Category Fundamentals of Aerodynamics

Historical Note: Prandtl—The Man

The modern science of aerodynamics rests on a strong fundamental foundation, a large percentage of which was established in one place by one man—at the Univer­sity of Gottingen by Ludwig Prandtl. Prandtl never received a Noble Prize, although his contributions to aerodynamics and fluid mechanics are felt by many to be of that caliber. Throughout this book, you will encounter his name in conjunction with ma­jor advances in aerodynamics: thin airfoil theory in Chapter 4, finite-wing theory in Chapter 5, supersonic shock – and expansion-wave theory in Chapter 9, compress­ibility corrections in Chapter 11, and what may be his most important contribution, namely, the boundary-layer concept in Chapter 17. Who was this man who has had such a major impact on fluid dynamics? Let us take a closer look.

Ludwig Prandtl was bom on February 4, 1874, in Freising, Bavaria. His father was Alexander Prandtl, a professor of surveying and engineering at the agricultural college at Weihenstephan, near Freising. Although three children were born into the Prandtl family, two died at birth, and Ludwig grew up as an only child. His mother, the former Magdalene Ostermann, had a protracted illness, and partly as a result of this, Prandtl became very close to his father. At an early age, Prandtl became interested in his father’s books on physics, machinery, and instmments. Much of Prandtl’s remarkable ability to go intuitively to the heart of a physical problem can be traced to his environment at home as a child, where his father, a great lover of nature, induced Ludwig to observe natural phenomena and to reflect on them.

In 1894, Prandtl began his formal scientific studies at the Technische Hochschule in Munich, where his principal teacher was the well-known mechanics professor, August Foppl. Six years later, he graduated from the University of Munich with a Ph. D., with Foppl as his advisor. However, by this time Prandtl was alone, his father having died in 1896 and his mother in 1898.

By 1900, Prandtl had not done any work or shown any interest in fluid mechanics. Indeed, his Ph. D. thesis at Munich was in solid mechanics, dealing with unstable elastic equilibrium in which bending and distortion acted together. (It is not generally recognized by people in fluid dynamics that Prandtl continued his interest and research in solid mechanics through most of his life—this work is eclipsed, however, by his major contributions to the study of fluid flow.) However, soon after graduation from

Munich, Prandtl had his first major encounter with fluid mechanics. Joining the Nuremburg works of the Maschinenfabrick Augsburg as an engineer, Prandtl worked in an office designing mechanical equipment for the new factory. He was made responsible for redesigning an apparatus for removing machine shavings by suction. Finding no reliable information in the scientific literature about the fluid mechanics of suction, Prandtl arranged his own experiments to answer a few fundamental questions about the flow. The result of this work was bis new design for shavings’ cleaners. The apparatus was modified with pipes of improved shape and size, and carried out satisfactory operation at one-third its original power consumption. Prandtl’s contributions in fluid mechanics had begun.

One year later, in 1901, he became Professor of Mechanics in the Mathematical Engineering Department at the Technische Hochschule in Hanover. (Please note that in Germany a “technical high school” is equivalent to a technical university in the United States.) It was at Hanover that Prandtl enhanced and continued his new-found interest in fluid mechanics. He also developed his boundary-layer theory and became interested in supersonic flow through nozzles at Hanover. In 1904, Prandtl delivered his famous paper on the concept of the boundary layer to the Third Congress on Mathematicians at Heidelberg. Entitled “Uber Flussigkeitsbewegung bei sehr kleiner Reibung,” Prandtl’s Heidelberg paper established the basis for most modem calculations of skin friction, heat transfer, and flow separation (see Chapters 15 to 20). From that time on, the star of Prandtl was to rise meteorically. Later that year, he moved to the prestigious University of Gottingen to become Director of the Institute for Technical Physics, later to be renamed Applied Mechanics. Prandtl spent the remainder of his life at Gottingen, building his laboratory into the world’s greatest aerodynamic research center of the 1904-1930 time period.

At Gottingen, during 1905-1908 Prandtl carried out numerous experiments on supersonic flow through nozzles and developed oblique shock – and expansion-wave theory (see Chapter 9). He took the first photographs of the supersonic flow through nozzles, using a special schlieren optical system (see chapter 4 of Reference 21). From 1910 to 1920, he devoted most of his efforts to low-speed aerodynamics, principally airfoil and wing theory, developing the famous lifting-line theory for finite wings (see Section 5.3). Prandtl returned to high-speed flows in the 1920s, during which he contributed to the evolution of the famous Prandtl-Glauert compressibility correction (see Sections 11.4 and 11.11).

