Prandtl-Meyer Expansion Waves
When a supersonic flow is turned around a corner, an expansion fan occurs
producing a higher speed, lower pressure, etc. in an isentropic process.
fan
Forward
Mach line
Rearward
Mach line
µ1
M1 > 1
µ2
θ
waves
∗ Just as we saw with shock waves, if we apply conservation of mass and
momentum across a single wave, the tangential velocity is unchanged.
∗ Unlike a shock wave, an expansion wave is isentropic.
So let’s pick out a single wave:
Mach wave
w
V
u
µ
V
V + dv
From law of sines:
⎞
⎛π
sin ⎜ + µ ⎟
V + dV
⎠
⎝2
=
V
⎞
⎛π
sin ⎜ − µ − dθ ⎟
⎠
⎝2
1
, we can find :
M
dV
dθ = M 2 − 1
V
Using dθ → 0 & sin µ =
dθ
du
Prandtl-Meyer Expansion Waves
dv dM da
=
+
v
M
a
Using adiabatic relationships, we can re-write:
Next, v = M a ⇒
γRTo
γRT
=
ao
Tc
∂ −1 2
M
=
= 1+
a
T
∂
−1
da
⎛ γ −1⎞ ⎛ γ −1 2 ⎞
⇒
= −⎜
M ⎟⎟ dM
⎟ M ⎜⎜1 +
a
γ
⎝ 2 ⎠ ⎝
⎠
dv
dM
1
M 2 − 1 dM
⇒
=
⇒
dθ =
γ −1 2 M
v
γ −1 2 M
M
1+
M
1+
2
2
Finally, integrating dθ we find:
θ = v( M 2 ) − v( M 1 )
Where
γ + 1 −1 γ − 1 2
v( M ) =
tan
( M − 1) − tan −1 M 2 − 1
γ −1
γ −1
Prandtl-Meyer function
Problem: Estimate the rates of the pressure inside the pitot probe to the
freestream static pressure, pa p∞ .
M∞ = 3
16.100 2002
15°
Pressure
in pitot
probe
= pa
2