operates at a low Re during takeoff and landing. For streamlined struts with fairings, it decreases to 0.5 to 0.6, depending on the type.

Torenbeek [10] suggests using an empirical formula if details of undercarriage sizes are not known at an early conceptual design phase. This formula is given in the FPS system as follows:

Understandably, it could result in a slightly higher value (see the following example).

Continue with the previous example using the largest in the design (i.e., MTOM = 24,200 lb and SW = 323 ft2) for the undercarriage size. It has a twin-wheel, single-strut length of 2 ft (i.e., diameter of 6 inches, Aπ strul ≈ 0.2 ft2) and a main wheel size with a 22-inch diameter and a 6.6-inch width (i.e., wheel aspect ratio = 3.33, Aπ wheel ≈ 1 ft2). From Table 9.6, a typical value of CDπ wheel = 0.18, based on the frontal area and increased by 50% for the twin-wheel (i.e., CD0 = 0.27). Including the nose wheel (although it is smaller and a single wheel, it is better to be liberal in drag estimation), the total frontal area is about 3 ft2:

f wheel = 0.27 × 3 = 0.81 ft2

f strut = 1.0 × 3 × 2 × 0.2 = 1.2 ft2

Total fUC = 2 × (0.81 + 1.2) = 4.02 ft2 (100% increase due to interference,

doors, tubing, and so on) in terms of CDpmin UC = 4.02/323 = 0.0124. Checking the empirical relation in Equation 9.36, CD UC = 0.00403 × (24,2000.785)/323 =

0.034, a higher value that is acceptable when details are not known.

9.14.4 One-Engine Inoperative Drag

Mandatory requirements by certifying agencies (e.g., FAA and CAA) specify that multiengine commercial aircraft must be able to climb at a minimum specified gradient with one engine inoperative at “dirty” configuration. This immediately safeguards an aircraft in the rare event of an engine failure; and, in certain cases, after liftoff. Certifying agencies require backup for mission-critical failures to provide safety regardless of the probability of an event occurring.

Asymmetric drag produced by the loss of an engine would make an aircraft yaw, requiring a rudder to fly straight by compensating for the yawing moment caused by the inoperative engine. Both the failed engine and rudder deflection substantially increase drag, expressed by CD engine out+ruddert. Typical values for coursework are in Table 9.7.

9.15 Propeller-Driven Aircraft Drag

Drag estimation of propeller-driven aircraft involves additional considerations. The slipstream of a tractor propeller blows over the nacelle, which blocks the resisting flow. Also, the faster flowing slipstream causes a higher level of skin friction over the downstream bodies. This is accounted for as a loss of thrust, thereby keeping

the drag polar unchanged. The following two factors arrest the propeller effects with piston engines (see Chapter 10 for calculating propeller thrust):

1.Blockage factor, fb, for tractor-type propeller: 0.96 to 0.98 applied to thrust (for the pusher type, there is no blockage; therefore, this factor is not required – i.e., fb = 1.0)

2.A factor, fh, as an additional profile drag of a nacelle: 0.96 to 0.98 applied to thrust (this is the slipstream effect applicable to both types of propellers)

Turboprop nacelles have a slightly higher value of fb than piston-engine types because of a more streamlined shape. However, the slipstream from a turboprop is higher and therefore has a lower value of fh.

9.16 Military Aircraft Drag

Although military aircraft topics are not discussed here, and instead are found in the Web at www.cambridge.org/Kundu, this important topic of military aircraft drag estimation is kept here.

Military aircraft drag estimation requires additional considerations to account for the weapon system because few are carried inside the aircraft mould lines (e.g., guns, ammunition, and bombs inside the fuselage bomb bay, if any); most are external stores (e.g., missiles, bombs, drop-tanks, and flares and chaff launchers). Without external carriages, military aircraft are considered at typical configuration (the pylons are not removed – part of a typical configuration). Internal guns without their consumables is considered a typical configuration; with armaments, the aircraft is considered to be in a loaded configuration. In addition, most combat aircraft have a supersonic-speed capability, which requires additional supersonic-wave drag.

Rather than drag due to passenger doors and windows as in a civil aircraft, military aircraft have additional excrescence drag (e.g., gun ports, extra blisters and antennas, and pylons) that requires a drag increment. To account for these additional excrescences, [3] suggests an increment of the clean flat-plate equivalent drag, f, by 28.4%.

Streamlined external-store drag is shown in Table 9.8 based on the frontal maximum cross-sectional area.

Table 9.8. External-store drag

Bombs and missiles flush with the aircraft contour line have minor interference drag and may be ignored at this stage. Pylons and bomb racks create interference, and Equation 9.17 is used to estimate interference on both sides (i.e., the aircraft and the store). These values are highly simplified at the expense of unspecified inaccuracy; readers should be aware that these simplified values are not far from reality (see [1], [4], and [5] for more details).