By the 1930s, Prandtl was recognized worldwide as the “elder statesman” of fluid dynamics. Although he continued to do research in various areas, including structural mechanics and meteorology, his “Nobel Prize-level” contributions to fluid dynamics had all been made. Prandtl remained at Gottingen throughout the turmoil of World War II, engrossed in his work and seemingly insulated from the intense political and physical disruptions brought about by Nazi Germany. In fact, the German Air Ministry provided Prandtl’s laboratory with new equipment and financial support. Prandtl’s attitude at the end of the war is reflected in his comments to a U. S. Army interrogation team which swept through Gottingen in 1945; he complained about bomb damage to the roof of his house, and he asked how the Americans planned to support his current and future research. Prandtl was 70 at the time and was still going strong. However, the fate of Prandtl’s laboratory at this time is summed up in the words of Irmgard Flugge-Lotz and Wilhelm Flugge, colleagues of Prandtl, who wrote 28 years later in the Annual Review of Fluid Mechanics (Vol. 5, 1973):

World War II swept over all of us. At its end some of the research equipment was dismantled, and most of the research staff was scattered with the winds. Many are now in this country (the United States) and in England, some have returned. The seeds sown by Prandtl have sprouted in many places, and there are now many “second growth” Gottingers who do not even know that they are.

What type of person was Prandtl? By all accounts he was a gracious man, studious, likable, friendly, and totally focused on those things that interested him. He enjoyed music and was an accomplished pianist. Figure 5.47 shows a rather introspective man busily at work. One of Prandtl’s most famous students, Theodore von Karman, wrote in his autobiography The Wind and Beyond (Little, Brown and Company, Boston, 1967) that Prandtl bordered on being naive. A favorite story along these lines is that, in 1909, Prandtl decided that he should be married, but he did not


Figure 5.47 Ludwig Prandtl (1875-1953).

know quite what to do. He finally wrote to Mrs. Foppl, the wife of his respected teacher, asking permission to marry one of her two daughters. Prandtl and Foppl’s daughters were acquainted, but nothing more than that. Moreover, Prandtl did not stipulate which daughter. The Foppl’s made a family decision that Prandtl should marry the elder daughter, Gertrude. The marriage took place, leading to a happy relationship. The Prandtl’s had two daughters, bom in 1914 and 1917.

Prandtl was considered a tedious lecturer because he could hardly make a state­ment without qualifying it. However, he attracted excellent students who later went on to distinguish themselves in fluid mechanics—such as Jakob Ackeret in Zurich, Switzerland, Adolf Busemann in Germany, and Theodore von Karman at Aachen, Germany, and later at Cal Tech in the United States.

Prandtl died in 1953. He was clearly the father of modern aerodynamics—a monumental figure in fluid dynamics. His impact will be felt for centuries to come.

5.9 Summary

Return to the chapter road map in Figure 5.5, and review the straightforward path we have taken during the development of finite-wing theory. Make certain that you feel comfortable with the flow of ideas before proceeding further.

A brief summary of the important results of this chapter follows:

The wing-tip vortices from a finite wing induce a downwash which reduces the angle of attack effectively seen by a local airfoil section:

acff = a — o’, [5.1]

In turn, the presence of downwash results in a component of drag defined as induced drag D,.


Vortex sheets and vortex filaments are useful in modeling the aerodynamics of finite wings. The velocity induced by a directed segment dl of a vortex filament is given by the Biot-Savart law:


Historical Note: Prandtl—The Man



In Prandtl’s classical lifting-line theory, the finite wing is replaced by a single spanwise lifting line along which the circulation F(y) varies. A system of vortices trails downstream from the lifting line, which induces a downwash at the lifting line. The circulation distribution is determined from the fundamental equation


Historical Note: Prandtl—The Man

(dF jdy) dy >’0 – V





Types of Flow

An understanding of aerodynamics, like that of any other physical science, is obtained through a “building-block” approach—we dissect the discipline, form the parts into nice polished blocks of knowledge, and then later attempt to reassemble the blocks to form an understanding of the whole. An example of this process is the way that different types of aerodynamic flows are categorized and visualized. Although nature has no trouble setting up the most detailed and complex flow with a whole spectrum of interacting physical phenomena, we must attempt to understand such flows by modeling them with less detail, and neglecting some of the (hopefully) less significant phenomena. As a result, a study of aerodynamics has evolved into a study of numerous and distinct types of flow. The purpose of this section is to itemize and contrast these types of flow, and to briefly describe their most important physical phenomena.

