The Vortex System for a Wing

A wing’s lift is the result of a generally higher pressure acting on its lower surface compared with the pressure on the upper surface. This pressure difference causes a spanwise flow of air outward toward the tips on the lower surface, around the tips, and inward toward the center of the wing. Combined with the free-stream velocity, this spanwise flow produces a swirling motion of the air trailing downstream of the wing, as illustrated in Figure 3.51. This motion, first perceived by Lanchester, is referred to as the wing’s trailing vortex system.

Immediately behind the wing the vortex system is shed in the form of a vortex sheet, which rolls up rapidly within a few chord lengths to form a pair

of oppositely rotating line vortices. Looking in the direction of flight, the vortex from the left wing tip rotates in a clockwise direction; the right tip vortex rotates in the opposite direction.

The trailing vortex system, not present with a two-dimensional airfoil, induces an additional velocity field at the wing that must be considered in calculating the behavior of each section of the wing.

If the aspect ratio of the wing is large, approximately 5 or higher, the principal effect of the trailing vortex system is to reduce the angle of attack of each section by a small decrement known as the induced angle of attack, a,. In this case Prandtl’s classical lifting line theory (Ref. 3.28) applies fairly well. As shown in Figure 3.52, the wing is replaced by a single equivalent vortex line, known as the “bound vortex,” since it is in a sense bound to the wing. The strength of this vortex, Г(у), is related to the lift distribution along the wing by the Kutta-Joukowski relationship.

Figure 3.52 Lifting line model of a wing and trailing vortex system.

Expressing the lift per unit length of span in terms of the section chord length, c(y), and section lift coefficient, C/(y), leads to

Г(у) = іс(у)С,(у)У (3.65)

With no physical surface outboard of the wing tips to sustain a pressure difference, the lift, and hence Г, must vanish at the tips.

According to the Helmholtz theorem regarding vortex continuity (Ref.

1.3, p. 120), a vortex line or filament can neither begin nor end in a fluid; hence it appears as a closed loop, ends on a boundary, or extends to infinity. Thus, it follows that if in going from у to у + dy the bound circulation around the wing increases from Г to Г + dr, a free vortex filament of strength dT, lying in the direction of the free-stream velocity, must be feeding into Г in order to satisfy vortex continuity. This statement may be clarified by reference to Figure 3.53.

The entire vortex system shown in Figure 3.52 can be visualized as being closed infinitely far downstream by a “starting” vortex. This vortex, opposite

Figure 3.53 Illustration of vortex continuity.

in direction to the bound vortex, would be shed from the trailing edge of the wing as its angle of attack is increased from zero.

The trailing vortex system of strength dr induces a downwash, w(y), at the lifting line, as mentioned earlier. As pictured in Figure 3.54, this reduces the angle of attack by the small angle a,. Thus the section lift coefficient will be given by

Ci = Cia(a — a,) (3.66)

a being measured relative to the section zero lift line.

To a small angle approximation, the induced angle of attack, at, is given by wl V. The downwash, w, can be determined by integrating the contributions of the elemental trailing vortices of strength dr. If the vortex strength dr trails from the wing at a location of y, its contribution to the downwash at another location y0 can be found by Equation 2.64 to be

Thus, af becomes

Equations 3.65, 3.66, and 3.68 together relate Г(у) to c(y) and a(y) so that, given the wing geometry and angle of attack, one should theoretically be able to solve for Г and hence the wing lift. In order to accomplish the solution, it is expedient to make the coordinate transformation pictured in Figure 3.55.

у = — cos в

Hence, Equation 3.67 becomes

Since more elaborate and comprehensive treatments of wing theory can be found in texts devoted specifically to the subject (e. g., see Ref. 3.29), only the classical solution for the eliptic Г distribution will be covered here. This particular case is easily handled and results in the essence of the general problem.

Assume that Г is of the form

r=r"V’-©!

This transforms to

Г = Г0 sin в

Here, Г0 is obviously the midspan value of the bound circulation. Thus Equation 3.69 becomes

Гр f" cos в d0 2irbV J0 cos в – cos e0

The preceding integral was encountered previously in thin airfoil theory and has a value of it. Thus, for an elliptic Г distribution, a, and hence the downwash is found to be a constant independent of y.

… Гр

2 bV

If the wing is untwisted so that a is also not a function of у then, from

Equation 3.66, it follows that the section Ct is constant along the span. Thus,

Thus it is found that, according to lifting line theory, an untwisted wing with an elliptical planform will produce an elliptic Г distribution. Such a wing will have a constant downwash and section С/.

Since Ci is constant and equal to CL, Equations 3.65, 3.66, and 3.71 can be applied at the midspan position in order to determine the slope of the wing lift curve. First, from Equations 3.71 and 3.65,

CqCl 4 b But, for the planform given by Equation 3.66, c0 and b are related to the aspect ratio by a-*L 7ГС о Thus, a, becomes Й II (3.72) Inserted into Equation 3.66, the preceding results in

c-Sr)

or,

Cl = C’*[(l + QJItta] “ [(1 + COIita]

Using the theoretical value of 2ir Ci/rad derived earlier, the preceding becomes

С‘=^(лТ2)“ <373>

Equations 3.72 and 3.73 are important results. The induced angle of attack is seen to increase with decreasing aspect ratio which, in turn, reduces the slope of the lift curve, CLa. A wing having a low aspect ratio will require a higher angle of attack than a wing with a greater aspect ratio in order to produce the same CL.

