Cylindrical Airfoils

For the moment, let us work in the complex $\zeta$-plane, where $\zeta=\xi+{\rm i}\,\eta$. Consider a cylindrical airfoil with a circular cross-section of radius $a$, centered on the origin, that is situated in a uniform, high Reynolds number wind of speed $V$ whose direction subtends a (clockwise) angle $\alpha$ with the negative $\xi$-axis. Let ${\mit\Gamma}$ be the circulation of air around the airfoil. A slight generalization of the analysis of Section 6.4 reveals that the appropriate complex velocity potential is

$$F(\zeta) = V\left(\zeta\,{\rm e}^{\,{\rm i}\,\alpha} + \frac... ...,\frac{{\mit\Gamma}}{2\pi}\,\ln\!\left(\frac{\zeta}{a}\right),$$ (856)

whereas the associated stream function takes the form

$$\psi(r,\theta) = V\left(r-\frac{a^2}{r}\right)\sin(\theta+\alpha) + \frac{{\mit\Gamma}}{2\pi}\,\ln\!\left(\frac{r}{a}\right),$$ (857)

where $\zeta=r\,{\rm e}^{\,{\rm i}\,\theta}$. It follows that

$$\frac{dF}{d\zeta} = V\,{\rm e}^{\,{\rm i}\,\alpha} + {\rm i}... ...a} - \frac{V\,a^2\,{\rm e}^{\,-{\rm i}\,\alpha}}{\zeta^{\,2}}.$$ (858)

Comparison with Equation (848) (with $z=\zeta$) reveals that

$$B = - V\,a^2\,{\rm e}^{-{\rm i}\,\alpha}.$$ (859)

Hence, Equations (852) and (855) yield

$${\bf V} = V\left(-\cos\alpha\,{\bf e}_\xi + \sin\alpha\,{\bf e}_\eta\right),$$ (860)

$${\bf L} = \rho\,V\,{\mit\Gamma}\left(\sin\alpha\,{\bf e}_\xi+\cos\alpha\,{\bf e}_\eta\right),$$ (861)

$$M = 0,$$ (862)

where ${\bf V}$ is the wind vector, ${\bf L}$ the lift vector, and $M$ the moment of the lift vector about the origin. We conclude that, for a cylindrical airfoil of circular cross-section, the lift vector is normal to the wind vector, and the line of action of the lift passes through the centroid of the cross-section (since the lift generates zero moment about the origin). See Figure 64.

Figure 64: A cylindrical airfoil of circular cross-section.

Of course, a cylindrical airfoil of circular cross-section is completely unrealistic, since its back side (i.e., the side opposite to that from which the wind is incident) is not sufficiently streamlined to prevent boundary layer separation. (See Chapter 7.) However, as described in Section 6.7, we can use the conformal transformation

$$z = \zeta+ \frac{l^{\,2}}{\zeta}$$ (863)

to transform a cylinder of circular cross-section in the $\zeta$-plane to a cylinder of elliptical cross-section in the $z$-plane. (Note that both cross-sections have centroids located at the origin.) Moreover, a cylindrical airfoil of elliptical cross-section that is sufficiently elongated, and whose major axis subtends a sufficiently small angle with the incident wind direction, constitutes a realistic airfoil, since its back side is, for the most part, closely aligned with the external flow.

Figure 65: A cylindrical airfoil of elliptical cross-section.

An elliptical airfoil of width $c$ and thickness $\delta<c$, as shown in Figure 65, is obtained when the parameters $a$ and $l$ are given the following values:

$$a = \frac{1}{4}\,(c+\delta),$$ (864)

$$l = \frac{1}{4}\,(c^2-\delta^{\,2})^{1/2}.$$ (865)

In this case, the surface of the airfoil satisfies the parametric equations

$$x = \frac{c}{2}\,\cos\theta,$$ (866)

$$y = \frac{\delta}{2}\,\sin\theta.$$ (867)

In particular, the airfoil's leading and trailing edges correspond to $\theta=0$ and $\theta=\pi$, respectively.

According to Equations (858) and (863), the complex velocity in the $z$-plane is given by

$$\frac{dF}{dz} = \frac{dF}{d\zeta}\,\frac{d\zeta}{dz} = \left... ...}}\right]\left(\frac{\zeta^{\,2}}{\zeta^{\,2}-l^{\,2}}\right).$$ (868)

Thus, on the airfoil surface, where $\zeta=a\,{\rm e}^{\,{\rm i}\,\theta}$, we obtain

$$\frac{dF}{dz} = {\rm i}\left[V\,\sin(\alpha+\theta) + \frac{... ...lta)}{\left(\delta\,\cos\theta+{\rm i}\,c\,\sin\theta\right)}.$$ (869)

A long way from the airfoil, $\zeta\simeq z-l^{\,2}/z$, so that Equation (868) reduces to

$$\frac{dF}{dz}\simeq V\,{\rm e}^{\,{\rm i}\,\alpha} + {\rm i}... ...\,{\rm i}\,\alpha}-a^2\,{\rm e}^{-{\rm i}\,\alpha})}{z^{\,2}}.$$ (870)

A comparison with Equation (848) reveals that the circulation of air around the airfoil takes the same value, ${\mit\Gamma}$, in both the complex $\zeta$- and $z$-planes. In other words, the conformal transformation (863) does not modify the circulation. Note that the transformation also does not modify the external wind speed or direction [since, from (858) and (870), $dF/d\zeta= dF/dz=V\,{\rm e}^{\,{\rm i}\,\alpha}$ at very large $\vert\zeta\vert$ and $\vert z\vert$]. On the other hand, it is clear that the constant $B$, which takes the value zero in the complex $\zeta$-plane, takes the value

$$B = V\,(l^{\,2}\,{\rm e}^{\,{\rm i}\,\alpha} -a^2\,{\rm e}^{-{\rm i}\,\alpha})$$ (871)

in the complex $z$-plane. Hence, Equations (852) and (855) reveal that

$${\bf V} = V\,{\bf e}_\parallel,$$ (872)

$${\bf L} = \rho\,V\,{\mit\Gamma}\,{\bf e}_\perp,$$ (873)

$$M = \frac{\pi}{8}\,\rho\,V^{\,2}\,(c^2-\delta^{\,2})\,\sin(2\,\alpha),$$ (874)

where ${\bf V}$ is the wind vector, ${\bf L}$ the lift vector, and $M$ the moment of the lift vector about the origin. Here,

$${\bf e}_\parallel = -\cos\alpha\,{\bf e}_x + \sin\alpha\,{\bf e}_y$$ (875)

is a unit vector parallel to the incident wind direction, and

$${\bf e}_\perp= {\bf e}_\parallel \times{\bf e}_z= \sin\alpha\,{\bf e}_x+\cos\alpha\,{\bf e}_y$$ (876)

is a unit vector perpendicular to the wind direction. We conclude that, for a cylindrical airfoil of elliptic cross-section, the lift vector is normal to the wind vector, but the line of action of the lift intersects the major axis of the airfoil a distance

$$d = \frac{M}{L\,\cos\alpha} = \frac{1}{4}\,(c-\delta)\,\frac{\sin\alpha}{\widehat{\mit\Gamma}}$$ (877)

in front of the cross-section's centroid, $C$, where $\widehat{\mit\Gamma}= {\mit\Gamma}/[\pi\,V\,(c+\delta)]$. See Figure 65. Incidentally, the point, $F$, at which the line of action of the lift intersects the airfoil's major axis is conventionally termed the focus of the airfoil.