Lifting-line theory

Airfoils in two dimensions are easier to understand, but they do not directly map to three-dimensional finite wings

An unrealistic lift distribution that neglects three-dimensional effects

A lift distribution as observed over a (finite) trapezoidal wing

It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate. When analyzing a three-dimensional finite wing, the first approximation to understanding is to consider slicing the wing into cross-sections and analyzing each cross-section independently as a wing in a two-dimensional world. Each of these slices is called an airfoil, and it is easier to understand an airfoil than a complete three-dimensional wing.

One might expect that understanding the full wing simply involves adding up the independently calculated forces from each airfoil segment. However, it turns out that this approximation is grossly incorrect: on a real wing, the lift over each wing segment (local lift per unit span, ${\displaystyle l}$ or ${\displaystyle {\tilde {L}}}$) does not correspond simply to what two-dimensional analysis predicts. In reality, the local amount of lift on each cross-section is not independent and is strongly affected by neighboring wing sections.

The lifting-line theory corrects some of the errors in the naive two-dimensional approach by including some of the interactions between the wing slices. It produces the lift distribution along the span-wise direction, ${\displaystyle {\tilde {L}}_{(y)}}$ based on the wing geometry (span-wise distribution of chord, airfoil, and twist) and flow conditions (${\displaystyle \rho }$, ${\displaystyle V_{\infty }}$, ${\displaystyle \alpha _{\infty }}$).

The lifting-line theory applies the concept of circulation and the Kutta–Joukowski theorem,

${\displaystyle {\tilde {L}}_{(y)}=\rho V\Gamma _{(y)},}$

so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span, ${\displaystyle \Gamma _{(y)}}$.

Modeling the local lift (unknown and sought-after) with the local circulation (also unknown) allows us to account for the influence of one section over its neighbors. In this view, any span-wise change in lift is equivalent to a span-wise change of circulation. According to Helmholtz's theorems, a vortex filament cannot begin or terminate in the air. Any span-wise change in lift can be modeled as the shedding of a vortex filament down the flow, behind the wing.

This shed vortex, whose strength is the derivative of the (unknown) local wing circulation distribution, ${\displaystyle d\Gamma /dy}$, influences the flow left and right of the wing section.

This sideways influence (upwash on the outboard, downwash on the inboard) is the key to the lifting-line theory. Now, if the change in lift distribution is known at given lift section, it is possible to predict how that section influences the lift over its neighbors: the vertical induced velocity (upwash or downwash, ${\displaystyle \omega _{i}}$) can be quantified using the velocity distribution within a vortex and related to a change in effective angle of attack over neighboring sections.

In mathematical terms, the local induced change of angle of attack ${\displaystyle \alpha _{i}}$ on a given section can be quantified with the integral sum of the downwash induced by every other wing section. In turn, the integral sum of the lift on each downwashed wing section is equal to the (known) total desired amount of lift.

This leads to an integro-differential equation in the form of ${\displaystyle L_{\text{total}}=\rho V_{\infty }\int _{\text{tip}}^{\text{tip}}\Gamma _{(y)}\,dy}$, where ${\displaystyle \Gamma _{(y)}}$ is expressed solely in terms of the wing geometry and its own span-wise variation ${\displaystyle d\Gamma _{(y)}/dy}$. The solution to this equation is a function ${\displaystyle \Gamma _{(y)}}$ that accurately describes the circulation (and therefore lift) distribution over a finite wing of known geometry.

(Based on.[8])

Nomenclature:

• ${\displaystyle \ \Gamma }$ is the circulation over the entire wing (m²/s)
• ${\displaystyle \ C_{L}}$ is the 3D lift coefficient (for the entire wing)
• ${\displaystyle \ AR}$ is the aspect ratio
• ${\displaystyle \ \alpha _{\infty }}$ is the freestream angle of attack (rad)
• ${\displaystyle \ V_{\infty }}$ is the freestream velocity (m/s)
• ${\displaystyle \ C_{D_{i}}}$ is the drag coefficient for induced drag
• ${\displaystyle \ e}$ is the planform efficiency factor

