Category Fundamentals of Aerodynamics

Historical Note: The Illusive Center of Pressure

The center of pressure of an airfoil was an important matter during the development of aeronautics. It was recognized in the nineteenth century that, for a heavier-than – air machine to fly at stable, equilibrium conditions (e. g., straight-and-level flight), the moment about the vehicle’s center of gravity must be zero (see Chapter 7 of Reference 2). The wing lift acting at the center of pressure, which is generally a distance away from the center of gravity, contributes substantially to this moment. Hence, the understanding and prediction of the center of pressure was felt to be absolutely necessary in order to design a vehicle with proper equilibrium. On the other hand, the early experimenters had difficulty measuring the center of pressure, and much confusion reigned. Let us examine this matter further.

The first experiments to investigate the center of pressure of a lifting surface were conducted by the Englishman George Cayley (1773-1857) in 1808. Cayley was the inventor of the modem concept of the airplane, namely, a vehicle with fixed wings, a fuselage, and a tail. He was the first to separate conceptually the functions of lift and propulsion; prior to Cayley, much thought had gone into omithopters—machines that flapped their wings for both lift and thrust. Cayley rejected this idea, and in 1799, on a silver disk now in the collection of the Science Museum in London, he inscribed a sketch of a rudimentary airplane with all the basic elements we recognize

today. Cayley was an active, inventive, and long-lived man, who conducted numerous pioneering aerodynamic experiments and fervently believed that powered, heavier – than-air, manned flight was inevitable. (See Chapter 1 of Reference 2 for an extensive discussion of Cayley’s contributions to aeronautics.)

In 1808, Cayley reported on experiments of a winged model which he tested as a glider and as a kite. His comments on the center of pressure are as follows:

By an experiment made with a large kite formed of an hexagon with wings extended from it, all so constructed as to present a hollow curve to the current, I found that when loaded nearly to 1 lb to a foot and 1/2, it required the center of gravity to be suspended so as to leave the anterior and posterior portions of the surface in the ratio of 3 to 7. But as this included the tail operating with a double leverage behind, I think such hollow surfaces relieve about an equal pressure on each part, when they are divided in the ratio of 5 to 12, 5 being the anterior portion. It is really surprising to find so great a difference, and it obliges the center of gravity of flying machines to be much forwarder of the center of bulk (the centroid) than could be supposed a priori.

Here, Cayley is saying that the center of pressure is 5 units from the leading edge and 12 units from the trailing edge; i. e., xcp = 5/1 7c. Later, he states in addition: “I tried a small square sail in one plane, with the weight nearly the same, and I could not perceive that the center-of-resistance differed from the center of bulk.” That is, Cayley is stating that the center of pressure in this case is 1 /2c.

There is no indication from Cayley’s notes that he recognized that center of pressure moves when the lift, or angle of attack, is changed. However, there is no doubt that he was clearly concerned with the location of the center of pressure and its effect on aircraft stability.

The center of pressure on a flat surface inclined at a small angle to the flow was studied by Samuel R Langley during the period 1887-1896. Langley was the secretary of the Smithsonian at that time, and devoted virtually all his time and much of the Smithsonian’s resources to the advancement of powered flight. Langley was a highly respected physicist and astronomer, and he approached the problem of powered flight with the systematic and structured mind of a scientist. Using a whirling arm apparatus as well as scores of rubber-band powered models, he collected a large bulk of aerodynamic information with which he subsequently designed a full-scale aircraft. The efforts of Langley to build and fly a successful airplane resulted in two dismal failures in which his machine fell into the Potomac River—the last attempt being just 9 days before the Wright brothers’ historic first flight on December 17, 1903. In spite of these failures, the work of Langley helped in many ways to advance powered flight. (See Chapter 1 of Reference 2 for more details.)

Langley’s observations on the center of pressure for a flat surface inclined to the flow are found in the Langley Memoir on Mechanical Flight, Part I, 1887 to 1896, by Samuel P. Langley, and published by the Smithsonian Institution in 1911—5 years after Langley’s death. In this paper, Langley states:

The center-of-pressure in an advancing plane in soaring flight is always in advance of the center of figure, and moves forward as the angle-of-inclination of the sustaining

surfaces diminishes, and, to a less extent, as horizontal flight increases in velocity. These facts furnish the elementary ideas necessary in discussing the problem of equilibrium, whose solution is of the most vital importance to successful flight.

