Category Fundamentals of Aerodynamics


Refer again to the road map for Chapter 1 given in Figure 1.6. Read again each block in this diagram as a reminder of the material we have covered. If you feel uncomfortable about some of the concepts, or if your memory is slightly “foggy” on certain points, go back and reread the pertinent sections until you have mastered the material.

This chapter has been primarily qualitative, emphasizing definitions and basic concepts. However, some of the more important quantitative relations are summarized below:



The center of pressure is obtained from










[1.80] and [1.91]


The criteria for two or more flows to be dynamically similar are:

1. The bodies and any other solid boundaries must be geometrically similar.

2. The similarity parameters must be the same. Two important similarity parameters are Mach number M = V/a and Reynolds number Re = p V с/ц..

If two or more flows are dynamically similar, then the force coefficients Cl, Cd, etc., are the same.



In fluid statics, the governing equation is the hydrostatic equation:

dp = ~gp dy


For a constant density medium, this integrates to

p + pgh = constant


or p і + pgh = P2 + pgh2

Such equations govern, among other things, the operation of a manometer, and also lead to Archimedes’ principle that the buoyancy force on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.

Bernoulli’s Equation

As will be portrayed in Section 3.19, the early part of the eighteenth century saw the flowering of theoretical fluid dynamics, paced by the work of Johann and Daniel Bernoulli and, in particular, by Leonhard Euler. It was at this time that the relation between pressure and velocity in an inviscid, incompressible flow was first understood.

The resulting equation is


Bernoulli’s Equation

Equation (3.12) is called Euler’s equation. It applies to an inviscid flow with no body forces, and it relates the change in velocity along a streamline d V to the change in pressure dp along the same streamline.

Equation (3.12) takes on a very special and important form for incompressible flow. In such a case, p — constant, and Equation (3.12) can be easily integrated between any two points 1 and 2 along a streamline. From Equation (3.12), with p = constant, we have


Подпись: or[3.13]

Equation (3.13) is Bernoulli’s equation, which relates pi and Vt at point 1 on a streamline to pz and V2 at another point 2 on the same streamline. Equation (3.13)

Подпись: p + pV2 = const along a streamline Подпись: [3.14]

can also be written as

Подпись: p + ~pV2 = const throughout the flow Подпись: [3.15]

In the derivation of Equations (3.13) and (3.14), no stipulation has been made as to whether the flow is rotational or irrotational—these equations hold along a streamline in either case. For a general, rotational flow, the value of the constant in Equation (3.14) will change from one streamline to the next. Flowever, if the flow is irrotational, then Bernoulli’s equation holds between any two points in the flow, not necessarily just on the same streamline. For an irrotational flow, the constant in Equation (3.14) is the same for all streamlines, and

The proof of this statement is given as Problem 3.1.

The physical significance of Bernoulli’s equation is obvious from Equations

(3.13) to (3.15); namely, when the velocity increases, the pressure decreases, and when the velocity decreases, the pressure increases.

Note that Bernoulli’s equation was derived from the momentum equation; hence, it is a statement of Newton’s second law for an inviscid, incompressible flow with no body forces. Flowever, note that the dimensions of Equations (3.13) to (3.15) are energy per unit volume (pV2 is the kinetic energy per unit volume). Flence, Bernoulli’s equation is also a relation for mechanical energy in an incompressible flow; it states that the work done on a fluid by pressure forces is equal to the change in kinetic energy of the flow. Indeed, Bernoulli’s equation can be derived from the general energy equation, such as Equation (2.114). This derivation is left to the reader. The fact that Bernoulli’s equation can be interpreted as either Newton’s second law or an energy equation simply illustrates that the energy equation is redundant for the analysis of inviscid, incompressible flow. For such flows, the continuity and momentum equations suffice. (You may wish to review the opening comments of Section 2.7 on this same subject.)

The strategy for solving most problems in inviscid, incompressible flow is as follows:

1. Obtain the velocity field from the governing equations. These equations, appro­priate for an inviscid, incompressible flow, are discussed in detail in Sections 3.6

and 3.7.

2. Once the velocity field is known, obtain the corresponding pressure field from Bernoulli’s equation.

However, before treating the general approach to the solution of such flows (Section 3.7), several applications of the continuity equation and Bernoulli’s equation are made to flows in ducts (Section 3.3) and to the measurement of airspeed using a Pitot tube (Section 3.4).

Example 3.1 I Consider an airfoil in a flow at standard sea level conditions with a freestream velocity of 50 m/s. At a given point on the airfoil, the pressure is 0.9 x 105 N/m2. Calculate the velocity at this point.


At standard sea level conditions, рх = 1.23 kg/m3 and px = 1.01 x 105 N/m2. Hence,

Pcо + pVl, = p + pV2

v – 01 x ‘O’ 7^1

U = 142.8 m/s

Modern Low-Speed Airfoils

The nomenclature and aerodynamic characteristics of standard NACA airfoils are discussed in Sections 4.2 and 4.3; before progressing further, you should review these sections in order to reinforce your knowledge of airfoil behavior, especially in light of our discussions on airfoil theory. Indeed, the purpose of this section is to provide a modem sequel to the airfoils discussed in Sections 4.2 and 4.3.


During the 1970s, NASA designed a series of low-speed airfoils that have perfor­mance superior to the earlier NACA airfoils. The standard NACA airfoils were based almost exclusively on experimental data obtained during the 1930s and 1940s. In con­trast, the new NASA airfoils were designed on a computer using a numerical technique similar to the source and vortex panel methods discussed earlier, along with numerical predictions of the viscous flow behavior (skin friction and flow separation). Wind – tunnel tests were then conducted to verify the computer-designed profiles and to obtain the definitive airfoil properties. Out of this work first came the general aviation— Whitcomb [GA(W) — 1] airfoil, which has since been redesignated the LS(1)-0417 airfoil. The shape of this airfoil is given in Figure 4.30, obtained from Reference 16. Note that it has a large leading-edge radius (0.08c in comparison to the standard 0.02c) in order to flatten the usual peak in pressure coefficient near the nose. Also, note that the bottom surface near the trailing edge is cusped in order to increase the camber and

Подпись: Figure 4.30 Profile for the NASA LS(1)-0417 airfoil.When first introduced, this airfoil was labeled the GA (W)-l airfoil, a nomenclature which has now been superseded. (From Reference 16.)

hence the aerodynamic loading in that region. Both design features tend to discourage flow separation over the top surface at high angle of attack, hence yielding higher values of the maximum lift coefficient. The experimentally measured lift and moment properties (from Reference 16) are given in Figure 4.31, where they are compared with the properties for an NACA 2412 airfoil, obtained from Reference 11. Note that Q. max for the NASA LS(1)-0417 is considerably higher than for the NACA 2412.

The NASA LS(1)-0417 airfoil has a maximum thickness of 17 percent and a design lift coefficient of 0.4. Using the same camber line, NASA has extended this airfoil into a family of low-speed airfoils of different thicknesses, for example, the NASA LS(l)-0409 and the LS(1)-0413. (See Reference 17 for more details.) In comparison with the standard NACA airfoils having the same thicknesses, these new LS(l)-04xx airfoils all have:

1. Approximately 30 percent higher c/imax•

2. Approximately a 50 percent increase in the ratio of lift to drag (L/D) at a lift coefficient of 1.0. This value of q = 1.0 is typical of the climb lift coefficient for general aviation aircraft, and a high value of L/D greatly improves the climb

Подпись: 2.4 2.0 -16 -12 -8 -4 0 4 8 12 16 20 a, degrees Figure 4.31 Comparison of the modern NASA LS(1)-0417 airfoil with the standard NACA 2412 airfoil.

© NASA LS(1)0417 (ref. 16), Re = 6.3 X 106 0 NACA 2412 (ref. 11), Re = 5.7 X 106

performance. (See Reference 2 for a general introduction to airplane performance

and the importance of a high L/D ratio to airplane efficiency.)

