Category AIRCRAF DESIGN

For a given wing loading (i. e., the wing area and maximum takeoff mass [MTOM] invariant), aerodynamicists prefer a large wingspan to improve the aspect ratio in order to reduce induced drag at the cost of a large wing root bending moment. Structural engineers prefer to see a lower span resulting in a lower aspect ratio.

The BWB (see Figure 1.15) design for larger aircraft has proven merits over conven­tional designs but awaits technological and market readiness. Interesting deductions are made in the following sections.

The lower the speed at landing, the safer is the aircraft in case of any inadvertent mishap. An aircraft landing occurs near the wing stall condition when the aileron effectiveness should be retained to avoid a wing tip hitting the ground. In other words, when approaching the stall condition, its gradual development should start from the wing root, which allows the aileron at the wing tip to retain its ability to maintain level flight. Figure 3.40 (see also Figure 3.18) shows typical wing stall prop­agation patterns on various types of wing planforms.

Figure 3.38. Thickness-to-chord ratio for various aircraft

Because a swept-back wing tends to stall at the tip first, twisting of the wing tip nose downward (i. e., washout) is necessary to force the root section to stall first, thereby retaining roll control during the landing.

A good way to ensure the delay of the wing tip stall is to twist the wing about the У-axis so that the tip LE is lower than the wing root LE (see Figure 3.32). Typical twist-angle values are 1 to 2 deg and rarely exceed 3 deg.

Section 3.7.1 explains the transonic effect resulting from the thickness distribution along an aircraft body. On the wing, the same phenomenon can occur, most impor­tantly along the wing chord but altered due to the 3D wing tip influence. A local shock interacting with the boundary layer can trigger early separation, resulting in unsteady vibration and – in extreme cases – even causing the wing to stall. A typical consequence is a rapid drag increase due to the compressibility effect resulting from the transonic-flow regime. Military aircraft in hard maneuver can enter into such an undesirable situation even at a lower speed. As much as possible, designers try to

avoid, delay, or minimize the onset of flow separation over the wing due to local shocks.

Drag divergence is a sudden increase in drag. A 20-count drag rise (CD = 0.002) at the Mach number is known as the drag divergence mach (MDD), shown in Figure 3.36b. The critical Mach (Mcrit) is the onset of the transonic-flow field and is lower than the MDD. Some texts use Mcrit with a 20-count drag increase.

Structural engineers prefer aerofoil sections to be as thick as possible, which favors structural integrity at lower weights and allows the storage of more fuel onboard. However, aerodynamicists prefer the aerofoil to be as thin as possible to minimize the transonic-flow regime in order to keep the wave drag rise lower. One way to delay the Mcrit is to sweep the wing (Figure 3.36a) either backward (see Figure 3.31, Boeing 737) or forward (see Figure 4.37e, SU47 [at www. cambridge .org/Kundu]), which thins the aerofoil t/c ratio and delays the sudden drag rise (Figure 3.36b). The former is by far more prevalent because of structural considera­tions. Wing slide (i. e., in which the chord length remains the same) is different from wing sweep, in which the chord length is longer by the secant of the sweep angle.

Shown here is the relationship between the sweep angle and wing geometries. The chord length of a swept wing increases, resulting in a decrease in the t/c ratio:

chordswept = (chordunswept)/CosA (3.39)

This results in:

(thickness/chordswept) < (thickness/chordunswept) (3.40)

This directly benefits the drag divergence Mach number, divided by the cosine of the sweep angle:

Л1/4;that is, Machdivj;wept = Machdiv_unswept/C°sA1/4 (341)

The sweep also degrades the Cbmax by the cosine of the sweep angle, Л^; that is:

CLmax^swept = CLmax-unswept x СтеЛ/ (3.42)

If the trailing edge can remain unswept, then flap effectiveness is less degraded due to a quarter-chord sweep.

Qualitative characteristics between the wing sweep and the t/c ratio variation are shown in Figure 3.37.

Figure 3.38 shows typical values used in various aircraft. Another effect of speed gain is a change in CLmax, as shown in Figure 3.39. For a particular wing, the ratio

of CLmax. compressible/CLmaxJncompressible decreases to approximately 0П, as shown in

Figure 3.39.

