negative 1 Baseline left-parenthesis StartFraction w Over upper V EndFraction right-parenthesis"/>
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
Figure 3.17 Typical boundary layer over a wing/tail airfoil
(3.10)
where ρ is fluid (here, air) density, V is air velocity, μ is air viscosity, and l is a characteristic length.
In aeronautical work, the characteristic length is usually taken as the chord of a wing surface. The Reynolds number is an important indicator of whether the boundary layer is in a laminar or turbulent condition. Laminar flow creates considerably less drag than turbulent, but nevertheless causes difficulties with small surfaces, as we shall learn later. Typical Reynolds numbers for various air vehicles – including a bird and an insect – are shown in Table 3.1.
Laminar flow causes drag by virtue of the friction between layers and is particularly sensitive to the surface condition. Normally, laminar flow results in less drag and is desirable. The drag of the turbulent boundary layer is caused by a completely different mechanism (e.g., vortex) that depends on knowledge of viscous flow.
In any flow, two fundamental laws are always applicable: (1) energy conservation law and (2) mass conservation law. For an incompressible flow (M < 0.3), the energy conservation law indicates that for an ideal fluid (no friction), the sum of the static pressure (P) and the dynamic pressure (q), where
(3.11)
Applying this principle to flow in a duct (e.g., convergent–divergent duct), with the first half representing the first part of an airplane wing, the distribution of pressure and velocity in a boundary layer can be analyzed. The flow inside a duct is very similar to a flow over and under a wing.
In an incompressible flow where the air density remains constant along the flow, this equation – for two arbitrary points (1 and 2) – is expanded as
(3.12)
This equation is referred to as Bernoulli’s equation. As the fluid (assumed to be incompressible) moves through the duct or over a wing, its velocity increases (because of the law of conservation of mass) and, as a consequence of Bernoulli’s equation, its pressure decreases, causing what is known as a favorable pressure gradient. The pressure gradient is favorable because it helps push the fluid in the boundary layer on its way.
Table 3.1 Typical Reynolds numbers
| No. | Air Vehicle Type | Reynolds Number |
|---|---|---|
| 1 | Large subsonic UAVs | 5,000,000 |
| 2 | Small UAVs | 400,000 |
| 3 | Mini‐UAVs – Quadcopters | 50,000 |
| 4 | A Seagull | 100,000 |
| 5 | A Gliding Butterfly | 7,000 |
After reaching a maximum velocity (usually at the maximum
