CHAPTER 11
SUBSONIC COMPRESSIBLE FLOW OVER
AIRFOILS,LINEAR THEORY
11.1 Introduction
This chapter mainly deal with the properties of
two- dimensional airfoils at Mach number above
0.3 but below 1,where the compressibility must
be considered,
Velocity potential equation
Linearized velocity
potential equation
Prandtl-Glauet
Compressibilty correction
Improved compressibilty
Correction
Critical Mach number
The area rule for transonic flow
Supercritical airfoils
Figure 11.1
Road Map for Chap.11,
11.2 The Velocity Potential Equation
For two-dimensional,steady,irrotational,isentropic flow,a
velocity potential can be defined such that,
),( yx?? ?
???V
The introduction of velocity potential can greatly simplify the
governing equations,we can derive the velocity potential
equation from continuity,momentum,energy equations,
The continuity equation for steady,two-dimensional flow
is,
0)()( ?????? y vx u ??
0???????????? yvyvxuxu ????or
Substituting into it,we get
yvxu ?
??
?
?? ??,
0)( 2
2
2
2
?
?
?
?
??
?
?
?
??
?
??
?
?
yyxxyx
???????
To eliminate from above equation,we consider the
momentum equation, ?
V d Vdp ???
)(22 222 vuddVV d Vdp ??????? ???
??
?
??
?
?
??
?
??? 22 )()(
2 yx
ddp ???
Since the flow we are considering is isentropic,so
2ap
d
dp
s
???
?
?
???
?
?
??
??
??
?
??
?
?
??
?
??? 22
2 )()(2 yxdad
????
?
?
?
?
?
?
?
??
?
?
?
???
?
? 22
2 )()(2 yxxax
????
so
?
?
?
?
?
?
?
??
?
?
?
???
?
? 22
2 )()(2 yxyay
????
0)( 2
2
2
2
?
?
?
?
??
?
?
?
??
?
??
?
?
yyxxyx
???????
0))((
2
)(
1
1)(
1
1
2
2
2
2
2
22
2
2
2
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
yxyxa
yyaxxa
???
????
We get the velocity potential equation,
In this equation,the speed of sound is also the function
of,
?
?
?
?
?
?
?
?
?
?
?
??
?? 22202 )()(
2
1
yx
aa
???
(11.12)
For subsonic flow,Eq,11.12 is an elliptic partial differential
equation,For supersonic flow,Eq.11.12 is a hyperbolic partial
differential equation,For transonic flow,Eq.11.12 is mixed type
equation,
Eq,11.12 represents a combination of continuity,momentum,
energy equations,In principle,it can be solved to obtain
for the flow field around any two-dimensional flow,
The infinite boundary condition is
???
?? V
xu
?
0???? yv ?
The wall boundary condition is
0???n?
?
?
Once is known,all the other value flow variables are directly
obtained as follows,
1,Calculate u and v,and
xu ?
?? ?
yv ?
?? ?
2.Calculate a,
??
?
??
?
?
??
?
???? 222
0 )()(2
1
yxaa
???
3.Calculate M,
a
vu
a
VM 22 ???
4,Calculate T,p,,?
1
1
2
0
12
0
12
0
)
2
1
1(
)
2
1
1(
)
2
1
1(
?
?
?
?
?
?
??
?
??
?
??
?
?
?
?
??
?
?
M
Mpp
MTT
11.3 THE LINEARIZED VELOCITY POTENTIAL EQUATION
If we define,,we call and the
perturbation velocities,
uVu ??? ? vv ?? v?u?
So we can define a perturbation velocity potential,??
?? ??? ? xVwith
0
?
)
?
)(
?
(2
?
)
?
(
?
)
?
(
2
2
2
22
2
2
22
?
??
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
??
?
?
yxyx
V
yy
a
xx
Va
???
????
Substituting to Eq,11.12,and multiplying by, 2a
(11.14)
This equation is called the perturbation velocity potential equation
0??)?(2?)?(?)]?([ 222 ?