It is interesting to note that the shape of the airfoil in Figure 4.30 is very similar to the supercritical airfoils to be discussed in Chapter 11. The development of the supercritical airfoil by NASA aerodynamicist Richard Whitcomb in 1965 resulted in a major improvement in airfoil drag behavior at high subsonic speeds, near Mach 1. The supercritical airfoil was a major breakthrough in high-speed aerodynamics. The LS(1)-0417 low-speed airfoil shown in Figure 4.30, first introduced as the GA(W)-1 airfoil, was a later spin-off from supercritical airfoil research. It is also interesting to note that the first production aircraft to use the NASA LS( 1 )-0417 airfoil was the Piper PA-38 Tomahawk, introduced in the late 1970s.

Подпись: This chapter deals with incompressible flow over airfoils. Moreover, the analytical thin airfoil theory and the numerical panel methods discussed here are techniques for calculating the aerodynamic characteristics for a given airfoil of specified shape. Such an approach is frequently called the direct problem, wherein the shape of the body is given, and the surface pressure distribution (for example) is calculated. For design purposes, it is desirable to turn this process inside-out; it is desirable to specify the surface pressure distribution—a pressure distribution that will achieve enhanced airfoil performance—and calculate the shape of the airfoil that will produce the specified pressure distribution. This approach is called the inverse problem. Before the advent of the high-speed digital computer, and the concurrent rise of the discipline of computational fluid dynamics in the 1970s (see Section 2.17.2), the analytical solution of the inverse problem was difficult, and was not used by the practical airplane designer. Instead, for most of the airplanes designed before and during the twentieth century, the choice of an airfoil shape was based on reasonable experimental data (at best), and guesswork (at worst). This story is told in some detail in Reference 62. The design problem was made more comfortable with the introduction of the various families of NACA airfoils, beginning in the early 1930s. A logical method was used for the geometrical design of these airfoils, and definitive experimental data on the NACA airfoils were made available (such as shown in Figures 4.5, 4.6, and 4.22). For this reason, many airplanes designed during the middle of the twentieth century used standard NACA airfoil sections. Even today, the NACA airfoils are sometimes the most expeditious choice of the airplane designer, as indicated by the tabulation (by no means complete) in Section 4.2 of airplanes using such airfoils.

In summary, new airfoil development is alive and well in the aeronautics of the late twentieth century. Moreover, in contrast to the purely experimental development of the earlier airfoils, we now enjoy the benefit of powerful computer programs using panel methods and advanced viscous flow solutions for the design of new airfoils. Indeed, in the 1980s NASA established an official Airfoil Design Center at The Ohio State University, which services the entire general aviation industry with over 30 dif­ferent computer programs for airfoil design and analysis. For additional information on such new low-speed airfoil development, you are urged to read Reference 16, which is the classic first publication dealing with these airfoils, as well as the concise review given in Reference 17.

However, today the power of computational fluid dynamics (CFD) is revolutionizing airfoil design and anal­ysis. The inverse problem, and indeed the next step—the overall automated procedure that results in a completely optimized airfoil shape for a given design point—are being made tractable by CFD. An example of such work is illustrated in Figures 4.32 and 4.33, taken from the recent work of Kyle Anderson and Daryl Bonhaus (Refer­ence 68). Here, CFD solutions of the continuity, momentum, and energy equations for a compressible, viscous flow (the Navier-Stokes equations, as denoted in Section 2.17.2) are carried out for the purpose of airfoil design. Using a finite volume CFD technique, and the grid shown in Figure 4.32, the inverse problem is solved. The specified pressure distribution over the top and bottom surfaces of the airfoil is given by the circles in Figure 4.33a. The optimization technique is iterative and requires starting with a pressure distribution that is not the desired, specified one; the initial distribution is given by the solid curves in Figure 4.33a, and the airfoil shape corresponding to this initial pressure distribution is shown by the solid curve in Figure 4.33b. (In Figure 4.33b, the airfoil shape appears distorted because an expanded scale is used for the ordinate.) After 10 design cycles, the optimized airfoil shape



Figure 4.32 Unstructured mesh for the numerical calculation of the flow over an airfoil. (Source: Anderson ond Bonhaus, Reference 68.)


(a) Pressure coefficient distributions


(,b) Airfoil shapes


Figure 4.33 An example of airfoil optimized design using computational fluid dynamics (Reference 68).



that supports the specified pressure distribution is obtained, as given by the circles in Figure 4.33b. The initial airfoil shape is also shown in constant scale in Figure 4.32.

The results given in Figures 4.32 and 4.33 are shown here simply to provide the flavor of modern airfoil design and analysis. This is reflective of the wave of future airfoil design procedures, and you are encouraged to read the contemporary literature in order to keep up with this rapidly evolving field. However, keep in mind that the simpler analytical approach of thin airfoil theory discussed in the present chapter, and especially the simple practical results of this theory, will continue to be part of the whole “toolbox” of procedures to be used by the designer in the future. The fundamentals embodied in thin airfoil theory will continue to be part of the fundamentals of aerodynamics and will always be there as a partner with the modern CFD techniques.

The Basic Normal Shock Equations

Consider the normal shock wave sketched in Figure 8.3. Region 1 is a uniform flow upstream of the shock, and region 2 is a different uniform flow downstream of the shock. The pressure, density, temperature, Mach number, velocity, total pressure, total enthalpy, total temperature, and entropy in region 1 are p, p, 7), M, u, po,, ho, і, 7’o. i, and ^|, respectively. The corresponding variables in region 2 are denoted by p2, Pi, T2, M2, u2, po,2, ho,2, ?o,2, and s2. (Note that we are denoting the magnitude of the flow velocity by и rather than V; reasons for this will become obvious as we progress.) The problem of the normal shock wave is simply stated as follows: given the flow properties upstream of the wave (p, Tu M, etc.), calculate the flow properties (p2, T2, M2, etc.) downstream of the wave. Let us proceed.

Consider the rectangular control volume abed given by the dashed line in Figure 8.3. The shock wave is inside the control volume, as shown. Side ab is the edge view of the left face of the control volume; this left face is perpendicular to the flow, and its area is A. Side cd is the edge view of the right face of the control volume; this right face is also perpendicular to the flow, and its area is Л. We apply the integral form of conservation equations to this control volume. In the process, we observe three important physical facts about the flow given in Figure 8.3:

1. The flow is steady, that is, 9/9f = 0.

2. The flow is adiabatic, that is, q = 0. We are not adding or taking away heat from the control volume (we are not heating the shock wave with a Bunsen burner, for

Figure 8*3 Sketch of a normal wave.

example). The temperature increases across the shock wave, not because heat is being added, but rather, because kinetic energy is converted to internal energy across the shock wave.

3. There are no viscous effects on the sides of the control volume. The shock wave itself is a thin region of extremely high velocity and temperature gradients; hence, friction and thermal conduction play an important role on the flow structure inside the wave. However, the wave itself is buried inside the control volume, and with the integral form of the conservation equations, we are not concerned about the details of what goes on inside the control volume.

4. There are no body forces; f = 0.

The Basic Normal Shock Equations Подпись: [8.1]

Consider the continuity equation in the form of Equation (7.39). For the condi­tions described above, Equation (7.39) becomes

To evaluate Equation (8.1) over the face ab, note that V is pointing into the control volume whereas dS by definition is pointing out of the control volume, in the opposite direction of V; hence, V • dS is negative. Moreover, p and |V| are uniform over the face ab and equal to p and u, respectively. Hence, the contribution of face ab to the surface integral in Equation (8.1) is simply — pUA. Over the right face cd both V and dS are in the same direction, and hence V • dS is positive. Moreover, p and | V| are uniform over the face cd and equal to pn and «2, respectively. Thus, the contribution of face cd to the surface integral is P2U2A. On sides be and ad, V and dS are always perpendicular; hence, V • dS = 0, and these sides make no contribution to the surface

integral. Hence, for the control volume shown in Figure 8.3, Equation (8.1) becomes

Подпись: or Подпись: Pi «1 = p2u2 Подпись: [8.2]

Pi Mi A + p2u2A = 0

Equation (8.2) is the continuity equation for normal shock waves.

