Nonplanar Lifting Surfaces in Supersonic Flow
For analyzing the corresponding problem by linearized theory at supersonic Mach numbers, we rely entirely on the technique of supersonic aerodynamic influence coefficients (AIC’s) and present only the general outlines of a feasible computational approach. Their only special virtue over other techniques is that they have been practically and successfully mechanized for a variety of steady and unsteady problems.
It should be pointed out first that there exists a formula for a supersonic pressure doublet analogous to the one that has been worked with in the preceding section [see Watkins and Berman (1956)]. This is quite difficult for manipulation, however, because of the law of forbidden signals and other wavelike discontinuities which occur. As a result, even the case of a planar lifting surface at supersonic speed has not been worked through completely and in all generality using pressure doublets. Rather, the problem has been handled by the sorts of special techniques described in Chapter 8. Examples of numerical generalization for arbitrary distributions of incidence will be found in Etkin (1955), Beane et al. (1963), and Pines et al. (1955).
For nonplanar wings, the method of AIC’s has been developed and mechanized for the highspeed computer, but only in cases where two or
more individuallyplane surfaces intersect or otherwise interfere. Thus a threedimensional biplane, Ttail or Vtail can be handled, but ring or channel wings remain to be studied.
In preparation for the interference problem, let us describe more thoroughly the procedure for a single surface that was outlined in Section 85. For simplicity, let the trailing edges be supersonic; but in the case of a subsonic leading edge the forward disturbed region of the x,«/plane is assumed to be extended by a diaphragm, where a condition of zero pressure (or potential) discontinuity must be enforced. This situation is illustrated in Fig. 116.
For the disturbance potential anywhere in z > 0+, (86) gives
The region 2 is the portion of the wing plus diaphragm area intercepted by the upstream Mach cone from (x, y, z). When z —» 0+ so that <p is being calculated for a point P on the upper wing surface, 2 reduces to the area between the two forward Mach lines (see the figure).
Let us restrict ourselves to the lifting problem and let the wing have zero thickness. We know w(xu yi) from the given meansurface slope over that portion of 2 that does not consist of diaphragm; on the latter, w is unknown but a boundary condition ip = 0 applies. As illustrated in Figs. 815 and 816, let the wing and diaphragm be overlaid to the closest possible approximation with rectangular elementary areas (“boxes”) having a chordwise dimension bi and spanwise dimension bx/B. In (1129), introduce the transformation [cf. Zartarian and Hsu (1955)]
One thus obtains something of the form
This simultaneously employs dimensionless independent variables and converts all supersonic flows to equivalent cases at M = /2. The Mach lines now lie at 45° to the flight direction. The elementary areas, which had their diagonals parallel to the Mach lines in the x,//plane, are thus deformed into squares.
Next let it be assumed that w is a constant over each area element and equal to the value wVtlI at the center. Both’ v and ц are integers used to count the positions of these areas rearward and to the right from v = 0 and p = 0 at the origin of coordinates. With this further approximation, the potential may be written
v((, «>».А. ЛЇ, v, Г), (П32)
v, n
where the summation extends over all boxes and portions of boxes in the forward Mach cone. The definition of Ф„і(, as an integral over an area element is fairly obvious. For example, for a complete box
("[8]/2) y/(n — £i)2 — [(m — rji)2 + l2] 
This can easily be worked out in closed form. The computation is mechanized by choosing £, ri (and possibly f) to be integers, corresponding to the centers of “receiving boxes. ” Thus, if we choose £ = n, ij = m, f = l, we get
where p. = m — n, v = n ~ v. (Special forms apply for combinations of v, //, l on the upstream Machcone boundary, which may be determined by taking the real part of the integral. Also Фу;;,; = 0 when is imaginary throughout the range of integration.)
In a similar way, we can work out, for the vertical and horizontal velocity components in the field at a point (n, m, l), expressions of the following forms:
v(n, m, l) = ^2
v. ft
w(n, m, l) = "22 wVtliWytfi, t
The AIC’s V and W involve differentiation of the Фц,^,і formula with respect to q and f, respectively, but they can be worked out without difficulty.
