Category Fundamentals of Aerodynamics

The Kutta-Joukowski Theorem and the Generation of Lift

Although the result given by Equation (3.140) was derived for a circular cylinder, it applies in general to cylindrical bodies of arbitrary cross section. For example,

consider the incompressible flow over an airfoil section, as sketched in Figure 3.37. Let curve A be any curve in the flow enclosing the airfoil. If the airfoil is producing lift, the velocity field around the airfoil will be such that the line integral of velocity around A will be finite, that is, the circulation

Подпись:f> Vds

is finite. In turn, the lift per unit span L’ on the airfoil will be given by the Kutta – Joukowski theorem, as embodied in Equation (3.140):

Подпись: L' = P»oVxr[3.140]

This result underscores the importance of the concept of circulation, defined in Section 2.13. The Kutta-Joukowski theorem states that lift per unit span on a two-dimensional body is directly proportional to the circulation around the body. Indeed, the concept of circulation is so important at this stage of our discussion that you should reread Section 2.13 before proceeding further.

The general derivation of Equation (3.140) for bodies of arbitrary cross section can be carried out using the method of complex variables. Such mathematics is beyond the scope of this book. (It can be shown that arbitrary functions of complex variables are general solutions of Laplace’s equation, which in turn governs incompressible potential flow. Hence, more advanced treatments of such flows utilize the mathematics of complex variables as an important tool. See Reference 9 for a particularly lucid treatment of inviscid, incompressible flow at a more advanced level.)

In Section 3.15, the lifting flow over a circular cylinder was synthesized by superimposing a uniform flow, a doublet, and a vortex. Recall that all three elementary flows are irrotational at all points, except for the vortex, which has infinite vorticity at the origin. Therefore, the lifting flow over a cylinder as shown in Figure 3.33 is

image275

Figure 3.37 Circulation around a lifting airfoil.

 

irrotational at every point except at the origin. If we take the circulation around any curve not enclosing the origin, we obtain from Equation (2.137) the result that Г = 0. It is only when we choose a curve that encloses the origin, where V x V is infinite, that Equation (2.137) yields a finite Г, equal to the strength of the vortex. The same can be said about the flow over the airfoil in Figure 3.37. As we show in Chapter 4, the flow outside the airfoil is irrotational, and the circulation around any closed curve not enclosing the airfoil (such as curve В in Figure 3.37) is consequently zero. On the other hand, we also show in Chapter 4 that the flow over an airfoil is synthesized by distributing vortices either on the surface or inside the airfoil. These vortices have the usual singularities in V x V, and therefore, if we choose a curve that encloses the airfoil (such as curve A in Figure 3.37), Equation (2.137) yields a finite value of Г, equal to the sum of the vortex strengths distributed on or inside the airfoil. The important point here is that, in the Kutta-Joukowski theorem, the value of Г used in Equation (3.140) must be evaluated around a closed curve that encloses the body, the curve can be otherwise arbitrary, but it must have the body inside it. r

At this stage, let us pause and assess our thoughts. The approach we have dis­cussed above—the definition of circulation and the use of Equation (3.140) to obtain the lift—is the essence of the circulation theory of lift in aerodynamics. Its devel­opment at the turn of the twentieth century created a breakthrough in aerodynamics. However, let us keep things in perspective. The circulation theory of lift is an alter­native way of thinking about the generation of lift on an aerodynamic body. Keep in mind that the true physical sources of aerodynamic force on a body are the pres­sure and shear stress distributions exerted on the surface of the body, as explained in Section 1.5. The Kutta-Joukowski theorem is simply an alternative way of ex­pressing the consequences of the surface pressure distribution; it is a mathematical expression that is consistent with the special tools we have developed for the analysis of inviscid, incompressible flow. Indeed, recall that Equation (3.140) was derived in Section 3.15 by integrating the pressure distribution over the surface. Therefore, it is not quite proper to say that circulation “causes” lift. Rather, lift is “caused” by the net imbalance of the surface pressure distribution, and circulation is simply a defined quantity determined from the same pressures. The relation between the surface pres­sure distribution (which produces lift L’) and circulation is given by Equation (3.140), However, in the theory of incompressible, potential flow, it is generally much easier to determine the circulation around the body rather than calculate the detailed surface pressure distribution. Therein lies the power of the circulation theory of lift.

