25 Aerodynamics of Finite Wings

Introduction

When the flow passes over a finite wing, i.e., a wing with a definitive span from tip to tip, the downstream flow is characterized by the formation of a trailing wake system comprised of swirling flows called wing tip vortices, an example being shown in the photograph below. These vortices are like horizontal tornados and contain high rotational “induced” flow velocities, particularly near their centers, extending out for more than a wing span. Wing-generated vortices will form along the edges of the wing tips (or at the tip of the winglet) or wherever else on the wing there is a significant spanwise change in the pressure and lift distribution, such as at the side edges of the flaps.

A Boeing A330 Aircraft flying away from the camera on a clear sky background. White water vapor is trailing off the wings and the winglet tips.
The flow rendered visible by natural condensation showing the wing wake and the wing tip vortices trailing from the tips of the winglets. The vortices are left in the wake behind the wing and have significant effects on the aerodynamics of the aircraft as a whole, primarily increasing its drag.

The most significant aerodynamic effects of these wing tip vortices or tip vortices are on the aircraft’s wing, producing a vertical downwash flow velocity over its surface, especially near the wing tips. This downwash is of sufficient magnitude to alter the angle of attack of every wing section and, subsequently, the amount of aerodynamic lift and drag produced on the entire wing. Consequently, finite-span wings have different aerodynamic characteristics from those of two-dimensional airfoils. The effects produced on the wing’s aerodynamic characteristics depend on the wing’s shape, including its chord distribution, span and aspect ratio, spanwise twist, and other factors, such as if the wing has a winglet.

Objectives of this Lesson

  • Appreciate the physical nature and effects of the trailed wake system behind a wing of finite-span.
  • Understand the basic aerodynamic characteristics of finite wings, including the effects of wing aspect ratio on the lift and drag.
  • Know how to interpret and use a drag polar for a finite wing and an aircraft.

Origin of Trailing Vortices

The formation of vortices behind finite wings has been studied for over a century. Some of the first investigations were conducted by Frederick Lanchester in the early 1900s, whose book on the subject eventually led to the development of modern wing theory. By any aerodynamic standard, tip vortex formation is a complex physical process. It is known that when combined with the free-stream flow then, the flow in the tip region flows from the lower surface to the upper surface and begins to rotate and form a vortical flow that is trailed behind the aircraft. This significant aerodynamic behavior is shown in the photograph below, which in this case, was visualized by using smoke ejected into the wake behind a Boeing 747. The strengths of these vortical flows depend on the magnitude of the pressure gradient, as well as the tip shape, which in turn is related to the total lift and lift distribution on the wing. A short distance downstream of the wing’s trailing edge, the outer wing tip vortices in the photograph can be seen to be rolled up rather tightly.

A Boeing 747 taking part in a NASA trailing wake study. Two dominant swirling trailing vortices, one from each wingtip, can be seen to stay tightly rolled-up behind the aircraft.

To explain the underlying flow physics of tip vortex formation, it should be remembered that there is a static pressure difference between the upper and lower surfaces, which is the source of the lift on the wing. Research has shown that the tip vortex originates from the tendency of flow to curl around the wing tips under the action of this pressure gradient at the wing tips, as suggested by the schematic below. In this regard, the pressure naturally wants to equalize, so the flow moves in a direction from the high pressure on the lower surface to the low pressure on the upper surface. The consequence of this behavior is to initiate a rotational motion in the flow, which is the beginning of the formation of the wing tip vortex.

The wing tip vortices trail from behind the wing and will produce a downwash flow over the wing that will affect its aerodynamic lift and drag.

The inherent three-dimensionality of the flow at the wing tip and the formation of the tip vortex has been researched over many decades employing experiments and computationally utilizing CFD, an example of the latter being shown below. These results, which can be plotted in terms of flow velocity, pressure, etc., have revealed the complex and intricate aerodynamics of the vortex roll-up. This roll-up process is related to the presence and interaction of three-dimensional boundary layers, flow separation, and shear layers on the wing tip.

The origin of the trailing vortex system from behind a wing is a complex evolutionary process involving the effects of pressure gradients, boundary layers, and flow separation. The tip vortex is usually tightly rolled-up within 2 to 3 chord lengths behind the wing.

One research finding is that the tip vortex formation is an evolutionary process, and it takes some downstream distance before the vortices become fully formed behind a wing. Once they are formed, however, they are persistent in that they spin down only slowly under viscosity and turbulence. One other concern here is, then besides the aerodynamic effects of the vortices on the aircraft itself, is the possible effects on a following aircraft, such as near an airport, this latter issue being discussed at the end of this lesson.

Effects of Trailing Vortices

The effects of these rolled-up vortices (one from each wingtip) are significant. Because of their swirling-induced flow field, they will produce a downwash at each section of the wing. The principle is shown in the figure below. 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, which is called the induced angle of attack. It will be apparent from this figure that the resultant flow now approaches the wing at a different angle, and so creates an “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, and so effectively tilting of the lift vector aft to produce a component of lift force in the direction of drag.
From the geometry of the problem shown above, then the induced angle \alpha_i as given by

(1)   \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, then it is sufficient to write that

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

or in general that

(3)   \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

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

or in general that

(5)   \begin{equation*} \alpha_{\rm eff} (y) = \alpha (y) - \alpha_i(y) = \alpha(y)  - \frac{w(y)}{V_{\infty}} \end{equation*}

which means now 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 can be assumed to be V_{\infty}, i.e.,

