Aerofoils and Wings
It is usually necessary to have wings to fly, and the design of the wings (and the cross-sectional geometry of them, i.e. aerofoils) largely dictates the aerodynamic performance and handling characteristics of aeroplanes.
You do not have to be an aeronautical engineer to fly gliders (though a considerable number of CUGC members actually are), but it generally helps the training and examinations if the pilot has a sound understanding of some of the fundamental physical concepts. This article aims to provide a correct understanding of how wings work, through a rather painless physical discussion.
Contents
Definitions
A cross section of a wing has the geometry of an aerofoil.
The foremost point of the aerofoil is the leading edge, where the aftmost point is the trailing edge. A straight line connecting these two is the chord line.
The concept of leading edge and trailing edge extends naturally into three dimensions.
In the aeroplane frame of reference, the angle between the chord line and the incident flow is the angle of attack.
What a wing actually does
The fundamental purpose of having wings is to produce lift, which balances the gravitational force on the aeroplane so that it can stay airborne.
Lift does not come for free, and it can be theoretically proven that, whenever lift is created by means of a wing (or indeed by any other geometry moving through unconfined fluid), a drag force will also be created. Drag is a force that constantly does negative work on an aeroplane thereby consuming its mechanical energy, which must be balanced by using either an engine (in case of a powered aeroplane) or by releasing gravitational potential (height, in case of a glider). This is why gliders must always descend relative to the surrounding air to maintain its airspeed.
Wherever forces are involved, a resultant moment can be defined. By definition, an aerofoil produces a constant moment irrespective of angle of attack around its aerodynamic centre.
How lift is created
Consider an object moving along a curved path. Elementary Newtonian physics dictates that force (at least a component of it) must be exerted on the object in the direction normal to its path pointing into the concave side of the path, i.e. the centripetal force.
Now observe the flow around the aerofoil. The streamlines are effectively the path of fluid particles when they move around the aerofoil. Note that, because of the presence of the aerofoil, the streamlines are bent. Using the argument made above, a force must be exerted on the fluid (by the aerofoil) pointing into the concave side of the streamlines, i.e. downwards. By Newton’s third law, the fluid exerts on the aerofoil a force of equal magnitude but in the opposite direction, i.e. upwards. This is the lift force on an aerofoil.
The reason why the streamlines follow the geometry of the aerofoil and bends into the way we see is not trivial, and nature enforces that this be the flow field by viscosity: it can be theoretically proven that no lift may be generated should the fluid be inviscid.
An alternative method of explaining lift creation is available if the reader understands the basics of the control volume method: examine a control volume enclosing the aerofoil and the immediate vicinity, air enters the control volume going horizontal or slightly upwards, but exits going downwards. Because the aerofoil pushes the air down, the air pushes the aerofoil up.
Note that the energy approach is generally not applicable to the explanation of lift: no work is being done nor is any energy being transferred between the aeroplane and the air in the process.
Creation of lift: fake physics
The following argument is usually quoted in an attempt to explain the creation of lift:
“Air meets the wing and separates into two streams, but the particles must meet again at the trailing edge in the same time. Therefore, the airflow on the upper surface is faster because the route to be travelled in the same period is longer. According to Bernoulli’s Equation, fluid that travels faster has lower pressure. Therefore, there is a pressure difference between the two surfaces which integrates into the force of lift.”
This explanation is, disappointingly, being used by science textbooks in various countries, by flying instructors, and (so it is said) by the RAF. It cannot be stressed enough that this explanation is wrong, despite the fact that it invokes Bernoulli’s Equation which is of fundamental importance in the theory of potential flow. To be specific, this explanation contains two points of error:
1. The Bernoulli’s Equations is a streamline equation. It can only be applied along a streamline and not otherwise, unless adequate treatment is given to the Bernoulli Constant to prove that it is the same for the two points of interest. This can, however, be done in this case. Furthermore, Bernoulli’s Equation is only strictly true for inviscid, incompressible fluids, and air is neither (but it is often assumed to be so).
2. Air particles do not meet at the trailing edge in the same time: this is easily demonstrated by using a pulse of smoke in a wind tunnel. In fact, the air over the upper surface flows faster and overtakes the same air used to be alongside it by a considerable amount by the time the trailing edge is reached. It can be proven that, if the air takes the same time to travel along the wing, no lift is generated.
For the same reason as in (1), blowing air over a sheet of paper is not a demonstration of the Bernoulli’s Equation (the paper will not bend sideways if held vertical). The sheet of paper works almost exactly as an aerofoil as explained before: it causes the streamlines to bend, thereby harvesting the reaction by the fluid.
People with some qualitative aerodynamic knowledge often argues that it is the “Kutta condition” that the air meets at the trailing edge in the same time. However, the Kutta condition, despite a lack of precise mathematical formulation, requires nothing more than the trailing edge being a stagnation point (in 2D). In other words, it requires that the streamlines meet, just like two carriage ways merge into one, but the vehicles on the carriage ways can travel at very different speeds before reaching the junction.