Consider the momentum equation in the form of Equation (7.41). For the flow we are treating here, Equation (7.41) becomes

Подпись: sПодпись: simage525[8.3]

Equation (8.3) is a vector equation. Note that in Figure 8.3, the flow is moving only in one direction (i. e., in the x direction). Hence, we need to consider only the scalar x component of Equation (8.3), which is


In Equation (8.4), (p dS)x is the x component of the vector (p dS). Note that over the face ab, dS points to the left (i. e., in the negative x direction). Hence, (p dS)x is negative over face ab. By similar reasoning, (p dS)x is positive over the face cd. Again noting that all the flow variables are uniform over the faces ab and cd, the surface integrals in Equation (8.4) become

Подпись: [8.5]

Подпись: or Подпись: Pi + Pll = p2 + p2u Подпись: [8.6]

P(-uA)u + p2(u2A)u2 — —(—pA + p2A)

Equation (8.6) is the momentum equation for normal shock waves.

Consider the energy equation in the form of Equation (7.43). For steady, adia­batic, inviscid flow with no body forces, this equation becomes

Подпись: s [8.7]


Evaluating Equation (8.7) for the control surface shown in Figure 8.3, we have

Rearranging, we obtain

Подпись: El Pi Подпись: ■ ei Подпись: P2 . «2 — + Є2 + ~ Рг 2 Подпись: [8.9]

Dividing by Equation (8.2), that is, dividing the left-hand side of Equation (8.8) by PU and the right-hand side by P2U2, we have

From the definition of enthalpy, h = e + pv = e + р/р. Hence, Equation (8.9) becomes


Equation (8.10) is the energy equation for normal shock waves. Equation (8.10) should come as no surprise; the flow through a shock wave is adiabatic, and we derived in Section 7.5 the fact that for a steady, adiabatic flow, ho = h + Vі/2 = const. Equation (8.10) simply states that ho (hence, for a calorically perfect gas Го) is constant across the shock wave. Therefore, Equation (8.10) is consistent with the general results obtained in Section 7.5.

Подпись: Continuity: Momentum: Energy: The Basic Normal Shock Equations Подпись: [8.8] [8.6] [8.10]

Repeating the above results for clarity, the basic normal shock equations are

Examine these equations closely. Recall from Figure 8.3 that all conditions upstream of the wave, pi, «і, Pi, etc., are known. Thus, the above equations are a system of three algebraic equations in four unknowns, p2, U2, P2, and /12- However, if we add the following thermodynamic relations

Enthalpy: h2 = cpT2

Equation of state: p2 — P2RT2

we have five equations for five unknowns, namely, P2, U2, P2, ^2, and T2. In Section 8.6, we explicitly solve these equations for the unknown quantities behind the shock. However, rather than going directly to that solution, we first take three side trips as shown in the road map in Figure 8.2. These side trips involve discussions of the speed of sound (Section 8.3), alternate forms of the energy equation (Section 8.4), and compressibility (Section 8.5)—all of which are necessary for a viable discussion of shock-wave properties in Section 8.6.

Finally, we note that Equations (8.2), (8.6), and (8.10) are not limited to normal shock waves; they describe the changes that take place in any steady, adiabatic, inviscid flow where only one direction is involved. That is, in Figure 8.3, the flow is in the x direction only. This type of flow, where the flow-field variables are functions of x only [ p = p(x), и = u(x), etc.], is defined as one-dimensional flow. Thus, Equations (8.2), (8.6), and (8.10) are governing equations for one-dimensional, steady, adiabatic, inviscid flow.

Continuity Equation

In Section 2.3, we discussed several models which can be used to study the motion of a fluid. Following the philosophy set forth at the beginning of Section 2.3, we now apply the fundamental physical principles to such models. Unlike the above derivation of the physical significance of V • V wherein we used the model of a moving finite control volume, we now employ the model of a fixed finite control volume as sketched on the left side of Figure 2.11. Here, the control volume is fixed in space, with the flow moving through it. Unlike our previous derivation, the volume V and control surface S are now constant with time, and the mass of fluid contained within the control volume can change as a function of time (due to unsteady fluctuations of the flow field).

Before starting the derivation of the fundamental equations of aerodynamics, we must examine a concept vital to those equations, namely, the concept of mass flow. Consider a given area A arbitrarily oriented in a flow field as shown in Figure 2.16. In Figure 2.16, we are looking at an edge view of area A. Let A be small enough such that the flow velocity V is uniform across A. Consider the fluid elements with velocity V that pass through A. In time dt after crossing A, they have moved a distance V dt and have swept out the shaded volume shown in Figure 2.16. This volume is equal to the base area A times the height of the cylinder V„ dt, where V„ is the component of velocity normal to A; i. e.,

Volume = (V„dt)A

The mass inside the shaded volume is therefore

Подпись: [2.42]Mass = p(Vn dt)A

This is the mass that has swept past A in time dt. By definition, the mass flow through A is the mass crossing A per second (e. g., kilograms per second, slugs per second). Let m denote mass flow. From Equation (2.42).

. p{Vndt)A

m =——— :——



m = pVnA





Figure 2.1 6 Sketch for discussion of mass flow through area A in a flow field.


Подпись: Area x density x component of flow velocity normal to the area

Equation (2.43) demonstrates that mass flow through A is given by the product

Подпись: Mass flux = — — pVn A Подпись: [2.44]

A related concept is that of mass flux, defined as the mass flow per unit area.

Typical units of mass flux are kg/(s • m2) and slug/(s • ft2).

The concepts of mass flow and mass flux are important. Note from Equation

(2.44) that mass flux across a surface is equal to the product of density times the component of velocity perpendicular to the surface. Many of the equations of aero­dynamics involve products of density and velocity. For example, in cartesian coor­dinates, V = Vxi + Vyj + Т, к = ui + uj 4- i/jk, where u, v, and w denote the x, y, and z components of velocity, respectively. (The use of u, v, and w rather than Vx, Vy, and V, to symbolize the x, у, and z components of velocity is quite common in aerodynamic literature; we henceforth adopt the u, v, and w notation.) In many of the equations of aerodynamics, you will find the products pu, pv, and pw always remember that these products are the mass fluxes in the x, y, and z directions, re­spectively. In a more general sense, if V is the magnitude of velocity in an arbitrary direction, the product p V is physically the mass flux (mass flow per unit area) across an area oriented perpendicular to the direction of V.

We are now ready to apply our first physical principle to a finite control volume fixed in space.

Doublet Flow: Our Third Elementary Flow

There is a special, degenerate case of a source-sink pair that leads to a singularity called a doublet. The doublet is frequently used in the theory of incompressible flow; the purpose of this section is to describe its properties.

Consider a source of strength Л and a sink of equal (but opposite) strength —A separated by a distance /, as shown in Figure 3.24a. At any point P in the flow, the stream function is

A A r ,

t/r = —(0, – в2) =———— Ав [3.84]

2jt 2jt

where Ав = 02 — в as seen from Figure 3.24a. Equation (3.84) is the stream func­tion for a source-sink pair separated by the distance /.

