Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

The parasite drag of an airplane can be estimated by estimating the drag of each component and then totaling the component drag while accounting for some interference drag. If CA. and Si are the drag coefficient and reference area, respectively, for the ith component, then the total drag will be

D = ipV2CDtSi + ipV^C^Sj + • • • + ipV2CDiS, + • • •

= pV2(CDtSi + CbjS 2 + • • • + CD, S, + • • •) (4.34)

Obviously, the drag coefficients of the components cannot be added since the reference areas are different. However, from Equation 4.34, the products CA. S, can be added. Such a product is referred to as the equivalent flat-plate area, /. One will also hear it referred to as the “parasite area” or simply, the “flat-plate area.” The connotation “flat plate” is misleading, since it is not the area of a flat plate with the same drag. Instead, it is the reference area of a fictitious shape having a Cd of 1.0, which has the same drag as the shape in question. / is therefore simply D/q. It is a convenient way of handling the •hag, since the /’s of the drag components can be added to give the total f of an airplane.

^=ад + ад+

4

or

f = 2 cDiSi

i= 1

= 2/. (4.35)

І = 1

This notation indicates that the flat-plate areas are to be summed for the ith component, from і = 1 to n where n is the total number of components.

It was stated earlier that the profile drag of an airfoil section increases approximately with the square of the section Q. Combined with the induced drag, given by Equation 4.20, the total CD for a wing can be written approximately as

Си = + kCi2 + (1 + S)

where к us the constant of proportionality giving the rate of increase of Cd with C2.

Equation 4.31 can be rewritten as

where

1

1 + 8 + кттА

The factor e is known as Oswald’s efficiency factor (see Ref. 4.2). The Ifoduct Ae is referred to as the “effective aspect ratio” and is sometimes written as Ae.

Consider data from References 3.1 and 3.27 in light of Equation 4.32. Figure 4.25 presents CD as a function of CL for the finite wing tested in Reference 3.27 and Cd versus Q from Reference 3.1 for the 65-210 airfoil.

This particular airfoil section is conducive to laminar flow for Q values between approximately 0.2 and 0.6, as reflected in the “drag bucket” in the lower curve of this figure. The “bucket” is not evident in the wing test results of Reference 3.27, either as the result of wing. surface roughness or wind tunnel flow disturbances. Neglecting the bucket in the airfoil section Cd curve, the constant, k, is found to be 0.0038. From Figure 4.22, 8 = 0.01. Thus, from the airfoil Cd curve and lifting line theory, the wing CD curve is predicted to be –

CD = 0.0055 + 0.0394CL2 (4^3)

This equation is included on Figure 4.25, where it can be seen to agree closely with the test results. Thus one can conclude that the difference in the drag between an airfoil and a wing is satisfactorily explained by the induced drag. In this particular Case, Oswald’s efficiency factor is 0.89.

Generally, for a complete airplane configuration, e is not this high because of wing-fuselage interference and contributions from the tail and other components.

High-wing and low-wing airplanes show a measurable difference in Oswald’s efficiency factor. Most likely as the result of interference between the boundary layer on the wing’s upper surface with that on the fuselage, e values for low-wing airplanes are lower than those for high-wing airplanes. The boundary layer on the upper surface of a wing is considerably thicker than the one on the lower surface. Combining with the boundary layer over the sides of the fuselage, the wing’s upper surface boundary layer, for the low-wing airplane, can cause a rapid increase in the wing and fuselage parasite drag as the angle of attack increases. For a high-wing airplane, the relatively thin boundary layer on the lower surface of the wing interferes only slightly with the fuselage boundary layer. Typically, e is equal approximately to 0.6 for low-wing airplanes and 0.8 for high-wing airplanes. These values are

confirmed by the flight tests reported in Reference 4.3 and in other data from isolated sources.

Referring again to Figure 3.54, the lift vector for a wing section is seen to be tilted rearward through the induced angle of attack, a,. As a result, а component of the lift is produced in the streamwise direction. This com­ponent, integrated over the wingspan, results in the induced drag. For a differential element,

dDi _ dL dy dy .

Defining the induced drag coefficient as

For the special case of an untwisted elliptic wing, a, and Q are constant over the span, so that Equation 4.17 becomes

Сц = a, Cl

The induced angle of attack for this case was given previously by Equation 3.7^. Thus,

(4.19)

This is a well-known and often-used relationship that applies fairly well to other than elliptic platforms. For a given aspect ratio and wing lift coefficient, it can be shown (Ref. 4.1) that Equation 4.19 represents the minimum achievable induced drag for a wing. In other words, the elliptic lift distribution is optimum from the viewpoint of induced drag.

In order to account for departures from the elliptic lift distribution and the dependence of the parasite drag on angle of attack, Equation 4.19 is modified in practice in several different ways. Theoretically one can calculate the downwash and section lift coefficients, either analytically or numerically, according to the methods of the previous chapter. These results can then be substituted into Equation 4.17 to solve for CDi. The final result for an arbitrary planform is usually compared to Equation 4.19 and expressed in the form

(4.20)

8, for a given planform shape, is a constant that is normally small by comparison to unity. It therefore represents, for a given wing, the fractional increase in the induced drag over the optimum elliptic case.

