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INDUCED DRAG
Another important influence of the induced flow is the orientation of the actual lift on a wing. Figure 1.30 illustrates the fact that the lift produced by the wing sections is perpendicular to the average relative wind. Since the average relative wind is inclined downward, the section lift is inclined aft by the same amount—the induced angle of attack, The lift and drag of a wing must continue to be referred perpendicular and parallel to the remote free stream ahead of the wing. In this respect, the lift on the wing has a component of force parallel to the remote free stream. This component of lift in the drag direction is the undesirable—but unavoidable—conse-
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qucncc of developing lift with a finite wing and is termed INDUCED DRAG, D,. Induced drag is separate from the drag due to form and friction and is due simply to the development of lift.
By inspection of the force diagram of figure
1.30, a relationship between induced drag, lift, and induced angle of attack is apparent. The induced drag coefficient, CDi, will vary directly with the wing lift coefficient, CL, and the induced angle of attack, The effective lift is the vertical component of the actual lift and, if the induced angle of attack is small, will be essentially the same as the actual lift. The J horizontal and vertical component of drag is insignificant under the same conditions. By a detailed study of the factors involved, the following relationships can be derived for a wing with an elliptical lift distribution:
(1) The induced drag equation follows the same form as atmlied to anv other aerodv-
А А Ф *
namic force.
Di—CDiqS
where
D{— induced drag, lbs. q=dynamic pressures, psf trV1
295
CD.=induced drag coefficient S= wing area, sq. ft.
(2) The induced drag coefficient can be derived as: or where
CL—lih coefficient
sin a,=natural sine of the induced angle of attack, degrees и-=3.141б, constant ЛИ=wing aspect ratio
(3) The induced angle of attack can be derived as:
<*i= 18.24 (degrees)
(Note: the derivation of these relationships may be found in any of the standard engineering aerodynamics textbooks.)
These relationships facilitate an understanding and appreciation of induced drag.
The induced angle of attack
depends on the lift coefficient and aspect ratio. Flight at high lift conditions such as low speed or maneuvering flight will create high induced angles of attack while high speed, low lift flight will create very small induced angles of attack. The inference is that high lift coefficients require large downwash and result in large induced angles of attack. The effect of aspect ratio is significant since a very high aspect ratio would produce a negligible induced angle of attack. If the aspect ratio were infinite, the induced angle of attack would be zero and the aerodynamic characteristics of the wing would be identical with the airfoil section properties. On the other hand, if the wing aspect ratio is low, the induced angle of attack will be large and the low aspect ratio airplane must operate at high angles of attack at maximum lift. Essentially, the low aspect ratio wing affects a relatively small mass of air and consequently must provide a large deflection (downwash) to produce lift.
EFFECT OF LIFT. The induced drag co
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ilar effects of lift coefficient and aspect ratio. Because of thepower of variation of induced drag coefficient with lift coefficient, high lift coefficients provide very high induced drag and low lift coefficients very low induced drag. The direct effect of CL can be best appreciated by assuming an airplane is flying at a given weight, altitude, and airspeed. If the airplane is maneuvered from steady level flight to a load factor of two,
favliMf January 1965
the lift coefficient is doubled and the induced drag is four times as great. If the flight load factor is changed from one to five, the induced drag is twenty-five times as great. If all other factors are held constant to single out this effect, it could be stated that “induced drag varies as the square of the lift”
where
Dix = induced drag corresponding to some original lift, Li Z),4= induced drag corresponding to some new lift, La
(and q (or EAS~), S, AR are constant)
This expression defines the effect of gross weight, maneuvers, and steep turns on the induced drag, e. g., 10 percent higher gross weight increases induced drag 21 percent, 4G maneuvers cause 16 times as much induced drag, a turn with 45° bank requires a load factor of 1.41 and this doubles the induced drag.
EFFECT OF ALTITUDE. The effect of altitude on induced drag can be appreciated by holding all other factors constant. The general effect of altitude is expressed by:
Dij_ /ffA
Dii <г2/
where
Dii=induced drag corresponding to some original altitude density ratio, <n
Di2= induced drag corresponding to some new altitude density ratio, <r2
(and L, S, AR, V are constant)
This relationship implies that induced drag would increase with altitude, e. g., a given airplane flying in level flight at a given TAS at 40,000 ft. (ir=0.25) would have four times as much induced drag than when at sea level (<r=1.00). This effect results when the lower air density requires a greater deflection of the airstream to produce the same lift. However, if the airplane is flown at the same EAS, the dynamic pressure will be the same and induced drag will not vary. In this case, the TAS would be higher at altitude to provide the same EAS.
