# Category Fundamentals of Aerodynamics

## Source Flow: Our Second Elementary Flow

Consider a two-dimensional, incompressible flow where all the streamlines are straight lines emanating from a central point O, as shown at the left of Figure 3.20. Moreover, let the velocity along each of the streamlines vary inversely with distance from point O. Such a flow is called a source flow. Examining Figure 3.20, we see that the velocity components in the radial and tangential directions are Vr and Vg, respectively, where Vg = 0. The coordinate system in Figure 3.20 is a cylindrical

 Source flow

 Sink flow

 Figure 3.20 Source and sink flows

coordinate system, with the г axis perpendicular to the page. (Note that polar co­ordinates are simply the cylindrical coordinates r and в confined to a single plane given by г = constant.) It is easily shown (see Problem 3.9) that (1) source flow is a physically possible incompressible flow, that is, V • V = 0, at every point except the origin, where V • V becomes infinite, and (2) source flow is irrotational at every point.

In a source flow, the streamlines are directed away from the origin, as shown at the left of Figure 3.20. The opposite case is that of a sink flow, where by definition the streamlines are directed toward the origin, as shown at the right of Figure 3.20. For sink flow, the streamlines are still radial lines from a common origin, along which the flow velocity varies inversely with distance from point О. Indeed, a sink flow is simply a negative source flow.

The flows in Figure 3.20 have an alternate, somewhat philosophical interpreta­tion. Consider the origin, point О, as a discrete source or sink. Moreover, interpret the radial flow surrounding the origin as simply being induced by the presence of the discrete source or sink at the origin (much like a magnetic field is induced in the space surrounding a current-carrying wire). Recall that, for a source flow, V • V = 0 everywhere except at the origin, where it is infinite. Thus, the origin is a singular point, and we can interpret this singular point as a discrete source or sink of a given strength, with a corresponding induced flow field about the point. This interpreta­tion is very convenient and is used frequently. Other types of singularities, such as doublets and vortices, are introduced in subsequent sections. Indeed, the irrotational, incompressible flow field about an arbitrary body can be visualized as a flow induced by a proper distribution of such singularities over the surface of the body. This concept is fundamental to many theoretical solutions of incompressible flow over airfoils and other aerodynamic shapes, and it is the very heart of modem numerical techniques for the solution of such flows. You will obtain a greater appreciation for the concept of distributed singularities for the solution of incompressible flow in Chapters 4 through

6. At this stage, however, simply visualize a discrete source (or sink) as a singularity that induces the flows shown in Figure 3.20.

Let us look more closely at the velocity field induced by a source or sink. By definition, the velocity is inversely proportional to the radial distance r. As stated earlier, this velocity variation is a physically possible flow, because it yields V • V =

0. Moreover, it is the only such velocity variation for which the relation V • V = 0 is satisfied for the radial flows shown in Figure 3.20. Hence,

Vr = – [3.59a]

r

and Ve = 0 [3.59b]

where c is constant. The value of the constant is related to the volume flow from the source, as follows. In Figure 3.20, consider a depth of length l perpendicular to the page, that is, a length l along the z axis. This is sketched in three-dimensional perspective in Figure 3.21. In Figure 3.21, we can visualize an entire line of sources along the z axis, of which the source О is just part. Therefore, in a two-dimensional flow, the discrete source, sketched in Figure 3.20, is simply a single point on the line source shown in Figure 3.21. The two-dimensional flow shown in Figure 3.20 is the

 у

 rd6

 X

 z

 Figure 3.21 Volume flow rate from a line source.

same in any plane perpendicular to the z, axis, that is, for any plane given by z, = constant. Consider the mass flow across the surface of the cylinder of radius r and height l as shown in Figure 3.21. The elemental mass flow across the surface element dS shown in Figure 3.21 is pV • dS = pVr( r d6){l). Hence, noting that Vr is the same value at any в location for the fixed radius r, the total mass flow across the surface of the cylinder is

[3.60]

Since p is defined as the mass per unit volume and m is mass per second, then m/p is the volume flow per second. Denote this rate of volume flow by v. Thus, from Equation (3.60), we have

m

v = — = 2nrlVr P

Moreover, the rate of volume flow per unit length along the cylinder is Ь/1. Denote this volume flow rate per unit length (which is the same as per unit depth perpendicular to the page in Figure 3.20) as Л. Hence, from Equation 3.61, we obtain

Hence, comparing Equations (3.59a) and (3.62), we see that the constant in Equation (3.59a)isc = А/2л. In Equation (3.62), Л defines the source length: it is physically the rate of volume flow from the source, per unit depth perpendicular to the page

of Figure 3.20. Typical units of Л are square meters per second or square feet per second. In Equation (3.62), a positive value of Л represents a source, whereas a negative value represents a sink.

The velocity potential for a source can be obtained as follows. From Equations

(2.157) , (3.5%), and (3.62),

 Э Ф A — = Vr =———- dr 2nr [3.63] 1 Э ф and ——————————————— = Ve = 0 г дв [3.64] Integrating Equation (3.63) with respect to r, we have Ф = ^Inr + f(6) L7Z [3.65] Integrating Equation (3.64) with respect to в, we have ф = const + f{r) [3.66]

Comparing Equations (3.65) and (3.66), we see that fir) = (h/2n) In r and f(6) = constant. As explained in Section 3.9, the constant can be dropped without loss of rigor, and hence Equation (3.65) yields

Equation (3.67) is the velocity potential for a two-dimensional source flow.

The stream function can be obtained as follows. From Equations (2.151a and b), (3.59b), and (3.62),

Integrating Equation (3.68) with respect to в, we obtain

ф = ^в + f(r) [3.70]

in

Integrating Equation (3.69) with respect to r, we have

ф = const + f(6) [3.71]

Comparing Equations (3.70) and (3.71) and dropping the constant, we obtain

[3.72]

Equation (3.72) is the stream function for a two-dimensional source flow.

The equation of the streamlines can be obtained by setting Equation (3.72) equal to a constant:

Л

f — —в = const [3.73]

2tt

From Equation (3.73), we see that в — constant, which, in polar coordinates, is the equation of a straight line from the origin. Hence, Equation (3.73) is consistent with the picture of the source flow sketched in Figure 3.20. Moreover, Equation (3.67) gives an equipotential line as r = constant, that is, a circle with its center at the origin, as shown by the dashed line in Figure 3.20. Once again, we see that streamlines and equipotential lines are mutually perpendicular.

To evaluate the circulation for source flow, recall the V x V = 0 everywhere. In turn, from Equation (2.137),

Г = ~ JJ(V x V) – dS = 0

5

for any closed curve C chosen in the flow field. Hence, as in the case of uniform flow discussed in Section 3.9, there is no circulation associated with the source flow.

It is straightforward to show that Equations (3.67) and (3.72) satisfy Laplace’s equation, simply by substitution into V20 = 0 and V2i/r = 0 written in terms of cylindrical coordinates [see Equation (3.42)]. Therefore, source flow is a viable elementary flow for use in building more complex flows.

## Physical Significance

Consider again the basic model underlying Prandtl’s lifting-line theory. Return to Fig­ure 5.13 and study it carefully. An infinite number of infinitesimally weak horseshoe vortices are superimposed in such a fashion as to generate a lifting line which spans the wing, along with a vortex sheet which trails downstream. This trailing-vortex sheet is the instrument that induces downwash at the lifting line. At first thought, you might consider this model to be somewhat abstract—a mathematical convenience that somehow produces surprisingly useful results. However, to the contrary, the model shown in Figure 5.13 has real physical significance. To see this more clearly, return to Figure 5.1. Note that in the three-dimensional flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions; that is, there is a discontinuity in the tangential velocity at the trailing edge. We know from Chapter 4 that a discontinuous change in tangential velocity is theoretically allowed across a vortex sheet. In real life, such discontinuities do not exist; rather, the different velocities at the trailing edge generate a thin region of large velocity gradients—a thin region of shear flow with very large vorticity. Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing. This sheet

 tends to roll up at the edges and helps to form the wing-tip vortices sketched in Fig­ure 5.2. Thus, Prandd’s lifting-line model with its trailing-vortex sheet is physically consistent with the actual flow downstream of a finite wing.

 Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8. The airfoil section is thin and symmetric. Calculate the lift and induced drag coefficients for the wing when it is at an angle of attack of 5°. Assume that 5 = r. Solution From Figure 5.18, 5 = 0.055. Hence, from the stated assumption, r also equals 0.055. From Equation (5.70), assuming ao = 2n from thin airfoil theory,

 Example 5.1

 _ Uo 1 + a0/+rAR(l + r) = 0.0867 degree-1

 Since the airfoil is symmetric, ctL=o = 0°. Thus,

 CL = act = (0.0867 degree 1 (5°) =

 0.4335

 From Equation (5.61),

 (0.4335)2(1 +0.055)8л

 0.00789

 Cn

Consider a rectangular wing with an aspect ratio of 6, an induced drag factor 5 = 0.055, and a zero-lift angle of attack of —2°. At an angle of attack of 3.4°, the induced drag coefficient for this wing is 0.01. Calculate the induced drag coefficient for a similar wing (a rectangular wing with the same airfoil section) at the same angle of attack, but with an aspect ratio of 10. Assume that the induced factors for drag and the lift slope, S and r, respectively, are equal to each other (i. e., 5 = г). Also, for AR = 10, 5 = 0.105.

Solution

We must recall that although the angle of attack is the same for the two cases compared here (AR = 6 and 10), the value of Cl is different because of the aspect-ratio effect on the lift slope. First, let us calculate Cl for the wing with aspect ratio 6. From Equation (5.61),

Hence, CL = 0.423

The lift slope of this wing is therefore dCL 0.423

—– = —————- = 0.078/degree = 4.485/rad

da 3.4° – (-2°) ‘ 6 ‘

 The lift slope for the airfoil (the infinite wing) can be obtained from Equation (5.70): dCi ^ «о da 1 + (a0/7rAR)(l + r)

 a0 _ a0 1 + [(1.055)«0/л-(6)] 1 + 0.056a0

 4.485

 Solving for ao, we find that this yields ao = 5.989/rad. Since the second wing (with AR = 10) has the same airfoil section, then a0 is the same. The lift slope of the second wing is given by

 a0 5.989 1 + (a0/TrAR)(l + r) _ 1 + [(5.989)(1.105)/л-(103ї = 0.086/degree

 a

 4.95/rad

 The lift coefficient for the second wing is therefore

 CL = a (a – aL=0) = 0.086[3.4° – (-2°)] = 0.464

 In turn, the induced drag coefficient is

 0.0076

Note: This problem would have been more straightforward if the lift coefficients had been stipulated to be the same between the two wings rather than the angle of attack. Then Equation (5.61) would have yielded the induced drag coefficient directly. A purpose of this example is to reinforce the rationale behind Equation (5.65), which readily allows the scaling of drag coefficients from one aspect ratio to another, as long as the lift coefficient is the same. This allows the scaled drag-coefficient data to be plotted versus CL (not the angle of attack) as in Figure 5.20. However, in the present example where the angle of attack is the same between both cases, the effect of aspect ratio on the lift slope must be explicitly considered, as we have done above.

Consider the twin-jet executive transport discussed in Example 1.6. In addition to the infor­mation given in Example 1.6, for this airplane the zero-lift angle of attack is —2°, the lift slope of the airfoil section is 0.1 per degree, the lift efficiency factor r = 0.04, and the wing aspect ratio is 7.96. At the cruising condition treated in Example 1.6, calculate the angle of attack of the airplane.

Solution

The lift slope of the airfoil section in radians is

a0 = 0.1 per degree = 0.1 (57.3) = 5.73 rad From Equation (5.70) repeated below

_ _____ "o____

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

 lift distribution reaching farther away from the root. Such wings require heavier internal structure. Hence, as the aspect ratio of a wing increases, so does the structural weight of the wing. As a result of this compromise between aerodynamics and structures, typical aspect ratios for conventional subsonic airplanes are on the order of 6 to 8. However, examine the three-view of the Lockheed U-2 high altitude reconnaissance aircraft shown in Figure 5.24. This airplane has the unusually high aspect ratio of 14.3. Why? The answer is keyed to its mission. The U-2 was essentially a point design; it was to cruise at the exceptionally high altitude of 70,000 ft or higher in order to not be reached by interceptor aircraft or ground-to-air-missiles during overflights of the Soviet Union in the 1950s. To achieve this mission, the need for incorporating a very high aspect ratio wing was paramount, for the following reason. In steady, level flight, where the airplane lift L must equal its weight W, L = W = q. xSCL = p^VlSCL [5.71] As the airplane flies higher, px decreases and hence from Equation (5.71) С/. must be increased in order to keep the lift equal to the weight. As its high-altitude cruise design point, the U-2 flies at a high value of C;, just on the verge of stalling. (This is in stark contrast to the normal cruise conditions of conventional airplanes at conventional altitudes, where the cruise lift coefficient is relatively small.) At the high value of С/, for the U-2 at cruising altitude, its induced drag coefficient [which from Equation (5.62) varies as C} would be unacceptably high if a conventional aspect ratio were used. Hence, the Lockheed design group (at the Lockheed Skunk Works) had to opt for as high an aspect ratio as possible to keep the induced drag coefficient within reasonable bounds. The wing design shown in Figure 5.24 was the result. We made an observation about induced drag Д itself, in contrast to the induced drag coefficient CD, . We have emphasized, based on Equation (5.62), that Cdj can be reduced by increasing the aspect ratio. For an airplane in steady, level flight, however, the induced drag force itself is governed by another design parameter, rather than the aspect ratio per se, as follows. From Equation (5.62), we have

 Three-view of the Lockheed U-2 high-altitude reconnaissance airplane.

 Figure 5.34

## Aerodynamic Forces and Moments

At first glance, the generation of the aerodynamic force on a giant Boeing 747 may seem complex, especially in light of the complicated three-dimensional flow field over the wings, fuselage, engine nacelles, tail, etc. Similarly, the aerodynamic resistance on an automobile traveling at 55 mi/h on the highway involves a complex interaction of the body, the air, and the ground. However, in these and all other cases, the aerodynamic forces and moments on the body are due to only two basic sources:

1. Pressure distribution over the body surface

2. Shear stress distribution over the body surface

No matter how complex the body shape may be, the aerodynamic forces and moments on the body are due entirely to the above two basic sources. The only mechanisms nature has for communicating a force to a body moving through a fluid are pressure and shear stress distributions on the body surface. Both pressure p and shear stress r have dimensions of force per unit area (pounds per square foot or newtons per square meter). As sketched in Figure 1.8, p acts normal to the surface, and r acts tangential to the surface. Shear stress is due to the “tugging action” on the surface, which is caused by friction between the body and the air (and is studied in great detail in Chapters 15 to 20).

The net effect of the p and r distributions integrated over the complete body surface is a resultant aerodynamic force R and moment M on the body, as sketched in Figure 1.9. In turn, the resultant R can be split into components, two sets of which are shown in Figure 1.10. In Figure 1.10, V0c is the relative wind, defined as the

 p – pis) – surface pressure distribution t = t(s) = surface shear stress distribution Figure 1.8 Illustration of pressure and shear stress on an aerodynamic surface.

 Figure 1.9 Resultant aerodynamic force and moment on the body.

 N Figure 1.10 Resultant aerodynamic force and the components into which it splits.

flow velocity far ahead of the body. The flow far away from the body is called the freestream, and hence is also called the freestream velocity. In Figure 1.10, by

definition,

L = lift = component of R perpendicular to Vx D = drag = component of R parallel to

The chord c is the linear distance from the leading edge to the trailing edge of the body. Sometimes, R is split into components perpendicular and parallel to the chord, as also shown in Figure 1.10. By definition,

N = normal force = component of R perpendicular to c A = axial force = component of R parallel to c

The angle of attack a is defined as the angle between c and Vx. Hence, a is also the angle between L and N and between D and A. The geometrical relation between these two sets of components is, from Figure 1.10,

L = N cos a — A sin a [1.1]

D = N sin a + A cos a [1.2]

Let us examine in more detail the integration of the pressure and shear stress distributions to obtain the aerodynamic forces and moments. Consider the two­dimensional body sketched in Figure 1.11. The chord line is drawn horizontally, and hence the relative wind is inclined relative to the horizontal by the angle of attack a. An xy coordinate system is oriented parallel and perpendicular, respectively, to the chord. The distance from the leading edge measured along the body surface to an arbitrary point A on the upper surface is su; similarly, the distance to an arbitrary point В on the lower surface is si. The pressure and shear stress on the upper surface are denoted by pu and ru, respectively; both pu and zu are functions of su. Similarly, Pi and Т/ are the corresponding quantities on the lower surface and are functions of

 Figure 1.11 Nomenclature for the integration of pressure and shear stress distributions over a two-dimensional body surface.

