Category Fundamentals of Aerodynamics

The Aerodynamic Center: Additional Considerations

The definition of the aerodynamic center is given in Section 4.3; it is that point on a body about which the aerodynamically generated moment is independent of angle of attack. At first thought, it is hard to imagine that such a point could exist. However, the moment coefficient data in Figure 4.6, which is constant with angle of attack, experimentally proves the existence of the aerodynamic center. Moreover, thin airfoil theory as derived in Sections 4.7 and 4.8 clearly shows that, within the assumptions embodied in the theory, not only does the aerodynamic center exist but that it is located at the quarter-chord point on the airfoil. Therefore, to Figure 1.19 which illustrates three different ways of stating the force and moment system on an airfoil, we can now add a fourth way, namely, the specification of the lift and drag acting through the aerodynamic center, and the value of the moment about the aerodynamic center. This is illustrated in Figure 4.23.


Figure 4.23 Lift, drag, and moments about the aerodynamic center.


Figure 4.24 Lift and moments about the

quarter-chord point, and a sketch useful for locating the aerodynamic center.


For most conventional airfoils, the aerodynamic center is close to, but not neces­sarily exactly at, the quarter-chord point. Given data for the shape of the lift coefficient curve and the moment coefficient curve taken around an arbitrary point, we can calcu­late the location of the aerodynamic center as follows. Consider the lift and moment system taken about the quarter-chord point, as shown in Figure 4.24. We designate the location of the aerodynamic center by ciac measured from the leading edge. Here, xac is the location of the aerodynamic center as a fraction of the chord length c. Taking moments about the aerodynamic center designated by ac in Figure 4.24, we have

Подпись: [4.67]M’c = L'(cxac – cl A) + M’c/4 Dividing Equation (4.67) by q^Sc, we have

The Aerodynamic Center: Additional Considerations





The Aerodynamic Center: Additional Considerations

„ dci dcmcj 4

0 = — (xac – 0.25) H——— —-L-

da da




For airfoils below the stalling angle of attack, the slopes of the lift coefficient and


The Aerodynamic Center: Additional Considerations The Aerodynamic Center: Additional Considerations

moment coefficient curves are constant. Designating these slopes by

Equation (4.70) becomes

Подпись: or Подпись: ttln xx = - + 0.25 a0 Подпись: [4.711

0 = a0(*ac – 0.25) + m0

Hence, Equation (4.71) proves that, for a body with linear lift and moment curves, that is, where ao and mo are fixed values, the aerodynamic center exists as a fixed point on the airfoil. Moreover, Equation (4.71) allows the calculation of the location of this point.

Подпись: Example 4.3Consider the NACA 23012 airfoil studied in Example 4.2. Experimental data for this airfoil is plotted in Figure 4.22, and can be obtained from Reference 11. It shows that, at a = 4°, с-; = 0.55 and c„, t/4 = —0.005. The zero-lift angle of attack is —1.1°. Also, at a = —4°, cm cj4 = —0.0125. (Note that the “experimental” value of cmx/4 = —0.01 tabulated at the end of Example 4.2 is an average value over a range of angle of attack. Since the calculated value of 4 from thin airfoil theory states that the quarter-chord point is the aerodynamic center, it makes sense in Example 4.2 to compare the calculated c,„ ,./4 with an experimental value averaged over a range of angle of attack. However, in the present example, because c„,,( /4 in reality varies with angle of attack, we use the actual data at two different angles of attack.) From the given information, calculate the location of the aerodynamic center for the NACA 23012 airfoil.


Подпись: ao The Aerodynamic Center: Additional Considerations

Since ci = 0.55 at a = 4° and q = 0 at a = —1.1°, the lift slope is

Подпись: m0 The Aerodynamic Center: Additional Considerations

The slope of the moment coefficient curve is

Подпись: mo a0 Подпись: ■ 0.25 = Подпись: 9.375 x 10~4 0.1078 Подпись: 0.25 Подпись: 0.241

From Equation (4.71),

The result agrees exactly with the measured value quoted on page 183 of Abbott and Von Doenhoff (Reference 11).

The Aerodynamic Center: Additional Considerations

Definition of Total (Stagnation) Conditions

At the beginning of Section 3.4, the concept of static pressure p was discussed in some detail. Static pressure is a measure of the purely random motion of molecules in a gas; it is the pressure you feel when you ride along with the gas at the local flow velocity. In contrast, the total (or stagnation) pressure was defined in Section 3.4 as the pressure existing at a point (or points) in the flow where V = 0. Let us now define the concept of total conditions more precisely.

