# 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.