7. DRAG ESTIMATION

7.1 Introduction The total drag of an airplane is considered to consist of three different types in the subsonic regime of flight. These are:

zero-lift drag, or parasite drag, CD,0

induced drag, or drag-due-to-lift, CD,i = kCL2 where k= 1/Ae, and e

= 0.85 (M < 0.7) compressibility, or wave, drag

The zero-lift drag has two components, form drag and skin-friction drag. The former is due to the pressure field around the body and the latter is due to the shear stresses at the surface of the body. The induced drag arises as a consequence of the production of lift and represents the “cost” of producing lift by pushing a body through a fluid. The wave drag is a pressure drag that arises due to the effects of compressibility that form compression and expansion waves due to the body shape. The DATCOM (Ref. 7-1) methods will be most often followed and the pertinent material is presented in Appendices F and G.

7.2 Skin Friction Drag The skin friction drag can be produced by either a laminar or a turbulent boundary layer flow. At the flight speeds and altitudes at which aircraft fly it is usually conservative to assume that the boundary layer flow is fully turbulent over the entire airplane. On some of the smaller aircraft, made of fiberglass honeycomb or molded plywood construction, a considerable extent of laminar boundary layer flow exists prior to the transition point and for such aircraft the above assumption may be overly conservative. Since most aircraft, small or large, are still made using riveted aluminum skin/stringer/former construction for more than 85% of the total airframe weight, it is considered prudent in a preliminary design to assume the fully turbulent skin friction value over the entire airplane.

It is worth noting that next generation airliners like the Boeing 787 and the Airbus 380 will use much more composite materials in their design. Latest estimates suggest that the B787 will use up to 50% composites in the airframe and the A380 will use up to 25%. The major motivation for using advanced composite materials like Carbon Fiber Reinforced Plastics (CFRP), however, is to save weight, not to reduce skin friction drag.

The aeronautical literature usually provides the engineer with two forms of the skin friction coefficients and these can easily be confused by the novice. They are the local skin friction coefficient and the average skin friction coefficient, as calculated for a flat plate. The local value is the skin friction coefficient at a particular point on the plate and the average, as the name suggests, is integrated value of the skin friction coefficient from the leading edge of the plate to any point x = l along the plate. These coefficients are defined first for laminar flow. The local skin friction coefficient is

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The average, or integrated, skin friction coefficient is

Note that for the local skin friction coefficient the Reynolds number is based on the local distance, x, from the leading edge of the plate, while for the average skin friction coefficient the Reynolds number is based on the length, l, of the plate. The skin friction coefficient in laminar flow is a function also of Mach number, but for the range of flight Mach numbers typical of commercial jet transports, M<1, the differences due to compressibility are slight and are neglected for preliminary design purposes.

The average turbulent skin friction coefficient for a flat plate is presented in Fig. 4.1.5.1-26 in Appendix F. It can be seen that the skin friction coefficient depends both upon the value of the Reynolds number and the Mach number, but as stated previously the Mach number dependence is small in the subsonic flight regime. For our purposes we will consider the average turbulent skin friction coefficient to depend on the overall Reynolds number Rel where l is the characteristic length of the component under consideration. For example, , the mean aerodynamic chord for a wing, l=c for an airfoil, and l=lf for a fuselage.

For preliminary design purposes the approximation is made that the boundary layer develops over an airplane component, i.e., wing or fuselage, at exactly the same rate it would if that component were a flat plate. Therefore the skin friction drag is calculated as the product of the average (turbulent) skin friction coefficient, the total exposed surface area of the component (the "wetted" area), and the free stream dynamic pressure, q, as given below:

In detailed studies of the airplane performance more elaborate calculations may be made. However for preliminary design purposes the above is satisfactory. Von Karman determined an expression for the incompressible average skin friction coefficient for a flat plate as follows:

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Schlichting (Ref. 7-2) fitted the following more manageable equation:

; 106<Rel <109

This relation agrees with the values presented in Fig.4.1.5.1-26 of Appendix F. Note that the Reynolds number based on the length l, Rel, is a function of Mach number and altitude. Atmospheric properties for the 1976 U.S. Standard Atmosphere (Ref. 7-3) should be used in all calculations for consistency.

7.3 Form Drag In the preliminary design stage the form drag, or profile drag, is generally not calculated in detail but is obtained from wind tunnel measurements and empirical equations based upon correlations of those measurements. The most thorough collection of these results is presented by Hoerner (Ref. 7-4).