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

It is of particular significance in this situation that the lift vector takes on a new orientation and is tilted slightly rearward from its original orientation in two-dimensional flow. Remember that the lift force, by definition, acts perpendicular to the relative wind, so in this case, it can be seen in the figure above that the lift vector rotates rearward through a slight angle \alpha_i. The consequence is that there is now a component of the lift that acts in the downstream direction, which is called the induced drag, D'_{i}, i.e.,

(7)   \begin{equation*} D'_{i} = L' \, \alpha_i \end{equation*}

The induced drag component, D'_{i}, is often called “the drag due to lift” because its origin is primarily a consequence of the formation of the wing tip vortices, and these vortices will only form when the wing creates lift. Remember that L' and D'_{i} represent forces per unit length at each section of the wing, which, in general, will be different, i.e., L' = L' (y) and D'_{i}= D'_{i}(y). Therefore, the total lift and drag on the wing must be obtained by spanwise integration. Consequently, the total wing lift, L, will be given by

(8)   \begin{equation*} L = \int_{-s}^{s} L' (y) \, dy \end{equation*}

where y runs from y = -s at the left wing tip to y = s at the right wing tip.  Similarly, the total induced drag on the wing is given by

(9)   \begin{equation*} D_i = \int_{-s}^{s} D'_{i}(y) \, dy = \int_{-s}^{s} L' (y) \alpha_i (y)\, dy \end{equation*}

Wing Force & Moment Coefficients

When force and moment coefficients are defined for finite wings, not only is a reference length needed such as chord length, but also a reference area. The reference area is usually the projected or planform wing area, S. The dimensionless coefficients for a finite wing are defined as:

Lift coefficient, C_{L} = \displaystyle{\frac{L}{q_{\infty} S}}

Drag coefficient, C_{D} = \displaystyle{\frac{D}{q_{\infty} S}}

Normal force coefficient, C_{N} = \displaystyle{\frac{N}{q_{\infty} S}}

Axial (chord) force coefficient, C_{A} = \displaystyle{\frac{A}{q_{\infty} S}}

Moment coefficient about some point a, C_{M_{a}} = \displaystyle{\frac{M_a}{q_{\infty} S \b \overline{\overline{c}} }}

Recall that for finite wings, it is generally the convention that the “wing area,” S, is based on the projected planform wing area and not the actual surface area. Wing area is obtained by integrating the distribution of wing chord along the span from one wing tip to the other, i.e.,

(10)   \begin{equation*} S = \int_{-s}^{s} c \, dy = \int_{-b/2}^{b/2} c \, dy = 2 \int_{0}^{s} c \, dy = 2 \int_{0}^{b/2} c \, dy \end{equation*}

where c = c(y). It is important not to confuse the wing area (symbol capital S) with the semi-span of the wing (symbol lowercase s).

For bodies such as aircraft fuselages or road vehicles, the value of S may have a definition such as the body’s maximum cross-sectional area. In general, for other than wings it should be remembered that the reference area may not be unique, and standard conventions should always be used when defining force and moment coefficients.

Notice in these foregoing definitions of the coefficients the use of a reference length \overline{c} in the definition of the moment coefficient, which for a finite wing is usually the mean aerodynamic chord (MAC). Recall that the MAC is the chord of an equivalent wing that would experience aerodynamic forces identical to those of the actual wing. The MAC is generally defined as

(11)   \begin{equation*} {\rm MAC} = \overline{\overline{c}} = \frac{2 \displaystyle{\int_{0}^{s} c^2 dy}}{S} \end{equation*}

which like the wing area must be obtained by spanwise integration.

Drag Polar for a Finite Wing

The net effect of these wing tip vortices on the entire wing, therefore, is a reduction in the lift (for a given angle of attack) and an increase in drag, the primary dependency being the effects of wing span and, specifically, the aspect ratio, as shown in the figure below in terms of the drag polar. These classic results, which are wind tunnel measurements from Ludwig Prandtl’s original work on the subject with his students over a century ago, show that the aspect ratio of the wing has a significant effect on both lift and drag. Recall that the aspect ratio of the wing, A\!R, is defined as the ratio of the square of the wing span to the wing reference area, i.e.,

(12)   \begin{equation*} A\!R = \frac{{\rm span}^2}{\rm area} = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{2 \displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}

Classic results showing the effects of wing aspect ratio on the drag polar of a finite wing.
The Wright brothers were also to notice the importance of aspect ratio in their wind tunnel experiments in 1900, the results from which were reflected in the design of their “Flier” in 1903. Unlike others at the time, including Whitehead and Langley, they understood the the ability to build and fly a high aspect ratio wing was one key to successful flight on the limited engine power that they had.

The results in the figure above confirm that the aspect ratio of a wing is important in aerodynamic analysis because a higher lift-to-drag ratio can be obtained with wings of the higher aspect ratio, i.e., a long slender wing with a large span relative to its average chord. The physical reason is that the higher the aspect ratio of the wing, then the further the tip vortices are away from more of the wing. Therefore, their effects produce a lower downwash and have less of an effect on the three-dimensional aerodynamics.

A reminder of the significance of the drag polar, in this case for a finite wing, is now appropriate because it forms a basis for understanding total aircraft performance. An annotated version of a representative polar is shown in the figure below. Notice that the slope of a straight line running from the origin of the graph at (0, 0) to any point on the polar curve is the lift-to-drag ratio of the wing at that operating point.