?
???
?
???
?
???
?? y
uvuV
y
vva
x
uuVa
2
?)?(
121
22222 vuVaVa ??
?
?
??
?
???
??
)]
??
)(
?
1(
?
[
?
]
?
2
1?
2
1?
)1[(
?
]
?
2
1?
2
1?
)1[(
??
)1(
2
2
2
2
2
2
2
2
2
2
22
y
v
x
u
V
u
V
v
M
x
v
V
u
V
v
V
u
M
x
u
V
v
V
u
V
u
M
y
v
y
u
M
?
?
?
?
?
??
?
??
?
?
???
?
??
?
?
???
?
?
?
?
?
?
??
?
???
?
???
??
??
?
??
?
(11.16)
Consider small perturbation situation,
1
?
,
?
,1
?
,
?
2
2
2
2
? ? ???
???? V
v
V
u
V
v
V
u
The small perturbation’s meaning is assume the body is a
slender body at small angle of attack,
0
??
)1( 2 ?
?
?
?
?
?
? ?
y
v
x
u
M
So,for subsonic or supersonic flow,the small perturbation
equation is,
For transonic small perturbation,the small perturbation equation
is,
x
u
V
u
M
y
v
x
u
M
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
??
??
)1(
??
)1( 22 ?
0
??
)]
?
)1(1(1[ 2 ?
?
?
?
?
?
???
?
? y
v
x
u
V
u
M ?
or
0
??
)1( 2
2
2
2
2 ?
?
?
?
?
?
? ?
yx
M
??
0
??
)]
?1
1(1[
22
2
2 ?
?
?
?
?
?
?
??
??
?
? yxxVM
????
Equa tio n I nvi sc id I r rota tio na l S mall
P e rturba tio n
I n c om pre ssi -
ble
Note s
Na vier -St okes - - - - Homo g e -
ne ous
Re y nol ds-
A ve ra g e d
Na vier -St okes
- - - -
Modele d
T urbulenc e
Eule r X - - -
F ull P otentia l X X - -
T ra nso nic
S mall
P e rturba tio n
X X
X
-
S ubs oni c,S up
e rsonic,Sm a ll
P e rturba tio n
X
X X -
Ac ous tic X X X -
L a pl a c e X X - X
Summary of commonly-used equations and the corresponding
assumption,
Derivation of the pressure coefficient expression,
?
???
q
ppC
p
Since
2
2
2
2
22
22
/
1
2
2
1
2
1
??
?
?
?
?
??
?
??
?
?
???
??
?
??
Mp
a
V
p
V
p
p
V
p
p
Vq
??
??
?
?
?
?
?
so
)1(2 2 ??
?? p
p
M
C p
?
pp c
V
T
c
V
T
22
22
?
? ???
1?? ?
?Rc
p
2
2222
2
1
2
11
?
?
?
?
?
???????
a
VV
RT
VV
T
T ?
?
?
pc
VVTT
2
22
?
?
???
222 ?)?( vuVV ???
?
)???2(
2
1
1 22
2
vuVu
aT
T
??
?
?? ?
??
?
1
22
2
1
)???2(
2
1
1
)(
?
?
?
?
??
?
?
?
?
?
?
??
?
??
?
?
?
?
?
?
vuVu
a
T
T
p
p
1
2
22
2
)
???2
(
2
1
1
?
??
?
?
?
?
?
?
?
? ?
?
?
??
?
?
?
V
vu
V
u
M
p
p
????
?
???
??
?
?
)
???2
(
2
1
2
22
2
V
vu
V
u
M
p
p ?
2
22 ???2
??
?
???
V
vu
V
u
C p
?
??
V
u
C p
?2 (11.32)
The Boundary Condition of small perturbation equation,
At Infinity,
0??;c o n s t a n t?
??
?
vuor
?
At the body surface,;
?
?
t a n
uV
v
u
v
?
??
?
?
Corresponding to small perturbation assumption,
?
?
t a n
?
???
?
V
y
(11.34)