Category BASIC AERODYNAMICS

When designing a flight vehicle, it is necessary to clearly understand how aerody­namic forces are related to the following:

• the shape, size, and orientation of a body

• the properties of the airstream in which a body moves

Flow-field properties may include density, flight speed, and pressure. Common experience tells us that the force on a body moving through a fluid depends in some way on variables of this type.

Conduct the following simple aeronautical experiment: Hold your hand out the window of a moving automobile and observe that a force can be perceived by the need to exert muscular forces to oppose those created by the airflow. Notice that if your hand is held nearly perpendicular to the air motion, there is a strong drag effect. If you hold it at a shallow angle, you feel both lift and drag. There is an obvious dependence on the orientation of your hand as well as the speed of the car, and so on. If you were to guess at the set of contributing physical variables involved based on your observations, you might choose to represent the aerodynamic force F as a mathematical function such as:

F = F (speed, density, pressure, size, shape). (2.8)

The first three variables describe the airflow; the last two indicate that the size and orientation of your hand (or another object) with respect to airflow probably affect the forces.

The basic problem of aerodynamics is defined in Chapter 1 as the determination of mathematical details of physical relationships of this type. To accomplish this, it is necessary to understand all of the important interactions of all of the participating variables and their influence on the force system. The initial difficulty is determining which of myriad possible variables matter. In the following subsections, we explore practical means to make this crucial determination.

Systems of Units

The first step is to establish precise definitions for all of the variables likely to be important. It must be possible to express each of these and their interactions in math­ematical form. Of vital importance are the dimensions and associated systems of units needed to describe the variables. Such considerations likely have already had a major role in a student’s technical training, therefore, only a brief review is necessary.

A basic set of physical variables significant in aerodynamics is defined in Table 2.2. The dimensions of each property are given in terms of familiar mass-based (M, L, T) and force-based (F, L, T) systems of units. M represents mass (in kg and slugs in Systeme International [SI] and English systems of units, respectively).[2] For example, F represents the magnitude of the force (N or lbf) and t is the time in seconds.

For the most part, these are familiar thermodynamic variables; therefore, only a brief review is necessary. Other variables may be needed to describe the fun­damental characteristics of a flowing liquid or a gas. They are defined by basic physical relationships; they are sometimes empirical in nature, requiring knowledge of experimentally determined parameters. For example, in the case of an ideal gas, the thermodynamic variables are related by the familiar gas law, sometimes called the equation of state or constitutive equation, expressed in Eq. 2.1. The gas constant, R = WM, that appears in this equation is known from extensive experimental studies and from statistical mechanics and the kinetic theory of gases. ^ is the universal gas constant and M is the molecular weight.

Similarly, forces and acceleration of gas particles are related by Newton’s Second Law of Motion, so that (from the point of view of physical dimensions) for a gas par­ticle of mass m, we write:

dV

F = m—V. (2.9)

dt

If we must account for viscous stresses, t, we need to know that a relationship of the type deduced in Eq. 2.4 might be involved. Table 2.2 illustrates how the cor­rect dimensions for possibly unfamiliar physical variables such as the coefficient of viscosity, p, in Eq. 2.4 can be determined by means of the principle of dimensional homogeneity. For Eqs. 2.4 and 2.9 to be correct, the dimensions of quantities repre­sented on the left and right sides of the equation must be identical.

EXAMPLE 2.1 Required: Verify the dimensions of force in a mass-based system of units as listed in Table 2.2. Determine the corresponding standard metric SI and English engineering units for force.

Approach: Use the principle of dimensional homogeneity and Newton’s Second Law (Eq. 2.9).

Solution: The defining equation and the corresponding dimensions are:

dV

m——

where the brackets denote dimensions of the enclosed variable.

Discussion: Often, the symbols denoting basic units are the same as those appearing in the defining equation. Here, the symbols for generic mass, m, and time, t, are the same as the corresponding quantities in Newton’s Second Law.

From this, it is clear that 1 unit of force in the SI system is expressed (in mass-based units) as:

„kg m „

1 &. =1 newton

s2

because mass is measured in kilograms, length in meters, and time in seconds. Similarly, 1 unit of force in the English engineering system is:

iSuS^.nbf

sec2

because the basic unit of mass is the slug in English units. The most common abbreviations for various quantities are used.

