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36 Lifting-Line Theory

Introduction

When a free-stream flow approaches a finite wing, characterized by a definitive span from tip to tip, the downstream flow develops a trailing wake system containing swirling flows known as wingtip vortices, as shown in the figure below. These vortices induce a vertical downwash velocity over the wing’s surface, particularly near the wing tips.[1] This downwash reduces the effective angle of attack of the wing, diminishes the lift, and increases drag, which is called induced drag.

The vortices trailing from the wing tips produce a downwash over the wing, affecting its lift and drag.

A wing’s aerodynamic performance, including its lift and drag characteristics, depends significantly on its geometric design. Factors such as planform shape (chord distribution), span and aspect ratio, and spanwise twist are all essential in determining the wing’s aerodynamic efficiency. The wing design process must carefully balance these elements to optimize lift generation and minimize induced drag, amongst other factors, thereby giving the wing its best aerodynamic performance for the intended application.

Lifting line theory is another cornerstone of classical aerodynamics that can explain and predict the aerodynamic behavior of finite wings at low speeds, i.e., without considering compressibility effects. Unlike the two-dimensional airfoil theory, which assumes infinite span and aspect ratio, the lifting line theory accounts for the three-dimensional effects of finite wings, including the formation of trailing vortices and the creation of induced drag. By modeling the wing as a bound vortex line with a spanwise variation in circulation, with the associated effects of the trailed vortex wake, this theory provides a sound mathematical framework for determining the spanwise lift distribution, the induced drag, and the overall aerodynamic efficiency of a finite wing.

Learning Objectives

  • Mathematically understand how trailing vortices generate downwash and affect the angle of attack of a wing.
  • Predict how and why the lift varies along the span of a finite wing and its impact on total lift and induced drag.
  • Understand why an elliptical lift distribution minimizes the induced drag.
  • Be able to determine the total lift and induced drag of a finite wing of arbitrary shape.

History

Lifting line theory, initially developed by Ludwig Prandtl and his students in the early 1900s, revolutionized the understanding of finite wing aerodynamics and marked a significant milestone in aeronautical developments. Before Prandtl’s work, aerodynamic theories were primarily based on two-dimensional airfoil analysis, which assumed a wing of infinite span and ignored the effects of the wing tip vortices and the three-dimensional impact caused by the wake. The work of Frederick Lanchester on the rollup of wing tip vortices, a figure from his book shown below, was influential in establishing the importance of adequately modeling finite wing aerodynamics. Furthermore, as aviation advanced, particularly during WWI, including the transition from biplanes to higher-performance monoplanes by the 1920s, the predictive limitations of existing aerodynamic theories quickly became evident. Designers needed a way to predict better and optimize the performance of monoplane wings of finite span, especially regarding their lift and drag, as well as other factors such as their aeroelastic characteristics and stalling behavior.

An idealization of the roll-up of a wing tip vortex sketched by Frederick Lanchester in 1908.

Prandtl’s lifting line theory[2] addressed this need by modeling a finite wing as a bound vortex line with a spanwise circulation distribution. Prandtl’s advance over Lanchester was in the theoretical modeling of the wing, with its wake represented as a sheet of vortices trailing behind it. This wake produced a downwash flow, decreased the effective angle of attack along the entire wing, and created induced drag, a drag component directly associated with lift production. Prandtl also demonstrated that an elliptical lift distribution over the wing span minimized the induced drag, setting a practical goal for efficient wing design. Prandtl’s original published works also show an “ideal” elliptical wing planform, which may have influenced the design of the Supermarine Spitfire in the early 1930s.

In the 1920s, Hermann Glauert[3] further refined and extended Prandtl’s work, making it more accessible and applicable to practical engineering problem-solving.[4] The use of straight vortex lines, as shown in the figure below from his book, allowed the induced velocity over the wing to be readily calculated using the Biot-Savart law, which formed the cornerstone of the lifting line theory. He then clarified the fundamental importance of the elliptical lift distribution and introduced systematic methods for analyzing wings of arbitrary planform and twist using a Fourier series, much like what was done with the thin airfoil theory.

Glauert’s mathematical model of a lifting finite wing comprised a series of interlaced horseshoe vortices.

Glauert’s detailed methods for calculating induced drag helped bridge the gap between theoretical predictions and practical wing designs, making lifting line theory one of the first valuable approaches for three-dimensional aerodynamic analysis. Additionally, the theory was written in a form that later allowed it to be adapted to swept wings and non-planar configurations and extended to lifting surface methods that model the chordwise distribution of circulation. Though more advanced techniques, such as computational fluid dynamics (CFD), are now widely used, the lifting line theory remains a cornerstone of aerodynamic modeling. Its mathematical simplicity and elegance make it a valuable tool for preliminary wing design and an essential part of aerodynamic education.

Theory of Vortex Lines

The line vortex is a fundamental singularity in fluid dynamics used to model velocity fields induced by a circulation, \Gamma. The ideas of circulation and the effects associated with vortices have been previously introduced. A vortex line is a curve of any shape and tangent to the vortex, as shown in the figure below. Notice that a line vortex in three dimensions is analogous to a point vortex in two dimensions, but instead of being a singular point, it extends along a curve. The circulation around any closed curve is equal to the sum of the strengths of the line vortices intersecting the surface bounded by the curve. Consequently, a line vortex cannot terminate within a fluid. It must form a closed loop, end at a solid boundary, or extend to infinity, the principles formally embodied in Helmholtz’s theorems.

The idea of a downwash velocity induced by a vortex line.

The Biot-Savart law describes the velocity field induced by such a circulation distribution, \Gamma. Regarding the situation shown in the figure above, the increment in the downwash, dw, from the vortex element ds can be mathematically expressed as

(1)   \begin{equation*} dw = \dfrac{\Gamma \, ds}{4 \pi \, r^2} \, \sin \theta \end{equation*}

where \Gamma is the circulation or vortex strength, ds is a differential element of the line vortex, r is the distance of the point P from the element, and \theta is the angle between the direction of the element and the straight line joining the element to the point P. Notice that the element ds of a line vortex cannot exist by itself and only forms the basis for integrating the effects along a line vortex of finite length.

Straight Line Vortices

Straight-line vortices are used in the development of the finite-wing theory. Consider the evaluation of the induced velocity of a straight-line vortex of finite length AB, as shown in the figure below. If PN has a perpendicular length {h}, then the induced velocity at P from the element ds at point Q is

(2)   \begin{equation*} dw = \dfrac{\Gamma \, ds}{4 \pi \, r^2} \, \sin \theta = \dfrac{\Gamma \, h \, ds}{4 \pi \, r^3} \end{equation*}

Induced velocity from a straight line vortex of finite length.

To show this result, the element ds may be geometrically expressed as

(3)   \begin{equation*} ds = d (h \tan \phi ) = \dfrac{h}{\cos^2 \phi} \, d \phi \end{equation*}

where the angle \phi is as shown in the figure above, so that

(4)   \begin{equation*} dw = \frac{\Gamma}{4\pi \, h} \cos \phi \, d\phi \end{equation*}

The total induced velocity is then obtained by integration along the length of the vortex line using

(5)   \begin{equation*} w = \bigintss_{-\pi/2- \alpha}^{\pi/2- \beta} \frac{\Gamma}{4\pi \, h} \cos \phi \, d\phi \end{equation*}

which gives

(6)   \begin{equation*} w = \frac{\Gamma}{4\pi \, h} \left( \cos \alpha + \cos \beta \right) \end{equation*}

Doubly-Infinite Straight Line Vortex

If, as a special case, the line is of doubly infinite length, as shown in the figure below, then it will be apparent that \cos \alpha = \cos \beta = 1, so the general result reduces to

(7)   \begin{equation*} w = \frac{\Gamma}{2\pi \, h} \end{equation*}

which is a result consistent with a two-dimensional point vortex.