What does the lift force depend on
In 2D, the lift force produced by an aerofoil depends only on two factors: the angle of attack, and the geometry of the aerofoil. To be specific, the only factor about an aerofoil that matters is the camber of the aerofoil, i.e. how bent it is (the mathematical definition will not be introduced in this elementary article). The thickness of the aerofoil has zero (in theory, and very little in practice) effect on lift, which is not easy to understand without working through the continuum mechanics. However, thickness is a useful design tool to modify the pressure distribution around the aerofoil, thereby improving the stalling characteristics.
Any symmetric aerofoil has a lift coefficient (lift force per unit chord and unit depth) of 2πα, where α is the angle of attack. A cambered aerofoil has an additional lift coefficient at zero angle of attack added to this value.
In 3D things are more complicated. Consider the tip of a wing: little pressure difference on the upper and lower surfaces can be sustained at the tip, otherwise the flow will accelerate to very high speeds because an escape from the lower to the upper surface is possible. Therefore, it is necessary that the flow around a 3D wing has a spanwise variation, despite the wing might have the geometry of a uniform extrusion. As a result, it is expected that the aspect ratio of the wing (how slender or stubby it is) to have an effect on aerodynamic performance.
A real wing has a larger lift coefficient (lift force per unit area) at the same angle of attack if the aspect ratio is higher. When the aspect ratio tends to infinity, the wing (3D) lift coefficient is the same as the aerofoil (2D) lift coefficient. If the aspect ratio is finite, the wing lift coefficient is generally smaller than that of an aerofoil would achieve in 2D.
This is the first reason why high aspect ratio (slender) wings are aerodynamically desirable.
Boundary layer and stall
Glider pilots usually are introduced the phenomenon of stall within the first several flights. This section explains the fundamental physics of stall.
Stalling is a viscous phenomenon: it cannot be predicted using the inviscid flow theory. As far as the pilot is concerned, stalling matters because there is a loss of lift (which results in a high rate of descent), ineffective controls, and a possibility of spinning (which will not be discussed here).
Review how lift is created: it is necessary that the streamlines follow the shape of the aerofoil and bends accordingly. If the streamlines cease to follow the aerofoil, i.e. the flow detaches from the aerofoil, lift will be reduced very significantly. This is fundamentally what a stall is. It is observed that wings stall once a critical angle of attack is reached. To understand the physics of stall, the concept of boundary layers must first be introduced.
Consider pouring honey out of a jar: can you empty the jar completely? This is not possible because the honey sticks on the inside wall of the jar and it will not come off completely no matter how long you allow the jar to drain. This is the no-slip condition of viscous flow: honey is a viscous fluid, and whenever it contacts a wall, it will not slip on it but stick onto it.
Air is not as viscous as honey, but it still has some viscosity, which means it will stick onto the surface of a wing. Now consider the air on top of the wing: at some distance away the air is flowing at the flying speed, while on the surface it sticks, i.e. flowing at zero speed. As a result, there must be some distance where the flow speed increases gradually from zero to maximum, and in this region the flow speed is less than the outer bulk flow because of the retardation effect of the wing surface. This layer is referred to as the boundary layer.
Just like glider pilots, air can trade freely between two forms of energy, namely pressure potential and kinetic energy. By releasing an inflated balloon the pressure potential is traded for kinetic energy, thus the air accelerates and forms a jet which propels the balloon forward. When air flows around an aerofoil, these two forms of energy is constantly traded for each other: air accelerates by going from high pressure to low pressure and vice versa. The quantitative description of this trading is the Bernoulli’s Equation.
Flow in the boundary layer, however, is in a less favourable position, because its kinetic energy is constantly being robbed by the friction effect of the wall. When it moves from high pressure to low pressure, it gains some kinetic energy, but when it has to go back to the same high pressure again, it finds itself having less kinetic energy than what it would need to do so. When the air has exhausted its kinetic energy but still has a pressure mountain to climb (referred to as an adverse pressure gradient), it has no means of doing so and it will refuse and go away. This is the phenomenon of boundary layer separation.
A separation of the boundary layer means the flow will cease to follow the aerofoil. If the separated region is significant, it is characterised as a stall.
Things are, unfortunately, further complicated by the fact that, while the wing can remove energy from the boundary layer, the outer flow can help the boundary layer by “dragging it along”. The interacting factors become such a mess that a mathematical description of the precise point of separation is not yet possible. However, it is generally observed that, there are two methods to make a boundary layer separate, namely a very steep adverse pressure gradient, or a less steep adverse pressure gradient over a prolonged distance.
Here it is necessary to quote without proof that the pressure gradients over the upper surface of an aerofoil is generally proportional to the angle of attack.
A leading edge stall happens when the angle of attack is so large that the boundary layer separates straight away at the leading edge under a very steep adverse pressure gradient. This stalling behaviour is very unpleasant as very little warning is given and the loss of lift is sudden and drastic. This is usually avoided by designing a thick aerofoil where the steep adverse pressure gradient around the leading edge is smoothed out.
A trailing edge stall happens when the boundary layer separation point near the trailing edge moves forward because the adverse pressure gradient is increased due to an increased angle of attack. This type of stall is more gentle with plenty of warning signs and a gradual loss of lift. This is the stall behaviour observed on training gliders, e.g. K21s.