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Implications of Linearized Supersonic Flow on Ai rfoil Li ft & Drag

To begin, we will divide the airfoil geometry into camber and thicknessdistributions:

)(2

1)()(

)(2

1)()(

sincos

xxyxy

xxyxy

jUiuUU

c

cu

=

+=

+=

l

vvv

Note:The x -axis, which is the axis defined through the leading edge and trailing edge

points, is used to define the freestream angle of attack. The chord line is the lineconnecting ..ei to ..et and, the chord length is the distance between these points.

To calculate the lift and drag, we need to integrate the pressure forces aroundthe airfoil.

= airfoil dsnpF vv

when nv

points out of the airfoil surface.

Now, we begin specializing this formula for the assumptions of linearized flow,i.e.

(small thickness)

(small camber)

(small )

The normal on the upper surface is

y

ix

1

)(xyc

)(xyu

)(x

)(xyl

1

1

1

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Implications of Linearized Supersonic Flow on Airfoil Lift & Drag

16.100 2002 2

jds

dxi

ds

dyn

jds

dyi

ds

dxt

u

u

u

u

vvv

vvv

+=

+=

But, since we have thin airfoils dxds , thus jidx

dynu

vvv l 1+=

Similarly, on the lower surface ji

dx

dyn

vvv l

l1=

Thus, the force may be written:

( )

+

=

+=

=

c

u

cu

u

c cu

u

c c

uu

dxppjdxdx

dyp

dx

dypiF

dyjidx

dypdxji

dx

dypF

dxnpdxnpF

00

0 0

0 0

l

l

l

l

l

ll

vvv

vvvvv

vvv

The drag component is in the freestream direction:

( )

+=

+=

=

jFiFD

jiFu

u

FD

vvvv

vvvv

v

sincos

Similarly, the lift is:( )

jFiF

jiFL

vvvv

vvv

+=

+=

cossin

y

x

unv

utv

)(xyu

1

( ) +

=

c

u

c

u dxppdxdx

dyp

dx

dypD

00

0l

l

l

1

( ) +

=

c c

u

u

u dxppdxdx

dyp

dx

dypL

0 0 l

l

l

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Implications of Linearized Supersonic Flow on Airfoil Lift & Drag

16.100 2002 4

+

=

+

+

+

+

=

cu

c

cu

uc

u

cdxdx

dydx

dx

dy

Mc

dxdx

dy

dx

dy

Mcc

dxdx

dy

dx

dy

Mdx

dy

dx

dy

Mcc

002

02

0 22

21

2

21

2

11

21

1

21

l

l

l

ll

l

But 000)0()(0

=== lll ycydx

dx

dyc

And, similarly, =c

u dxdx

dy0

0

Important result!!

lc is linear with but note the slope is different than subsonic case

=

21

2

Md

dc

l .

lc does not depend on camber! All the y -dependence has disappeared in this

result. Thus,l

c also does not depend on thickness.

Now, lets look at dc :

dxdx

dy

dx

dy

Mcc

dxdx

dy

dx

dy

dx

dy

dx

dy

Mc

dxdx

dy

dx

dy

Mcc

cu

d

cuu

cu

d

+

+

=

+

+

+

=

+

=

0

2

22

2

0

2

22

2

0

22

2

21

2

221

2

1

2

l

ll

l

small small

1

4

2 =

Mc

l

This term willintegrate to 0

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