Category Fundamentals of Aerodynamics

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

Подпись: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

image114

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]

Physical principle Mass can be neither created nor destroyed Подпись: [2.50]

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

Подпись: or Physical principle Mass can be neither created nor destroyed Подпись: [2.51]

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

Подпись: ^ + V • (pV) = 0 at Подпись: [2.52]

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

Подпись: к sinf? 2 JT r

Подпись: or Подпись: 1jr — Voor sin в Подпись: 2nVx Подпись: [3.91]

l/r = УооГ sin в —

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

image243[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.

image244

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.

image245

(V<xr sin0)—— + r*

 

(T^ sin 9)

 

image246

Подпись: Voo sin вimage247[3.94]

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

image248"image249[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.

image250

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

image251[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.

Подпись: and Подпись: К =0 Vg = —2VQO sin# Подпись: [3.99] [3.100]

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

Подпись: VQ is positive in the direction of increasing в
image252

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

Подпись: VQ in polar coordinates.

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

Подпись: [3.101]

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

 

image254

Подпись: Example 3.9

Подпись: в = 30°, 150°, 210°, 330°

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

 

image255

. 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.

image452

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

image453

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

image455,image457
image454,image456

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

Подпись:
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-

image459

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

image460

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.)

image461

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.

image462

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.)

image463

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.)

There are two other phenomena that are reflected by the data in Figure 5.44. The first is the effect of greatly rounding the leading edges of the delta wing. In our previous discussions, we have treated the case of a sharp leading edge; such sharp edges cause the flow to separate at the leading edge, forming the leading-edge vortices. On the other hand, if the leading-edge radius is large, the flow separation will be minimized, or possibly will not occur. In turn, the drag penalty discussed above will not be present, and hence the L/D ratio will increase. The dashed curve in Figure 5.44 is the case for a 60° delta wing with well-rounded leading edges. Note that (L/D)mm for this case is about 16.5, almost a factor of 2 higher than the sharp leading-edge case. However, keep in mind that these are results for subsonic speeds. There is a major design compromise reflected in these results. At the beginning of this section, we mentioned that the delta-wing planform with sharp leading edges is advantageous for supersonic flight—its highly swept shape in combination with sharp leading edges has a low supersonic drag. However, at supersonic speeds this advantage will be negated if the leading edges are rounded to any great extent. We will find in our study of supersonic flow in Part 3 that a blunt-nosed body creates very large values of wave drag. Therefore, leading edges with large radii are not appropriate for supersonic aircraft; indeed, it is desirable to have as sharp a leading edge as is practically possible for supersonic airplanes. A singular exception is the design of the space shuttle. The leading-edge radius of the space shuttle is large; this is due to three features that combine to make such blunt leading edges advantageous for the shuttle. First, the shuttle must slow down early during reentry into the earth’s atmosphere to avoid massive aerodynamic heating (aspects of aerodynamic heating are discussed in Part 4). Therefore, in order to obtain this deceleration, a high drag is desirable for the space shuttle; indeed, the maximum L/D ratio of the space shuttle during reentry is about 2. A large leading-edge radius, with its attendant high drag, is therefore advantageous. Second, as we will see in Part 4, the rate of aerodynamic heating to the leading edge itself—a region of high heating—is inversely proportional to the square root of the leading-edge radius. Hence, the larger the radius, the smaller will be the heating rate to the leading edge. Third, as already explained above, a highly rounded leading edge is certainly advantageous to the shuttle’s subsonic aerodynamic characteristics. Hence, a well-rounded leading edge is an important design feature for the space shuttle on all accounts. However, we must be reminded that this is not the case for more conventional supersonic aircraft, which demand very sharp leading edges. For these aircraft, a delta wing with a sharp leading edge has relatively poor subsonic performance.

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

image464

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.

 

image38

Flow Similarity

image43

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

image44

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.

 

Flow Similarity

image45

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

 

image46

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

Подпись: [1.48]Подпись:D _ T _ IT

Чос s qxs PocV^S

Solving Equation (1.48) for 14c,

2 T

Рос SC о

Flow Similarity Подпись: [1.50]

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,

Flow Similarity

image47

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

 

image48image49

Flow Similarity

image50

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

 

image51image52image53

Flow Similarity

image54

Figure 1.28

 

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

 

image55

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.