Consequently, the theoretical analysis of lift on two-dimensional bodies in in­compressible, inviscid flow focuses on the calculation of the circulation about the body. Once Г is obtained, then the lift per unit span follows directly from the Kutta- Joukowski theorem. As a result, in subsequent sections we constantly address the question; How can we calculate the circulation for a given body in a given incom­pressible, inviscid flow?

Three-Dimensional. Incompressible Flow

Подпись: Paul Cezanne, 1890

Treat nature in terms of the cylinder, the sphere, the cone, all in perspective.

6.1 Introduction

To this point in our aerodynamic discussions, we have been working mainly in a two-dimensional world; the flows over the bodies treated in Chapter 3 and the airfoils in Chapter 4 involved only two dimensions in a single plane—so-called planar flows. In Chapter 5, the analyses of a finite wing were carried out in the plane of the wing, in spite of the fact that the detailed flow over a finite wing is truly three-dimensional. The relative simplicity of dealing with two dimensions, (i. e., having only two independent variables), is self-evident and is the reason why a large bulk of aerodynamic theory deals with two-dimensional flows. Fortunately, the two-dimensional analyses go a long way toward understanding many practical flows, but they also have distinct limitations.

The real world of aerodynamic applications is three-dimensional. However, because of the addition of one more independent variable, the analyses generally become more complex. The accurate calculation of three-dimensional flow fields has been, and still is, one of the most active areas of aerodynamic research.

The purpose of this book is to present the fundamentals of aerodynamics. There­fore, it is important to recognize the predominance of three-dimensional flows, al­though it is beyond our scope to go into detail. Therefore, the purpose of this chapter is to introduce some very basic considerations of three-dimensional incompressible

flow. This chapter is short; we do not even need a road map to guide us through it. Its function is simply to open the door to the analysis of three-dimensional flow.

The governing fluid flow equations have already been developed in three dimen­sions in Chapters 2 and 3. In particular, if the flow is irrotational, Equation (2.154) states that

V = V0 [2.154]

where, if the flow is also incompressible, the velocity potential is given by Laplace’s equation:

V20 = 0 [3.40]

Solutions of Equation (3.40) for flow over a body must satisfy the flow-tangency boundary condition on the body, that is,

V • n = 0 [3.48a]

where n is a unit vector normal to the body surface. In all of the above equations, ф is, in general, a function of three-dimensional space; for example, in spherical coordinates ф = ф(г, в, Ф). Let us use these equations to treat some elementary three-dimensional incompressible flows.

Inviscid Versus Viscous Flow

A major facet of a gas or liquid is the ability of the molecules to move rather freely, as explained in Section 1.2. When the molecules move, even in a very random fashion, they obviously transport their mass, momentum, and energy from one location to another in the fluid. This transport on a molecular scale gives rise to the phenomena

of mass diffusion, viscosity (friction), and thermal conduction. Such “transport phe­nomena” will be discussed in detail in Chapter 15. For our purposes here, we need only to recognize that all real flows exhibit the effects of these transport phenomena; such flows are called viscous flows. In contrast, a flow that is assumed to involve no friction, thermal conduction, or diffusion is called an inviscidflow. Inviscid flows do not truly exist in nature; however, there are many practical aerodynamic flows (more than you would think) where the influence of transport phenomena is small, and we can model the flow as being inviscid. For this reason, more than 70 percent of this book (Chapters 3 to 14) deals with inviscid flows.

Theoretically, inviscid flow is approached in the limit as the Reynolds number goes to infinity (to be proved in Chapter 15). However, for practical problems, many flows with high but finite Re can be assumed to be inviscid. For such flows, the influence of friction, thermal conduction, and diffusion is limited to a very thin region adjacent to the body surface (the boundary layer, to be defined in Chapter 17), and the remainder of the flow outside this thin region is essentially inviscid. This division of the flow into two regions is illustrated in Figure 1.35. Hence, the material discussed in Chapters 3 to 14 applies to the flow outside the boundary layer. For flows over slender bodies, such as the airfoil sketched in Figure 1.35, inviscid theory adequately predicts the pressure distribution and lift on the body and gives a valid representation of the streamlines and flow field away from the body. However, because friction (shear stress) is a major source of aerodynamic drag, inviscid theories by themselves cannot adequately predict total drag.

