Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

The preceding material on lift and drag was limited primarily to incompressible flows, that is, to Mach numbers less than approximately 0.4. Compressibility becomes more and more important as the Mach number increases. In the vicinity of a Mach number of unity, airfoils and wings undergo a radical change in their behavior. It is not too surprising, therefore, to find that the equations covering the flow of air undergo a similar change at around M«, = 1.0.

Many textbooks are devoted entirely to the subject of compressible flows. References 5.1 and 5.2 are two such examples. Here, the several equations and techniques for the study of gas dynamics are developed in Considerable detail. An excellent, lucid, qualitative explanation of com­pressibility effects on wings and airfoils is found in Reference 5.3.

QUALITATIVE BEHAVIOR OF AIRFOILS AS A FUNCTION OF MACH NUMBER

We will consider three regimes of flow around an airfoil. In the first, the flow is everywhere subsonic with a relatively high Mach number. The second regime is referred to as transonic flow. Here the free-stream Mach number is less than unity, but sufficiently high so that the flow locally, as it accelerates over the airfoil, exceeds the local speed of sound; that is, locally the flow becomes supersonic. The lowest free-stream Mach number at which the local flow at some point on the airfoil becomes supersonic is known as the critical Mach number. The third regime is the supersonic flow regime, in which the free-stream Mach number exceeds unity. Even here, a small region of subsonic flow may exist near the leading edge of the airfoil immediately behind a shock wave depending on the bluntness of the leading edge. The sharper the leading edge, the smaller is the extent of the subsonic flow region.

FIVE

The foregoing material has considered the separate sources of drag that contribute to the total drag of an airplane. Adding these together, let us now

Thus the drag is composed of two parts: the parasite drag, which varies directly with V2, and the induced drag, which varies inversely with V2. The drag has a minimum value when the parasite, and induced drag are equal. This minimum value is easily determined from Equation 4.58 and is equal to

and occurs at a velocity equal to

(4.60)

Observe that the minimum drag is independent of density and depends on the span loading instead of the wing loading.

D, Dmi„, and can be combined and expressed in a general manner as

The first term on the right-hand side of Equation 4.61 represents the parasite drag; the second term is proportional to the induced drag. Both of these terms and the total drag ratio are presented in Figure 4.52.

Laminar flow control has the potential for achieving significant drag reductions. However, it has yet to be proven on an operational aircraft. Even without LFC, the parasite drag of many of today’s aircraft could be significantly reduced by cleaning up many small drag items that are negligible individually but are appreciable collectively.

Figure 4.51, based on full-scale wind tunnel tests, illustrates how the drag of an aircraft can deteriorate as items are £dded to the airframe. In Figure 4.51a, the airplane is shown in the faired and sealed condition. Then, as the

Table 4.6 Drag Items as Shown in Figure 4.51 (Ref.

4.14)______________________________________

Power plant installation

Open cowling inlet and exit 18.6%

Unfaired carburetor air scoop 3.6%

Accessory cooling airflow 3.0%

Exhaust stacks and holes 3.6%

Intercooler 6.6%

Oil cooler, 10.2%

4 Total 45.6%

Other items for service condition Remove’ seals from cowl flaps 5.4%

Opening case and link ejector 1.8%

Opening seals around landing gear doors 1.2%

Sanded walkway 4.2%

Radio aerials 4.8%

Guns and blast tubes 1.8%

Total 19.2%

items tabulated in Table 4.6 were added, drag increments were measured. These are expressed as a percentage of the original clean airplane drag.

Table 4.6 shows that the drag of the original clean airplane is increased by nearly 65% by the total effect of these drag items. Some of this additional drag is, of course, necessary, but more than half of it is not. Additional tests and analysis of this particular airplane showed that the drag of the power plant items could be reduced to 26.6% of the initial drag.

The moral of the foregoing and other material contained in this chapter is that, with regard to drag, attention should be paid to detail. Surfaces should be smooth and protuberances streamlined or avoided if possible. Tight seals should be provided around wheel wells, door openings, and other cutouts. It is exactly this attention to detail (or lack olit) that explains the wide disparity in the CF values tabulated in Table 4.2 forairplanes of the same class.

Possibly the ultimate in aerodynamic cleanliness is represented by the latest generation of sailplanes. Employing molded fiberglass or other types of plastics, ultrasmooth surfaces are achieved. Using very high aspect ratios, ranging from 10 to 36, and laminar flow airfoils, mainly of the Wortmann design (Ref. 4.20), lift-to-drag ratios as high as 40 have been accomplished.