Representative drag polar for a finite wing. A straight line running from the origin of the graph at (0, 0) to any point on the polar curve is the lift-to-drag ratio.

The tangent point of the line to the drag polar represents the highest slope and so the operating point for the best lift-to-drag ratio, i.e., the best L/D or C_L/C_D, which can be seen to occur at one specific value of the lift coefficient for a given wing as well as its operating Reynolds number and Mach number. The value of C_{D_0}} represents the part of the drag that is relatively constant and independent of the lift coefficient, i,e., the sum of skin friction and pressure drag or the so-called “parasitic drag, which is sometimes just called “form drag.”

Calculating the Lift & Drag on a Finite Wing

The lift and induced drag of a finite wing in an ideal “potential” inviscid flow can be calculated using a theoretical formulation known as Prandtl’s lifting line theory. The original formulation assumes an “elliptically loaded” finite wing, which gives as a consequence a special case of uniform downwash across the wing span and so the production of the minimum induced drag. The result for the drag can be expressed in coefficient form as

(13)   \begin{equation*} C_{D_{i}} = \frac{{C_L}^2}{\pi A\!R} \end{equation*}

this much cited equation showing that the “induced” drag coefficient, C_{D_{i}}, is proportional to the square of the lift coefficient of the wing, C_L, and inversely proportional to the aspect ratio of the wing, A\!R. Prandtl’s results, however, as will be described, can also be generalized to a wing with any form of spanwise lift loading.

The lift and drag on a finite wing can also be deduced from the use of the conservation laws in integral form, which is a more physically tangible result rather than approaching the problem from the lifting line theory. An elliptically spanwise loaded wing with an induced angle of attack \alpha_i produces a flow with a downwash angle \epsilon = 2 \alpha_i in the trailing wake, i.e., there is a vertical change in time rate of change of momentum of the flow as it passes about the wing, and it is turned through an angle 2 \alpha_i.

Using these principles, the force on the fluid to increase its vertical momentum can be established and the reaction force then is the lift on the wing. It is found that the lift on the wing is given by

(14)   \begin{equation*} L = \left[ \rho_{\infty} \left( \frac{\pi b^2}{2} \right) V_{\infty} \right] V_{\infty} \sin \epsilon \end{equation*}

Therefore, proceeding using the standard representation of the lift in terms of lift coefficient C_L then

(15)   \begin{equation*} L = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S C_L = \rho \left( \frac{\pi b^2}{4} \right) V_{\infty}^2 \epsilon = \rho_{\infty} \left( \frac{\pi b^2}{4} \right) \end{equation*}

Solving for the induced angle of attack of the wing gives

(16)   \begin{equation*} \alpha_i = \frac{C_L}{\pi A\!R} \end{equation*}

The rearward pointing component of the lift, which is the induced drag D_i, is then obtained from force resolution (assuming small angles) giving

(17)   \begin{equation*} D_i = L \, \alpha_i = L \left( \frac{C_L}{\pi A\!R} \right) = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S C_L \left( \frac{C_L}{\pi A\!R} \right) = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S \left( \frac{{C_L}^2}{\pi A\!R} \right) \end{equation*}

or simply in coefficient form that

(18)   \begin{equation*} C_{D_{i}} = \frac{{C_L}^2}{\pi A\!R} \end{equation*}

which is another way of deriving Prandtl’s classic formula for the induced drag coefficient of a finite wing with an ideal elliptical spanwise loading.

In the more general case where the wing is not elliptically loaded then \alpha_i is not the same at all points over the wing. In this case, the induced drag coefficient is given by

(19)   \begin{equation*} C_{D_{i}} = \frac{(1 + \delta) \, {C_L}^2}{\pi A\!R} \end{equation*}

where \delta > 0, which is known as the wing spanwise efficiency factor. This \delta parameter has been known as a span factor. In the lifting-line theory, the values of \delta can be calculated exactly by assuming that the spanwise loading over the wing comprises a series of “modes” expressed as a Fourier series.
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., thrust 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.

In practice, 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 a poorly designed wing. 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 > 6, and also uses some wing twist or washout to control the spanwise lift distribution.

Values of spanwise efficiency factor “\delta” factor for several wing planform shapes. The idea is to use variations in wing chord along the span, as well as perhaps wing twist and airfoil sections, to approximate closely the ideal elliptical spanwise loading and so minimize the value of \delta.

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 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.

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 can be seen to be nominally elliptically distributed. The highest lift (highest pressure difference) is 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 deviations can occur because of interference effects on the wing from the fuselage, engines, undercarriage, external stores, etc.

The nominally elliptical form of the spanwise loading on a wing, in this case for a F-15, which is rendered visible by natural condensation in the low pressure area above the wing.

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 a value of 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.

Spanwise flow interference from the effects of the fuselage, engines, etc., tend to spoil the more ideal spanwise lift distribution and so increase the value of \delta.

