Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

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

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

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.

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.

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

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,

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°.

A two-dimensional airfoil with its zero lift line at an angle of attack of 10° will deliver a lift coefficient, Ct, of approximately 1.0. When incorporated into a wing of finite aspect ratio, however, this same airfoil at the same angle of attack will produce a wing lift coefficient, CL, significantly less than 1.0. The effect of aspect ratio is to decrease the slope of the lift curve CLa as the aspect ratio decreases. Figure 3.50 illustrates the principal differences in the

lift behavior of a wing and an airfoil. First, because the wing is twisted so that the tip is at a lower angle of attack than the root (washout), the angle for zero lift, measured at the root, is higher for the wing by approximately 0.6°. Next, the slope of the wing’s lift curve, CLa, is approximately 0.79 of the slope for the airfoil. Finally, is only slightly less than C^, in the ratio of

approximately 0.94. These three differences are almost exactly what one would expect on the basis of wing theory, which will now be developed.

Figure 3.45 (taken from Ref. 3.26) presents the growth of over the years since the Wright Brothers’ success. The two points labeled К and L are somewhat misleading, since these two aircraft were experimental in nature and used distributed suction over the wing to delay separation. From this figure and the preceding information on flapped and plain airfoils, C, ^ of

slightly over 3 is probably the best that can be achieved without the addition of power. Although two-dimensional airfoils with double-slotted flaps can do better than this, as will be seen later, their full potential cannot be achieved when applied to an airplane. Generally, the flaps cannot be applied over the entire span of the wing. In addition to this loss in СУ1[[ц. an added penalty results from the fuselage and tail download required for trim.

‘ values considerably above those achievable with flap systems dis­

cussed so far are possible by the expenditure of power. Most of the powered – flap systems presently under consideration bear a resemblance, or can be

related in their performance, to the jet flap. Thus, in order to understand better the performance of systems such as upper surface blowing (USB), – externally blown flaps (EBF), augmentor wing, and circulation control, we will begin with the jet flap shown in Figure 3.46. A thin sheet of air exits the trailing edge at a downward angle of S relative to the airfoil zero lift line. This line is shown at an angle of attack a. If Гс is the total circulation around the airfoil then, assuming a and S to be small, the total lift on the airfoil will be

where itij is the mass flux in the jet and r, is the jet velocity.

As the jet leaves the airfoil, it gets turned in the direction of the free-stream velocity. In order to redirect the flux of jet momentum, it follows that a pressure difference must exist across the jet. This pressure difference, Др, can be related to mjVj and the radius of curvature R by the use of Figure 3.46b. Applying the momentum theorem in the direction of curvature,

Д pR6 = mjVj6 or

Figure 3.44b Effect of transition location on the lift and drag of a Liebeck airfoil. (R. H. Liebeck, “Class of Airfoils Designed for High Lift", AIAA Journal of Aircraft, 1973. Reprinted from the Journal of Aircraft by permission of the American Institute of Aeronautics and Astronautics.)

Since the jet exerts a force on the fluid, it can be replaced by an equivalent continuous vortex sheet that exerts the same force. Letting – y, be the strength per unit length of the sheet,

A pRO = pVyRO or

A p

VI = ф (3.54)

Measuring the position of the jet, y, positively downward, the radius of curvature and у for a nearly horizontal jet are related by

1 d2y

R dx2

Combining Equations 3.53, 3.54, and 3.55 gives

mjVj d2y pV ~d?

Equation 3.56 relates the jet vortex strength to the shape of the sheet and the jet momentum flux.

The total circulation of the jet vortex sheet can be obtained by integrating Equation 3.56 from x = 0 to °°.

Г,= Г у, d,

Jo

– _ mivi dy l” pV dx о

= ^(a + 5) (3.57)

Combining Equations 3.52 and 3.57 shows that the Kutta-Joukowski rela­tionship holds for the jet-flapped airfoil if the circulation is taken around both the airfoil and the jet.

L = PV( Гс + Гу) (3.58)

The boundary value problem is then posed where the airfoil and jet sheet are each replaced by an unknown vortex distribution. Distributions must then be found that will induce a velocity at each point on the airfoil and combining with the free-stream velocity to give a resultant velocity tangent to the airfoil. Along the sheet the following must hold.

w(x)_ dy V ~ dx

The details of the solution are beyond the scope of this text and can be found in Reference 3.19.

Although the solution of Reference 3.19 is not in closed form, the results

can be expressed in a relatively simple way. As with a physical flap, the increment in Ci because of a change in angle of attack and-flap angle can be expressed as a linear combination of the two angles.

Ci = Ci a + C,,8 (3.59)

where

^ – dC,

C|“ da

r _ dC,

С,°~І8

The derivatives Cta and Ct, are a function of the ratio of the jet momentum flux to the product of the free-stream dynamic pressure and a reference area. This ratio, known as the momentum coefficient, C„, is defined for a two-dimensional airfoil by

(3.60)

If, in addition, the jet exits ahead of the trailing edge and blows over and is deflected by a physical flap having a chord of cf (see the blown flap of Figure 3.24), then Ct, is also a function of cflc. For a pure jet flap (Q/C = 0), Cta and Q, are given by

Ci, = [4ТТ-СД1 + 0.151С/2 + 0.139См)]1/2 (3.61)

Cia = 2тг(1 + 0.151Q1/2 + 0.219Q) (3.62)

For c^c values other than zero, Q, is given in Figure 3.47. The curve ^labeled Cf/c = 1.0 in this figure corresponds to Equation 3.62 since, for this case, Ci, = Cia.

