• Section 9.2 presents a handy formulation to calculate various properties of trapezoidal surfaces, which is the most common planform shape for lifting surfaces such as wings, and horizontal and vertical tails.

• Section 9.3 presents various concepts, topics, and methods for laying out the wing planform. These include aspect ratio, taper ratio, washout, and wing incidence angle.

• Section 9.4 is intended to help with the selection of a suitable planform shape. It introduces a number of different geometries and presents information about their pros and cons, in addition to giving ideas of the distribution of section lift coefficients.

• Section 9.5 presents a number of methods to evaluate the lift and pitching moment characteristics of the wing (methods to evaluate drag are presented in Chapter 15, Aircraft drag analysis).

• Section 9.6 presents a number of issues that have to do with wing stall characteristics and how to improve them.

• Section 9.7 presents Prandtl’s Lifting Line Theory, which is a numerical method used to estimate the aerodynamic properties of the wing. A computer function, written in Visual Basic for Applications, intended for use with Microsoft Excel is also presented.

The Establishment of Basic Datum

The positioning of the wing is the determination of where, with respect to some reference point in space, the wing shall be located. During the design of the aircraft it is usually decided that the reference datum for the airplane may be a point on the fuselage or the wing or a point in space (typically ahead of the nose of the aircraft as discussed in Section 6.6.11, Center of gravity envelope).

Among those are evaluation of dynamic stability and compliance with the design checklist of Section 23.3, GA aircraft design checklist. Typically, this will consider the following:

Wing area:

(9-1)

Aspect ratio – general:

(9-2)

Aspect ratio – constant-chord:

(9-3)

Taper ratio:

(9-4)

Average chord:

(9-5)

Mean geometric chord:

(9-6)

Mean aerodynamic chord:

(9-7)

Y-location of MGC_LE:

(9-8)

X-location of MGC_LE:

(9-9)

Angle of quarter-chord line:

(9-10)

Angle of an arbitrary chord line:

(9-11)

Aspect Ratio for HT and VT

The aspect ratio for a horizontal tail (HT) is calculated exactly as it is for the wing. The aspect ratio for a vertical tail (VT) is calculated using the following expression:

The aspect ratio for a twin tail is calculated by applying Equation (9-12) to one tail only, where S_VT refers to one half the total area of the VT.

Aspect Ratio for Multi-wing Configurations

The aspect ratio for multi-wing aircraft, such as biplanes and triplanes, is given below:

Derivation of Equations (9-6), (9-8), and (9-10)

Equation (9-6): see Section D.5.3, Derivation of MGC.

Equation (9-8): see Section D.5.2, Spanwise location of the MGC.

Equation (9-10): refer to the Figure 9-3 to define the dimensions a, b/2, c, d, and h.

QED

From

(9-15)

From

(9-16)

Average chord:

(9-17)

From

(9-18)

From

(9-19)

Mean geometric chord:

(9-20)

Chord for a straight tapered wing:

(9-21)

Derivation of Equation (9-20)

Using Equation (9-18), we can calculate the root chord, C_r, using any set of values for the wingspan, b, and aspect ratio, AR:

We then insert this into Equation (9-6) and manipulate as follows:

QED

Derivation of Equation (9-21)

Parametric formulation of the chord c as a function of spanwise station y for a straight tapered wing whose taper ratio is λ, root chord C_r, and span is b is obtained by inserting the geometric parameters as follows (where the parameter t is 2y/b and ranges from 0 to 1):

QED

Total area:

(9-22)

where

C = airfoil chord, in ft or m

k = location of the airfoil’s maximum thickness as a fraction of C

t = airfoil thickness, in ft or m

Total area:

(9-23)

Derivation of Equations (9-22) and (9-23)

Consider Figure 9-7, which shows an airfoil of chord C approximated by a parabolic D-cell and a triangular section. It is assumed the two sections join at the chord station of maximum thickness, t, whose location is given by k·C, where k is the location of the airfoil’s maximum thickness as a fraction of C. The cross-sectional areas of the two sections and the total area are given by the following expressions:

Therefore, the internal area of the airfoil can be approximated by adding the two as shown below:

Then, multiply by C/C (=1) to get Equation (9-23).

QED

Monoplane:

(9-24)

Biplane:

(9-25)

where

b = wingspan

S = wing area

Illustrating the Difference Between Low and High AR Wings

The difference between a low and high AR is illustrated in Figure 9-9, which shows two wings of equal wing areas and taper ratios. The upper wing has a relatively low AR (6.67) while the lower one has a relatively large one (18.57). The airspeed in both cases is 100 KTAS and both are positioned at an AOA such they generate 2000 lb_f of lift. The figure shows that the low AR wing requires an AOA of 1.55° but the high AR one 0.66°. In accordance with Section 15.3.4, The lift-induced drag coefficient: C_Di, the higher AOA means the low AR wing generates higher induced drag. The greater flow field disruption is even visible by the larger wingtip vortex generated by the low AR wing in Figure 9-9. Here, a lift-induced drag value of 71 lb_f was estimated for the low AR wing and 45 lb_f for the high AR one. Additionally, the lift curve slope, C_Lα, for the low AR wing (4.475 per radian) is less than that of the high AR wing (5.459 per radian). Therefore, the low AR wing will stall at a higher AOA and airspeed than the high AR wing. However, it is far more responsive as its roll damping coefficient, C_lp, is lower.

