Elliptical Lift Distribution

Consider a circulation distribution, \[\Gamma(y) = \Gamma_0 \sqrt{1-(\frac{2y}{b})^2}\]

Clearly \[\frac{d\Gamma}{dy} = -\frac{4\Gamma_0}{b^2}\frac{y}{(1-4y^2/b^2)^{1/2}}\]

where \(b\) is the span of the wing and the origin is located at the root chord of the wing. Since, \[L'(y) = \rho_{\infty}U_{\infty}\Gamma(y),\]

lift distribution is also elliptical. Any wing that has this kind of circulation is called as elliptical wing. It should be noted that the elliptical lift distribution is symmetric about the origin. Also, it is maximum at the root chord and decreases elliptically to zero at the wing tips. This satisfies the conditions on the acceptable lift distributions as laid down in the derivation of the lift distribution for an arbitrary wing.

Now let us look at the downwash distribution for this elliptical lift distribution.

Downwash \(w(y_0)\), at a point \(y_0\) on the wing, is given by,

\[\begin{align*} w(y_0) &= -\frac{1}{4\pi}\int_{-b/2}^{b/2}\frac{(\frac{d\Gamma}{dy})dy}{y_0-y}\\ w(y_0) &= -\frac{\Gamma_0}{\pi b^2}\int_{-b/2}^{b/2}\frac{y}{(1-4y^2/b^2)^{1/2}(y_0-y)}dy\\ \end{align*}\]

Substituting \(y = \frac{b}{2}\cos{\theta}\), we get

\[\begin{align*} w(\theta_0) &= -\frac{\Gamma_0}{2\pi b} \int_{\pi}^{0}\frac{\cos{\theta}}{\cos{\theta_0}-\cos{\theta}} d\theta\\ &= -\frac{\Gamma_0}{2\pi b} \int_{0}^{\pi}\frac{\cos{\theta}}{\cos{\theta}-\cos{\theta_0}} d\theta\\ &= -\frac{\Gamma_0}{2b}\\ \end{align*}\]

This implies that the downwash is constant over the span for an elliptical lift distribution.

Therefore the induced angle of attack is also constant along the span.

\[\alpha_i = -\frac{w}{U_{\infty}} = \frac{\Gamma_0}{2bU_{\infty}}\]

Clearly,

\[\lim_{b \rightarrow \infty} \alpha_i = 0\]

As the wing span tends to infinity, the downwash and generated induced angle of attack tend to zero.

A wing with a very large span will have a very small downwash and behave just like an airfoil.

We can calculate the total lift generated by an elliptical wing by integrating the sectional lift obtained using Kutta-Joukowski theorem.

\[\begin{align*} L &= \int_{-b/2}^{b/2} L'(y) dy\\ &= \rho_{\infty} U_{\infty}\int_{-b/2}^{b/2} \Gamma(y) dy\\ &= \rho_{\infty}U_{\infty}\Gamma_0 \int_{-b/2}^{b/2}{(1-\frac{4y^2}{b^2})}^{1/2}dy\\ &= \rho_{\infty}U_{\infty}\Gamma_0 \frac{b}{2}\int_{0}^{\pi} \sin^2{\theta}d\theta = \rho_{\infty}U_{\infty}\Gamma_0 \frac{b}{4} \pi\\ \end{align*}\]

We could have calculated the sectional (\(L'(y)\)) in a different way as follows.

\[L = \int_{-b/2}^{b/2} L'(y) dy = q_{\infty} \int_{-b/2}^{b/2} c(y) C_l(y) dy\]

Now, the sectional lift coefficient is nothing but \(C_l = C_{l_\alpha} \alpha_{eff} = C_{l_\alpha} (\alpha(c) - \alpha_i(y))\).

Here, \(\alpha\) is the geometric angle of attack which, in general, is a function of span location. It can be modified by introducing

• Aerodynamic twist or
• Geometric twist.

