Section 7, Lecture 2:

Supersonic Flow on Finite Thickness Wings

• Concept of Wave Drag

• Anderson, Chapter 4, pp. 174-179

1
MAE 5420 - Compressible Fluid Flow
 Supersonic Airfoils
• Recall that for a leading edge in Supersonic flow there is
γ=1.1
a maximum wedge angle at which the oblique shock wave
remains attached
γ=1.3 • Beyond that angle

γ=1.4 shock wave becomes

γ=1.1
γ=1.05

γ=1.2 detached from

γ=1.3

γ=1.4
leading edge

! max = tan
%
)
"1 {
2 M 12 sin 2 ( # max ) " 1 } '
*
) tan ( # max ) & 2 + M 1 %&$ + cos ( 2 # max ) '( ( *
% 2
'
& (

# max
1
= cos )
"1 ( )
% M 12 ($ " 1) + 4 " 16($ + 1) + 8M 12 ($ 2 " 1) + M 14 ($ + 1)2 '
*
2 )& 2$ M 12 *(

2
MAE 5420 - Compressible Fluid Flow
 Supersonic Airfoils (cont’d)
• Normal Shock wave formed off theγ=1.1
front of a blunt leading
causes significant drag

Detached shock waveγ=1.3

Localized normal shock wave

Credit: Selkirk College Professional Aviation Program

3
MAE 5420 - Compressible Fluid Flow
 Supersonic Airfoils (cont’d)
• To eliminate this leading edge drag caused by detached bow wave
γ=1.1
Supersonic wings are typically quite sharp at the leading edge

• Design feature allows oblique wave to attach to the leading edge

γ=1.3
eliminating the area of high pressure ahead of the wing.

• Double wedge or “diamond”

Airfoil section
Credit: Selkirk College Professional Aviation Program

4
MAE 5420 - Compressible Fluid Flow
 Supersonic Airfoils (cont’d)
•When supersonic airfoil is at positive
angle of attack shock wave
γ=1.1
at the top leading edge is weakened
and the one at the bottom is intensified.

γ=1.3 • Result is the airflow over the top

of the wing is now faster.

• Airflow will also be accelerated

through the expansion fans.

• The fan on the top will be stronger

than the one on the bottom.

• Result is faster flow and therefore

Credit: Selkirk College Professional Aviation Program
lower pressure on the top of the airfoil.
• We already have all of the tools we need to analyze the flow on this wing

5
MAE 5420 - Compressible Fluid Flow
 Supersonic Flow on
Finite Thickness Wings at zero α

Drag = b [ p2 ! p3 ] t

• Symmetrical Diamond-wedge airfoil, zero angle of attack

t /2
Drag = 2b [ p2l sin(! ) " p3l sin(! )] # sin(! ) =
l
Drag = b [ p2 " p3 ] t
6
MAE 5420 - Compressible Fluid Flow
 Supersonic Flow on
Finite Thickness Wings at zero α

Drag = b [ p2 ! p3 ] t

• Look at pressure difference, p2-p3

• Oblique shock wave from region 1->2 .. p2 >p∞
¥ Expansion fan from region 2---> 3 ... p3 < p2
... p2 >p∞ −−> Drag > 0 7
MAE 5420 - Compressible Fluid Flow
 Supersonic Flow on
Finite Thickness Wings at zero α
Lift Fp

α
Drag

• Compare to Flat Plate

# pl pu &
%p " p (
CL = $ ! ! '
cos 0 CD = 0
*) 2-
,+ M /
2 ! .

8
MAE 5420 - Compressible Fluid Flow
 Compare to wing in subsonic flow
2
Mach 1
3

A0 A1 A0

A0 A1 A0

• from Bernoulli 2 2
! dx
" %
! pdx = P0 !
_
P0 = p + Cpmax q = p $1 + Cpmax M 2 ' (
# 2 & 1 1
f [M ]
3 3 2 3
P0 P0 dx dx
p=
"
1 + Cp M '
=
! 2 % f [M ] ! pdx = P !
0
f [M ]
= "P0 !
f [M ]
# ! pdx = 0
$# max
2 &
2 2 1 1

9
MAE 5420 - Compressible Fluid Flow
 Compare to wing in subsonic flow (cont’d)

2 2
dx • at zero lift .. Subsonic wings in
! pdx = P !
1
0
1
f [M ] inviscid flow have No-drag!