The solution would be comparatively simple if the position of the center-of- pressure could be accurately known beforehand, but how difficult the solution is may be realized from a consideration of one of the facts just stated, namely, that the position of the center-of – pressure in horizontal flight shifts with velocity of the flight itself.

Here, we see that Langley is fully aware that the center of pressure moves over a lifting surface, but that its location is hard to pin down. Also, he notes the correct variation for a flat plate, namely, xcp moves forward as the angle of attack decreases. However, he is puzzled by the behavior of xcp for a curved (cambered) airfoil. In his own words:

Later experiments conducted under my direction indicate that upon the curved sur­faces I employed, the center-of-pressure moves forward with an increase in the angle of elevation, and backward with a decrease, so that it may lie even behind the center of the surface. Since for some surfaces the center-of-pressure moves backward, and for others forward, it would seem that there might be some other surface for which it will be fixed.

Here, Langley is noting the totally opposite behavior of the travel of the center of pressure on a cambered airfoil in comparison to a flat surface, and is indicating ever so slightly some of his frustration in not being able to explain his results in a rational scientific way.

Three-hundred-fifty miles to the west of Langley, in Dayton, Ohio, Orville and Wilbur Wright were also experimenting with airfoils. As described in Section 1.1, the Wrights had constructed a small wind tunnel in their bicycle shop with which they conducted aerodynamic tests on hundreds of different airfoil and wing shapes during the fall, winter, and spring of 1901-1902. Clearly, the Wrights had an appreciation of the center of pressure, and their successful airfoil design used on the 1903 Wright Flyer is a testimonial to their mastery of the problem. Interestingly enough, in the written correspondence of the Wright brothers, only one set of results for the center of pressure can be found. This appears in Wilbur’s notebook, dated July 25, 1905, in the form of a table and a graph. The graph is shown in Figure 1.49—the original form as plotted by Wilbur. Here, the center of pressure, given in terms of the percentage of distance from the leading edge, is plotted versus angle of attack. The data for two airfoils are given, one with large curvature (maximum height to chord ratio = 1/12) and one with more moderate curvature (maximum height to chord ratio = 1/20). These results show the now familiar travel of the center of pressure for a curved airfoil, namely, xcp moves forward as the angle of attack is increased, at least for small to moderate values of a. However, the most forward excursion of xcp in Figure 1.49 is 33 percent behind the leading edge—the center of pressure is always behind the quarter-chord point.

The first practical airfoil theory, valid for thin airfoils, was developed by Ludwig Prandtl and his colleagues at Gottingen, Germany, during the period just prior to and

Подпись: Figure 1 .4© Wright brothers' measurements of the center of pressure as a function of angle of attack for a curved (cambered) airfoil. Center of pressure is plotted on the ordinate in terms of percentage distance along the chord from the leading edge. This figure shows the actual data as hand plotted by Wilbur Wright, which appears in Wilbur's notebook dated July 25, 1905.

during World War I. This thin airfoil theory is described in detail in Chapter 4. The result for the center of pressure for a curved (cambered) airfoil is given by Equation

(4.66) , and shows that xcp moves forward as the angle of attack (hence q) increases, and that it is always behind the quarter-chord point for finite, positive values of q. This theory, in concert with more sophisticated wind-tunnel measurements that were being made during the period 1915-1925, finally brought the understanding and prediction of the location of the center of pressure for a cambered airfoil well into focus.