Aerodynamic twist is referred to the scenario where the airfoil section changes along the span. As the airfoil shape changes, each section has a different zero lift angle of attack (\(\alpha_0\)). Hence \(\alpha(y) = \alpha_g(y) - \alpha_0(y)\) where \(\alpha_g\) is the sectional geometric angle of attack.

If we consider an untwisted wing, then \(\alpha_g\) is not a function of \(y\). But it is simply the angle between the aircraft FRL and the freestream velocity component in the aircraft symmetry plane (\(U_\infty \cos{\beta}\)). However, if we were to twist the wing about the Y axis then the geometric AOA at a given section location will be \(\alpha_g(y) = \alpha_{FRL} - \alpha_{twist}(y)\). Most commonly the geometric twist is zero at the root chord and increases linearly (or some mild nonlinear function) to tips. This is called as geometric twist.

FIND OUT THE TWIST OF 787 AND A320

We get \(\Gamma_0\) as,

\[\Gamma_0 = \frac{4L}{\rho_{\infty}U_{\infty}b\pi}\]

However,

\[L = \frac{1}{2}\rho_{\infty}U_{\infty}^2 S C_L\]

Hence,

\[\begin{align*} \Gamma_0 &= \frac{2 U_{\infty}S C_L}{b\pi}\\ \therefore \alpha_i &= \frac{2 U_{\infty}S C_L}{b\pi}\frac{1}{2bU_{\infty}} = \frac{S C_L}{\pi b^2}\\ \end{align*}\]

Aspect ratio (\(AR\)) is an important geometric property of a finite wing, given by, \(AR=\frac{b^2}{S}\)\

Thus,

\[\alpha_i = \frac{C_L}{\pi AR}\]

is an useful expression for induced angle of attack. And the induced drag coefficient is given by (note \(\alpha_i = constant\)),

\[\begin{align*} C_{D,i} &= \frac{2\alpha_i}{U_{\infty}S}\int_{-b/2}^{b/2} \Gamma(y) dy = \frac{2\alpha_i \Gamma_0}{U_{\infty}S}\frac{b}{2}\int_{0}^{\pi} \sin^2{\theta}d\theta - \frac{\pi \alpha_i \Gamma_0 b}{2 U_{\infty} S}\\ C_{D,i} &=\frac{\pi b}{2 U_{\infty} S}(\frac{C_L}{\pi AR})\frac{2 U_{\infty}S C_L}{b\pi}\\ C_{D,i} &= \frac{C_L^2}{\pi AR}\\ \end{align*}\]

This result states that the induced drag coefficient is directly proportional to the square of the lift coefficient.Wing-tip vortices, which are brought on by the pressure differential between the lower and top wing surfaces, are what cause induced drag. This pressure differential is also what causes the lift. In turn, induced drag is frequently referred to as the drag due to lift. As a result, induced drag is closely linked to the generation of lift on a finite wing.

caption{Elliptical lift distribution,planform and constant downwash}

The fact that \(C_{D ,i}\) has an inverse relationship to aspect ratio is another crucial element of induced drag. Consequently, we seek a finite wing with the highest aspect ratio feasible in order to lessen the induced drag.

Consider a wing that has no aerodynamic twist (\(\alpha_L =0\) is constant along the span) and no geometry twist (\(\alpha\) is constant along the span). We are aware that \(\alpha_i\) is constant throughout the span. As a result, \(\alpha_{eff} = \alpha - \alpha_i\) is also constant over the period. Given that \(c_l\) is the local section lift rate,

\[C_l = a_0(\alpha_{eff}-\alpha_{L=0})\]

assuming \(a_0\) is same for all sections (\(a_0 = 2\pi\)), implying \(c_l\) must be constant along the span. Lift per unit span is given by

\[L'(y) = q_{\infty}c c_l\]

Solving for chord,

\[c(y) = \frac{L'(y)}{q_{\infty}c_l}\]

This equation demonstrates that \(q_{\infty}\) and \(C_l\) remain constant over the period. \(L'(y)\), however, fluctuates elliptically over the period(Fig.1). The wing planform is elliptic for the circumstances stated above because this equation requires that for such an elliptic lift distribution, the chord must vary elliptically along the span.

