Category Fundamentals of Aerodynamics

Governing Equations for Inviscid, Compressible Flow

In Chapters 3 to 6, we studied inviscid, incompressible flow; recall that the primary dependent variables for such flows are p and V, and hence we need only two basic equations, namely, the continuity and momentum equations, to solve for these two unknowns. Indeed, the basic equations are combined to obtain Laplace’s equation and

Bernoulli’s equation, which are the primary tools used for the applications discussed in Chapters 3 to 6. Note that both p and T are assumed to be constant throughout such inviscid, incompressible flows. As a result, no additional governing equations are required; in particular, there is no need for the energy equation or energy concepts in general. Basically, incompressible flow obeys purely mechanical laws and does not require thermodynamic considerations.

In contrast, for compressible flow, p is variable and becomes an unknown. Hence, we need an additional governing equation—the energy equation—which in turn in­troduces internal energy e as an unknown. Since e is related to temperature, then T also becomes an important variable. Therefore, the primary dependent variables for the study of compressible flow are p, V, p, e, and T; to solve for these five variables, we need five governing equations. Let us examine this situation further.

To begin with, the flow of a compressible fluid is governed by the basic equations derived in Chapter 2. At this point in our discussion, it is most important for you to be familiar with these equations as well as their derivation. Therefore, before proceeding further, return to Chapter 2 and carefully review the basic ideas and relations contained therein. This is a serious study tip, and if you follow it, the material in our next seven chapters will flow much easier for you. In particular, review the integral and differential forms of the continuity equation (Section 2.4), the momentum equation (Section 2.5), and the energy equation (Section 2.7); indeed, pay particular attention to the energy equation because this is an important aspect which sets compressible flow apart from incompressible flow.

For convenience, some of the more important forms of the governing equations for an inviscid, compressible flow from Chapter 2 are repeated below:

Continuity: From Equation (2.48),

Подпись: vПодпись: simage499[7.39]

From Equation (2.52),

Подпись: [7.40]77 + V • PV = 0


Momentum: From Equation (2.64),

Governing Equations for Inviscid, Compressible Flow

From Equations (2.113a to c),


Du dp







Dw dp






Governing Equations for Inviscid, Compressible Flow

Energy: From Equation (2.95),

In regard to the basic equations for compressible flow, please note that Bernoulli’s equation as derived in Section 3.2 and given by Equation (3.13) does not hold for compressible flow; it clearly contains the assumption of constant density, and hence is invalid for compressible flow. This warning is necessary because experience shows that a certain number of students of aerodynamics, apparently attracted by the sim­plicity of Bernoulli’s equation, attempt to use it for all situations, compressible as well as incompressible. Do not do it! Always remember that Bernoulli’s equation in the form of Equation (3.13) holds for incompressible flow only, and we must dismiss it from our thinking when dealing with compressible flow.

As a final note, we use both the integral and differential forms of the above equations in our subsequent discussions. Make certain that you feel comfortable with these equations before proceeding further.

Models of the Fluid: Control Volumes and Fluid Elements

Aerodynamics is a fundamental science, steeped in physical observation. As you proceed through this book, make every effort to gradually develop a “physical feel” for the material. An important virtue of all successful aerodynamicists (indeed, of all successful engineers and scientists) is that they have good “physical intuition,” based on thought and experience, which allows them to make reasonable judgments on difficult problems. Although this chapter is full of equations and (seemingly) esoteric concepts, now is the time for you to start developing this physical feel.

With this section, we begin to build the basic equations of aerodynamics. There is a certain philosophical procedure involved with the development of these equations, as follows:

1. Invoke three fundamental physical principles which are deeply entrenched in our macroscopic observations of nature, namely,

a. Mass is conserved, i. e., mass can be neither created nor destroyed.

b. Newton’s second law: force = mass x acceleration.

c. Energy is conserved; it can only change from one form to another.

2. Determine a suitable model of the fluid. Remember that a fluid is a squishy substance, and therefore it is usually more difficult to describe than a well-defined solid body. Hence, we have to adopt a reasonable model of the fluid to which we can apply the fundamental principles stated in item 1.

3. Apply the fundamental physical principles listed in item 1 to the model of the fluid determined in item 2 in order to obtain mathematical equations which properly describe the physics of the flow. In turn, use these fundamental equations to analyze any particular aerodynamic flow problem of interest.

In this section, we concentrate on item 2; namely, we ask the question: What is a suitable model of the fluid? How do we visualize this squishy substance in order to apply the three fundamental physical principles to it? There is no single answer to this question; rather, three different models have been used successfully throughout the modern evolution of aerodynamics. They are (1) finite control volume, (2) infinitesimal fluid element, and (3) molecular. Let us examine what these models involve and how they are applied.

Wall Boundary Conditions

Подпись: or image226 Подпись: [3.48a] [3.48Ы

If the body in Figure 3.18 has a solid surface, then it is impossible for the flow to penetrate the surface. Instead, if the flow is viscous, the influence of friction between the fluid and the solid surface creates a zero velocity at the surface. Such viscous flows are discussed in Chapters 15 to 20. In contrast, for inviscid flows the velocity at the surface can be finite, but because the flow cannot penetrate the surface, the velocity vector must be tangent to the surface. This “wall tangency” condition is illustrated in Figure 3.18, which shows V tangent to the body surface. If the flow is tangent to the surface, then the component of velocity normal to the surface must be zero. Fet n be a unit vector normal to the surface as shown in Figure 3.18. The wall boundary condition can be written as

Equation (3.48a or b) gives the boundary condition for velocity at the wall; it is expressed in terms of ф. If we are dealing with ф rather than ф, then the wall boundary condition is

Wall Boundary Conditions



where 5 is the distance measured along the body surface, as shown in Figure 3.18. Note that the body contour is a streamline of the flow, as also shown in Figure 3.18. Recall that ф = constant is the equation of a streamline. Thus, if the shape of the body in Figure 3.18 is given by уъ = fix), then

Подпись: [3.484]^surface — Ф у=}h — Const

is an alternative expression for the boundary condition given in Equation (3.48c).

If we are dealing with neither ф nor ф, but rather with the velocity components и and v themselves, then the wall boundary condition is obtained from the equation of a streamline, Equation (2.118), evaluated at the body surface; that is,


Equation (3.48e) states simply that the body surface is a streamline of the flow. The form given in Equation (3.48e) for the flow tangency condition at the body surface is used for all inviscid flows, incompressible to hypersonic, and does not depend on the formulation of the problem in terms of ф or ф (or ф).

General Lift Distribution

Consider the transformation

у — —— cos в [5.46]


where the coordinate in the spanwise direction is now given by в, with 0 < в < л. In terms of в, the elliptic lift distribution given by Equation (5.31) is written as

Г(0) = Г0 sin0 [5.47]

Equation (5.47) hints that a Fourier sine series would be an appropriate expression for the general circulation distribution along an arbitrary finite wing. Hence, assume for the general case that


Г(0) = 2bV, A„ sin пв [5.48]


where as many terms N in the series can be taken as we desire for accuracy. The coefficients An (where n = 1,…, N) in Equation (5.48) are unknowns; however, they must satisfy the fundamental equation of Prandtl’s lifting-line theory; that is, the An’s must satisfy Equation (5.23). Differentiating Equation (5.48), we obtain

General Lift Distribution




dr de

d9 dy


General Lift Distribution



Подпись: [5.54]Substituting Equations (5.48) and (5.49) into (5.23), we obtain

2b ■Дл 1 fn y’f «A„ cos «0 , ,

а (во) = —V A„ sin «ft, + аь=о(во) + ————- ^— — dQ [5.50]

7Tc(0o) , 7T Jq cos в — COS Oq

General Lift Distribution General Lift Distribution Подпись: [5.51]

The integral in Equation (5.50) is the standard form given by Equation (4.26). Hence, Equation (5.50) becomes

Examine Equation (5.51) closely. It is evaluated at a given spanwise location; hence, во is specified. In turn, b, с(во), and а/=о(во) are known quantities from the geometry and airfoil section of the finite wing. The only unknowns in Equation (5.51) are the A„’s. Hence, written at a given spanwise location (a specified во), Equation (5.51) is one algebraic equation with N unknowns, Ab A2,…, A„. However, let us choose N different spanwise stations, and let us evaluate Equation (5.51) at each of these N stations. We then obtain a system of N independent algebraic equations with N unknowns, namely, А,, Аг,…, A, y. In this fashion, actual numerical values are obtained for the A„’s—numerical values that ensure that the general circulation distribution given by Equation (5.48) satisfies the fundamental equation of finite-wing theory, Equation (5.23).

