*The compressibilty correction rule for thin wing
The effect of compressibility in 3-D flows is somewhat less
dramatic than with 2-D flows,but many of the same effects
become important.Many of the same techniques for predicting
linear compressibility effects work in 3-D too,For example,
we can transform the 3-D Prandtl-Glauert equation into the 3-D
Laplace equation for incompressible flow by changing variables
just as in 2-D,
0
???
2
2
2
2
2
2
2 ?
?
?
?
?
?
?
?
?
zyx
???
?
Defining the geometry of a finite wing,y=f(x,z)
So the boundary condition is,
x
y
V
y ?
?
?
?
?
?
??
0
2
2
?
2
2
2
?
2
2
2
?
22
?
?
?
?
?
?
?
?
?
?
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?
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??
???
zyx
Transform the (x,y,z) and in the following way,??
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?
z
y
x
z
y
x
If
???
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??
?
2
?
2
?
22
zyx ??
02
2
2
2
2
2
?
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??
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??
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y
x
y
x
y
V
y
?
?Derive the boundary
condition,
x
y
V
y ?
?
?
?
?
?
??
?
?
?
??
?
? ?
?
??
?
?
?V
y
x
2
?
12 ? ?
y
x
?
?? ?If Then
?
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??
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?
?V
???
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??
?
2
?
2
?
22
zyx ??
12 ? ?
y
x
?
?? ?
???
??
??
?
?2
?
?
?
?
z
y
x
???
?
?
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?
?
1
?
?
?
?
z
y
x
The Geometry relation,
?
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?
?
??
???
???
???
t a n
1
t a n
0
0
0
0
0
0
?
?
?
?
?
?
AA
?
?
?
?
??
???
???
???
t a n
1
t a n
0
0
0
0
0
0
?
?
?
?
?
?
AA
The relation of Aerodynamic coefficients,
0,2
2
1
12?2
p
x
p
C
VV
C
?
?
?
?
?
?
?
?
????
??
),t a n
1
,,,,,0(
1
),t a n,,,,,(
2
??
?
???????
?
?????
AMC
AMC
p
p
?
?
0,2
2
1
12?2
p
x
p
C
VV
C
?
?
?
?
?
?
?
?
????
??
This correction rule is Goethert Rule,
??
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????
??
?
???????
?
??
?
????
?????
,t a n
1
,,,,,0(
),t a n
1
,,,,,0(
1
,t a n
1
,,,,,0(
),t a n,,,,,(
2
AMC
AMC
AMC
AMC
p
p
p
p
?
?
?
?
Derivation of the 3-D Prandtl-Glauert
correction rule from Goethert rule,
),t a n
1
,,,,,0(
1
),t a n,,,,,(
??
?
????
?
?????
AMC
AMC
p
p
?
?
This is the Prandtl-Glauert correction Rule
Nonetheless,changing the lift curve slope just by the
Prandtl-Glauert factor does not do too badly,
A somewhat better approximation is obtained by
applying the Prandtl-Glauert correction to the 2-D
lift curve slope,then applying the downwash
correction from lift line theory,
Applying the 2-D Prandtl-Glauert correction to sections based
on the normal component of the freestream Mach number is better
yet and this is the basic idea behind a formula for lift curve slope
that is widely used for preliminary design calculations,
The DATCOM formula shown below includes the effects of finite
aspect ratio,sweep,and Mach number as well as a correction factor
for viscous effects on the 2-D lift curve slope of the section,
CHAPTER 12 LINERIZED SUPERSONIC FLOW
0
??
)1( 2
2
2
2
2 ?
?
?
?
?
?
? ?
yx
M
??
12 ?? ?M?For the case of supersonic flow,define We can rewrite the linearized perturbation velocity equation,
0
??
2
2
2
2
2 ?
?
?
?
?
?
yx
??
?
This Equation has a basic solution,
)()(? yxgyxf ??? ????
Along
.,,c o n s tfc o n s tyx ??? ?
.,,c o n s tgc o n s tyx ??? ?
Along
,c o n s tyx ?? ?
1
11
2 ?
??
?M
dx
dy
?
1
1
t a n
2 ?
?
?M
?
For the supersonic flow over a surface with a small
hump in the middle,
0?g
)(? yxf ?? ??
so
'
?
? f
x
u ?
?
?? ?
'
?
? f
y
v ?
?
??
?
?
?
?
vu ?? ??
?
?
t a n
?
? ??
?
?
? V
y
v
??? Vv?
?
???? Vu?
?
??
V
uC
p
2
1
2
2 ?
?
?M
C p
?
Conclusion,The linearized supersonic pressure coefficient
is directly proportional to the local surface inclination with
respect to the free stream,
The pressure is higher on the front section of
the hump,and lower on the rear section,So
there is a drag force exerting on the hump,
This drag is called wave drag,
12.3 APPLICATION TO SUPERSONIC AIRFOILS
The definition of sign of,
When the surface is inclined into the freestream direction,is taken positive,
When the surface is inclined away from the freestream direction,is taken
negative,
?
?
?
Example,a flat plate at small angle of attck
1
2
2,?
?
?M
C lp ?
1
2
2,?
??
?M
C up ?
21
2
1
4
1
4
2
,
2
2
2
l
LEm
d
l
c
M
c
M
c
M
c
??
?
??
?
?
?
?
?
?
?
?
?
?
For a thin airfoil of arbitrary shape at small angle of attack,
1
4
1
2
)(
1
4
1
4
2
1
2
,
222
2
2
?
?
?
??
??
?
?
?
?
??
?
?
M
K
M
c
gg
M
c
M
c
LEm
tcd
l
?
?
?
1
4
2
1
?
?
?M
Kc
m a c
*Small perturbation velocity potential equation for
three dimensional Supersonic flow,
0
???
2
2
2
2
2
2
2 ?
?
?
?
?
?
?
?
?
zyx
???
?
])()[()(
),(
2
1
2222 ????
???
?
?
?????
??
zyx
Law of forbidden signals,
The disturbances originating at (A) can only affect the
darker shaded area,Similarly,points outside the
forward-going Mach cone (lightly shaded area) cannot
affect the flow at point A,
Dependence domain
of A
Influence domain
of A
Two Dimensional Domain and three Dimensional
Domain in the supersonic flow,
The lift coefficient of a rectangular flat wing,
)
12
1
1(
1
4
22 ?
?
?
?
?? MAM
c l
?