In contrast, there are some flows that are dominated by viscous effects. For example, if the airfoil in Figure 1.35 is inclined to a high incidence angle to the flow (high angle of attack), then the boundary layer will tend to separate from the top surface, and a large wake is formed downstream. The separated flow is sketched at the top of Figure 1.36; it is characteristic of the flow field over a “stalled” airfoil. Separated flow also dominates the aerodynamics of blunt bodies, such as the cylinder at the bottom of Figure 1.36. Here, the flow expands around the front face of the cylinder, but separates from the surface on the rear face, forming a rather fat wake downstream. The types of flow illustrated in Figure 1.36 are dominated by viscous

Flow outside the boundary

layer is inviscid Thin boundary layer of

image63

Figure 1 .35 The division of a flow into two regions: (1) the thin viscous boundary layer adjacent to the body surface and (2) the inviscid flow outside the boundary layer.

 

image64

flow.

 

effects; no inviscid theory can independently predict the aerodynamics of such flows. They require the inclusion of viscous effects, to be presented in Part 4.

Relationship Between the Stream Function and Velocity Potential

In Section 2.15, we demonstrated that for an irrotational flow Y = Уф. At this stage, take a moment and review some of the nomenclature introduced in Section 2.2.5 for the gradient of a scalar field. We see that a line of constant ф is an isoline of ф; since ф is the velocity potential, we give this isoline a specific name, equipotential line. In addition, a line drawn in space such that Уф is tangent at every point is defined as a gradient line; however, since Уф = V, this gradient line is a streamline. In turn, from Section 2.14, a streamline is a line of constant Ф (for a two-dimensional flow). Because gradient lines and isolines are perpendicular (see Section 2.2.5, Gradient of a Scalar Field), then equipotential lines (ф = constant) and streamlines (ф = constant) are mutually perpendicular.

To illustrate this result more clearly, consider a two-dimensional, irrotational, incompressible flow in cartesian coordinates. For a streamline, ф(х, у) = constant. Hence, the differential of ф along the streamline is zero; i. e.,

image181[2.159]

From Equation (2.150a and b), Equation (2.159) can be written as

Подпись:йф = —v dx + и dy = 0

2 ф (or тД) can be defined for axisymmetric flows, such as the flow over a cone at zero degrees angle of attack. However, for such flows, only two spatial coordinates are needed to describe the flow field (see Chapter 6).

Relationship Between the Stream Function and Velocity Potential Relationship Between the Stream Function and Velocity Potential

Solve Equation (2.160) for dy/dx, which is the slope of the ф = constant line, i. e., the slope of the streamline:

Подпись: [2.163]dф = и dx + v dy = 0

Relationship Between the Stream Function and Velocity Potential Relationship Between the Stream Function and Velocity Potential

Solving Equation (2.163) for dy /dx, which is the slope of the ф — constant line, i. e., the slope of the equipotential line, we obtain

Equation (2.165) shows that the slope of а ф = constant line is the negative reciprocal of the slope of а ф = constant line, i. e., streamlines and equipotential lines are mutually perpendicular.

Kelvin’s Circulation Theorem and the Starting Vortex

In this section, we put the finishing touch to the overall philosophy of airfoil theory before developing the quantitative aspects of the theory itself in subsequent sections. This section also ties up a loose end introduced by the Kutta condition described in the previous section. Specifically, the Kutta condition states that the circulation around an airfoil is just the right value to ensure that the flow smoothly leaves the trailing edge. Question: How does nature generate this circulation? Does it come from nowhere, or is circulation somehow conserved over the whole flow field? Let us examine these matters more closely.

Consider an arbitrary inviscid, incompressible flow as sketched in Figure 4.15. Assume that all body forces f are zero. Choose an arbitrary curve C, and identify the fluid elements that are on this curve at a given instant in time t{. Also, by definition the circulation around curve Cj is Ті = — fc V • ds. Now let these specific fluid elements move downstream. At some later time, t2, these same fluid elements will form another curve C2, around which the circulation is Г2 = – fr У • ds. For the conditions stated above, we can readily show that Tt = Г2. In fact, since we are following a set of specific fluid elements, we can state that circulation around a closed curve formed by a set of contiguous fluid elements remains constant as the fluid elements move throughout the flow. Recall from Section 2.9 that the substantial derivative gives the time rate of change following a given fluid element. Hence, a mathematical statement of the above discussion is simply

[4.11]

which says that the time rate of change of circulation around a closed curve consisting of the same fluid elements is zero. Equation (4.11) along with its supporting discussion

image319

Figure 4.15 Kelvin’s theorem.

is called Kelvin’s circulation theorem.[15] Its derivation from first principles is left as Problem 4.3. Also, recall our definition and discussion of a vortex sheet in Section 4.4. An interesting consequence of Kelvin’s circulation theorem is proof that a stream surface which is a vortex sheet at some instant in time remains a vortex sheet for all times.