The contribution to the drag of a streamlined shape from skin friction can be reduced appreciably if transition from laminar to turbulent flow can be delayed. An estimate of the gains to be realized can be seen from Figure 4.40. This flgurd has befen prepared based on Figure 4.1 and the methods outlined earlier for calculating the drag of a flat plate over which the boundary layer is partly laminar and partly turbulent. In this case, the total CF is calculated from

Cf = CFt(R) ~ x,[CPT(Rt) — CpL(R,)] (4.55)

In this equation, a subscript T refers to turbulent flow and a subscript L to a laminar flow. R, is the transition Reynolds number based on the transition length, /,, shown in Figure 4.40. x, is the relative distance from the leading edge to the transition point expressed as a fraction of the total length. The notation CFt(R,), for example, does not indicate a product but, instead, shows that Cft is to be evaluated at the Reynolds number R,.

From Figure 4.40, it is obvious that the skin friction can be reduced significantly if some means can be found to stabilize the laminar layer so as to prevent or delay transition to a turbulent layer. Such a procedure is known as boundary layer control (BLC) or laminar flow control (LFC). BLC is the more

general term and encompasses other purposes such as controlling the boun­dary layer, laminar or turbulent, in order to delay separation. LFC is therefore preferred when reference is made to stabilizing the laminar boundary layer.

‘It is not the purpose of this textbook to consider in detail the fluid mechanics involved with stabilizing the laminar layer. Generally, the problem is that of maintaining a boundary layer that is thin with a full velocity profile. This latter statement is clarified in Figure 4.41.

A passive method of maintaining laminar flow is by shape alone. A good example of this is the family of airfoils, the NACA 6-series airfoils discussed briefly in Chapter Three. One of these, the NACA 662-015 airfoil, is pictured in Figure 4.42 along with its chordwise pressure distribution. Note that because

of its shape, the pressure decreases with distance all the way back to the 65% chord position. This favorable pressure gradient is conducive to maintaining a thin boundary layer with a stable velocity profile. One might assume, as a first estimate, that its transition point is close to the 65% chord position.

For comparison, consider the NACA0015 airfoil having the same thick­ness ratio but with its maximum thickness located further forward than the 662-OI5 airfoil. This airfoil, together with its pressure distribution, is shown in Figure 4.43. For this airfoil, one might expect transition to occur at around the 20% chorck where*the flow first encounters an adverse pressure gradient.

Both the 662-OI5 and 0015 airfoils lie within the families of airfoils considered in Figure 4.44. For rough surfaces, Cd is approximately the same for both airfoil families. The roughness causes transition in both cases to occur near the leading edge. The picture is different in the case of smooth surfaces. Here Cd equals 0.0064 for the 0015 airfoil but only 0.0036 for the laminar flow airfoil. These correspond to Q values of approximately 0.0032 and 0.0018 for the respective airfoils. Using Figure 4.40 and the transition points of 0.2C and 0.65C, values of Q of 0.0026 and 0.0014, respectively, are obtained corresponding to Cd values of 0.0052 and 0.0028. The difference between these values and the experimental results may be attributable to errors in the estimated transition locations. Most likely, however, the difference is attributable to form drag. For both airfoils, the differences are

Figure 4.44 Variation of Section Cdmln with thickness ratio for conventional and laminar flow NACA airfoils, (a) NACA four and five-digit series, (b) NACA 66- series.

close to the estimates of form drag that one obtains from examining the increase in total Cd with thickness ratio.

The favorable pressure distribution of the series-66 airfoils undoubtedly delays transition, thereby reducing the skin friction drag. For a particular airfoil, however, extensive laminar flow can only be maintained over a limited range of Q values and for Reynolds numbers that are not too large. The Cd
versus C, curve, known as the drag polar, for a laminar flow airfoil has the rather unusual shape typified by Figure 4.45 (or earlier by Figure 3.8). This drag bucket results from the fact that for C, values between approximately + or -0.2, the chordwise pressure distribution is sufficiently favorable to main­tain laminar flow over most of the airfoil. Without this “bucket,!’ the drag curve extrapolates to a Cd value at a zero C, close to that for a more conventiopal airfoil having this same thickness.

With careful attention to surface waviness and roughness, appreciable laminar flow can be achieved with airfoils up to Reynolds numbers in excess of 20 million, as shown by Figure 4.46 (Ref. 3.1). This same figure emphasizes the importance of surface finish. Unimproved paint is seen to be rough. enough to cause premature transition at a Reynolds number of approximately 20 x 106. The result is a doubling in the drag coefficient for this particular airfoil.

One has to be somewhat careful in interpreting this figure. At first glance, it might appear that transition is being significantly delayed up to a Reynolds number of 60 x 106, since the drag coefficient is nearly constant up to this Reynolds number. A closer look shows the Cd to decrease up to an R of approximately 32 x Ю6. It then increases up until an R of approximately 54 x 106. Above this value of R, it appears that Cd is tending to decrease.

Obviously, from Figure 4.40, a constant Cd as R increases requires that the transition point move forward. This is assuming that the form drag is not dependent on R. This is a valid assumption; if anything, the form Cd tends to decrease with R.

