SP-367 Introduction to the Aerodynamics of Flight





Airfoils and Wings


[59] The airfoil section.- Figure 13 showed that by taking a slice out of an airplane wing and viewing it from the side, one has the shape of the airfoil called the airfoil cross section or more simply airfoil section. The question arises as to how this shape is determined.


The ultimate objective of an airfoil is to obtain the lift necessary to keep an airplane in the air. A flat plate at an angle of attack, for example, could be used to create the lift but the drag is excessive. Sir George Cayley and Otto Lilienthal in the 1800's demonstrated that curved surfaces produced more lift and less drag than flat surfaces. Figure 13 shows the airfoil section used by the Wright Brothers in their 1903 airplane.


In those early days of canvas and wood wings, few airfoil shapes evolved from theory. The usual procedure at that time was the "cut and try" method. Improvements came from experimentation. If the modification helped performance, it was adopted. Early tests showed, in addition to a curved surface, the desirability of a rounded leading edge and a sharp trailing edge.


The hit and miss methods of these early days were replaced by much better, systematic methods used at Gottingen, by the Royal Air Force, and finally by the National Advisory Committee for Aeronautics (NACA). The purpose here was to determine as much information as possible about "families" of airfoil shapes. During World War II, NACA investigations produced results that are still in use or influence the design of most of today's airplanes. The discussions that follow are based considerably on these NACA results.


The following six terms are essential in determining the shape of a typical airfoil:


(1) The leading edge
(2) The trailing edge
(3) The chord line
(4) The camber line (or mean line)
(5) The upper surface
(6) The lower surface

Figure 40 illustrates the step-by-step geometric construction of an airfoil section:

(1) the desired length of the airfoil section is determined by placing the leading and...



Figure 40 - Geometric construction of an airfoil

Figure 40.- Geometric construction of an airfoil.


....trailing edges their desired distance apart. The chord line is drawn connecting the two points together, (2) the amount of curvature is determined by the camber line. This curvature greatly aids an airfoil section's lifting abilities, (3) a thickness function is "wrapped" about the camber line, that is, one adds the same amount of thickness above and below the camber line; this thickness determines the upper and lower surfaces, (4) the last step shows the final result- a typical airfoil shape. It has a specific set of aerodynamic characteristics all its own which may be determined from wind-tunnel testing.


Figure 41 illustrates all the aforementioned terms for several differently shaped airfoil sections. Figure 42 illustrates an important aspect of the camber line (or mean line). If the camber line is the same as the chord line, one has a symmetric airfoil (the upper surface is a mirror image of the lower surface about the chord line). When the free-stream velocity of the oncoming airstream is alined along the chord line, no lift is produced. The angle of attack a [Greek letter alpha] is the angle between the chord line....



Figure 41 - Airfoil terminology

Figure 41.- Airfoil terminology.

Figure 42 - Airfoil camber line variations

Figure 42. Airfoil camber line variations.


[62]....and the free-stream velocity vector. It is zero in this case, that is, a = 0° [a = Greek letter alpha]. Thus, the angle of attack for zero lift is zero, or aL=0 = 0°

If the camber line lies above the chord line, then an asymmetrical airfoil section results. (Upper surface is not a mirror image of the lower surface.) When the free-stream velocity is alined along the chord line (a = 0°), a positive lift results. The chord line must be negatively inclined with respect to the free stream to obtain zero lift (that is, the angle of zero lift aL=0 is less than 0° . In a similar manner negative camber yields an asymmetrical airfoil where the angle of zero lift aL=0 is greater than 0°.


The two-dimensional wing.- A two-dimensional wing has no variation of aerodynamic characteristics in a spanwise direction. In figure 43(a) the airfoil section at station A is the same as at station B or anywhere along the span, and the wing is limitless in span. The point of this is to prevent air from flowing around the wing tips and causing three-dimensional effects (to be discussed later). One is trying to separate the airfoil's aerodynamic characteristics from the wing's three-dimensional effects.


Of course, no wing is infinite in length but a close simulation may be obtained by insuring that the model of the airfoil section, when placed in the wind tunnel for measurements, spans the wind tunnel from one wall to the other. (See fig. 43(b).) In this case (except for minor tunnel- wall effects that can be corrected for), the wing behaves two dimensionally, that is, there is no spanwise variation of the airfoil section aerodynamic characteristics. A later discussion will show the influence that limiting the wing span has on the aerodynamic characteristics.


Circulation about a two-dimensional wing.- The fluid flow about an airfoil may be viewed as consisting of two superimposed patterns -one is the free-stream motion of the fluid about the airfoil (see fig. 44(a)) and the other is a circulatory flow, or circulation, around the airfoil (see fig. 44(b)). These two flows coexist to give the total flow pattern. The question is, if the free-stream flow is prescribed, can the circulation, represented by q[Greek letter capital Gamma] be of any value? A physical condition provides the answer. The flow about the pointed trailing, edge cannot turn a sharp corner without the velocity becoming infinite. As this is not possible with a real fluid. the flow instead leaves the trailing edge tangentially and smoothly (fig. 44(c)). This is the Kutta condition and it sets the required value of q such that the rear stagnation point moves to the trailing edge. The Kutta- Joukowsky theorem relates the circulation to the section lift by


Kutta- Joukowsky theorem




Figure 43 - an illustration of the aerodynamic characteristics of an airfoil section's by using a two dimensional wing

(a) Two-dimensional (2D) wing.
(b) Testing for airfoil section's aerodynamic characteristics by using a two dimensional (2D) wing.
Figure 43. Two dimensional wing testing.


Figure 44 - chart graphicly illustrating the flow around a 2D wing

(a) Flow with no circulation.
(b) Circulatory flow only.
(c) Flow with circulation.
Figure 44. Circulation about 2D wing.



lift/unit span of two-dimensional wing.

         letter rho

free-stream air density. [Greek letter rho]


free-stream velocity.