EFFECT OF SPEED. The general effect of speed on induced drag is unusual since low airspeeds are associated with high lift coefficients and high lift coefficients create high induced drag coefficients. The immediate implication is that induced drag increases with decreasing airspeed. If all other factors are held constant to single out the effect of airspeed, a rearrangement of the previous equations would predict that “induced drag varies inversely as the square of the airspeed.’’
DhjVA*
Dh Vj
where
Dii=induced drag corresponding to some original speed, Vі
Dt2 = induced drag corresponding to some new speed, V2
(and L, S, AR, в are constant)
Such an effect would imply that a given airplane in steady flight would incur one-fourth as great an induced drag at twice as great a speed or four times as great an induced drag at half the original speed. This variation may be illustrated by assuming that an airplane in steady level flight is slowed from 300 to 150 knots. The dynamic pressure at 150 knots is one-fourth the dynamic pressure at 300 knots and the wing must deflect the airstream four times as greatly to create the same lift. The same lift force is then slanted aft four times as greatly and the induced drag is four times as great.
The expressed variation of induced drag with speed points out that induced drag will be of
greatest importance at low speeds and practically insignificant in flight at high dynamic pressures. For example, a typical single engine jet airplane at low altitude and maximum level flight airspeed has an induced drag which is less than 1 percent of the total drag. However, this same airplane in steady flight just above the stall speed could have an induced drag which is approximately 75 percent of the total drag.
EFFECT OF ASPECT RATIO, The effect of aspect ratio on the induced drag
^=0.318^0
is the principal effect of the wing planform. The relationship for induced drag coefficient emphasizes the need of a high aspect ratio for the airplane which is continually operated at high lift coefficients. In other words, airplane configurations designed to operate at high lift coefficients during the major portion of their flight (sailplanes, cargo, transport, patrol, and antisubmarine types) demand a high aspect ratio wing to minimize the induced drag. While the high aspect ratio wing will minimize induced drag, long, thin wings increase structural weight and have relatively poor stiffness characteristics. This fact will temper the preference for a very high aspect ratio. Airplane configurations which are developed for very high speed flight (es- specially supersonic flight) operate at relatively low lift coefficients and demand great aerodynamic cleanness. These configurations of airplanes do not have the same preference for high aspect ratio as the airplanes which operate continually at high lift coefficients. This usually results in the development of low aspect ratio planforms for these airplane configurations.
The effect of aspect ratio on the lift and drag characteristics is shown in figure 1.31 for wings of a basic 9 percent symmetrical section. The basic airfoil section properties are shown on these curves and these properties would be
typical only of a wing planform of extremely high (infinite) aspect ratio. When a wing of some finite aspect ratio is constructed of this basic section, the principal differences will be in the lift and drag characteristics—the moment characteristics remain essentially the same. The effect of decreasing aspect ratio on the lift curve is to increase the wing angle of attack necessary to produce a given lift coefficient. The difference between the wing angle of attack and the section angle of attack
Q
is the induced angle of attack, ai—18.24
ЛД
which increases with decreasing aspect ratio. The wing with the lower aspect ratio is less sensitive to changes in angle of attack and requires higher angles of attack for maximum lift. When the aspect ratio is very low (below 5 or 6) the induced angles of attack are not accurately predicted by the elementary equation for а і and the graph of CL versus a develops distinct curvature. This effect is especially true at high lift coefficients where the lift curve for the very low aspect ratio wing is very shallow and CLmgx and stall angle of attack are less sharply defined.
The effect of aspect ratio on wing drag characteristics may be appreciated from inspection of figure 1.31. The basic section properties are shown as the drag characteristics of an infinite aspect ratio wing. When a planform of some finite aspect ratio is constructed, the wing drag coefficient is the sum of the induced drag coeffi, and the section drag со efficient. Decreasing aspect ratio increases the wing drag coefficient at any lift coefficient since the induced drag coefficient varies inversely with aspect ratio. When the aspect ratio is very low, the induced drag varies greatly with lift and at high lift coefficients, the induced drag is very high and increases very rapidly with lift coefficient.
While the effect of aspect ratio on lift curve slope and drag due to lift is an important relationship, it must be realized that design for
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WING LIFT COEFFICIENT, CL WING LIFT COEFFICIENT, CL |
very high speed flight does not favor the use of high aspect ratio planforms. Low aspect ratio planforms have structural advantages and allow the use of thin, low drag sections for high speed flight. The aerodynamics of transonic and supersonic flight also favor short span, low aspect ratio surfaces. Thus, the modern configuration of airplane designed for high speed flight will have a low aspect ratio planform with characteristic aspect ratios of two to four. The most important impression that should result is that the typical modem configuration will have high angles of attack for maximum lift and very prodigious drag due to lift at low flight speeds. This fact is of importance to the Naval Aviator because the majority of pilot – caused accidents occur during this regime of flight—during takeoff, approach, and landing. Induced drag predominates in these regimes of flight.
