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.
For simplicity, this article only considers flight at gliding speed (Mach ~0.1), so any compressible flow effect is ignored.
- 1 Definitions
- 2 Fundamental physical concepts
- 3 What a wing actually does
- 4 How lift is created
- 5 Creation of lift: fake physics
- 6 What does the lift force depend on
- 7 Boundary layer and stall
- 8 Drag
- 9 Downwash (advanced topic)
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.
Fundamental physical concepts
Force and moment
Force is the action on a body by a body. If you jump, you fall back onto the ground, and it is the gravitational force that pulls you down. Gravity is the action by the Earth on you.
A force combined with a leverage gives a moment. When you use a wrench to tighten a nut, you exert a force, and, combined with the leverage of the wrench, creates a "twisting force" on the nut which tightens it. The moment is proportional to the leverage, so if you cannot tighten a nut sufficiently, the solution is to use a longer wrench.
Mechanical work and mechanical energy
Work is a transfer of energy. When you lift a stone, your lifting force does work on it, in which process energy goes from you to the stone (and as a result, you will feel tired). If, as a result of doing work, energy goes into A, we say positive work is done on A, conversely, if energy leaves B, we say negative work is done on B. In this case, your lifting force do positive work on the stone, and the weight of the stone does negative work on you.
Energy can take many forms, such as kinetic energy (energy associated with moving with a speed) and potential energy. In this aerodynamic discussion we consider gravitational potential and pressure potential.
Gravitational potential is the energy associated with being in a (physically) high place. When you lift the stone, the energy that you give the stone becomes its gravitational potential because it has moved to a higher place. Pressure potential is the energy associated with being in a high pressure: you will feel tired after blowing up a hundred balloons, which is because the air inside a balloon has pressure potential, which ultimately comes from you, if you do not blow, the balloon does not inflate itself.
Energy is conserved, which means it cannot be created nor destroyed. It can only go from one body to another body, or go from one form into another form. If you drop the stone, its gravitational potential will become kinetic energy, so it will pick up speed. When the stone hits the ground and comes to a stop, we say its mechanical energy has been dissipated, but it has really gone into heat: we just do not notice.
A fluid has viscosity if it is "sticky". When you spill ketchup or syrup, it does not come off your clothes voluntarily: it sticks onto it. A fluid which has viscosity is said to be viscid. If a fluid has zero viscosity, it is said to be inviscid. Air is viscid, although its viscosity is so small that we seldom notice.
Being normal is being orthogonal (at an angle of 90 degrees) to the tangent line of the curve drawn at the location of interest.
A coefficient is a non-dimensional quantity. It is obtained by multiplying (or dividing) something on the quantity of interest to eliminate the dimensions. If you are 1.8m tall, your height has a unit of metres and a dimension of length. If your height is divided by a quantity (which can be chosen rather randomly), say the wingspan of a K-21 (18 metres), we can say you have a "height coefficient" of 0.1, which has no unit and no dimension. So long as this random quantity is kept constant, the coefficient is an equivalent representation of the actual value.
A field is a spacial distribution of some quantity of interest. If a field is known, effectively the quantity is known at every location within that field (we say the field evaluates to something at that point). For example, there is a temperature field in your house, which might evaluate to 25 degrees on the armchair in the living room, 20 degrees on the bedroom floor, but only 15 degrees somewhere on the balcony.
A gradient is how quickly something (a field variable, i.e. a quantity that can be represented by a field) changes with respect to distance. The most notable example being the slope of the ground. On a hill, the height varies with position, so we can define a height gradient. If the height gradient is large, this means the height changes quickly with respect to distance, so we say (the slope of) the hill is steep.
The aspect ratio of something is (roughly) the ratio between the length and the width. It describes how slender or stubby something is. A square has the aspect ratio of 1 because its width is the same as its length. A (new) pencil has a larger aspect ratio than a piece of rubber.
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.
To keep the argument rigorous, it is easier to understand why gliders descend relative to air by following a force equilibrium argument (See also: How Gliders Fly). In this process, gravitational potential may be lost (usually) or gained (in case of soaring). When a glider soars, the energy it harvests from the air flow is greater than the rate of dissipation due to drag, thereby allowing the glider to climb without losing 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
An incorrect explanation
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:
- 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).
- 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 two dimensions (aerofoils)
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 three dimensions (wings)
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.
Boundary layer physics
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.
Stalling in 2D
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.
3D stalling behaviour: planform and washout
The stalling of wings in 3D is slightly more complicated than the 2D case. As discussed before, we would expect the spanwise variation of the flow field to have an effect.
Assume in this section that the aerofoil shape does not vary with span: it only scales larger or smaller.