3 3 2 3
dx dx
! pdx = P !
2
0
2
f [M ]
= "P0 !
1
f [M ]
# ! pdx = 0
1

• Known as d'Alembert's Paradox …

• Subsonic flow over wings

… induced drag (drag due to lift) + viscous drag

10
MAE 5420 - Compressible Fluid Flow
 Supersonic Wave Drag
• Finite Wings in Supersonic Flow have drag .. Even
at zero angle of attack and no lift and no viscosity…. “wave drag”

• Wave Drag coefficient is proportional to thickness ratio (t/c)

C D wave =
Drag
=
[ p
2! p3 ] $ t '
" & ) c
2 % c(
_
bc q p# M
2
t

• Supersonic flow over wings

… induced drag (drag due to lift) +
viscous drag + wave drag
11
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
• 5° ramps, M∞ = 2.00 Airfoil
• Total chord: 2 meters • Look at flow properties across
each of the numbered regions
• Fineness ratio (t/c): 0.08749

3 5 7
1
4 6
2

12
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil (cont’d)
• Across region 1-2 … Oblique shock
δ2 = 5°−−>β2=34.302°, p2/p1=1.3154, M2=1.8213

M∞ = 2.00

3 5 7
1
4 6
2

13
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil (cont’d)
• Across region 2-4 … expansion fan M2=1.8213
δ4 = -10°−−> p4/p2=0.5685, p4/p1=0.7478, M4=2.1848

M∞ = 2.00

3 5 7
1
4 6
2

14
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil (cont’d)
• Across region 1-3 … Oblique shock
δ3 = 5°−−>β3=34.302°, p3/p1=1.3154, M3=1.8213

M∞ = 2.00

3 5 7
1
4 6
2

15
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil (cont’d)
• Across region 3-5 … expansion fan M3=1.8213
δ5 = -10°−−> p5/p3=0.5685, p5/p1=0.7478, M5=2.1848

M∞ = 2.00

3 5 7
1
4 6
2

16
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag
• Calculate Drag Coefficient
p2/p1=1.3154 M∞ = 2.00
p3/p1=1.3154
p4/p1=0.7478 3 5 7
p5/p1=0.7478 1
2 4 6

) 1 " p2 p3 % 1 " p4 p5 % , $ 1.3154 + 1.3154 ! 0.7478 + 0.7478 % 0.08749

+ ( $ + " #
+ $ ' ' . 2 2
* 2 # p! p! & 2 # p! p! & - " t %
C D wave = $# '& = 1.4 2
/ 2 c 2 =0.01774
M 2
2

17
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag (cont’d)
• Plot wave drag Mach 2.0
versus thickness ratio 0° α

C D wave =
Drag
=
[ p
2! p3 ] $ t '
_
" &% )(
bc q p# M 2 c
2

Thickness ratio
18
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag (cont’d)
• Look at mach number
Effect on wave drag

Increasing mach • Mach Number tends

to suppress wave drag

Thickness ratio 19
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag (cont’d)
• How About The
Compression effect of angle of
“Induced” drag attack on drag

Wave drag

+ α=0°

=
20
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag (cont’d)
Mach 2.0 Total drag

Increasing t/c

21
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Lift
• Calculate Zero α Lift Coefficient
p2/p1=1.3154
M∞ = 2.00
p3/p1=1.3154
p4/p1=0.7478
p5/p1=0.7478 3 5 7
1
4 6
2

) 1 " p2 p4 % 1 " p3 p5 % ,
+ $ + ' ( $ + '& .
2 # p p & 2 # p p
( )
Lift
= _ =*
! ! ! ! - •Makes sense
C L wave cos[5 o ] = 0
/ 2
bc q M
2
22
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (1)
Consider airfoil
@ positive
δ = wing half angle angle of attack,
… length … L

• Lower forward
Ramp Angle
… θ=δ + α

• Upper Forward
Ramp Angle
… θ=δ − α
L
θ > 0 “oblique shock”
θ < 0“expansion fan”
23
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (2)
• Angle from
δ = wing half angle 3-5 and 2-4
Is always convex
(expansion), 2δ

• Use Shock-Expansion
Theory to calculate
Ramp pressures

Upper: { P3, P5}

Lower: {P2, P4}

24
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (3)
δ = wing half angle
• Upper: { P3, P5}
Lower: {P2, P4}