The numerical determination of 8 will now be outlined for the simplified lifting line model and some typical results will be presented. As an exercise, develop your own program; it is not too lengthy if a subroutine is available for solving simultaneous linear algebraic equations.

A lifting line model composed of discrete vortex line elements is pictured in Figure 4.21. For clarity only five trailing vortices are shown on each side. Their strengths and positions are symmetrical about the centerline. These can be equally spaced but, with the loading dropping off faster at the tips, it is better to have a closer spacing in this region.

A particular trailing vortex is located a distance of yv from the centerline. Control points are then chosen midway between the vortex lines. Generally,

Ус, = kyv, + y„(_,) 1 < і < n

Ус, = 0

At a control point, the bound circulation, Г, is equal to the sum of the strengths of the vortex lines shed outboard of the point. Thus, for n trailing vortices,

or, generally,

г,= Іт,

i=j

For computational efficiency, it should be recognized that

Г.+І = Г, – 7,

Thus it is expedient to determine Г, first and then apply this recursion relationship in order to determine Г2, Г3,…, Г„.

The section lift is given by

f-ipWC

It can also be obtained from the Kutta-Joukowski law as

dL т/p

drpVT

Comparing these two relationships for dL/dy, it follows that

Г = 2 cQV

(4.23)

Thus Equations 4.21 to 4.23 are interrelated, since the downwash, w, depends on the Г distribution.

By applying the Biot-Savart law to our discrete vortex line model, w can be determined along the lifting line (neglecting any contribution from the lifting line itself). The downwash at the j’th control point produced by the ith trailing vortex is

In the parentheses, the first term arises from the downwash at a point on the right side of the wing produced by a vortex trailing from the left. The second term is from the vortex of opposite rotation and symmetric position trailing from the right wing. This expression reduces to

(4-24)

or

(4.25)

The total downwash at yCj is then found by summing Equation 4.13 over і from 1 to n.

(4.26)

Without any loss of generality, it is convenient to set the free-stream velocity, V, the density, p, and the wing semispan, b/2, equal to unity. It then follows that

(4.27)

or

This can be rewritten as

where

The preceding is of the form

E Д.7, = Bj j = 1,2,3,…, n ‘ (4.28)

i = l

and represents n simultaneous linear algebraic equations that can be solved for the unknown vortex strengths yt, y2,…. у(,…, yn.

Since b/2 is set equal to unity, the chord lengths, c„ must be expressed as a fraction of b/2. The values of cf are defined at the control point locations. Having determined the – y, values, the wing lift coefficient can be calculated from

L = 2[Г, уг, + Г2(у„2 – yv) + • • • + Г i(y„. – у и,.,) + • • • + Г„(у„„ – ус„_,)] or, in coefficient form,

CL = A jr, yC| + 2Гi(yVi ~ (4.29)

The aspect ratio, A, appears, since the dimensionless reference wing area equals the actual area divided by (b/2)2. w, is calculated from Equation 4.26 so that the induced drag follows from Equation 4.15.

Di = 2[r1y„1w, + r2(yt,2-y„1)w2+ • • – + Г;(у„,.-yc.^)Wi + • • • + Г„(у„л – y„„.,)w„] or, in coefficient form,

Cd, = A JViy0|w, + 2 Г,-Wj(yVi – y„, ,)] (4.30)

The accuracy of the foregoing numerical formulation of the lifting line model can be tested by applying it to the elliptical planform. Using 25 vortices (n = 25) trailing from each side of the wing, Equations 4.26 to 4.30 are evaluated for flat, untwisted elliptical planform shapes having aspect ratios of 4, 6, 8, and 10 using a theoretical section lift curve slope of 2rrC//rad. The numerical results are presented in Table 4.1 for CL and CDi where they are

compared with corresponding analytical values. The CL values are seen to agree within W of each other, while the CDi values differ by only 2% at the most.

With confidence established in the numerical program, the calculations presented in Figures 4.22, 4.23, and 4.24 can be performed. Here unswept wings with linearly tapered planforms are investigated. For this, family of wings, the span wise chord distribution is defined by

c

or

-= 1-(1-A)X

Co

where A equals the taper ratio, cr/c0, cT and c0 being the tip and root chord, respectively. X is the distance along the span measured out from the centerline as a fraction of the semispan.

Figure 4.22 presents 8 as a function of the taper ratio for aspect ratio values of 4, 6, 8, and 10. For all of the aspect ratios, 8 is seen to be a minimum at a taper ratio of around 0.3, being less than 1% higher than the ideal elliptic case at this value of A. The rectangular wing is represented by a A value of 1.0. For this planform shape, used on many light, single-engine

aircraft, the induced drag is seen to be 6% or higher than that for the elliptic wing for aspect ratios of 6 or higher.

*• The results of Figure 4.22 can be explained by reference to Figure 4.23, which presents spanwise distributions of Г for the elliptic, rectangular, and

0. 25 taper ratio wings. Observe that the distribution for A = 0.25 is close to the elliptic distribution. Thus the kinetic energies of the trailing vortex systems shed from these two Г distributions are about the same. On the other hand, the Г distribution for the rectangular planform is nearly constant inboard out to about 70% of the semispan and then drops off more rapidly than the elliptic distribution toward the tip. Thus the kinetic energy per unit length of the trailing vortex system shed from the rectangular ■ wing is approximately 6% higher than the energy left in the wake by the tapered or elliptic wing.

