SME
Compressible Flow
Supersonic aerodynamics
Use this for “A thin flat plate at α = 3° flies at M∞ = 2. Find cₗ, the wave drag cd and L/D.”
Diagram
Above, the flow speeds up to M = 2.111 through the fan; below, it slows to M = 1.892 behind the shock.
The pressure difference lifts the plate ⟂ the stream and drags it back.
This is an example. Type your own values to start a new problem.
αrad (rad)
=
α (°)
×
π
180
cₗ
=
4
×
αrad (rad)
M∞
2
−
1
cd
=
4
×
(
αrad (rad)
)
2
M∞
2
−
1
L/D
Lift-to-drag ratio
calculated
Formulas
αrad
=
α
×
π
180
0.05236
=
3
×
π
180
cₗ
=
4
×
αrad
÷
√
M∞² − 1
0.1209
=
4
×
0.05236
÷
√
2
2
− 1
cd
=
4
×
αrad
2
÷
√
M∞² − 1
0.006331
=
4
×
0.05236
2
÷
√
2
2
− 1
L/D
=
cₗ
cd
19.1
=
0.1209
0.006331
Assumptions
•
Thin airfoil at small α in linearized supersonic flow (Ackeret), good for M∞ from about 1.2 to 5.
•
No friction: cd here is the wave drag of the shocks and expansions only.
•
Thickness and camber add wave drag but leave cₗ the same; the plate has neither.
Step-by-step
WE KNOW
M∞ = 2, α = 3°
FIND
angle in radians (αrad), lift coefficient (cₗ), wave-drag coefficient (cd), lift-to-drag ratio (L/D)
Step 1 · Find angle in radians (αrad)
Use αrad = α × π ÷ 180
The
theory
works
in
radians:
multiply
the
degrees
by
π
180
.
αrad
=
3
×
π
180
αrad = 0.05236 rad
Step 2 · Find lift coefficient (cₗ)
Use cₗ = 4 × αrad ÷ √(M∞² − 1)
Ackeret’s
lift:
each
face’s
pressure
change
is
2
α
÷
√
M∞² − 1
,
and
the
two
faces
add.
cₗ
=
4
×
0.05236
÷
√
2
2
− 1
cₗ
=
4
×
0.05236
÷
√
4 − 1
cₗ
=
4
×
0.05236
÷
√
3
cₗ
=
4
×
0.05236
1.732
cₗ
=
0.2094
1.732
cₗ = 0.1209
Step 3 · Find wave-drag coefficient (cd)
Use cd = 4 × αrad² ÷ √(M∞² − 1)
Wave
drag:
the
lift
tilted
back
by
α
,
so
cd
=
cₗ
×
α
.
cd
=
4
×
0.05236
2
÷
√
2
2
− 1
cd
=
4
×
0.05236
2
÷
√
4 − 1
cd
=
4
×
0.05236
2
÷
√
3
cd
=
4
×
0.0027416
1.7321
cd
=
0.010966
1.7321
cd = 0.006331
Step 4 · Find lift-to-drag ratio (L/D)
Use L/D = cₗ ÷ cd
Lift
over
wave
drag;
in
this
theory
it
is
1
α
in
radians.
L/D
=
0.1209
0.006331
L/D = 19.1
Check
Put the numbers back in. Both sides should match.
0.05236
=
3
×
π
180
✓
0.1209
=
4
×
0.05236
÷
√
2
2
− 1
✓
0.006331
=
4
×
0.05236
2
÷
√
2
2
− 1
✓
19.1
=
0.1209
0.006331
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.