Page 190 - Aerodynamics for Engineering Students

Two-dimensional wing theory 173

Cumbered plate at incidence Symmetric aerofoil at zero
(circulatory flow ) incidence
( non-circulatory flow)
Fig. 4.11 Cambered thin aerofoil at incidence as superposition of a circulatory and non-circulatory flow

Thus the non-circulatory flow is given by the solution of potential flow subject to
the boundary condition v' = f U dyt/dx which is applied at y = 0 for 0 5 x 5 c. The
solution of this problem is discussed in Section 4.9. The lifting characteristics of the
aerofoil are determined solely by the circulatory flow. Consequently, it is the solution
of this problem that is of primary importance. Turn now to the formulation and
solution of the mathematical problem for the circulatory flow.
It may be seen from Sections 4.1 and 4.2 that vortices can be used to represent
lifting flow. In the present case, the lifting flow generated by an infinitely thin
cambered plate at incidence is represented by a string of line vortices, each of
infinitesimal strength, along the camber line as shown in Fig. 4.12. Thus the camber
line is replaced by a line of variable vorticity so that the total circulation about the
chord is the sum of the vortex elements. This can be written as

r = L'kds (4.13)

Fig. 4.12 Insert shows velocity and pressure above and below 6s