Kelvin’s theorem helps to explain the generation of circulation around an airfoil, as follows. Consider an airfoil in a fluid at rest, as shown in Figure 4.16a. Because V = 0 everywhere, the circulation around curve C is zero. Now start the flow in motion over the airfoil. Initially, the flow will tend to curl around the trailing edge, as explained in Section 4.5 and illustrated at the left of Figure 4.12. In so doing, the velocity at the trailing edge theoretically becomes infinite. In real life, the velocity tends toward a very large finite number. Consequently, during the very first moments after the flow is started, a thin region of very large velocity gradients (and therefore high vorticity) is formed at the trailing edge. This high-vorticity region is fixed to the same fluid elements, and consequently it is flushed downstream as the fluid elements begin to move downstream from the trailing edge. As it moves downstream, this thin sheet of intense vorticity is unstable, and it tends to roll up and form a picture similar to a point vortex. This vortex is called the starting vortex and is sketched in Figure 4.16b. After the flow around the airfoil has come to a steady state where the flow leaves the trailing edge smoothly (the Kutta condition), the high velocity gradients at the trailing edge disappear and vorticity is no longer produced at that point. However, the starting vortex has already been formed during the starting process, and it moves steadily downstream with the flow forever after. Figure 4.16b

image320

(a) Fluid at rest relative to the airfoil

 

Cj

image321

(b) Picture some moments after the start of the flow

Figure 4.1 6 The creation of the starting vortex and the

resulting generation of circulation around the airfoil.

 

shows the flow field sometime after steady flow has been achieved over the airfoil, with the starting vortex somewhere downstream. The fluid elements that initially made up curve C i in Figure 4.16a have moved downstream and now make up curve C2, which is the complete circuit abcda shown in Figure 4.16b. Thus, from Kelvin’s theorem, the circulation Г2 around curve C2 (which encloses both the airfoil and the starting vortex) is the same as that around curve C1, namely, zero. Г2 = F| = 0. Now let us subdivide C2 into two loops by making the cut bd, thus forming curves C3 (circuit bcdb) and C4 (circuit abda). Curve C3 encloses the starting vortex, and curve C4 encloses the airfoil. The circulation Г3 around curve C3 is due to the starting vortex; by inspecting Figure 4.16b, we see that Г3 is in the counterclockwise direction (i. e., a negative value). The circulation around curve C4 enclosing the airfoil is Г4. Since the cut bd is common to both C3 and C4, the sum of the circulations around C3 and C4 is simply equal to the circulation around C2:

r3 + Г4 = Г2

However, we have already established that Г2 = 0. Hence,

Г4 = – r3

that is, the circulation around the airfoil is equal and opposite to the circulation around the starting vortex.

This brings us to the summary as well as the crux of this section. As the flow over an airfoil is started, the large velocity gradients at the sharp trailing edge result in the

formation of a region of intense vorticity which rolls up downstream of the trailing edge, forming the starting vortex. This starting vortex has associated with it a coun­terclockwise circulation. Therefore, as an equal-and-opposite reaction, a clockwise circulation around the airfoil is generated. As the starting process continues, vorticity from the trailing edge is constantly fed into the starting vortex, making it stronger with a consequent larger counterclockwise circulation. In turn, the clockwise circu­lation around the airfoil becomes stronger, making the flow at the trailing edge more closely approach the Kutta condition, thus weakening the vorticity shed from the trailing edge. Finally, the starting vortex builds up to just the right strength such that the equal-and-opposite clockwise circulation around the airfoil leads to smooth flow from the trailing edge (the Kutta condition is exactly satisfied). When this happens, the vorticity shed from the trailing edge becomes zero, the starting vortex no longer grows in strength, and a steady circulation exists around the airfoil.