1.10.1 Continuum Versus Free Molecule Flow

Consider the flow over a body, say, e. g., a circular cylinder of diameter d. Also, consider the fluid to consist of individual molecules, which are moving about in random motion. The mean distance that a molecule travels between collisions with neighboring molecules is defined as the mean-free path X. If X is orders of magnitude smaller than the scale of the body measured by d, then the flow appears to the body as a continuous substance. The molecules impact the body surface so frequently that the body cannot distinguish the individual molecular collisions, and the surface feels the fluid as a continuous medium. Such flow is called continuum flow. The other extreme is where X is on the same order as the body scale; here the gas molecules are spaced so far apart (relative to d) that collisions with the body surface occur only infrequently, and the body surface can feel distinctly each molecular impact. Such flow is calledfree molecular flow. For manned flight, vehicles such as the space shuttle encounter free molecular flow at the extreme outer edge of the atmosphere, where the air density is so low that X becomes on the order of the shuttle size. There are intermediate cases, where flows can exhibit some characteristics of both continuum and free molecule flows; such flows are generally labeled “low-density flows” in contrast to continuum flow. By far, the vast majority of practical aerodynamic applications involve continuum flows. Low-density and free molecule flows are just a small part of the total spectrum of aerodynamics. Therefore, in this book we will always deal with continuum flow;

i. e., we will always treat the fluid as a continuous medium.

Velocity Potential

Recall from Section 2.12 that an irrotational flow is defined as a flow where the vorticity is zero at every point. From Equation (2.129), for an irrotational flow,

§ = V x V = 0 [2.152]

Consider the following vector identity: if ф is a scalar function, then

V x (V0) = 0 [2.153]

i. e., the curl of the gradient of a scalar function is identically zero. Comparing Equations (2.152) and (2.153), we see that

Подпись: Y = V0[2.154]

Equation (2.154) states that for an irrotational flow, there exists a scalar function ф such that the velocity is given by the gradient of ф. We denote ф as the velocity poten­tial. ф is a function of the spatial coordinates; i. e., ф = ф(х, у, z), or ф = ф(г, в, z), or ф = ф(г, в, Ф). From the definition of the gradient in cartesian coordinates given by Equation (2.16), we have, from Equation (2.154),

дф дф дф

мі + uj + шк = -^-i + тр-j + тг“к [2.155]

dx dy dz

The coefficients of like unit vectors must be the same on both sides of Equation

(2.155) . Thus, in cartesian coordinates,

Подпись: dф _ dф dx V dyimage179[2.156]

Velocity Potential Подпись: dф

In a similar fashion, from the definition of the gradient in cylindrical and spherical coordinates given by Equations (2.17) and (2.18), we have, in cylindrical coordinates,

and in spherical coordinates,

Подпись:dф _ 1 dф _ 1 dф

dr 6 г дв Ф r sin# ЗФ

The velocity potential is analogous to the stream function in the sense that deriva­tives of ф yield the flow-field velocities. However, there are distinct differences between ф and ф (or ф):

1. The flow-field velocities are obtained by differentiating ф in the same direction as the velocities [see Equations (2.156) to (2.158)], whereas ф (or ф) is differ­entiated normal to the velocity direction [see Equations (2.147) and (2.148), or Equations (2.150) and (2.151)].

2. The velocity potential is defined for irrotational flow only. In contrast, the stream function can be used in either rotational or irrotational flows.

3. The velocity potential applies to three-dimensional flows, whereas the stream function is defined for two-dimensional flows only.2

When a flow field is irrotational, hence allowing a velocity potential to be defined, there is a tremendous simplification. Instead of dealing with the velocity components (say, u, v, and w) as unknowns, hence requiring three equations for these three unknowns, we can deal with the velocity potential as one unknown, therefore requiring the solution of only one equation for the flow field. Once іjr is known for a given problem, the velocities are obtained directly from Equations (2.156) to (2.158). This is why, in theoretical aerodynamics, we make a distinction between irrotational and rotational flows and why the analysis of irrotational flows is simpler than that of rotational flows.

Because irrotational flows can be described by the velocity potential ф, such flows are called potential flows.

In this section, we have not yet discussed how ф can be obtained in the first place; we are assuming that it is known. The actual determination of ф for various problems is discussed in Chapters 3, 6, 11, and 12.

Without Friction Could We Have Lift?

