1
Use of Aerodynamic Forces to Control the Trajectory of an Aircraft

In order to steer an aircraft in space, we use aerodynamic forces (lift and drag) created on small surfaces called control surfaces to produce rotations on the three axes of the aircraft (roll, pitch and yaw).

1.1. Definitions

Consider a thin, flat plate placed in a rapid air current (Figure 1.1).

Figure 1.1. Horizontal plate. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

In the diagram, the plate is horizontal: it does not tend to go up or down, but only moves back. If this plate were moving in still air instead of being held still in moving air, it would be slowed by a force called drag, conventionally denoted RX. This force is due to the viscosity of the air, which sticks to the plate as it passes over and under it.

The plate is now tilted up. We gave it an angle with respect to the initial direction of the air streams (called the incidence a) (Figure 1.2).

Figure 1.2. Inclined plate. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

The air streams are deflected around the plate, which tends to rise and move back more sharply. The drag increases and a lift RZ appears, pulling it upward. The combination of the two forces, lift and drag, combine in an aerodynamic resultant R.

Figure 1.3. Aerodynamic resultant. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

NOTE.- The indices x and z assigned to the aerodynamic forces come from a trihedron defined by convention around an aircraft in flight (Figure 1.4).

G being the center of gravity of the aircraft, there are three axes:

• - the Gx axis, or roll axis;
• - the Gy axis, or pitch axis;
• - the Gz axis, or yaw axis.

Figure 1.4. Trihedron related to the aircraft's center of gravity

As the aircraft turns on the Gx, Gy or Gz axis, it rolls, pitches or yaws, respectively.

If we consider a wing section (Figure 1.5) or profile, we can define some important terms for what follows.

Figure 1.5. Wing section. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

1.1.1. Lift

The movement of the air around the wing creates, depending on the profile and the angle of attack, a weak depression or overpressure under the wing, on the undersurface, and especially a strong depression above, on the upper surface.

It is the large pressure difference between the upper surface and undersurface that creates the lift.

Figure 1.6. Lift

The wing moves forward with a certain angle of attack, which has the effect of deflecting the air downward. The air deflected violently downwards creates a void above, on the upper surface of the wing, which is thus sucked upwards. Air is a fluid that has a certain viscosity. This viscosity requires it to follow the profile; therefore, the air streams are separated by the wing, but they end up meeting after the trailing edge. When the wing passes, the air is therefore split in two: above and below.

The air below is deflected downwards, channeled as it is through the sloping undersurface. However, due to the viscosity of the air, the top streams also follow the profile. They stick to the upper surface, in exactly the same way a trickle of water follows the circular outline of a bottle placed under a faucet.

We have seen that lift is due to the intense downward deflection of the air and the viscosity of the air. Another phenomenon that contributes to the creation of dorsal wing depression is Bernoulli's principle, which says that accelerated air induces a drop in pressure (depression). This is what happens on the upper surface, where the air streams are accelerated as they try to travel a longer path (relative to the lower surface) in the same time.

1.1.2. Drag

The air flows along the surfaces and this friction produces significant braking. The airspeed is zero on the "skin" of the aircraft and it increases rapidly as soon as we move away from the surface. The layer of air close to the surface of the profile, where the air is slowed down, is called the boundary layer.

The shape of an object moving through air has a huge influence on induced drag. Anything that is smooth and tapered has little drag, because the air can move around the obstacle smoothly, effortlessly. Anything that disturbs the air and generates eddies is bad and increases drag.

The wing is a vortex generator. The pressure on the undersurface is much greater than on the upper surface. As a result, air passes from the lower surface to the upper surface at the ends and naturally to the trailing edge, to fill the void.

At the level of the wing tips, we can observe marginal vortices, which have a shape similar to that shown in Figure 1.7.

Figure 1.7. Drag

The passage of air from the bottom to the top of the wing creates a deflection of the air streams, outward of the wing for the lower surface and inward for the upper surface (Figure 1.8).

Figure 1.8. Deflection of air streams. For a color version of this figure, see www.iste.co.uk/louis/flight.zip

The intersecting air streams at the trailing edge also create vortices. To limit these marginal vortices, the wingtips are generally thinner. Thus, the passage of air from the bottom to the top is less abrupt. Some aircraft are equipped with flettners, kinds of vertical planes which reduce marginal losses and thus simulate a longer wing (the greater the aspect ratio, the more the influence of marginal losses decreases).

The marginal vortices increase drag and lower lift, creating airflow that rises from the lower surface to the upper surface and down to the wing. This descending air is animated by an induced velocity which is maximal at the wing tip and which decreases as we approach the center of the wing.

This induced velocity Vi has the effect of reducing the effective angle of attack of the wing towards the tips of the wings, which reduces lift.

Figure 1.9. Effects of induced velocity

NOTE ON FIGURE 1.9.-

• - V0: initial velocity;
• - Ve: effective velocity;
• - Vi: induced velocity;
• - a: initial incidence;
• - ae: effective incidence.

This ultimately results in a lift distribution over the wingspan in the shape of a semi-ellipse: the aircraft has a lot of lift in the center, and less and less the closer you get to the wing tips.

Figure 1.10. Lift distribution

1.1.3. Equilibrium in horizontal flight

Propeller pull opposes drag, just as lift opposes aircraft weight. In horizontal flight, we therefore have the equilibrium situation shown in Figure 1.11.

Figure 1.11. Equilibrium in horizontal flight

The lift is given only by the wing; on the other hand, the drag is induced by the wing itself as well as the fuselage, the antennas, the landing gear, etc. Ultimately, the wing is a small contributor to drag compared to everything else. Hence, we have the interest in the construction of aircraft of limiting as much as possible the relative importance of everything that does not produce lift. Making very long wings and a very thin, smooth fuselage increases the importance of lift over drag.

Here we come to the notion of the gliding ratio, which is the lift/drag ratio (or Rz/Rx). It is also the maximum horizontal distance that a dropped aircraft can travel from a given altitude, divided by this same altitude: if the aircraft can travel a distance of 10,000 m when you stop the engine at an altitude of 1,000 m in calm weather, then it has a gliding ratio of 10,000 ÷ 1,000 = 10. This means that the lift of this aircraft is 10 times its drag.

1.1.4. Aerodynamic moments

A force F which is exerted at the level of the center of gravity of any solid body does not turn that body. If the point of application is moved away by a distance d, the aforementioned force produces an effect which will tend to set the body in rotation: it is said that it applies the moment F*d to the solid body in question.

In order to operate in all three dimensions, the aircraft experiences roll, pitch and yaw moments, which cause the latter to rotate around the axes defined in the introduction (section I.1) and which pass through its center of gravity.

We have seen that there are aerodynamic coefficients that apply to lift and drag forces. Likewise, there are moment coefficients that apply to aerodynamic moments. The control surfaces of an aircraft generate moments of roll (the ailerons of an aircraft), pitch (the front wing flaps or rear horizontal plane of an aircraft) or yaw (the rudder or rear horizontal plane of an aircraft). Wind tunnel testing quantifies...