Linearized Compressible Potential Flow Governing Equation
Recall the 2-D full potential eqn is:
1
1
2
⎡
⎡
2⎤
2⎤
1
(
φ
)
φ
1
(
φ
)
φ
−
φ xφ yφ xy = 0
+
−
−
x
xx
y
yy
2
2
⎢⎣ a
⎥⎦
⎢⎣ a
⎥⎦
a2
γ −1
Where a 2 = a o2 −
(φ x ) 2 + (φ y ) 2
γ
[
]
As you saw, for small perturbations to a uniform flow, the linearized form of the
equation was:
φˆ = perturbation
(1 − M ∞2 )φˆxx + φˆyy = 0
potential
Where v
v
u = u ∞iv + uˆ
v
uˆ = φˆxvj + φˆyJv ⇒ uˆ = φˆx
vˆ = φˆy
perturbation
velocity
This equation is valid for both M ∞ < 1 and M ∞ > 1 . Note, though, it is not correct
for M ∞ → 1 or for M ∞ large, say greater than about 2 or so.
So what happens to the linearized potential equation for M ∞ > 1 :
Subsonic Flow
M∞ < 1
Supersonic
M∞ > 1
1 − M ∞2 > 0
⇒ Elliptic PDE
(Laplace’s eqn)
1 − M ∞2 < 0
⇒ Hyperbolic PDE
(wave eqn)
It also turns out that (1 − M ∞2 )φˆxx + φˆyy = 0 is much easier to solve when M ∞ > 1 .
Define
λ = M ∞2 − 1
⇒ λ2φˆxx − φˆyy = 0
Then,
φˆ( x, y ) = φˆ(η ) where η = x − λy
is a solution to the linearized potential. To see this:
Linearized Compressible Potential Flow Governing Equation
φˆx = φˆ′η x = φˆ′ ← φˆ′ ≡
dφˆ
dη
φˆy = φˆ′η y = −λφˆ′
Similarly,
φˆxx = φˆ′′
⇒ λ2φˆxx − φˆyy = λ2φˆ′′ − λ2φˆ′′
= 0√
φˆyy = λ φˆ′′
2
φˆ(η ) is solution to linearized
potential!
This means that φˆ is constant for lines described by η = x − λy = const. For
example, consider an airfoil:
φˆ = const.
η = x − λy = const
M∞ > 1
λ = M ∞2 − 1
V∞
dy
=
dx
1
M ∞2 − 1
So, the values of φˆ will be determined by the boundary conditions on the
surface. Recall:
v v
u • n = 0 on boundary
v
n
For linearized flow, this b.c. reduces to:
dy
v̂ = V∞θ
θ
dx
Also, note that:
uˆ = φˆx = φˆ′
vˆ = φˆy = −λφˆ′ ⇒ vˆ = −λuˆ
⇒
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Vθ
)
u=− ∞
λ
on boundary
2
Linearized Compressible Potential Flow Governing Equation
This is very useful because the linearized pressure coefficient is:
Cp =
⇒
p − p∞
2 uˆ
=−
1
V∞
ρ ∞ V ∞2
2
Cp =
2θ
on boundary!
M ∞2 − 1
⇒ C p > 0 for θ > 0 (i.e. a compression)
C p < 0 for θ < 0 (i.e. an expansion)
Using linear potential theory, let’s calculate the lift and drag coefficients for a flat
plate at α , M ∞ ,
C Ppu
y
α
M∞ > 1
C
x
C pl
Find Cl & Cd :
C pu =
C pl =
2θ u
M ∞2 − 1
=
2θ l
=
M ∞2 − 1
Then, the result is a force in the y - direction:
c⎛ 1
⎞
Fy = ∫ ⎜ ρ ∞V∞2 ⎟ C pl − C pu dx
o 2
⎝
⎠
(
)
(
)
⎛1
⎞
Fy = ⎜ ρ ∞V∞2 C ⎟ C pl − C pu
⎝2
⎠
Fy
⇒ C fy ≡
= C p l − C pu
1
ρ ∞V∞2 c
2
Finally, we need to rotate this into lift and drag directions:
C l = −C f x sin α + C f y cos α
C d = C f x cos α = C f y sin α
But, in our case, C f x = 0 & α << 1
⇒
Cl ≅ C f y
Cd ≅ C f y α
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3
Linearized Compressible Potential Flow Governing Equation
Cd
(linear
potential
theory)
0
0
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M∞
4