In Section 1.5 we emphasized that the resultant aerodynamic force on a body immersed in a flow is due to the net integrated effect of the pressure and shear stress distributions over the body surface. Moreover, in Section 4.1 we noted that lift on an airfoil is primarily due to the surface pressure distribution, and that shear stress has virtually no effect on lift. It is easy to see why. Look at the airfoil shapes in Figures 4.12 and 4.13, for example. Recall that pressure acts normal to the surface, and for these airfoils the direction of this normal pressure is essentially in the vertical direction, that is, the lift direction. In contrast the shear stress acts tangential to the surface, and for these airfoils the direction of this tangential shear stress is mainly in the horizontal direction, that is, the drag direction. Hence, pressure is the dominant player in the generation of lift, and shear stress has a negligible effect on lift. It is for this reason that the lift on an airfoil below the stall can be accurately predicted by inviscid theories such as that discussed in this chapter.

However, if we lived in a perfectly inviscid world, an airfoil could not produce lift. Indeed, the presence of friction is the very reason why we have lift. These sound like strange, even contradictory statements to our discussion in the preceding
paragraph. What is going on here? The answer is that in real life, the way that nature insures that the flow will leave smoothly at the trailing edge, that is, the mechanism that nature uses to choose the flow shown in Figure 4.13c, is that the viscous boundary layer remains attached to the surface all the way to the trailing edge. Nature enforces the Kutta condition by means of friction. If there were no boundary layer (i. e., no friction), there would be no physical mechanism in the real world to achieve the Kutta condition.

So we are led to the most ironic situation that lift, which is created by the surface pressure distribution—an inviscid phenomenon, would not exist in a frictionless (in­viscid) world. In this regard, we can say that without friction we could not have lift. However, we say this in the informed manner as discussed above.

Entropy and the Second Law of Thermodynamics

Consider a block of ice in contact with a red-hot plate of steel. Experience tells us that the ice will warm up (and probably melt) and the steel plate will cool down. However, Equation (7.11) does not necessarily say this will happen. Indeed, the first law allows that the ice may get cooler and the steel plate hotter—just as long as energy is conserved during the process. Obviously, in real life this does not happen; instead, nature imposes another condition on the process, a condition that tells us which direction a process will take. To ascertain the proper direction of a process, let us define a new state variable, the entropy, as follows:


where. s’ is the entropy of the system, Sq, cv is an incremental amount of heat added reversibly to the system, and T is the system temperature. Do not be confused by the above definition. It defines a change in entropy in terms of a reversible addition of heat SqKv. However, entropy is a state variable, and it can be used in conjunction with any type of process, reversible or irreversible. The quantity SqKV in Equation

(7.13) is just an artifice; an effective value of SqKy can always be assigned to relate the initial and end points of an irreversible process, where the actual amount of heat

added is 8q. Indeed, an alternative and probably more lucid relation is

Entropy and the Second Law of Thermodynamics

Подпись: 'irrevimage491[7.14]

In Equation (7.14), &q is the actual amount of heat added to the system during an actual irreversible process, and dsmv is the generation of entropy due to the irreversible, dis­sipative phenomena of viscosity, thermal conductivity, and mass diffusion occurring within the system. These dissipative phenomena always increase the entropy:

Подпись: 'irrev[7.15]

In Equation (7.15), the equals sign denotes a reversible process, where by definition no dissipative phenomena occur within the system. Combining Equations (7.14) and

(7.15) , we have


Furthermore, if the process is adiabatic, &q = 0, and Equation (7.16) becomes

Подпись: ds > 0[7.17]

Equations (7.16) and (7.17) are forms of the second law of thermodynamics. The second law tells us in what direction a process will take place. A process will proceed in a direction such that the entropy of the system plus that of its surroundings always increases or, at best, stays the same. In our example of the ice in contact with hot steel, consider the system to be both the ice and steel plate combined. The simultaneous heating of the ice and cooling of the plate yield a net increase in entropy for the system. On the other hand, the impossible situation of the ice getting cooler and the plate hotter would yield a net decrease in entropy, a situation forbidden by the second law. In summary, the concept of entropy in combination with the second law allows us to predict the direction that nature takes.