The difference between the two configurations is further illustrated in Figure 9-10. The graphs were generated for three wings using the potential flow theory. The wings all have a constant TR (0.5), but varying AR. The left graph shows the spanwise distribution of section lift coefficients at an AOA of 10°. The tip loading that results from the TR (and will be discussed next) is evident. It can be seen that the high AR of 16 generates the highest C_l at the give AOA, as is to be expected since its C_lα is the highest. Other than the magnitude of the section lift coefficients, it is also evident that the AR does not have major changes on the general shape of the distribution; in other words, the AR has great effects on the magnitude of section lift coefficients at a given AOA, and relatively small influence on the shape of the distribution.

The left graph in Figure 9-10 shows how the AR affects the lift induced drag. This is illustrated for three AOAs; 2°, 6°, and 10°. The graph shows substantial reduction in lift-induced drag is to be had with larger AR.

Determining AR Based on a Desired Range

It is possible to evaluate and recommend AR for some special design cases. One such is the determination of AR based on a desired range, R, at some desired cruising airspeed, V_C. If the aircraft being designed belongs to a class of aircraft for which C_Dmin can be estimated with reasonable accuracy and the expected initial and final cruise weights are known, denoted by W_ini and W_fin, respectively, then the average cruise lift coefficient can be used with Equation (20-12) to extract the effective aspect ratio, AR_e, for the vehicle using the following expression:

(9-26)

where

The denominator in Equation (9-26) results in a singularity when . In fact, it returns negative values if the airspeed is lower than obtained by the following expression:

(9-27)

For this reason, Equation (9-26) should only be applied to airspeeds that are greater than the value obtained by the above expression. It should be used to guide the selection of the AR rather than obtain it directly, for instance by plotting isobars.

Note that AR_e is the product of the AR and Oswald’s span efficiency, AR·e. Once this product has been determined, it is the responsibility of the designer to devise a proper combination of AR and planform geometry to achieve this value. Also note that if the airplane features a swept back planform that the AR limits of Equation (9-94) must be considered as well.

Although the low AR surface is less efficient than the high AR, it is much better suited as a control surface for tail-aft configurations (i.e. as a horizontal and vertical tail). This results from the higher stall AOA, which is a consequence of the low AR. Delayed stall introduces a certain level of safety to the operation of tail-aft airplanes because it will require them to reach very high AOA or AOY before the stabilizing moments begin to drop. This is the reason why the horizontal and vertical tail surfaces on airplanes typically feature low aspect ratios.

Derivation of Equation (9-26)

Assuming the simplified drag model, Equation (20-12) can be solved for the drag coefficient as follows:

This can then be solved for the effective AR as follows:

Note that this is a generic case and the subscript C is used to denote some desired cruise conditions.

QED

Determining AR Based on Desired Endurance

The AR can also be determined based on a desired endurance, E, at some desired cruising airspeed, V_C, similar to what was done for range and using the same basic variables. This time the method uses Equation (20-22) to extract the effective aspect ratio, AR_e, for the vehicle using the following expression:

(9-28)

where E = endurance in seconds.

Like Equation (9-26), the denominator in Equation (9-28) has similar limitations and should be used with cruise lift coefficients greater than those obtained with the following expression:

(9-29)

Derivation of Equation (9-28)

Assuming the simplified drag model, Equation (20-22) can be solved for the drag coefficient as follows:

Note that this is a generic case and the subscript C is used to denote some desired cruise conditions.

QED

Estimating AR Based on Minimum Drag

The AR can also be set based on the minimum drag of the airplane. This approach requires the design lift coefficient to be determined (e.g. see Equation [9-49]). Then, assuming constant wing area, a relatively sophisticated estimation of the total drag coefficient of the airplane is accomplished. The estimation takes into account changes in the wing geometry as a function of AR (wing chords change since S is constant) and evaluates skin friction assuming either a fully turbulent or a mixed boundary layer. Then, using the appropriate form and interference factors, the minimum drag coefficient is estimated (see Chapter 15, Aircraft drag analysis). The lift-induced drag coefficient is also calculated, using an appropriate model of the Oswald’s span efficiency. Adding the two coefficients comprises the total drag coefficient, C_D. This can then be plotted versus the AR, in a carpet plot, similar to that of Figure 9-12. The carpet plot is usually prepared using a range of lift coefficients. Such a plot reveals the location of the optimum AR. The upper and lower bounds in the figure are simply the optimum AR ± 2, but the designer can widen or narrow this range depending on the project. Figure 9-12 shows that for the hypothetical aircraft being presented, practically any AR between the two limits will yield the lowest value of the total drag coefficient. This gives the designer room to accommodate other concerns such as structural weight.

Estimating AR for Sailplane Class Aircraft

A plausible AR for powered and unpowered sailplanes can be established using the historical data in Figure 9-13. The data points represent contemporary manufacturers’ information collected from Ref. [2]. The maximum lift-to-drag ratios (LD_max) presented includes the drag of the fuselage and stabilizing surfaces and demonstrates the sophistication of the modern sailplane. The graph shows trends for both regular and powered sailplanes and presents accompanying least-squares curves. Note that these estimates do not replace analyses of the kind presented by Ref. [3], but rather supply initial estimates based on historical sailplanes.

Also plotted are theoretical predictions of the LD_max using the simplified drag model (see Chapter 15, Aircraft drag analysis) and C_Dmin of 0.01 (typical of many sailplanes). This model is often the first choice of the novice airplane designer, but it is inaccurate for sailplanes designed for extensive NLF as it does not model the drag bucket associated with such aircraft. These curves are merely presented here to demonstrate why the simplified drag model is a poor predictor.

The theoretical LD_max was calculated using Equation (19-18), where the Oswald’s span efficiency was calculated per Equations (9-91) and (9-89) and. The curves shown assume a minimum drag coefficient of 0.01, which is a typical value for clean sailplanes. The graph shows clearly the limitation of the simplified drag model.

