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Fluids – Lecture 12 Notes
- Stream Function
- Velocity Potential
Reading: Anderson 2.14, 2.
Stream Function
Definition
Consider defining the components of the 2-D mass flux vector ρ
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
V.
∂ψ ∂ψ
u = , v = −
∂y ∂x
Example
Suppose we specify the constant-density streamfunction to be
�
1 2
ψ(x, y) = ln x
2
2 = ln(x
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
2
, v = − =
∂x x
2
2
which corresponds to a vortex flow around the origin.
x
y
vortex flow example
�
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. Hence, lines of constant ψ(x, y) are
streamlines of the flow. Similarly, for the constant-density case, lines of constant ψ(x, y) are
streamlines of the flow. In the example above, the streamline defined by
ln x
2
2 = ψ 1
can be seen to be a circle of radius exp(ψ 1
V
d
dx
dy
−
u
v
− −
y 1
x
Mass flow interpretation
Consider two streamlines along which ψ has constant values of ψ 1
and ψ 2
. The constant
mass flow between these streamlines can be computed by integrating the mass flux along
any curve AB spanning them. First we note the geometric relation along the curve,
nˆ dA = ˆı dy − ˆ dx
and the mass flow integration then proceeds as follows.
� B
� B
� B
m = ρ
˙ V · ˆn dA = (ρu dy − ρv dx) = dψ = ψ 2 − ψ 1
A A A
Hence, the mass flow between any two streamlines is given simply by the difference of their
stream function values.
x
y
vortex flow example
which corresponds to a vortex flow around the origin. Note that this is exactly the same
velocity field as in the previous example using the stream function.
Irrotationality
If we attempt to compute the vorticity of the potential-derived velocity field by taking its
curl, we find that the vorticity vector is identically zero. For example, for the vorticity
x-component we find
∂w ∂v ∂ ∂φ ∂ ∂φ ∂
2 φ ∂
2 φ
ξ x
∂y ∂z ∂y ∂z ∂z ∂y ∂y∂z ∂z∂y
and similarly we can also show that ξ y
= 0 and ξ z
= 0. This is of course just a manifestation
of the general vector identity curl(grad) = 0. Hence, any velocity field defined in terms
of a velocity potential is automatically an irrotational flow. Often the synonymous term
potential flow is also used.
Directional Derivative
In many situations, only one particular component of the velocity is required. For example,
for computing the mass flow across a surface, we only require the normal velocity component.
This is typically computed via the dot product
V · nˆ. In terms of the velocity potential, we
have
∂φ
~ V · ˆn = ∇φ · ˆn =
∂n
where the final partial derivative ∂φ/∂n is called the directional derivative of the potential
along the normal coordinate n. The figure illustrates the relations.
n
1
2
3
4
V n =
^ 6 φ
6 n
V =
In general, the component of the velocity along any direction can be obtained simply by
taking the directional derivative of the potential along that same direction.