Подпись: ф = lim /-*<) K—l AsaCOIlSt Подпись: A 2JT Подпись: [3.85]

Now in Figure 3.24a, let the distance / approach zero while the absolute magni­tudes of the strengths of the source and sink increase in such a fashion that the product lA remains constant. This limiting process is shown in Figure 3.24b. In the limit, as / -> 0 while lA remains constant, we obtain a special flow pattern defined as a doublet. The strength of the doublet is denoted by к and is defined as к = l A. The stream function for a doublet is obtained from Equation (3.84) as follows:


where in the limit A6 d6 -> 0. (Note that the source strength A approaches an infinite value in the limit.) In Figure 3.24i>, let r and b denote the distances to point P from the source and sink, respectively. Draw a line from the sink perpendicular to r, and denote the length along this line by a. For an infinitesimal dG, the geometry

is a circle with a diameter d on the vertical axis and with the center located d/2 directly above the origin. Comparing Equations (3.89) and (3.90), we see that the streamlines for a doublet are a family of circles with diameter к/Ътс, as sketched in Figure 3.25. The different circles correspond to different values of the parameter c. Note that in Figure 3.24 we placed the source to the left of the sink; hence, in Figure 3.25 the direction of flow is out of the origin to the left and back into the origin from the right. In Figure 3.24, we could just as well have placed the sink to the left of the source. In such a case, the signs in Equations (3.87) and (3.88) would be reversed, and the flow in Figure 3.25 would be in the opposite direction. Therefore, a doublet has associated with it a sense of direction—the direction with which the flow


Figure 3.25 Doublet flow with strength к.

moves around the circular streamlines. By convention, we designate the direction of the doublet by an arrow drawn from the sink to the source, as shown in Figure 3.25. In Figure 3.25, the arrow points to the left, which is consistent with the form of Equations (3.87) and (3.88). If the arrow would point to the right, the sense of rotation would be reversed, Equation (3.87) would have a positive sign, and Equation (3.88) would have a negative sign.

Returning to Figure 3.24, note that in the limit as l —> 0, the source and sink fall on top of each other. However, they do not extinguish each other, because the absolute magnitude of their strengths becomes infinitely large in the limit, and we have a singularity of strength (oo — oo); this is an indeterminate form which can have a finite value.

As in the case of a source or sink, it is useful to interpret the doublet flow shown in Figure 3.25 as being induced by a discrete doublet of strength к placed at the origin. Therefore, a doublet is a singularity that induces about it the double-lobed circular flow pattern shown in Figure 3.25.

The Lifting-Surface Theory and the Vortex Lattice Numerical Method

Prandtl’s classical lifting-line theory (Section 5.3) gives reasonable results for straight wings at moderate to high aspect ratio. However, for low-aspect-ratio straight wings, swept wings, and delta wings, classical lifting-line theory is inappropriate. For such planforms, sketched in Figure 5.30, a more sophisticated model must be used. The purpose of this section is to introduce such a model and to discuss its numerical im­plementation. However, it is beyond the scope of this book to elaborate on the details of such higher-order models; rather, only the flavor is given here. You are encouraged to pursue this subject by reading the literature and by taking more advanced studies in aerodynamics.

Return to Figure 5.13. Here, a simple lifting line spans the wing, with its asso­ciated trailing vortices. The circulation Г varies with у along the lifting line. Let us extend this model by placing a series of lifting lines on the plane of the wing, at different chordwise stations; that is, consider a large number of lifting lines all parallel to the у axis, located at different values of jc, as shown in Figure 5.31. In the limit of an infinite number of lines of infinitesimal strength, we obtain a vortex sheet, where the vortex lines run parallel to the у axis. The strength of this sheet (per unit length in the л direction) is denoted by у, where у varies in the >• direction, analogous to the variation of Г for the single lifting line in Figure 5.13. Moreover, each lifting line will have, in general, a different overall strength, so that у varies with x also. Hence, у = y(x, v) as shown in Figure 5.31. In addition, recall that each lifting line has a system of trailing vortices; hence, the series of lifting lines is crossed

The Lifting-Surface Theory and the Vortex Lattice Numerical Method

Types of wing planforms for which classical lifting-line theory is not appropriate.


Figure 5.30



by a series of superimposed trailing vortices parallel to the x axis. In the limit of an infinite number of infinitesimally weak vortices, these trailing vortices form another vortex sheet of strength S (per unit length in the у direction). [Note that this S is different from the S used in Equation (5.61); the use of the same symbol in both cases is standard, and there should be no confusion since the meanings and context are completely different.] To see this more clearly, consider a single line parallel to the x axis. As we move along this line from the leading edge to the trailing edge, we pick up an additional superimposed trailing vortex each time we cross a lifting line. Hence,

S must vary with x. Moreover, the trailing vortices are simply parts of the horseshoe vortex systems, the leading edges of which make up the various lifting lines. Since the circulation about each lifting line varies in the у direction, the strengths of different trailing vortices will, in general, be different. Hence, & also varies in the у direction, that is, & = S(x, y), as shown in Figure 5.31. The two vortex sheets—the one with vortex lines running parallel to у with strength у (per unit length in the x direction) and the other with vortex lines running parallel to x with strength <5 (per unit length in the у direction)—result in a lifting surface distributed over the entire planform of the wing, as shown in Figure 5.31. At any given point on the surface, the strength of the lifting surface is given by both у and 8, which are functions of x and y. We denote у = y(x, у) as the spanwise vortex strength distribution and 3 = 8(x, y) as the chordwise vortex strength distribution.

Note that downstream of the trailing edge we have no spanwise vortex lines, only trailing vortices. Hence, the wake consists of only chordwise vortices. The strength of this wake vortex sheet is given by 8W (per unit length in the у direction). Since in the wake the trailing vortices do not cross any vortex lines, the strength of any given trailing vortex is constant with x. Hence, <$„, depends only on у and, throughout the wake, 8w(y) is equal to its value at the trailing edge.

Now that we have defined the lifting surface, of what use is it? Consider point P located at (x, y) on the wing, as shown in Figure 5.31. The lifting surface and the wake vortex sheet both induce a normal component of velocity at point P. Denote this normal velocity by w(x, y). We want the wing planform to be a stream surface of the flow; that is, we want the sum of the induced w{x, y) and the normal component of the freestream velocity to be zero at point P and for all points on the wing—this is the flow-tangency condition on the wing surface. (Keep in mind that we are treating the wing as a flat surface in this discussion.) The central theme of lifting-surface theory is to find у (x, y) and <$(x, y) such that the flow-tangency condition is satisfied at all points on the wing. [Recall that in the wake, <$„,(>•) is fixed by the trailing – edge values of 8(x, y); hence, 8w(y) is not, strictly speaking, one of the unknown dependent variables.!

Подпись: |dV| Подпись: Г dl x x 4л- |rp Подпись: у іif (dri)r sin# 4л- r3 Подпись: [5.77]

Let us obtain an expression for the induced normal velocity w(x, y) in terms of y, S, and Sw. Consider the sketch given in Figure 5.32, which shows a portion of the planview of a finite wing. Consider the point given by the coordinates (£, t]). At this point, the spanwise vortex strength is rj). Consider a thin ribbon, or filament, of the spanwise vortex sheet of incremental length d£ in the x direction. Hence, the strength of this filament is у d^, and the filament stretches in the у (or rj) direction. Also, consider point P located at (x, y) and removed a distance r from the point (§, rj). From the Biot-Savart law, Equation (5.5), the incremental velocity induced at P by a segment drj of this vortex filament of strength у d£ is

Examining Figure 5.32, and following the right-hand rule for the strength y, note that |dV| is induced downward, into the plane of the wing (i. e., in the negative z direction). Following the usual sign convention that w is positive in the upward direction (i. e., in



Figure 5.33 Velocity induced at point P by an infinitesimal segment of the lifting surface. The velocity is perpendicular to the plane of the paper.


