Physical Significance

tends to roll up at the edges and helps to form the wing-tip vortices sketched in Fig­ure 5.2. Thus, Prandd’s lifting-line model with its trailing-vortex sheet is physically consistent with the actual flow downstream of a finite wing.

Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8. The airfoil section is thin and symmetric. Calculate the lift and induced drag coefficients for the wing when it is at an angle of attack of 5°. Assume that 5 = r.

Solution

From Figure 5.18, 5 = 0.055. Hence, from the stated assumption, r also equals 0.055. From Equation (5.70), assuming ao = 2n from thin airfoil theory,

Example 5.1

_ Uo

1 + a0/+rAR(l + r) = 0.0867 degree-1

Since the airfoil is symmetric, ctL=o = 0°. Thus,

CL = act = (0.0867 degree 1 (5°) =

0.4335

From Equation (5.61),

(0.4335)2(1 +0.055)
8л

0.00789

Cn

The lift slope for the airfoil (the infinite wing) can be obtained from Equation (5.70):

dCi ^ «о

da 1 + (a0/7rAR)(l + r)

a0 _ a0

1 + [(1.055)«0/л-(6)] 1 + 0.056a0

4.485

Solving for ao, we find that this yields ao = 5.989/rad. Since the second wing (with AR = 10) has the same airfoil section, then a0 is the same. The lift slope of the second wing is given by

a0 5.989

1 + (a0/TrAR)(l + r) _ 1 + [(5.989)(1.105)/л-(103ї

= 0.086/degree

a

4.95/rad

The lift coefficient for the second wing is therefore

CL = a (a – aL=0) = 0.086[3.4° – (-2°)] = 0.464

In turn, the induced drag coefficient is

0.0076

lift distribution reaching farther away from the root. Such wings require heavier internal structure. Hence, as the aspect ratio of a wing increases, so does the structural weight of the wing. As a result of this compromise between aerodynamics and structures, typical aspect ratios for conventional subsonic airplanes are on the order of 6 to 8.

However, examine the three-view of the Lockheed U-2 high altitude reconnaissance aircraft shown in Figure 5.24. This airplane has the unusually high aspect ratio of 14.3. Why? The answer is keyed to its mission. The U-2 was essentially a point design; it was to cruise at the exceptionally high altitude of 70,000 ft or higher in order to not be reached by interceptor aircraft or ground-to-air-missiles during overflights of the Soviet Union in the 1950s. To achieve this mission, the need for incorporating a very high aspect ratio wing was paramount, for the following reason. In steady, level flight, where the airplane lift L must equal its weight W,

L = W = q. xSCL = p^VlSCL [5.71]

As the airplane flies higher, px decreases and hence from Equation (5.71) С/. must be increased in order to keep the lift equal to the weight. As its high-altitude cruise design point, the U-2 flies at a high value of C;, just on the verge of stalling. (This is in stark contrast to the normal cruise conditions of conventional airplanes at conventional altitudes, where the cruise lift coefficient is relatively small.) At the high value of С/, for the U-2 at cruising altitude, its induced drag coefficient [which from Equation (5.62) varies as C} would be unacceptably high if a conventional aspect ratio were used. Hence, the Lockheed design group (at the Lockheed Skunk Works) had to opt for as high an aspect ratio as possible to keep the induced drag coefficient within reasonable bounds. The wing design shown in Figure 5.24 was the result.

We made an observation about induced drag Д itself, in contrast to the induced drag coefficient CD, . We have emphasized, based on Equation (5.62), that Cdj can be reduced by increasing the aspect ratio. For an airplane in steady, level flight, however, the induced drag force itself is governed by another design parameter, rather than the aspect ratio per se, as follows. From Equation (5.62), we have

Three-view of the Lockheed U-2 high-altitude reconnaissance airplane.

Figure 5.34