Category Fundamentals of Aerodynamics

Review of Vector Relations

Aerodynamics is full of quantities that have both magnitude and direction, such as force and velocity. These are vector quantities, and as such, the mathematics of aerodynamics is most conveniently expressed in vector notation. The purpose of this section is to set forth the basic relations we need from vector algebra and vector calculus. If you are familiar with vector analysis, this section will serve as a concise review. If you are not conversant with vector analysis, this section will help you establish some vector notation, and will serve as a skeleton from which you can fill in more information from the many existing texts on vector analysis (see, e. g., References 4 to 6).

2.2.1 Some Vector Algebra

Consider a vector quantity A; both its magnitude and direction are given by the arrow labeled A in Figure 2.2. The absolute magnitude of A is |A|, and is a scalar quantity. The unit vector n, defined by n = A/|A|, has a magnitude of unity and a direction equal to that of A. Let В represent another vector. The vector addition of A and В yields a third vector C,

Подпись: [2.1]A + B = C

which is formed by connecting the tail of A with the head of B, as shown in Figure 2.2. Now consider —B, which is equal in magnitude to B, but opposite in direction. The vector subtraction of В and A yields vector D,

Подпись: [2.2]A – В = D

Vector subtraction

 

Vector

 

Vector addition

 

A

 

image86

Scalar product

 

G

 

Vector product

 

image85image87

Figure 2.2 Vector diagrams.

which is formed by connecting the tail of A with the head of —B, as shown in Figure 2.2.

There are two forms of vector multiplication. Consider two vectors A and В at an angle в to each other, as shown in Figure 2.2. The scalar product (dot product) of the two vectors A and В is defined as

A • В = IAI |B I cos0 [2.3]

= magnitude of A x magnitude of the component of В along the direction of A

Note that the scalar product of two vectors is a scalar. In contrast, the vector product (cross product) of the two vectors A and В is defined as

A x В = (|A||B| sin0)e = G [2.4]

where G is perpendicular to the plane of A and В and in a direction which obeys the “right-hand rule.” (Rotate A into B, as shown in Figure 2.2. Now curl the fingers of your right hand in the direction of the rotation. Your right thumb will be pointing in the direction of G.) In Equation (2.4), e is a unit vector in the direction of G, as also shown in Figure 2.2. Note that the vector product of two vectors is a vector.

Pitot Tube: Measurement of Airspeed

In 1732, the Frenchman Henri Pitot was busy trying to measure the flow velocity of the Seine River in Paris. One of the instruments he used was his own invention—a strange-looking tube bent into an L shape, as shown in Figure 3.11. Pitot oriented one of the open ends of the tube so that it faced directly into the flow. In turn, he used the pressure inside this tube to measure the water flow velocity. This was the first time in history that a proper measurement of fluid velocity was made, and Pitot’s invention has carried through to the present day as the Pitot tube—one of the most common and frequently used instruments in any modern aerodynamic laboratory. Moreover, a Pitot tube is the most common device for measuring flight velocities of airplanes. The purpose of this section is to describe the basic principle of the Pitot tube.5

Consider a flow with pressure p moving with velocity V. as sketched at the left of Figure 3.11. Let us consider the significance of the pressure p more closely. In Section 1.4, the pressure is associated with the time rate of change of momentum of the gas molecules impacting on or crossing a surface; that is, pressure is clearly related to the motion of the molecules. This motion is very random, with molecules moving in all directions with various velocities. Now imagine that you hop on a fluid

5 See chapter 4 of Reference 2 for a detailed discussion of the history of the Pitot tube, how Pitot used it to overturn о basic theory in civil engineering, how it created some controversy in engineering, and how it finally found application in aeronautics.

Pitot Tube: Measurement of Airspeed

image213

Static pressure A /measured here

yy>//777;77^ " 7^7777777^

Подпись: ^ Static-pressure orifice

Figure 3.1 1 Pitot tube and a static pressure orifice.

element of the flow and ride with it at the velocity V. The gas molecules, because of their random motion, will still bump into you, and you will feel the pressure p of the gas. We now give this pressure a specific name: the static pressure. Static pressure is a measure of the purely random motion of molecules in the gas; it is the pressure you feel when you ride along with the gas at the local flow velocity. All pressures used in this book so far have been static pressures; the pressure p appearing in all our previous equations has been the static pressure. In engineering, whenever a reference is made to “pressure” without further qualification, that pressure is always interpreted as the static pressure. Furthermore, consider a boundary of the flow, such as a wall, where a small hole is drilled perpendicular to the surface. The plane of the hole is parallel to the flow, as shown at point A in Figure 3.11. Because the flow moves over the opening, the pressure felt at point A is due only to the random motion of the molecules; that is, at point A, the static pressure is measured. Such a small hole in the surface is called a static pressure orifice, or a static pressure tap.

In contrast, consider that a Pitot tube is now inserted into the flow, with an open end facing directly into the flow. That is, the plane of the opening of the tube is perpendicular to the flow, as shown at point В in Figure 3.11. The other end of the Pitot tube is connected to a pressure gage, such as point C in Figure 3.11 (i. e., the Pitot tube is closed at point C). For the first few milliseconds after the Pitot tube is inserted into the flow, the gas will rush into the open end and will fill the tube. However, the tube is closed at point C; there is no place for the gas to go, and hence after a brief period of adjustment, the gas inside the tube will stagnate; that is, the gas velocity inside the tube will go to zero. Indeed, the gas will eventually pile up

and stagnate everywhere inside the tube, including at the open mouth at point B. As a result, the streamline of the flow that impinges directly at the open face of the tube (streamline DB in Figure 3.11) sees this face as an obstruction to the flow. The fluid elements along streamline DB slow down as they get closer to the Pitot tube and go to zero velocity right at point B. Any point in a flow where V = 0 is called a stagnation point of the flow; hence, point В at the open face of the Pitot tube is a stagnation point, where VB = 0. In turn, from Bernoulli’s equation we know the pressure increases as the velocity decreases. Hence, pB > p. The pressure at a stagnation point is called the stagnation pressure, or total pressure, denoted by po- Hence, at point В, рв = po-

From the above discussion, we see that two types of pressure can be defined for a given flow: static pressure, which is the pressure you feel by moving with the flow at its local velocity Vj, and total pressure, which is the pressure that the flow achieves when the velocity is reduced to zero. In aerodynamics, the distinction between total and static pressure is important; we have discussed this distinction at some length, and you should make yourself comfortable with the above paragraphs before proceeding further. (Further elaboration on the meaning and significance of total and static pressure will be made in Chapter 7.)

Подпись: or Подпись: PA + pV = PB + pV Pi + рУ = Po + 0 Подпись: [3.33]

How is the Pitot tube used to measure flow velocity? To answer this question, first note that the total pressure po exerted by the flow at the tube inlet (point В) is impressed throughout the tube (there is no flow inside the tube; hence, the pressure everywhere inside the tube is po). Therefore, the pressure gage at point C reads Po – This measurement, in conjunction with a measurement of the static pressure p at point A, yields the difference between total and static pressure, po — Pi, and it is this pressure difference that allows the calculation of Vj via Bernoulli’s equation. In particular, apply Bernoulli’s equation between point A, where the pressure and velocity are p and V, respectively, and point B, where the pressure and velocity are po and V = 0, respectively:

Solving Equation (3.33) for Vj, we have

image214[3.34]

Equation (3.34) allows the calculation of velocity simply from the measured difference between total and static pressure. The total pressure po is obtained from the Pitot tube, and the static pressure p is obtained from a suitably placed static pressure tap.

It is possible to combine the measurement of both total and static pressure in one instrument, a Pitot-static probe, as sketched in Figure 3.12. A Pitot-static probe measures po at the nose of the probe and p at a suitably placed static pressure tap on the probe surface downstream of the nose.

In Equation (3.33), the term Vj2 is called the dynamic pressure and is denoted by the symbol q. The grouping pV2 is called the dynamic pressure by definition

image215

Figure 3.12 Pitot-static probe.

 

and is used in all flows, incompressible to hypersonic:

q = pV2

However, for incompressible flow, the dynamic pressure has special meaning; it is precisely the difference between total and static pressure. Repeating Equation (3.33), we obtain

P +

pVi =

Po

static

dynamic

total

pressure

pressure

pressure

or

pi

+ q – po

or

qі = Po ~ Pi

[3.35]

It is important to keep in mind that Equation (3.35) comes from Bernoulli’s equation, and thus holds for incompressible flow only. For compressible flow, where Bernoulli’s equation is not valid, the pressure difference p0 — p is not equal to q. Moreover, Equation (3.34) is valid for incompressible flow only. The velocities of compressible flows, both subsonic and supersonic, can be measured by means of a Pitot tube, but the equations are different from Equation (3.34). (Velocity measurements in subsonic and supersonic compressible flows are discussed in Chapter 8.)

At this stage, it is important to repeat that Bernoulli’s equation holds for incom­pressible flow only, and therefore any result derived from Bernoulli’s equation also holds for incompressible flow only, such as Equations (3.26), (3.32), and (3.34). Ex­perience has shown that some students when first introduced to aerodynamics seem to adopt Bernoulli’s equation as the gospel and tend to use it for all applications, including many cases where it is not valid. Hopefully, the repetitive warnings given above will squelch such tendencies.

Подпись: Example 3.5An airplane is flying at standard sea level. The measurement obtained from a Pitot tube mounted on the wing tip reads 2190 lb/ft2. What is the velocity of the airplane?

Solution

Standard sea level pressure is 2116 lb/ft2. From Equation (3.34), we have

image216Подпись: 250 ft/s2(po-Pi) = /2(2190 — 21167 p V 0.002377

In the wind-tunnel flow described in Example 3.4, a small Pitot tube is mounted in the flow just upstream of the model. Calculate the pressure measured by the Pitot tube for the same flow conditions as in Example 3.4.