The practical calculation of entropy is carried out as follows. In Equation (7.12), assume that heat is added reversibly; then the definition of entropy, Equation (7.13), substituted in Equation (7.12) yields

Подпись: or Подпись: T ds = de + p dv Подпись: [7.18]

T ds — p dv = de

From the definition of enthalpy, Equation (7.3), we have

Подпись: [7.19]dh = de + p dv + v dp

Combining Equations (7.18) and (7.19), we obtain

Entropy and the Second Law of Thermodynamics

T ds — dh — v dp




Equations (7.18) and (7.20) are important; they are essentially alternate forms of the first law expressed in terms of entropy. For a perfect gas, recall Equations (7.5a and b), namely, de = cvdT and dh = cp dT. Substituting these relations into Equations

(7.18) and (7.20), we obtain

dT pdv

ds = cv— + — [7.21]

dT v dp

and ds=c„—————— [7.22]

p T t

Working with Equation (7.22), substitute the equation of state pv = RT, or v/T = R/p, into the last term:

ds=c„———- R— [7.23]


Consider a thermodynamic process with initial and end states denoted by 1 and 2, respectively. Equation (7.23), integrated between states 1 and 2, becomes

Подпись: 52 - 5)image493[7.24]

Подпись: Pi Подпись: [7.25] Подпись: Ti p2 S2 - Si = cp In — - R In — 11

For a calorically perfect gas, both R and cp are constants; hence, Equation (7.24) becomes

Подпись: T2 V2 S2 — 5] = c„ In R In — Ti vi Подпись: [7.26]

In a similar fashion, Equation (7.21) leads to

Equations (7.25) and (7.26) are practical expressions for the calculation of the entropy change of a calorically perfect gas between two states. Note from these equations that 5 is a function of two thermodynamic variables, for example, 5 = s(p, T), s = 5(u, T).

Line Integrals

Consider a vector field

A = A(x, y, z) = A(r, в, z) = A (г, в, Ф)

Also, consider a curve C in space connecting two points a and b as shown on the left side of Figure 2.8. Let ds be an elemental length of the curve, and n be a unit vector tangent to the curve. Define the vector ds = n ds. Then, the line integral of A along curve C from point a to point b is

A • ds

If the curve C is closed, as shown at the right of Figure 2.8, then the line integral is given by

J. A • ds

where the counterclockwise direction around C is considered positive. (The positive direction around a closed curve is, by convention, that direction you would move such that the area enclosed by C is always on your left.)


Figure 2.8 Sketch for line integrals.


Condition on Velocity for Incompressible Flow

Consulting our chapter road map in Figure 3.4, we have completed the left branch dealing with Bernoulli’s equation. We now begin a more general consideration of incompressible flow, given by the center branch in Figure 3.4. However, before intro­ducing Laplace’s equation, it is important to establish a basic condition on velocity in an incompressible flow, as follows.

First, consider the physical definition of incompressible flow, namely, p = con­stant. Since p is the mass per unit volume and p is constant, then a fluid element of fixed mass moving through an incompressible flow field must also have a fixed, con­stant volume. Recall Equation (2.32), which shows that V • V is physically the time rate of change of the volume of a moving fluid element per unit volume. However, for an incompressible flow, we have just stated that the volume of a fluid element is constant [e. g., in Equation (2.32), D(SV)/Dt = 0]. Therefore, for an incompressible flow,

Подпись: V • V = 0[3.39]

The fact that the divergence of velocity is zero for an incompressible flow can also be shown directly from the continuity equation, Equation (2.52):

Condition on Velocity for Incompressible Flow



For incompressible flow, p = constant. Hence, dp/dt = 0 and V • (pV) = pV ■ V.

Equation (2.52) then becomes

O + pV • V = 0

or V • V = 0

which is precisely Equation (3.39).

Incompressible Flow. over Finite Wings

The one who has most carefully watched the soaring birds of prey sees man with wings and the faculty of using them.

James Means, Editor of the Aeronautical Annual, 1895

5.1 Introduction: Downwash and Induced Drag

In Chapter 4 we discussed the properties of airfoils, which are the same as the proper­ties of a wing of infinite span; indeed, airfoil data are frequently denoted as “infinite wing” data. However, all real airplanes have wings of finite span, and the purpose of the present chapter is to apply our knowledge of airfoil properties to the analysis of such finite wings. This is the second step in Prandtl’s philosophy of wing theory, as described in Section 4.1. You should review Section 4.1 before proceeding further.

