Steady Aerodynamics of Thin Wings

The finite wing aerodynamics, for special wing geometries, can yield analytical expressions for the aerodynamic coefficients in terms of the sectional properties of the wing. A special case is the elliptical span wise loading of the wing which is proportional to Jl2 — y2, where y is the span wise coordinate and l is the half span. For the wings with large span, using the Prandtl’s lifting line theory the wing’s lift coefficient CL becomes equal to the constant sectional lift coeffi­cient C[. Hence,

Cl = ci (1.9)

Another interesting aspect of the finite wing theory is the effect of the tip vortices on the overall performance of the wing. The tip vortices induce a vertical velocity which in turn induces additional drag on the wing. Hence, the total drag coefficient of the wing reads

Here the aspect ratio is AR = l2/S, and S is the wing area. For the symmetric and untwisted wings to have elliptical loading the plan form geometry also should be elliptical as shown in Fig. 1.3.

For the case of non-elliptical wings, we use the Glauert’s Fourier series expansion of the span wise variation of the circulation given by the lifting line theory. The integration of the numerically obtained span wise distribution of the circulation gives us the total lift coefficient.

If the aspect ratio of a wing is not so large and the sweep angle is larger than 15o, then we use the Weissenger’s L-Method to evaluate the lift coefficient of the wing.

For slender delta wings and for very low aspect ratio slender wings, analytical expressions for the lift and drag coefficients are also available. The lift coefficient for a delta wing without a camber in spanwise direction is

CL = — n AR a

The induced drag coefficient for delta wings having elliptical load distribution along their span is given as

Co, = Cl a/2

The lift and drag coefficients for slender delta wings are almost unaffected from the cross flow. Therefore, even at high speeds the cross flow behaves incom­pressible and the expressions given by Eqs. 1.11-1.12 are valid even for the supersonic ranges. In Chap. 4, the Weissenger’s L-Method and the derivation of Eqs. 1.11-1.12 will be seen in a detail.