Because л:ср makes such a large excursion over the airfoil as the angle of attack is varied, its importance as a basic and practical airfoil property has diminished. Beginning in the early 1930s, the National Advisory Committee for Aeronautics (NACA), at its Langley Memorial Aeronautical Laboratory in Virginia, measured the properties of several systematically designed families of airfoils—airfoils which became a standard in aeronautical engineering. These NACA airfoils are discussed in Sections 4.2 and 4.3. Instead of giving the airfoil data in terms of lift, drag, and center of pressure, the NACA chose the alternate systems of reporting lift, drag, and moments about either the quarter-chord point or the aerodynamic center. These are totally appropriate alternative methods of defining the force-and-moment system on an airfoil, as discussed in Section 1.6 and illustrated in Figure 1.19. As a result, the

center of pressure is rarely given as part of modem airfoil data. On the other hand, for three-dimensional bodies, such as slender projectiles and missiles, the location of the center of pressure still remains an important quantity, and modem missile data frequently include xcp. Therefore, a consideration of center of pressure still retains its importance when viewed over the whole spectmm of flight vehicles.

The Bigger Picture

The evolution of our intellectual understanding of aerodynamics is over 2500 years old, going all the way back to ancient Greek science. The aerodynamics you are studying in this book is the product of this evolution. (See Reference 62 for an in­depth study of the history of aerodynamics.) Relevant to our current discussion is the development of the experimental tradition in fluid dynamics, which took place in the middle of the seventeenth century, principally in France, and the introduction of rational analysis in mechanics pioneered by Isaac Newton towards the end of the same century. Since that time, up until the middle of the twentieth century, the study and practice of fluid dynamics, including aerodynamics, has dealt with pure experiment on one hand and pure theory on the other. If you were learning aerodynamics as recently as, say 1960, you would have been operating in the “two-approach world” of theory and experiment. However, computational fluid dynamics has revolutionized the way we study and practice aerodynamics today. As sketched in Figure 2.44, CFD is today an equal partner with pure theory and pure experiment in the analysis and solution of aerodynamic problems. This is no flash in the pan—CFD will continue to play this role indefinitely, for as long as our advanced human civilization exists. Also, the double arrows in Figure 2.44 imply that today each of the equal partners constantly interact with each other—they do not stand alone, but rather help each other to continue to resolve and better understand the “big picture” of aerodynamics.


Figure 2.44 The three equal partners of modern aerodynamics.

2.18 Summary

Return to the road map for this chapter, as given in Figure 2.1. We have now covered both the left and right branches of this map and are ready to launch into the solution of practical aerodynamic problems in subsequent chapters. Look at each block in Figure 2.1; let your mind flash over the important equations and concepts represented by each block. If the flashes are dim, return to the appropriate sections of this chapter and review the material until you feel comfortable with these aerodynamic tools.

For your convenience, the most important results are summarized below:

Design Box   The result of Example 4.3 shows that the aerodynamic center for the NACA 23012 airfoil is located ahead of, but very close to, the quarter-chord point. For some other families of airfoils, the aerodynamic center is located behind, but similarly close to, the quarter-chord point. For a given airfoil family, the location of the aerodynamic center depends on the airfoil thickness, as shown in Figure 4.25. The variation of xac with thickness for the NACA 230XX family is given in Figure 4.25a. Here, the aerodynamic center is ahead of the quarter-chord point, and becomes progressively farther ahead as the airfoil thickness is increased. In contrast, the variation of iac with thickness for the NACA 64-2XX family is given in Figure 4.25b. Here, the aerodynamic center is behind the quarter-chord point, and becomes progressively farther behind as the airfoil thickness is increased. From the point of view of purely aerodynamics, the existence of the aerodynamic center is interesting, but the specification of the force and moment system on the airfoil by placing the lift and drag at the aerodynamic center and giving the value of M’c as illustrated in Figure 4.23, is not more useful than placing the lift and drag at any other point on the airfoil and giving the value of M’ at that point, such as shown in Figure 1.19. However, in flight   0 4 8 12 16 20 24 Airfoil thickness, percent of chord (a) NACA 230XX Airfoil   0 4 8 12 16 20 24 Airfoil thickness, percent of chord (b) NACA 64-2XX Airfoil Figure 4.35 Variation of the location of the aerodynamic center with airfoil thickness, (a) NACA 230XX airfoil, (b) NACA 64-2XX airfoil.   (continued) &nbsp

