Chapter 20 The light aeroplane as an example of the use of aerofoils
Read as one text
1.1 Stability in pitch
1.3 The response of low-winged aeroplanes to a disturbance in roll
2.2 Relationship between rate of turn, centripetal acceleration, angle of bank and airspeed in a turn.
2.3.1 Flight relative to the wind and to the ground.
2.3.2 Turning in a wind
3.3.1 Use of a twist in the wing (washout)
3.3.2 Use of a change in section from root to tip
3.3.3 The use of flaps
3.3.4 Using flaps to descend
Section 4 Taking off and the climb to cruising height.
4.1 Acceleration to rotation
4.2 The use of flaps
4.5 Climbing to the ceiling
Section 5 Aerobatic flying
Limits to the location of the centre of gravity of an aeroplane
How an aeroplane flies
I have spent a whole long chapter on the aerofoil and reached the point where I need to explain how aerofoil data may be used to select an aerofoil for a proposed application. The most obvious application of the aerofoil is the aeroplane wing indeed one might argue that aeroplane wings existed before we began to think about aerofoils and their behaviour.
The goal of the Wright Brothers was to achieve safe, controlled flight in any way that it could be done but now things have advanced to the point where we expect heavy aeroplanes to fly at high speed and to be able to take off and land at much lower speeds. Somehow we have to select aerofoils for use in our wings that will behave predictably and manageably for what we now know to be the various phases of flight. We must decide what demands will be made on the wings and by extension the aerofoil, during these several phases and this means that we have to understand how an aeroplane flies.
It seems to me that it would be best for me to use the light aeroplane by way of illustration not least because anyone can look at light aeroplanes close-up from the viewing enclosure at local airfields.
For various reasons most light aircraft have evolved to have only one basic control arrangement and this arrangement is shown in figure 20-1 where alternative versions for the stabiliser are shown. One is the original fixed stabiliser with inset elevator and the other is the more recent (post-war) all-moving stabiliser which is now quite common. The primary controls are the ailerons and either the all-moving stabiliser or the elevator.
The ailerons are interconnected so that as one goes up the other goes down. This produces forces on the outboard sections of the wings that are in opposite directions so that they make the aeroplane roll. The elevator or all-moving stabiliser is used to control the force on the stabiliser either to lift the aft end of the fuselage up or to push it down. We can see why these are the primary controls if we consider an aeroplane with an engine that is powerful enough for the aeroplane to loop from level flight. All that is needed is to apply “up” elevator and the aeroplane will loop. If now this aeroplane in level flight is made to roll through 90º by the use of the ailerons and then, after the ailerons are centred, immediately performs a horizontal half loop followed by a half roll through 90º in the opposite direction the aeroplane will make a change of course of 180° in a “pylon” turn. Any other turn can be made by rolling through any angle that is less than 90° and using much less elevator movement. Then the aeroplane flies round with one wing low. This is the type of turn made by aeroplanes with engines of "normal" power.
The other flight control is the rudder which is a part of the fin. The fin is set up at right angles to the stabiliser and provides a “weathercock” effect. The deflection of the rudder makes the fin exert a transverse force on the tail of the aeroplane. This is not generally used for directional control but has several functions when used with the two primary controls especially during crosswind landings to align the aeroplane with the runway at the time of touchdown and during turns to keep the fuselage aligned with the intended direction of flight to minimise drag.
Most aeroplanes of latter years have been fitted with some sort of flap. The flaps are fitted to the wings inboard of the ailerons. Mechanically they are very like ailerons but on powered aeroplanes they can only be deflected downwards. Then they have two main functions. They are lowered by 5° to 10° to give a high lift coefficient for take-off (at the expense of increased drag) and they are lowered by perhaps 30° to give high lift and high drag at low speed for landing.
In pictures 20-2 and 20-3 I have shown typical designs for light aircraft. The Cessna is a high-winged aeroplane with inset elevator, the Robin is low-winged with an all-moving stabiliser. Both have flaps.
Here, when we are talking about an aeroplane, we are talking about a device that is unconnected to the Earth and is free to move in the air in any direction yet must be made to move in a controlled manner. The brief description that I have given of the controls of a light aeroplane is totally inadequate and now I must deal with the phases of flight and the function of the controls in detail. In order to do so I need to give names to the three coordinate axes about which an aeroplane can rotate.
If we are to look at the behaviour of an aeroplane in level flight in response to disturbances in the air we shall need some reference axes. These could be any three axes but for our purpose it makes sense to choose the axes shown in figure 20-4. As the aeroplane can rotate about its fore-and-aft axis just as a ship can this is called the “roll” axis. Ships also pitch and so rotation about a transverse axis is called “pitch”. The third axis is one at right angles to the first two and is, for an aeroplane, when it swings to left or right and this is called “yaw".
Aircraft have five phases of flight, take-off and climb, descent and landing, level flight in a straight line, turning and aerobatics. At this early stage the phase that must interest us is level flight in a straight line. We can start by looking at the forces and moments that act on an aeroplane when it is in level flight. Figure 20-5 shows a light aeroplane in level flight. On it I have shown the five forces that must be in equilibrium. They are in realistic positions. The forces are the weight, the thrust, the total drag, the force on the stabiliser and the lift. I have ignored the two couples on the wing and the stabiliser and the torque on the aeroplane that is the reaction to the rotation of the propeller. The couples are small when compared with the accuracy of aerofoil data and the engine torque must be dealt with separately by trimming the aeroplane.
The dominant force is that due to gravity, that is the weight of the aeroplane. It acts at the centre of gravity and no matter what attitude the aeroplane may adopt it will always act vertically downwards. If this aeroplane is to fly, the wings must generate an upward force that is equal to the weight less the stabilising force on the tail-plane (supposing that to be upwards). The lift will not be generated unless the aeroplane is moving through the air and for this to happen the propeller must generate a thrust. The thrust, that acts along the axis of the engine but not necessarily along the axis of the aeroplane, has to overcome the drag of the air on the aeroplane.
There is no inbuilt reason why the lift and the weight should be in line or at any desired distance apart. If we want them to be in some special place we must arrange for them to be so. The centre of gravity must be in front of the main wheels but its actual position is affected by the position and weight of the occupants. The aeroplane will have been designed so that the wings, the centre line of the engine, and the stabiliser are all attached to the fuselage at the correct positions and angles to make the aeroplane efficient in level flight.
I have shown the thrust acting above the line of the drag and, to balance out the resulting couple, I have let the weight act behind the lift and shown a small balancing force on the stabiliser.