EXAMPLE 2.2 Required: Find the weight of an object with a mass of 1 slug at the earth’s surface.

Approach: Use Newton’s Second Law in English engineering units.

Solution: Because the acceleration due to gravity at the earth’s surface is approximately 32.17 ft/sec2, the gravitational force (i. e., the weight) is:

W = 1 slug x 32.17 ft/sec2 = 32.17 lbf.

Discussion: It is useful to use the abbreviation Ibf to denote forces to distinguish between two common uses for the pound (Zb). The pound-mass (lbm) often is used in some technical fields: 1 Ibm is the amount of mass that would weigh 1 pound at the earth’s surface. In compatible units, this is equivalent to 1/32.17 ~ 0.031 slug. At the earth’s surface, the weight of an object with a mass of 1 slug is about 32.17 lbf.

Notice that English units are a force-based system using the pound as the basic unit. However, Newton’s Second Law (Eq. 2.9) indicates that 1 lbf can be written as:

Notice that this defines the mass unit that is compatible with the chosen units for acceleration. It is incorrect to replace slug with lbm because, by definition, 1 lbm is the mass that weighs 1 lbf at the earth’s surface.

Of course, equations can be written that contain the conversion factors needed to adjust the units. In some fields of study (e. g., thermodynamics and heat transfer), it is traditional to use lbm as the mass unit when working in English engineering units. It is then necessary to insert the factor go = 32.17 lbm/slug in any numerical calculations.

It is better to use compatible units so that serious numerical errors can be avoided. It was once common practice in engineering textbooks to display constants such as go or J (i. e., the number of ft-lb/sec per BTU) in the fundamental equations to remind users to adjust the units. This is not done in this book! Dimensional numerical constants should not appear in the fundamental equations. It is necessary to check dimensional homogeneity only to be sure that the correct units are applied to represent each variable involved.

EXAMPLE 2.3 Required: Find the dimensions of the coefficient of viscosity using Newton’s Viscous Force Law (Eq. 2.4). Then, determine the corresponding units in standard SI and English units.

Approach: Use the principle of dimensional homogeneity.

Solution: From Table 2.2, the shearing stress (i. e., force per unit area) in mass – based units has the following dimensions:

[t]=[ml/ t2 ]/[l2 ]=[m/lt2 ],

and the dimensions of the velocity gradient are:

3u

0y

Then, for dimensional homogeneity, the correct dimensions for the viscosity coefficient must be:

M

LT ‘

Thus, in the SI system, the viscosity coefficient must have kg/m-sec units. In the force-based engineering system, the units are lbf sec/ft2.

The application of Newton’s Second Law shows that viscosity also can be written in terms of the units slug/ft sec. By similar reasoning, it is correct to write the viscosity coefficient with N-s/m2 units.

In Section 2.1, we encountered two fundamental forces: (1) the force perpen­dicular to a surface due to normal stress, or pressure; and (2) the tangential force due to viscous-shearing stress. These two forces are clearly related to the lift and drag on a body. That is, integration of the differential force contributions over the wetted surface of a body moving through air leads to the force system dis­cussed in Chapter 1. The integrated force components can be resolved into lift and drag and moments about the center of gravity, as previously defined. The emphasis in Section 2.1 was on associating these forces with processes occurring at the molecular level. The benefits of using a macroscopic point of view based on a continuum model were also described. This section explores this concept in more detail. Rather than concentrating on microscopic interactions, we attempt to discover how much can be learned by applying the simple ideas of dimensional analysis. In this process, we identify important properties, both physical and geo­metrical, that influence the generation of the force system on a body moving through a fluid.

Many benefits accrue from the process. For instance, we discover key scaling parameters, or similarity parameters. Understanding the physical content of these parameters is a major step in classifying types of aerodynamic problems in ways that simplify the modeling process and lead to approximate formulations of prac­tical flow problems that are both accurate and easy to use. The results also have an acritical application in guiding the design of experimental procedures and in the correct interpretation and correlation of experimental measurements in aeronautics as well as other fields.

All gases and even solids are compressible. The speed of sound, a, represents the speed of propagation of weak disturbances due to compressibility of the medium. These weak waves often are referred to as acoustic waves. For example, sound waves in water propagate at the acoustic speed—or sound speed—characteristic of that liquid.