Подпись: Example 3.6Solution

From Equation (3.35),

Po — Poo “Ь Чоо — Poo “h 2 Poo

= 2116+ 4(0.002377)(328.4)2

Подпись: 2244 lb/ft2= 2116+ 128.2 =

Note in this example that the dynamic pressure is ^Дх> = 128.2 lb/ft2. This is only 8 percent larger than the pressure difference (p — /+), calculated in Example 3.4, that is required to produce the test-section velocity in the wind tunnel. Why is (pi — po) so close to the test-section dynamic pressure? Answer: Because the velocity in the settling chamber Vi is so small that p is close to the total pressure of the flow. Indeed, from Equation (3.22),

v’ = J[Vl = (їг) (328’4) = 213 ft/s

Compared to the test-section velocity of 328.4 ft/s, V is seen to be small. In regions of a flow where the velocity is finite but small, the local static pressure is close to the total pressure. (Indeed, in the limiting case of a fluid with zero velocity, the local static pressure is the same as the total pressure; here, the concepts of static pressure and total pressure are redundant. For example, consider the air in the room around you. Assuming the air is motionless, and assuming standard sea level conditions, the pressure is 2116 lb/ft2, namely, 1 atm. Is this pressure a static pressure or a total pressure? Answer: It is both. By the definition of total pressure given in the present section, when the local flow velocity is itself zero, then the local static pressure and the local total pressure are exactly the same.)

Design Box

The configuration of the Pitot-static probe shown in Figure 3.12 is a schematic only. The design of an actual Pitot-static probe is an example of careful engineering, intended to provide as accurate an instrument as possible. Let us examine some of the overall features of Pitot-static probe design.

Above all, the probe should be a long, streamlined shape such that the surface pressure over a substantial portion of the probe is essentially equal to the free stream static pressure. Such a shape is given in Figure 3.13a. The head of the probe, the nose at which the total pressure is measured, is usually a smooth hemispherical shape in order to encourage smooth, streamlined flow downstream of the nose. The diameter of the tube is denoted by d. A number of static pressure taps are arrayed radially around the circumference of the tube at a station that should be from 8d to 16d downstream of the nose, and at least 16<з? ahead of the downstream support stem. The reason for

 

– 14 d

 

20 d

 

Pitot Tube: Measurement of Airspeed

Stagnation point Total pressure measured here

 

(«)

image218

Figure 3.13 (a) Pitot-static tube, (b) Schematic of the pressure distribution along

the outer surface of the tube.

 

this is shown in Figure 3.13b, which gives the axial distribution of the pressure coefficient along the surface of the tube. From the definition of pressure coefficient given in Section 1.5, and from Bernoulli’s equation in the form of Equation (3.35), the pressure coefficient at a stagnation point for incompressible flow is given by

C, = = ^ = 1.0

с Цэc

Hence, in Figure 3.13b the Cp distribution starts out at the value of 1.0 at the nose, and rapidly drops as the flow expands around the nose. The pressure decreases below px, yielding a minimum value of Cp ~ —1.25 just downstream of the nose. Further downstream the pressure tries to recover and approaches a value nearly equal to px at some distance (typically about 8d) from the nose. There follows a region where the static pressure along the surface of the tube is very close to px, illustrated by the region where Cp = 0 in Figure 3.13b. This is the region where the static pressure taps should be located, because the surface pressure measured at these taps will be essentially equal to the freestream static pressure px. Further downstream, as the flow approaches the support stem, the pressure starts to increase above px. This starts at a distance of about 16d ahead of the support stem. In Figure 3.13a, the static pressure taps are shown at a station 14d downstream of the nose and 20d ahead of the support stem.

The design of the static pressure taps themselves is critical. The surface around the taps should be smooth to insure that the pressure sensed inside the tap is indeed the surface pressure along the tube. Examples of poor design as well as the proper design of the pressure taps are shown in Figure 3.14. In Figure 3.14a, the surface has a burr on the upstream side; the local flow will expand around this burr, causing the pressure sensed at point a inside the tap to be less than px. In 3.14 b, the surface has a burr on the downstream side; the local flow will be

 

image217

Pitot Tube: Measurement of Airspeed

 

slowed in this region, causing the pressure sensed at point b inside the tap to be greater than p^. The correct design is shown in Figure 3.14c; here, the opening of the tap is exactly flush with the surface, allowing the pressure sensed at point c inside the tap to be equal to p

When a Pitot-static tube is used to measure the speed of an airplane, it should be located on the airplane in a position where it is essentially immersed in the freestream flow, away from any major influence of the local flow field around the airplane itself. An example of this can be seen in Figure 3.2, where a Pitot-static probe can be seen mounted near the right wing tip of the P-35, extending into the freestream ahead of the wing. A similar wing-mounted probe is shown in the planview (top view) of the North American F-86 in Figure 3.15.

Today, many modem airplanes have a Pitot tube mounted at some location on the fuselage, and the measurement of poo is obtained independently from a properly placed static pressure tap somewhere else on the fuselage. Figure 3.16 illustrates a fuselage-mounted Pitot tube in the nose region of the Boeing Stratoliner, a 1940s vintage airliner. When only a Pitot measurement is required, the probe can be much shorter than a Pitot-static tube, as can be seen in Figure 3.16. In this type of arrangement, the location of the static pressure tap on the surface of the fuselage is critical; it must be located in a region where the surface pressure on the fuselage is equal to pa0. We have a pretty good idea where to locate the static pressure taps on a Pitot-static tube, as shown in Figure 3.13a. But the proper location on the fuselage of a given airplane must be found experimentally, and it is different for different airplanes. However, the basic idea is illustrated in Figure 3.17, which shows the measured pressure coefficient distribution over a streamlined body at zero angle of attack. There are two axial stations where Cp = 0 (i. e., where the surface pressure on the body equals p~,_). If this body were an airplane fuselage, the static pressure tap should be placed at one of these two locations. In practice, the forward location, near the nose, is usually chosen.

Finally, we must be aware that none of these instruments, no matter where they are located, are perfectly accurate. In particular, misalignment of the probe with respect to the freestream direction causes an error which must be assessed for each particular case. Fortunately, the measurement of the total pressure by means of a Pitot tube is relatively insensitive to misalignment. Pitot tubes with hemispherical noses, such as shown in Figure 3.13a, are insensitive to the mean flow direction up to a few degrees. Pitot tubes with flat faces, such as illustrated in Figure 3.12, are least sensitive. For these tubes, the total pressure measurement varies only 1 percent for misalignment as large as 20°. For more details on this matter, see Reference 65.

 

image219

image220

Figure 3.15 Three-view of the North American F-86H. Note the wing-mounted Pitot-static tube.

Pitot tube

 

Figure 3.16 Nose-mounted Pitot tube on the Boeing Stratoliner. (Stratoliner detail

courtesy of Paul Matt, Alan and Drina Abel, and Aviation Heritage, Inc., with permission.)

image222

 

Experimentally measured pressure coefficient distribution over a streamlined body with a fineness ratio (length-to-diameter ratio) of 3. Zero angle of attack. Low-speed flow.

 

Figure 3.17

 

image221

Historical Note: Early Airplane Design and the Role of Airfoil Thickness

In 1804, the first modem configuration aircraft was conceived and built by Sir George Cayley in England—it was an elementary hand-launched glider, about a meter in length, and with a kitelike shape for a wing as shown in Figure 4.44. (For the pivotal role played by George Cayley in the development of the airplane, see the exten­sive historical discussion in chapter 1 of Reference 2.) Note that right from the beginning of the modem configuration aircraft, the wing sections were very thin— whatever thickness was present, it was strictly for structural stiffness of the wing. Extremely thin airfoil sections were perpetuated by the work of Horatio Phillips in England. Phillips carried out the first serious wind-tunnel experiments in which the aerodynamic characteristics of a number of different airfoil shapes were measured. (See section 5.20 of Reference 2 for a presentation of the historical development of airfoils.) Some of Phillips airfoil sections are shown in Figure 4.45—note that they are the epitome of exceptionally thin airfoils. The early pioneers of aviation such as Otto Lilienthal in Germany and Samuel Pierpont Langley in America (see

image384

Figure 4.44 The first modern configuration airplane in history: George Cayley’s model glider of 1804.

 

No. 1

 

No. 2

 

No. 3

 

^2222^

 

No. 4

 

No. 5

image385

Figure 4.45 Double-surface airfoil sections by Horatio Phillips. The six upper shapes were patented by Phillips in 1 884; the lower airfoil was patented in 1891. Note the thin profile shapes.

 

Historical Note: Early Airplane Design and the Role of Airfoil Thickness

chapter 1 of Reference 2) continued this thin airfoil tradition. This was especially true of the Wright brothers, who in the period of 1901-1902 tested hundreds of different wing sections and planform shapes in their wind tunnel in Dayton, Ohio (recall our discussion in Section 1.1 and the models shown in Figure 1.2). A sketch of some of the Wrights’ airfoil sections is given in Figure 4.46—for the most part, very thin sections. Indeed, such a thin airfoil section was used on the 1903 Wright Flyer, as can be readily seen in the side view of the Flyer shown in Figure 4.47. The important point here is that all of the early pioneering aircraft, and especially the Wright Flyer, incorporated very thin airfoil sections—airfoil sections that performed essentially like the flat plate results discussed in Section 4.12, and as shown in Figure 4.36 (the dashed curve) and by the streamline pictures in Figure 4.37. Conclusion: These early airfoil sections suffered flow-field separation at small angles of attack and, consequently, had low values of c; max. By the standards we apply today, these were simply very poor airfoil sections for the production of high lift.

-VZV77777,

image386

Figure 4.46 Some typical airfoil shapes tested by the Wright brothers in their wind tunnel during 1902-1903.

image387

image388

Figure 4.47 Front and side views of the 1903 Wright Flyer. Note the thin airfoil sections. ICourtesy of the National Air and Space Museum.)