It has already been discussed why the lift on the wing is reduced by the effects of the trailing vortices. Because a result for the induced angle of attack has been obtained then the lift coefficient for a plain finite wing can now be written as

(20)   \begin{equation*} C_L = C_{l_{\alpha}} \alpha_{\rm eff} = C_{l_{\alpha}} \left( \alpha - \alpha_i \right) = C_{l_{\alpha}} \left( \alpha - \frac{(1 + \delta) \, C_L}{\pi A\!R } \right) \end{equation*}

where C_{l_{\alpha}} is the two-dimensional lift curve slope of the airfoil section that comprises the wing. Rearranging to solve for C_L gives that

(21)   \begin{equation*} C_L = \frac{C_{l_{\alpha}} \alpha}{1 + \displaystyle{\frac{(1 + \delta)  \, C_{l_{\alpha}}}{\pi A\!R}}} = \left( \frac{C_{l_{\alpha}}}{1 + \displaystyle{\frac{(1 + \delta)  \, C_{l_{\alpha}}}{\pi A\!R}}}\right) \alpha \end{equation*}

so the lift curve slope of the finite wing is reduced to

(22)   \begin{equation*} \frac{d C_L}{d \alpha} = \left( \frac{C_{l_{\alpha}}}{1 + \displaystyle{\frac{(1 + \delta) \  C_{l_{\alpha}}}{\pi A\!R}}} \right) \end{equation*}

the effects being shown in the figure below; the significant reduction of the lift curve slope with decreasing wing aspect ratio will be apparent

Results showing the effects of wing aspect ratio on the lift coefficient of a finite wing as a function of its angle of attack.

This foregoing result establishes that the lift curve slope of a finite wing is reduced by decreasing its aspect ratio. It also confirms that in the limiting case where the aspect ratio becomes large and approaches infinity, then the lift curve slope approaches the two-dimensional lift curve slope of the airfoil section comprising the wing.

Example #1 – Calculating the Lift Curve Slope of a Finite Wing

Consider a finite wing with an aspect ratio of 7.2 and a spanwise efficiency factor \delta = 0.1. The wing is comprised of an airfoil with a two-dimensional lift curve slope of 0.10 per degree. Calculate the lift curve slope of the finite wing.

Notice that a  lift curve slope 0.1 per degree is equal to 0.1 \times 180/\pi = 5.73 per radian angle of attack. The three-dimensional lift curve slope is calculated using

    \[ \frac{d C_L}{d \alpha} = \left( \frac{C_{l_{\alpha}}}{1 + \displaystyle{ \frac{ ( 1 + \delta) \  C_{l_{\alpha}}}{\pi A\!R}}} \right) =\frac{ 5.73}{1 + \displaystyle{ \frac{(1 + 0.1)  \times 5.73}{\pi \times 7.2}} } = 5.029~\mbox{/rad.} = 0.087~\mbox{/deg.} \]

Drag Polar for an Airplane

Having developed results for the lift and drag of a finite wing, an approximate result for the drag of an entire airplane can now be established. In the first instance, the best approach is to develop the most straightforward possible representation (i.e., in the form of a simple but representative equation) for use in various forms of analysis. The simplest form can be assumed to comprise the sum of the non-lifting and lifting components of drag.

Although there are other lifting surfaces on the airplane, such as the horizontal tail, these contributions can still be included into a single drag contribution, so that the drag equation can be represented approximately by

(23)   \begin{equation*} C_D = C_{D_{0}} + \frac{ {C_L}^2}{\pi A\!R \, e} = C_{D_{0}} + \left( \frac{1}{\pi A\!R \, e} \right) {C_L}^2 = C_{D_{0}} + K {C_L}^2 \end{equation*}

where C_{D_{0}} is the non-lifting part and K {C_L}^2 is the lifting part. This latter equation would be the simplest possible representation of the drag polar for the aircraft.

The factor “e” is often known as Oswald’s efficiency factor (after William Bailey Oswald), which can be interpreted as the loss of aerodynamic efficiency that can be associated with non-ideal effects associated with a non-elliptical spanwise lift distribution, as well as the growth in profile drag on the airfoil sections comprising the wing (in aggregate) with increasing {C_L}^2.

Recall from previously that in the lower angle of attack regime, the profile drag coefficient on an airfoil section can be represented by the equation

(24)   \begin{equation*} C_d = C_{d_{0}} + d_1 \, C_l + d_2 \, {C_l}^2 \end{equation*}

For a symmetric airfoil (where \alpha_0 = 0) then d_1  will be zero. In the case of a finite wing, then the net profile drag from the airfoils that comprise the wing can be written as

(25)   \begin{equation*} C_D = C_{D_{0}} + D_1 C_L + D_2 {C_L}^2 \end{equation*}

With the addition of the induced drag then the total wing drag becomes

(26)   \begin{equation*} C_D = C_{D_{0}} + D_1 C_L  + D_2 {C_L}^2 + \frac{ (1 + \delta) \, {C_L}^2}{\pi A\!R} \end{equation*}

If D_1 is assumed to be zero or small (typical), then

(27)   \begin{equation*} C_D = C_{D_{0}} + D_2 {C_L}^2 + \frac{ (1 + \delta) \, {C_L}^2}{\pi \, A\!R} = C_{D_{0}} + \frac{ {C_L}^2}{\pi A\!R \, e} \end{equation*}

This means that Oswald’s efficiency factor will be given by

(28)   \begin{equation*} e = \frac{1}{\pi \, A\!R  \, D_2 + (1+\delta)} \end{equation*}

and so the representation for C_D as an equation becomes

(29)   \begin{equation*} C_D = C_{D_{0}} + K {C_L}^2 \end{equation*}

where K is given by

(30)   \begin{equation*} K = \pi \, A\!R  \, D_2 + (1 + \delta) \end{equation*}

Values of K for practical aircraft would typically range from about 1.1 to 1.4, with corresponding values of e varying from about 0.7 (average) to 0.9 (very good). At low angles of attack, such as when the aircraft is in cruise, then

(31)   \begin{equation*} K = (1 + \delta) \end{equation*}

and so for these conditions then

(32)   \begin{equation*} e \approx \frac{1}{ 1 + \delta} \end{equation*}

A representative form of the drag polar is shown in the figure below. The non-lifting part depends primarily on the overall shape of the airplane is usually called the form drag or parasitic drag coefficient. The lifting part is called the induced drag coefficient or, more generally, “drag due to lift,” and it can be seen that this value depends significantly on the wing’s operating lift coefficient. 