Greek letter
         Capital Gamma

circulation strength. [Greek letter Capital Gamma]

[65] Thus, the circulation strength qis set by a necessary physical condition, and the lift l is uniquely determined. For a perfect fluid, the drag per unit length is zero. However, in a viscous fluid flow one must include a skin-friction drag and a pressure drag along with a resulting loss of lift Later the changes that occur when a finite wing is considered will be shown


The two-dimensional coefficients.- Figure 45(a) shows the resultant aerodynamic force acting on an airfoil. The point of intersection of the chord line and the line of action of this resultant force is the center of pressure. The resultant aerodynamic force may be resolved into lift and drag components as shown in figure 45(b). The lift, drag, and center of pressure are for the cambered airfoil shown to vary as the angle of attack is changed. No aerodynamic moments are present at the center of pressure because the line of action of the aerodynamic force passes through this point. If one has the airfoil mounted at some fixed point along the chord, for example, a quarter of a chord length behind the leading edge, the moment is not zero unless the resultant aerodynamic force is zero or the point corresponds to the center of pressure.


The moment about the quarter-chord point is generally a function of angle of attack. Figure 45(c) shows a system of reporting a lift, drag, and moment about the quarter-chord point- all are functions of angle of attack.


There is a point, the aerodynamic center, where the moment is independent of the angle of attack. Figure 45(d) shows the lift, drag, and moment about the aerodynamic center. This system of reporting is convenient for a number of aerodynamic calculations.


The data obtained by wind-tunnel testing of NACA families of airfoil sections ar two- dimensional data. Aerodynamic characteristics recorded include the lift coefficient cl , the drag coefficient cd, the moment coefficient about the quarter-chord point (cm)0.25c, and the moment coefficient about the aerodynamic center (cm)ac. These coefficients are obtained by measuring, in wind-tunnel tests, the forces and moments per unit length of the airfoil wing and nondimensionalizing as follows:

cl = l/qc


where l is the measured lift per unit length of the airfoil wing, q is the testing dynamic pressure or 1/2 pV2 , and c is the chord length of the airfoil section. Similarly,

cd= d/qc


where d is the measured drag per unit length of the airfoil wing...




Figure 45 - chart illustrating the effect on the center of pressure as air foil angle is adjusted

Figure 45.- Airfoil aerodynamic characteristics.




cm = m/qc2



where m is the measured moment per unit length acting on the airfoil (whether at the quarter-chord point or the aerodynamic center or any other point desired).


It has previously been shown that the aerodynamic coefficients are dependent on body shape (airfoil section chosen), attitude (angle of attack a [Greek letter alpha]), Reynolds number, Mach number, surface roughness, and air turbulence. For low subsonic flow, Mach number effects are negligible and air turbulence is dependent on the Reynolds number and surface roughness and need not be indicated as a separate dependency. Figure 46 shows data reported for a particular airfoil shape, namely, an NACA 2415 airfoil. The main point of this figure is to show the dependence of the aerodynamic coefficients on angle of attack, Reynolds number, and surface roughness.




Figure 46 - chart and graphical representation of aerodynamic coefficient

Figure 46.- Aerodynamic coefficient dependencies. c denotes chord length, x and y denote distances along X and Y axes, respectively.


It is best at this point to examine, in a general manner, the variation of the coefficients with angle of attack and to discuss some typical features often found in the informative graphs of these results.


Figure 47 is a typical graph of coefficient of lift cl against the angle of attack a [Greek letter alpha] of the airfoil section. One of the first things noticed is the fact that at an angle of attack of 0°, there is a positive coefficient of lift, and, hence, positive lift. This is the case of most cambered airfoils and was discussed earlier. One must move to a negative angle of attack to obtain zero lift coefficient (hence zero lift). It will be remembered that this angle is called the angle of zero lift. A symmetric airfoil was shown to have an angle of zero lift equal to 0° as might be expected.


Notice next that from 0° up to about 10° or 12° the "lift curve" is almost a straight line. There is a linear increase in the coefficient of lift with angle of attack. Above this angle, however, the lift coefficient reaches a peak and then declines. The angle at which the lift coefficient (or lift) reaches a maximum is called the stall angle.




Figure 47 - graph illustrating the relationship between the Coefficient of lift and angle of attack

Figure 47.- Coefficient of lift as a function of angle of attack.


The coefficient of lift at the stall angle is the maximum lift coefficient cl,max Beyond the stall angle, one may state that the airfoil is stalled and a remarkable change in the flow pattern has occurred. Figure 48 shows an airfoil whose angle of attack is being raised from 0° to past the stall angle of attack. Note that below the stall angle, the separation points on the airfoil move forward slowly but remain relatively close to the trailing edge. Near the stall angle the separation points move rapidly forward and the pressure drag rises abruptly. Past the stall angle, the effects of the greatly increased separated flow is to decrease the lift.


It is interesting to note (fig. 47) that the "lift curve" continues through negative angles of attack and that a negative stall angle occurs also. In general, however, an aircraft will be operating at a positive angle of attack to obtain the lift necessary for flight.



Figure 48 - illustration of the relationship between air foil angle and stall effect

Figure 48.- Stall formation.


Figure 49 is a typical graph of the coefficient of drag cd as a function of angle of attack of the airfoil section. Usually, the minimum drag coefficient occurs at a small positive angle of attack corresponding to a positive lift coefficient and builds only gradually at the lower angles. As one nears the stall angle, however, the increase in cd is rapid because of the greater amount of turbulent and separated flow occurring. The drag coefficient curve may also be plotted as a function of the lift coefficient as shown in figure 46. Since the lift coefficient up to the stall angle is a near-linear function of angle of attack, the cd curve appears much the same as before and the same comments apply.


The coefficient of moment is an important parameter in the stability and control of an aircraft and will be discussed when that subject is introduced.


Two-dimensional wing compared with three-dimensional wing.- The wing shown in figure 50 is a finite-span three-dimensional (3D) version of the infinite-span two...




Figure 49 - grap illustrating the relationship between angle of attack and the coefficient of drag

Figure 49.- Coefficient of drag as a function of angle of attack of airfoil section.