The following are all functions of the wings span-wise station ${\displaystyle y}$ (i.e. they can all vary along the wing)

• ${\displaystyle \ C_{l}}$ is the 2D lift coefficient (units/m)
• ${\displaystyle \ \gamma }$ is the 2D circulation at a section (m/s)
• ${\displaystyle \ c}$ is the chord length of the local section
• ${\displaystyle \ \alpha _{geo}}$ is the local change in angle of attack due to geometric twist of the wing
• ${\displaystyle \ \alpha _{0}}$ is zero-lift angle of attack of that section (depends on the airfoil geometry)
• ${\displaystyle \ C_{l_{\alpha }}}$ is the 2D lift coefficient slope (units/m⋅rad, and depends on airfoil geometry, see Thin airfoil theory)
• ${\displaystyle \ \alpha _{i}}$ is change in angle of attack due to downwash
• ${\displaystyle \ w_{i}}$ is the local downwash velocity

To derive the model we start with the assumption that the circulation of the wing varies as a function of the spanwise locations. The function assumed is a Fourier function. Firstly, the coordinate for the spanwise location ${\displaystyle y}$ is transformed by ${\displaystyle y=s\cos(\theta )}$, where y is spanwise location, and s is the semi-span of the wing.

and so the circulation is assumed to be:

${\displaystyle \Gamma (y)=\Gamma (\theta )=\gamma =4sV_{\infty }\sum _{n}{A_{n}\sin(n\theta })\qquad (1)}$

Since the circulation of a section is related to the ${\displaystyle C_{l}}$ by the equation:

${\displaystyle C_{l}={\frac {2\gamma }{V_{\infty }c}}\qquad (2)}$

but since the coefficient of lift is a function of angle of attack:

${\displaystyle C_{l}=C_{l_{\alpha }}(\alpha _{\infty }+\alpha _{geo}-\alpha _{0}-\alpha _{i})\qquad (3)}$

hence the vortex strength at any particular spanwise station can be given by the equations:

${\displaystyle \gamma ={\frac {1}{2}}V_{\infty }cC_{l_{\alpha }}(\alpha _{\infty }+\alpha _{geo}-\alpha _{0}-\alpha _{i})\qquad (4)}$

This one equation has two unknowns: the value for ${\displaystyle \gamma }$ and the value for ${\displaystyle \alpha _{i}}$. However, the downwash is purely a function of the circulation only. So we can determine the value ${\displaystyle \alpha _{i}}$ in terms of ${\displaystyle \Gamma (y)}$, bring this term across to the left hand side of the equation and solve. The downwash at any given station is a function of the entire shed vortex system. This is determined by integrating the influence of each differential shed vortex over the span of the wing.

Differential element of circulation:

${\displaystyle d\Gamma =4sV_{\infty }\sum _{n=1}^{\infty }nA_{n}\cos(n\theta )\qquad (5)}$

Differential downwash due to the differential element of circulation (acts like half an infinite vortex line):

${\displaystyle dw_{i}={\frac {d\Gamma }{4\pi r}}\qquad (6)}$

The integral equation over the span of the wing to determine the downwash at a particular location is:

${\displaystyle w_{i}=\int _{-s}^{s}{\frac {1}{y-y_{0}}}d\Gamma \qquad (7)}$

After appropriate substitutions and integrations we get:

${\displaystyle w_{i}=V_{\infty }\sum _{n=1}^{\infty }{\frac {nA_{n}\sin(n\theta )}{\sin(\theta )}}\qquad (8)}$

And so the change in angle attack is determined by (assuming small angles):

${\displaystyle \alpha _{i}={\frac {w_{i}}{V_{\infty }}}\qquad (9)}$

By substituting equations 8 and 9 into RHS of equation 4 and equation 1 into the LHS of equation 4, we then get:

${\displaystyle 4sV_{\infty }\sum _{n=1}^{\infty }A_{n}\sin(n\theta )={\frac {1}{2}}V_{\infty }cC_{l_{\alpha }}\left[\alpha _{\infty }+\alpha _{geo}-\alpha _{0}-\sum _{n=1}^{\infty }{\frac {nA_{n}\sin(n\theta )}{\sin(\theta )}}\right]\qquad (10)}$