Isentropic Relations

We have defined an isentropic process as one which is both adiabatic and reversible. Consider Equation (7.14). For an adiabatic process, Sq = 0. Also, for a reversible process, 6/.virrev = 0. Thus, for an adiabatic, reversible process, Equation (7.14) yields ds = 0, or entropy is constant; hence, the word “isentropic.” For such an isentropic

Equation (7.32) is very important; it relates pressure, density, and temperature for an isentropic process. We use this equation frequently, so make certain to brand it on

your mind. Also, keep in mind the source of Equation (7.32); it stems from the first law and the definition of entropy. Therefore, Equation (7.32) is basically an energy relation for an isentropic process.

Why is Equation (7.32) so important? Why is it frequently used? Why are we so interested in an isentropic process when it seems so restrictive—requiring both adiabatic and reversible conditions? The answers rest on the fact that a large number of practical compressible flow problems can be assumed to be isentropic—contrary to what you might initially think. For example, consider the flow over an airfoil or through a rocket engine. In the regions adjacent to the airfoil surface and the rocket nozzle walls, a boundary layer is formed wherein the dissipative mechanisms of viscosity, thermal conduction, and diffusion are strong. Hence, the entropy increases within these boundary layers. However, consider the fluid elements moving outside the boundary layer. Here, the dissipative effects of viscosity, etc., are very small and can be neglected. Moreover, no heat is being transferred to or from the fluid element (i. e., we are not heating the fluid element with a Bunsen burner or cooling it in a refrigerator); thus, the flow outside the boundary layer is adiabatic. Consequently, the fluid elements outside the boundary layer are experiencing an adiabatic reversible process—namely, isentropic flow. In the vast majority of practical applications, the viscous boundary layer adjacent to the surface is thin compared with the entire flow field, and hence large regions of the flow can be assumed isentropic. This is why a study of isentropic flow is directly applicable to many types of practical compressible flow problems. In turn, Equation (7.32) is a powerful relation for such flows, valid for a calorically perfect gas.

This ends our brief review of thermodynamics. Its purpose has been to give a quick summary of ideas and equations which will be employed throughout our subsequent discussions of compressible flow. For a more thorough discussion of the power and beauty of thermodynamics, see any good thermodynamics text, such as References 22 to 24.

Consider a Boeing 747 flying at a standard altitude of 36,000 ft. The pressure at a point on the | Example 7.1

wing is 400 lb/ft2. Assuming isentropic flow over the wing, calculate the temperature at this

point.

Solution

Подпись: or
image495,image496

At a standard altitude of 36,000 ft, px = 476 lb/ft2 and Тх = 391 °R. From Equation (7.32),

Surface Integrals

Consider an open surface S bounded by the closed curve C, as shown in Figure 2.9. At point P on the surface, let dS be an elemental area of the surface and n be a unit vector normal to the surface. The orientation of n is in the direction according to the right-hand rule for movement along C. (Curl the fingers of your right hand in the direction of movement around C; your thumb will then point in the general direction of n.) Define a vector elemental area as dS = ndS. In terms of dS, the surface integral over the surface S can be defined in three ways:

її p dS = surface integral of a scalar p over the 5 open surface S (the result is a vector)

If A • dS = surface integral of a vector A over the 5 open surface S (the result is a scalar)

N A x dS = surface integral of a vector A over the 5 open surface S (the result is a vector)

Surface Integrals image97 Подпись: s

If the surface S is closed (e. g., the surface of a sphere or a cube), n points out of the surface, away from the enclosed volume, as shown in Figure 2.10. The surface integrals over the closed surface are

Governing Equation for Irrotational, Incompressible Flow: Laplace’s Equation

We have seen in Section 3.6 that the principle of mass conservation for an incom­pressible flow can take the form of Equation (3.39):

V • V = 0 [3.39]

In addition, for an irrotational flow we have seen in Section 2.15 that a velocity potential ф can be defined such that [from Equation (2.154)]

[2.154]

Therefore, for a flow that is both incompressible and irrotational, Equations (3.39) and (2.154) can be combined to yield

Подпись: or Подпись: V20 = 0 Подпись: [3.40]

V • (V</>) = 0

Equation (3.40) is Laplace’s equation—one of the most famous and extensively stud­ied equations in mathematical physics. Solutions of Laplace’s equation are called harmonic functions, for which there is a huge bulk of existing literature. Therefore, it is most fortuitous that incompressible, irrotational flow is described by Laplace’s equation, for which numerous solutions exist and are well understood.