Induced velocity from a doubly-infinite straight-line vortex.

Singly-Infinite Straight Line Vortex

For a singly infinite vortex, which starts at point N and then extends to infinity only in one direction, as shown in the figure below, then \cos \alpha = 0 and \cos \beta = 1, and the downwash is given by

(8)   \begin{equation*} w = \frac{\Gamma}{4\pi \, h} \end{equation*}

Induced velocity from a singly-infinite straight-line vortex.

This latter result for the downwash from a semi-infinite line vortex is extensively used in developing the lifting line theory, in which several line vortices are combined appropriately to represent the physics of the flow produced by a finite wing.

Lifting Line Theory

The following exposition of the lifting line theory and the development of the monoplane wing equation follows the work of Gluert. In addressing the problem of a wing of finite span in three-dimensional flow, the following assumptions were made:

  1. The wing’s chord is small compared to its span, so it has a high aspect ratio.
  2. The wing’s spanwise axis can be considered a straight line perpendicular to the free-stream flow, i.e., the wing is unswept.
  3. The wing planform is symmetrical laterally about its centerline, i.e., the left and right wing panels mirror each other.

Apart from these restrictions, the planform (chord), pitch angle (including any twist angle), and shape of the airfoil section may vary arbitrarily across the wing’s span.

Bound & Trailed Vortices

If a wing generates lift, there must be a flow circulation around each airfoil section comprising the wing, effectively creating a line vortex or a set of line vortices along the wing’s span. These line vortices, which are attached to the wing, as shown in the figure below, are referred to as the bound vortices and create lift in accordance with the Kutta-Joukowski theorem. Physically, these bound vortices are formed by the integrated effect of the vorticity in the boundary layer or equivalent vortex sheet surrounding the airfoil’s surface, as in the manner used to develop the thin airfoil theory. According to the general theory of vortex flows and Helmholtz’s theorems, these bound vortices cannot terminate at the tips of the wing but must extend off the wing tips into the fluid as free line vortices. These are called the trailing or trailed vortices, as shown in the figure below.

A finite wing can be represented by bound and trailed vortices.

This entire vortex system, which extends to infinity far behind the wing, is completed by a transverse vortex parallel to the span. However, this vortex has no consequence in the steady state as its induced effects are asymptotically zero. For practical purposes, the trailing vortices can be considered to align with the free-stream flow and extend downstream to infinity as a singly infinite line vortex.

Horseshoe Vortex System

Glauert describes that the simplest type of vortex system occurs when the circulation around each airfoil section has a constant value \Gamma across the wing’s span. The bound vortex system can then be represented as a single “lumped” line vortex with strength \Gamma, which, to be consistent with the thin airfoil theory, can be positioned at the aerodynamic center along the 1/4-chord of the wing, as shown in the figure below. The trailing vortices will consist of two line vortices, each with the same strength, originating from the tips of the wing and extending downstream in the direction of the free-stream flow. This representation is called a “horseshoe vortex system.”

A simple horseshoe vortex system used to represent a lifting finite wing.

In practice, the vortices trailing from the wing tips will generally not be straight lines because of variations of the downwash at different distances behind the wing. Experiments show that as they develop downstream, the trailing vortices tend to descend below the wing and contract inward. However, as Glauert explains, they can be approximated as straight lines parallel to the direction of motion for most practical purposes. This assumption leads to the simplified representation of the wing and its wake as a horseshoe vortex system, which forms the fundamental basis of the lifting line theory.

Nested Horseshoe System

In reality, the vortex system of an aerofoil is more complex because the circulation is not uniform across the span. Instead, it typically reaches its maximum value at the center and tapers to zero at the tips. This distribution can be represented by superimposing multiple simple “horseshoe vortex systems,” as shown in the figure below. This setup results in a more representative model of the vortex system over the wing and behind it in the wake.

The nested horseshoe vortex system forms the basis of the lifting line model.

This model then comprises interlaced bound vortices, all colocated at the aerodynamic center on the 1/4-chord axis, and a sheet of trailing vortices extending from the wing’s trailing edge. Glauert used this representation of the vortex system as the fundamental premise in developing the mathematics of the lifting line model.

Wake Rollup

As Glauert explains, the origin of the trailing vortex wake system can also be understood from the perspective of the difference in pressure between the upper and lower surfaces of the wing. Suppose the lift distribution across the span of a wing reaches a maximum at its centerline. In that case, there will be a significant decrease in pressure above the wing and an increase in pressure below it, as shown in the figure below. These pressure differences decrease towards the tips of the wing because of the influence of the trailing wing tip vortices.

 

While wake formation and roll-up are complex physical problems, they can be modeled adequately using straight vortices.

As the streamlines pass above the wing and flow inward toward the center, while those below the wing flow outward, they leave the trailing edge and form a discontinuity surface, as shown in the figure above. The trailing vortices from the wing then represent the vorticity of this discontinuity surface. The sheet of trailing vortices rolls up into a pair of concentrated vortices downstream, forming the characteristic pair of vortices behind a wing, which is seen in practice using flow visualization methods.

Nearer to the wing, the influence of the trailing vortex system can be modeled by assuming that the individual trailing vortices extend downstream as straight lines. For the wake farther from the wing, which tends to contract somewhat, it is perhaps more accurate to assume a horseshoe vortex system with a span a little shorter than the wing span. Both representations are sufficiently representative of the actual flow physics for modeling purposes.

Calculating the Induced Velocity

In general, the circulation \Gamma(y) will vary across the span of a wing, being symmetrical about the centerline and decreasing to zero at the tips. Consistent with the nested horseshoe vortex model, between the points {y} and (y + dy) of the span of the wing, the circulation can be assumed to decrease by an amount

    \[ \Delta \Gamma = \Gamma - (\Gamma - d\Gamma) = \Gamma - \dfrac{d\Gamma}{dy} dy = -\frac{d\Gamma}{dy} dy \]

Therefore, a trailed wake filament of this strength originates from the element dy of the span, as shown in the figure below.

Calculating the induced velocity from the vortex sheet is related to the change in circulation along the wing’s span.

Therefore, in the aggregate, a sheet of trailing vortices will extend behind the entire wing span because of the continuous change in \Gamma along the wing, i.e., \Gamma(y). The induced downwash at any point of the span will then be obtained as the sum of the effects of all the trailing vortices of this sheet. For example, consider some reference point on the wing, say y_1. Using Eq. 8, the downwash at this point from all of the trailing line vortices comprising the sheet will be

(9)   \begin{equation*} w(y_1) = \bigints_{-s}^s \dfrac{-\left( \dfrac{d\Gamma}{dy} \right)}{4\pi (y - y_1)} \, dy = \dfrac{1}{4\pi} \bigints_{-s}^s \dfrac{\dfrac{d\Gamma}{dy}}{y - y_1} \, dy \end{equation*}

This latter result will then apply at every section, y_1, along the span of the wing, and different values of w will be obtained depending on the distribution, \Gamma(y), and the proximity to the wing tips. It will be apparent from the previous discussion that the circulation gradient, i.e., d\Gamma/dy, is much higher at the wing tips, so the trailed wake will have a greater circulation here. This behavior is consistent with Lanchester’s interpretation of the wake generated behind a wing. The question now is how this downwash distribution can be calculated as it affects the lift and drag at each wing section, {y}, and the overall lift and drag of the wing.

Sectional Effects of the Induced Velocity

The induced velocity from the trailed wake will affect the aerodynamic angle of attack at each section across the wing. Adding the downwash flow vector {w} to the free-stream flow vector V_{\infty} produces a resultant flow velocity (or local relative wind) that turns through an angle \alpha_i, called the induced angle of attack, as shown in the figure below. It will be apparent that the resultant flow now approaches the wing at a different angle and so creates a reduced “effective” angle of attack, \alpha_{\rm eff}.

