3.15 Aspect Ratio Correction of 2D Aerofoil Characteristics for 3D Finite Wing

forces over the span gives the following:

CL = Lcos ε/qSW and CDi = Lsin ε/qSW (the induced-drag coefficient)

For small angles, ε, it reduces to:

CDi is the drag generated from the downwash angle, ε, 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:

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, λ ≈ 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 λ = 1.0 and a delta wing has a ratio of λ = 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.

3.14.1 Induced Drag and Total Aircraft Drag

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, Di. Therefore, the total aircraft drag in incompressible flow would be as follows:

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

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, CDPmin, 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:

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

3.15 Aspect Ratio Correction of 2D Aerofoil Characteristics for 3D Finite Wing

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 αeff at an attitude when the aerofoil is at the angle of attack, α. Because αeff is always less than α, the wing produces less CL corresponding to aerofoil Cl (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, α, relationship. Within the linear variation, dCL/dα needs to be evaluated at low angles (e.g., from −2 to 8 deg).

where α = angle of attack (incidence).

The 2D aerofoil will generate the same lift at a lower α of αeff (see Equation 3.25) than what the wing will generate at α (α3D > α2D). Therefore, using the 2D aerofoil data, the wing lift coefficient CL can be worked at the angle of attack, α, as shown here (all angles are in degrees). The wing lift at an angle of attack, α, is as

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 αmax. Note that αmax is the shift of CLmax; this value αmax is determined experimentally. In this book, the empirical relationship of αmax = 2 deg, for AR > 5 to 12, αmax = 1 deg, for AR > 12 to 20, and αmax = 0 deg, for AR > 20.

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

Figure 3.30. Wing planform definition (half wing shown)

Re = 1.5 × 106. Finally, for AR = 7, the αmax 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 αmax + 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 α 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, α, can be constructed (see Figure 3.27).

3.16 Wing Definitions

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, ; and wing taper ratio, λ (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.

3.16.1 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