Other variables become important when additional physical effects must be accounted for, such as compressibility. For instance, as the speed of a vehicle approaches a significant percentage of the speed of sound, several changes take place in the flow-field characteristics and in the accompanying forces on the vehicle. If the representative aircraft shown in Figs. 1.2-1.12 in Chapter 1 are studied, it becomes apparent that drastic changes in design accompany the increases in speed. Compare the pictures of the high-speed seaplanes used in the Schneider Cup races with the supersonic F-22 and the Concorde transport; quite different shapes are required when the speed is higher than the speed of sound. Similarly, there are important design differences between low-speed aircraft and those that must operate in the transonic range, which begins when the speed is about 80% of the speed of sound. Other families of shapes are demanded when a vehicle must operate at hypersonic speeds; that is, when the velocity is five or more times higher than the acoustic speed.

The dimensionless ratio of the flow speed to the speed of sound represents a measure of the relative importance of compressibility. The thermodynamics and gas – dynamics of this situation are carefully worked out as needed in subsequent chap­ters. It suffices at this point to introduce the speed of sound for isentropic wave propagation in air:

a = ^Jp, (2.5)

where y is the ratio of specific heats. For air, this parameter has a value of у = 1.4. For an ideal gas, it is easy to see that by substituting Eq. 2.1, the speed of sound depends on only the temperature, T. We find:

a = ї/yRT. (2.6)

For sea-level air, the speed of sound is approximately a = 1,116 ft/sec = 340 m/s.

An important parameter is the ratio of vehicle speed to speed of sound—that is, the Mach number:

Whether M is zero or less than or greater than unity determines which of sev­eral flow “regimes” within which a vehicle operates. Each regime is characterized by distinct flow-field features, which are summarized in Table 2.1. The ramifications of several flow regimes are examined later in great detail because they are important in determining aerodynamic performance and in solving practical vehicle-design problems.

The part of the interaction force parallel to the surface, dFt, often is described as the shear force or viscous force. As the latter term suggests, it is the result of frictional effects and is a major source of drag force on aerodynamic vehicles.

As in the case of the pressure, the simple ideas of kinetic theory based on random molecular motion provide a useful physical picture of the origin of shearing stresses in a fluid or a gas. The random motion tends to eliminate discontinuities that may form in a gas-velocity distribution. For example, consider the gas flow over a surface, as shown in Fig. 2.2. The parallel velocity distribution is expected to look somewhat like that shown in Fig. 2.3.

The molecules in the immediate vicinity of the surface must be brought to rest. The result is known as the no-slip condition and is satisfied in almost all practical situ­ations.[1] Gas particles lose momentum in the tangential direction at the surface due to collisions with the irregular-surface molecular lattice, as depicted in the blowup of a small region near the surface in Fig. 2.3. Clearly, if the mean free path is short, as it almost always is, there will be molecular collisions between the crystal-lattice structures of the surface (which, on the microscopic scale, represent a rough surface texture) involving lateral-momentum transfer. Gas particles must be brought to rest in the tangential direction at the surface.

Consider this simple thought experiment based on events leading up to the steady-state situation illustrated in Fig. 2.3. The surface is suddenly introduced into uniform flow parallel to the surface moving at velocity U. This initially creates a discontinuity in the velocity distribution because the no-slip condition applies only

to gas particles close to the surface. This discontinuity must be adjusted by the mol­ecular motion. Gas molecules at a distance have yet to be affected by the pres­ence of the surface. Molecules moving outward from the surface vicinity have zero momentum parallel to the surface. The mixing of these two types of molecules must finally produce a velocity distribution similar to that shown in Fig. 2.3. (It is the task of subsequent chapters to determine the exact form of this distribution.) The effect can be visualized as the dragging-along of slower fluid particles located closer to the surface by those at a greater distance, and a similar retardation of molecules farther from the surface by those that have been affected already by its presence. A careful measurement of the force on the surface would show that there is a shearing stress, t, that tends to move the plate in the original direction of the fluid. There is, of course, an equal and opposite retarding stress on the fluid.