It is difficult to divide the total drag into form and skin friction drag because of the dependence of the skin friction drag on the transition location. However, based on the potential flow pressure distribution, it is reasonable to assume that transition occurs at around the 50% chord point at the lower Reynolds numbers. With this assumption, the same form drag coefficient is obtained at R values of 12 x 106 and 30 x 106, that is, a form Cd, of 0.0013. For the same transition location and form Cd, Figure 4.40 leads to a predicted Cd of

0. 0036. This is close to what one might expect if the data for Figure 4.46 a are extrapolated beyond an R of 32 million.

Using the form Cd of 0.0013 and Figure 4.40, the peak Cd of 0.0050 at an R of 54 x 106 leads to a transition location at this higher Reynolds number of 18% of the chord. Thus, it is concluded that the shape of the 65(42i)-420 airfoil is able to stabilize the laminar boundary layer up to the midchord point for Reynolds numbers as high as 30 million. For higher Reynolds numbers, the transition point moves progressively forward.

The size of roughness that can be tolerated without causing transition can be estimated from Figure 4.47 (Ref. 3.1). It is somewhat surprising to find that the results do not depend significantly on the chordwise position of the roughness. In fact, it appears that the downstream positions are less tolerant to roughness height than positions near the leading edge.

Figure 4.47 Variation of boundary layer Reynolds number with projection fineness ratio for two low drag airfoils. [Як, = transition Reynolds number based on height (which will cause transition) of protuberance, к, and local velocity outside transition of boundary layer.] (I. H. Abbott and A. E. VonDoenhoff, Theory of Wing Sections, Dover Publications, Inc., 1959. Reprinted by permission of Dover Publications, Inc.)

Since one can never be sure of the shape of a particle, based on Figure 4.47, a value of Rkt of 1400 is recommended as being reasonable. In the case of Figure 4.46, this criterion leads to a roughness as small as 0.004 in. in height as the cause of the drag rise at an R value of 20 million.

Figure 4.48 is a convenient graph for quickly determining Reynolds numbers at a given speed and altitude. For example, a typical light airplane operating at 10,000 ft at a speed of 150 mph has a unit Reynolds number of

1.1 x 10* or, for a chord of 5.5 ft, a Reynolds number of 6.05 x Ю6. A jet transport cruising at 35,000 ft at 500 mph las a unit R of 1.8 x Ю6. This results in an R of 27 x 106 for a chord length of 15 ft.

For the light plane, an Rof 1400 gives an allowable roughness height of

0. 015 in. A height of 0.009 in. or less should not cause transition on the jet transport’s wing. These may not be difficult criteria to meet for a wind tunnel model’or an isolated panel. On an operational, full-scale aircraft with rivets, access panels, deicers, gas caps, wheel-well covers, and the like, the achievement of this degree of smoothness is a real challenge. Even if such smoothness is attained, a few bugs smashed on the leading edge can easily destroy the aerodynamic cleanliness.

An active method of providing LFC involves removing the boundary layer as it develops so as to keep it thin with a stable velocity profile. This requires that power be expended to apply suction to the boundary layer either through a porous surface or across closely spaced thin slots transverse to the flow, as shown in Figure 4.49. The latter method has received the most attention. One of the earliest investigations of LFC using discrete spanwise slots was reported in Reference 4.18. Here, laminar flow was achieved up to a Reynolds number of 7.0 x 106 on NACA 18-212, 27-215, and 0007-34 air­foils. This result is not very impressive in comparison to Figure 4.46, where transition is apparently delayed up to R values of 30 x Ю6 for a smooth surface and 20 x 106 for the painted surface. However, the airfoils tested by Reference 4.18 were prior to the series-6 airfoils and had pressure gradients less favorable than the laminar flow series developed later. It was found that, with only a small expenditure of power, the boundary layer could be stabil­ized over an extensive region having an Adverse pressure gradient. Somewhat discouraging was the fact that the use of suction did not reduce the sensitivity of transition to roughness.

Flight testing performed in the mid-1960s provided more encouraging results, as reported in Reference 4.19. Two WB-66 airplanes were modified and redesignated X-21A. These airplanes had 30° swept wings with an aspect ratio of 7. The boundary layer was removed by approximately 120 slots on each surface. The slots varied in width from about 0.0035 to 0.01 in.

I Difficulties were encountered with instabilities in the skewed boundary layer along the swept leading edge produced by the spanwise flow. However, the use of fences and chorwise suction slots spaced along the leading edge

V fps

Figure 4.48 Reynolds number as a function of velocity and altitude.

alleviated this problem. The final result was the attainment of full-chord laminar flow at a Reynolds number of 45.7 x 106.

