Page 239 - Aerodynamics for Engineering Students

222 Aerodynamics for Engineering Students

. .
The core
Fig. 5.13 Velocity distribution in a real vortex with a core

5.3 The simplified horseshoe vortex

A simplified system may replace the complete vortex system of a wing when con-
sidering the influence of the lifting system on distant points in the flow. Many such
problems do exist and simple solutions, although not all exact, can be readily
obtained using the suggested simplification. This necessitates replacing the wing by
a single bound spanwise vortex of constant strength that is turned through 90" at
each end to form the trailing vortices that extend effectively to infinity behind the
wing. The general vortex system and its simplified equivalent must have two things in
common:
(i) each must provide the same total lift
(ii) each must have the same value of circulation about the trailing vortices and
hence the same circulation at mid-span.
These equalities provide for the complete definition of the simplified system.
The spanwise distributions created for the general vortex system and its simplified
equivalent are shown in Fig. 5.14. Both have the same mid-span circulation ro that
is now constant along part of the span of the simplified equivalent case. For
equivalence in area under the curve, which is proportional to the total lift, the span
length of the single vortex must be less than that of the wing.

T / ////////I/

TO
Total IiftipV
1
//////I/
(a Normal loading (b ) Equivalent simplified
loading

Fig. 5.14