Question: Why are the aerodynamic characteristics of a finite wing any different from the properties of its airfoil sections? Indeed, an airfoil is simply a section of a wing, and at first thought, you might expect the wing to behave exactly the same as the airfoil. However, as studied in Chapter 4, the flow over an airfoil is two-dimensional. In contrast, a finite wing is a three-dimensional body, and consequently the flow over the finite wing is three-dimensional; that is, there is a component of flow in the spanwise direction. To see this more clearly, examine Figure 5.1, which gives the top and front views of a finite wing. The physical mechanism for generating lift on the wing is the existence of a high pressure on the bottom surface and a low pressure




High pressure

Figure 5.1 Finite wing. In this figure, the curvature of the

streamlines over the top and bottom of the wing is exaggerated for clarity.

on the top surface. The net imbalance of the pressure distribution creates the lift, as discussed in Section 1.5. However, as a by-product of this pressure imbalance, the flow near the wing tips tends to curl around the tips, being forced from the high – pressure region just underneath the tips to the low-pressure region on top. This flow around the wing tips is shown in the front view of the wing in Figure 5.1. Asa result, on the top surface of the wing, there is generally a spanwise component of flow from the tip toward the wing root, causing the streamlines over the top surface to bend toward the root, as sketched on the top view shown in Figure 5.1. Similarly, on the bottom surface of the wing, there is generally a spanwise component of flow from the root toward the tip, causing the streamlines over the bottom surface to bend toward the tip. Clearly, the flow over the finite wing is three-dimensional, and therefore you would expect the overall aerodynamic properties of such a wing to differ from those of its airfoil sections.

The tendency for the flow to “leak” around the wing tips has another important effect on the aerodynamics of the wing. This flow establishes a circulatory motion that trails downstream of the wing; that is, a trailing vortex is created at each wing tip. These wing-tip vortices are sketched in Figure 5.2 and are illustrated in Figure 5.3. The tip vortices are essentially weak “tornadoes” that trail downstream of the finite wing. (For large airplanes such as a Boeing 747, these tip vortices can be powerful


Figure 5.2 Schematic of wing-tip vortices.


Figure 5.3 Wing-tip vortices from a rectangular wing. The wing is in a smoke tunnel,

where individual streamtubes are made visible by means of smoke filaments.

(Source: Head, M. R., in Flow Visualization II, W. Merzkirch (Ed.), Hemisphere Publishing Co., New York, 1 982, pp. 399-403. Also available in Van Dyke, Milton, An Album of Fluid Motion, The Parabolic Press,

Stanford, CA, 1982.)

enough to cause light airplanes following too closely to go out of control. Such accidents have occurred, and this is one reason for large spacings between aircraft landing or taking off consecutively at airports.) These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighborhood

of the wing itself. This can be seen by inspecting Figure 5.3; the two vortices tend to drag the surrounding air around with them, and this secondary movement induces a small velocity component in the downward direction at the wing. This downward component is called downwash, denoted by the symbol w. In turn, the downwash combines with the freestream velocity Voo to produce a local relative wind which is canted downward in the vicinity of each airfoil section of the wing, as sketched in Figure 5.4.

Examine Figure 5.4 closely. The angle between the chord line and the direction of Vqo is Ле angle of attack a, as defined in Section 1.5 and as used throughout our discussion of airfoil theory in Chapter 4. We now more precisely define a as the geometric angle of attack. In Figure 5.4, the local relative wind is inclined below the direction of Vqo by the angle a,, called the induced angle of attack. The presence of downwash, and its effect on inclining the local relative wind in the downward direction, has two important effects on the local airfoil section, as follows:

1. The angle of attack actually seen by the local airfoil section is the angle between the chord line and the local relative wind. This angle is given by «cff in Figure

5.4 and is defined as the effective angle of attack. Hence, although the wing is at a geometric angle of attack a, the local airfoil section is seeing a smaller angle, namely, the effective angle of attack aes. From Figure 5.4,

Qfeff = a — a, [5.1 ]

2. The local lift vector is aligned perpendicular to the local relative wind, and hence is inclined behind the vertical by the angle a,, as shown in Figure 5.4.



Consequently, there is a component of the local lift vector in the direction of V,*,; that is, there is a drag created by the presence of downwash. This drag is defined as induced drag, denoted by Ц in Figure 5.4.

Hence, we see that the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag—the induced drag D,. Keep in mind that we are still dealing with an inviscid, incompressible flow, where there is no skin friction or flow separation. For such a flow, there is a finite drag—the induced drag—on a finite wing. D’Alembert’s paradox does not occur for a finite wing.

The tilting backward of the lift vector shown in Figure 5.4 is one way of visual­izing the physical generation of induced drag. Two alternate ways are as follows:

1. The three-dimensional flow induced by the wing-tip vortices shown in Figures

5.2 and 5.3 simply alters the pressure distribution on the finite wing in such a fashion that a net pressure imbalance exists in the direction of (i. e., drag is created). In this sense, induced drag is a type of “pressure drag.”

