EFFECT OF MACH NUMBER ON THE ZERO LIFT DRAG OF TWO – AND THREE-DIMENSIONAL SHAPES

In subsonic flow, the drag of nonlifting shapes is relatively unaffected by Mach number until a critical value is reached. Below Mcr, the drag, excluding that due to lift, results from skin friction and the unbalance of normal pressures integrated around the body. The estimation of this drag has been covered in some detail in the preceding chapter. As the Mach number is increased, a value is reached where local shock waves of sufficient strength to produce s€paration are generated. At this point, the drag coefficient begins to rise. As the Mach number continues to increase, CD will increase through the transonic flow region until supersonic flow is established. Depending on the particular shape, the rate of increase of CD with M diminishes. CD may continue to increase with M, but at a lower rate, it can remain fairly constant, or it can actually decrease with increasing Mach number. The behavior of CD in the supersonic flow regime depends on the composition of the drag. Excluding the drag caused by lift, the remainder of the drag is composed of skin friction drag, wave drag, and base drag.

Base drag is a term not yet used. It refers to the drag produced by the pressure acting on the blunt rear end (base) of a body, such as that pictured in Figure 5.49. The base drag was not stressed in Chapter Four since, in subsonic flow, the shape of the base affects the flow over the rest of the body ahead of

Figure 5.49 Origin of base drag.

it, so the base was viewed as simply an integral part of the overall pressure drag. In supersonic flow, however, the base does not affect the flow ahead of it, so it is convenient to treat it separately.

An upper limit on the base drag can be easily obtained by noting that the base pressure can never be less than zero. Hence the base pressure coefficient is bounded by

P _ Рв — P о Pn (l/2)p0 Vі

-Po ,

(U2)pV2

2

> уM2

Therefore, the base drag coefficient, based on the base area, must satisfy the inequality

(5.110)

There appears to be no accepted method available for calculating CD„- Generally, the base drag is affected by the thickness of the boundary layer just ahead of the base. Hence, CDb depends on the body shape, surface condition, and Reynolds number, as well as on the Mach number. In the supersonic regime, experimental data presented as a ratio to the upper limit (Equation 5.110) show the trends pictured in Figure 5.50.

The total drag coefficients of some basic shapes are presented in Figures 5.51 to 5.53. These curves are based on data from a number of sources. In Figure 5.51, CD as a function of M is presented for circular disks and 2-dimensional flat plates, while Figure 5.52 presents the corresponding graphs for spheres and cylinders. In both cases, for the two-dimensional bluff shapes, Cd peaks at a Mach number of unity and then decreases, with increasing M ^reaching a value at approximately M = 2, which is equal to or slightly less th^tn the low-speed value. Above M = 2, CD remains nearly constant as M increases. The three-dimensional values, however, begin to rise at an M of approximately 0.7 and continue to rise until an M of approximately 2.0 is reached.

Figure 5.53 presents the drag of various conical heads having different apex angles as a function of Mach number. This is only the drag resulting from the pressure on the forward surface of the cone.

Drag data on a number of bodies of revolution are presented in Reference 5.25. Most of these data were derived by differentiating the velocity time history obtained by radar of free-flying, fin-stabilized models as

Drag coefficient,

2.8

Mach number, M

Figure 5.51 Drag of flat plates and discs normal to the flow as a function of Mach number.

they decelerated from supersonic to high subsonic Mach numbers. The data were then reduced, assuming that the effects of shape and fineness ratio could be considered separately. Skin friction drag was estimated on the basis of Figure 5.54, which presents Q for flat plates as a function of Reynolds number for constant Mach numbers. This figure appears to be consistent with Figure 5.34 and Equations 5.87 and 5.88, and it is applicable to bodies of revolution, provided the length-to-diameter ratio, lid, is sufficiently large.

Most of the shapes that were flown had afterbodies (i. e., a base diameter smaller than the maximum body diameter). For such bodies, the base drag is only a small fraction of the total drag and varies approximately as the third power of the ratio of the base diameter to the body diameter.

Figure 5.55 presents CD as a function of Mach number for parabolic bodies having different fineness ratios and positions of maximum diameter. Similar data for degrees of nose roundness are given in Figure 5.56. The effect of afterbody shape on CD can be estimated from Figure 5.57. Finally, for these data, the effect of shape on the pressure drag of noses is shown in Figure 5.58. Draw your own conclusions regarding an optimum shape from these data. The CD values are all based on the maximum projected frontal area. Normally one is concerned with packaging a given payload; therefore, a CD based on volume to the 2/3 power might be more informative.

Area Rule for Transonic Flow

Reference 5.27 represents a significant contribution to the aerodynamics of high-speed aircraft. Whitcomb (Ref. 5.27) experimentally investigated the zero lift drag of wing-body combinations through the transonic flow regime. Based on analyses by Hayes (Ref. 5.28), Busemann (Ref. 5.29), and others, Whitcomb formulated some general guidelines for the design of wing-body combinations to have minimum wave drag that are reflected in most of today’s aircraft designed to operate near or in excess of Mach 1. These guidelines are included in the general designation “area rule.”

The essence of the area rule is contained in the data of Figures 5.59 to 5.62 (taken from Whitcomb’s original NACA report). These figures present Co at zero lift as a function of M for a cylindrical body alone and in combination with a triangular wing and a tapered swept wing. Сц, is the total CD measured for zero lift, while Д Сц, is obtained by simply subtracting Сц at M = 0.85

Figure 5.58 Drag coefficients due to pressure on noses at M = 1.4.

from the other C* values. Сц, at M = 0.85 is approximately equal to the skin friction CD over the range of M values tested, so that ДСц, represents the wave drag coefficient.

In these figures, there are essentially four different combinations.

1. Basic body alone.

2. Wing attached to the unaltered basic body.

3. A “flattened” body alone.

4. Wing attached to a “thinned” body.

For combinations 1 and 4, the longitudinal distribution of the total (wing

plus body) cross-sectional area in any transverse plane is the same. This is also true in comparing combinations 2 and 3.

After considering these results and some others not presented here, Whitcomb states:

“Comparisons of the shock phenomena and drag-rise increments for representative wing and central-body combinations with those for bodies of

revolution having the same axial developments and cross-sectional areas normal to the airstream have indicated the following conclusions:

1. The shock phenomena and drag-rise increments measured for these representative wing and central-body combinations at zero lift near the speed of sound are essentially the same as those for the comparable bodies of revolution.

2. Near the speed of sound, the zero-lift drag rise of a low-aspect-ratio thin-wing-body combination is primarily dependent on the axial develop­ment of the cross-sectional areas normal to the airstream. Therefore, it follows that the drag rise for any such configuration is approximately the same as that for any other with the same development of cross-sectional areas. ‘<

Further results have indicated that indenting the bodies of three represen­tative wing-body combinations, so that the axial developments of cross – sectional areas for the combinations were the same as for the original body alone, greatly reduced or eliminated the zero-lift drag-rise increments asso­ciated with wings near the speed of sound. ”

As a corollary to the conclusions stated by Whitcomb, the longitudinal distribution of the total cross-sectional area from the wing and fuselage should be a smooth one for minimum wave drag (i. e., significant increases or decreases of the total area over short distances in the streamwise direction should be avoided).