Some Aspects of Supersonic Flow: Shock Waves

Return to the different regimes of flow sketched in Figure 1.37. Note that subsonic compressible flow is qualitatively (but not quantitatively) the same as incompressible flow; Figure 1.37a shows a subsonic flow with a smoothly varying streamline pattern, where the flow far ahead of the body is forewarned about the presence of the body and begins to adjust accordingly. In contrast, supersonic flow is quite different, as sketched in Figure 1.37<i and e. Here, the flow is dominated by shock waves, and the flow upstream of the body does not know about the presence of the body until it encounters the leading-edge shock wave. In fact, any flow with a supersonic region, such as those sketched in Figure 1.37ft to e, is subject to shock waves. Thus, an essential ingredient of a study of supersonic flow is the calculation of the shape and strength of shock waves. This is the main thrust of Chapters 8 and 9.

A shock wave is an extremely thin region, typically on the order of 1СГ5 cm, across which the flow properties can change drastically. The shock wave is usually at an oblique angle to the flow, such as sketched in Figure 7.5a; however, there are many cases where we are interested in a shock wave normal to the flow, as sketched in Figure 7.5b. Normal shock waves are discussed at length in Chapter 8, whereas oblique shocks are considered in Chapter 9. In both cases, the shock wave is an almost explosive compression process, where the pressure increases almost discontinuously across the wave. Examine Figure 7.5 closely. In region 1 ahead of the shock, the Mach number, flow velocity, pressure, density, temperature, entropy, total pressure, and total enthalpy are denoted by p, p,T, s, ро. ь and /год, respectively.

The analogous quantities in region 2 behind the shock are М2, V2, pi, Pi> Ti, si, po,2> and /го,2, respectively. The qualitative change across the wave are noted in Figure 7.5. The pressure, density, temperature, and entropy increase across the shock, whereas the total pressure, Mach number, and velocity decrease. Physically, the flow across a shock wave is adiabatic (we are not heating the gas with a laser beam or cooling it in a refrigerator, for example). Therefore, recalling the discussion in Section 7.5, the total enthalpy is constant across the wave. In both the oblique shock and normal shock cases, the flow ahead of the shock wave must be supersonic (i. e., Mx > 1). Behind the oblique shock, the flow usually remains supersonic (i. e., М2 > 1), but at a reduced Mach number (i. e., М2 < M). However, as discussed in Chapter 9, there are special cases where the oblique shock is strong enough to decelerate the downstream flow to a subsonic Mach number; hence, M2 < 1 can occur behind an oblique shock. For the

Some Aspects of Supersonic Flow: Shock Waves

Some Aspects of Supersonic Flow: Shock Waves





Figure 7.6 (continued) (c) Space Shuttle Orbiter model at Mach 6. This photo also shows regions of high aerodynamic heating on the model surface by means of the visible phase-change paint pattern, (c/) A conceptual hypersonic aircraft at Mach 6. (Courtesy of the NASA Langley Research Center.)

In summary, compressible flows introduce some very exciting physical phenom­ena into our aerodynamic studies. Moreover, as the flow changes from subsonic to supersonic, the complete nature of the flow changes, not the least of which is the occurrence of shock waves. The purpose of the next seven chapters is to describe and analyze these flows.

Molecular Approach

In actuality, of course, the motion of a fluid is a ramification of the mean motion of its atoms and molecules. Therefore, a third model of the flow can be a microscopic approach wherein the fundamental laws of nature are applied directly to the atoms and molecules, using suitable statistical averaging to define the resulting fluid properties. This approach is in the purview of kinetic theory, which is a very elegant method with many advantages in the long run. However, it is beyond the scope of the present book.

In summary, although many variations on the theme can be found in different texts for the derivation of the general equations of fluid flow, the flow model can usually be categorized under one of the approaches described above.