....dimensional (2D) wing tested in the wind tunnel (fig. 43). The wing area is S and is the chord length c times the wing span b . Thus


S = bc



This is also known as the planform area. If one measures the lift, drag, and moment on this 3D wing and nondimensionalizes by using the wing area, free-stream dynamic pressure, and chord length, one obtains the 3D aerodynamic characteristics of the wing; CL, CD, and Cm where


CL = L/qS

(L = Total lift on wing) (25)

CD = D/qS

(D = Total drag on wing) (26)

Cm= M/qSc

(M = Total moment acting on wing) (27)



Figure 50 - illustration of two and three dimensional air foil characteristics

Figure 50.- Two-dimensional compared with three-dimensional conditions.


Notice that the coefficients for 3D flow are capitalized whereas the coefficients for 2D flow are lower case letters. This is the notation used to distinguish the finite-span coefficients from the infinite-span coefficients.


The important question now arises: How can one use experimental NACA 2D airfoil characteristics data to obtain the lift, drag, and moments on a real, finite 3D wing? Or to put it another way, how are cl, cd, and cm related to CL, CD, and Cm. In figure 50 the wing has simply been moved out in the free stream so that the wing tips are freely exposed. At first glance one might conclude that cl = CL, cd = CD, and cm = Cm. But this is wrong! Why? Where does the problem lie? The answer is that the 2D wing tested in the wind tunnel spanned the tunnel walls and did not allow for the possibility of airflow about the wing tips, that is, spanwise flow of air. But the 3D wing is freely exposed in the free stream and spanwise flow may occur. The two-dimensional results must be modified to account for the effects of three dimensional flow.


Circulation and the vortex system of a finite wing.- As was shown earlier in the discussion of a two-dimensional wing the airfoil could be represented by a free-stream [72] flow and a circulation of strength qdetermined by the Kutta- Joukowsky condition. For an infinite, two-dimensional wing, at subsonic speeds, the upflow in front of the wing exactly balances the downflow at the rear of the wing, so that there exists no net downward movement of air past wing caused by the circulation q. This is not the case for a finite three-dimensional wing.


By a theorem of Helmholtz, a line of circulation or vortex line cannot end in midair. For an infinite wing, the vortex extends to infinity which is permissible but for a finite wing, the vortex line cannot simply end at the wingtips. Instead, the vortex continues outside the wing tips where the free-stream flow forces them to trail back from the wing tips, hence, the names "trailing vortices" or "wing-tip vortices." These tip vortices have the same circulation strength q. Physically, the formation of these trailing vortices can be explained as follows.


The static pressure on the upper surface of a wing is for the most part lower than that on the lower surface of the wing operating at a normally positive lift. As shown in figure 51(a), the pressures must become equal at the wing tips since pressure is a continuous function. A pressure gradient exists between the upper and lower surfaces. The tendency of the air is to equalize any pressure differences so that particles of air tend to move from the lower wing surface around the wing tip to the upper surface (from the region of high pressure to the region of low pressure). In addition, there exists the oncoming free-stream flow approaching the wing. (See fig. 51(b).) If these two movements of air are combined (superimpose the spanwise flow about the wing on the oncoming free stream), one has an inclined inward flow of air on the upper wing surface and an inclined outward flow of air on the lower wing surface. The spanwise flow is strongest at the wing tips and decreases to zero at the midspan point as evidenced by the flow direction there being parallel to the free-stream direction (fig. 52(a)).


Figure 51 - two illustration of how pressure is applied to all sides of a wing

Figure 51.- Finite-wing flow tendencies.


Figure 52 - illustration of the formation of wing-tip vortices

Figure 52.- Formation of wing-tip vortices.


When the air leaves the trailing edge of the wing, the air from the upper surface is inclined to that from the lower surface and helical paths or vortices result. A whole line of vortices trail back from the wing, the vortex "strength" being strongest at the tips and decreasing rapidly to zero at midspan. (See fig. 52(b).) A short distance downstream the vortices roll up and combine into two distinct cylindrical vortices which constitute the so-called "tip vortices." Figure 52(c) shows the simplified picture of the tip vortex system replacing the vortex distribution just discussed.


[74] An account of the tip-vortex effects constitutes the modifications of the 2D airfoil aerodynamic coefficients into their 3D counterparts.


The vortex in the wing (which is equivalent to the lift of the wing (fig. 53(a)) is called the bound vortex. Figure 53(b) shows the bound vortex and the tip vortices for a finite wing (sometimes known as the horseshoe vortex system). Also, the vortex system must be closed in some manner and is accomplished by the so-called starting vortex which is left behind the wing when the wing starts from rest in the case of constant vorticity (fig. 53(c)). If the lift of the wing is being continually changed, new starting vortices are shed. Generally, the starting vortices are soon dissipated because of the air's viscosity. Also, their influence on the flow behind a wing decays rapidly the further back they are left.


Figure 53 - illustration of the complete-wing vortex system

Figure 53.- Complete-wing vortex system.


[75] The tip vortices trail back from the wing tips and they have a tendency to sink and roll toward each other downstream of the wing. Again, eventually the tip vortices dissipate, their energy being transformed by viscosity. As will be discussed later, this change may take some time and may prove to be dangerous to other aircraft.


The important effects of the vortex system are shown in figure 54. Indicated are the directions of air movement due to the vortex system. The left-tip vortex rotates clockwise, the right-tip vortex rotates counterclockwise (when viewed from behind), and the bound vortex rotates clockwise (when viewed from the left side).


The bound vortex is directly related to the lift on the wing as in the dimensional case. For a finite wing the relation becomes


formula for bound vortex





lift on three-dimensional wing

Greek letter rho

free-stream air density [Greek letter rho]


free-stream velocity


wing span

         letter Capital Gamma

circulation (spin strength) [Greek letter Capital Gamma]


In both the 2D and 3D cases the upflow (or upwash) in front of the wing balanced the downflow (or downwash) in back of the wing caused by the bound vortex. But, in the finite-wing case one must also take into account the Lip vortices (assuming that the influence of the starting vortex is negligible). The tip vortices cause additional down...