This situation carried into the early part of World War I. In Figure 4.48, we see four airfoil sections that were employed on World War I aircraft. The top three sections had thickness ratios of about 4 to 5 percent and are representative of the type of sections used on all aircraft until 1917. For example, the SPAD XIII (shown in Figure 3.50),

image390

RAF 14, British

 

Gottingen 298, German

 

Some examples of different airfoil shapes used on World War I aircraft, jSource: Loftin, Reference 48.)

 

Figure 4.48

 

image389image391image392

the fastest of all World War I fighters, had a thin airfoil section like the Eiffel section shown in Figure 4.48. Why were such thin airfoil sections considered to be the best by most designers of World War I aircraft? The historical tradition described above might be part of the answer—a tradition that started with Cayley. Also, there was quite clearly a mistaken notion at that time that thick airfoils would produce high drag. Of course, today we know the opposite to be true; review our discussion of streamlined shapes in Section 1.11 for this fact. Laurence Loftin in Reference 48 surmises that the mistaken notion might have been fostered by early wind-tunnel tests. By the nature of the early wind tunnels in use—small sizes and very low speeds—the data were obtained at very low Reynolds numbers, less than 100,000 based on the airfoil-chord length. These Reynolds numbers are to be compared with typical values in the millions for actual airplane flight. Modem studies of low Reynolds number flows over conventional thick airfoils (e. g., see Reference 51) clearly show high-drag coefficients, in contrast to the lower values that occur for the high Reynolds number associated with the flight of full-scale aircraft. Also, the reason for the World War I airplane designer’s preference for thin airfoils might be as simple as the tendency to follow the example of the wings of birds, which are quite thin. In any event, the design of all English, French, and American World War I aircraft incorporated thin airfoils and, consequently, suffered from poor high-lift performance. The fundamentals of airfoil aerodynamics as we know them today (and as being presented in this book) were simply not sufficiently understood by the designers during World War I. In turn, they never appreciated what they were losing.

This situation changed dramatically in 1917. Work carried out in Germany at the famous Gottingen aerodynamic laboratory of Ludwig Prandtl (see Section 5.8 for a biographical sketch of Prandtl) demonstrated the superiority of a thick airfoil section, such as the Gottingen 298 section shown at the bottom of Figure 4.48. This revolutionary development was immediately picked up by the famous designer An­thony Fokker, who incorporated the 13-percent-thick Gottingen 298 profile in his new Fokker Dr-1—the famous triplane flown by the “Red Baron,” Rittmeister Manfred Freiher von Richthofen. A photograph of the Fokker Dr-1 is shown in Figure 4.49. The major benefits derived from Fokker’s use of the thick airfoil were:

1. The wing structure could be completely internal; that is the wings of the Dr – 1 were a cantilever design, which removed the need for the conventional wire bracing that was used in other aircraft. This, in turn, eliminated the high drag associated with these interwing wires, as discussed at the end of Section 1.11. For this reason, the Dr-1 had a zero-lift drag coefficient of 0.032, among the lowest of World War I airplanes. (By comparison the zero-lift drag coefficient of the French SPAD XIII was 0.037.)

2. The thick airfoil provided the Fokker Dr-1 with a high maximum lift coefficient. Its performance was analogous to the upper curves shown in Figure 4.36. This in turn, provided the Dr-1 with an exceptionally high rate-of-climb as well as enhanced maneuverability—characteristics that were dominant in dog-fighting combat.

Anthony Fokker continued the use of a thick airfoil in his design of the D-VII, as shown in Figure 4.50. This gave the D-VII a much greater rate-of-climb than its two principal opponents at the end of the war—the English Sopwith Camel and the

image393

image394

Figure 4.50 The World War I Fokker D-VII, one of the most effective fighters of the war, due in part to its superior aerodynamic performance allowed by a thick airfoil section.

French SPAD XIII, both of which still used very thin airfoil sections. This rate-of – climb performance, as well as its excellent handling characteristics, singled out the Fokker D-VII as the most effective of all German World War I fighters. The respect given by the Allies to this machine is no more clearly indicated than by a paragraph in article IV of the armistice agreement, which lists war material to be handed over to the Allies by Germany. In this article, the Fokker D-VII is specifically listed—the only airplane of any type to be explicitly mentioned in the armistice. To this author’s knowledge, this is the one and only time where a breakthrough in airfoil technology is essentially reflected in any major political document, though somewhat implicitly.

Special Forms of the Energy Equation

Special Forms of the Energy Equation Подпись: [8.38]
image531

In this section, we elaborate upon the energy equation for adiabatic flow, as originally given by Equation (7.44). In Section 7.5, we obtained for a steady, adiabatic, inviscid flow the result that

where V and VA are velocities at any two points along a three-dimensional streamline. For the sake of consistency in our current discussion of one-dimensional flow, let us use и і and м2 in Equation (8.28):

2 2

MT ui, „

hi + – j – = h2 + у [8.39]

However, keep in mind that all the subsequent results in this section hold in general

along a streamline and are by no means limited to just one-dimensional flows.

Specializing Equation (8.29) to a calorically perfect gas, where h = cpT, we

Подпись: u u cpT + у = срТг + у Подпись: [8.30]

have

From Equation (7.9), Equation (8.30) becomes

Подпись: [8.31]yRTi _ yRT2 и| y-l+2~y-l+2

Special Forms of the Energy Equation Подпись: [8.33]
image532 image533 image534

Since a = л/уТТГ, Equation (8.31) can be written as

Special Forms of the Energy Equation Подпись: [8.33]
image535 image536

If we consider point 2 in Equation (8.32) to be a stagnation point, where the stagnation speed of sound is denoted by ao, then, with u2 = 0, Equation (8.32) yields (dropping the subscript 1)

Подпись: 1 + 2 у Special Forms of the Energy Equation Подпись: [8.34]

In Equation (8.33), a and и are the speed of sound and flow velocity, respectively, at any given point in the flow, and ao is the stagnation (or total) speed of sound associated with that same point. Equivalently, if we have any two points along a streamline, Equation (8.33) states that

Подпись: or image537,image538,image539 Подпись: [8.35]

Recalling the definition of a* given at the end of Section 7.5, let point 2 in Equation (8.32) represent sonic flow, where и = a*. Then

Подпись: 7 + T Подпись: a , 4 У- 1 2 Special Forms of the Energy Equation Подпись: [8.36]

In Equation (8.35), a and и are the speed of sound and flow velocity, respectively, at any given point in the flow, and a* is a characteristic value associated with that same point. Equivalently, if we have any two points along a streamline, Equation (8.35) states that

Comparing the right-hand sides of Equations (8.34) and (8.36), the two properties a0 and a* associated with the flow are related by

V + 1 *2 flQ r ,

——— – a*2 = —5— = const Ї8.37І

2(y – 1) У – 1

Clearly, these defined quantities, a0 and a*, are both constants along a given streamline in a steady, adiabatic, inviscid flow. If all the streamlines emanate from the same uniform freestream conditions, then a(l and a* are constants throughout the entire flow field.

Подпись: CPT + Y = CpTo Подпись: [8.38]

Recall the definition of total temperature To, as discussed in Section 7.5. In Equation (8.30), let u2 = 0; hence T2 = 70. Dropping the subscript 1, we have

Equation (8.38) provides a formula from which the defined total temperature TQ can be calculated from the given actual conditions of T and и at any given point in a general flow field. Equivalently, if we have any two points along a streamline in a steady, adiabatic, inviscid flow, Equation (8.38) states that

2 2

CPT + Y = cpT2 + Y = CpTo = const [8.39]

If all the streamlines emanate from the same uniform freestream, then Equation (8.39) holds throughout the entire flow, not just along a streamline.

For a calorically perfect gas, the ratio of total temperature to static temperature То/T is a function of Mach number only, as follows. From Equations (8.38) and

(7.9) , we have

To і і и2 1 i и2 1 | и2

7 = 1 + 2c~^T = 1 + lyRT/iy – О = 1 + 2a2/(у – 1)

Подпись: Y- 1Подпись: 2image540

Подпись: Hence, Подпись: [8.40]
image541 image542

= 1 +

Equation (8.40) is very important; it states that only M (and, of course, the value of y) dictates the ratio of total temperature to static temperature.

Recall the definition of total pressure po and total density p0, as discussed in Section 7.5. These definitions involve an isentropic compression of the flow to zero velocity. From Equation (7.32), we have

Special Forms of the Energy Equation

Y

 

yPy – 1)

 

Po

P

 

To

T

 

Po

P

 

[8.41]

 

Combining Equations (8.40) and (8.41), we obtain

image543[8.42]

[8.43]

Similar to the case of Tq/T, we see from Equations (8.42) and (8.43) that the total – to-static ratios po/p and po/p are determined by M and у only. Hence, for a given gas (i. e., given y), the ratios T0/T, po/p, and po/p depend only on Mach number.

Equations (8.40), (8.42), and (8.43) are very important; they should be branded on your mind. They provide formulas from which the defined quantities 7b, po, and Po can be calculated from the actual conditions of M, T, p, and p at a given point in a general flow field (assuming a calorically perfect gas). They are so important that values of T0/T, p0/p, and po/p obtained from Equations (8.40), (8.42), and (8.43), respectively, are tabulated as functions of M in Appendix A for у = 1.4 (which corresponds to air at standard conditions).

Consider a point in a general flow where the velocity is exactly sonic (i. e., where M = 1). Denote the static temperature, pressure, and density at this sonic condition as T*, p*, and p*, respectively. Inserting M = 1 into Equations (8.40), (8.42), and

(8.43) , we obtain

image544[8.44]

[8.45]

[8.48]

For у = 1.4, these ratios are

у *

— = 0.833

p*

— = 0.528

P*

— = 0.634

To

Po

Po

which are useful numbers to keep in mind for subsequent discussions.

