# Category Fundamentals of Aerodynamics

## A Brief Review of Thermodynamics

The importance of thermodynamics in the analysis and understanding of compressible flow was underscored in Section 7.1. Hence, the purpose of the present section is to review those aspects of thermodynamics that are important to compressible flows. This is in no way indended to be an exhaustive discussion of thermodynamics; rather, it is a review of only those fundamental ideas and equations that will be of direct use in subsequent chapters. If you have studied thermodynamics, this review should serve as a ready reminder of some important relations. If you are not familiar with thermodynamics, this section is somewhat self-contained so as to give you a feeling for the fundamental ideas and equations that we use frequently in subsequent chapters.

7.2.1 Perfect Gas

As described in Section 1.2, a gas is a collection of particles (molecules, atoms, ions, electrons, etc.) which are in more or less random motion. Due to the electronic structure of these particles, a force field pervades the space around them. The force field due to one particle reaches out and interacts with neighboring particles, and vice versa. Hence, these fields are called intermolecular forces. However, if the particles of the gas are far enough apart, the influence of the intermolecular forces is small and can be neglected. A gas in which the intermolecular forces are neglected is defined as a perfect gas. For a perfect gas, p, p, and T are related through the following equation of state:

[7.1]

where R is the specific gas constant, which is a different value for different gases. For air at standard conditions, R = 287 J/(kg • K) = 1716 (ft • lb)/(slug • °R).

At the temperatures and pressures characteristic of many compressible flow ap­plications, the gas particles are, on the average, more than 10 molecular diameters apart; this is far enough to justify the assumption of a perfect gas. Therefore, through­out the remainder of this book, we use the equation of state in the form of Equation (7.1), or its counterpart,

[7.2]

where v is the specific volume, that is, the volume per unit mass; v — 1/p. (Please note: Starting with this chapter, we use the symbol v to denote both specific volume and the у component of velocity. This usage is standard, and in all cases it should be obvious and cause no confusion.)

## Aerodynamics: Some Fundamental. Principles and Equations

There is so great a difference between a fluid and a collection of solid particles that the laws of pressure’ and of equilibrium of fluids are very different from the laws of the pressure and equilibrium of solids.

Jean Le Rond d’Alembert, 1768

The principle is most important, not the detail.

Theodore von Karman, 1954

To be a good craftsperson, one must have good tools and know how to use them effectively. Similarly, a good aerodynamicist must have good aerodynamic tools and must know how to use them for a variety of applications. The purpose of this chapter is “tool-building”; we develop some of the concepts and equations that are vital to the study of aerodynamic flows. However, please be cautioned; A craftsperson usually derives his or her pleasure from the works of art created with the use of the tools; the actual building of the tools themselves is sometimes considered a mundane chore. You may derive a similar feeling here. As we proceed to build our aerodynamic tools, you may wonder from time to time why such tools are necessary and what possible value they may have in the solution of practical problems. Rest assured, however, that every aerodynamic tool we develop in this and subsequent chapters is important for the analysis and understanding of practical problems to be discussed later. So, as

we move through this chapter, do not get lost or disoriented; rather, as we develop each tool, simply put it away in the store box of your mind for future use.

To help you keep track of our tool building, and to give you some orientation, the road map in Figure 2.1 is provided for your reference. As we progress through each section of this chapter, use Figure 2.1 to help you maintain a perspective of our work. You will note that Figure 2.1 is full of strange-sounding terms, such as “substantial derivative,” “circulation,” and “velocity potential.” However, when you finish this chapter, and look back at Figure 2.1, all these terms will be second nature to you.

 Figure 2.1 Road map for Chapter 2.

## Incompressible Flow in a Duct: the Venturi and Low-Speed Wind Tunnel

Consider the flow through a duct, such as that sketched in Figure 3.5. In general, the duct will be a three-dimensional shape, such as a tube with elliptical or rectangular cross sections which vary in area from one location to another. The flow through such a duct is three-dimensional and, strictly speaking, should be analyzed by means of the full three-dimensional conservation equations derived in Chapter 2. However, in many applications, the variation of area A = A(x) is moderate, and for such cases it is reasonable to assume that the flow-field properties are uniform across any cross section, and hence vary only in the x direction. In Figure 3.5, uniform flow is sketched at station 1, and another but different uniform flow is shown at station 2. Such flow, where the area changes as a function of x and all the flow-field variables are assumed to be functions of x only, that is, A = A(x), V — V(x), p = p(x), etc., is called quasi-one-dimensional flow. Although such flow is only an approximation of the truly three-dimensional flow in ducts, the results are sufficiently accurate for many aerodynamic applications. Such quasi-one-dimensional flow calculations are frequently used in engineering. They are the subject of this section.

 Figure 3.5 Quasi-one-dimensional flow in a duct.

Consider the integral form of the continuity equation written below:

 9 dt

 p dV ■

 [2.48]

 v

Apply Equation (3.16) to the duct shown in Figure 3.5, where the control volume is bounded by At on the left, A2 on the right, and the upper and lower walls of the duct. Hence, Equation (3.16) is

Along the walls, the flow velocity is tangent to the wall. Since by definition dS is perpendicular to the wall, then along the wall, V • dS = 0, and the integral over the wall surface is zero; that is, in Equation (3.17),

JJ pY • dS = 0 [3.18]

wall

At station 1, the flow is uniform across A. Noting that dS and V are in opposite directions at station 1 (dS always points out of the control volume by definition), we have in Equation (3.17)

//"v-dS = —pA V] [3.19]

-A

At station 2, the flow is uniform across A2, and since dS and V are in the same direction, we have, in Equation (3.17),

JJ pY • dS = p2A2V2 [3.20]

Лі

Substituting Equations (3.18) to (3.20) into (3.17), we obtain

—pAV — p2A2V2 0 — 0

Equation (3.21) is the quasi-one-dimensional continuity equation; it applies to both compressible and incompressible flow.[6] In physical terms, it states that the mass flow

through the duct is constant (i. e., what goes in must come out). Compare Equation (3.21) with Equation (2.43) for mass flow.

Consider incompressible flow only, where p = constant. In Equation (3.21), Pi = p2, and we have

[3.22]

Equation (3.22) is the quasi-one-dimensional continuity equation for incompressible flow. In physical terms, it states that the volume flow (cubic feet per second or cubic meters per second) through the duct is constant. From Equation (3.22), we see that if the area decreases along the flow (convergent duct), the velocity increases; conversely, if the area increases (divergent duct), the velocity decreases. These variations are shown in Figure 3.6; they are fundamental consequences of the incompressible con­tinuity equation, and you should fully understand them. Moreover, from Bernoulli’s equation, Equation (3.15), we see that when the velocity increases in a convergent duct, the pressure decreases; conversely, when the velocity decreases in a divergent duct, the pressure increases. These pressure variations are also shown in Figure 3.6.

Consider the incompressible flow through a convergent-divergent duct, shown in Figure 3.7. The flow enters the duct with velocity V) and pressure p. The velocity increases in the convergent portion of the duct, reaching a maximum value V2 at the minimum area of the duct. This minimum area is called the throat. Also, in the convergent section, the pressure decreases, as sketched in Figure 3.7. At the throat, the pressure reaches a minimum value p2- In the divergent section downstream of the throat, the velocity decreases and the pressure increases. The duct shown in Figure 3.7 is called a venturi-, it is a device that finds many applications in engineering, and its use dates back more than a century. Its primary characteristic is that the pressure P2 is lower at the throat than the ambient pressure p outside the venturi. This pressure difference p — p2 is used to advantage in several applications. For example, in the carburetor of an automobile engine, there is a venturi through which the incoming air is mixed with fuel. The fuel line opens into the venturi at the throat. Because P2 is less than the surrounding ambient pressure p, the pressure difference p — P2 helps to force the fuel into the airstream and mix it with the air downstream of the throat.

In an application closer to aerodynamics, a venturi can be used to measure air­speeds . Consider a venturi with a given inlet-to-throat area ratio A/A2, as shown in Figure 3.7. Assume that the venturi is inserted into an airstream that has an unknown

V,

 Figure 3.7 Flow through a venturi.

velocity V. We wish to use the venturi to measure this velocity. With regard to the venturi itself, the most direct quantity that can be measured is the pressure differ­ence p і — P2- This can be accomplished by placing a small hole (a pressure tap) in the wall of the venturi at both the inlet and the throat and connecting the pressure leads (tubes) from these holes across a differential pressure gage, or to both sides of a U-tube manometer (see Section 1.9). In such a fashion, the pressure difference Pi — P2 can be obtained directly. This measured pressure difference can be related to the unknown velocity V) as follows. From Bernoulli’s equation, Equation (3.13), we have

Vf = ~(P2 – Pi) + V22 p

From the continuity equations, Equation (3.22), we have

[3.24]

Substituting Equation (3.24) into (3.23), we obtain

v’=h-p’)+(j-Sv‘

Solving Equation (3.25) for Vj, we obtain

 2(pi – p2) p[(A, M2)2- 1]

 [3.26]

Equation (3.26) is the desired result; it gives the inlet air velocity V in terms of the measured pressure difference p— pi and the known density p and area ratio A i / A 2. In this fashion, a venturi can be used to measure airspeeds. Indeed, historically the first practical airspeed indicator on an airplane was a venturi used by the French Captain A. Eteve in January 1911, more than 7 years after the Wright brothers’ first powered flight. Today, the most common airspeed-measuring instrument is the Pitot tube (to be discussed in Section 3.4); however, the venturi is still found on some general aviation airplanes, including home-built and simple experimental aircraft.