Figure 54 - illustration of a jet and the vortex it produces

Figure 54.- Vortex flow effects. Note that upwash and downwash are due to both the bound vortex and the tip vortices.


...wash behind the wing within the wing span. One can see that, for an observer fixed in the air (fig. 55) all the air within the vortex system is moving downwards (this is called downwash) whereas all the air outside the vortex system is moving upwards (this is called upwash). Note that an aircraft flying perpendicular to the flight path of the airplane creating the vortex pattern will encounter upwash, downwash, and upwash in that order. The gradient, or change of downwash to upwash, can become very large at the tip vortices and cause extreme motions in the airplane flying through it. Also shown is an airplane flying into a tip vortex. Note that there is a large tendency for the airplane to roll over. If the control surfaces of the airplane are not effective enough to counteract the airplane roll tendency, the pilot may lose control or in a violent case experience structural failure.


The problems of severe tip vortices are compounded by the take-off and landings of the new generation of jumbo jets. During these times the speed of the airplane is low and the airplane is operating at high lift coefficients to maintain night. The Federal Aviation Agency has shown that for a 0.27 MN (600 000 lb) plane, the tip...



Figure 55 - illustration of the turbulance created by a plane in flight

Figure 55.- Upwash and downwash fields around an airplane.


....vortices may extend back strongly for 5 miles from the airplane and the downwash may approach 160 meters per minute (500 ft/min). Tests also show that a small light aircraft flying into a vortex could be rolled over at rates exceeding 90°/sec. Between 1964 and 1969 at least 100 airplane accidents and countless other incidents could be traced to this vortex phenomenom. Realizing this, the FAA is requesting much greater separation times and distances between the large jets and small aircraft especially during take-offs and landings.


The tip vortices contribute to the downwash field at and behind the wing. To create downwash due to finite-wing contribution requires the expenditure of energy per unit time or power. The power required to induce this component of downwash may be associated with an additional drag force known as induced drag. Additionally, the net lift on the wing is decreased by the tip vortex effects.


It may be noted at this point that induced drag is an ideal fluid effect not in any way associated with a fluid's viscosity. A finite wing operating in an ideal fluid will not possess a skin- friction or pressure drag (taken together often called parasitic drag) but will still possess an induced drag if generating lift and thus a circulation Greek letter gamma.


The aspect ratio may be defined as


Aspect ratio = (Wing span)2/Wing area





wing aspect ratio


for any wing.


[78] For the special case of a rectangular wing


S = b x c


so that

AR = b/c = Wing span = Chord length



for a rectangular wing. Aspect ratio is a measure of the slenderness of a wing; a long thin wing has a high aspect ratio compared with a short stubby wing of low aspect ratio.


With this in mind, return to the case of the 2D and 3D wings shown in figure 50. The 2D wing is the equivalent of an infinite span wing and, as such, one can say it has an infinite aspect ratio. The 3D wing has a finite aspect ratio whose value is determined by equation (30). Figure 56 shows the coefficient of lift curves ('lift curves") obtained for both wings by experiment. Readily evident is the effect that the tip vortices have in creating the additional downwash w at the wing; the lift curve is flattened out so that at the same angle of attack less lift is obtained for the smaller aspect ratio wing. This is not a beneficial effect.


Consider the case where one wants to get the same lift from the finite wing as predicted by the 2D aerodynamic characteristics, or namely, CL = cl . From figure 56, this is achieved by raising the angle of attack of the finite wing by a small amount over that of the 2D wing, that is,



CL = cl (31)


This increase in angle of attack to obtain the same lift is due to the effect of tip vortices on the downwash in changing the relative flow seen by the wing where for small angles





It may be stated that the drag coefficient for the finite 3D wing is the infinite wing 2D drag coefficient plus the induced drag coefficient or....

CD = cd + (CD)induced




Figure 56 - graph illustrating the effect of aspect ratio on coefficient of lift

Figure 56.- Effect of aspect ratio on coefficient of lift.


....where cd here is the parasitic drag coefficient (skin-friction drag plus pressure drag coefficient) of the 2D wing operating at the higher angle of attack a3D necessary to get CL. This is greater than the original value of cd operating at a2D. (CD)induced is inversely proportional to the aspect ratio AR and an "efficiency factor" e relating how close one comes to achieving an ideal elliptic spanwise lift distribution shown by theory to give minimum induced drag (e = 1). Also, (CD)induced is proportional to CL2. Thus

(CD)induced = KCL2



where K is related to the aspect ratio and the efficiency factor.


To summarize to this point, a finite-span version of a 2D wing will give the same lift coefficient (CL = cl ) only if its angle of attack is raised slightly. This increase in angle of attack causes an increase in the parasitic drag coefficient and, in addition, yields an induced drag coefficient. In this way, the 2D wind-tunnel data are modified.


It is important not to confuse the drag coefficient with the actual drag force it represents. If one converts the coefficient to actual induced drag, one finds that it is [80] inversely proportional to (1) the span efficiency factor e, (2) the wing span squared b2, and (3) the free-stream velocity squared free-stream velocity squared.


Methods of reducing induced drag.- From the stated results it can be seen that NACA 2D wind-tunnel data may be used to obtain the lift and drag acting on the entire 3D wing if proper corrections for the tip vortices are included. One would like these corrections to be as small as possible to get the most lift with the least drag. How may the induced drag be reduced? From the previous discussion one may (1) increase the span efficiency factor to as close to e = 1 as possible, (2) increase the wing span b (or aspect ratio AR), and (3) increase the free-stream velocity free-stream velocity. This last fact points up that induced drag is a small component at high speeds (cruising flight) and relatively unimportant since it constitutes only about 5 to 15 percent of the total drag at those speeds. At low speeds (take-off or landing) it is a considerable component since it accounts for up to 70 percent of the total drag.