The effect of a downwash in the flow {w} is to cause a realignment of the relative wind to the wing section, effectively tilting the lift vector aft to produce a component of drag.

From the geometry, as shown above, the induced angle, \alpha_i, is given by

(10)   \begin{equation*} \alpha_i = \tan^{-1} \left( \frac{w}{V_{\infty}} \right) \end{equation*}

which will, in general, be different at each station on the wing, i.e., w = w(y) and \alpha_i = \alpha_i (y). For small angles, which is typical for a wing, it is sufficient to write that

(11)   \begin{equation*} \alpha_i (y) = \frac{w(y)}{V_{\infty}} \end{equation*}

Therefore, the effective (and lower) angle of attack of the wing section is now

(12)   \begin{equation*} \alpha_{\rm eff} (y) = \alpha (y) - \alpha_i(y) = \alpha(y) - \left( \dfrac{w(y)}{V_{\infty}} \right) \end{equation*}

This outcome means that the corresponding lift per unit span will be reduced from its two-dimensional value, i.e., the lift obtained without the effects of the downwash flow. Notice that because {w} will typically be much smaller than V_{\infty}, then the resultant velocity, V_R, can be assumed to be equal to V_{\infty}, i.e.,

(13)   \begin{equation*} V_R = \sqrt{ V_{\infty}^2 + w^2} \approx V_{\infty} \end{equation*}

It is of particular significance in this situation to note that the lift vector at each section takes on a new orientation and is tilted slightly rearward from its original direction in two-dimensional flow, thereby reducing the vertical component of the lift for a given angle of attack. The consequence is that there is now a component of the lift that acts in the downstream direction, which explains the origin of the induced drag, {D'_{i}}, i.e.,

(14)   \begin{equation*} D'_{i} = L' \, \sin \alpha_i \approx L' \, \alpha_i = L' \left( \frac{w}{V_{\infty}} \right) \end{equation*}

Development of the Monoplane Wing Equation

With an understanding of the aerodynamic elements that make up the problem of modeling a finite wing, the development of the fundamental equation of lifting line theory can be established, which is called the monoplane wing equation.

Connection Between Circulation and Downwash

As previously shown, if \Gamma is the circulation produced around any section of the wing, the downwash from the trailing wake system at a point y_1 of the span is determined using

(15)   \begin{equation*} w(y_1) = \frac{1}{4\pi} \bigints \frac{\dfrac{d\Gamma}{dy}}{y_1 - y'} \, dy \end{equation*}

A representative wing section, therefore, experiences the lift force corresponding to the two-dimensional conditions at the effective angle of attack, i.e.,

(16)   \begin{equation*} \alpha_{\rm eff} = \alpha - \alpha_{i} = \alpha - \frac{w}{V_{\infty}} \end{equation*}

If the angles of attack {\alpha} and \alpha_{\rm eff} are measured from the angle of zero lift of the airfoil section,\alpha_0, then

(17)   \begin{equation*} \alpha_{\rm eff} = \alpha - \frac{w}{V_{\infty}} - \alpha_0 \end{equation*}

Therefore, the lift coefficient of the wing section will be

(18)   \begin{equation*} C_l = C_{l_{\alpha}} \alpha_{\rm eff} \end{equation*}

where C_{l_{\alpha}} is the slope of the curve of lift coefficient against the angle of attack for the wing section in two-dimensional flow.

Furthermore, the circulation \Gamma around the wing section can be connected to the lift coefficient, C_l, using

(19)   \begin{equation*} L' = \dfrac{1}{2} \varrho V_{\infty}^2 \, c \, C_l = \varrho \, V_{\infty} \, \Gamma \end{equation*}

This means that

(20)   \begin{equation*} \Gamma = \dfrac{1}{2} V_{\infty} \, c \, C_l = \dfrac{1}{2} V_{\infty} \, c \, C_{l_{\alpha}} \alpha_{\rm eff} = \dfrac{1}{2} V_{\infty} \, c \, C_{l_{\alpha}} \left( \alpha - \frac{w}{V} - \alpha_0 \right) \end{equation*}

Therefore, it is possible to determine the circulation and the downwash for any wing in terms of the chord and angle of attack of the wing sections, which may vary across its span.

When the circulation \Gamma and the downwash w of any monoplane wing have been determined, the lift and induced drag are obtained by evaluating the integrals

(21)   \begin{equation*} L = \bigintsss_{-s}^{s} \varrho \, V_{\infty} \, \Gamma \, dy \quad \text{and} \quad D_i = \bigintsss_{-s}^{s} \varrho \, w \, \Gamma \, dy \end{equation*}

Use of sectional lift curve slope

Strictly, the lift curve slope, C_{l_{\alpha}}, for which Glauert assigns the symbol a, depends on the shape(s) of the wing section(s). However, two-dimensional thin airfoil theory has shown that C_{l_{\alpha}} = a_{\infty} = 2 \pi per radian angle of attack. It is known from experiments that C_{l_{\alpha}} is approximately equal to 2 \pi for all practical wing sections at low Mach numbers, and hence any normal variations in C_{l_{\alpha}} may be neglected without any appreciable loss of accuracy. Nevertheless, because a wing section may differ from the theoretical value C_{l_{\alpha}} = 2\pi, it is best to retain the value of C_{l_{\alpha}} as a variable, and the theoretical value of C_{l_{\alpha}} = 2\pi is generally only used only in theoretical solutions.

Method of Solution

Glauert used a transformation[5] to replace the coordinate {y}, measured to the left wing tip along the span of the wing from its center, by the angle \theta, as defined by

(22)   \begin{equation*} y = -s \cos \theta \end{equation*}

where s is the semi-span. It will be apparent that as {y} varies from -s to s, then \theta varies from 0 to \pi across the span from the left tip to the right. The circulation, \Gamma, which is a function of {y}, can then be expressed as the Fourier series

(23)   \begin{equation*} \Gamma = 4 \, s \, V_{\infty} \, \sum_{n=1}^\infty A_n \sin n \theta \end{equation*}

where the values of the coefficients A_n must be determined by connecting the distribution of \Gamma(y) “bound” to the wing with w(y) produced by the trailed wake in a consistent manner.

Notice that the series chosen for the circulation \Gamma satisfies the condition that the circulation decreases to zero at the wing tips. As shown in the figure below, if the wing is flying in level flight, then the spanwise loading will be symmetrical about its centerline, and only odd integral values of n will occur in the Fourier series.

 

Example of the first three symmetric loading modes comprising the solution to the lifting line model.

It will be apparent that the primary mode is the n = 1 or \sin \theta mode, corresponding to an elliptical circulation distribution over the wing. Because \cos\theta = -y/s, then

(24)   \begin{equation*} \sin^2\theta + \cos^2\theta = \sin^2\theta + \left(-\frac{y}{s}\right)^2 = 1 \end{equation*}

Simplifying gives

(25)   \begin{equation*} \sin^2\theta + \frac{y^2}{s^2} = 1 - \frac{y^2}{s^2} \end{equation*}

and so

(26)   \begin{equation*} \sin\theta = \sqrt{1 - \frac{y^2}{s^2}} \end{equation*}

which is elliptical. The higher symmetric modes, i.e., n = 3, 5, 7, ..., for which n =3 and n = 5 are shown in the figure below, represent a symmetric modification to the primary loading, i.e., a deviation from the elliptically distributed form.