If two layers of fluid separated by a vertical distance, dy, are considered, a change of velocity, dU, is evident between the layers, which can be interpreted as the net result of molecular mixing at that particular location. That is, it is a measure of the local shearing stress in the fluid. The force per unit area parallel to the surface, the shear stress, can be written asf

dU

dy

where the constant of proportionality is the coefficient of viscosity, ц. The region in which the adjustment of the velocity is made by molecular diffusion or mixing is called a boundary layer and is studied in a subsequent chapter. The value of the shearing stress, t, at the surface has the maximum value because the velocity gradient

f This equation often is referred to as Newton’s Law of Viscosity. Gases or liquids that behave as

indicated are called Newtonian fluids; those that do not are called non-Newtonian fluids and are rarely encountered in aeronautics.

is largest there. The viscosity coefficient, p, mainly depends on the temperature of gases; it also may be sensitive to pressure in other media.

As an example of the molecular basis of important properties, we first examine the origin of the thermodyamic pressure. Consider a uniform gas contained in a box.

From kinetic theory, we know that the pressure is the reaction force generated by the interactions of the molecules with the box surfaces. Pressure is defined as the force per unit area arising from these collisions. Because the interaction is (in the first approximation) an elastic collisions, then the net reaction force vector is per­pendicular to the surface at any point. If one-third of the molecules are traveling in each of three mutually perpendicular directions (e. g., axes parallel to the edges of the box) at a given time, and half of those going in any particular direction are traveling away from the surface on average, then one-sixth of the molecules in a layer of thickness dx strikes the surface in a time dt = dx/c, where c is the average velocity, as already defined. If m is the mass of each molecule and the momentum is reversed in the elastic collision with the surface, then the reaction force in each collision is dF = 2mc. Thus, if n is the number of molecules per unit volume, the pressure is:

(2.3)

because p = nm is the density, or mass per unit volume. This shows that the pressure is proportional to the kinetic energy per unit mass due to random molecular motion.

Throughout this book, a continuum assumption is implicit, which means that no matter how small the fluid particle being considered, it is assumed to contain many molecules of the fluid or the gas such that the behavior of individual molecules is not important. Another way to say this is that the mean free path of the molecules (i. e., the distance between molecular collisions) is assumed to be small when compared with any length scale considered, such as a body length or diameter or an average airfoil-chord length—even if the length scale is infinitesimal.

Size of a Particle in the Continuum Model

When applying the continuum idea, we often refer to the motion of a fluid particle, which represents an element of the medium that is small compared with the basic scale of the problem. For example, it could represent a particle of air moving over the surface of a wing. In that case, we assume that the particle is small compared with the distance traveled from the leading to the trailing edge—that is, the chord length. It is important to form an accurate picture of just how small the particle is. If it were the size of a molecule, then the concept of a continuum would fail because it would be necessary to account for individual molecular motion, as in the previous example. Figure 2.1 illustrates a “thought experiment” to select the correct size for a particle. Consider a particle in the form of a cubical box of finite length L on each side and imagine measuring the variation of a property of the fluid across the element. The pressure is a convenient property to use in this thought experiment. We want to choose a particle size such that the property in question exhibits a constant value throughout the element. Thus, if L were the chord length of the wing, it would not work because the pressure changes radically in moving from the leading to the trailing edge. As Fig. 2.1 shows, if L is too large, then there are large variations in the value of the property across the particle. Conversely, if the particle is too small (e. g., the size of a few molecules), then the concept of pressure is lost. There would be huge oscillations in the pressure, depending on how many molecules happened to be inside the test volume at a given instant. On this basis, it appears that a prac­tical lower limit for the size of a valid particle is a fairly large number of mean free paths, X.

We must ensure that there is a sufficient number of molecules in the particle so that the influence of individual molecular motion is “smoothed out.” Therefore, the practical answer to the question, “How large is a fluid particle?,” is “Very, very small, but not too small.”

Resolution of Aerodynamic Forces in Normal and Parallel Components

In addition to the pressure, other fluid properties are needed to model aerodynamic forces. In particular, the effects of energy dissipation or friction must be considered. In the simplest picture, forces normal to a surface are created by elastic molecular collisions. This simplified model suffices in some situations; however, the effects of forces parallel to the surface arising from viscous interactions can be of crucial importance in many aerovehicle-design problems.