The simplest form of a representative drag polar for an airplane, which is comprised of a non-lifting part plus a lifting (induced) part.

A table of representative values of C_{D_{0}} and e for several different aircraft is shown below. Notice that because of the diversity of aircraft designs within one group, it is only possible to give a range of values based on historical data. Such values are useful in preliminary design studies of new aircraft where the actual values may not be known until more detailed analyses are conducted, including wind tunnel and flight testing.

Representative values of C_{D_{0}} and e based on historical data.
Aircraft type C_{D_{0}} e
Twin-engine piston prop 0.022–0.028 0.75–0.8
Large turboprop 0.018–0.024 0.8–0.85
GA airplane w/retractable gear 0.02–0.03 0.75–0.8
GA airplane w/fixed gear 0.025–0.04 0.65–0.8
Subsonic jet 0.014–0.02 0.75–0.85
Supersonic jet 0.02–0.04 0.6–0.8
Sailplane 0.012–0.015 0.8–0.9
Drones & model aircraft 0.025–0.045 0.75–0.85

Total Aircraft Drag & Thrust Required for Flight

As previously derived, if the simplest form of drag coefficient variation for an airplane is assumed, i.e., using

(33)   \begin{equation*} C_D = C_{D_{0}} + K {C_L}^2 = C_{D_{0}} + \frac{ {C_L}^2}{\pi A\!R \, e} \end{equation*}

then the total drag on an airplane can now be calculated as a function of its airspeed. Remember that A\!R is the aspect ratio of the wing and the value of e (Oswald’s efficiency factor) is always less than unity in any practical case.

Another way of writing this latter equation is

(34)   \begin{equation*} C_D = C_{D_{0}} + C_{D_{i}} \end{equation*}

where C_{D_{i}} is called the induced drag coefficient or the drag coefficient on the wing resulting from the creation of lift or “lift due to drag,” i.e.,

(35)   \begin{equation*} C_{D_{i}} = \frac{ {C_L}^2}{\pi A\!R \, e} \end{equation*}

The total (dimensional) drag on the airplane is then

(36)   \begin{equation*} D = \frac{1}{2} \rho V_{\infty}^2 S \left( C_{D_{0}} + \frac{ {C_L}^2}{\pi A\!R \, e} \right) \end{equation*}

which must be equal to the thrust needed from the propulsive system when lift equals weight, i.e., in trimmed, strady, unaccelerated flight then lift = weight, and thrust = drag, i.e.,

(37)   \begin{equation*} L = W\quad \mbox{and} \quad T = D \end{equation*}

The lift coefficient C_L can be calculated because

(38)   \begin{equation*} L = W = \frac{1}{2} \rho V_{\infty}^2 S C_L \end{equation*}

so solving for C_L gives

(39)   \begin{equation*} C_L = \frac{2 W}{\rho V_{\infty}^2 S} \end{equation*}

This latter equation shows that for a given wing, then the value of C_L is higher at low airspeeds and lower at higher airspeeds, and C_L also increases with aircraft weight. Of course, the maximum attainable lift coefficient is determined by the type of wing, which will reach a point at higher angles of attack when it will stall; the corresponding airspeed is called the stall speed.

Therefore, the drag on the airplane (and hence the thrust required for flight) is

(40)   \begin{equation*} D = T = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S C_{D_{0}} + \frac{1}{2} \rho_{\infty} V_{\infty}^2 S \left( \frac{2 W}{\rho_{\infty} V_{\infty}^2 S} \right)^2 \left( \frac{1}{\pi A\!R \, e} \right) \end{equation*}

and after some rearrangement then

(41)   \begin{equation*} D = T = \left( \frac{1}{2} \rho_{\infty} S C_{D_{0}} \right) V_{\infty}^2 + \left( \frac{4 W^2}{\rho_{\infty} S (\pi A\!R \, e)} \right) \frac{1}{V_{\infty}^2 } \end{equation*}

For a constant weight W and density (altitude) then \rho_{\infty}, it will be apparent that this equation is of the form

(42)   \begin{equation*} D = T = A V_{\infty}^2 + \frac{B}{V_{\infty}^2} \end{equation*}

where the values of A and B are now considered to be constants.

This form of the variation of drag, and hence the corresponding thrust required for flight, is shown in the figure below. Notice that according to Eq. 42, then the profile/parasitic (non-lifting) drag increases with the square of the airspeed, and the induced drag decreases inversely with the square of the airspeed. The resulting drag curve takes on a distinctive “U-shape,” with the minimum drag obtained at some intermediate airspeed.

Representative drag curve for an airplane in level flight. It takes on a very distinctive U-shape, with a minimum drag at some intermediate airspeed. Notice that after this minimum point is reached, then drag grows quickly with increasing airspeed.

The corresponding power required for flight is

(43)   \begin{equation*} P_{\rm req} = T V_{\infty} \end{equation*}

which is of the form

(44)   \begin{equation*} P= A V_{\infty}^3 + \frac{B}{V_{\infty}} \end{equation*}

It is apparent that much more power (and hence more fuel) is required for flight as airspeed increases because it increases with the cube of airspeed.