For convenience, Laplace’s equation is written below in terms of the three com­mon orthogonal coordinate systems employed in Section 2.2:

From Equations (3.40) and (3.46), we make the following obvious and important conclusions:

1. Any irrotational, incompressible flow has a velocity potential and stream function (for two-dimensional flow) that both satisfy Laplace’s equation.

2. Conversely, any solution of Laplace’s equation represents the velocity potential or stream function (two-dimensional) for an irrotational, incompressible flow.

Note that Laplace’s equation is a second-order linear partial differential equation. The fact that it is linear is particularly important, because the sum of any particular solutions of a linear differential equation is also a solution of the equation. For example, if ф], <p2, 03, …, фп represent n separate solutions of Equation (3.40), then the sum

Ф — Ф + Фі + • • • + Фп

is also a solution of Equation (3.40). Since irrotational, incompressible flow is gov­erned by Laplace’s equation and Laplace’s equation is linear, we conclude that a complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows that are also irrotational and in­compressible. Indeed, this establishes the grand strategy for the remainder of our discussions on inviscid, incompressible flow. We develop flow-field solutions for several different elementary flows, which by themselves may not seem to be practical flows in real life. However, we then proceed to add (i. e., superimpose) these elemen­tary flows in different ways such that the resulting flow fields do pertain to practical problems.

Before proceeding further, consider the irrotational, incompressible flow fields over different aerodynamic shapes, such as a sphere, cone, or airplane wing. Clearly, each flow is going to be distinctly different; the streamlines and pressure distribution over a sphere are quite different from those over a cone. However, these different flows are all governed by the same equation, namely, V2</> = 0. How, then, do we obtain different flows for the different bodies? The answer is found in the boundary conditions. Although the governing equation for the different flows is the same, the boundary conditions for the equation must conform to the different geometric shapes, and hence yield different flow-field solutions. Boundary conditions are therefore of vital concern in aerodynamic analysis. Let us examine the nature of boundary conditions further.

Consider the external aerodynamic flow over a stationary body, such as the airfoil sketched in Figure 3.18. The flow is bounded by (1) the freestream flow that occurs (theoretically) an infinite distance away from the body and (2) the surface of the body itself. Therefore, two sets of boundary conditions apply as follows.

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems

To establish a rational aerodynamic theory for a finite wing, we need to introduce a few additional aerodynamic tools. To begin with, we expand the concept of a vortex filament first introduced in Section 4.4. In Section 4.4, we discussed a straight vortex filament extending to ±oo. (Review the first paragraph of Section 4.4 before proceeding further.)

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems Подпись: [5.5]

In general, a vortex filament can be curved, as shown in Figure 5.6. Here, only a portion of the filament is illustrated. The filament induces a flow field in the surrounding space. If the circulation is taken about any path enclosing the filament, a constant value Г is obtained. Hence, the strength of the vortex filament is defined as Г. Consider a directed segment of the filament dl, as shown in Figure 5.6. The radius vector from dl to an arbitrary point P in space is r. The segment dl induces a velocity at P equal to

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems Подпись: [5.6]

Equation (5.5) is called the Biot-Savart law and is one of the most fundamental re­lations in the theory of inviscid, incompressible flow. Its derivation is given in more advanced books (see, e. g., Reference 9). Here, we must accept it without proof. How­ever, you might feel more comfortable if we draw an analogy with electromagnetic theory. If the vortex filament in Figure 5.6 were instead visualized as a wire carrying an electrical current /, then the magnetic field strength dB induced at point P by a segment of the wire dl with the current moving in the direction of dl is

where ц, is the permeability of the medium surrounding the wire. Equation (5.6) is identical in form to Equation (5.5). Indeed, the Biot-Savart law is a general result of potential theory, and potential theory describes electromagnetic fields as well as

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems

image406

Figure 5.6 Vortex filament and illustration of the Biot-Savart law.

 

inviscid, incompressible flows. In fact, our use of the word “induced” in describing velocities generated by the presence of vortices, sources, etc. is a carry-over from the study of electromagnetic fields induced by electrical currents. When developing their finite-wing theory during the period 1911-1918, Prandtl and his colleagues even carried the electrical terminology over to the generation of drag, hence the term “induced” drag.