2. The wing-tip vortices contain a large amount of translational and rotational ki­netic energy. This energy has to come from somewhere; indeed, it is ultimately provided by the aircraft engine, which is the only source of power associated with the airplane. Since the energy of the vortices serves no useful purpose, this power is essentially lost. In effect, the extra power provided by the engine that goes into the vortices is the extra power required from the engine to overcome the induced drag.

Clearly, from the discussion in this section, the characteristics of a finite wing are not identical to the characteristics of its airfoil sections. Therefore, let us proceed to develop a theory that will enable us to analyze the aerodynamic properties of finite wings. In the process, we follow the road map given in Figure 5.5—keep in touch with this road map as we progress through the present chapter.

In this chapter, we note a difference in nomenclature. For the two-dimensional bodies considered in the previous chapters, the lift, drag, and moments per unit span have been denoted with primes, for example, V, ІУ. and M’, and the correspond­ing lift, drag, and moment coefficients have been denoted by lowercase letters, for example, ci, Cd, and cm. In contrast, the lift, drag, and moments on a complete three­dimensional body such as a finite wing are given without primes, for example, L, D, and M, and the corresponding lift, drag, and moment coefficients are given by capital letters, for example, CL, CD, and CM – This distinction has already been mentioned in Section 1.5.

Finally, we note that the total drag on a subsonic finite wing in real life is the sum of the induced drag /),, the skin friction drag I) /, and the pressure drag Dp due to flow separation. The latter two contributions are due to viscous effects, which are discussed in Chapters 15 to 20. The sum of these two viscous-dominated drag contributions is called profile drag, as discussed in Section 4.3. The profile drag coefficient cj for an NACA 2412 airfoil was given in Figure 4.6. At moderate angle of attack, the profile drag coefficient for a finite wing is essentially the same as for its


Figure 5.5 Road map for Chapter 5.


Подпись: Cd Подпись: Df + Dp Подпись: [5.3]

airfoil sections. Hence, defining the profile drag coefficient as

Подпись: CD, І Подпись: Pi QccS Подпись: [5.3]

and the induced drag coefficient as

the total drag coefficient for the finite wing Co is given by

Подпись: [5.4]Cd = Cd + Co, i

In Equation (5.4), the value of Q is usually obtained from airfoil data, such as given in Figure 4.6. The value of Cd, і can be obtained from finite-wing theory as presented in this chapter. Indeed, one of the central objectives of the present chapter is to obtain an expression for induced drag and to study its variation with certain design characteristics of the finite wing. (See Chapter 5 of Reference 2 for an additional discussion of the characteristics of finite wings.)

Preface to the First Edition

This book is for students—to be read, understood, and enjoyed. It is consciously written in a clear, informal, and direct style designed to talk to the reader and to gain his or her immediate interest in the challenging and yet beautiful discipline of aerodynamics. The explanation of each topic is carefully constructed to make sense to the reader. Moreover, the structure of each chapter is highly organized in order to keep the reader aware of where we are, where we were, and where we are going. Too frequently the student of aerodynamics loses sight of what is trying to be accomplished; to avoid this, we attempt to keep the reader informed of our intent at all times. For example, virtually each chapter contains a road map—a block diagram designed to keep the reader well aware of the proper flow of ideas and concepts. The use of such chapter road maps is one of the unique features of this book. Also, to help organize the reader’s thoughts, there are special summary sections at the end of most chapters.

The material in this book is at the level of college juniors and seniors in aerospace or mechanical engineering. It assumes no prior knowledge of fluid dynamics in general, or aerodynamics in particular. It does assume a familiarity with differential and integral calculus, as well as the usual physics background common to most students of science and engineering. Also, the language of vector analysis is used liberally; a compact review of the necessary elements of vector algebra and vector calculus is given in Chapter 2 in such a fashion that it can either educate or refresh the reader, whichever may be the case for each individual.

This book is designed for a 1-year course in aerodynamics. Chapters 1 to 6 constitute a solid semester emphasizing inviscid, incompressible flow. Chapters 7 to 14 occupy a second semester dealing with inviscid, compressible flow. Finally, Chapters 15 to 18 introduce some basic elements of viscous flow, mainly to serve as a contrast to and comparison with the inviscid flows treated throughout the bulk of the text.

