CHAPTER 11
SUBSONIC COMPRESSIBLE FLOW OVER
AIRFOILS,LINEAR THEORY
11.4 PRANDTL-GLAUERT COMPRESSIBILITY
CORRECTION
The methods that approximately take into account of
the effects of compressibility by correct the
incompressible flow results is called compressible
corrections,
We will derive the most widely known correction of Prandtl-Glauert
compressibility correction in this section,
Since the Prandtl-Glauert method is based on the linearized
perturbation velocity potential equation,
0
??
)1( 2
2
2
2
2 ?
?
?
?
?
?
? ?
yx
M
??
So it has restrictions,thin airfoil at small angle of attack;
purely subsonic;
give inappropriate results at 7.0?
?M
0
??
2
2
2
2
2 ?
?
?
?
?
?
yx
??
?
)1( 22 ??? M?
x??
y?? ?
),(?),( yx????? ?
?
?
?
?
?
??
?
??
?
??
?
?
?
??
?
?
?
??
?
? 1???
xxx
?
??
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
yyy
???
2
2
2
2 11?
?
?
?
?
?
?
??
?
?
?
?
?
?
??
?
?
??
?
?
?
?
?
?
?
?
?
xx
2
2
2
2 ?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
??
?
?
?
?
?
?
?
?
?
yy
0
1
??
2
2
2
2
2
2
2
2
2
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
yx
02
2
2
2
?
?
?
?
?
?
?
?
?
?
Boundary Condition,
?
?
?
?
?
?
?
??
?
??
?
??
?
?1?
ydx
dfV
?
?
?
??
? dx
dqV
In (x,y) space,
In transformed space,
dx
df
dx
dq ?So
This Equation implies that the shape of the airfoil in the transformed
space is the same as the physical space,Hence,the above tranforma-
tion relates the compressible flow over an airfoil in (x,y) space to the
in space over the same airfoil,),( ??
)
2
(
1
?12?2?2
?
?
?
?
?
?
?
?
??
?
?
??
?
?
????
?
???
V
xVxVV
u
C
p
u?
?
?
?
?
)2(1
?
??
V
uC
p ?
?
0,p
p
CC ?
2
0,
1 ??
?
M
C
C pp
2
0,
2
0,
1
1
?
?
?
?
?
?
M
c
c
M
c
c
m
m
l
l
(11.51)
11.5 IMPROVED COMPRESSIBILITY
CORRECTIONS
2/)]11/([1 0,222
0,
p
p
p
CMMM
C
C
??? ????
?
0,
2222
0,
)]12/
2
1
1/([1 p
p
p
CMMMM
C
C
???? ?
?
???
?
?
(11.54)
(11.55)
11.6 CRITICAL MACH NUMBER
In this section we deal with several aspects of transonic flow
from a qualitative point of view,
What is the definition of
Critical Mach Number?
The critical Mach number
is that free stream Mach
number at which sonic flow
is first achieved on the
airfoil surface,
Derivation of critical pressure coefficient,
?
?
?
?
?
?
?
?
???
?
?
??
?
?
??
???
?
?
?
1
]2/)1[(1
]2/)1[(11
)1(
2
2
2,
??
?
?
? AAp M
M
M
C
?
?
?
?
?
? ??
??
11 2,
p
p
M
C AAp
?
)1(
2
2
0
0
]2/)1[(1
]2/)1[(1
?
?
??
???
?
???
?
??
????
??
?
?
A
AA
M
M
pp
pp
p
p
?
?
?
?
?
?
?
?
???
?
?
??
?
?
??
???
?
1
2/)1(1
]2/)1[(11
)1(2
2,
??
?
?
?
cr
cr
crp
M
M
C
?
?
?
?
?
?
?
?
???
?
?
??
?
?
??
???
?
?
?
1
2/)1(1
]2/)1[(11
)1(2
2,
??
?
?
?
M
M
C crp
(11.6)
Estimation of, crM
)(,crcrp MfC ?
Eq.(11.51),(11.54),or(11.55)
11.7 DRAG-DIVERGENCE MACH NUMBE,
THE SOUND BARRER
The definition of drag divergence Mach number,
The value of at which the sudden increase in drag
starts is defined the as drag divergence Mach number,?
M
11.8 The Area Rule
The Area Rule is a design concept which has effectively reduced
the drag rise near Mach 1 for complete airplane,
FIGURE 11.10 FIGURE 11.11
The area rule for transonic flow,
The cross-sectional area distribution of an airplane,
including fuselage,wing,and tail,should have a smooth
distribution along the axis of the airplane,
11.9 THE SUPERCRITICAL AIRFOIL
The purpose of a supercritical airfoil is increase the value of
drag-divergence Mach number,
Supercritical airfoils are specially designed profiles to increase the
drag-divergence Mach number,