The Maximum Lift of a Finite Wing

The maximum lift coefficient of a finite wing is influenced by several factors. Obviously, СЪт1х is strongly dependent on C^; that is, the wing’s performance depends on it’s airfoil performance. Second, the spanwise extent to which the wing is flapped has a significant influence on Also in

estimating Cl^, one must account for the presence of the fuselage, the tail download required to trim the aerodynamic pitching moment, and the span – wise distribution of loading over the wing.

The effect of aspect ratio on CLmai is slight, as one might expect from the preceding considerations on the elliptic wing. The wing lift coefficient and section, lift coefficients are nearly equal.

The detailed estimation of a wing’s Ci _ begins with a calculation of its spanwise load distribution. There are several methods to be found in the literature for doing this. Many of these fall into a class known as vortex lattice methods. One of the first of these can be found in Reference 3.30.

The vortex lattice method is similar to lifting line theory except that discrete vortex lines are also distributed in the chordwise direction. As illustrated in Figure 3.56, the wing is covered with a mesh of spanwise and

chordwise bound vortex lines with free vortex lines shed downstream from the trailing edge. At every juncture of the vortex lines, Helmholtz’s law of vortex continuity must hold. In addition, control points are chosen over the wing equal in number to the number of unknown vortex line strengths. The unknown vortex strengths are then adjusted to assure that the resultant flow at each control point is tangent to the mean surface of the wing.

The lattice model shown in Figure 3.56, in the manner of Reference 3.30, is formed by the superposition of horseshoe-shaped line vortices. One such vortex is shown dashed for illustrative purposes. Note that the downstream control points are aft of the last bound vortex lines; some points are even slightly off of the wing’s surface. Stipulating that the flow at these points parallels the camber surface at the trailing edge will satisfy approximately the Kutta condition.

Trends iq the behavior of as related to wing geometry can be seen by the application of an approximate method for determining spanwise load distribution known as Schrenk’s approximation (Ref. 3.31).

This method deals with two distributions: a basic lift distribution, and an additional lift distribution. The basic distribution is that which exists along the span when the total lift is equal to zero. Approximately, this lift distribution is taken as the average of a constant “zero” distribution and one obtained by neglecting any induced velocities. Thus,

the basic section lift coefficient, Cib> will be

(3.78)

The additional lift distribution results from an angle of attack different from and is assumed to be the average of an elliptic distribution and one proportional to the planform, both having the same total lift. For the latter,

cCi « c = kc

But

so that

L-£>>/•-(¥) *

The constant of proportionality, K, in this case becomes

К-ЦCL

The additional section lift coefficient then becomes

c’.-T[1+^V’-(y)1 <3-79’

Usually C, a is defined as the value of Equation 3.79 for a CL of unity; thus,

c, = c, b + ClaCL (3.80)

The manner in which this equation is used to estimate is best

explained by an example. Consider the wing of Figure 3.50. This particular wing has a taper ratio of 0.4 and a washout of 2°. Using Equations 3.78 and 3.79, the basic and additional section lift coefficient distributions given in Figure 3.57 were calculated. Also graphed on this figure is Qmax as a function of spanwise location. In this instance, is taken from Figure 3.50 to be a constant. In many cases, Ci^ decreases toward the tip as the airfoil becomes relatively thinner or as the chord lengths become smaller.

Combining Cib and Cta in the form of Equation 3.80, Figure 3.57 shows that a wing CL of 1.22 results in a section Q halfway out along the span, which is just equal to the section at that location. Any attempt to increase CL above this value will therefore cause the wing to stall at this location. Since the Ci curve is rather flat in this location, the stalling would be expected to spread to either side of 2ylb equal to 0.5. Thus, to estimate the of a wing, one finds the wing CL that results in a section Ct somewhere along the span, which is just equal to the section C(max. In this instance, the Cl^ value of 1.22 compares favorably with the experimental results. Generally, however, the C, predicted by this method will be somewhat conservative, since the total wing Cl may still increase somewhat, even though a section of it begins to stall.

