Page 251 - Aerodynamics for Engineering Students

234 Aerodynamics for Engineering Students

Strength, ksxl
/, Strength, (kt all ak 6z,)8xl ,Strength, kSx,
--___
I
I
1
I
k Z n g t h ,
- 81, sx,
1

(a ) Horseshoe vortices (b) L-shaped vortices
Fig. 5.24 Equivalence between distributions of (a) horseshoe and (b) L-shaped vortices

when calculating the induced velocity. These problems can be overcome by recom-
bining the elements in the way depicted in Fig. 5.24. Here it is recognized that partial
cancellation occurs for two elemental horseshoe vortices occupying adjacent span-
wise positions, z and z + 6z. Accordingly, the horseshoe-vortex element can be
replaced by the L-shaped vortex element shown in Fig. 5.24. Note that although this
arrangement appears to violate Helmholtz’s second theorem, it is merely a math-
ematically convenient way of expressing the model depicted in Fig. 5.23 which fully
satisfies this theorem.

5.5 Relationship between spanwise loading
and trailing vorticity

It is shown below in Section 5.5.1 how to calculate the velocity induced by
the elements of the vortex sheet that notionally replace the wing. This is an essential
step in the development of a general wing theory. Initially, the general case
is considered. Then it is shown how the general case can be very considerably
simplified in the special case of wings of high aspect ratio. The general case is
then dropped, to be taken up again in Section 5.8, and the assumption of large aspect
ratio is made for Section 5.6 and the remainder of the present section. Accordingly,
some readers may wish to pass over the material immediately below and go
directly to the alternative derivation of Eqn (5.32) given at the end of the present
section.

5.5.1 Induced velocity (downwash)
Suppose that it is required to calculate the velocity induced at the point Pl(x1, zl) in
the y = 0 plane by the L-shaped vortex element associated with the element of wing
surface located at point P (x, z) now relabelled A (Fig. 5.25).