The Aerofoil

tl;dr

Aerofoils are specially designed and constructed to produce a force called Lift as they move through the air. Other forces, such as Drag, Thrust and Weight also act on the aerofoil as it moves.

The Lift an aerofoil produces depends on a number of factors, some of which are beyond our control as quadcopter operators. Others, such as the rotational speed of the Rotors and hence the velocity of the air passing the aerofoil, can be adjusted and this is how we control the quadcopter as we fly it.

Intro

In this chapter, I’m not going to attempt to establish how Aerofoils work and Lift is produced in any great detail. However, I will summarise the concepts and I suggest some links for you to visit if you are interested in finding out more. I encourage you to do this.

What is an Aerofoil

An Aerofoil (or ‘airfoil’ for North Americans) describes the cross section of a wing, rotor, rudder, propeller or other lifting surfaces. They come in many shapes and sizes and are usually designed for a specific purpose or application. For a broad description and more information go to the Airfoil page on Wikipedia.

If you take some time to read that Wikipedia page you will see that there are a few characteristics of aerofoils that help define it. While they may vary in size, two key qualities help define the type of aerofoil:

  • The Chord Line is defined as a straight line joining the leading edge with the trailing edge.
  • The Camber Line is defined as the line equidistant between the upper and lower surfaces.

On the diagram below you can see pictorial descriptions of the Chord and Camber lines. This aerofoil has an Angle of Attack of zero degrees, which is defined as the angle between the Relative Air Flow (RAF) and the Chord Line. This means it is streamlined to the direction of travel which usually implies it is producing the least Drag or wind resistance.

Fig4.1_Clean_Aerofoil_0aoa_Watermarked_1
Fig 4.1 – Aerofoil at Zero Degrees

How Lift is Produced

As an Aerofoil moves forward through the air, it displaces some of that air. Unless the aerofoil is perfectly symmetrical and perfectly aligned with the direction of the RAF, some of the air is forced away from the aerofoil.

In this case, the force applied to move the air is opposed, in and equal and opposite sense, to the aerofoil. Those familiar with basic physics will have heard of Newton’s Third Law; this effect on the aerofoil is an example of that Law.

Basic Forces

This force that moves the Aerofoil is called the ‘Total Reaction’ or the ‘Total Aerodynamic Force’. For convenience, aerodynamicists resolve this force into two vectors:

  • Lift – the vector that opposes gravity in normal circumstances, and
  • Drag – the vector that opposes the movement of the Aerofoil through the air.

To keep things in balance at this stage, two further ‘equal and opposite’ forces are needed to keep the aerofoil in equilibrium. They are:

  • Weight – the force applied to the aerofoil due to gravity, and
  • Thrust – the force required to oppose the Drag and keep the aerofoil moving at a constant velocity.

This diagram shows these four forces in equilibrium. In this case, the aerofoil has an Angle of Attack of 10 degrees. You can see that the Chord Line is at 10 degrees relative to the horizontal, which is the direction of the RAF.

Fig4.2_Forces_Aerofoil_10aoa_Watermarked_1
Fig 4.2 – Basic Forces on an Aerofoil

 

What Affects the Lift

The Lift produced by an Aerofoil can vary due to a number of reasons. Some are within our control, either in the design and contraction phase or when operating the quadcopter. Others are due to environmental factors beyond our control.

There is a lot of physics and maths theory involved in understanding how lift is produced and controlled. The general concepts are covered well in the Lift Wikipedia page for those who want more info. I suggest you continue reading here then follow up there for any concepts that you need further explained.

Aerofoil Design

For our purposes, the main design criterion an Aerofoil needs for its specific task is its profile or Camber. The actual cross-sectional shape of the aerofoil can be symmetrical, that is the Camber Line and the Chord Line are coincident, or asymmetric, where the Camber varies, sometimes significantly, from the Chord Line.

An aerofoil’s thickness, or more specifically its thickness to chord (or t/c) ratio, also affects its performance. A ‘fatter’ aerofoil has to force the air further away from its resting position as it passes. In principle, this means that a fatter aerofoil may produce more Lift, but there is also the penalty of the Drag or wind resistance, to consider and hence the thickness to chord ratio is often a compromise.

Most quadcopters have an aerofoil that is moderately cambered with a low t/c ratio, or ‘thin’ profile. This provides a compromise between Lift production at the relatively high speed of rotation of the blades and the Drag that is produced and hence the power of the motors required to spin them.

For those really keen to learn more, try the NACA Airfoil Wikipedia page.

Angle of Attack

As a fixed Aerofoil design moves through the air at a constant velocity it would be beneficial to be able to vary the Lift produced. This is done by changing the Angle of Attack (AoA) of the aerofoil to the Relative Air Flow (RAF) and thus changing the amount and direction that the aerofoil moves the air out of its path.

As we have discussed above, moving more air increases the ‘equal and opposite’ force on the aerofoil which we have described as the Total Reaction, and which we resolve as Lift and Drag vectors.

So, increasing the AoA, up to a specific angle, proportionally increases the amount of lift the aerofoil produces. In scientific terms, the AoA is often referred to as ‘alpha’ and its Greek letter ‘α’ is used to denote AoA in aerodynamic formulas. Therefore we can say that Lift is proportional to α.

So in this next figure, you can see that increasing the α to 22 degrees gives a corresponding increasing in Lift. Because to is actually the Total Reaction that is increased, we also get a corresponding increase in Drag. Thrust must be increased to counter the Drag if we want to maintain equilibrium. The increased Lift would normally cause the aerofoil (and the attached aircraft) to increase altitude or climb. In this illustration, the Weight has increased and so the aerofoil remains at equilibrium.