Figure 2.2 illustrates the basic concept for resolving the interaction-force incre­ment at a point into components normal and parallel to the surface. Consider the force dF acting on a small area element, dA, at any location on the body shown. The normal component dFn most often is due to the action of pressure on an element of the surface:

dF„ = -(pdA)n. (2.2)

The negative sign indicates that the local pressure force acts toward the surface— that is, in a direction opposite to the outward-pointing unit normal vector, n. The pressure can be due to momentum exchange from both the random molecular motion (i. e., the thermodynamic pressure) and the directed motion of the particles due to flow of the gas in the continuum sense. The latter pressure component often is referred to as the dynamic pressure and is an important element in subsequent chap­ters, which discuss control of dynamic pressure by the shape and orientation of the body. This is crucially important in the generation and control of lift (e. g., on a wing surface in external flows) and in the production of thrust (e. g., in a ducted propulsion system with internal flow).

2.1 Aerodynamic Forces

Because the objective of aerodynamics is the determination of forces acting on a flying object, it is necessary that we clearly identify their source. Lift and drag forces, for example, are the result of interactions between the airflow and vehicle surfaces. Part of the force must be a result of pressure variations from point to point along the surface; another part must be related to friction of gas particles as they scrub the surface. Clearly, the key to understanding these forces is found in details of the fluid motions. The application of simple molecular concepts provides considerable insight into these motions.

Modeling of Gas Motion

As a branch of fluid mechanics, aerodynamics is concerned with the motion of a continuously deformable medium. That is, when acted on by a constant shear force, a body of liquid or gas changes shape continuously until the force is removed. This is unlike a solid body, which only deforms until internal stresses come into equilibrium with the applied force; that is, a solid does not deform continuously.

To understand the motion of a fluid, it is necessary to apply a set of basic physical laws, which consist of some or all of the following:

• conservation of mass (the continuity equation)

• Newton’s Second Law of Motion (the momentum equation)

• First Law of Thermodynamics (the energy equation)

• Second Law of Thermodynamics (the entropy equation)

One skill that the student must develop is the effective application of approxi­mation and simplification methods. A proper set of approximations may make unnecessary the use of some laws to produce a practical yet accurate solution for a given problem. This approach is possible only if a clear understanding of the physics of a fluid motion is attained. It is, of course, possible to construct a mathematically correct solution to an incorrectly formulated problem or a solution that is based

on an inappropriate set of assumptions. In such cases, the results can be confusing or misleading or can even lead to costly mistakes. There is no substitute in aero­dynamics for a sound understanding of the fundamental physical laws on which fluid mechanics is based.

It also may be necessary to introduce additional mathematical models or relationships to supplement those in the preceding list. For example, it often is necessary to use equations that characterize a working fluid. These equations of state, or constitutive equations, describe the physical attributes of a fluid. For example, it often is the case in practical problems that the fluid is an ideal or perfect gas, for which the equation of state is:

p = pRT, (2.1)

which is a special relationship among the thermodynamic-state variables, pressure p, density p, and temperature T, needed to describe the behavior of a gas. R is the gas constant—a constant of proportionality—that is determined by the molecular con­figuration of the gas. This and other equations are discussed in considerable detail as needed throughout this chapter.

With regard to details of the molecular structure of a working fluid, we can choose to approach problems from the standpoint of the molecular motion, or a continuum model can be applied that does not attempt to address directly the actual small-scale particle motion. The former is called statistical mechanics, or the kinetic theory of gases. These are fascinating disciplines; however, we do not need a full description of molecular motion to predict forces on an aerodynamic vehicle unless it is flying at an extremely high altitude. In this book, we concentrate almost exclu­sively on a continuum model, which takes advantage of simplifications that result from the relatively large geometrical scale in realistic engineering situations when compared with an atomic or molecular-length scale.

In contrast, it is important to be aware of the molecular origin of physical quantities. In an ideal gas, the size of individual molecules is small when com­pared with the average distance between them. On this basis, the change in kinetic energy due to mutual attraction is negligible. Also, collisions between molecules can be considered as perfectly elastic so that there is no loss of energy due to per­manent deformation of the molecular configuration. The average distance traveled between collisions is the mean free path, X, which is important in the kinetic theory of gases. The mean random velocity of molecules is represented herein by the symbol c.