si. At a given point, the pressure is normal to the surface and is oriented at an angle 9 relative to the perpendicular; shear stress is tangential to the surface and is oriented at the same angle 9 relative to the horizontal. In Figure 1.11, the sign convention for в is positive when measured clockwise from the vertical line to the direction of p and from the horizontal line to the direction of r. In Figure 1.11, all thetas are shown in their positive direction. Now consider the two-dimensional shape in Figure 1.11 as a cross section of an infinitely long cylinder of uniform section. A unit span of such a cylinder is shown in Figure 1.12. Consider an elemental surface area dS of this cylinder, where dS — (ds)( ) as shown by the shaded area in Figure 1.12. We are interested in the contribution to the total normal force N’ and the total axial force A’ due to the pressure and shear stress on the elemental area dS. The primes on N’ and A’ denote force per unit span. Examining both Figures 1.11 and 1.12, we see that the elemental normal and axial forces acting on the elemental surface dS on the upper body surface are

dN’u — — pudsu cos0 — rudsu sin0 [1.3]

dA’u = —pudsu sin 9 + Tudsu cos 9 [1.4]

On the lower body surface, we have

dN{ = pidsi cos9 — Tidsisind [1.5]

dA = pidsi sin 9 + Tidsi cos 9 [1.6]

In Equations (1.3) to (1.6), the positive directions of N’ and A’ are those shown in Figure 1.10. In these equations, the positive clockwise convention for 9 must be followed. For example, consider again Figure 1.11. Near the leading edge of the body, where the slope of the upper body surface is positive, r is inclined upward, and hence it gives a positive contribution to N1. For an upward inclined г, в would

 Figure 1.13 Aerodynamic force on an element of the body surface.

be counterclockwise, hence negative. Therefore, in Equation (1.3), sin 0 would be negative, making the shear stress term (the last term) a positive value, as it should be in this instance. Hence, Equations (1.3) to (1.6) hold in general (for both the forward and rearward portions of the body) as long as the above sign convention for в is consistently applied.

The total normal and axial forces per unit span are obtained by integrating Equa­tions (1.3) to (1.6) from the leading edge (LE) to the trailing edge (ТЕ):

In turn, the total lift and drag per unit span can be obtained by inserting Equations (1.7) and (1.8) into (1.1) and (1.2); note that Equations (1.1) and (1.2) hold for forces on an arbitrarily shaped body (unprimed) and for the forces per unit span (primed).

The aerodynamic moment exerted on the body depends on the point about which moments are taken. Consider moments taken about the leading edge. By convention, moments which tend to increase a (pitch up) are positive, and moments which tend to decrease a (pitch down) are negative. This convention is illustrated in Figure 1.13. Returning again to Figures 1.11 and 1.12, the moment per unit span about the leading edge due to p and г on the elemental area dS on the upper surface is

dM’u = (pu cos в + zu sin0)x dsu + (—pu sin# + xu cos 9)y dsu [1.9] On the bottom surface,

dM[ = {—pi cos в + tі sin 0)x dsi + (pi sin в + Т/ cosd)y dsi [1.10]

In Equations (1.9) and (1.10), note that the same sign convention for в applies as before and that у is a positive number above the chord and a negative number below the chord. Integrating Equations (1.9) and (1.10) from the leading to the trailing edges, we obtain for the moment about the leading edge per unit span

/■ТЕ

+ / [(—р/ cos в + n sin в)х + (pi sin в + Ті cos в)у] dsi

J LE

 (+)

 (-)

 Figure 1.13 Sign convention for aerodynamic moments.

In Equations (1.7), (1.8), and (1.11), 9, x, and у are known functions of і for a given body shape. Hence, if pu, pi, t„, and г/ are known as functions of і (from theory or experiment), the integrals in these equations can be evaluated. Clearly, Equations (1.7), (1.8), and (1.11) demonstrate the principle stated earlier, namely, the sources of the aerodynamic lift, drag, and moments on a body are the pressure and shear stress distributions integrated over the body. A major goal of theoretical aerodynamics is to calculate p{s) and г(.v) for a given body shape and freestream conditions, thus yielding the aerodynamic forces and moments via Equations (1.7), (1.8), and (1.11).

As our discussions of aerodynamics progress, it will become clear that there are quantities of an even more fundamental nature than the aerodynamic forces and moments themselves. These are dimensionless force and moment coefficients, defined as follows. Let poo and be the density and velocity, respectively, in the freestream, far ahead of the body. We define a dimensional quantity called the freestream dynamic pressure as

Dynamic pressure: qx = ^p^V^

The dynamic pressure has the units of pressure (i. e., pounds per square foot or newtons per square meter). In addition, let S be a reference area and l be a reference length. The dimensionless force and moment coefficients are defined as follows:

L

ЯоqS

QooS N

Cn =——–

qooS

_ A

Ca q<*>s

_ M

CM=q^Sl

In the above coefficients, the reference area S and reference length l are chosen to pertain to the given geometric body shape; for different shapes, S and l may be different things. For example, for an airplane wing, S is the planform area, and l is the mean chord length, as illustrated in Figure 1.14a. However, for a sphere, S is the cross-sectional area, and l is the diameter, as shown in Figure 1.14b. The particular choice of reference area and length is not critical; however, when using force and moment coefficient data, you must always know what reference quantities the particular data are based upon.

The symbols in capital letters listed above, i. e., CL, CD, Cm, and CA, denote the force and moment coefficients for a complete three-dimensional body such as an airplane or a finite wing. In contrast, for a two-dimensional body, such as given in Figures 1.11 and 1.12, the forces and moments are per unit span. For these two-

Цс f c *1 nd2

її j (l j S – cross-sectional area =

( j 1 = d = diameter

(b)

Figure 1.14 Some reference areas and lengths.

dimensional bodies, it is conventional to denote the aerodynamic coefficients by lowercase letters; e. g.,

_ L’ _ D’ _ M’

qooC qooC m qooC2

where the reference area S = c(l) = c.

Two additional dimensionless quantities of immediate use are

Pressure coefficient: C„ = ——^

*700 r

Skin friction coefficient: c t = —

qoo

where poo is the freestream pressure.

The most useful forms of Equations (1.7), (1.8), and (1.11) are in terms of the dimensionless coefficients introduced above. From the geometry shown in Figure 1.15,

dx = dx cos в [1.12]

dy = — (ds sin#) [1.13]

S — c(l) [1.14]

Substituting Equations (1.12) and (1.13) into Equations (1.7), (1.8), and (1.11), di­viding by qoo, and further dividing by S in the form of Equation (1.14), we obtain the following integral forms for the force and moment coefficients:

 Figure 1.15 Geometrical relationship of differential lengths.

 [1.15] [1.15] [1.17]

The simple algebraic steps are left as an exercise for the reader. When evaluating these integrals, keep in mind that yu is directed above the x axis, and hence is positive, whereas yi is directed below the x axis, and hence is negative. Also, dy/dx on both the upper and lower surfaces follow the usual rule from calculus, i. e., positive for those portions of the body with a positive slope and negative for those portions with a negative slope.

The lift and drag coefficients can be obtained from Equations (1.1) and (1.2) cast in coefficient form:

q = cn cos a — ca sin a [1.18]

Cd — cn sin a + ca cos a [1.19]

Integral forms for c; and cj are obtained by substituting Equations (1.15) and (1.16) into (1.18) and (1.19).

It is important to note from Equations (1.15) through (1.19) that the aerody­namic force and moment coefficients can be obtained by integrating the pressure and skin friction coefficients over the body. This is a common procedure in both theoretical and experimental aerodynamics. In addition, although our derivations have used a two-dimensional body, an analogous development can be presented for

three-dimensional bodies—the geometry and equations only get more complex and involved—the principle is the same.

Consider the supersonic flow over a 5° half-angle wedge at zero angle of attack, as sketched in Figure 1.16a. The freestream Mach number ahead of the wedge is 2.0, and the freestream pressure and density are 1.01 x 105 N/m2 and 1.23 kg/m3, respectively (this corresponds to standard sea level conditions). The pressures on the upper and lower surfaces of the wedge are constant with distance s and equal to each other, namely, p„ = pi = 1.31 x 10s N/m2, as shown in Figure 1.16b. The pressure exerted on the base of the wedge is equal to px. As seen in Figure 1.16c, the shear stress varies over both the upper and lower surfaces as r„, =431 s 2. The chord length, c, of the wedge is 2 m. Calculate the drag coefficient for the wedge.

Solution

We will carry out this calculation in two equivalent ways. First, we calculate the drag from Equation (1.8), and then obtain the drag coefficient. In turn, as an illustration of an alternate approach, we convert the pressure and shear stress to pressure coefficient and skin friction coefficient, and then use Equation (1.16) to obtain the drag coefficient.

