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Ellipsoidal Airfoils

Consider an ellipsoidal airfoil whose outer surface is specified by the parametric equations

$\displaystyle x$ $\displaystyle =\frac{c_0}{2}\,\sin\phi\,\cos\theta,$ (9.91)
$\displaystyle y$ $\displaystyle =\frac{\delta_0}{2}\,\sin\phi\,\sin\theta,$ (9.92)
$\displaystyle z$ $\displaystyle =-\frac{b}{2}\,\cos\phi,$ (9.93)

where $ 0\leq \phi\leq \pi$ and $ 0\leq \theta\leq 2\pi$ . Here, $ b$ is the wingspan, $ c_0$ the maximum wing width, and $ \delta_0$ the maximum wing thickness. Note that the wing's cross-section is elliptical both in the $ x$ -$ y$ and the $ x$ -$ z$ planes. It is assumed that $ b>c_0\gg\delta_0$ : that is, the wingspan is greater than the wing width, which in turn is much greater than the wing thickness. At fixed $ \phi$ (i.e., fixed $ z$ ), the width and thickness of the airfoil are $ c(\phi)=c_0\,\sin\phi$ and $ \delta(\phi)=\delta_0\,\sin\phi$ , respectively.

Assuming that the two-dimensional result (9.44) holds at fixed $ z$ , we deduce that the air circulation about the wing satisfies

$\displaystyle {\mit\Gamma}(z)= \pi\,V\,c(z)\,\sin\alpha = {\mit\Gamma}_0\,\sin\phi,$ (9.94)


$\displaystyle {\mit\Gamma}_0\simeq \pi\,V\,c_0\,\alpha.$ (9.95)

Here, the angle of attack, $ \alpha $ , is assumed to be small. From Equations (9.90) and (9.94), the downwash velocity in the region $ \vert z\vert<b/2$ is given by

$\displaystyle w(\phi) = \frac{{\mit\Gamma}_0}{2\pi\,b}\int_0^\pi\frac{\cos\phi'...
...left( 1+\frac{\cos\phi}{\pi}\int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$ (9.96)

The integrand appearing in the integral

$\displaystyle \int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}$ (9.97)

is singular when $ \phi'=\phi$ . However, we can still obtain a finite value for the integral by taking its Cauchy principal part: that is,

$\displaystyle \lim_{\epsilon\rightarrow 0}\left(\int_0^{\phi-\epsilon}\frac{d\p...
...hi'-\cos\phi}+\int_{\phi+\epsilon}^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$ (9.98)

Physically, this is equivalent to omitting the contribution of the local free vortex at a given point on the airfoil's trailing edge to the downwash velocity induced at that point, which is reasonable because a vortex induces zero velocity at its center. Hence, we obtain

$\displaystyle \int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}$ $\displaystyle =\lim_{\epsilon\rightarrow 0}\left\{\left(\frac{1}{\sin\phi}\,\ln...
...\phi+\phi')} {\sin\,(1/2)\,(\phi-\phi')}\right]\right)_0^{\phi-\epsilon}\right.$    
  $\displaystyle \phantom{=}\left.+\left(\frac{1}{\sin\phi}\,\ln\left[\frac{\sin\,...
...i'+\phi)} {\sin\,(1/2)\,(\phi'-\phi)}\right]\right)_\pi^{\phi+\epsilon}\right\}$    
  $\displaystyle =\lim_{\epsilon\rightarrow 0}\left(\frac{1}{\sin\phi}\,\ln\left[\frac{\sin(\phi-\epsilon/2)}{\sin(\phi+\epsilon/2)}\right]\right) = 0,$ (9.99)

which implies that

$\displaystyle w(\phi) = \frac{{\mit\Gamma}_0}{2\,b}.$ (9.100)

In the region $ \vert z\vert>b/2$ , we can write $ \eta=2\,z/b$ , so that

$\displaystyle w(\eta) = \frac{{\mit\Gamma}_0}{2\,b}\left(1-\frac{\eta}{\pi}\int...
...{\mit\Gamma}_0}{2\,b}\left(1-\frac{\vert\eta\vert}{\sqrt{\eta^{\,2}-1}}\right).$ (9.101)

Hence, we conclude that the downwash velocity profile induced by an ellipsoidal airfoil takes the form

$\displaystyle w(z) = \frac{{\mit\Gamma}_0}{2\,b}\left\{ \begin{array}{lll} 1&\m...
... 1-\vert z\vert/(z^{\,2}-b^{\,2}/4)^{1/2}&&\vert z\vert>b/2 \end{array}\right..$ (9.102)

This profile is shown in Figure 9.13. It can be seen that the downwash velocity is uniform and positive in the region between the wingtips (i.e., $ -b/a<z<b/2$ ), but negative and decaying in the region outside the wingtips. Hence, we conclude that as air passes over an airfoil subject to an upward lift it acquires a net downward velocity component, which, of course, is a consequence of the reaction to the lift. On the other hand, the air immediately behind and to the sides of the airfoil acquires a net upward velocity component. In other words, the lift acting on the airfoil is associated with a downwash of air directly behind, and an upwash behind and to either side of, the airfoil. The existence of upwash slightly behind and to the side of a flying object allows us to explain the V-formation adopted by wild geese--a bird flying in the upwash of another bird needs to generate less lift in order to stay in the air, and, consequently, experiences less induced drag.

Figure 9.13: Downwash velocity profile induced at the trailing edge by an ellipsoidal airfoil.

It follows from Equation (9.93) and (9.94) that

$\displaystyle \int_{-b/2}^{b/2} {\mit\Gamma}(z)\,dz = \frac{{\mit\Gamma}_0\,b}{2}\int_0^\pi\sin^2\phi\,d\phi = \frac{\pi}{4}\,{\mit\Gamma}_0\,b.$ (9.103)

Hence, Equation (9.80), (9.82), and (9.100) yield the following expression for the lift and induced drag acting on an ellipsoidal airfoil,

$\displaystyle L$ $\displaystyle =\frac{\pi}{4}\,\rho\,V\,b\,{\mit\Gamma}_0,$ (9.104)
$\displaystyle D$ $\displaystyle = \frac{\pi}{8}\,\rho\,{\mit\Gamma}_0^{\,2}.$ (9.105)

The surface area of the airfoil in the $ x$ -$ z$ plane is

$\displaystyle S = \frac{\pi}{4}\,b\,c_0.$ (9.106)

Moreover, the airfoil's aspect-ratio is conventionally defined as the length to width ratio for a rectangle of length $ b$ that has the same area as the airfoil: that is,

$\displaystyle A = \frac{b^{\,2}}{S}= \frac{4}{\pi}\,\frac{b}{c_0}.$ (9.107)

It thus follows from Equation (9.95) that

$\displaystyle L$ $\displaystyle = L_0\,\alpha,$ (9.108)
$\displaystyle D$ $\displaystyle = \frac{2}{A}\,L_0\,\alpha^2,$ (9.109)


$\displaystyle L_0 = \pi\,\rho\,V^{\,2}\,S.$ (9.110)

next up previous
Next: Simple Flight Problems Up: Incompressible Aerodynamics Previous: Aerodynamic Forces
Richard Fitzpatrick 2016-03-31