Finite Wing Theory

2 Lift And Drag Estimation In A Finite Wing

We know an aeroplane is in the air because of the lift created by its wings as a result of higher pressure on the lower (bottom) surface and lower pressure on the suction (top) surface. The lift is produced by the pressure differential between the lower and higher surfaces. It also causes the flow to curve around at the wingtips as it is driven from the pressure surface to the suction surface. As a result, there will be a spanwise component of flow from the wingtip to the wing root, causing the streamlines on the upper surface to bend toward the wing root, as seen in Figure 1. Similarly, the spanwise component of flow on the bottom surface will be in the direction of the wing root to the wingtip. As a result, it is clear that the flow across the wings is three-dimensional, and their aerodynamic characteristics differ significantly from those of their airfoil sections (Figure 2).

Figure 1.

Front and top views of a three-dimensional wing's flow pattern

The curling of the flow at the wing tips has additional effect on the wing's aerodynamic properties. This “flow slip” creates a circulatory motion that trails downstream of the wing, resulting in a trailing vortex at each wing tip, shedding downstream of the wing. These trailing vortices, which are emitted at the wing tip, cause a minor downward component of air velocity (opposite to the direction of lift) in the region of the wing. This downward component of air velocity is known as downwash, and it is generally represented by the symbol ‘w’. The downwash may alternatively be interpreted as the effect of the lifting wing “pressing down” on the air, causing the air to have more downward motion in the vicinity of the wing. As seen in Figure 3, downwash mixes with freestream velocity (U_f) to generate a local relative wind that is canted downward in the area of each airfoil segment.

Figure 2.

Diagram showing the shedding of wingtip vortices and the downwash

Figure 3.

a) Two-dimensional airfoil section; b) three-dimensional wing are shown in a schematic diagram.

The geometric angle of attack (𝛼) is defined as the angle formed by the chord line and the freestream direction. However, as seen in Figure 3, the local wind speed is inclined below the freestream velocity (U_f) by the angle (𝛼_i) which is referred to as the induced angle of attack. When the induced downwash is w_i and the freestream velocity is U_f, the induced angle of attack (𝛼_i) is expressed as