The following empirical formulation can be used to estimate the AR for the conceptual design of unpowered and powered sailplanes (as long as AR < 36):

Naturally, the calculated AR will not guarantee that a desired LD_max will be achieved. Rather it indicates that, historically, sailplanes with that AR have achieved the said LD_max. Achieving it will require the utmost attention to anything on the airplane that generates drag.

Impact of Sweep Angle on the Critical Mach Number

The greatest benefit of wing sweep is a reduction in the strength of and delay in the onset of shock formation. The shock formation will not only cause a sharp increase in drag; it also changes the chordwise pressure distribution on the airfoil, causing the center of lift to move from approximately the airfoil’s quarter-chord to mid-chord. The consequence of this is called “Mach-tuck,” a severe increase in nose-down pitching moment. The change in drag due to this effect is detailed in Section 15.4.2, The effect of Mach number. Also, Figure 15-28 shows how the leading edge sweep increases the critical Mach number, M_crit, and delays the onset of the peak of the compressibility drag coefficient. This is helpful as it allows thicker and more structurally efficient airfoils to be used in the wing.

Consider the swept wing in Figure 9-15 and the deconstruction of the far-field airspeed into two components: one parallel to and the other perpendicular to the leading edge. In its simplest form the wing sweep theory contends that it is the airspeed component normal to the leading edge that dictates when shockwaves begin to form. This allows the following correction to the critical Mach number to be made (see Section 8.3.7, The critical Mach number, M_crit).

(9-32)

In practice, only half of this reduction is experienced.

Impact of Sweep Angle on the C_Lmax

The maximum lift coefficient is reduced with an increase in wing sweep angle. This is detailed in Section 9.5.10, Step-by-step: Rapid C_Lmax estimation.

Impact of Sweep Angle on Structural Loads

The sweep has very important effect on structural loads. As discussed in Chapter 8, The anatomy of the airfoil, cambered airfoils inherently generate a pitching moment about their quarter chords whose tendency is to rotate the airfoil LE down. All flapped airfoils, cambered or not, generate extra pitching moment (torsion), which is added to the baseline moment. The total moment, regardless of composition, is reacted as shear flow in the wing structure. If the wing is swept aft, an additional and usually much larger pitching moment is generated because of the center of lift being moved a large distance aft. Consequently, the swept wing structure will be heavier than a straight wing designed for the same flight condition. Similarly, a forward-swept wing may also introduce a large LE up torsion, if the sweep angle is large enough, although this is reduced by an amount corresponding to the innate torsion of the airfoils of the wing.

Impact of Sweep Angle on Stall Characteristics

Sweep angle has very detrimental effects on the stall characteristics for reasons detailed in Sections 9.6.5, Cause of spanwise flow for a swept-back wing planform and 9.6.6, Pitch-up stall boundary for a swept-back wing planform.

Impact on AOA:

(9-33)

Similarly, the lift generated in the plane of symmetry is the product of the lift normal to the wing surface, denoted by L_N, reduced by the same factor, cos Γ.

Tilting of lift force:

(9-34)

Since L_N can be written as , the lift in the plane of symmetry can be presented as follows:

Lift of a wing with dihedral:

(9-35)

where

C_Lα = lift curve slope of the wing assuming Γ = 0°

q = dynamic pressure

S = reference wing area

1. To prevent the wingtip region from stalling before the wing root. If the wingtip stalls before the root, there is an increased risk the aircraft will roll abruptly and uncontrollably at stall, a condition that may lead to incipient stall. Roll tendency at stall contributes significantly to fatal accidents¹ as it will often take place at low altitudes when an airplane banks to establish final approach before landing. An aircraft with such a roll-off tendency may stall if the pilot banks steeply and it may, ultimately, result in a crash as lack of altitude will prevent successful recovery. In the real world, manufacturing tolerances inevitably lead to airplanes not being perfectly symmetrical. This promotes one side of the airplane to stall before the other one, generating a rolling moment at stall. A wing twist can be used to build a “buffer” so the wingtips remain un-stalled when the rest of the wing stalls, greatly improving roll stability.

2. To modify the spanwise lift distribution to help achieve minimum drag at mission condition. The wing is most efficient when the section lift coefficients are constant along the span, as it is for an elliptical wing planform. Twisting the wing will allow the spanwise distribution of section lift coefficients to be modified for a tip-loaded wing, bringing the general distribution closer to that of an elliptical wing. A byproduct of this is a reduction in wing bending moments as the center of lift is brought closer to the plane of symmetry.

Geometric Washout

Geometric washout usually refers to the difference in angles of incidence of the root and tip airfoils (see Figure 9-16). It is denoted by ϕ_G. Generally the wing twist ranges from 0° to −4°, where the negative sign indicates that the leading edge of the tip is lower than that of the root. Thus, if we say the “washout is 3°” we mean that ϕ_G = −3° and the LE of the tip is lower than that of the root. If we say the “washin is 3°” we mean that ϕ_G = +3° and the LE of the tip is higher than that of the root. For instance, the inserted image of the root and tip airfoils in Figure 9-16 displays a washout. However, sometimes twist is highly complicated along the wingspan as in the case of the McDonnell-Douglas AV-8B, whose twist varies in a segmented fashion to a maximum of −8° at the tip [8].

Note that if the twist is linear, the relative angle of twist at any spanwise station can be determined using the following expression:

(9-36)

The expression assumes the reference angle is 0 when y = 0 (the plane of symmetry) and becomes ϕ_G at the wingtip (where y = b/2).