Since the wedge in Figure 1.16 is at zero angle of attach, then D’ = A’. Thus, the drag can be obtained from Equation (1.8) as

/>[’[■ pTE

D’ = ( —p„sin0 + T„COS0)c/.V„ + (Pi Sind + T/COS0)^.V/

Jle Jle

Referring to Figure 1.16c, recalling the sign convention for в, and noting that integration over the upper surface goes from. q to s2 on the inclined surface and from s2 to + on the base, whereas integration over the bottom surface goes from я to v4 on the inclined surface and from s4 to л’з on the base, we find that the above integrals become

f — pu sin в dsu = ( —(1.31 x 105)sin(—5“)z2.v„

JLE Js і

+ j —(1.01 x 105) sin 90° dsu

= 1.142 x 104(,v2 – .я) – 1.01 x 105(+ – s2)

= 1.142 x 104 (—!—) – 1.01 x 105(c)(tan 5°)

Vcos5°/

= 1.142 x 104(2.008) – 1.01 x 10s(0.175) = 5260 N

Г ТЕ p. У4 p S}

pisinedsi = (1.31 x 10s)sin(5°)ds,+ (1.01 x 105) sin(-90°)dst

JLE Js i */.s-4

= 1.142 x 104(.v4 — я) +1.01 x 105(—l)(.v3 – 44)

= 1.142 x 104 (——) – 1.01 x 105(c)(tan 5°)

Vcos5°z

= 2.293 x 104 – 1.767 x 104 = 5260 N

 Figure 1.16 Illustration for Example 1.1.

 Shear Stress Distribution

Note that the integrals of the pressure over the top and bottom surfaces, respectively, yield the same contribution to the drag—a result to be expected from the symmetry of the configuration in Figure 1.16:

Г ТЕ rs2

r„cos в dsu = I 431s~02cos(—5°)dsu J LE Л,

/ C Vі.

= 429 (——– ) — = 936.5 N

Vcos5°/ 0.8

r/cos0dsi = I 43b °-2cos(—5°)ds.

4′ ~si

0. 8

/ c y1» 1

= 429 (—– -) — = 936.5 N

Vcos5°/ 0.8

Again, it is no surprise that the shear stress acting over the upper and lower surfaces, respectively, give equal contributions to the drag; this is to be expected due to the symmetry of the wedge shown in Figure 1.16. Adding the pressure integrals, and then adding the shear stress integrals, we have for total drag

D’ = 1.052 x 104 + 0.1873 x 104 =

4——- v — 1 v ‘ 11 "

pressure skin friction

drag drag

Note that, for this rather slender body, but at a supersonic speed, most of the drag is pressure drag. Referring to Figure 1.16a, we see that this is due to the presence of an oblique shock wave from the nose of the body, which acts to create pressure drag (sometimes called “wave drag”). In this example, only 15 percent of the drag is skin friction drag; the other 85 percent is the pressure drag (wave drag). This is typical of the drag of slender supersonic bodies. In contrast, as we will see later, the drag of a slender body at subsonic speed, where there is no shock wave, is mainly skin friction drag.

The drag coefficient is obtained as follows. The velocity of the freestream is twice the sonic speed, which is given by

«СС = Vyrt*c = v/(l-4)(287)(288) = 340.2 m/s

(See Chapter 8 for a derivation of this expression for the speed of sound.) Note that, in the above, the standard sea level temperature of 288 К is used. Hence, V^ = 2(340.2) = 680.4 m/s. Thus,

qx = IpxV^ = (0.5)(1.23)(680.4)2 = 2.847 x 105 N/m2 Also, S = c(l) = 2.0m2

An alternate solution to this problem is to use Equation (1.16), integrating the pressure coefficients and skin friction coefficients to obtain directly the drag coefficient. We proceed as follows:

 C.

On the lower surface, we have the same value for C„, i. e.,

■■ 0.1054

Also,

= 0.009223x I2 +0.00189x°’8|p

= 0.01854 + 0.00329 :

This is the same result as obtained earlier.

Example 1 .2 | Consider a cone at zero angle of attack in a hypersonic flow. (Hypersonic flow is very high­speed flow, generally defined as any flow above a Mach number of 5; hypersonic flow is further defined in Section 1.10.) The half-angle of the cone is вс, as shown in Figure 1.17. An approximate expression for the pressure coefficient on the surface of a hypersonic body is given by the newtonian sine-squared law (to be derived in Chapter 14):

Cp = 2 sin2 вс

Note that Cp, hence, p; is constant along the inclined surface of the cone. Along the base of the body, we assume that p = px. Neglecting the effect of friction, obtain an expression for the drag coefficient of the cone, where Co is based on the area of the base St,.

Solution

We cannot use Equations (1.15) to (1.17) here. These equations are expressed for a two­dimensional body, such as the airfoil shown in Figure 1.15, whereas the cone in Figure 1.17 is a shape in three-dimensional space. Hence, we must treat this three-dimensional body as follows. From Figure 1.17, the drag force on the shaded strip of surface area is

(p sin6>c.)(2jrг)- =2nrpdr

sin вс

 Figure 1.17 Illustration for Example 1.2.

The total drag due to the pressure acting over the total surface area of the cone is

rrb rn>

D= І Inrpdr — I Ілр-^dr

Jo Jo

The first integral is the horizontal force on the inclined surface of the cone, and the second integral is the force on the base of the cone. Combining the integrals, we have

D=f 2лг(р – PoJr/r = 7T(p – px)rl Jo

Referenced to the base area, nrft, the drag coefficient is

D ЛГЬІР ~ Poo)

(Note: The drag coefficient for a cone is equal to its surface pressure coefficient.) Flence, using the newtonian sine-squared law, we obtain

## Fundamental Equations in Terms of the Substantial Derivative

In this section, we express the continuity, momentum, and energy equations in terms of the substantial derivative. In the process, we make use of the following vector identity:

v. (pV) =pV-V + V-Vp [2.106]

In words, this identity states that the divergence of a scalar times a vector is equal to the scalar times the divergence of the vector plus the dot product of the vector and the gradient of the scalar.

First, consider the continuity equation given in the form of Equation (2.52):

^ + V • (pV) = 0 [2.52]

ot

Using the vector identity given by Equation (2.106), Equation (2.52) becomes

— +V-Vp + pV-V = 0 [2.107]

31

However, the sum of the first two terms of Equation (2.107) is the substantial derivative of p [see Equation (2.104)]. Thus, from Equation (2.107),

Equation (2.108) is the form of the continuity equation written in terms of the sub­stantial derivative.

Next, consider the x component of the momentum equation given in the form of Equation (2.70a):

 [2.70a]

The first terms can be expanded as

 d{pu) dt

 [2.109]

In the second term of Equation (2.70a), treat the scalar quantity as и and the vector quantity as pV. Then the term can be expanded using the vector identity in Equation (2.106):

V • (puV) = V • [a(pV)] = mV • (pV) + (pV)) • Vm [2.1 lO]

Substituting Equations (2.109) and (2.110) into (2.70a), we obtain

or

+ (pV) • Vm = – —- + pfx + (Fx) viscous [2.1 11] dx

Examine the two terms inside the square brackets; they are precisely the left side of the continuity equation, Equation (2.52). Since the right side of Equation (2.52) is zero, the sum inside the square brackets is zero. Hence, Equation (2.11) becomes

Эм dp _

P~—b pV • Vm = — ——- — h pfx + (Fx(viscous

dt dx

/Эм dp

or P ( — +V • VmJ =-—+ pfx + (^(viscous [2.112]

Examine the two terms inside the parentheses in Equation (2.12); their sum is precisely the substantial derivative Du/Dt. Hence, Equation (2.12) becomes

 Du P~Dt ~

In a similar manner, Equations (2.10b and c) yield

 [2.113b]

[2.113c]

Equations (2.113a to c) are the x, y, and z components of the momentum equation written in terms of the substantial derivative. Compare these equations with Equations (2.70a to c). Note that the right sides of both sets of equations are unchanged; only the left sides are different.

In an analogous fashion, the energy equation given in the form of Equation

(2.96) can be expressed in terms of the substantial derivative. The derivation is left as a homework problem; the result is

[2.114]

Again, the right-hand sides of Equations (2.96) and (2.114) are the same; only the form of the left sides is different.

