introduction to Aerodynamics of a 
Compressible Fluid; John Wiley & Sons, Inc., New York, 1947. 
2
 Laitone, E. V., and Pardee, Otway O., Location of Detached Shock Wave 
in Front of a Body Moving at Supersonic Speeds, N.A.C.A. R.M. No. A7B10, 
May 6, 1947. 
3
 Laitone, E. V., An Experimental Investigation of Transonic and Acceler-
ated Supersonic Flow by the Hydraulic Analogy, University of California, 
Institute of Engineering Research, Berkeley, Series No. 3, Issue No. 315, 
July 3, 1950. 
4
 von Karman, Theodore, Compressibility Effects in Aerodynamics, Journal 
of the Aeronautical Sciences, Vol. 8, No. 9, pp. 337-356, July, 1941. 
Note on the Minimum Critical Reynolds 
Number and the Form Parameter* 
John C. Freemdn, Jr. 
Department of Meteorology, The University of Chicago 
February 9, 1951 
IN T H E T H E O R Y O F T H E B O U N D A R Y L A Y E R of a n incompress ib l e fluid, t h e " fo rm p a r a m e t e r " H = <5iAV w h e r e 8X is t h e d i sp lace -
m e n t t h i ckness a n d & is t h e m o m e n t u m th i cknes s , h a s been found 
useful a s a p a r a m e t e r t o c h a r a c t e r i z e t u r b u l e n t b o u n d a r y l a y e r s 1 
a n d ( empi r i ca l ly ) as a p a r a m e t e r t h a t d e t e r m i n e s Ri ( t h e m i n i -
m u m cr i t ica l R e y n o l d s N u m b e r b a s e d on t h e d i s p l a c e m e n t t h i ck -
ness) for l a m i n a r b o u n d a r y - l a y e r profiles on a p o r o u s flat p l a t e 
* This work was started at Brown University. 
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