Category Fundamentals of Aerodynamics

Preface to the Third Edition

The purpose of this third edition is the same as the first two—to be read, understood, and enjoyed. Due to the extremely favorable comments from readers and users of the first two editions, virtually all of the earlier editions have been carried over intact to the third edition. Therefore, all the basic philosophy, approach, and content discussed and itemized by the author in the Preface to the First Edition is equally applicable now. Since that preface was repeated earlier, no further elaboration will be given here.

Question: What distinguishes the third edition from the first two? Answer: Much new material has been added in order to enhance and expand that covered in the earlier editions. In particular, the third edition has:

1. A series of Design Boxes scattered throughout the book. These design boxes are special sections for the purpose of discussing design aspects associated with the fundamental material covered throughout the book. These sections are literally placed in boxes to set them apart from the mainline text. Modern engineering education is placing more emphasis on design, and the design boxes in this book are in this spirit. They are a means of making the fundamental material more relevant, and making the whole process of learning aerodynamics more fun.

2. Additional sections highlighting the role of computational fluid dynamics (CFD). In the practice of modern aerodynamics, CFD has become a new “third dimen­sion” existing side-by-side with the previous classic dimensions of pure theory and pure experiment. In recognition of the growing significance of CFD, new material has been added to give the reader a broader image of aerodynamics in today’s world.

3. More material on viscous flow. Part 4 on viscous flow has been somewhat rear­ranged and expanded, and now contains two additional chapters in comparison to the previous editions. The new material includes aspects of stagnation point aerodynamic heating, engineering methods of calculation such as the reference temperature method, turbulence modeling, and expanded coverage of modern CFD Navier-Stokes solutions. However, every effort has been made to keep this material within reasonable bounds both in respect to its space and the level of its presentation.

4. Additional historical content scattered throughout the book. It is the author’s opinion that knowledge of the history of aerodynamics plays an important role in the overall education and practice of modern aerodynamics. This additional historical content simply complements the historical material already contained in the previous editions.

5. Many additional worked examples. When learning new technical material, es­pecially material of a fundamental nature as emphasized in this book, one can

never have too many examples of how the fundamentals can be applied to the solution of problems.

6. A large number of new, additional figures and illustrations. The additional ma­terial just itemized is heavily supported with visual figures. I vigorously believe that “a picture is worth a thousand words.”

7. New homework problems added to those carried over from the second edition.

Much of the new material in this third edition is motivated by my experiences over the decade that has elapsed since the second edition. In particular, the design boxes follow the objectives and philosophy that dominate my new book Aircraft Per­formance and Design, McGraw-Hill, Boston, 1999. Moreover, the feature of design boxes has recently been introduced in my new edition of Introduction to Flight, 4th ed., McGraw-Hill, Boston, 2000, and has already met with success. The importance of CFD reflected in this third edition is part of my efforts to introduce aspects of CFD into undergraduate education; my book Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill, New York, 1995, is intended to provide a window into the subject of CFD at a level suitable for advanced undergraduates. Finally, the new historical notes contained here are a product of my research and maturity gained while writing A History of Aerodynamics, and Its Impact on Flying Machines, Cam­bridge University Press, New York, 1997 (hardback), 1998 (paperback). I would like to think this third edition of Fundamentals of Aerodynamics has benefited from the above experience.

All the new additions not withstanding, the main thrust of this book remains the presentation of the fundamentals of aerodynamics; the new material is simply intended to enhance and support this thrust. The book is organized along classical lines, dealing with inviscid incompressible flow, inviscid compressible flow, and viscous flow in sequence. My experience in teaching this material to undergraduates finds that it nicely divides into a two-semester course, with Parts 1 and 2 in the first semester, and Parts 3 and 4 in the second semester. Also, for the past eight years I have taught the entire book in a fast-paced, first-semester graduate course intended to introduce the fundamentals of aerodynamics to new graduate students who have not had this material as part of their undergraduate education. The book works well in such a mode.

I would like to thank the McGraw-Hill editorial staff for their excellent help in producing this book, especially Jonathan Plant and Kristen Druffner in Boston, and Kay Brimeyer in Dubuque. Also, special thanks go to my long-time friend and associate, Sue Cunningham, whose expertise as a scientific typist is beyond comparison, and who has typed all my book manuscripts for me, including this one, with great care and precision.

As a final comment, aerodynamics is a subject of intellectual beauty, composed and drawn by many great minds over the centuries. Fundamentals of Aerodynamics is intended to portray and convey this beauty. Do you feel challenged and interested by these thoughts? If so, then read on, and enjoy!

John D. Anderson, Jr.

Energy Equation

For an incompressible flow, where p is constant, the primary flow-field variables are p and V. The continuity and momentum equations obtained earlier are two equations

in terms of the two unknowns p and V. Hence, for a study of incompressible flow, the continuity and momentum equations are sufficient tools to do the job.

However, for a compressible flow, p is an additional variable, and therefore we need an additional fundamental equation to complete the system. This fundamental relation is the energy equation, to be derived in this section. In the process, two additional flow-field variables arise, namely, the internal energy e and temperature T. Additional equations must also be introduced for these variables, as will be mentioned later in this section.