The adoption of a powered LFC system represents a challenging exercise in systems analysis and design. The saving in drag must be measured against the weight and initial cost of the ducting, pumps, and double skin required to remove the boundary layer. According to Reference 4.19, performance analyses showed that the required engine size for a jet transport in the 300,000-lb class is smaller than that for the turbulent counterpart. This smaller engine results in a weight saving that offsets the weight penalty of the pumping equipment. With both the inner and outer skins contributing to the

wing’s structural integrity, the weight of all of the pumping equipment is estimated at between 1.3 and 1.4 psf. Considering the weight, drag, and specific fuel consumption, an optimized design incorporating LFC shows an increase of one-third in the range for a fixed payload or in the payload for a fixed range at a design range of 5000 nmi. These predicted performance gains are shown in Figure 4.50 (taken from Ref. 4.19).

*

A recent development that holds some promise for reducing induced drag, short of increasing the aspect ratio, is the so-called winglet. The details of a winglet (studied in Ref. 4.15) are shown in Figure 4.35; Figure 4.36 pictures the winglet mounted on the wing tip of a first-generation jet transport (such as a Boeing 707).

The winglet is reminiscent of the tip plate, which has been tried over the years for the same purpose. These plates have never proven very successful for reasons that will become clear as the details of the winglet design are discussed.

Placing the winglet on an existing wing will alter the spanwise dis­tribution of circulation along the wingspan and hence the structure of its trailing vortex system far downstream. One can calculate the reduction in the induced drag afforded by the winglet solely by reference to the ultimate wake, or so-called Treffetz plane. This is the method used by Reference 4.17 together with a numerical vortex-lattice, lifting-surface theory. Examining only the ultimate wake is not very satisfying, however, from a physical standpoint. Instead, consider the flow field into which the winglet is inserted.

Figure 4.37 qualitatively illustrates the situation. Outboard of the tip, the flow is nearly circular as air from beneath the wing flows outward along the span, around the tip, and inward on the upper surface. The velocities induced by the wing are shown. To these the free-stream velocity is vectorially added. The magnitudes of the induced velocities generally increase toward the tip. At a given spanwise location, the induced velocities are highest close to the surface of the wing, just outside of the boundary layer.

Consider a section of the winglet as shown in Figure 4.37c. The induced velocity Vi produced by the main wing combines with the free-stream velocity, V, to produce an angle of attack, a. Assuming a to be a small angle, a net forward component of force, – dD, results from the differential section lift and drag on the winglet. Denotin^winglet quantities by a subscript w,

— dD = dLwaw — dDw

Observe that the same result is obtained if the winglet is mounted below the wing, where the induced velocity is outward.

Since we do not know the induced flow in sufficient detail to integrate along the span of the winglet, let us assume an average 17,- acting over the

figure 4.37 Generation of negative drag by winglet section, (a) Looking in direction of flight. (b) Planview. (c) Forces acting on winglet.

The induced angle, aw, must be proportional to CL. Therefore let

aw = KCL

Also, approximately,

Cr =277

Therefore, Acd becomes

This approximate analysis indicates that:

1. The reduction in CD increases linearly with Cl-

2. At low CL values, CD will be increased by the addition of a winglet.

3. High winglet aspect ratios are desirable.

The severe limitations inherent in the assumptions leading to Equation 4.54 must be recognized. For a given value of SJS, it would appear that increasing Aw would always result in a greater reduction of Co – This is not true, since increasing the winglet span will result in a smaller constant of proportionality, K. The same can be said for increasing SJS. Despite these limitations, the foregoing discloses the basic elements that are necessary for the design of an effective winglet. Its profile drag (including interference with the wing) must be low. Its aspect ratio should be fairly high to assure a high lift curve slope and low induced draggfor the winglet. Not as apparent, the winglet should be mounted as near the trailing edge as possible in order to experience the highest induced velocities possible for a given wing CL■ Also, in this regard, a winglet would be expected to produce a larger decrement in CD for a wing having a relatively higher loading near its tips.

Figure 4.38 presents experimental measurements of ДCd as reported in References 4.15 and 4.16. In the case of the second-generation jet transport

(such as a DC-10), the loading is relatively lower near the wing tips, so the winglets are less effective. As predicted, ДCD is seen to vary nearly linearly with CL2. In the case of the first-generation jet transports, a decrement in CD is achieved for CL values greater than 0.22. This number increases to 0.30 for the second-generation jet transports.

