A full analysis for lift and drag must consider not only the contribution of the wing but also by the tail and fuselage and must account for varying airfoil cross‐section characteristics and twist along the span. Determining the three‐dimensional moment coefficient also is a complex procedure that must take into account the contributions from all parts of the aircraft.
A crude estimate (given without proof) of the three‐dimensional wing lift coefficient, indicated by an uppercase subscript (CL), in terms of the “infinite wing” lift coefficient is
(3.7)
where Cl is also the two‐dimensional airfoil lift coefficient. From this point onward, we will use uppercase subscripts, and assume that we are using coefficients that apply to the 3d wing and aircraft.
3.8 Induced Drag
Drag of the three‐dimensional airplane wing plays a particularly important role in airplane design because of the influence of drag on performance and its relationship to the size and shape of the wing planform.
The most important element of drag introduced by a wing – at high angles of attack – is the “induced drag,” which is drag that is inseparably related to the lift provided by the wing. For this reason, the source of induced drag and the derivation of an equation that relates its magnitude to the lift of the wing will be described in some detail, although only in its simplest form.
The lowest induced drag is generated when the lift distribution over the wing is elliptical (Figure 3.14), which provides a constant downwash along the span, as shown in Figure 3.14. Aerodynamicists in the past – including Ludwig Prandtl – believed that a wing whose planform is elliptical would have an elliptical lift distribution. But further research has not proven this idea. The notion of a constant downwash velocity (w) along the span will be the starting point for the development of the effect of three‐dimensional drag.
Considering the geometry of the flow with downwash, as shown in Figure 3.15, it can be seen that the downward velocity component for the airflow over the wing (w) results in a local “relative wind” flow that is deflected downward. This is shown at the bottom, where w is added to the velocity of the air mass passing over the wing (V) to determine the effective local relative wind (Veff) over the wing. Therefore, the wing “sees” an angle of attack that is less than it would have had there been no downwash.
Figure 3.14 Elliptical lift distribution
Figure 3.15 Induced drag diagram
The lift (L) is perpendicular to V and the net force on the wing is perpendicular to Veff. The difference between these two vectors, which is parallel to the velocity of the wing through the air mass, but opposed to it in direction, is the induced drag (Di). This reduction in the angle of attack is
(3.8)
Then, the induced drag coefficient (CDi) is given by
This expression reveals to us that air vehicles with short stubby wings (small AR) will have relatively high‐induced drag and therefore suffer in range and endurance. Air vehicles that are required to stay aloft for long periods of time and/or have limited power, as, for instance, most electric‐motor‐driven UAVs, will have long (high AR) thin wings.
3.9 Boundary Layer
A fundamental axiom of fluid dynamics and aerodynamics is the notion that a fluid flowing over a surface has a very thin layer adjacent to the surface that sticks to it and therefore has a zero velocity. The next layer (or lamina) adjacent to the first has a very small velocity differential, relative to the first layer, whose magnitude depends on the viscosity of the fluid. The more viscous the fluid, the lower the velocity differential between each succeeding layer. At some distance δ, measured perpendicular to the surface, the velocity is equal to the free‐stream velocity of the fluid. The distance δ is defined as the thickness of the boundary layer (BL).
The boundary layer – at subsonic speeds – is often composed of three regions beginning at the leading edge of a surface: (1) the laminar region where each layer or lamina slips over the adjacent layer in an orderly manner, creating a well‐defined shear force in the fluid, (2) a transition region, and (3) a turbulent region, where the particles of fluid mix with each other in a random way, creating turbulence and eddies. The transition region is where the laminar region begins to become turbulent. The shear force in the laminar region and the swirls and eddies in the turbulent region both create drag, but with different physical processes. The cross‐section of a typical boundary layer might look like Figure 3.16.
Figure 3.17 illustrates a typical boundary layer (BL) over a wing/tail airfoil, where two laminar and turbulent portions are distinguished. As we move along the flow, the thickness of the BL is increased, and the flow becomes more and more turbulent.
The shearing stress that the fluid exerts on the surface is called skin friction and is an important component of the overall drag. The two distinct regions in the boundary layer (laminar and turbulent) depend on the velocity of the fluid, the surface roughness, the fluid density, and the fluid viscosity. These factors, with the exception of the surface roughness, were combined by Osborne Reynolds in 1883 into a formula that has become known as the Reynolds number, which mathematically is expressed as
Figure 3.16 Typical boundary layer over a flat surface