We have one final item of business in this section. In Chapter 1, we defined the Mach number as M = V/a (or, following the one-dimensional notation in this chapter, M = и/a). In turn, this allowed us to define several regimes of flow, among them being

M < 1 (subsonic flow)

M = 1 (sonic flow)

M > 1 (supersonic flow)

In the definition of M, a is the local speed of sound, a = л/yRT. In the theory of supersonic flow, it is sometimes convenient to introduce a “characteristic” Mach number M* defined as

Подпись: M*и

a* where a* is the value of the speed of sound at sonic conditions, not the actual local value. This is the same a* introduced at the end of Section 7.5 and used in Equation

(8.35) . The value of a* is given by a* = *JyRT*. Let us now obtain a relation between the actual Mach number M and this defined characteristic Mach number M*. Dividing Equation (8.35) by u2, we have

(a/u)2 1 _ у + 1 / a*2

у – 1 2 “ 2(y – 1) V и )

(1/M)2 = y + 1 /J_2 _ 1

у – 1 2(y – 1) M*/ 2

Подпись:2 2

M2 = —————— Z————-

(у + 1 )/M*2 – (у – 1)

Подпись: _Jy±l)M2_ 2 + (y - 1)M2 Подпись: [8.48]

Equation (8.47) gives M as a function of M*. Solving Equation (8.47) for M*2, we have

which gives M* as a function of M. As can be shown by inserting numbers into Equation (8.48) (try some yourself),

M* = 1

if M = 1

M* < 1

if M < 1

M* > 1

if M > 1

Iy + 1

M* It———– r

if M —>■ oo

у – 1

Therefore, M* acts qualitatively in the same fashion as M except that M* approaches a finite value when the actual Mach number approaches infinity.

In summary, a number of equations have been derived in this section, all of which stem in one fashion or another from the basic energy equation for steady, inviscid, adiabatic flow. Make certain that you understand these equations and become very

2 Foe

[1] Strictly speaking, dA can never achieve the limit of zero, because there would be no molecules at point В in that case. The above limit should be interpreted as dA approaching a very small value, near zero in

terms of our macroscopic thinking, but sufficiently larger than the average spacing between molecules on a microscopic basis.

[3] The specific heat of a fluid is defined as the amount of heat added to a system, Sq, per unit increase in temperature; cv = Sq/dT if Sq is added at constant volume, and similarly, for cp if Sq is added at constant pressure. Specific heats are discussed in detail in Section 7.2. The thermo! conductivity relates heat flux to temperature gradients in the fluid. For example, if qx is the heat transferred in the x direction per second per unit area and aT/ax is the temperature gradient in the x direction, then thermal conductivity к is defined by qx = —k(a = T/ax). Thermal conductivity is discussed in detail in Section 15.3.

[4] Some books do not use the minus sign in the definition of circulation. In such cases, the positive sense of both the line integral and Г is in the same direction. This causes no problem as long as the reader is aware of the convention used in a particular book or paper.

[5] An inviscid, incompressible fluid is sometimes called an ideal fluid, or perfect fluid. This terminology will not be used here because of the confusion it sometimes causes with "ideal gases" or "perfect gases" from thermodynamics. This author prefers to use the more precise descriptor "inviscid, incompressible flow," rother than ideal fluid or perfect fluid.

[6] For a simpler, more rudimentary derivation of Equation (3.21), see chapter 4 of Reference 2. In the present discussion, we have established a more rigorous derivation of Equation (3.21), consistent with the general integral form of the continuity equation.

The above example illustrates two aspects of such a flow, as follows:

[8] Consider a given point on the airfoil surface. The Cp is given at this point and, from the statement of the problem. Cp is obviously unchanged when the velocity is increased from 80 to 300 ft/s. Why? The answer underscores part of our discussion on dimensional analysis in Section 1.7, namely, Cp should depend only on the Mach number, Reynolds number, shape and orientation of the body, and location on the body. For the low-speed inviscid flow considered here, the Mach number and Reynolds number are not in the picture. For this type of flow, the variation of Cp is a function only of location on the surface of the body, and the body shape and orientation. Hence, Cp will not change with or p^ as long as the flow can be considered inviscid and incompressible. For such a flow, once the Cp distribution over the body has been determined by some means, the same Cp distribution will exist for all freestream values of and

[9] In part ib) of Example 3.8, the velocity at the point where Cp is a peak (negative) value is a large value, namely, 753 ft/s. Is Equation (3.38) valid for this case? The answer is essentially no. Equation (3.38) assumes incompressible flow. The

Consider the nonlifting flow over a circular cylinder. Calculate the locations on the surface of the cylinder where the surface pressure equals the freestream pressure.

Solution

When p = pco, then Cp = 0. From Equation (3.101),

Cp = 0 = 1 — 4 sin2 в

Hence, sin# =

Aerodynamics, The Science of Air in Motion, McGraw-Hill, New York, 1982.

[12] For more details, see the interesting discussion on forest aerodynamics in the book by John E. Allen entitled

[13] The design lift coefficient is the theoretical lift coefficient for the airfoil when the angle of attack is such that the slope of the mean camber line at the leading edge is parallel to the freestream velocity.

[14] In many references, such as Reference 1 1, if is common to plot versus С/, rather than versus a. A plot of versus Cj is called a drag polar. For the sake of consistency with Figure 4.5, we choose to plot versus a here.

[15] Kelvin’s theorem also holds for an inviscid compressible flow in the special case where p — p[p); that is, the density is some single-valued function of pressure. Such is the case for isentropic flow, to be treated in later chapters.

[16] fh!2

CDJ = -—- r(y)oCj(y)dy

vcoJ J-b/2

Momentum Equation

Newton’s second law is frequently written as

F = та [2.55]

where F is the force exerted on a body of mass m and a is the acceleration. However, a more general form of Equation (2.55) is

F = — (mV) [2.56]

at

which reduces to Equation (2.55) for a body of constant mass. In Equation (2.56), mV is the momentum of a body of mass m. Equation (2.56) represents the second fundamental principle upon which theoretical fluid dynamics is based.

Physical principle Force = time rate of change of momentum

We will apply this principle [in the form of Equation (2.56)] to the model of a finite control volume fixed in space as sketched in Figure 2.17. Our objective is to obtain expressions for both the left and right sides of Equation (2.56) in terms of the familiar flow-field variables p, p, V, etc. First, let us concentrate on the left side of Equation

(2.56) , i. e., obtain an expression for F, which is the force exerted on the fluid as it flows through the control volume. This force comes from two sources:

1. Body forces: gravity, electromagnetic forces, or any other forces which “act at a distance” on the fluid inside V.

2. Surface forces: pressure and shear stress acting on the control surface S.

Let f represent the net body force per unit mass exerted on the fluid inside V. The body force on the elemental volume dV in Figure 2.17 is therefore

pi dV

Momentum Equation

and the total body force exerted on the fluid in the control volume is the summation of the above over the volume V:

where the negative sign indicates that the force is in the direction opposite of dS. That is, the control surface is experiencing a pressure force which is directed into the control volume and which is due to the pressure from the surroundings, and examination of Figure 2.17 shows that such an inward-directed force is in the direction opposite of
dS. The complete pressure force is the summation of the elemental forces over the entire control surface:

Подпись: Pressure force =image119[3.58]

In a viscous flow, the shear and normal viscous stresses also exert a surface force. A detailed evaluation of these viscous stresses is not warranted at this stage of our discussion. Let us simply recognize this effect by letting Fuscous denote the total viscous force exerted on the control surface. We are now ready to write an expression for the left-hand side of Equation (2.56). The total force experienced by the fluid as it is sweeping through the fixed control volume is given by the sum of Equations (2.57) and (2.58) and Fviscous:

F=j^pfdV-^pdS + Fviscous [3.59]

v s

Now consider the right side of Equation (2.56). The time rate of change of momentum of the fluid as it sweeps through the fixed control volume is the sum of two terms:

Подпись: [2.60a]Net flow of momentum out
of control volume across surface S

Momentum Equation Подпись: [2.60Ы

and

Consider the term denoted by G in Equation (2.60a). The flow has a certain momen­tum as it enters the control volume in Figure 2.17, and, in general, it has a different momentum as it leaves the control volume (due in part to the force F that is exerted on the fluid as it is sweeping through V). The net flow of momentum out of the control volume across the surface S is simply this outflow minus the inflow of momentum across the control surface. This change in momentum is denoted by G, as noted above. To obtain an expression for G, recall that the mass flow across the elemental area dS is (pV • dS); hence, the flow of momentum per second across dS is

(pV • dS)V

Momentum Equation Подпись: [2.61]

The net flow of momentum out of the control volume through S is the summation of the above elemental contributions, namely,

In Equation (2.61), recall that positive values of (pV • dS) represent mass flow out of the control volume, and negative values represent mass flow into the control volume. Hence, in Equation (2.61) the integral over the whole control surface is a combination of positive contributions (outflow of momentum) and negative contributions (inflow
of momentum), with the resulting value of the integral representing the net outflow of momentum. If G has a positive value, there is more momentum flowing out of the control volume per second than flowing in; conversely, if G has a negative value, there is more momentum flowing into the control volume per second than flowing out.

Now consider H from Equation (2.60b). The momentum of the fluid in the elemental volume dV shown in Figure 2.17 is

(,pdV)

The momentum contained at any instant inside the control volume is therefore

v

and its time rate of change due to unsteady flow fluctuations is

pVdV [2.62]

v

Combining Equations (2.61) and (2.62), we obtain an expression for the total time rate of change of momentum of the fluid as it sweeps through the fixed control volume, which in turn represents the right-hand side of Equation (2.56):

(mV) = G + H = <jlj> (pV • dS)V + Pv dV [2.63]

S V

Hence, from Equations (2.59) and (2.63), Newton’s second law,

d

— (mV) = F dt

applied to a fluid flow is

image120

[2.64]

Equation (2.64) is the momentum equation in integral form. Note that it is a vector equation. Just as in the case of the integral form of the continuity equation, Equation

(2.64) has the advantage of relating aerodynamic phenomena over a finite region of space without being concerned with the details of precisely what is happening at a given distinct point in the flow. This advantage is illustrated in Section 2.6.