The efficiency factor e and wing span are physical factors that may be controlled by proper design. Figure 57 shows two airplanes with rectangular wings. Both wings are at the same lift coefficient and they have the same wing area. The only difference is that the wing span of the second wing is twice that of the first wing. Ideally, both wings should produce the same lift and drag. But the longer span wing (higher aspect ratio) has one-fourth the induced drag, and therefore, a greater efficiency. The thought arises that with less induced drag with longer wing spans, why not make a wing with an extremely long wing span (high AR). This would give nearly idealized flow, the tip vortex effects being very small. In fact, as figure 58...



Figure 57 - illustration of two planes with uneven wing lengths

Figure 57.- Wing-span effect on induced drag for airplanes having same wing area, same lift coefficient, and same dynamic pressure.


[81]...illustrates, sailplanes that must rely on high efficiencies, do have very long slender wings. But structural considerations become a dominant factor. A very thin long wing requires a large structural weight to support it.


There comes a point where the disadvantage of increasing structural weight necessary to support increased wing span counteracts the advantage of decreased drag due to smaller vortex effects. A compromise aspect ratio would give the optimum performance. This is necessarily also dependent on factors such as fuel capacity, control characteristics, size allowances, and numerous other factors. A survey of airplane categories show sailplanes with an aspect ratio of 15 or more, single-engine light airplanes with an aspect ratio of about 6, and supersonic fighter airplanes with an aspect ratio of high aspect wing ratio.


Figure 58 - overhead illustration of a high aspect wing ratio design

Figure 58.- High-aspect-ratio wing. AR



Another interesting way of reducing induced drag is by the use of tip plates or tip tanks as shown in figure 59. This arrangement tends to promote a 2D flow by inhibiting the formation of tip vortices. They have the same physical effect as an increase in wing span (or aspect ratio). Normally, these are not used since there are other more valuable drag reduction methods.


Before looking further at methods of reducing induced drag, it is necessary to define taper and twist for a wing. For a general wing, the airfoil sections may vary in three distinct ways along the wing. First, the size or chord length may change; second, the shape of the airfoil section may change as one moves along the wing, and lastly, the angles of attack of the airfoil sections may change along the wing. These variations give the wing taper and twist- terms that are now considered.


Planform taper is the reduction of the chord length and thickness as one proceeds from the root section (near the fuselage) to the tip section (at the wing tip) so that the airfoil sections also remain geometrically similar. (See fig. 60(a).)



Figure 59 - drawing of a plane with wing tanks, a P-80

Figure 59.- Tip plates and tip tanks.

Figure 60 - illustration of wing design differences

(a) Planform taper.
(b) Thickness taper.
(c) Planform and thickness taper.
(d) Inverse taper in planform and thickness
Figure 60.- Planform and thickness taper.


Thickness taper s the reduction of only the airfoil s thickness as one proceeds from the root section to the tip section-this reduction results in thinner airfoil sections at the wing tip. (See fig. 60(b).) The chord remains constant. Figure 60(c) shows a typical wing with both planform and thickness taper. One notable exception to this normal taper was the XF-91 fighter which has inverse taper in planform and thickness so that the wing tips were thicker and wider than the inboard stations. (See fig. 60(d).)


[83] Wings are given twist so that the angle of attack varies along the span. A decrease in angle of attack toward the wing tip is called washout whereas an increase in angle of attack toward the wing tip is called washin. Geometric twist (fig. 61(a)) represents a geometric method of changing the lift distribution, whereas aerodynamic twist, by using different airfoil sections along 'he span represents an aerodynamic method of changing the lift distribution in a spanwise manner (fig. 61(b)). To give minimum induced drag it was demonstrated that the spanwise efficiency factor e should be as close to 1 as possible. This is the case of an elliptic spanwise lift distribution. A number of methods are available to modify the spanwise distribution of lift.


Figure 61 - illustration of geometric and aerodynamic twist

(a) Geometric twist.
(b) Aerodynamic twist.
Figure 61.- Geometric and aerodynamic twist.


[84] These methods include (1) planform taper to obtain an elliptic planform as shown in figure 62(a) for the Spitfire wing which is remarkably elliptic; (2) a geometric twist and/or aerodynamic twist to obtain elliptic lift distribution; or (3) a combination of all of these methods.


An elliptical planform is hard to manufacture and is costly. Of course, from the point of view of construction, the best type of wing is the untapered, untwisted wing. This is used considerably by light plane manufacturers. (See fig. 62(b).) Surprisingly, there are data that indicate that a square- tipped rectangular wing is very nearly as efficient as the elliptic wing so that the gains in reduced induced drag may be insignificant. This result may be traced to the fact that for a real wing the lift distribution does fall off to zero at the wing tips and approximates an elliptical distribution.


The wing-tip shape, being at the point of production of the tip vortices, appears to be of more importance in minimizing tip vortex formation and thus minimizing induced drag. Figure 62(c) presents a good wing-tip shape, whereas figure 62(d) indicates a less favorable one.


Taper and twist are perhaps of greater importance when the problem of stalling is discussed later.


Aerodynamic Devices


One of the most fascinating subjects of aerodynamics of flight is the vast number of, for want of a better term, "aerodynamic devices" affixed to a simple wing to achieve increases or decreases in lift and drag such as slats, slots, flaps, spoilers, and dive brakes. With all these devices hanging on a wing, the unsuspecting air traveler might well think that the wing is a piece of modern art. The sound of flaps and slats opening as one approaches for a landing combined with a visual inspection of the wing "coming apart at the seams" may unnerve the unknowledgeable. But a purpose exists for all of these devices and the safety and economy of air travel is dependent on them.


It is in the interest of safety to perform take-off and landing maneuvers at as low a speed as possible. But also, one does not want the normal flying characteristics to be affected. Consider a near- level flight condition in which the airplane weight is equal to the lift (L = W). For minimum flying speed (take-off or landing) the wing would be operating at maximum lift or CL,max. From equation (25) after some manipulation, solving for the minimum flight velocity, Vmin yields


minimum flight velocity equation



Figure 62 - illustration of the eliptical wing design of a Supermarine Spitfire

(a) Elliptic wing - Supermarine Spitfire Mk.1
(b) Untapered, untwisted wing.
(c) Good tip shape.
(d) Poor tip shape.
Figure 62.- Reduction of induced drag.