Подпись: Example 1.6Solution

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

Flow Similarity Подпись: 0.21

From Equation (1.45),

From Equation (1.51),

Подпись: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.

Flow Similarity

image56

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.

Circulation Подпись: [2.136]

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.

 

image171

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)],

image172[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

Подпись: or Подпись: (V x V) • n = Подпись: dr ~dS Подпись: [2.138]

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.

Circulation

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

Circulation

Figure 3.37 Relation between vorticity and
circulation.

 

Example 2.6

 

Circulation

1

г

 

= Vrdr + rVed6 =0 + r

 

– Id6 = – dd

 

Hence, Г = — <j) • ds= — j —dO =

Note that we never used the 5-m diameter of the circular path; in this case, the value of Г is independent of the diameter of the path.

 

The Vortex Sheet

In Section 3.14, the concept of vortex flow was introduced; refer to Figure 3.31 for a schematic of the flow induced by a point vortex of strength Г located at a given point О. (Recall that Figure 3.31, with its counterclockwise flow, corresponds to a negative value of Г. By convention, a positive Г induces a clockwise flow.) Let us now expand our concept of a point vortex. Referring to Figure 3.31, imagine a straight line perpendicular to the page, going through point O, and extending to infinity both out of and into the page. This line is a straight vortex filament of strength Г. A straight vortex filament is drawn in perspective in Figure 4.7. (Here, we show a clockwise

flow, which corresponds to a positive value of Г.) The flow induced in any plane perpendicular to the straight vortex filament by the filament itself is identical to that induced by a point vortex of strength Г; that is, in Figure 4.7, the flows in the planes perpendicular to the vortex filament at О and O’ are identical to each other and are identical to the flow induced by a point vortex of strength Г. Indeed, the point vortex described in Section 3.14 is simply a section of a straight vortex filament.

In Section 3.17, we introduced the concept of a source sheet, which is an infinite number of line sources side by side, with the strength of each line source being infinitesimally small. For vortex flow, consider an analogous situation. Imagine an infinite number of straight vortex filaments side by side, where the strength of each filament is infinitesimally small. These side-by-side vortex filaments form a vortex sheet, as shown in perspective in the upper left of Figure 4.8. If we look along the series of vortex filaments (looking along the у axis in Figure 4.8), the vortex sheet will appear as sketched at the lower right of Figure 4.8. Here, we are looking at an edge view of the sheet; the vortex filaments are all perpendicular to the page. Let і be the distance measured along the vortex sheet in the edge view. Define у = y(s) as the strength of the vortex sheet, per unit length along s. Thus, the strength of an infinitesimal portion ds of the sheet is у ds. This small section of the vortex sheet can be treated as a distinct vortex of strength у ds. Now consider point P in the flow,

image305

Figure 4.7 Vortex filament.

image306

A

The Vortex Sheet Подпись: [4.1]

located a distance r from ds; the Cartesian coordinates of P are (x, z). The small section of the vortex sheet of strength у ds induces an infinitesimally small velocity dV at point P. From Equation (3.105), dV is given by

Подпись: йф Подпись: у ds в 2тс Подпись: [4.2]

and is in a direction perpendicular to r, as shown in Figure 4.8. The velocity at P induced by the entire vortex sheet is the summation of Equation (4.1) from point a to point b. Note that dV, which is perpendicular to r, changes direction at point P as we sum from a to b; hence, the incremental velocities induced at P by different sections of the vortex sheet must be added vectorally. Because of this, it is sometimes more convenient to deal with the velocity potential. Again referring to Figure 4.8, the increment in velocity potential d<p induced at point P by the elemental vortex у ds is, from Equation (3.112),

In turn, the velocity potential at P due to the entire vortex sheet from a to b is

Подпись: ф{х, z) — —Подпись: 1 2TC image307[4.3]

Equation (4.1) is particularly useful for our discussion of classical thin airfoil theory, whereas Equation (4.3) is important for the numerical vortex panel method.

