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The Centre Section Shape of Swept Tapered Wings with a Linear Chordwise Load Distribution

The main purpose of this paper is to put on record the Values of
integrals required in the computation of the downwash at the centre section
of tapered swept wings with constant spanwise load distribution, where the
standard methods of linear theory lead to singularities and so break down.
Unfortunately the exact linear load distribution cannot be used without
producing very complicated integrals. Consequently an approximation has
been made by means of which the integrals are calculable, though the work is
tedious. The approximation could have been further extended to terms of
higher than the second order in e (: tan QPLE wtan (PTE) and the integrals

could still have been evaluated, but the extensions seemed scarcely
necessary at the Mach numbers and sweeps and tapers of interest in this
connection. Indeed, Fig.3 shows that over the first 40% of the chord the
untapered case (8 = 0) gives a sufficiently accurate approximation. The
second correction term proportional to 83 only gives a correction greater
than 0.5% of the untapered value at points very close to the trailing edge,
where there is a logarithmic singularity in any case. The first correction
term (preportional to 8) gives very much greater corrections amounting to
about 5079 at 0.9 chord, and rising omrards from there. It accounts almost
entirely for the divergencies of the curves from the untapered case.

Fig.3 shows the dowrm'ash for an R.A.E. 10’] section with thickness/chord
ratio of 670, at a Mach number of 1.2 for various sweeps and tapers. Fig.4
shows the corresponding angles of twist required. It appears that taper
increases the angle of twist required to maintain the linear load distribution
for a given mean sweep.