Fluids
– Lecture 12
Notes
1. Stream Function
2. Velocity
Potential
Reading: Anderson 2.14, 2.15
Stream Function
Definition
Consider
defining
the components of
the 2D
mass ﬂux
vector
ρ
vector
V
as the partial derivatives
¯
of
a scalar
stream
function
, denoted by
ψ
(
x,
y
):
¯
¯
∂ψ
∂ψ
ρu
=
,
ρv
=
−
∂y
∂x
For
low
speed ﬂows,
ρ
is just
a
known constant, and it
is more convenient
to
work
with a
scaled stream function
¯
ψ
ψ
(
x,
y
) =
ρ
which then
gives the components of the velocity
vector
vector
V
.
∂ψ
∂ψ
u
=
,
v
=
−
∂y
∂x
Example
Suppose
we specify
the constantdensity
streamfunction to
be
�
1
2
ψ
(
x,
y
)
= ln
x
2
+
y
2
=
ln(
x
2
+
y
)
2
which has a circular
“funnel”
shape as shown in the figure. The implied velocity components
are
then
∂ψ
y
∂ψ
−
x
u
=
∂y
=
x
2
+
y
2
,
v
=
−
=
∂x
x
2
+
y
2
which corresponds to
a
vortex
ﬂow
around the origin.
ψ
x
y
vortex flow example
1
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Streamline interpretation
The stream function
can
be interpreted
in
a number of
ways.
First we
determine
the
differ
¯
ential
of
ψ
as follows.
¯
¯
∂ψ
∂ψ
¯
dψ
=
dx
+
dy
∂x
∂y
¯
dψ
=
ρu
dy
−
ρv
dx
¯
¯
Now
consider a
line along
which
ψ
is some
constant
ψ
1
.
¯
¯
ψ
(
x,
y
) =
ψ
1
¯
¯
Along this
line,
we can
state that
dψ
=
dψ
1
=
d
(constant) = 0,
or
dy
v
ρu
dy
−
ρv
dx
= 0
→
=
dx
u
¯
which is
recognized
as the equation
for a streamline.
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 Fall '05
 MarkDrela
 Derivative, Velocity, velocity potential, dΨ, implied velocity components

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