In view of the preceding one might ask why rectangular planforms are used in many general aviation airplanes instead of tapered planforms. Part of the answer lies with the relative cost of manufacture. Obviously, the rec­tangular planform with an untapered spar and constant rib sections is less costly to fabricate. Figure 4.24 discloses a second advantage to the rec­tangular planform. Here, the section lift coefficient is presented as a ratio to the wing lift coefficient for untwisted elliptic, rectangular, and linearly tapered planforms. For the elliptic wing, the section Q is seen to be constant and equal to the wing CL except in the very region of the tip, where numerical errors show an increase in Cj contrary to the analytical solution. The rec­tangular planform shows the section Q to be higher than the wing Cl at the

centerline and gradually decreasing to zero at the tip. The tapered planform, however, has a section Q that is lower than the wing Cl at midspan. Its Q then increases out to approximately the 75% station before decreasing rapidly to zero at the tip. Thus, again reiterating the discussions of the previous chapter, the tapered planform, unless twisted, will stall first outboard, resul­ting in a possible loss of lateral control.

The two major components of the total drag of an airplane are the induced drag and the parasite drag. The parasite drag is the drag not directly associated with the production of lift. This drag, expressed as a coefficient, is

nearly constant and approximately equal to the drag for an airplane lift coefficient of zero. As the lift coefficient takes on a value different than zero, the drag coefficient will increase. This increment in Cd is defined as the induced drag coefficient, Q. Thus, for an airplane,

Cd = Сц, + Сц (4.14)

Here Сц, is the parasite drag coefficient and is not a function of CL. On the other hand, the induced drag coefficient, CD|, varies approximately as the square of CL. This dependence will be derived later.

Strictly speaking, this definition of Сц is not correct. Although it has become practice to charge to CDi any drag increase associated with CL, some of this increase results from the dependency of the parasite drag on the angle of atfack. What, then, is a more precise definition of Сц? Very simply, the induced drag at a given CL can be defined as the drag that the wing would experience in an inviscid flow at the same Cl. D’Alembert’s paradox assures us that a closed body can experience no drag in an inviscid flow. However, as we saw in the last chapter, a wing of finite aspect ratio generates a trailing vortex system that extends infinitely far downstream. Thus, the system in effect is not closed, because of the trailing vortex system that continuously transports energy across any control surface enclosing the wing, no matter how far downstream of the wing this surface is chosen.

In addition to skin friction drag, a body generally experiences some form drag. Unlike the skin friction drag that results from viscous shearing forces tangential to a body’s surface, form drag results from the distribution of pressure normal to the body’s surface. The extreme case of a flat plate normal to the flow is pictured in Figure 4.3. Here the drag is totally the result of an unbalance in the normal pressure distribution. There is no skin friction drag present in this case.

Generally, form drag is difficult to predict. For that matter, so is skin friction drag except for the simplest cases. Thus, in general cases, such as that pictured in Figure 4.4, where the total drag results from both normal and tangential stresses (or pressures) one must usually resort to experimental data to estimate the drag.

v

As with skin friction drag, form drag is generally dependent on Reynolds number.‘To see why, consider the flow around the circular cylinder pictured in Figure 4.5. In Figure 4.5a flow is pictured at a low Reynolds number. Here, beginning at the stagnation point, a laminar boundary layer develops. On the surface of the cylinder, the static pressure (normal) is highest at the stag­nation point and decreases to a minimum at the top and bottom. Moving around toward the rear, beyond these points, the static pressure increases, tending toward the stagnation pressure at the very rear. In the absence of viscosity the normal pressure distribution would be symmetrical (Equation 2.78) and there would be no drag. This is a clear example of D’Alembert’s paradox, which states that a body in a inviscid fluid will experience no drag. As the slower moving fluid in the laminar boundary layer moves beyond the minimum pressure point on the cylinder, its momentum is insufficient to move against the positive pressure gradient, referred to as an adverse gradient, and thus the flow separates just past the top and bottom locations on the cylinder. In the separated region over most of the rear portion of the cylinder the static pressure is constant and equal to the low pressure at the top and bottom. Thus the high pressure acting over the front and the low pressure over the rear result in a high form drag.

The high-Reynolds number case is shown in Figure 4.5b. Here the laminar boundary layer undergoes transition to a turbulent boundary layer before separating. The subsequent turbulent mixing increases the momentum and energy of the boundary layer so that it remains attached toward the rear of the cylinder, well beyond the separation point of the laminar layer. Thus, in this case, the separation region is much smaller and the static pressure is higher on the rear of the cylinder than for the laminar case. Therefore, because of reduced form drag, the drag coefficient of a cylinder isjower at higher Reynolds numbers.

Cd as a function of Reynolds number is presented in Figure 4.6 for both spheres and two-dimensional circular cylinders. Here, Q is based on the projected frontal area. Note the rapid drop in Q above an R value of approximately 2x 105. This is the so-called critical Reynolds number^ whet? the transition point is nearly coincident with the separation point. “Sub – critical” refers to flow at Reynolds numbers that are less than critical; “supercritical” denotes R values that are higher than critical. A body shape having a well-defined separation point will not exhibit a critical Reynolds number; neither will streamlined shapes.

