Page 280 - Aerodynamics for Engineering Students

Finite wing theov 263

For the delta wing b = 2x tan A so that
LC
1‘ bg dx = 4 tan’ A x dx = 2c’ tan’ A

Eqn (5.75) then gives

L
CL = = 2ir tan A sin a cos a (5.76)
ipU&cztanA
The drag is found in a similar fashion except that now the pressure force has to be resolved
in the direction of the free stream, so that CD oc sin a whereas CL 3; cos a therefore
CD = CL tan a (5.77)

For small a, sin a tan a N a. Note also that the aspect ratio (AR) = 4 tan A and that for
small a Eqn (5.76) can be rearranged to give

a=- CL
27~ tan A

Thus for small a Eqn (5.77) can also be written in the form

(5.78)

Note that this is exactly twice the corresponding drag coefficient given in Eqn (5.43) for an
elliptic wing of high aspect ratio.

At first sight the procedure outlined above seems to violate d’Alembert’s Law (see
Section 4.1) that states that no net force is generated by a purely potential flow
around a body. For aerofoils and wings it has been found necessary to introduce
circulation in order to generate lift and induced drag. Circulation has not been
introduced in the above procedure in any apparent way. However, it should be noted
that although the flow around each spanwise wing section is assumed to be non-
circulatory potential flow, the integrated effect of summing the contributions of each
wing section will not, necessarily, approximate the non-circulatory potential flow
around the wing as a whole. In fact, the purely non-circulatory potential flow around
a chordwise wing section, at the centre-line for example, will look something like that
shown in Fig. 4.la above. By constructing the flow around the wing in the way
described above it has been ensured that there is no flow reversal at the trailing edge
and, in fact, a kind of Kutta condition has been implicitly imposed, implying that the
flow as a whole does indeed possess circulation.
The so-called slender wing theory described above is of limited usefulness because
for wings of small aspect ratio the ‘wing-tip’ vortices tend to roll up and dominate the
flow field for all but very small angles of incidence. For example, see the flow field
around a slender delta wing as depicted in Fig. 5.40. In this case, the flow separates
from the leading edges and rolls up to form a pair of stable vortices over the upper
surface. The vortices first appear at the apex of the wing and increase in strength on
moving downstream, becoming fully developed by the time the trailing edge is
reached. The low pressures generated by these vortices contribute much of the lift.