• Calculate Normal
And Axial Pressure
Forces on Wing …

Normal Force

*$ L/2 L /2' $ L/2 L /2' -

Fnormal = b ! ,& P2 ! # P3 ! ) cos " + &% 4
P ! # P ! ) cos " /
+% cos " cos " ( cos " cos " (
5
.
* L L- b!L
= b ! ,( P2 # P3 ) + ( P4 # P5 ) / = *+( P2 + P4 ) # ( P3 + P5 ) -.
+ 2 2. 2
25
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (4)
δ = wing half angle
• Upper: { P3, P5}
Lower: {P2, P4}

• Calculate Normal
And Axial Pressure
Forces on Wing …

Axial Force

*$ L/2 L /2' $ L/2 L /2' -

Faxial = b ! ,& P2 ! # P4 ! ) sin " + &% 3
P ! # P ! ) sin " /
+% cos " cos " ( cos " cos " (
5
.
* L L- b!L
= b ! ,( P2 # P4 ) + ( P3 # P5 ) / tan " = *+( P2 + P3 ) # ( P4 + P5 ) -. tan "
+ 2 2. 2
26
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (5)
δ = wing half angle b&L
Fnormal = "#( P2 + P4 ) ! ( P3 + P5 ) $%
2
b&L
Faxial = "#( P2 + P3 ) ! ( P4 + P5 ) $% tan '
2

• Transform to wind axis

To calculate Lift, Drag

Drag = Faxial cos(! ) + Fnormal sin(! )

Lift = Fnormal cos(! ) " Faxial sin(! )

27
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (6)
δ = wing half angle b&L
Fnormal = "#( P2 + P4 ) ! ( P3 + P5 ) $%
2
b&L
Faxial = "#( P2 + P3 ) ! ( P4 + P5 ) $% tan '
2

• Transform to wind axis

To calculate Lift, Drag

Drag = Faxial cos(! ) + Fnormal sin(! )

Lift = Fnormal cos(! ) " Faxial sin(! )

( b&L + ( b&L+
Drag = * "#( P2 + P3 ) ! ( P4 + P5 ) $% tan ' - & cos(. ) + * "#( P2 + P4 ) ! ( P3 + P5 ) $% - & sin(. )
) 2 , ) 2 ,
( b&L+ ( b&L +
Lift = * "#( P2 + P4 ) ! ( P3 + P5 ) $% - & cos(. ) ! * "
# ( P + P ) ! ( P + P )
5 %$ tan ' - & sin(. )
) 2 , ) 2 3 4
2 ,

28
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (7)
δ = Terms
• Collect wing half angleEquation
on Drag
( b&L + ( b&L+
Drag = * "#( P2 + P3 ) ! ( P4 + P5 ) $% tan ' - & cos(. ) + * "#( P2 + P4 ) ! ( P3 + P5 ) $% - & sin(. )
) 2 , ) 2 ,
/
# b!L & # b!L& # b!L & # b!L&
Drag = % P2 tan " ( ! cos() ) + % P2 ( ! sin() ) + % P3 tan " ( ! cos() ) * % P3 ( ! sin() )
$ 2 ' $ 2 ' $ 2 ' $ 2 '
# b!L& # b!L & # b!L & # b!L&
+ % P4 ( ! sin() ) * % P4 tan " ( ! cos() ) * % P5 tan " ( ! cos() ) * % P5 ( ! sin() ) =
$ 2 ' $ 2 ' $ 2 ' $ 2 '
# b!L& # b!L&
%$ 2
P ( %$ P3 (
2 ' 2 '
cos "
[sin " ! cos() ) + cos " sin() )] + cos " [sin " ! cos() ) * cos " sin() )] +
# b!L& # b!L&
%$ 4
P ( %$ P5 (
2 ' 2 '
cos "
[ cos " ! sin() ) * sin " cos() )] * cos " [sin(" ) cos() ) + cos(" ) ! sin() )]

29
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (8)
• CollectδTerms
= wing half angle
" b!L% " b!L%
$# 2
P ' $# P3 '
2 & 2 &
Drag =
cos (
[sin ( ! cos() ) + cos ( sin() )] + cos ( [sin ( ! cos() ) * cos ( sin() )] +
" b!L% " b!L%
$# 4P ' $# P5 '
2 & 2 &
cos (
[ cos ( ! sin() ) * sin ( cos() )] * cos ( [sin(( ) cos() ) + cos(( ) ! sin() )]