Consider a fluid element passing through a given point in a flow where the local pressure, temperature, density, Mach number, and velocity are p, T, p, M, and V, respectively. Here, p, T, and p are static quantities (i. e., static pressure, static

temperature, and static density, respectively); they are the pressure, temperature, and density you feel when you ride along with the gas at the local flow velocity. Now imagine that you grab hold of the fluid element and adiabatically slow it down to zero velocity. Clearly, you would expect (correctly) that the values of p, T, and p would change as the fluid element is brought to rest. In particular, the value of the temperature of the fluid element after it has been brought to rest adiabatically is defined as the total temperature, denoted by Tq. The corresponding value of enthalpy is defined as the total enthalpy h0, where ho = cpT0 for a calorically perfect gas. Keep in mind that we do not actually have to bring the flow to rest in real life in order to talk about the total temperature or total enthalpy; rather, they are defined quantities that would exist at a point in a flow if (in our imagination) the fluid element passing through that point were brought to rest adiabatically. Therefore, at a given point in a flow, where the static temperature and enthalpy are T and h, respectively, we can also assign a value of total temperature To and a value of total enthalpy ho defined as above.

For such a flow, Equation (7.56) can be used as a form of the governing energy equation.

Keep in mind that the above discussion marbled two trains of thought: On the one hand, we dealt with the general concept of an adiabatic flow field [which led to Equations (7.51) to (7.53)], and on the other hand, we dealt with the definition of total enthalpy [which led to Equation (7.54)]. These two trains of thought are really separate and should not be confused. Consider, for example, a general nonadiabatic flow, such as a viscous boundary layer with heat transfer. A generic non-adiabatic flow is sketched in Figure 7.4a. Clearly, Equations (7.51) to (7.53) do not hold for such a flow. However, Equation (7.54) holds locally at each point in the flow, because the assumption of an adiabatic flow contained in Equation (7.54) is made through the definition of ho and has nothing to do with the general overall flow field. For example, consider two different points, 1 and 2, in the general flow, as shown in Figure 7.4a. At point 1, the local static enthalpy and velocity are h i and V, respectively. Hence, the local total enthalpy at point 1 is ho, і = h + V2/2. At point 2, the local static enthalpy and velocity are h2 and V2, respectively. Hence, the local total enthalpy at point 2 is hop = h2+ V}/2. If the flow between points 1 and 2 is nonadiabatic, then ho, Ф hop. Only for the special case where the flow is adiabatic between the two points would ho, — ho<2. This case is illustrated in Figure l. Ab. Of course, this is the special case treated by Equations (7.55) and (7.56).

Return to the beginning of this section, where we considered a fluid element passing through a point in a flow where the local properties are p. T, p, M, and V. Once again, imagine that you grab hold of the fluid element and slow it down to zero velocity, but this time, let us slow it down both adiabatically and reversibly. That is, let us slow the fluid element down to zero velocity isentropically. When the fluid element is brought to rest isentropically, the resulting pressure and density are defined as the total pressure po and total density po. (Since an isentropic process is also adiabatic, the resulting temperature is the same total temperature Го as discussed earlier.) As before, keep in mind that we do not have to actually bring the flow to rest in real life in order to talk about total pressure and total density; rather, they are defined quantities that would exist at a point in a flow if (in our imagination) the fluid element passing through that point were brought to rest isentropically. Therefore, at a given point in a flow, where the static pressure and static density are p and p, respectively, we can also assign a value of total pressure po, and total density po defined as above.

The definition of po and po deals with an isentropic compression to zero velocity. Keep in mind that the isentropic assumption is involved with the definition only. The concept of total pressure and density can be applied throughout any general nonisentropic flow. For example, consider two different points, 1 and 2, in a general

Definition of Total (Stagnation) Conditions


Definition of Total (Stagnation) Conditions

flow field, as sketched in Figure 7.4c. At point 1, the local static pressure and static density are p and f>, respectively; also the local total pressure and total density are Po. i and (>{) і, respectively, defined as above. Similarly, at point 2, the local static pressure and static density are p2 and /ь. respectively, and the local total pressure and total density are po,2 and po.2. respectively. If the flow is nonisentropic between points 1 and 2, then p0.i ф p0 2 and po. i ф Ли, as shown in Figure 7.4c. On the other hand, if the flow is isentropic between points 1 and 2, then po. i = Po,2 and Po. i = Po.2, as shown in Figure 7.4d. Indeed, if the general flow field is isentropic throughout, then both po and po are constant values throughout the flow.

As a corollary to the above considerations, we need another defined temperature, denoted by T*, and defined as follows. Consider a point in a subsonic flow where the local static temperature is T. At this point, imagine that the fluid element is

speeded up to sonic velocity, adiabatically. The temperature it would have at such sonic conditions is denoted as T*. Similarly, consider a point in a supersonic flow, where the local static temperature is T. At this point, imagine that the fluid element is slowed down to sonic velocity, adiabatically. Again, the temperature it would have at such sonic conditions is denoted as T*. The quantity T* is simply a defined quantity at a given point in a flow, in exactly the same vein as T0, p0, and p0 are defined quantities. Also, a* = yJyRT*.

