This preview shows pages 1–2. Sign up to view the full content.
10
CHAPTER 11. POTENTIAL FLOW
11.8
Chapter 11, Problem 8
Problem:
For steady, compressible, twodimensional flows with (variable) density
ρ
,
we can define a “streamfunction” according to
u
=
1
ρ
∂
˜
ψ
∂
y
,v
=
−
1
ρ
∂
˜
ψ
∂
x
(a) What are the dimensions of
˜
ψ
?
(b) Verify that conservation of mass is satisfied.
(c) If the flow is irrotational, verify that
∇
2
˜
ψ
=
u
×
∇
ρ
where
˜
ψ
≡
˜
ψ
k
.
Solution: (a)
To determine the dimensions of
˜
ψ
, note that from its definition,
[
u
]=
1
[
ρ
]
^
∂
˜
ψ
∂
y
±
=
⇒
L
T
=
1
M/L
3
[
˜
ψ
]
L
=
L
2
M
[
˜
ψ
]
Therefore, we conclude that
[
˜
ψ
M
LT
(b)
Substituting into the continuity equation, we find
∂
∂
x
(
ρ
u
)+
∂
∂
y
(
ρ
v
)=
∂
∂
x
X
∂
˜
ψ
∂
y
~
+
∂
∂
y
X
−
∂
˜
ψ
∂
x
~
=
∂
2
˜
ψ
∂
x
∂
y
−
∂
2
˜
ψ
∂
y
∂
x
=0
(c)
If the flow is irrotational, we have
∂
v
∂
x
−
∂
u
∂
y
=
⇒
∂
∂
x
^
−
1
ρ
∂
˜
ψ
∂
x
±
−
∂
∂
y
^
1
ρ
∂
˜
ψ
∂
y
±
Performing the indicated differentiations and multiplying through by
(
−
1)
,
1
ρ
∂
2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '06
 Phares

Click to edit the document details