2.3.2 Physical Meaning of the Divergence of Velocity

In the equations to follow, the divergence of velocity, V • V, occurs frequently. Before leaving this section, let us prove the statement made earlier (Section 2.2) that V • V is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element. Consider a control volume moving with the fluid (the case shown on the right of Figure 2.11). This control volume is always made up of the same fluid particles as it moves with the flow; hence, its mass is fixed, invariant with time. However, its volume V and control surface S are changing with time as it moves to different regions of the flow where different values of p exist. That is, this moving control volume of fixed mass is constantly increasing or decreasing its volume and is changing its shape, depending on the characteristics of the flow. This control volume is shown in Figure 2.13 at some instant in time. Consider an infinitesimal element of the surface d S moving at the local velocity V, as shown in Figure 2.13. The change in the volume of the control volume AV, due to just the


Figure 2.1 3 Moving control volume used for the physical interpretation of the divergence of velocity.


movement of dS over a time increment At, is, from Figure 2.13, equal to the volume of the long, thin cylinder with base area dS and altitude (V At) • n; i. e.,

ДУ = [(VAf) • n]dS = (VAt) ■ dS [2.28]

Over the time increment At, the total change in volume of the whole control volume is equal to the summation of Equation (2.28) over the total control surface. In the limit as dS —>■ 0, the sum becomes the surface integral

(VAt) • dS

Подпись: DV ~Dt Molecular Approach Подпись: [2.29]

If this integral is divided by At, the result is physically the time rate of change of the control volume, denoted by DV/Df, i. e.,

(The significance of the notation D/Dt is revealed in Section 2.9.) Applying the divergence theorem, Equation (2.26), to the right side of Equation (2.29), we have

^ – V)dV [2.30]


Now let us imagine that the moving control volume in Figure 2.13 is shrunk to a very small volume 8V, essentially becoming an infinitesimal moving fluid element as sketched on the right of Figure 2.12. Then Equation (2.30) can be written as


Assume that 8 V is small enough such that V • V is essentially the same value through­out (5 V. Then the integral in Equation (2.31) can be approximated as (V • V)i5V. From

Equation (2.31), we have


Подпись: D(SV) Dt

Подпись: or Подпись: V Подпись: 1 £>(<5V) SV Dt Подпись: [2.32]

= (V • V)5V

Examine Equation (2.32). It states that V • V is physically the time rate of change of the volume of a moving fluid element, per unit volume. Hence, the interpretation of V • V, first given in Section 2.2.6, Divergence of a Vector Field, is now proved.

Source Flow: Our Second Elementary Flow

Consider a two-dimensional, incompressible flow where all the streamlines are straight lines emanating from a central point O, as shown at the left of Figure 3.20. Moreover, let the velocity along each of the streamlines vary inversely with distance from point O. Such a flow is called a source flow. Examining Figure 3.20, we see that the velocity components in the radial and tangential directions are Vr and Vg, respectively, where Vg = 0. The coordinate system in Figure 3.20 is a cylindrical

Source flow


Sink flow


Figure 3.20 Source and sink flows



coordinate system, with the г axis perpendicular to the page. (Note that polar co­ordinates are simply the cylindrical coordinates r and в confined to a single plane given by г = constant.) It is easily shown (see Problem 3.9) that (1) source flow is a physically possible incompressible flow, that is, V • V = 0, at every point except the origin, where V • V becomes infinite, and (2) source flow is irrotational at every point.

In a source flow, the streamlines are directed away from the origin, as shown at the left of Figure 3.20. The opposite case is that of a sink flow, where by definition the streamlines are directed toward the origin, as shown at the right of Figure 3.20. For sink flow, the streamlines are still radial lines from a common origin, along which the flow velocity varies inversely with distance from point О. Indeed, a sink flow is simply a negative source flow.

The flows in Figure 3.20 have an alternate, somewhat philosophical interpreta­tion. Consider the origin, point О, as a discrete source or sink. Moreover, interpret the radial flow surrounding the origin as simply being induced by the presence of the discrete source or sink at the origin (much like a magnetic field is induced in the space surrounding a current-carrying wire). Recall that, for a source flow, V • V = 0 everywhere except at the origin, where it is infinite. Thus, the origin is a singular point, and we can interpret this singular point as a discrete source or sink of a given strength, with a corresponding induced flow field about the point. This interpreta­tion is very convenient and is used frequently. Other types of singularities, such as doublets and vortices, are introduced in subsequent sections. Indeed, the irrotational, incompressible flow field about an arbitrary body can be visualized as a flow induced by a proper distribution of such singularities over the surface of the body. This concept is fundamental to many theoretical solutions of incompressible flow over airfoils and other aerodynamic shapes, and it is the very heart of modem numerical techniques for the solution of such flows. You will obtain a greater appreciation for the concept of distributed singularities for the solution of incompressible flow in Chapters 4 through

6. At this stage, however, simply visualize a discrete source (or sink) as a singularity that induces the flows shown in Figure 3.20.