From Equation (2.64), we now proceed to a partial differential equation which relates flow-field properties at a point in space. Such an equation is a counterpart to the differential form of the continuity equation given in Equation (2.52). Apply

Momentum Equation

where the subscripts у and z on / and T denote the у and z components of the body and viscous forces, respectively. Equations (2.70a to c) are the scalar x, y, and г components of the momentum equation, respectively; they are partial differential equations that relate flow-field properties at any point in the flow.

Подпись: and image124 Momentum Equation

Note that Equations (2.64) and (2.70a to c) apply to the unsteady, three-dimensional flow of any fluid, compressible or incompressible, viscous or inviscid. Specialized to a steady (d/dt = 0), inviscid (FviSCous = 0) flow with no body forces (f = 0), these equations become

Since most of the material in Chapters 3 through 14 assumes steady, inviscid flow with no body forces, we will have frequent occasion to use the momentum equation in the forms of Equations (2.71) and (2.72a to c).

The momentum equations for an inviscid flow [such as Equations (2.72a to c)] are called the Euler equations. The momentum equations for a viscous flow [such as Equations (2.70a to c) are called the Navier-Stokes equations. We will encounter this terminology in subsequent chapters.

Vortex Flow: Our Fourth Elementary Flow

Again, consulting our chapter road map in Figure 3.4, we have discussed three ele­mentary flows—uniform flow, source flow, and doublet flow—and have superimposed these elementary flows to obtain the nonlifting flow over several body shapes, such as ovals and circular cylinders. In this section, we introduce our fourth, and last, elementary flow; vortex flow. In turn, in Sections 3.15 and 3.16, we see how the superposition of flows involving such vortices leads to cases with finite lift.

Consider a flow where all the streamlines are concentric circles about a given point, as sketched in Figure 3.31. Moreover, let the velocity along any given circular streamline be constant, but let it vary from one streamline to another inversely with distance from the common center. Such a flow is called a vortex flow. Examine Figure 3.31; the velocity components in the radial and tangential directions are Vr and Vg, respectively, where Vr = 0 and Vg = constant/r. It is easily shown (try it yourself) that (1) vortex flow is a physically possible incompressible flow, that is, V • V = 0 at every point, and (2) vortex flow is irrotational, that is, V x V = 0, at every point except the origin.

From the definition of vortex flow, we have

Vortex Flow: Our Fourth Elementary Flow

[3.104]

 

image256

Подпись: or Подпись: Г image257 Подпись: [3.105]

To evaluate the constant C, take the circulation around a given circular streamline of radius r

Comparing Equations (3.104) and (3.105), we see that

Г

C =———- [3.106]

2jr

Therefore, for vortex flow, Equation (3.106) demonstrates that the circulation taken about all streamlines is the same value, namely, Г = —2л C. By convention, Г is called the strength of the vortex flow, and Equation (3.105) gives the velocity field for a vortex flow of strength Г. Note from Equation (3.105) that Vg is negative when Г is positive; that is, a vortex of positive strength rotates in the clockwise direction. (This is a consequence of our sign convention on circulation defined in Section 2.13, namely, positive circulation is clockwise.)

We stated earlier that vortex flow is irrotational except at the origin. What happens at r = 0? What is the value of V x V at r = 0? To answer these questions, recall Equation (2.137) relating circulation to vorticity:

Г = ~jj(v x V)- dS [2.137]

Combining Equations (3.106) and (2.137), we obtain

Подпись:2л C = j j (V x V) • dS

Подпись: 2 лC Vortex Flow: Our Fourth Elementary Flow Подпись: [3.108]

Since we are dealing with two-dimensional flow, the flow sketched in Figure 3.31 takes place in the plane of the paper. Hence, in Equation (3.107), both V x V and dS are in the same direction, both perpendicular to the plane of the flow. Thus, Equation (3.107) can be written as

In Equation (3.108), the surface integral is taken over the circular area inside the streamline along which the circulation Г = —2ттС is evaluated. However, Г is the same for all the circulation streamlines. In particular, choose a circle as close to the origin as we wish (i. e., let r —»■ 0). The circulation will still remain Г = —2лС. However, the area inside this small circle around the origin will become infinitesimally small, and

JJV x\dS ^ V x\dS [3.109]

s

Combining Equations (3.108) and (3.109), in the limit as r —>■ 0, we have

2тсС = |V x V| dS

2 тсС r,

or |VxV| = ——- [3.110]

dS

However, as r —>■ 0, dS —»■ 0. Therefore, in the limit as r —»■ 0, from Equation (3.110), we have

Подпись: oo|V x V|

Conclusion: Vortex flow is irrotational everywhere except at the point r = 0. where the vorticity is infinite. Therefore, the origin, r = 0, is a singular point in the flow field. We see that, along with sources, sinks, and doublets, the vortex flow contains a singularity. Hence, we can interpret the singularity itself, that is, point О in Figure

3.31, to be a point vortex which induces about it the circular vortex flow shown in Figure 3.31.

Vortex Flow: Our Fourth Elementary Flow Подпись: [3.111a] [3.111Ы [3.1 12]

The velocity potential for vortex flow can be obtained as follows:

Equation (3.112) is the velocity potential for vortex flow.

Подпись: Table 3.1 Type of flow Velocity Ф * Uniform flow in x direction U = Voc V<x,x Vxy Source A Vr _ 2TT7 Л , — lnr 2тг І9 Vortex Ve = ~r~ г ~Ътв S*' Doublet К cos # Vr = ~2^~^ к cos 9 2тт г к sin# 2тг г к sin# V9 = - —— In r1

1 Зф

Подпись: The stream function is determined in a similar manner: 0 Подпись:—- — = Vr

г Зв

Зф Г

dr в 2лг

Подпись: ф — — In r In Подпись: [3.114]

Integrating Equations (3.113а and b), we have

Equation (3.114) is the stream function for vortex flow. Note that since ф = constant is the equation of the streamline, Equation (3.114) states that the streamlines of vortex flow are given by r = constant (i. e., the streamlines are circles). Thus, Equation

(3.114) is consistent with our definition of vortex flow. Also, note from Equation (3.112) that equipotential lines are given by в = constant, that is, straight radial lines from the origin. Once again, we see that equipotential lines and streamlines are mutually perpendicular.

At this stage, we summarize the pertinent results for our four elementary flows in Table 3.1.

Prandtl—The Early Development of Finite-Wing Theory

On June 27, 1866, in a paper entitled “Aerial Locomotion” given to the Aeronautical Society of Great Britain, the Englishman Francis Wenham expressed for the first time in history the effect of aspect ratio on finite-wing aerodynamics. He theorized (correctly) that most of the lift of a wing occurs from the portion near the leading edge, and hence a long, narrow wing would be most efficient. He suggested stacking a number of long thin wings above each other to generate the required lift, and he built two full-size gliders in 1858, both with five wings each, to demonstrate (successfully) his ideas. (Wenham is also known for designing and building the first wind tunnel in history, at Greenwich, England, in 1871.)

However, the true understanding of finite-wing aerodynamics, as well as ideas for the theoretical analysis of finite wings, did not come until 1907. In that year, Frederick W. Lanchester published his now famous book entitled Aerodynamics. We have met Lanchester before—in Section 4.14 concerning his role in the development of the circulation theory of lift. Here, we examine his contributions to finite-wing theory.

In Lanchester’s Aerodynamics, we find the first mention of vortices that trail downstream of the wing tips. Figure 5.46 is one of Lanchester’s own drawings from his 1907 book, showing the “vortex trunk” which forms at the wing tip. Moreover, he knew that a vortex filament could not end in space (see Section 5.2), and he theorized that the vortex filaments that constituted the two wing-tip vortices must cross the wing along its span—the first concept of bound vortices in the spanwise direction. Hence, the essence of the horseshoe vortex concept originated with Lanchester. In his own words:

Thus the author regards the two trailed vortices as a definite proof of the existence

of a cyclic component of equal strength in the motion surrounding the airfoil itself.

Considering the foresight and originality of Lanchester’s thinking, let us pause for a moment and look at the man himself. Lanchester was bom on October 23,1868,

image465

Figure 5.46 A figure from Lanchester’s Aerodynamics, 1 907; this is his own drawing of the wing-tip vortex on a finite wing.

in Lewisham, England. The son of an architect, Lanchester became interested in engineering at an early age. (He was told by his family that his mind was made up at the age of 4.) He studied engineering and mining during the years 1886-1889 at the Royal College of Science in South Kensington, London, but never officially graduated. He was a quick-minded and innovative thinker and became a designer at the Forward Gas Engine Company in 1889, specializing in internal combustion engines. He rose to the post of assistant works manager. In the early 1890s, Lanchester became very interested in aeronautics, and along with his development of high-speed engines, he also carried out numerous aerodynamics experiments. It was during this period that he formulated his ideas on both the circulation theory of lift and the finite-wing vortex concepts. A serious paper written by Lanchester first for the Royal Society, and then for the Physical Society, was turned down for publication—something Lanchester never forgot. Finally, his aeronautical concepts were published in his two books Aerodynamics and Aerodonelics in 1907 and 1908, respectively. To his detriment, Lanchester had a style of writing and a means of explanation that were not easy to follow and his works were not immediately seized upon by other researchers. Lanchester’s bitter feelings about the public’s receipt of his papers and books are graphically seen in his letter to the Daniel Guggenheim Medal Fund decades later. In a letter dated June 6, 1931, Lanchester writes:

So far as aeronautical science is concerned, I cannot say that I experienced any­thing but discouragement; in the early days my theoretical work (backed by a certain amount of experimental verification), mainly concerning the vortex theory of sus – tentation and the screw propeller, was refused by the two leading scientific societies in this country, and I was seriously warned that my profession as an engineer would suffer if I dabbled in a subject that was merely a dream of madmen! When I pub­lished my two volumes in 1907 and 1908 they were well received on the whole, but this was mainly due to the success of the brothers Wright, and the general interest aroused on the subject.