The density q is considered to be constant and if the weight is considered a fixed characteristic of the airplane, then it is obvious that the only way to reduce Vmin is to increase CL,max and/or the wing area S. Slots and flaps are used for this purpose.


Slots.- The maximum coefficient of lift may be increased through the use of a slot formed by a leading-edge auxiliary airfoil called a slat. Figure 63(a) illustrates the operating principle. When the slot is open, the air flows through the slot and over the airfoil. The slot is a boundary-layer control device and the air thus channeled energizes the boundary layer about the wing and retards the separation. The airfoil can then be flown at a higher angle of attack before stall occurs and thus get a higher...



Figure 63 - chart illustrating the effects of moveable slats on wing aerodynamics

(a) Slat
(b) Slat aerodynamic effects.
Figure 63.- Slat-slot operation.


... CL,max value. A curve showing CL as a function of a [Greek letter alpha] for the normal and the slotted airfoil is given in figure 63(b). Notice particularly that for angles of attack less than the stall angle, the airfoil lift curve is relatively unaffected whether the slot is opened or closed.


There are two types of slots -fixed and automatic. The fixed slot is self explanatory; the leading-edge slat is mounted a fixed distance from the airfoil. Its main disadvantage is that it creates excessive drag at high speeds. Figure 64 shows a German World War II designed rocket fighter - the Me-163 with fixed slots in the wing. The automatic slot depends on air pressure lifting the slat away from the wing [87] at high angles of attack to open the slot. At low angles of attack the slat is flush against the wing leading edge and reduces drag at high speeds compared with the fixed slot. Its main disadvantages are its added weight, complexity, and cost.


One main disadvantage of both types of slots is the high stall angle created. The airplane must approach for a landing in an extreme nose-up attitude which promotes reduced visibility.


Figure 64 - illustration of aerodynamic effects of  fixed wing slats

Figure 64.- Fixed slots.


Flaps.- Flaps may be used to increase the maximum lift coefficient, increase the wing area, or both. A change in the maximum lift coefficient may be realized by a change in the shape of the airfoil section or by increased camber. The trailing-edge flap is one method of accomplishing this. Figure 65(a) shows a normal airfoil and the same shaped airfoil with a simple flap in the down position for increased camber. The maximum lift coefficient for the airfoil with the simple flap is greater than that for the unflapped airfoil. Also, the coefficients of lift are increased over the entire angle-of-attack range. This is shown in figure 65(b). Note also that the stall angle is essentially unchanged from that of the unflapped airfoil. This is opposed to the slot operation where a higher stall angle was obtained. The flapped airfoil reduces the disadvantage that the slot has in high landing angles.




Figure 65 - an illustration of the aerodynamic effect of moveable wing flaps

(a) Flap.
(b) Flap aerodynamic effects.
Figure 65.- Simple flap operation.

Figure 66 - an illustration of Fowler flaps and the Complex slotted flap of Boeing 737

(a) Fowler flaps.
(b) Complex slotted flap of Boeing 737.
Figure 66.- Types of flaps.


[89] Figure 66(a) shows a Fowler flap which is hinged such that it can move back and increase the airplane wing area. Also, it may be rotated down to increase the camber. A very large increase in maximum lift coefficient is realized.


There are many combinations of slots and flaps available for use on airplanes. Figure 66(b) shows the arrangement on a Boeing 737 airplane which utilizes a leading-edge slat and a triple- slotted trailing-edge flap. This combination is a highly efficient lift-increasing arrangement. The slots in the flaps help retard separation over the flap segments and thus enhance lift.


It may also be noted that flaps in an extreme down position (50° to 90°) act as a high-drag device and can retard the speed of an airplane before and after landing.


Boundary-layer control.- Another method of increasing CL,max is by boundary-layer control. The idea is to either remove the low-energy segment of the boundary layer and let it be replaced by high-energy flow from above or by adding kinetic energy to the boundary layer directly. Both of these methods maintain a laminar flow for a longer distance over the airfoil, delay separation, and allow one to get a larger angle of attack before stall occurs, and thus a higher CL,max The slot was shown to be one means of passing high-energy flow over the top surface of a wing.


The low-energy boundary layer may be sucked through slots or holes in the wing as shown in figure 67(a) or high-energy air may be blown into the boundary layer through backward facing holes or slots as shown in figure 67(b).


Figure 67 - an illustration of the suction of boundary layer and the reenergizing the boundary layer

(a) Suction of boundary layer.
(b) Reenergizing the boundary layer.
Figure 67.- Forms of boundary-layer control.


[90] Spoilers.- Spoilers are devices used to reduce the lift on the airplane wing. They may serve the purpose as on gliders to vary the total lift and control the glide angle. Or on large commercial jets they may be used to help the aileron control by "dumping" lift on one wing and thus help to roll the airplane. Also, on landing, with spoilers up, the lift is quickly destroyed and the airplane may quickly settle on its landing gear without bouncing. Figure 68 shows the spoiler arrangement on a Boeing 707 wing.



Figure 68 - an illustration of a Boeing 707 with flaps up and the aerodynamic effect

Figure 68.- Use of spoilers.


Dive brakes.- Dive (or speed) brakes are used in airplanes to control descent speed. Whether slowing down quickly when approaching for a landing, after landing, or in a dive, these aerodynamic brakes are helpful. Essentially, they promote a large separation wake and increase the pressure drag. Figure 69 shows a civilian aircraft speed-brake arrangement and two military aircraft dive brake arrangements.


Stall characteristics.- The present discussion has concentrated considerably on operating near or at the stall condition CL,max. A further word about stalling is in order. A wing should possess favorable stall characteristics so that (1) the pilot has adequate warning of the stall, (2) the stall is gradual, and (3) there is little tendency to spin after a stall. This may be achieved by "forcing" the stall to occur at the wingroot section first and let it progress toward the wing tips. The outboard, wing-tip stations should be the last to stall so that the ailerons remain effective (are not immersed in a turbulent "dead air" wake). Use of twist, namely washout, is often employed so...