Let us look more closely at the velocity field induced by a source or sink. By definition, the velocity is inversely proportional to the radial distance r. As stated earlier, this velocity variation is a physically possible flow, because it yields V • V =

0. Moreover, it is the only such velocity variation for which the relation V • V = 0 is satisfied for the radial flows shown in Figure 3.20. Hence,

Vr = – [3.59a]


and Ve = 0 [3.59b]

where c is constant. The value of the constant is related to the volume flow from the source, as follows. In Figure 3.20, consider a depth of length l perpendicular to the page, that is, a length l along the z axis. This is sketched in three-dimensional perspective in Figure 3.21. In Figure 3.21, we can visualize an entire line of sources along the z axis, of which the source О is just part. Therefore, in a two-dimensional flow, the discrete source, sketched in Figure 3.20, is simply a single point on the line source shown in Figure 3.21. The two-dimensional flow shown in Figure 3.20 is the









Figure 3.21 Volume flow rate from a line source.



same in any plane perpendicular to the z, axis, that is, for any plane given by z, = constant. Consider the mass flow across the surface of the cylinder of radius r and height l as shown in Figure 3.21. The elemental mass flow across the surface element dS shown in Figure 3.21 is pV • dS = pVr( r d6){l). Hence, noting that Vr is the same value at any в location for the fixed radius r, the total mass flow across the surface of the cylinder is


Since p is defined as the mass per unit volume and m is mass per second, then m/p is the volume flow per second. Denote this rate of volume flow by v. Thus, from Equation (3.60), we have

Подпись: [3.61]m

v = — = 2nrlVr P

Подпись: or image234 Подпись: [3.62]

Moreover, the rate of volume flow per unit length along the cylinder is Ь/1. Denote this volume flow rate per unit length (which is the same as per unit depth perpendicular to the page in Figure 3.20) as Л. Hence, from Equation 3.61, we obtain

Hence, comparing Equations (3.59a) and (3.62), we see that the constant in Equation (3.59a)isc = А/2л. In Equation (3.62), Л defines the source length: it is physically the rate of volume flow from the source, per unit depth perpendicular to the page

of Figure 3.20. Typical units of Л are square meters per second or square feet per second. In Equation (3.62), a positive value of Л represents a source, whereas a negative value represents a sink.

The velocity potential for a source can be obtained as follows. From Equations

(2.157) , (3.5%), and (3.62),


— = Vr =———-

dr 2nr


1 Э ф

and ——————————————— = Ve = 0

г дв


Integrating Equation (3.63) with respect to r, we have

Ф = ^Inr + f(6)



Integrating Equation (3.64) with respect to в, we have

ф = const + f{r)


Подпись: A ф = — In r 2n Подпись: [3.67]

Comparing Equations (3.65) and (3.66), we see that fir) = (h/2n) In r and f(6) = constant. As explained in Section 3.9, the constant can be dropped without loss of rigor, and hence Equation (3.65) yields

Equation (3.67) is the velocity potential for a two-dimensional source flow.

Подпись: and Source Flow: Our Second Elementary Flow Подпись: [3.68] [3.69]

The stream function can be obtained as follows. From Equations (2.151a and b), (3.59b), and (3.62),

Integrating Equation (3.68) with respect to в, we obtain

ф = ^в + f(r) [3.70]


Integrating Equation (3.69) with respect to r, we have

ф = const + f(6) [3.71]

Comparing Equations (3.70) and (3.71) and dropping the constant, we obtain


Equation (3.72) is the stream function for a two-dimensional source flow.