Consider a circulation distribution, \[\Gamma(y) = \Gamma_0 \sqrt{1-(\frac{2y}{b})^2}\]

Clearly \[\frac{d\Gamma}{dy} = -\frac{4\Gamma_0}{b^2}\frac{y}{(1-4y^2/b^2)^{1/2}}\]

where \(b\) is the span of the wing and the origin is located at the root chord of the wing. Since, \[L'(y) = \rho_{\infty}U_{\infty}\Gamma(y),\]

lift distribution is also elliptical. Any wing that has this kind of circulation is called as elliptical wing. It should be noted that the elliptical lift distribution is symmetric about the origin. Also, it is maximum at the root chord and decreases elliptically to zero at the wing tips. This satisfies the conditions on the acceptable lift distributions as laid down in the derivation of the lift distribution for an arbitrary wing.

Now let us look at the downwash distribution for this elliptical lift distribution.

Downwash \(w(y_0)\), at a point \(y_0\) on the wing, is given by,

\[\begin{align*} w(y_0) &= -\frac{1}{4\pi}\int_{-b/2}^{b/2}\frac{(\frac{d\Gamma}{dy})dy}{y_0-y}\\ w(y_0) &= -\frac{\Gamma_0}{\pi b^2}\int_{-b/2}^{b/2}\frac{y}{(1-4y^2/b^2)^{1/2}(y_0-y)}dy\\ \end{align*}\]

Substituting \(y = \frac{b}{2}\cos{\theta}\), we get

\[\begin{align*} w(\theta_0) &= -\frac{\Gamma_0}{2\pi b} \int_{\pi}^{0}\frac{\cos{\theta}}{\cos{\theta_0}-\cos{\theta}} d\theta\\ &= -\frac{\Gamma_0}{2\pi b} \int_{0}^{\pi}\frac{\cos{\theta}}{\cos{\theta}-\cos{\theta_0}} d\theta\\ &= -\frac{\Gamma_0}{2b}\\ \end{align*}\]

This implies that the downwash is constant over the span for an elliptical lift distribution.

Therefore the induced angle of attack is also constant along the span.

\[\alpha_i = -\frac{w}{U_{\infty}} = \frac{\Gamma_0}{2bU_{\infty}}\]

Clearly,

\[\lim_{b \rightarrow \infty} \alpha_i = 0\]

As the wing span tends to infinity, the downwash and generated induced angle of attack tend to zero.

A wing with a very large span will have a very small downwash and behave just like an airfoil.

We can calculate the total lift generated by an elliptical wing by integrating the sectional lift obtained using Kutta-Joukowski theorem.

\[\begin{align*} L &= \int_{-b/2}^{b/2} L'(y) dy\\ &= \rho_{\infty} U_{\infty}\int_{-b/2}^{b/2} \Gamma(y) dy\\ &= \rho_{\infty}U_{\infty}\Gamma_0 \int_{-b/2}^{b/2}{(1-\frac{4y^2}{b^2})}^{1/2}dy\\ &= \rho_{\infty}U_{\infty}\Gamma_0 \frac{b}{2}\int_{0}^{\pi} \sin^2{\theta}d\theta = \rho_{\infty}U_{\infty}\Gamma_0 \frac{b}{4} \pi\\ \end{align*}\]

We could have calculated the sectional (\(L'(y)\)) in a different way as follows.

\[L = \int_{-b/2}^{b/2} L'(y) dy = q_{\infty} \int_{-b/2}^{b/2} c(y) C_l(y) dy\]

Now, the sectional lift coefficient is nothing but \(C_l = C_{l_\alpha} \alpha_{eff} = C_{l_\alpha} (\alpha(c) - \alpha_i(y))\).

Here, \(\alpha\) is the geometric angle of attack which, in general, is a function of span location. It can be modified by introducing

• Aerodynamic twist or
• Geometric twist.

Aerodynamic twist is referred to the scenario where the airfoil section changes along the span. As the airfoil shape changes, each section has a different zero lift angle of attack (\(\alpha_0\)). Hence \(\alpha(y) = \alpha_g(y) - \alpha_0(y)\) where \(\alpha_g\) is the sectional geometric angle of attack.