Aerodynamic Washout

Aerodynamic washout is another way of designing roll stability at stall into the wing. In this case, two different airfoils are selected for the root and tip that are specifically based on one of two parameters: (1) their individual zero-lift AOA or (2) their two-dimensional stall AOA or maximum section lift coefficient. The former is favored when analyzing wings using methods such as the lifting line theory. Typically the idea is to provide roll stability at high AOA by ensuring the tip airfoil remains un-stalled before the inboard airfoil.

The primary reason for selecting an aerodynamic washout is that it allows the main wing spar caps to be straight, something that is important for the construction of composite aircraft. A spar cap in a composite aircraft mostly consists of unidirectional fibers, whose strength is very sensitive to fiber alignment. Once two airfoils have been selected, the aerodynamic washout is defined as follows (see Figure 9-17):

Note that there is an inherent problem with the definition based on Equation (9-37) and this is displayed in Figure 9-18. To begin with, the left graph of Figure 9-18 depicts the idea behind the aerodynamic washout: that the root airfoil stalls at an AOA less than the tip. It assumes that (1) α_{ZL tip} – α_{ZL root} = α_{stall root} – α_{stall tip}, which is not necessarily achieved in practice, and (2) requires C_{Lmax tip} = C_{Lmax root} (otherwise assumption (1) will not hold). This allows the aerodynamic washout can effectively be represented as the difference in the zero-lift AOAs.

One of the problems with this definition is shown in the right graph of Figure 9-18, which depicts the common case of the root airfoil being thicker than the tip airfoil. Consequently, its C_lmax may be larger than that of the tip airfoil and its stall AOA is greater. Furthermore, if the wing is tapered, the difference in Reynolds number between the two can make the C_{Lmax tip} and α_{stall tip} substantially smaller than that of the root. The figure effectively depicts a scenario in which the true washout is zero and possibly a washin, whereas relying on the zero-lift AOAs might indicate ample washout and therefore is misleading. Additional complexity must be considered in the magnitude of the section lift coefficients, which vary along the span. The above definition, thus, only gives a part of the whole picture; the remainder calls for analysis of the distribution of section lift coefficients along the span.

A Combined Geometric and Aerodynamic Washout

If two different airfoils are selected for the root and tip, in addition to a geometric washout, the combined effect can be calculated from:

(9-39)

Panknin and Culver Wing Twist Formulas

The wing twist of a flying wing is such a fundamental parameter that it should be determined early on in the design phase – effectively as soon as the design lift coefficient has been determined. The designer of tailless aircraft can determine the proper washout of the wing using one of two special formulas specifically devised to help with the layout of such configurations. They are called the Panknin and Culver twist formulas. They are presented here without derivation for completeness of the discussion in this article.

The Panknin twist formula is attributed to Dr. Walter Panknin, who in 1989 presented a method to the designers of radio-controlled (RC) aircraft intended to help them properly lay out the washout of flying wings so the elevons are in trail at cruise. The formulation has been applied with great success to many RC flying wing designs [2]. It determines the geometric twist between the inboard and outboard airfoils of the wing (see Figure 9-19) using the following expression:

(9-40)

where

The formula determines the geometric twist angle based on the selected static margin, but allows the designer to select airfoils, AR, and quarter-chord sweep angle. It is applicable to both forward- and aft-swept wings. Note that one must exercise care in the application of units for the quarter-chord sweep. Its units must be in terms of degrees.

The Culver twist formula is attributed to Mr. Irv Culver, a former engineer at Lockheed Skunk Works. Like the Panknin formula, his formula is widely used by designers of small RC tailless aircraft. It is intended for flying wings of moderate sweep angles (typically ≈20°) and design lift coefficients of 0.9–1.2, where the lower value indicates a high-speed glider and the higher is for high-performance sailplanes [8]. Note that the presentation here differs slightly from that of Ref. [8] in the simplification of terms.

(9-41)

where

Once the washout is known its distribution along the span can be found using the following expression:

(9-42)

Step 1: Determine the Optimum AOA for the Fuselage

The first step will be highly dependent on the geometry of the aircraft, although when considering fuselages for conventional commercial aircraft it is good to assume it is parallel to the centerline of the tubular structure that forms the passenger cabin (see the datum in Figure 9-20). A wind tunnel testing or a CFD analysis should be used to determine the optimum orientation of the fuselage as its influence on the three-dimensional flow field may be quite complicated. The result from this step would be denoted by α_Fopt, for AOA of optimum fuselage orientation, using the following stipulation:

Step 2: Determine the Representative AOA at Cruise

The value of the midpoint cruise AOA, α_C, can be determined from Equation (9-43) below:

(9-43)

where

Step 3: Determine the Recommended Wing Angle-of-incidence

The angle-of-incidence (see Figure 9-21) at the root airfoil, denoted by i_W, can now be determined from:

(9-44)

where

Note that since the first two terms of the above expression really return the AOI at the MGC rather than at the root, a correction must be introduced if the wing features a washout. This is denoted by the term Δϕ_MGC, which is the difference in angular incidences between the root and MGC airfoils. If the wing has a linear twist the value of Δϕ_MGC can be determined as follows by inserting the expression for the spanwise location of the MGC, y_MGC, of Equation (9-8) into Equation (9-36). Note that the minus sign in Equation (9-44) is required to ensure that a washout will be added to determine:

(9-45)

Step 4: Determine the Recommended HT Angle-of-incidence

Once a recommended wing AOI has been determined, the incidence of the HT can be determined. Again using the representative cruise mission weight (W₁ + W₂)/2 the horizontal tail should be mounted so the elevator deflection at this point will be neutral. This is called flying the horizontal in trail and simply means neutral deflection. This will result in a minimum trim drag.

