Page 187 - Aerodynamics for Engineering Students

170 Aerodynamics for Engineering Students

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Fig. 4.8 Zhukovsky transformation, of the flow around a circular cylinder with circulation, to that around
an aerofoil generating lift
members of this family of shapes have a cusped trailing edge whereas the aerofoils
used in practical aerodynamics have trailing edges with finite angles. Kkrmkn and
Trefftz* later devised a more general conformal transformation that gave a family of
aerofoils with trailing edges of finite angle. Aerofoil theory based on conformal
transformation became a practical tool for aerodynamic design in 1931 when the
American engineer Theodorsen' developed a method for aerofoils of arbitrary shape.
The method has continued to be developed well into the second half of the twentieth
century. Advanced versions of the method exploited modern computing techniques
like the Fast Fourier Transform.**
If aerodynamic design were to involve only two-dimensional flows at low speeds,
design methods based on conformal transformation would be a good choice. How-
ever, the technique cannot be extended to three-dimensional or high-speed flows. For
this reason it is no longer widely used in aerodynamic design. Methods based on
conformal transformation are not discussed further here. Instead two approaches,
namely thin aerofoil theory and computational boundary element (or panel) methods,
which can be extended to three-dimensional flows will be described.
The Zhukovsky theory was of little or no direct use in practical aerofoil design.
Nevertheless it introduced some features that are basic to any aerofoil theory. Firstly,
the overall lift is proportional to the circulation generated, and secondly, the magni-
tude of the circulation must be such as to keep the velocity finite at the trailing edge,
in accordance with the Kutta condition.
It is not necessary to suppose the vorticity that gives rise to the circulation be due
to a single vortex. Instead the vorticity can be distributed throughout the region
enclosed by the aerofoil profile or even on the aerofoil surface. But the magnitude of
circulation generated by all this vorticity must still be such as to satisfy the Kutta
condition. A simple version of this concept is to concentrate the vortex distribution
on the camber line as suggested by Fig. 4.9. In this form, it becomes the basis of the
classic thin aerofoil theory developed by Munk' and G1auert.O
Glauert's version of the theory was based on a sort of reverse Zhukovsky trans-
formation that exploited the not unreasonable assumption that practical aerofoils are

* 2. Fhgtech. Motorluftschiffahrt, 9, 11 1 (1918).
NACA Report, No. 411 (1931).
** N.D. Halsey (1979) Potential flow analysis of multi-element airfoils using conformal mapping, AZAA J.,
12, 1281.
NACA Report, No. 142 (1922).
Aeronautical Research Council, Reports and Memoranda No. 910 (1924).