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Pressure instruments are widely used in aviation. This article introduces the basic components of pressure as relevant to gliding. We shall then discuss how the pressure varies in the atmosphere, and the implications of this on the functioning of pressure instruments.

The author finds it very hard to discuss such topics without the aid of mathematical formulae. Every effort shall be made to explain the physical background of all the definitions and theorems used. If there is something you cannot understand, please consult an instructor who may determine what level of mathematical understanding (if any) is necessary. You do not need to be a physicist or engineer to fly gliders.

== Fundamental Thermodynamics ==

Thermodynamics is a basic science that deals with energy. Its most notable application being the generation of power through heat engines. However, the only bit of thermodynamics we need to understand pressure is the '''Ideal Gas Law'''.

=== States and thermodynamic properties ===

Air can be liquified, but for all aviation purposes air is a '''gas'''. We shall not define a gas but the reader should have a good understanding of the concept through life experience.

Any substance of engineering interest can have '''states'''. For example, water can be ice, liquid water, or water vapour. If you are asked the question "What is the state of water at room temperature?" you may intuitively answer "liquid water". However, believe it or not, by reaching this conclusion you are making the assumption that the pressure that this water is at is at everyday atmospheric level. If the same water is put into vacuum while maintained at the same temperature, it will boil.

Therefore, to fully define the state that a substance is at, two thermodynamic properties must be specified. The most common thermodynamic properties can be:
*Temperature \(T\)
*Pressure \(p\)
*Density \( \rho \)
and a bunch of others. At least two must be known to fully define the state of a substance.

It is '''very important''' to understand that, by calling a thermodynamic property a '''property''', it is implied that it belongs to the substance. It does '''not''' depend on the way you look at it, or, in physical language, it is independent of the frame of reference. For example, when you stand still you think the air is stationary at a temperature of 25°C, but if you drive past in a car you will think the air is moving. This does not, however, affect the fact that the air is still at  25°C. We shall see later that we can define some other things that depends on motion, known as the stagnation '''quantities''', sometimes incorrectly referred to as "stagnation properties".

=== Ideal gas ===

Air can be modelled as an ideal gas. An '''ideal gas''' is a gas for which the following relationship holds:

\[ p = \rho R T \]

where:
*\(p\) is the pressure
*\(\rho\) is the density
*\(R\) is a constant that is a property of the gas itself, and
*\(T\) is the temperature.

Therefore, if we want to find the density of some air, both the pressure and the temperature needs to be specified (it is possible to substitute one or both with something else, but we shall not investigate this complexity). For example, atmospheric air and the air in a tyre are at the same temperature, but since the air in the tyre has a higher pressure (if you remember pumping it in), it has a higher density than atmospheric air.

In this equation, the temperature and pressure must be absolute quantities. In other words, the temperature (with the unit of Kelvins) needs to be measured against absolute zero (where the entropy of any perfect crystal is zero), and the pressure is measured against absolute vacuum (the pressure to be found in a finite volume in which there is no substance). You may have come across the term "gauge pressure" where the pressure is measured against something else: these relative quantities are incompatible with thermodynamics.

== Pressure ==

Three types of pressure are used in the physics of flying. This section introduces the concepts, which will be used subsequently in later sections.

Gliding is a kind of low speed flying, so we shall ignore compressiblility effects where applicable: a note will be made when this happens.

Pressure is a potential. It measures the ability of some gas to do useful mechanical work. Compressed gas can be used to move a piston and start an engine, for example, because that gas is at high pressure. Thus, a gas at some pressure also has some energy associated with it. Understanding this part will show that some definitions are not invented in random.

=== Definitions ===

==== Static pressure ====

The static pressure, denoted as \(p\), is a thermodynamic property of the air. By the ideal gas Equation of State, the static pressure can be determined if the density and temperature are known.

==== Dynamic pressure ====

The dynamic pressure is a quantity (i.e. not a thermodynamic property) that describes the energy due to the motion of a fluid. If a body of mass \(m\) is moving at speed \(V\), then the kinetic energy it has is found to be:

\[E_K = \frac{1}{2} m V^2 \]

This can be shown by considering the work input to accelerate the body from rest to the speed \(V\).

We need something similar to describe the kinetic energy of a moving fluid, and we need this to have the dimensions of pressure. It is found that the following quantity is suitable:

\[p_D = \frac{1}{2} \rho V^2 \]

where \(p_D\) is defined as the '''dynamic pressure'''. It is the kinetic energy of one unit volume of fluid moving at the speed \(V\).

==== Total pressure ====

The '''total pressure''', also known as the '''stagnation pressure''', is a sum of the static and dynamic pressures.

\[ p_0=p+p_D \]

It is the '''static''' pressure that the fluid will reach if it is '''stagnated''', i.e. brought to a rest from motion. If you stick an object into water flow (try this with a finger, be sure not to use hot water), you will observe the water level increase a bit in front of that object, which is a consequence of stagnating the water increasing its pressure, reflected in depth. 