Data concerning C/max for jet-flapped airfoils is sparse. Generally, the jet flap follows the predictions of Figure 3.41 fairly closely, since the jet fixes the Kutta condition and provides some control over the boundary layer to prevent separation. As a preliminary estimate for Cu- Reference 3.3 recommends the use of the relationship presented in Figure 3.48. Here the difference in the angle of attack for stall, with and without blowing, is presented as a function of C„.

The negative pitching moment of the jet flaps is higher than the moment for conventional flaps for two reasons. First, the jet reaction acts at the trailing edge; second, the jet-flapped airfoil maintains lift all the way back to the trailing edge. As with the lift, CM can be written as a linear combination of a and S.

См = См„а + См,8

or

In this equation dC^dCi can be obtained from Figure 3.49.

To illustrate the use of the foregoing relationship for the jet-flapped airfoil, consider the prediction of for the NACA 63,4-421 airfoil (Figure 3.30) equipped with a 25% blown flap deflected 50°. The jet expands isen – tropically from a reservoir pressure of 25 psia and a temperature of 70 °F. The airfoil is operating at 50 mph at SSL conditions. It has a chord of 5 ft and the jet thickness is 0.2 in.

We begin by calculating the jet velocity from the compressible Bernoulli equation (Equation 2.31).

or, with the use of Equation 2.30,

(т-о/щ У2

Цу – l)po L W From the equation of state (Equation 2.1),

Po/po= RT0

Thus,

p(1.4)(1716)(529.7) ^ 14/7^ 0 286Jj1,2

= 946.9 fps

The mass density of the expanded jet will be

p0 is calculated from the equation of state.

25(144)

p0 1716(529.7)

= 0.00396 slugs/ft3

Thus,

Pi = 0.00271 slugs/ft3

The jet mass flux will be equal to

Щ = PAvi

= 0.00271(0.2/12)(946.9)

= 0.0428 slugs/s

The free-stream dynamic pressure is

q = I pV2

= 0.002378(50 x 1.467)2/2 = 6.397 psf

Thus,

(0.0428)(946.9)

6.397(5)

= 1.27

From Figures 3.47 and 3.48,

C, a = 9.09/rad C, s = 6.5/rad

<*171M — <*max(CM = 0) = — 2

From Figure 3.30, for the unblown airfoil without a flap is equal approximately to 1.4. Using Figures 3.33 and 3.34, AC, due to the plain flap (CM = 0) is estimated as

ДС, = 2тг(0.6)(.41)(^) = 1.35 so that, using Figure 3.34

~ (0.72) (1.35) = 0.97

Thus, for Q, = 0 with the flap deflected, C(max is estimated to be 2.37. At a = 0, Q is estimated to be 1.5. Thus, with Cla = 0.109 (from Figure 3.30),

or, relative to the zero lift line, flaps up,

«ШИ = 9.7°

For the operating CM, the angle of attack for the zero lift line at stall is – estimated to equal 7.1°.

Thus,

CU = Cl«amax + C|, S

= 6.8

The preceding answer must, of course, be further corrected, using Figure 3.43 and Equation 3.46, to account for trimming tail loads. Also, it should be emphasized that the preceding is, at best, an estimate for preliminary design purposes or relative parametric design studies. In the final analysis, model and prototype component testing of the blowing system must be performed.

Credit for the practical application of the jet flap must be given to John Attinello. Prior to 1951, all blown systems utilized pressure ratios less than critical in order to avoid supersonic flow in the jet. Such systems required large and heavy ducting. For his honors thesis at Lafayette College, Attinello demonstrated with “homemade” equipment that a supersonic jet would adhere to a deflected flap. This was contrary to the thinking of the day that not only would a supersonic jet separate from a blown flap, but the losses associated with the shock wave system downstream of the nozzle would be prohibitive. Later, more sophisticated testing performed by the David Taylor Model Basin confirmed Attinello’s predictions (Ref. 3.38) of high lift coefficients for supersonic jet flaps. This led to the development of compact, lightweight systems using bleed air from the turbojet engine compressor section. The Attinello flap system was flight tested on an F9F-4 and produced a significant decrease in the stalling speed for an added weight of only 50 lb. Following this success, the Attinello flap went into production on the F-109, F-4, F8K, A5, and other aircraft, including several foreign models.

Stratford, in References 3.23 and 3.24, examined both theoretically and experimentally the possibility of diffusing a turbulent boundary layer in such a way as to produce zero wall shear. Known as “imminent separation pressure recovery,” Stratford found that it is indeed possible, with the proper pressure gradient, to maintain a velocity profile along a diffuser such that du(y)ldy is equal to zero at the wall. н(у) is the velocity in the boundary layer parallel to the wall and is a function of the distance, y, from the wall. With the velocity gradient at the wall equal to zero, the boundary layer is just on the verge of separating, since a negative value of this gradient will result in reverse flow, as illustrated in Figure 3.41.

In the abstract to Reference 3.24, Stratford states:

“No fundamental difficulty was encountered in establishing the flow and it had, moreover, a good margin of stability. The dynamic head in the zero skin friction boundary layer was found to be linear at the wall (i. e., и °c ym), as predicted theoretically in the previous paper. (Author’s note, Stratford is referring to Ref. 3.23.)

The flow appears to achieve any specified pressure rise in the shortest possible distance and with probably the least possible dissipation of energy for a given initial boundary layer. Thus, an airfoil which could utilize it immediately after transition from laminar flow would be expected to have a very low drag."