Return again to our picture of the vortex filament in Figure 5.6. Keep in mind that this single vortex filament and the associated Biot-Savart law [Equation (5.5)] are simply conceptual aerodynamic tools to be used for synthesizing more complex flows of an inviscid, incompressible fluid. They are, for all practical purposes, a solution of the governing equation for inviscid, incompressible flow—Laplace’s equation (see Section 3.7)—and, by themselves, are not of particular value. However, when a number of vortex filaments are used in conjunction with a uniform freestream, it is possible to synthesize a flow which has a practical application. The flow over a finite wing is one such example, as we will soon see.

Let us apply the Biot-Savart law to a straight vortex filament of infinite length, as sketched in Figure 5.7. The strength of the filament is Г. The velocity induced at point P by the directed segment of the vortex filament dl is given by Equation (5.5). Hence, the velocity induced at P by the entire vortex filament is

image407[5.7]

From the definition of the vector cross product (see Section 2.2), the direction of V

Подпись: is downward in Figure 5.7. The magnitude of the velocity, V = |V|, is given by

image408[5.8]

image410

In Figure 5.7, let h be the perpendicular distance from point P to the vortex filament.

by an infinite, straight vortex filament.

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems

Then, from the geometry shown in Figure 5.7,

Подпись: or The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems Подпись: [5.10]

Substituting Equations (5.9a to c) in Equation (5.8), we have

Thus, the velocity induced at a given point P by an infinite, straight vortex filament at a perpendicular distance h from P is simply Г/2лІг, which is precisely the result given by Equation (3.105) for a point vortex in two-dimensional flow. [Note that the minus sign in Equation (3.105) does not appear in Equation (5.10); this is because V in Equation (5.10) is simply the absolute magnitude of V, and hence it is positive by definition.]

Consider the semi-infinite vortex filament shown in Figure 5.8. The filament extends from point A to oo. Point A can be considered a boundary of the flow. Let P be a point in the plane through A perpendicular to the filament. Then, by an integration similar to that above (try it yourself), the velocity induced at P by the semi-infinite vortex filament is

We use Equation (5.11) in the next section.

The great German mathematician, physicist, and physician Hermann von Helm­holtz (1821-1894) was the first to make use of the vortex filament concept in the analysis of inviscid, incompressible flow. In the process, he established several

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems

z

 

A

 

The Vortex Filament, the Biot-Savart Law, and Helmholtz’s Theorems

image413

image414

Figure 5.0 Sketch of the lift distribution along the span of a wing.

basic principles of vortex behavior which have become known as Helmholtz’s vortex theorems:

1. The strength of a vortex filament is constant along its length.

2. A vortex filament cannot end in a fluid; it must extend to the boundaries of the

fluid (which can be ±oo) or form a closed path.

We make use of these theorems in the following sections.

Finally, let us introduce the concept of lift distribution along the span of a finite wing. Consider a given spanwise location yb where the local chord is c, the local geometric angle of attack is a, and the airfoil section is a given shape. The lift per unit span at this location is L'(yj). Now consider another location vy along the span, where c, a, and the airfoil shape may be different. (Most finite wings have a variable chord, with the exception of a simple rectangular wing. Also, many wings are geometrically twisted so that a is different at different spanwise locations—so – called geometric twist. If the tip is at a lower a than the root, the wing is said to have washout; if the tip is at a higher a than the root, the wing has washin. In addition, the wings on a number of modern airplanes have different airfoil sections along the span, with different values of ac=o; Ibis is called aerodynamic twist.) Consequently, the lift per unit span at this different location, L'(y2), will, in general, be different from L'(y). Therefore, there is a distribution of lift per unit span along the wing, that is, L’ = L'{y), as sketched in Figure 5.9. In turn, the circulation is also a function of y, Г (у) = L'(y)/PoqVoo. Note from Figure 5.9 that the lift distribution goes to zero at the tips; that is because there is a pressure equalization from the bottom to the top of the wing precisely at у = —b/2 and b/2, and hence no lift is created at these points. The calculation of the lift distribution L(y) [or the circulation distriution Г (у)] is one of the central problems of finite-wing theory. It is addressed in the following sections.

In summary, we wish to calculate the induced drag, the total lift, and the lift distribution for a finite wing. This is the purpose of the remainder of this chapter.

Preface to the Third Edition

The purpose of this third edition is the same as the first two—to be read, understood, and enjoyed. Due to the extremely favorable comments from readers and users of the first two editions, virtually all of the earlier editions have been carried over intact to the third edition. Therefore, all the basic philosophy, approach, and content discussed and itemized by the author in the Preface to the First Edition is equally applicable now. Since that preface was repeated earlier, no further elaboration will be given here.