Light aeroplanes have to be flown by ordinary people who will never be test pilots and this means that they must be relatively easy to control. What is required is an aeroplane that responds reasonably quickly to any control change that may be made by the pilot but also does not require continuous input from the pilot in order to climb or descend and to fly more or less straight and level. We need to examine the behaviour of aeroplanes in general.
The air in which aeroplanes fly is moving as a whole i.e. as a wind and, within that motion, there are much smaller disturbances produced by uneven solar heating and by interaction between terrestrial features and the general movement of the air. When a light aeroplane flies in such conditions it continually alters its attitude in response to these small-scale movements because its speed is not so great compared to the speed of the wind and it is light when compared with a fighter aeroplane or an airliner. It is desirable for the light aeroplane to be able to fly through these disturbances without continuous control by the pilot of the aeroplane. One might think that this means that the condition of equilibrium shown in figure 20-5 must be stable and by stable we mean that it will return to its original attitude following a disturbance of this equilibrium. In fact whilst we would like our aeroplanes to automatically generate forces that tend to restore the attitude following a disturbance we do not want these forces to be so great that the aeroplane becomes slow to respond to control movements. We are looking for a limited stability.
Unfortunately, in the case of aeroplanes, this limited stability cannot be made to be foolproof and pilots of aeroplanes should understand the methods that are used to give an acceptable level of stability to the aircraft that they fly.
1.1 Stability in pitch
It seems to me that the best place to start is the stability in pitch because it applies to all aircraft. Stability in pitch is really about the way in which a wing and its stabiliser behave so we must look carefully at this system.
The data that has been gathered in Chapter 19 was gathered from tests on aerofoils in wind tunnels. There, the models are fixed in position and the fixings are adequate to resist any forces that may be generated by the models. Once the aerofoil becomes a wing and the wing is to be used to support a free-flying aeroplane these forces take on a new significance. Somehow the forces on the wing and the inevitable moment that is generated have to be balanced out so that the wing can retain some desired angle of attack and be controllable by the pilot. We have to look at the systems that make this possible.
We have already seen that aeroplane wings do not normally work unaided but as one element in a combination of a wing and a stabiliser. Figure 20-6 shows a model of a wing and stabiliser in the working section of a wind tunnel. The wing is of parallel chord and is pivoted at its tips in the sides of the tunnel and the pivots are at the quarter point of the chord. A rod, acting as a fuselage, is fixed to the wing to support the stabiliser at one end and a balance weight at the other. Provision is made for the adjustment of the angle that the stabiliser makes with the axis of the rod. I have shown both wing and stabiliser with symmetrical sections but the wing can obviously have an asymmetrical section.
Let us start with the stabiliser aligned with the rod. When the tunnel is running neither of the two "flying" surfaces will experience a vertical force although both are subject to a skin drag. Our interest starts when the stabiliser is set at a small angle of just a few degrees to the rod.
The immediate effect, as shown in figure 20-7, is to produce a force on the stabiliser that deflects the stabiliser downward and tilts the wing upwards to give it an angle of attack. This movement brings two forces and two moments into existence. The aerodynamic force on the wing is exerted directly on the pivots. As we have seen in Chapter 19 this force is usually regarded as the combination of a lift and a drag and it is inclined towards the trailing edge. As we have also seen, a moment about the pivot will be exerted on the wing. These will both change with the angle of attack. A similar force and a moment of smaller magnitude will be exerted on the stabiliser. The final position adopted by the model is as shown in figure 20-7. In this position the wing and stabiliser are in equilibrium with the stabiliser providing an upward force to balance out the combined moments on the wing and on itself and to do this it must make an angle of attack (probably smaller) in the same direction as that of the wing. The net force exerted by the two aerodynamic surfaces will be exerted on the pivots. It must be evident that this system permits the control of the angle of attack of the wing, and therefore the force exerted on it by the air flowing over it, by adjusting the angle that the stabiliser makes with the fuselage.
However we want to see whether this system is in stable equilibrium and the test for that is to try to decide what happens when the system is disturbed. Let us start by supposing that the model is deliberately tipped upwards by a few degrees at the nose. In this position the angle of attack of both the main wing and the stabiliser will be greater than that needed for equilibrium. The moment on the wing may have increased but the stabiliser, acting as it does on a long arm, will produce a much greater moment in the opposite direction. When the system is released the tail will rise and the position for equilibrium will be re-established. It is stable in this direction and a similar test in the other direction will show that it stable both ways. It is a matter of experience that this system is sufficiently responsive for practical flying.
As soon as one contemplates the design of a low winged aeroplane it becomes evident that the centre of gravity could easily be above the point of action of the lift. If it is, the aeroplane will be unstable in roll. Figure 20-9a shows the outline of a “Jodel”. Such a design puts all the heavy parts of the aeroplane like the strength members for the wing and the fuselage low down. The front seat passengers sit on the wing and the rear seat passengers on the fuel tank. With its fixed undercarriage one must expect the centre of gravity of the aeroplane to be almost as low as it can be. When we considered the stability of the “Cub” the lift was shown to act at a point in the centre section of the wing. This is in accordance with figure 20-8. If we plan to draw a matching diagram for the “Jodel” with its cranked wings we have to decide on a point of action for the lift. The wing of the “Jodel” has about one third of its area in the outboard panels and the lift on these must act above the centre section. It follows that the force on the whole wing must act at a point that is either near to the top surface of the centre section or a little above it. The effect of the cranking of the wing is to raise the point of application of the lift to be fairly close to the centre of gravity. Even if the lift does act below the centre of gravity the resulting instability will be a small effect that can be controlled. In figure 20-9a I have drawn the “Jodel” banked at 10° and in order to draw the forces so that they can be seen I have drawn an enlarged view of the centre of the wing and the fuselage omitting other details in figure 20-9b. On this I have drawn the equivalent forces to those on the “Cub”. It becomes clear that the couple exerted on the aeroplane can be quite small and the aeroplane will respond readily to aileron control without going seriously unstable.
1.3 The response of low-winged aeroplanes to a disturbance in roll
This is my best guess. I think that if you get, say a Robin, in straight and level flight and roll it through 5° and centre the ailerons I think that the aeroplane will then roll very slowly either to return to level flight or to roll to ever greater angles. Which it does will depend on whether the centre of gravity is above or below the point of application of the lift. If it recovers the aeroplane is stable in roll.