Finite Control Volume Approach

Consider a general flow field as represented by the streamlines in Figure 2.11. Let us imagine a closed volume drawn within a finite region of the flow. This volume defines a control volume V, and a control surface S is defined as the closed surface which bounds the control volume. The control volume may be fixed in space with the fluid moving through it, as shown at the left of Figure 2.11. Alternatively, the control volume may be moving with the fluid such that the same fluid particles are always inside it, as shown at the right of Figure 2.11. In either case, the control volume is a reasonably large, finite region of the flow. The fundamental physical principles are applied to the fluid inside the control volume, and to the fluid crossing the control surface (if the control volume is fixed in space). Therefore, instead of looking at the whole flow field at once, with the control volume model we limit our attention to just the fluid in the finite region of the volume itself.

2.3.1 Infinitesimal Fluid Element Approach


Consider a general flow field as represented by the streamlines in Figure 2.12. Let us imagine an infinitesimally small fluid element in the flow, with a differential volume dV. The fluid element is infinitesimal in the same sense as differential calculus; however, it is large enough to contain a huge number of molecules so that it can be viewed as a continuous medium. The fluid element may be fixed in space with the fluid moving through it, as shown at the left of Figure 2.12. Alternatively, it may be moving along a streamline with velocity V equal to the flow velocity at each point. Again, instead of looking at the whole flow field at once, the fundamental physical principles are applied to just the fluid element itself.

Figure 2.1 1 Finite control volume approach.





Volume d°V


Подпись: Infinitesimal fluid element moving along a streamline with the velocity V equal to the local flow velocity at each point Infinitesimal fluid element fixed in space with the fluid moving through it

Figure 2.12 Infinitesimal fluid element approach.

Uniform Flow: Our First Elementary Flow

In this section, we present the first of a series of elementary incompressible flows which later will be superimposed to synthesize more complex incompressible flows. For the remainder of this chapter and in Chapter 4, we deal with two-dimensional steady flows; three-dimensional steady flows are treated in Chapters 5 and 6.

Consider a uniform flow with velocity V7^ oriented in the positive a direction, as sketched in Figure 3.19. It is easily shown (see Problem 3.8) that a uniform flow is a physically possible incompressible flow (i. e., it satisfies V • V = 0) and that it is irrotational (i. e., it satisfies V x V = 0). Hence, a velocity potential for uniform flow can be obtained such that Уф = V. Examining Figure 3.19, and recalling Equation

(2.156) , we have

3 ф r,

— = и = Voo [3.49a]


3 ф

and — = u = 0 [3.49b]


Integrating Equation (3.49a) with respect to a, we have

Ф — УЖХ + f(y) [3.50]

where f(y) is a function of у only. Integrating Equation (3.49b) with respect to y, we obtain

ф = const + g(x) [3.51]

where g(x) is a function of a only. In Equations (3.50) and (3.51), ф is the same function; hence, by comparing these equations, g(x) must be V~^ a, and /(>’) must be constant. Thus,

Подпись: [3.52]ф = VooA + const

Note that in a practical aerodynamic problem, the actual value of ф is not significant; rather, ф is always used to obtain the velocity by differentiation; that is, Уф = V. Since the derivative of a constant is zero, we can drop the constant from Equation (3.52) without any loss of rigor. Hence, Equation (3.52) can be written as

Подпись: -©■ ф = const


Figure 3.19 Uniform flow.

Equation (3.53) is the velocity potential for a uniform flow with velocity Vqo oriented in the positive x direction. Note that the derivation of Equation (3.53) does not depend on the assumption of incompressibility; it applies to any uniform flow, compressible or incompressible.

Consider the incompressible stream function 1js. From Figure 3.19 and Equations (2.150a and b), we have

w v


[3.54 a]


dfi _ v__Q



Integrating Equation (3.54a) with respect to у and Equation (3.54b) with respect to x, and comparing the results, wc obtain

Подпись: Ф = УооУ[3.55]

Equation (3.55) is the stream function for an incompressible uniform flow oriented in the positive x direction.

From Section 2.14, the equation of a streamline is given by i/r = constant. Therefore, from Equation (3.55), the streamlines for the uniform flow are given by i/f = Vooy = constant. Because is itself constant, the streamlines are thus given mathematically as у = constant (i. e., as lines of constant y). This result is consistent with Figure 3.19, which shows the streamlines as horizontal lines (i. e., as lines of constant y). Also, note from Equation (3.53) that the equipotential lines are lines of constant*, as shown by the dashed line in Figure 3.19. Consistent with our discussion in Section 2.16, note that the lines of ijr = constant and ф = constant are mutually perpendicular.

