Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 357
Further rearranging equation (10.154) provides
⎛ ⎞
=0
r
{ }} 2
{ }} {
{
⎜
⎝x 2 + y 2 ⎟
⎠ = Q r sin θ ( )
0 {}}{
2 ( ) 2 Q0 Q0
y − +
(10.155)
2 πψ 2 πψ 2 πψ
and converting to the standard equation of circles as
y− Q 0
2 πψ
{ }} {
y 2 − Q ( ) 2
0
2 πψ y + Q0
+ x 2 =
2 πψ
( ) 2 Q0
(10.156)
2 πψ
End Caution: mathematical details
The equation (10.153) (or (10.156)) represents a circle with a radius of
Q 0
2 πψ
with location at x =0and y = ± Q 0
2 πψ
. The identical derivations can be done for the
potential function. It can be noticed that the difference between the functions results
from difference of r sin θ the instead of the term is r cos θ. Thus, the potential functions
are made from circles that the centers are at same distance as their radius from origin
on the x coordinate. It can be noticed that the stream function and the potential
function can have positive and negative values and hence there are family on both sides
of coordinates. Figure 10.12 displays the stream functions (cyan to green color) and
potential functions (gold to crimson color). Notice the larger the value of the stream
function the smaller the circle and the same for the potential functions.
It must be noted that in the derivations above it was assumed that the sink is
on the left and source is on the right. Clear similar results will obtained if the sink and
source were oriented differently. Hence the dipole (even though) potential and stream
functions are scalar functions have a direction. In this stage this topic will not be treated
but must be kept in question form.
Example 10.6:
This academic example is provided mostly for practice of the mathematics. Built the
stream function of dipole with angle. Start with a source and a sink distance r from
origin on the line with a angle β from x coordinates. Let the distance shrink to zero.
Write the stream function.
10.3.1 Flow Around a Circular Cylinder
After several elements of the potential flow were built earlier, the first use of these
elements can be demonstrated. Perhaps the most celebrated and useful example is the
flow past a cylinder which this section will be dealing with. The stream function made
by superimposing a uniform flow and a doublet is
ψ = U 0 y + Q 0 sin θ
= U 0 r sin θ + Q 0 r sin θ
2 π r
2 π r 2 (10.157)