A wing seldomly stall simultaneously (i.e. stalling at the same time for all spanwise locations). Instead, depending on the design of a wing (to be specific, the planform and twist), a stall firstly develops at a particular spanwise location. This location can be at the root, at the tip, or somewhere in the middle. These stalling behaviours are referred to as root stall, tip stall, and midspan stall respectively.
In addition, the stall of an aeroplane is almost never symmetric, i.e. the two wings almost always stall differently, this can be a difference of how deep the stall is, or one stalls but the other one does not. The reader probably has been told by an instructor that this is the fundamental cause of a wing drop and a spin.
Tip stalling is a characteristic of strongly tapered wings. A tip stall is unpleasant and difficult to handle. This is because the imbalanced force from the stalling sites (which are at the wing tips) can create a large moment around the roll axis of the aeroplane, because the leverage is large. This means the aeroplane can enter a wing drop very quickly and a spin will readily develop. In addition, because the ailerons are typically located at the wing tip rather than the root, a tip stall means the ailerons will be completely ineffective.
Rectangular wings or slightly tapered wings have the root stall characteristic. A root stall is the safe kind of stall: the leverage of the imbalanced forces is small, so the aeroplane will be slow to enter a wing drop. Furthermore, the aileron may still have some effectiveness even when the root of the wing is stalled, so the aeroplane is less tricky to handle (consider the exercise to maintain a glider in a straight stall using ailerons). This is the reason why trainer aeroplanes tend to have rectangular wings, e.g. the Cessna Skyhawk.
A special case is an elliptical wing, which will stall simultaneously. This is part of the reason why a Spitfire is difficult to handle.
An aeroplane designer will usually aim to give the aeroplane the root stall characteristic. When a rectangular wing is not suitable for use, there is another method to move the stall site inboard, which is to add a twist along the spanwise direction, referred to as “washout”. This makes the tip installation incidence less than the root, so when the aeroplane increases its angle of attack, the critical value is firstly reached at the root instead of the tip. A K-21 has this feature.
Washout can also be used to modify the lift distribution and reduce the induced drag: this is not the primary consideration of this article.
A classic problem in hydrodynamics is called the “d'Alembert’s Paradox”. It describes the very bizarre result that, despite every care being taken and a wide range of cases and methods considered, theoretically anything moving in a fluid will have precisely, absolutely, identically zero drag. Any child who has waved a tennis bat will be able to tell that this is complete rubbish, but for hundreds of years fluid dynamists could not tell what had gone wrong.
Until in 1904, the superstar of modern engineering mechanics, Ludwig Prandtl, published the most important paper ever in fluid mechanics: On the Motion of a Fluid with Very Small Viscosity, which established the theory of boundary layers as introduced earlier. Prandtl pointed out that the old assumption that, since the object moves sufficiently fast the viscosity effects can be neglected, does not apply in the immediate vicinity of a solid surface (i.e. within a boundary layer). It is necessary to consider the viscous effects in the boundary layer to predict correctly the drag on a body.
Viscosity is both beloved and hated in the field of aeronautics. On one hand, no lift can be generated without viscosity, and on the other hand, viscosity creates all the drag that causes a glider the loss of altitude. The holy grail of glider design is to minimise the drag introduced for the same amount of lift, i.e. to maximise the lift-to-drag ratio (L/D) of the aeroplane, which is identical to the glide ratio of a glider. In short, less drag means a glider goes further, higher, and longer. It is, therefore, necessary to understand the physics of drag.
An aeroplane experiences three kinds of drag, namely skin-friction drag, form drag, and induced drag.
Laminar and turbulent boundary layers
Unfortunately, before the discussion of drag creation, it is necessary to introduce the concept of laminar and turbulent boundary layers. In short, a flow can be either laminar or turbulent, and going from laminar to turbulent is called the “transition”. This is most easily visualised by observing a burning cigarette in still air: the smoke initially goes straight up nicely (laminar), but after some distance the column suddenly goes unstable (transition) and the smoke wraps up into some unpredictable shape (turbulent flow).
When a boundary layer is laminar, there is less friction between the surface and the flow, but a laminar boundary layer separates easily. In contrast, a turbulent boundary layer causes more friction, but it is more reluctant to separate.
A laminar boundary layer can naturally transit into a turbulent one due to inherent instability, typically when the flow speed is sufficiently high. If natural transition does not occur, it is possible to use a “tripping device” to force premature transition at lower speeds.
Drag forces explained
Form drag, also known as pressure drag, is the drag caused by the pressure in front of the body being higher than the rear pressure. This is most obvious for a “bluff body”, i.e. a body that does not have a streamlined shape. It is difficult to cycle or walk into strong head wind, because a human body is a bluff body with a lot of form drag. The pressure in front of you is greater than the pressure on your back, and this pressure difference tries to push you backwards. Going forward and you have to continuously fight against this form drag.
