Decrease of pressure and density with altitude

The rate at which the pressure decreases is much greater near the earth’s surface than at altitude. This is easily seen by reference to Fig. 2.2 (overleaf); between sea-level and 10 000 ft (3480 m) the pressure has been reduced from 1013 mb to 697 mb, a drop of 316 mb; whereas for the corresponding increase of 10 000 ft between 20 000 ft (6096 m) and 30 000 ft (9144 m), the decrease of pressure is from 466 mb to 301 mb, a drop of only 165 mb; and between 70 000 ft (21 336 m) and 80 000 ft (24 384 m) the drop is only 17 mb.

This is because air is compressible; the air near the earth’s surface is com­pressed by the air above it, and as we go higher the pressure becomes less, the air becomes less dense, so that if we could see a cross-section of the atmos­phere it would not appear homogeneous – i. e. of uniform density – but it would become thinner from the earth’s surface upwards, the final change from atmosphere to space being so gradual as to be indistinguishable. In this respect air differs from liquids such as water; in liquids there is a definite dividing line or surface at the top; and beneath the surface of a liquid the pressure increases in direct proportion to the depth because the liquid, being practically incom­pressible, remains of the same density at all depths.

Inertia of the air

It will now be easy to understand that air must also possess, in common with other substances, the property of inertia and the tendency to obey the laws of mechanics. Thus air which is still will tend to remain still, while air which is moving will tend to remain moving and will resist any change of speed or direction (First Taw); secondly, if we wish to alter the state of rest or uniform motion of air, or to change the direction of the airflow, we must apply a force to the air, and the more sudden the change of speed or direction and the greater the mass of air affected, the greater must be the force applied (Second Taw); and, thirdly, the application of such a force upon the air will cause an equal and opposite reaction upon the surface which produces the force (Third Taw).

Pressure of the atmosphere

As explained in Chapter 1, the weight of air above any surface produces a pressure at that surface – i. e. a force of so many newtons per square metre of surface. The average pressure at sea-level due to the weight of the atmosphere is about 101kN/m2, a pressure which causes the mercury in a barometer to rise about 760 mm. This pressure is sometimes referred to as ‘one atmosphere’, and high pressures are then spoken of in terms of ‘atmospheres’. The higher we ascend in the atmosphere, the less will be the weight of air above us, and so the less will be the pressure.

Density of the air

Another property of air which is apt to give us misleading ideas when we first begin to study flight is its low density. The air feels thin, it is difficult for us to obtain any grip upon it, and if it has any mass at all we usually consider it as negligible for all practical purposes. Ask anyone who has not studied the ques­tion, ‘What is the mass of air in any ordinary room?’ – you will probably receive answers varying from ‘almost nothing’ up to ‘about 5 kilograms.’ Yet the real answer will be nearer 150 kilograms, and in a large hall may be over a metric tonne! Again, most of us who have tried to dive have experienced the sensation of coming down ‘flat’ onto the surface of the water; since then we

Density of the air

Density of the air

have treated water with respect, realising that it has substance, that it can exert forces which have to be reckoned with. We have probably had no such experi­ence with air, yet if we ever try we shall find that the opening of a parachute after a long drop will cause just such a jerk as when we encountered the surface of the water. It is, of course, true that the density of air – i. e. the mass per unit volume – is low compared with water (the mass of a cubic metre of air at ground level is roughly 1.226 kg – whereas the mass of a cubic metre of water is a metric tonne, 1000 kg, nearly 800 times as much); yet it is this very property of air – its density – which makes all flight possible, or perhaps we should say airborne flight possible, because this does not apply to rockets. The balloon, the kite, the parachute and the aeroplane – all of them are supported in the air by forces which are entirely dependent on its density; the less the density, the more difficult does flight become; and for all of them flight becomes impossible in a vacuum. So let us realise the fact that, however thin the air may seem to be, it possesses the property of density.

Invisibility of the atmosphere

Air is invisible, and this fact in itself makes flight difficult to understand. When a ship passes through water we can see the ‘bow wave’, the ‘wash’ astern, and all the turbulence which is caused; when an aeroplane makes its way through

air nothing appears to happen – yet in reality there has been even more com­motion (Fig. 2.1).