In modem aerodynamics, it is conventional to call the form of Equations (2.52), (2.70a to c), and (2.96) the conservation form of the fundamental equations (some­times these equations are labeled as the divergence form because of the divergence terms on the left side). In contrast, the form of Equations (2.108), (2.113a to c), and (2.114), which deals with the substantial derivative on the left side, is called the nonconservation form. Both forms are equally valid statements of the fundamental principles, and in most cases, there is no particular reason to choose one form over the other. The nonconservation form is frequently found in textbooks and in aerodynamic theory. However, for the numerical solution of some aerodynamic problems, the con­servation form sometimes leads to more accurate results. Hence, the distinction between the conservation form and the nonconservation form has become important in the modem discipline of computational fluid dynamics. (See Reference 7 for more details.)

## Incompressible Flow over Airfoils

Of the many problems now engaging attention, the following are considered of immediate importance and will be considered by the committee as rapidly as funds can be secured for the purpose…. The evolution of more efficient wing sections of practical form, embodying suitable dimensions for an economical structure, with moderate travel of the center-of-pressure and still affording a large range of angle-of-attack combined with efficient action.

From the first Annual Report of the NACA, 1915

4,1 Introduction

With the advent of successful powered flight at the turn of the twentieth century, the importance of aerodynamics ballooned almost overnight. In turn, interest grew in the understanding of the aerodynamic action of such lifting surfaces as fixed wings on airplanes and, later, rotors on helicopters. In the period 1912-1918, the analy­sis of airplane wings took a giant step forward when Ludwig Prandtl and his col­leagues at Gottingen, Germany, showed that the aerodynamic consideration of wings could be split into two parts: (1) the study of the section of a wing—an airfoil—and (2) the modification of such airfoil properties to account for the complete, finite wing. This approach is still used today; indeed, the theoretical calculation and experimental measurement of modem airfoil properties have been a major part of the aeronautics research carried out by the National Aeronautics and Space Administration (NASA) in the 1970s and 1980s. (See Chapter 5 of Reference 2 for a historical sketch on airfoil development and Reference 10 for a description of modem airfoil research.) Following Prandtl’s philosophy, the present chapter deals exclusively with airfoils,

 Figure 4.1 Definition of an airfoil.

whereas Chapter 5 treats the case of a complete, finite wing. Therefore, in this chapter and Chapter 5, we make a major excursion into aerodynamics as applied to airplanes.

What is an airfoil? Consider a wing as drawn in perspective in Figure 4.1. The wing extends in the у direction (the span direction). The freestream velocity Voo is parallel to the xz plane. Any section of the wing cut by a plane parallel to the xz plane is called an airfoil. The purpose of this chapter is to present theoretical methods for the calculation of airfoil properties. Since we are dealing with inviscid flow, we are not able to predict airfoil drag; indeed, d’Alembert’s paradox says that the drag on an airfoil is zero—clearly not a realistic answer. We will have to wait until Chapter 15 and a discussion of viscous flow before predictions of drag can be made. However, the lift and moments on the airfoil are due mainly to the pressure distribution, which (below the stall) is dictated by inviscid flow. Therefore, this chapter concentrates on the theoretical prediction of airfoil lift and moments.

The road map for this chapter is given in Figure 4.2. After some initial discussion on airfoil nomenclature and characteristics, we present two approaches to low-speed airfoil theory. One is the classical thin airfoil theory developed during the period 1910-1920 (the right-hand branch of Figure 4.2). The other is the modem numerical approach for arbitrary airfoils using vortex panels (the left-hand branch of Figure 4.2). Please refer to this road map as you work your way through this chapter.

## Applied Aerodynamics: The Flow Over a Sphere—The Real Case

The present section is a complement to Section 3.18, in which the real flow over a cir­cular cylinder was discussed. Since the present chapter deals with three-dimensional flows, it is fitting at this stage to discuss the three-dimensional analog of the circular cylinder, namely, the sphere. The qualitative features of the real flow over a sphere are similar to those discussed for a cylinder in Section 3.18—the phenomenon of flow separation, the variation of drag coefficient with a Reynolds number, the precip­itous drop in drag coefficient when the flow transits from laminar to turbulent ahead of the separation point at the critical Reynolds number, and the general structure of the wake. These items are similar for both cases. However, because of the three­dimensional relieving effect, the flow over a sphere is quantitatively different from that for a cylinder. These differences are the subject of the present section.

The laminar flow over a sphere is shown in Figure 6.9. Here, the Reynolds number is 15,000, certainly low enough to maintain laminar flow over the spherical

 Figure 6.9 Laminar flow case: Instantaneous flow past a sphere in water. Re = 15,000. Flow is made visible by dye in the water. (Courtesy of H. Werle, ONERA, France. Also in Van Dyke, Milton, An Album of Fluid Motion, The Parabolic Press, Stanford, CA, 1982.)

surface. However, in response to the adverse pressure gradient on the back surface of the sphere predicted by inviscid, incompressible flow theory (see Section 6.4 and Figure 6.6), the laminar flow readily separates from the surface. Indeed, in Figure 6.9, separation is clearly seen on the. forward surface, slightly ahead of the vertical equator of the sphere. Thus, a large, fat wake trails downstream of the sphere, with a consequent large pressure drag on the body (analogous to that discussed in Section 3.18 for a cylinder.) In contrast, the turbulent flow case is shown in Figure 6.10. Here, the Reynolds number is 30,000, still a low number normally conducive to laminar flow. However, in this case, turbulent flow is induced artificially by the presence of a wire loop in a vertical plane on the forward face. (Trip wires are frequently used in experimental aerodynamics to induce transition to turbulent flow; this is in order to study such turbulent flows under conditions where they would not naturally exist.) Because the flow is turbulent, separation takes place much farther over the back surface, resulting in a thinner wake, as can be seen by comparing Figures 6.9 and 6.10. Consequently, the pressure drag is less for the turbulent case.

The variation of drag coefficient Co with the Reynolds number for a sphere, is shown in Figure 6.11. Compare this figure with Figure 3.39 for a circular cylinder;

 Figure 6.1 О Turbulent flow case: Instantaneous flow past a sphere in water. Re = 30,000. The turbulent flow is forced by a trip wire hoop ahead of the equator, causing the laminar flow to become turbulent suddenly. The flow is made visible by air bubbles in wafer. (Courtesy of H. Werle, ONERA, France. Also in Van Dyke, Milton, An Album of Fluid Motion, The Parabolic Press, Stanford, CA, I982.j
 Figure 6.1 1 Variation of drag coefficient with Reynolds number for a sphere. (From Schlichting, Reference 42.)

the С о variations are qualitatively similar, both with a precipitous decrease in Cp near a critical Reynolds number of 300,000, coinciding with natural transition from laminar to turbulent flow. However, quantitatively the two curves are quite different. In the Reynolds number range most appropriate to practical problems, that is, for Re > 1000, the values of CD for the sphere are considerably smaller than those for a cylinder—a classic example of the three-dimensional relieving effect. Reflecting on Figure 3.39 for the cylinder, note that the value of CD for Re slightly less than the critical value is about 1 and drops to 0.3 for Re slightly above the critical value. In contrast, for the sphere as shown in Figure 6.11, Cp is about 0.4 in the Reynolds number range below the critical value and drops to about 0.1 for Reynolds numbers above the critical value. These variations in Co for both the cylinder and sphere are classic results in aerodynamics; you should keep the actual Co values in mind for future reference and comparisons.

As a final point in regard to both Figures 3.39 and 6.11, the value of the critical Reynolds number at which transition to turbulent flow takes place upstream of the separation point is not a fixed, universal number. Quite the contrary, transition is influenced by many factors, as will be discussed in Part 4. Among these is the amount of turbulence in the freestream; the higher the freestream turbulence, the more readily transition takes place. In turn, the higher the freestream turbulence, the lower is the value of the critical Reynolds number. Because of this trend, calibrated spheres are used in wind-tunnel testing actually to assess the degree of freestream turbulence in the test section, simply by measuring the value of the critical Reynolds number on the sphere.

## Historical Note: Aerodynamic Coefficients

In Section 1.5, we introduced the convention of expressing aerodynamic force in terms of an aerodynamic coefficient, such as

L = ^PocVISCl

D = pooVlSCD

where L and D are lift and drag, respectively, and Cl and Co are the lift coefficient and drag coefficient, respectively. This convention, expressed in the form shown above, dates from about 1920. But the use of some type of aerodynamic coefficients goes back much further. In this section, let us briefly trace the genealogy of aerodynamic coefficients. For more details, see the author’s recent book, A History of Aerodynamics and Its Impact on Flying Machines (Reference 62).