Another application of incompressible flow in a duct is the low-speed wind tunnel. The desire to build ground-based experimental facilities designed to produce flows of air in the laboratory which simulate actual flight in the atmosphere dates back to 1871, when Francis Wenham in England built and used the first wind tunnel in history.4 From that date to the mid-1930s, almost all wind tunnels were designed to produce airflows with velocities from 0 to 250 mi/h. Such low-speed wind tunnels are still much in use today, along with a complement of transonic, supersonic, and hypersonic tunnels. The principles developed in this section allow us to examine the basic aspects of low-speed wind tunnels, as follows.

In essence, a low-speed wind tunnel is a large venturi where the airflow is driven by a fan connected to some type of motor drive. The wind-tunnel fan blades are similar to airplane propellers and are designed to draw the airflow through the tunnel circuit. The wind tunnel may be open circuit, where the air is drawn in the front directly from the atmosphere and exhausted out the back, again directly to the atmosphere, as shown in Figure 3.8a; or the wind tunnel may be closed circuit, where the air from the exhaust is returned directly to the front of the tunnel via a closed duct forming a loop, as shown in Figure 3.8b. In either case, the airflow with pressure p enters the nozzle at a low velocity Vi, where the area is Ai. The nozzle converges to a smaller area A2 at the test section, where the velocity has increased to 33 and the pressure has decreased to p2- After flowing over an aerodynamic model (which may be a model of a complete airplane or part of an airplane such as a wing, tail, engine, or nacelle), the air passes into a diverging duct called a diffuser, where the area increases to A3, the velocity decreases to V3, and the pressure increases to рз. From the continuity equation (3.22), the test-section air velocity is

[3.27]

In turn, the velocity at the exit of the diffuser is

[3.28]

The pressure at various locations in the wind tunnel is related to the velocity by Bernoulli’s equation:

P + pVf = P2 + pV% = Рз + рУз

I 4 For a discussion on the history of wind tunnels, see chapter 4 of Reference 2.

 (reservoir)

 (a) Open-circuit tunnel Figure 3.8 (a) Open-circuit tunnel, (b) Closed-circuit tunnel.

The basic factor that controls the air velocity in the test section of a given low – speed wind tunnel is the pressure difference p — /ъ. To see this more clearly, rewrite Equation (3.29) as

Vi = — (Pi – Pi) + v,2 [3.30]

p

From Equation (3.27), V) = (Аг/АОУг – Substituting into the right-hand side of Equation (3.30), we have

Solving Equation (3.31) for we obtain

 2(pi – P2) PW -(a2/ao2]

 v2

 [3.32]

The area ratio A2/A1 is a fixed quantity for a wind tunnel of given design. More­over, the density is a known constant for incompressible flow. Therefore, Equation (3.32) demonstrates conclusively that the test-section velocity Vi is governed by the pressure difference p — pi. The fan driving the wind-tunnel flow creates this pres­sure difference by doing work on the air. When the wind-tunnel operator turns the “control knob” of the wind tunnel and adjusts the power to the fan, he or she is es­sentially adjusting the pressure difference p — P2 and, in turn, adjusting the velocity via Equation (3.32).

In low-speed wind tunnels, a method of measuring the pressure difference p — pi, hence of measuring Vi via Equation (3.32), is by means of a manometer as discussed in Section 1.9. In Equation (1.56), the density is the density of the liquid in the manometer (not the density of the air in the tunnel). The product of density and the acceleration of gravity g in Equation (1.56) is the weight per unit volume of the manometer fluid. Denote this weight per unit volume by w. Referring to Equation

(1.56) , if the side of the manometer associated with pa is connected to a pressure tap in the settling chamber of the wind tunnel, where the pressure is p, and if the other side of the manometer (associated with рь) is connected to a pressure tap in the test section, where the pressure is P2, then, from Equation (1.56),

p^—pi — wAh

where Ah is the difference in heights of the liquid between the two sides of the manometer. In turn, Equation (3.32) can be expressed as

In many low-speed wind tunnels, the test section is vented to the surrounding atmosphere by means of slots in the wall; in others, the test section is not a duct at all, but rather, an open area between the nozzle exit and the diffuser inlet. In both cases, the pressure in the surrounding atmosphere is impressed on the test-section flow; hence, pi = 1 atm. (In subsonic flow, a jet that is dumped freely into the surrounding air takes on the same pressure as the surroundings; in contrast, a supersonic free jet may have completely different pressures than the surrounding atmosphere, as we see in Chapter 10.)

Keep in mind that the basic equations used in this section have certain limita­tions—we are assuming a quasi-one-dimensional inviscid flow. Such equations can sometimes lead to misleading results when the neglected phenomena are in reality important. For example, if A3 = A1 (inlet area of the tunnel is equal to the exit area), then Equations (3.27) and (3.28) yield V3 = V. In turn, from Equation (3.29), ръ = /з,; that is, there is no pressure difference across the entire tunnel circuit. If this were true, the tunnel would run without the application of any power—we would have a perpetual motion machine. In reality, there are losses in the airflow due to friction at the tunnel walls and drag on the aerodynamic model in the test section. Bernoulli’s equation, Equation (3.29), does not take such losses into account. (Review the derivation of Bernoulli’s equation in Section 3.2; note that viscous effects are neglected.) Thus, in an actual wind tunnel, there is a pressure loss due to viscous and drag effects, and p3 < p. The function of the wind-tunnel motor and fan is to

add power to the airflow in order to increase the pressure of the flow coming out of the diffuser so that it can be exhausted into the atmosphere (Figure 3.8a) or returned to the inlet of the nozzle at the higher pressure p (Figure 3.8/;). Photographs of a typical subsonic wind tunnel are shown in Figure 3.9a and b.

(a)

Consider a low-speed subsonic wind tunnel with a 12/1 contraction ratio for the nozzle. If the flow in the test section is at standard sea level conditions with a velocity of 50 m/s, calculate the height difference in a U-tube mercury manometer with one side connected to the nozzle inlet and the other to the test section.

Solution

At standard sea level, p = 1.23 kg/m3. From Equation (3.32),

 P~ P2 = ^Pv2

 1527 N/nr

However, p, — p2 = wAh. The density of liquid mercury is 1.36 x 104 kg/m Hence,

w = (1.36 x 104 kg/m3)(9.8 m/s2) = 1.33 x!05 N/nr

Consider a model of an airplane mounted in a subsonic wind tunnel, such as shown in Figure 3.10. The wind-tunnel nozzle has a 12-to-l contraction ratio. The maximum lift coefficient of

 Figure 3.1 О Typical model installation in the test section of a large wind tunnel. The Glenn L. Martin Wind Tunnel at the University of Maryland.

the airplane model is 1.3. The wing planform area of the model is 6 ft2. The lift is measured with a mechanical balance that is rated at a maximum force of 1000 lb; that is, if the lift of the airplane model exceeds 1000 lb, the balance will be damaged. During a given test of this airplane model, the plan is to rotate the model through its whole range of angle of attack, including up to that for maximum CL. Calculate the maximum pressure difference allowable between the wind-tunnel settling chamber and the test section, assuming standard sea level density in the test section (i. e., px = 0.002377 slug/ft3).

 117.5 lb/ft2

## Applied Aerodynamics: The Flow over an Airfoil—The Real Case

In this chapter, we have studied the inviscid, incompressible flow over airfoils. When compared with actual experimental lift and moment data for airfoils in low-speed flows, we have seen that our theoretical results based on the assumption of inviscid flow are quite good—with one glaring exception. In the real case, flow separation occurs over the top surface of the airfoil when the angle of attack exceeds a certain value—the “stalling” angle of attack. As described in Section 4.3, this is a viscous effect. As shown in Figure 4.4, the lift coefficient reaches a local maximum denoted by c’/.max, and the angle of attack at which q max is achieved is the stalling angle of attack. An increase in a beyond this value usually results in a (sometimes rather precipitous) drop in lift. At angles of attack well below the stalling angle, the experimental data

clearly show a linear increase in q with increasing a—a result that is predicted by the theory presented in this chapter. Indeed, in this linear region, the inviscid flow theory is in excellent agreement with the experiment, as reflected in Figure 4.5 and as demonstrated by Example 4.2. However, the inviscid theory does not predict flow separation, and consequently the prediction of Q, max and the stalling angle of attack must be treated in some fashion by viscous flow theory. Such viscous flow analyses are the purview of Part 4. On the other hand, the purpose of this section is to examine the physical features of the real flow over an airfoil, and flow separation is an inherent part of this real flow. Therefore, let us take a more detailed look at how the flow field over an airfoil changes as the angle of attack is increased, and how the lift coefficient is affected by such changes.

The flow fields over an NACA 4412 airfoil at different angles of attack are shown in Figure 4.34. Here, the streamlines are drawn to scale as obtained from the experimental results of Hikaru Ito given in Reference 50. The experimental streamline patterns were made visible by a smoke wire technique, wherein metallic wires spread with oil over their surfaces were heated by an electric pulse and the resulting white smoke creates visible streaklines in the flow field. In Figure 4.34, the angle of attack is progressively increased as we scan from Figure 4.34a to e; to the right of each streamline picture is an arrow, the length of which is proportional to the value of the lift coefficient at the given angle of attack. The actual experimentally measured lift curve for the airfoil is given in Figure 4.34/. Note that at low angle of attack, such as a — 2° in Figure 4.34a, the streamlines are relatively undisturbed from their freestream shapes and q is small. As a is increased to 5°, as shown in Figure 4.34£>, and then to 10°, as shown in Figure 4.34c, the streamlines exhibit a pronounced upward deflection in the region of the leading edge, and a subsequent downward deflection in the region of the trailing edge. Note that the stagnation point progressively moves downstream of the leading edge over the bottom surface of the airfoil as a is increased. Of course, c; increases as a is increased, and, in this region, the increase is linear, as seen in Figure 4.34/. When a is increased to slightly less than 15°, as shown in Figure 4.34d, the curvature of the streamlines is particularly apparent. In Figure 4.34/ the flow field is still attached over the top surface of the airfoil. However, as a is further increased slightly above 15°, massive flow-field separation occurs over the top surface, as shown in Figure 4.34e. By slightly increasing a from that shown in Figure 434d to that in Figure 4.34e, the flow quite suddenly separates from the leading edge and the lift coefficient experiences a precipitous decrease, as seen in Figure 4.34/.

