Maximum Lift of Wings

In the previous sections of this chapter, the fluid was considered to be incompressible and inviscid when establishing the theory of lift. The wing theory based on this concept is in good agreement with measurements as long as the angle of attack is small to moderate; see, for example, Figs. 3-38, 3-39, 3-42-3-44, and

3- 49. Only in the range of large angles of attack does the effect of friction have significance for the lift. In particular, the maximum lift of a wing is not only determined by its geometry, but it is also considerably affected by friction. Determination of the maximum lift of a wing by strictly theoretical methods is not yet possible. Cooke and Brebner [11] report on flow separation from wings in general terms. Schlichting [73] presents the aerodynamic problems of maximum lift of wings in comprehensive form.

From measurements it is known that the maximum lift coefficient is strongly dependent on the geometric profile parameters (thickness, camber, nose radius) and on the Reynolds number. In Sec. 2-5-1 this relationship was discussed briefly; see, for example, Figs. 2-39 and 2-42-2-44. These previously reported results should be supplemented by the statement that the maximum lift of an unswept wing is essentially a problem of two-dimensional flow. A large aspect ratio of unswept wings of finite span cannot have an important effect on flow separation and consequently on the maximum lift because in this case the flow over the major portion of the wing deviates only a little from plane flow. Quite different are the conditions for wings of small aspect ratio. Here the flow around the wing tips reaches to the middle of the wing. For strongly swept-back wings, which includes delta wings, the flow conditions are particularly complex because the leading edge acts in a similar way as the tips of an unswept wing. For these kinds of wings, even the attached flow is much harder to assess than that for unswept wings, because the flow directions in the boundary layer may deviate from that of the outside flow (departure of the boundary layer to the wing tips, boundary-layer fence).

Contrary to unswept wings, the flow over strongly swept-back wings without twist separates locally first at the wing tips because the lift load has its maximum there (see Fig. 347).

When the angle of attack increases, the separated region expands inward in span direction. This behavior is discussed in more detail in [26]. A very comprehensive compilation of material on this behavior of swept-back wings at large angles of attack and high Reynolds numbers has been given by Furlong and McHugh [16].

The effect of the aspect ratio and the sweepback angle on the maximum lift coefficient will now be examined using some test results.

In Fig. 3-53, results are plotted for the maximum lift coefficient of rectangular wings and swept-back wings of constant chord (</? = 45°). The Reynolds numbers of these measurements are Re ^ 106. Figure 3-53c confirms that the maximum lift coefficient for A > 2 is almost independent of the angle of attack. For very small aspect ratios, cLmax is somewhat larger than for large aspect ratios. Particularly noteworthy in Fig. 3-53b is, for aspect ratios Л < 2, the strong increase to values of а 30° in the angle of attack for which the maximum lift coefficient is obtained.

In Fig. 3-54 curves are given for the lift coefficients of a series of delta wings plotted against the angle of attack. When the aspect ratio A decreases, the lift slope becomes considerably smaller, while the maximum lift coefficient and the corresponding angles of attack increase. The lift slopes dcLjda of these wings have been presented earlier in Fig. 3-38. Maximum lift coefficients cLmzx for these and additional delta wings are plotted in Fig. 3-55 against the aspect ratio. Comparison

Figure 3-53 Maximum lift coefficients of rec­tangular wings = 0) and swept-back wings of constant chord <> #0), Reynolds number Re » 106. (a) Maximum lift coefficient C£,max vs. aspect ratio л. (b) angle of attack a for max vs. aspect ratio A. Curve 1, = 0°; profile NACA

0015, from Bussmann and Kopfermann [25]. Curve 2, = 45°; profile NACA 0012, from

Truckenbrodt [85]. Curve 3, ip = 0°; 8 « 0.10, mean values of various measurements. Curve 4, ^=35°; 6 « 0.10, mean values of various mea­surements.

A=0.83 1.61 2.38

Figure 3-54 Lift coefficients ci vs, angle of attack a. for delta wings of various aspect ratios – t; taper = 2> thickness ratio 5 = 0,12, Reynolds number Re ^ 7 * 10s, from Truckenbrodt [85].

with Fig. 3-53д shows that the increase in cLraaK for small aspect ratios is considerably larger than for rectangular and swept-back wings. Also, Fig. 3-556 shows a strong increase of &cLmax at small aspect ratios in agreement with Fig. 3-53b. Experimental studies on the separation characteristic of delta wings have been carried out by Truckenbrodt and Feindt [85] by means of simple wake measurements.

Figure 3-55 Maximum lift coef­ficients of delta wings, Reynolds number Re « 106. (a) Maximum lift coefficient C£max vs. aspect ratio Л. (b) Angle of attack a for C£,max vs – aspect ratio A. Curve 1, delta wing; A = 0; pro­file NACA 0012, from Lange and Wacke [25]. Curve 2, delta wing; A = profile NACA 0012, from Truckenbrodt [85]. Curve 3, mean values of various mea­surements.