2 Ch/2 2h[16]

Cl = 77—- / Г(y)dy = —J2A4 si

J-b/2 J , JO

General Lift Distribution General Lift Distribution Подпись: [5.52]

Now that Г(0) is known via Equation (5.48), the lift coefficient for the finite wing follows immediately from the substitution of Equation (5.48) into (5.26):

Hence, Equation (5.52) becomes

Подпись: [5.53]CL = Аіяу = Ai7tAR

Note that Cl depends only on the leading coefficient of the Fourier series expan­sion. (However, although Cl depends on A i only, we must solve for all the A„’s simultaneously in order to obtain A.)

The induced drag coefficient is obtained from the substitution of Equation (5.48) into Equation (5.30) as follows:

General Lift Distribution

2 b2



at (в) sin в dd



General Lift Distribution Подпись: [5.55]

The induced angle of attack a, (в) in Equation (5.54) is obtained from the substitution of Equations (5.46) and (5.49) into (5.18), which yields

The integral in Equation (5.55) is the standard form given by Equation (4.26). Hence, Equation (5.55) becomes


sin nSo

nA„ ——— [5.56]

, Sin во

In Equation (5.56), 60 is simply a dummy variable which ranges from 0 to ж across the span of the wing; it can therefore be replaced by в, and Equation (5.56) can be written as


Подпись:Подпись: sin0sin пв пА„ —



General Lift Distribution Подпись: [5.58]

Substituting Equation (5.57) into (5.54), we have

General Lift Distribution Подпись: 0 for m ф к ж/2 for m = к Подпись: [5.59]

Examine Equation (5.58) closely; it involves the product of two summations. Also, note that, from the standard integral,

Hence, in Equation (5.58), the mixed product terms involving unequal subscripts (such as A1A2, A2A4) are, from Equation (5.59), equal to zero. Hence, Equation

(5.58) becomes

9,2 / N V

Подпись: = л-AR A Подпись: I + E»(^7 Подпись: [5.60]

cD, i – (J2 nAln) у = лARZ! иЛ»

Substituting Equation (5.53) for Cl into Equation (5.60), we obtain

General Lift Distribution




Cd. i




where S = Y^2 n{An/A] )2. Note that S > 0; hence, the factor 1 + S in Equation


General Lift Distribution Подпись: г2 7reAR Подпись: [5.62]

is either greater than 1 or at least equal to 1. Let us define a span efficiency factor, e, as e = (1 + <5)_1. Then Equation (5.61) can be written as

where e < 1. Comparing Equations (5.61) and (5.62) for the general lift distribution with Equation (5.43) for the elliptical lift distribution, note that 5 = 0 and e = 1 for the elliptical lift distribution. Hence, the lift distribution which yields minimum induced drag is the elliptical lift distribution. This is why we have a practical interest in the elliptical lift distribution.

Recall that for a wing with no aerodynamic twist and no geometric twist, an elliptical lift distribution is generated by a wing with an elliptical planform, as sketched at the top of Figure 5.16. Several aircraft have been designed in the past with elliptical wings; the most famous, perhaps, being the British Spitfire from World War II, shown in Figure 5.17. However, elliptic planforms are more expensive to manufacture than, say, a simple rectangular wing as sketched in the middle of Figure 5.16. On the other hand, a rectangular wing generates a lift distribution far from optimum. A compromise is the tapered wing shown at the bottom of Figure 5.16. The tapered wing can be designed with a taper ratio, that is, tip chord/root chord ее ct/cr, such that the lift distribution closely approximates the elliptic case. The variation of S as a function of taper ratio for wings of different aspect ratio is illustrated in Figure 5.18. Such calculations of S were first performed by the famous English aerodynamicist, Hermann Glauert and published in Reference 18 in the year 1926. Note from Figure 5.18 that a tapered wing can be designed with an induced drag coefficient reasonably close to the minimum value. In addition, tapered wings with straight leading and trailing

Elliptic wing

Rectangular wing


Tapered wing

Figure 5*16 Various planforms for straight wings.




Figure 5.1 7 Three views of the Supermarine Spitfire, a famous British World War II fighter.

edges are considerably easier to manufacture than elliptic planforms. Therefore, most conventional aircraft employ tapered rather than elliptical wing planforms.

Aerodynamics: Classification and Practical Objectives

A distinction between solids, liquids, and gases can be made in a simplistic sense as follows. Put a solid object inside a larger, closed container. The solid object will not change; its shape and boundaries will remain the same. Now put a liquid inside the container. The liquid will change its shape to conform to that of the container and will take on the same boundaries as the container up to the maximum depth of the liquid. Now put a gas inside the container. The gas will completely fill the container, taking on the same boundaries as the container.

The word “fluid” is used to denote either a liquid or a gas. A more technical distinction between a solid and a fluid can be made as follows. When a force is applied tangentially to the surface of a solid, the solid will experience a finite deformation, and the tangential force per unit area—the shear stress—will usually be proportional to the amount of deformation. In contrast, when a tangential shear stress is applied to the surface of a fluid, the fluid will experience a continuously increasing deformation, and the shear stress usually will be proportional to the rate of change of the deformation.

The most fundamental distinction between solids, liquids, and gases is at the atomic and molecular level. In a solid, the molecules are packed so closely together that their nuclei and electrons form a rigid geometric structure, “glued” together by powerful intermolecular forces. In a liquid, the spacing between molecules is larger, and although intermolecular forces are still strong they allow enough movement of the molecules to give the liquid its “fluidity.” In a gas, the spacing between molecules is much larger (for air at standard conditions, the spacing between molecules is, on the average, about 10 times the molecular diameter). Hence, the influence of intermolecular forces is much weaker, and the motion of the molecules occurs rather freely throughout the gas. This movement of molecules in both gases and liquids leads to similar physical characteristics, the characteristics of a fluid—quite different from those of a solid. Therefore, it makes sense to classify the study of the dynamics of both liquids and gases under the same general heading, called fluid dynamics. On the other hand, certain differences exist between the flow of liquids and the flow of gases; also, different species of gases (say, N2, He, etc.) have different properties. Therefore, fluid dynamics is subdivided into three areas as follows:

Hydrodynamics—flow of liquids Gas dynamics—flow of gases Aerodynamics—flow of air

These areas are by no means mutually exclusive; there are many similarities and identical phenomena between them. Also, the word “aerodynamics” has taken on a popular usage that sometimes covers the other two areas. As a result, this author tends to interpret the word “aerodynamics” very liberally, and its use throughout this book does not always limit our discussions just to air.

Aerodynamics is an applied science with many practical applications in engineer­ing. No matter how elegant an aerodynamic theory may be, or how mathematically complex a numerical solution may be, or how sophisticated an aerodynamic exper-

iment may be, all such efforts are usually aimed at one or more of the following practical objectives:

1. The prediction of forces and moments on, and heat transfer to, bodies moving through a fluid (usually air). For example, we are concerned with the generation of lift, drag, and moments on airfoils, wings, fuselages, engine nacelles, and most importantly, whole airplane configurations. We want to estimate the wind force on buildings, ships, and other surface vehicles. We are concerned with the hydrodynamic forces on surface ships, submarines, and torpedoes. We need to be able to calculate the aerodynamic heating of flight vehicles ranging from the supersonic transport to a planetary probe entering the atmosphere of Jupiter. These are but a few examples.