The modern configuration of high speed airplane usually has a low aspect ratio planform with high wing loading. When wing sweep – back is coupled with low aspect ratio, the wing lift curve has distinct curvature and is very flat at high angles of attack, i. e., at high CL, CL increases very slowly with an increase in a. In addition, the drag curve shows extremely rapid rise at high lift coefficients since the drag due to lift is so very large. These effects produce flying qualities which are distinctly different from a more "conventional” high aspect ratio airplane configuration.
Some of the most important ramifications of the modern high speed configuration are:
(1) During takeoff where the airplane must not be over-rotated to an excessive angle of attack. Any given airplane will have some fixed angle of attack (and Cff) which produces the best takeoff performance and this angle of attack will not vary with weight, density altitude, or temperature. An excessive angle of attack produces additional induced drag and may have an undesirable effect on takeoff performance. Takeoff acceleration may be seriously reduced and a large increase in
takeoff distance may occur. Also, the initial climb performance may be marginal at an excessively low airspeed. There are modern configurations of airplanes of very low aspect ratio (plus sweepback) which—if overrotated during a high altitude, high gross weight takeoff—cannot fly out of ground effect. With the more conventional airplane configuration, an excess angle of attack produces a well defined stall. However, the modern airplane configuration at an excessive angle of attack has no sharply defined stall but developes an excessive amount of induced drag. To be sure that it will not go unsaid, an excessively low angle of attack on takeoff creates its own problems—excess takeoff speed and distance and critical tire loads.
(2) During approach where the pilot must exercise proper technique to control the flight path. “Attitude plus power equals performance. ” The modern high speed configuration at low speeds will have low lift – drag ratios due to the high induced drag | and can require relatively high power settings during the power approach. If the pilot interprets that his airplane is below the desired glide path, his first reaction must not be to just ease the nose up. An increase in angle of attack without an increase in power will lower the airspeed and greatly increase the induced drag. Such a reaction could create a high rate of descent and lead to very undesirable consequences. The angle of attack indicator coupled with the mirror landing system provides reference to the pilot and emphasizes that during the steady approach “angle of attack is the primary control of airspeed and power is the primary control of rate of climb or descent.” Steep turns during approach at low airspeed are always undesirable in any type of airplane because of the increased stall speed and induced drag. Steep turns at low airspeeds in a low aspect ratio airplane can create extremely high induced drag and can incur dangerous sink rates.
(3) During the landing phase where an excessive angle of attack (or excessively low airspeed) would create high induced drag and a high power setting to control rate of descent. A common error in the technique of landing modern configurations is a steep, low power approach to landing. The steep flight path requires considerable maneuver to flare the airplane for touchdown and necessitates a definite increase in angle of attack. Since the maneuver of the flare is a transient condition, the variation of both lift and drag with angle of attack must be considered. The lift and drag curves for a high aspect ratio wing (fig. 1.31) show continued strong increase in CL with a up to stall and large changes in CD only at the point of stall. These characteristics imply that the high aspect ratio airplane is usually capable of flare without unusual results. The increase in angle of attack at flare provides the increase in lift to change the flight path direction without large changes in drag to decelerate the airplane.
The lift and drag curves for a low aspect ratio wing (fig. 1.31) show that at high angles of attack the lift curve is shallow, i. e., small changes in Cl with increased a. This implies a large rotation needed to provide the lift to flare the airplane from a steep approach. The drag curve for the low aspect ratio wing shows large, powerful increases in CD with Cl well below the stall. These lift and drag characteristics of the low aspect ratio wing create a distinct change in the flare characteristics. If a flare is attempted from a steep approach at low airspeed, the increased angle of attack may provide such increased induced drag and rapid loss of airspeed that the airplane does not actually flare. A possible result is that an even higher sink rate may be incurred. This is one factor favoring the use of the ‘‘no-flare” or “minimum flare” type landing technique for certain modern configurations. These same aerodynamic properties set the best glide speeds of low aspect ratio airplanes above the speed for (L/D)^. The additional speed provides a more favorable margin of flare capability for flameout landing from a steep glide path (low aspect ratio, low (L/D)^, low glide ratio).
The landing technique must emphasize proper control of angle of attack and rate of descent to prevent high sink rates and hard landings. As before, to be sure that it will not go unsaid, excessive airspeed at landing creates its own problems—excessive wear and tear on tires and brakes, excessive landing distance, etc.
The effect of the low aspect ratio planform of modern airplanes emphasizes the need for proper flying techniques at low airspeeds. Excessive angles of attack create enormous induced drag which can hinder takeoff performance and incur high sink rates at landing. Since such aircraft have intrinsic high minimum flying speeds, an excessively low angle of attack at takeoff or landing creates its own problems. These facts underscore the importance of a “thread-the-needle,” professional flying technique.
