Critical Mach Number
We can estimate the freestream Mach number at which the flow first accelerates
above M > 1 (locally) using the Prandtl-Glauert scaling and isentropic
relationships.
Recall from P-G:
On the airfoil
surface:
C p (M ∞ ) =
C p ( M ∞ = 0)
1 − M ∞2
If we have C p ( M ∞ = 0) say from an incompressible panel solution, we could then
find C p anywhere on the airfoil for higher M ∞ under the assumptions of P-G
(linearized flow, M ∞ < 1) .
We can also use isentropic relationships:
Cp = −
2
p
(
− 1)
2
γM ∞ p ∞
γ
⎤
⎡
γ −1
1
⎛
⎞
⎢⎜ 1 + (γ − 1) M 2 ⎟
⎥
∞
2 ⎢⎜
2
⎟ − 1⎥
⇒ Cp =
2 ⎢
⎥
1
γM ∞ ⎜ 1 + (γ − 1) M 2 ⎟
⎢⎜
⎥
⎟
2
⎠
⎢⎣⎝
⎥⎦
The C p for M = 1 at a given M ∞ is:
C pcrit
⎡
⎢⎛⎜ 1 + 1 (γ − 1) M 2
∞
2 ⎢⎜
2
= C p ( M = 1, M ∞ ) =
γM ∞2 ⎢⎜ 1 + 1 (γ − 1)
⎢⎜
2
⎢⎣⎝
⎞
⎟
⎟
⎟
⎟
⎠
γ
γ −1
⎤
⎥
− 1⎥⎥
⎥
⎥⎦
This critical freestream M ∞ occurs when C pP −G ( M cr ) = C pcrit ( M cr ).
This critical M ∞ can be found graphically or can be solved for with a root-finding
method. Let’s look at what happens graphically:
Critical Mach Number
− Cp
C pmin
C pcrit
M cr
M∞
1. Find minimum C p at M ∞ = 0
2. Plot
C pmin P −G vs. M ∞
3. Plot C pcrit from isentropic relationships
16.100 2002
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