The material discussed in this section is germane to the study of compressible flow. For those readers interested only in the study of incompressible flow for the time being, you may bypass this section and return to it at a later stage.

Nonlifting Flows over Arbitrary Bodies: The Numerical Source Panel Method

In this section, we return to the consideration of nonlifting flows. Recall that we have already dealt with the nonlifting flows over a semi-infinite body and a Rankine oval and both the nonlifting and the lifting flows over a circular cylinder. For those cases, we added our elementary flows in certain ways and discovered that the dividing streamlines turned out to fit the shapes of such special bodies. However, this indirect method of starting with a given combination of elementary flows and seeing what body shape comes out of it can hardly be used in a practical sense for bodies of arbitrary shape. For example, consider the airfoil in Figure 3.37. Do we know in advance the correct combination of elementary flows to synthesize the flow over this specified body? The answer is no. Rather, what we want is a direct method; that is, let us specify the shape of an arbitrary body and solve for the distribution of singularities which, in combination with a uniform stream, produce the flow over the given body. The purpose of this section is to present such a direct method, limited for the present to nonlifting flows. We consider a numerical method appropriate for solution on a high-speed digital computer. The technique is called the source panel method, which, since the late 1960s, has become a standard aerodynamic tool in industry and most research laboratories. In fact, the numerical solution of potential flows by both source and vortex panel techniques has revolutionized the analysis of low-speed flows. We return to various numerical panel techniques in Chapters 4 through 6. As a modem student of aerodynamics, it is necessary for you to become familiar with the fundamentals of such panel methods. The purpose of the present section is to introduce the basic ideas of the source panel method, which is a technique for the numerical solution of nonlifting flows over arbitrary bodies.

First, let us extend the concept of a source or sink introduced in Section 3.10. In that section, we dealt with a single line source, as sketched in Figure 3.21. Now imagine that we have an infinite number of such line sources side by side, where the strength of each line source is infinitesimally small. These side-by-side line sources form a source sheet, as shown in perspective in the upper left of Figure 3.38. If we look along the series of line sources (looking along the z axis in Figure 3.38), the source sheet will appear as sketched at the lower right of Figure 3.38. Here, we are looking at an edge view of the sheet; the line sources are all perpendicular to the page. Let s be the distance measured along the source sheet in the edge view. Define X = X(s) to be the source strength per unit length along s. [To keep things in perspective, recall from Section 3.10 that the strength of a single line source Л was defined as the volume flow rate per unit depth, that is, per unit length in the z direction. Typical units for Л are square meters per second or square feet per second. In turn, the strength of a source sheet A.(.v) is the volume flow rate per unit depth (in the z direction) and per unit length (in the s direction). Typical units for X are meters per second or feet per second.] Therefore, the strength of an infinitesimal portion ds



Figure 3.38 Source sheet.



of the sheet, as shown in Figure 3.38, is Xds. This small section of the source sheet can be treated as a distinct source of strength X ds. Now consider point P in the flow, located a distance r from ds the cartesian coordinates of P are (x, у). The small section of the source sheet of strength X ds induces an infinitesimally small potential d<p at point P. From Equation (3.67), d<p is given by

Подпись: 2n [3.141]

Подпись: ф(х,у)= • lnr image278

The complete velocity potential at point P, induced by the entire source sheet from a to b, is obtained by integrating Equation (3.141):

Note that, in general, X(s) can change from positive to negative along the sheet; that is, the “source” sheet is really a combination of line sources and line sinks.

Next, consider a given body of arbitrary shape in a flow with freestream velocity Vqo, as shown in Figure 3.39. Let us cover the surface of the prescribed body with a source sheet, where the strength X(s) varies in such a fashion that the combined action of the uniform flow and the source sheet makes the airfoil surface a streamline of the flow. Our problem now becomes one of finding the appropriate X(s). The solution of this problem is carried out numerically, as follows.

Let us approximate the source sheet by a series of straight panels, as shown in Figure 3.40. Moreover, let the source strength X per unit length be constant over a given panel, but allow it to vary from one panel to the next. That is, if there are a total of n panels, the source panel strengths per unit length are X ], X2,…, Xj…, Xn. These panel strengths are unknown; the main thrust of the panel technique is to solve for Xj, j — 1 to n, such that the body surface becomes a streamline of the flow. This boundary condition is imposed numerically by defining the midpoint of each panel


Uniform flow Source sheet on surface of body, with (s) calculated to make the body surface a streamline



Flow over the body of given shape


Figure 3.39


Superposition of a uniform flow and a source sheet on a body of given shape, to produce the flow over the body.


Figure 3.40


Source panel distribution over the surface of a body of arbitrary shape.



Nonlifting Flows over Arbitrary Bodies: The Numerical Source Panel Method

to be a control point and by determining the kj ’s such that the normal component of the flow velocity is zero at each control point. Let us now quantify this strategy.

Let P be a point located at (x, y) in the flow, and let r[4 be the distance from any point on the / th panel to P, as shown in Figure 3.40. The velocity potential induced at P due to the y’th panel A<f>j is, from Equation (3.142),

Equation (3.146) is physically the contribution of all the panels to the potential at the control point of the ith panel.