Equations (3.53) and (3.55) can be expressed in terms of polar coordinates, where x = r cos 9 and у = r sin 9, as shown in Figure 3.19. Hence,


Подпись: 1jr = V^r sin 9 Подпись: [3.57] Подпись: and


Consider the circulation in a uniform flow. The definition of circulation is given by

Г = – j> Vds [2.136]

Let the closed curve C in Equation (2.136) be the rectangle shown at the left of Figure 3.19; h and l are the lengths of the vertical and horizontal sides, respectively, of the rectangle. Then

V • ds = – Уооl – 0(h) + V^l + 0(h) = 0



Г = 0





Equation (3.58) is true for any arbitrary closed curve in the uniform flow. To show this, note that Voo is constant in both magnitude and direction, and hence

because the line integral of ds around a closed curve is identically zero. Therefore, from Equation (3.58), we state that circulation around any closed curve in a uniform flow is zero.

The above result is consistent with Equation (2.137), which states that

Подпись: 5 [2.137]

We stated earlier that a uniform flow is irrotational; that is, V x V = 0 everywhere. Hence, Equation (2.137) yields Г = 0.

Note that Equations (3.53) and (3.55) satisfy Laplace’s equation [see Equation

(3.41) ], which can be easily proved by simple substitution. Therefore, uniform flow is a viable elementary flow for use in building more complex flows.

Effect of Aspect Ratio

Returning to Equations (5.61) and (5.62), note that the induced drag coefficient for a finite wing with a general lift distribution is inversely proportional to the aspect ratio, as was discussed earlier in conjunction with the case of the elliptic lift distribution. Note that AR, which typically varies from 6 to 22 for standard subsonic airplanes


Figure 5.1 8 Induced drag factor S as a function of taper ratio. (Source: McCormick, B. W., Aerodynamics, Aeronautics, and Flight Mechanics, John Wiley & Sons, New York, 1979.)


and sailplanes, has a much stronger effect on Cdj than the value of <5, which from Figure 5.18 varies only by about 10 percent over the practical range of taper ratio. Hence, the primary design factor for minimizing induced drag is not the closeness to an elliptical lift distribution, but rather, the ability to make the aspect ratio as large as possible. The determination that Cdj is inversely proportional to AR was one of the great victories of Prandtl’s lifting-line theory. In 1915, Prandtl verified this result with a series of classic experiments wherein the lift and drag of seven rectangular wings with different aspect ratios were measured. The data are given in Figure 5.19. Recall from Equation (5.4), that the total drag of a finite wing is given by

Подпись:Подпись: CD — cd +C2



Подпись: and Effect of Aspect Ratio Подпись: [5.64a] [5.64b]

The parabolic variation of CD with CL as expressed in Equation (5.63) is reflected in the data of Figure 5.19. If we consider two wings with different aspect ratios ARi and AR2, Equation (5.63) gives the drag coefficients CDд and C0,2 for the two wings as

Assume that the wings are at the same Cl. Also, since the airfoil section is the same for both wings, Cd is essentially the same. Moreover, the variation of e between the


Figure 5.1 9 Prandtl’s classic rectangular wing data for seven different aspect ratios from 1 to 7; variation of lift coefficient versus drag coefficient. For historical interest, we reproduce here Prandtl’s actual graphs. Note that, in his nomenclature, Ca = lift coefficient and Cw = drag coefficient. Also, the numbers on both the ordinate and abscissa are 100 times the actual values of the coefficients. (Source: Prandtl, L., "Applications of Modern Hydrodynamics to Aeronautics,"

NACA Report No. 116, 1921.)


Подпись: C2, Cn. і — Co,2 H 7ГЄ Подпись: 1 AR^ Подпись: 1 AR2 Подпись: [5.65]

wings is only a few percent and can be ignored. Hence, subtracting Equation (5.64b) from (5.64a), we obtain

Equation (5.65) can be used to scale the data of a wing with aspect ratio AR2 to correspond to the case of another aspect ratio AR|. For example, Prandtl scaled the data of Figure 5.19 to correspond to a wing with an aspect ratio of 5. For this case,

Equation (5.65) becomes

Сд, і = Co,2 H———– f – — —c—^ [5.66]

же 5 AR2/

Inserting the respective values of Cd,2 and AR2 from Figure 5.19 into Equation (5.66), Prandtl found that the resulting data for Сд, ь versus Cl collapsed to essentially the same curve, as shown in Figure 5.20. Hence, the inverse dependence of Cd, і on AR was substantially verified as early as 1915.

