SME
Compressible Flow · Problem type
Prandtl–Meyer expansion fan
Use this for “Air at M₁ = 2 turns round a 10° convex corner. Find ν₁, ν₂ and M₂.”
Diagram
ν rises by exactly θ: 26.38° + 10° = 36.38°.
The fan runs from μ₁ off the flow before to μ₂ off the flow after.
This is an example. Type your own values to start a new problem.
ν(
M₁
)
=
ν₁ (°)
ν₂ (°)
=
ν₁ (°)
+
θ (°)
ν(
M₂
)
=
ν₂ (°)
Formulas
ν₁
=
√
6
×
tan⁻¹(√(
M₁
2
− 1
6
))
−
tan⁻¹(
√
M₁
2
− 1
)
26.38
=
√
6
×
tan⁻¹(√(
2
2
− 1
6
))
−
tan⁻¹(
√
2
2
− 1
)
ν₂
=
ν₁
+
θ
36.38 = 26.38 + 10
ν₂
=
√
6
×
tan⁻¹(√(
M₂
2
− 1
6
))
−
tan⁻¹(
√
M₂
2
− 1
)
36.38
=
√
6
×
tan⁻¹(√(
2.385
2
− 1
6
))
−
tan⁻¹(
√
2.385
2
− 1
)
Assumptions
•
Isentropic expansion through a centred fan of Mach waves; air γ = 1.4.
•
The flow turns by the corner’s θ, and ν rises by exactly θ.
•
Angles are in degrees throughout, so set the calculator to degrees.
Step-by-step
WE KNOW
M₁ = 2, θ = 10°
FIND
Prandtl–Meyer angle before (ν₁), Prandtl–Meyer angle after (ν₂), Mach number after (M₂)
Step 1 · Find Prandtl–Meyer angle before (ν₁)
Use ν₁ = √6 × tan⁻¹(√((M₁² − 1) ÷ 6)) − tan⁻¹(√(M₁² − 1))
The
Prandtl–Meyer
function,
in
degrees;
for
air
√(
γ + 1
γ − 1
)
=
√
6
.
ν₁
=
√
6
×
tan⁻¹(√(
2
2
− 1
6
))
−
tan⁻¹(
√
2
2
− 1
)
ν₁
=
26.38°
Step 2 · Find Prandtl–Meyer angle after (ν₂)
Use ν₂ = ν₁ + θ
Turning
the
flow
away
by
θ
raises
ν
by
θ
.
ν₂
=
26.38
+
10
ν₂
=
36.38°
Step 3 · Find Mach number after (M₂)
Use ν₂ = √6 × tan⁻¹(√((M₂² − 1) ÷ 6)) − tan⁻¹(√(M₂² − 1))
No
rearrangement
puts
M₂
on
its
own,
so
it
is
found
by
trial:
each
next
try
is
where
a
straight
line
through
the
last
two
meets
the
target.
Try
M₂
=
2:
√
6
×
tan⁻¹(√(
2
2
− 1
6
))
−
tan⁻¹(
√
2
2
− 1
)
=
26.38
(want
36.38)
Try
M₂
=
3:
√
6
×
tan⁻¹(√(
3
2
− 1
6
))
−
tan⁻¹(
√
3
2
− 1
)
=
49.757
(want
36.38)
Try
M₂
=
2.4278:
√
6
×
tan⁻¹(√(
2.4278
2
− 1
6
))
−
tan⁻¹(
√
2.4278
2
− 1
)
=
37.416
(want
36.38)
Try
M₂
=
2.3798:
√
6
×
tan⁻¹(√(
2.3798
2
− 1
6
))
−
tan⁻¹(
√
2.3798
2
− 1
)
=
36.256
(want
36.38)
Try
M₂
=
2.3849:
√
6
×
tan⁻¹(√(
2.3849
2
− 1
6
))
−
tan⁻¹(
√
2.3849
2
− 1
)
=
36.38
(want
36.38)
M₂
=
2.385
Check
Put the numbers back in. Both sides should match.
26.38
=
√
6
×
tan⁻¹(√(
2
2
− 1
6
))
−
tan⁻¹(
√
2
2
− 1
)
✓
36.38 = 26.38 + 10 ✓
36.38
=
√
6
×
tan⁻¹(√(
2.3849
2
− 1
6
))
−
tan⁻¹(
√
2.3849
2
− 1
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.