Form drag largely depends on the wake size, which in turn depends on the geometry of the body. Objects such as spheres and cubes create very large wakes. An aeroplane is usually streamlined, which means it does not trail a significant wake behind it in flight. The form drag associated with an aeroplane is generally not dominating if not small.
A wake is created by boundary layer separation. Recall that a turbulent boundary layer is more reluctant to separate compared with a laminar one, so somehow making the boundary layer turbulent can reduce the form drag considerably. This is the reason why golf balls have dimples: the dimples can trip the boundary layer so that it is turbulent.
Pulling a teaspoon out of a cup of tea can be effortless, but pulling it out of a jar of honey can be more difficult. This is because the teaspoon moving in honey experiences a significant skin-friction drag. Skin-friction drag is created by the viscous fluid sticking onto the solid object, and it largely depends on the surface area submerged (or in case of an aeroplane, indeed almost all the surface area). Unfortunately, streamlining a body to reduce form drag usually means creating a large surface area, so there is a subtle balance between form drag and skin-friction drag for the aeroplane designer to get right.
Most of the skin-friction drag on a glider comes from the wings. Recall that a laminar boundary layer causes less skin-friction than a turbulent boundary layer, so a lot of gliders use what is called “laminar flow aerofoils” which are aerofoils designed to keep the boundary layer laminar for as long as possible. The problem being laminar boundary layers separate easily, so the post hoc fix is to add a turbulent trip on the surface of the wing, just before the point where it would otherwise separate. Such a device can be found on the lower surface of the wing of DM (a CGC single seater).
Induced drag only exists in 3D: there is no 2D equivalent. When a lifting wing flies through the air, it is well known that a pair of tip vortices will be created behind the aeroplane. This is generally explained by considering the air escaping from the under side to the upper side around the tip: a more precise explanation is very involved and shall not be discussed here. However, by creating these vortices, extra kinetic energy is transferred to the air, which now has an additional swirl motion. This energy must come from the wing, and the result to the aeroplane is known as induced drag.
The magnitude of the induced drag coefficient is proportional to the lift coefficient to the second power, and inversely proportional to the aspect ratio of the wing. When the aspect ratio tends to infinity, the induced drag is zero. This is the second reason that, for aeroplanes whose drag is critical, high aspect ratio wings are desirable. Almost all gliders have large aspect ratio wings: the most extreme example being the ETA, which has a record-breaking glide ratio of 70:1.
It can be shown that, for a glider, the aeroplane lift-to-drag ratio is identical to the glide ratio. It just needs to be noted that the aeroplane lift-to-drag ratio is not the same as the wing L/D, because the fuselage contributes only to drag but not lift.
Downwash (advanced topic)
Warning: readers whose working knowledge on fluid mechanics is limited are advised to Google “tip vortices” and jump to the Implications section.
Circulation results in lift. For a finite span wing, at each section there must be a defined circulation value which can be determined by a closed line integral around the aerofoil. By the Stokes’s theorem (vector calculus), this implies that there is a distribution of vorticity within the integration loop. However, the loop encloses potential flow (which, by definition, can have no vorticity), and within the aerofoil contour the flow speed is zero so there cannot be vorticity either (the curl of a constant zero is zero). Therefore, this vorticity is distributed on the surface of the aerofoil. In the real flow, this corresponds to the vorticity in the boundary layer.
We have already argued that at the tip of the wing, the local wing loading is zero. If for an aerofoil section there is zero loading, this implies that the circulation for this section is zero. By the same argument, we find that the vorticity is zero for a section plane taken at the tip of the wing.
According to Helmholtz’s second theorem, a vortex filament cannot end in a fluid. If there is a vortex line going spanwise when a section plane in the middle is examined, but it is not found going outwards on the section plane at the tip, the only possibility is that it has been deflected to some other direction. But in which direction? Again, according to Helmholtz’s theorems, vortex lines move with the fluid. Because the aeroplane flies forward, the flow around it is effectively going backwards. Therefore, the vortex line has been deflected into going backwards by the flow.
In a sense, we have shown that, for a wing of finite span, a sheet of vorticity is trailed behind it because vorticity is being shed off from the root to the tip of the wing. But this does not remain as a sheet forever. For the third time according to Helmholtz’s theorems, these vortex lines move with the fluid, but do not forget that the vortices can introduce swirling flow themselves. The result is that this vortex sheet rolls up into a single vortex with all the strength there is. For an aeroplane, because of the obvious symmetry, a pair of vortices are formed. These are known as tip vortices. Despite the name, it is good to understand that they are not produced by flow escaping around the tip, but it is a result of a finite span wing generating lift in general. A more detailed examination of the rolling up process reveals that the distance between these tip vortices is less than the geometrical span of the wing, i.e. the tip vortices are closer to the fuselage than the tips are.