Ellipsoidal Airfoils

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

$$x = \frac{c_0}{2}\,\sin\phi\,\cos\theta,$$ (933)

$$y = \frac{\delta_0}{2}\,\sin\phi\,\sin\theta,$$ (934)

$$z = -\frac{b}{2}\,\cos\phi,$$ (935)

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 $b>c_0\gg\delta_0$: i.e., 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 (886) holds at fixed $z$, we deduce that the air circulation about the wing satisfies

$${\mit\Gamma}(z)= \pi\,V\,c(z)\,\sin\alpha = {\mit\Gamma}_0\,\sin\phi,$$ (936)

where

$${\mit\Gamma}_0\simeq \pi\,V\,c_0\,\alpha.$$ (937)

Here, the angle of attack, $\alpha$, is assumed to be small. From Equations (932) and (936), the downwash velocity in the region $\vert z\vert<b/2$ is given by

$$w(\phi) = \frac{{\mit\Gamma}_0}{2\pi\,b}\int_0^\pi\frac{\cos... ...s\phi}{\pi}\int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$$ (938)

Now, the integrand in the integral

$$\int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}$$ (939)

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

$$\lim_{\epsilon\rightarrow 0}\left(\int_0^{\phi-\epsilon}\fra... ...t_{\phi+\epsilon}^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$$ (940)

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

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

$$= \lim_{\epsilon\rightarrow 0}\left(\frac{1}{\sin\phi}\,\ln\left[\frac{\sin(\phi-\epsilon/2)}{\sin(\phi+\epsilon/2)}\right]\right) = 0,$$ (941)

which implies that

$$w(\phi) = \frac{{\mit\Gamma}_0}{2\,b}.$$ (942)

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

$$w(\eta) = \frac{{\mit\Gamma}_0}{2\,b}\left(1-\frac{\eta}{\pi... ...0}{2\,b}\left(1-\frac{\vert\eta\vert}{\sqrt{\eta^2-1}}\right).$$ (943)

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

$$w(z) = \frac{{\mit\Gamma}_0}{2\,b}\left\{ \begin{array}{lll}... ...z\vert/(z^2-b^2/4)^{1/2}&&\vert z\vert>b/2 \end{array}\right..$$ (944)

This profile is shown in Figure 76. 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 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 76: Downwash velocity profile induced at the trailing edge by an ellipsoidal airfoil.

It follows from Equation (935) and (936) that

$$\int_{-b/2}^{b/2} {\mit\Gamma}(z)\,dz = \frac{{\mit\Gamma}_0... ...int_0^\pi\sin^2\phi\,d\phi = \frac{\pi}{4}\,{\mit\Gamma}_0\,b.$$ (945)

Hence, Equation (922), (924), and (942) yield the following expression for the lift and induced drag acting on an ellipsoidal airfoil,

$$L = \frac{\pi}{4}\,\rho\,V\,b\,{\mit\Gamma}_0,$$ (946)

$$D = \frac{\pi}{8}\,\rho\,{\mit\Gamma}_0^{\,2}.$$ (947)

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

$$S = \frac{\pi}{4}\,b\,c_0.$$ (948)

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: i.e.,

$$A = \frac{b^2}{S}= \frac{4}{\pi}\,\frac{b}{c_0}.$$ (949)

It thus follows from Equation (937) that

$$L = L_0\,\alpha,$$ (950)

$$D = \frac{2}{A}\,L_0\,\alpha^2,$$ (951)

where

$$L_0 = \pi\,\rho\,V^{\,2}\,S.$$ (952)