The downwash at the point y_1 or \theta_1 of the wing now becomes

(27)   \begin{equation*} w(\theta_1) = \displaystyle{\frac{V}{\pi} \frac{\bigints_{~0}^{\pi }\displaystyle{\sum_{n=1}^\infty } n \, A_n \cos n\theta }{\cos \theta - \cos \theta_1} } d\theta = V_{\infty} \, \sum_{n=1}^\infty n \, A_n \frac{\sin n\theta_1}{\sin \theta_1} \end{equation*}

because

(28)   \begin{equation*} \bigintss_0^\pi \frac{\cos n\theta \, d\theta}{\cos \theta - \cos \phi} = \frac{\sin n\phi}{\sin \phi} \end{equation*}

Therefore, at the general point \theta of the wing, then the downwash is

(29)   \begin{equation*} \dfrac{w}{V_{\infty}} = \dfrac{\displaystyle{\sum_{n=1}^\infty} n \, A_n \, \sin n\theta}{\sin \theta} \end{equation*}

Monoplane Wing Equation

The equation connecting the circulation and the downwash velocity now becomes

(30)   \begin{equation*} \Gamma = 4s \, V_{\infty} \, \sum_{n=1}^\infty A_n \, \sin n\theta = C_{l_{\alpha}} \, c \, V_{\infty} \left( (\alpha - \alpha_0) - \frac{\displaystyle{\sum_{n=1}^\infty} n \, A_n \sin n\theta}{\sin \theta} \right) \end{equation*}

which gives

(31)   \begin{equation*} \sum_{n=1}^\infty A_n \, \sin n\theta \left( n\mu + \sin \theta \right) = \mu (\alpha - \alpha_0) \sin \theta \end{equation*}

and finally

(32)   \begin{equation*} \sum_{n=1}^\infty A_n \, \sin n\theta \bigg( \dfrac{n\mu}{\sin \theta} + 1 \bigg) = \mu (\alpha - \alpha_0), \quad \text{where} \,\, \mu = \frac{C_{l_{\alpha}} c}{8s} \end{equation*}

Equation 32 is called the fundamental equation of lifting line theory or the monoplane wing equation and can be used for determining the values of the coefficients A_n for any monoplane[6] wing. It is generally solved numerically. The equation must be satisfied at all points of the wing, but because the wing is symmetrical about its mid-point in level flight, it is usually sufficient to consider values of \theta between 0 and \dfrac{\pi}{2}. In this case, only the symmetric modes, i.e., n = 3, 5, 7, ..., must be evaluated. The dimensionless Glauert parameter \mu, which is proportional to the chord {c}, and the angle of attack {\alpha}, will generally depend on wing twist, e.g., if the wing has wash-out.

Lift and Induced Drag

The lift on the wing is given by

(33)   \begin{equation*} L = \bigintsss_{-s}^s \varrho V_\infty \Gamma(y) \, dy \end{equation*}

Switching to the spanwise angle coordinate, \theta, where y = s \sin\theta and dy = s \cos\theta \, d\theta, the lift becomes

(34)   \begin{equation*} L = \bigintsss_0^\pi \varrho \, V_\infty \, \Gamma(\theta) s \sin\theta \, d\theta \end{equation*}

Substituting the general series expression for circulation gives

(35)   \begin{equation*} \Gamma(\theta) = 2 V_\infty \, s \sum_{n=1}^\infty A_n \sin(n\theta) \end{equation*}

and so

(36)   \begin{equation*} L = \bigintsss_0^\pi \, \varrho \,V_\infty^2 \, s^2 \left( \sum_{n=1}^\infty A_n \sin(n\theta) \right) \sin\theta \, d\theta \end{equation*}

Using the orthogonality property of sine functions, i.e.,

(37)   \begin{equation*} \bigintsss_0^\pi \sin(n\theta) \, \sin(\theta) \, d\theta = \begin{cases} \dfrac{\pi}{2}, & n = 1 \\[12pt] 0, & n \neq 1 \end{cases} \end{equation*}

then the only term that contributes to the integral is n = 1, so

(38)   \begin{equation*} L = 2\pi \, s^2 \, \varrho \, V_\infty^2 \, A_1 \end{equation*}

The coefficient, A_1, which is the elliptical loading mode, is related to the lift coefficient, C_L, using

(39)   \begin{equation*} A_1 = \frac{C_L}{\pi \, AR}, \quad \text{where } AR = \frac{s^2}{S} \end{equation*}

The induced drag is determined using

(40)   \begin{equation*} D_i = \bigintsss_{-s}^s \varrho \, w \, \Gamma(y) \, dy \end{equation*}

Switching to the \theta coordinate gives

(41)   \begin{equation*} D_i = \bigintsss_0^\pi \varrho \, V_\infty \, w(\theta) \, \Gamma(\theta) \, s \, \sin\theta \, d\theta \end{equation*}

Recall that the downwash velocity is given by

(42)   \begin{equation*} w(\theta) = \frac{V_\infty}{2s} \sum_{n=1}^\infty n \, A_n \sin(n\theta) \end{equation*}

Substituting \Gamma(\theta) and w(\theta) gives

(43)   \begin{equation*} D_i = \varrho \, V_\infty^2 \, s^2 \bigintsss_0^\pi \left( \sum_{n=1}^\infty n \, A_n \sin(n\theta) \right) \left( \sum_{m=1}^\infty A_m \, \sin(m\theta) \right) \sin\theta \, d\theta \end{equation*}

Expanding the summation gives

(44)   \begin{equation*} D_i = \varrho \, V_\infty^2 \, s^2 \sum_{n=1}^\infty \sum_{m=1}^\infty n \, A_n \, A_m \bigintss_0^\pi \sin(n\theta) \, \sin(m\theta) \, \sin\theta \, d\theta \end{equation*}

Again, the orthogonality of \sin(n\theta) functions ensures that only terms where n = m are retained, so that

(45)   \begin{equation*} D_i = \frac{1}{2} \varrho \, V_\infty^2 \, s^2 \, \pi \sum_{n=1}^\infty n \, A_n^2 \end{equation*}

It is convenient to write

(46)   \begin{equation*} 1 + \delta = \dfrac{1}{A_1^2} \, \sum_{n=1}^\infty n \, A_n^2 \end{equation*}

so the induced drag, D_i, is

(47)   \begin{equation*} D_i = \dfrac{(1+\delta) L}{2\pi s^2 \varrho V_\infty^2} \end{equation*}

In terms of drag coefficient, then

(48)   \begin{equation*} C_{D_{i}} = \frac{(1+\delta) C_L^2}{\pi \, AR} \end{equation*}

The total drag coefficient includes both profile drag and induced drag. If the profile drag coefficient C_{d_0} varies across the span, the profile drag coefficient for the wing is

(49)   \begin{equation*} C_{D_{0}} = \frac{1}{S} \bigintsss_{-s}^s C_{d_0} c \, dy \end{equation*}

Adding the induced drag gives

(50)   \begin{equation*} C_D = C_{D_{0}} + \frac{(1+\delta) \, C_L^2}{\pi \, AR} \end{equation*}

Asymmetric Loading Modes

The even modes in the lifting line theory represent an asymmetric lift distribution. These modes can arise from aileron deflections and angular velocity of the wing in roll, yaw, sideslip, or other asymmetries along the wing’s span. This situation then leads to lift asymmetry and influences rolling moments, possible yawing moments, and induced drag. For example, ailerons are control surfaces near the wingtips, deflected asymmetrically to induce a rolling moment about the aircraft’s longitudinal axis. The figure below shows that their operation produces asymmetric modes in the lifting-line theory, particularly A_2 and other higher even-numbered modes.

 

Example of the first two asymmetric loading modes comprising the solution to the lifting line model.

