AE 3003 Chapter III  Handout #2
Sources, Sinks and Doublets  the Building Blocks of Potential Flow
In the previous handout we developed the following equation for the velocity potential:
0
0
2
2
2
2
2
2
2
=
∇
=
∂
∂
+
∂
∂
+
∂
∂
φ
Or
z
y
x
(1)
where the operator
2
∇
is called the Laplacian operator. This equation holds for 2D and
3D inviscid irrotational flows. If we are only interested in 2D irrotational inviscid
flows, we may also solve for:
0
2
=
∇
ψ
(2)
where
is the stream function.
After we have solved for the velocity potential or the stream function, we can compute
the velocities. In a Cartesian coordinate system, for 2D flows,
we will use:
x
y
v
y
x
u
∂
∂

=
∂
∂
=
∂
∂
=
∂
∂
=
(3)
In a polar coordinate system, for 2D flows we will use:
r
v
r
r
v
r
∂
∂

=
∂
∂
=
=
∂
∂
=
∂
∂
=
=
θ
r
1
velocity
Tangential
1
velocity
Radial
(4)
In 3D, the velocities are given only in terms of the velocity potential, as follows:
z
w
y
v
x
u
Or
V
∂
∂
=
∂
∂
=
∂
∂
=
∇
=
,
(5)
Once the velocity is known, we can find pressure from the Bernoulli's equation.
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View Full DocumentIn this section, we consider some simple solutions to the Laplace's equation (1 or
2). Since equation 91) and 92) are linear, we can superpose many such simple solutions
to arrive at a more complex flow field. This is like building a complex configuration
using Lego blocks. The individual simple solutions are the individual Lego pieces, which
on their own, are not very interesting. Together, however, they can solve some very
interesting flows, including flow over airfoils and wings.
Building Block #1: 2D Sources and Sinks:
A source is like a lawn sprinkler. It sprays
the water (or air) radially, and equally, in all the directions, at the rate of Q units per unit
time. If this is a sink (e.g. a drain hole on a concrete pavement) the velocity vectors will
still be radial, but directed inwards towards the center. The sign of Q will be positive for
a source, and negative for a sink.
Consider a circle of radius r enclosing this source. Let v
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 Fall '08
 Yeung
 Cartesian Coordinate System, Polar coordinate system, potential flow, uniform flow, stream function, Doublets

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