It was stated previously that the comparative performance between the wing and airfoil shown in Figure 3.50 could be explained theoretically. In this case, A = 9.02 so that, on the basis of Equation 3.73,

Cl„ — 0.819 C;o

This result is within about 2% of the experimental results presented in Figure 3.50.

As the aspect ratio decreases, the lifting line becomes progressively less accurate. For example, for an aspect ratio of 4.0, Equation 3.73 is ap­proximately 1 % higher than that predicted by more exact methods.

As described in Reference 3.3, a more accurate estimate of CLa is obtained from

Cl° = C’“ A + [2(A + 4)l(A + 2)] (3.14a)

An alternate to Equation 3.74a is offered by Reference 3.35 and is referred to as the Helmbold equation, after the original source noted in the reference. The Helmbold equation reads

c =c л

^ ‘“(C,» + V(C(»2tA2

Replacing Cta by 2v in the denominator,

Equation 3.74a and 3.74b agree within a couple of percent over the range of practical aspect ratios and approach each other in the limits of A = 0 or A = oo. This holds for high or low aspect ratios and is based on an approximate lifting surface theory, which accounts for the chordwise distribution of bound circulation as well as the spanwise distribution.

Let us visualize an aerodynamically untwisted wing, that is, one for which the zero lift lines all lie in a plane. Imagine this wing to be at a zero angle of attack and hence operating at zero CL. Holding the midspan section fixed, let us now twist the tip up through an angle eT. We will assume that the twist along the wing is linear, so that at any spanwise location y, the twist relative to the midspan section is given by

Obviously, the wing will now develop a positive CL, since every section except the midspan is at a positive angle of attack. If we define the angle of attack of the wing to be that of the zero lift line at the root, the angle of attack of the wing for zero lift will be negative; that is, we must rotate the entire wing nose downward in order to return to the zero lift condition. For a CL of zero, Equation 3.72 shows that on the average, a, equals zero; thus, at any section,

Сі = Сі«{ЄтьІ2 +^

where ащ is the angle of attack of the wing for zero lift. To find this angle, an expression is written for the total wing lift and is equated to zero.

г Ы2

1 L = I qcCi dy

J – Ы2

‘ OcG-(‘r W2+ ‘■’h

Equating this to zero and taking q and Cla to be constant (this for Cla, but close) leads to,

Л, (Ы2

If e is not linear,

1 fm

04= c cedy (3.76)

* J-bl2

Now consider a linearly tapered wing for which the chord distribution is given by

C — Co — (Co — Cr)

Defining the taper ratio Л as the ratio of the tip chord, cT, to the midspan chord, C0, the preceding equation can be written as

C = C0[l-(l-A)^]

Substituting this into Equation 3.75 and integrating results in the angle of attack of the wing for zero lift as a function of twist and taper ratio.

€f (1 + 2A)

T (1 + Л)

Most wings employ a negative twist referred to as “washout.” Generally, eT is of the order of -3 or 4° to assure that the inboard sections of the wing stall before the tip sections. Thus, as the wing begins to stall, the turbulent separated flow from the inboard portion of the wing flows aft over the horizontal tail, providing a warning of impending stall as the pilot feels the resulting buffeting. In addition, with the wing tips still unstalled, the pilot has aileron control available to keep the wings level in order to prevent the airplane from dropping into a spin. At the present time, the stall-spin is one of the major causes of light airplane accidents.

The wing of the airplane in Figure 3.50 has a 2° washout and a taper ratio of 0.4. According to Equation 3.77, aWfj for this wing will be +0.8°. This is close to the results presented in Figure 3.50, where the angle of attack of the wing for zero lift is seen to be approximately 0.6° greater than the cor­responding angle for the airfoil. Thus, knowing an airfoil lift curve, one can estimate with reasonable accuracy the lift curve of a wing incorporating that airfoil by calculating the slope and angle for zero lift with the use of Equations 3.74 and 3.77 respectively.

As a further example, in the use of these equations, consider a wing having an NACA 63,4-421 airfoil (Figure 3.30) at its midspan that fairs linearly into an NACA 0012 airfoil at the tip. The wing has no geometric twist; that is, the section chord lines all lie in the same plane. The wing’s aspect ratio is equal to 6.0, and it has a taper ratio of 0.5. The problem is to find the angle of attack of the wing measured relative to the midspan chord, which will result in a CL value of 0.8.

To begin, we note from Figure 3.30 that the 63,4-421 airfoil has an angle of zero lift of -3°; thus, the zero lift line at the midspan is up 3° from the chord line. It follows that the aerodynamic twist eT is -3°, since the tip airfoil is symmetrical. Inserting this and the taper ratio into Equation 3.77 results in <*„„= 1.3°. From Equation 3.74, for an aspect ratio of 6.0,

CLa = 0.706 C, a

From Figure 3.30 or Reference 3.1, Cta is nearly the same for the midspan and tip sections and is equal approximately to 0.107 CJdeg. Hence, CL„ =

0. 076 CJ deg. Therefore, over the linear portion of the lift curve, the equation for the wing Cl relative to the midspan zero lift line becomes

Cl = 0.076(a – 1.3)

For Cl of 0.8, a is found to equal 11.8°. Thus, to answer the original problem, the angle of attack of the midspan chord, aw, will equal 11.8-3, or 8.8°.