There is an extra requirement that the stagnation happens in an isentropic manner, but this is usually true when we consider air at low speeds. In fact, the stagnation of air is usually treated as isentropic unless we know it is not (for example, if we know a shock wave is present). Not being isentropic means mechanical energy is dissipated in the process, if you spill water on the ground and let itself come to a rest, you will not observe the level rise, for example.

==== Compressible effects ====

You may ask, if the static pressure is to increase, then by the ideal gas Equation of State, definitely the density and the temperature are to change? You are correct and this is the compressible effect which, up to now, the author has tried to avoid like the plague. Some people say that, because the flow is incompressible, the density remains constant and the temperature increases, but this is wrong because by invoking the Equation of State the assumption of incompressibility is voided automatically. In other words, the Equation of State has its foundation on the fact that gas molecules can get closer or further apart as governed by some rules.

What will happen is that the density and temperature will increase simultaneously. The precise amount we cannot solve for based on the equations presented in this article: other thermodynamic formulae relating temperature to pressure must be introduced. The exact solutions will be presented as a function of '''Mach number''' (the ratio between velocity and the local speed of sound).

If you are still interested, you may contact the author. You are encouraged to become an aeronautical engineer.

=== Pressure coefficient ===

The '''pressure coefficient''' at point A (around an aeroplane), by definition, is:

\[ C_p(A) = \frac{p_A-p_{\infty}}{\frac{1}{2}\rho V_{\infty}^2} \]

where:
*\(p_A\) is the static pressure at point A
*\(p_{\infty}\) is the static pressure far away, and
*\(V_{\infty}\) is the flow velocity far away

By far away (known as '''free-stream quantities''') we mean at a sufficient distance away from the aeroplane that effectively the flow is not disturbed by the aeroplane flying through it. Consider a car that passes in front of you at several feet away, you will feel some wind which is the flow disturbed by the presence of the car. However, you do not feel a lot of disturbance even if there are millions of cars driving in the United States.

Note that the velocity is measured in an aeroplane frame of reference so we can treat the problem as flow going around the aeroplane which makes things simpler. If the frame of reference is changed in this way, even the air above the United States can come at you at a velocity \(V\) depending on how fast you fly, so the author is just half joking when the above example is made.

The pressure coefficient is non-dimensional: it is a pressure divided by a pressure so no unit emerge from this algebra. However, if the flying speed is kept constant, i.e. the denominator is kept constant, the pressure coefficient is a representation of \(p_A\) given that the pressure in United States does not depend on the way you fly. If \(C_p\) increases, it means \(p_A\) is higher.

The maximum possible value of \(C_p\) is unity, which corresponds to stagnating the flow. If all the kinetic energy has been transformed into pressure potential, there is nothing else whatsoever we can do to increase the flow pressure further (with the exception of adding some work by mechanical means, which we shall not consider). There is, in theory, no minimum value pf \(C_p\), so long as \(p_A\) is above zero.

=== How to measure pressure ===

A '''pitot tube''' is a device used to measure total pressure. It works by pointing a bent tube directly into flow so that the flow is brought to a rest when the bend is reached (i.e. stagnated). As a result, the static pressure raises to the total pressure value of the free stream. A pitot tube is usually found at the nose of an aeroplane (e.g. the K-21s), but it can be elsewhere. There are designs where the pitot tube sticks out of the vertical stabliser (the fin) or on the side of the fuselage.

A '''static port''' is used to measure static pressure. This port must be located on an aeroplane where the pressure coefficient is zero. In addition, it needs to be perpendicular to the flow so that no dynamic pressure is converted into static pressure by the slowing down of the flow. On an aeroplane a static port (a small hole) is usually found on the side of the fuselage. You can ask an instructor to show you where this is.

The difference between the pitot tube reading and the static port reading is the dynamic pressure.

Note that there are restrictions on both of these regarding the relative direction with respect to the flow. Given that most of the times an aeroplane flies straight and level, the designers will use this attitude to design the pitot tube and the static port. If a significant amount of yaw is present, or if the angle of attack is extreme (such as when an aeroplane is stalled), these pressure readings will be unreliable.

For reasons that should be obvious by now, it is important that the pitot tube and the static port are not blocked. This should be a part of the daily inspection of an aeroplane. Furthermore, there is a chance that the pitot tube may ice up in flight, for example, if the aeroplane is flown in rain at low temperatures.

== Atmosphere ==

=== Simplified model ===

=== Compressible effects ===

=== The International Standard Atmosphere ===

== Instrumentation ==

=== Air speed indicator ===

==== Function ====

==== Errors: TAS & IAS ====

==== Stall speed and \(V_{NE}\) ====

=== Altimeter ===

==== Function ====

==== Setting ====

=== Vertical speed indicator (power) ===

=== Total energy in gliding: a more interesting quantity ===

==== How to measure total energy ====

=== Variometer (gliders) ===