The equation of the streamlines can be obtained by setting Equation (3.72) equal to a constant:


f — —в = const [3.73]


From Equation (3.73), we see that в — constant, which, in polar coordinates, is the equation of a straight line from the origin. Hence, Equation (3.73) is consistent with the picture of the source flow sketched in Figure 3.20. Moreover, Equation (3.67) gives an equipotential line as r = constant, that is, a circle with its center at the origin, as shown by the dashed line in Figure 3.20. Once again, we see that streamlines and equipotential lines are mutually perpendicular.

To evaluate the circulation for source flow, recall the V x V = 0 everywhere. In turn, from Equation (2.137),

Г = ~ JJ(V x V) – dS = 0


for any closed curve C chosen in the flow field. Hence, as in the case of uniform flow discussed in Section 3.9, there is no circulation associated with the source flow.

It is straightforward to show that Equations (3.67) and (3.72) satisfy Laplace’s equation, simply by substitution into V20 = 0 and V2i/r = 0 written in terms of cylindrical coordinates [see Equation (3.42)]. Therefore, source flow is a viable elementary flow for use in building more complex flows.

Physical Significance

Consider again the basic model underlying Prandtl’s lifting-line theory. Return to Fig­ure 5.13 and study it carefully. An infinite number of infinitesimally weak horseshoe vortices are superimposed in such a fashion as to generate a lifting line which spans the wing, along with a vortex sheet which trails downstream. This trailing-vortex sheet is the instrument that induces downwash at the lifting line. At first thought, you might consider this model to be somewhat abstract—a mathematical convenience that somehow produces surprisingly useful results. However, to the contrary, the model shown in Figure 5.13 has real physical significance. To see this more clearly, return to Figure 5.1. Note that in the three-dimensional flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions; that is, there is a discontinuity in the tangential velocity at the trailing edge. We know from Chapter 4 that a discontinuous change in tangential velocity is theoretically allowed across a vortex sheet. In real life, such discontinuities do not exist; rather, the different velocities at the trailing edge generate a thin region of large velocity gradients—a thin region of shear flow with very large vorticity. Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing. This sheet

tends to roll up at the edges and helps to form the wing-tip vortices sketched in Fig­ure 5.2. Thus, Prandd’s lifting-line model with its trailing-vortex sheet is physically consistent with the actual flow downstream of a finite wing.


Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8. The airfoil section is thin and symmetric. Calculate the lift and induced drag coefficients for the wing when it is at an angle of attack of 5°. Assume that 5 = r.


From Figure 5.18, 5 = 0.055. Hence, from the stated assumption, r also equals 0.055. From Equation (5.70), assuming ao = 2n from thin airfoil theory,


Example 5.1


Physical Significance

_ Uo

1 + a0/+rAR(l + r) = 0.0867 degree-1


Since the airfoil is symmetric, ctL=o = 0°. Thus,


CL = act = (0.0867 degree 1 (5°) =




From Equation (5.61),


(0.4335)2(1 +0.055)


Physical Significance





Physical Significance

Подпись: Example 5.2Consider a rectangular wing with an aspect ratio of 6, an induced drag factor 5 = 0.055, and a zero-lift angle of attack of —2°. At an angle of attack of 3.4°, the induced drag coefficient for this wing is 0.01. Calculate the induced drag coefficient for a similar wing (a rectangular wing with the same airfoil section) at the same angle of attack, but with an aspect ratio of 10. Assume that the induced factors for drag and the lift slope, S and r, respectively, are equal to each other (i. e., 5 = г). Also, for AR = 10, 5 = 0.105.