A good deal of time is required to estimate the drag of an airplane considering the basic form or profile drag, the mutual interference of the various airplane components, and especially the drag of irregularities on the surface contours, protuberances, etc. For example, North American Aviation (NAA), now Northrop Grumman, correlated the drag of a number of airplanes and developed a method for estimating the form drag of the various components as a fraction of the skin friction drag. Experience has shown that the total zero-lift drag coefficient (incompressible) calculated in this fashion provides a reasonable estimate of the airplane drag for preliminary design purposes. It should be pointed out that the NAA method (Ref. 7-5) works best for well constructed airplanes typified by military fighters and commercial airliners. It is slightly optimistic, i.e. it yields values of zero-lift drag coefficients which are generally too low, for the older types of general aviation aircraft. With the increase in fuel prices, all airplane manufacturers are paying considerably greater attention to drag producing items and the drag of these may be estimated using the NAA method. The DATCOM (Ref. 7-1) method for wing drag, given in Appendix F, and for body drag, given in Appendix G, is based on this approach and will be employed here.

7.4 Drag Build-up by Components The zero-lift-drag of the typical airplane component, say the wing, is estimated as the sum of friction and profile drag as follows:

DW,0 = DW,f + DW,p

The basis of the drag build-up method is the assumption that the form or profile drag may be expressed as a multiple of the friction drag as follows:

DW,0 = CF,turbSwet q + k CF,turbSwet q

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DW,0 = (1 + k) CF,turbSwet q

DW,0 = KWCFSwet q

Here, KW = (1+k) is a form factor for the wing which includes form and skin friction drag, and CF is the average turbulent skin friction coefficient of a flat plate of length equal to the mean aerodynamic chord of the wing, . The quantity Swet is the wetted area of the wing and q is the free stream dynamic pressure, q = ½ V2 = ½ pM2. The zero-lift-drag coefficient of the wing is defined as

Then the zero-lift drag coefficient of the wing is related to the skin friction coefficient as follows:

This equation is now generalized to apply to the entire aircraft as follows:

The individual aircraft components considered are identified by the subscript i. In the cruise configuration the components usually number five (n = 5) corresponding to (1) wing, (2) fuselage, (3) horizontal tail, (4) vertical tail, and (5) the engine nacelles, including pylons and their interference effects. In the landing and take off configuration the landing gears (nose and main), wing flap drag, and cowl flap drag components would be added as well. The list increases for externally carried antenna, externally carried fuel or stores, etc.

It should be realized that the actual zero-lift drag coefficient may be substituted for any one of the components used above if this is known. For example, if standard NACA airfoil sections are used in the wing design the airfoil drag coefficient may be obtained from the NACA Summary of Airfoil Data (Ref. 7-7), or from Abbott and von Doenhoff (Ref. 7-8). Note that the data shown in those references are typically for Rel =3, 6, and 9 million for smooth surfaces and 6 million for standard surface roughness. Flight values for Rel are often much larger, and this distinction must be kept in mind when determining drag. For the case of a tapered wing the wing drag can be calculated by integrating the local airfoil drag coefficient over the span:

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The local zero-lift drag coefficient of the airfoil, cD,0, varies in correspondence with the variation of Reynolds number along the span; the Reynolds number is directly proportional to the local chord c and it therefore varies with distance along the span for a tapered wing. The local zero-lift drag coefficient of the airfoil, cD,0, can also change in the spanwise direction if different airfoil sections are used from station to station. It is obvious that if a rectangular wing planform is chosen, and only a single airfoil used along the span, the wing and airfoil drag coefficients are equal provided the Reynolds numbers are equal and the fuselage is not considered.

7.5Fuselage DragThe elements comprising the fuselage drag have been discussed in some detail in Chapter 3 and Eq. (3-12) was developed to estimate the drag contribution of the fuselage and is repeated again below in terms of the drag coefficient based on frontal area:

(7-1)

The parameter k represents the ratio of wetted area to frontal area and for conventional fuselages with circular cabin cross-section may be approximated, according to Torenbeek (Ref. 7-6, p.447), by the following expression:

(7-2)

In Eq. (7-2) F >4.5, but since the fineness ratios for typical airliners is always greater than this value the equation should be applicable. There are other estimates available but they all have the same basic structure. Torenbeek (Ref. 7-6) offers the following form:

(7-3)

The DATCOM approach (Ref. 7-1), which is reproduced in Appendix G, has the form

(7-4)

All three equations are shown in Fig. 7-1 for the case of equal skin friction coefficient value of CF=0.002. It is clear that they all yield essentially the same result for fineness ratios typical of airliner fuselages.