Question: What distinguishes the third edition from the first two? Answer: Much new material has been added in order to enhance and expand that covered in the earlier editions. In particular, the third edition has:

1. A series of Design Boxes scattered throughout the book. These design boxes are special sections for the purpose of discussing design aspects associated with the fundamental material covered throughout the book. These sections are literally placed in boxes to set them apart from the mainline text. Modern engineering education is placing more emphasis on design, and the design boxes in this book are in this spirit. They are a means of making the fundamental material more relevant, and making the whole process of learning aerodynamics more fun.

2. Additional sections highlighting the role of computational fluid dynamics (CFD). In the practice of modern aerodynamics, CFD has become a new “third dimen­sion” existing side-by-side with the previous classic dimensions of pure theory and pure experiment. In recognition of the growing significance of CFD, new material has been added to give the reader a broader image of aerodynamics in today’s world.

3. More material on viscous flow. Part 4 on viscous flow has been somewhat rear­ranged and expanded, and now contains two additional chapters in comparison to the previous editions. The new material includes aspects of stagnation point aerodynamic heating, engineering methods of calculation such as the reference temperature method, turbulence modeling, and expanded coverage of modern CFD Navier-Stokes solutions. However, every effort has been made to keep this material within reasonable bounds both in respect to its space and the level of its presentation.

4. Additional historical content scattered throughout the book. It is the author’s opinion that knowledge of the history of aerodynamics plays an important role in the overall education and practice of modern aerodynamics. This additional historical content simply complements the historical material already contained in the previous editions.

5. Many additional worked examples. When learning new technical material, es­pecially material of a fundamental nature as emphasized in this book, one can

never have too many examples of how the fundamentals can be applied to the solution of problems.

6. A large number of new, additional figures and illustrations. The additional ma­terial just itemized is heavily supported with visual figures. I vigorously believe that “a picture is worth a thousand words.”

7. New homework problems added to those carried over from the second edition.

Much of the new material in this third edition is motivated by my experiences over the decade that has elapsed since the second edition. In particular, the design boxes follow the objectives and philosophy that dominate my new book Aircraft Per­formance and Design, McGraw-Hill, Boston, 1999. Moreover, the feature of design boxes has recently been introduced in my new edition of Introduction to Flight, 4th ed., McGraw-Hill, Boston, 2000, and has already met with success. The importance of CFD reflected in this third edition is part of my efforts to introduce aspects of CFD into undergraduate education; my book Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill, New York, 1995, is intended to provide a window into the subject of CFD at a level suitable for advanced undergraduates. Finally, the new historical notes contained here are a product of my research and maturity gained while writing A History of Aerodynamics, and Its Impact on Flying Machines, Cam­bridge University Press, New York, 1997 (hardback), 1998 (paperback). I would like to think this third edition of Fundamentals of Aerodynamics has benefited from the above experience.

All the new additions not withstanding, the main thrust of this book remains the presentation of the fundamentals of aerodynamics; the new material is simply intended to enhance and support this thrust. The book is organized along classical lines, dealing with inviscid incompressible flow, inviscid compressible flow, and viscous flow in sequence. My experience in teaching this material to undergraduates finds that it nicely divides into a two-semester course, with Parts 1 and 2 in the first semester, and Parts 3 and 4 in the second semester. Also, for the past eight years I have taught the entire book in a fast-paced, first-semester graduate course intended to introduce the fundamentals of aerodynamics to new graduate students who have not had this material as part of their undergraduate education. The book works well in such a mode.

I would like to thank the McGraw-Hill editorial staff for their excellent help in producing this book, especially Jonathan Plant and Kristen Druffner in Boston, and Kay Brimeyer in Dubuque. Also, special thanks go to my long-time friend and associate, Sue Cunningham, whose expertise as a scientific typist is beyond comparison, and who has typed all my book manuscripts for me, including this one, with great care and precision.

As a final comment, aerodynamics is a subject of intellectual beauty, composed and drawn by many great minds over the centuries. Fundamentals of Aerodynamics is intended to portray and convey this beauty. Do you feel challenged and interested by these thoughts? If so, then read on, and enjoy!

John D. Anderson, Jr.