The induced drag of a wing can also be reduced simply by extending its tip and thereby increasing its aspect ratio. Reference 4.17 considers this possibility and compares the savings in drag to be gained from extending the tips with those obtained by the use of winglets. Since either method will result in greater root bending moments and hence increased wing structure and weight, both the induced drag and wing root bending moments are treated by tiie reference. Typical results from this study are presented in Figure 4.39. It is emphasized that these results are from potential flow calculations and thus do not include the profile drag of the winglet or any interference drag. The trends determined by the reference are probably valid but somewhat optimis­tic with regard to the winglets. For identical increases in bending moment, the winglet can provide a greater reduction in induced drag than can be achieved with a tip extension. Referring to Figure 4.39, the ratio «with/^without is simply

the ratio of the induced drag coefficient of the original wing to the coefficient with a winglet or tip extension. Consider, for example, an untwisted wing with an aspect ratio of 8 and a taper ratio of 0.5. The leading edge, as with all the wing studies in Reference 4.17, is swept back 30°. With a winglet, the induced drag can be reduced by 24% with only a 2.6% increase in bending moment. For the same bending moment increase, extending the tip would save only 6% in the induced drag. To achieve the same reduction in the induced drag with a

tip extension as with the winglet would require a 13% increase in the bending moment.

Skin friction drag and induced drag are the major contributors to the total drag of an airplane, at least for a modern jet transport. For lower-speed, general aviation aircraft, form drag assumes more relative importance. A typical drag buildup (or breakdown, depending on your outlook) is presented in the bar graph of Figure 4.34 for a jet transport, as reproduced from

Reference 4.11. It is seen that skin friction drag and induced drag account for approximately 75% of the total drag. Although the remaining 25% is not to be taken lightly, the potential for real savings in power or fuel rests with reducing the skin friction drag and induced drag.

Cylinder heads, oil coolers, and other heat exchangers require a flow of air through them for purposes of cooling. Usually, the source of this cooling air is the free stream, possibly alimented to some extent by a propeller slipstream or bleed air from the compressor section of a turbojet. As the air flows through the baffling, it experiences a loss in total pressure, Дp, thus extracting energy from the flow. At the same time, however, heat is added to

the flow. If the rate at which the heat is being added to the flow is less than the rate at which energy is being extracted from the flow, the energy and momentum flux in the exiting flow after it has expanded to the free-stream ambient pressure will be less than that of the entering flow. The result is a drag force known as cooling drag.

It is a matter of “bookkeeping” as to whether to penalize the airframe or the engine for this drag. Some manufacturers prefer to estimate the net power lost to the flow and subtract this from the engine power. Thus, no drag increment is added to the airplane. Typically, for a piston engine, the engine power is reduced by as much as approximately 6% in order to account for the cooling losses.

Because of the complexity of the internal flow through a typical engine installation, current methods for estimating cooling losses are semiempirical in nature, as exemplified by the Lycoming installation manual (Ref. 4.12). Before considering an example from that manual, let us examine the basic fundamentals of the problem. A cowling installation is schematically pictured in Figure 4.30. Far ahead of the cowling, free-stream conditions exist. Just ahead of the baffle, the flow is slowed so that the static pressure, PB, and the temperature, TB, are both higher than their free-stream values. As the flow passes through the baffle, PB drops by an amount Дp because of the friction in the restricted passages. At the same time, heat is added at the rate Q, which increases TB by the amount AT. The flow then exits with a velocity of VE and a pressure of PE, where PE is determined by the flow external to the cowling. The exit area, AE, and the pressure, pE, can both be controlled by the use of cowl flaps, as pictured in Figure 4.31. As the cowl flaps are opened, the amount of cooling flow increases rapidly because of the decreased pressure and the increased area. Downstream of the exit, the flow continues to accelerate (or decelerate) until the free-stream static pressure is reached. Corresponding to this state, the cooling air attains an ultimate velocity denoted by V„.

If m represents the mass flow rate through the system, the cooling drag,

Dc, will be given by the momentum theorem as

Dc=tn(Vo-V„) (4.45)

The rate, Д W, at which work is extracted from the flow is

AW = i2m(V02-Vj) (4.46)

Observe that the cooling’drag is obtained by dividing the increment in the

energy by a velocity that is the average between V„ and V»; that is,

For a fixed-baffle geometry, the pressure drop through the baffle will be proportional to the dynamic pressure ahead of the baffle.

Ap <x pVB (4.48)

Vb can be related to the mass flow, m, through the baffle and an average flow area through the baffle.

m = pABVB (4.49)

Substitution of Equation 4.49 into Equation 4.48 results in

2

Др a — (4.50)

P

In the preceding, for a fixed baffle, the fictitious area, AB, has been absorbed into the constant of proportionality.

The rate at which heat is conducted away from the baffles by the cooling air must depend on the difference between the temperature of the baffles and that of the entering cooling air. For a piston engine, the baffle temperature is the cylinder head temperature (CHT). Denoting the cooling air temperature just ahead of the baffles by TB, the rate of heat rejection by the engine can be written as

Q « CHT – TB (4.51)

In Equations 4.50 and 4.51, the constants of proportionality will depend on the particular engine geometry and power. These constants or appropriate graphs must be obtained from the manufacturer. One is tempted to scale Q in direct proportion to the engine power, P, and to scale AB with the two-thirds power of the engine power, in which case Equations 4.50 and 4.51 become

Др oc (4.52)

Q « (CHT – TB)P (4.53)

However, these are speculative relationships on my part; they are not substantiated by data and therefore should be used with caution.