This book contains several unique features:

1. The use of chapter road maps to help organize the material in the mind of the reader, as discussed earlier.

2. An introduction to computational fluid dynamics as an integral part of the begin­ning study of aerodynamics. Computational fluid dynamics (CFD) has recently become a third dimension in aerodynamics, complementing the previously ex­isting dimensions of pure experiment and pure theory. It is absolutely necessary that the modem student of aerodynamics be introduced to some of the basic ideas of CFD—he or she will most certainly come face to face with either its “machinery” or its results after entering the professional ranks of practicing aero – dynamicists. Hence, such subjects as the source and vortex panel techniques, the method of characteristics, and explicit finite-difference solutions are introduced

and discussed as they naturally arise during the course of our discussions. In particular, Chapter 13 is devoted exclusively to numerical techniques, couched at a level suitable to an introductory aerodynamics text.

3. A short chapter is devoted entirely to hypersonic flow. Although hypersonics is at one extreme end of the flight spectrum, it has current important applications to the design of the space shuttle, hypervelocity missiles, and planetary entry vehicles. Therefore, hypersonic flow deserves some attention in any modern presentation of aerodynamics. This is the purpose of Chapter 14.

4. Historical notes are placed at the end of many of the chapters. This follows in the tradition of the author’s previous books, Introduction to Flight: Its Engineering and History (McGraw-Hill, 1978), and Modem Compressible Flow: With His­torical Perspective (McGraw-Hill, 1982). Although aerodynamics is a rapidly evolving subject, its foundations are deeply rooted in the history of science and technology. It is important for the modem student of aerodynamics to have an appreciation for the historical origin of the tools of the trade. Therefore, this book addresses such questions as who were Bernoulli, Euler, d’Alembert, Kutta, Joukowski, and Prandtl; how was the circulation theory of lift developed; and what excitement surrounded the early development of high-speed aerodynamics? The author wishes to thank various members of the staff of the National Air and Space Museum of the Smithsonian Institution for opening their extensive files for some of the historical research behind these history sections. Also, a con­stant biographical reference was the Dictionary of Scientific Biography, edited by С. C. Gillespie, Charles Schribner’s Sons, New York, 1980. This is a 16-volume set of books which is a valuable source of biographic information on the leading scientists in history.

This book has developed from the author’s experience in teaching both incom­pressible and compressible flow to undergraduate students at the University of Mary­land. Such courses require careful attention to the structure and sequence of the presentation of basic material, and to the manner in which sophisticated subjects are described to the uninitiated reader. This book meets the author’s needs at Maryland; it is hoped that it will also meet the needs of others, both in the formal atmosphere of the classroom and in the informal pleasure of self-study.

Readers who are already familiar with the author’s Introduction to Flight will find the present book to be a logical sequel. Many of the aerodynamic concepts first introduced in the most elementary sense in Introduction to Flight are revisited and greatly expanded in the present book. For example, at Maryland, Introduction to Flight is used in a sophomore-level introductory course, followed by the material of the present book in junior – and senior-level courses in incompressible and com­pressible flow. On the other hand, the present book is entirely self-contained; no prior familiarity with aerodynamics on the part of the reader is assumed. All basic principles and concepts are introduced and developed from their beginnings.

The author wishes to thank his students for many stimulating discussions on the subject of aerodynamics—discussions which ultimately resulted in the present book. Special thanks go to two of the author’s graduate students, Tae-Hwan Cho and

Kevin Bowcutt, who provided illustrative results from the source and vortex panel techniques. Of course, all of the author’s efforts would have gone for nought if it had not been for the excellent preparation of the typed manuscript by Ms. Sue Osborn.

Finally, special thanks go to two institutions: (1) the University of Maryland for providing a challenging intellectual atmosphere in which the author has basked for the past 9 years and (2) the Anderson household—Sarah-Alien, Katherine, and Elizabeth—who have been patient and understanding while their husband and father was in his ivory tower.

John D. Anderson, Jr.


In this section, we have applied the momentum principle (Newton’s second law) to large, fixed control volumes in flows. On one hand, we demonstrated that, by know­ing the detailed flow properties along the control surface, this application led to an accurate result for an overall quantity such as drag on a body, namely, Equation (2.83) for a compressible flow and Equation (2.84) for an incompressible flow. On the other hand, in Example 2.2, we have shown that, by knowing an overall quantity such as the net drag on a flat plate, the finite control volume concept by itself does not necessarily provide an accurate calculation of detailed flow-field properties along the control sur­face (in this case, the velocity profile), although the momentum principle is certainly satisfied in the aggregate. Example 2.2 is designed specifically to demonstrate this fact. The weakness here is the need to assume some form for the variation of flow properties over the control surface; in Example 2.2, the assumption of the particular power-law profile proved to be unsatisfactory.