1
Introduction and Approach

1.1 Introduction

1.1.1 Wing Aerodynamics

Airplanes feature many external components, but by far the most important ones are the wings that provide the lift necessary to remain airborne. Wings also produce pitching moments, and much of the drag on the airplane, therefore, this book focuses on the aerodynamics of the wing. In our approach, we examine flows that surround the airfoil with thin viscous layers that create minimum drag as it generates lift. Under fully immersed conditions, moving objects may be studied with irrotational flow descriptions (e.g. Bernoulli's equation) in relatively conventional ways. As will also be evident, the lift along with portions of the drag and other key aerodynamic phenomena can be analyzed with a flow-induced vorticity model known as circulation, and this important concept is properly introduced in Chapter 3 and applied to airfoils beginning with Chapter 5.

Our most frequent aerodynamic situation focuses on airfoils restricted to operating angles of attack where no flow separation develops. Airfoils are assumed to be thin and either symmetric or of low curvature in subsonic flows so that we may replace them by their mean line because effects of thickness are known to play a secondary role. Low-viscosity air flows develop mostly "thin shearing boundary layers," and while these remain attached to the shape of the body closely resembles the displaced inviscid region surrounding it. Viscosity, however, plays an indispensable role in generating the above-mentioned circulation that surrounds lifting airfoils because it depends on a stagnation point developing at the airfoil's sharp trailing edge (the so-called the Kutta condition). Airfoils cannot produce much lift whenever their boundary layers detach and lift does not occur on moving immersed objects without any embedded circulation. Flows of interest in aerodynamics, however, while never really inviscid may be tailored so as to isolate the effects of viscosity while allowing the generation of lift through purposely streamlined airfoil configurations.

The background necessary to study aerodynamics assumes you have had previous formal introduction to fluid mechanics preferably followed by gas dynamics, together with the needed exposure to basic thermodynamics, but an effort will be made to briefly review such concepts as they are presented. Courses on fundamental calculus and modern physics are needed prerequisites. Our listings under Selected References contain many of the textbooks that cover these subjects. At the end of this chapter, we present a Glossary of Terms and Symbols which are commonly used in aerodynamics intended to be a useful and handy reference for the rest of the book.

1.2 Necessary Assumptions

While we shall endeavor to develop wing-aerodynamics with some generality, for most situations we restrict ourselves to one or more of the following conditions:

1. Steady flows - effects of mass inertia due to transients are neglected as we concentrate on wings traveling with a time-independent velocity so aircraft are moving under non-maneuvering situations.
2. One-dimensional or two-dimensional flows - flow descriptions primarily based on (x, y) or (r, ?) coordinates.
3. Negligible body forces - the weight of the air mass surrounding the airfoil is reflected in pressure and density variations with elevation in the Standard Atmosphere table.
4. Incompressible flows - here changes of density can be neglected in the fluid equations; formulations resemble those in hydrodynamic flows (i.e. objects immersed in water).
5. Compressible flows - here effects of compressibility become important and are separated into subsonic, transonic, supersonic, and hypersonic regimes depending on Mach number.
6. Inviscid or ideal flows - these flows operate where shearing forces may be neglected (outside of a region labelled the boundary layers).
7. Boundary layer flows - viscous effects are localized in a thin envelope around the airfoil where the flow is largely incompressible even at high Mach numbers.
8. Streamlined flows - airfoils with properly rounded leading edges and sharp trailing edges in subsonic flows and pointed leading and trailing edges in supersonic flows.

Briefly then, our default approach will be the study of thin, streamlined airfoils operating with attached boundary layers. Whenever we extrapolate our discussion to an entire aircraft, we will consider a cruising or force equilibrium situation where lift equals weight and thrust equals drag as shown with the vector lengths in Figure 1.1.

1.3 Units

Aerodynamics uses concepts from both mechanics and thermodynamics so it is necessary to be proficient with two systems of units, namely, the metric or International System (SI) and the English Engineering (EE) system. You will need to manage gas properties in both systems, even if we work mostly with the SI system. In aerodynamics, we develop many important expressions with dimensionless coefficients because they are devoid of units and more general than their dimensional counterparts.

Gas properties such as pressure, density, and the speed of sound, consist of several basic units, namely, length, time, mass, and temperature. These units are listed in Table 1.1. Lengths exist in up to three dimensions as vectors, whereas the rest are scalars.

1.3.1 Additional Systems of Units with Their Conversion Factors

We also need to be aware of Gravitational Units such as the "lbm" and the "kgf?" together with their equivalent mechanical units the "slug" and the "newton." Thermodynamics often uses units not generic to fluid mechanics, called "caloric," which require conversion factors to SI or EE units. Another factor is gc as it appears in Newton's second law (i.e. F =?ma/gc). In the gravitational system, mass and weight amounts are equivalent (often undistinguished) because the numerical value of the acceleration due to gravity at sea level (i.e. the g in W =?mg/gc) cancels out.

Figure 1.1 Generic commercial aircraft on cruise mode with principal forces acting on it.

Table 1.1 Basic systems of units.

Another conversion factor needed when dealing with the units of work and heat with the first law of thermodynamics is "J," the mechanical equivalent of heat.

We should also note here that among thermodynamic units the "joule or J" is defined as a N-m (or 1.0?kg-m2/s2) in the SI system, whereas the "Btu" is defined as 778.16?ft-lbf in the EE system. While thermodynamic work and heat transfer often share the same energy units, they carry different entropy contents.

Notation

A considerable number of special terms are needed to describe and analyze flows around airfoils (see the Glossary). Here we mention a few wing related ones:

• The cruising or x-direction - the direction of the relative wind velocity vector ().
• Wingspan (b) - the span or width of the wing in the y-direction. This span is the distance between wing tips without the aircraft's fuselage so as to only represent the lifting surface of the aircraft. (Note under two-dimensional conditions we use the coordinate y instead of z for elevation.)
• Wing chord (c) - this is the line connecting the leading and trailing edges of a wing. For trapezoidal and other non-rectangular wings, there can be a root chord and a tip chord along with a taper ratio (?). The chord dimension and the span characterize the airfoil.
• Planform area (S) - the total wing area. There are many different wing configurations that include rectangular, elliptical, tapered, swept-back, and delta. For rectangular planforms, S =?bc.
• Aspect ratio (AR) - defined as the ratio b2/S; for a rectangular wing, it simplifies to b/c.
• Angle of attack (a) - represents the orientation of the wing planform area to the relative wind velocity vector ().
• Mean camber line - a line halfway between the upper and lower surfaces of the airfoil. When this line is not straight but "concave down," the airfoil is said to be cambered, a...