If sufficient information is provided by the engine manufacturer to relate Ap, m, and p, and to estimate Q for a given temperature difference between CHT and the cooling air, then one is in a position to calculate the cooling drag. The details of this are best illustrated by means of an example given in Reference 4.12. In accordance with the reference, the English system of units will be used for this example.

This example will consider a horizontally opposed engine delivering 340 bhp operating at a 25,000 ft pressure altitude and a true airspeed of 275 mph. The outside air temperature is taken to be 20 °F higher than staifdard for this altitude, which means an OAT of -10°F. Lycoming recommends 435 °F as a maximum continuous cylinder head temperature for maximum engine life. This example is to represent a cruise operation with cowl flaps closed. (This does not mean that the exit is closed; see Figure 4.31.) It is therefore assumed that the exit static pressure is equal to the ambient static pressure. Given the foregoing, the problem is to size the area of the exit and to calculate the cooling drag.

The mass density, p, can be calculated from the equation of state (Equation 2.1).

h

P Ps J.

where subscript s refers to standard values. From this, using Figure 2.3,

p = 0.00102 slugs/ft3

The free-stream dynamic pressure, q0, is thus

q0 = 83 psf

Neglecting any contribution from the propeller slipstream for the cruise condition, the reference, on the basis of experience, calculates the tem­perature at the engine face by assuming a 75% recovery of the dynamic pressure with a resulting adiabatic temperature rise. From Equations 2.1 and 2.30,

T

pHy-nivi ~ constant

or

= дсп Г786-3 + 0 75(83)1° L 786.3 J

= 460 °R

Thus, the temperature at the engine face is 0 °F.

With the assumed 25% loss in dynamic pressure resulting from the diffusion by the cowling, the static pressure at the engine face, assuming the flow there to have a negligible velocity, will be

pB = 848 psf

This corresponds to a pressure altitude of 23,200 ft. Next, Figure 4.32 (pro­vided by the manufacturer), is entered with the cooling air temperature at the

Щип 4.32 Required cooling airflow as a function of cooling air temperature at ehgine face.

engine face. For a CHT of 435 °F, this figure leads to a required cooling airflow of 2.55 lb/sec.

The baffle pressure drop is determined next from Figure 4.33 (also provided by the manufacturer). It would seem, in accordance with Equation 4.50, that this graph should be in terms of the density altitude instead of the pressure altitude. Nevertheless, it is presented here as taken from Reference

4.12. Entering this graph with an airflow of 2.55 lb/sec at a pressure altitude of 23,200 ft results in a pressure drop of 47 psf. Thus, downstream of the cylinders, the total pressure will be

pB – Ap = 801 psf

Instead of using a heat flow, Q, per se, the reference assumes (presum­ably based on experience) that the cooling air will experience a temperature rise of approximately 150 °F across the cylinders. For this particular example, this temperature rise can be expressed in terms of Q by using the specific heat at constant pressure.

Q = CpATm

For air Cp — 6000 ft/lb/slug/°R. Thus, for the airflow of 2.55 lb/sec,

Q = 6000(150)^|

= 71,273 ft-lb/sec = 129.6 hp

Notice that the above rejected heat amounts to 38% of the engine power. This is close to the 40 to 50% qupted in other sources.

The cooling air density just downstream of the cylinder heads after the temperature rise can be calculated from the equation of state.

Рв-Ьр

RT

801

1711(460+150)

= 0.000767 slugs/ft3

This flow is then assumed to expand adiabatically to the ambient static pressure of 786 psf. Thus the density at the exit, from Equation 2.30, is found to be

pE = 0.00076 slugs/ft3

The corresponding velocity is determined by the use of Equation 2.31.

= 205 fps

This velocity, together with the density of the cooling flow at the exit and the required cooling flow, leads to a required exit area of 0.510 ft2. This number, as well as VE, differs slightly from Reference 4.12 as the result of calculating the flow state following the addition of heat in a manner somewhat different from the reference.

The resulting cooling drag for this example can be calculated from Equation 4.45.

Dc =— (403 – 205)

= 15.7 lb

This corresponds to an increment in the flat-plate area of 0.19 ft2. In terms of engine bhp, assuming a propeller efficiency of 85%, this represents a loss of

10.2 bhp, or 3% of the engine power.