Figure 69 - an illustration of the various locations of 'air breaks' and their effect

Figure 69.- Dive (speed) brake arrangements.


....that the wing-root section reaches the stall angle first. (See fig. 70(a).) Also, airfoil sections with gradual stall characteristics are more favorable than ones with quick stall characteristics. (See fig. 70(b).)


As the inboard root stations stall, turbulent flow from the wing strikes the tailplane and buffets the pilot's controls. This condition is an adequate stall warning device. With a gradual stall on both wings, the plane should maintain a level attitude with few spin tendencies.


Total Drag of Airplane


Up to now the drag acting on a finite wing has been considered. It has been shown that three components are present: (1) skin-friction drag, (2) pressure drag, and (3) induced drag. Of course, an airplane is composed of many other components and each will introduce a total drag of its own. Possible airplane component drags....



Figure 70 - illustration and graph demonstrating increase of drag effect as wing angle of attack increases

(a) Twist and stall. Note that stalled region moves toward wing tip as wing angle of attack increases.
(b) Gradual stall.
Figure 70.- Stall characteristics.


....include (1) drag of wing, wing flaps, (2) drag of fuselage, (3) drag of tail surfaces, (4) drag of nacelles, (5) drag of landing gear, (6) drag of wing tanks and external stores, (7) drag of engines, and (8) drag of miscellaneous parts. The net drag of an aircraft is not simply the sum of the drag of the components. When the components are combined into a complete aircraft, one component can affect the flow field, and hence, the drag of another. These effects are called interference effects, and the change in the sum of the component drags is called interference drag. Thus,


(Drag)l+2 = (Drag)1 + (Drag)2 + (Drag)interference


[93] Generally, interference drag will add to the component drags but in a few cases, for example, adding tip tanks to a wing, total drag will be less than the sum of the two component drags because of reduced induced drag.


Interference drag can be minimized by proper fairing and filleting which induces smooth mixing of air past the components. Figure 71 shows a Grumman F9F Panther Jet with a large degree of filleting. No adequate theoretical method will predict interference drag; thus, wind-tunnel or flight- test measurements are required. For rough computational purposes a figure of 5 percent to 10 percent can be attributed to interference drag on a total aircraft.


Figure 71 - drawing of a F9F jet with wing fillets

Figure 71.- Wing fillets.


Small items also add to the total aircraft drag and although seemingly trivial, they can greatly reduce the aircraft's top speed. Figure 72 shows a TBF Avenger aircraft from World War II and shows the increase in drag coefficient as these small changes and components are accounted for.


Figure 73 shows a Me-109G German fighter from World War II. Shown is the percentage breakdown of the drag (includes interference drag) of the components.


Figure 74 presents a graph of how the total airplane drag coefficient has decreased over the years. Doing away with bracing wires, shielding engines behind streamline cowls, use of flush- riveting techniques, and use of polished surfaces have all aided in the reduction of drag. It is beyond the scope of this discussion to expand upon what has been introduced about total airplane drag at subsonic speeds. Although prediction of drag and wind-tunnel drag measurements of models does yield good results, final drag evaluation must be obtained by flight tests. Even here, however, the accuracy of the measurements is dependent on flight-test equipment, pilot technique, and subsequent proper evaluation of test data.



illustration of an airplane's possible configurations relative to drag effect

Airplane configuration
CD at CL = 0.245
Reference condition
(see column 1)


Airplane completely sealed and faired





Flat plate removed from nose





Seals removed from flapped-cowling air exits





Seals removed from cowling-flap hinge-line gaps





Exhaust stacks replaced





Canopy fiaring removed, turret leaks sealed





Tail wheel and arresting-hook openings uncovered





Aerial, mast, and trailing antenna tube installed





Canopy and turret leak seals removed





Leak seals removed from shock strut, cover plate, and wing-fold axis





Leak seals removed from bomb-bay doors and miscellaneous leak seals removed





Fairings over catapult hooks removed





Wheel-well cover plates removed





Seals removed from tail-surface gaps





Plates over wing-tip slot openings removed. Airplane in servive condition.




Total drag change


Figure 72. Small item influence on total airplane drag.


Figure 73 - front, underneath and side view of a fighter plane and drag values

Figure 73.- Typical fighter-drag breakdown.

Figure 74 - chart illustrating the reduction in drag coefficient with improvments in design from 1900 to 1960

Figure 74.- Decrease in airplane drag coefficient with time.


[96] Propellers and Rotors


Propellers.- The propeller converts the turning power of an engine's crankshaft into the thrust force. This thrust is equal to the mass of air forced backward by the propeller per second times the added velocity imparted to this air. If one has ever stood behind a spinning propeller while the airplane was at rest on the ground, this backward moving air, the slipstream, is very noticeable.


Figure 75 shows a variety of propeller configurations used on military and civilian airplanes. Basically, a propeller blade is a small wing producing a resultant aerodynamic force which for the purposes of this discussion, may be resolved into a force pointing along the axis of the airplane (thrust), and a force in the plane of the propeller blades (the torque force). (See fig. 76.) The torque force opposes the rotary motion of the engine by acting as a "drag" on it. In equilibrium, the propeller rotates at a constant rate determined by the engine torque equal and opposite to the propeller torque.


Figure 75 - an illustration of 4 popular propeller configurations

(a) Two-bladed propeller on Beagle pup.
(b) Three-bladed propeller on Me-109G.
(c) Four-bladed propeller on B-29.
(d) Eight-bladed propeller contrarotating propellers on Antonov AN-22.
Figure 75.- Various propeller configurations.


Figure 76 - drawing of a propeller indicating  thrust and torque produced

Figure 76.- Thrust and torque of a propeller.