The Stratford imminent separation pressure recovery was adopted for airfoils by Liebeck and Ormsbee (Ref. 3.25) and was extended later by Liebeck (Ref. 3.22). Using variational calculus, optimum chordwise pressure distributions for the upper and lower surfaces are prescribed that are modified slightly by additional constraints not present in the optimization process. Specifically, the optimum C„ distributions are modified in order to (1) close the airfoil contour at the trailing edge, (2) round the leading edge to allow operation over an angle-of-attack range, and (3) satisfy the Kutta cohdition at the trailing edge.

The resulting modified form of the optimum pressure distribution is compared with the optimum distribution in Figure 3.42. Beginning at the stagnation point, the flow is accelerated up to a so-called rooftop region over which the velocity, and hence the pressure, is constant. Following the rooftop region, the Stratford pressure recovery distribution is employed to reduce the velocity over the upper surface to its value at the trailing edge.

One such airfoil design is presented in Figure 3.43 (taken from Ref. 3.22). Included on the figure is the pressure distribution to which the airfoil was designed. Test data on this airfoil obtained at a Reynolds number of 3 x 106 are presented in Figure 3.44a and 3.44b. Although this configuration is referred to by the reference as a-“turbulent rooftop” case, transition does not occur until the start of the Stratford pressure recovery. In this case the performance of the airfoil is seen to be good from the standpoint of Cimax and Cd. The drag coefficient remains below the value of 0.01 over a wide range of Ci values from 0.6 to 1.6.

Artificially producing transition near the leading edge severely com­promises and Cd, as shown in Figure 3.441». Still, by comparison with the standard NACA airfoils, the Liebeck airfoil appears to offer superior per­formance at low speeds and, in the future, may find application to general aviation aircraft. One possible drawback in this regard is the sharp drop in its lift curve at stall.

In order to avoid leading edge separation, particularly at low Reynolds numbers or for airfoils with relatively sharp leading edges, special high-lift devices can also be incorporated into the leading edge to supplement the benefits of trailing edge flaps. These are illustrated in Figure 3.35. The fixed slot and extensible slat have been in use for some time, whereas the Kruger-type nose flap was first employed on the turbojet transport.

As the name implies, the fixed slot is just that—a narrow channel through which the air can flow from the lower surface to the upper surface. This channeling of the flow allows the airfoil to operate at higher angles of attack before the upper surface of the leading edge separates than otherwise would be the case. Increments in C(inax of approximately 0.1 or 0.2 are achieved by the fixed slot. It is a moot question as to why this delay in the separation occurs. As in the case of slots with trailing edge flaps, the explanation has been offered in the past that the flow through the slot feeds energy into the slower moving boundary layer, thereby decreasing its tendency to separate. More recently, however, in a Wright Brothers’ Lecture (Ref. 3.16) Smith, in examining numerical results on multielement airfoils, concluded that im­proved stall performance from slots is most likely the result of more favor­able pressure gradients being produced on one airfoil element by the other.

The extensible slat is similar in its performance to the slot, but it is

considerably more efficient because it can be positioned to optimize its contribution to C|max. The mechanically extended slat is finding increased application, particularly with the use of thinner airfoil sections for high-speed applications. Figure 3.36 presents some data on slats taken from Reference

3.17. Here a NACA 64A010 airfoil was tested using a slat in combination with split and double-slotted trailing edge flaps. The slat is seen to improve C(max significantly, producing increments in C;max of approximately 0.9, 0.8, and 0.6 for the no-flap, split-flap, and double-slotted flap configurations, respectively. Unlike the trailing edge flap, the primary effect of the slat is seen to be an extension of the lift curve without the slat; that is, opening the slat does not change Ci by a large increment at a fixed angle of attack. The same is true of leading edge flaps and is not unexpected in view of Figure 3.32.

The performance of a leading edge flap is presented in Figure 3.37 for the same airfoil as for Figure 3.36. Comparing the two figures, it is obvious that the two leading edge devices are nearly comparable in performance.

Figure 3.38 shows a section of a sophisticated Kruger-type flap. As this flap swings down and forward, it assumes the curved shape that is shown. With this optimum shaping, its performance probably exceeds to some extent

the data presented in Figures 3.39 and 3.40. Figure 3.39 (taken from Ref. 3.18) shows that without a trailing edge flap the Kruger flap gives an increment in Cw to the 64-012 airfoil of only 0.4. However, the plain airfoil has a higher Cw to begin with than that of Figures 3.36 and 3.37. Hence, the total C(max for the Kruger-flapped airfoil without a trailing edge flap is about the same as for the other two leading edge devices. However, with the split flap, the Kruger flap produces a combined Clmax equal to 3.0, which is 0.3 to 0.4 higher than the corresponding data of Figures 3.36 and 3.37.

The data of Figure 3.40 (taken from Ref. 3.21) are based on Kruger’s original work.

An examination of all of the airfoil data presented in Reference 3.1 discloses that the greatest value of Cimax one can expect at a high Reynolds number from an ordinary airfoil is around 1.8. This maximum value is achieved by the NACA 23012 airfoil. Another 12% thick airfoil, the 2412, delivers the second highest value, 1.7.

In order to achieve higher Cimax values for takeoff and landing without unduly penalizing an airplane’s cruising performance, one resorts to the use of mechanical devices to alter temporarily the geometry of the airfoil. These devices, known as flaps, exist in many different configurations, the most common of which are illustrated in Figure 3.24. In addition to the purely mechanical flaps, this figure depicts flaps that can be formed by sheets of air exiting at the trailing edge. These “jet flaps” can produce C(max values in excess of those from mechanical flaps, provided sufficient energy and momentum are contained in the jet. Frequently one uses the terms “powered” and “unpowered” to distinguish between jet and mechanical flaps.