Derivation of Equation (9-43)

Assuming the general lift properties of the wing are known, the lift coefficient can be written as follows combining either Equation (9-43) or (9-50):

(i)

The weight of the airplane at the midpoint during the cruise is the average of the initial cruise weight, W₁, and the weight at the end of cruise, W₂:

(ii)

Where α_C is the midpoint cruise AOA. Solving for α_C leads to:

The last term is simply the zero lift AOA, α_ZL, which can be seen from:

Derivation of Equation (9-45)

Assuming the general lift properties of the wing are known, the lift coefficient can either be written as follows combining Equations (9-43) or (9-50):

QED

Decalage Angle for a Monoplane

On a monoplane, a decalage angle is the difference between the incidence angles of the wing and horizontal tail: Figure 9-23. The angle is an important indicator of the stability of the aircraft and generally requires substantial stability analyses to determine. Strictly speaking, the AOI of the HT, i_HT, is determined based on the wing AOI, i_W, and then the decalage angle can be calculated as shown below:

(9-46)

Pros

The configuration has two primary advantages. The first is forgiving stall characteristics, attributed to the reduction in section lift coefficients toward the wingtip (see Figure 9-29). This promotes a very favorable stall progression (growth of flow separation) that begins at the root and moves outboard to the tip, rendering it the last part of the wing to stall. This effect is very desirable as it gives the configuration important roll stability at stall.

The other important advantage is the lower manufacturing costs because all ribs have the same basic geometry and the spar is a constant-thickness beam. This simplicity affects not only the wing ribs and spar, but also the control surfaces.

This configuration is ideal for trainer aircraft or airplanes for which manufacturing cost is imperative.

Cons

Since the wingtip contributes less to the lift generation than the root (in terms of section lift coefficients), the planform is particularly inefficient. For this reason, the planform should never be used for efficient aircraft, such as sailplanes or long-range aircraft. This includes wings and stabilizing surfaces.

Pros

Its primary advantage is the uniform distribution of section lift coefficients (see Figure 9-30). This distribution makes the planform very efficient when it comes to utilizing the entire span and results in the least amount of lift-induced drag of any planform.

Cons

The planform’s greatest drawback is its producibility, which is severely impaired by the complex compound surfaces. It is very difficult to manufacture using aluminum, as this would call for sheets to be stretched through hydroforming (or similar). However, it is much easier to produce using modern composites.

Another serious drawback is that the section lift coefficients are mostly uniform along the span, which causes the entire wing to stall at once (again assuming a constant airfoil and no washout). This can pose a serious problem for low-speed (or high AOA) operations and requires a decisive washout or airfoil selection to remedy.

Pros

The primary advantage of tapered wings is reduction in bending moments and lift-induced drag. Straight tapered wings offer improved efficiency over the Hershey-bar wing as the section lift coefficients are higher toward the tip. Consequently, the wingtips contribute more to the total lift coefficient. This, combined with the relative geometric simplicity of the linear taper, which is easier to manufacture than the elliptical planform, renders the planform ideal for most airplanes. The improved efficiency of the configuration usually warrants the increased production complications.

Cons

The taper compromises the stall characteristics and requires a geometric or aerodynamic wing washout, or a combination of both, to be employed. An additional solution might be to select an airfoil for the tip that has a higher stall AOA than the root. It is common to locate root and tip airfoils such that their quarter-chords are on a line perpendicular to the plane of symmetry. This allows a straight main spar to be placed at the quarter-chord of the wing, but this is an effective location as it results in good structural depth that provides the potential for reasonably large fuel tanks behind the spar. The configuration causes manufacturing complications due to the different geometry of each rib.

Straight Leading or Trailing Edges

Other airplanes feature straight tapered wings whose leading or trailing edge is perpendicular to the plane of symmetry (see Figure 9-34). This means that a spanwise line going through the quarter-chord of the root and tip is swept forward or back.

Pros

Improved structural and aerodynamic efficiency over the constant-chord configuration. The straight LE planform may be advantageous if it is foreseen that the operational CG will be too far forward. The opposite holds for an operational CG that turns out to be too far aft.

Cons

It is a drawback that if the wing features a spar that is perpendicular to the plane of symmetry and extends from tip to tip, its caps will be curved. This can be eliminated by sweeping the spar forward, which makes the use of a single-piece spar more challenging.

The hinge line of control surfaces can have a significant forward sweep, which makes them less efficient aerodynamically and can complicate the design of the control system. Among a number of examples of aircraft that feature the straight LE are the Arado Ar-79, Cessna 177 Cardinal, Commonwealth Ca-12 Boomerang, and Fairey Barracuda, and a series of Mooney aircraft feature the planform shape for both the wing and stabilizing surfaces.

On the Mooney aircraft, the straight LE of the horizontal and vertical tails arguably renders them more efficient aerodynamically than the more popular aft-swept tails and, in theory, this allows the tail be a tad smaller, with less wetted area. This is an argument often heard among laypeople. However, an investigation reveals this is not true. Some of that efficiency is simply lost in lowered control authority of the less effective control surfaces whose hinge lines are inevitably swept forward. Of course, the same argument can be made for aft-swept stabilizers – they also have hinge lines that are swept aft. The difference is that the aft sweep adds arm that makes up for the loss in aerodynamic efficiency. A VLM analysis of the Mooney M-20R revealed that its directional stability derivative, C_nβ, is in the range of 0.054 per radian. Introducing a 45° aft sweep in the LE of the VT (no change in surface or wetted area and root chord precisely the same place) increased it to 0.060 per radian, directly contradicting any claims of greater effectiveness of the straight LE tail. This simply means that the reduction in C_Lα of the VT reduces less than the tail arm increases. The important point is that in the case of the Mooney, the tail is a signature feature for the brand. One can always recognize a Mooney tail and for the brand this is important. But more effective it is not.