Подпись: (dw)y = Подпись: У (x — %)d% dr) An r3 Подпись: [5.78]

the positive z direction), we denote the contribution of Equation (5.77) to the induced velocity w as (dw)y = — |dV|. Also, note that sin в = (x — §)/r. Hence, Equation (5.77) becomes

Подпись: (dw)s Подпись: My ~ dr) An r3 Подпись: [5.79]

Considering the contribution of the elemental chordwise vortex of strength S dri to the induced velocity at P, we find by an analogous argument that

To obtain the velocity induced at P by the entire lifting surface, Equations (5.78) and

(4.79) must be integrated over the wing planform, designated as region S in Figure

5.32. Moreover, the velocity induced at P by the complete wake is given by an equation analogous to Equation (5.79), but with Sw instead of S, and integrated over the wake, designated as region W in Figure 5.32. Noting that

r = s/(x -£)2 + (y – r])2


the normal velocity induced at P by both the lifting surface and the wake is

The central problem of lifting-surface theory is to solve Equation (5.80) for y(£, rf) and 5(£, T]) such that the sum of w(x, y) and the normal component of the freestream is zero, that is, such that the flow is tangent to the planform surface S. The details of various lifting-surface solutions are beyond the scope of this book; rather, our purpose here was simply to present the flavor of the basic model.

The advent of the high-speed digital computer has made possible the implemen­tation of numerical solutions based on the lifting-surface concept. These solutions are similar to the panel solutions for two-dimensional flow discussed in Chapters 3 and 4 in that the wing planform is divided into a number of panels, or elements. On each panel, either constant or prescribed variations of both у and S can be made. Control points on the panels can be chosen, where the net normal flow velocity is zero. The evaluation of equations like Equation (5.80) at these control points results in a system of simultaneous algebraic equations that can be solved for the values of the y’s and 5’s on all the panels.

A related but somewhat simpler approach is to superimpose a finite number of horseshoe vortices of different strengths F„ on the wing surface. For example, consider Figure 5.33, which shows part of a finite wing. The dashed lines define a panel on the wing planform, where l is the length of the panel in the flow direction. The panel is a trapezoid; it does not have to be a square, or even a rectangle. A horseshoe vortex abed of strength Г,, is placed on the panel such that the segment be is a distance 1/4 from the front of the panel. A control point is placed on the





Figure 5.33 Schematic of a single horseshoe vortex, which is part of a vortex system on the wing.




Figure 5.34 Vortex lattice system on a finite wing.


centerline of the panel at a distance | / from the front. The velocity induced at an arbitrary point P only by the single horseshoe vortex can be calculated from the Biot-Savart law by treating each of the vortex filaments ab, be, and cd separately. Now consider the entire wing covered by a finite number of panels, as sketched in Figure 5.34. A series of horseshoe vortices is now superimposed. For example, on one panel at the leading edge, we have the horseshoe vortex abed. On the panel behind it, we have the horseshoe vortex aefd. On the next panel, we have aghd, and on the next, aijd, etc. The entire wing is covered by this lattice of horseshoe vortices, each of different unknown strength Г„. At any control point P, the normal velocity induced by all the horseshoe vortices can be obtained from the Biot-Savart law. When the flow-tangency condition is applied at all the control points, a system of simultaneous algebraic equations results which can be solved for the unknown r„’s. This numerical approach is called the vortex lattice method and is in wide use today for the analysis of finite-wing properties. Once again, only the flavor of the method is given above; you are encouraged to read the volumes of literature that now exist on various versions of the vortex lattice method. In particular, Reference 13 has an excellent introductory discussion on the vortex lattice method, including a worked example that clearly illustrates the salient points of the technique.

The Lifting-Surface Theory and the Vortex Lattice Numerical Method

This relation, and others like it, is useful for the coneeptual design process, where simple formulas, albeit approx­imate, can lead to fast, back-of-the-envelope calculations. However, Equation (5.69), like all results from simple lifting-line theory, is valid only for high-aspect-ratio straight wings (AR > 4, as a rule of thumb).

The German aerodynamicist H. B. Helmbold in 1942 modified Equation (5.69) to obtain the following form applicable to low-aspect-ratio straight wings:


__________ Oo_________

v7! + (ао/л-AR)2 + a0/(7rAR)


low-aspect-ratio straight wing




Equation (5.81) is remarkably accurate for wings with AR < 4. This is demonstrated in Figure 5.35, which gives experimental data for the lift slope for rectangular wings as a function of AR from 0.5 to 6; these data are compared with the predictions from Prandtl’s lifting-line theory, Equation (5.69), and Helmbold’s equation, Equation (5.81). Note from Figure 5.35 that Helmbold’s equation gives excellent agreement with the data for AR < 4, and that Equation (5.69) is preferable for AR > 6.

For swept wings, Kuchemann (Reference 70) suggests the following modification to Helmbold’s equation:


ao cos A

у/l + [(a0cosA)/(7rAR)]2 + (a0 cosA/(7rAR))




swept wing


where A is the sweep angle of the wing, referenced to the half-chord line, as shown in Figure 5.36.



Figure 5.35 Lift slope versus aspect ratio for straight wings in low-speed flow.


Dimensional Analysis: The Buckingham Pi Theorem

The aerodynamic forces and moments on a body, and the corresponding force and moment coefficients, have been defined and discussed in Section 1.5. Question: What physical quantities determine the variation of these forces and moments? The answer can be found from the powerful method of dimensional analysis, which is introduced in this section.3

Consider a body of given shape at a given angle of attack, e. g., the airfoil sketched in Figure 1.10. The resultant aerodynamic force is R. On a physical, intuitive basis, we expect R to depend on:

1. Freestream velocity Voo.

2. Freestream density Poo.

3. Viscosity of the fluid. We have seen that shear stress r contributes to the aero­dynamic forces and moments. In turn, in Chapter 15, we will see that r is proportional to the velocity gradients in the flow. For example, if the velocity gradient is given by du/dy, then r = /іди/ду. The constant of proportionality is the viscosity coefficient fi. Hence, let us represent the influence of viscosity on aerodynamic forces and moments by the freestream viscosity coefficient fi<*,.

4. The size of the body, represented by some chosen reference length. In Figure 1.10, the convenient reference length is the chord length c.

5. The compressibility of the fluid. The technical definition of compressibility is given in Chapter 7. For our present purposes, let us just say that compressibility is related to the variation of density throughout the flow field, and certainly the aerodynamic forces and moments should be sensitive to any such variation. In turn, compressibility is related to the speed of sound a in the fluid, as shown in Chapter 8.4 Therefore, let us represent the influence of compressibility on aerodynamic forces and moments by the freestream speed of sound, ax.

3 For a more elementary treatment of dimensional analysis, see Chapter 5 of Reference 2.

4 Common experience tells us that sound waves propagate through air at some finite velocity, much slower than the speed of light; you see a flash of lightning in the distance, and hear the thunder moments later. The speed of sound is an important physical quantity in aerodynamics and is discussed in detail in Section 8.3.

In light of the above, and without any a priori knowledge about the variation of R, we can use common sense to write

Подпись: [1.33]R — f (Pco, Fqo, C, fJ, oc, r^oc)

Equation (1.23) is a general functional relation, and as such is not very practical for the direct calculation of R. In principle, we could mount the given body in a wind tunnel, incline it at the given angle of attack, and then systematically measure the variation of R due to variations of px, V^, с, /xTO, and ax, taken one at a time. By cross-plotting the vast bulk of data thus obtained, we might be able to extract a precise functional relation for Equation (1.23). However, it would be hard work, and it would certainly be costly in terms of a huge amount of required wind-tunnel time. Fortunately, we can simplify the problem and considerably reduce our time and effort by first employing the method of dimensional analysis. This method will define a set of dimensionless parameters which governs the aerodynamic forces and moments; this set will considerably reduce the number of independent variables as presently occurs in Equation (1.23).

Dimensional analysis is based on the obvious fact that in an equation dealing with the real physical world, each term must have the same dimensions. For example, if

ф + г} + ї =Ф

Подпись: ф ф Ф

is a physical relation, then ifr, t], £, and ф must have the same dimensions. Oth­erwise we would be adding apples and oranges. The above equation can be made dimensionless by dividing by any one of the terms, say, ф:

These ideas are formally embodied in the Buckingham pi theorem, stated below without derivation. (See Reference 3, pages 21-28, for such a derivation.)

