Subsonic Leading Edges

Reference 5.22 is a linearized treatment of delta wings with subsonic leading edges. The method is valid for large values of the apex angle up to and coincident with the Mach angle. Reference 5.23 extends the method of Reference 5.22 to include swept wings with pointed tips, but with trailing edges that are also swept. The trailing edges in Reference 5.22 are restricted only by the requirement that they be supersonic. Hence their sweep angle must be less than the “sweep” angle of the Mach wave. Thus, swept forward trailing edges are allowed. It is interesting to note that any triangular wing having subsonic leading edges and supersonic trailing edges will have the same pressure distribution along a ray from the apex as any other triangular wing having the same leading edge sweep angle and operating at the same Mach number. Thus, if one finds the pressure distribution over a “basic” delta wing, the effect of sweeping its trailing edges can be readily determined. The conclusion follows from the fact that disturbances cannot propagate ahead of the Mach cone in linearized supersonic flow. Figure 5.41 illustrates the foregoing principle.

For delta wings only, the following expression was obtained by Brown (Ref. 5.22) for the lift curve slope.

(5.102)

A is a function of the ratio of apex angle tangent to that of the Mach angle; it is presented in Figure 5.42. Note that when the leading edge of the wing and the Mach line are coincident, CLa reduces to equation 5.76; that is, the slope of the lift curves are the same for a supersonic two-dimensional airfoil and a delta wing, the leading edge of which is coincident with the Mach line. This also holds if the leading edge is supersonic. Thus, for values of tan e/tan ft greater than unity, the value of the A function in Figure 5.42 is constant and equal to 1.793. Note also that Equation 5.102 reduces to the results from slender wing theory as the apex angle approaches zero.

For subsonic leading edges, Brown (Ref. 5.22) shows that a leading edge suction force will exist. As a result, an induced drag is obtained that is less than the streamwise component of the wing’s normal force.

(5.103)

It is emphasized that Equations 5.102 and 5.103 hold only for delta wings with subsonic leading edges. A in these particular equations refers to the function shown in Figure 5.42 and does not stand for the taper ratio. The taper ratio of a delta wing is zero. Since tan fi = І/В and A is a function of tan el tan д, we can write

Cd,

I it A)

;/(tane

•4 tan д/

Referencing CLa to the two-dimensional value, we can also write

(5.105)

The functions, / and g, are presented in Figure 5.43. With regard to the drag, note that the leading edge suction force vanishes when the leading edge and Mach line are coincident. Thus, for д values equal to or less than e, the drag is simply equal to the streamwise component of the wing normal force. In coefficient form,

CDi = C[ tan а

C[ — “ — 4 tan д and A = 4 tan 6 H

Thus, for д < e,

CD

(Cl 177A) 17

This is the value shown in Figure 5.43 for tan e/tan д = 1.

Figure 5.44 depicts the planform shapes treated in Reference 5.23. They might be described as truncated deltas with trailing edge sweep. Trends of CLa with M, A, A, and taper ratio are presented in Figure 5.45 (taken from Ref. 5.23). For this family of wings, A, A, e, and 8 (see Figure 5.44) are related by

Before leaving the subject of wings with subsonic leading edges, it should be emphasized once again that such wings can develop leading edge suction

forces. Thus, to prevent leading edge separation in order to maintain this suction force, it is beneficial (from a drag standpoint) to round the leading edges of these wings, even though they are operating at a supersonic free – stream Mach number. Experimental proof of this (Ref. 5.26) is offered by Figure 5.46a. Here the lift-to-drag ratio as a function of CL is presented for a delta wing with its leading edges lying within the Mach cone. Three different airfoil sections were tested, including one with a rounded leading edge. This latter section is seen to have a maximum LID value approximately 8% higher than the other two.

Other data from the same reference are presented in Figure 5.46b to 5.46/, which shows the effects of aspect ratio and sweep on lift, moment, and drag for a family of tapered wings. These wings all have a taper ratio of 0.5 and employ the cambered wedge section having a maximum thickness ratio of 5%, pictured in Figure 5.466.

Figure 5.44 Supersonic wings and subsonic leading edges, (a) Sweptback trailing edge, (b) Sweptforward trailing edge.

The moment coefficients for this set of graphs are about the centroid of the planform area, with the mean aerodynamic chord as the reference length. If the aerodynamic center of a wing is a distance of x ahead of the centroid and M is the moment about the centroid, then

Mac = M — xL

In coefficient form,

x Смас = Cm — Cl~

Differentiating with respect to Cl and recalling that, by definition, dCMJdCL – 0, gives

£= dCV c dCL

Thus Figure 5.46d and 5.46e represents, in effect, the distance of the aerody­namic center ahead of the centroid.

0 0.2 0.4 0.6 0.8 45 50 55 60