Another version of the straight tapered wing is the straight trailing edge planform in Figure 9-34. Many of the same arguments for or against hold for this planform. It is featured on a number of aircraft, among others the Aermacchi MB-326 and MB-339, Bücker Bü-180 Student, Zlin Z-526, 726, and Z-26.

The Compound Tapered Planform

The compound tapered wing planform consists of two tapered sections; the inboard section is an inverse taper and the outboard section has standard taper (see Figure 9-35). The lift distribution for this planform is also shown. It can be seen that section lift coefficients peak inboard, indicating this is where stall would initiate. This is significantly reduced midspan and then increases toward the tip. The analysis indicates the planform is not particularly efficient although the discontinuity at the leading edge may provide some viscous benefits. Among well-known aircraft featuring this planform is the Westland Lysander, a British single-engine observation aircraft with short take-off and landing (STOL) designed in the 1930s. The reason for this choice was pilot visibility – its STOL capability can be attributed to the high-lift system it was equipped with and not the planform itself.

The Aft-swept Planform

The invention of the planform is generally thought to date back to the 1930s. However, it was not possible without the work of scientists who preceded the era. Meier [11] takes this history back to Isaac Newton (1643–1727), Pierre-Simon, Marquis de Laplace (1749–1827), Ernst Mach (1838–1916), and many others, up to Adolf Busemann (1901–1986), who is generally credited with the invention of the swept-back wing. Busemann belonged to a group of very famous German scientists who were led by Ludwig Prandtl (1875–1953). The best known were Theodore von Kármán (1881–1963), Max M. Munk (1890–1986), and Jakob Ackeret (1898–1981) [12]. The result of Busemann’s work was first published at the 5th Volta conference in Rome in 1935 [13]. The Volta conference was an annual conference of physics, mathematics, history, and philosophy, named to honor the great Italian physicist Alessandro Volta (1745–1827). In the USA, Robert T. Jones tested swept-back wings toward the end of 1944. It was the first such work in the USA. It was published in 1945 in NACA TR-863 [14]. In it, Jones cites Busemann’s research from 1935 as one of his references.

Pros

One of the primary advantages of the configuration is that when used with high-speed aircraft, it delays the formation of shockwaves to a higher Mach number (see Section 9.3.3, Leading edge and quarter chord sweep Angles, Λ_LE and Λ_C/4).

The aft-swept configuration is less susceptible to flutter than either the straight or the forward-swept configurations. This results from the tendency of a positive (vertical upward) lift force to reduce the AOA of the tip through aeroelastic effects.

The aft-swept (or forward-swept) configuration is also a possible solution when the CG is expected to end up too far aft or forward on a proposed configuration. The Messerschmitt Me-262 Schwalbe was the first jet aircraft to feature a swept-back wing. However, the modest sweep angle of 18.5° was insufficient to greatly impact divergence drag number and was a consequence of its Junkers Jumo jet engines being heavier than expected. For this reason its originally straight wings were swept aft to solve an issue with the location of its center of gravity [15].

Cons

Among drawbacks is the deterioration of airflow close to the tip of the planform with an increase in AOA. The reason for this deterioration is explained in Section 9.6.5, Cause of spanwise flow for a swept-back wing planform. Two important consequences of this phenomenon is a powerful nose pitch-up moment that develops as the AOA approaches stall and an accompanying deterioration in roll stability and aileron effectiveness.

Since the center of lift is positioned behind the wing root attachment there is an increase in wing torsion, often substantial, that increases the weight of the airframe.

The configuration is more susceptible to control reversal as a result of the tendency of the lift force to reduce the AOA of the tip through aeroelastic effects. The same effect moves the center of lift forward with AOA if the wing is flexible and regardless of flexibility when the wing stalls. It is the cause of the powerful and dangerous nose pitch-up moment at stall.

Aircraft with swept wings often experience issues with the wing fuel tanks. For instance, as the airplane rotates to take-off and begins to climb, fuel will flow toward the outboard and aft side of the tank. This may cause the CG to shift aft. This is solved using so-called baffle check valves, which are one-way flapper valves that allow fuel only to flow inboard [16]. Large transport aircraft, some carrying as much as 40% of their gross weight in fuel, can experience large changes in CG location as fuel is consumed. This often calls for a computer-controlled fuel management system that monitors and controls how fuel from the wing tanks is used.

In order to understand the advantages and disadvantages of the swept aft configuration, it is helpful to break it down depending on taper ratio, λ. This way it is possible to consider three classes; λ < 1, λ = 1, λ > 1 (see Figure 9-36 and Figure 9-37). Most aircraft with aft-swept wings fall into the first category. In fact, there as so many different types of aircraft in this class that it would be unfair to name any particular one. Almost all high-speed military and passenger transport aircraft feature the configuration. These are particularly easy to locate.

The other two classes contain considerably fewer members. When it comes to the second class, for which λ = 1, there is a handful of examples. In current times, the Boeing B-2 Spirit is probably the best known. However, there are others. Among those are the Dunne Monoplane and Biplane, designed in the 1910s by John W. Dunne (1875–1949). The English Electric Lightning, MiG-8 Outcka, MiG I-320, MiG-17, Sukhoï Su-15P, and Yakovlev Yak-25 (Flashlight) have a λ close to 1. And in the third category, the Republic XF-91 Thunderceptor is the only contender. Designed with the tip chord larger than the root chord, the idea was to reduce the section lift coefficients at the tip in order to improve low-speed handling.