The type of stalling phenomenon shown in Figure 4.34 is called leading-edge stall; it is characteristic of relatively thin airfoils with thickness ratios between 10 and 16 percent of the chord length. As seen above, flow separation takes place rather suddenly and abruptly over the entire top surface of the airfoil, with the origin of this separation occurring at the leading edge. Note that the lift curve shown in Figure 4.34/ is rather sharp-peaked in the vicinity of Q max with a rapid decrease in q above the stall.

A second category of stall is the trailing-edge stall. This behavior is character­istic of thicker airfoils such as the NACA 4421 shown in Figure 4.35. Here, we see a progressive and gradual movement of separation from the trailing edge toward the

 (a)

 0= 1-2 /4

 ci

 Figure 4.34 Example of leading-edge stall. Streamline patterns for an NACA 441 2 airfoil at different angles of attack. (The streamlines are drawn to scale from experimental data given by Hikaru Ito in Reference 50.) Re = 2.1 x 105 and Vx = 8 m/s in air. The corresponding experimentally measured lift coefficients are indicated by arrows at the right of each streamline picture, where the lenqth of each arrow indicates the relative maqnitude of the lift. The lift coefficient is also shown in part (f).

leading edge as a is increased. The lift curve for this case is shown in Figure 4.36. The solid curve in Figure 4.36 is a repeat of the results for the NACA 4412 airfoil shown earlier in Figure 4.34/—an airfoil with a leading-edge stall. The dot-dashed curve is the lift curve for the NACA 4421 airfoil—an airfoil with a trailing-edge stall. In comparing these two curves, note that:

1. The trailing-edge stall yields a gradual bending-over of the lift curve at maximum lift, in contrast to the sharp, precipitous drop in q for the leading-edge stall. The stall is “soft” for the trailing-edge stall.

2. The value of c/.max is not so large for the trailing-edge stall.

3. For both the NACA 4412 and 4421 airfoils, the shape of the mean camber line is the same. From the thin airfoil theory discussed in this chapter, the linear lift slope and the zero-lift angle of attack should be the same for both airfoils; this is confirmed by the experimental data in Figure 4.36. The only difference between the two airfoils is that one is thicker than the other. Hence, comparing results shown in Figures 4.34 to 4.36, we conclude that the major effect of thickness of the airfoil is its effect on the value of ct max, and this effect is mirrored by the leading-edge stall behavior of the thinner airfoil versus the trailing-edge stall behavior of the thicker airfoil.

There is a third type of stall behavior, namely, behavior associated with the extreme thinness of an airfoil. This is sometimes labeled as “thin airfoil stall.” An extreme example of a very thin airfoil is a flat plate; the lift curve for a flat plate is

 Figure 4.36 Lift-coefficient curves for three airfoils with different aerodynamic behavior: trailing-edge stall (NACA4421 airfoil), leading-edge stall (NACA4412 airfoil), thin airfoil stall (flat plate).

shown as the dashed curve in Figure 4.36 labeled “thin airfoil stall.” The streamline patterns for the flow over a flat plate at various angles of attack are given in Figure 4.37. The thickness of the flat plate is 2 percent of the chord length. Inviscid, incompressible flow theory shows that the velocity becomes infinitely large at a sharp convex comer; the leading edge of a flat plate at an angle of attack is such a case. In the real flow over the plate as shown in Figure 4.37, nature addresses this singular behavior by having the flow separate at the leading edge, even for very low values of a. Examining Figure 4.37a, where a = 3°, we observe a small region of separated flow at the leading edge. This separated flow reattaches to the surface further downstream, forming a separation bubble in the region near the leading edge. As a is increased, the reattachment point moves further downstream; that is, the separation bubble becomes larger. This is illustrated in Figure 4.37/? where a = 7°. At a = 9° (Figure 4.37c), the separation bubble extends over almost the complete flat plate. Referring back to Figure 4.36, we note that this angle of attack corresponds to c/ max for the flat plate. When a is increased further, total flow separation is present, such as shown in Figure 4.37d. The lift curve for the flat plate in Figure 4.36 shows an early departure from its linear variation at about a = 3°; this corresponds to the formation of the leading-edge separation bubble. The lift curve gradually bends over as a is increased further and exhibits a very gradual and “soft” stall. This is a trend similar to the

Example of thin airfoil stall. Streamline patterns for a flat plate at angle of attack. (The streamlines are drawn to scale from the experimental data of Hikaru Ito in Reference 50.)

case of the trailing-edge stall, although the physical aspects of the flow are quite different between the two cases. Of particular importance is the fact that Q, max for the flat plate is considerably smaller than that for the two NACA airfoils compared in Figure 4.36. Hence, we can conclude from Figure 4.36 that the value of с/іШах is critically dependent on airfoil thickness. In particular, by comparing the flat plate with the two NACA airfoils, we see that some thickness is vital to obtaining a high value of Q max. However, beyond that, the amount of thickness will influence the type of stall (leading-edge versus trailing-edge), and airfoils that are very thick tend to exhibit reduced values of c; max as the thickness increases. Hence, if we plot Q. max versus thickness ratio, we expect to see a local maximum. Such is indeed the case, as shown in Figure 4.38. Here, experimental data for c/,max for the NACA 63- 2XX series of airfoils is shown as a function of the thickness ratio. Note that as the thickness ratio increases from a small value, c/ max first increases, reaches a maximum value at a thickness ratio of about 12 percent, and then decreases at larger thickness ratios. The experimental data in Figure 4.38 is plotted with the Reynolds number as a parameter. Note that Q max for a given airfoil is clearly a function of Re, with higher values of c;?max corresponding to higher Reynolds numbers. Since flow separation is responsible for the lift coefficient exhibiting a local maximum, since flow separation is a viscous phenomenon, and since a viscous phenomenon is governed by a Reynolds number, it is no surprise that q max exhibits some sensitivity to Re.

When was the significance of airfoil thickness first understood and appreciated? This question is addressed in the historical note in Section 4.13, where we will see

that the aerodynamic properties of thick airfoils even transcended technology during World War I and impacted the politics of the armistice.

Let us examine some other aspects of airfoil aerodynamics—aspects that are not always appreciated in a first study of the subject. The simple generation of lift by an airfoil is not the prime consideration in its design—even a barn door at an angle of attack produces lift. Rather, there are two figures of merit that are primarily used to judge the quality of a given airfoil:

1. The lift-to-drag ratio L/D. An efficient airfoil produces lift with a minimum of drag; that is, the ratio of lift-to-drag is a measure of the aerodynamic efficiency of an airfoil. The standard airfoils discussed in this chapter have high L/D ratios— much higher than that of a barn door. The L/D ratio for a complete flight vehicle has an important impact on its flight performance; for example, the range of the vehicle is directly proportional to the L/D ratio. (See Reference 2 for an extensive discussion of the role of L/D on flight performance of an airplane.)

2. The maximum lift coefficient c;>max. An effective airfoil produces a high value of ci, max—much higher than that produced by a bam door.

The maximum lift coefficient is worth some additional discussion here. For a complete flight vehicle, the maximum lift coefficient CT max determines the stalling speed of the aircraft as discussed in the Design Box at the end of Section 1.8. From Equation (1.47), repeated below:

 2 W

 [1.47]

 Tstall —

 PocSCl .max

Therefore, a tremendous incentive exists to increase the maximum lift coefficient of an airfoil, in order to obtain either lower stalling speeds or higher payload weights at the same speed, as reflected in Equation (1.47). Moreover, the maneuverability of an airplane (i. e„ the smallest possible turn radius and the fastest possible turn rate) depends on a large value of C7,max (see Section 6.17 of Reference 2). On the other hand, for an airfoil at a given Reynolds number, the value of c; max is a function primarily of its shape. Once the shape is specified, the value of Q, max is what nature dictates, as we have already seen. Therefore, to increase c/>max beyond such a value, we must carry out some special measures. Such special measures include the use of flaps and/or leading-edge slats to increase c/,max above that for the reference airfoil itself. These are called high-lift devices, and are discussed in more detail below.