Military aircraft engines are buried into the fuselage and do not have nacelles and associated pylons. Intake represents the air-inhalation duct. Skin friction drag and other associated 3D effects are integral to fuselage drag, but their intakes must accommodate large variations of intake air-mass flow. Military aircraft intakes operate supersonically; their power plants are very low bypass turbofans (i.e., on the order of less than 3.0 – earlier designs did not have any bypass). For speed capabilities higher than Mach 1.9, most intakes and exhaust nozzles have an adjustable mechanism to match the flow demand in order to extract the best results. In general, the adjustment aims to keep the Vintake/V∞ ratio more than 0.8 over operational flight conditions, thereby practically eliminating spillage drag (see Figure 9.7). Supersonic flight is associated with shock-wave drag.

9.17 Supersonic Drag

A well-substantiated reference for industrial use is [3], which was prepared by Lockheed as a NASA contract for the National Information Service, published in 1978. A comprehensive method for estimating supersonic drag that is suitable for coursework is derived from this exercise. The empirical methodology (called the Delta Method) is based on regression analyses of eighteen subsonic and supersonic military aircraft (i.e., the T-2B, T37B, KA-3B, A-4F, TA-4F, RA-5C, A-6A, A-7A, F4E, F5A, F8C, F-11F, F100, F101, F104G, F105B, F106A, and XB70) and fifteen advanced (i.e., supercritical) aerofoils. The empirical approach includes the effects of the following:

wing geometry (AR, , t/c, and aerofoil section)

cross-sectional area distribution

CD variation with CL and Mach number

The methodology presented herein follows [3], modified to simplify CDp estimation resulting in minor discrepancies. The method is limited and may not be suitable to analyze more exotic aircraft configurations. However, this method is a learning tool for understanding the parameters that affect supersonic aircraft drag buildup. Results can be improved when more information is available.

The introduction to this chapter highlights that aircraft with supersonic capabilities require estimation of CDpmin at three speeds: (1) at a speed before the onset of wave drag, (2) at Mcrit, and (3) at maximum speed. The first two speeds follow the same procedure as for the high-subsonic aircraft discussed in Sections 9.7 through 9.14. In the subsonic drag estimation method, the viscous-dependent CDp varying with the CL is separated from the wave drag, CDw (i.e., transonic effects), which also varies with the CL but independent of viscosity.

For bookkeeping purposes in supersonic flight, such a division between theCDp and the CDw is not clear with the CL variation. In supersonic speed, there is little complex transonic flow over the body even when the CL is varied. It is not

clear how shock waves affect the induced drag with a change in the angle of attack. For simplicity, however, in the empirical approach presented here, it is assumed that supersonic drag estimation can use the same approach as the subsonic drag estimation by keeping CDp and CDw separate. The CDp values for the workedout example are listed in Table 9.13. Here, drag due to shock waves is computed at CL = 0, and CDw is the additional shock-wave drag due to compressibility varying with CL > 0. The total supersonic aircraft drag coefficient can then be expressed as follows:

It is recommended that in current practice, CFD analysis should be used to obtain the variation of CDp and CDw with CL. Reference [3] was published in 1978 using aircraft data before the advent of CFD. Readers are referred to [1], [4], and [5] for other methods. The industry has advanced methodologies, which are naturally more involved.

The aircraft cross-section area distribution should be as smooth as possible, as discussed in Section 3.13 (see Figure 3.23). It may not always be possible to use narrowing of the fuselage when appropriate distribution of areas may be carried out.

The stepwise empirical approach to estimate supersonic drag is as follows:

Step 1: Progress in the same manner as for subsonic aircraft to obtain the aircraft-component Re for the cruise flight condition and the incompressible CFcomponent.

Step 2: Increase drag in Step 1 by 28.4% as the military aircraft excrescence effect.

Step 3: Compute CDpmin at the three speeds discussed previously. Step 4: Compute induced drag using CDi = CL2 /π AR.

Step 5: Obtain CDp from the CFD and tests or from empirical relations. Step 6: Plot the fuselage cross-section area versus the length and obtain the

maximum area, Sл, and base area, Sb (see example in Figure 9.17). Step 7: Compute the supersonic wave drag at zero lift for the fuselage and

the empennage using graphs; use the parameters obtained in Step 6. Step 8: Obtain the design CL and the design Mach number using graphs (see

example in Figure 9.19).

Step 9: Obtain the wave drag, CDw , for the wing using graphs.

Step 10: Obtain the wing-fuselage interference drag at supersonic flight using graphs.

Step 11: Total all the drags to obtain the total aircraft drag and plot as CD versus CL.

The worked-out example for the North American RA-5C Vigilante aircraft is a worthwhile coursework exercise. Details of the Vigilante aircraft drag are in [3]. The subsonic drag estimation methodology described in this book differs with what is presented in [3] yet is in agreement with it. The supersonic drag estimation follows the methodology described in [3]. A typical combat aircraft of today is not too different than the Vigilante in configuration details, and similar logic can be applied. Exotic shapes (e.g., the F117 Nighthawk) should depend more on information generated from CFD and tests along with the empirical relations. For this reason, the