2. Determination of flows moving internally through ducts. We wish to calculate and measure the flow properties inside rocket and air-breathing jet engines and to calculate the engine thrust. We need to know the flow conditions in the test section of a wind tunnel. We must know how much fluid can flow through pipes under various conditions. A recent, very interesting application of aerodynamics is high-energy chemical and gas-dynamic lasers (see Reference 1), which are nothing more than specialized wind tunnels that can produce extremely powerful laser beams. Figure 1.5 is a photograph of an early gas-dynamic laser designed in the late 1960s.


Figure 1.5 A CO2-N2 gas-dynamic laser, circa 1969. (Courtesy of the Avco-Everett Research Laboratory.!

The applications in item 1 come under the heading of external aerodynamics since they deal with external flows over a body. In contrast, the applications in item 2 involve internal aerodynamics because they deal with flows internally within ducts. In external aerodynamics, in addition to forces, moments, and aerodynamic heating associated with a body, we are frequently interested in the details of the flow field away from the body. For example, the communication blackout experienced by the space shuttle during a portion of its reentry trajectory is due to a concentration of free electrons in the hot shock layer around the body. We need to calculate the variation of electron density throughout such flow fields. Another example is the propagation of shock waves in a supersonic flow; for instance, does the shock wave from the wing of a supersonic airplane impinge upon and interfere with the tail surfaces? Yet another example is the flow associated with the strong vortices trailing downstream from the wing tips of large subsonic airplanes such as the Boeing 747. What are the properties of these vortices, and how do they affect smaller aircraft which happen to fly through them?

The above is just a sample of the myriad applications of aerodynamics. One purpose of this book is to provide the reader with the technical background necessary to fully understand the nature of such practical aerodynamic applications.

Interim Summary

At this stage, let us pause and think about the various equations we have developed. Do not fall into the trap of seeing these equations as just a jumble of mathematical symbols that, by now, might look all the same to you. Quite the contrary, these equations speak words: e. g., Equations (2.48), (2.52), (2.53), and (2.54) all say that mass is conserved; Equations (2.64), (2.70a to c), (2.71), and (2.72a to c) are statements of Newton’s second law applied to a fluid flow; Equations (2.95) to (2.98) say that energy is conserved. It is very important to be able to see the physical principles behind these equations. When you look at an equation, try to develop the ability to see past a collection of mathematical symbols and, instead, to read the physics that the equation represents.

The equations listed above are fundamental to all of aerodynamics. Take the time to go back over them. Become familiar with the way they are developed, and make yourself comfortable with their final forms. In this way, you will find our subsequent aerodynamic applications that much easier to understand.

Also, note our location on the road map shown in Figure 2.1. We have finished the items on the left branch of the map—we have obtained the basic flow equations containing the fundamental physics of fluid flow. We now start with the branch on the right, which is a collection of useful concepts helpful in the application of the basic flow equations.

Euler—The Origins of Theoretical Fluid Dynamics

Bernoulli’s equation, expressed by Equations (3.14) and (3.15), is historically the most famous equation in fluid dynamics. Moreover, we derived Bernoulli’s equation from the general momentum equation in partial differential equation form. The momentum equation is just one of the three fundamental equations of fluid dynamics—the others being continuity and energy. These equations are derived and discussed in Chapter 2 and applied to an incompressible flow in Chapter 3. Where did these equations first originate? How old are they, and who is responsible for them? Considering the fact that all of fluid dynamics in general, and aerodynamics in particular, is built on these fundamental equations, it is important to pause for a moment and examine their historical roots.

As discussed in Section 1.1, Isaac Newton, in his Principia of 1687, was the first to establish on a rational basis the relationships between force, momentum, and acceleration. Although he tried, he was unable to apply these concepts properly to a moving fluid. The real foundations of theoretical fluid dynamics were not laid until the next century—developed by a triumvirate consisting of Daniel Bernoulli, Leonhard Euler, and Jean Le Rond d’Alembert.

First, consider Bernoulli. Actually, we must consider the whole family of Bernoulli’s because Daniel Bernoulli was a member of a prestigious family that dom­inated European mathematics and physics during the early part of the eighteenth century. Figure 3.51 is a portion of the Bernoulli family tree. It starts with Nikolaus Bernoulli, who was a successful merchant and druggist in Basel, Switzerland, during the seventeenth century. With one eye on this family tree, let us simply list some of the subsequent members of this highly accomplished family:

1. Jakob—Daniel’s uncle. Mathematician and physicist, he was professor of math­ematics at the University of Basel. He made major contributions to the develop­ment of calculus and coined the term “integral.”

2. Johann—Daniel’s father. He was a professor of mathematics at Groningen, Netherlands, and later at the University of Basel. He taught the famous French mathematician L’Hospital the elements of calculus, and after the death of Newton in 1727 he was considered Europe’s leading mathematician at that time.

3. Nikolaus—Daniel’s cousin. He studied mathematics under his uncles and held a master’s degree in mathematics and a doctor of jurisprudence.

4. Nikolaus—Daniel’s brother. He was Johann’s favorite son. He held a master of arts degree, and assisted with much of Johann’s correspondence to Newton and Liebniz concerning the development of calculus.

5. Daniel himself—to be discussed below.

6. Johann—Daniel’s other brother. He succeeded his father in the Chair of Math­ematics at Basel and won the prize of the Paris Academy four times for his work.

7. Johann—Daniel’s nephew. A gifted child, he earned the master of jurisprudence at the age of 14. When he was 20, he was invited by Frederick II to reorganize the astronomical observatory at the Berlin Academy.

8. Jakob—Daniel’s other nephew. He graduated in jurisprudence but worked in mathematics and physics. He was appointed to the Academy in St. Petersburg, Russia, but he had a promising career prematurely ended when he drowned in the river Neva at the age of 30.

With such a family pedigree, Daniel Bernoulli was destined for success.

Daniel Bernoulli was bom in Groningen, Netherlands, on February 8, 1700. His father, Johann, was a professor at Groningen but returned to Basel, Switzerland, in 1705 to occupy the Chair of Mathematics which had been vacated by the death of Jacob Bernoulli. At the University of Basel, Daniel obtained a master’s degree in 1716 in philosophy and logic. He went on to study medicine in Basel, Heidelburg, and Strasbourg, obtaining his Ph. D. in anatomy and botany in 1721. During these studies, he maintained an active interest in mathematics. He followed this interest by moving briefly to Venice, where he published an important work entitled Exerci – tationes Mathematicae in 1724. This earned him much attention and resulted in his winning the prize awarded by the Paris Academy—the first of 10 he was eventually to receive. In 1725, Daniel moved to St. Petersburg, Russia, to join the academy. The St. Petersburg Academy had gained a substantial reputation for scholarship and intellectual accomplishment at that time. During the next 8 years, Bernoulli experi­enced his most creative period. While at St. Petersburg, he wrote his famous book Hydrodynamica, completed in 1734, but not published until 1738. In 1733, Daniel returned to Basel to occupy the Chair of Anatomy and Botany, and in 1750 moved to the Chair of Physics created exclusively for him. He continued to write, give very popular and well-attended lectures in physics, and make contributions to mathematics and physics until his death in Basel on March 17, 1782.

Daniel Bernoulli was famous in his own time. He was a member of virtually all the existing learned societies and academies, such as Bologna, St. Petersburg, Berlin, Paris, London, Bern, Turin, Zurich, and Mannheim. His importance to fluid dynamics is centered on his book Hydrodynamica (1738). (With this book, Daniel introduced the term “hydrodynamics” to the literature.) In this book, he ranged over such topics as jet propulsion, manometers, and flow in pipes. Of most importance, he attempted to obtain a relationship between pressure and velocity. Unfortunately, his derivation was somewhat obscure, and Bernoulli’s equation, ascribed by history to Daniel via his Hydrodynamica, is not to be found in this book, at least not in the form we see it today [such as Equations (3.14) and (3.15)]. The propriety of Equations (3.14) and (3.15) is further complicated by his father, Johann, who also published a book in 1743 entitled Hydraulica. It is clear from this latter book that the father understood Bernoulli’s theorem better than his son; Daniel thought of pressure strictly in terms of the height of a manometer column, whereas Johann had the more fundamental understanding that pressure was a force acting on the fluid. (It is interesting to note the Johann Bernoulli was a person of some sensitivity and irritability, with an overpowering drive for recognition. He tried to undercut the impact of Daniel’s Hydrodynamica by predating the publication date of Hydraulica to 1728, to make it appear to have been the first of the two. There was little love lost between son and father.)