Designers require this body of information for the aerofoil selection. The choice decides the extent of wing sweep required to lower the t/c ratio to achieve the desired result (i. e., to minimize the compressible drag increase for the cruise Mach number) while also satisfying the structural requirements. To standardize drag-rise characteristics, the flow behavior is considered to be nearly incompressible up to Mcrit and can tolerate up to MDD, allowing a 20-count drag increase (ACD = 0.002).

Aircraft in a yaw/roll motion have a cross-flow over the wing affecting the aircraft roll stability (see Chapter 12). The dihedral angle (i. e., the wing tip chord raised above the wing root chord) assists roll stability. A typical dihedral angle is between 2 and 3 deg and rarely exceeds 5 deg. Figure 3.34a shows that the dihedral angle with a low-wing configuration also permits more ground clearance for the wing tip. The opposite of a dihedral angle is an anhedral angle, which lowers the wing tip with respect to the wing root and is typically associated with high-wing aircraft (Figure 3.34b). The dihedral or anhedral angle also can be applied to the horizontal tail.

3.8 Mean Aerodynamic Chord

Various wing reference geometries and parameters are used in aerodynamic compu­tations. A most important parameter is the mean aerodynamic chord (MAC), which is the chord-weighted average chord length of the wing, defined as follows:

2 fb/2 2

MAC = c2dy, (3.37)

Sw 0

where c is the local wing chord and SW is the wing reference area:

trapezoidal wing reference area, SW (with sweep) (Figure 3.35) half wing area = rectangle – two triangles

= A x B – i/j(A – CR) x B – i/j(A – CT) x B = A x B – i/2(A x B) + i/2(B x CR) – i/2(A x B) + ^(B x CT)

= /2(Cr + CT) x B

For the full wing when the span b = 2B:

wing area, SW = /2 (CR + CT) x b Evaluating Equation 3.20 for the linear trapezoidal wing results in:

c = Croot 2(Croot Ctip)y/b

When substituting the integral, Equation 3.37 becomes:

= (2/SW)[bCr2/2 – bCR(CR – Ct)/2 – b(CR – Ct)2/6]

When substituting for the SW:

MAC = [2/(Cr + Ct)][c2 – C2 + CrCT + C2/3 + CT/3 – 2CrCT/3]

= [2/(Cr + Ct)][c2/3 + CT/3 – CrCT/3]

2 Г (Cr + Ct)2 CrCt ]

3 (Cr + Ct) (Cr + Ct)

For a linearly tapered (trapezoidal) wing, this integral is equal to:

MAC = 2/3[Croot + Ctip – CrootCtip/(Croot + Ctip)] (3.38)

For wings with a glove/yehudi, the MAC may be computed by evaluating each lin­early tapered portion and then taking an average, weighted by the area of each por­tion. In many cases, however, the MAC of the reference trapezoidal wing is used. The MAC is often used in the nondimensionalization of pitching moments as well as to compute the reference length for calculating the Re as part of the wing drag estimation. The MAC is preferred for computation over the simpler mean geomet­ric chord for aerodynamic quantities whose values are weighted more by the local chord, which are reflected by their contribution to the area.

This is defined as the ratio of the wing tip chord to the wing root chord (ctip/croot). The best taper ratio is in the range from 0.3 to 0.6. The taper ratio improves the wing efficiency by giving a higher Oswald’s efficiency factor (see Section 3.10).

3.16.6

Wing Twist

The wing can be twisted by making the wing tip nose down (i. e., washout) relative to the wing root (Figure 3.32), which causes the wing root to stall earlier (i. e., retain aileron effectiveness). Typically, a 1- to 2-deg washout twist is sufficient. Twisting the wing tip upward is known as washin.

3.16.7 High/LowWing

Depending on the design drivers, an aircraft configuration can place the wing any­where from the top (i. e., high wing) to the bottom (i. e., low wing) of the fuselage or in between (i. e., midwing), as shown in Figure 3.33. Structural considerations of the wing attachment to the fuselage comprise a strong design driver, although in the civil aircraft market, the choice could be dictated by customer preference. The wing cen­ter section should not interfere with the cabin passage-height clearance – especially critical for smaller aircraft. A fairing is shown for low-wing aircraft (Figure 3.33a, Cessna Citation) or high-wing aircraft (Figure 3.33c, Dornier 328), where the wing passes under or over the fuselage, respectively. Both cases have a generous fair­ing that conceals the fuselage mould-line kink (i. e., drag-reduction measure), which would otherwise be visible. Midwing (or near-midwing) designs are more appropri­ate to larger aircraft with a passenger cabin floorboard high enough to allow the wing box positioned underneath it.