The effect of a mechanical flap can be seen by referring once again to Figure 3.6. Deflecting the flap, in this case a split flap, is seen to shift the lift curve upward without changing the slope. This is as one might expect from Equation 3.26, since deflecting the flap is comparable to adding camber to the airfoil.

Some flap configurations appear to be significantly better than others simply because, when deflected, they extend the original chord on which the lift coefficient is based. One can determine if the flap is extensible, such as the Fowler or Zap flaps in Figure 3.24, by noting whether or not the slope of the lift curve is increased with the flap deflected. Consider a flap that, when deflected, extends the original chord by the fraction x. The physical lift curve would have a slope given by

~ = ^PV2( + x)cCla (3.43)

since (l + x)c is the actual chord. Qa does not depend significantly on thickness or camber; hence, the lift curve slope of the flapped airfoil based on the unflapped chord, c, would be

Cia (flapped) = (1 + jc)Q_ (unflapped)

Now the maximum lift, expressed in terms of the extended chord and C;maXe

■Lmax = 2 РУ2(1 + *)cCLxe

based on the original chord becomes

C, =(1 + x)C, ~~

•max v ‘ ‘max.

Figure 3.25 Performance of plain flaps, (a) Variation of maximum section lift coefficient with flap deflection for several airfoil sections equipped with plain flaps, (b) Variation of optimum increment of maximum section lift coefficient with flap chord ratio for several airfoil sections equipped with plain flaps, (c) Effect of gap seal on maximum lift coefficient of a rectangular Clark-Y wing equipped with a full-span 0.20c plain flap. A = 6, Я = 0.6 x Ю4.

Figures 3.25 to 3.30 and Tables 3.1 and 3.2 present section data on plain, split, and slotted flaps as reported in Reference 3.15. With these data one should be able to estimate reasonably accurately the characteristics of an airfoil section equipped with flaps.

A study of this data suggests the following:

Plain Flaps

1. The optimum flap chord ratio is approximately 0.25.

2. The optimum flap angle is approximately 60°.

3. Leakage through gap at flap nose can decrease C(max by approximately 0.4.

4. The maximum achievable increment in C|max is approximately 0.9.

Split Flaps

1. The optimum flap chord ratio is approximately 0.3 for 12% thick airfoils, increasing to 0.4 or higher for thicker airfoils.

2. The optimum flap angle is approximately 70°.

3. The maximum achievable increment in С/тал is approximately 0.9.

4. CL, increases nearly linearly with log R for 0.7 x 106 < R < 6 x 106.

5. The optimum thickness ratio is approximately 18%.

Slotted Flaps

1. The optimum flap chord ratio is approximately 0.3.

2. The optimum flap angle is approximately 40° for single slots and 70° for double-slotted flaps.

3. The optimum thickness ratio is approximately 16%.

4- Cw is sensitive to flap (and vane) position.

5. The maximum achievable increment in C(max is approximately 1.5 for single slots and 1.9 for double slotted flaps.

Figure 3.28 Contours of flap and vane positions for maximum section lift coefficient for several airfoil sections equipped with double-slotted flaps, (a) NACA 23012 airfoil section; 5, = 60°. (b) NACA 23021 airfoil section; Sf = 60°. (c) NACA 611-212 airfoil section.

Referring to Equation 3.44 and Figure 3.30, it is obvious that some of the superior performance of the double-slotted flap results from the extension of the chord. From the figure, C(a (flapped) is equal to 0.12 C(/deg as compared to the expected unflapped value of approximately 0.1. Hence, based on the actual chord, the increment in for the double-slotted flap is only 1.6. However, this is still almost twice that afforded by plain or split flaps and points to the beneficial effect of the slot in delaying separation.

Figure 3.31 (taken from Ref. 3.15) presents pitching moment data for flapped airfoil sections. The lift and moment are taken to act at the aerody­namic center of the airfoil, located approximately 25% of the chord back from the leading edge. The moment is positive if it tends to increase the angle of attack.

From Figure 3.31, the lowering of a flap results in an incremental pitching moment. In order to trim the airplane a download must /be produced on the horizontal tail. The wing must now support this download in addition to the aircraft’s weight. Hence the effective increment in lift due to the flap is less than that which the wing-flap combination produces alone. This correction can typically reduce by 0.1 to 0.3.

oment coefficient, (a)

In a high-wing airplane, lowering the flaps can cause the nose to pitch up. This is due to the moment produced about the center of gravity from the increase in wing drag because of the flaps. Based on the wing area, the increment in wing drag coefficient, Д Co, due to the flaps is given ap­proximately by,

ДCD = 1.7(cflc)l3S(SflS) sin2S/ (plain and split) (3.45)

= 0.9(cflc)’ x(SfIS) sin2Sf (slotted) (3.46)

If the wing is located a height of h above the center of gravity, a balancing upload is required on the tail. The effect of trim on for a complete airplane will be discussed in more detail later.

Flap Effectiveness in the Linear Range

Frequently one needs to estimate the increment in C, below stall, ДО, produced by a flap deflection. Not only is this needed in connection with the

wing lift, but A Ci is required also in analyzing the effectiveness of movable control surfaces, which frequently resemble plain flaps.

If an airfoil section has a lift curve slope of Cla and lowering its flap produces an increment of Д Ci, the angle of zero lift, a0i, is decreased by

(3.47)

The rate of decrease of a0/ per unit increase in the flap angle Sf is referred to as the flap effectiveness factor, r. Thus, for a flapped airfoil, the lift coefficient can be written as

Ct – C(„(a + rSf)

where/ a is the angle of attack of the airfoil’s zero lift line with the flap undeflected.