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Figure 7-1 Drag coefficient based on frontal area as a function of fineness ratio for conventional fuselages as given by different authors. The case shown is for a skin friction coefficient CF=0.002.

It must be remembered that the fuselage drag coefficients shown are based on frontal area. The drag coefficient for the complete aircraft is always based on the wing planform area S, so the fuselage drag coefficients based on frontal area must be corrected accordingly. The drag of the fuselage may be written as

(7.5)

Here q is the dynamic pressureV 2/2, CD,fuselage,o is the drag coefficient based on fuselage frontal area Ao=(dfuselage)2/4 while CD,fuselage is the drag coefficient based on wing planform area S. Therefore, the drag coefficient required for the drag build-up by components is

(7-6)

7.6 Wing and Tail DragThe drag of the wing s and the tail surfaces may be found in a similar fashion to that for the fuselage. The drag is developed on the basis of a correction to the skin friction drag by a general coefficient 1 and a shape factor 2 involving the maximum thickness ratio of the wing or tail section so as to account for the pressure drag so that the drag coefficient based on wing planform area is of the form

(7-7)

Several different methods have been proposed and the coefficients for them are presented in Table 7-1. A comparison of the coefficients 2 for the various methods is shown in Fig. 7-2.

Table 7-1 Coefficients for Eq. (7-7)

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Source 1 2

DATCOM (Ref. 7-1) RL.S., See Fig. 4.1.5.1-28b in Appendix F

Hoerner (Ref.7-4) 1.0

NAA (Ref. 7-5) 1.0

Torenbeek (Ref. 7-6) 1.0

Figure 7-2 Pressure drag modifying factor 2 for wing and tail surfaces as a function of section maximum thickness ratio (t/c)max according to several sources

A comparison of the drag coefficient as estimated by Eq. (7-7) and the coefficients in Table 7-1 is shown in Fig. 7-3. For the purposes of the comparison a value of CF=0.003 was chosen and it was assumed that the wetted area of the wing Sw is equal to twice the planform area S. In addition f1 was estimated for typical flight conditions to be 1.25 from Appendix F. It is apparent that the results are reasonably close with the DATCOM results being the most optimistic. However, the general approach for all aerodynamic

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characteristics is based on the DATCOM method so it will also be used for the drag unless otherwise indicated.

Figure 7-3 Drag coefficients for wings as predicted by several methods. The comparison is based on Sw=2S and CF=0.003

7.7 Calculation of the Zero-Lift Drag Coefficient The drag coefficient may be found in a systematic fashion by following the DATCOM procedures provided in Appendices F and G. A general outline of how one might proceed to develop the drag of the entire airplane is illustrated in Table 7-2 and described in the following steps. Note that 3 tables must be prepared; one each for flight configuration: cruise, landing, and take-off operations. The data in the table shown is merely representative.

1. A three-view CAD drawing of the airplane has been prepared as part of the previous chapter.

2. The wetted area is the contour surface area in contact with the fluid. The wing and tailplane wetted area includes only the exposed wing area and is approximately equal to twice this exposed area. (For thick airfoils the perimeter is greater than twice the length of the chord line). The fuselage wetted area excludes those areas covered by the wing and empennage airfoil cross-sections. The nacelle dimensions can be obtained from aircraft using similar power plants and/or the engine dimensions. Simple relations for estimating wetted areas are given in Appendix B of Torenbeek (Ref. 7- 6).

3. The characteristic length of each component should be included in the table.

4. The Reynolds numbers of the components should be calculated and presented in this table also. The Reynolds numbers should be based upon the velocity and

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altitude of cruising flight since this condition determines the amount of fuel carried primarily.

5. The turbulent values of the skin friction coefficients can be calculated next. Do not forget to take into account the cutoff Reynolds number, which depends on surface roughness, in calculating the skin friction for each component.

6. The form factor, K, for the wing and empennage are presented in Eq. 4.1.5.1-a in Appendix F. These factors are given as a function of the wing and tail thickness ratios and a correction factor RL.S..