In 1899, he formed the Lanchester Motor Company, Limited, and sold automobiles of his own design. He married in 1919, but had no children. Lanchester maintained

his interest in automobiles and related mechanical devices until his death on March 8, 1946, at the age of 77.

In 1908, Lanchester visited Gottingen, Germany, and fully discussed his wing theory with Ludwig Prandtl and his student Theodore von Karman. Prandtl spoke no English, Lanchester spoke no German, and in light of Lanchester’s unclear way of explaining his ideas, there appeared to be little chance of understanding between the two parties. However, shortly after, Prandtl began to develop his own wing theory, using a bound vortex along the span and assuming that the vortex trails downstream from both wing tips. The first mention of Prandtl’s work on finite-wing theory was made in a paper by O. Foppl in 1911, discussing some of Foppl’s experimental work on finite wings. Commenting on his results, Foppl says:

They agree very closely with the theoretical investigation by Professor Prandtl on the current around an airplane with a finite span wing. Already Lanchester in his work, “Aerodynamics” (translated into German by C. and A. Runge), indicated that to the two extremities of an airplane wing are attached two vortex ropes (Wirbelzopfe) which make possible the transition from the flow around the airplane, which occurs nearly according to Kutta’s theory, to the flow of the undisturbed fluid at both sides. These two vortex ropes continue the vortex which, according to Kutta’s theory, takes place on the lamina.

We are led to admit this owing to the Helmholtz theorem that vortices cannot end in the fluid. At any rate these two vortex ropes have been made visible in the Gottingen Institute by emitting an ammonia cloud into the air. Prandtl’s theory is constructed on the consideration of this current in reality existing.

In the same year, Prandtl expressed his own first published words on the subject. In a paper given at a meeting of the Representatives of Aeronautical Science in Gottingen in November 1911, entitled “Results and Purposes of the Model Experimental Institute of Gottingen,” Prandtl states:

Another theoretical research relates to the conditions of the current which is formed by the air behind an airplane. The lift generated by the airplane is, on account of the principle of action and reaction, necessarily connected with a descending current behind the airplane. Now it seemed very useful to investigate this descending current in all its details. It appears that the descending current is formed by a pair of vortices, the vortex filaments of which start from the airplane wing tips. The distance of the two vortices is equal to the span of the airplane, their strength is equal to the circulation of the current around the airplane and the current in the vicinity of the airplane is fully given by the superposition of the uniform current with that of a vortex consisting of three rectilinear sections.

In discussing the results of his theory, Prandtl goes on to state in the same paper:

The same theory supplies, taking into account the variations of the current on the airplane which came from the lateral vortices, a relationship showing the depen­dence of the airplane lift on the aspect ratio; in particular it gives the possibility of extrapolating the results thus obtained experimentally to the airplane of infinite span wing. From the maximum aspect ratios measured by us (1:9 to that of 1 :oo) the lifts increase further in marked degree—by some 30 or 40 percent. I would add here a

remarkable result of this extrapolation, which is, that the results of Kutta’s theory of the infinite wing, at least so far as we are dealing with small cambers and small angles of incidence, have been confirmed by these experimental results.

Starting from this line of thought we can attack the problem of calculating the surface of an airplane so that lift is distributed along its span in a determined manner, previously fixed. The experimental trial of these calculations has not yet been made, but it will be in the near future.

It is clear from the above comments that Prandtl was definitely following the model proposed earlier by Lanchester. Moreover, the major concern of the finite-wing theory was first in the calculation of lift—no mention is made of induced drag. It is interesting to note that Prandtl’s theory first began with a single horseshoe vortex, such as sketched in Figure 5.11. The results were not entirely satisfactory. During the period 1911­1918, Prandtl and his colleagues expanded and refined his finite-wing theory, which evolved to the concept of a lifting line consisting of an infinite number of horseshoe vortices, as sketched in Figure 5.13. In 1918, the term “induced drag’’ was coined by Max Munk, a colleague of Prandtl at Gottingen. Much of Prandtl’s development of finite-wing theory was classified secret by the German government during World War I. Finally, his lifting-line theory was released to the outside world, and his ideas were published in English in a special NACA report written by Prandtl and published in 1922, entitled “Applications of Modern Hydrodynamics to Aeronautics” (NACA TR 116). Hence, the theory we have outlined in Section 5.3 was well-established more than 80 years ago.

One of Prandtl’s strengths was the ability to base his thinking on sound ideas, and to apply intuition that resulted in relatively straightforward theories that most engineers could understand and appreciate. This is in contrast to the difficult writ­ings of Lanchester. As a result, the lifting theory for finite wings has come down through the years identified as Prandtl’s lifting-line theory, although we have seen that Lanchester was the first to propose the basic model on which lifting-line theory is built.

In light of Lanchester’s 1908 visit with Prandtl and Prandtl’s subsequent de­velopment of the lifting-line theory, there has been some discussion over the years that Prandtl basically stole Lanchester’s ideas. However, this is clearly not the case. We have seen in the above quotes that Prandtl’s group at Gottingen was giving full credit to Lanchester as early as 1911. Moreover, Lanchester never gave the world a clear and practical theory with which results could be readily obtained—Prandtl did. Therefore, in this book we have continued the tradition of identifying the lifting-line theory with Prandtl’s name. On the other hand, for very good reasons, in England and various places in western Europe, the theory is labeled the Lanchester-Prandtl theory.

To help put the propriety in perspective, Lanchester was awarded the Daniel Guggenheim Medal in 1936 (Prandtl had received this award some years earlier). In the medal citation, we find the following words:

Lanchester was the foremost person to propound the now famous theory of flight based on the Vortex theory, so brilliantly followed up by Prandtl and others. He first

put forward his theory in a paper read before the Birmingham Natural History and Philosophical Society on 19th June, 1894. In a second paper in 1897, in his two books published in 1907 and 1908, and in his paper read before the Institution of Automobile Engineers in 1916, he further developed this doctrine.

Perhaps the best final words on Lanchester are contained in this excerpt from his obituary found in the British periodical Flight in March 1946:

And now Lanchester has passed from our ken but not from our thoughts. It is to be hoped that the nation which neglected him during much of his lifetime will at any rate perpetuate his work by a memorial worthy of the “Grand Old Man” of aerodynamics.

Fluid Statics: Buoyancy Force

In aerodynamics, we are concerned about fluids in motion, and the resulting forces and moments on bodies due to such motion. However, in this section, we consider the special case of no fluid motion, i. e., fluid statics. A body immersed in a fluid will still experience a force even if there is no relative motion between the body and the fluid. Let us see why.

To begin, we must first consider the force on an element of fluid itself. Consider a stagnant fluid above the xz plane, as shown in Figure 1.31. The vertical direction is given by y. Consider an infinitesimally small fluid element with sides of length dx, dy, and dz. There are two types of forces acting on this fluid element: pressure forces from the surrounding fluid exerted on the surface of the element, and the gravity force due to the weight of the fluid inside the element. Consider forces in the у direction. The pressure on the bottom surface of the element is p, and hence the force on the bottom face is p(dx dz) in the upward direction, as shown in Figure

(p + ^dy)dx dz

image57

fluid.

 

1.31. The pressure on the top surface of the element will be slightly different from the pressure on the bottom because the top surface is at a different location in the fluid. Let dp/dy denote the rate of change of p with respect to y. Then the pressure exerted on the top surface will be p + (dp/dy) dy, and the pressure force on the top of the element will be [p + (dp/dy) dy](dx dz) in the downward direction, as shown in Figure 1.31. Hence, letting upward force be positive, we have

Net pressure force = p(dx dz) — ( p + —dy ) (dx dz)

dy J

= — — (dx dy dz) dy

Let p be the mean density of the fluid element. The total mass of the element is p(dx dy dz). Therefore,

Gravity force = —p(dx dy dz)g

where g is the acceleration of gravity. Since the fluid element is stationary (in equi­librium), the sum of the forces exerted on it must be zero:

Подпись: or Подпись: dp = ~gp dy Подпись: [1.52]

-—(dx dy dz) – gp(dxdydz) = 0 dy

Equation (1.52) is called the Hydrostatic equation; it is a differential equation which relates the change in pressure dp in a fluid with a change in vertical height dy.

The net force on the element acts only in the vertical direction. The pressure forces on the front and back faces are equal and opposite and hence cancel; the same is true for the left and right faces. Also, the pressure forces shown in Figure 1.31 act at the center of the top and bottom faces, and the center of gravity is at the center of the elemental volume (assuming the fluid is homogeneous); hence, the forces in Figure 1.31 are colinear, and as a result, there is no moment on the element.

Equation (1.52) governs the variation of atmospheric properties as a function of altitude in the air above us. It is also used to estimate the properties of other planetary atmospheres such as for Venus, Mars, and Jupiter. The use of Equation (1.52) in the analysis and calculation of the “standard atmosphere” is given in detail in Reference 2; hence, it will not be repeated here.

Let the fluid be a liquid, for which we can assume p is constant. Consider points 1 and 2 separated by the vertical distance Ah as sketched on the right side of Figure

1.31. The pressure and у locations at these points are pi, h, and /?2, hi. respectively. Integrating Equation (1.52) between points 1 and 2, we have

Г P2 rhl

dp = – pg dy J p J h

Подпись: [1.53]Подпись: orPi – Pi = ~Pg(hi — hj) — pg Ah

where Ah = h — hi. Equation (1.46) can be more conveniently expressed as

Подпись: or Подпись: p + pgh = constant Подпись: [1.54]

P2 + pgh2 = pi + pghi

Note that in Equations (1.53) and (1.54), increasing values of h are in the positive (upward) у direction.