Aircraft with a high wing allow better ground clearance (see Figures 3.33c and 3.49) and the fuselage to be closer to the ground, which makes cargo-loading eas­ier – especially with a rear-fuselage cargo door. Turboprops favor a high-wing con­figuration to allow sufficient ground clearance for the propeller. The main under­carriage is mounted on the fuselage sides with the bulbous fairing causing some additional drag. However, this configuration provides better aerodynamics (e. g., the BAe RJ100 and Dornier 328 are successful high-wing designs). The dominant con­figuration for civil transport aircraft has been a low wing, which provides a wider

main-undercarriage wheel track (see Chapter 7), allowing better ground maneuver­ing. A low wing also offers a better crashworthy safety feature in the extremely rare emergency situation of a belly landing. However, the author believes more high-winged, large commercial transport aircraft could be developed. Design trends shows that military transport aircraft have predominantly high wings with large rear- mounted cargo doors.

In the simplest rectangular wing planform area, the aspect ratio is defined as aspect ratio, AR = (span, b)/(chord, c). For a generalized trapezoidal wing planform area:

aspect ratio, AR = (b x b)/(b x c) = (b2)/(SW) (3.36)

3.7.3 Wing Sweep Angle, Л

The wing quarter-chord line is the locus of one fourth of the chord of the refer­ence wing planform area measured from the LE, as shown in Figure 3.31. The wing sweep is measured by the angle of the quarter-chord line extended from the line perpendicular to the centerline.

3.7.4 Wing Root (croot) and Tip (ctip) Chord

These are the aerofoil chords parallel to the aircraft at the centerline and the tip, respectively, of the trapezoidal reference area.

This section defines the parameters used in wing design and explains their role. The parameters are the wing planform area (also known as the wing reference area, SW); wing-sweep angle, Л; and wing taper ratio, X (dihedral and twist angles are given after the reference area is established). Also, the reference area generally does not include any extension area at the leading and trailing edges. Reference areas are concerned with the projected rectangular/trapezoidal area of the wing.

3.7.2 Planform Area, Sw

The wing planform area acts as a reference area for computational purposes. The wing planform reference area is the projected area, including the area buried in the fuselage shown as a dashed line in Figure 3.30. However, the definition of the wing planform area differs among manufacturers. In commercial transport aircraft design, there are primarily two types of definitions practiced (in general) on either side of the Atlantic. The difference in planform area definition is irrelevant as long as the type is known and adhered to. This book uses the first type (Figure 3.30a), which is prevalent in the United States and has straight edges extending to the fuse­lage centerline. Some European definitions show the part buried inside the fuselage

Figure 3.31. Wing geometry definition (Boeing 737 half wing)

as a rectangle (Figure 3.30b); that is, the edges are not straight up to centerline unless it is a rectangular wing normal to the centerline. Section 4.8 describes the various options available from which to choose a wing planform.

A typical subsonic commercial transport-type wing is shown in Figure 3.31. An extension at the LE of the wing root is called a glove and an extension at the trail­ing edge is called a yehudi (this is Boeing terminology). The yehudi’s low-sweep trailing edge offers better flap characteristics. These extensions can originate in the baseline design or on the existing platform to accommodate a larger wing area. A glove and/or a yehudi can be added later as modifications; however, this is not easy because the aerofoil geometry would be affected.

To incorporate the tip effects of a 3D wing, 2D test data need to be corrected for Re and span. This section describes an example of the methodology.

Equation 3.25 indicates that a 3D wing will produce aeff at an attitude when the aerofoil is at the angle of attack, a. Because aeff is always less than a, the wing pro­duces less CL corresponding to aerofoil Ci (see Figure 3.28). This section describes

Figure 3.27. Lift-curve-slope correction for aspect ratio

how to correct the 2D aerofoil data to obtain the 3D wing lift coefficient, CL, versus the angle of attack, a, relationship. Within the linear variation, dCL/da needs to be evaluated at low angles (e. g., from -2 to 8 deg).

The 2D aerofoil lift-curve slope a0 = (dCL/da), (3.32)

where a = angle of attack (incidence).