If it does not recover then the roll has to be stopped in some way. The obvious way is to use the ailerons but it is possible to use the rudder. If we choose to use the rudder another feature of dihedral comes into action. The effect of applying the rudder is to make the aeroplane yaw. Then the air flows onto the wings at an angle. This effectively increases the angle of attack of the forward wing and reduces the angle of attack of the trailing wing. Figure 20-10 shows how it works. It is a picture of a biscuit tin on a glass plate and photographed along an axis that makes 5° to that of the tin. If wind were to be blowing from the camera towards the tin the lower part of the left hand side has a positive angle of attack and the right hand side a negative angle. We can see that the angle depends on where we choose to look but it is clear that a dihedral angle of 7.5° would give useful angle of attack as a result of quite modest yaw. This will lift one wing and depress the other and the aeroplane will recover its level position when the rudder can be centred. There might well be a change of course as a result of all this.
In pitch and roll we have the vertical direction of the gravitational field to provide a reference direction but in yaw there is no equivalent reference at all. We cannot devise an airframe that will be stable in yaw. The best that we can manage is to fit the fin to keep the centreline of the fuselage in line with the direction of flight relative to the wind and to use the rudder and ailerons to make the aeroplane perform a level turn and it might also be needed for a cross-wind landing.
It may be that the basic mode of flight is level flight in a straight line but inevitably intentional changes of course are required and this involves turning. An explanation of turning is relatively easy if the aeroplane is flying in still air but when it flies in a wind turning becomes much more complicated. I think that it makes sense to start by considering turning in still air.
Aeroplanes do not use the rudder as the primary control during a turn in the horizontal plane. Turning normally involves flying round a horizontal arc of a circle with the aeroplane "banked" at some angle between the two extremes of a “flat” turn and a “pylon” turn. It is easy to see why this is the case.
Any object that follows a curved path has an acceleration towards the centre of its arc of motion. This is the centripetal acceleration and for this acceleration to exist there must be a force towards the centre of the arc. This is the centripetal force. Our aeroplane must somehow produce such a force if it is to turn. The force cannot come from the weight because that always acts downwards so the aeroplane must roll to some angle so that the lift can acquire a component acting towards the centre.
Such a turn would normally be made using the ailerons to initiate the turn and the elevator to make it level and possibly some rudder control to stop the nose dropping. We shall see that things are more complicated for low-winged aeroplanes.
The pylon turn can only be performed by an aeroplane that can loop from level flight. It starts with a roll through nearly 90° and, ailerons are centred and up elevator applied to produce what amounts to a half loop in the horizontal plane. the elevator is then centred and followed by a near 90° roll to regain level flight. During this manœuvre the aeroplane will have turned through 180° but throughout the “half loop” there will be little lift and the aeroplane will drop through some height that depends in the time taken to perform the half loop. This may be avoided by the use of the rudder to produce yaw in the appropriate direction and lead to lift from the fuselage.
Normally turns are made with the wings at some angle of bank between the small angle of the flat turn and the near 90°of the pylon turn.
I have shown an aeroplane in a fairly tight turn in figure 20-10. It is all obvious but it should be noted that the balance is now between the weight and the vertical component of lift and the lift must increase to keep the balance. The angle of attack will increase as a result of elevator action and engine speed must increase if the turn is not to involve loss of height.
So a turn in still air is a relatively simple procedure.
It happens that it is quite easy to relate the speed of the aeroplane, the angle of bank and the rate of turning. In order to do so we have to return to our aeroplane in a banked attitude. I have drawn the “Jodel” banked at 30° and, because the centre of gravity and the point of action of the lift are so close, I have shown them acting at the same point. The lift still acts at right angles to the centre section of the wing and the weight acts vertically downwards. The vertical component of the lift can support the weight of the aeroplane and the horizontal component is available to provide the centripetal acceleration. If now the aeroplane, at this angle of bank, turns at the correct rate for the speed at which it is flying, the forces will be in equilibrium and vertical component of the lift will equal the weight and the horizontal component be just right to give the centripetal acceleration. The diagram has been drawn for this condition.
2.2 Relationship between rate of turn, centripetal acceleration, angle of bank and airspeed in a turn.
It is not difficult to get an idea of these relationships. We must decide on a suitable range of speed for light aeroplanes. I have chosen a range of 50 to 125 knots. Let us start with the rate of turn. It is no more than a calculation of the angular velocity given the tangential speed and the radius. It probably will be best to calculate it is in degrees/second for a speed in knots.
We can write :- degrees/second where is the speed in knots and is the radius of the turn in metres. (1 knot = 0.5147 m/s). This can be plotted for a range of . It is the graph in figure 20-11. When looking at this graph keep in mind that at 30° per second a 180º turn takes only 6 seconds!
It is a short step to plotting a graph of centripetal acceleration against the radius of turning for the same airspeeds. Centripetal acceleration is given by :- metres/second/ second where is in knots and is in metres. The figures that are generated from this expression are not easy to visualise. Others have attempted to make it more comprehensible by dividing it by the acceleration that is exerted by the gravitational field of the Earth on a body that is falling freely. This is 9.81 metres/sec/sec and then the acceleration is in multiples of “g”. I have actually plotted and the vertical axis gives the number of times that the centripetal acceleration is greater than g. It is easy to see that turns having a quite small radius are needed before the centripetal acceleration reaches 2g. I will expand on this later in this chapter.
The next useful step is to plot a graph of the angle of bank, , against the radius of the turn that goes with the four airspeeds. For this we need a diagram of forces and we can use figure 20-10. We know the centripetal acceleration and from that the centripetal force = where is the weight of the aeroplane. But the centripetal force also equals from figure 20-10. From these two expressions . This is a very small step on from the graph for acceleration and the graph of the angle of bank against radius of turn for the four airspeeds is given in figure 20-13. Figure 20-10 is drawn for an angle of bank of 30º and does not seem to be excessive but, at 75 knots which is not a high speed, it corresponds to a turning radius of about 300 metres.
There is an unlikely further relationship that seems to me to be useful. There is a very simple relationship between the angle of bank and the apparent weight of the pilot. The matchstick pilot in figure 20-14 feels no sideways force if the aeroplane is banked at the correct angle for its airspeed and radius of turn. Nevertheless his apparent weight is greater than his rest weight (ie his weight as read by the bathroom scales). His apparent weight is the vector sum of his rest weight and the inertial reaction to the centripetal acceleration acting on his body. The ratio of his rest weight to his apparent weight is clearly equal to where is the angle of bank but we are more interested in the way in which his apparent weight changes with the angle of bank and it is better to write:
The graph of this is given in figure 20-15. This graph is correct for the aeroplane banking at the correct angle for its speed and radius of turn. The pilot must be able to sense this condition. Up to a bank angle of 60° the apparent weight of the pilot is less than twice his rest weight (2g). The apparent weight then rises very rapidly for greater angles of bank. (This accounts for the need for pressure suits for the pilots of modern fighter aircraft.)