During Daniel Bernoulli’s most productive years, partial differential equations had not yet been introduced into mathematics and physics; hence, he could not ap­proach the derivation of Bernoulli’s equation in the same fashion as we have in Section 3.2. The introduction of partial differential equations to mathematical physics was due to d’Alembert in 1747. d’Alembert’s role in fluid mechanics is detailed in Section 3.20. Suffice it to say here that his contributions were equally if not more important than Bernoulli’s, and d’Alembert represents the second member of the triumvirate which molded the foundations of theoretical fluid dynamics in the eighteenth century.

The third and probably pivotal member of this triumvirate was Leonhard Euler. He was a giant among the eighteenth-century mathematicians and scientists. As a result of his contributions, his name is associated with numerous equations and tech­niques, for example, the Euler numerical solution of ordinary differential equations, eulerian angles in geometry, and the momentum equations for inviscid fluid flow [see Equation (3.12)].

Leonhard Euler was born on April 15, 1707, in Basel, Switzerland. His father was a Protestant minister who enjoyed mathematics as a pastime. Therefore, Euler grew up in a family atmosphere that encouraged intellectual activity. At the age of 13, Euler entered the University of Basel which at that time had about 100 students and 19 professors. One of those professors was Johann Bernoulli, who tutored Euler in mathematics. Three years later, Euler received his master’s degree in philosophy.

It is interesting that three of the people most responsible for the early develop­ment of theoretical fluid dynamics—Johann and Daniel Bernoulli and Euler—lived in the same town of Basel, were associated with the same university, and were con­temporaries. Indeed, Euler and the Bernoulli’s were close and respected friends—so much that, when Daniel Bernoulli moved to teach and study at the St. Petersburg Academy in 1725, he was able to convince the academy to hire Euler as well. At this

invitation, Euler left Basel for Russia; he never returned to Switzerland, although he remained a Swiss citizen throughout his life.

Euler’s interaction with Daniel Bernoulli in the development of fluid mechanics grew strong during these years at St. Petersburg. It was here that Euler conceived of pressure as a point property that can vary from point to point throughout a fluid and obtained a differential equation relating pressure and velocity, that is, Euler’s equation given by Equation (3.12). In turn, Euler integrated the differential equation to obtain, for the first time in history, Bernoulli’s equation in the form of Equations

(3.14) and (3.15). Hence, we see that Bernoulli’s equation is really a misnomer; credit for it is legitimately shared by Euler.

When Daniel Bernoulli returned to Basel in 1733, Euler succeeded him at St. Petersburg as aprofessor of physics. Euler was a dynamic and prolific man; by 1741 he had prepared 90 papers for publication and written the two-volume book Mechanica. The atmosphere surrounding St. Petersburg was conducive to such achievement. Euler wrote in 1749: “I and all others who had the good fortune to be for some time with the Russian Imperial Academy cannot but acknowledge that we owe everything which we are and possess to the favorable conditions which we had there.”

However, in 1740, political unrest in St. Petersburg caused Euler to leave for the Berlin Society of Sciences, at that time just formed by Frederick the Great. Euler lived in Berlin for the next 25 years, where he transformed the society into a major academy. In Berlin, Euler continued his dynamic mode of working, preparing at least 380 papers for publication. Here, as a competitor with d’Alembert (see Section 3.20), Euler formulated the basis for mathematical physics.

In 1766, after a major disagreement with Frederick the Great over some financial aspects of the academy, Euler moved back to St. Petersburg. This second period of his life in Russia became one of physical suffering. In that same year, he became blind in one eye after a short illness. An operation in 1771 resulted in restoration of his sight, but only for a few days. He did not take proper precautions after the operation, and within a few days, he was completely blind. However, with the help of others, he continued his work. His mind was sharp as ever, and his spirit did not diminish. His literary output even increased—about half of his total papers were written after 1765!

On September 18, 1783, Euler conducted business as usual—giving a mathe­matics lesson, making calculations of the motion of balloons, and discussing with friends the planet of Uranus, which had recently been discovered. At about 5 p. m., he suffered a brain hemorrhage. His only words before losing consciousness were “I am dying.” By 11 p. m., one of the greatest minds in history had ceased to exist.

With the lives of Bernoulli, Euler, and d’Alembert (see Section 3.20) as back­ground, let us now trace the geneology of the basic equations of fluid dynamics. For example, consider the continuity equation in the form of Equation (2.52). Although Newton had postulated the obvious fact that the mass of a specified object was con­stant, this principle was not appropriately applied to fluid mechanics until 1749. In this year, d’Alembert gave a paper in Paris, entitled “Essai d’une nouvelle theorie de la resistance des fluides,” in which he formulated differential equations for the conservation of mass in special applications to plane and axisymmetric flows. Euler

took d’Alembert’s results and, 8 years later, generalized them in a series of three basic papers on fluid mechanics. In these papers, Euler published, for the first time in history, the continuity equation in the form of Equation (2.52) and the momentum equations in the form of Equations (2.113a and c), without the viscous terms. Hence, two of the three basic conservation equations used today in modem fluid dynamics were well established long before the American Revolutionary War—such equations were contemporary with the time of George Washington and Thomas Jefferson!

The origin of the energy equation in the form of Equation (2.96) without viscous terms has its roots in the development of thermodynamics in the nineteenth century. Its precise first use is obscure and is buried somewhere in the rapid development of physical science in the nineteenth century.

The purpose of this section has been to give you some feeling for the historical development of the fundamental equations of fluid dynamics. Maybe we can appre­ciate these equations more when we recognize that they have been with us for quite some time and that they are the product of much thought from some of the greatest minds of the eighteenth century.

Flow Over A Sphere

Consider again the flow induced by the three-dimensional doublet illustrated in Figure 6.3. Superimpose on this flow a uniform velocity field of magnitude in the negative

г direction. Since we are more comfortable visualizing a freestream which moves horizontally, say, from left to right, let us flip the coordinate system in Figure 6.3 on its side. The picture shown in Figure 6.4 results.

Examining Figure 6.4, the spherical coordinates of the freestream are

Vr = —Voo cos в [6.11a]

Vg = Voo sin в [6. lib]

Уф = 0 [6.11«]

Flow Over A Sphere

Adding Vr, Vg, and УФ for the free stream, Equations (6.11 a to c), to the representative components for the doublet given in Equation (6.10), we obtain, for the combined flow,

To find the stagnation points in the flow, set Vr — Vg = 0 in Equations (6.12) and

(6.13) . From Equation (6.13), Vg = 0 gives sin в = 0; hence, the stagnation points

Flow Over A Sphere


Figure 6.4 The superposition of a uniform flow and a three-dimensional doublet.


are located at в = 0 and n. From Equation (6.12), with Vr = 0, we obtain


where r = R is the radial coordinate of the stagnation points. Solving Equation


Подпись: R = Подпись: 2 nV* Подпись: 1/3 Подпись: [6.16]

for R, we obtain

Flow Over A Sphere Подпись: and Подпись: 2 JTV* Flow Over A Sphere

Hence, there are two stagnation points, both on the z axis, with (r, 6) coordinates

Подпись: V' = ‘(V““2^)COSe = = —(Voo - Voo)cos6 = 0 Подпись: JL 2n Подпись: 27rVoo ft Подпись: cos в

Insert the value of r = R from Equation (6.16) into the expression for V, given by Equation (6.12). We obtain

Thus, Vr = 0 when r = R for all values of в and Ф. This is precisely the flow – tangency condition for flow over a sphere of radius R. Hence, the velocity field given by Equations (6.12) to (6.14) is the incompressible flow over a sphere of radius R. This flow is shown in Figure 6.5; it is qualitatively similar to the flow over the cylinder shown in Figure 3.19, but quantitatively the two flows are different.