Buckingham pi theorem. Let К equal the number of fundamental dimensions required to describe the physical variables. (In mechanics, all physical variables can be expressed in terms of the dimensions of mass, length, and time’, hence, К = 3.)

Let P, P2, ■ ■ ■, PN represent N physical variables in the physical relation

Подпись: [1.34]fi(Pi, P2,…,PN)= 0

Then, the physical relation Equation (1.24) may be reexpressed as a relation of (N — K) dimensionless products (called П products),

Подпись: [1.35]/2(П,,П2……….. Пм) = 0

where each П product is a dimensionless product of a set of К physical variables plus one other physical variable. Let P, P2, …, Pk be the selected set of К physical variables. Then

Подпись: [1.36]n, = h(Pi, P2,…,PK, PK+,) П2 = f4(Pi, P2,…, PK, PK+2)

П/1-r — fs{P, P2, ■ ■ ■, Pk, Pn)

The choice of the repeating variables, P, P2, ■ ■ ■, Pk should be such that they include all the К dimensions used in the problem. Also, the dependent variable [such as R in Equation (1.23)] should appear in only one of the П products.

Returning to our consideration of the aerodynamic force on a given body at a given angle of attack, Equation (1.23) can be written in the form of Equation (1.24):

§(R> Росі koo> C, oo, ttoo) — 0 [1 *27]

Following the Buckingham pi theorem, the fundamental dimensions are

m = dimensions of mass l = dimension of length t = dimension of time

Hence, К — 3. The physical variables and their dimensions are

[R] = mlt~2 iPod = ml~3

[c] = l

[доо] = ml~xt~l

[floo] = If1

Hence, N = 6. In the above, the dimensions of the force R are obtained from Newton’s second law, force = mass x acceleration; hence, [RJ = mlt~2. The dimensions of p. x are obtained from its definition, e. g., i =■ т/(ди/ду), and from Newton’s second law. (Show for yourself that [доо] = ml~lt~x.) Choose px, and c as the arbitrarily selected sets of К physical variables. Then Equation (1.27) can be reexpressed in terms of N — К = 6 — 3 = 3 dimensionless П products in the form of Equation (1.25);

/2(ПьП2,Пз) = 0 [1.28]

From Equation (1.26), these П products are

Пі =f3(p0O, V00,c, R) [1.29a]

П2 — /4(Roo 1 кос5 f, Доо) [ 1.298]

П3 = /5(Poo, ^ocChOoc) [1.29c]

For the time being, concentrate on Пь from Equation (1.29a). Assume that

Пі =piV^ceR [1.30]

where d, b, and e are exponents to be found. In dimensional terms, Equation (1.30) is

Because П, is dimensionless, the right side of Equation (1.31) must also be dimen­sionless. This means that the exponents of m must add to zero, and similarly for the exponents of / and t. Hence,

Form: d. + 1=0

For/: -3d + b + e + 1 = 0

For t: – b – 2 = 0

Solving the above equations, we find that d — – l, b — -2, ande = -2. Substituting these values into Equation (1.30), we have

Пі = fip^V"2^2 [1.32]

_ R ~ PooVfr*

Подпись: Пі Подпись: R Подпись: [1.33]

The quantity R/px F^c2 is a dimensionless parameter in which c2 has the dimensions of an area. We can replace c2 with any reference area we wish (such as the planform area of a wing S), and Пі will still be dimensionless. Moreover, we can multiply Пі by a pure number, and it will still be dimensionless. Thus, from Equation (1.32), П! can be redefined as

Hence, Пі is a force coefficient CR, as defined in Section 1.5. In Equation (1.33), S is a reference area germane to the given body shape.

The remaining П products can be found as follows. From Equation (1.29b), assume

n2 = /OooV£c’V [1.34]

Paralleling the above analysis, we obtain

[П21 = (тГ3)(Іг’У’(іУ(тГ’г1У


Form: 1 + j — 0

For/: —3 + h + i— j = 0

For t: – h – j — 0

Dimensional Analysis: The Buckingham Pi Theorem Подпись: [1.35]

Thus, j = — 1, h = 1, and і = 1. Substitution into Equation (1.34) gives

The dimensionless combination in Equation (1.35) is defined as the freestream Rey­nolds number Re = Poo Vooc/P-ao – The Reynolds number is physically a measure of the ratio of inertia forces to viscous forces in a flow and is one of the most powerful parameters in fluid dynamics. Its importance is emphasized in Chapters 15 to 20.

Returning to Equation (1.29c), assume

Подпись: [1.36]n3 = Voo p^c’a^

[П3] = (ІГ1)(тГ3)к(.іУ(ІГ1У

For m: к — О

For 1: 1— 3fc + r + s = 0

For P. — 1 — s = 0

Hence, к = 0, s = — 1, and r = 0. Substituting Equation (1.36), we have

Пе = — [1.37]


The dimensionless combination in Equation (1.37) is defined as the freestream Mach number M = Vqo/Ooo – The Mach number is the ratio of the flow velocity to the speed of sound; it is a powerful parameter in the study of gas dynamics. Its importance is emphasized in subsequent chapters.

Подпись: /2 Подпись: R PooVooC VQO aoo Подпись: = 0

The results of our dimensionless analysis may be organized as follows. Inserting Equations (1.33), (1.35), and (1.37) into (1.28), we have

Подпись: or

Подпись: [1.38] Dimensional Analysis: The Buckingham Pi Theorem

fi(CR, Re, Moo) = 0

This is an important result! Compare Equations (1.23) and (1.38). In Equation (1.23), R is expressed as a general function of five independent variables. However, our dimensional analysis has shown that:

1. R can be expressed in terms of a dimensionless force coefficient,

cR = R/Poovls.

2. CR is a function of only Re and M„, from Equation (1.38).

Therefore, by using the Buckingham pi theorem, we have reduced the number of independent variables from five in Equation (1.23) to two in Equation (1.38). Now, if we wish to run a series of wind-tunnel tests for a given body at a given angle of attack, we need only to vary the Reynolds and Mach numbers in order to obtain data for the direct formulation of R through Equation (1.38). With a small amount of analysis, we have saved a huge amount of effort and wind-tunnel time. More importantly, we have defined two dimensionless parameters, Re and М^, which govern the flow. They are called similarity parameters, for reasons to be discussed in the following section. Other similarity parameters are introduced as our aerodynamic discussions progress.

Since the lift and drag are components of the resultant force, corollaries to Equa­tion (1.38) are

Cl = /7 (Re, Mx) [1.39]

CD = MRe, Moo) [1.40]

Moreover, a relation similar to Equation (1.23) holds for the aerodynamic moments, and dimensional analysis yields

CM = /9(Re, Mx) [1.41]

Подпись: CL = /10(Re, Moo, a) CD = /11 (Re, Moo, a) CM = /i2(Re, Moo, a) Подпись: [1.42] [1.43] [1.44]

Keep in mind that the above analysis was for a given body shape at a given angle of attack a. If a is allowed to vary, then CL, CD, and CM will in general depend on the value of a. Hence, Equations (1.39) to (1.41) can be generalized to

Equations (1.42) to (1.44) assume a given body shape. Much of theoretical and experimental aerodynamics is focused on obtaining explicit expressions for Equations (1.42) to (1.44) for specific body shapes. This is one of the practical applications of aerodynamics mentioned in Section 1.2, and it is one of the major thrusts of this book.