Recall that the boundary condition is applied at the control points; that is, the normal component of the flow velocity is zero at the control points. To evaluate this component, first consider the component of freestream velocity perpendicular to the panel. Let n, be the unit vector normal to the ith panel, directed out of the body, as shown in Figure 3.40. Also, note that the slope of the ith panel is (dy/dx)j. In general, the freestream velocity will be at some incidence angle a to the x axis, as shown in Figure 3.40. Therefore, inspection of the geometry of Figure 3.40 reveals that the component of Voo normal to the ith panel is

Подпись: [3.148]koo, n — Vco • n, — Voo CDS

where Pi is the angle between and n, . Note that V7^ ,, is positive when directed away from the body, and negative when directed toward the body.

The normal component of velocity induced at (x,, y,) by the source panels is, from Equation (3.146),

= — [ф(Хі, уі)] [3.14P]


where the derivative is taken in the direction of the outward unit normal vector, and hence, again, V„ is positive when directed away from the body. When the derivative in Equation (3.149) is carried out, appears in the denominator. Consequently, a singular point arises on the ith panel because when j — i, at the control point itself rtj = 0. It can be shown that when j = і, the contribution to the derivative is simply

A.,-/2. Hence, Equation (3.149) combined with Equation (3.146) becomes


In Equation (3.150), the first term A,/2 is the normal velocity induced at the і th control point by the г th panel itself, and the summation is the normal velocity induced at the (th control point by all the other panels.

The normal component of the flow velocity at the г th control point is the sum of that due to the freestream [Equation (3.148)] and that due to the source panels [Equation (3.150)]. The boundary condition states that this sum must be zero:

Eoo," + vn = 0 [3.151]

Substituting Equations (3.148) and (3.150) into (3.151), we obtain

A. ■ n к’ С d

Т + (lnru)dsi + Voccosft = 0 [3.152]

2 7^ 2tt Jj dn, J


Equation (3.152) is the crux of the source panel method. The values of the integrals in Equation (3.152) depend simply on the panel geometry; they are not properties of the flow. Let Ijj be the value of this integral when the control point is on the (th panel and the integral is over the jth panel. Then Equation (3.152) can be written as

к’ П к ■

j + U-i + V°° C0S ft = 0 [3.153]

J = [


Equation (3.153) is a linear algebraic equation with n unknowns A,, A2,…, A„. It represents the flow boundary condition evaluated at the control point of the (th panel. Now apply the boundary condition to the control points of all the panels; that is, in Equation (3.153), let ( = 1,2, ,n. The results will be a system of n linear algebraic

equations with n unknowns (Aj, A2, …, A„), which can be solved simultaneously by conventional numerical methods.

Look what has happened! After solving the system of equation represented by Equation (3.153) with і = 1, 2, …, n, we now have the distribution of source panel strengths which, in an appropriate fashion, cause the body surface in Figure 3.40 to be a streamline of the flow. This approximation can be made more accurate by increasing the number of panels, hence more closely representing the source sheet of continuously varying strength A(.v) shown in Figure 3.39. Indeed, the accuracy of the source panel method is amazingly good; a circular cylinder can be accurately represented by as few as 8 panels, and most airfoil shapes, by 50 to 100 panels. (For an airfoil, it is desirable to cover the leading-edge region with a number of small panels to represent accurately the rapid surface curvature and to use larger panels over the relatively flat portions of the body. Note that, in general, all the panels in Figure 3.40 can be different lengths.)

Once the A; ’s (i = 1,2,…, n) are obtained, the velocity tangent to the surface at each control point can be calculated as follows. Let s be the distance along the body surface, measured positive from front to rear, as shown in Figure 3.40. The component of freestream velocity tangent to the surface is

Voo, s = VooSinft [3.154]

The tangential velocity V, at the control point of the /th panel induced by all the panels is obtained by differentiating Equation (3.146) with respect to s:

30 Ая, г a r, = 13,511

7 = 1 J

[The tangential velocity on a flat source panel induced by the panel itself is zero; hence, in Equation (3.155), the term corresponding to j = і is zero. This is easily seen by intuition, because the panel can only emit volume flow from its surface in a direction perpendicular to the panel itself.] The total surface velocity at the г th control point Vj is the sum of the contribution from the freestream [Equation (3.154)] and from the source panels [Equation (3.155)]:

n X • C 3

Vi = Уоо, і + vs = Loosing + —(Inrij)dsj [3.156]

7 = 1 j

In turn, the pressure coefficient at the г th control point is obtained from Equation (3.38):

Cw -1′

In this fashion, the source panel method gives the pressure distribution over the surface of a nonlifting body of arbitrary shape.

When you carry out a source panel solution as described above, the accuracy of your results can be tested as follows. Let Sj be the length of the yth panel. Recall that Xj is the strength of the у th panel per unit length. Hence, the strength of the yth panel itself is X, Sj. For a closed body, such as in Figure 3.40, the sum of all the source and sink strengths must be zero, or else the body itself would be adding or absorbing mass from the flow—an impossible situation for the case we are considering here. Hence, the values of the Xj’s obtained above should obey the relation


J2>-jSj= о [3.157]

7 = 1

Equation (3.157) provides an independent check on the accuracy of the numerical results.