Recall from Section 3.14 that the circulation Г around a point vortex is equal to the strength of the vortex. Similarly, the circulation around the vortex sheet in

Figure 4.8 is the sum of the strengths of the elemental vortices; that is

The Vortex Sheet

image308[4.4]

Recall that the source sheet introduced in Section 3.17 has a discontinuous change in the direction of the normal component of velocity across the sheet (from Fig­ure 3.38, note that the normal component of velocity changes direction by 180° in crossing the sheet), whereas the tangential component of velocity is the same im­mediately above and below the source sheet. In contrast, for a vortex sheet, there is a discontinuous change in the tangential component of velocity across the sheet, whereas the normal component of velocity is preserved across the sheet. This change in tangential velocity across the vortex sheet is related to the strength of the sheet as follows. Consider a vortex sheet as sketched in Figure 4.9. Consider the rectangular dashed path enclosing a section of the sheet of length ds. The velocity components tangential to the top and bottom of this rectangular path are u and иг, respectively, and the velocity components tangential to the left and right sides are r j and i>2, re­spectively. The top and bottom of the path are separated by the distance dn. From the definition of circulation given by Equation (2.36), the circulation around the dashed path is

Г = — (v2 dn — uds — i>i dn + иг ds)

or Г = (иі — иг) ds + (i>i — иг) dn [4.5]

However, since the strength of the vortex sheet contained inside the dashed path is у ds, we also have

Г = у ds [4.6]

Therefore, from Equations (4.5) and (4.6),

у ds = (u — иг) ds + (uj — і>г) dn [4.7]

image310

Let the top and bottom of the dashed line approach the vortex sheet; that is, let dn —»• 0. In the limit, и i and иг become the velocity components tangential to the

vortex sheet immediately above and below the sheet, respectively, and Equation (4.7) becomes

Подпись: or Подпись: у — ll — ІІ2 Подпись: [4.8]

у ds = (u і — и 2 ) ds

Equation (4.8) is important; it states that the local jump in tangential velocity across the vortex sheet is equal to the local sheet strength.

The Vortex Sheet

We have now defined and discussed the properties of a vortex sheet. The concept of a vortex sheet is instrumental in the analysis of the low-speed characteristics of an airfoil. A philosophy of airfoil theory of inviscid, incompressible flow is as follows. Consider an airfoil of arbitrary shape and thickness in a freestream with velocity Vx, as sketched in Figure 4.10. Replace the airfoil surface with a vortex sheet of variable strength y(x), as also shown in Figure 4.10. Calculate the variation of у as a function of і such that the induced velocity field from the vortex sheet when added to the uniform velocity of magnitude Vж will make the vortex sheet (hence the airfoil surface) a streamline of the flow. In turn, the circulation around the airfoil will be given by

where the integral is taken around the complete surface of the airfoil. Finally, the resulting lift is given by the Kutta-Joukowski theorem:

L’ = Рос Too Г

This philosophy is not new. It was first espoused by Ludwig Prandtl and his col­leagues at Gottingen, Germany, during the period 1912-1922. However, no general analytical solution for у = у (s) exists for an airfoil of arbitrary shape and thickness. Rather, the strength of the vortex sheet must be found numerically, and the practical implementation of the above philosophy had to wait until the 1960s with the advent of large digital computers. Today, the above philosophy is the foundation of the modern vortex panel method, to be discussed in Section 4.9.

The concept of replacing the airfoil surface in Figure 4.10 with a vortex sheet is more than just a mathematical device; it also has physical significance. In real life, there is a thin boundary layer on the surface, due to the action of friction between

image311image312Airfoil of arbitrary shape and thickness

Figure 4.10 Simulation of an arbitrary airfoil by distributing a vortex sheet over the airfoil surface.

The Vortex Sheet

Vortex sheet on camber line

 

Кос

 

О

 

C.