Physical Significance Подпись: TZARCDJ 1+5 Physical Significance

We must recall that although the angle of attack is the same for the two cases compared here (AR = 6 and 10), the value of Cl is different because of the aspect-ratio effect on the lift slope. First, let us calculate Cl for the wing with aspect ratio 6. From Equation (5.61),

Hence, CL = 0.423

The lift slope of this wing is therefore dCL 0.423

—– = —————- = 0.078/degree = 4.485/rad

da 3.4° – (-2°) ‘ 6 ‘

Physical Significance

The lift slope for the airfoil (the infinite wing) can be obtained from Equation (5.70):

dCi ^ «о

da 1 + (a0/7rAR)(l + r)


a0 _ a0

1 + [(1.055)«0/л-(6)] 1 + 0.056a0




Solving for ao, we find that this yields ao = 5.989/rad. Since the second wing (with AR = 10) has the same airfoil section, then a0 is the same. The lift slope of the second wing is given by


a0 5.989

1 + (a0/TrAR)(l + r) _ 1 + [(5.989)(1.105)/л-(103ї

= 0.086/degree






The lift coefficient for the second wing is therefore


CL = a (a – aL=0) = 0.086[3.4° – (-2°)] = 0.464


In turn, the induced drag coefficient is


Physical Significance



Physical Significance

Note: This problem would have been more straightforward if the lift coefficients had been stipulated to be the same between the two wings rather than the angle of attack. Then Equation (5.61) would have yielded the induced drag coefficient directly. A purpose of this example is to reinforce the rationale behind Equation (5.65), which readily allows the scaling of drag coefficients from one aspect ratio to another, as long as the lift coefficient is the same. This allows the scaled drag-coefficient data to be plotted versus CL (not the angle of attack) as in Figure 5.20. However, in the present example where the angle of attack is the same between both cases, the effect of aspect ratio on the lift slope must be explicitly considered, as we have done above.

Подпись: Example 5.3Consider the twin-jet executive transport discussed in Example 1.6. In addition to the infor­mation given in Example 1.6, for this airplane the zero-lift angle of attack is —2°, the lift slope of the airfoil section is 0.1 per degree, the lift efficiency factor r = 0.04, and the wing aspect ratio is 7.96. At the cruising condition treated in Example 1.6, calculate the angle of attack of the airplane.


The lift slope of the airfoil section in radians is

a0 = 0.1 per degree = 0.1 (57.3) = 5.73 rad From Equation (5.70) repeated below

_ _____ "o____

1 + (a0/7rAR)(l + r)

Physical Significance

lift distribution reaching farther away from the root. Such wings require heavier internal structure. Hence, as the aspect ratio of a wing increases, so does the structural weight of the wing. As a result of this compromise between aerodynamics and structures, typical aspect ratios for conventional subsonic airplanes are on the order of 6 to 8.

However, examine the three-view of the Lockheed U-2 high altitude reconnaissance aircraft shown in Figure 5.24. This airplane has the unusually high aspect ratio of 14.3. Why? The answer is keyed to its mission. The U-2 was essentially a point design; it was to cruise at the exceptionally high altitude of 70,000 ft or higher in order to not be reached by interceptor aircraft or ground-to-air-missiles during overflights of the Soviet Union in the 1950s. To achieve this mission, the need for incorporating a very high aspect ratio wing was paramount, for the following reason. In steady, level flight, where the airplane lift L must equal its weight W,

L = W = q. xSCL = p^VlSCL [5.71]

As the airplane flies higher, px decreases and hence from Equation (5.71) С/. must be increased in order to keep the lift equal to the weight. As its high-altitude cruise design point, the U-2 flies at a high value of C;, just on the verge of stalling. (This is in stark contrast to the normal cruise conditions of conventional airplanes at conventional altitudes, where the cruise lift coefficient is relatively small.) At the high value of С/, for the U-2 at cruising altitude, its induced drag coefficient [which from Equation (5.62) varies as C} would be unacceptably high if a conventional aspect ratio were used. Hence, the Lockheed design group (at the Lockheed Skunk Works) had to opt for as high an aspect ratio as possible to keep the induced drag coefficient within reasonable bounds. The wing design shown in Figure 5.24 was the result.

We made an observation about induced drag Д itself, in contrast to the induced drag coefficient CD, . We have emphasized, based on Equation (5.62), that Cdj can be reduced by increasing the aspect ratio. For an airplane in steady, level flight, however, the induced drag force itself is governed by another design parameter, rather than the aspect ratio per se, as follows. From Equation (5.62), we have


Three-view of the Lockheed U-2 high-altitude reconnaissance airplane.


Figure 5.34



Physical Significance