Курсовая работа: Airfoils and Lift

Fig 9 Coefficient of lift versus the effective angle of attack.

Typically, the lift begins to decrease at an angle of attack of about 15 degrees. The forces necessary to bend the air to such a steep angle are greater than the viscosity of the air will support, and the air begins to separate from the wing. This separation of the airflow from the top of the wing is a stall.

The wing as air "scoop"

We now would like to introduce a new mental image of a wing. One is used to thinking of a wing as a thin blade that slices though the air and develops lift somewhat by magic. The new image that we would like you to adopt is that of the wing as a scoop diverting a certain amount of air from the horizontal to roughly the angle of attack, as depicted in figure 10. The scoop can be pictured as an invisible structure put on the wing at the factory. The length of the scoop is equal to the length of the wing and the height is somewhat related to the chord length (distance from the leading edge of the wing to the trailing edge). The amount of air intercepted by this scoop is proportional to the speed of the plane and the density of the air, and nothing else.

Fig 10 The wing as a scoop.

As stated before, the lift of a wing is proportional to the amount of air diverted down times the vertical velocity of that air. As a plane increases speed, the scoop diverts more air. Since the load on the wing, which is the weight of the plane, does not increase the vertical speed of the diverted air must be decreased proportionately. Thus, the angle of attack is reduced to maintain a constant lift. When the plane goes higher, the air becomes less dense so the scoop diverts less air for the same speed. Thus, to compensate the angle of attack must be increased. The concepts of this section will be used to understand lift in a way not possible with the popular explanation.

Lift requires power

When a plane passes overhead the formerly still air ends up with a downward velocity. Thus, the air is left in motion after the plane leaves. The air has been given energy. Power is energy, or work, per time. So, lift must require power. This power is supplied by the airplane’s engine (or by gravity and thermals for a sailplane).

How much power will we need to fly? The power needed for lift is the work (energy) per unit time and so is proportional to the amount of air diverted down times the velocity squared of that diverted air. We have already stated that the lift of a wing is proportional to the amount of air diverted down times the downward velocity of that air. Thus, the power needed to lift the airplane is proportional to the load (or weight) times the vertical velocity of the air . If the speed of the plane is doubled the amount of air diverted down doubles. Thus the angle of attack must be reduced to give a vertical velocity that is half the original to give the same lift. The power required for lift has been cut in half. This shows that the power required for lift becomes less as the airplane's speed increases. In fact, we have shown that this power to create lift is proportional to one over the speed of the plane.

But, we all know that to go faster (in cruise) we must apply more power. So there must be more to power than the power required for lift. The power associated with lift, described above, is often called the "induced" power. Power is also needed to overcome what is called "parasitic" drag, which is the drag associated with moving the wheels, struts, antenna, etc. through the air. The energy the airplane imparts to an air molecule on impact is proportional to the speed squared. The number of molecules struck per time is proportional to the speed. Thus the parasitic power required to overcome parasitic drag increases as the speed cubed.

Figure 11 shows the power curves for induced power, parasitic power, and total power which is the sum of induced power and parasitic power. Again, the induced power goes as one over the speed and the parasitic power goes as the speed cubed. At low speed the power requirements of flight are dominated by the induced power. The slower one flies the less air is diverted and thus the angle of attack must be increased to maintain lift. Pilots practice flying on the "backside of the power curve" so that they recognize that the angle of attack and the power required to stay in the air at very low speeds are considerable.

Fig 11 Power requirements versus speed.

At cruise, the power requirement is dominated by parasitic power. Since this goes as the speed cubed an increase in engine size gives one a faster rate of climb but does little to improve the cruise speed of the plane.

Since we now know how the power requirements vary with speed, we can understand drag, which is a force. Drag is simply power divided by speed. Figure 12 shows the induced, parasitic, and total drag as a function of speed. Here the induced drag varies as one over speed squared and parasitic drag varies as the speed squared. Taking a look at these curves one can deduce a few things about how airplanes are designed. Slower airplanes, such as gliders, are designed to minimize induced drag (or induced power), which dominates at lower speeds. Faster airplanes are more concerned with parasitic drag (or parasitic power).

Fig 12 Drag versus speed.