A trailing-edge flap is simply a portion of the trailing-edge section of the airfoil that is hinged and which can be deflected upward or downward, as sketched in the insert in Figure 4.39a. When the flap is deflected downward (a positive angle <5 in Figure 4.39a, the lift coefficient is increased, as shown in Figure 4.39a. This increase is due to an effective increase in the camber of the airfoil as the flap is deflected

 Figure 4.39 Effect of flap deflection on streamline shapes. (The streamlines are drawn to scale from the experimental data of Hikaru Ito in Reference 50.j (o) Effect of flap deflection on lift coefficient, (b) Streamline pattern with no flap deflection, (c) Streamline pattern with a 15° flop deflection.

downward. The thin airfoil theory presented in this chapter clearly shows that the zero-lift angle of attack is a function of the amount of camber [see Equation (4.61)], with о!^=о becoming more negative as the camber is increased. In terms of the lift curve shown in Figure 4.39a, the original curve for no flap deflection goes through the origin because the airfoil is symmetric; however, as the flap is deflected downward, this lift curve simply translates to the left because aL=0 is becoming more negative. In Figure 4.39a, the results are given for flap deflections of ±10°. Comparing the case for 5 = 10° with the no-deflection case, we see that, at a given angle of attack, the lift coefficient is increased by an amount Де/ due to flap deflection. Moreover, the actual value of Cf, max is increased by flap deflection, although the angle of attack at which С/ max occurs is slightly decreased. The change in the streamline pattern when the flap is deflected is shown in Figure 4.39b and c. Figure 4.39b is the case for a = 0 and 5 = 0 (i. e., a symmetric flow). However, when a is held fixed at zero, but the flap is deflected by 15°, as shown in Figure 4.39c, the flow field becomes unsymmetrical and resembles the lifting flows shown (e. g., in Figure 4.34). That is, the streamlines in Figure 4.39c are deflected upward in the vicinity of the leading edge and downward near the trailing edge, and the stagnation point moves to the lower surface of the airfoil—just by deflecting the flap downward.

High-lift devices can also be applied to the leading edge of the airfoil, as shown in the insert in Figure 4.40. These can take the form of a leading-edge slat, leading – edge droop, or a leading-edge flap. Let us concentrate on the leading-edge slat,

 Figure 4.40 Effect of leading-edge flap on lift coefficient.

Figure 4.41 Effect of a leading-edge slat on the streamline pattern over an NACA 4412 airfoil.

(The streamlines are drawn to scale from the experimental data in Reference 50.J

The high-lift devices used on modern, high-performance aircraft are usually a combination of leading-edge slats (or flaps) and multielement trailing-edge flaps. Typical airfoil configurations with these devices are sketched in Figure 4.42. Three configurations including the high-lift devices are shown: A—the cruise configuration, with no deployment of the high-lift devices; В—a typical configuration at takeoff, with both the leading – and trailing-edge devices partially deployed; and C—a typical configuration at landing, with all devices fully extended. Note that for configuration C, there is a gap between the slat and the leading edge and several gaps between the different elements of the multielement trailing-edge flap. The streamline pattern for the flow over such a configuration is shown in Figure 4.43. Flere, the leading-edge slat and the multielement trailing-edge flap are fully extended. The angle of attack is 25°. Although the main flow over the top surface of the airfoil is essentially separated, the local flow through the gaps in the multi-element flap is locally attached to the top surface of the flap; because of this locally attached flow, the lift coefficient is still quite high, on the order of 4.5.

With this, we end our discussion of the real flow over airfoils. In retrospect, we can say that the real flow at high angles of attack is dominated by flow separation—a phenomenon that is not properly modeled by the inviscid theories presented in this chapter. On the other hand, at lower angles of attack, such as those associated with the cruise conditions of an airplane, the inviscid theories presented here do an excellent job of predicting both lift and moments on an airfoil. Moreover, in this section, we have clearly seen the importance of airfoil thickness in determining the angle of

 A: Cruise configuration

 Figure 4.42 Airfoil with leading-edge and trailing-edge high-lift mechanisms. The trailing-edge device is a multi-element flap.
 Figure 4.43 Effect of leading-edge and multi-element flaps on the streamline pattern around an airfoil at angle of attack of 25°. (The streamlines are drawn to scale from the experimental data of Reference 50.)

attack at which flow separation will occur, and hence greatly affecting the maximum lift coefficient.

## Speed of Sound

Common experience tells us that sound travels through air at some finite velocity. For example, you see a flash of lightning in the distance, but you hear the corresponding thunder at some later moment. What is the physical mechanism of the propagation of sound waves? How can we calculate the speed of sound? What properties of the gas does it depend on? The speed of sound is an extremely important quantity which dominates the physical properties of compressible flow, and hence the answers to the above questions are vital to our subsequent discussions. The purpose of this section is to address these questions.

The physical mechanism of sound propagation in a gas is based on molecular motion. For example, imagine that you are sitting in a room, and suppose that a firecracker goes off in one comer. When the firecracker detonates, chemical energy (basically a form of heat release) is transferred to the air molecules adjacent to the firecracker. These energized molecules are moving about in a random fashion. They eventually collide with some of their neighboring molecules and transfer their high energy to these neighbors. In turn, these neighboring molecules eventually collide with their neighbors and transfer energy in the process. By means of this “domino” effect, the energy released by the firecracker is propagated through the air by molecular collisions. Moreover, because T, p, and p for a gas are macroscopic averages of the detailed microscopic molecular motion, the regions of energized molecules are also regions of slight variations in the local temperature, pressure, and density. Hence, as this energy wave from the firecracker passes over our eardrums, we “hear” the slight pressure changes in the wave. This is sound, and the propagation of the energy wave is simply the propagation of a sound wave through the gas.

Because a sound wave is propagated by molecular collisions, and because the molecules of a gas are moving with an average velocity of lit given by ki­

netic theory, then we would expect the velocity of propagation of a sound wave to be approximately the average molecular velocity. Indeed, the speed of sound is about three-quarters of the average molecular velocity. In turn, because the kinetic theory expression given above for the average molecular velocity depends only on the tem­perature of the gas, we might expect the speed of sound to also depend on temperature only. Let us explore this matter further; indeed, let us now derive an equation for the speed of sound in a gas. Although the propagation of sound is due to molecular col­lisions, we do not use such a microscopic picture for our derivation. Rather, we take advantage of the fact that the macroscopic properties p, T, p, etc., change across the wave, and we use our macroscopic equations of continuity, momentum, and energy to analyze these changes.

Consider a sound wave propagating through a gas with velocity a, as sketched in Figure 8.4a. Here, the sound wave is moving from right to left into a stagnant gas (region 1), where the local pressure, temperature, and density are p, T, and p, respectively. Behind the sound wave (region 2), the gas properties are slightly different and are given by p + dp, T + dT, and p + dp, respectively. Now imagine that you hop on the wave and ride with it. When you look upstream, into region 1,

you see the gas moving toward you with a relative velocity a, as sketched in Figure 8.4b. When you look downstream, into region 2, you see the gas receding away from you with a relative velocity a + da, as also shown in Figure 8.4. (We have enough fluid-dynamic intuition by now to realize that because the pressure changes across the wave by the amount dp, then the relative flow velocity must also change across the wave by some amount da. Hence, the relative flow velocity behind the wave is a + da.) Consequently, in Figure 8.4b, we have a picture of a stationary sound wave, with the flow ahead of it moving left to right with velocity a. The pictures in Figure 8.4a and b are analogous; only the perspective is different. For purposes of analysis, we use Figure 8.4b.

(Note: Figure 8.4b is similar to the picture of a normal shock wave shown in Figure 8.3. In Figure 8.3, the normal shock wave is stationary, and the upstream flow is moving left to right at a velocity u. If the upstream flow were to be suddenly shut off, then the normal shock wave in Figure 8.3 would suddenly propagate to the left with a wave velocity of u, similar to the moving sound wave shown in Figure 8.4a. The analysis of moving waves is slightly more subtle than the analysis of stationary waves; hence, it is simpler to begin a study of shock waves and sound waves with the pictures of stationary waves as shown in Figures 8.3 and 8.4b. Also, please note that the sound wave in Figure 8.4b is nothing more than an infinitely weak normal shock wave.)

Examine closely the flow through the sound wave sketched in Figure 8.4b. The flow is one-dimensional. Moreover, it is adiabatic, because we have no source of heat transfer into or out of the wave (e. g., we are not “zapping” the wave with a laser beam or heating it with a torch). Finally, the gradients within the wave are very small—the changes dp, dT, dp, and da are infinitesimal. Therefore, the influence of dissipative phenomena (viscosity and thermal conduction) is negligible. As a result, the flow through the sound wave is both adiabatic and reversible—the flow is isentropic. Since we have now established that the flow is one-dimensional and isentropic, let us apply the appropriate governing equations to the picture shown in Figure 8.4b.

## Physical principle Mass can be neither created nor destroyed

Consider a flow field wherein all properties vary with spatial location and time, e. g., p — p(x, y,z, t). In this flow field, consider the fixed finite control volume shown in Figure 2.17. At a point on the control surface, the flow velocity is V and the vector elemental surface area is dS. Also dV is an elemental volume inside the control volume. Applied to this control volume, the above physical principle means

Net mass flow out of control _ time rate of decrease of volume through surface S mass inside control volume V

or В = C where В and C are just convenient symbols for the left and right sides, respectively, of Equation (2.45a). First, let us obtain an expression for В in terms of the quantities shown in Figure 2.17. From Equation (2.43), the elemental mass flow across the area dS is

pVn dS = p • dS

Examining Figure 2.17, note that by convention, dS always points in a direction out of the control volume. Hence, when V also points out of the control volume (as shown

 Figure 2.1 7 Finite control volume fixed in space.

in Figure 2.17), the product pV • dS is positive. Moreover, when V points out of the control volume, the mass flow is physically leaving the control volume; i. e., it is an outflow. Hence, a positive pV • dS denotes an outflow. In turn, when V points into the control volume, pV • dS is negative. Moreover, when V points inward, the mass flow is physically entering the control volume; i. e., it is an inflow. Hence, a negative pV • dS denotes an inflow. The net mass flow out of the entire control surface S is the summation over S of the elemental mass flows. In the limit, this becomes a surface integral, which is physically the left side of Equations (2.45a and b) i. e.,

physical principle of the conservation of mass to a finite control volume fixed in space. Equation (2.48) is called the continuity equation. It is one of the most fundamental equations of fluid dynamics.