After rearranging, we get the series of simultaneous equations:

${\displaystyle \sum _{n=1}^{\infty }A_{n}\sin(n\theta ){\bigg (}\sin(\theta )+{\frac {nC_{l\alpha }c}{8s}}{\bigg )}={\frac {C_{l\alpha }c}{8s}}\sin(\theta )(\alpha _{\infty }+\alpha _{geo}-\alpha _{0})\qquad (11)}$

By taking a finite number of terms, equation 11 can be expressed in matrix form and solved for coefficients A. Note the left-hand side of the equation represents each element in the matrix, and the terms on the RHS of equation 11 represent the RHS of the matrix form. Each row in the matrix form represents a different span-wise station, and each column represents a different value for n.

Appropriate choices for ${\displaystyle \theta }$ are as a linear variation between ${\displaystyle (0,\pi )}$. Note that this range does not include the values for 0 and ${\displaystyle \pi }$, as this leads to a singular matrix, which can't be solved.

The lift can be determined by integrating the circulation terms:

${\displaystyle {\text{Lift}}=\rho V_{\infty }\int _{-s}^{s}\Gamma (y)\cos {(\alpha _{i}(y))}dy\approx \rho V_{\infty }\int _{-s}^{s}\Gamma (y)dy}$

which can be reduced to:

${\displaystyle C_{L}=\pi A\!RA_{1}}$

where ${\displaystyle A_{1}}$ is the first term of the solution of the simultaneous equations shown above.

The induced drag can be determined from

${\displaystyle {\text{D}}_{\text{i}}=\rho V_{\infty }\int _{-s}^{s}\Gamma (y)\sin {(\alpha _{i}(y))}dy\approx \rho V_{\infty }\int _{-s}^{s}\Gamma (y)\alpha _{i}(y)dy}$

which can also be reduced to:

${\displaystyle C_{D_{\text{induced}}}=\pi A\!R\sum _{n=1}^{\infty }nA_{n}^{2}}$

where ${\displaystyle A_{n}}$ are all the terms of the solution of the simultaneous equations shown above.

Moreover, this expression may be arranged as a function of ${\displaystyle C_{L}}$ in the following way :

${\displaystyle C_{D_{\text{induced}}}=\pi A\!RA_{1}^{2}+\pi A\!R\sum _{n=2}^{\infty }nA_{n}^{2}}$

${\displaystyle C_{D_{\text{induced}}}=\pi A\!RA_{1}^{2}*{\frac {\pi A\!R}{\pi A\!R}}+\pi A\!R\sum _{n=2}^{\infty }nA_{n}^{2}*{\frac {\pi A\!RA_{1}^{2}}{\pi A\!RA_{1}^{2}}}}$

${\displaystyle C_{D_{\text{induced}}}={\frac {\pi ^{2}A\!R^{2}A_{1}^{2}}{\pi A\!R}}+{\frac {\pi ^{2}A\!R^{2}A_{1}^{2}}{\pi A\!R}}*{\frac {\sum _{n=2}^{\infty }nA_{n}^{2}}{A_{1}^{2}}}}$

${\displaystyle C_{D_{\text{induced}}}={\frac {C_{L}^{2}}{\pi A\!R}}(1+{\frac {\sum _{n=2}^{\infty }nA_{n}^{2}}{A_{1}^{2}}})={\frac {C_{L}^{2}}{\pi A\!Re}}}$

where

${\displaystyle \delta ={\frac {\sum _{n=2}^{\infty }nA_{n}^{2}}{A_{1}^{2}}}}$

${\displaystyle e={\frac {1}{1+\delta }}}$ is the span efficiency factor

For a symmetric wing, the even terms of the series coefficients are identically equal to 0, and so can be dropped.

When the aircraft is rolling, an additional term can be added that adds the wing station distance multiplied by the rate of roll to give additional angle of attack change. Equation 3 then becomes:

${\displaystyle C_{l}=C_{l_{\alpha }}\left(\alpha _{\infty }+\alpha _{geo}-\alpha _{0}-\alpha _{i}+{\frac {py}{s}}\right)\qquad (3)}$

where

• ${\displaystyle \ p}$ is the rate of roll in rad/sec,

Note that y can be negative, which introduces non-zero even coefficients in the equation that must be accounted for.

