Page 245 - Aerodynamics for Engineering Students

228 Aerodynamics for Engineering Students

Fig. 5.19

filaments. In order to satisfy Helmholtz’s second theorem (Section 5.2.1) each fila-
ment must either be part of a closed loop or form a horseshoe vortex with trailing
vortex filaments running to infinity. Even with this restriction there are still infinitely
many ways of arranging such vortex elements for the purposes of modelling the flow
field associated with a lifting wing. For illustrative purposes consider the simple
arrangement where there is a sheet of vortex filaments passing in the spanwise
direction through a given wing section (Fig. 5.19). It should be noted, however, that
at two, here unspecified, spanwise locations each of these filaments must be turned
back to form trailing vortex filaments.
Consider the flow in the vicinity of a sheet of fluid moving irrotationally in the xy
plane, Fig. 5.19. In this stylized figure the ‘sheet’ is seen to have a section curved in
the xy plane and to be of thickness Sn, and the vorticity is represented by a number of
vortex filaments normal to the xy plane. The circulation around the element of fluid
having sides Ss, Sn is, by definition, AI? = 56s. Sn where 5 is the vorticity of the fluid
within the area SsSn.
Now for a sheet Sn -0 and if 5 is so large that the product [Sn remains finite, the
sheet is termed a vortex sheet of strength k = CSn. The circulation around the
element can now be written
AI? = kSs (5.21)
An alternative way of finding the circulation around the element is to integrate the
tangential flow components. Thus
AI?= (UZ - u~)SS (5.22)

Comparison of Eqns (5.21) and (5.22) shows that the local strength k of the vortex
sheet is the tangential velocity jump through the sheet.
Alternatively, a flow situation in which the tangential velocity changes discontinu-
ously in the normal direction may be mathematically represented by a vortex sheet of
strength proportional to the velocity change.
The vortex sheet concept has important applications in wing theory.