The
"curl" of the gradient of a scalar function is zero.
Objective: Get a method to describe flow velocity fields and relate them to surface shapes consistently.
Strategy: Describe the flow field as the effect of variations of one quantity.
Vector Identity:
The
"curl" of the gradient of a scalar function is zero.
Now, we defined the "vorticity" of flow as
This is a
measure of the Rotation of the flow.
means "irrotational
flow."
Now, we can define 
such
that
is the "velocity potential."
The definition implies:
Steady Incompressible Potential Flow
Continuity Eqn. for a fluid element
Incompressible: no change in 
due
to changes in velocity:
Steady: 
=> 
or: Continuity eqn. for a control volume:
Steady: 
const: 
Laplace's Equation
i.e., 
i.e., 
Note: Laplace's eqn. is simply the statement that mass is conserved, applied to a steady incompressible potential [no viscosity] flow.
Superposition
If 
is
each a solution of 
,
then 
is also a solution of ,
where A, B ... X are constants.
Thus, the solution for a complex problem can be expressed as the sum of solutions of several simpler problems.
Let's use these building blocks to analyze several flows:
1. Uniform flow + source:
2. Uniform flow + source + sink.
3. As source and sink come close together, you get
a doublet.
4. Uniform flow + vortex.
This looks like flow over a cylinder.
We see that (1) –> (3) are all symmetric about the
freestream direction. (4) is not.
Boundary Conditions
This is what specifies the details of the problem
in precise mathematical terms.
(i) Along the surface of the airfoil, the flow must
be tangential to the surface: Component of velocity 
to
the surface is zero. i.e., 
at the line describing the surface.
Another thing about the streamline closest to the
surface: it has to be tangential to the surface. 
.
This is another way to express condition (1).
(ii) The disturbance due to an object must die away as you go far from the surface [any solution which says otherwise is physically unrealistic].
Thus, as 
Elementary Solutions of the Laplace Equation
(i) Uniform flow:
If we are always using
,
drop the constant; it makes no difference.
(ii) Source or sink (note: a sink is simply a negative source).
Except at the origin, 
At the origin, there is mass flow being added or
subtracted ["singular point"]. [ There's flow out of the plane if 2-D.
]
c is a constant, related to the volume flow from/to the source/sink.
where 
is
the source strength: 
(iii) Doublet
(iv) Vortex
Let us take the "circulation" around the vortex
at radius r.
Note that if you take the circulation at any radius,
you'll get the same value of 
.
The only vorticity is concentrated at the center: the flow is irrotational
everywhere except at the center.
What is special about the fourth one? Let's try computing
the circulation for the first 3: we get zero 
.
However, for the 4th, the circulation 
.
Consider a segment of a vortex filament as shown;
with strength .
Velocity
induced at point P by the segment 
is
Note: This is a general result for potential
fields, taken by analogy from the result for the magnetic field 
induced
by a segment 
of
a conductor carrying current I, in a medium of permeability µ:
is 
where 
is
the angle between 
and 
.
Thus, velocity induced at P by an infinite, straight vortex filament is
or
where
h is the 
distance to the
vortex sheet. Thus, the induced velocity drops off as 1/h and h increases.
A vortex sheet is a continuous sheet of vortices.
It is used to represent
a "shear layer", across which the velocity changes, as shown below.
Define 
as
the strength of the vortex sheet per unit length of the vortex sheet. Thus,
strength of an infinitesimal portion 
of
the sheet is 
.
Velocity induced at P by this segment is
Velocity potential at P due to vortex sheet segment is
where is
the angle of r from your axis reference.
Circulation around the segment a –> b is
Circulation is also 
As dn –> 0, 
Local jump in tangential vel. across 
vortex sheet = local sheet strength 
Example 1 Question: How can we put in a set
of simple flows so that the streamlines
look the same as if there were an airfoil
present?
Overall, what happens is
Close up, we see other effects:
Example 2