The first person to define and use aerodynamic force coefficients was Otto Lilien – thal, the famous German aviation pioneer at the end of the nineteenth century. Inter­ested in heavier-than-flight from his childhood, Lilienthal carried out the first defini­tive series of aerodynamic force measurements on cambered (curved) airfoil shapes using a whirling arm. His measurements were obtained over a period of 23 years, cul­minating in the publication of his book Der Vogelflug als Grundlage der Fliegekunst (Birdflight as the Basis of Aviation) in 1889. Many of the graphs in his book are plotted in the form that today we identify as a drag polar, i. e., a plot of drag coeffi­cient versus lift coefficient, with the different data points being measured at angles of attack ranging from below zero to 90°. Lilienthal had a degree in Mechanical Engineering, and his work reflected a technical professionalism greater than most at that time. Beginning in 1891, he put his research into practice by designing several gliders, and executing over 2000 successful glider flights before his untimely death in a crash on August 9, 1896. At the time of his death, Lilienthal was working on the design of an engine to power his machines. Had he lived, there is some conjecture that he would have beaten the Wright brothers in the race for the first heavier-than-air, piloted, powered flight.

In his book, Lilienthal introduced the following equations for the normal and axial forces, which he denoted by N and T, respectively (for normal and “tangential”)

 N = 0.3r)FV2 T = 0A30FV2

 [1.60]

 and

 [1.61]

where, in Lilienthal’s notation, F was the reference planform area of the wing in m2, V is the freestream velocity in m/s, and /V and T are in units of kilogram force (the force exerted on one kilogram of mass by gravity at sea level). The number 0.13 is Smeaton’s coefficient, a concept and quantity stemming from measurements made in the eighteenth century on flat plates oriented perpendicular to the flow. Smeaton’s coefficient is proportional to the density of the freestream; its use is archaic, and it went out of favor at the beginning of the twentieth century. By means of Equations

(1.60) and (1.61) Lilienthal introduced the “normal” and “tangential” coefficients, tj and в versus angle of attack. A copy of this table, reproduced in a paper by Octave Chanute in 1897, is shown in Figure 1.50. This became famous as the “Lilienthal Tables,” and was used by the Wright brothers for the design of their early gliders. It is proven in Reference 62 that Lilienthal did not use Equations (1.60) and (1.61) explicitly to reduce his experimental data to coefficient form, but rather determined his experimental values for i] and в by dividing the experimental measurements for N and T by his measured force on the wing at 90° angle of attack. In so doing, he divided out the influence of uncertainties in Smeaton’s coefficient and the veloc­ity, the former being particularly important because the classical value of Smeaton’s coefficient of 0.13 was in error by almost 40 percent. (See Reference 62 for more de­tails.) Nevertheless, we have Otto Lilienthal to thank for the concept of aerodynamic force coefficients, a tradition that has been followed in various modified forms to the present time.

Following on the heals of Lilienthal, Samuel Langley at the Smithsonian Institu­tion published whirling arm data for the resultant aerodynamic force R on a flat plate as a function of angle of attack, using the following equation:

R = kSV2F(ct) [1.62]

where S is the planform area, к is the more accurate value of Smeaton’s coefficient (explicitly measured by Langley on his whirling arm), and F (a ) was the correspond­ing force coefficient, a function of angle of attack.

The Wright brothers preferred to deal in terms of lift and drag, and used expres­sions patterned after Lilienthal and Langley to define lift and drag coefficients:

L = kSV2CL [1.63]

D = kSV2CD [1.64]

The Wrights were among the last to use expressions written explicitly in terms of Smeaton’s coefficient k. Gustave Eiffel in 1909 defined a “unit force coefficient” Ki as

R = KjSV2 [1.65]

In Equation (1.65), Smeaton’s coefficient is nowhere to be seen; it is buried in the direct measurement of A’,. (Eiffel, of Eiffel Tower fame, built a large wind tunnel in 1909, and for the next 14 years reigned as France’s leading aerodynamicist until his death in 1923.) After Eiffel’s work, Smeaton’s coefficient was never used in the aerodynamic literature—it was totally passe.

TABLE OF NORMAL AND TANGENTIAL PRESSURES

Deduced by Lilienthal from the diagrams on Plate VI., in his book “ Bird-flight as the Basis of the Flying Art”

 a Angle, * Normal. a Tangential. a Angle. 4 Normal. » Tangential. -9°……………………………………………… OjOOO + 01070 i6°…………… 0.909 — 0X275 — 8°………………………………………… охцо + OX267 17°……………………………………… 0.915 — 0X273 -7°……………………………………………… 0x280 + ОЛ64 18°………………………………………. 0.919 — ОЛ7О — 6°………………………………………… a 120 + 0uo6o *9°………………………………………. 0921 — OX265 -s°……………………………….. 0.160 + ОЛ55 ao°……………………………………… 0922 — OX259 -4°………………………………………… 0.200 + OX249 ai°……………………………………….. 0933 — OX253 -3°…………………………………………….. 0.242 + 0X243 aa°……………………………………… 0934 — 0X247 – 2°……………………………………………. 0.286 + 0X237 *3°………………………………………. 0.934 — 0X241 — 1°………….. 0.33a + OX23I 4°……………………………………… 0923 — ОЛ36 0°…………. 0.381 + 0X224 4°…………………………………………. 0^23 — 0X231 + 1°………………………………………….. 0434 + OX2I6 26°……………………………………… 0.920 — 0X226 + 3°………………………………………… 0489 + 0Л08 27°………….. 0.918 — 0031 + 3°…………… 0.546 OdOOO 28°………….. 0.915 — 0x216 + 4°…………. 0.600 — OU007 39а………….. Q.9I2 — 0012 + 5°…………. 0.650 — 0014 30°………….. 0910 —ox»8 + 6°…………. 0.696 — 0031 33°………….. 0906 + 7°…………. 0.737 — ftprf 35°………….. 0896 +0x210 + 8°…………. 0.771 —OX23S 40°………….. 0^90 +0x216 + 9°…………. O.8OO — 0043 45°………….. ОІИМ +0x220 up……… 0.825 — 0050 s°°…. 0888 +0x223 n°………. 0.846 — 0x258 55a………….. 0^90 +0x226 i2°……… 0864 — 0x264 6o°………….. 0900 + 0x228 «3°. 0879 — 0070 70°………….. 0.930 +0x230 •4*……… 0Л91 —0x274 8o°…………. 0960 +0.015 «5°….. 0901 — 0x276 90°……… 1.000 0.000

Figure 1.50 The Lilienthal Table of normal and axial force coefficients. This is a facsimile of the actual table that was published by Octave Chanute in an article entitled "Sailing Flight," The Aeronautical Annual,

1 897, which was subsequently used by the Wright Brothers.

Gorrell and Martin, in wind tunnel tests carried out in 1917 at MIT on various airfoil shapes, adopted Eiffel’s approach, giving expressions for lift and drag:

L = K, AV2 [1.66]

D = KXAV2 [1.67]

where A denoted planform area and K, and Kx were the lift and drag coefficients, respectively. For a short period, the use of Kx and Kx became popular in the United States.

However, also by 1917 the density p began to appear explicitly in expressions for force coefficients. In NACA Technical Report No. 20, entitled “Aerodynamic Coefficients and Transformation Tables,” we find the following expression:

F = CpSV2

where F is the total force acting on the body, p is the freestream density, and C is the force coefficient, which was described as “an abstract number, varying for a given airfoil with its angle of incidence, independent of the choice of units, provided these are consistently used for all four quantities (F, p, S, and V).”

Finally, by the end of World War I, Ludwig Prandtl at Gottingen University in Germany established the nomenclature that is accepted as standard today. Prandtl was already famous by 1918 for his pioneering work on airfoil and wing aerodynamics, and for his conception and development of boundary layer theory. (See Section 5.8 for a biographical description of Prandtl.) Prandtl reasoned that the dynamic pressure, (he called it “dynamical pressure”), was well suited to describe aerodynamic force. In his 1921 English-language review of works performed at Gottingen before and during World War I (Reference 63), he wrote for the aerodynamic force,

W — cFq [1.68]

where W is the force, F is the area of the surface, q is the dynamic pressure, and c is a “pure number,” i. e., the force coefficient. It was only a short, quick step to express lift and drag as

L=qxSCL І1.69]

and D=qocSCo [1.70]

where CL and C n are the “pure numbers” referred to by Prandtl (i. e., the lift and drag coefficients). And this is the way it has been ever since.

## Fundamentals of Inviscid,. Incompressible Flow

Theoretical fluid dynamics, being a difficult subject, is for convenience, commonly divided into two branches, one treating of frictionless or perfect fluids, the other treating of viscous or imperfect fluids. The frictionless fluid has no existence in nature, but is hypothesized by mathematicians in order to facilitate the investigation of important laws and principles that may be approximately true of viscous or natural fluids.