Flow Over A Sphere Подпись: [6.17]

On the surface of the sphere, where r = R, the tangential velocity is obtained from Equation (6.13) as follows:

From Equation (6.16),

Подпись: [6.18]fi = 27Г Я3

Flow Over A Sphere


Figure 6.5 Schematic of the incompressible flow over a sphere.


Substituting Equation (6.18) into (6.17), we have


Vg = Voo Sin в


Подпись: or[6.19]

The maximum velocity occurs at the top and bottom points of the sphere, and its magnitude is | V~^. Compare these results with the two-dimensional circular cylinder case given by Equation (3.100). For the two-dimensional flow, the maximum velocity is IVoq. Hence, for the same V^, the maximum surface velocity on a sphere is less than that for a cylinder. The flow over a sphere is somewhat “relieved” in comparison with the flow over a cylinder. The flow over a sphere has an extra dimension in which to move out of the way of the solid body; the flow can move sideways as well as up and down. In contrast, the flow over a cylinder is more constrained; it can only move up and down. Hence, the maximum velocity on a sphere is less than that on a cylinder. This is an example of the three-dimensional relieving effect, which is a general phenomenon for all types of three-dimensional flows.

Подпись: Cp = 1 — I sin2 в

The pressure distribution on the surface of the sphere is given by Equations (3.38) and (6.19) as follows:


Compare Equation (6.20) with the analogous result for a circular cylinder given by Equation (3.101). Note that the absolute magnitude of the pressure coefficient on a sphere is less than that for a cylinder—again, an example of the three-dimensional relieving effect. The pressure distributions over a sphere and a cylinder are compared in Figure 6.6, which dramatically illustrates the three-dimensional relieving effect.

Applied Aerodynamics: The Aerodynamic Coefficients—Their Magnitudes and Variations

With the present section, we begin a series of special sections in this book under the general heading of “applied aerodynamics.” The main thrust of this book is to present the fundamentals of aerodynamics, as is reflected in the book’s title. How­ever, applications of these fundamentals are liberally sprinkled throughout the book, in the text material, in the worked examples, and in the homework problems. The term applied aerodynamics normally implies the application of aerodynamics to the practical evaluation of the aerodynamic characteristics of real configurations such as airplanes, missiles, and space vehicles moving through an atmosphere (the earth’s, or that of another planet). Therefore, to enhance the reader’s appreciation of such applications, sections on applied aerodynamics will appear near the end of many of the chapters. To be specific, in this section, we address the matter of the aerody­namic coefficients defined in Section 1.5; in particular, we focus on lift, drag, and moment coefficients. These nondimensional coefficients are the primary language of applications in external aerodynamics (the distinction between external and internal aerodynamics was made in Section 1.2). It is important for you to obtain a feeling for
typical values of the aerodynamic coefficients. (For example, do you expect a drag coefficient to be as low as 10 5, or maybe as high as 1000—does this make sense?) The purpose of this section is to begin to provide you with such a feeling, at least for some common aerodynamic body shapes. As you progress through the remainder of this book, make every effort to note the typical magnitudes of the aerodynamic coefficients that are discussed in various sections. Having a realistic feel for these magnitudes is part of your technical maturity.

Question: What are some typical drag coefficients for various aerodynamic con­figurations? Some basic values are shown in Figure 1.39. The dimensional analysis described in Section 1.7 proved that Co — /(M, Re). In Figure 1.39, the drag – coefficient values are for low speeds, essentially incompressible flow; therefore, the Mach number does not come into the picture. (For all practical purposes, for an incompressible flow, the Mach number is theoretically zero, not because the velocity goes to zero, but rather because the speed of sound is infinitely large. This will be made clear in Section 8.3.) Thus, for a low-speed flow, the aerodynamic coefficients

Flat plate

(Broadside) length = d Cq = 2.0


Cylinder diameter – d CD = 1.2




thickness – d CD = 0.12


Cylinder j

diameter = —d Q> = 1.2




diameter = d CD = 0.6


Figure 1.39 Drag coefficients for various aerodynamic shapes. (Source:

Talay, T. A., Introduction to the Aerodynamics of Flight, NASA SP-367, 1975.)



for a fixed shape at a fixed orientation to the flow are functions of just the Reynolds number. In Figure 1.39, the Reynolds numbers are listed at the left and the drag – coefficient values at the right. In Figure 1.39a, a flat plate is oriented perpendicular to the flow; this configuration produces the largest possible drag coefficient of any con­ventional configuration, namely, Co = D’/q^S = 2.0, where. S’ is the frontal area per unit span, i. e., S = (d)(1), where d is the height of the plate. The Reynolds number is based on the height d i. e., Re = рж V^d//i^ = 105. Figure 1.39b illustrates flow over a circular cylinder of diameter d; here, Co = 1.2, considerably smaller than the vertical plate value in Figure 1.39a. The drag coefficient can be reduced dramatically by streamlining the body, as shown in Figure 1.39c. Flere, Co = 0.12; this is an order of magnitude smaller than the circular cylinder in Figure 1.39£>. The Reynolds numbers for Figure 1.39a, b, and c are all the same value, based on d (diameter). The drag coefficients are all defined the same, based on a reference area per unit span of (d)(1). Note that the flow fields over the configurations in Figure 1.39a, b, and c show a wake downstream of the body; the wake is caused by the flow separating from the body surface, with a low-energy, recirculating flow inside the wake. The phenomenon of flow separation will be discussed in detail in Part 4 of this book, dealing with viscous flows. Flowever, it is clear that the wakes diminish in size as we progressively go from Figure 1.39a, b, and c. The fact that CD also diminishes progressively from Figure 1.39a, b, and c is no accident—it is a direct result of the regions of separated flow becoming progressively smaller. Why is this so? Simply consider this as one of the many interesting questions in aerodynamics—a question that will be answered in due time in this book. Note, in particular that the physical effect of the streamlining in Figure 1.39c results in a very small wake, hence a small value for the drag coefficient.

Consider Figure 1.39d, where once again a circular cylinder is shown, but of much smaller diameter. Since the diameter here is 0. Id, the Reynolds number is now 104 (based on the same freestream Too, px, and /i^ as Figure 1.39a, b, and c). It will be shown in Chapter 3 that Co for a circular cylinder is relatively independent of Re between Re = 104 and 105. Since the body shape is the same between Figure 1.39c/ and b, namely, a circular cylinder, then Co is the same value of 1.2 as shown in the figure. Flowever, since the drag is given by D’ = qxSCo, and S is one-tenth smaller in Figure 1.39 d, then the drag force on the small cylinder in Figure 139 d is one-tenth smaller than that in Figure 1.39b.

Another comparison is illustrated in Figure 1.39c and d. Here we are comparing a large streamlined body of thickness d with a small circular cylinder of diameter

O. ld. For the large streamlined body in Figure 1.39c,

D’ = qooSCo = 0.12 q^d

For the small circular cylinder in Figure 1.39d,

D’ = qooSCD — qoo(0.d)(.2) — O. Hq^d

The drag values are the same! Thus, Figure 1.3c and d illustrate that the drag on a circular cylinder is the same as that on the streamlined body which is ten times thicker—another way of stating the aerodynamic value of streamlining.

As a final note in regard to Figure 1.39, the flow over a circular cylinder is again shown in Figure 1.39c. However, now the Reynolds number has been increased to 107, and the cylinder drag coefficient has decreased to 0.6—a dramatic factor of two less than in Figure 139b and d. Why has Co decreased so precipitously at the higher Reynolds number? The answer must somehow be connected with the smaller wake behind the cylinder in Figure 1.39e compared to Figure 1.39b. What is going on here? This is one of the fascinating questions we will answer as we progress through our discussions of aerodynamics in this book—an answer that will begin with Section 3.18 and culminate in Part 4 dealing with viscous flow.

At this stage, pause for a moment and note the values of Co for the aerodynamic shapes in Figure 1.39. With Co based on the frontal projected area (S = <i(l) per unit span), the values of Cd range from a maximum of 2 to numbers as low as 0.12. These are typical values of Co for aerodynamic bodies.

