Page 163 - Aerodynamics for Engineering Students

146 Aerodynamics for Engineering Students

fdl?

Fig. 3.33

small (Le. drb/dz << 1) then the perpendicular and radial velocity components may be
considered the same. Thus the requirement that the net normal velocity be zero
becomes (see Fig. 3.33)

-

Sources Oncoming flow
So that the source strength per unit length and body shape are related as follows

dS
o(z) = u- (3.85)
dz
where S is the frontal area of a cross-section and is given by S = ~4.
In the limit as r + 0 Eqn (3.84) simplifies to

(3.86)

Thus once the variation of source strength per unit length has been determined
according to Eqn (3.85) the axial velocity can be obtained by evaluating Eqn (3.86)
and hence the pressure evaluated from the Bernoulli equation.
It can be seen from the derivation of Eqn (3.86) that both rb and drbldz must be
very small. Plainly the latter requirement would be violated in the vicinity of z = 0 if
the body had a rounded nose. This is a major drawback of the method.
The slender-body theory was extended by Munk* to the case of a body at an
angle of incidence or yaw. This case is treated as a superposition of two distinct
flows as shown in Fig. 3.34. One of these is the slender body at zero angle of
incidence as discussed above. The other is the slender body in a crossflow. For
such a slender body the flow around a particular cross-section is closely analogous
to that around a circular cylinder (see Section 3.3.9). Accordingly this flow can
be modelled by a distribution of point doublets with axes aligned in the direction

*Munk, M.M. (1934), Fluid Mechanics, Part VI, Section Q, in Aerodynamic Theory, volume 1 (ed.
W. Durand), Springer, Berlin; Dover, New York.