The total circulation distribution can be expressed as

(51)   \begin{equation*} \Gamma(\theta) = 2s\, V_{\infty} \sum_{n=1}^\infty A_n \, \sin(n\theta) \end{equation*}

where A_1 represents the symmetric lift distribution from the free-stream velocity, and A_2 captures the antisymmetric lift contribution from aileron deflection, as shown in the figure below. The primary contribution from the ailerons is

(52)   \begin{equation*} \Gamma_\text{aileron} = 4s \, A_2 \, \sin(2\theta) \end{equation*}

The coefficient A_2 depends on the aileron deflection angle \delta_a, which modifies the local angle of attack near the wingtips, i.e.,

(53)   \begin{equation*} A_2 \, \propto \, \frac{\delta_a}{AR} \end{equation*}

where AR is the aspect ratio. The constant of proportionality depends on the actual wing’s geometry and aerodynamic characteristics. For small deflection angles, A_2 is approximately linear in \delta_a.

The application of ailerons will produce asymmetric modes in the lifting line theory.

The resulting rolling moment, L_r, about the longitudinal axis, is directly related to this antisymmetric lift distribution, i.e.,

(54)   \begin{equation*} L_r = \int_{-s}^s y \, \Delta L(y) \, dy \end{equation*}

where \Delta L(y) is the lift difference from aileron deflection at spanwise location {y}. This rolling moment scales with A_2, i.e.,

(55)   \begin{equation*} L_r \, \propto \, A_2 \, s^2 \end{equation*}

From a design perspective, proper sizing and placement of ailerons are crucial to ensure sufficient rolling authority without excessive induced drag or adverse yaw effects. To this end, the lifting line theory can provide much insight into what aileron designs may be required.

Worked Example #1 – Further understanding of circulation modes

A wing with an untwisted elliptical wing planform is installed symmetrically in a wind tunnel with its centerline along the tunnel axis. If the air in the wind tunnel has a free-stream axial velocity V_{\infty} and also has a small angular velocity \omega about the tunnel axis, show that there will be an extra asymmetric distribution of circulation along the wing given by

    \[ \Gamma = 4 s \, A_2 \, \sin(2\theta) \]

Determine A_2 in terms of \omega and the wing parameters.

Show solution/hide solution.

From the lifting-line theory, the spanwise circulation distribution \Gamma(\theta) can be expressed as a Fourier sine series, i.e.,

    \[ \Gamma(\theta) = 2s \, V_{\infty} \sum_{n=1}^\infty A_n \, \sin(n\theta) \]

where s is the semi-span of the wing, \theta is the spanwise angular coordinate related to the spanwise position by y = s \cos\theta, and A_n are the Fourier coefficients determined by the flow and wing properties. For an elliptic wing, the baseline lift distribution from the free-stream flow, V_{\infty}, corresponding to the n=1 term and perturbations introduced by \omega that will manifest as higher harmonics, starting with n=2.The angular velocity \omega induces a spanwise velocity component v_y at a location y = s \cos\theta, given by

    \[ v_y = \omega \, y = \omega \, s \, \cos\theta \]

This spanwise velocity modifies the local effective angle of attack \alpha_\text{eff}, and the change in angle of attack \Delta\alpha(y) is

    \[ \Delta\alpha(y) = \frac{v_y}{V_{\infty}} = \frac{\omega y}{V_{\infty}} = \frac{\omega s \cos\theta}{V_{\infty}} \]

Therefore, the angular velocity introduces a spanwise perturbation in \Delta\alpha, which varies linearly with \cos\theta. The local circulation \Gamma (y) or \Gamma(\theta) is proportional to the effective angle of attack \alpha_\text{eff}, so the perturbation in circulation \Delta\Gamma from \Delta\alpha will be

    \[ \Delta\Gamma(y) \, \propto \, \Delta\alpha(y) \]

Substituting \Delta\alpha(y) = \dfrac{\omega \, s \, \cos\theta}{V_{\infty}} gives

    \[ \Delta\Gamma(y) \, \propto \, \frac{\omega \, s \, \cos\theta}{V_{\infty}} \]

For an elliptical wing, the spanwise variation \cos\theta contributes to the n = 2 term in the series expansion. This means that

    \[ \Delta\Gamma(\theta) = 4s \, A_2 \, \sin(2\theta) \]

where A_2 is the second harmonic, specifically

    \[ A_2 \, \propto \, \frac{\Delta\alpha(y)}{C_{l_{\alpha}}} \]

where C_{l_{\alpha}} is the lift curve slope (per radian). Substituting \Delta \alpha(y) = \dfrac{\omega \, s \, \cos\theta}{V_{\infty}}, then

    \[ A_2 = \frac{\omega s}{4 V_{\infty}} \]

Elliptic Spanwise Loading

The lift and induced drag of a wing are given by

(56)   \begin{equation*} L = 2 \pi \, s^2 \, \varrho \, V_{\infty}^2 \, A_1 \quad \text{and} \quad D_i = 2 \pi \, s^2 \, \varrho \, V_{\infty}^2 \sum_{n=1}^\infty n \, A_n^2 \end{equation*}

For a wing of a given size, the coefficient A_1 has a fixed value independent of the wing’s shape. The induced drag will be minimized when all higher-order coefficients A_n (n \geq 2) in the series for circulation are zero, i.e., \delta = 0. In this case, the distribution of circulation across the span of the wing simplifies to

(57)   \begin{equation*} \Gamma(\theta) = 4s \,V_\infty \, A_1 \, \sin\theta = 4s \, V_\infty \, A_1 \sqrt{1 - \frac{y^2}{s^2}} \end{equation*}

so the circulation is elliptically loaded.

Putting A_n = 0, n \ne 1 into Eq. 32 gives

(58)   \begin{equation*} A_1 \, \sin \theta \bigg( \dfrac{\mu}{\sin \theta} + 1 \bigg) = \mu (\alpha - \alpha_0) \end{equation*}

and so

(59)   \begin{equation*} A_1 = \frac{\mu (\alpha - \alpha_0)}{\sin \theta + \mu} \end{equation*}

The corresponding induced drag coefficient will be

(60)   \begin{equation*} C_{D_{i}} = \frac{C_L^2}{\pi \, AR} \end{equation*}

Notice that the result of the downwash becomes

(61)   \begin{equation*} w = V_{\infty} \dfrac{A_1 \, \sin \theta}{\sin \theta} = V_{\infty} \, A_1 = \text{constant} \end{equation*}

and the equivalent angle of attack is

(62)   \begin{equation*} \alpha_{\rm eff} = \alpha - \alpha_0 - \frac{w}{V_\infty} = \alpha - \alpha_0 - A_1 = \text{constant} \end{equation*}

along the span. This implies that the local lift coefficient, C_l, is constant across the span, as shown in the figure below.

 

An untwisted wing with an elliptical planform has a uniform downwash distribution, an elliptical lift distribution, and a uniform lift coefficient.

To produce a constant C_L and an elliptic circulation distribution, the chord length {c} must vary elliptically along the span. This result can be seen by using

(63)   \begin{equation*} L'(y) = \frac{1}{2} \, \varrho V_{\infty}^2 C_l \, c(y) = \varrho \, V_{\infty} \Gamma(y) = 4 s \, \varrho V_\infty^2 \, A_1 \sqrt{1 - \frac{y^2}{s^2}} \end{equation*}

where it will be apparent that

(64)   \begin{equation*} c(y) \, \propto \, \sqrt{1 - \frac{y^2}{s^2}} \end{equation*}

This result can also be written as

(65)   \begin{equation*} c(y) = c_0 \sqrt{1 - \frac{y^2}{s^2}} \end{equation*}

where c_0 is the chord length at the wing root. This special case of elliptic loading is significant for several reasons:

  1. It results in the minimum possible induced drag for a given total lift, i.e., \delta = 0.
  2. Most conventional wing designs closely approximate an elliptic load distribution, making it a valid first-order approximation.
  3. The results derived from the assumption of elliptic loading represent the optimal scale for minimizing induced drag while remaining applicable to practical wing designs.
  4. Elliptic loading (or almost elliptic) can also be achieved with wings of non-elliptic planforms by suitably varying the twist angle along the span.

