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Vortex Sheets
A vortex sheet is
defined as a planar array of parallel vortex filaments. Consider a uniform vortex sheet, lying in the

plane, in which the
vortex filaments run parallel to the
axis. (See Figure 9.7.) The vorticity within the sheet can be written

(9.54) 
where
is a Dirac delta function.
Here,
is the sheet's vortex intensity per unit length. Let
and
be the
component of the fluid velocity immediately above and
below the sheet, respectively. Consider a small rectangular loop in the

plane that straddles the sheet, as shown in the figure.
Integration of
around the loop (making use of the curl theorem) yields

(9.55) 
In other words, a vortex sheet induces a discontinuity in the tangential flow across the sheet. The previous expression can easily be generalized to
give

(9.56) 
where
is the sheet's vortex intensity per unit length,
is a unit vector normal to the sheet, and
is the jump in tangential velocity across the sheet (traveling in the direction of
). Furthermore, it is reasonable to assume that the previous
relation holds locally for nonplanar and nonuniform vortex sheets.
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Up: Incompressible Aerodynamics
Previous: Zhukovskii's Hypothesis
Richard Fitzpatrick
20160331