Albert F. Zahm, 1912 (Professor of aeronautics, and developer of the first aeronautical laboratory in a U. S. university, The Catholic University of America)

3.1 Introduction and Road Map

The world of practical aviation was born on December 17, 1903, when, at 10:35 A. M., and in the face of cold, stiff, dangerous winds, Orville Wright piloted the Wright Flyer on its historic 12-s, 120-ft first flight. Figure 3.1 shows a photograph of the Wright Flyer at the instant of lift-off, with Wilbur Wright running along the right side of the machine, supporting the wing tip so that it will not drag the sand. This photograph is the most important picture in aviation history; the event it depicts launched the profession of aeronautical engineering into the mainstream of the twentieth century.1

I 1 See Reference 2 for historical details leading to the first flight by the Wright brothers.

The flight velocity of the Wright Flyer was about 30 mi/h. Over the ensuing decades, the flight velocities of airplanes steadily increased. By means of more powerful engines and attention to drag reduction, the flight velocities of airplanes rose to approximately 300 mi/h just prior to World War II. Figure 3.2 shows a typical fighter airplane of the immediate pre-World War II era. From an aerodynamic point of view, at air velocities between 0 and 300 mi/h the air density remains essentially constant, varying by only a few percent. Hence, the aerodynamics of the family of airplanes spanning the period between the two photographs shown in Figures 3.1 and 3.2 could be described by incompressible flow. As a result, a huge bulk of experimental and theoretical aerodynamic results was acquired over the 40-year period beginning with the Wright Flyer—results that applied to incompressible flow. Today, we are still very interested in incompressible aerodynamics because most modern general aviation aircraft still fly at speeds below 300 mi/h; a typical light general aviation airplane is shown in Figure 3.3. In addition to low-speed aeronautical applications, the principles of incompressible flow apply to the flow of fluids, for example, water flow through pipes, the motion of submarines and ships through the ocean, the design of wind turbines (the modem term for windmills), and many other important applications.

For all the above reasons, the study of incompressible flow is as relevant today as it was at the time of the Wright brothers. Therefore, Chapters 3 to 6 deal exclusively with incompressible flow. Moreover, for the most part, we ignore any effects of friction, thermal conduction, or diffusion; that is, we deal with inviscid incompressible flow in these chapters.[5] Looking at our spectrum of aerodynamic flows as shown in Figure 1.38, the material contained in Chapters 3 to 6 falls within the combined blocks D and E.

 Figure 3.2 The Seversky P-35, fCourtesy of the U. S. Air Force.]
 Figure 3.3 The Beechcraft Bonanza F33A. (Courtesy of Beechcraft. j

The purpose of this chapter is to establish some fundamental relations applicable to inviscid, incompressible flows and to discuss some simple but important flow fields and applications. The material in this chapter is then used as a launching pad for the airfoil theory of Chapter 4 and the finite wing theory of Chapter 5.

A road map for this chapter is given in Figure 3.4. There are three main avenues: (1) a development of Bernoulli’s equation, with some straightforward applications;

 Figure 3.4 Road map for Chapter 3.

(2) a discussion of Laplace’s equation, which is the governing equation for inviscid, incompressible, irrotational flow; (3) the presentation of some elementary flow pat­terns, how they can be superimposed to synthesize both the nonlifting and lifting flow over a circular cylinder, and how they form the basis of a general numerical technique, called the panel technique, for the solution of flows over bodies of general shape. As you progress through this chapter, occasionally refer to this road map so that you can maintain your orientation and see how the various sections are related.

## . Lifting Flows over Arbitrary Bodies: The Vortex Panel Numerical Method

The thin airfoil theory described in Sections 4.7 and 4.8 is just what it says—it ap­plies only to thin airfoils at small angles of attack. (Make certain that you understand exactly where in the development of thin airfoil theory these assumptions are made and the reasons for making them.) The advantage of thin airfoil theory is that closed – form expressions are obtained for the aerodynamic coefficients. Moreover, the results compare favorably with experimental data for airfoils of about 12 percent thickness or less. However, the airfoils on many low-speed airplanes are thicker than 12 percent. Moreover, we are frequently interested in high angles of attack, such as occur during takeoff and landing. Finally, we are sometimes concerned with the generation of aerodynamic lift on other body shapes, such as automobiles or submarines. Hence, thin airfoil theory is quite restrictive when we consider the whole spectrum of aero­dynamic applications. We need a method that allows us to calculate the aerodynamic characteristics of bodies of arbitrary shape, thickness, and orientation. Such a method is described in this section. Specifically, we treat the vortex panel method, which is a numerical technique that has come into widespread use since the early 1970s. In reference to our road map in Figure 4.2, we now move to the left-hand branch. Also, since this chapter deals with airfoils, we limit our attention to two-dimensional bodies.

The vortex panel method is directly analogous to the source panel method de­scribed in Section 3.17. However, because a source has zero circulation, source panels are useful only for nonlifting cases. In contrast, vortices have circulation, and hence vortex panels can be used for lifting cases. (Because of the similarities between source and vortex panel methods, return to Section 3.17 and review the basic philosophy of the source panel method before proceeding further.)

The philosophy of covering a body surface with a vortex sheet of such a strength to make the surface a streamline of the flow was discussed in Section 4.4 We then went on to simplify this idea by placing the vortex sheet on the camber line of the airfoil as shown in Figure 4.11, thus establishing the basis for thin airfoil theory. We

now return to the original idea of wrapping the vortex sheet over the complete surface of the body, as shown in Figure 4.10. We wish to find у (s) such that the body surface becomes a streamline of the flow. There exists no closed-form analytical solution for y(s); rather, the solution must be obtained numerically. This is the purpose of the vortex panel method.

Let us approximate the vortex sheet shown in Figure 4.10 by a series of straight panels, as shown earlier in Figure 3.40. (In Chapter 3, Figure 3.40 was used to discuss source panels; here, we use the same sketch for discussion of vortex panels.) Let the vortex strength у (s) per unit length be constant over a given panel, but allow it to vary from one panel to the next. That is, for the n panels shown in Figure 3.40, the vortex panel strengths per unit length are y, y2,…, yj,…, y„. These panel strengths are unknowns; the main thrust of the panel technique is to solve for yj, j = 1 to n, such that the body surface becomes a streamline of the flow and such that the Kutta condition is satisfied. As explained in Section 3.17, the midpoint of each panel is a control point at which the boundary condition is applied; that is, at each control point, the normal component of the flow velocity is zero.

Let P be a point located at (x, y) in the flow, and let rpj be the distance from any point on the yth panel to P, as shown in Figure 3.40. The radius rpl makes the angle 0Pj with respect to the. r axis. The velocity potential induced at P due to the y’th panel, Дг/),, is, from Equation (4.3),

In Equation (4.72), yj is constant over the y’th panel, and the integral is taken over the jth panel only. The angle 6PJ is given by

[4.73]

In turn, the potential at P due to all the panels is Equation (4.72) summed over all the panels:

[4.74]

Since point P is just an arbitrary point in the flow, let us put P at the control point of the ith panel shown in Figure 3.40. The coordinates of this control point are (*,-, уi). Then Equations (4.73) and (4.74) become

-і Уі – Уі Xi-Xj

Equation (4.75) is physically the contribution of all the panels to the potential at the control point of the ith panel.

At the control points, the normal component of the velocity is zero; this velocity is the superposition of the uniform flow velocity and the velocity induced by all the vortex panels. The component of VA normal to the / th panel is given by Equation (3.148):

V0o. n = V0C cos p, [3.148]

The normal component of velocity induced at (a, , v, ) by the vortex panels is

v„ = [4.76]

an.

Combining Equations (4.75) and (4.76), we have

where the summation is over all the panels. The normal component of the flow velocity at the ith control point is the sum of that due to the freestream [Equation (3.148)] and that due to the vortex panels [Equation (4.77)]. The boundary condition states that this sum must be zero:

[4.78]

Substituting Equations (3.148) and (4.77) into (4.78), we obtain

[4.79]

Equation (4.79) is the crux of the vortex panel method. The values of the integrals in Equation (4.79) depend simply on the panel geometry; they are not properties of the flow. Let Jij be the value of this integral when the control point is on the ith panel. Then Equation (4.79) can be written as

П

Езо cos p, – ~ Ji. j = 0 [4.80]

i=і 171

Equation (4.80) is a linear algebraic equation with n unknowns, jq, Y2- ■ ■ ■■ Yn• It represents the flow boundary condition evaluated at the control point of the ith panel. If Equation (4.80) is applied to the control points of all the panels, we obtain a system of n linear equations with n unknowns.