7. The form factor for the fuselage is presented in Eq. 4.2.3.1-a in Appendix G as a function of the effective fineness ratio. The fineness ratio of the body of revolution fuselage is defined in terms of the overall length L as . If the cross section is other than circular, an equivalent diameter is obtained using

the maximum cross sectional area . The base drag

contribution of the fuselage in Eq. 4.2.3.1 is considered to be zero because the body is a closed body. The nacelle drag for turbofan engines of moderate to high bypass ratio is considered to be the drag of the fan shroud or cowl alone because the drag of the center body, which encloses the hot section of the engine, is already included in the thrust rating. Following Torenbeek’s (Ref. 7-6, Appendix F) suggestion, one may use a value of K=1.25 as the form factor for the fan nacelle in Table 7-2. The characteristic length to be used in calculating the skin friction is the length of the fan cowl. Since the quoted static thrust of the engine includes the effects of the fan flow over the inside of the cowl, the wetted area for the nacelle should be the just outside of the fan cowl because that portion is actually wetted by the free stream flow.

8. The equation for CD of the flaps given subsequently may be used. There is then no need to fill in columns 2 through 8 of the drag summary table.

9. Use an appropriate equation to obtain CD for the landing gear contribution. Again, columns 2 through 8 of the drag summary table may be left blank.

10. For interference drag use 5% of sum of previous entries. There is no need to fill in columns 2 through 8.

The drag coefficient calculation table must be repeated twice more, once for the take-off

configuration at the take-off speed, , and one for the landing configuration at V

= Vl. There is an apparent contradiction on the above table, namely, the flight condition shown is M = 0.8 but the CD value has been calculated for incompressible flow. The incompressible drag values must be corrected for the effects of compressibility and this is done subsequently.

Table 7-2 Incompressible Zero-Lift Airplane Drag Coefficient

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Configuration: CruiseAircraft Designation: Smith 2000-150Take-off Weight: 150,000 lbsWing area: 1200 ft2

Aspect Ratio: 7.2Oswald span efficiency: 0.85Flight condition: Mach 0.8 at 37,000 ft altitude

1 2 3 4 5 6 7 8 9component Ref.

lengthWetted area, Sw

Rel Fineness ratio

K CF,turb KCFSw CD,0

WingFlap - - - - - - -Horiz. TailVert. TailFuselage Nacelles Land. gear - - - - - - -Interference

- - - - - - -

Summation - - - - - - -

7.8 Flap and Landing Gear Drag The wing flap and landing gear drag may be obtained (for example) from curves in Perkins and Hage (Ref. 7-9, Fig. 2-68). They use the equivalent parasite area:

f = CDS = CD,cAc

where CD is the zero lift drag coefficient of the component in question (here the landing gear) based on wing planform area, S, while CD,c is its drag coefficient based on some component reference area. Then

CD,landing gear=flanding gear/S

Perkins and Hage (Ref. 7-9) show a curve for estimating the parasite drag area for tricycle landing gear based on take-off weight, which can be fitted by the following equation:

CD,landing gear=4.05x10-3WTO0.785 /S

This equation is also given by Torenbeek (Ref. 7-6, Eq. G-66, p.550), while Mair and Birdsall (Ref. 7-14) quote data from Ref. 7-15 which give the same form of the equation, but with different constants, that is

CD,landing gear=3.30x10-3WTO0.785 /S (zero flap deflection)

CD,landing gear=1.79x10-3WTO0.785 /S (full flap deflection)

The reduced drag coefficient at full flaps is supported by some flight tests on commercial

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airliners and is thought to be a result of reduced airspeeds over the landing gear.

A preferable method for estimating the drag coefficient increment due to flaps is that presented by B. W. McCormick (Ref. 7-10). The equations used are shown below, with cf /c representing the flap chord to wing chord ratio and Sf /S representing the flap planform area to wing area ratio. Based on the wing area, the increment in wing drag coefficient, CD, due to the flaps is given approximately by

CD = 1.7 (cf /c)1.38 (Sf /S) sin2 (plain and split)

CD = 0.9 (cf /c)1.38 (Sf /S) sin2 (slotted)

Other sources of information on drag estimation may be found in the books by Roskam (Ref. 7-11) and Nicolai (Ref. 7-12).Some further remarks on the calculation of the wing wetted area are warranted. It was previously indicated that the wetted area of the wing is approximately equal to twice the exposed wing area. This statement applies to aircraft with mid-wing configurations only. Extreme high-and low-wing configurations have the uppermost and lowermost surfaces, respectively, the central section of the wing totally exposed, while the opposite surface is of course shielded by the fuselage. The calculation of the overlapped area would require a detailed drawing. It is not necessary to perform such a calculation for this study. A reasonable estimate of the overlap area should be made and subtracted from the wing and fuselage areas.