Now, if the ratio of apparent weight of the pilot to his rest weight can be related to the angle of bank in this simple way so can the ratio of the apparent weight of the aeroplane to its static weight. As the apparent weight of the aeroplane increases with angle of bank so must the lift increase to equal to apparent weight. If the aeroplane flies round the turn at a steady speed the only way to increase the lift is to increase the angle of attack of the wings. Mechanically this is possible using the elevator but whether it is possible in practice depends on the angle of attack when the aeroplane was flying straight and level. If the increase in angle of attack takes the wing into a stalled condition the aeroplane cannot make the turn. The behaviour of the aerofoil now becomes important.
We have and in this case we want to use it in the form . For our aeroplane is constant so, if the apparent weight must change so must the angle of attack. We have seen in Chapter 19 that the slope of the graph of coefficient of lift versus angle of attack is effectively constant and equal to about 1/10 and that there is a maximum value of the angle of attack for every aerofoil. This means that it is easy to try to make a turn that has too small a radius and take some part of the wing into a stalled condition. For reasons that escape me this is called "tip stalling". There is no certainty that it will be the outer wing that stalls first but it is best if it does because the aeroplane rolls out of the turn and tends to drop its outer wing which is safe. If the inner wing stalls the aeroplane tends to roll on to a wing tip and drop its nose and this is very inconvenient if it occurs near to the ground.
2.3 Flight relative to the wind and to the ground.
Any magazine that is connected with flight will get involved over and over again with the thorny problem of flight in a wind. It is all about whether an aeroplane flies relative to the wind or to the ground. As usual in such things the answer is determined by the form of the question and here the question is no help. The aircraft flies relative to both and we have two effects to sort out.
There is no problem when an aeroplane flies in a straight line in a wind. Suppose that we have an aeroplane that flies in a wind of 20 knots at 80 knots relative to the air. When it flies directly up wind its speed relative to the ground is 60 knots. When it flies directly down wind its speed relative to the ground is 100 knots. When it flies at an angle to the wind its speed relative to the ground will be somewhere between 60 and 100 knots but in some direction that is a combination of the speed of the air relative to the ground and the speed of the aeroplane relative to the air.
In figure 20-16 I have drawn velocity triangles for an aeroplane that flies at 80 knots in a steady wind of 20 knots for a series of headings at 30° intervals from flying up wind to flying down wind. Vectors representing speeds relative to the air are in blue and vectors for speeds relative to the ground are in red with the speed of the wind being in black. I have shown the vectors for flying up wind and down wind separately. Clearly these are 60 knots and 100 knots. It is easy to see how the course relative to the ground varies with different headings for the change from flying up-wind to flying down-wind.
2.4 Turning in a wind
Now we can look at the process of making a 180° turn from flying up-wind to flying down-wind. Let us start by looking at making a level turn through 180° in a semicircle in still air. Then we can see that throughout the turn the speed and therefore the kinetic energy remains unchanged in magnitude although it does change its direction. As we have seen, a centripetal force has to be created during the turn and this force produces the change in the direction of the kinetic energy. The reference frame for kinetic energy is the distant stars or, for this purpose, the ground and it cannot be the air.
However when we try to do the same in a wind there is a change in the speed of the aeroplane in figure 20-16 from 60 knots to 100 knots relative to the ground. So the kinetic energy of our aeroplane increases during the turn by the ratio of the squares of the speeds i.e. from 602 to 1002 or in the ratio of nearly 2.8 to 1. This kinetic energy has to come from somewhere and we must decide how the increase is brought about.
We can make progress by looking at the path that would be taken by an aeroplane if it could fly a semicircular path relative to the wind from flying up-wind to down-wind. The path is shown in figure 20-17 where points on the curve have been found by drawing first the semicircular path that the aeroplane would follow in still air. Then for any point A on the semicircle it is possible to make a good approximation to the time taken to fly from O to A in the absence of wind and the distance AB that the aeroplane would move down wind in that time. The point can then be shifted down wind by this amount to give a point on the path of the aeroplane. I have done this at 30° intervals round the curve for an aeroplane flying at 80 knots in a wind of 20 knots.
The first part of the curve has a tight radius and, recalling the first part of this chapter this will require a high centripetal force and, in order to get it, a greater angle of bank. Then the lift is not big enough and either the aeroplane must be allowed to fall or the thrust must be increased. The kinetic energy increases either by giving up potential energy as it falls or, from the engine, if the thrust is increased. During the later stages of the turn the radius increases and the centripetal force can be reduced by reducing the angle of bank. So, if we choose to let the aeroplane lose height, the turn will start with a large angle of bank and then progressively reduce the bank to nothing as the turn is completed at a new lower altitude. If one chose to keep the thrust unchanged it might be possible to increase the angle of attack and reduce the airspeed to finish at the same height at a much lower speed. If the thrust is increased at the start of the turn and then reduced as the turn progresses the outcome could be a level turn.
It is possible to calculate loss of height that is needed to produce a given increase in velocity. We relate the increase in kinetic energy and the loss of potential energy through where is the velocity at height and the velocity at height . For the aeroplane flying at 80 knots in wind of 20 knots the loss of height in a 180° turn that began and ended at the same airspeed without additional input from the engine would be about 90 metres.
This has serious implications in some circumstances. Occasionally pilots suffer an engine failure soon after take-off and have to make an emergency landing. There appears to be a choice between continuing in the same direction with limited opportunity to land or turning back and landing on the runway. If there is a wind blowing the explanation above shows how much height will be lost in trying to make even the 180° turn let alone the landing and the laws of physics are telling us that there is no choice because the 180° turn with the engine out is impossible and it is probably impossible with no wind as well.
Pilots probably make turns in a wind in their own way to suit what they want to do without conscious effort.
It may seem to be out of order to deal with descent and landing before taking-off but I want to look at the way in which the descent affects the design and use of the wings.
Suppose that we have an aeroplane flying at a height of 500 metres and we want to make a landing. Landing involves losing height in a controlled manner so that an approach to the runway and, ultimately, contact with the ground can be made at what is deemed to be the “correct” point on the runway. At altitude the aeroplane will be flying at its cruising speed and at the point of touch down it will be flying as slowly as has been found to be safe. If nothing else this reduces tyre wear caused by the sudden spinning-up of the wheels. Two things must happen during the descent and landing, the aeroplane must lose speed and it must lose height. Another and more informative statement of these two requirements is than the aeroplane must lose perhaps three quarters of its kinetic energy and lose all of its potential energy. The problem we now face is deciding how to lose this energy.
