SP-367 Introduction to the Aerodynamics of Flight

VIII. PERFORMANCE

[137] In the earlier discussions, the concepts of lift and drag were explored extensively to discover how these forces arise. With these basic ideas in mind, it is relatively easy to follow the results of the application of the fundamental forces on a complete airplane.

As indicated earlier, there are four basic forces that act on an airplane - these include lift, drag, weight, and thrust. Additionally, in curved flight another force, the centrifugal force, appears. Performance, to be considered first, is basically the effects that the application of these forces have on the flight path of the airplane. Stability and control, considered later, is the effect that these forces have over a short term on the attitude of the airplane itself. For performance purposes the airplane is assumed to possess stability and a workable control system.

Motions of an Airplane

Figure 116 illustrates the various flight conditions encountered by an airplane. All the motions may be grouped into one of three classes: (1) unaccelerated linear flight, (2) accelerated and/or curved flight, and (3) hovering flight.

Performance of an airplane is a very broad subject and much could be written on it alone. In the interest of brevity, therefore, only the simplest, but probably the most important, aspects of airplane flight are considered.

Class 1 Motion

Straight and level unaccelerated flight (cruise flight).- Although straight and level flight may occur only over a small section of the total flight, it is very important since it is usually considered the standard condition in the design of an airplane. This condition has been touched on before but some additional comments will be made.

Figure 117 shows the force system for straight and level flight. The flight path is horizontal to the Earth's surface and for simplicity it is assumed that the thrust always acts along this horizontal plane. For the flight to be horizontal, or constant altitude, it is easily seen that lift must equal weight. To fly at constant velocity (unaccelerated) the thrust must equal the drag.

The velocity of the airplane must be sufficient to produce a lift equivalent to the weight. If one examines this statement closely, it says that there is a range of velocities over which the plane may fly straight and level. Expanding equation (25) and combining it with the condition that Lift = Weight, one obtains..[138]

 (36)

If it is assumed that the weight, air density , and wing area S are constant, one easily observes that as the velocity increases, the wing lift coefficient CL decreases, which may be accomplished by a decrease in the wing angle of attack. Minimum flying speed for straight and level night occurs when the wing is operating at CL,max, that is, near the stall angle. The maximum flying speed for straight and level night is limited by the thrust available from the engine. This condition also requires a small value of CL and hence a small angle of attack.

Figure 116.- Airplane flight conditions.

[139] In conclusion, at low speeds to fly straight and level the airplane angle of attack is large (fig. 118(a)) whereas for high speeds the airplane angle of attack is small (fig. 118(b)).

Figure 117.- Straight and level night. Lift = Weight; Thrust = Drag.

(a) Straight and level-low speed.
(b) Straight and level-high speed.

Figure 118.- Speed effects on straight and level flight.

Straight, unaccelerated ascent (climb) or descent (dive).- Figure 119 illustrates the force systems for the cases of an airplane in a straight, constant-velocity climb or dive. It has been assumed that the thrust line lies along the free-stream direction or flight path. The climb or descent angle is given by +y or -y [Greek letter gamma], respectively. If the...

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(a) Climb, unaccelerated.
(b) Dive, unaccelerated.

Figure 119.- Unaccelerated ascent and descent.

....forces are summed parallel and perpendicular to the flight path, it is seen that the weight force is resolved into two components. One obtains

 L = W cos y = W cos (-y ) (Climb or dive) (37)

 T = D + W sin y (Climb) (38)

 T = D + W sin (-y ) = D - W sin y (Dive) (39)

To maintain a straight climbing (or diving) flight path, the lift equals the component of weight perpendicular to the flight path (eq. (37)). In the case of the climb condition to maintain a constant velocity the thrust must equal the drag plus a weight component retarding the forward motion of the airplane. In the case of the dive condition the weight component along the flight path helps the thrust by reducing the drag component for constant velocity.

[141] The conclusion is that one must use an increased thrust to climb at constant velocity and use less thrust to dive at constant velocity. This is analogous to the situation of a car where one must "give it the gas" (apply more thrust) to prevent the car from slowing down in going up a hill and "let up on the gas" (use less thrust) to prevent the car from speeding up when going down a hill.

It is interesting also to examine three special cases of the use of equations (37), (38), and (39). First, in straight and level flight, the climb angle y is zero, hence sin y = 0 and cos y = 1. This yields the previously derived conditions that Lift = Weight (L = W) and Thrust = Drag (T = D). Secondly in a vertical climb y = 90°, and hence sin y = 1 and cos y = 0. Thus, the thrust necessary to climb vertically is equal to the drag plus the airplane weight (T = D + W). Also, for a vertical climb, the lift equals zero (L = 0). This condition is shown in figure 119(c).