Note that Equation (2.48) expresses the continuity equation in integral form. We will have numerous opportunities to use this form; it has the advantage of relating aerodynamic phenomena over a finite region of space without being concerned about the details of precisely what is happening at a given distinct point in the flow. On the other hand, there are many times when we are concerned with the details of a flow and we want to have equations that relate flow properties at a given point. In such a case, the integral form as expressed in Equation (2.48) is not particularly useful. However, Equation (2.48) can be reduced to another form that does relate flow properties at a given point, as follows. To begin with, since the control volume used to obtain Equation (2.48) is fixed in space, the limits of integration are also fixed. Hence, the time derivative can be placed inside the volume integral and Equation (2.48) can be written as

[2.49]

Applying the divergence theorem, Equation (2.26), we can express the right-hand term of Equation (2.49) as

Substituting Equation (2.50) into (2.49), we obtain

Examine the integrand of Equation (2.51). If the integrand were a finite number, then Equation (2.51) would require that the integral over part of the control volume be equal and opposite in sign to the integral over the remainder of the control volume, such that the net integration would be zero. However, the finite control volume is arbitrarily drawn in space; there is no reason to expect cancellation of one region by the other. Hence, the only way for the integral in Equation (2.51) to be zero for an

arbitrary control volume is for the integrand to be zero at all points within the control volume. Thus, from Equation (2.51), we have

Equation (2.52) is the continuity equation in the form of a partial differential equation. This equation relates the flow field variables at a point in the flow, as opposed to Equation (2.48), which deals with a finite space.

It is important to keep in mind that Equations (2.48) and (2.52) are equally valid statements of the physical principle of conservation of mass. They are mathematical representations, but always remember that they speak words—they say that mass can be neither created nor destroyed.

Note that in the derivation of the above equations, the only assumption about the nature of the fluid is that it is a continuum. Therefore, Equations (2.48) and (2.52) hold in general for the three-dimensional, unsteady flow of any type of fluid, inviscid or viscous, compressible or incompressible. (Note: It is important to keep track of all assumptions that are used in the derivation of any equation because they tell you the limitations on the final result, and therefore prevent you from using an equation for a situation in which it is not valid. In all our future derivations, develop the habit of noting all assumptions that go with the resulting equations.)

It is important to emphasize the difference between unsteady and steady flows. In an unsteady flow, the flow-field variables are a function of both spatial location and time, e. g.,

P – P(x, У, z, t)

This means that if you lock your eyes on one fixed point in space, the density at that point will change with time. Such unsteady fluctuations can be caused by time – varying boundaries (e. g., an airfoil pitching up and down with time or the supply valves of a wind tunnel being turned off and on). Equations (2.48) and (2.52) hold for such unsteady flows. On the other hand, the vast majority of practical aerodynamic problems involve steady flow. Here, the flow-field variables are a function of spatial location only, e. g.,

P = P(x, y, z)

## Nonlifting Flow Over a Circular Cylinder

Consulting our road map given in Figure 3.4, we see that we are well into the third column, having already discussed uniform flow, sources and sinks, and doublets. Along the way, we have seen how the flow over a semi-infinite body can be simulated by the combination of a uniform flow with a source, and the flow over an oval-shaped body can be constructed by superimposing a uniform flow and a source-sink pair. In this section, we demonstrate that the combination of a uniform flow and a doublet produces the flow over a circular cylinder. A circular cylinder is one of the most

basic geometric shapes available, and the study of the flow around such a cylinder is a classic problem in aerodynamics.

Consider the addition of a uniform flow with velocity and a doublet of strength

к, as shown in Figure 3.26. The direction of the doublet is upstream, facing into the uniform flow. From Equations (3.57) and (3.87), the stream function for the combined flow is

l/r = УооГ sin в —

Let R2 = к/2nVoa. Then Equation (3.91) can be written as

[3.92]

Equation (3.92) is the stream function for a uniform flow-doublet combination; it is also the stream function for the flow over a circular cylinder of radius R as shown in Figure 3.26 and as demonstrated below.

The velocity field is obtained by differentiating Equation (3.92), as follows:

Figure 3.26 Superposition of a uniform flow and a doublet; nonlifting flow over a circular cylinder.

 (V

 (T^ sin 9)

[3.94]

To locate the stagnation points, set Equations (3.93) and (3.94) equal to zero:

[3.95]

[3.96]

Simultaneously solving Equations (3.95) and (3.96) for r and в, we find that there are two stagnation points, located at (г, в) = (R, 0) and (R, ж). These points are denoted as A and B, respectively, in Figure 3.26.

The equation of the streamline that passes through the stagnation point В is obtained by inserting the coordinates of В into Equation (3.92). For r = R and в = ж, Equation (3.92) yields r/r — 0. Similarly, inserting the coordinates of point A into Equation (3.92), we also find that = 0. Hence, the same streamline goes through both stagnation points. Moreover, the equation of this streamline, from Equation (3.92), is

Note that Equation (3.97) is satisfied by r = R for all values of 9. However, recall that R2 = к/2ж Уж, which is a constant. Moreover, in polar coordinates, r = constant = R is the equation of a circle of radius R with its center at the origin. Therefore, Equation (3.97) describes a circle with radius R, as shown in Figure 3.26. Moreover, Equation (3.97) is satisfied by в — ж and в — 0 for all values of r; hence, the entire horizontal axis through points A and B, extending infinitely far upstream and downstream, is part of the stagnation streamline.

Note that the [r — 0 streamline, since it goes through the stagnation points, is the dividing streamline. That is, all the flow inside xjr = 0 (inside the circle) comes from the doublet, and all the flow outside ij/ = 0 (outside the circle) comes from the uniform flow. Therefore, we can replace the flow inside the circle by a solid body, and the external flow will not know the difference. Consequently, the inviscid irrotational, incompressible flow over a circular cylinder of radius R can be synthesized by adding a uniform flow with velocity and a doublet of strength к, where R is related to Voo and к through

[3.98]

Note from Equations (3.92) to (3.94) that the entire flow field is symmetrical about both the horizontal and vertical axes through the center of the cylinder, as clearly seen by the streamline pattern sketched in Figure 3.26. Hence, the pressure

distribution is also symmetrical about both axes. As a result, the pressure distribution over the top of the cylinder is exactly balanced by the pressure distribution over the bottom of the cylinder (i. e., there is no net lift). Similarly, the pressure distribution over the front of the cylinder is exactly balanced by the pressure distribution over the back of the cylinder (i. e., there is no net drag). In real life, the result of zero lift is easy to accept, but the result of zero drag makes no sense. We know that any aerodynamic body immersed in a real flow will experience a drag. This paradox between the theoretical result of zero drag, and the knowledge that in real life the drag is finite, was encountered in the year 1744 by the Frenchman Jean Le Rond d’Alembert—and it has been known as d’Alembert’s paradox ever since. For d’Alembert and other fluid dynamic researchers during the eighteenth and nineteenth centuries, this paradox was unexplained and perplexing. Of course, today we know that the drag is due to viscous effects which generate frictional shear stress at the body surface and which cause the flow to separate from the surface on the back of the body, thus creating a large wake downstream of the body and destroying the symmetry of the flow about the vertical axis through the cylinder. These viscous effects are discussed in detail in Chapters 15 through 20. However, such viscous effects are not included in our present analysis of the inviscid flow over the cylinder. As a result, the inviscid theory predicts that the flow closes smoothly and completely behind the body, as sketched in Figure 3.26. It predicts no wake, and no asymmetries, resulting in the theoretical result of zero drag.

Let us quantify the above discussion. The velocity distribution on the surface of the cylinder is given by Equations (3.93) and (3.94) with r — R, resulting in

Note that at the surface of the cylinder, Vr is geometrically normal to the surface; hence, Equation (3.99) is consistent with the physical boundary condition that the component of velocity normal to a stationary solid surface must be zero. Equation (3.100) gives the tangential velocity, which is the full magnitude of velocity on the surface of the cylinder, that is, V = Vg = —2 V» sin 0 on the surface. The minus sign in Equation (3.100) is consistent with the sign convention in polar coordinates that Vg is positive in the direction of increasing в, that is, in the counterclockwise direction as shown in Figure 3.27. However, in Figure 3.26, the surface velocity for

Figure 3.27 Sign convention for

0 < в < jt is obviously in the opposite direction of increasing 0; hence, the minus sign in Equation (3.100) is proper. For it < 0 < In, the surface flow is in the same direction as increasing 0, but sin 0 is itself negative; hence, once again the minus sign in Equation (3.100) is proper. Note from Equation (3.100) that the velocity at the surface reaches a maximum value of 2Voo at the top and the bottom of the cylinder (where 0 = тг/2 and Зтг/2, respectively), as shown in Figure 3.28. Indeed, these are the points of maximum velocity for the entire flow field around the cylinder, as can be seen from Equations (3.93) and (3.94).

The pressure coefficient is given by Equation (3.38):

c„=,-(£)2 13.3.1

 Cp = 1—4 sin[10] 0

Combining Equations (3.100) and (3.38), we find that the surface pressure coefficient over a circular cylinder is

Note that Cp varies from 1.0 at the stagnation points to —3.0 at the points of maximum velocity. The pressure coefficient distribution over the surface is sketched in Figure 3.29. The regions corresponding to the top and bottom halves of the cylinder are identified at the top of Figure 3.29. Clearly, the pressure distribution over the top half of the cylinder is equal to the pressure distribution over the bottom half, and hence the lift must be zero, as discussed earlier. Moreover, the regions corresponding to the front and rear halves of the cylinder are identified at the bottom of Figure 3.29. Clearly, the pressure distributions over the front and rear halves are the same, and hence the drag is theoretically zero, as also discussed previously. These results are confirmed by Equations (1.15) and (1.16). Since Cf — 0 (we are dealing with an inviscid flow), Equations (1.15) and (1.16) become, respectively,

Figure 3.28

 Pressure coefficient distribution over the surface of a circular cylinder; theoretical results for inviscid, incompressible flow.