For operating conditions other than cruise, it may be necessary to open the cowl flaps. Reference 4.12 states that a pressure coefficient at the cowl exit as low as —0.5 can be produced by opening the flaps to an angle of approximately 15°. By so doing, a relatively higher cooling flow can be generated at a lower speed, such as during a climb. Even though the engine power may be higher during climb, operating with open cowl flaps and with a richer fuel mixture can hold the CHT down to an acceptable value. Also, CHT values higher than the maximum continuous rating are allowed by the manufacturer for a limited period of time.

It will not be repeated here, but Reference 4.12 also performs a cal­culation similar to the foregoing but for climb conditions at 19,000 ft pressure altitude, 450 bhp, mixture rich, 130 mph true airspeed, and an OAT of 31 °R A CHT of 475 °F is allowed with the cowl flaps open to give an exit Cp of -0.5.

For this case, the required airflow is determined to be 1.95 lb/sec with an exit velocity of 73.8 fps. Thus, for this case,

Dc = 7.1 lb fc = 0.32 ft2

The equivalent power loss is only approximately 0.5% of the engine power for this case, even though the increment in equivalent flat-plate area is appreci­ably higher than that in cruise.

Basically, trim drag is not any different from the types of drag already discussed. It arises mainly as the result of having to produce a horizontal tail load in order to balance the airplane around its pitching axis. Thus, any drag increment that can be attributed to a finite lift on the horizontal tail con­tributes to the trim drag. Such increments mainly represent changes in the induced drag of the tail. To examine this further, we again write that the sum of the lifts developed by the wing and tail must equal the aircraft’s weight.

L + LT=W

Solving for the wing lift and dividing by qS leads to

Cl.

Here, CLis the wing lift coefficient, CL is the lift coefficient based on the weight and wing area, and CLt is the horizontal tail lift coefficient. The CDi of the wing, accounting for the tail lift, thus becomes

г-* _ ^ Lw

D’w irAew

Cl 2 Cl CLt St (4.38)

u ігАе ттАе CL S

The term [CLi.(St/S)]2 has been dropped as being of higher order. Since СьІтгАе is the term normally defined as CDp it follows from Equation 4.38 that the increment in the induced drag coefficient contributed by the wing because of trim is

ДСд = -2ССі^^: (4.39)

Added to Equation 4.39 the total increment in the induced drag coefficient becomes

In order to gain further insight into the trim drag, consider the simplified configuration shown in Figure 4.28. For the airplane to be in equilibrium, it follows that

Lw + LT = W xLw = (/ – x)LT

where l is the distance from the aerodynamic center of the wing to the aerodynamic center of the tail, x is the distance of the center of gravity aft of the wing’s aerodynamic center. Soling for LT gives

LT = ±W

Thus,

Substituting Equation 4.43 into Equation 4.42 leads to

The ratio of the wingspan to tailspan is of the order of 3, while e is equal approximately to eT. With these magnitudes in mind, Figure 4.29 was pre­pared; it presents the trim drag as a fraction of the original induced drag as influenced by x/l.

Notice that the possibility of a negative trim drag exists for small, positive, center-of-gravity positions. This results from the slight reduction in wing lift, and hence its induced drag, for aft center-of-gravity positions. However, as the center of gravity moves further aft, the induced drag from the tail overrides the saving from the wing so that the net trim drag becomes more positive.

The aerodynamic moment about the airplane’s aerodynamic center was neglected in this analysis. By comparison to the moment contributed by the tail, Mac should be small. Qualitatively, the results of Figure 4.29 should be relatively unaffected by the inclusion of Mac.

The trim drag is usually small, amounting to only 1 or 2% of the total drag of an airplane for the cruise condition. Reference 4.10, for example, lists the trim drag for the Learjet Model 25 as being only 1.5% of the total drag for the cruise condition. As another example, consider the Cherokee once again at an indicated airspeed of 135 mph. At its gross weight of 2400 lb, this corresponds to a CL of 0.322. For this weight the most forward center of gravity allowed by the flight manual is 3% of the chord ahead of the quarter-chord point. With a chord of 63 in. and the distance between the wing and tail aerodynamic center of approximately 13 ft, xll has a value of -0.012. Since b — 3bT, Figure 4.29 gives

= 0.028

With an effective aspect ratio of approximately 3.38, CDi will he equal to

0. 0098, so that = 0 .0003. The parasite drag coefficient is approximately

0. 0037, so that the total CD equals 0.0138. Thus, for the Cherokee in cruise, the trim drag amounts to only 2% of the total drag.

The use of Equation 4.35 is illustrated in Table 4.4, where I have performed an estimate of the drag breakdown for the Cherokee (Figure 3.62). Armed with a tape measure, I made a visual inspection of the airplane, noting the dimensions of all drag-producing appendages. What you see here is a first estimate without any iteration. The total value for / of 0.36 m2 (3.9 ft2) is obviously too low and should be approximately 50% higher. This airplane has a total wetted area of approximately 58.06 m2 (625 ft2).