There are two primary differences between airfoil and finite-wing properties. We have discussed one difference, namely, a finite wing generates induced drag. However, a second major difference appears in the lift slope. In Figure 4.4, the lift slope for an airfoil was defined as a0 = dci/da. Let us denote the lift slope for a finite wing as a = dCL/da. When the lift slope of a finite wing is compared with that of its airfoil section, we find that a < a0- To see this more clearly, return to Figure 5.4, which illustrates the influence of downwash on the flow over a local airfoil section of a finite wing. Note that although the geometric angle of attack of the finite wing is a, the airfoil section effectively senses a smaller angle of attack, namely, aetf, where acff = a — a,-. For the time being, consider an elliptic wing with no twist; hence, a, and acff are both constant along the span. Moreover, q is also constant along the span, and therefore CL = ci. Assume that we plot Cl for the finite wing versus aeff, as shown at the top of Figure 5.21. Because we are using аеп the lift slope corresponds

Figure 5.30 Data of Figure 5.19

scaled by Prandtl to an aspect ratio of 5.

to that for an infinite wing ao. However, in real life, our naked eyes cannot see aeff; instead, what we actually observe is a finite wing with a certain angle between the chord line and the relative wind; that is, in practice, we always observe the geometric angle of attack a. Hence, CL for a finite wing is generally given as a function of a, as sketched at the bottom of Figure 5.21. Since a > aeff, the bottom abscissa is stretched, and hence the bottom lift curve is less inclined; it has a slope equal to a, and Figure 5.21 clearly shows that a < a(). The effect of a finite wing is to reduce the lift slope. Also, recall that at zero lift, there are no induced effects; i. e., a, = CDJ = 0. Thus, when Cl = 0, a = aeff. As a result, a/ =0 is the same for the finite and the infinite wings, as shown in Figure 5.21.

The values of a о and a are related as follows. From the top of Figure 5.21,


d(a — ctj)

Integrating, we find

Подпись: [5.67]CL = ao(a — at) + const

Substituting Equation (5.42) into (5.67), we obtain


image435,image436 Подпись: Infinite wing


Подпись: Figure 5.21Lift curves for an infinite wing versus a finite elliptic wing.

Differentiating Equation (5.68) with respect to a, and solving for dCi/da, we obtain

Подпись: [5.09]dCb _ _ do

da 1 + qq/^AR

Equation (5.69) gives the desired relation between a0 and a for an elliptic finite wing. For a finite wing of general planform, Equation (5.69) is slightly modified, as given below:

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

In Equation (5.70), г is a function of the Fourier coefficients A„. Values of r were first calculated by Glauert in the early 1920s and were published in Reference 18, which should be consulted for more details. Values of r typically range between 0.05 and 0.25.

Подпись: Figure 5.22 Prandtl's classic rectangular wing data. Variation of lift coefficient with angle of attack for seven different aspect ratios from 1 to 7. Nomenclature and scale are the same as given in Figure 5.19.

Of most importance in Equations (5.69) and (5.70) is the aspect-ratio variation. Note that for low-AR wings, a substantial difference can exist between a0 and a. However, as AR oo, a ao – The effect of aspect ratio on the lift curve is dramat­ically shown in Figure 5.22, which gives classic data obtained on rectangular wings byPrandtlin 1915. Note the reduction in dCi/da as AR is reduced. Moreover, using the equations obtained above, Prandtl scaled the data in Figure 5.22 to correspond to an aspect ratio of 5; his results collapsed to essentially the same curve, as shown in

Figure 5.23. In this manner, the aspect-ratio variation given in Equations (5.69) and (5.70) was confirmed as early as the year 1915.

Road Map for This Chapter

When learning a new subject, it is important for you to know where you are, where you are going, and how you can get there. Therefore, at the beginning of each chapter in this book, a road map will be given to help guide you through the material of that chapter and to help you obtain a perspective as to how the material fits within the general framework of aerodynamics. For example, a road map for Chapter 1 is given in Figure 1.6. You will want to frequently refer back to these road maps as you progress through the individual chapters. When you reach the end of each chapter, look back over the road map to see where you started, where you are now, and what you learned in between.

1.3 Some Fundamental Aerodynamic Variables

A prerequisite to understanding physical science and engineering is simply learn­ing the vocabulary used to describe concepts and phenomena. Aerodynamics is no exception. Throughout this book, and throughout your working career, you will be adding to your technical vocabulary list. Let us start by defining four of the most frequently used words in aerodynamics: “pressure,” “density,” “temperature,” and “flow velocity.”1

Consider a surface immersed in a fluid. The surface can be a real, solid surface such as the wall of a duct or the surface of a body; it can also be a free surface which we simply imagine drawn somewhere in the middle of a fluid. Also, keep in mind that


Figure 1 .6 Road map for Chapter 1.


the molecules of the fluid are constantly in motion. Pressure is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting on (or crossing) that surface. It is important to note that even though pressure is defined as force “per unit area,” you do not need a surface that is exactly 1 ft2 or 1 m2 to talk about pressure. In fact, pressure is usually defined at a point in the fluid or a point on a solid surface and can vary from one point to another. To see this more clearly, consider a point В in a volume of fluid. Let

dA — elemental area at В

dF = force on one side of dA due to pressure

Then, the pressure at point В in the fluid is defined as




p = lim






The pressure p is the limiting form of the force per unit area, where the area of interest has shrunk to nearly zero at the point B.[1] Clearly, you can see that pressure is a point property and can have a different value from one point to another in the fluid.