7.9 Compressibility Drag at High Subsonic and Low Transonic Speeds The method presented is based on one of the earliest corrections (Ref. 7-9, Figs. 2-71 and 2-72) for adding the compressibility drag rise to the base value of CD,0,inc . It is sufficient for our purposes and, furthermore, since the drag is so important to fuel economy the newer methods are considered proprietary information and therefore closely guarded. However, DATCOM methods for transonic flow are included for wings and tails in Appendix F and for fuselages and nacelles in Appendix G, and may be used if desired.

The aspect ratios and thickness ratios of the horizontal and vertical tails are typically chosen to be lower than the wing values. This has the effect of raising their critical Mach numbers and delaying the compressibility drag rise compared to the wing. The fuselage, being a very slender body of high fineness ratio, is analogous to a very small aspect ratio wing and therefore has a high critical Mach number. Therefore the wing is primarily responsible for the drag rise of the aircraft. Consider a wing whose quarter-chord line is swept back at an angle c/4 as shown in Fig. 7-4. For simplicity and clarity we show a simple rectangular wing.

An important consideration here is the variation of the cross-sectional area of the complete airplane as a function of axial distance from the nose. As a first approximation, at high subsonic speeds the aircraft will affect the flow field in much the same manner as would an axially symmetric body with the same area distribution. If that area distribution

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is relatively smooth, without abrupt changes in slope, then the assumption that the wing is the primary element in producing compressibility drag is reasonable. On the other hand, rapid changes in area distribution will lead to important interference effects and the drag increase will be large. This “area rule” was an important lesson learned in the early efforts at transonic flight and led to the development of the “coke-bottle” fuselage shape of early century series fighter aircraft. The reduction of the “waist size” of the fuselage was to smooth out the cross-sectional area distribution of the aircraft. A general discussion of the area rule and its effects is given in Ref. 7-10. Therefore it is important to estimate the cross-sectional area distribution of the design aircraft to ensure that the assumption of negligible interference drag is allowable.

Figure 7-4 A swept wing in a free stream flow at Mach number M

If the wing and fuselage combination have a poor cross sectional area distribution in the longitudinal direction, their interference drag will be very large and will prove an exception to the last statement. We will assume a low interference drag (negligible).

The drag force pertinent to the component of Mach number normal to the wing (the x-direction here) is denoted by the subscript x and is given by

Dx = CD,xqxcxby = CD,xqcb(qx/q)(cx/c)(by/b)

On the other hand, the drag force pertinent to the free stream Mach number M is given by

D=Dx/(cosc/4)=CD,x(qx/q)(cx/c)(by/b)qcb/(cosc/4)

Substituting the relations between the chord and the span in the direction normal to the leading edge and in the free stream direction yields

D=CD,x(qx/q)q(cosc/4)c(1/cosc/4)b/(cosc/4)

Mc/4=M

Mc/4=0 c/4

Dx = Dc/4=0

D=Dc/4

dy

yxcx

by

b

c

c/4

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Now the dynamic pressure ratio qx/q=(Vx/V)2=cos2c/4 and therefore

D=CD,xqcb(qx/q)/(cosc/4)=CD,x (cosc/4)qcb

Since the definition of the drag in the free stream direction is D=CDqS, the drag coefficient of the swept wing is CD=CD,xcosc/4. However, the swept wing has the same thickness independent of the sweep while the thickness-to-chord ratio depends on the angle of sweep according to the relation t/cx=t/c(cosc/4). For example, a 12% thick airfoil in a wing that is swept back 30o has an effective t/cx=13.86% in the direction normal to the quarter-chord line.