Example 3.12

Calculate the pressure coefficient distribution around a circular cylinder using the source panel technique.


We choose to cover the body with eight panels of equal length, as shown in Figure 3.41. This choice is arbitrary; however, experience has shown that, for the case of a circular cylinder, the arrangement shown in Figure 3.41 provides sufficient accuracy. The panels are numbered from 1 to 8, and the control points are shown by the dots in the center of each panel.

Let us evaluate the integrals which appear in Equation (3.153). Consider Figure 3.42, which illustrates two arbitrary chosen panels. In Figure 3.42, (x,. y,) are the coordinates of the


Figure 3.41 Source panel distribution around a circular cylinder.


Figure 3.42 Geometry required for the evaluation of Іц.

control point of the ith panel and (xh yj) are the running coordinates over the entire jth panel. The coordinates of the boundary points for the і th panel are (X, , Yt) and (Xi+1, F,+i); similarly, the coordinates of the boundary points for the jth panel are (Ху, У,) and (XJ+] In this

problem, Vcc is in the x direction; hence, the angles between the x axis and the unit vectors n, and itj are ft and ft, respectively. Note that, in general, both ft and ft vary from 0 to 2л Recall that the integral ft; is evaluated at the (th control point and the integral is taken over the complete jth panel:

Nonlifting Flows over Arbitrary Bodies: The Numerical Source Panel Method



r4 =•/(■*(- Xj)2 + (yf – Уі)1

9 , 1 drtj

-—(In Лу) = — T—

orii r(j drii

= — ^[(a-( – Xjf + (Уі – уj)2r’/2

nj 2

dxi dy.

■ 2{Х‘-Х^+2(У‘-У^

9 (Xi – xj) cos ft + (у, – уj) sin ft

9 Пі(ППі)~ (x.-Xj)2 + (y.-yj)2










Figure 3.43 Pressure distribution over a circular cylinder; comparison of the source panel results and theory.


Substituting the values for the X’s obtained into Equation (3.157), we see that the equation is identically satisfied.

Nonlifting Flows over Arbitrary Bodies: The Numerical Source Panel Method Nonlifting Flows over Arbitrary Bodies: The Numerical Source Panel Method Подпись: [3.165]

The velocity at the control point of the ith panel can be obtained from Equation (3.156). In that equation, the integral over the jth panel is a geometric quantity which is evaluated in a similar manner as before. The result is


With the integrals in Equation (3.156) evaluated by Equation (3.165), and with the values for 7-і, 7-2,. • •, 7.8 obtained above inserted into Equation (3.156), we obtain the velocities Vi, V2,…, Vs. In turn, the pressure coefficients Cp 1, Cp>2,…, Cp, g are obtained directly from

Results for the pressure coefficients obtained from this calculation are compared with the exact analytical result, Equation (3.101) in Figure 3.43. Amazingly enough, in spite of the relatively crude paneling shown in Figure 3.41, the numerical pressure coefficient results are excellent.

Three-Dimensional Source

Return to Laplace’s equation written in spherical coordinates, as (3.43). Consider the velocity potential given by

given by Equation


ф =——



where C is a constant and r is the radial coordinate from the origin. Equation (6.1) satisfies Equation (3.43), and hence it describes a physically possible incompressible, irrotational three-dimensional flow. Combining Equation (6.1) with the definition of the gradient in spherical coordinates, Equation (2.18), we obtain

V = V0 =



In terms of the velocity components, we have


V’ = ~ r1


■ о







Clearly, Equation (6.2), or Equations (6.3a to c), describes a flow with straight stream­lines emanating from the origin, as sketched in Figure 6.1. Moreover, from Equation

(6.2) or (6.3a), the velocity varies inversely as the square of the distance from the origin. Such a flow is defined as a three-dimensional source. Sometimes it is called

Подпись: Figure 6.1 Three-dimensional (point) source.

simply a point source, in contrast to the two-dimensional line source discussed in Section 3.10.

Подпись: Mass flow Подпись: pV-dS

To evaluate the constant C in Equation (6.3a), consider a sphere of radius r and surface S centered at the origin. From Equation (2.46), the mass flow across the surface of this sphere is

Подпись: A = Подпись: V • dS Подпись: [6.4]

Hence, the volume flow, denoted by A., is

On the surface of the sphere, the velocity is a constant value equal to Vr = С/г2 and is normal to the surface. Hence, Equation (6.4) becomes

Подпись: A =C 7

—г 4тсг2 = 4лС rz

Подпись:TT A

Hence, C = —


Three-Dimensional Source Подпись: [6.6]

Substituting Equation (6.5) into (6.3a), we find

Compare Equation (6.6) with its counterpart for a two-dimensional source given by Equation (3.62). Note that the three-dimensional effect is to cause an inverse r – squared variation and that the quantity 4л appears rather than 2л. Also, substituting

Подпись: X ф = A nr Подпись: [6.7]

Equation (6.5) into (6.1), we obtain, for a point source,

In the above equations, X is defined as the strength of the source. When X is a negative quantity, we have a point sink.