Another important aerodynamic variable is density, defined as the mass per unit volume. Analogous to our discussion on pressure, the definition of density does not require an actual volume of 1 ft[2] or 1 m3. Rather, it is a point property that can vary from point to point in the fluid. Again, consider a point В in the fluid. Let

dv = elemental volume around В dm — mass of fluid inside dv Then, the density at point В is


p = lim—— dv —»■ 0


Therefore, the density p is the limiting form of the mass per unit volume, where the volume of interest has shrunk to nearly zero around point B. (Note that dv cannot achieve the value of zero for the reason discussed in the footnote concerning dA in the definition of pressure.)

Temperature takes on an important role in high-speed aerodynamics (introduced in Chapter 7). The temperature Г of a gas is directly proportional to the average kinetic energy of the molecules of the fluid. In fact, if KE is the mean molecular kinetic energy, then temperature is given by ICE = |kT, where к is the Boltzmann constant. Hence, we can qualitatively visualize a high-temperature gas as one in which the molecules and atoms are randomly rattling about at high speeds, whereas in a low-temperature gas, the random motion of the molecules is relatively slow. Temperature is also a point property, which can vary from point to point in the gas.

The principal focus of aerodynamics is fluids in motion. Hence, flow velocity is an extremely important consideration. The concept of the velocity of a fluid is slightly more subtle than that of a solid body in motion. Consider a solid object in translational motion, say, moving at 30 m/s. Then all parts of the solid are simul­taneously translating at the same 30 m/s velocity. In contrast, a fluid is a “squishy” substance, and for a fluid in motion, one part of the fluid may be traveling at a different velocity from another part. Hence, we have to adopt a certain perspective, as follows. Consider the flow of air over an airfoil, as shown in Figure 1.7. Lock your eyes on a specific, infinitesimally small element of mass in the gas, called a fluid element, and watch this element move with time. Both the speed and direction of this fluid element can vary as it moves from point to point in the gas. Now fix your eyes on a specific fixed point in space, say, point В in Figure 1.7. Flow velocity can now be defined as follows: The velocity of a flowing gas at any fixed point В in space is the velocity of an infinitesimally small fluid element as it sweeps through B. The flow velocity V has both magnitude and direction; hence, it is a vector quantity. This is in contrast to p, p, and T, which are scalar variables. The scalar magnitude of V is frequently



Figure 1.7

Illustration of flow velocity and streamlines.


used and is denoted by V. Again, we emphasize that velocity is a point property and can vary from point to point in the flow.

Referring again to Figure 1.7, a moving fluid element traces out a fixed path in space. As long as the flow is steady, i. e., as long as it does not fluctuate with time, this path is called a streamline of the flow. Drawing the streamlines of the flow field is an important way of visualizing the motion of the gas; we will frequently be sketching the streamlines of the flow about various objects. A more rigorous discussion of streamlines is given in Chapter 2.

Substantial Derivative

Consider a small fluid element moving through a flow field, as shown in Figure 2.24. This figure is basically an extension of Figure 2.14, in which we introduced the concept of a fluid element moving through a specified flow field. The velocity field is given by V = ui + vj + uik, where

и — u(x, y, z, t) v = v(x, y, z, t) w — w(x, y, z, t)

In addition, the density field is given by

p = p(x, y, z, t)

At time t, the fluid element is located at point 1 in the flow (see Figure 2.24), and its density is

Pi = P(x, Vi, Zi, fi)


Figure 2.24 Fluid element moving in a flow field —illustration for the substantial derivative.


