Page 127 - Aerodynamics for Engineering Students

1 10 Aerodynamics for Engineering Students

demonstrating the validity of (iv), Le. a flow described by a unique velocity potential
must be irrotational.
Alternatively substituting (iii) in (ii) and (iv) in (i) the criteria for irrotational
continuous flow are that

a24 a24
+-
-+-=o=- a=+ a=+
8x2 ay2 8x2 ay= (3.4)
also written as V2q5 = V2$ = 0, where the operator nabla squared

a2 a2
v =-+-
ax= ay=
Eqn (3.4) is Laplace's equation.

3.3 Standard flows in terms of w and @

There are three basic two-dimensional flow fields, from combinations of which all
other steady flow conditions may be modelled. These are the uniform parallelflow,
source (sink) and point vortex.
The three flows, the source (sink), vortex and uniform stream, form standard flow
states, from combinations of which a number of other useful flows may be derived.

3.3.1 Two-dimensional flow from a source
(or towards a sink)
A source (sink) of strength m(-m) is a point at, which fluid is appearing (or
disappearing) at a uniform rate of m(-m)m2 s- . Consider the analogy of a
small hole in a large flat plate through which fluid is welling (the source). If there
is no obstruction and the plate is perfectly flat and level, the fluid puddle will get
larger and larger all the while remaining circular in shape. The path that any particle
of fluid will trace out as it emerges from the hole and travels outwards is a purely
radial one, since it cannot go sideways, because its fellow particles are also moving
outwards.
Also its velocity must get less as it goes outwards. Fluid issues from the hole at a
rate of mm2 s- . The velocity of flow over a circular boundary of 1 m radius is
m/27rm s-I. Over a circular boundary of 2m radius it is m/(27r x 2), i.e. half as much,
and over a circle of diameter 2r the velocity is m/27rr m s-'. Therefore the velocity of
flow is inversely proportional to the distance of the particle from the source.
All the above applies to a sink except that fluid is being drained away through the
hole and is moving towards the sink radially, increasing in speed as the sink is
approached. Hence the particles all move radially, and the streamlines must be radial
lines with their origin at the source (or sink).

To find the stream function w of a source
Place the source for convenience at the origin of a system of axes, to which the point
P has ordinates (x, y) and (r, 0) (Fig. 3.6). Putting the line along the x-axis as $ = 0