ESDU TM 169:2012

ESDU TM 169 provides a brief background to the development of
the compressible form of the Helmbold-Diederich equation for wing
lift-curve slope, a simple analytical equation which was in common
use prior to the advent of more reliable methods such as the
lifting-surface theory which was used to derive the data in ESDU
70011.

In its original form, involving only wing aspect ratio and sweep
to define the planform, the chordwise position of the wing sweep
was undefined, consistent with its infinite swept wing basis. The
earliest uses of the Helmbold-Diederich equation adopted the sweep
of the quarter-chord line (commonly used to define planform sweep)
whereas later uses employed the half-chord sweep in an attempt to
eliminate the effects of wing taper, not otherwise accounted for.
Comparisons are given of predictions using the Helmbold-Diederich
equation with both quarter-chord sweep and half-chord sweep against
the lifting-surface theory data for 80 planforms used in developing
ESDU 70011. The comparisons show that neither form has a maximum
error much less than 10%, although the half-chord sweep form is
rather better, albeit with some bias.

The residual effects of wing taper are shown to be largely
accounted for using empirical corrections similar to those
determined by Isaacs. The improved version of the
Helmbold-Diederich equation, using half-chord sweep, provides an
error band of about +2% to –3% when compared with the data from
ESDU 70011. Removal of just 10% of the 80 planforms (those with the
most extreme sweeps) corrects the slight bias to give an accuracy
of ±2%.