In the chapters that follow, we present the theory of aerodynamics mainly by means of a set of increasingly more realistic models of key flow phenomena. Chapter 2 discusses basic flow behavior and indicates strategies to represent the physical prob­lems in mathematical form. Each chapter contains many examples that illustrate the techniques for solving key problems. An important aid to learning the material is a set of computer programs that demonstrate application of the solutions and numerical approaches required in the solution of more complex situations. The problem set at the end of each chapter provides students the opportunity to develop their skill with the material.

Organization of Text Material

It is demonstrated in Chapter 2 that a convenient way to study aerodynamics is to organize it into a set of flow regimes, each of which focuses on an important special

case governed by special features of the fluid motion. Chapter 3 is a review of the fundamental physics and modeling strategies needed to describe aerodynamic flows. These concepts are developed in the mathematical language needed to formulate and solve the key problems of aerodynamics.

Chapter 3 emphasizes the need for several methods of approach in problem formulation and solution. A major goal is to instruct the student regarding the cor­rect choice of the most effective strategies for problem solution. Features of real – life aeronautical situations calling for either analytical, numerical, or experimental approaches (or combinations of these fundamental problem-solving methods) are introduced throughout the book by using examples and the problem sets based on them.

Figure 1.16 presents the book layout in terms of the flow regimes, which are organized so that each part forms the foundation for those that follow. Thus, we begin in Chapter 4 to study problems that can be described in terms of a flow field that is both frictionless and incompressible. This leads to the simplest type of math­ematical formulation, which can be treated using many of the engineering math­ematics tools acquired in other courses. For instance, methods of linear differential equation theory, vector analysis, and integral theorems are applied in an elegant solution method usually referred to as potential-flow theory. This forms the basis for the solution of many problems of major importance in aeronautics. A key example is the treatment of flow around an airfoil, which leads to an understanding of the generation of lift—a central theme of aerodynamics.

The fluid-mechanics solutions are progressively extended to include problems that are affected strongly by compressibility and viscous effects. The former study leads us into the important flow regimes of transonic, supersonic, and hypersonic flight. Inclusion of viscous forces leads finally to a comprehensive model in which all important features of the flow about an object at any speed or any altitude can be accommodated. Numerical techniques are introduced to solve these problems when the complexity precludes an analytical approach.

In addition to the categories shown in Figure 1.16, we could add further unsteady and low-density or free-molecular flows in each regime illustrated. In this book, however, these latter types of problems do not have a central role.

PROBLEMS

1.1. Referring to the free-body diagram in Fig. 1.13, write the three equations for static equilibrium. Show that the center of pressure must coincide with the center of mass so that the system is in equilibrium flight.

1.2. Draw the free-body diagram of a glider in steady flight (i. e., constant velocity) in still air. Note that the velocity vector must lie at a downward angle, y, to the local horizon so that the force system is in equilibrium. By examining the com­ponents of the lift and drag, show that the tangent of this angle is proportional to the inverse of the L/D ratio. Also show that the ratio of the distance covered along the ground to the altitude lost is proportional to the L/D ratio.

1.3. Compare the gliding distance in still air of the following three airplanes if power is shut off at an altitude of 10,000 feet:

(a) Curtis Jenny with a glide ratio of L/D = 8

(b) Boeing 767 with a glide ratio of L/D = 14

(c) high-performance sailplane with a glide ratio of L/D = 55

1.4. If the measured range of a certain airplane in its aerodynamically unrefined form is 800 miles, what percentage reduction in the drag, D is needed if the range must be increased to 1,200 miles?

1.5. A small airplane is to be designed to fly nonstop around the earth. Describe some of the design features that you would incorporate. Then, compare your design to the Rutan Voyager that actually accomplished this task. Why does it look like a sailplane?

REFERENCES AND SUGGESTED READING

Anderson, John D., Fundamental Aerodynamics, McGraw-Hill Book Company, New York, 1984.

Crouch, Tom D., A Dream of Wings, Americans and the Airplane, 1875-1905, W. W. Norton & Co., New York, 1981.

Gabrielli, G., and von Karman, T, “What Price Speed?,” Mechanical Engineering, Vol. 72, 1950, pp. 775-781.

Hallion, Richard P., Supersonic Flight: Breaking the Sound Barrier and Beyond, The Smithsonian Institution, 1972.

