Page 138 - Aerodynamics for Engineering Students

Potential flow 121

or
rn
v=-
27r x2 + y2
and this is upwards.
This expression also shows, by comparing it, in the rearranged form x2 +y2-
(m/27rv)y = 0, with the general equation of a circle (x2 + y2 + 2gx + 2hy +f = 0),
that lines of constant vertical velocity are circles with centres (0, rn/47rv) and
radii rnl47rv.
The ultimate thickness, 2h (or height of cliff h) of the shape given by $ = 0 for this
combination is found by putting y = h and 0 = 7r in the general expression, i.e.
substituting the appropriate data in Eqn (3.18):

Therefore
h = m/2U (3.22)

Note that when 0 = ~/2, = h/2.
y
The position of the stagnation point
By finding the stagnation point, the distance of the foot of the cliff, or the front of the
fairing, from the source can be found. A stagnation point is given by u = 0, v = 0, i.e.
w rnx
u = - = 0 =--- U (3.23)
dY 27rx2 + y2
(3.24)

From Eqn (3.24) v = 0 when y = 0, and substituting in Eqn (3.23) when y = 0 and
x = xo:

when
xo = rn/2.rrU (3.25)

The local velocity
The local velocity q = dm.
w rn
jy=- and $ = -tan-' - Uy
dY 27r X
Therefore

rn 1/x
u=-
2-u
27r 1 + (y/x)