 Figure 3.29

For the circular cylinder, the chord c is the horizontal diameter. From Figure 3.29, Cp j = Cp u for corresponding stations measured along the chord, and hence the integrands in Equations (3.102) and (3.103) are identically zero, yielding cn = ca = 0. In turn, the lift and drag are zero, thus again confirming our previous conclusions.

These points, as well as the stagnation points and points of minimum pressure, are illustrated in Figure 3.30. Note that at the stagnation point, where Cp = 1, the pressure is p^, + q-s. the pressure decreases to pco in the first 30° of expansion around the body, and the minimum pressure at the top and bottom of the cylinder, consistent with Cp = —3, is px — 3qca-

 Values of pressure at various locations on the surface of a circular cylinder; nonlifting case.

 Figure 3.30

## . Applied Aerodynamics: The Delta Wing

In Part 3 of this book, we will see that supersonic flow is dramatically different from subsonic flow in virtually all respects—the mathematics and physics of these two flow regimes are totally different. Such differences impact the design philosophy of aircraft for supersonic flight in comparison to aircraft for subsonic flight. In particular, supersonic airplanes usually have highly swept wings (the reasons for this are discussed in Part 3). A special case of swept wings is those aircraft with a triangular planform—called delta wings. A comparison of the planform of a conventional swept wing was shown in Figure 5.30. Two classic examples of aircraft with delta wings are the Convair F-102A, the first operational jet airplane in the United States to be designed with a delta wing, shown in Figure 5.37a, and the space shuttle, basically a hypersonic airplane, shown in Figure 5.31b. In reality, the planform of the space shuttle is more correctly denoted as a double-delta shape. Indeed, there are several variants of the basic delta wing used on modem aircraft; these are shown in Figure 5.38. Delta wings are used on many different types of high-speed airplanes around the world; hence, the delta planform is an important aerodynamic configuration.

 Ui) Figure 5.37 Some delta-winged vehicles, (a) The Convair F-l 02A. (Courtesy of the U. S. Air Force.}

Question: Since delta-winged aircraft are high-speed vehicles, why are we dis­cussing this topic in the present chapter, which deals with the low-speed, incompress­ible flow over finite wings? The obvious answer is that all high-speed aircraft fly at low speeds for takeoff and landing; moreover, in most cases, these aircraft spend the

 Figure 5.37 (continued) Some delta-winged vehicles, (b) The space shuttle. (Courtesy of NASA.)

Figure 5.38 Four versions of a delta-wing planform.

(From Loftin, Reference 48.)

vast majority of their flight time at subsonic speeds, using their supersonic capability for short “supersonic dashes,” depending on their mission. Several exceptions are, of course, the Concorde supersonic transport which cruises at supersonic speeds across oceans, and the space shuttle, which is hypersonic for most of its reentry into the earth’s atmosphere. However, the vast majority of delta-winged aircraft spend a great deal of their flight time at subsonic speeds. For this reason, the low-speed aerodynamic characteristics of delta wings are of great importance; this is accentuated by the rather different and unique aerodynamic aspects associated with such delta wings. Therefore, the low-speed aerodynamics of delta wings has been a subject of much serious study over the past years, going back as far as the early work on delta wings by Alexander Lippisch in Germany during the 1930s. This is the answer to our question posed above—in the context of our discussion on finite wings, we must give the delta wing some special attention.

The subsonic flow pattern over the top of a delta wing at angle of attack is sketched in Figure 5.39. The dominant aspect of this flow are the two vortex patterns that occur in the vicinity of the highly swept leading edges. These vortex patterns are created by the following mechanism. The pressure on the bottom surface of the wing at the angle of attack is higher than the pressure on the top surface. Thus, the flow on the bottom surface in the vicinity of the leading edge tries to curl around the leading edge from the bottom to the top. If the leading edge is sharp, the flow will

separate along its entire length. (We have already mentioned several times that when low-speed, subsonic flow passes over a sharp convex comer, inviscid flow theory predicts an infinite velocity at the corner, and that nature copes with this situation by having the flow separate at the corner. The leading edge of a delta wing is such a case.) This separated flow curls into a primary vortex which exists above the wing just inboard of each leading edge, as sketched in Figure 5.39. The stream surface which has separated at the leading edge (the primary separation line Л) in Figure 5.39) loops above the wing and then reattaches along the primary attachment line (line A in Figure 5.39). The primary vortex is contained within this loop. A secondary vortex is formed underneath the primary vortex, with its own separation line, denoted by S2 in Figure 5.39, and its own reattachment line A%. Notice that the surface streamlines flow away from the attachment lines A and Аг on both sides of these lines, whereas the surface streamlines tend to flow toward the separation fines S) and Sj and then simply lift off the surface along these lines. Inboard of the leading-edge vortices, the surface streamlines are attached, and flow downstream virtually is undisturbed along a series of straight-line rays emanating from the vertex of the triangular shape. A graphic illustration of the leading-edge vortices is shown in both Figures 5.40 and 5.41. In Figure 5.40, we see a highly swept delta wing mounted in a water tunnel. Filaments of colored dye are introduced at two locations along each leading edge. This photograph, taken from an angle looking down on the top of the wing, clearly shows the entrainment of the colored dye in the vortices. Figure 5.41 is a photograph of the vortex pattern in the crossflow plane (the crossflow plane is shown in Figure 5.39). From the photographs in Figures 5.40 and 5.41, we clearly see that the leading-

 Figure 5.40 Leading-edge vortices over the top surface of a delta wing at angle of attack. The vortices are made visible by dye streaks in water flow. (Courtesy of H. Werle, ONERA, France. Also in Van Dyke, Milton, An Album of Fluid Motion, The Parabolic Press, Stanford, CA, 1982.)

edge vortex is real and is positioned above and somewhat inboard of the leading edge itself.

The leading-edge vortices are strong and stable. Being a source of high energy, relatively high-vorticity flow, the local static pressure in the vicinity of the vortices is small. Hence, the surface pressure on the top surface of the delta wing is reduced near the leading edge and is higher and reasonably constant over the middle of the wing. The qualitative variation of the pressure coefficient in the spanwise direction (the у direction as shown in Figure 5.39) is sketched in Figure 5.42. The spanwise variation of pressure over the bottom surface is essentially constant and higher than the freestream pressure (a positive Cp). Over the top surface, the spanwise variation in the midsection of the wing is essentially constant and lower than the freestream pressure (a negative Cp). However, near the leading edges the static pressure drops considerably (the values of Cp become more negative). The leading-edge vortices are literally creating a strong “suction” on the top surface near the leading edges. In Figure 5.42, vertical arrows are shown to indicate further the effect on the spanwise lift distribution; the upward direction of these arrows as well as their relative length show the local contribution of each section of the wing to the normal force distribution. The suction effect of the leading-edge vortices is clearly shown by these arrows.

The suction effect of the leading-edge vortices enhances the lift; for this reason, the lift coefficient curve for a delta wing exhibits an increase in С/ for values of a at

 Figure 5.41 The flow field in the crossflow plane above a delta wing at angle of attack, showing the two primary leading-edge vortices. The vortices are made visible by small air bubbles in water. (Courtesy of H. Werle, ONERA, France. Also in Van Dyke, Milton, An Album of Fluid Motion, The Parabolic Press, Stanford, CA, 1982.)

 Figure 5.42 Schematic of the spanwise pressure coefficient distribution across a delta wing. (Courtesy of John Stollery, Cranfield Institute of Technology, England. j

which conventional wing planforms would be stalled. A typical variation of С/ with a for a 60° delta wing is shown in Figure 5.43. Note the following characteristics:

1. The lift slope is small, on the order of 0.05/degree.

2. However, the lift continues to increase to large values of a; in Figure 5.43, the stalling angle of attack is on the order of 35°. The net result is a reasonable value of ax, on the order of 1.3.

The next time you have an opportunity to watch a delta-winged airplane take off or land, say, for example, the televised landing of the space shuttle, note the large angle of attack of the vehicle. Moreover, you will understand why the angle of attack is large—because the lift slope is small, and hence the angle of attack must be large enough to generate the high values of Cl required for low-speed flight.

The suction effect of the leading-edge vortices, in acting to increase the normal force, consequently, increases the drag at the same time it increases the lift. Hence, the aerodynamic effect of these vortices is not necessarily advantageous. In fact, the lift-to-drag ratio L/D for a delta planform is not so high as conventional wings. The typical variation of L/D with Cl for a delta wing is shown in Figure 5.44; the results for the sharp leading edge, 60° delta wing are given by the lower curve. Note that the maximum value of L/D for this case is about 9.3—not a particularly exciting value for a low-speed aircraft.

 Figure 5.43 Variation of lift coefficient for a flat delta wing with angle of attack. (Courtesy of John Stollery, Cranfield Institute of Technology England.)
 CL Figure 5.44 The effect of leading-edge shape on the lift-to-drag ratio for a delta wing of aspect ratio 2.31. The two solid curves apply to a sharp leading edge, and the dashed curve applies to a rounded leading edge. LEVF denotes a wing with a leading-edge vortex flap. (Courtesy of John Stollery, Cranfield Institute of Technology, England.)