The Forward-swept Planform

One of the aerodynamic advantages of the forward-swept wing can be seen in Figure 9-38: the reduced section lift coefficients at the tip. The distribution ensures the inboard wing stalls first and gives the configuration great roll stability at stall (remember this also means while pulling high gs), making it almost impossible to tip stall. The reduced tip loading also means that the ailerons retain far more authority at high AOA than on an aft-swept configuration. This improves controllability during critical phases of the flight, such as during landing. The shape of the lift distribution places the center of lift closer to the plane of symmetry, reducing bending moments (although it does not prevent divergent torsional tendency). Additionally, spanwise flow is directed inboard rather than outboard, where the fuselage more or less acts like an endplate or a fence that prevents early separation.

The primary disadvantage of the geometry is divergent aeroelastic deformation. As the AOA of the wing is increased, the elastic torsional deformation twists the wing, increasing the AOA further. Forward-swept wings must be especially reinforced to keep the divergent deformation to a minimum, inevitably increasing their weight.

The first major development of a forward-swept wing configuration was the German Junker Ju-287. An interesting discussion of its development is given in Ref. [11]. The configuration has been used in a few other designs, including the Grumman X-29, the HFB-340 Hansajet, and the Sukhoi Su-47. A modest forward sweep is relatively common, for instance in sailplanes and a number of GA aircraft. However, it is almost always a solution to a CG problem and not for compressibility fixes.

Variable Swept Planform

The variable sweep planform (see schematic in Figure 9-39) dates back to the German Messerschmitt P.1101, whose incomplete prototype was discovered at the end of WWII. It featured a swiveling wing design whose sweep angle was to be selected and set manually before each flight. Later, this innovative design led to fighter and bomber models, such as the General Dynamics F-111, Grumman F-14, Sukhoi Su-17, Panavia Tornado, and others. The configuration is not used in any GA aircraft, as it inevitably leads to a heavier airframe, control system complexity, and other complications. It is really a configuration suited for supersonic aircraft, where it solves problems at the high- and low-speed extremes of the flight envelope. It is included here for the sake of completion.

Semi-tapered Planform

The semi-tapered planform is one that has a constant-chord inboard section and a tapered outboard section (see Figure 9-40). The planform increases the section lift coefficients on the outboard wing and improves its efficiency over that of the Hershey-bar wing. A side benefit of this is improvement in aileron effectiveness. The drawback is the added complexity of construction, and reduction in the aileron effectiveness if its hingeline becomes highly swept.

The configuration is best known on a family of single-engine Cessna aircraft: the 150, 152, 172, 182, 206, and many others. It is also used on a number of Piper PA-28 Warrior and Cherokee Archer aircraft.

Crescent Planform

The crescent wing planform (Figure 9-41) has seen very limited use. It is best known for its use on the British Handley-Page Victor, where the highly swept inboard section of the wing allowed the use of a thick airfoil without introducing early shock formation at high Mach numbers. The thick airfoil was needed to accommodate its four jet engines, which were buried in the wing root. The wing of the Victor varied in quarter-chord sweep and thickness, progressively from 22° and 4% t/c at the tip to 53° and 16% at the root. This allowed a constant M_crit to be maintained along the wing and resulted in more efficient reaction of bending moments. A consequence of this geometry was improved aileron control authority and reduced tendency for tip stall and subsequent nose pitch-up.

Schuemann Planform

The Schuemann planform shape has already been thoroughly introduced in the Appendix C1.5.2, The Schuemann wing. It is featured on a number of sailplanes, for instance, the Stemme S-10 and the DG-1000. The wing style has also been introduced on commuter aircraft such as the Dornier Do-228, Dornier 328, and Fairchild-Dornier 328JET.

Figure 9-42 shows the fundamental dimensions of the planform shape and the distribution of the section lift coefficients compared to those of a Hershey-bar and elliptical planform shapes. It can be seen that the distribution approximates that of the elliptical planform, leading to a reduction in lift-induced drag. This explains the wing’s popularity for sailplanes. The figure also reveals that a potential problem with the wing is an early wing tip stall, not unlike the elliptical wing and is caused by the sharp outboard taper. This is of course complicated by viscous effects at the wingtip, as highly swept wingtips may position the tip vortex so it suppresses early tip stall tendency, not unlike delta wings. Sailplane wingtips at a high AOA will flex substantially due to aeroelastic effects. This effectively unloads the wingtip and loads up the center portion of the wing, which then stalls before the wingtip. For this reason, sailplanes usually have good stall characteristics. A short and stiff wing, likelier to be used for a GA aircraft, will not flex nearly as much as the high AR wing and should thus be expected to feature decisive wing washout at the tip, unless the tip segment of the wing features a high enough sweep.

The Delta Planform Shape³

A discussion of the pros and cons of delta wings (see schematic in Figure 9-43) is given in Appendix C1.5.5, The delta wing, and will not be elaborated on here, other than that they are planform shapes intended for high-subsonic or supersonic aircraft and not low-subsonic airplanes. Even though it is certainly possible to use delta wings for low-speed airplanes (as evident by the Dyke Delta) it is a choice that is hard to justify for reasons other than fun flying; with reduced storage space (thanks to a short wingspan), being another drawback. Deltas stall at very high angles of attack and generate low C_Lmax compared to straight un-swept wings; they will therefore require high approach speed and deck angles. As an example, the 2000 lb_f Dyke JD-2 Delta reportedly stalls at about 61–65 KCAS [17]. This means that the 173 ft² wing area generates a C_Lmax around 0.8–0.9. That is 50–65% less than conventional straight wings. They also have poor LD_max, which is of concern for engine-out emergencies. The best-known aircraft that uses the configuration is the Dassault Mirage III and its various derivative aircraft (e.g. Mirage IV, 2000, Rafale).