For mechanical problems that also involve thermodynamics and heat transfer, the temperature, specific heat, and thermal conductivity of the fluid, as well as the tem­perature of the body surface (wall temperature), must be added to the list of physical variables, and the unit of temperature (say, kelvin or degree Rankine) must be added to the list of fundamental dimensions. For such cases, dimensional analysis yields additional dimensionless products such as heat transfer coefficients, and additional similarity parameters such as the ratio of specific heat at constant pressure to that at constant volume cp/cv, the ratio of wall temperature to freestream temperature Tw/Too, and the Prandtl number Pr = k^_. where кж is the thermal conduc­

tivity of the freestream.[3] Thermodynamics is essential to the study of compressible flow (Chapters 7 to 14), and heat transfer is part of the study of viscous flow (Chapters 15 to 20). Hence, these additional similarity parameters will be emphasized when they appear logically in our subsequent discussions. For the time being, however, the Mach and Reynolds numbers will suffice as the dominant similarity parameters for our present considerations.

Angular Velocity, Vorticity, and Strain

In several of our previous discussions, we made use of the concept of a fluid element moving through the flow field. In this section, we examine this motion more closely, paying particular attention to the orientation of the element and its change in shape as it moves along a streamline. In the process, we introduce the concept of vorticity, one of the most powerful quantities in theoretical aerodynamics.

Consider an infinitesimal fluid element moving in a flow field. As it translates along a streamline, it may also rotate, and in addition its shape may become distorted as sketched in Figure 2.30. The amount of rotation and distortion depends on the velocity field; the purpose of this section is to quantify this dependency.


Element at time 11

Figure 2.30 The motion of a fluid element along a


streamline is a combination of translation and rotation; in addition, the shape of the element can become distorted.

Consider a two-dimensional flow in the xy plane. Also, consider an infinitesimal fluid element in this flow. Assume that at time t the shape of this fluid element is rectangular, as shown at the left of Figure 2.31. Assume that the fluid element is moving upward and to the right; its position and shape at time t + At are shown at the right in Figure 2.31. Note that during the time increment At, the sides AB and AC have rotated through the angular displacements — Дві and Аві, respectively. (Counterclockwise rotations by convention are considered positive; since line A В is shown with a clockwise rotation in Figure 2.31, the angular displacement is negative, — Дві.) At present, consider just the line AC. It has rotated because during the time increment At, point C has moved differently from point A. Consider the velocity in the у direction. At point A at time t, this velocity is v, as shown in Figure 2.31. Point C is a distance dx from point A; hence, at time t the vertical component of velocity of point C is given by v + (dv/dx) dx. Hence,

Distance in у direction that A moves

= vAt

during time increment At



Distance in у direction that C moves

This net displacement is shown at the right of Figure 2.31. From the geometry of Figure 2.31,



Figure 2.31 Rotation and distortion of a fluid element.


Angular Velocity, Vorticity, and Strain Подпись: [2.120]

Since A92 is a small angle, tan A92 ~ A92. Hence, Equation (2.119) reduces to

Now consider line AB. The x component of the velocity at point A at time t is u, as shown in Figure 2.31. Because point В is a distance dy from point A, the horizontal component of velocity of point В at time t is и + (du/dy)dy. By reasoning similar to that above, the net displacement in the x direction of В relative to A over the time increment At is [(du/dy) dy] At, as shown in Figure 2.31. Hence,

Angular Velocity, Vorticity, and Strain




A9 =——- At





Consider the angular velocities of lines AB and AC, defined as d9/dt and d92/dt, respectively. From Equation (2.122), we have


d9 j. A9 du dt At—>o At dy

From Equation (2.120), we have

d92 A92 dv

dt Ar->o At dx






Подпись: (X>z Подпись: 1 /de1 2 l ~dt Подпись: d92 dt Подпись: [2.125]

By definition, the angular velocity of the fluid element as seen in the xy plane is the average of the angular velocities of lines AB and AC. Let a>z denote this angular velocity. Therefore, by definition,

Combining Equations (2.123) to (2.125) yields

1 Подпись:/Эи Эи

z 2 Эх ду

Подпись: /9 w dv. /Эи dw. / dv Эи^  Эу dz) ”*9z 9JC/^”*"9JC Эу / Подпись: [2.127]

In the above discussion, we have considered motion in the xy plane only. How­ever, the fluid element is generally moving in three-dimensional space, and its an­gular velocity is a vector ш that is oriented in some general direction, as shown in Figure 2.32. In Equation (2.126), we have obtained only the component of ш in the z direction; this explains the subscript z in Equations (2.125) and (2.126). The x and у components of <о can be obtained in a similar fashion. The resulting angular velocity of the fluid element in three-dimensional space is

Equation (2.127) is the desired result; it expresses the angular velocity of the fluid element in terms of the velocity field, or more precisely, in terms of derivatives of the velocity field.

The angular velocity of a fluid element plays an important role in theoretical aerodynamics, as we shall soon see. However, the expression 2со appears frequently, and therefore we define a new quantity, vorticity, which is simply twice the angular velocity. Denote vorticity by the vector |:

Подпись: Figure 2.32 Angular velocity of a fluid element in three-dimensional space.

£ = 2<u

Подпись: / d w dv V /Эм V ду dz/I+dz Подпись: dv dx Подпись: [2.128]
image162 image163

Hence, from Equation (2.127),

Recall Equation (2.22) for V x V in cartesian coordinates. Since u, v, and w denote the x, y, and z components of velocity, respectively, note that the right sides of Equations (2.22) and (2.128) are identical. Hence, we have the important result that

Подпись: ? = VxV[2.129]

In a velocity field, the curl of the velocity is equal to the vorticity.

The above leads to two important definitions:

1. If V x V ф 0 at every point in a flow, the flow is called rotational. This implies that the fluid elements have a finite angular velocity.

2. If V x V = 0 at every point in a flow, the flow is called irrotational. This implies that the fluid elements have no angular velocity; rather, their motion through space is a pure translation.

The case of rotational flow is illustrated in Figure 2.33. Here, fluid elements moving along two different streamlines are shown in various modes of rotation. In contrast, the case of irrotational flow is illustrated in Figure 2.34. Here, the upper streamline shows a fluid element where the angular velocities of its sides are zero. The lower streamline shows a fluid element where the angular velocities of two intersecting sides are finite but equal and opposite to each other, and so their sum is identically zero. In both cases, the angular velocity of the fluid element is zero; i. e., the flow is irrotational.

If the flow is two-dimensional (say, in the xy plane), then from Equation (2.128),

Подпись: dv dx image164[2.130]

Angular Velocity, Vorticity, and Strain


Figure 2.33 Fluid elements in a rotational flow.



Figure 2.34 Fluid elements in an irrotational flow.


Also, if the flow is irrotational, £ = 0. Hence, from Equation (2.130),

Подпись: [2.131]dv 3 и dx dy

Equation (2.131) is the condition of irrotationality for two-dimensional flow. We will have frequent occasion to use Equation (2.131).

Why is it so important to make a distinction between rotational and irrotational flows? The answer becomes blatantly obvious as we progress in our study of aero­dynamics; we find that irrotational flows are much easier to analyze than rotational flows. However, irrotational flow may at first glance appear to be so special that its applications are limited. Amazingly enough, such is not the case. There are a large number of practical aerodynamic problems where the flow field is essentially irrotational, e. g., the subsonic flow over airfoils, the supersonic flow over slender bodies at small angle of attack, and the subsonic-supersonic flow through nozzles. For such cases, there is generally a thin boundary layer of viscous flow immediately adjacent to the surface; in this viscous region the flow is highly rotational. However, outside this boundary layer, the flow is frequently irrotational. As a result, the study of irrotational flow is an important aspect of aerodynamics.

Return to the fluid element shown in Figure 2.31. Let the angle between sides A В and AC be denoted by к. As the fluid element moves through the flow field, к will change. In Figure 2.31, at time t, к is initially 90°. At time t + At, к has changed by the amount А к, where

Подпись: [2.132]А к = – Д02 – (-Д0О

By definition, the strain of the fluid element as seen in the xy plane is the change in к, where positive strain corresponds to a decreasing к. Hence, from Equation (2.132),

Strain = —Лк = Л02 — A#i [2.133]

In viscous flows (to be discussed in Chapters 15 to 20), the time rate of strain is an important quantity. Denote the time rate of strain by єху, where in conjunction with

" 3 и















_ dx



The sum of the diagonal terms is simply equal to V • V, which from Section 2.3 is equal to the time rate of change of volume of a fluid element; hence, the diagonal terms represent the dilatation of a fluid element. The off-diagonal terms are cross derivatives which appear in Equations (2.127), (2.128), and (2.135a to c). Hence, the off-diagonal terms are associated with rotation and strain of a fluid element.