The 2D aerofoil will generate the same lift at a lower a of aeff (see Equation 3.25) than what the wing will generate at a (a3D > a2D). Therefore, using the 2D aerofoil data, the wing lift coefficient CL can be worked at the angle of attack, a, as shown here (all angles are in degrees). The wing lift at an angle of attack, a, is as follows:

CL = a0 x aeff + constant = a0 x (a – e) + constant (3.33)

or

CL = a0 x (a – 57.3CL/ep./AR) + constant

or

CL + (57.3 CL x a0/e^AR) = a0 x a + constant

or

CL = (a0 x a)/[1 + (57.3 x a0/e^AR)] + constant/[1 + (57.3 x a0/e^AR] (3.34)

Differentiating with respect to a, it becomes:

dCL/da = a0/[1 + (57.3/e^AR)] = a = lift – curve slope of the wing (3.35)

The wing tip effect delays the stall by a few degrees because the outer-wing flow distortion reduces the local angle of attack; it is shown as A «max. Note that A «max is the shift of CLmax; this value Aamax is determined experimentally. In this book, the empirical relationship of Aamax = 2 deg, for AR > 5 to 12, Aamax = 1 deg, for AR > 12 to 20, and Aamax = 0 deg, for AR > 20.

Evidently, the wing-lift-curve slope, dCL/da = a, is less than the 2D aerofoil- lift-curve slope, a0. Figure 3.27 shows the degradation of the wing-lift-curve slope, dCL/da, from its 2D aerofoil value.

Figure 3.28. Effect of t/c on dCb/da

The 2D test data offer the advantage of representing any 3D wing when cor­rected for its aspect ratio. The effect of the wing sweep and aspect ratio on dCb/da is shown in Figures 3.28 and 3.29 (taken from NASA).

If the flight Re is different from the experimental Re, then the correction for Cbmax must be made using linear interpolation. In general, experimental data pro­vide Cbmax for several Res to facilitate interpolation and extrapolation.

Example: Given the NACA 2412 aerofoil data (see test data in Appendix D), construct wing Cb versus a graph for a rectangular wing planform of aspect ratio 7 having an Oswald’s efficiency factor, e = 0.75, at a flight Re = 1.5 x 106.

From the 2D aerofoil test data at Re = 6 x 106, find dC/da = a0 = 0.095 per degree (evaluate within the linear range: -2 to 8 deg). Cmax is at a = 16 deg.

Use Equation 3.26 to obtain the 3D wing-lift-curve slope:

dCb/da = a = ao/[1 + (57.3/eqAR)] = 0.095/[1 + (57.3/0.75 x 3.14 x 7)]

= 0.095/1.348 = 0.067

From the 2D test data, Cmax for three Res for smooth aerofoils and one for a rough surface, interpolation results in a wing Cmax = 1.25 at flight

Figure 3.29. Effect of sweep on dCb/da

4 0

Wing quarter-chord sweep (deg)

Re = 1.5 x 106. Finally, for AR = 7, the Aomax increment is 1 deg, which means that the wing is stalling at (16 + 1) = 17 deg.

The wing has lost some lift-curve slope (i. e., less lift for the same angle of attack) and stalls at a slightly higher angle of attack compared to the 2D test data. Draw a vertical line from the 2D stall amax + 1 deg (the point where the wing maximum lift is reached). Then, draw a horizontal line with CLmax = 1.25. Finally, translate the 2D stalling characteristic of A a to the 3D wing-lift-curve slope joining the portion to the CLmax point following the test-data pattern.

This demonstrates that the wing CL versus the angle of attack, a, can be constructed (see Figure 3.27).

Equation 3.19 gives the basic definition of drag, which is viscous-dependent. The previous section showed that the tip effects of a 3D wing generate additional drag for an aircraft that appears as induced drag, mi. Therefore, the total aircraft drag in incompressible flow would be as follows:

aircraft drag = skin-friction drag + pressure drag + induced drag

= parasite drag + induced drag (3.30)

Most of the first two terms does not contribute to the lift and is considered para­sitic in nature; hence, it is called the parasite drag. In coefficient form, it is referred to as CDp. It changes slightly with lift and therefore has a minimum value. In coefficient form, it is called the minimum parasite drag coefficient, CmPmm, or CD0. The induced drag is associated with the generation of lift and must be tolerated. Incorporating this new definition, Equation 3.30 can be written in coefficient form as follows:

Cm = CDp + CDi (3.31)

Chapter 9 addresses aircraft drag in more detail and the contribution to drag due to the compressibility effect also is presented.