Although not concerned with drag per se, Figure 4.6a also includes the

I

(b)

Figure 4.6 Drag coefficients of cylinders and spheres versus Reynolds number,

(a) Two-dimensional circular cylinders, (b) Spheres.

quantity fD/V, known as the Strouhal number, S. S characterizes an interes­ting behavior of bluff bodies with rounded trailing edges. As such a body first begins to move through a fluid, the vorticity in the boundary layer is shed symmetrically from the upper and lower surfaces to form two vortices of opposite rotation. However, the symmetrical placement of the vortex pair is unstable, so that succeeding vortices are then shed alternately from the upper and lower surfaces. The resulting flow pattern of periodically spaced vortices downstream of the body is known as a Karman vortex street.

In the definition of Strouhal number, / is the frequency at which the

vortices are shed. As a vortex is shed from one surface of the cylinder, it produces a momentary circulation around the cylinder opposite in direction to the vortex. From the Kutta-Joukowski law, a force on the cylinder normal to V results. As the next vortex is shed, the force reverses its direction, resulting in an alternating force on the cylinder. This particular phenomenon is the cause for the “singing” of telephone wires in the wind.

As an example of the use of Figure 4.6a, consider a wire 2 cm in diameter in a wind blowing at a speed of 8 m/s. Assuming standard sea level conditions,

R = ™ v

8(0.02)

1.456 X 10 5

= 1.099 x 104

From Figure 4.6a, for this Reynolds number,

Figure 4.7a Examples of shapes having Ca values nearly independent of Rey­nolds number.

If the ratio of the span to the height (or diameter) of a flat plate (or cylinder) normal to the flow is approximately 5 or less, Cd is nearly constant and equal to the 3-D value. For aspect ratios greater than 5, Cd varies approximately in the manner given by the normalized curve of Figure 4.7 b. This curve is based on data from several sources, including Reference 4.4.

A qualitative evaluation of the drag coefficient for a given shape can be made using some “educated intuition.” Referring to Figure 4.8, the drag

Crf = 1.55

X cd – 0.3

jected frontal areas becomes

Q(3-D) Sw(3-D) 4 Cd(2-D) S„(2-D) irD

where D is the maximum three-dimensional body diameter or the maximum thickness of the two-dimensional shape. For an elliptical two-dimensional shape compared to an ellipsoid, this becomes

Cd(3-D) – гг CA2-D) 2

This is close to the ratio from Figure 4.11 for a finess ratio of 8 and only slightly lower than the corresponding ratio given earlier for the form drag.

Minimum profile drag coefficients for NACA four – and five-digit airfoils are presented in Figure 4.12 as a function of thickness ratio at a Reynolds number of 6 x 106. Here, as is usual for airfoils, Q is based on the chord length. The several data points at each thickness ratio result from airfoils of different camber ratios. Note that Cdmi„ does not vary significantly with

camber. CdmiI1 appears to vary almost linearly with t/c and extrapolates to a value of 0.004 for a tic of zero. This corresponds to a Q value of 0.002. According to Figure 4.1, this would require laminar flow over these sections more extensive than one would expect. Probably, transition is delayed until approximately the 25% chord point, the location of maximum thickness. One would then expect a Cdmin value of about 0.005.

Figure 4.13 presents three-dimensional drag data directly comparable to Figure 4.11, but with more detail. Data representing practical fuselage and nacelle construction are included in Figure 4.13 together with Cd results from torpedo-shaped bodies. Assuming a reasonable relationship between the frontal and wetted areas of such bodies, expected Cd values for various values of Cf are also included on the figure. For a given Q value, the experimental results should approach one of these lines as the fineness ratio gets large.

For fully turbulent flow at an R of 25 x 106, Q for a flat plate would be 0.0026, whereas the data appears to be approaching a Q of 0.0032 to 0.0034. The higher skin friction drag on the bodies is probably the result of surface roughness.

It is interesting to examine the data of Figure 4.13 in terms of minimum drag for a given body volume. This is particularly important for airship and underwater applications. It is also of interest to the design of tip tanks, where minimum drag for a given volume of fuel is desirable. Denoting the volume by Vm, we will define another drag coefficient.

(4.12)

Cdv is related to Cd in Figure 4.13 by

Obviously, the ratio of the frontal area, A, to the 2/3 power of the volume depends, on the particular body shape. We will assume the body to be composed approximately of a hemispherical nose, a cylindrical midbody extending to the middle of the body, and a tail cone. For this particular shape,

(4.13)

Using this relationship and Figure 4.13, the graphs presented in Figure 4.14 were obtained. From this figure it can be seen that to enclose a given volume with a minimum drag body, its fineness ratio should be higher than the optimum values from Figure 4.13. Indeed, for fuselages, the drag for a given volume is nearly constant for lid values from 4 to 10.

I* Figure 4.14 Drag coefficients based on volume for bodies as a function of fineness ratio.