In summary, in this section, we have examined the rotation and deformation of a fluid element moving in a flow field. The angular velocity of a fluid element and the corresponding vorticity at a point in the flow are concepts which are useful in the analysis of both inviscid and viscous flows; in particular, the absence of vorticity— irrotational flow—greatly simplifies the analysis of the flow, as we will see. We take advantage of this simplification in much of our treatment of inviscid flows in subsequent chapters. On the other hand, we do not make use of the time rate of strain until we discuss viscous flow, beginning with Chapter 15.

Example 2.4


For the velocity field given in Example 2.3, calculate the vorticity. Solution


Angular Velocity, Vorticity, and Strain

(x2 + y2) – y(2y)
X2 + y2


= 0i + Oj + Ok = 0


The flow field is irrotational at every point except at the origin, where x2 + y2 = 0.


Consider the boundary-layer velocity profile used in Example 2.2, namely, u/V, x = (y/S)025. Is this flow rotational or irrotational?


For a two-dimensional flow, the irrotationality condition is given by Equation (2.131), namely


Example 2.5


dv du Эх dy


Does this relation hold for the viscous boundary-layer flow in Example 2.2? Let us examine this question. From the boundary-layer velocity profile given by


и /у 0-25

= U/


Angular Velocity, Vorticity, and Strain

[E. l]


we obtain


In Example 2.5, we demonstrated a basic result which holds in general for viscous flows, namely, viscous flows are rotational. This is almost intuitive. For example, consider an infinitesimally small fluid element moving along a streamline, as sketched in Figure 2.35. If this is a viscous flow, and assuming that the velocity increases in the upward direction (i. e., the velocity is higher on the neighboring streamline above and lower on the neighboring streamline below), then the shear stresses on the upper and lower faces of the fluid element will be in the directions shown. Such shear stresses will be discussed at length in Chapter 15. Examining Figure 2.35, we see clearly that the shear stresses exert a rotational moment about the center of the element, thus providing a mechanism for setting the fluid element into rotation. Although this picture is overly simplistic, it serves to emphasize that viscous flows are rotational flows. On the other hand, as stated earlier in this section, there are numerous

Angular Velocity, Vorticity, and Strain


Figure 2.35 Shear stress and the consequent rotation of a fluid element.


inviscid flow problems that are irrotational, with the attendant simplifications to be explained later. Some inviscid flows are rotational, but there exists such a large number of practical aerodynamic problems described by inviscid, irrotational flows that the distinction between rotational and irrotational flow is an important consideration.

Airfoil Characteristics

Before discussing the theoretical calculation of airfoil properties, let us examine some typical results. During the 1930s and 1940s, the NACA carried out numerous measurements of the lift, drag, and moment coefficients on the standard NACA air­foils. These experiments were performed at low speeds in a wind tunnel where the constant-chord wing spanned the entire test section from one sidewall to the other. In this fashion, the flow “sees” a wing without wing tips—a so-called infinite wing,

which theoretically stretches to infinity along the span (in the у direction in Fig­ure 4.1). Because the airfoil section is the same at any spanwise location along the infinite wing, the properties of the airfoil and the infinite wing are identical. Hence, airfoil data are frequently called infinite wing data. (In contrast, we see in Chapter 5 that the properties of a finite wing are somewhat different from its airfoil properties.)

The typical variation of lift coefficient with angle of attack for an airfoil is sketched in Figure 4.4. At low-to-moderate angles of attack, с/ varies linearly with a; the slope of this straight line is denoted by ao and is called the lift slope. In this region, the flow moves smoothly over the airfoil and is attached over most of the surface, as shown in the streamline picture at the left of Figure 4.4. However, as a becomes large, the flow tends to separate from the top surface of the airfoil, creating a large wake of relatively “dead air” behind the airfoil as shown at the right of Figure 4.4. Inside this separated region, the flow is recirculating, and part of the flow is actually moving in a direction opposite to the freestream—so-called reversed flow. (Refer also to Figure 1.36.) This separated flow is due to viscous effects and is discussed in Chapter 15. The consequence of this separated flow at high a is a precipitous decrease in lift and a large increase in drag; under such conditions the airfoil is said to be stalled. The maximum value of С/, which occurs just prior to the stall, is denoted by ci, max; it is one of the most important aspects of airfoil performance, because it determines the stalling speed of an airplane. The higher is c/>max, the lower is the stalling speed. A great deal of modem airfoil research has been directed toward in­creasing c/,max. Again examining Figure 4.4, we see that c; increases linearly with а until flow separation begins to have an effect. Then the curve becomes nonlinear, q reaches a maximum value, and finally the airfoil stalls. At the other extreme of the curve, noting Figure 4.4, the lift at a = 0 is finite; indeed, the lift goes to zero only when the airfoil is pitched to some negative angle of attack. The value of a when lift equals zero is called the zero-lift angle of attack and is denoted by chl=o- For a

Stall due to flow separation


Figure 4.4 Schematic of lift-coefficient variation with angle of attack for an airfoil.


symmetric airfoil, aL=0 — 0, whereas for all airfoils with positive camber (camber above the chord line), aL=0 is a negative value, usually on the order of —2 or —3°.

The inviscid flow airfoil theory discussed in this chapter allows us to predict the lift slope ao and aL=0 for a given airfoil. It does not allow us to calculate c/.max, which is a difficult viscous flow problem, to be discussed in Chapters 15 to 20.

Experimental results for lift and moment coefficients for the NACA 2412 airfoil are given in Figure 4.5. Here, the moment coefficient is taken about the quarter-chord point. Recall from Section 1.6 that the force-and-moment system on an airfoil can be transferred to any convenient point; however, the quarter-chord point is commonly used. (Refresh your mind on this concept by reviewing Section 1.6, especially Fig­ure 1.19.) Also shown in Figure 4.5 are theoretical results to be discussed later. Note that the experimental data are given for two different Reynolds numbers. The lift slope ao is not influenced by Re; however, c/,max is dependent upon Re. This makes sense, because c/,max is governed by viscous effects, and Re is a similarity parameter that governs the strength of inertia forces relative to viscous forces in the flow. [See


Figure 4.5 Experimental data for lift coefficient and moment

coefficient about the quarter-chord point for an NACA 2412 airfoil. (Source: Data obtained from Abbott and von Doenhoff, Reference 1 1.) Also shown is a comparison with theory described in Section 4.8.

Section 1.7 and Equation (1.35).] The moment coefficient is also insensitive to Re except at large a. The NACA 2412 airfoil is a commonly used airfoil, and the results given in Figure 4.5 are quite typical of airfoil characteristics. For example, note from Figure 4.5 that (Xl=о = —2.1°, c/ max ~ 1 -6, and the stall occurs at a ~ 16°.

This chapter deals with airfoil theory for an inviscid, incompressible flow; such theory is incapable of predicting airfoil drag, as noted earlier. However, for the sake of completeness, experimental data for the drag coefficient cd for the NACA 2412 airfoil are given in Figure 4.6 as a function of the angle of attack.[14] The physical source of this drag coefficient is both skin friction drag and pressure drag due to flow separation (so-called form drag). The sum of these two effects yields the profile drag coefficient cd for the airfoil, which is plotted in Figure 4.6. Note that cd is sensitive to Re, which is to be expected since both skin friction and flow separation are viscous effects. Again, we must wait until Chapters 15 to 20 to obtain some tools for theoretically predicting cd.