+
[sin ( ! cos() ) + cos ( sin() )] = sin (( + ) ) +
[ cos ( ! sin() ) * sin ( cos() )] = sin (( * ) ) Trigonometric Identity
b!L
Drag = 2 P2 sin (( + ) ) + P3 sin (( * ) ) * P4 sin(( * ) ) * P5 sin (( + ) ) =
cos (
b!L
,( P2 * P5 ) sin (( + ) ) + ( P3 * P4 ) sin (( * ) ) ./
2 cos ( -

30
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (9)
• Similarδ Reduction forangle
= wing half Lift

' b&L* ' b&L *

Lift = ) "#( P2 + P4 ) ! ( P3 + P5 ) $% , & cos(- ) ! ) "
# ( P + P ) ! ( P + P )
5 %$ tan . ,+ & sin(- )
( 2 + ( 2 3 4
2
/
" P2 & [ cos . cos(- ) ! sin . & sin(- )] + $ " P2 & cos (. + - ) + $
0 1 0 1
b&L 0 3 !P [ cos . cos(- ) + sin . & sin(- ) ] + 1 b&L 0 3!P cos (. ! - ) + 1=
= =
2 cos . 0 P4 [ cos . cos(- ) + sin . & sin(- )] + 1 2 cos . 0 P cos (. ! - ) + 1
0 1 0 4 1
0# !P5 & [ cos . cos(- ) ! sin . & sin(- )] 1% 0# !P5 & cos (. + - ) 1%
b&L
"#( P2 ! P )5 & cos (. + - ) + ( P4 ! P3 ) cos (. ! - ) $%
2 cos .

!
[ cos " # cos($ ) + sin " sin($ )] = sin (" % $ ) !
Trigonometric Identity
[ cos " # cos($ ) % sin " sin($ ) ] = sin (" + $ )
31
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (10)
δ = Coefficients
• Lift/Drag wing half angle
b!L
Drag = %( P2 # P5 ) sin (" + $ ) + ( P3 # P4 ) sin (" # $ ) '(
2 cos " &
b!L
Lift = %&( P2 # P5 ) ! cos (" + $ ) + ( P4 # P3 ) cos (" # $ ) '(
2 cos "

!CD $ 1 ! Drag $ b)L 1 ! ( P2 + P5 ) sin (* + , ) + ( P3 + P4 ) sin (* + , ) $

#C & '= # Lift & = 2 cos * ) ' # &=
" L% P( M ( ) ( b ) L ) "
2 % (
P( M ( ) ( b ) L ) " 2
2 P + P5 ) ) cos (* + , ) + ( P4 + P3 ) cos (* + , ) %
2 2
! -P P 0 -P P 0 $
- 0 # / 2 + 5 2 sin (* + , ) + / 3 + 4 2 sin (* + , ) &
!CD $ / 1 1 2 # . P( P( 1 . P( P( 1 &
= )
# C & / 2 cos * ' 2 # &
" L% / M (2 2 #- P2 + P5 0 ) cos (* + , ) + - P4 + P3 0 cos (* + , ) &
. 1 /
2 #". P( P( 21 /. P
( P( 21 &%

32
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (11)
δ+ = wing half. !#angle
+ P2 P5 . +P
sin (' + 2 ) + 3
P4 . $
1
-, P 0/ -, P 1 P 0/ sin (' 1 2 ) &
! D$ - 1
C 1 0# * P* * * &
= (
# C & - 2 cos ' ) 0 # &
" L% - M *2 0 #+ P2 1 P5 . ( cos (' + 2 ) + + P4 1 P3 . cos (' 1 2 ) &
, / -
2 #", P* P* 0/ -, P
* P* 0/ &%

33
MAE 5420 - Compressible Fluid Flow
 Pulling it All Together (12)
δ+ = wing half. !#angle
+ P2 P5 . +P
sin (' + 2 ) + 3
P4 . $
1
-, P 0/ -, P 1 P 0/ sin (' 1 2 ) &
! D$ - 1
C 1 0# * P* * * &
= (
# C & - 2 cos ' ) 0 # &
" L% - M *2 0 #+ P2 1 P5 . ( cos (' + 2 ) + + P4 1 P3 . cos (' 1 2 ) &
, / -
2 #", P* P* 0/ -, P
* P* 0/ &%

