# 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, Л, 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 ACDp 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 CDpmi„ 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 ACDp 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 the ACDp 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 ACDp and CDw separate. The ACDp values for the worked – out 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:

Cd = Cupmin + ACup + C2L/n AR + (Cu^hock@CL = 0) + Cdw (9.37)

It is recommended that in current practice, CFD analysis should be used to obtain the variation of ACDp and ACDw 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 = C2JnAR.

Step 5: Obtain ACDp 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, Бл, 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

exposed areas

ate shaded

author does not recommend undertaking coursework on exotic-aircraft configurations unless the results can be substantiated. Learning with a familiar design that can be substantiated gives confidence to practitioners. Those in the industry are fortunate to have access to more accurate in-house data.

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