The Double-delta Planform Shape

The double delta (see Figure 9-44) is also known as the compound delta. They have an advantage over conventional delta wings in that they produce a vortex pair, rather than a single vortex over each wing that mutually interfere with each other. The resulting system increases the lift of the double delta over that of the conventional delta, rendering supersonic fighter aircraft far more maneuverable [18]. When it comes to subsonic aircraft, the configuration suffers from similar limitations to the single delta. The configuration has been used on military aircraft like the SAAB J-35 Draken and General Dynamics F-16E cranked arrow.

Disc- or Circular-shaped Planform

In short, the disc-shaped planform (see Figure 9-45) is something no self-respecting aircraft designer should propose as a primary lifting surface. Perhaps inspired by the fad of UFO sightings in the 1950s, the well-respected but long-defunct Avro Canada Ltd, a company that at one time employed some 50,000 people, dared to develop the AVRO Canada VZ-9A, nicknamed the AVRO car. A secret military project, the Avrocar employed the Coandǎ effect to generate lift and thrust. The project was cancelled in 1961 when it became clear that insufficient thrust and stability issues would hamper its success.

Figure 9-45 shows a comparison of spanwise lift distribution between the disc planform and an elliptical and a Hershey-bar wing. In spite of the uniform distribution of section lift coefficients, the graph clearly shows how inefficient this planform shape is when compared to the Hershey-bar or elliptical wings; it barely ekes out one-third of the lift at the selected AOA (10°). From an efficiency standpoint, the disc-shaped wing is simply beyond objectionable. However, it is presented here as it is an ideal shape for a radar disc of the kind installed on a reconnaissance or early warning and control system (AWACS) military aircraft such as the Boeing E-3 Sentry (a modified Boeing 707) or Grumman E-2 Hawkeye. For such applications a shallow lift curve slope is essential for stability and control reasons.

Other Configurations

Four other configurations are shown in Figure 9-46: an ogival, joined wing, blended wing-body, and a rhomboid planform. Lift distribution curves for these shapes will not be presented here, since these planform shapes are highly specialized or experimental.

Lift Curve Slope, C_Lα

The lift curve slope is a measure of how rapidly the wing generates lift with change in AOA. As stated in Section 8.1.4, the theoretical maximum is 2π, although real airfoils deviate from it. The lift curve slope of a three-dimensional wing is always less than that of the airfoils it features (see Section 9.5.3, Determination of lift curve slope, C_Lα, for a 3D lifting surface). Once a certain AOA has been achieved the wing will display a pronounced reduction in the lift curve slope (see Figure 9-47). This point is called stall and, although not shown in the figure, occurs both at a positive and negative angle-of-attack. The lift at stall dictates how much wing area the aircraft must feature for a desired stalling speed.

Maximum and Minimum Lift Coefficients, C_Lmax and C_Lmin

The largest and smallest magnitudes of the lift coefficient are denoted by C_Lmax and C_Lmin, respectively. It indicates at what angle-of-attack the airplane will achieve its minimum airspeed (stalling speed), or what wing area is required for a desired stalling speed. As for the airfoil, the stall is defined as the flow conditions that follow the first lift curve peak, which is where the C_Lmax (or C_Lmin) occur (see, for instance, Ref. [19]). Both values are required when generating aerodynamic loads for the structures group.

C_L at Zero AOA, C_Lo

C_Lo is the value of the lift coefficient of the wing at zero AOA. It is of great importance in the scheme of things, because it affects the angle-of-incidence at which the wing must be mounted. Generally this value ranges from 0.0 (for symmetric airfoils) to 0.6 (for highly cambered airfoils). It is negative for under-cambered airfoils (e.g. airfoils used near the root of high subsonic jet aircraft).

Angle-of-attack at Zero Lift, α_ZL

This is the angle at which the wing generates no lift. For positively cambered airfoils this angle is always negative, unless some specific components (e.g. cambered fuselage) affect it greatly. For symmetrical airfoils it is always 0°.

Linear Range

The linear range is analogous to that of the airfoil, except it applies to the entire aircraft. In this range, the following equation of a line can be used to describe how lift varies with AOA.

(9-48)

Angle-of-attack where Lift Curve Becomes Non-linear, α_NL

Once a certain AOA is reached the wing begins to display a pronounced reduction in the lift curve slope. This always happens before the stall AOA is reached.

Angle-of-attack for Maximum Lift Coefficient, α_stall

Once a certain AOA is reached, a pronounced reduction in lift curve takes place; this is the stall.

Design Lift Coefficient, C_LC

The design lift coefficient is the C_L at which the aircraft is expected to operate during the mission for which it is designed. This is usually the lift coefficient during cruise. It is important to select an airfoil that has been designed to generate the least amount of drag at that lift coefficient (see Section 8.3.15, Airfoil selection how-to), as well as ensure flow separation areas are completely suppressed for minimum drag. In the case of a constant cruising speed (or loiter) mission, if the target airspeed is known, it is possible to estimate this design cruising speed using the weight of the airplane at the beginning and end of the mission using the following expression:

(9-49)

where

Angle-of-attack for Design Lift Coefficient, α_C

Ideally, the airplane should be flying at an AOA that generates the least amount of drag during the intended mission for which it is designed. For instance, this could be a cruise or long-range performance point. In the case of the design mission being a cruise or loiter mission, this AOA can be calculated from Equation (9-43).

Derivation of Equation (9-49)

We start with Equation (9-43) for the design cruise AOA, repeated here for convenience:

Then we insert it into the Equation (9-51) and manipulate algebraically:

QED