There is only one place for it to go and that is into the air. This may sound strange but that is where all the work done by the engine of an aeroplane goes during a flight. The aeroplane may start and finish a flight in exactly the same position on the ground yet may have burnt off a whole tank full of fuel. The chemical energy that was in that fuel is in the air somewhere. Initially some is stored in the air as heat and some in kinetic energy of the air in the wake. This kinetic energy gradually dies away to give more heat to the air until the only effect is heat. We have produced the same effect as simply burning the fuel.
The process by which the potential energy and the kinetic energy is to be lost during descent is the creation of a wake behind the aeroplane. This is a process that we cannot avoid but, in normal flight, we try to minimise. When it comes to descending we want to increase the drag but always retain control of the aeroplane and not give up the option of returning to level flight.
This was relatively easy in the early days of flying because aeroplanes bristled with struts and wires and wheels and other non-streamlined lumps. All that was needed was to throttle back the engine and the drag would do the rest. Design moved on and aeroplanes were cleaned up and used only one wing and suddenly losing height became tricky. Let us see what options we have.
3.2 Descent by diving
Diving involves flying downwards at an angle to the horizontal. Figure 20-18 shows a light aeroplane diving at an angle of 20°. On it I have shown the various forces that act on the aeroplane when they are in equilibrium. Before we can consider these forces we have to decide how they came to be in equilibrium. From level flight the application of down elevator will lift the tail and reduce the angle of attack of the wings. This will temporarily reduce the lift, the nose will drop in response, and the aeroplane will start to dive. As soon as it does the gravitational force on the aeroplane will acquire a component in the direction of flight that will add to the propeller thrust and make the aeroplane accelerate. As the speed increases the lift at the new angle of attack will increase until the vertical component of the lift is equal to the weight. If the drag on the aeroplane as a whole now equals the sum of the thrust and the component of the weight the aeroplane will fly in equilibrium at this nose-down attitude. It will dive. However this condition of equilibrium will not be reached automatically and the pilot will have to adjust the elevator and perhaps the throttle to achieve the desired angle of the dive and then the dive will continue at a steady speed. The important observation to make is that the forces on the aeroplane are in equilibrium once the transition between level flight and diving has taken place. Diving at this angle does not seem to be an option as a way of approaching a landing because there is an increase in speed and not a decrease.
A dive can be started by reducing the engine speed. This will lead to a reduction of airspeed. The pilot must now either to raise the elevator to increase the angle of attack of the wings to restore the lift or lower the elevator to make the nose drop and decrease the angle of attack of the wings. If the elevator is lowered the aeroplane behaves as before and enters a dive but, now that the thrust is less the speed is lower and the angle of attack greater. Otherwise the net effect is the same. However by this method the aeroplane could be put into a shallow dive and lose height with all the forces in equilibrium. There is still a problem because adjustments will have to be made to the flight path so that the point of touch-down is correct and the use of a high angle of attack will reduce the margins available for control of the aeroplane.
Suppose that the wing is not twisted and has no change in section so that, apart from the effects of aspect ratio, the angle of attack is the same from root to tip. If the elevator is raised to increase the angle of attack of the wings this might permit the aeroplane to sink gently but it also reduces the authority of the ailerons because there is only a very few degrees of movement possible on the down-going one before the outboard panel stalls. Such a stall might easily be catastrophic.
Diving for such a wing does not seem to be the best way to approach a landing.
3.3 Methods of controlling a descent
Various methods are in use and we should try to see how each is intended to work.
3.3 Maintaining control during descent
3.3.1 Use of a twist in the wing (washout)
If the only problem with descending in a shallow dive is the loss of aileron authority one solution is to build the wings with a twist of just a few degrees. The twist is only perhaps 1 to 2 degrees in a direction to raise the trailing edge of the tip relative to the root. It is called washout and it is shown in figure 20-19.
The value of CL at the root will be between 0.1 and 0.2 greater than that at the tip for 1° or 2° of washout. The twist will ensure that the angle of attack of the tip is always less than that of the root (not for inverted flight). This gives sufficient margin to retain aileron authority during a shallow dive and it has other benefits as well.
The sudden and complete stall of either or both wings resulting from flying too slowly is not a normal part of general flying. However it may occur as a result of an error in controlling the aeroplane e.g. in a tight turn. If a wing has washout the wing root will reach the angle of attack at which it stalls before the rest of the wing and the stall will be progressive and not sudden. This may give time for corrective action to be taken.
The disadvantage with this wing twist is that the whole wing operates at a net lower angle of attack for a given weight and speed and so must have a larger area. Also it cannot be operating at the condition of least drag and so the efficiency of the wing is reduced. A compromise is involved here so that the wing is suitable for use by "average" pilots and that compromise has to be made at the design stage.
3.3.2 Use of a change in section from root to tip
In figure 20-20 I have drawn the graphs of CL against angle of attack for a typical asymmetrical section and for a typical symmetrical section. Wings have been made with an asymmetrical section at the root and a symmetrical section at the tip with a progressive change from one section to the other. NACA chose to draw a chord line for asymmetrical sections that passed through the centre of the nose radius and this decision leads to the angle of attack being negative when the coefficient of lift is zero.
The plot in figure 20-20 suggests that, if a wing were to be made with the asymmetrical section at the root and the symmetrical section at the tip with the chord lines of all the different sections in the same plane, all the intermediate sections might be expected to lie in between these two graphs. We are not at all certain that these graphs, that were for high quality test models, can be used directly but any real wing would not see a change in slope of the graphs only a lowering of the angles of attack at the stall for the two sections. The major change is in the drag. So a wing with a progressive change in section like this would behave like a wing with washout. I have drawn the end view of a wing with an asymmetrical section at the root and a symmetrical section at the tip. In figure 20-21 I have shown the wing with all the chord lines in the same plane but there is no reason to be restricted in this way; the wing could also be twisted.
When the wing as a whole changes its attitude to the flow of air both sections change their angles of attack by the same amount. This means that the root section will always operate with a lower angle of attack but with a higher value of CL which must be seen as a desirable arrangement. We can have aileron authority at high angles of attack and have the wing roots stall first. Such a wing could be used in a shallow dive.
3.3.3 The use of flaps
The two methods that involve special design of the wings with washout do not attempt to make provision to add to the drag of the wing only to make it safe to use. The use of inboard flaps gives a way of increasing the drag without interfering with the successful operation of the ailerons and we need to look carefully at this method.