Effect of Aspect Ratio

The results developed for the lift and drag coefficients of an elliptic wing can be used to calculate the effect of a change in aspect ratio. If the aspect ratio is reduced from A to A', the changes in the drag coefficient at a given value of the lift coefficient is

(66)   \begin{equation*} \Delta C_D = C_D' - C_D = \frac{2}{\pi} \left(\frac{1}{AR'} - \frac{1}{AR}\right) C_L^2 \end{equation*}

It will be apparent that wings with a higher aspect ratio have aerodynamic advantages, all other factors being equal, e.g., wing area, and planform shape.

While this formula for the drag coefficient applies only to wings with elliptic loading, the lift distribution curves for rectangular and most airplane wings do not differ significantly from the elliptic form. Therefore, the transformation formula may be used more generally to calculate the effect of a modest aspect ratio change.

Effect on Lift Curve Slope

The assumptions are: 1. The two-dimensional lift curve slope for the airfoil is a_{\infty}, which assumes no three-dimensional effects. 2. The lift curve slope for the finite wing is a, which includes the effects of finite aspect ratio (AR).

The lifting line theory introduces an induced angle of attack \alpha_i caused by the downwash from the wing’s trailing vortices. The total angle of attack \alpha is related to the effective angle of attack (\alpha_{\text{eff}}) and \alpha_i, i.e.,

(67)   \begin{equation*} \alpha_{\text{eff}} = \alpha - \alpha_i \end{equation*}

The total lift coefficient is then

(68)   \begin{equation*} C_L = a_{\infty} \, \alpha_{\text{eff}} \end{equation*}

Substituting \alpha_{\text{eff}} from the first equation

(69)   \begin{equation*} C_L = a_{\infty} (\alpha - \alpha_i) \end{equation*}

The induced angle of attack is related to the lift coefficient and aspect ratio

(70)   \begin{equation*} \alpha_i = \frac{C_L}{\pi AR} \end{equation*}

Substituting this back into the lift equation

(71)   \begin{equation*} C_L = a_{\infty} \left(\alpha - \frac{C_L}{\pi AR}\right) \end{equation*}

Rearranging gives

(72)   \begin{equation*} C_L = a_{\infty} \alpha - \frac{a_{\infty} C_L}{\pi AR} = C_L \left(1 + \frac{a_{\infty}}{\pi AR}\right) = a_{\infty} \alpha \end{equation*}

Dividing through by \alpha gives

(73)   \begin{equation*} \frac{C_L}{\alpha} = a = \frac{a_{\infty}}{1 + \dfrac{a_{\infty}}{\pi AR}} \end{equation*}

Therefore, for wings with a high aspect ratio or small a_{\infty}, the lift curve slope approaches the two-dimensional value, corrected by a small term for three-dimensional effects.

Worked Example #2 – Aerodynamic characteristics of an elliptical wing

An elliptical wing has the following characteristics:

  • Span, b = 10 \, \text{m}.
  • Planform area, S = 8 \, \text{m}^2.
  • Angle of attack, {\alpha} = 5^\circ.
  • Zero-lift angle of attack, \alpha_0 = -0.5^\circ.
  • 2-dimensional lift curve slope, a_{\infty} = 2\pi \, \text{(per radian)}.

Compute:
1. The wing’s lift coefficient, C_L.
2. The corresponding induced drag coefficient, {C_{D_{i}}}.

Show solution/hide solution.

The aspect ratio of the wing is

    \[ AR = \dfrac{b^2}{S} = \dfrac{10^2}{8} = 12.5 \]

The lift curve slope of the wing adjusted for 3-dimensional effects is

    \[ a = \dfrac{a_{\infty}}{1 + \dfrac{a_{\infty}}{\pi \, AR}} = \dfrac{2\pi}{1 + \dfrac{2\pi}{\pi \times 12.5}} = \dfrac{2\pi}{1 + \dfrac{2}{12.5}} = \dfrac{2\pi}{1.16} = 5.418 \]

The lift coefficient is

    \[ C_L = a (\alpha - \alpha_0) = 5.418 \left(\dfrac{5\pi}{180} - \dfrac{-0.5\pi}{180}\right) = 5.418 \times \dfrac{\pi}{32.727} = 0.520 \]

For an elliptical wing, the induced drag coefficient is

    \[ C_{D_i} = \dfrac{C_L^2}{\pi \, AR} = \dfrac{(0.520)^2}{\pi \times 12.5 \times 1} = 0.00688 \]

Numerical Solution to the Monoplane Wing Equation

The governing equation for the monoplane wing is

(74)   \begin{equation*} \sum_{n=1}^\infty A_n \, \sin n\theta \left( n\mu + \sin \theta \right) = \mu \, \alpha \, \sin \theta \end{equation*}

where:

  1. A_n are the coefficients of the circulation expansion.
  2. \mu = \dfrac{C_{l_\alpha} c}{4s} is Glauert’s non-dimensional parameter.
  3. {\alpha} is the geometric angle of attack, or more correctly, pitch angle.
  4. {c} is the chord length (may vary with \theta).
  5. {s} is the semi-span of the wing.

To solve numerically, the domain \theta \in [0, \pi] is divided into N discrete points, i.e.,

(75)   \begin{equation*} \theta_i = \frac{i\pi}{N}, \quad i = 1, 2, \dots, N \end{equation*}

At each discrete point \theta_i, the governing equation becomes

(76)   \begin{equation*} \sum_{n=1}^N A_n \, \sin n\theta_i \left( n\mu + \sin \theta_i \right) = \mu \, \alpha \, \sin \theta_i \end{equation*}

This equation can be rewritten in matrix form as

(77)   \begin{equation*} \mathbf{M} \mathbf{A} = \mathbf{b} \end{equation*}

where \mathbf{A} = [A_1, A_2, \dots, A_N]^T is the vector of unknown coefficients; \mathbf{b} = [b_1, b_2, \dots, b_N]^T is the right-hand side vector containing the the wing properties; \mathbf{M} is the influence coefficient matrix.

The components of the matrix equation are

(78)   \begin{equation*} b_i = \mu \, \alpha \, \sin \theta_i, \quad i = 1, 2, \dots, N \end{equation*}

and

(79)   \begin{equation*} M_{ij} = \sin(j\theta_i) \left( j \,\mu + \sin \theta_i \right), \quad i, j = 1, 2, \dots, N \end{equation*}

The matrix equation is expanded as

    \[ \begin{bmatrix} M_{11} & M_{12} & \dots & M_{1N} \\[8pt] M_{21} & M_{22} & \dots & M_{2N} \\[8pt] \vdots & \vdots & \ddots & \vdots \\[8pt] M_{N1} & M_{N2} & \dots & M_{NN} \end{bmatrix} \begin{bmatrix} A_1 \\[8pt] A_2 \\[8pt] \vdots \\[8pt] A_N \end{bmatrix} = \begin{bmatrix} b_1 \\[8pt] b_2 \\[8pt] \vdots \\[8pt] b_N \end{bmatrix} \]

where M_{ij} = \sin(j\theta_i) \left( j\mu + \sin \theta_i \right) and b_i = \mu \alpha \sin \theta_i

The unknown coefficients \mathbf{A} = [A_1, A_2, \dots, A_N]^T are found by solving

(80)   \begin{equation*} \mathbf{A} = \mathbf{M}^{-1} \mathbf{b} \end{equation*}

Efficient numerical methods, such as LU decomposition, are typically used for inverting \mathbf{M} when N is large.