A simple application of Equation (1.54) is the calculation of the pressure distri­bution on the walls of a container holding a liquid, and open to the atmosphere at the top. This is illustrated in Figure 1.32, where the top of the liquid is at a heght hi. The atmospheric pressure pa is impressed on the top of the liquid; hence, the pressure at h is simply pa. Applying Equation (1.54) between the top (where h = h) and an arbitrary height h, we have

P + Pgh = pі + pghi = pa+ Pgh і

or p = Pa + pg(h – h) [1.55]

Equation (1.55) gives the pressure distribution on the vertical sidewall of the container as a function of h. Note that the pressure is a linear function of h as sketched on the right of Figure 1.32, and that p increases with depth below the surface.

image58

Another simple and very common application of Equation (1.54) is the liquid – filled U-tube manometer used for measuring pressure differences, as sketched in Figure 1.33. The manometer is usually made from hollow glass tubing bent in the shape of the letter U. Imagine that we have an aerodynamic body immersed in an airflow (such as in a wind tunnel), and we wish to use a manometer to measure the surface pressure at point b on the body. A small pressure orifice (hole) at point b is connected to one side of the manometer via a long (usually flexible) pressure tube. The other side of the manometer is open to the atmosphere, where the pressure pa is a known value. The U tube is partially filled with a liquid of known density p. The tops of the liquid on the left and right sides of the U tube are at points 1 and 2, with heights h and h2, respectively. The body surface pressure pt, is transmitted through the pressure tube and impressed on the top of the liquid at point 1. The atmospheric pressure pa is impressed on the top of the liquid at point 2. Because in general рь ф pa, the tops of the liquid will be at different heights; i. e., the two sides of the manometer will show a displacement Ah = hi — h2 of the fluid. We wish to

The use of a U-tube manometer.

obtain the value of the surface pressure at point b on the body by reading the value of Ah from the manometer. From Equation (1.54) applied between points 1 and 2,

Pb + pghi = pa + pgh2

or Pb = Pa – pg{h – h2)

or pb = Pa – pg Ah [1.56]

In Equation (1.56), pa, p, and g are known, and Ah is read from the U tube, thus allowing рь to be measured.

At the beginning of this section, we stated that a solid body immersed in a fluid will experience a force even if there is no relative motion between the body and the fluid. We are now in a position to derive an expression for this force, henceforth called the buoyancy force. We will consider a body immersed in either a stagnant gas or liquid, hence p can be a variable. For simplicity, consider a rectangular body of unit width, length/, and height {h—h2), as shown in Figure 1.34. Examining Figure 1.34, we see that the vertical force F on the body due to the pressure distribution over the surface is

F = (p2~ pi)l(l) [1.57]

There is no horizontal force because the pressure distributions over the vertical faces of the rectangular body lead to equal and opposite forces which cancel each other. In Equation (1.57), an expression for p2 — p can be obtained by integrating the hydrostatic equation, Equation (1.52), between the top and bottom faces:

Подпись: Pg dy

image60

ГР2 phi ph

P2 ~ Pi = dp = — I pgdy = I

J pi J h і «/ hi

image61

Figure 1.34 Source of the buoyancy force on a body immersed in a fluid.

 

Substituting this result into Equation (1.57), we obtain for the buoyancy force

Подпись: F = K 1) / Jhh

pgdy [1.58]

2

Подпись: Pg dy
image62

Consider the physical meaning of the integral in Equation (1.58). The weight of a small element of fluid of height dy and width and length of unity as shown at the right of Figure 1.34 is pg dy (1)(1). In turn, the weight of a column of fluid with a base of unit area and a height (h — /12) is

which is precisely the integral in Equation (1.58). Moreover, if we place l of these fluid columns side by side, we would have a volume of fluid equal to the volume of the body on the left of Figure 1.34, and the weight of this total volume of fluid would be

l pgdy J /12

which is precisely the right-hand side of Equation (1.58). Therefore, Equation (1.58) states in words that

Buoyancy force

weight of fluid

on body

— displaced by body

We have just proved the well-known Archimedes principle, first advanced by the Greek scientist, Archimedes of Syracuse (287-212 B. C.). Although we have used a rectangular body to simplify our derivation, the Archimedes principle holds for bodies of any general shape. (See Problem 1.14 at the end of this chapter.) Also, note from our derivation that the Archimedes principle holds for both gases and liquids and does not require that the density be constant.

The density of liquids is usually several orders of magnitude larger than the density of gases; e. g., for water p = 103 kg/m3, whereas for air p = 1.23 kg/m3. Therefore, a given body will experience a buoyancy force a thousand times greater in water than in air. Obviously, for naval vehicles buoyancy force is all important, whereas for airplanes it is negligible. On the other hand, lighter-than-air vehicles, such as blimps and hot-air balloons, rely on buoyancy force for sustenation; they obtain sufficient buoyancy force simply by displacing huge volumes of air. For most problems in aerodynamics, however, buoyancy force is so small that it can be readily neglected.

A hot-air balloon with an inflated diameter of 30 ft is carrying a weight of 800 lb, which includes the weight of the hot air inside the balloon. Calculate (a) its upward acceleration at sea level the instant the restraining ropes are released and (b) the maximum altitude it can achieve. Assume that the variation of density in the standard atmosphere is given by p = 0.002377(1 — 7 x 10~6/t)4-21, where h is the altitude in feet and p is in slug/ft3.

Solution

(a) At sea level, where h = 0, p = 0.002377 slug/ft3. The volume of the inflated balloon is |я(15)3 = 14,137 ft3. Hence,

Buoyancy force = weight of displaced air

= gpV

where g is the acceleration of gravity and V is the volume.

Buoyancy force = В = (32.2)(0.002377)(14,137) = 1082 lb

The net upward force at sea level is F = В — W, where W is the weight. From Newton’s second law,

F = В — W = та

where m is the mass, m = Щ = 24.8 slug. Hence,

Подпись: 11.4 ft/s2Подпись: Example 1.8_ B-W 1082 – 800 a ~ m ~ 24Я

Подпись: P Подпись: В gv Подпись: 800 (32.2) (14,137) Подпись: 0.00176 slug/ft3

(b) The maximum altitude occurs when В = W = 800 lb. Since В = gpV, and assuming the balloon volume does not change,

From the given variation of p with altitude, h,

p = 0.002377(1 – 7 x 10~6/t)4-21 = 0.00176

Solving for h, we obtain

1

0.00176 1/4 21

h =————

1 – (————-

9842 ft

7 x 10~6

V 0.002377 j

Stream Function

In this section, we consider two-dimensional steady flow. Recall from Section 2.11 that the differential equation for a streamline in such a flow is given by Equation

 

image173

(2.118), repeated below

Подпись: [2.1 18] Equation (2.118) can be integrated Подпись:dy v dx и

If и and v are known functions of x and y, then to yield the algebraic equation for a streamline:

f(x, y) = c

where c is an arbitrary constant of integration, with different values for different streamlines. In Equation (2.139), denote the function of x and у by the symbol f. Hence, Equation (2.139) is written as

Подпись: [2.140]Ф(х, y)=C

The function jr(x, y) is called the stream function. From Equation (2.140) we see that the equation for a streamline is given by setting the stream function equal to a constant, i. e., сі, c2, сз, etc. Two different streamlines are illustrated in Figure 2.38; streamlines ab and cd are given by jr = a and iJr = C2, respectively.

There is a certain arbitrariness in Equations (2.139) and (2.140) via the arbitrary constant of integration c. Let us define the stream function more precisely in order to reduce this arbitrariness. Referring to Figure 2.38, let us define the numerical value of іJr such that the difference AiJr between f = ci for streamline cd and jr = c for streamline ab is equal to the mass flow between the two streamlines. Since Figure 2.38 is a two-dimensional flow, the mass flow between two streamlines is defined per unit depth perpendicular to the page. That is, in Figure 2.38 we are considering the mass flow inside a streamtube bounded by streamlines ab and cd, with a rectangular cross­sectional area equal to An times a unit depth perpendicular to the page. Here, An is the normal distance between ab and cd, as shown in Figure 2.38. Hence, mass flow between streamlines ab and cd per unit depth perpendicular to the page is

Ді/r = C2 — Ci [2.141]

Подпись: Figure 2.38 Different streamlines are given by different values of the stream function.

The above definition does not completely remove the arbitrariness of the constant of integration in Equations (2.139) and (2.140), but it does make things a bit more precise. For example, consider a given two-dimensional flow field. Choose one streamline of the flow, and give it an arbitrary value of the stream function, say, fr = a. Then, the value of the stream function for any other streamline in the flow, say, fr = сг,

is fixed by the definition given in Equation (2.141). Which streamline you choose to designate as ф = c and what numerical value you give c usually depend on the geometry of the given flow field, as we see in Chapter 3.

The equivalence between ф = constant designating a streamline, and Аф equal­ing mass flow (per unit depth) between streamlines, is natural. For a steady flow, the mass flow inside a given streamtube is constant along the tube; the mass flow across any cross section of the tube is the same. Since by definition Д ф is equal to this mass flow, then Аф itself is constant for a given streamtube. In Figure 2.38, if ф = C designates the streamline on the bottom of the streamtube, then ф2 = <"2 = c + Аф is also constant along the top of the streamtube. Since by definition of a streamtube (see Section 2.11) the upper boundary of the streamtube is a streamline itself, then ф2 = C2 = constant must designate this streamline.