Theoretically т is a constant for a given flap geometry but, unfortunately, flap behavior with Sf is rather nonlinear and hence т must be empirically corrected by a factor 17 to account for the effects of viscosity. Including 17, Equation 3.48 becomes,

Ci = Ci„(a + T17 Sf)

The functions t and 17 can be obtained from Figures 3.32 and 3.33. Figure 3.33 is empirical and is based on data from References 3.15, 3.17, 3.19, and 3.20. Although there is some scatter in the data, as faired, the comparisons between the various types of flaps are consistent. The double-slotted flap delays separation on the upper surface, so that the decrease in flap effective­ness occurs at higher flap angles than for the other flap types. The same can be said of the slotted flap relative to the plain and split flaps. The plain flap is fairly good out to about 20° and then apparently the flow separates from the upper surface and the effectiveness drops rapidly, approaching the curve for split flaps at the higher flap angles. In A sense the flow is always separated on the upper surface of a split flap. Thus, even for small flap angles, the effective angular movement of the mean camber line at the trailing edge of an airfoil with a split flap would only be about half of the flap displacement.

In the case of the double-slotted flap it should be emphasized that this curve in Figure 3.33 is for an optimum flap geometry. The trailing segment of the flap is referred to as the main flap and the leading segment is called the vane. In applying Equation 3.49 and Figures 3.32 and^>.33 to the double – slotted flap, the total flap chord should be used together’with the flap angle of the main flap. Usually, the deflection angle of the vane’is less than that for the main flap for maximum lift performance.

Figure 3.32 is based on the thin airfoil theory^represented by Equation 3.39. As an exercise, derive the expression for т given on the figure, r can also be obtained using the numerical methods that led to Figures 3.17 and

3.18. As another exercise, apply Weissinger’s approximation to the flapped airfoil using only two point vortices to represent the airfoil. Placing one vortex on the quarter chord of the flap and the other on the quarter chord of the remainder of the airfoil leads to

3(3-2 Oc, 4(l-c/)c/ + 3

where Cf is the fraction of chord that is flapped. Equation 3.50 is ap­proximately 10% lower than Figure 3.32 for cflc values of around 0.25.

The angle of attack at which the flapped airfoil stalls is generally less than that for the plain airfoil. Hence, the increment in because of the flap is not as great as the increment in Q at an angle below the stall. Denoting these increments by ДС^ and ДС/, respectively, it is obvious that the ratio ДС^/ДС, must depend on cflc. If cflc, for example, is equal to 1.0, in a sense the entire airfoil is the flap and Дmust be zero. Systematic data on ДС^/ДС/ are sparse. Figure 3.34 has been drawn based on a limited number of data points and should be used with discretion.

As an example, in using Figures 3.32, 3.33, and 3.34, consider the prediction of Clni„ for a 23012 airfoil equipped with a 30% chord split flap deflected 60° and operating at a Reynold’s number of 3.5 x Ю6. From Figure 3.32, t = 0.66 for Cflc = 0.3 and from Figure 3.33, tj = 0.35 for a split flap deflected 60°. Hence from Equation 3.49, ДQ is equal to

Д C, = Qtij S

= (0.105)(0.66)(0.35)(60) (3.51)

= 1.46

In Equation 3.51, C(a of 0.105 is obtained from Reference 3.1. Using Figure 3.34, the ratio of ДC/max to Д Ct is obtained as 0.66. Hence,

ДС/_ = 0.96

According to Figure 3.27, C/max for a plain 23012 airfoil equals 1.65 at R = 3.5 x 106. Thus, for the flapped airfoil, C|max is predicted to be 1.65 + 0.96, or 2.61. This result compares closely with Figure 3.26a. If the procedure is repeated for other flap angles, close agreement is also obtained with the figure. However for a flap chord ratio of 0.1, the predicted values of C(max based on Figures 3.32 to 3.34 are higher than those shown in Figure 3.26a.

Airfoil theory based on potential flow methods predicts the lift of an airfoil in its linear range but does not provide any information concerning maximum lift capability. As discussed previously, Cimiu, is determined by flow separation, which is a “real fluid” effect. Separation is difficult to predict analytically, so the following material on Qmax is mainly empirical.

Typically, conventional airfoils without any special high-lift devices will deliver C|mai values of approximately 1.3 to 1.7, depending on Reynolds number, camber, and thickness distribution. The appreciable dependence of Cimiu on Я shown in Figure 3.12 for the GA(W)-1 airfoil is typical of other airfoils. Figure 3.23 presents Qmia as a function of R and thickness ratio for NACA four-digit airfoils having a maximum camber of 2%, located 40% of the chord back from the leading edge. At intermediate thickness ratios of around 0.12, the variation of C(m„ with R parallels that of the 17% thick GA(W)-1 airfoil. Note, at least for this camber function, that a thickness ratio of 12% is about optimum. This figure is taken from Reference 3.14. This same reference presents the following empirical formula for C/max for NACA four – digit airfoils at an R of 8 x 106.

„ „ (0.123 + 0.022P – 0.5z – tf

-2.6 рз

t, z, and p are thickness, maximum camber, and position of maximum camber, respectively, expressed as a fraction of the chord. For example, for a 2415 airfoil,

t = 0.15 z = 0.02 p =0.40

so that according to Equation 3.42,

For a plain wing (unflapped), there is little effect of aspect ratio or taper ratio on Cw Even the presence of a fuselage does not seem to have much effect. As the angle of attack of. a wing increases, is reached and any further increase in a will result in a loss of lift. Beyond the wing is said to be stalled. Although taper ratio does not significantly affect the overall wing Cw it (and wing twist) significantly affect what portion of the wing stalls first. As the taper ratio is decreased, the spanwise position of initial stall moves progressively outboard. This tendency is undesirable and can be compensated for by “washing out” (negative twist) the tips. One usually wants a wing to stall inboard initially for two reasons. First, with inboard stall, the turbulence shed from the stalled region can shake the tail, providing a built-in stall warning device. Second, the outboard region being unstalled will still provide

aileron roll control even though the wing has begun to stall. The lift charac­teristics of three-dimensional wings will be treated in more detail later.