Incompressible Versus Compressible Flows

A flow in which the density p is constant is called incompressible. In contrast, a flow where the density is variable is called compressible. A more precise definition of compressibility will be given in Chapter 7. For our purposes here, we will simply note that all flows, to a greater or lesser extent, are compressible; truly incompressible flow, where the density is precisely constant, does not occur in nature. However, analogous to our discussion of inviscid flow, there are a number of aerodynamic problems that can be modeled as being incompressible without any detrimental loss of accuracy. For example, the flow of homogeneous liquids is treated as incompressible, and hence most problems involving hydrodynamics assume p = constant. Also, the flow of gases at a low Mach number is essentially incompressible; for M < 0.3, it is always safe to assume p = constant. (We will prove this in Chapter 8.) This was the flight regime of all airplanes from the Wright brothers’ first flight in 1903 to just prior to World War II. It is still the flight regime of most small, general aviation aircraft of today. Hence, there exists a large bulk of aerodynamic experimental and theoretical data for incompressible flows. Such flows are the subject of Chapters 3 to 6. On the other hand, high-speed flow (near Mach 1 and above) must be treated as compressible; for such flows p can vary over wide latitudes. Compressible flow is the subject of Chapters 7 to 14.

How Do We Solve the Equations?

This chapter is full of mathematical equations—equations that represent the basic physical fundamentals that dictate the characteristics of aerodynamic flow fields. For the most part, the equations are either in partial differential form or integral form. These equations are powerful and by themselves represent a sophisticated intellectual construct of our understanding of the fundamentals of a fluid flow. However, the equations by themselves are not very practical. They must be solved in order to obtain the actual flow fields over specific body shapes with specific flow conditions. For example, if we are interested in calculating the flow field around a Boeing 777 jet transport flying at a velocity of 800 ft/s at an altitude of 30,000 ft, we have to obtain a solution of the governing equations for this case—a solution that will give us the results for the dependent flow-field variables p, p, V, etc. as a function of the independent variables of spatial location and time. Then we have to squeeze this solution for extra practical information, such as lift, drag, and moments exerted on the vehicle. How do we do this? The purpose of the present section is to discuss two philosophical answers to this question. As for practical solutions to specific problems of interest, there are literally hundreds of different answers to this question, many of which make up the content of the rest of this book. However, all these solutions fall under one or the other of the two philosophical approaches described next.

Classical Thin Airfoil Theory: The Symmetric Airfoil

Some experimentally observed characteristics of airfoils and a philosophy for the theoretical prediction of these characteristics have been discussed in the preceding sections. Referring to our chapter road map in Figure 4.2, we have now completed the central branch. In this section, we move to the right-hand branch of Figure 4.2, namely a quantitative development of thin airfoil theory. The basic equations necessary for the calculation of airfoil lift and moments are established in this section, with an application to symmetric airfoils. The case of cambered airfoils will be treated in Section 4.8.

For the time being, we deal with thin airfoils; for such a case, the airfoil can be simulated by a vortex sheet placed along the camber line, as discussed in Section 4.4. Our purpose is to calculate the variation of у (s) such that the camber line becomes a streamline of the flow and such that the Kutta condition is satisfied at the trailing edge; that is, у (ТЕ) = 0 [see Equation (4.10)]. Once we have found the particular y(s) that satisfies these conditions, then the total circulation Г around the airfoil is found by integrating y(s) from the leading edge to the trailing edge. In turn, the lift is calculated from Г via the Kutta-Joukowski theorem.

Consider a vortex sheet placed on the camber line of an airfoil, as sketched in Figure 4.17a. The freestream velocity is Voo, and the airfoil is at the angle of attack a. The x axis is oriented along the chord line, and the z axis is perpendicular to the chord. The distance measured along the camber line is denoted by 5. The shape of the camber line is given by z = z(x). The chord length is c. In Figure 4.17a, w’ is the component of velocity normal to the camber line induced by the vortex sheet; w’ = w'(s). For a thin airfoil, we rationalized in Section 4.4 that the distribution of a vortex sheet over the surface of the airfoil, when viewed from a distance, looks almost the same as a vortex sheet placed on the camber line. Let us stand back once again and view Figure 4.17a from a distance. If the airfoil is thin, the camber line is close to the



(b) Vortex sheet on the chord line Figure 4.1 7 Placement of the vortex sheet for thin airfoil analysis.

chord line, and viewed from a distance, the vortex sheet appears to fall approximately on the chord line. Therefore, once again, let us reorient our thinking and place the vortex sheet on the chord line, as sketched in Figure 4.17b. Here, у = у (дг). We still wish the camber line to be a streamline of the flow, and у = y(x) is calculated to satisfy this condition as well as the Kutta condition y(c) = 0. That is, the strength of the vortex sheet on the chord line is determined such that the camber line (not the chord line) is a streamline.