Hirsch, Robert S., Schneider Trophy Racers, Motorbooks International, 1993.

Kinnert, Reed, Racing Planes and Air Races, Aero Publishers, 1967.

Yeager, J., and Rutan, D., Voyager, Alfred A. Knopf, Inc., 1987.

Simple concepts from thermodynamics make it clear that creation of lift to balance the weight of the vehicle in flight is not without a penalty. The rule (i. e., First Law of Thermodynamics) that “you cannot get something for nothing” applies here. The drag force is a measure of the cost or penalty function for atmospheric flight. Drag results from complex interactions involving not only friction but also other fun­damental loss effects involved in the lift-generation process. The aerodynamic pen­alty for lift generation is the production of what is called induced drag.

Figure 1.14(a) shows the force balance for a vehicle in level, unaccelerated flight. It is clear that if a particular level flight speed is to be maintained, a force must be introduced to balance the drag force. This is provided by the propulsion system in the form of thrust, T. If the vehicle is to climb or accelerate to a yet higher speed, additional energy must be expended in producing an even higher thrust force, T, to overcome the additional drag. The drag force always acts to retard the motion through the air. Thus, in producing sufficient lift to balance the weight in level flight, energy must be expended to counter the drag. Therefore, a measure of the efficiency of the aerodynamic design is the ratio:

It is obvious that we strive to make this value as large as possible within a set of design constraints and the mission requirements for the vehicle. The range, speed, rate of climb, and many other performance factors depend on achievement of high values for this ratio.

Key performance elements depend on how well a designer addresses the many competing requirements. For example, it is shown readily by computing the power required for level flight and determining the rate at which fuel must be consumed to produce this power that the range of a propeller-driven airplane is:

the famous Breguet range equation. In this formula, n is the efficiency of the pro­peller, c is the specific fuel consumption (i. e., rate at which fuel is used to produce the motor power output), and Winitial/Wfinal is the ratio of the initial weight of the airplane to the weight after the fuel has been expended. This result clearly indicates the importance of aerodynamic efficiency in attaining long-range flight.

In the special field of sailplane (i. e., motorless airplane or glider; cf. Figs. 1.14(b) and 1.15) competition, recent research in drag reduction has led to the achievement of lift-to-drag (L/D) ratios approaching 60. The motivations for a high L/D ratio in this application are obvious. In Fig. 1.14(b), notice that in equilibrium, unpow­ered flight, the airplane must fly downward relative to the airmass at the glide angle Y. Therefore, the lift must balance only the cosine component of the weight. The other weight component (proportional to sin y) is required to counteract the drag. A measure of the sailplane performance is the “flatness” of the glide (i. e., smallness of angle y). By equating the balancing-force components, we see that:

L _ Wcosу_ 1 (1.3)

D Wsin у tan y

Therefore, a high L/D ratio means a very flat glide. A sailplane with L/D = 60 can cover a distance of 60 miles while losing only 1 mile of altitude. The glide-slope angle y is less than 1 degree! Thus, the L/D ratio often is referred to as the glide ratio. Much of the work needed to achieve this phenomenal performance was done using mathematical tools and concepts of the type studied in detail in this book, as well as careful wind-tunnel testing of aerodynamic refinements.

Similar attention to drag reduction has led to more economical commercial flight in the transonic flight regime and to higher cruising speeds in the supersonic regime. More complex dependence on lift and drag is involved in those cases.

In Fig. 1.13, the point of view is assumed to be that of an observer fixed with respect to the atmosphere with the vehicle moving past with speed V = |V| in the direction of its velocity vector V. It often is simpler for an observer to be moving with the vehicle. Then, the airflow relative to the vehicle is in the direction – V at a sufficiently great distance upstream of the body—that is, far enough upstream that the effect of the presence of the vehicle has not yet affected the relative flow of the air particles. This change in point of view is useful and often referred to as a Galilean Transformation. As long as there are no acceleration effects present (i. e., no vehicle acceleration rela­tive to the airmass or angular motion about the mass center), then the force system on the body can be taken to be independent of the choice of coordinate frame. This is a great convenience in aerodynamic modeling because it is often the case that the flow problem is best described in terms of the gas motion relative to the body.