As a further and more extreme example of the method, consider the wing of Figure 3.50 equipped with 60% span, 20% chord split flaps deflected 60°. From Equation 3.43, ДC, is estimated to be

ДС/ = Сіатг) S

= 0.108 (0.545)(0.35)(60)

= 1.236

ДС^, empirically, is approximately 0.83 of the preceding equation (Figure

Flgurg 3.57 Predicted lift coefficient distributions for wing of Figure 3.50.

3.34) or 1.026. Thus, Figure 3.57 is revised as shown in Figure 3.58 to increase the section C;max to 2.30 over the inner 60% of the span.

Aerodynamically the twist of the wing is changed by lowering the flaps. Relative to the midspan chord, the zero lift lines of the sections outboard of the flaps are reduced in angle of attack by ACJCia, or 11.4°. Thus, for this flapped wing, e, in degrees, becomes

0<|yUo.6

€ = -2 Щ -11.4 |yl>0.6

For this twist distribution and a taper ratio of 0.4, the angle of attack of the midspan zero lift line, aW(), for zero lift becomes 4.24°. Thus

C, b = 0.054(4.24+ e)

The additional lift distribution remains the same, since the planform is

Figure 3.58 Predicted lift coefficient distributions for wing of Figure 3.50 with 60% span, 20% chord split flaps deflected 60°.

unchanged. The predicted С/ distributions with the partial span split flaps are presented in Figure 3.58.

Figure 3.58 predicts that the wing will begin to stall just outboard of the flaps at a wing CL of 1.63. This result agrees exactly with Reference 3.27 with regard to both C(.max and the location of the initial stall. This agreement is somewhat fortuitous in view of Shrenk’s approximation, which is obviously inexact, since it allows a finite loading at the tip and other discontinuities in the cCt distribution. Nevertheless, for preliminary design studies, or in lieu of more exact lifting surface methods, Shrenk’s approximation is a useful tool.

Effect of Fuselage on CLmay

In working with a wing-fuselage combination, one normally defines the wing planform area to include the portion submerged within the fuselage. When a lift coefficient is quoted for the combination, it is based on this total wing planform area obtained by extrapolating the leading and trailing edges into the fuselage centerline. Generally, the fuselage will effect a decrease in

2

Figure 3.59 Effect of fuselage on spanwise lift distribution.

the lift per unit span over the portion of the wing covered by the fuselage. This is illustrated in Figure 3.59. The upper distribution is without the fuselage. The dashed line on the lower figure is the qualitative drop in cC( due to the fuselage. As an approximation, let us assume that the fuselage effects a constant drop in cCt over its width proportional to the midspan value of cCt. Thus, the lift decrement resulting from the fuselage will be

Sfuse is the wing planform area submerged in the fuselage, and is the nearly constant section Q near the center of the wing, к is the constant of proportionality. Thus the total CL with the fuselage, С^им, can be written in terms of CL before the fuselage is added as

(3.81)

In Reference 3.27, two wings equipped with partial and full-span, split, single-slotted, and double-slotted flaps were tested with and without a fuselage. The fuselage was circular in cross-section and the wing was mounted slightly above the middle of the fuselage. The ratio SfUse/S was equal to 0.083. The results of these tests are plotted in Figure 3.60 and compared with Equation 3.79 using kCJCL= 1.0. Also plotted on Figure 3.60 are test results from References 3.32 and 3.33. The ratio SfusJS was nearly the same for these

Figure 3.60 Effect of fuselage on CLmax.

two references as for Reference 3.27. These data support the form of Equation 3.81, at least to the extent that the correction to Сц^ for the fuselage appears to increase linearly with Ci-max of the wing alone. The correction depends on the cross-sectional shape of the fuselage and seems to vanish or even be slightly favorable for a rectangularly shaped section. Reference 3.34 also shows the correction to be slight for elliptical shapes where the height is greater than the width.

The decrement in Сі^т also depends on wing position and appears to be a maximum for the midwing configuration.