For certain applications, it is desirable to keep the rear portion of a fuselage as wide and bluff as possible without paying too much of a drag penalty. If the afterbody is tapered too abruptly, flow separation will occur over the rear, resulting in an unduly high form drag. Some guidance in this regard is provided by Figure 4.15 (taken from Ref. 4.8). Here, the increment in Q (based on frontal area) resulting from afterbody contraction is presented as a function of afterbody geometry. From this figure it appears that the ratio of the afterbody length to the equivalent diameter should be no less than approximately 2.0.

The importance of streamlining is graphically illustrated in Figure 4.16, which is drawn to scale. Conservatively (supercritical flow), the ratio of Cd for a circular cylinder to a two-dimensional streamlined shape having a fineness ratio of 4 is approximately 7.5. Thus, as shown in Figure 4.16, the height of the streamlined shape can be 7.5 times greater than the circular cylinder for the same drag. For subcritical flow the comparison becomes even more impressive, with the ratio increasing to approximately 25.

When two shapes intersect or are placed in proximity, their pressure distributions and boundary layers can interact with each other, resulting in a net drag of the combination that is higher than the sum of the separate drags. This increment in the drag is known as interference drag. Except for specific cases where data are available, interference drag is difficult to estimate accurately. Some examples of interference drag are presented in Figures 4.17,

4.18, and 4.19.

Figure 4.17 illustrates the drag penalty that is paid for placing an engine nacelle in proximity to a rear pylon on a tandem helicopter (like a CH-47). In this particular instance, the interference drag is nearly equal to the drag of the nacelle alone, because the nacelle is mounted very close to the pylon. For spacings greater than approximately one-half of a nacelle diameter, the interference drag vanishes.

Figure 4.18 presents the interference drag between the rotor hub and pylon for a helicopter. The trends shown in this figure are similar to those in the previous figure. In both instances the added interference drag is not necessarily on the appended member; probably, it is on the pylon.

Figure 4.19 shows a wing abutting the side of a fuselage. At the fuselage­wing juncture a drag increment results as the boundary layers from the two airplane components interact and thicken locally at the junction. This type of drag penalty will become more severe if surfaces meet at an angle other than 90°. In particular, acute angles between intersecting surfaces should be avoided. Reference 4.4, for example, shows that the interference drag of a 45% thick strut abutting a plane wall doubles as the angle decreases from 90°

to approximately 60°. If acute angles cannot be avoided, filleting should be used at the juncture.

In the case of a high-wing configuration, interference drag results prin­cipally from the interaction of the fuselage boundary layer with that from the wing’s lower surface. This latter layer is relatively thin at positive angles of attack. On the other hand, it is the boundary layer on the upper surface of a

*•

low wing that interferes with the fuselage boundary layer. This upper surface layer is appreciably thicker than the lower surface layer. Thus the wing – fuselage interference drag for a low-wing configuration is usually greater than for a high-wing configuration.

The available data on wing-fuselage interference drag are sparse. Reference 4.4 presents a limited amount but, even here, there is no correlation with wing position or lift coefficient. Based on this reference, an approximate drag increment caused by wing-fuselage interference is estimated to equal 4% of the wing’s profile drag for a typical aspect ratio and wing thickness.

Although data such as those in Reference 4.4 may be helpful in estimat­ing interference drag, an accurate estimate of this quantity is nearly im­possible. For example, a wing protruding from a fuselage just forward of the station where the fuselage begins to taper may trigger separation over the rear portion of the fuselage.

Sometimes interference drag can be favorable as, for example, when one body operates in the wake of another. Race car drivers frequently use this to their advantage in the practice of “drafting.” Some indication of this favor-

able interference is provided by Figure 4.20, based on data obtained in Pennsylvania State University’s subsonic wind tunnel. Here the drag on one rectangular cylinder in tandem with another is presented as a function of the distance between the cylinders. The cylinders have a 2:1 fineness ratio. Tests were performed with the long side oriented both with and normal to the free-stream velocity. The drag is referenced with respect to D„, the drag on the one cylinder alone. The spacing is made dimensionless with respect to the dimension of the cylinder normal to the flow. The spacing, x, is positive when the cylinder on which the drag is being measured is downstream of the other. Notice that the cylinder’s drag is reduced significantly for positive x values and even becomes negative for small positive values of x. For small negative Values of x, the drag is increased slightly. Similar data for circular cylinders presented in Reference 4.4 show somewhat similar results, except that inter­ference on the forward cylinder is slightly favorable for spacings less than three diameters. For the downstream cylinder, the drag is reduced by a factor ®f 0.3 for spacings between three and nine diameters. For less than three diameters, the downstream drag is even less, and becomes negative for spacings less than approximately two diameters.

Figure 4.1 depicts a thin, flat plate aligned with the free-stream velocity. Frequently the drag of a very streamlined shape such as this is expressed in terms of a skin friction drag coefficient, Q, defined by,

(4.1)

where Sw is the wetted surface area that is exposed to the flow. This coefficient is presented in Figure 4.1 as a function of Reynolds number for the two cases where the flow in the boundary layer is entirely laminar or entirely turbulent over the plate. Here the Reynolds number is based on the total length of the plate in the direction of the velocity. In a usual application, the boundary layer is normally laminar near the leading edge of the plate undergoing transition to a turbulent layer at some distance back along the surface, as described in Chapter Two. The situation is pictured in Figure 4.1, where the velocity profile through the layer is shown. In order to illustrate it, the thickness of the layer is shown much greater than it actually is.