For the camber line to be a streamline, the component of velocity normal to the camber line must be zero at all points along the camber line. The velocity at any point in the flow is the sum of the uniform freestream velocity and the velocity induced by the vortex sheet. Let V^,, be the component of the freestream velocity normal to the camber line. Thus, for the camber line to be a streamline,

Voc. n + u/(.v) = 0 [4.12]

at every point along the camber line.

An expression for Vco. n in Equation (4.12) is obtained by the inspection of Fig­ure 4.18. At any point P on the camber line, where the slope of the camber line is





/ I


Classical Thin Airfoil Theory: The Symmetric Airfoil

Classical Thin Airfoil Theory: The Symmetric Airfoil

dz/dx, the geometry of Figure 4.18 yields

Подпись: Voo.n = So sin a + tanimage325[4.13]

For a thin airfoil at small angle of attack, both a and tan-1 (—dz/dx) are small values. Using the approximation that sin в ~ tan 0 яа 9 for small 9, where 9 is in radians, Equation (4.13) reduces to


Equation (4.14) gives the expression for to be used in Equation (4.12). Keep in mind that, in Equation (4.14), a is in radians.

Returning to Equation (4.12), let us develop an expression for w'(s) in terms of the strength of the vortex sheet. Refer again to Figure 4.17b. Here, the vortex sheet is along the chord line, and w'(s) is the component of velocity normal to the camber line induced by the vortex sheet. Let w(x) denote the component of velocity normal to the chord line induced by the vortex sheet, as also shown in Figure 4.17Й. If the airfoil is thin, the camber line is close to the chord line, and it is consistent with thin airfoil theory to make the approximation that

Подпись: [4.15]w'(s) ЯИ w(x)

An expression for w(x) in terms of the strength of the vortex sheet is easily obtained from Equation (4.1), as follows. Consider Figure 4.19, which shows the vortex sheet along the chord line. We wish to calculate the value of w(x) at the location*. Consider an elemental vortex of strength у d% located at a distance f from the origin along the chord line, as shown in Figure 4.19. The strength of the vortex sheet у varies with



* w

£—0-^0—О – e—e4-e—0-

Figure 4.1 9 Calculation of the induced velocity at the chord line.

the distance along the chord; that is, у = у (£). The velocity dw at point x induced by the elemental vortex at point § is given by Equation (4.1) as

Подпись: [4.16]Y<M)dS

2tz (x — I)

Подпись: w(x) = Подпись: Г У (£)<*$ о 2n(x-%) Подпись: [4.17]

In turn, the velocity w(x) induced at point x by all the elemental vortices along the chord line is obtained by integrating Equation (4,16) from the leading edge (£ = 0) to the trailing edge (| = c):

Combined with the approximation stated by Equation (4.15), Equation (4.17) gives the expression for w'(s) to be used in Equation (4.12).

Подпись: or Подпись: [4.18]

Recall that Equation (4.12) is the boundary condition necessary for the camber line to be a streamline. Substituting Equations (4.14), (4.15), and (4.17) into (4.12), we obtain

the fundamental equation of thin airfoil theory, it is simply a statement that the camber line is a streamline of the flow.

Note that Equation (4.18) is written at a given point x on the chord line, and that dz/dx is evaluated at that point x. The variable § is simply a dummy variable of integration which varies from 0 to c along the chord line, as shown in Figure 4.19. The vortex strength у = у (|) is a variable along the chord line. For a given airfoil at a given angle of attack, both a and dz/dx are known values in Equation (4.18). Indeed, the only unknown in Equation (4.18) is the vortex strength y(£). Hence, Equation (4.18) is an integral equation, the solution of which yields the variation of у (I) such that the camber line is a streamline of the flow. The central problem of

thin airfoil theory is to solve Equation (4.18) for у (f), subject to the Kutta condition, namely, y(c) = 0.

In this section, we treat the case of a symmetric airfoil. As stated in Section 4.2, a symmetric airfoil has no camber; the camber line is coincident with the chord line. Hence, for this case, dz/dx = 0, and Equation (4.18) becomes

Classical Thin Airfoil Theory: The Symmetric Airfoil



In essence, within the framework of thin airfoil theory, a symmetric airfoil is treated the same as a flat plate; note that our theoretical development does not account for the airfoil thickness distribution. Equation (4.19) is an exact expression for the inviscid, incompressible flow over a flat plate at angle of attack.

To help deal with the integral in Equations (4.18) and (4.19), let us transform f into 0 via the following transformation;

£ = ^(1 — cos 0) [4.20]

Since x is a fixed point in Equations (4.18) and (4.19), it corresponds to a particular value of 0, namely, 0q, such that

Подпись: COS во)image328[4.21]

Also, from Equation (4.20),

Подпись: [4.22]– sin в dd 2

Substituting Equations (4.20) to (4.22) into (4.19), and noting that the limits of inte­gration become в = 0 at the leading edge (where £ = 0) and в = л at the trailing edge (where £ = c), we obtain

1 Подпись: [4.23]f71 y(e)sine dd

2л Jo cos в — cos e0

A rigorous solution of Equation (4.23) for у (в) can be obtained from the mathematical theory of integral equations, which is beyond the scope of this book. Instead, we simply state that the solution is

