The properties associated with an airfoil section which are indeed the same as the properties of a wing of infinite span. These properties are different for the wings of finite span attached to a real aircraft. This is because, unlike an airfoil which is a two-dimensional object, a wing is essentially a three-dimensional body, that is, there will be a component of flow in the spanwise direction. That is, the flow over the wings is three-dimensional in nature and hence their aerodynamic properties are quite different from those of its airfoil sections.

The downwash produced by shedding trailing edge vortices from the wing tips and its effect on the inclination of the local relative wind has two major consequences on the local airfoil section. The actual angle of attack as seen by the airfoil locally is, in fact, lower than the geometric angle of attack

\(\mathrm {\left( \alpha \right) }\). This angle of attack is referred to as effective angle of attack

\(\mathrm {\left( \alpha _{eff}\right) }\) for a three-dimensional wing. That is,

$$ \mathrm {\alpha _{eff}=\alpha -\alpha _{i}} $$

Also, the effective freestream velocity

\(\mathrm {\left( U_{eff}\right) }\) will now become

However, for small downwash

\(\mathrm {\left( w\approx 0\right) }\)$$\begin{aligned} \mathrm {U_{eff}}&\mathrm {\approx U_{a}} \end{aligned}$$

Besides, the downwash induced by these trailing edge vortices from the wing tips leads to an additional component of drag known as induced drag.

The concepts of vortex sheets and vortex filaments are advantageous in evaluating the aerodynamic characteristics of wings of finite span. From a directed line segment

\(\mathrm {\left( dl\right) }\) of a vortex filament, the induced velocity can be calculated by using the below mentioned Biot–Savart law.

$$\begin{aligned} \mathrm {\mathrm {\overrightarrow{\mathrm {v}}\left( r\right) }}\,&\mathrm {=\mathrm {\frac{\Gamma }{4\pi }\int }\left( \frac{\overrightarrow{\mathrm {dl}}\times \left( \overrightarrow{\mathrm {r}}-\overrightarrow{\mathrm {s}}\right) }{\left| \overrightarrow{\mathrm {r}}-\overrightarrow{\mathrm {s}}\right| ^{3}}\right) } \end{aligned}$$

The velocity induced by a straight vortex filament of the finite length is given as

$$\begin{aligned} \mathrm {v_{i}}\,&\mathrm {=\frac{\Gamma }{4\pi d}\left( \cos \theta _{1}-\cos \theta _{2}\right) } \end{aligned}$$

In the lifting line model, developed by Ludwig Prandtl, a wing is numerically described by an infinite number of horseshoe vortices and these bound vortices pass through the aerodynamic centers of the airfoils, which in turn creates the lifting line. Besides, the trailing edge vortices starting at the lifting line and shed downstream toward the infinity are basically responsible for inducing the downwash at the lifting line, and consequently, modify the local angles of attack. The circulation distribution

\(\mathrm {\Gamma \left( y\right) }\) is calculated from the accompanying relation

$$ \mathrm {\alpha \left( \beta \right) =}\mathrm {\frac{\Gamma \left( \beta \right) }{\pi U_{a}c\left( \beta \right) }+\alpha _{L=0}\left( \beta \right) +\frac{1}{4\pi U_{a}}\intop _{-b}^{b}\frac{\left[ \frac{d\Gamma \left( y\right) }{dy}\right] dy}{\left( \beta -y\right) }} $$

For the symmetric aerodynamic load distribution, defined as

\(\mathrm {\Gamma \left( y\right) =\Gamma _{max}\left[ 1-\left( \frac{y}{b}\right) ^{2}\right] ^{\frac{1}{2}}}\), a summary of important relations is described below.

For a symmetric elliptical lift distribution over the wingspan, both induced downwash and induced angle are constant along the span.

$$ \mathrm {w_{i}\left( \theta \right) =-\frac{\Gamma _{max}}{4b}} $$

$$ \mathrm {\alpha _{i}=\frac{\Gamma _{max}}{4bU_{a}}} $$

The total lift acting on the complete wingspan for a symmetric elliptic lift distribution is

$$\begin{aligned} \mathrm {L}\,&\mathrm {=\frac{\pi b}{4}\rho U_{a}\Gamma _{max}} \end{aligned}$$

and the expression for

\(\mathrm {C_{L}}\) is

$$\begin{aligned} \mathrm {C_{L}}&\mathrm {=\frac{\pi }{2}\frac{b}{S}\frac{\Gamma _{max}}{U_{a}}} \end{aligned}$$

The overall induced drag for a symmetric elliptical loading is given by

$$\begin{aligned} \mathrm {D_{i}}\,&\mathrm {=\frac{\pi }{8}\rho \Gamma _{max}^{2}} \end{aligned}$$

In addition, the coefficient of induced drag is

$$\begin{aligned} \mathrm {C_{D_{i}}}\,&\mathrm {=\frac{\pi }{4}\frac{\Gamma _{max}^{2}}{SU_{a}^{2}}=\frac{1}{\pi }\frac{S}{b^{2}}C_{L}^{2}} \end{aligned}$$

This relation can also be written as

$$\begin{aligned} \mathrm {C_{D_{i}}}\,&\mathrm {=\frac{C_{L}^{2}}{\pi AR}} \end{aligned}$$

where

\(\mathrm {AR=\frac{b^{2}}{S}}\) is the aspect ratio of a finite wing.

For the symmetric general aerodynamic load distribution, given by Open image in new window , a summary of important relations is described as follows.

The generalized expression for the induced angle

\(\mathrm {\left( \alpha _{i}\right) }\) is

For a wing of finite span, the lift coefficient is given by

$$\begin{aligned} \mathrm {C_{L}}\,&\mathrm {=\pi A_{1}AR} \end{aligned}$$

and the coefficient of induced drag is

$$\begin{aligned} \mathrm {C_{D_{i}}}\,&\mathrm {=\pi AR\sum _{n=1}^{\infty }nA_{n}^{2}=\frac{C_{L}^{2}}{\pi AR}\left( 1+\delta \right) =\frac{C_{L}^{2}}{\pi eAR}} \end{aligned}$$

where

\(\mathrm {\delta =\sum _{n=2}^{\infty }n\left( \frac{A_{n}^{2}}{A_{1}^{2}}\right) }\) and

\(\mathrm {e=\left( \frac{1}{1+\delta }\right) }\) is the span efficiency factor.