Example #2 – Calculating the Drag on a Wing

Consider a flying wing with a wing area of 210 m2, an aspect ratio of 10, and an Oswald’s efficiency factor of 0.90. The airfoil section on the wing has a profile drag coefficient of 0.015. The mass of the airplane is 50,000 kg. If the aircraft is flying at a density altitude of 3 km and the true airspeed is 230 m/s, then calculate the total drag on the aircraft.

The density of air at a density altitude of 3 km is 0.90925 kg m-3 when using the ISA model.  For vertical force equilibrium, the lift on the wing L is equal to the weight of the aircraft, W, i.e.,

    \[ L = M g = 50,000 \times 9.81 =490,500 {\mbox  N} \]

The lift is given by

    \[ L = \frac{1}{2} \rho V^2 S C_L \]

so the operating lift coefficient of the wing is

    \[ C_L = \frac{L}{ \frac{1}{2} \rho V^2 S} \]

Inserting the known values gives

    \[ C_L = \frac{490,500 }{0.5 \times 0.90925 \times 230.0^2 \times 210.0} = 0.097 \]

The drag coefficient is

    \[ C_D = C_{D_{0}} + \frac{{C_L}^2}{\pi \, A\!R \, e} = 0.015 + \frac{0.097^2}{\pi \times 10.0 \times 0.90} = 0.0153 \]

where the non-lifting part C_{D_{0}} = 0.015, i.e., C_{D_{0}} = C_{d_{0}}.
Therefore, the total drag force is

    \[ D = \frac{1}{2} \rho V^2 S C_D = 0.5 \times 0.90925 \times 230.0^2 \times 210.0 \times 0.0153 = 77,271.6  \mbox{ N} \]

which gives a corresponding lift-to-drag ratio of about 6.

Generalization of the Aircraft Drag Polar

In general, the drag produced on an airplane will also be a function of the Reynolds number and flight Mach number. Therefore, another generalization of the prior approach is to write that

(45)   \begin{equation*} C_D = C_{D_{0}}(Re, M) + D_1 (Re, M) C_L + D_2 (Re, M) {C_L}^2 \end{equation*}

although, again, the challenge is actually evaluating the values of the coefficients; this will usually involve a combination of wind tunnel measurements and flight test.

Normally for higher Mach numbers above 0.3, the effects of Mach number on the aerodynamics are more important than the effects of Reynolds number, and so Reynolds number variations can be ignored, i.e.,

(46)   \begin{equation*} C_D = C_{D_{0}}(M) + D_1 (M) C_L + D_2 (M) {C_L}^2 \end{equation*}

Finally, these forgoing equations will apply to an airplane where the minimum drag is obtained at zero lift conditions. If this is not the case, then a further generalization is to use

(47)   \begin{equation*} C_D = C_{D_{\rm min}} + K \left( C_L - C_{L_{\rm min~drag}} \right)^2 \end{equation*}

as shown in the figure below.

Generalization of the drag polar for an airplane. This form is usually more appropriate for flight at transonic and supersonic Mach numbers.

Winglets

The purpose of winglets is to increase the effective aspect ratio of the wing but without significantly increasing the span of the wing. The classic “Whitcomb” winglet moves the tip vortices from the wing tip to the top of the winglet, as confirmed in the photograph below. The consequence is that the tip vortices are further away from more of the wing, thereby reducing the magnitude of the induced downwash over the span of the wing and so decreasing its induced drag. The flow velocities induced by the tip vortices decreases inversely with distance, so only a relatively small winglet can decrease the downwash velocities over the wing, hence significantly reducing drag.

Confirmation that a “tip vortex” forms at the tip of a winglet, which in this case is rendered visible by natural condensation.

Aircraft today either have a winglet by design or are retrofitted with winglets after delivery. The motivation is evident, in that a winglet can reduce drag and save fuel. While the winglet adds some structural weight and produces a small increment in skin friction drag, the reductions on induced drag outweigh these concerns. Fuel savings can offset the cost of retrofitting a winglet to an airliner in as little as two years. Therefore, the use of a winglet makes sense compared to a wing redesign with a higher wing span and aspect ratio.

A large “Jumbo” airliner’s feasible wingspan and aspect ratio may be limited for several reasons, such as higher wing weight and the possibility of aeroelastic issues. There are also factors imposed by airport operations such as taxiway access, parking, hangar size, etc. The Boeing 777X is unique because of its extremely high wingspan and aspect ratio, it uses a folding wing tip design. The idea is to give an ultra-high aspect ratio wing for flight while having a reduced span on the ground; in this regard, a winglet to boost the effective aspect ratio of the wing is not needed.

The Boeing 777X does not have a winglet but an ultra-high aspect ratio wing. The higher wing span requires that the wingtips must be folded up to allow safe ground operations at airports.

There are no validated formulas that can be used to calculate the effects of winglets, but one common approach is to use an aspect ratio correction of the form

(48)   \begin{equation*} A\!R_{\rm eff} = A\!R \left( 1 + K_{\rm wl} \, \frac{h_{\rm wl}}{s} \right) \end{equation*}

where A\!R_{\rm eff} is the “effective” or corrected aspect ratio, h_{\rm wl} is the height of the winglet, as shown in the figure below, and K_{\rm} is an empirical coefficient that depends on the type of winglet.

The effect of a winglet is to decrease the induced drag by giving an effective increase in the aspect ratio of the wing.