If only we could see this commotion, many of the phenomena of flight would need much less explanation, and certainly if the turbulence formed in the atmosphere were visible no one could have doubted the improvement to be gained by such inventions as streamlining. After some experience it is poss­ible to cultivate the habit of ‘seeing the air’ as it flows past bodies of different shapes, and the ability to do this is made easier by introducing smoke into the air or by watching the flow of water, which exhibits many characteristics similar to those of air.

Air and airflow – subsonic speeds

Introduction – significance of the speed of sound

As was explained in Chapter 1, the remainder of the book will be concerned almost entirely with fluids in motion or, what comes to much the same thing, with motion through fluids. But it would be misleading even to start explaining the subject without a mention of the significance of the speed of sound.

The simple fact is that fluids behave quite differently when they move, or when bodies move through them, at speeds below and at speeds above the speed at which sound travels in that fluid. This virtually means that in order to understand modern flight we have to study two subjects – flight at speeds below that at which sound travels in air, and flight at speeds above that speed – in other words, flight at subsonic and flight at supersonic speeds.

To complicate things still further, the airflow at speeds near the speed of sound, transonic speeds, is complex enough to be yet a third subject in its own right. We shall cover these subjects as fully as we can, but we must not let our natural interest in supersonic flight tempt us to try to run before we can walk, and in the early chapters the emphasis will be on flight at subsonic speeds, though we shall point out from time to time where we may expect to find dif­ferences at supersonic speeds.

But first let us have a closer look at the fluid, air, with which we are most concerned.

Mechanics of flight

We do not pretend to have covered all the principles of mechanics, nor even to have explained fully those that have been covered. All we have done has been to select some aspects of the subject which seem to form the chief stumbling blocks in the understanding of how an aeroplane flies; we have attempted to remove them as stumbling blocks, and perhaps even so to arrange them that, instead, they become stepping stones to the remainder of the subject. In the next chapter we will turn to our real subject – the Mechanics of Flight.

Before continuing, try to answer some of the questions below, and the numerical questions in Appendix 3.

Can you answer these?

These questions are tests not so much of mechanical knowledge as of mechan­ical sense. Try to puzzle them out. Some of them are easy, some difficult; the answers are given in Appendix 5.

1 A lift is descending, and is stopping at the ground floor. In what direction is the acceleration?

2 What is the difference between –

(a) Pressure and Force?

(b) Moment and Momentum?

(c) Energy and Work?

3 Why does it require less force to pull a body up an inclined plane than lift it vertically? Is the same work done in each case?

4 Distinguish between the mass and weight of a body.

5 If the drag of an aeroplane is equal to the thrust of the propeller in straight and level flight, what makes the aeroplane go forward?

6 Is the thrust greater than the drag during take-off?

7 Can the centre of gravity of a body be outside the body itself?

8 Is an aeroplane in a state of equilibrium during –

(a) A steady climb?

(b) Take-off?

9 Are the following the same, or less, or more, on the surface of the moon as on the surface of the earth –

(a) The weight of a given body as measured on a spring balance?

(b) The apparent weight of a given body as measured on a weigh-bridge (using standard set of weights)?

(c) The time of fall of a body from 100 m?

(d) The time of swing of the same pendulum?

(e) The thrust given by a rocket?

10 In a tug-o’-war does the winning team exert more force on the rope than the losing team?

11 Are the following in equilibrium –

(a) A book resting on a table?

(b) A train ascending an incline at a steady speed?

12 A flag is flying from a vertical flag pole mounted on the top of a large balloon. If the balloon is flying in a strong but steady east wind, in what direction will the flag point?

For solutions see Appendix 5.

For numerical examples on mechanics see Appendix 3.

Moments, couples and the principles of moments

The moment of a force about any point is the product of the force and the per­pendicular distance from the point to the line of action of the force.

Thus the moment of a force of ION about a point whose shortest distance from the line of action is 3 m (Fig. 1.11) is 10 X 3 = 30 N-m. Notice that, though both are measured by force X distance, there is a subtle but important distinction between a moment (unit N-m) and the work done by a force (unit Nm, or joules).

The distance in the moment is merely a leverage and no movement is involved; moments cannot be measured in joules.