[142] The final condition to be discussed is gliding flight. In gliding flight the thrust equals zero. It is therefore necessary to balance the aerodynamic reaction forces of lift and drag with the weight. Equation (37) remains unchanged but equation (39) is simplified. In a glide...

 L = W cosyg (40)

 D = W sinyg (41)

..as shown in figure 120(a). If one divides equation (40) by equation (41), the result is...

 L/D = 1/tanyg (42)

In nonmathematical language this means that the smallest glide angle, and hence maximum gliding range, is obtained when the lift-drag ratio is the maximum. The liftdrag ratio is a measure of the aerodynamic efficiency of the airplane. Sailplanes possess the greatest lift-drag ratios with excellent aerodynamic design since they rely on air currents to keep them aloft. For a particular airplane, as shown in figure 120(b), the lift-drag ratio varies with the angle of attack of the airplane (not to be confused with the glide angle of the flight path). There is a particular angle of attack for which this ratio is a maximum. This is then the angle of attack for minimum glide angle and maximum range. For any other angle of attack, the lift-drag ratio is less and the glide angle is increased; hence, a steeper glide results. It is a natural tendency for a pilot to raise the airplane nose (increase the angle of attack) to try to get maximum range but unless this gives the maximum lift- drag ratio, the descent will be steeper instead.

Class 2 Motion

Class 2 accelerated motion and curved flight is considered, specifically for the cases of take- off, landing, and the constant-altitude banked turn.

Take-off.- The take-off of an airplane is a case of accelerated motion. From the instant the airplane begins its take-off roll to the time it begins its climbout after leaving the ground, it is under continuous acceleration. (See fig. 121.) The total takeoff distance needed may be considered to consist of three parts: (1) the ground-roll distance, (2) the transition distance, and (3) the climbout distance over, say, a 15.25-m (50-ft) obstacle.

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(a) Unaccelerated glide conditions.
(b) Glide aerodynamic characteristics.

Figure 120. Glide characteristics.

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Figure 121.- Total take-off distance.

Figure 122 shows the forces acting during the ground roll. In addition to thrust, weight, drag, and lift, there is a rolling frictional force due to the landing gear. The sum of the forces in a horizontal direction is equal to the net force acting to accelerate the airplane down the runway. At the beginning of the ground roll, lift and drag are zero as dynamic pressure is still zero (assuming no winds). Acting under the net acceleration (thrust exceeding the total retarding force), the velocity increases and lift and drag build. The airplane remains in a horizontal attitude until some velocity (about 10 percent above the airplane stall velocity for safety) is reached at which point the airplane is "rotated" or pitched up. The pitch increases the airplane angle of attack, the lift quickly exceeds the weight, and the airplane leaves the ground. Rolling friction forces drop to zero at liftoff, and the airplane's total drag decreases greatly as the landing gear is retracted. At the end of transition, about 20 percent above the stall velocity, the airplane begins its climbout usually at constant velocity. The ordinary equations for climb (eqs. (37) and (38)) apply in this case.

Figure 122.- Forces acting during take-off ground roll.

[145] The total distance for the airplane to clear 15.25 m (50 ft) from the start of its roll is important and determines the amount of runway required for design purposes. Additionally, the pilot should know the maximum speed from which the take-off may be aborted so that sufficient runway exists for deceleration to a stop.

The take-off distance may be reduced by the use of flaps and other high lift devices. However, there is a limit to their use since they also contribute to increased drag and retard the airplane's acceleration. There is usually an optimum flap setting for an airplane which will minimize the take-off distance. Some airplanes may also use rocket-assisted units to take off in the minimum distance. These units represent a transitory increase in thrust and provide a means of high acceleration for short periods. On board an aircraft carrier, this method takes the form of a catapult, where flying speed is achieved in a matter of a second or two.

Landing.- Landing an airplane consists of touching down at the lowest possible vertical and horizontal velocities. The approach phase and its associated techniques to a landing will not be considered, but only the two terminal phases, namely, the touchdown and ground rollout.

Under touchdown conditions it is assumed that the vertical velocity is near zero and that the lift equals the weight. The previous discussion about flaps indicates that they are used advantageously to decrease the landing velocity. Indeed, they increase the maximum lift coefficient and decrease the landing velocity as indicated by equation (35).