At a later time f2 the same fluid element has moved to a different location in the flow field, such as point 2 in Figure 2.24. At this new time and location, the density of the fluid element is

Pi = P(x2, У2, Z2, tl)

Since p = p{x, у, г, t), we can expand this function in a Taylor series about point 1 as follows:

Подпись: 8p . . Tx 1Подпись:Подпись: Pi = Pi +(Уі ~ Уі) + (Z2 – Z)


+ ( — I (h — fi) + higher-order terms dt ) у

Dividing by t2 — t, and ignoring the higher-order terms, we have Pi – Pi / dp x2 x / dp / У2 ~ Уі / dp Z2-Z1 / Эр

t2-t dx)xt2-ti уЗуД V t2 – ft / 3z / ] t2 t 3f/j


Consider the physical meaning of the left side of Equation (2.101). The term (p2 — P)/(h — t) is the average time rate of change in density of the fluid element as it moves from point 1 to point 2. In the limit, as t2 approaches t, this term becomes

p2 – pi Dp

hm ———- = —

h—’t2 — t Dt

Here, Dp/Dt is a symbol for the instantaneous time rate of change of density of the fluid element as it moves through point 1. By definition, this symbol is called the substantial derivative D/Dt. Note that Dp/Dt is the time rate of change of density

of a given fluid element as it moves through space. Here, our eyes are locked on the fluid element as it is moving, and we are watching the density of the element change as it moves through point 1. This is different from (dp/dt), which is physically the time rate of change of density at the fixed point 1. For (dp/dt), we fix our eyes on the stationary point 1, and watch the density change due to transient fluctuations in the flow field. Thus, Dp/Dt and dpjdt are physically and numerically different quantities.

Returning to Equation (2.101), note that

Substantial Derivative

Thus, taking the limit of Equation (2.101) as t2 t, we obtain

Substantial Derivative

dp dp dz + dt




Examine Equation (2.102) closely. From it, we can obtain an expression for the substantial derivative in cartesian coordinates:


Furthermore, in cartesian coordinates, the vector operator V is defined as

Подпись:Подпись: + j3 3

——- 1- к —

dy dz

Hence, Equation (2.103) can be written as






Equation (2.104) represents a definition of the substantial derivative in vector notation; thus, it is valid for any coordinate system.

Focusing on Equation (2.104), we once again emphasize that D/Dt is the sub­stantial derivative, which is physically the time rate of change following a moving fluid element; 3/31 is called the local derivative, which is physically the time rate of change at a fixed point; V • V is called the convective derivative, which is physically the time rate of change due to the movement of the fluid element from one location to another in the flow field where the flow properties are spatially different. The sub­stantial derivative applies to any flow-field variable, e. g., Dp/Dt, DT/Dt, Du/Dt. For example,

DT З T dT dT d T dT

—– =——— b (V • V)T =——— f – и——- b V—— b w —

Dt dt dt dx dy dz

local comcclive

derivative derivative

Again, Equation (2.105) states physically that the temperature of the fluid element is changing as the element sweeps past a point in the flow because at that point the flow-field temperature itself may be fluctuating with time (the local derivative) and because the fluid element is simply on its way to another point in the flow field where the temperature is different (the convective derivative).

Consider an example which will help to reinforce the physical meaning of the substantial derivative. Imagine that you are hiking in the mountains, and you are about to enter a cave. The temperature inside the cave is cooler than outside. Thus, as you walk through the mouth of the cave, you feel a temperature decrease—this is analogous to the convective derivative in Equation (2.105). However, imagine that, at the same time, a friend throws a snowball at you such that the snowball hits you just at the same instant you pass through the mouth of the cave. You will feel an additional, but momentary, temperature drop when the snowball hits you—this is analogous to the local derivative in Equation (2.105). The net temperature drop you feel as you walk through the mouth of the cave is therefore a combination of both the act of moving into the cave, where it is cooler, and being struck by the snowball at the same instant—this net temperature drop is analogous to the substantial derivative in Equation (2.105).

Historical Note: d’Alembert and His Paradox

You can well imagine the frustration that Jean le Rond d’Alembert felt in 1744 when, in a paper entitled “Traite de l’equilibre et des mouvements de fluids pour servir de siute au traite de dynamique,” he obtained the result of zero drag for the inviscid, incompressible flow over a closed two-dimensional body. Using different approaches, d’Alembert encountered this result again in 1752 in his paper entitled “Essai sur la resistance” and again in 1768 in his “Opuscules mathematiques.” In this last paper can be found the quote given at the beginning of Chapter 15; in essence, he had given up trying to explain the cause of this paradox. Even though the prediction of fluid – dynamic drag was a very important problem in d’Alembert’s time, and in spite of the number of great minds that addressed it, the fact that viscosity is responsible for drag was not appreciated. Instead, d’Alembert’s analyses used momentum principles in a frictionless flow, and quite naturally he found that the flow field closed smoothly around the downstream portion of the bodies, resulting in zero drag. Who was this man, d’Alembert? Considering the role his paradox played in the development of fluid dynamics, it is worth our time to take a closer look at the man himself.