When the wing is rolling, the induced drag is altered because the lift vector is rotated at each spanwise station due to rolling rate.[9] The resulting induced drag for a wing with a rolling rate is

${\displaystyle C_{D_{\text{induced}}}={\frac {C_{L}^{2}}{\pi A\!R}}(1+{\frac {\sum _{n=2}^{\infty }nA_{n}^{2}}{A_{1}^{2}}})-{\frac {\pi A\!R{\bar {p}}}{2}}A_{2}={\frac {C_{L}^{2}}{\pi A\!Re}}-{\frac {\pi A\!R{\bar {p}}}{2}}A_{2}}$

where

• ${\displaystyle {\bar {p}}={\frac {pb}{2V}}}$ is the dimensionless rolling rate.

A similar change in induced drag is also present when the wing is flapping, and comprises the main production of thrust for flapping wings.[9]

The effects of control surface deflection can be accounted for by simply changing the ${\displaystyle \alpha _{0}}$ term in Equation 3. For non-symmetric controls such as ailerons the ${\displaystyle \alpha _{0}}$ term changes on each side of the wing.

For an elliptical wing with no twist, with:

${\displaystyle y(\theta )=s\cos(\theta )}$

The chord length is given as a function of span location as:

${\displaystyle c(\theta )=c_{root}\sin(\theta )}$

Also,

${\displaystyle e=1}$

This yields the famous equation for the elliptic induced drag coefficient:

${\displaystyle C_{D_{induced}}={\frac {C_{L}^{2}}{\pi A\!R}}}$

where

• ${\displaystyle s}$ is the value of the wing span,
• ${\displaystyle y(\theta )}$ is the position on the wing span, and
• ${\displaystyle c(\theta )}$ is the chord.

A decomposed Fourier series solution can be used to individually study the effects of planform, twist, control deflection, and rolling rate.[10][11]

A useful approximation is that

${\displaystyle \ C_{L}=C_{l_{\alpha }}\left({\frac {\text{AR}}{{\text{AR}}+2}}\right)\alpha }$

where

• ${\displaystyle \ C_{\text{L}}}$ is the 3D lift coefficient for elliptical circulation distribution,
• ${\displaystyle \ C_{l_{\alpha }}}$ is the 2D lift coefficient slope (see thin airfoil theory),
• ${\displaystyle \ {\text{AR}}}$ is the aspect ratio, and
• ${\displaystyle \ \alpha }$ is the angle of attack in radians.

The theoretical value for ${\displaystyle \ C_{l_{\alpha }}}$ is 2${\displaystyle \pi }$. Note that this equation becomes the thin airfoil equation if AR goes to infinity.[12]

As seen above, the lifting-line theory also states an equation for induced drag:[13][14]

${\displaystyle \ C_{D_{i}}={\frac {{C_{L}}^{2}}{\pi {\text{AR}}e}}}$

where

• ${\displaystyle \ C_{D_{i}}}$ is the induced drag component of the drag coefficient,
• ${\displaystyle \ C_{L}}$ is the 3D lift coefficient,
• ${\displaystyle \ {\text{AR}}}$ is the aspect ratio,
• ${\displaystyle \ e}$ is the Oswald efficiency number (or span efficiency factor.) This is equal to 1 for elliptical circulation distribution, and usually tabulated for other distributions.

According to lifting-line theory, any wing planform can be twisted to produce an elliptic lift distribution.[11]

The lifting line theory does not take into account the following:

• Compressible flow
• Viscous flow
• Swept wings
• Low aspect ratio wings
• Unsteady flows

• Horseshoe vortex
• Kutta condition
• Thin airfoil theory
• Vortex lattice method

• L. J. Clancy (1975), Aerodynamics, Pitman Publishing Limited, London. ISBN 0-273-01120-0
• Abbott, Ira H., and Von Doenhoff, Albert E. (1959), Theory of Wing Sections, Dover Publications Inc., New York. Standard Book Number 486-60586-8