Moments, couples and the principles of moments

Moments, couples and the principles of moments

Fig 1.11 Moment of a force (Anti-clockwise)

A moment is normally taken to be positive if it is in a clockwise direction, and negative if it is in an anti-clockwise direction.

If a body is in equilibrium under the influence of several forces in the same plane, the sum of the clockwise moments about any point is equal to the sum of the anti-clockwise moments about that point, or, what amounts to the same thing and is much shorter to express, the total moment is zero. This is called the principle of moments, and applies whether the forces are parallel or not.

When considering the forces acting on a body, the weight of the body itself is often one of the most important forces to be considered. The weight may be taken as acting through the centre of gravity, which is defined as the point through which the resultant weight acts whatever position the body may be in.

Two equal and opposite parallel forces are called a couple. The moment of a couple is one of the forces multiplied by the distance between the two, i. e. by the arm of the couple. Notice that the moment is the same about any point (Fig. 1.12), and a couple has no resultant.

Moments about O. P10Xl = 10 clock, Q10Xl = 10 clock,

Total 20 clock.

Moments about A. P zero, Q 10 X 2 = 20 clock,

Total 20 clock.

Moments about В. P 10 X 2 = 20 clock, Q zero,

Total 20 clock.

Moments about С. P 10 X 6 = 60 clock, Q 10 X 4 = 40 anti,

Total 20 clock.

Подпись: 2 metres ------ О Подпись: 4 metres

P 10 newtons

Q 10 newtons

The triangle, parallelogram, and polygon of forces

If three forces which act at a point are in equilibrium, they can be represented by the sides of a triangle taken in order (Fig. 1.10). This is called the principle of the triangle of forces, and the so-called parallelogram of forces is really the same thing, two sides and the diagonal of the parallelogram corresponding to the triangle.

If there are more than three forces, the principle of the polygon of forces is used – when any number of forces acting at a point are in equilibrium, the polygon formed by the vectors representing the forces and taken in order will form a closed figure, or, conversely, if the polygon is a closed figure the forces are in equilibrium.

Composition and resolution of forces, velocities, etc

Подпись: в
Composition and resolution of forces, velocities, etc

A force is a vector quantity – it has magnitude and direction, and can be rep­resented by a straight line, passing through the point at which the force is applied, its length representing the magnitude of the force, and its direction corresponding to that in which the force is acting. Forces can be added, or sub­tracted, to form a resultant force, or they can be resolved, that is to say, split into component parts, by drawing the vectors to represent them (Fig. 1.9).

Fig 1.9 Composition and resolution of vector quantities

Note that velocity and momentum are also vector quantities and can be rep­resented in the same way by straight lines. Mass, on the other hand, is not; a mass has no direction, and this is yet another distinction between a force and a mass.

The behaviour of gases

In the study of the flight of aircraft, we are really only interested in the behav­iour of one particular gas, air. The most important relationship that we need to know is called the gas law, which can be written as:

Подпись:= RT

where p is the pressure, p is the density, T is the temperature measured relative to absolute zero (i. e. in degrees Kelvin in the SI system), and К is a constant called the gas constant.

If the gas is compressed its density increases, so either or both the other quantities, temperature or pressure, must change. The way that they change depends on how the compression takes place. If the compression is very slow, and the gas is contained in a poorly insulated vessel so that heat is trans­ferred out of the system, then the temperature will stay constant, and the pressure change will be directly proportional to the density change. This is called an isothermal process, and it involves a heat transfer from the gas to its surroundings. In this case the relationship between pressure and density is given by:

Подпись: t = P a constant

If the compression takes place with no transfer of heat, which commonly occurs when compression is very rapid, then the change is said to be adiabatic. If the change also takes place without any increase in turbulence, so there is no increase in the disorder (entropy) of the system, then the process is called isentropic, and the relationship between pressure and density is given by:


рУ = a constant

where у is the ratio of the specific heat at constant pressure to the specific heat at constant volume, and has a value of approximately 1.4 for air.

We cannot go much further down this path without becoming embroiled in the complexities of thermodynamics, however, and as the relationships above are the only ones that are relevant to the understanding of the contents of this book, we will not pursue the subject any further.