Example Problem: Diamond-Wedge Airfoil in Supersonic Flow A diamond –wedge airfoil with a half-angle ε = 10° is placed at an angle of attack α = 15° in a Mach 3 freestream. Calculate the lift and wave-drag coefficients for the airfoil. Solution

For region 2

.51 1

.52 1 2

.1 11

1

.1 22

2

3 49.76

49.76 5 54.76 3.27

3 36.73

3.27 54.76

TableA

TableA

TableA o

TableA o

M

MpMppMp

ν

ν ν θ

= → =

= + = + = → =

= → =

= → =

For region 3

.53 2 3

.1 33

3

54.76 20 74.76 4.78

4.78 407.83

TableA

TableA o

MpMp

ν ν θ= + = + = → =

= → =

For region 4

1 0

.2 441 1 4

1 1

.544 4

.1 44

4

344

25

sin 3sin44 2.08 4.881, 0.5643, 0.6835

0.5643 1.733 18.69sin( ) sin(44 25)

1.733 5.165

M

TableA on n

o

TableAn

TableA o

M

ppM M Mp p

MM

pMp

θ β βθ

β

νβ θ

− −= → =

=

= = = → = = =

= = = → =− −

= → =

For region 5 .5 .1 5

5 4 55

18.69 20 38.69 2.48 16.56TableA TableA opMp

ν ν θ= + = + = → = → =

1 12 2

1 2 1 1

3 3 3 22

1 1 3 3 2

4

1

5 5 5 4 1

1 5 4 01 1

1 (1)(36.73) 0.670754.76

10.6707 (1)(54.76) 0.09407.83

4.881

1 (1)(0.6835)(36.73) 1.51616.56

o o

o o

o o

o

o o o

o o

p pp pp p p pp p p ppp p p p pppp p p p pp p p p p

= = =

= = =

=

= = =

[ ]

4 5 2 3

4 3 5 2

4 3 5 22

2 1 1 11 1

2

' cos25 cos5 cos5 cos25' ( ) cos25 ( ) cos5

' ' 2 cos25 cos5

22 (4.881 0.09)cos25 (1.516 0.6707)cos5 0.823

1.4(3)/ 2 cos10

l

l

L p l p l p l p lL p p l p p l

p p p pL L lCq S M c p ppM c

l lCc c

c lwherel c

γ γ∞

= + − −= − + −

− −= = = + =

= − + − =

= ⇒

1 0.50772cos10

0.418l

so

C

= =

=

( )

4 5 2 3

4 3 5 2

3 54 22

2 1 1 1 1 11 1

2

' sin25 sin5 sin5 sin25' ( ) sin25 ( ) sin5

' ' 2 sin25 sin5

22 14.881 0.09 sin25 (1.516 0.6707)sin5

1.4(3) 3

0

d

d

d

D p l p l p l p lD p p l p p l

p pD D l p pCq S M c p p p ppM c

l lCc c

C

γ γ∞

= + − −= − + −

= = = − + −

= − + − =

=

.169