Figure 123 presents the forces acting on an airplane during the landing rollout. They are the same as during the take-off except for their magnitude and direction. The rolling friction is greater as the brakes are applied. For safe operation this condition occurs near the end of the rollout. Spoilers on the wings are used to "dump" the airplane lift to prevent the airplane from rebounding into the air after touchdown. This condition increases the rolling friction as the normal force is increased. The engine thrust is zero or, more usually for large commercial and military airplanes, is negative. This condition is accomplished by using reversible pitch propellers or thrust reversers. For ground roll during landing the thrust force is retarding. The airplane drag may be increased by setting the flaps for maximum drag. From figure 123, therefore, there is a net deceleration acting on the airplane to slow it to a stop. Another favorite braking device used by military airplanes is the parachute which is opened at touchdown. On board aircraft carriers, the usual landing brake is mechanical in the form of the arresting hook on the airplane engaging a cable laid across the flight deck. Deceleration is exceedingly swift and the airplane is subjected to large structural forces.

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Figure 123.- Forces acting after landing.

Constant-altitude banked turn.- As shown in figure 116, not all motions of an airplane are in a straight line. There are ample cases of curved flight paths. These cases include the climbing and descending turns, maneuvers in combat and aerobatics. One of the basic maneuvers required to change the flight-path heading is the constant altitude banked turn.

In the previous discussions of motions of an airplane, accelerations due to a change of direction of flight were insignificant. But in a turn they acquire added significance. By Newton's first law, a body in motion in a straight line will continue in motion in that same line unless acted upon by an external force. To maintain an airplane in a curved path requires that an acceleration be supplied toward the center of the curve. By Newton's second law the force required to perform this, called centripetal force, is proportional to the acceleration required to maintain the curved flight. By Newton's third law there is a reactive force by the body, opposite the centripetal force, called the centrifugal force. The centrifugal force is given by:

 (43)

where m is the mass of the airplane, is the velocity of the airplane in the curve, and R is the radius of the turn or curved flight path. From this equation one sees that the highest centrifugal forces occur for massive airplanes at high speeds in tight turns.

Figure 124 shows the disposition of forces in a properly executed turn. Notice particularly that the wings are banked at an angle ø [Greek letter theta] to the horizontal. This angle causes the resultant lift on the wings to bank also. When resolved into vertical and horizontal components, it is seen that it is the horizontal component of lift that is the centripetal force needed to maintain the curved flight path. This force is balanced by the reaction centrifugal force. For a constant-altitude turn the vertical component of lift must equal the weight. Thus, the total lift must be increased to maintain constant altitude when entering a banked turn.

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Figure 124.- Forces in a properly banked turn. Vertical lift = Weight;

Horizontal lift = Centrifugal force.

The smaller the turning radius is or the greater the velocity in a turn, the larger the banking angle must be. This is required to produce a large enough horizontal lift component to hold the airplane in the turn.

Class 3 Motion-Hovering Flight

Class 3 motion has been assigned to a special flight condition; that of hovering flight. In hovering flight there is no motion of the aircraft with respect to the atmosphere. As such, this results in no aerodynamic reaction forces of the aircraft on the whole, that is, no lift and drag forces. In equilibrium, the remaining forces, thrust and weight, must be balanced as shown in figure 125. Hence, for hovering flight,

 Thrust = Weight (44)

By properly controlling the thrust, the aircraft may be made to rise and descend vertically as shown in figure 126. The chief advantage of such aircraft is their ability...

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Figure 125.- Hovering flight. Thrust = Weight.

Figure 126.- VTOL ascent and descent.

[149] ...to land and take-off in small spaces without the use of long runways. Since they land and take- off vertically they are called VTOL aircraft. They have the added distinction of being able to perform at high speeds as a conventional airplane in flight. This is why helicopters, although capable of hovering flight, are usually not included in this grouping. They are, at present, incapable of the speeds and maneuvers of conventional airplanes.

The first concepts to be tried were three "tail sitting" airplanes, the Lockheed XFV-1, the Convair XFY-1, and the Ryan X-13 Vertijet as shown in figure 127. The first two used turboprop- powered contrarotating propellers to supply the vertical thrust needed whereas the X-13 was jet powered. The main problems with these VTOL airplanes were the tricky piloting maneuvering required in the take-off and landing and the need to tilt the entire aircraft over into conventional flight. The next concept tried was to keep the main body of the aircraft in a conventional sense but tilt the wing and engines from the vertical to the horizontal. The LTV-Hiller-Ryan XC-142A in figure 128(a) was such an aircraft.

Another concept was to use separate powerplants for vertical take-off and landing and conventional level flight. But this added dead weight to each flight regime. For simplicity and efficiency, the Hawker Siddeley Harrier (fig. 128(b)) is one of the best present-day VTOL aircraft. This plane uses the concept of "vectored thrust" where four rotating exhaust nozzles are used to deflect the exhaust from vertically down to directly behind as shown in figure 128(c). Control at low flight velocities and in hovering flight is supplied by reaction jets in the wing tips, nose, and tail.

Figure 127.- Figure 127. Early VTOL airplanes.

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Figure 128. VTOL concepts.