Similar to a bird’s wing, an aircraft’s wing is the lifting surface with the chosen aerofoil section, which can vary spanwise. The lift generated by the wing sustains the weight of the aircraft to make flight possible. Proper wing planform shape and size are crucial to improving aircraft efficiency and performance; however, aerofoil parameters are often compromised with the cost involved.

Downwash

Figure 3.24. Wing tip vortex

A 3D finite wing produces vortex flow as a result of tip effects, as shown in Figure 3.24 and explained in Figure 3.25. The high pressure from the lower surface rolls up at the free end of the finite wing, creating the tip vortex.

The direction of vortex flow is such that it generates downwash, which is dis­tributed spanwise at varying strengths. A reaction force of this downwash is the lift generated by the wing. Energy consumed by the downwash appears as lift- dependent induced drag, D, and its minimization is a goal of aircraft designers.

The physics explained thus far is represented in geometrical definitions, as shown in Figure 3.26. This is used in formulations, as discussed herein. An elliptical wing planform (e. g., the Spitfire fighter of World War II) creates a uniform spanwise

downwash at its lowest magnitude and leads to minimum induced drag. Figure 3.26 shows that the downwash effect of a 3D wing deflects free streamflow, VTO, by an angle, e, to Viocai. It can be interpreted as if the section of 3D wing behaves as a 2D infinite wing with:

effective angle of incidence, aeff = (a – e), (3.25)

where a is the angle of attack at the aerofoil section, by VTO.

Aerodynamics textbooks may be consulted to derive theoretically the down – wash angle:

e = Cl/^AR (in radians) = 57.3Cl/^AR (in deg) (3.26)

For a nonelliptical wing planform, the downwash will be higher and a semi-empirical correction factor, e, called Oswald’s efficiency factor (always < 1) is applied, as follows:

average downwash angle, e = CL/e^AR (in radians) = 57.3CL/eцAR (in deg)

(3.27)

The extent of downwash is lift-dependent; that is, it increases with an increase in Cl. Strictly speaking, Oswald’s efficiency factor, e, varies with wing incidence; however, the values used are considered an average of those found in the cruise segment and remain constant. In that case, for a particular aircraft design, the average downwash angle, e, is treated as a constant taken at the midcruise condition. Advanced wings of commercial transport aircraft can be designed in such a way that at the design point, e & 1.0.

Equations 3.26 and 3.27 show that the downwash decreases with an increase in the aspect ratio, AR. When the aspect ratio reaches infinity, there is no down – wash and the wing becomes a 2D infinite wing (i. e., no tip effects) and its sectional characteristics are represented by aerofoil characteristics. The downwash angle, e, is small – in general, less than 5 deg for aircraft with a small aspect ratio. The aerofoil section of the 3D wing apparently would produce less lift than the equivalent 2D aerofoil. Therefore, 2D aerofoil test results would require correction for a 3D wing application, as explained in the following section.

Local lift, Llocal, produced by a 3D wing, is resolved into components perpen­dicular and parallel to free streamflow, VTO. In coefficient form, the integral of these forces over the span gives the following:

CL = Lcos e/qSW and Cm = Lsin e/qSW (the induced-drag coefficient)

For small angles, e, it reduces to:

Cl = L/qSw and Cm = Le/qSw=CLe (3.28)

CDi is the drag generated from the downwash angle, e, and is lift-dependent (i. e., induced); hence, it is called the induced-drag coefficient. For a wing planform, Equa­tions 3.27 and 3.28 become:

Cm = CLe = Cl x Cl/єцAR = Cp /epAR (3.29)

Induced drag is lowest for an elliptical wing planform, when e = 1; however, it is costly to manufacture. In general, the industry uses a trapezoidal planform with a taper ratio, X & 0.4 to 0.5, resulting in an e value ranging from 0.85 to 0.98 (an optimal design approaches 1.0). A rectangular wing has a ratio of X = 1.0 and a delta wing has a ratio of X = 0, which result in an average e below 0.8. A rectangular wing with its constant chord is the least expensive planform to manufacture for having the same-sized ribs along the span.