If we consider an untwisted wing, then \(\alpha_g\) is not a function of \(y\). But it is simply the angle between the aircraft FRL and the freestream velocity component in the aircraft symmetry plane (\(U_\infty \cos{\beta}\)). However, if we were to twist the wing about the Y axis then the geometric AOA at a given section location will be \(\alpha_g(y) = \alpha_{FRL} - \alpha_{twist}(y)\). Most commonly the geometric twist is zero at the root chord and increases linearly (or some mild nonlinear function) to tips. This is called as geometric twist.

FIND OUT THE TWIST OF 787 AND A320

We get \(\Gamma_0\) as,

\[\Gamma_0 = \frac{4L}{\rho_{\infty}U_{\infty}b\pi}\]

However,

\[L = \frac{1}{2}\rho_{\infty}U_{\infty}^2 S C_L\]

Hence,

\[\begin{align*} \Gamma_0 &= \frac{2 U_{\infty}S C_L}{b\pi}\\ \therefore \alpha_i &= \frac{2 U_{\infty}S C_L}{b\pi}\frac{1}{2bU_{\infty}} = \frac{S C_L}{\pi b^2}\\ \end{align*}\]

Aspect ratio (\(AR\)) is an important geometric property of a finite wing, given by, \(AR=\frac{b^2}{S}\)\

Thus,

\[\alpha_i = \frac{C_L}{\pi AR}\]

is an useful expression for induced angle of attack. And the induced drag coefficient is given by (note \(\alpha_i = constant\)),

\[\begin{align*} C_{D,i} &= \frac{2\alpha_i}{U_{\infty}S}\int_{-b/2}^{b/2} \Gamma(y) dy = \frac{2\alpha_i \Gamma_0}{U_{\infty}S}\frac{b}{2}\int_{0}^{\pi} \sin^2{\theta}d\theta - \frac{\pi \alpha_i \Gamma_0 b}{2 U_{\infty} S}\\ C_{D,i} &=\frac{\pi b}{2 U_{\infty} S}(\frac{C_L}{\pi AR})\frac{2 U_{\infty}S C_L}{b\pi}\\ C_{D,i} &= \frac{C_L^2}{\pi AR}\\ \end{align*}\]

This result states that the induced drag coefficient is directly proportional to the square of the lift coefficient.Wing-tip vortices, which are brought on by the pressure differential between the lower and top wing surfaces, are what cause induced drag. This pressure differential is also what causes the lift. In turn, induced drag is frequently referred to as the drag due to lift. As a result, induced drag is closely linked to the generation of lift on a finite wing.

caption{Elliptical lift distribution,planform and constant downwash}

The fact that \(C_{D ,i}\) has an inverse relationship to aspect ratio is another crucial element of induced drag. Consequently, we seek a finite wing with the highest aspect ratio feasible in order to lessen the induced drag.

Consider a wing that has no aerodynamic twist (\(\alpha_L =0\) is constant along the span) and no geometry twist (\(\alpha\) is constant along the span). We are aware that \(\alpha_i\) is constant throughout the span. As a result, \(\alpha_{eff} = \alpha - \alpha_i\) is also constant over the period. Given that \(c_l\) is the local section lift rate,

\[C_l = a_0(\alpha_{eff}-\alpha_{L=0})\]

assuming \(a_0\) is same for all sections (\(a_0 = 2\pi\)), implying \(c_l\) must be constant along the span. Lift per unit span is given by

\[L'(y) = q_{\infty}c c_l\]

Solving for chord,

\[c(y) = \frac{L'(y)}{q_{\infty}c_l}\]

This equation demonstrates that \(q_{\infty}\) and \(C_l\) remain constant over the period. \(L'(y)\), however, fluctuates elliptically over the period(Fig.1). The wing planform is elliptic for the circumstances stated above because this equation requires that for such an elliptic lift distribution, the chord must vary elliptically along the span.