Thus it seems appropriate to determine the increment in drag coefficient due to compressibility as described in Table 7-3. That is, compute the drag coefficient increment (column 5) for the wing at a series of Mach numbers normal to the quarter-chord line. Then compute the effective drag coefficient increment in the free stream direction (column 7) by multiplying the former drag coefficient increment by the cosine of the quarter-chord sweepback angle. The appropriate Mach number to use would be, according to the analysis presented previously, the normal Mach number divided by the cosine of the sweepback angle. However, Table 7-3 (column 6) divides by cos1/2c/4

because this value, rather than the ideal value, gives more realistic answers in practical cases. It should be pointed out that care should be taken in using either the “peaky” airfoil results of Fig. 7-6 or the NACA 6 series airfoil data in Ref. 7-7 because the effective thickness to chord ratio is larger in the direction normal to the quarter-chord line and this should be taken into account when determining the Mcr’ values for each lift coefficient for use in Table 7-3.

The procedure for estimating the drag rise using the approach in Perkins and Hage (Ref. 7-9) is as follows:

1. Obtain the appropriate curve of MCR for the (average) design airfoil from Ref. 7-7. The pertinent curves are given here in in Appendix H. Bisect the angle the "roof top" makes with the horizontal through the point "A" shown in Figure 7-5. Then form the dash-dot line defined as Mcr

’. A copy of the graph used should also appear in the design report. Shevell (Ref. 7-16, p.199) presents curves of critical Mach numbers for "peaky" airfoils of the type used in commercial airliners of that period. They are shown in Fig. 7-6 and are appreciably broader than the curves for the NACA 6-series airfoils in Ref. 7-7. One may use these curves to represent the critical Mach numbers of more modern supercritical airfoils, according to Shevell, by using the curves shown in Fig. 7-5 and adding 0.06 to the critical Mach number so found. He also notes that the thickness to chord ratio, t/c, shown in the

figure is a weighted one and is given by the ratio of to that of , both

taken over the half-span. For a linear thickness distribution, t(y)~y, the result is (t/c)=(troot + ttip)/(croot + ctip)

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2. Make a table of Mcr’ obtained from this curve vs CL :

CL Mcr’

0 _____0.1 _____0.2 _____0.3 _____0.4 _____

The actual CL values selected need not be the ones illustrated. Instead one should select the range of values to be expected in the cruise condition.

Figure 7-5 Typical Mcr versus CL curve from Ref. 7-7

3. For each CL value used above, namely 0, 0.1, 0.2, etc. construct a table like Table 7-3 based on Figs. 7-5 or 7-6 and 7-7. Note that in Table 2 the quantity k=(eA)-1

M’cr

Mcr

CL

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Figure 7-6 Critical Mach number curves for “peaky” airfoils. Adapted from Ref. 7-16

Figure 7-7 Compressibility correction curve

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Table 7-3 Corrected Drag Coefficients for Each Lift Coefficient

1 2 3 4 5 6 7 8

CL Mcr’ M M (CD)

atc/4=0

Mc/4 (CD)c/4 CD

=M-

Mcr’

from Fig. 7-7

=(3)(cosc/4)-1/2 =(5)cosc/4 =CD0+k(1)2+(7)

0 0.723 0.600 -0.123 0.00070.650 -0.073 0.00120.700 -0.023 0.00200.750 0.027 0.00250.800 0.077 0.01000.825 0.102 0.01900.850 0.127 0.0350

4. Plot CD versus M for each value of CL. An example of such a plot is shown in Fig 7-8. The low speed values of CD correspond to the incompressible values calculated earlier (CD = CD,0 + kCL

2) for the cruise configuration. They are independent of Mach number until the drag rise starts at high subsonic speeds. A cross-plot of the data shown provides a drag polar for each Mach number.

Figure 7-8 Total CD versus Mach number for various values of CL

5. The curves in Fig. 7-8 are used to determine the thrust required for your performance studies, i.e., Treq = D = CDqS. The designer calculates the thrust required using spreadsheet tables of the form shown in Table 7-4. Note that the table is divided into two Mach number ranges, 0.25 < M < 0.6 and 0.65 < M < 0.85. The first is simplified since compressibility effects can be ignored. Note

CL=0.4

CL=0.3

CL=0.2

CL=0.1

CL=0

CD

M

0.850 0.6

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also that the lower limit on the Mach number (M = 0.25) is a function of altitude, because the lift coefficient will exceed CLmax at low Mach numbers and high altitudes.

Table 7-4 Thrust Required

Aircraft designation: Smith 2000-150Take-off weight: 150,000 lbsWing Area: 1200 ft2

Altitude: 35,000 ftp/ps.l.=: 0.2351CD,0 (incompressible): 0.0197k=1/eA: 0.0468

1 2 3 4 5M q

= ½ ps.l.M2

CL

=(W/S)/q

CD

See notes in 1st column

Treq’d

=CDqS

0.25<M<0.6(incompressible)CD=CD,0+kCL

2

.25

.30

.35

.40

.45

.50

.55

.600.6<M<0.85(compressible)CD from graph of CD vs M for various CL. Use finer increments in M.