For a classic Whitcomb winglet then it is usually assumed in Eq. 48 that K_{\rm wl} = 0.95, at least for preliminary design or aircraft performance estimates. The lifting or induced component of the drag can then be written as

(49)   \begin{equation*} C_{D_{i}} = \frac{ {C_L}^2}{\pi A\!R_{\rm eff} \, e} \end{equation*}

Effects of Compressibility & Wing Sweep

One issue that must be remembered is that the drag of a wing increases quickly as the flight Mach number approaches transonic conditions, a result of the development of shock waves and wave drag. However, such effects are difficult to generalize because the drag depends critically on the specific wing geometry (especially its thickness and sweepback angle) and its operating lift coefficient.

The figures below show representative wind tunnel measurements of the minimum (non-lifting) drag on semi-span wings with different sweepback angles, \Lambda, and different thickness-to-chord ratios, t/c. The transonic and supersonic drag will be minimized for a given sweepback by using as thin a wing as possible, bearing in mind that this may not be possible in practice for structural reasons. However, even with a thin wing, there are still significant benefits to be realized by using sweepback, especially in the supersonic regime.

The effects of thickness sweepback angle on the non-lifting drag of a semi-span wing in transonic and supersonic flow.

 

The effects of thickness to chord ratio on the non-lifting drag of a semi-span wing in transonic and supersonic flow.

Examples of Aircraft Drag Polars

A combination of calculations (maybe even with CFD), wind tunnel measurements, and flight tests can most likely define the drag polar to an acceptable degree of fidelity for performance evaluations and other applications. Drag polars have been published for various aircraft, including general aviation airplanes, gliders, commercial airplanes, and even some military airplanes. One purpose of publishing such results is to allow engineers and analysts to validate their modeling and use it for instructional purposes.

An example of a complete drag polar for a legacy commercial airplane, shown in the figure below, at higher subsonic Mach numbers and into the transonic domain. First, notice the increase in the zero-lift drag coefficient C_{D_{0}} with increasing Mach number. Second, notice the more rapid growth in the drag as transonic conditions are approached (i.e., for M > 0.76), this being the consequence of the development of shock waves and an increase in wave drag.

Drag polars for a commercial airplane at subsonic and transonic flight Mach numbers.

Another example of a drag polar is for a legacy military fighter airplane, shown below. In this case, the results are for subsonic and transonic flight, as well as for supersonic flight. Again, the rapid increase in drag is apparent as transonic conditions are encountered and supersonic flight is approached. Notice that in supersonic flight, the aircraft has a lift-to-drag ratio of only about 3. For a military aircraft, it is the usual practice to determine another form of the drag polar as a function of the load factor, i.e., for maneuvering flight.

Drag polars for a military fighter airplane for subsonic, transonic and supersonic flight Mach numbers.

Example #3 – Finding the Lift to Drag Ratio of a Wing

The drag coefficient C_D of a particular airplane design is described by the equation

    \[C_D = 0.04 + \left( \frac{1.26}{\pi AR} \right) {C_L}^2\]

where AR is the aspect ratio of the wing. For aspect ratios of 10 and 20, determine the best lift-to-drag ratio of the wing and the lift coefficient at which this occurs.

The drag coefficient for the aircraft is given by

    \[ C_D = 0.04 + \left( \frac{1.26}{\pi AR} \right) {C_L}^2 = A + B {C_L}^2 \]

This means that

    \[ \frac{C_D}{C_L} = \frac{A}{C_L} + B C_L \]

Differentiating the foregoing expression gives

    \[ \frac{d (C_D/C_L)}{C_L} = -\frac{A}{{C_L}^2} + B = 0 \mbox{~~for a maximum or minimum} \]

Therefore, solving for C_L at this condition gives

    \[ C_L = \sqrt{ \frac{A}{B} } = \sqrt{ \frac{0.04 \pi AR}{1.26} } \]

and this will be the lift coefficient to obtain the maximum lift-to-drag ratio for a wing with a given aspect ratio.

For AR = 10 then the C_L for best C_L/C_D is

    \[ C_L = \sqrt{ \frac{0.04 \pi AR}{1.26} } = \sqrt{ \frac{0.04 \times \pi \times  10 }{1.26} } = 1.0 \]

and substituting values with C_L = 1.0 gives the best lift-to-drag ratio as

    \[ \frac{C_L}{C_D} = \frac{C_L}{0.04 + (1.26/\pi/AR){C_L}^2} = \frac{1.0}{0.04 + (1.26/\pi/10) 1.0^2} = 12.48 \]

For AR = 20 then the C_L for best C_L/C_D is

    \[ C_L = \sqrt{ \frac{0.04 \pi AR}{1.26} } = \sqrt{ \frac{0.04 \times \pi \times 20 }{1.26} } = 1.41 \]

and substituting values with C_L = 1.41 gives the best lift-to-drag ratio as

    \[ \frac{C_L}{C_D} = \frac{C_L}{0.04 + (1.26/\pi/AR){C_L}^2} = \frac{1.41}{0.04+ (1.26/\pi/20) 1.41^2} = 17.65 \]

Center of Pressure & Aerodynamic Center

The procedures for finding the center of pressure and aerodynamic center on a finite wing are similar to that for two-dimensional airfoils. Recall that, by definition, the center of pressure on a wing is a point about which the pitching moments are zero, i.e., a point where the resultant forces can be assumed to act. The aerodynamic center is a point where the moment is constant and independent of the angle of attack. Determining the aerodynamic center, like that for the center of pressure, requires values of the lift and moment coefficient versus the angle of attack about any other point, which by default is usually the 1/4-chord.