Classical Thin Airfoil Theory: The Symmetric Airfoil



We can verify this solution by substituting Equation (4.24) into (4.23) yielding

Classical Thin Airfoil Theory: The Symmetric Airfoil

Подпись: Л sin «00 sin 00 cosпв dd
о cos в — cos во

Using Equation (4.26) in the right-hand side of Equation (4.25), we find that

Eoo a f71 (1+cos 6)d6 V^a / f71 d0 f71 cos Odd

Jo cos в – cos 90 7t Jo

Classical Thin Airfoil Theory: The Symmetric Airfoil

COS в — COS во




Classical Thin Airfoil Theory: The Symmetric Airfoil





Using Equations (4.20) and (4.22), Equation (4.28) transforms to

Г = – f у (в) sin в dd 2 Jo

Substituting Equation (4.24) into (4.29), we obtain

Г = acVoo / (1 +cos0) dd = TtacVoo Jo






Substituting Equation (4.30) into the Kutta-Joukowski theorem, we find that the lift per unit span is


L’ = РсоУсоГ = nacpocV^




The lift coefficient is


Classical Thin Airfoil Theory: The Symmetric Airfoil

Substituting Equation (4.31) into (4.32), we have _ тгасрорУ^

1 і Pocked)

Подпись: [4.33]Подпись:

Classical Thin Airfoil Theory: The Symmetric Airfoil Подпись: 2л Подпись: [4.34] Подпись: Lift slope :

сі = 2ла

Equations (4.33) and (4.34) are important results; they state the theoretical result that the lift coefficient is linearly proportional to angle of attack, which is supported by the experimental results discussed in Section 4.3. They also state that the theoret­ical lift slope is equal to 2л rad-1, which is 0.11 degree-1. The experimental lift coefficient data for an NACA 0012 symmetric airfoil are given in Figure 4.20; note that Equation (4.33) accurately predicts с/ over a large range of angle of attack. (The NACA 0012 airfoil section is commonly used on airplane tails and helicopter blades.)

Classical Thin Airfoil Theory: The Symmetric Airfoil


Figure 4.30 Comparison between theory and experiment for the lift and moment coefficients for an NACA 0012 airfoil. (Source: Abbott and von Doenhoff, Reference 11.)


The moment about the leading edge can be calculated as follows. Consider the elemental vortex of strength у (£) located a distance £ from the leading edge, as sketched in Figure 4.21. The circulation associated with this elemental vortex is dF = y(%)d%. In turn, the increment of lift dl. contributed by the elemental vortex is dL = Poo Voc This increment of lift creates a moment about the leading edge dM = — £ (dL). The total moment about the leading edge (LE) (per unit span) due to the entire vortex sheet is therefore

Classical Thin Airfoil Theory: The Symmetric Airfoil

, 2 7ta

MLE = —q0qC —-




The moment coefficient is


_ Кь



where S = c(l). Hence,


= MLE = _™

qxc2 2




However, from Equation (4.33),



жа — —


Combining Equations (4.37) and (4.38), we obtain










Classical Thin Airfoil Theory: The Symmetric Airfoil

From Equation (1.22), the moment coefficient about the quarter-chord point is

Подпись:Cm, с/4 — Cm. le T ^

Combining Equations (4.39) and (4.40), we have


In Section 1.6, a definition is given for the center of pressure as that point about which the moments are zero. Clearly, Equation (4.41) demonstrates the theoretical result that the center of pressure is at the quarter-chord point for a symmetric airfoil.

By the definition given in Section 4.3, that point on an airfoil where moments are independent of angle of attack is called the aerodynamic center. From Equation (4.41), the moment about the quarter chord is zero for all values of a. Hence, for a symmetric airfoil, we have the theoretical result that the quarter-chord point is both the center of pressure and the aerodynamic center.

The above theoretical result for cm>c/4 = 0 is supported by the experimental data given in Figure 4.20. Also, note that the experimental value of cmc/4 is constant over a wide range of a, thus demonstrating that the real aerodynamic center is essentially at the quarter chord.

Let us summarize the above results. The essence of thin airfoil theory is to find a distribution of vortex sheet strength along the chord line that will make the camber line a streamline of the flow while satisfying the Kutta condition у (ТЕ) = 0. Such a vortex distribution is obtained by solving Equation (4.18) for y(%), or in terms of the transformed independent variable в, solving Equation (4.23) for у (в) [recall that Equation (4.23) is written for a symmetric airfoil]. The resulting vortex distribution у (в) for a symmetric airfoil is given by Equation (4.24). In turn, this vortex distribu­tion, when inserted into the Kutta-Joukowski theorem, gives the following important theoretical results for a symmetric airfoil:

1. сі = 2л a.

2. Lift slope = 27Г.

3. The center of pressure and the aerodynamic center are both located at the quarter-

chord point.

Definition of Compressibility

All real substances are compressible to some greater or lesser extent; that is, when you squeeze or press on them, their density will change. This is particularly true of gases, much less so for liquids, and virtually unnoticeable for solids. The amount by which a substance can be compressed is given by a specific property of the substance called the compressibility, defined below.