 

7(s)

 

image313

The Vortex Sheet

the surface and the airflow (see Figure 1.35). This boundary layer is a highly viscous region in which the large velocity gradients produce substantial vorticity; that is, V x V is finite within the boundary layer. (Review Section 2.12 for a discussion of vorticity.) Hence, in real life, there is a distribution of vorticity along the airfoil surface due to viscous effects, and our philosophy of replacing the airfoil surface with a vortex sheet (such as in Figure 4.10) can be construed as a way of modeling this effect in an inviscid flow.3

Imagine that the airfoil in Figure 4.10 is made very thin. If you were to stand back and look at such a thin airfoil from a distance, the portions of the vortex sheet on the top and bottom surface of the airfoil would almost coincide. This gives rise to a method of approximating a thin airfoil by replacing it with a single vortex sheet distributed over the camber line of the airfoil, as sketched in Figure 4.11. The strength of this vortex sheet у (s) is calculated such that, in combination with the freestream, the camber line becomes a streamline of the flow. Although the approach shown in Figure 4.11 is approximate in comparison with the case shown in Figure 4.10, it has the advantage of yielding a closed-form analytical solution. This philosophy of thin airfoil theory was first developed by Max Munk, a colleague of Prandtl, in 1922 (see Reference 12). It is discussed in Sections 4.7 and 4.8.

Internal Energy and Enthalpy

Consider an individual molecule of a gas, say, an O2 molecule in air. This molecule is moving through space in a random fashion, occasionally colliding with a neighboring molecule. Because of its velocity through space, the molecule has translational kinetic energy. In addition, the molecule is made up of individual atoms which we can visualize as connected to each other along various axes; for example, we can visualize the O2 molecule as a “dumbbell” shape, with an О atom at each end of a connecting axis. In addition to its translational motion, such a molecule can execute a rotational motion in space; the kinetic energy of this rotation contributes to the net energy of the molecule. Also, the atoms of a given molecule can vibrate back and forth along and across the molecular axis, thus contributing a potential and kinetic energy of vibration to the molecule. Finally, the motion of the electrons around each of the nuclei of the molecule contributes an “electronic” energy to the molecule. Hence, the energy of a given molecule is the sum of its translational, rotational, vibrational, and electronic energies.

Now consider a finite volume of gas consisting of a large number of molecules. The sum of the energies of all the molecules in this volume is defined as the internal energy of the gas. The internal energy per unit mass of gas is defined as the specific internal energy, denoted by e. A related quantity is the specific enthalpy, denoted by h and defined as

h = e + pv [7.3]

For a perfect gas, both e and h are functions of temperature only:

e = e(T) [7.4a]

h=h(T) [7.4b]

Let de and dh represent differentials of e and h, respectively. Then, for a perfect gas,

de = cvdT [7.5a]

dh=cpdT [7.5b]

where cv and cp are the specific heats at constant volume and constant pressure, respectively. In Equations (7.5a and b), cv and cp can themselves be functions of T. However, for moderate temperatures (for air, for T < 1000 K), the specific heats are reasonably constant. A perfect gas where cv and cp are constants is defined as a calorically perfect gas, for which Equations (1.5a and b) becomes

Подпись: e — cvT h - cpT[7.6 a] [7.6b]

For a large number of practical compressible flow problems, the temperatures are moderate; for this reason, in this book we always treat the gas as calorically perfect; that is, we assume that the specific heats are constant. For a discussion of compressible flow problems where the specific heats are not constant (such as the high-temperature chemically reacting flow over a high-speed atmospheric entry vehicle, that is, the space shuttle), see Reference 21.

Note that e and h in Equations (7.3) through (7.6) are thermodynamic state variables—they depend only on the state of the gas and are independent of any process. Although cv and cp appear in these equations, there is no restriction to just a constant volume or a constant pressure process. Rather, Equations (1.5a and b) and (1.6a and b) are relations for thermodynamic state variables, namely, e and h as functions of /’. and have nothing to do with the process that may be taking place.

For a specific gas, cp and cv are related through the equation

cp – cv = R [7.7]

Dividing Equation (7.7) by cp, we obtain

Подпись: [7.8], cv R

Define у = cp/cv. For air at standard conditions, у = 1.4. Then Equation (7.8) becomes

Internal Energy and Enthalpy

R

cp

 

yR

 

or

 

[7.9]

 

cp

 

У ~ 1

 

image488

Internal Energy and Enthalpy

Internal Energy and Enthalpy Подпись: [7.10]

Similarly, dividing Equation (7.7) by c„, we obtain

Equations (7.9) and (7.10) are particularly useful in our subsequent discussion of compressible flow.