Wing efficiency

At cruise, a non-negligible amount of the drag of a modern wing is induced drag. Parasitic drag, which dominates at cruise, of a Boeing 747 wing is only equivalent to that of a 1/2-inch cable of the same length. One might ask what affects the efficiency of a wing. We saw that the induced power of a wing is proportional to the vertical velocity of the air. If the length of a wing were to be doubled, the size of our scoop would also double, diverting twice as much air. So, for the same lift the vertical velocity (and thus the angle of attack) would have to be halved. Since the induced power is proportional to the vertical velocity of the air, it too is reduced by half. Thus, the lifting efficiency of a wing is proportional to one over the length of the wing. The longer the wing the less induced power required to produce the same lift, though this is achieved with an increase in parasitic drag. Low speed airplanes are affected more by induced drag than fast airplanes and so have longer wings. That is why sailplanes, which fly at low speeds, have such long wings. High-speed fighters, on the other hand, feel the effects of parasitic drag more than our low speed trainers. Therefore, fast airplanes have shorter wings to lower parasite drag.

There is a misconception held by some that lift does not require power. This comes from aeronautics in the study of the idealized theory of wing sections (airfoils). When dealing with an airfoil, the picture is actually that of a wing with infinite span. Since we have seen that the power necessary for lift is proportional to one over the length of the wing, a wing of infinite span does not require power for lift. If lift did not require power airplanes would have the same range full as they do empty, and helicopters could hover at any altitude and load. Best of all, propellers (which are rotating wings) would not require power to produce thrust. Unfortunately, we live in the real world where both lift and propulsion require power.

Power and wing loading

Let us now consider the relationship between wing loading and power. Does it take more power to fly more passengers and cargo? And, does loading affect stall speed? At a constant speed, if the wing loading is increased the vertical velocity must be increased to compensate. This is done by increasing the angle of attack. If the total weight of the airplane were doubled (say, in a 2-g turn) the vertical velocity of the air is doubled to compensate for the increased wing loading. The induced power is proportional to the load times the vertical velocity of the diverted air, which have both doubled. Thus the induced power requirement has increased by a factor of four! The same thing would be true if the airplane’s weight were doubled by adding more fuel, etc.

One way to measure the total power is to look at the rate of fuel consumption. Figure 13 shows the fuel consumption versus gross weight for a large transport airplane traveling at a constant speed (obtained from actual data). Since the speed is constant the change in fuel consumption is due to the change in induced power. The data are fitted by a constant (parasitic power) and a term that goes as the load squared. This second term is just what was predicted in our Newtonian discussion of the effect of load on induced power.

Fig 13 Fuel consumption versus load for a large transport airplane traveling at a constant speed.

The increase in the angle of attack with increased load has a downside other than just the need for more power. As shown in figure 9 a wing will eventually stall when the air can no longer follow the upper surface, that is, when the critical angle is reached. Figure 14 shows the angle of attack as a function of airspeed for a fixed load and for a 2-g turn. The angle of attack at which the plane stalls is constant and is not a function of wing loading. The stall speed increases as the square root of the load. Thus, increasing the load in a 2-g turn increases the speed at which the wing will stall by 40%. An increase in altitude will further increase the angle of attack in a 2-g turn. This is why pilots practice "accelerated stalls" which illustrate that an airplane can stall at any speed. For any speed there is a load that will induce a stall.

Fig 14 Angle of attack versus speed for straight and level flight and for a 2-g turn.

Wing vortices

One might ask what the downwash from a wing looks like. The downwash comes off the wing as a sheet and is related to the details of the load distribution on the wing. Figure 15 shows, through condensation, the distribution of lift on an airplane during a high-g maneuver. From the figure one can see that the distribution of load changes from the root of the wing to the tip. Thus, the amount of air in the downwash must also change along the wing. The wing near the root is "scooping" up much more air than the tip. Since the root is diverting so much air the net effect is that the downwash sheet will begin to curl outward around itself, just as the air bends around the top of the wing because of the change in the velocity of the air. This is the wing vortex. The tightness of the curling of the wing vortex is proportional to the rate of change in lift along the wing. At the wing tip the lift must rapidly become zero causing the tightest curl. This is the wing tip vortex and is just a small (though often most visible) part of the wing vortex. Returning to figure 6 one can clearly see the development of the wing vortices in the downwash as well as the wing tip vortices.

Fig 15 Condensation showing the distribution of lift along a wing. The wingtip vortices are also seen. (from Patterns in the Sky, J.F. Campbell and J.R. Chambers, NASA SP-514.)