Applied Aerodynamics: The Aerodynamic Coefficients—Their Magnitudes and Variations

Also, note the values of Reynolds number given in Figure 1.39. Consider a circular cylinder of diameter 1 m in a flow at standard sea level conditions = 1.23 kg/m3 and р. ж = 1.789 x 10-5 kg/m • s) with a velocity of 45 m/s (close to 100 mi/h). For this case,

Note that the Reynolds number is over 3 million; values of Re in the millions are typical of practical applications in aerodynamics. Therefore, the large numbers given for Re in Figure 1.39 are appropriate.

Applied Aerodynamics: The Aerodynamic Coefficients—Their Magnitudes and Variations Подпись: [1.59]

Let us examine more closely the nature of the drag exerted on the various bodies in Figure 1.39. Since these bodies are at zero angle of attack, the drag is equal to the axial force. Hence, from Equation (1.8) the drag per unit span can be written as

рТЕ a ТЕ

I rM cos 9dsu– I ti cos 0 ds/


skin friction drag

That is, the drag on any aerodynamic body is composed of pressure drag and skin friction drag; this is totally consistent with our discussion in Section 1.5, where it is emphasized that the only two basic sources of aerodynamic force on a body are the pressure and shear stress distributions exerted on the body surface. The division of total drag onto its components of pressure and skin friction drag is frequently useful in analyzing aerodynamic phenomena. For example, Figure 1.40 illustrates the comparison of skin friction drag and pressure drag for the cases shown in Figure 1.39. In Figure 1.40, the bar charts at the right of the figure give the relative drag force on each body; the cross-hatched region denotes the amount of skin friction drag, and the blank region is the amount of pressure drag. The freestream density and viscosity are the same for Figure 1.40a to e; however, the freestream velocity Vx is varied by

Applied Aerodynamics: The Aerodynamic Coefficients—Their Magnitudes and Variations



Pressure drag


Figure 1.40 The relative comparison between skin friction drag and pressure drag for various aerodynamic shapes. (Source: Talay, T. A., Introduction to the Aerodynamics of Flight, NASA SP-367, 1975.)



the necessary amount to achieve the Reynolds numbers shown. That is, comparing Figure 1.40b and e, the value of is much larger for Figure 1.40c. Since the drag force is given by

D’ = PoovlscD

then the drag for Figure 1,40c is much larger than for Figure 1.40b. Also shown in the bar chart is the equal drag between the streamlined body of thickness d and

the circular cylinder of diameter 0. Id—a comparison discussed earlier in conjunction with Figure 1.39. Of most importance in Figure 1.40, however, is the relative amounts of skin friction and pressure drag for each body. Note that the drag of the vertical flat plate and the circular cylinders is dominated by pressure drag, whereas, in contrast, most of the drag of the streamlined body is due to skin friction. Indeed, this type of comparison leads to the definition of two generic body shapes in aerodynamics, as follows:

Blunt body = a body where most of the drag is pressure drag Streamlined body = a body where most of the drag is skin friction drag

In Figures 1.39 and 1.40, the vertical flat plate and the circular cylinder are clearly blunt bodies.

The large pressure drag of blunt bodies is due to the massive regions of flow sep­aration which can be seen in Figures 1.39 and 1.40. The reason why flow separation causes drag will become clear as we progress through our subsequent discussions. Hence, the pressure drag shown in Figure 1.40 is more precisely denoted as “pres­sure drag due to flow separation”; this drag is frequently called form drag. (For an elementary discussion of form drag and its physical nature, see Reference 2.)

Let us now examine the drag on a flat plate at zero angle of attack, as sketched in Figure 1.41. Here, the drag is completely due to shear stress; there is no pressure force in the drag direction. The skin friction drag coefficient is defined as

D’ D’

Cf =——— =————-

4oo$ 9ooC(l)

where the reference area is the planform area per unit span, i. e., the surface area as seen by looking down on the plate from above. C/ will be discussed further in Chapter 16. However, the purpose of Figure 1.41 is to demonstrate that:


Figure 1.41 Variation of laminar and turbulent skin friction coefficient for a flat plate as a function of Reynolds number based on the chord length of the plate. The intermediate dashed curves are associated with various transition paths from laminar flow to turbulent flow.

1. Cf is a strong function of Re, where Re is based on the chord length of the plate, Re = PooVooc/^oo – Note that C/ decreases as Re increases.

2. The value of Cf depends on whether the flow over the plate surface is laminar or turbulent, with the turbulent C/ being higher than the laminar Cf at the same Re. What is going on here? What is laminar flow? What is turbulent flow? Why does it affect Cfl The answers to these questions will be addressed in Chapters 15, 17, and 18.

3. The magnitudes of Cf range typically from 0.001 to 0.01 over a large range of Re. These numbers are considerably smaller than the drag coefficients listed in Figure 1.39. This is mainly due to the different reference areas used. In Figure 1.39, the reference area is a cross-sectional area normal to the flow; in Figure 1.41, the reference area is the planform area.

A flat plate is not a very practical aerodynamic body—it simply has no volume. Let us now consider a body with thickness, namely, an airfoil section. An NACA 63-210 airfoil section is one such example. The variation of the drag coefficient, cj, with angie of attack is shown in Figure і.42. Here, as usual, cj is defined as


Cd – ———-


where D’ is the drag per unit span. Note that the lowest value of cd is about 0.0045. The NACA 63-210 airfoil is classified as a “iaminar-flow airfoil” because it is designed to promote such a flow at small a. This is the reason for the “bucketlike” appearance of the Cd curve at low a; at higher a, transition to turbulent flow occurs over the airfoil

0.02 – 0.016 –

image73cd 0.012 – 0.008 – 0.004 –

-1.2 -0.8 -0.4 0 0.4 0.8 1.2


______ I______ I I I I

-8 -4 0 4 8

a, degrees

Figure 1.42 Variation of section drag coefficient for an NACA 63-210 airfoil. Re = 3 x 106.

surface, causing a sharp increase in cd. Hence, the value of cd = 0.0045 occurs in a laminar flow. Note that the Reynolds number is 3 million. Once again, a reminder is given that the various aspects of laminar and turbulent flows will be discussed in Part

4. The main point here is to demonstrate that typical airfoil drag-coefficient values are on the order of 0.004 to 0.006. As in the case of the streamlined body in Figures 1.39 and 1.40, most of this drag is due to skin friction. However, at higher values of a, flow separation over the top surface of the airfoil begins to appear and pressure drag due to flow separation (form drag) begins to increase. This is why cd increases with increasing a in Figure 1.42.

Let us now consider a complete airplane. In Chapter 3, Figure 3.2 is a photograph of the Seversky P-35, a typical fighter aircraft of the late 1930s. Figure 1.43 is a detailed drag breakdown for this type of aircraft. Configuration 1 in Figure 1.43 is the stripped-down, aerodynamically cleanest version of this aircraft; its drag coefficient (measured at an angle of attack corresponding to a lift coefficient of CL = 0.15) is







(Cl= 0.15)



Completely faired condition, long nose fairing



Completely faired condition, blunt nose fairing



Original cowling added, no airflow through cowling





Landing-gear seals and fairing removed





Oil cooler installed





Canopy fairing removed





Carburetor air scoop added





Sanded walkway added





Ejector chute added





Exhaust stacks added





Intercooler added





Cowling exit opened





Accessory exit opened





Cowling fairing and seals removed





Cockpit ventilator opened





Cowling venturi installed





Blast tubes added





Antenna installed






“Percentages based on completely faired condition with long nose fairing.

Подпись: Airplane condition

Figure 1.43 The breakdown of various sources of drag on a late 1930s airplane, the Seversky XP-41 (derived from the Seversky P-35 shown in Figure 3.2). [Source: Experimental data from Coe, Paul J., "Review of Drag Cleanup Tests in Langley Full-Scale Tunnel (From 1935 to 1 945) Applicable to Current General Aviation Airplanes," NASA TN-D-8206, 1976.]