As figure 77 shows, the propeller blade consists of a set of airfoil-shaped sections which may vary in shape from the tip to the root of the blade. Although a wing is fixed with respect to the airplane and sees only the relative free-stream flow of air (fig. 78(a)), the propeller is also rotating with respect to the airplane, and it sees an oncoming relative flow of air which is the vector sum of the airplane free-stream velocity and the propeller rotational velocity. (See fig. 78(b).) The angle between this relative velocity and the plane of the propeller rotation is called the helix angle or angle of advance. For a particular airplane velocity this helix angle varies from the root to the tip since the tip sections of the propeller are revolving faster than the root sections. As one approaches the root, the relative velocity vector comes inclined closer to the oncoming free-stream velocity of the aircraft, that is, the helix angle approaches 90°.


To obtain an aerodynamic force, the airfoil blade section is placed at an angle of attack to the relative velocity vector. Thus, the total angle from the plane of the propeller rotation to the chord line of the blade section is the sum of the helix angle and angle of attack for that section as shown in figure 78(c). This is known as the blade angle. A propeller blade appears to be twisted with the tip sections with small blade angles and the root sections with large blade angles due in main to the increase in helix angle.


The blade angle is also called the pitch angle. This pitch angle may be fixed for a propeller blade, hence a fixed-pitch propeller. or may be adjustable by hand on the ground (adjustable pitch propeller) or controlled automatically in the air (controllable pitch propeller). The efficiency of a propeller is power output divided by power input...



Figure 77 - sectional drawing of a typical propeller

Figure 77.- Propeller blade sections. a [Greek letter alpha] denotes angle of attack of airfoil sections.

Figure 78 - propeller diagram illustrating helix angle

Figure 78.- Propeller terminology.


[99] ....and would desirably be as close to a value of one (or 100 percent) as possible. The efficiency is proportional to the free-stream velocity and for maximum efficiency requires a different pitch-angle setting. For take-off, an airplane uses a fine or low pitch (flat blade angle or small angle of attack) to provide high revolutions per minute. A coarse or high pitch is used for cruising and gives low revolutions per minute. This effect is illustrated in figure 79(a).



Figure 79 - diagram of a propeller in vaious modes and resulting effect

(a) Pitch control.
(b) Feathered propeller.
(c) Landing brake.
Figure 79.- Use of pitch control.


Some propellers may be feathered in flight. This means that the blades are turned so that the leading edges of the airfoil sections are alined to the free-stream velocity. Feathering is used on a stopped propeller to avoid damaging an engine and to decrease the propeller drag. (See fig. 79(b).) Some propellers have reversible pitch for use as a landing brake. (See fig. 79(c).) In this case, negative thrust is obtained by turning the blade to a large negative angle of attack.


The design of a propeller, like an airplane, is influenced by many factors, some of which cause contradictions in design. The overall shape is determined by compromise and is largely dependent on the mission to be performed. For low speeds the propellor blade is usually slender with rounded tips. For high speeds larger paddleshaped blades are used or more propeller blades are used.


[100] The slipstream is produced by a propeller producing thrust by forcing air backwards. It is a cylindrical core of spiraling air that flows back over the fuselage and sailplane. The fact that it strikes the sailplane has important effects - some detrimental and some beneficial. The slipstream flow is faster than the free-stream flow; this means that the drag of the fuselage, sailplane, and other parts exposed to it is larger. The slipstream moves over the sailplane and is beneficial in providing for effective control by the tail surfaces since the aerodynamic forces produced by these surfaces are dependent on the square of the velocity of the air moving over them. This is important in the cases of taxiing or false-off when the free-stream velocities may be low.


The rotary motion of the slipstream, however, causes the air to strike the tailplane at an angle and not headon. This may have an effect on the stability and control of the airplane (considered in a later discussion). The effects of the rotary motion of the propeller may be counteracted by using contrarotating, propellers (spinning in "opposite directions). Figure 80 shows three aircraft that used this form of a thrust-producing device.


Figure 80 - drawing of 3 airplanes using 2 propellers

(a) Westland Wyvern TF Mk 2.
(b) Douglas XB-42.
(c) Lockheed XFV-1 VTO.
Figure 80.- Contrarotating propellers.

[101] Lifting rotors.- For a helicopter the rotor is the lift-producing device. The blades of the rotor are airfoil shaped and are long and slender (large aspect ratio). The number of blades vary with the design. Figure 81 shows three helicopters, each employing a different number of blades. Generally, for the heavier helicopter, more blades are used to reduce the load that each must carry.


drawing of a single bladed helicopter

drawing of a two bladed helicopter

(a) Bell AH-1G Huey Cobra. Two-bladed.

(b) Hughes OH-6A. Four-bladed.

drawing of a six bladed helicopter

(c) Sikorsky CH-3C. Six-bladed.

Figure 81.- Helicopters with varying blade numbers.


As for the airplane propeller, the helicopter rotor blades have a pitch angle defined as the angle between the plane of rotation of the blades and the chord line of the blades. The pitch of the blades may be controlled collectively (collective pitch) or controlled individually (cyclic pitch).


[102] Collective pitch changes the pitch of all blades together and with changes in engine power settings, produces the lift necessary for the helicopter to take-off, hover, climb, and descend.


Cyclic pitch is controlled by the swashplate of the rotor head which allows the pitch of individual blades to vary as they rotate about the hub. When a pilot wishes to fly forward, the swashplate is tilted forward. As each rotor blade approaches the forward position (toward the direction of flight) of its cycle, its pitch decreases, the blade lift is reduced, and its flight path descends. As the blade rotates to the rear, the pitch is increased, the blade lift is increased, and the flight path ascends. The net effect is to tilt the whole rotor disk forward to the desired angle, the total lift vector is rotated (see fig. 82) and a forward thrust component is given to the helicopter.


The rotation of the main rotor blades produces a reactive torque which tends to rotate the helicopter body in the opposite direction. Directional control is accomplished by a tail rotor. It provides sufficient thrust to counteract this rotational tendency. Additionally, by controlling the thrust of the tail rotor, the heading of the helicopter may be controlled.


Figure 82 - drawing of a helicopter illustrating the foward motion produced by tilting the rotor axis foward

Figure 82.- Helicopter forward motion.