This leads to the second of the phenomena reflected in Figure 5.44. The middle curve in Figure 5.44 is labeled LEVF, which denotes the case for a leading-edge vortex flap. This pertains to a mechanical configuration where the leading edges can be deflected downward through a variable angle, analogous to the deflection of a conventional trailing-edge flap. The spanwise pressure-coefficient distribution for this case is sketched in Figure 5.45; note that the direction of the suction due to the

 Figure 5.45 A schematic of the spanwise pressure coefficient distribution over the top of a delta wing as modified by leading-edge vortex flaps. (Courtesy of John Stollery, Cranfield Institute of Technology, England.!

leading-edge vortice is now modified in comparison to the case with no leading-edge flap shown earlier in Figure 5.42. Also, returning to Figure 5.39, you can visualize what the wing geometry would look like with the leading edge drooped down; a front view of the downward deflected flap would actually show some projected frontal area. Since the pressure is low over this frontal area, the net drag can decrease. This phenomenon is illustrated by the middle curve in Figure 5.44, which shows a generally higher L/D for the leading-edge vortex flap in comparison to the case with no flap (the flat delta wing).

In summary, the delta wing is a common planform for supersonic aircraft. In this section, we have examined the low-speed aerodynamic characteristics of such wings and have found that these characteristics are in some ways quite different from a conventional planform.

## Flow Similarity

Consider two different flow fields over two different bodies. By definition, different flows are dynamically similar if:

1. The streamline patterns are geometrically similar.

2. The distributions of V/ Voo, p/p^, T/Tetc., throughout the flow field are the same when plotted against common nondimensional coordinates.

3. The force coefficients are the same.

Actually, item 3 is a consequence of item 2; if the nondimensional pressure and shear stress distributions over different bodies are the same, then the nondimensional force coefficients will be the same.

The definition of dynamic similarity was given above. Question: What are the criteria to ensure that two flows are dynamically similar? The answer comes from the results of the dimensional analysis in Section 1.7. Two flows will be dynamically similar if:

1. The bodies and any other solid boundaries are geometrically similar for both flows.

2. The similarity parameters are the same for both flows.

So far, we have emphasized two parameters, Re and Mx. For many aerodynamic applications, these are by far the dominant similarity parameters. Therefore, in a lim­ited sense, but applicable to many problems, we can say that flows over geometrically similar bodies at the same Mach and Reynolds numbers are dynamically similar, and hence the lift, drag, and moment coefficients will be identical for the bodies. This is a key point in the validity of wind-tunnel testing. If a scale model of a flight vehicle is tested in a wind tunnel, the measured lift, drag, and moment coefficients will be the same as for free flight as long as the Mach and Reynolds numbers of the wind-tunnel test-section flow are the same as for the free-flight case. As we will see in subse­quent chapters, this statement is not quite precise because there are other similarity parameters that influence the flow. In addition, differences in freestream turbulence between the wind tunnel and free flight can have an important effect on Co and the maximum value of CL. However, direct simulation of the free-flight Re and Мж is the primary goal of many wind-tunnel tests.

Example 1.4

Consider the flow over two circular cylinders, one having four times the diameter of the other, as shown in Figure 1.20. The flow over the smaller cylinder has a freestream density, velocity and temperature given by p, V, and Гь respectively. The flow over the larger cylinder has a freestream density, velocity, and temperature given by p2, V2, and T2, respectively, where p2 = pi/4, V2 = 2Vi, and T2 = AT. Assume that both ц and a are proportional to T1’2. Show that the two flows are dynamically similar.

 Geometrically similar bodies

 Figure 1.20 Example of dynamic flow similarity. Note that as part of the definition of dynamic similarity, the streamlines (lines along which the flow velocity is tangent at each point) are geometrically similar between the two flows.

 Figure 1.22 The NACA variable density tunnel (VDT). Authorized in March of 1921, the VDT was operational in October 1922 at the NACA Langley Memorial Laboratory at Hampton, Virginia. It is essentially a large, subsonic wind tunnel entirely contained within an 85-ton pressure shell, capable of 20 atm. This tunnel was instrumental in the development of the various families of NACA airfoil shapes in the 1920s and 1930s. In the early 1940s, it was decommissioned as a wind tunnel and used as a high-pressure air storage tank. In 1983, due to its age and outdated riveted construction, its use was discontinued altogether. Today, the VDT remains at the NASA Langley Research Center; it has been officially designated as a National Historic Landmark. (Courtesy of NASA.}

 Screen Figure 1.23 Schematic of the variable density tunnel. (From Baals, D. D. and Carliss, W. R„ Wind Tunnels of NASA, NASA SP-440, 1981.}

 Note that for most conventional flight situations, the magnitude of L and W is much larger than the magnitude of T and D, as indicated by the sketch in Figure 1.23. Typically, for conventional cruising flight, L/D ~ 15 to 20. For an airplane of given shape, such as that sketched in Figure 1.24, at given Mach and Reynolds number, Cl and CD are simply functions of the angle of attack, a of the airplane. This is the message conveyed by Equations (1.42) and (1.43). It is a simple and basic message—part of the beauty of nature—that the actual values of CL and Cd for a given body shape just depend on the orientation of the body in the flow, i. e., angle of attack. Generic variations for CL and Cd versus a are sketched in Figure 1.25. Note that CL increases linearly with a until an angle of attack is reached when the wing stalls, the lift coefficient reaches a peak value, and then drops off as a is further increased. The maximum value of the lift coefficient is denoted by Ci milx, as noted in Figure 1.25. The lowest possible velocity at which the airplane can maintain steady, level flight is the stalling velocity, Vstaii; it is dictated by the value of Cl, max, as follows.6 From the definition of lift coefficient given in Section 1.5, applied for the case of level flight where L = W, we have

 extreme measures sometimes taken in order to simulate simultaneously the free-flight values of the important similarity parameters in a wind tunnel. Today, for the most part, we do not attempt to simulate all the parameters simultaneously; rather, Mach number simulation is achieved in one wind tunnel, and Reynolds number simulation in another tunnel. The results from both tunnels are then analyzed and correlated to obtain reasonable values for CL and CD appropriate for free flight. In any event, this example serves to illustrate the difficulty of full free-flight simulation in a given wind tunnel and underscores the importance given to dynamically similar flows in experimental aerodynamics.

 Design Box I Lift and drag coefficients play a strong role in the preliminary design and performance analysis of airplanes. The purpose of this design box is to enforce the importance of CL and Ct) in aeronautical engineering; they are much more than just the conveniently defined terms discussed so far—they are fundamental quantities, which make the difference between intelligent engineering and simply groping in the dark. Consider an airplane in steady, level (horizontal) flight, as illustrated in Figure 1.24. For this case, the weight W acts vertically downward. The lift L acts vertically upward, perpendicular to the relative wind Vx (by definition). In order to sustain the airplane in level flight,

 L = W The thrust T from the propulsive mechanism and the drag D are both parallel to Vk,. For steady (unaccelerated) flight,

 T = D

 L _ W _ 2 W qxS qxS PocV^S

 [1.45]

 6 The lowest velocity may instead by dictated by the power required to maintain level flight exceeding the power available from the powerplant. This occurs on the "back side of the power curve." The velocity at which this occurs is usually less than the stalling velocity, so is of academic interest only. See Anderson, Aircraft Performance and Design, McGraw-Hill, 1999, for more details.

 Figure 1.24 The four forces acting on an airplane in flight.

 Figure 1.25 Schematic of lift and drag coefficients versus angle of attack; illustration of maximum lift coefficient and minimum drag coefficient.

 Solving Equation (1.45) for V4*.,

 2W PooSCL

 [1.46]

 For a given airplane flying at a given altitude, W, p, and S are fixed values; hence from Equation (1.46) each value of velocity corresponds to a specific value of CL. In particular, will be the smallest when CL is a maximum. Hence, the stalling velocity for a given airplane is determined by C;. max from Equation (1.46)

 2W

 [1.47]

 К..П

 PooSCl.1

For a given airplane, without the aid of any artificial devices, Cz,,max is determined purely by nature, through the physical laws for the aerodynamic flowfield over the airplane. However, the airplane designer has some devices available that artificially increase CL, mm beyond that for the basic airplane shape. These mechanical devices are called high-lift devices-, examples are flaps, slats, and slots on the wing which, when deployed by the pilot, serve to increase CLлшх, and hence decrease the stalling speed. High-lift devices are usually deployed for landing and take-off; they are discussed in more detail in Section 4.11.

On the other extreme of flight velocity, the maximum velocity for a given airplane with a given maximum thrust from the engine is determined by the value of minimum drag coefficient, CD min, where Со, тіп is marked in Figure 1.25. From the definition of drag coefficient in Section 1.5, applied for the case of steady, level flight where T = D, we have

D _ T _ IT

Чос s qxs PocV^S

Solving Equation (1.48) for 14c,

2 T

Рос SC о

For a given airplane flying at maximum thrust Гтах and a given altitude, from Equation (1.49) the maximum value of Vx corresponds to flight at CD, min

From the above discussion, it is clear that the aerodynamic coefficients are important engineering quantities that dictate the performance and design of airplanes. For example, stalling velocity is determined in part by Ci max, and maximum velocity is determined in part by C0 min.

Broadening our discussion to the whole range of flight velocity for a given airplane, note from Equation (1.45) that each value of Vx corresponds to a specific value of CL. Therefore, over the whole range of flight velocity from Tstaii to Fmax, the airplane lift coefficient varies as shown genetically in Figure 1.26. The values of CL given by the curve in Figure 1.26 are what are needed to maintain level flight over the whole range of velocity at a given altitude. The airplane designer must design the airplane to achieve these values of CL for an airplane of given weight and wing area. Note that the required values of Cl decrease as Vx increases. Examining the lift coefficient variation with angle of attack shown in Figure 1.26, note that as the airplane flies faster, the angle of attack must be smaller, as also shown in Figure 1.26. Hence, at high speeds, airplanes are at low a, and at low speeds, airplanes are at high a; the specific angle of attack which the airplane must have at a specific Vx is dictated by the specific value of CL required at that velocity.