Any search for information on flaps will show that flaps have been the subject of considerable evolution. They seem to have started as a form of air brake at a time when other developments were being made in the field of high-lift devices. Gradually it became evident that the flap could evolve to be an airbrake and a high lift device and most light aeroplanes that are fitted with flaps use them in both roles. Typically the flaps will be lowered by 5° for take-off and by 15° during the landing approach and by 30° for landing.
There are many quite different designs for devices that can give high lift or act as an airbrake each of which has its own concept and characteristics. As this chapter is about light aeroplanes I will restrict my interest to the two basic types that are in use. These designs are heavily influenced by first-cost considerations as an important design criterion. I have drawn the section of a wing that is fitted with flaps with the flaps lowered by 5°, 15°, and 30°. In the first three figures the flap is hinged outside the wing so that, as the flap is lowered, a gap is formed between the flap and the rest of the wing.
Now we must look at these three sections and decide how to think about them. The section with 5° of flap shown in figure 20-22a is clearly an asymmetrical aerofoil with a greater camber than the basic section with a small gap joining the flow between the two surfaces. The flow pattern over the underside of the aerofoil will not change much in appearance. The presence of the lowered flap will increase the pressure on the underside and deflect the approach flow more towards the upper surface. The flow over the nose on the upper surface will probably remain attached but the emergence of air leaking through the gap will certainly initiate separation on the upper surface. However the combination will behave like any other aerofoil of significant camber.
Let me now look at figure 20-22c. The flap has produced a significant change from the basic section. I do not think that one should imagine this flap to be a sort of brake just sticking out into the airflow and causing drag. I have made an attempt in figure 20-23 to draw a flow pattern round the aerofoil with its flap lowered. Where necessary I have added short lengths of flow line to the basic pattern. It is not drawn from a photograph of a real flow but it does not contravene the rules of fluid flow. I think that it will show the essential features correctly. The flap will cause the air pressure ahead of it and under the wing to rise significantly. It will deflect the approach flow upwards much more than the section would with the flap raised. In addition the flow along the underside as it approaches the flap will be diverted up into the gap, which is now quite large, and then it will join with the flow over the upper surface to flow over the flap. In effect the flap is a small aerofoil at a high angle of attack. It will be stalled and produce an eddying wake that will join up with the thin wake from the upper surface of the wing to occupy the area that I have shaded. The flap will produce lift and a high drag (its drag coefficient may be as high as 0.2). The two parts acting together behave like an aerofoil with a graph of CL against angle of attack like the dotted line in figure 20-24.
The flap has two squarish corners and the wake produced by the flap will have rotating wakes from each corner. It is an ideal device for losing energy from the aeroplane to the air.
Summing up, the flap with 5° behaves like an ordinary aerofoil and at 30° it produces a high lift at high drag. This is just what is required. However there are intermediate angles and the fact that nominal angles 5°, 15° and 30° are specified in aeroplane handbooks suggests that these angles are not critical and that the action of flaps is predictable. Obviously there can be a graph of CL against angle of attack for an aerofoil with its flap lowered and I have added a typical graph to Graph 2‑2 to give figure 2--24 The effect of progressively lowering the flap is to shift the graph from something like the graph for NACA 641-412 to something like the dotted line. The graph for the section in figure 20-22b will lie in this gap. Of course, as the flap is lowered the coefficient of drag increases quite dramatically.
The other type of flap is the most simple mechanically. The flap is hinged to the wing at its lower edge. In figure 20-25 such a flap is shown in the same three positions as in figure 20-22.
The design decisions are clear. The inevitable breakdown in flow at the gap in the upper surface even for a closed flap is accepted in order to have the mechanical simplicity of a simple hinge. The flap cannot act as a second aerofoil in any of the lowered positions. The aerofoil with the flap lowered will have a graph of CL against angle of attack like the one above but have a larger coefficient of drag.
3.3.4 Using flaps to descend
If the flaps are lowered on the wings of a powered aeroplane the effect is to increase the angle of attack of the part of the wing that incorporates the flaps whilst leaving the outboard panel with the ailerons unchanged. The engine speed can be reduced to bring the weight and the lift into equilibrium again. This will give a way to get rid of the unwanted kinetic energy. Now, if the engine is throttled back still more and the nose lowered to initiate a dive, the aeroplane can fly in equilibrium on its downward path with the much higher drag resisting the engine thrust. With care the aeroplane can dive at quite high angles without gaining speed. Should it be necessary to increase airspeed the engine speed can be increased and this gives a condition where the aeroplane is under control.
We have a simple way of losing both the potential energy and the kinetic energy.
3.3.5 Movement of the centre of pressure resulting from the use of flaps
In Chapter 19 I pointed out that the various pressures on any aerofoil can be reduced to one force and a couple. People call the point of application of this force on a wing the centre of pressure. It is a convenient name. Unfortunately the centre of pressure is hard to locate because of the complication of the moment and also, unfortunately, it moves as the angle of attack changes. The amount of the movement varies with the profile of the aerofoil tending to become greater with increasing camber and as the position of maximum thickness is moved forwards.
The centre of pressure moves significantly as the flaps are lowered and usually the aeroplane must be re-trimmed by adjustment of the elevator or its tab. This may be automatic as the flaps are lowered.
4 Taking off and the climb to cruising height.
Regardless of aeroplane size it is desirable to take off in as short a run as is safe in order to leave the maximum length of unused runway in which to land should the engine fail. The length of run is reduced if take-off is made into wind although, as the wind is seldom aligned with the runway, the acceleration to the point of becoming airborne has to be done in a cross wind. This can be done best if, during the run, there is a load on the wheels to keep them in contact with the ground and permit steering.
An aeroplane is prepared for take-off from a runway by pointing it into the general direction of the wind. The take-off is started by increasing the engine speed in the manner specified by the manufacturer and then there are three phases to the take-off and climb. The first phase is the acceleration to the speed at which flight is possible, then comes the rotation to give the wing an angle of attack and lift off, and finally the climb to cruising height. I will deal with them in sequence.
4.1 Acceleration to rotation
We have seen that there is a lowest speed at which the wing of an aeroplane can generate enough lift to permit flight. The absolute lowest speed is when the angle of attack of the wing is at its maximum value which is, of course, very close to the stall. Clearly this is not a suitable attitude for flying close to the ground. When the point of leaving the ground is reached the angle of attack should be considerably less than the stalling angle at a speed that is somewhat higher than the minimum.
