Wing of Finite Thickness. in Supersonic Flow
General statements In the previous sections, the inclined wing of finite span in supersonic flow was treated (lift problem). Now, the special case of a wing of finite
thickness with zero lift (displacement problem) will be discussed in more detail. Of interest here are the pressure distribution over the wing contour and the resulting wave drag. The latter is a strong function of the profile thickness, as was discussed for the plane problem in Sec. 4-3-3. The most general method of determining the pressure distribution of wings of finite thickness at zero lift is the source-sink method of von Karman . The fundamentals of this method for the wing with supersonic incident flow were furnished in Sec. 4-5-3. The basic concept of this method is to cover the planform area of the given wing with a source distribution q(x, y) in the xy plane. From this, the x component of the velocity on the wing surface u(x, y) is obtained from Eq. (4-103) and the z component w(x, y) from Eq. (4-104). By describing the wing contour by z^x, y) = z(x, y), the kinematic flow condition is expressed by Eq. (3-173b). Introducing this into Eq. (4-104) yields the source distribution Eq. (3-176). Introducing this result into Eq. (4-103) furnishes the pressure. coefficient cp = —2и/иж as
~ [x’, 7/0 dx dy’ _______________ dx
І(х – –1[Ma^ – ЇМ*/ – УТ
Here, A’ is the influence range of the point x, y, as indicated in Fig. 4-58 by cross-hatching. The pressure distribution for a given wing contour z(x, y) can thus be determined.
Wave drag The coefficient of wave drag of the wing at zero lift is obtained through integration of the pressure distribution over the wing area A as
This formula is applicable to sharp-edged profiles only. The dependence of the drag coefficient on profile thickness ratio, taper, aspect ratio, sweepback angle, and Mach number of the incident flow is given according to the supersonic similarity rule by Eq. (4-27). This relationship is of great value for a systematic presentation of theoretical and experimental results.
Rectangular wing For the wing of rectangular planform and spanwise constant profile z(x, y) = z(x), introducing Eq. (4-124) into Eq. (4-125) and integrating twice yield (see Dorfner )
Note that, for A’ = As/MaL — 1 >1, the drag formula for the rectangular wing of finite span is identical to that of the rectangular wing of infinite span (see Table 4-2).
For a convex parabolic profile Z = z/c = 2bX( — X) with X = x/c, the integration yields
= 1 (A’ > 1)
(4-1276) where cDo* is given by Eq. (4-50a). The numerical evaluation is given in Fig, 4-94.
Delta wing A few results will be added for delta (triangular) wings. Delta wings with double-wedge profiles have been computed ‘ by Puckett , those with biconvex parabolic profiles by Beane . Coefficients of the wave drag at zero lift for double-wedge and biconvex parabolic profiles of 50% relative thickness position are shown in Fig, 4-95 as a function of the parameter m = /MaL — Ы/4. For the double-wedge profile, cDQO* is expressed by Eq. (4-51). For supersonic leading edges
Figure 4-94 Drag coefficient (wave drag) at zero lift for rectangular wings at supersonic incident flow vs. Mach number. Biconvex parabolic profile cd0«, from Eq. (4-5 Oc).
(m> 1), cD0/cDQoo is almost independent of Mach number, whereas it changes strongly with Mach number for subsonic leading edges (m < 1). Both curves have pronounced breaks at m = 1, that is, when the Mach line coincides with the leading edge. The curve for the double-wedge profile has another break at m = , that is, when the Mach line is parallel to the line of greatest thickness.
In Fig. 4-96, a number of measurements on delta wings with double-wedge profiles and 19% relative thickness position are plotted from . Similar to Fig. 4-86, different representations have been chosen for m < 1 and m> 1. At the kind of presentation chosen here, these measurements on 11 wings at Mach numbers Ma„ = 1.62, 1.92, and 2.40 fall very well on a single curve. Hence, the supersonic similarity rule of Eq. (4-27) has been confirmed again. The theoretical curve from
Puckett  for the relative thickness position X*=0.18 shows a high peak at m = 1 that is not confirmed by measurements, as would be expected because the incident flow velocity at the leading edge is just sonic. Comparison between theory and experiment suffers from the uncertainty in the determination of the friction drag, which has to be subtracted from the measured values.
The treatment of the thickness problem of a delta wing with sonic leading edge has been compared with transonic flow theory by Sun .
Swept-back wing The wave drag coefficients of swept-back wings of constant chord are illustrated in Fig. 4-97. The corresponding information for the lift slope was given in Fig. 4-88. The wing has a double-wedge profile, of which the drag coefficient in plane flow cDo is obtained from Eq. (4-51). The curves show a pronounced break at m = 1, that is, when the Mach line and leading edge fall together. It should be noted that, according to ,
-02- = —— for m > 1 + —(4-128)
Cdqoo iml — 1 /loot у
is obtained in the range of the supersonic leading edge if the Mach line originating at the apex (line g) intersects the trailing edge.
Arbitrary wing planforms To conclude this discussion, the total drag coefficient at zero lift (wave drag + friction drag) of the three wings (trapezoidal, swept-back, and delta) treated earlier (Figs. 4-89-4-91) is plotted in Fig. 4-98 against the Mach number. These three wings have double-wedge profiles with a thickness ratio t/c = 0.05 and an aspect ratio A = 3. Within the Mach number range presented, the
Figure 4-98 Total drag coefficient (wave drag 4- friction drag) vs. Mach number for a trapezoidal, a swept-back, and a delta wing of aspect ratio л = 3. Double-wedge profile tjc = 0.05, xtjc =
0. 50, from .
wave drag is two to three times larger than the friction drag. The latter has been determined from Fig. 4-4 for Reynolds numbers Re ^ 107. Since the wave drag at supersonic incident flow is proportional to (t/c)2, this contribution, and thus the total wing drag at zero lift, can be reduced considerably by keeping t/c small. This fact is taken into account in airplane design by choosing extremely small thickness ratios for supersonic airplanes; compare Fig. 3-4a.
Concluding remarks In addition to the references included in the text, attention should be directed toward summary reports and reports dealing with various theories on the aerodynamics of the wing in supersonic flow [6, 11, 19, 22, 23, 40, 51, 92, 105-107] – The special case of the aerodynamics of the wing of small aspect ratios, first studied by Jones , has been investigated comprehensively as the “slender-body theory” for both lift and drag problems [2, 13, 14, 41, 108]. The aerodynamics of slender bodies is treated in Sec. 6-4. The influence of vortex shedding at the lateral wing edges of rectangular wings, and the leading-edge separation on swept-back and delta wings at supersonic flow, are treated in [12, 72, 91], based on the understanding of incompressible flow. Based on a suggestion of Jones, questions concerning the minimum wing drag have been investigated by several authors [36, 61, 97, 110]. In this connection, the investigations on the design aerodynamics of wings at high flight velocities, promoted mainly by Kuchemann, play an important role [9, 38, 46, 60].