To this point, we have been deliberately paralleling the discussion of the source panel method given in Section 3.17; however, the similarity stops here. For the source panel method, the n equations for the n unknown source strengths are routinely solved, giving the flow over a nonlifting body. In contrast, for the lifting case with vortex panels, in addition to the n equations given by Equation (4.80) applied at all the panels, we must also satisfy the Kutta condition. This can be done in several ways. For example, consider Figure 4.26, which illustrates a detail of the vortex panel distribution at the trailing edge. Note that the length of each panel can be different; their length and distribution over the body are up to your discretion. Let the two panels at the trailing edge (panels і and і — 1 in Figure 4.26) be very small. The

Kutta condition is applied precisely at the trailing edge and is given by у (ТЕ) = 0. To approximate this numerically, if points і and г — 1 are close enough to the trailing edge, we can write

Yi = ~Yi- [4.81]

such that the strengths of the two vortex panels і and і — 1 exactly cancel at the point where they touch at the trailing edge. Thus, in order to impose the Kutta condition on the solution of the flow, Equation (4.81) (or an equivalent expression) must be included. Note that Equation (4.80) evaluated at all the panels and Equation (4.81) constitute an overdetermined system of n unknowns with n + 1 equations. Therefore, to obtain a determined system, Equation (4.80) is not evaluated at one of the control points on the body. That is, we choose to ignore one of the control points, and we evaluate Equation (4.80) at the other n — 1 control points. This, in combination with Equation (4.81), now gives a system of n linear algebraic equations with n unknowns, which can be solved by standard techniques.

At this stage, we have conceptually obtained the values of y, y2,…, yn which make the body surface a streamline of the flow and which also satisfy the Kutta condition. In turn, the flow velocity tangent to the surface can be obtained directly from y. To see this more clearly, consider the airfoil shown in Figure 4.27. We are concerned only with the flow outside the airfoil and on its surface. Therefore, let the velocity be zero at every point inside the body, as shown in Figure 4.27. In particular, the velocity just inside the vortex sheet on the surface is zero. This corresponds to U2 = 0 in Equation (4.8). Hence, the velocity just outside the vortex sheet is, from Equation (4.8),

у = U — U2 = U — 0 = Ml

In Equation (4.8), и denotes the velocity tangential to the vortex sheet. In terms of the picture shown in Figure 4.27, we obtain Va = ya at point а, V/, = yh at point b, etc. Therefore, the local velocities tangential to the airfoil surface are equal to the local values of у. In turn, the local pressure distribution can be obtained from Bernoulli’s equation.

The total circulation and the resulting lift are obtained as follows. Let sj be the length of the yth panel. Then the circulation due to the у th panel is y, s r In turn, the

 Figure 4.27 Airfoil as a solid body, with zero velocity inside the profile.

total circulation due to all the panels is

n

г = &л [4-821

./=1

Hence, the lift per unit span is obtained from

П

L’ = Poo Too y. isJ [4.83]

./=>

The presentation in this section is intended to give only the general flavor of the vortex panel method. There are many variations of the method in use today, and you are encouraged to read the modem literature, especially as it appears in the A/A4 Journal and the Journal of Aircraft since 1970. The vortex panel method as described in this section is termed a “first-order” method because it assumes a constant value of у over a given panel. Although the method may appear to be straightforward, its numerical implementation can sometimes be frustrating. For example, the results for a given body are sensitive to the number of panels used, their various sizes, and the way they are distributed over the body surface (i. e., it is usually advantageous to place a large number of small panels near the leading and trailing edges of an airfoil and a smaller number of larger panels in the middle). The need to ignore one of the control points in order to have a determined system in n equations for n unknowns also introduces some arbitrariness in the numerical solution. Which control point do you ignore? Different choices sometimes yield different numerical answers for the distribution of у over the surface. Moreover, the resulting numerical distributions for у are not always smooth, but rather, they have oscillations from one panel to the next as a result of numerical inaccuracies. The problems mentioned above are usually overcome in different ways by different groups who have developed relatively sophisticated panel programs for practical use. For example, what is more common today is to use a combination of both source and vortex panels (source panels to basically simulate the airfoil thickness and vortex panels to introduce circulation) in a panel solution. This combination helps to mitigate some of the practical numerical problems just discussed. Again, you are encouraged to consult the literature for more information.

Such accuracy problems have also encouraged the development of higher-order panel techniques. For example, a “second-order” panel method assumes a linear variation of у over a given panel, as sketched in Figure 4.28. Here, the value of у at the edges of each panel is matched to its neighbors, and the values y, y2, уз, etc. at the boundary points become the unknowns to be solved. The flow-tangency boundary condition is still applied at the control point of each panel, as before. Some results using a second-order vortex panel technique are given in Figure 4.29, which shows

 Figure 4.28 Linear distribution of у over each panel—a second-order panel method.

 Figure 4.29 Pressure coefficient distribution over an NACA 0012 airfoil; comparison between second-order vortex panel method and NACA theoretical results from Reference 1 1. The numerical panel results were obtained by one of the author’s graduate students, Mr. Tae-Hwan Cho.

the distribution of pressure coefficients over the upper and lower surfaces of an NACA 0012 airfoil at a 9° angle of attack. The circles and squares are numerical results from a second-order vortex panel technique developed at the University of Maryland, and the solid lines are from NACA results given in Reference 11. Excellent agreement is obtained.

Again, you are encouraged to consult the literature before embarking on any serious panel solutions of your own. For example, Reference 14 is a classic paper on panel methods, and Reference 15 highlights many of the basic concepts of panel methods along with actual computer program statement listings for simple applica­tions. Reference 66 is a modern compilation of papers, several of which deal with current panel techniques. Finally, Katz and Plotkin (Reference 67) give perhaps the most thorough discussion of panel techniques and their foundations to date.

## Normal Shock Waves. and Related Topics

Shock wave: A large-amplitude compression wave, such as that produced by an explosion, caused by supersonic motion of a body in a medium.

From the American Heritage Dictionary of the English Language, 1969

8.1 Introduction

The purpose of this chapter and Chapter 9 is to develop shock-wave theory, thus giving us the means to calculate the changes in the flow properties across a wave. These changes were discussed qualitatively in Section 7.6; make certain that you are familiar with these changes before continuing.

The focus of this chapter is on normal shock waves, as sketched in Figure lAb. At first thought, a shock wave that is normal to the upstream flow may seem to be a very special case—and therefore a case of little practical interest—but nothing could be further from the truth. Normal shocks occur frequently in nature. Two such ex­amples are sketched in Figure 8.1; there are many more. The supersonic flow over a blunt body is shown at the left of Figure 8.1. Here, a strong bow shock wave exists in front of the body. (We study such bow shocks in Chapter 9.) Although this wave is curved, the region of the shock closest to the nose is essentially nor­mal to the flow. Moreover, the streamline that passes through this normal portion of the bow shock later impinges on the nose of the body and controls the values

 Figure 8.1 Two examples where normal shock waves are of interest.

 Normal shock inside the nozzle Overexpanded flow through a nozzle

of stagnation (total) pressure and temperature at the nose. Since the nose region of high-speed blunt bodies is of practical interest in the calculation of drag and aerodynamic heating, the properties of the flow behind the normal portion of the shock wave take on some importance. In another example, shown at the right of Figure 8.1, supersonic flow is established inside a nozzle (which can be a super­sonic wind tunnel, a rocket engine, etc.) where the back pressure is high enough to cause a normal shock wave to stand inside the nozzle. (We discuss such “over­expanded” nozzle flows in Chapter 10.) The conditions under which this shock wave will occur and the determination of flow properties at the nozzle exit down­stream of the normal shock are both important questions to be answered. In sum­mary, for these and many other applications, the study of normal shock waves is important.

Finally, we will find that many of the normal shock relations derived in this chapter carry over directly to the analysis of oblique shock waves, as discussed in Chapter 9. So once again, time spent on normal shock waves is time well spent.

The road map for this chapter is given in Figure 8.2. As you can see, our objectives are fairly short and straightforward. We start with a derivation of the basic continuity, momentum, and energy equations for normal shock waves, and then we employ these basic relations to obtain detailed equations for the calculation of flow properties across the shock wave. In addition, we emphasize the physical trends indicated by the equations. On the way toward this objective, we take three side streets having to do with (1) the speed of sound, (2) special forms of the energy equation, and (3) a further discussion of the criteria used to judge when a flow must be treated as compressible. Finally, we apply the results of this chapter to the measurement of airspeed in a compressible flow using a Pitot tube. Keep the road map in Figure 8.2 in mind as you progress through the chapter.

 Figure 8.2 Road map for Chapter 8.