• Check against flat plate

Formulae … (δ=0)
# - Pl Pu 0 - Pu Pl 0 &
% / , 2 sin (3 ) , / , sin (3 ) (
# P3 = P5 = Pu & #C D & 1 1 % . P+ P+ 1 . P+ P+ 21 (!
!" = 0! % ( ! % (! )
$ P2 = P4 = Pl ' $ C L ' 2 * M 2 %- Pl Pu 0 - Pl Pu 0 (
2
+ %
/ , 2 ) cos (3 ) + / , 2 cos (3 ) (
%$. P+ P+ 1 . P+ P+ 1 ('

# - Pl Pu 0 &
%/ , 2 sin (3 ) (
#CD & 1 % . P+ P+ 1 ( .........Q.E.D
!% ( =
$ C L '" =0 * 2 %- P P 0 (
M + % l , u ) cos (3 ) (
2 %$/. P+ P+ 21 ('

34
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … Drag (cont’d)
Lift Coefficient Climbs • How About The
Almost Linearly with α effect of angle of
attack on Lift

+
=
35
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … L/D
• For Inviscid flow
Supersonic
t/c = 0.035 Lift to drag ratio
almost infinite
for very thin
airfoil

• But airfoils do not

+ fly in inviscid flows

=
36
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
Airfoil … L/D (cont’d)
• Friction effects
t/c = 0.035 have small effect
on Nozzle flow
or flow in “large
“ducts”

• But contribute

+ significantly
to reduce the

=
performance of
supersonic wings

37
MAE 5420 - Compressible Fluid Flow
 Effect of Wing Sweep on Supersonic Airfoils
• Supersonic airflow turns corners very easily.
That is why sharp corners are OK on a supersonic
aircraft such as the F117 shown to the left, whereas
such corners would cause flow separation and a lot
of drag in subsonic flow.

38
MAE 5420 - Compressible Fluid Flow
 Effect of Wing Sweep on Supersonic Airfoils
(cont’d)

• Problem with sharp leading edges is poor performance in

subsonic flight.

• Lead to very high stall speeds, poor subsonic handling

qualities, and poor take off and landing performance for
conventional aircraft

• Why JSC uses

VSTOL thrust
vectoring and Lift
Fan

39
MAE 5420 - Compressible Fluid Flow
 Wing Design 101

• Subsonic Wing in Subsonic Flow • Subsonic Wing in Supersonic Flow

M>1

M<1

• Supersonic Wing in Subsonic Flow • Supersonic Wing in Supersonic Flow

Flow Separation

M>1

M>1
• Wings that work well sub-sonically generally
Don’t work well supersonically, and vice-versa

40
MAE 5420 - Compressible Fluid Flow
 Wing Design 101 (cont’d)

• Compromise Delta design

generates lift at low speeds by • The exception is the Highly-Swept
increasing the angle-of-attack, Delta-Wing design … works “pretty well”
but also has sufficient sweepback in both flow regimes
and slenderness to perform very
efficiently at high speeds. Supersonic Subsonic
• On a traditional aircraft wing
a trailing vortex is formed
only at the wing tips.

• On a delta-wing at low speeds,

the vortex is formed along the
entire wing surface and produces
most of the lift.

• This vortical-lift generation at

high angles-of-attack is
fundamental for reentry vehicles
like the Space Shuttle
to be able to fly at slow speeds. 41
MAE 5420 - Compressible Fluid Flow
 Wing Sweep Reduces Wave Drag
• One way to augment the performance of supersonic
aircraft is with wing sweep …
• Lowers the speed of flow
Normal to the wing …

• Decreasing the strength

Of the oblique shock wave

• Result is a Decrease in wave

Drag and enhanced L/D

42
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing

• Freestream Mach number resolved into 3 components

i) vertical to wing …
ii) in plane of wing, but tangent to leading edge
iii) in plane of wing, but normal to leading edge
43
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

i)M vert = M ! sin "

ii)M || = M ! cos " sin #
ii)M $ = M ! cos " cos #
44
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Equivalent Mach Number normal to leading edge

M eq = M ! + M vert =
2 2
( M " sin # ) + ( M " cos # cos $ )
2 2
=

M" (1 % cos # ) + cos # (1 % sin $ ) = M

2 2 2
" 1 % sin 2 $ cos 2 #
45
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Equivalent angle of attack normal to leading edge

M # sin ! tan (! )
( )
tan ! eq =
M vert
M"
= =
M # cos ! cos $ cos $
46
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Equivalent chord and span

ceq = c [ cos ! ]
• Chord is shortened

b • Span is lengthened
beq =
cos !
47
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Equivalent 2-D Lift Coefficient