What happens during this acceleration phase depends to some extent on the type of undercarriage that is fitted to the aeroplane. In figure 20-26a I have shown the “Chipmunk” with its undercarriage comprising two main wheels and a tail wheel. and, in 20-16b an “Aircruiser” with a tricycle undercarriage. The figures show the attitudes of the aeroplanes when they are standing on the ground ready for take off. Clearly the Aircruiser is standing more or less in the attitude of flight and the Chipmunk is nose up at about 13°. The Aircruiser can accelerate to take-off speed with its slightly nose-down attitude which keeps the load on the wheels and then, just before take-off, lift the nose by using up elevator and so increase the angle of attack of the wing. By comparison the Chipmunk must start its run with too great an angle of attack, wait for the speed to increase enough for the stabiliser to gain authority, and then lift the tail which puts weight back onto the main wheels and continue in much the same way as the Aircruiser. The tail-dragging aeroplane is vulnerable during the lifting of its tail and the need to correct for the cross wind at the same time adds to the pilot workload.
Aeroplanes with powerful engines reduce this workload because the power of the engine can be used to shorten the time taken for take-off and to reduce the effects of crosswinds etc.
4.2 The use of flaps
We have already seen that the use of flaps during landing is so helpful that flaps are fitted to almost every aeroplane. These flaps are often designed so that they can be used for take-off as well. Of course the aeroplane must be fitted with an engine of adequate power if flaps are to be used.
In general the aeroplane reaches its take-off speed most quickly when the drag of the aeroplane is least. For the overall drag to be least the aeroplane should be moving with its wings at the angle of attack that produces the least drag and the fuselage should also be at an attitude that offers the least drag.
We have already seen in figure 20-22a how a flap that has been lowered by 5° produces a quite modest but useful increase the camber of the wing section and, at the same time, a small increase the angle of attack. A look at figure 20-26b and at the attitude of the Aircruiser as it stands on the runway shows how the wing with its flap lowered would be at an angle that produces little lift and a low drag until just before take-off when the aeroplane can be “rotated” to increase the angle of attack. This is just what is needed.
By comparison the “Chipmunk” will start its take-off run with the wing making a high angle of attack when no lift is required and the fuselage also at an angle that produces significant drag. Once the speed at which the tail lifts has been reached the attitude becomes that of low drag and there is little difference between it and an aeroplane with a tricycle undercarriage.
The aeroplane reaches the correct speed for rotation with almost no lift on the wings and possibly some down force. Use of the elevator will raise the nose and increase the angle of attack of the wings and the aeroplane will take off. As most aeroplane engines are not designed for prolonged running at maximum power most pilots throttle back a little and start to climb to a safe height. At some point the flaps may be raised and the aeroplane fly on its designed wing. Now we need to look more closely at this phase of climbing.
When the aeroplane rotates to take off its whole attitude changes so that the angle of attack of the wings increases and the centre line of the engine points upwards. The engine now makes a contribution to the lift. In figure 20-27 I have shown an aeroplane in the attitude that it would have when climbing. It would be flying with the wings having a high angle of attack (8° perhaps) and the direction of flight would make a small angle to the line of thrust of the engine. We have already seen in figure 20-5 that the forces on the aeroplane do not all act at the same point and I have shown the forces acting at appropriate places and a force on the stabiliser to put all the major forces into equilibrium.
Now I want to simplify the system of forces and draw them as if they act through one point, I want to ignore the force on the stabiliser, and also subtract the drag from the thrust as if they were in line and call the result “net thrust”. This then leads to figures 20-28 and 20-29. For my purpose here, it is sufficient to show the forces acting at one point and this I have done in figure 20-28 where the forces are shown acting at a point and in a triangle of forces.
The triangle of forces tells us that the weight is supported by the vertical components of both the aerodynamic lift and the net thrust. At this small angle of climb (15°) the vertical component of the thrust is small as a proportion of the whole.
Figure 20-29 the angle of climb is greater (30°) and now the thrust is making a significant contribution to the total lift.
Airscrews are really rotating aerofoils and small aerofoils are much less efficient than large ones. It follows that, as devices for generating vertical lift, airscrews are much less efficient than wings. Given the fact that wings are normally fixed to fuselages and cannot move independently of the engine the share of the lift provided by the engine increases as the angle of climb increases until in a vertical climb the whole of the lift is provided by the engine. So, as the angle of climb increases, the efficiency of the aeroplane as a lifting device decreases.
This leads pilots to climb rapidly to a safe height and then to throttle back to unload the engine and reduce the rate of climb to an efficient value. This has an added advantage of reducing the area on the ground that is directly affected by the engine noise.
4.5 Climbing to the ceiling
Aeroplanes cannot climb indefinitely not because they run out of atmosphere but because the density drops as they climb. As a result every aeroplane has a ceiling to which it might climb in the hands of a good pilot. It is instructive to see how this ceiling comes about.
There are two effects of the reducing density. The output of the engine falls as the weight of air entering it decreases and the lift of the wings falls. Regardless of this the weight of the aeroplane remains substantially unchanged and somehow the lift must be maintained if the aeroplane is to continue to fly.
We have expressions for lift: They are :-
Lift = and Drag =
We can look at these expressions in connection with climbing to the ceiling of say 3,000 metres when the density will have dropped by about a half. If the lift is to equal the weight, as it must, the most efficient method is to get as much lift as possible from the wings. So the expression for lift above becomes :
Weight = .
does not change so must remain unchanged. We know that the density drops as the aeroplane climbs so the value of must increase steadily as the aeroplane climbs. This really means that we have to increase the angle of attack as much as we dare and get as much speed from the engine as is prudent. Unfortunately the increase in angle of attack produces a large increase in drag on the wings and any increase in speed will increase the drag on the fuselage. In order to reach the ceiling the pilot will have to juggle between speed and angle of attack and avoid a stall.
No instrument reads angle of attack but the altimeter actually measures the value of even when the density falls despite being calibrated in knots.
Section 5 Aerobatic flying
This is a mode of flying all of its own. It is about making a small, strong, high-powered aeroplane perform manoeuvres that are usually essentially geometric shapes like straight lines combined with circles and helices to high standards of precision and to perform manoeuvres that involve sudden and complete stalls and flying in the stalled condition or flying with the wing vertical. Over the years of aerobatic competition the envelope of predictable manoeuvres has expanded to make possible almost any manoeuvre that can be described in a way that sets a performance goal for an experienced aerobatic pilot.