Consider a small element of fluid of volume v, as sketched in Figure 7.3. The pressure exerted on the sides of the element is p. Assume the pressure is now in­creased by an infinitesimal amount dp. The volume of the element will change by a corresponding amount d v; here, the volume will decrease; hence, d v shown in Figure

7.3 is a negative quantity. By definition, the compressibility r of the fluid is

Подпись: [7.33]1 d v v dp

Physically, the compressibility is the fractional change in volume of the fluid element per unit change in pressure. However, Equation (7.33) is not precise enough. We know from experience that when a gas is compressed (say, in a bicycle pump), its temperature tends to increase, depending on the amount of heat transferred into or out of the gas through the boundaries of the system. If the temperature of the fluid element in Figure 7.3 is held constant (due to some heat transfer mechanism), then r is identified as the isothermal compressibility rt, defined from Equation (7.33) as

Подпись: 1 Tj = V image497[7.34]

On the other hand, if no heat is added to or taken away from the fluid element, and if friction is ignored, the compression of the fluid element takes place isentropically, and t is identified as the isentropic compressibility rs, defined from Equation (7.33) as

Definition of Compressibility

1 /dv’




?s =


v dPJs


where the subscript 5 denotes that the partial derivative is taken at constant entropy. Both Xj and r, are precise thermodynamic properties of the fluid; their values for

Подпись: p Figure 7.3 Definition of compressibility.

different gases and liquids can be obtained from various handbooks of physical prop­erties. In general, the compressibility of gases is several orders of magnitude larger than that of liquids.

The role of the compressibility r in determining the properties of a fluid in motion is seen as follows. Define v as the specific volume (i. e., the volume per unit mass). Hence, v = 1 / p. Substituting this definition into Equation (7.33), we obtain

Подпись: [7.36]1 dp P dp

Thus, whenever the fluid experiences a change in pressure dp, the corresponding change in density dp from Equation (7.36) is

Подпись: [7.37]dp = p x dp

Consider a fluid flow, say, for example, the flow over an airfoil. If the fluid is a liquid, where the compressibility r is very small, then for a given pressure change dp from one point to another in the flow, Equation (7.37) states that dp will be negligibly small. In turn, we can reasonably assume that p is constant and that the flow of a liquid is incompressible. On the other hand, if the fluid is a gas, where the compressibility r is large, then for a given pressure change dp from one point to another in the flow, Equation (7.37) states that dp can be large. Thus, p is not constant, and in general, the flow of a gas is a compressible flow. The exception to this is the low-speed flow of a gas; in such flows, the actual magnitude of the pressure changes throughout the flow field is small compared with the pressure itself. Thus, for a low-speed flow, dp in Equation (7.37) is small, and even though r is large, the value of dp can be dominated by the small dp. In such cases, p can be assumed to be constant, hence allowing us to analyze low-speed gas flows as incompressible flows (such as discussed in Chapters 3 to 6).

Later, we demonstrate that the most convenient index to gage whether a gas flow can be considered incompressible, or whether it must be treated as compressible, is the Mach number M, defined in Chapter 1 as the ratio of local flow velocity V to the local speed of sound a:


M = — [7.38]


We show that, when M > 0.3, the flow should be considered compressible, Also, we show that the speed of sound in a gas is related to the isentropic compressibility rs, given by Equation (7.35).

Volume Integrals

Consider a volume V in space. Let p be a scalar field in this space. The volume integral over the volume V of the quantity p is written as

image99p dV = volume integral of a scalar p over the
volume V (the result is a scalar)


Figure 2.9 Sketch for surface integrals. The

three-dimensional surface area S is bounded by the closed curve C.



Figure 2.10 Volume V enclosed by the closed surface S.


Let A be a vector field in space. The volume integral over the volume V of the quantity A is written as


A dV = volume integral of a vector A over the volume V (the result is a vector)


2.2.11 Relations Between Line, Surface, and Volume Integrals


Consider again the open area S bounded by the closed curve C, as shown in Figure 2.9. Let A be a vector field. The line integral of A over C is related to the surface integral of A over S by Stokes ’ theorem:




Consider again the volume V enclosed by the closed surface S, as shown in Figure 2.10. The surface and volume integrals of the vector field A are related through the divergence theorem:





(V. A) dV






If p represents a scalar field, a vector relationship analogous to Equation (2.26) is given by the gradient theorem:







2.2.12 Summary

This section has provided a concise review of those elements of vector analysis which we will use as tools in our subsequent discussions. Make certain to review these tools until you feel comfortable with them, especially the relations in boxes.

Infinity Boundary Conditions

Far away from the body (toward infinity), in all directions, the flow approaches the uniform freestream conditions. Let be aligned with the x direction as shown in Figure 3.18. Hence, at infinity,

Подпись: [3.47a]З Ф дф

и = — = — = Lx дх 3 у

Infinity Boundary Conditions

Подпись:Boundary conditions at infinity and on a body; inviscid flow.

Подпись: [3.47b]дф дф ду дх

Equations (3.47a and b) are the boundary conditions on velocity at infinity. They apply at an infinite distance from the body in all directions, above and below, and to the left and right of the body, as indicated in Figure 3.18.