We have yet to develop the most important property of the stream function, namely, derivatives of ф yield the flow-field velocities. To obtain this relationship, consider again the streamlines a b and cd in Figure 2.38. Assume that these streamlines are close together (i. e., assume An is small), such that the flow velocity V is a constant value across An. The mass flow through the streamtube per unit depth perpendicular to the page is

Подпись: [2.142]Подпись: [2.143]Подпись:Аф = pV An(l) Аф

~^=PV

An

Consider the limit of Equation (2.142) as An —» 0:

,, r АФ H

pV = lim ——— = —

An->0 An dn

Equation (2.143) states that if we know ф, then we can obtain the product (p V) by differentiating ф in the direction normal to V. To obtain a practical form of Equation (2.143) for cartesian coordinates, consider Figure 2.39. Notice that the directed normal distance An is equivalent first to moving upward in the у direction by the amount Ay and then to the left in the negative x direction by the amount — Ax. Due to conservation of mass, the mass flow through An (per unit depth) is equal to the sum of the mass flows through Ay and — Ax (per unit depth):

Mass flow = Аф = pV An = pu Ay + pv(-Ax) [2.144]

Letting cd approach ab, Equation (2.144) becomes in the limit

d’ijf = pudy — pvdx [2.145]

However, since ф = ф (x, y), the chain rule of calculus states

– 3 ф 3 ф

dф = —dx H——- dy [2.146]

dx dy

Comparing Equations (2.145) and (2.146), we have
[2.147a]

image175"

Подпись: Figure 2.39 Mass flow through An is the sum of the mass flows through Ay and - Ax.

[2.147b]

Equations (2.147a and b) are important. If ф(х, y) is known for a given flow field, then at any point in the flow the products pu and pv can be obtained by differentiating ф in the directions normal to и and v, respectively.

Stream Function Подпись: [2.148a] [2.148b]

If Figure 2.39 were to be redrawn in terms of polar coordinates, then a similar derivation yields

Such a derivation is left as a homework problem.

Note that the dimensions of ф are equal to mass flow per unit depth perpendicular to the page. That is, in SI units, ф is in terms of kilograms per second per meter perpendicular to the page, or simply kg/(s ■ m).

The stream function ф defined above applies to both compressible and incom­pressible flow. Now consider the case of incompressible flow only, where p = con-

stant. Equation (2.143) can be written as

Подпись:д(Ф/р)

dn

We define a new stream function, for incompressible flow only, as 1jr = i// /p. Then Equation (2.149) becomes

ЗіIr V = ~ dn

and Equations (2.147) and (2.148) become

image177[3.150a]

[3.150b]

and

image178[3.151a] [3.151b]

The incompressible stream function ф has characteristics analogous to its more gen­eral compressible counterpart iJr. For example, since ф(х, у) = c is the equation of a streamline, and since p is a constant for incompressible flow, then 1(r{x, y) = ф jp = constant is also the equation for a streamline (for incompressible flow only). In addition, since Д ф is mass flow between two streamlines (per unit depth perpendic­ular to the page), and since p is mass per unit volume, then physically Аф = Ді]r/p represents the volume flow (per unit depth) between two streamlines. In SI units, Аф is expressed as cubic meters per second per meter perpendicular to the page, or simply m2/s.

In summary, the concept of the stream function is a powerful tool in aerodynam­ics, for two primary reasons. Assuming that ф(х, y) [or ф(х, y)[ is known through the two-dimensional flow field, then:

1. ф = constant (or ф = constant) gives the equation of a streamline.

2. The flow velocity can be obtained by differentiating ф (or ф), as given by Equa­tions (2.147) and (2.148) for compressible flow and Equations (2.150) and (2.151) for incompressible flow. We have not yet discussed how ф(х, y) [or ф(х, у)] can be obtained in the first place; we are assuming that it is known. The ac­tual determination of the stream function for various problems is discussed in Chapter 3.

The Kutta Condition

The lifting flow over a circular cylinder was discussed in Section 3.15, where we ob­served that an infinite number of potential flow solutions were possible, corresponding to the infinite choice of Г. For example, Figure 3.28 illustrates three different flows over the cylinder, corresponding to three different values of Г. The same situation applies to the potential flow over an airfoil; for a given airfoil at a given angle of attack, there are an infinite number of valid theoretical solutions, corresponding to an

3 It is interesting to note that some recent research by NASA is hinting that even as complex a problem as flow separation, heretofore thought to be a completely viscous-dominated phenomenon, may in reality be an inviscid-dominated flow which requires only a rotational flow. For example, some inviscid flow-field numerical solutions for flow over a circular cylinder, when vorticity is introduced either by means of a nonuniform freestream or a curved shock wave, are accurately predicting the separated flow on the rearward side of the cylinder. However, as exciting as these results may be, they are too preliminary to be emphasized in this book. We continue to talk about flow separation in Chapters 15 to 20 as being a viscous-dominated effect, until definitely proved otherwise. This recent research is mentioned here only as another example of the physical connection between vorticity, vortex sheets, viscosity, and real life.

infinite choice of Г. For example, Figure 4.12 illustrates two different flows over the same airfoil at the same angle of attack but with different values of Г. At first, this may seem to pose a dilemma. We know from experience that a given airfoil at a given angle of attack produces a single value of lift (e. g., see Figure 4.5). So, although there is an infinite number of possible potential flow solutions, nature knows how to pick a particular solution. Clearly, the philosophy discussed in the previous section is not complete—we need an additional condition that fixes Г for a given airfoil at a given a.

To attempt to find this condition, let us examine some experimental results for the development of the flow field around an airfoil which is set into motion from an initial state of rest. Figure 4.13 shows a series of classic photographs of the flow over an airfoil, taken from Prandtl and Tietjens (Reference 8). In Figure 4.13a, the flow has just started, and the flow pattern is just beginning to develop around the airfoil. In these early moments of development, the flow tries to curl around the sharp trailing edge from the bottom surface to the top surface, similar to the sketch shown at the left of Figure 4.12. However, more advanced considerations of inviscid, incompressible flow (see, e. g., Reference 9) show the theoretical result that the velocity becomes infinitely large at a sharp comer. Hence, the type of flow sketched at the left of Figure 4.12, and shown in Figure 4.13a, is not tolerated very long by nature. Rather, as the real flow develops over the airfoil, the stagnation point on the upper surface (point 2 in Figure 4.12) moves toward the trailing edge. Figure 4.13b shows this intermediate stage. Finally, after the initial transient process dies out, the steady flow shown in Figure 4.13c is reached. This photograph demonstrates that the flow is smoothly leaving the top and the bottom surfaces of the airfoil at the trailing edge. This flow pattern is sketched at the right of Figure 4.12 and represents the type of pattern to be expected for the steady flow over an airfoil.

Reflecting on Figures 4.12 and 4.13, we emphasize again that in establishing the steady flow over a given airfoil at a given angle of attack, nature adopts that particular value of circulation (Г2 in Figure 4.12) which results in the flow leaving smoothly at the trailing edge. This observation was first made and used in a theoretical analysis by the German mathematician M. Wilhelm Kutta in 1902. Therefore, it has become known as the Kutta condition.

image314

Figure 4.12 Effect of different values of circulation on the potential flow over a given airfoil at a given angle of attack. Points 1 and 2 are stagnation points.

image315

(a)

image316

(b)

Figure 4.1 3 The development of steady flow over an airfoil; the airfoil is impulsively started from rest and attains a steady velocity through the fluid, (a) A moment just after starting, (b) An intermediate time. (Source: Prandtl and Tiejens, Reference 8.)

In order to apply the Kutta condition in a theoretical analysis, we need to be more precise about the nature of the flow at the trailing edge. The trailing edge can have a finite angle, as shown in Figures 4.12 and 4.13 and as sketched at the left of Figure 4.14, or it can be cusped, as shown at the right of Figure 4.14. First, consider the trailing edge with a finite angle, as shown at the left of Figure 4.14. Denote the velocities along the top surface and the bottom surface as V and to, respectively. Vt

image317

(f)

Figure 4.1 3 (continued) The development of steady flow over an airfoil; the airfoil is

impulsively started from rest and attains a steady velocity through the fluid, (c) The final steady flow. (Source: Prandtl and Tiejens, Reference 8.)

 

Finite angle Cusp

image318

At point a, Kj = V2 = 0 At point a: V1 = V2 Ф 0

Figure 4.1 4 Different possible shapes of the trailing edge and their relation to the Kutta condition.

 

is parallel to the top surface at point a, and V2 is parallel to the bottom surface at point a. For the finite-angle trailing edge, if these velocities were finite at point a, then we would have two velocities in two different directions at the same point, as shown at the left of Figure 4.14. Flowever, this is not physically possible, and the only recourse is for both Vj and V2 to be zero at point a. That is, for the finite trailing edge, point a is a stagnation point, where Vj = V2 = 0. In contrast, for the cusped trailing edge shown at the right of Figure 4.14, V and V2 are in the same direction at point a, and hence both Vj and V2 can be finite. Flowever, the pressure at point a, p2, is a single, unique value, and Bernoulli’s equation applied at both the top and bottom surfaces immediately adjacent to point a yields

Pa + pV i2 = Pa + pV2

Подпись:Vj = F2

Hence, for the cusped trailing edge, we see that the velocities leaving the top and bottom surfaces of the airfoil at the trailing edge are finite and equal in magnitude and direction.

We can summarize the statement of the Kutta condition as follows:

1. For a given airfoil at a given angle of attack, the value of Г around the airfoil is such that the flow leaves the trailing edge smoothly.

2. If the trailing-edge angle is finite, then the trailing edge is a stagnation point.

3. If the trailing edge is cusped, then the velocities leaving the top and bottom surfaces at the trailing edge are finite and equal in magnitude and direction.

Consider again the philosophy of simulating the airfoil with vortex sheets placed either on the surface or on the camber line, as discussed in Section 4.4. The strength of such a vortex sheet is variable along the sheet and is denoted by у (і). The statement of the Kutta condition in terms of the vortex sheet is as follows. At the trailing edge (ТЕ), from Equation (4.8), we have

У (ТЕ) = y(a) = V1- V2 [4.9]

However, for the finite-angle trailing edge, V = V2 = 0; hence, from Equation (4.9), у (ТЕ) = 0. For the cusped trailing edge, V = V2 Ф 0; hence, from Equation (4.9), we again obtain the result that у (ТЕ) = 0. Therefore, the Kutta condition expressed in terms of the strength of the vortex sheet is

[4.10]