As shown in this figure, a laminar boundary layer begins to develop at the leading edge and grows in thickness downstream. At some distance from the leading edge, the laminar boundary becomes unstable and is unable to suppress disturbances imposed on it by surface roughness or fluctuations in the free stream. In a short distance the boundary layer undergoes transition to a turbulent boundary layer. Here the layer suddenly increases in thickness and is characterized by a mean velocity profile on which a random fluctuating velocity component is superimposed. The distance, x, from the leading edge

•Turbulent boundary layer

Laminar boundary layer

of the plate to the transition point can be calculated from the transition Reynolds number, Rx. Rx is typically, for a flat plate, of the order of 3 x 105, Rx being defined by

For very smooth plates in a flow having a low level of ambient turbulence, Rx can exceed 1 x Ю6.

Since the velocity profile through the boundary layer approaches the velocity outside the layer asymptotically, the thickness of the layer is vague. To be more definitive, a displacement thickness, 8*, is frequently used to measure the thickness of the layer. 5* is illustrated in Figure 4.2 and is defined mathematically by

where у is the normal distance from the plate at any location such that, without any boundary layer, the total flow past that location would equal the flow for the original plate with a boundary layer. To clarify this further, let 8 be the boundary layer thickness where* for all intents and purposes, и = V.

Figure 4.2 Displacement thickness.

Then

Observe that relatively speaking, the turbulent boundary layer is more uniform, with 5* being only one-eighth of 8 as compared to one-third for the laminar layer.

In order to clarify the use of Figure 4.1 and Equations 4.5 to^4.8, let us consider the horizontal tail of the Cherokee pictured in Figure 3.62 at a velocity of 60.4 m/s (135 mph) at a 1524 m (5000 ft) standard altitude. We will assume that the tail can be approximately treated as a flat plate at zero angle of attack.

From Figure 3.62, the length of the plate is 30 in or 0.762 m. The total wetted area, taking both sides and neglecting the fuselage, is 4.65 m2 (50 ft2). From Figure 2.3, at an altitude of 1.52 km, p= 1.054 kg/m3 and v = 1.639 x 10~5 m2/s. We will assume that the transition Reynolds number is equal to 3 x 105.

The distance from the leading edge to the transition point is found from Equation 4.2.

x =

= 1.639 xl(T5x3x

= 0.0814 m (3.2 in.)

The Reynolds number based on the total length will be equal to

V

60.4 (0.762)

1.639 xlO’5

= 2.81 x 106

If the flow over the tail were entirely turbulent then, from Figure 4.1,

C, =0.455 (logioK,)’2 58 (4.9)

= 0.00371

The dynamic pressure q for this case is

q = pV2ll

1.054 (60.4)2
2

= 1923 N/m2

Hence the total skin friction drag would be

D = qSwCf

=4923^4.65X0.00371)

= 33.17 N

However, the leading portion of the plate is laminar. The wetted area of this portion is equal to 0.497 m2. For laminar flow over this portion,

Cf = 1.328R112 (4.10)

= 1.328 (3 x 105Г,/2 = 0.00242

Hence the drag of this portion of the plate is equal to

D = qCfSw

= 1923(0.0024^)(0.497)

-2.31 N

К the flow were turbulent over the leading portion of the plate, its Q would be

Cf = 0.455 (logw-R)"258 = 0.455 (togio 3 x 105r2 58 = 0.00566

Thus its drag for a turbulent boundary layer would be

D = qCfSw

= (1923)(0.00566)(0.497)

= 5.35 N

The above is 5.35-2.31, or 3.04 N higher than the actual drag for laminar flow. Hence this difference must be subtracted from the total drag of 33.17 N previously calculated assuming the boundary layer to be turbulent over the entire plate. Hence the final drag of the total horizontal tail is estimated to be

D = 33.17-3.04 = 30.13 N = 6.77 lb.

The thickness, 5, of the laminar boundary layer at the beginning of transition can be calculated from Equation 4.5.

8 = 5.2 (0.0814X3 x 105)1/2 = 7.728 xl0~4m = 0.0304 in.

The thickness of the turbulent layer right after transition is found from Equation 4.7 assuming the layer to have started at the leading edge.

8 = 0.37(0.0814)(3 x 105)1/5 = 2.418 x 10 3 m = 0.0952 in.

At the trailing edge, the thickness of the turbulent layer will be

8 = 0.37(0.762X2.81 x 106Г,/5 = 0.0145 m = 0.5696 in.

The displacement thickness at the trailing edge is thus only 0.0018 m (0.071 in.).

Before leaving the topic of skin friction drag, the importance of surface roughness should be discussed. Surface roughness can have either a beneficial or adverse effect on drag. If it causes premature transition, it can result in a

reduced form drag by delaying separation. This is explained more fully in the next section. Adversely, surface roughness increases the skin friction coefficient. First, by causing premature transition, the resulting turbulent Q is higher than Cf for laminar flow, in accordance with Figure 4.1. Second, for a given type of flow laminar or turbulent, Q increases as the surface is roughened.