CD — 0.0166. Here, CD is defined as


where D is the airplane drag and S is the planform area of the wing. For configurations 2 through 18, various changes are progressively made in order to bring the aircraft to its conventional, operational configuration. The incremental drag increases due to each one of these additions are tabulated in Figure 1.43. Note that the drag coefficient is increased by more than 65 percent by these additions; the value of CD for the aircraft in full operational condition is 0.0275. This is a typical airplane drag-coefficient value. The data shown in Figure 1.43 were obtained in the full-scale wind tunnel at the NACA Langley Memorial Laboratory just prior to World War II. (The full-scale wind tunnel has test-section dimensions of 30 by 60 ft, which can accommodate a whole airplane—hence the name “full-scale.”)

The values of drag coefficients discussed so far in this section have applied to low-speed flows. In some cases, their variation with the Reynolds number has been illustrated. Recall from the discussion of dimensional analysis in Section 1.7 that drag coefficient also varies with the Mach number. Question: What is the effect of increasing the Mach number on the drag coefficient of an airplane? Consider the answer to this question for a Northrop T-38A jet trainer, shown in Figure 1.44. The drag coefficient for this airplane is given in Figure 1.45 as a function of the Mach number ranging from low subsonic to supersonic. The aircraft is at a small negative angle of attack such that the lift is zero, hence the CD in Figure 1.45 is called the zero – lift drag coefficient. Note that the value of Сд is relatively constant from M — 0.1 to about 0.86. Why? At Mach numbers of about 0.86, the Сд rapidly increases. This large increase in Сд near Mach one is typical of all flight vehicles. Why? Stay tuned; the answers to these questions will become clear in Part 3 dealing with compressible flow. Also, note in Figure 1.45 that at low subsonic speeds, Сд is about 0.015. This is considerably lower than the 1930s-type airplane illustrated in Figure 1.43; of course, the T-38 is a more modem, sleek, streamlined airplane, and its drag coefficient should be smaller.

We now turn our attention to lift coefficient and examine some typical values. As a complement to the drag data shown in Figure 1.42 for an NACA 63-210 airfoil, the variation of lift coefficient versus angle of attack for the same airfoil is shown in Figure 1.46. Here, we see c; increasing linearly with a until a maximum value is obtained near a = 14°, beyond which there is a precipitous drop in lift. Why does ci vary with a in such a fashion—in particular, what causes the sudden drop in ci beyond a = 14°? An answer to this question will evolve over the ensuing chapters. For our purpose in the present section, observe the values of q; they vary from about —1.0 to a maximum of 1.5, covering a range of a from —12 to 14°. Conclusion: For an airfoil, the magnitude of с/ is about a factor of 100 larger than q. A particularly important figure of merit in aerodynamics is the ratio of lift to drag, the so-called L/D ratio; many aspects of the flight performance of a vehicle are directly related to the L/D ratio (see, e. g., Reference 2). Other things being equal, a higher L/D means better flight performance. For an airfoil—a configuration whose primary


Figure 1.44 Three-view of the Northrop T-38 jet trainer. (Courtesy of the U. S. Air Force.]

function is to produce lift with as little drag as possible—values of L/D are large. For example, from Figures 1.42 and 1.46, at a = 4°, q = 0.6 and q = 0.0046, yielding L/D = q0646 = 130. This value is much larger than those for a complete airplane, as we will soon see.

To illustrate the lift coefficient for a complete airplane, Figure 1.47 shows the variation of Cl with a for the T-38 in Figure 1.44. Three curves are shown, each for a different flap deflection angle. (Flaps are sections of the wing at the trailing edge which, when deflected downward, increase the lift of the wing. See Section 5.17 of Reference 2 for a discussion of the aerodynamic properties of flaps.) Note that at a given a, the deflection of the flaps increases CL. The values of Cl shown in Figure 1.47 are about the same as that for an airfoil—on the order of 1. On the other hand, the maximum L/D ratio of the T-38 is about 10—considerably smaller than that for an airfoil alone. Of course, an airplane has a fuselage, engine nacelles, etc., which are elements with other functions than just producing lift, and indeed produce only small amounts of lift while at the same time adding a lot of drag to the vehicle. Thus, the L/D ratio for an airplane is expected to be much less than that for an airfoil alone.


Figure 1.45 Zero-lift drag coefficient variation with Mach number for the T-38. (Courtesy of the U. S. Air Force.)


Figure 1.46 Variation of section lift coefficient for an NACA 63-210 airfoil.

Re = 3 x 106. No flap deflection.


Figure 1.47 Variation of lift coefficient with angle of attack for the T-38. Three curves are shown corresponding to three different flap deflections. Freestream Mach number is 0.4. (Courtesy of the U. S. Air Force.)

Moreover, the wing on an airplane experiences a much higher pressure drag than an airfoil due to the adverse aerodynamic effects of the wing tips (a topic for Chapter 5). This additional pressure drag is called induced drag, and for short, stubby wings, such as on the T-38, the induced drag can be large. (We must wait until Chapter 5 to find out about the nature of induced drag.) As a result, the L/D ratio of the T-38 is fairly small as most airplanes go. For example, the maximum L/D ratio for the Boeing B-52 strategic bomber is 21.5 (see Reference 48). However, this value is still considerably smaller than that for an airfoil alone.

Finally, we turn our attention to the values of moment coefficients. Figure 1.48 illustrates the variation of cm<e/4 for the NACA 63-210 airfoil. Note that this is a neg­ative quantity; all conventional airfoils produce negative, or “pitch-down,” moments. (Recall the sign convention for moments given in Section 1.5.) Also, notice that its value is on the order of —0.035. This value is typical of a moment coefficient—on the order of hundredths.

Variation of section moment coefficient about the quarter chord for an NACA 63-210 airfoil. Re = 3 x 106.


Figure 1.48



With this, we end our discussion of typical values of the aerodynamic coefficients defined in Section 1.5. At this stage, you should reread this section, now from the overall perspective provided by a first reading, and make certain to fix in your mind the typical values discussed—it will provide a useful “calibration” for our subsequent discussions.

Numerical Solutions—Computational Fluid Dynamics (CFD)

The other general approach to the solution of the governing equations is numerical. The advent of the modem high-speed digital computer in the last third of the twentieth century has revolutionized the solution of aerodynamic problems and has given rise to a whole new discipline—computational fluid dynamics. Recall that the basic governing equations of continuity, momentum, and energy derived in this chapter are either in integral or partial differential form. In Anderson, Computational Fluid Dynamics: The Basics With Applications, McGraw-Hill, 1995, computational fluid dynamics is defined as “the art of replacing the integrals or the partial derivatives (as the case may be) in these equations with discretized algebraic forms, which in turn are solved to obtain numbers for the flow field values at discrete points in time and/or space.” The end product of computational fluid dynamics (frequently identified by the acronym CFD) is indeed a collection of numbers, in contrast to a closed-form analytical solution. However, in the long run, the objective of most engineering analyses, closed form or otherwise, is a quantitative description of the problem, i. e., numbers.

The beauty of CFD is that it can deal with the full non-linear equations of con­tinuity, momentum, and energy, in principle, without resorting to any geometrical or physical approximations. Because of this, many complex aerodynamic flow fields have been solved by means of CFD which had never been solved before. An example of this is shown in Figure 2.40. Here we see the unsteady, viscous, turbulent, com­pressible, separated flow field over an airfoil at high angle of attack (14° in the case


Figure 3.40 Calculated streamline pattern for separated flow over an airfoil. Re = 300,000, = 0.5, angle of attack =14°.

shown), as obtained from Reference 56. The freestream Mach number is 0.5, and the Reynolds number based on the airfoil chord length (distance from the front to the back edges) is 300,000. An instantaneous streamline pattern that exists at a certain instant in time is shown, reflecting the complex nature of the separated, recirculating flow above the airfoil. This flow is obtained by means of a CFD solution of the two-dimensional, unsteady continuity, momentum, and energy equations, including the full effects of viscosity and thermal conduction, as developed in Sections 2.4, 2.5, and 2.7, without any further geometrical or physical simplifications. The equations with all the viscosity and thermal conduction terms explicitly shown are developed in Chapter 15; in this form, they are frequently labeled as the Navier-Stokes equations. There is no analytical solution for the flow shown in Figure 2.40; the solution can only be obtained by means of CFD.

