Instead of looking at a wing, we will simplify our model to two dimensions by building a wing of very large span (infinite, if we have enough material...) and by looking at a section close to the center of this wing: the result is a two dimensional airfoil.
When the airfoil is located in a stream of air of velocity , the flow has to part near the leading edge and pass along the upper and the lower airfoil surface. At the location, where the flow is splitting up, the flow velocity is reduced to zero. This point is called stagnation point.

Velocity distribution plot for both surfaces of the Eppler E 64 airfoil at 2 degrees angle of attack
(result from Eppler's high order panel method).

Let us follow the flow from the stagnation point, along both sides of the Eppler E 64 airfoil at 2º angle of attack, as shown above:

• Starting from v/=0 in the stagnation point, the velocity v increases rapidly to 1.38 times the velocity of the onset flow near the location x/c=0.1. Further downstream the velocity gradually decreases and reaches at the trailing edge approximately 85% of the free stream velocity .
• The velocity of the flow on the lower surface looks similar, but its level is considerably lower. In this example, it always stays below the free stream velocity.

Velocity and pressure are dependent on each other - Bernoulli's equation says that increasing the velocity decreases the local pressure and vice versa. Thus the higher velocities on the upper airfoil side result in lower than ambient pressure whereas the pressure on the lower side is higher that the ambient pressure. It is possible to plot a pressure distribution instead of the velocity distribution (usually not the pressure, but the ratio of the local pressure to the stagnation pressure is plotted and called pressure coefficient Cp):

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Summing up the pressure acting on the airfoil results in a total pressure force. Splitting up this total pressure force into a part normal to the flow and another one tangential to the flow direction, results in a lift force L and a drag force D. In some regions, the pressure force acting on the airfoil is of lower pressure than the surrounding pressure and in other regions, it is higher. The following image shows the pressure forces for the E64 at 2° angle of attack. It also shows, how the total force is split into lift and drag forces (the drag force due to pressure is usually small, see below).

Pressure force distribution the surface of the Eppler E 64 airfoil at 2 degrees angle of attack
(result from Drela's XFoil code).

If we had used a symmetrical airfoil at no incidence, the distribution of velocity and thus the pressures along both surfaces would have been exactly the same, cancelling each other to a resulting total lift force of zero. The calculation of the forces can be performed by summing (math: integrating) the pressure distribution of a plot of Cp versus chord. The total force corresponds to the area enclosed between the curves for the upper and the lower surface. It is easy to transform Bernoullis equation, written once for a point in space far away from the airfoil (where the velocity is ) and once for a point on the airfoil (v) to arrive at the actual pressure on the surface of the airfoil:

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A typical airfoil does not only create a lifting force, it also creates a moment, which tries to rotate the airfoil to a different, angle of attack. For comparisons it is generally assumed that the lift force and the torque moment are acting at a point, which is located 25% behind the leading edge on the x-Axis (often called «¼ chord point»). Typical, cambered airfoils create a moment, which tries to reduce the angle of attack - per definitionem its sign is negative. The moment acting at the ¼ chord point is roughly independent of angle of attack, as long as the flow stays attached to the airfoil.

If no separation or compressibility effects are present, the pressure field around a two dimensional airfoil creates a lifting force only, no drag. The drag of a two dimensional airfoil is created by the friction of the air particles moving close to the surface. The airfoil is surrounded by a boundary layer, which forms a thin sheet adjacent to the wall where the velocity is reduced from the free stream value down to zero on the wall. For the theoretical treatment of lift, the boundary layer effect is so small that can be neclected, as long as no separation occurs. This applies also (albeit to a lesser degree) to the moment of the airfoil, but, the boundary layer is the main source for the drag in two dimensional flow (airfoils).

The lift of a wing depends not only on the airfoil shape and its associated velocity distribution, but also on wing span and wing area. Experiments show, that doubling the wing area also doubles lift and drag, but doubling the air speed yields four times as much lift. The forces and moments also depend on the density of the air and on the shape of the wing. It is possible to compare the aerodynamic properties of different airfoils and wings, if all forces and moments are normalized. These dimensionless properties (coefficients) are defined as follows:

Lift Coefficient
Drag Coefficient
Moment Coefficient

Knowing these coefficients for a certain airfoil section at a certain angle of attack, makes it possible to calculate the forces acting on wing sections of different sizes, mounted between walls at different flow velocities and air densities, but at the same angle of attack. To calculate the lift of a "real wing", the local flow conditions, which change from root to tip, have to be taken into account although.
For "real wings" and aircraft the same coefficients are used, but the lower case identifiers l, d, and m are replaced by upper case characters L, D and M. If these coefficient are known for an aircraft, the total forces and moments for the complete aircraft can be calculated for different flow conditions.

Example:

Wind tunnel tests of an ultralight aircraft gave a lift coefficient CL = 0.1 at 0° angle of attack. What is the lift force L on the full scale aircraft, which has a wing area S of 12 m², at an air speed v of 252 km/h ? The air speed of the full scale aircraft in SAE units is 252/3.6 = 70 m/s. We assume the density of air to be = 1.225 kg/m³. The Lift force L can be calculated to be (lifts 367 kg) For a quarter scale model, flying at half the speed, we would obtain: (lifts 11.5 kg)

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