Once the coefficients \mathbf{A} are determined, the following aerodynamic quantities can be computed:

  1. The spanwise circulation distribution, \Gamma(y), i.e.,

    (81)   \begin{equation*} \Gamma(\theta) = 4s V_\infty \sum_{n=1}^N A_n \sin n\theta \end{equation*}

  2. The lift distribution, C_l(y), and the lift coefficient, C_L. The total lift coefficient is directly related to A_1, i.e.,

    (82)   \begin{equation*} C_L = \frac{2A_1}{\pi} \end{equation*}

  3. The induced drag coefficient, which is computed using

    (83)   \begin{equation*} C_{D_{i}} = \frac{2}{\pi} \sum_{n=1}^N n \, A_n^2 \end{equation*}

Worked Example #3 – Numerical solution of the monoplane wing equation

A rectangular wing has the following geometric and operating conditions:

  • Span, b = 10 \, \text{m}.
  • Chord, c = 1.0 \, \text{m}.
  • Angle of attack, \alpha = 12^\circ.
  • Zero-lift angle of attack, \alpha_0 = -0.5^\circ.
  • Lift curve slope, C_{l_\alpha} = 2\pi \, \text{(per radian)}.
  • Free-stream velocity, V_\infty = 50 \, \text{m/s}.
  • Air density, \varrho = 1.225 \, \text{kg/m}^3.

Set up a numerical solution of the monoplane wing equation to determine:
1. The spanwise circulation distribution, (\Gamma(\theta)\.
2. The lift coefficient, C_L.
3. The induced drag coefficient, {C_{D_{i}}}.

Use five spanwise modes to approximate your answer.

Show solution/hide solution.

The semi-span is

    \[ s = \frac{b}{2} = \frac{10}{2} = 5 \, \text{m} \]

so the non-dimensional parameter \mu is

    \[ \mu = \frac{C_{l_\alpha} c}{4s} = \frac{2\pi \times 1.0}{4 \times 5} = 0.3142 \]

The geometric angle of attack (allowing for the zero-lift angle) is

    \[ \alpha_{\text{geom}} = \alpha - \alpha_0 = 12^\circ - (-0.5^\circ) = 12.5^\circ = \frac{12.5\pi}{180} = 0.2182 \, \text{radians} \]

Dscretize \theta \in [0, \pi] into N = 5 into equally spaced angular points, i.e.,

    \[ \theta_i = \frac{i \, \pi}{N-1}, \quad i = 0, 1, 2, 3, 4 \]

Therefore,

    \[ \theta_0 = 0, \, \theta_1 = \frac{\pi}{4}, \, \theta_2 = \frac{\pi}{2}, \, \theta_3 = \frac{3\pi}{4}, \, \theta_4 = \pi \]

The governing equation with odd Fourier coefficients is

    \[ \sum_{n=1, 3, 5, 7, 9} A_n \, \sin n\theta_i \left( n \, \mu + \sin \theta_i \right) = \mu \, \alpha_{\text{geom}} \sin \theta_i \]

This is written in matrix form as

    \[ \mathbf{M} \mathbf{A} = \mathbf{b} \]

where

    \[ \mathbf{M}_{ij} = \sin(n_j \theta_i) \left( n_j \, \mu + \sin \theta_i \right) \quad \text{and} \quad \mathbf{A} = [A_1, A_3, A_5, A_7, A_9]^T \]

as well as

    \[ \mathbf{b}_i = \mu \, \alpha_{\text{geom}} \, \sin \theta_i \]

The right-hand side vector is

    \[ \mathbf{b} = \begin{bmatrix} 0 \\ 0.0484 \\ 0.0685 \\ 0.0484 \\ 0 \end{bmatrix} \]

The influence coefficient matrix is

    \[ \mathbf{M} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0.720 & 1.162 & 0.231 & 0.079 & 0.029 \\ 1.000 & 2.514 & 0.942 & 0.376 & 0.151 \\ 0.720 & 1.162 & 0.231 & 0.079 & 0.029 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

Solving \mathbf{M} \mathbf{A} = \mathbf{b} gives

    \[ \mathbf{A} = \begin{bmatrix} 0.137 \\ 0.042 \\ 0.010 \\ 0.003 \\ 0.001 \end{bmatrix} \]

The spanwise circulation distribution is

    \[ \Gamma(\theta) = 4s V_\infty \sum_{n=1, 3, 5, 7, 9} A_n \, \sin n\theta \]

Substituting the coefficients gives

    \[ \Gamma(\theta) = 1000 \big( 0.137 \, \sin \theta + 0.042 \, \sin 3\theta + 0.010 \, \sin 5\theta + 0.003 \, \sin 7\theta + 0.001 \, \sin 9\theta \big) \]

Therefore, the lift coefficient is

    \[ C_L = \frac{2A_1}{\pi} = \frac{2 \times 0.137}{\pi} = 1.11 \]

The corresponding induced drag coefficient is

    \[ C_{D_{i}} = \frac{2}{\pi} \sum_{n=1, 3, 5, 7, 9} n \, A_n^2 \]

and substituting the values of the coefficients gives

    \[ C_{D_{i}} = \frac{2}{\pi} \left( 1 \times 0.137^2 + 3 \times 0.042^2 + 5 \times 0.010^2 + 7 \times 0.003^2 + 9 \times 0.001^2 \right) = 0.024. \]

Wing Shape & Loading Distributions

The wing’s planform, twist, and interference effects strongly influence the spanwise lift distribution and overall wing performance. The planform shape, including the wing’s aspect ratio, taper ratio, and sweep angle, determines the baseline distribution of lift along the span and affects induced drag and aerodynamic efficiency. Whether geometric or aerodynamic, wing twists allow designers to tailor the angle of attack distribution, optimizing lift and reducing induced drag under specific flight conditions. Interference effects, such as those arising from wing-body junctions, nacelles, or wingtip devices, further modify the flow field around the wing, impacting both the spanwise lift distribution and the overall drag characteristics. These factors must be carefully considered in wing design to balance aerodynamic efficiency, structural feasibility, and operational requirements.

Planform Effects

The theoretically best aerodynamic efficiency (\delta = 0) and lowest induced drag are obtained with a wing planform that is elliptical in planform shape with no twist. This wing shape gives an elliptical spanwise aerodynamic loading (i.e., lift per unit span) and uniform downwash over the wing, which, as previously mentioned, is theoretically the minimum induced drag condition, and so \delta = 0. However, this value is unobtainable in any practical wing design. Values of \delta for a plain wing can range from about 0.01 to 0.1, as shown in the figure below; anything more than 0.1 would usually be classified as poorly designed. However, even a rectangular planform wing will have a reasonably good value of \delta (probably between 0.05 and 0.1) if it has a decent aspect ratio, i.e., A\!R > 5, and also uses some wing twist or wash-out to control the spanwise lift distribution.

Wing planform has an essential effect on the spanwise loading distribution and overall aerodynamic efficiency of the wing.

The shape of the wing (i.e., its planform) will affect the distribution of lift and the formation of the trailing vortex system, hence the magnitude of the induced drag on the wing. The idea is to use variations in wing chord along the span and perhaps wing twist and airfoil sections to approximate the ideal elliptical spanwise loading closely and minimize the value of \delta. The actual effective aspect ratio of the wing can also be improved somewhat by paying attention to the shape of the wing tips, which can lower the values of \delta if they are suitably shaped, e.g., with more rounded contours or perhaps with the addition of a winglet.