In the case of the center of pressure then

(50)   \begin{equation*} \frac{x_{cp}}{c} = \frac{1}{4} - \frac{C_{M_{c/4}}}{C_{L}} \end{equation*}

Because the center of pressure is a moving point , the center of pressure to resolve the forces and moments on a wing is not used much in practice.

If the aerodynamic center is assumed to at a distance x_{ac} behind the leading-edge then

(51)   \begin{equation*} C_{M_{a}} = C_{M_{\rm ac}} - C_{L} \left( \frac{x_{ac}}{c} - \frac{a}{c} \right) \end{equation*}

Differentiating the above equation with respect to C_{L} gives

(52)   \begin{equation*} \frac{d C_{M_{a}}}{dC_{l}} = \frac{d C_{M_{\rm ac}}}{dC_{L}} - \left( \frac{x_{\rm ac}}{c} - \frac{a}{c} \right) \end{equation*}

After rearrangement then

(53)   \begin{equation*} \frac{x_{\rm ac}}{c} = \frac{a}{c} - \frac{d C_{M_{a}}}{dC_{L}} \end{equation*}

The value of dC_{M_{\alpha}}/d C_L can be obtained by using

(54)   \begin{equation*} \frac{d C_{M_{\alpha}} }{ d C_L} = \left( \frac{ d C_{M_{\alpha}}}{d \alpha}\right) \left( \frac{d \alpha}{d C_{L}}\right) \end{equation*}

This latter process is performed by finding the slopes of the best straight line fit to the values on the graphs of C_L versus the angle of attack of the wing \alpha and also the C_{M_{1/4}} curve versus \alpha.

Vortex Wake Upsets

When viewed from in front of the aircraft, the tip vortex from the left wing tip rotates counterclockwise and the right one clockwise, as suggested by the photograph below. The vortices tend to be the strongest when the wing operates at high angles of attack, such as during takeoff and landing. As the tip vortices trail back from the wing tips, they begin to descend behind the wing under their own self-induced velocities.

Natural flow visualization of the wake behind a landing airplane, showing the signature of the wing tip vortices. A concern over the persistence of these vortices is their possible effects on a following aircraft.

While the wing tip vortices affect the aircraft that generates them, they may also affect other aircraft that encounter the downstream wake. The generic name used for the effects of the vortex wake, especially by pilots and the FAA, is called wake turbulence. Wing tip vortices tend to be relatively strong and persistent and can remain for up to several minutes and many miles after the passage of an aircraft through a given part of the sky. How persistent the vortices are also depends on the winds and other environmental factors, such as atmospheric turbulence.

Because of the persistence of the wing tip vortices, which can take many minutes to spin down, they can pose a concern for the following aircraft, particularly in the terminal area. In addition, the local downwash velocities may be a significant fraction of the aircraft’s airspeed with steep downwash gradients, so the potential hazard is significant in that it will affect the angle of attack and so the lift on the wing.

Notice from the figure below that an aircraft may end up flying more perpendicular or parallel to the wing tip vortices. On the one hand, flying nearly perpendicular to the vortices will cause the airplane to encounter an abrupt upwash followed by a downwash and then another upwash. This trajectory can cause notable changes in the airplane’s load factor when flying through it and may concern the passengers, who may feel a very distinct “bump.” On the other hand, if an airplane flies parallel to the vortices, there is a tendency for the airplane to roll, which is a bad outcome. Hence the name vortex wake upsets become clear.

The possibility and problems of wake upsets during takeoff and landing operations became much more acute starting in the 1970s with the introduction of wide-body or “jumbo jets” such as the Boeing 747 and McDonnell-Douglas DC-10. The higher weights of these aircraft produced intense and persistent wing tip vortices that caused many “wake turbulence” incidents. The most prevalent were for smaller aircraft, which in some cases of wake encounters, caused them to roll inverted and crash.

Subsequently, the FAA has mandated much greater separation distances between all aircraft in the terminal airspace regions, dramatically reducing the upsets from wake turbulence. However, increasing separation distances between aircraft limits the number of aircraft that can land and take off within a given time, which can seriously constrain aircraft movements at busy airports.

Summary & Closure

The aerodynamics of wings of finite span depend critically on their aspect ratio. The higher the aspect ratio, the lower the induced component of the drag. Induced drag is an inevitable consequence of lift generation and is highest when the wing operates at higher lift coefficients, thereby creating the strongest wing tip vortices. These vortices produce a downwash flow velocity over the wing, altering the lift and drag at every wing section. Therefore, the design of finite wings for minimum drag also depends on the spanwise lift distribution, which should be as close to elliptical as possible, and can be achieved by judicious variations in planform (local wing chord), wing twist, and airfoil section.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Calculate the potential improvement in lift-to-drag ratio for a wing when making a design change in the aspect ratio from 5 to 7 while keeping the same wing area. Make any reasonable assumptions.
  • It said that a winglet gives an increase in the wing’s effective aspect ratio without an increase in its span. Discuss why.
  • The drag polar for a high-speed aircraft shows a general reduction in the slope of the curves when approaching and exceeding Mach 1. Discuss why this behavior occurs.
  • Mandating greater separation distances between all aircraft in terminal airspace has reduced the likelihood of wake upsets but has led to other problems. Discuss.

Other Useful Online Resources

To understand more about the aerodynamics of finite wings then explore some of these online resources: