Earlier, We Defined Three Categories of Drag Produced by a Wing

INDUCED DRAG

Earlier, we defined three categories of drag produced by a wing:

• parasite drag
• shock wave drag
• induced drag

Induced drag is often called drag due to lift, but producing lift also creates parasite drag, as we have seen.

Parasite Drag Resulting from Producing Lift with an Infinite Span Wing

A finite span wing (wing with wingtips) generates wingtip vortices, when air flows from high static pressure on the lower surface around the wingtips into low pressure air on the upper surface. See Figure 5.4 (text p. 65)

Figure 2.37 Wing tip vortices.

As a consequence, the airflow pattern across an infinite span wing is different than the airflow pattern across a finite span wing. Specifically, a finite span wing bends the relative wind downward at the trailing edge, so that it has a different direction than the relative wind at the leading edge. By contrast, the relative wind at the trailing edge of an infinite span wing flows in illustrate this distinction.

Note that Figure 5.8 (text p. 67) may be a little misleading, since it seems to suggest (incorrectly) that the wing is operating at a negative angle of attack.

Downwash angle. For a finite span wing, the angle between the relative wind at the leading and trailing edges is known as the downwash angle. The average relative wind on such a wing is thus “tilted upward” by ½ the downwash angle. This angle i is called the induced angle of attack. This effect in turn tilts the lift vector “backwards” by i, creating induced drag.

Key Point. The induced angle of attack is half the downwash angle.

In essence, for a finite span wing, the aircraft must be operated at a higher pitch angle to get the same lift, thus creating additional drag. See Figures 5.9 (text p. 67) and 5.14(text p. 74)

Figure 5.9 Relative wind and force vectors on a finite wing

Figure 5.14 Drag vector diagram (for finite span wing)

Again, Figure 5.14 (like Figure 5.8) seems to suggest that the wing is operating at a negative angle of attack. This implication is incorrect.

Key Point. The total drag due to lift on a finite span wing has two components: one due to induced drag, and one due to parasite drag. The total drag produced by the wing is the vector addition of these two components.

Effect of Wing Planform on Induced Drag and Lift

In general, induced drag decreases as airspeed increases (and AOA decreases).

However, the planform and aspect ratio (AR) of a wing has a marked effect on induced drag.

Let b be the wing span, and c the chord of a rectangular wing. By definition,

AR = b / c;

i.e., aspect ratio is span divided by chord.

For other planforms, if cave is the average chord, then AR = b / cave. Since average chord is just wing area S divided by the average chord (Cave = S / b), we have

AR = b / cave = b / (S/b) = b2 / S.

Figure 5.3 (text p. 64) illustrates these and related concepts.

Figure 5.3 Wing planform terminology

Concepts illustrated on Figure 5.3include:

1. AR = b / cave = b / (S/b) = b2 / S (as explained above).
2. Taper ratio (small lambda) is tip chord ct divided by root chord cr: = ct / cr.
3. Sweepback angle (cap lambda) is the angle between the 25% chord line and a perpendicular to the root chord. (The 25% chord line is a straight line connecting points ¼--or 25%--of the way aft of the leading edge on the root and tip chords).
4. Mean aerodynamic chord (MAC) of a wing is the chord that divides a half wing (port or starboard wing) into equal areas.

Interesting Fact (of more interest to aeronautical engineers, perhaps, than to pilots): Suppose a wing with arbitrary planform has span b and MAC cmac. Then a rectangular planform wing of span b and chord cmac produces the same pitching moment at the same AOA.

In general, low aspect ratio means less lift and more induced drag, and vice versa, as illustrated in Figure 2.44 (text 1st edition, p.55):

Figure 2.44. Effect of aspect ratio on wing characteristics.

Also, sweepback angle  affects induced drag, since it also affects aspect ratio. Large  (high sweepback) is associated with low AR and high induced drag, particularly at high CL (e.g., in slow flight; i.e., for takeoff and landing).

L/D CURVES FOR AN ENTIRE AIRCRAFT

Lift and drag as a function of AOA can be computed in a wind tunnel for any object: a flat plate, a wing, or an entire aircraft. Here we are interested in lift and drag for an entire airplane.

From the lift and drag equations:

, where in the present context CL and CD are for the entire airplane.

L/D is called the lift-drag ratio. The L/D curve can be computed from CL and CD curves, either for the wing or for the entire aircraft, depending on which CL and CD curves are used. Figures 5.1 (text p. 62) and 5.2 (text p. 62) are relevant:

Figure 5.1. CL- and CD- curves.Figure 5.2. Typical lift to drag ratios.

(L/D)max—the highest point on the L/D curve—and the associated AOA are very important from the pilot’s point of view.

Key Point. (L/D)max is the highest point on the L/D curve.

Assume W = L for 1 G steady state flight. (This approximation is not far from correct.) Then L/D = W/D, and when L/D is maximum for a given weight, drag must be minimum. That is,

Dmin = .

Note, this identity applies for any arbitrary gross weight. The associated velocity V will vary with weight, but the AOA for (L/D)max is constant for a given aircraft and configuration.

Example: Suppose weight is 20,000# and (L/D)max = 11.0. Then

Dmin = W / (L/D)max = 20,000# / 11.0 = 1818.18#.

Key Point. Minimum drag in a given configuration varies directly with weight, but always occurs at the AOA corresponding to (L/D)max.

Also, we have already seen that the best power out glide ratio occurs at the AOA for (L/D)max. Recall from Unit 2 that when a is the glide angle, tan a = (w sin a) / (w cos a) = D/L = absolute altitude/glide distance, as shown below. This means a is minimum when (L/D) is maximum. We can conclude that the glide distance to altitude ratio is just (L/D)max.

Example: Suppose (L/D)max is 12.0 in the clean configuration and 5.5 in the dirty configuration. This means that the aircraft clean will glide 12 nm (or statute miles, or feet, or meters, or &c.) for every nm (or statute mile, or foot, or meter, or &c.) of altitude above ground level. Dirty it will glide 5.5 nm for every nm of altitude.

Key Point. (L/D)max gives the glide ratio for maximum range power off glide. Best glide velocity varies directly with weight, but AOA for best glide is constant for a given configuration.

TOTAL DRAG VS. VELOCITY CURVE

Induced drag, which is always a result of producing lift, decreases as AOA  decreases and velocity V increases. Parasite drag by contrast increases as V increases. Here we are still talking about lift and drag for the entire aircraft, not just the wing. 5.10 (text p. 68) and Figure 5.13 (text p. 74) are relevant:

Fig 5.10. Induced drag versus velocity.Figure 5.13. Parasite drag versus velocity.

Total drag for an aircraft (or a wing) is just the scalar addition of induced and parasite drag for the aircraft (or wing). Mathematically, Dt = Di + Dp. Figure 6.2 (text p. 82) shows the total drag curve for a unspecified aircraft (in fact, an Air Force T38 jet trainer at sea level and 10,000# gross weight). Ordinarily these curves are for an aircraft in the clean configuration.

Assuming thrust is equal to drag in level flight (again, an approximation not far from correct), each pound of drag requires a pound of thrust to offset it. Thus the total drag curve is also the thrust required curve (Figure 6.3, text p. 83). Again, this curve is for the T38 jet trainer at sea level and 10,000# gross weight.

Key Point. Total drag and thrust required curves are essentially the same, for the same aircraft, altitude, and gross weight.

Figure 6.2. Total drag curve. Figure 6.3. T38 thrust required.

Key Point. Total drag / thrust required curves are for a given gross weight and altitude, as indicated above.

Key Point. (L/D)max occurs at the minimum point on the drag curve, i.e., at minimum drag, as shown in the section on L/D curves.

SLOW FLIGHT (FLIGHT IN LANDING CONFIGURATION)

I. Effect of Sweepback on Stall Characteristics

Sweepback decreases aspect ratio for a constant planform area, hence increases induced drag and decreases lift for a given AOA. In addition, the CL curve for a sweptback airfoil is much flatter than for a straight wing aircraft, profoundly influencing flight characteristics at AOAs beyond CL(max) (stall onset). 11.1 (text p. 168) illustrates these claims.

Figure 11.1. Effect of sweepback on CL -  curve.

II. Other Effects of Wing Shape on Stall Behavior

A straight wing aircraft has a pronounced loss of lift at stall, while a swept wing aircraft loses lift gradually. However, high induced drag at stall on a sweptback wing may mean that insufficient power is available to “fly out of the stall,” and that altitude must be traded for airspeed to recover. Thus stalls past the ninety in swept wing aircraft are likely to result in a crash, even if there is no resultant roll/spin effect.

CL is the average coefficient of lift for a wing. Let Cl be the local coefficient of lift for any given point on the wing. The local wing loading ratio is defined to be Cl / CL. Beyond  for CL(max), an airfoil stalls first where the local wing loading coefficient is highest.

It has been found that an elliptical planform wing has the most even wing loading ratio. Specifically,

Cl / CL 1 over the whole wing, and the wing stalls evenly from root to tip on the upper aft surface. Figure 4.2 (text p. 127) gives the wing loading ratios for different wing planforms.

Recall that the taper ratio (small lambda) of a wing is just the tip chord divided by the root chord, i.e.,

 = tip chord / root chord. For example, for = 1.0, 0.5, and 0.25, the tip chord is respectively 1/1, ½, and ¼ of the root chord. When  = 0, since the root chord is finite, the tip chord must be zero; i.e., the wing tapers to a point. As  decreases, Cl / CL is increasingly higher at the wing tip compared to the wing root, as illustrated in Figures 11.2 and 11.3 (text pp. 168-169).

Key Point. Wings stall first where the wing loading ratio Cl / CL is highest.

Figures 11.2 and 11.3. Wing spanwise lift distribution

Since different planforms have different wing loading patterns, they have different stall characteristics. Figure 11.4 (text p. 170) illustrates stall characteristics of different planforms. For each shape, stall occurs first where the wing loading ratio is highest.

Figure 11.4. Stall patterns

For straight wing aircraft, root first stall means ailerons are effective well into the stall, and turbulent air contacts the fuselage and tailplane, giving a stall alarm. Neither is true for aircraft which stall at the tip first and have outboard ailerons. In this case, must use rudder to yaw/roll, avoiding aileron movement, which can induce spin.

In particular, swept wing aircraft stall at the tip first, and progress slowly into a stall, with the result that adequate stall warning many not occur. Sometimes, a horn or rudder shaker is used to indicate that stall AOA is about to be reached. Also, sometimes the angle of incidence at the tip is reduced in swept wing aircraft design, so that stall occurs more evenly over the entire wing, or at the root first. This however cannot improve the flat shape of the CL – AOA curve for a swept wing, so stall behavior is usually not improved.

III. Region of Reversed Command

We have seen on the thrust required curve for SS 1G flight that, except when thrust corresponds to (L/D)max, a given thrust setting can produce two often very different airspeeds. Figures 11.5 (text p. 171) and 11.6 (text p. 172) are relevant

Figure 11.5. Regions of normal and reversed command Figure 11.6. Climbing an aircraft, stick or throttle?

Flight in the region of reversed command is required for takeoff and landing, most importantly for jet powered swept wing aircraft. During takeoff, the duration of such flight is brief, but when landing, the time flying on the so-called “back side of the power curve” may be prolonged. (The book insists we must call it the “back side of the thrust curve,” but the former expression is more common in my experience.)

To control airspeed and altitude in the region of reversed command (SS level 1G flight):

• To slow down, increases AOA (back stick); it wills be necessary to add power to avoid descending, though initially a climb may result.
• To speed up, decrease AOA (forward stick); it will be necessary to reduce power to avoid climbing, although initially a sink rate may develop.

For approaches, coordinating stick and throttle is a little more complex, but in general:

Attitude (AOA) controls A/S. Throttle controls vertical speed (rate of climb or descent).

Note: you may hear a different analysis in AS310, Performance, involving the supposed difference between the way Air Force and Navy fighter pilots fly approaches. From my own experiences, it is true that Air Force pilots fly a “hotter” (faster) approach than Navy carrier pilots, who are trained to fly just a few knots above stall to minimize the energy which must be absorbed by the arresting gear during a carrier landing. However, both Navy and Air Force fighter aircraft approach on the back side of the power curve. On the other hand, even in the region of reversed command, excess speed may be traded for altitude, so Air Force pilots might use back stick rather than power as an initial response to low-on-glide-slope information, and sometimes push the nose down to remedy a high-on-glide-slope. All this means is they aren’t as concerned as Navy pilots about maintaining speed within 1-2 knots of the optimum. If you’re 20 knots above stall, losing 5 knots of airspeed isn’t very important. If you’re 5 knots above stall, then losing even 2-3 knots should be attention grabbing. And if you’re 5 knots fast, you’ll be waved off from the carrier because your airplane has too much kinetic energy to engage the arresting gear safely.

(The fact that Air Force pilots tend to fly a little faster on the glide slope for safety’s sake doesn’t mean they aren’t as good as Navy pilots, no matter what Navy people say. When Air Force pilots land on short runways, they slow down and fly just like Navy pilots.)

IV. Influence of Ground Effect

Recall that induced drag is caused when wingtip vortices create a downwash at the wing’s trailing edge, effectively changing the direction of the relative wind. Ground effect “neutralizes” downwash, increasing lift and decreasing drag. Ground effect starts within one wing span of the ground, and is more noticeable on low wing aircraft than on high wing aircraft. Figure 5.12 (text. 71) illustrates a number of important points about ground effect.

Figure 5.12. Ground Effect.

Some key points about ground effect.

• Light aircraft float more than heavy aircraft in ground effect.
• For props, can use ground effect over water to reduce DI and increase max range. Don’t get careless and fly into the water!
• For low horizontal stabilizer aircraft, downwash may hit stabilizer in normal flight, creating a nose up pitch which must be trimmed out. In ground effect, this nose up pitch disappears, resulting in a nose down pitch on close final.
• For low horizontal stabilizer aircraft, leaving ground effect on takeoff may cause a nose up pitch effect. See previous bullet for explantion.
• If the pitot-static system static port is low on an aircraft, static pressure may increase upon entering ground effect, giving a low airspeed reading.

A summary some of these effects is given in the following table.

AIR TURBULENCE

I. Wind Shear

Wind Shear is a change in wind velocity (speed or direction or both) at two points in close proximity in the atmosphere. Some facts about wind shear:

• 180o direction and 50 kt speed changes have been observed.
• Thunderstorms, fronts, and low level jet streams can cause wind shear.
• Downbursts from thunderstorms create wind shear.
• Headwind, tailwind, and crosswind bursts can occur when a downburst contacts the terrain.
• Wind shear is very dangerous, especially when experienced close to the ground. It is known to have caused a number of airline accidents involving the loss of many lives.

Figure 11.7 (text p. 174) and Figure 11.8 (text p. 174) illustrate how a thunderstorm can cause downbursts, and headwind, tailwind, and crosswind bursts.