Incorporating Wing Twist

In the lifting line theory, wing twist affects the circulation distribution by altering the local geometric angle of attack along the span, which modifies the local lift coefficient. The effective angle of attack at any spanwise location is now given by

(84)   \begin{equation*} \alpha_{\text{eff}}(y) = \alpha_{\text{geo}}(y) - \alpha_0 - \alpha_i(y) \end{equation*}

where \alpha_{\text{geo}}(y) is the local geometric angle of attack, including the effect of twist, and the induced angle of attack \alpha_i(y) is

(85)   \begin{equation*} \alpha_i(y) = \frac{w(y)}{V_{\infty}} \end{equation*}

Wing twist, denoted as \theta_{\text{tw}}(y), modifies the geometric angle of attack by

(86)   \begin{equation*} \alpha_{\text{geo}}(y) = \alpha_{\text{root}} - \theta_{\text{tw}}(y) \end{equation*}

where \alpha_{\text{root}} is the root angle of attack. Substituting this, the effective angle of attack becomes

(87)   \begin{equation*} \alpha_{\text{eff}}(y) = \alpha_{\text{root}} - \theta_{\text{tw}}(y) - \alpha_0 - \alpha_i(y) \end{equation*}

Notice that in the case of no twist, i.e., (\theta_{\text{tw}}(y) = 0), the geometric angle of attack is constant, and the circulation distribution depends only on the wing shape and induced effects. In the case of wash-out, i.e., \theta_{\text{tw}}(y) decreases toward the tip, this reduces lift at the wingtips, redistributes lift inboard, as shown in the figure below, and also decreases induced drag. If the wing has wash-in, i.e., \theta_{\text{tw}}(y) increases toward the tip (which is unusual), it increases lift at the wingtips, leading to higher induced drag. Twist allows designers to optimize the lift distribution and induced drag for specific flight conditions.

The effects of spanwise twist can be used to control the span loading and minimize the induced drag.

Interference Effects

The photograph below illustrates the nature of the spanwise loading over a wing from the process of natural condensation in the low-pressure zones, which is nominally elliptically distributed, even on this relatively low aspect ratio wing. The highest lift (highest pressure difference) is at midspan, and the lowest lift (lowest pressure difference) is at the wing tips. The problem is, however, that even minor deviations from the ideal elliptical form can result in higher induced drag from the wing. Such variations can occur because of interference effects on the wing from the fuselage, engines, undercarriage, external stores, etc.

Natural condensation in the low-pressure area renders visible the nominally elliptical form of the spanwise loading, in this case, for an F-15.

Examples of the effects of spanwise interference are shown in the figure below, which are for the same approximate value of total wing lift. Notice that the fuselage can produce significant deviations from the ideal elliptical form, perhaps increasing the value of \delta to over 1.15. However, local effects along the wing, such as because of the aerodynamic interference effects produced by an engine, tend to have more minor effects on the values of \delta.

Flow interference from the effects of the fuselage, engines, etc., tends to spoil the ideal spanwise lift distribution and increase the value of \delta.

What is aerodynamic twist?

Aerodynamic twist refers to the variation in the airfoil’s camber or thickness distribution along the wing span, which alters the local zero-lift angle of attack, \alpha_0, such as using a different airfoil section. Unlike geometric twist, which physically pitches the wing sections, aerodynamic twist modifies the airfoil shape to tailor the lift characteristics. By modifying \alpha_0, the effective angle of attack and, consequently, the spanwise lift distribution can be modified. In the context of lifting line theory, aerodynamic twist directly impacts the governing equation by introducing a spanwise variation in \alpha_0. This changes the effective angle of attack, which determines the local lift coefficient C_l and circulation distribution. The primary advantage of aerodynamic twist is its ability to achieve specific spanwise loading patterns, such as near-elliptic lift distributions, without changing the wing’s physical orientation. This minimizes induced drag, optimizes load redistribution, and enhances performance for non-elliptic planforms.

Additionally, aerodynamic twist improves stall behavior by reducing lift near the tips, ensuring the wing root stalls first. It also provides flexibility in wing design, as it does not require a physical twist of the wing sections like a geometric twist. By incorporating aerodynamic twist into lifting line theory, designers can better balance aerodynamic efficiency, structural feasibility, and flight performance.

Extension to Lifting Surface Theory

The progression from lifting line theory to lifting surface theory and then to incorporating thickness represents a series of refinements in aerodynamic modeling. Lifting line theory is the simplest model, where the wing is represented as a single line of circulation bound at the 1/4-chord along its span. As previously described, this theory assumes the wing is infinitely thin and has a high aspect ratio, giving a first-order prediction of the spanwise lift distribution and induced drag. However, the lifting line theory neglects variations along the chord and the effects of wing thickness, thereby limiting its broader application.

Lifting surface theory extends the concept by treating the wing as a two-dimensional surface, allowing for a more comprehensive representation of the flow around finite wings. This theory uses distributed vortex sheets to account for spanwise and chordwise lift variations. It also captures three-dimensional effects such as tip vortices and their influence on induced drag. These methods still model the flow as a potential flow but solve for the distribution of vorticity that satisfies the no-penetration boundary condition over the span and chord of the wing surface. Lifting surface theory significantly improves the accuracy of aerodynamic predictions for wings with moderate or low aspect ratios, but it still assumes that the wing is thin.

 

The concept of a lifting surface theory, one version without thickness and the other with thickness.

Wing thickness adds another layer of realism and complexity to the aerodynamic analysis. A wing with some finite thickness modifies the flow field, altering the pressure distribution and introducing form drag. The inclusion of thickness can also be addressed through various potential flow corrections, where source distributions are added to the vortex representation to account for the wing’s thickness and volume. Numerical methods, such as panel or boundary element methods, can model the wing as a three-dimensional body, capturing both lifting and thickness effects.

Summary & Closure

The lifting line theory provides a foundational framework for analyzing the aerodynamic behavior of finite wings. Accounting for the variation of circulation and downwash along the span allows for estimating key performance metrics such as lift distribution, induced drag, and downwash velocity. Despite its simplifying assumptions, the lifting line theory captures the essential physics governing wing aerodynamics with remarkable accuracy. It remains widely used in airplane design and performance analysis because of its relative simplicity and ability to approximate the wing’s aerodynamic behavior with good predictive capability. Moreover, the insights gained from lifting line theory, such as the benefits of elliptic load distribution for minimizing induced drag, continue to influence modern aerodynamic design practices, providing a basis for more advanced computational methods and wing optimization techniques.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • How might the assumptions used in lifting line theory (inviscid, incompressible flow) affect the quality of predictions of actual wings?
  • Why does the elliptical circulation distribution minimize induced drag, and what are the practical challenges in achieving it for non-elliptic wing planforms?
  • How does increasing a wing’s aspect ratio improve aerodynamic efficiency but introduce structural and design challenges?
  • How do wings with rectangular or linearly tapered shapes deviate from elliptic loading, and how does this impact induced drag and overall wing performance?
  • Why is the lifting line theory inapplicable in supersonic flows, and how are these issues addressed in modern aerodynamics?
  • How might winglets be modeled within the framework of the lifting line theory? What types of modifications, if any, may be required?
  • How does wing twist affect wing performance and allow for loading optimization under specific flight conditions?
  • What are some examples of aircraft that achieve near-elliptic wing loading?
  • How is the lifting line theory or its derivatives applied in other contexts, such as rotor blades, wind turbines, or hydrofoils?

Other Useful Online Resources

To understand more about the aerodynamics of finite wings and wing theory, explore some of these online resources:”

  1. The addition of a winglet will cause the tip vortex to form at the tip of the winglet. ↵
  2. Prandtl's original publication on the lifting-line theory was "Über Tragflügeltheorie" ("On Wing Theory"), which was published in 1904 in the proceedings of the Third International Congress of Mathematicians in Heidelberg, Germany. This paper introduced the fundamental ideas of the lifting-line theory. ↵
  3. Hermann Glauert (1892–1934) was a British aerodynamicist and a pioneer in theoretical aerodynamics. He worked at the Royal Aircraft Establishment (RAE) and is best known for his contributions to thin airfoil theory and lifting line theory. His influential book, "The Elements of Aerofoil and Airscrew Theory," remains a classic. ↵
  4. Glauert systematically presented this theory in his 1926 book The Elements of Aerofoil and Airscrew Theory. ↵
  5. This transformation is analogous to the one used in the thin airfoil theory. ↵
  6. There is also a biplane wing theory. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2025 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

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