Fig4.3_Forces_Aerofoil_22aoa_Watermarked_1
Fig 4.3 – Increasing Lift with increasing AoA

When a certain α (that is particular to the design of the aerofoil but usually between 20 and 25 degrees) is exceeded, the airflow over its surface breaks away and becomes turbulent. This means the aerofoil is unable to force the air downwards and hence Lift is dramatically reduced. In this case, the aerofoil is said to be ‘stalled’.

There are many Youtube videos available that show wind tunnel video of aerofoils in smoke streamlines that may help you visualise these concepts. Two good ones are here and here.

In a fixed wing aircraft, whilst the wings are set at a fixed angle to the aircraft fuselage, the Angle of Attack of its wings to the RAF can be varied by adjusting the tailplane and elevators. This ‘pitch attitude’ change adjusts the Angle of Attack and hence the Lift produced by the wings and determines whether the aircraft climbs or descends.

A helicopter works similarly with the pitch angle of the Rotor blades being able to be adjusted both collectively and cyclically by the pilot’s controls as the rotor spins.

In the case of our quadcopters, however, you will have noticed that the rotor (or propeller) blades are fixed to the hub and hence the Angle of Attack is fixed by the construction of the rotors. So how can we control the Lift?

Velocity

As we have learned above, as an Aerofoil moves through the air it displaces some of the air away from itself. If you have read the Wikipedia articles on Lift you will know that in a normal aviation application aerofoils are mounted in a way to deflect this air downwards and hence provide an upwards force to counter gravity.

It follows that the quicker the aerofoil moves through the air, more air is displaced and hence more lift is created. As the lift force exceeds the Weight (which is also a force, described as the mass of the aircraft multiplied by the acceleration due to gravity) the aircraft climbs. This fact is the reason why planes must accelerate along a runway to takeoff.

It happens that Velocity is so important to the production of lift that its effect is proportional to its square (that is, velocity times velocity). To learn more, read about it on the NASA website.

With normal planes and helicopters, whilst velocity is important for lift production, their sophistication allows other design features, such as controls that change their aerofoils AoA, to help control their lift so they can climb and descend. For quadcopters, with fixed angles of attack on their rotors, the only available method of lift control is velocity. The various motors that drive the quad’s rotors are able to increase and decrease their speed as required by the quad’s CPU to control the quad as commander by you as you pilot the aircraft around. We will see how this is done in further chapters.

Air Density

We have seen that an Aerofoil’s production of Lift, and by necessity Drag, is dependent on shifting a mass of air downwards (in the case of wings and rotors). The more air that is moved, the more lift is created. As we need this lift to exceed Weight (which, in the case of quadcopters, is a fixed amount), we need increasing amounts of lift to climb the quad and fly it around.

The density of air decreases as Pressure Altitude (that is, the height above sea level) increases. This is due to two main factors. Firstly, the closer the air molecules are to the centre of the earth, the greater the gravitational attraction, and secondly, the air near the surface of the earth has more molecules above it and the ‘weight’ of these molecules helps compress the air near the surface more. See this link for a greater explanation.

Air Density is also affected by the temperature (and to a lesser extent, the humidity) of the air. Simply put the hotter air molecules whizz around more and take up more volume than cooler ones. If the same number of molecules are in a larger space, the density is less. This principle explains how a hot air balloon works.

One of the main causes of an increase in air temperature is contact with the ground. It is no surprise that the air near a hot and high location, such as Denver in summer, is less dense than the air at 5,000 feet overhead San Francisco on the same day.

This relationship between the pressure, temperature and humidity of air is described as its Density Altitude. This concept gives us the best measure of the performance of an aerofoil as environmental factors change. It tells us that if we keep the Aerofoil, AoA and Velocity constant but increase the density altitude, the lift produced will be less. So, a quad that is able to hover at a lower altitude may not be able to hover at, say, 5,000 feet.

This concept of density altitude is why some manufacturers advise a maximum altitude for the operation of their quads. On a hot day, in a high place, the motors may not be able to spin the rotors fast enough to provide the lift to control the quad satisfactorily. We will explore this concept more in future chapters.

Area

Up until now, we have concentrated on an Aerofoil which is essentially a two-dimensional shape. If we extend an aerofoil in 3D we get an increase in lift that is proportional to the area of the wing or rotor. This is due to the 3D aerofoil interacting with a greater number of air molecules along its length.

The absolute length and hence area is a compromise of course, with rotor designers having to contend with the strength and tensile requirements of a longer rotor with the management of the mass of the component and therefore its effect on the weight of the aircraft.

Lift Formula

So we have seen that the Lift produced by an Aerofoil is dependent on a number of factors. They are:

  • Aerofoil design
  • Angle of attack
  • Velocity
  • Air density
  • Rotor or wing area

For the mathematically inclined, aerodynamicists have developed a Lift Formula which looks like this:

Lift = CL ½ ρ v2 A

where:

  • CL is the ‘Coefficient of Lift’ which combines all the fixed factors of a particular aerofoil, or rotor blade. These include:
    • The aerofoil design, including NACA profile, Chord, Camber, etc and
    • The aerofoil’s Angle of Attack
  • ρ (rho) is the symbol that represents Air Density
  • v is the velocity of the rotor through the air
  • A is the area of the rotor blade

The formula, therefore, advises us that the lift produced, and hence the performance of our quadcopter, is proportional to each of these factors. As we have seen, though, most of them are fixed, that is we cannot change them, for a particular day and place where we choose to operate.

Fortunately, we can control one key factor: the velocity of the blades as they travel through the air. How that works and how we can control our quad is what we will discuss in the next chapter.

Further Reading

Wikipedia

NASA

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