Let us explore the basic philosophy behind CFD. Again, keep in mind that CFD solutions are completely numerical solutions, and a high-speed digital computer must be used to carry them out. In a CFD solution, the flow field is divided into a number of discrete points. Coordinate lines through these points generate a grid, and the discrete points are called grid points. The grid used to solve the flow field shown in Figure 2.40 is given in Figure 2.41; here, the grid is wrapped around the airfoil, which is seen as the small white speck in the center-left of the figure, and the grid extends a very large distance out from the airfoil. This large extension of the grid into the main stream of the flow is necessary for a subsonic flow, because disturbances in a subsonic flow physically feed out large distances away from the body. We will learn why in subsequent chapters. The black region near the airfoil is simply the computer graphics way of showing a very large number of closely spaced grid points near the airfoil, for better definition of the viscous flow near the airfoil. The flow held properties, such as p, p, u, v, etc. are calculated just at the discrete grid points, and nowhere


else, by means of the numerical solution of the governing flow equations. This is an inherent property that distinguishes CFD solutions from closed-form analytical solutions. Analytical solutions, by their very nature, yield closed-form equations that describe the flow as a function of continuous time and/or space. So we can pick any one of the infinite number of points located in this continuous space, feed the coordinates into the closed-form equations, and obtain the flow variables at that point. Not so with CFD, where the flow-field variables are calculated only at discrete grid points. For a CFD solution, the partial derivatives or the integrals, as the case may be, in the governing flow equations are discretized, using the flow-field variables at grid points only.

How is this discretization carried out? There are many answers to this equation. We will look at just a few examples, to convey the ideas.

Let us consider a partial derivative, such as du/dx. How do we discretize this partial derivative? First, we choose a uniform rectangular array of grid points as shown in Figure 2.42. The points are identified by the index і in the x direction, and the index j in the у direction. Point P in Figure 2.42 is identified as point (i, j). The value of the variable и at point j is denoted by и, j. The value of и at the point immediately to the right of P is denoted by u,41,; and that immediately to the left of P by Ui-ij. The values of и at the points immediately above and below point P are denoted by Uij+i and respectively. And so forth. The grid points in the x

direction are separated by the increment Ax, and the у direction by the increment Ay. The increments Ax and Ay must be uniform between the grid points, but Ax can be

Подпись: v

Figure 2.42 An array of grid points in a uniform, rectangular grid.

Numerical Solutions—Computational Fluid Dynamics (CFD) Подпись: З 3M ЗІ3 Numerical Solutions—Computational Fluid Dynamics (CFD) Подпись: [2.166]

a different value than Ay. To obtain a discretized expression for 3u/dx evaluated at point P, we first write a Taylors series expansion for m,_ L ; expanded about point P as:

Numerical Solutions—Computational Fluid Dynamics (CFD) Подпись: [2.167]

Solving Equation (2.166) (Эм/Эх),-.,-, we have

Equation (2.167) is still a mathematically exact relationship. However, if we choose to represent (Эм/Эх); j just by the algebraic term on the right-hand side, namely,

image185M j _|_ ] j LI j j

—– 1——- — Forward difference [2.1 68]


then Equation (2.168) represents an approximation for the partial derivative, where the error introduced by this approximation is the truncation error, identified in Equation

(2.167) . Nevertheless, Equation (2.168) gives us an algebraic expression for the partial derivative; the partial derivative has been discretized because it is formed by the values m/+i j and n,. / at discrete grid points. The algebraic difference quotient in Equation (2.168) is called a forward difference, because it uses information ahead of joint (/, j), namely ui+[ j. Also, the forward difference given by Equation (2.168) has first-order accuracy because the leading term of the truncation error in Equation

(2.167) has Ax to the first power.

Equation (2.168) is not the only discretized form for (du/dx)ij. For example, let us write a Taylors series expansion for Uj-j expanded about point P as

I (du , л л I (д2ц (-~Ax’>2 , (93“ (~A*)3

Подпись: dX3Подпись:= Щ.1 + (j-x) (-^) + (ailj – y-

Подпись: Solving Equation (2.169) for (du/dx)ij, we have 2"x Ax (3 3u
Подпись: 3 и  dx)i,j Подпись: *i,j ui-,j Ax Подпись: 3 2u dx2); .-T Numerical Solutions—Computational Fluid Dynamics (CFD) Подпись: [2.170]


Подпись: Truncation errorRearward difference

Hence, we can represent the partial derivative by the rearward difference shown in Equation (2.170), namely,


Эи и; ; — и,_і і _ _

— ) = — ———— — Rearward difference [2.171]

Э xJij Ax

Equation (2.171) is an approximation for the partial derivative, where the error is given by the truncation error labeled in Equation (2.170). The rearward difference given by Equation (2.171) has first-order accuracy because the leading term in the truncation error in Equation (2.170) has Ax to the first power. The forward and rearward differences given by Equations (2.168) and (2.171), respectively, are equally valid representations of (du/dx)jj, each with first-order accuracy.

Подпись: R/+i,y Подпись: и i—j — 2 Подпись: Ax -- Подпись: Э3и (Ax)3 ^x3/ ij 3 Подпись: [2.1 72]

In most CFD solutions, first-order accuracy is not good enough; we need a discretization of (du/dx)jj that has at least second-order accuracy. This can be obtained by subtracting Equation (2.169) from Equation (2.166), yielding

Solving Equation (2.172) for (du/dx)ij, we have

Подпись: / Э3и (Ax)2 9*3/;j 3 + ' V Truncation error Подпись:/Эи Ui—j

3x / ■ . 2 Ax

4 7 C] ————- — —— ■

Central difference

Подпись: 3M  Uf +  j Ui—lj dx ) t j 2 Ax Подпись: Central difference Подпись: [2.174]

Hence, we can represent the partial derivative by the central difference shown in Equation (2.173), namely

Examining Equation (2.173), we see that the central difference expression given in Equation (2.174) has second-order accuracy, because the leading term in the trun­cation error in Equation (2.173) has (Ax)2. For most CFD solutions, second-order accuracy is sufficient.

So this is how it works—this is how the partial derivatives that appear in the gov­erning flow equations can be discretized. There are many other possible discretized

Numerical Solutions—Computational Fluid Dynamics (CFD)

forms for the derivatives; the forward, rearward, and central differences obtained above are just a few. Note that Taylor series have been used to obtain these discrete forms. Such Taylors series expressions are the basic foundation of finite-difference solutions in CFD. In contrast, if the integral form of the governing flow equations are used, such as Equations (2.48) and (2.64), the individual integrals terms can be discretized, leading again to algebraic equations that are the basic foundation of finite-volume solutions in CFD.


Example 2.7




This example is a simple illustration of how a CFD solution to a given flow can be set up, in this case for an unsteady, one-dimensional flow. Note that the unknown velocity and internal energy at grid point і at time t + At can be calculated in the same manner, writing the appropriate difference equation representations for the x component of the momentum equation, Equation (2.113a), and the energy equation, Equation (2.114).

The above example looks very straightforward, and indeed it is. It is given here only as an illustration of what is meant by a CFD technique. However, do not be misled. Computational fluid dynamics is a sophisticated and complex discipline. For example, we have said nothing here about the accuracy of the final solutions, whether or not a certain computational technique will be stable (some attempts at obtaining numerical solutions will go unstable—blow up—during the course of the calculations), and how much computer time a given technique will require to obtain the flow-field solution. Also, in our discussion we have given examples of some relatively simple grids. The generation of an appropriate grid for a given flow problem is frequently a challenge, and grid generation has emerged as a subdiscipline in its own right within CFD. For these reasons, CFD is usually taught only in graduate-level courses in most universities. However, in an effort to introduce some of the basic ideas of CFD at the undergraduate level, I have written a book, Reference 64, intended to present the subject at the most elementary level. Reference 64 is intended to be read before students go on to the more advanced books on CFD written at the graduate level. In the present book, we will, from time-to-time, discuss some applications of

CFD as part of the overall fundamentals of aerodynamics. However, this book is not about CFD; Reference 64 is.