L L
C L eq = = =
! $ b ' ! p M 2 cb
p" M eq c [ cos # ] &
2
) " eq
2 % cos # ( 2
L CL
=
!
p" M " cb (1 * sin # cos + )
2 2 2 (1 * sin 2
# cos 2
+)
2

48
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Equivalent 2-D
Drag Coefficient

D / cos ! D / cos !
C D eq = = =
" $ b ' " p M 2 cb
p# M eq c [ cos ! ] &
2
) # eq
2 % cos ! ( 2
D / cos ! C D / cos !
=
"
p# M # cb (1 * sin ! cos + )
2 2 2 (1 * sin 2
! cos 2
+)
2
49
MAE 5420 - Compressible Fluid Flow
 Equivalent 2-D Flow on Swept Wing
(cont’d)

• Solve for CL, CD, L/D

% L(
(
C L = C L eq 1 ! sin " cos # 2
L
2
) '& *)
D eq
$ =
C D = C D eq (
cos " 1 ! sin " cos #
2 2 D cos " )

50
MAE 5420 - Compressible Fluid Flow
 Example: Symmetric Double-wedge
• 5° ramps, M = 2.00
Airfoil
∞

• Total chord: 2 meters • 30 deg wing sweep

• Fineness ratio (t/c): 0.08749

• α = 5°

3 5 7
1
4 6
2

51
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
• 5° ramps, M∞ = 2.00 Airfoil
• Total chord: 2 meters

• Fineness ratio (t/c): 0.08749 • 30 deg wing sweep

• α = 5°

M eq = M ! 1 " sin # cos $ = 2 2

$ ! ! 2% 0.5
2 ' 1 & $" $" sin $" 30%# %# cos $" 5 %# %# ( = 1.7342
" 180 180 #

52
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
• 5° ramps, M∞ = 2.00 Airfoil (cont’d)
• Total chord: 2 meters

• Fineness ratio (t/c): 0.08749 • 30 deg wing sweep

• α = 5°
$ $ ! % %
$ tan (! ) ' & tan " 5# '
! eq = tan & "1
) = atan &
180 & 180 '
'
% cos # ( ! & cos $" !
30 %# '
" 180 #

= 5.769°

53
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
• 5° ramps, M∞ = 2.00 Airfoil (cont’d)
• Total chord: 2 meters

• Fineness ratio (t/c): 0.08749 • 30 deg wing sweep

• α = 5°

ceq = c [ cos ! ] = $
2 cos "
!
180
30%#

= 1.732

54
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
Airfoil (cont’d)
• Effective parameters

(
C L = C L eq 1 ! sin " cos # =
2 2
)
$ ! ! 2% 0.5
0.29212 ' 1 & $" $" sin $" 30%# %# cos $" 5 %# %# (
" 180 180 #

= 0.2533

55
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
Airfoil (cont’d)
• Effective parameters

(
C D = C D eq cos ! 1 " sin 2 ! cos 2 # = )
! $ ! ! % % 2%
0.5
$ $
0.05205 " cos " % % $ $ $
30# # ' 1 & " " sin " % % $
30# # cos " 5# # (
180 " 180 180 #

= 0.03909

L/D = 0.2533/0.03909=6.4799

56
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
Airfoil (cont’d)
• Compare to Unswept airfoil

• 5° ramps, M∞ = 2.00

• Total chord: 2 meters

• Fineness ratio (t/c): 0.08749

• α = 5°

• 0 deg wing sweep

57
MAE 5420 - Compressible Fluid Flow
 Example: Swept Symmetric Double-wedge
Airfoil (cont’d)
• Unswept Wing • 30° Swept Wing

C L: 0.205 C L: 0.2533
C D: 0.3606 C D: 0.03909
L/D: 5.68441 L/D: 6.4799

• WOW! … 14% IMPROVEMENT IN PERFORMANCE

58
MAE 5420 - Compressible Fluid Flow
 Next section: Overview of Turbulent
Compressible Boundary Layers and Friction
Induced Drag

59
MAE 5420 - Compressible Fluid Flow
 Section 7: Home Work

P"=26.436 kPa • Calculate Freestream

Mach Number
3
1 5
10°
2 10° 7
4 6 • Assume
γ=1.40

!=12° P04sensed=143.062kPa

Shock wave
P4=30.454 kPa

60
MAE 5420 - Compressible Fluid Flow