For the purposes of this chapter I think that the most important outcome is a range of aerofoil sections that stall very easily. They seem to have emerged from home-builts with symmetrical sections of about 16% thickness with the maximum thickness at about 20% of the chord. The section might easily comprise an ellipse and two tangents as I have shown in figure 20-30. Such a section would be dangerous if it were to be used on a conventional light aeroplane because the available power would be insufficient to recover from a stalled condition. The aerobatic aeroplane necessarily has an engine of high power and some can climb vertically and this permits the use of engine power to recover from a stall. What is interesting is that the wings can be operated deliberately with high angles of attack where data is not normally collected. I will look at this in Chapter 21 on the propeller that can also operate at high angles of attack.
Limits to the location of the centre of gravity of an aeroplane
To this point I have said nothing about the position of the centre of gravity of an aeroplane. It is very important that I should do so.
It is normal to arrange for the centre of gravity of the combined aeroplane and its load to be near to the point of application of the aerodynamic force on the wing as I showed it in figure 20-5. This gives a small couple to be balanced out by the stabiliser and the balancing out may be done using the trim tabs. But what happens if the centre of gravity is too far back? The unbalanced couple becomes larger and somehow the stabiliser has to generate an upward force to balance it out. This amounts to changing the system of forces on the aeroplane so that the weight is supported by both of the horizontal surfaces, that is, by the wing and the stabiliser. This involves increasing the angle of attack of the stabiliser.
The stabiliser is fitted to give stability in pitch and to control the angle of attack of the wing and it is sized to do this using only about 8º of its practical range of angle of attack. If now the stabiliser is required to produce a constant upward force as well, the range of angle of attack available for the control of the wing is seriously impaired.
Suppose that the centre of gravity is not so far back that it is possible for the aeroplane to take off. With care it can be flown but at some stage it must attempt to land. We have seen that in order to land we want to use the flaps and this requires re-trimming of the stabiliser to offset the increased moment of the wing. If that re-trimming makes the stabiliser stall the aeroplane will crash. The alternative is not to use the flaps and to fly in at high speed and attempt to touch down with sufficient length of runway left to come to rest safely. It would be a great feat of airmanship. It follows that there must be a mandatory limit for the distance aft for the centre of gravity.
There is a similar problem with a forward centre of gravity and there is a mandatory limit for the forward position of the centre of gravity.
This chapter has shown that the light aeroplane is quite a complex machine. Whilst one might look to use wings and stabilisers and fins that have been made to the standards envisaged by the NACA the reality is very different. Ailerons, flaps and other high-lift devices are essential for it to be flown safely and these take the wing away from the ideal aerofoil and the flying surfaces have to be made by cost-effective methods. It follows that the extraordinary performance levels that NACA have shown to be possible are not achievable in practice. One can sympathise with engineers who have not spent years on the study of mathematics and looking for help in deciding whether to use washout or different sections at root and tip consulting Abbott and von Doenhoff's book and wondering what was the point of it all. Chapter 18 in my book explains how it all comes about. The goals of engineers and scientists are quite different. One could be forgiven for contemplating what the output of NACA would have been if all the scientists had been pilots with 500 hours in their logbooks It might have tempered this abstruse scientific approach. However the data is there and bears the same relationship to real flying surfaces as say the Rankine cycle for steam does to real steam plant in that both set practical maxima as distinct from theoretical maxima.
An uncertainty that cannot resolve is that it seems to me that anyone choosing to design a light aeroplane will need to do some optimising to determine wing areas and so on and this would be made more simple if it was certain that the effect of discontinuities in the surfaces due to flaps and ailerons reduced the angle of attack at which the section stalled but did not alter the slope of the graph of coefficient of lift versus angle of attack. The effect of the artificial roughening by the NACA, that some might think was severe, certainly suggests that this will be the case.
In practice there must be very few opportunities for any given person to design a new aeroplane even a light aeroplane. Private aviation has never been large enough to support more than a very few manufacturers and aeroplanes have proved to have very long lives when compared with cars. So the need to select an aerofoil section for a new aeroplane almost never arises.
The uncertainties lead those who write manuals for the design of light aeroplanes to simply recommend a section that has proved to be successful and proceed. I have some sympathy with this approach.
However the performance of an aeroplane is partly in the attitude of mind of the designer. How could the same man have designed the winglet on a helicopter shown in figure 20-32 and the bracket that holds it? What would a sailor think if his keel were to be held on to the hull in this way?
Further to this the drag of the aeroplane as a whole can be divided into two components, that of the flying surfaces and that of the fuselage and all its appendages. Neither can be neglected as they may well be of similar magnitudes. There is probably more scope for reduction in drag of the fuselage than of the wings. Air flows both round and through the fuselage and casual inspection of engines and ducting leaves the impression that little thought is given to the internal flows. Perhaps the first cost criterion is prohibitive.
This is just one chapter about light aeroplanes as an application of the physics of the aerofoil. There is much more that could be written but I have said enough in the context of this book. One must conclude that the use of aerofoils on aeroplanes is much more complicated than just the physics of the flying surfaces and Chapter 18 gives an insight to some of the peripheral but important problems.
 Light aeroplanes usually have a tell tale piece of cord on the engine cowling and in view of the pilot to detect yaw. It is normal to apply control corrections using the rudder and/or ailerons to keep this cord aligned with the centreline of the aeroplane to minimise fuselage drag.
 Bombing aeroplanes and airliners turn out to be surprisingly aerobatic if they are flown very lightly loaded.
 In gliders the pilot lies on his back ahead of the wings and heavy pilots are balanced out by water in the vertical fin.
 The use of pivots ensures that the model can move only in two dimensions but it also introduces a simplification when compared with an aeroplane. For an aerofoil moving freely in the air, the forces caused by pressure on the wing reduce to a single force and a couple. The point of application of the force and its magnitude and that of the couple vary with the angle of attack. The use of pivots permits us to locate the force at the expense of changing the couple to a moment about the pivot.
 This system is used on some land yachts. Recently one set a speed record at 126 mph. I used the system on my wing sailed yacht reported on this web site.
 The lift appears to act through the centre of gravity but this is the result of the layout and not inevitable. When seen from the side the lift will not act through the centre of gravity
 Pilots of helicopters hovering and pointing into a wind must never expect to just rotate and set off down wind. The helicopter does not have the kinetic energy to do this and a 180 degree turn must be made to gain the kinetic energy.
 Streaming a very large parachute is not an option!
 Always supposing that the wings do not stall.
Lowering the windward aileron will make the windward wing lift downwards and add to the load on the windward wheel but the most important factor is the ground attitude of the aeroplane.
 If the take-off is from grass it may be possible to get completely in line with the wind.
 This was a major problem for the U2 spy plane.