Page 129 - Aerodynamics for Engineering Students

1 12 Aerodynamics for Engineering Students

Alternatively, since the velocity q is always radial (q = qn) it must be some function
of r only and the tangential component is zero. Now
qn=-=- m 84
27rr ar
Therefore
m m r
4 = Lor2md' = (3.7)

In Cartesian coordinates with 4 = 0 on the curve ro = 1

The equipotential pattern is given by 4 = constant. From Eqn (3.7)
m m
4 = -1nr - C where C = -1nro
27r 27r

which is the equation of a circle of centre at the origin and radius e2T($+o/m when 4 is
constant. Thus equipotentials for a source (or sink) are concentric circles and satisfy
the requirement of meeting the streamlines orthogonally.

3.3.2 Line (point) vortex
This flow is that associated with a straight line vortex. A line vortex can best be
described as a string of rotating particles. A chain of fluid particles are spinning on
their common axis and carrying around with them a swirl of fluid particles which flow
around in circles. A cross-section of such a string of particles and its associated flow
shows a spinningpoint outside of which is streamline flow in concentric circles (Fig. 3.7).
Vortices are common in nature, the difference between a real vortex as opposed to
a theoretical line (potential) vortex is that the former has a core of fluid which is
rotating as a solid, although the associated swirl outside is similar to the flow outside
the point vortex. The streamlines associated with a line vortex are circular and
therefore the particle velocity at any point must be tangential only.

.@

-3 A-3
0 Straight line Cross-section showing
a few of the associated
vortex
streamlines
Fig. 3.7