It is difficult to quantify the increment in Q as a function of roughness, since roughness comes in many forms. For some information on this, refer to the outstanding collection of drag data noted previously (e. g., Ref. 4.4). Generally, if a roughness lies well within the boundary layer thickness, say of the order of the displacement thickness, then its effect on Q will be minimal. Thus, for the preceding example of the horizontal tail for the Cherokee, the use of flush riveting near the trailing edge is probably not justified.

An approximate estimate of the effect of roughness, at least on stream­lined bodies, can be obtained by examining the airfoil data of Reference 3.1. Results are presented for airfoils having both smooth and rough surfaces. The NACA “standard” roughness for 0.61-m (2-ft) chords consisted of 0.028-cm (0.011-in.) carborundum grains applied to the model surface starting at the leading edge and extending 8% of the chord back on both the upper and lower surfaces. The grains were spread thinly to cover 5 to 10% of the area.

An examination of the drag data with and without the standard roughness discloses a 50 to 60% increase in airfoil drag resulting from the roughness. It is difficult to say how applicable these results are to production aircraft. Probably the NACA standard roughness is too severe for high-speed aircraft employing extensive flush riveting with particular attention to the surface finish. In the case of a production light aircraft for general aviation usage, the standard roughness could be quite appropriate.

As a child, it was fun to stick your hand out of the car window and feel the force of the moving, invisible air. To the aeronautical engineer, however, there is nothing very funny about aerodynamic drag. A continuing struggle for the practicing aerodynamicist is that of minimizing drag, whether it is for an airplane, missile, or ground-based vehicle such as an automobile or train. It takes power to move a vehicle through the air. This power is required to overcome the aerodynamic force on the vehicle opposite to its velocity vector. Any reduction of this force, known as the drag, represents either a direct saving in fuel or an increase in performance.

The estimation of the drag of a complete airplane is a difficult and challenging task, even for the simplest configurations. A list of the definitions of various types of drag partly reveals why this is so.

Induced Drag The drag that results from the generation of a trailing vortex system downstream of a lifting surface of finite aspect ratio.

Parasite Drag The total drag of an airplane minus the induced drag. Thus, it is the drag not directly associated with the production of lift. The parasite drag is composed of many drag components, the definitions of which follow.

Skin Friction Drag The drag on a body resulting from viscous shearing stresses over its wetted surface (see Equation 2.15).

Form Drag (Sometimes Called Pressure Drag) The drag on a body resulting from the integrated effect of the static pressure acting normal to its surface resolved in the drag direction.

Interference Drag The increment in drag resulting from bringing two bodies in proximity to each other. For example, the total drag of a wing-fuselage combination will usually be greater than the sum of the wing drag and fuselage drag independent of each other.

Trim Drag The increment in drag resulting from the aerodynamic forces required to trim the airplane about its center of gravity. Usually this takes the form of added induced and form drag on the horizontal tail.

Profile Drag Usually taken to mean the total of the skin friction drag and form drag for a two-dimensional airfoil section.

Cooling Drag The drag resulting from the momentum lost by the air that passes through the power plant installation for purposes of cooling the engine, oil, and accessories.

Base Drag The specific contribution to the pressure drag attributed to the blunt after-end of a body.

Wave Drag Limited to supersonic flow, this drag is a pressure drag resulting from noncanceling static pressure components to either side of a shock wave acting on the surface of the body from which the wave is emana­ting.

With the exception of wave drag, the material to follow will consider these various types of drag in detail and will present methods of reasonably estimating their magnitudes. Wave drag will be discussed in Chapter 6.

Occasionally one has the need for airfoil characteristics at Reynolds number values much lower than those used by the NACA and others to obtain the majority of the readily available airfoil data. Most of these data

were obtained at R values of 3 x 106 and higher. For remote-piloted vehicles (RPV), model airplanes, and the like, Reynolds numbers as low as 5 x 104-can be encountered. A search of the literature will show little airfoil data available in this Reynolds number range. The most reliable low-Reynolds number airfoil data appear to be those given in Reference 3.35, where tests of five different airfoil shapes are reported for R values as low as 42,000. These tests were conducted in a low-turbulence tunnel.

The five airfoil shapes that were tested in Reference 3.37 are shown in Figure 3.64. These are seen to comprise a thin, flat plate, a thin, cambered plate, two 12% thick airfoils with 3 and 4% camber, and one 20% thick airfoil with 6% camber. The airfoil shapes are similar in appearance to the NACA four-digit series.

The lift curves for these airfoils are presented in Figure 3.65 for four different Reynolds numbers. As one might expect, the flat-plate results are nearly independent of R since the separation point at the leading edge is well defined. To a slightly lesser degree, the same can be said for the cambered plate. The form of the lift curves for the three airfoils is seen to change substantially, however, over the R range from 4.2 x 105 down to 0.42 x 105. Particularly at the very lowest Reynolds number, the C, versus a curve is no longer linear. The flow apparently separates at all positive angles just down­stream of the minimum pressure point, near the maximum thickness location.

The N60R airfoil

Figure 3.64 Airfoil shapes tested at low Reynolds numbers.

Figure 3.68 Drag polar for the flat-plate airfoil at low Figure 3.69 Drag polar for the 625 airfo

Reynolds numbers. Reynolds numbers.