In Chapter Two it was noted that the concepts of a point vortex and a point source could be extended to a continuous distribution of the elementary flow functions. In that chapter a distribution of sources in a uniform flow was found to produce a nonlifting body of finite thickness. In the case of the circular cylinder, the addition of a vortex also produced lift.

Comparable to the continuous distribution of sources pictured in Figure 2.20, consider a similar distribution of vortices as illustrated in Figure 3.13. Such a distribution is referred to as a vortex sheet. If у is the strength per unit length of the sheet, у Ax will be the total strength enclosed by the dashed contour shown in the figure. The contour is taken to lie just above and below the sheet. Ax is sufficiently small so that the velocity tangent to the sheet, v, can be assumed to be constant. Because of the symmetry to the flow provided by any one segment of the sheet, the tangential velocity just below the sheet is equal in magnitude but opposite in direction to that just above the sheet. From Equation 2.55, relating circulation to the strength of a vortex, it follows that

у Ax = 2v Ax or

Note the similarity of this relationship to that expressed by Equation 2.82. However, in the case of Equation 3.14, the velocity is tangent to the vortex sheet whereas, for Equation 2.82, the velocity is normal to the line on which the sources lie.

Consider now the thin airfoil pictured in Figure 3.14. If the airfoil is producing a lift, the pressure on the lower surface is greater than that on the upper. Hence, from Bernoulli’s equation, the velocity on the upper surface is greater than the velocity on

).

velocity across the airfoil equal 2v, the upper and lower velocities can be written as

Vu = V+v and

V, = V-v

Thus the flow field around the airfoil is the same as that which would be produced by placing, in a uniform flow of velocity V, a vortex sheet of unit strength 2v along the airfoil.

The contribution to the lift of a differential length of the airfoil will be

dl=(pi – pu) dx

Or, using Bernoulli’s equation, this becomes,

dl = pV(2v) dx

Since 2v is the unit vortex strength, the Kutta-Joukowski law (Equation 2.81) is found to hold for the airfoil element.

dl = pVy dx

or, integrating the above equation over the entire chord,

l = pVT (3.15)

where Г is the total circulation around the airfoil given by

In order to predict the lift and moment on the airfoil, one must find the chordwise distribution of y(x) that will produce a resultant flow everywhere tangent to the mean camber line (thin airfoil approximation). In addition, the Kutta condition is applied at the trailing edge to assure that the flow leaves the trailing edge tangent to the mean camber line at that point. This is a

necessary condition; otherwise, the resulting flow will appear similar to Figure 3.4a with the lift being equal to zero.

An analytical solution to the thin airfoil will be obtained later but, first, let us consider a numerical approach to predicting the lift and moment of an airfoil.

As a gross approximation to the distributed vorticity along the airfoil, the distribution will be replaced by only one vortex of unknown strength, Г. However, Г will be placed at a particular point on the airfoil, at the quarter- chord point. The boundary condition and the Kutta condition will be satisfied at only one point, the three-quarter-chord point. This approximation, known as Weissinger’s approximation, is illustrated in Figure 3.15 for a flat-plate airfoil.

The velocity induced at 3c/4 by Г placed at cl4 will be

Г

Vj = —

TTC

Assuming a to be a small angle, it follows that

Vi = Via or

Г = ire Vat

m■

From the Kutta-Joukowski relationship, L = рУГ, so that

L = prrcV2a (3.19)

Expressing lift in terms of the lift coefficient and using Equation 3.19 leads to

Ci = lira (3.20)

where a is the angle of attack in radians.

The expression agrees identically with the theoretical solution of this problem that follows. Notice that the result predicts the slope of the lift curve, dCJda, to be 2ir/rad. Experimentally this figure is usually found to be

Flgun 3.15 Weissinger’s approximation to a thin airfoil.

somewhat less. Figures 3.5 and 3.6, for example, show a value of around

0. 105/deg or 6.02/rad.

The approximation of Figure 3.15 can be improved on by dividing the airfoil chord into a number of equal segments and placing a vortex of unknown strength at the quarter-chord point of each segment. The unknown strengths are determined by assuring that the normal velocity vanishes at the three-quarter-chord point of each segment. With the last control point down­stream of the last vortex singularity, the Kutta condition is assured.

To illustrate this numerical solution of the thin airfoil, consider Figure

3.16. Here, a circular arc airfoil having a unit chord length with a maximum camber ratio of z is operating at an angle of attack a.

If it is assumed that

z * 1 (3.21)

the radius of curvature, R, of the airfoil will be related to z approximately by

The slope of the camber line relative to the chord line (the angle ф in Figure 3.16) at any distance x can be determined from the geometry of the figure.

0 = (4-8x)z (3.22)

figure 3.16 A circular arc airfoil approximated by two vortices.