.65

.70

.775

.800

.825

.850

7.8 Thrust Available and Thrust Required To aid in the determination of maximum and minimum airspeeds at the cruise altitude,

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graphs of the required thrust, Treq, and the available thrust, Tavail, versus Mach number should be constructed, like that shown in Fig. 7-9. The maximum altitude may also be determined by considering such graphs for higher and higher altitudes. When the maximum and minimum Mach numbers are equal, i.e. when the curve of Treq is tangent to that of Tavail , one is at the maximum sustainable altitude, and is called the absolute ceiling. The service ceiling, on the other hand, is that altitude at which there is sufficient excess thrust to permit a rate of climb of 100 ft/min.

Figure 7-9 The variation of thrust available and thrust required with Mach number for a given altitude

The variation of thrust with altitude is often taken as being equal to that of the atmospheric density ratio, i.e. T(h) = Tto A review of data presented in Ref. 7-13 suggests that a better approximation is T(h) = Tto , where is the atmospheric pressure ratio. Of course if the actual value of thrust available at altitude is known from engine manufacturer data it should be used in the calculations.

It is important to note that most textbook treatments of drag effects on range performance tacitly assume that CD,0 is a constant, whereas above M = 0.6 it must be recognized that CD,0 = CD,0(M). Therefore one must be careful in using equations developed on the basis of constant zero-lift drag coefficient. One may also use the results of Table 7-4 to generate a graph of M(L/D) vs. M. The maximum value of M(L/D), rather than the

Thrust required

Thrust availableThrust

Mmin Mach Number Mmax

Excess thrust

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maximum value of L/D alone will determine the Mach number for maximum range. This issue arises in the calculation of the velocity for best range in the performance chapter of the design report. Note that at this point in the design, the drag polar (CL vs. CD) for each Mach number and altitude has been calculated so there is no need to resort to the approximate equations often quoted in textbooks on performance.

7.9 References

7-1 Hoak, D.E., et al: "USAF Stability and Control DATCOM", Flight Control Division, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, Sept. 1970.

7-2 Schlichting, H.: Boundary Layer Theory, McGraw-Hill, NY, 1979

7-3 U.S. Standard Atmosphere 1976: see, e.g., www.digitaldutch.com/atmoscalc/

7-4 Hoerner, S.F.: Fluid Dynamic Drag, published by the author, 1958.

7-5 Anon.: "External Drag Evaluation", North American Aviation, Inc. Report ADL-52-2, Oct. 1952.

7-6 Torenbeek, E.: Synthesis of Subsonic Airplane Design, Kluwer Academic Publishers, The Netherlands, 1982, Sections 5.3, 11.2, and 11.3 and Appendices F and G

7-7 Abbott, I. H., et al: "Summary of Airfoil Data", NACA Technical Report No. 824, 1945. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930090976_1993090976.pdf

7-8 Abbott, I.H. and VonDoenhoff, A.E.: Theory of Wing Sections, Dover, NY, 1959

7-9 Perkins & Hage Airplane Performance Stability and Control, Wiley 1949, see Chapter 2

7-10 McCormick, B.H.: Aerodynamics, Aeronautics, and Flight Mechanics, Wiley, 1995.

7-11 Roskam, Jan, Method for Estimating Drag Polars of Subsonic Airplanes, Roskam Aviation and Engineering Corp., Lawrence, Kansas, 66044, 1971

7-12 Nicolai, L.M., Fundamentals of Aircraft Design, University of Dayton Press, Dayton, Ohio, 1975, Chapters 2, 11, 12, and 13

7-13 Svoboda, C.: “Turbofan Engine Database as a Preliminary Design Tool”, Aircraft Design, Vol. 3, 2000, pp.17-31

7-14 Mair, W.A. and Birdsall, D.L.: Aircraft Performance, Cambridge University Press, NY, 1987

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7-15 Engineering Sciences Data Unit: “Undercarriage Drag Prediction Methods”, ESDU 79015, 1987

7-16 Shevell, R.: Fundamentals of Flight, Prentice-Hall, Englewood Cliffs, NJ, 1989.

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