d’Alembert was born illegitimately in Paris on November 17, 1717. His mother was Madame De Tenun, a famous salon hostess of that time, and his father was Cheva­lier Destouches-Canon, a cavalry officer. d’Alembert was immediately abandoned by his mother (she was an ex-nun who was afraid of being forcibly returned to the convent). However, his father quickly arranged for a home for d’Alembert—with a family of modest means named Rousseau. d’Alembert lived with this family for the next 47 years. Under the support of his father, d’Alembert was educated at the College de Quatre-Nations, where he studied law and medicine, and later turned to

mathematics. For the remainder of his life, d’Alembert would consider himself a mathematician. By a program of self-study, d’Alembert learned the works of Newton and the Bernoullis. His early mathematics caught the attention of the Paris Academy of Sciences, of which he became a member in 1741. d’Alembert published frequently and sometimes rather hastily, in order to be in print before his competition. However, he made substantial contributions to the science of his time. For example, he was (1) the first to formulate the wave equation of classical physics, (2) the first to express the concept of a partial differential equation, (3) the first to solve a partial differential equation—he used separation of variables—and (4) the first to express the differential equations of fluid dynamics in terms of a field. His contemporary, Leonhard Euler (see Sections 1.1 and 3.18) later expanded greatly on these equations and was responsible for developing them into a truly rational approach for fluid-dynamic analysis.

During the course of his life, d’Alembert became interested in many scientific and mathematical subjects, including vibrations, wave motion, and celestial mechanics. In the 1750s, he had the honored position of science editor for the Encyclopedia— a major French intellectual endeavor of the eighteenth century which attempted to compile all existing knowledge into a large series of books. As he grew older, he also wrote papers on nonscientific subjects, mainly musical structure, law, and religion.

In 1765, d’Alembert became very ill. He was helped to recover by the nursing of Mile. Julie de Lespinasse, the woman who was d’Alembert’s only love throughout his life. Although he never married, d’Alembert lived with Julie de Lespinasse until she died in 1776. d’Alembert had always been a charming gentleman, renowned for his intelligence, gaiety, and considerable conversational ability. However, after Mile, de Lespinasse’s death, he became frustrated and morose—living a life of despair. He died in this condition on October 29, 1783, in Paris.

d’Alembert was one of the great mathematicians and physicists of the eighteenth century. He maintained active communications and dialogue with both Bernoulli and Euler and ranks with them as one of the founders of modem fluid dynamics. This, then, is the man behind the paradox, which has existed as an integral part of fluid dynamics for the past two centuries.

General Three-Dimensional Flows: Panel Techniques

In modem aerodynamic applications, three-dimensional, inviscid, incompressible flows are almost always calculated by means of numerical panel techniques. The philosophy of the two-dimensional panel methods discussed in previous chapters is readily extended to three dimensions. The details are beyond the scope of this book—indeed, there are dozens of different variations, and the resulting computer programs are frequently long and sophisticated. However, the general idea behind all such panel programs is to cover the three-dimensional body with panels over which there is an unknown distribution of singularities (such as point sources, doublets, or vortices). Such paneling is illustrated in Figure 6.7. These unknowns are solved through a system of simultaneous linear algebraic equations generated by calculating the induced velocity at control points on the panels and applying the flow-tangency


Figure 6.6 The pressure distribution over the surface of a sphere and a cylinder. Illustration of the three-dimensional relieving effect.

condition. For a nonlifting body such as illustrated in Figure 6.7, a distribution of source panels is sufficient. However, for a lifting body, both source and vortex panels (or their equivalent) are necessary. A striking example of the extent to which panel methods are now used for three-dimensional lifting bodies is shown in Figure 6.8, which illustrates the paneling used for calculations made by the Boeing Company of the potential flow over a Boeing 747-space shuttle piggyback combination. Such applications are very impressive; moreover, they have become an industry standard and are today used routinely as part of the airplane design process by the major aircraft companies.

Examining Figures 6.7 and 6.8, one aspect stands out, namely, the geometric complexity of distributing panels over the three-dimensional bodies. How do you get the computer to “see” the precise shape of the body? How do you distribute the panels over the body; that is, do you put more at the wing leading edges and less on the fuselage, etc.? How many panels do you use? These are all nontrivial questions. It is not unusual for an aerodynamicist to spend weeks or even a few months determining the best geometric distribution of panels over a complex body.

We end this chapter on the following note. From the time they were introduced in the 1960s, panel techniques have revolutionized the calculation of three-dimensional potential flows. However, no matter how complex the application of these methods may be, the techniques are still based on the fundamentals we have discussed in this


Figure 6.7 Distribution of three-dimensional source panels over a general nonlifting body (Reference 14). (Courtesy of the McDonnell-Douglas Corp.)



Figure 6.8


Panel distribution for the analysis of the Boeing 747 carrying the space shuttle orbiter. (Courtesy of the Boeing Airplane Company.)

and all the preceding chapters. You are encouraged to pursue these matters further by reading the literature, particularly as it appears in such journals as the Journal of Aircraft and the AIAA Journal.

Historical Note: The Illusive Center of Pressure

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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