Obtaining raw lift on a body is relatively easy—even a barn door creates lift at angle of attack. The name of the game is to obtain the necessary lift with as low a drag as possible. That is, the values of CL required over the entire flight range for an airplane, as represented by Figure 1.26, can sometimes be obtained even for the least effective lifting shape—just make the angle of attack high enough. But CD also varies with as governed by Equation (1.48); the generic variation of Co with is sketched in Figure 1.27. A poor aerodynamic shape, even though it generates the necessary values of CL shown in Figure 1.26, will have inordinately high values of CD,

 a Decreasing

 Schematic of the variation of lift coefficient with flight velocity for level flight.

 Figure 1.26

 i. e., the CD curve in Figure 1.27 will ride high on the graph, as denoted by the dashed curve in Figure 1.27. An aerodynamically efficient shape, however, will produce the requisite values of CL prescribed by Figure 1.26 with much lower drag, as denoted by the solid curve in Figure 1.27. An undesirable by-product of the high-drag shape is a lower value of the maximum velocity for the same maximum thrust, as also indicated in Figure 1.27. Finally, we emphasize that a true measure of the aerodynamic efficiency of a body shape is its lift-to-drag ratio, given by

 L _ qocSCL _ Cl D qooSCp Co

 [1.51]

 Since the value of CL necessary for flight at a given velocity and altitude is determined by the airplane’s weight and wing area (actually, by the ratio of W/S, called the wing loading) through the relationship given by Equation (1.45), the value of L/D at this velocity is controlled by CD, the denominator in Equation (1.51). At any given velocity, we want L/D to be as high as possible; the higher is L/D, the more aerodynamically efficient is the body. For a given airplane at a given altitude, the variation of L/D as a function of velocity is sketched generically in Figure 1.28. Note that, as Voo increases from a low value, L/D first increases, reaches a maximum at some intermediate velocity, and then decreases. Note that, as increases, the angle of attack of the airplane decreases, as explained earlier. From a strictly aerodynamic consideration, L/D for a given body shape depends on angle of

 a Decreasing

 Figure 1.27 Schematic of the variation of drag coefficient with flight velocity for level flight. Comparison between high and low drag aerodynamic bodies, with the consequent effect on maximum velocity.

 attack. This can be seen from Figure 1.25, where Cl and Co are given as a function of a. If these two curves are ratioed, the result is L/D as a function of angle of attack, as sketched generically in Figure 1.29. The relationship of Figure 1.28 to Figure 1.29 is that, when the airplane is flying at the velocity that corresponds to (L/£>)raax as shown in Figure 1.28, it is at the angle of attack for (L/£>)max as shown in Figure 1.29. In summary, the purpose of this design box is to emphasize the important role played by the aerodynamic coefficients in the performance analysis and design of airplanes. In this discussion, what has been important is not the lift and drag per se, but rather CL and CD. These coefficients are a wonderful intellectual construct that helps us to better understand the aerodynamic characteristics of a body, and to make reasoned, intelligent calculations. Hence they are more than just conveniently defined quantities as might first appear when introduced in Section 1.5. For more insight to the engineering value of these coefficients, see Anderson, Aircraft Performance and Design, McGraw-Hill, 1999, and Anderson, Introduction to Flight, 4th edition, McGraw-Hill, 2000. Also, home­work problem 1.15 at the end of this chapter gives you the opportunity to construct specific curves for CL, CD, and L/D versus velocity for an actual airplane so that you can obtain a feel for some real numbers that have been only

 Figure 1.28

 Schematic of the variation of lift-to-drag ratio with flight velocity for level flight.

 Figure 1.29 Schematic of the variation of lift-to-drag ratio with angle of attack.

 generically indicated in the figures here. (In our present discussion, the use of generic figures has been intentional for pedagogic reasons.) Finally, an historical note on the origins of the use of aerodynamic coefficients is given in Section 1.13.

Consider an executive jet transport patterned after the Cessna 560 Citation V shown in three – view in Figure 1.30. The airplane is cruising at a velocity of 492 mph at an altitude of 33,000 ft, where the ambient air density is 7.9656 x 10~4 slug/ft3. The weight and wing planform areas of the airplane are 15,000 lb and 342.6 ft2, respectively. The drag coefficient at cruise is 0.015. Calculate the lift coefficient and the lift-to-drag ratio at cruise.

Solution

The units of miles per hour for velocity are not consistent units. In the English engineering system of units, feet per second are consistent units for velocity (see Section 2.4 of Reference 2). To convert between mph and ft/s, it is useful to remember that 88 ft/s = 60 mph. For the present example,

Vx> = 492(H) = 721.6 ft/s

From Equation (1.45),

From Equation (1.51),

L _ CL _ 0.21 Ъ ~ ~C~D ~~ 0.015

Remarks: For a conventional airplane such as shown in Figure 1.30, almost all the lift at cruising conditions is produced by the wing; the lift of the fuselage and tail are very small by comparison. Hence, the wing can be viewed as an aerodynamic “lever.” In this example, the lift-to-drag ratio is 14, which means that for the expenditure of one pound of thrust to overcome one pound of drag, the wing is lifting 14 pounds of weight—quite a nice leverage.

 Figure 1.30 Cessna 560 Citation V.

## Circulation

You are reminded again that this is a tool-building chapter. Taken individually, each aerodynamic tool we have developed so far may not be particularly exciting. However, taken collectively, these tools allow us to obtain solutions for some very practical and exciting aerodynamic problems.

In this section, we introduce a tool which is fundamental to the calculation of aerodynamic lift, namely, circulation. This tool was used independently by Frederick Lanchester (1878-1946) in England, Wilhelm Kutta (1867-1944) in Germany, and Nikolai Joukowski (1847-1921) in Russia to create a breakthrough in the theory of aerodynamic lift at the turn of the twentieth century. The relationship between circulation and lift and the historical circumstances surrounding this breakthrough are discussed in Chapters 3 and 4. The purpose of this section is only to define circulation and relate it to vorticity.

Consider a closed curve C in a flow field, as sketched in Figure 2.36. Let V and ds be the velocity and directed line segment, respectively, at a point on C. The circulation, denoted by Г, is defined as

The circulation is simply the negative of the line integral of velocity around a closed curve in the flow; it is a kinematic property depending only on the velocity field and the choice of the curve C. As discussed in Section 2.2.8, Line Integrals, by mathematical convention the positive sense of the line integral is counterclockwise. However, in aerodynamics, it is convenient to consider a positive circulation as being clockwise.

 Г = ~Фс V • ds

 Figure 2.36 Definition of circulation.

Hence, a minus sign appears in the definition given by Equation (2.136) to account for the positive-counterclockwise sense of the integral and the positive-clockwise sense of circulation.[4]

The use of the word “circulation” to label the integral in Equation (2.136) may be somewhat misleading because it leaves a general impression of something moving around in a loop. Indeed, according to the American Heritage Dictionary of the English Language, the first definition given to the word “circulation” is “movement in a circle or circuit.” However, in aerodynamics, circulation has a very precise technical meaning, namely, Equation (2.136). It does not necessarily mean that the fluid elements are moving around in circles within this flow field—a common early misconception of new students of aerodynamics. Rather, when circulation exists in a flow, it simply means that the line integral in Equation (2.136) is finite. For example, if the airfoil in Figure 2.26 is generating lift, the circulation taken around a closed curve enclosing the airfoil will be finite, although the fluid elements are by no means executing circles around the airfoil (as clearly seen from the streamlines sketched in Figure 2.26).

Circulation is also related to vorticity as follows. Refer back to Figure 2.9, which shows an open surface bounded by the closed curve C. Assume that the surface is in a flow field and the velocity at point P is V, where P is any point on the surface (including any point on curve C). From Stokes’ theorem [Equation (2.25)],

[2.137]

Hence, the circulation about a curve C is equal to the vorticity integrated over any open surface bounded by C. This leads to the immediate result that if the flow is irrotational everywhere within the contour of integration (i. e., if V x V = 0 over any surface bounded by C), then Г = 0. A related result is obtained by letting the curve C shrink to an infinitesimal size, and denoting the circulation around this infinitesimally small curve by dT. Then, in the limit as C becomes infinitesimally small, Equation

(2.137) yields

dF = -(V x V) – dS = -(V x V) • ndS

where dS is the infinitesimal area enclosed by the infinitesimal curve C. Referring to Figure 2.37, Equation (2.138) states that at a point P in a flow, the component of vorticity normal to dS is equal to the negative of the “circulation per unit area,” where the circulation is taken around the boundary of dS.

 For the velocity field given in Example 2.3, calculate the circulation around a circular path of radius 5 m. Assume that и and v given in Example 2.3 are in units of meters per second. Solution Since we are dealing with a circular path, it is easier to work this problem in polar coordinates, where x = r cosd, у = r sind, x2 + у2 = r2, Vr = и cost? + v sind, and Ve = —u sin# + vcosO. Therefore,

 у r sin 9 sin 9 X2 + у2 г2 г

 2тг m2/s

 Figure 3.37 Relation between vorticity andcirculation.

 Example 2.6

 1 г

 = Vrdr + rVed6 =0 + r

 – Id6 = – dd

 Hence, Г = —

 Tweet Copyright © 2019 Helicopters & Aircrafts - Everything about aircrafts and helicopters. News and events in aviation worldwide. Civil, transportation, military helicopters and airplanes.