The component of V normal to the mean camber line and directed upward is thus

vn = V(a – ф)

It follows that at control points 1 and 2 located at x values of 3c/8 and 7cIS, respectively, the two vortices similating the airfoil must induce velocities downward given by

Vi = V(a – z) at 1 (3.23a)

Vi = V(a + 3z) at 2

The problem is linearized by Equation 3.21 so that the vortices, – yi and y2, are taken to lie on the chord line. Thus, according to Equation 2.56, the total velocities induced at the two control points by the two vortices will be

Applying the Kutta-Joukowski law to each vortex results not only in a predicted total lift, but also in a moment. In coefficient form the lift and moment (about the leading edge) become

(3.26a)

(3.26b)

The moment coefficient about the leading edge can be transferred to the quarter-chord point by using

С = c

C"»LE

Thus,

This simple, two-point model results in several important observations that are in agreement with more exact solutions. First, note that Equation 3.26a shows the lift coefficient to be a linear combination of a and z. Thus,
cambering an airfoil will not change the slope of the lift curve. Second, it is predicted that the moment about the quarter chord will be independent of a. Hence, this point is predicted to be the aerodynamic center.

As one divides the airfoil into a greater and greater number of elements, the resulting у distributions will approach the theoretical pressure distribution predicted on the basis of continuous у distributions. The strength, y, of a vortex placed at the c/4 point of an element of length Ax will be related to the pressure jump, Дp, across the element by

(3.28)

Figure 3.17 presents a comparison, for the flat-plate airfoil, between the pressure distribution obtained using the foregoing numerical procedure with that based on a continuous distribution of у along the chord. It is seen that the numerical results rapidly converge to the continuous solution as the number of elements increases. In preparing this figure it should be noted that Дp, given by Equation 3.28, has been expressed in coefficient form and plotted at the location of each point vortex. Figure 3.18 presents a similar comparison for the circular arc airfoil. In this case a is taken to be zero, avoiding the infinitely negative Cp at the leading edge.

0.7 i—

The numerical model predicts the lift in exact agreement with more precise analytical models. However, the moment coefficient, given by Equa­tion 3.27, is only three-quarters of that obtained by analytical means. Figure 3.19 shows that the exact value is approached rapidly, however, as the number of segments increases.

As indicated by Figure 3.19, the exact value of the moment coefficient about the aerodynamic center (c/4) for the circular arc airfoil is given by

Стар 77" Z

Using Equation 3.13, the location of the center of pressure can be found as

Observe that as Q decreases, the center of pressure moves aft, ap­proaching infinity as C| goes to zero. This movement of the center of pressure is opposite to what was believed to be true by the early pioneers in aviation. The Wright Brothers were probably the first to recognize the true nature of the center-of-pressure movement as a result of their meticulous wind tunnel tests.

Analytical solutions to the thin airfoil can be found in several texts (e. g., Ref. 3.2 and 3.3). Here, the airfoil is replaced by a continuous distribution of vortices instead of discrete point vortices, as used with the numerical solu­tion.

Referring to Figure 3.20, without any loss of generality, the airfoil is taken to have a unit chord lying along the x-axis with the origin at the leading

edge. The shape of the camber line is given by z(x), and it is assumed that

z(x)«1

With this assumption the problem is linearized and made tractable by replac­ing the airfoil with a vortex sheet of unit strength y(x) lying along the chord line instead of along the camber line.

At the point Xo, the downward velocity induced by an elemental vortex of strength y(x) dx located at x, according to Equation 2.56, will be given by

y(s) dx 2tt(Xq – x)

or, integrating over the chord,

In order to satisfy the boundary condition that the flow be tangent everywhere to the mean camber line, it follows that, to a small angle approximation,

(3.32)

Thus, given a and z(x), the following integral equation must be solved for y(x).

(3.33)

In addition, y(x) must vanish at the trailing edge in order to satisfy the Kutta condition. Otherwise, the induced velocity will be infinite just down­stream of this point.

Equation 3.33 is solved by first transforming to polar coordinates.

Letting

x=|(l-cos0) (3.34)

Equation 3.33 becomes

1 (w_JiWde_ = a_(dz 2ttV Jo cos в – cos во dx4)

On the basis of the more sophisticated method of conformal mapping (e. g., see Ref. 3.4), it is known that y(x) is generally singular at the leading edge approaching infinity as 1/x. Thus we will assume a priori that Equation 3.35 can be satisfied by а у(в) distribution of the form

Г „ (1 + COS0) . V’ „ • J

lA°–sine +2*sm»eJ

Using the relationships

| j^cos (n – 1)0 – cos (n + 1)0 J = sin n0 sin 0 and

f" cos nOde _ sin n0о Jo COS 0 – COS 00 sin 00

Equation 3.35 becomes

A0 – 2 A„ cos nO = at

n = l

Multiplying both sides of the preceding equation by cos m0 (m =0, 1,2,…, n,…) and integrating from 0 to тт leads to

A‘-a~H’£de <3-38a)

A„ = ^J ^cosn0d0 (3.38b)

Thus, knowing the shape of the mean camber line, the coefficients A0, Au A2,… can be determined either in closed form or by graphical or numerical means (see Ref. 3.1). Having these coefficients, Q and Cm can then be easily determined from the Kutta-Joukowski relationship.

The lift and moment about the leading edge are given by

L = f pVy(x) dx Jo

MLE= – CPVy(x)xdx Jo

From these and using Equation 3.36,

Q = 2-nAo + itAi (3.39)

C"LE=-f (ao + A,-^) (3.40)

It follows that Cm about the quarter-chord point is independent of a, so that this point is the aerodynamic center, with the moment coefficient being given by

Cmac = – f(A.-A2) (3.41)

Since a is contained only in the A0 coefficient, it can be concluded immediately without considering the actual form of z(x) that Q is given by a

linear combination of a and a function of z. Thus, camber changes can be expected to affect the angle of zero lift but not the slope of the lift curve.

