Page 162 - Aerodynamics for Engineering Students

Potential flow 145
A/ rw

Sources

,xxxxxxxxxxq
U
X
)

Fig. 3.32 Flow over a slender body of revolution modelled by source distribution

For Munk's slender-body theory it is assumed that the radius of the body is very
much smaller than its total length. The flow is modelled by a distribution of sources
and sinks placed on the z axis as depicted in Fig. 3.32. In many respects this theory is
analogous to the theory for calculating the two-dimensional flow around symmetric
wing sections - the so-called thickness problem (see Section 4.9).
For an element of source distribution located at z = z1 the velocity induced at
point P (r, z) is

(3.82)

where a(z1) is the source strength per unit length and o(zl)dzl takes the place of Q in
Eqn (3.63). Thus to obtain the velocity components in the r and z directions at P due
to all the sources we resolve the velocity given by Eqn (3.82) in the two coordinate
directions and integrate along the length of the body. Thus
I
qr = 1 qRsinP

(3.83)

(3.84)

The source strength can be related to the body geometry by the following physical
argument. Consider the elemental length of the body as shown in Fig. 3.33. If the
body radius rb is very small compared to the length, 1, then the limit r --+ 0 can be
considered. For this limit the flow from the sources may be considered purely radial
so that the flow across the body surface of the element is entirely due to the sources
within the element itself. Accordingly
2rrq,dzl = a(z1)dzl at r = rb provided rb + 0
But the effects of the oncoming flow must also be considered as well as the sources.
The net perpendicular velocity on the body surface due to both the oncoming flow
and the sources must be zero. Provided that the slope of the body contour is very