This explanation is substantiated by Figure 3.66. Here Q versus a is given for the N60 airfoil. As a is first increased up to a value well beyond the stall and then decreased, a large hysteresis is seen to exist in the curves for the higher Reynolds numbers. Typically, as a is increased, complete separation on the upper surface occurs at around 12°. The angle of attack must then be decreased to around 5° before the flow will again reattach. At the lowest Reynolds number, the lift curve tends to follow the portion of the curves at the higher Reynolds numbers after stall has occurred and a is decreasing. Thus, above an a of approximately 0°, it would appear that the flow is entirely separated from the upper surface for the lower R values of 21,000 and 42,000.

Aerodynamic drag is considered in more detail in the following chapter. Nevertheless, the drag characteristics for these low-Reynolds number tests are presented now in Figures 3.67 to 3.71.

A Piper Cherokee PA-28 is pictured in Figure 3.62. Pertinent dimensions, areas, weights, and other data are tabulated on the figure. Extrapolating the

swept leading edge near the root into the fuselage centerline and accounting for the elliptically shaped tips gives a total wing area when the flaps are extended of 165.1ft2. The area of the wing within the fuselage is 25.3 ft2. Assuming beforehand, or by iteration, a reasonable value for the “stalling speed” of 60 mph leads to a Reynold’s number of approximately 3 x 106 for a wing section. For this Reynold’s number, Reference 3.1 shows a value for C(max of 1.45 for the plain 652-415 airfoil with a lift curve slope of 0.106 CJdeg.

Using an 18.5% chord, single-slotted flap deflected 40°, Figures 3.32, 3.33, and 3.34 predict а ДCt of 1.37 corresponding to an increase of 12.9° in the angle of attack of the zero lift line. ДС, тах is estimated at 1.33 giving a Cimax of 2.78 for the flapped wing sections. The derivative dCJdCi is estimated from Figure 3.31 to equal -0.20. Since CMac — —0.07 for the plain airfoil (Ref. 3.1), Смж— -0.34 for the flapped airfoil.

Accounting for the 2° of washout and the increment of айі caused by flaps leads to an aW(l value of 6.0° from Equation 3.76. This is the angle of attack of the zero lift line at midspan for a zero wing CL. Cta and C, h can then be calculated for the wing alone and are given in Table 3.3. The section Cimax

values are also included in the table. A small amount of trial and error will show that the wing stalls initially at 2ylb of around 0.3 at a wing Cl of 1.97. This estimated wing must next be corrected for the effect of the

fuselage.

However, since the cross section of the Cherokee’s fuselage is essentially rectangular, and with the low-wing configuration, the correction to for the fuselage is taken to be zero.

The aerodynamic moment of the wing is determined by integrating the section pitching moments from the wing-fuselage juncture to the wing tip.

fbl2

M = 2 I qc2Cm dy

J у fuse

Expressed as a moment coefficient,

M

qSc

■ f [IT (?) <-°-34) +L (?) (-oo7) *]

In this case be = S and c = 66 in., and CM becomes

Cm = -0.198

Assuming the increment in drag from the flaps to have a negligible effect on

the trim, Equation 3.82 becomes

= 1.86

In a similar manner, trim values were calculated for flap angles of 0, 10, and 25°. The resulting values were found to be 1.33, 1.42, and 1.70, respectively. These results are presented in Figure 3.63 together with experimental values. The points labeled “flight tests” were obtained by aerospace engineering students at The Pennsylvania State University as one of the experiments in a course on techniques of flight testing. The other two points were calculated from the stalling speeds quoted by the manufacturer in the airplane’s flight manual.

In order to calculate the stalling speed of an afrplane in steady flight, one must consider that, in addition to the weight, the wing’s lift must support any download on the horizontal tail required to trim the airplane around its pitching axis. In order to determine this additional trim load, refer to Figure

3.61. Here, tfae wing lift L, the tail lift, LT, the pitching moment about the wing’s aerodynamic center, Ми, and the weight are all shown in a positive sense. With the aerodynamic center of the tail located a distance of lT behind the center of gravity and the wing’s aerodynamic center a distance of (xcg-дсас)с ahead, the tail lift to trim is given by

г Afac T C. .

LT – —I – Lyixcg – xac)

It It

In addition, static equilibrium in the vertical direction requires that

L + LT = W

Therefore, it follows that

W = L[ l+^(*Cg-*ac)]+^

In coefficient form this becomes

CL = CLw [l + f<xct – xac)] + Cnjfc (3.82)

Here, CL is taken to mean the trim CL-

CLw refers to the untrimmed wing lift coefficient corrected for the fuselage.

It was mentioned earlier that the added drag caused by flaps must sometimes be considered in the trim of an airplane. If A CD denotes this increment in the drag coefficient and if the flaps are located a distance of h above the center of gravity, Equation 3.82 modified to account for the flap

(3.83)

ACd can be obtained experimentally or estimated on the basis of Equations 3.45 and 3.46.

CMac is normally negative and greater in magnitude than the moments resulting from CLw and ДC/> Thus CL is normally less than CLw. Since Equation 3.83 holds for maximum lift conditions, it follows that the trim is normally less than the wing C^.

In calculating CMac for use in Equation 3.83, the section aerodynamic moment determined from CMac, including the increment because of the flaps, is integrated over the wing excluding the part submerged in the fuselage.