Undoubtedly, an aerodynamicist working continuously with this type of aircraft would be able to make a drag breakdown more accurate than the one shown in Table 4.4. Based on one’s experience with his or her company’s aircraft, the aerodynamicist can make more certain allowances for surface roughness, interferences and leakage. For example, in the case of the Cherokee’s landing gear, the oleo struts have a bar linkage immediately behind them. The linkage, struts, wheel pants, and brake fittings produce a total drag for the entire landing gear that is probably significantly higher than the total / of 0.065 m2 (0.7 ft2) estimated for the wheels, wheel pants, and struts. The cylindrical oleo struts, in particular, being close to the wheel pants probably produce separation over the pants so that Cd for this item could be

Gas drain cocks

Rotating beacon

Tail tie-down Wing tie-downs

Five whip antennas

OAT gage Antenna fairing

Antenna supports Interference Fuse vertical tail Fuse horizontal tail Fuse wing

1 in. x I in. blunt (2 total)

4 in. D x 5 in.

semispherical 3| in. x g in. blunt 1 in. x I in. blunt (2)

з in. D x 20 in. each

3 in. D x 2 in. each 12 in. long, I in. thick, 2 in. chord 3 in. x I in. D blunt 3 in. deep x 6 in. width stream­lined

5 in. x I in. D (two)

Leakage?

Cooling?

more like 0.4 or even higher instead of 0.04, as listed in Table 4.4. This would add another 0.06 m2 (0.63 ft2) to /.

Another example of a drag breakdown is provided by Reference 4.10. In this case, the airplane is the Gates Learjet Model 25 pictured in Figure 4.27. Table 4.5 was prepared on the basis of Reference 4.10. The authors of the reference chose to base Cd for each item on the wing area. This is therefore the case for Table 4.5, since dimensions and areas for each item were not available. Also, the reference did not include interference or roughness and gap drag in the parasite drag. Why this was done is not clear, and these two items are included in Table 4.5. These two somewhat elusive drag items are estimated to account for 20% of the parasite drag. Although not related to its parasite drag, according to Reference 4.10, this airplane has an Oswald’s efficiency factor of 0.66.

As a measure of an airplane’s drag, in practice one will frequently hear the term “drag count” used. Usually, it is used in an incremental or decre – mental sense, such as “fairing the landing gear reduced the drag by 20 counts.” One drag count is defined simply as a change in the total airplane Co, based on the wing planform area of 0.0001. Hence a reduction in drag of 20 counts could mean a reduction in the CD from, say, 0.0065 down to 0.0045.

average skin friction coefficients

In examples to follow, one will see that several uncertainties arise in attempting to estimate, the absolute parasite drag coefficient (as opposed to incremental effects) of an airplane. These generally involve questions of interference drag and surface irregularities. In view of these difficulties, it is sometimes better4 to estimate the total drag of a new airplane on the basis of the known drag of existing airplanes having a similar appearance, that is, the same degree of streamlining and surface finish.

The most rational basis for such a comparison is the total wetted area and not the wing area, since Q depends only on the degree of streamlining and surface finish, whereas CD depends on the size of the wing in relation to the rest of the airplane. In terms of an average CF, the parasite drag at zero CL for the total airplane can be written as

D = qCpSw (4.36)

where Sw is the total wetted area of the airplane. Since

it follows that the ratio of the equivalent flat-plate area to the wetted area is

In order to provide a basis for estimating CF, Table 4.2 presents a tabulation of this quantity for 23 different airplanes having widely varying configurations. These range all the way from Piper’s popular light plane, the Cherokee, to Lockheed’s jumbo jet, the C-5A.

The data in this table were obtained from several sources and include results obtained by students taking a course in techniques of flight testing. Thus, the absolute value of CF for a given airplane may be in error by a few percent. For purposes of preliminary design, the CF ranges given in Table 4.3 are suggested for various types of airplanes. Where a particular airplane falls in the range of CF values for its type will depend on the attention given to surface finish, sealing (around cabin doors, wheel wells, etc.), external protu­berances, and other drag-producing items.

Additional drag data on a number of airplanes, including supersonic airplanes, are presented in Appendix A.3 as a function of Mach number and altitude.

Finally, with regard to average CF values, Figure 4.26 (taken from Ref. 4.11) is presented. Although only a few individual points are identified on this figure, its results agree generally with Table 4.2. This figure graphically depicts the dramatic improvement in aerodynamic cleanliness of airplanes that has been accomplished since the first flight of the Wright Brothers.

Table 4.3 Typical Total Skin Friction Coefficient Values for Different Air­plane Configurations

Airplane Configuration CF Range at Low Mach Numbers

Propeller driven, fixed gear 0.008-0.010

Propeller driven, retractable gear 0.0045-0.007

Jet propelled, engines pod-mounted щ 0.0035-0.0045

Jet propelled, engines internal 0.0030-0.0035