Page 160 - Aerodynamics for Engineering Students

Potential flow 143

omnidirectional. On the contrary the flow field is strongly directional. Moreover, the
case analysed above is something of a special case in that the source-sink pair lies on the
z axis. In fact the axis of the doublet can be in any direction in three-dimensional space.
For two-dimensional flow it was shown in Section 3.3.9 that the line doublet
placed in a uniform stream produces the potential flow around a circular cylinder.
Similarly it will be shown below that a point doublet placed in a uniform stream
corresponds to the potential flow around a sphere.
From Eqns (3.65) and (3.73) the velocity potential for a point doublet in a uniform
stream, with both the uniform stream and doublet axis aligned in the negative z
direction, is given by

4= -U Rcos~-- cosp (3.75)
47rR2
From Eqn (3.60) the velocity components are given by

(3.76)

(3.77)

The stagnation points are defined by qR = qp = 0. Let the coordinates of the
stagnation points be denoted by (&, ps). Then from Eqn (3.77) it can be seen that either
R3 =-- ’ or sinq, = o
47ru
The first of these two equations cannot be satisfied as it implies that R, is not a positive
number. Accordingly, the second of the two equations must hold implying that
ps=O and 7r (3.78a)
It now follows from Eqn (3.76) that

R, = (&)I” ( 3.78 b)
Thus there are two stagnation points on the z axis at equal distances from the origin.
From Eqns (3.66) and (3.74) the stream function for a point doublet in a uniform
flow is given by
$=-- R2 sin2 p + sin2 p (3.79)
2 47rR
It follows from substituting Eqns (3.78b) in Eqn (3.79) that at the stagnation points
$ = 0. So the streamlines passing through the stagnation points are described by

(3.80)

Equation (3.79) shows that when p # 0 or 7r the radius R of the stream-surface,
containing the streamlines that pass through the stagnation points, remains fixed
equal to R,. R can take any value when p = 0 or 7r. Thus these streamlines define the
surface of a sphere of radius R,. This is very similar to the two-dimensional case of
the flow over a circular cylinder described in Section 3.3.9.