Reference to airfoil data, such as that presented in Figures 3.5 and 3.6, will show that the predictions of thin airfoil theory are essentially correct. There is a range of angles of attack over which the lift coefficient varies linearly with a. The slope of this lift curve is usually not as high as the theory predicts, being approximately 4 to 8% less than the theoretical value. For many purposes an assumed value of 0.1 C(/deg is sufficiently accurate and is a useful number to remember. Experimental data also show the aerodynamic center to be close to the quarter-chord point. The effects of camber on Q, dQlda, and Cmac are also predicted well.

Recently large numerical programs have been developed to predict the performance of airfoils that incorporate Reynolds number and Mach number effects.’These are typified by Reference 3.5, which will be described briefly. This program begins by calculating the potential flow around the airfoil. In order to allow for both finite thickness and circulation, the airfoil contour is approximated by a closed polygon, as shown in Figure 3.21. A continuous distribution of vortices is then placed on each side of the polygon, with the vortex strength per unit length, y, varying linearly from one corner to the next and continuous across the corner. Figure 3.22 illustrates this model for two sides connecting corners 3, 4, and 5. Control points are chosen midway between the corners. The values of the vortex unit strengths at the corners are then found that will induce velocities at each control point tangent to the polygon side at that point. Note, however, that if there are n corners and hence n + 1 unknown у values at the corners, the n control points provide one less equation than unknowns. This situation is remedied by applying the Kutta condition at the trailing edge. This requires that yn+1 = – yb assuring that the velocities induced at the trailing edge are finite.

Flaving determined the vortex strengths, the velocity field and, hence, the

Уз

4

Figure 3.22 Vortex distributions representing airfoil contour.

pressure distribution around the airfoil can be calculated. This result is then used to calculate the boundary layer development over the airfoil, including the growth of the laminar layer, transition, the growth of the turbulent layer, and possible boundary layer separation. The airfoil shape is then enlarged slightly to allow for the boundary layer thickness and the potential flow solutions are repeated. The details of this iterative procedure are beyond the scope of this text.

Systematic series of airfoils have*given way, at least in part, to speci­alized airfoils designed to satisfy particular requirements. These airfoils are synthesized with the use of sophisticated computer programs such as the one described in Reference 3.5, which will be discussed in more detail later. One such special purpose airfoil is the so-called supercritical airfoil reported on in References 3.6 and 3.7. This airfoil has a well-rounded leading edge and is relatively flat on top with a drooped trailing edge. For a constant thickness of 12%, wind tunnel studies indicate a possible increase of approximately 15% in the drag-divergence Mach number for a supercritical airfoil as compared to a more conventional 6-series airfoil. In addition, the well-rounded leading edge provides an improvement in CJ^ at low speeds over the 6-series, which has sharper leading edges.

A qualitative explanation for the superior performance of the super­critical airfoil is found by reference to Figure 3.9. At a free-stream Mach number as low as 0.7 or so depending on the shape and CJ, a conventional airfoil will accelerate the flow to velocities that are locally supersonic over the forward or middle portion of its upper surface. The flow then decelerates rapidly through a relatively strong shock wave to subsonic conditions. This compression wave, with its steep positive pressure gradient,/causes the boundary layer to thicken and, depending on the strength of the shock, to separate. This, in turn, causes a significant increase in the drag. The minimum value of the free-stream Mach number for which the local flow becomes

supersonic is referred to as the critical Mach number. As this value is exceeded by a few hundredths, the shock wave strengthens sufficiently to cause the drag to rise suddenly. This free-stream Mach number is known as the drag-divergence Mach number.

The supercritical airfoil also accelerates the flow to locally supersonic conditions at free-stream Mach numbers comparable to the 1- or 6-series airfoils. However, the supercritical airfoil is shaped, so that around its design lift coefficient, the flow decelerates to subsonic conditions through a dis­tribution of weak compression waves instead of one strong one. In this way the drag-divergence Mach number is increased substantially.

Although the possibility of such airfoils was known for some time, their successful development in modern times is attributed to R. – T. Whitcomb. A Whitcomb-type supercritical airfoil is pictured in Figure 3.7.

Tested at low speeds, the supercritical airfoils were found to have good values as well as low Cd values at moderate lift coefficients. As a result, another family of airfoils evolved from the supercritical airfoils, but for low-speed applications. These are the “general aviation” airfoils, designated GA(W) for general aviation (Whitcomb). The GA(W)-1 airfoil is the last of the

airfoils pictured in Figure 3.7. Test results for this airfoil are reported in Reference 3.8, where its CtmtX values are shown to be about 30% higher than those for the older NACA 65-series airfoils. In addition, above Q values of around 0.6, its drag is lower than the older laminar flow series with standard roughness. These data are presented in Figure 3.10 for the GA(W)-1 airfoil. Comparisons of Qmax and Cd for this airfoil with similar coefficients, for other airfoils are presented in Figures 3.11 and 3.12.

Observe that the performance of the GA(W)-1 airfoil is very Reynolds number-dependent, particularly Clm„, which increases rapidly with Reynolds number from 2 to 6 million. At the time of this writing, the GA(W) airfoil is beginning to be employed on production aircraft. The same is true of the supercritical airfoil. Indeed, the supercritical airfoil is being used on both the Boeing YC-14 and McDonnell-Douglas YC-15 prototypes currently being tested for the advanced medium STOL transport (AMST) competition. At the time of this writing, NASA is adopting a new nomenclature for the GA(W) airfoils. They will be designated by LS (low speed) or MS (medium speed) followed by four digits. For example, the GA(W)-1 airfoil becomes LS(1)-0417. The (1) designates a family. The 04 refers to a design lift coefficient of 0.4, and 17 is the тяхітпт

thickness in percent of chord. For more information on modern airfoils, consult Reference 3.36.