Now to find the load distribution on a single plane surface, we set l = 0 and order the elementary areas from front to back in a suitable way. The values of disturbance potentials at the centers of all these areas can then be expressed in the matrix form
Wn. rni = – ^[Ф?,ї. о]{«*,,}• (1136)
A suitable ordering of the areas consists of making use of the law of forbidden signals to assure that all numbers in Фу^.о are zero to the left and below the principal diagonal. It is known that <pn, m = 0 at all box centers on the diaphragms, whereas w„i(1 is given at all points on the wing. The former information can be used to solve successively for the values of at the diaphragm, the computation being progressive and never requiring the inversion of a matrix.
Once ге„і(1 is known for all centers on the wing and diaphragm, the complete distribution of y> may be determined. From this, we can calculate the pressure distribution (which is antisymmetrical top to bottom) by the relatively inaccurate process of numerical differentiation. If only lift, moment, pressure drag, or some other generalized forces are needed, however, we find that these can be expressed entirely in terms of the potential discontinuity over the surface and along the trailing edge. Hence, the differentiation step can be avoided. Zartarian (1956) and Zar – tarian and Hsu (1955) provide many details.
Turning to the interfering surfaces, we illustrate the method by two examples. 1
Fig. 117. Two interferingplane supersonic wings with attached diaphragms. (Mach lines are at 45° in £, 7), pcoordinates.)
exercised in setting up the interference problem. We have found that the best way to avoid paradoxes is to focus on the two conditions:
(a) The streamlines must be parallel to the mean surface over the area of each wing.
(b) A<p = 0 must be enforced over each diaphragm (Дp = 0 on wake diaphragms).
These can best be handled in the biplane case by placing additional sources over each wingdiaphragm combination, whose purpose is to cancel the upwash induced over one particular wing area due to the presence of the other wing. There is no need to be concerned with interfering upwash over the diaphragms, since the diaphragm is not a physical barrier and the interference upwash there does not cause any discontinuity of potential.
Having placed suitable patterns of square area elements over Sy and Sl, we can write for the upwash induced at wing boxes on Sy due to the presence of Sl :
nij d Г) — (11—37)
V. M
The summation here extends over all wing and diaphragm boxes on Sl that can influence point (n, m, d). In a similar way, reasons of antisymmetry in the flow field produced by Sy lead to the upwash generated at wing box n, m, 0 on Sl due to Sy:
wLy(n, m, 0) = ^ (wc/),,MW7,?,(_d), (ll38a)
or
{wlu} wing = [Wi7,5,(_d)]{«!c/}. (ll38b)
only
When writing the matrix formulas for <py and <pl, Wul and wlv must be subtracted from the upwash that would be present at wing boxes on
Su and Sl, respectively, in the absence of the interfering partner. Thus we obtain
Wu} = — ^ [Фг. д,о](0"£/} — (1139)
where the last column covers wing and diaphragm boxes but zeros are inserted for the latter. Similarly,
{<Pl} = — ^ о]({и>п} — {w*Lu}) (1140)
Making appropriate substitutions for wul* and iolu*, we get
M = fc, ol(W – lWb, d]{wL} (ll41a)
Ш = ^[4f,;,oKW – IWlud)IW), (ll41b)
where the meaning of the notation for W* is obvious in the light of the foregoing remarks.
We now have a set of coupled equations in <p and w. The values of <p are equated to zero at all diaphragm boxes, whereas w is known at all wing boxes, so the system is determinate. A solution procedure, at least in principle, is straightforward.
2. Intersecting Vertical and Horizontal Stabilizers. Surfaces that interfere but are not parallel present no new conceptual difficulties. Once again, the source sheet representing the flow on one side of either surface is analyzed as if the other were not there, except that the meansurface “normalwash ” distribution must be modified to account for interference.
Thus, consider the empennage arrangement shown in Fig. 118. Diaphragms are shaded. The rightward sidewash at Sy due to Su and its diaphragm is
PVh(w, 0, l) = (WH)iMi^V. M,L (ll42a)
Fig. 118. Intersecting supersonic surfaces.
Here fl = m — д = 0 — ц = —fi. The matrix abbreviation is

Other supersonic interference problems can be handled in a similar manner. In all cases, a computational scheme can be found to make the solutions for the lifting pressures a determinate problem. Together with some numerical results, a list of references reporting progress towards mechanization of the foregoing supersonic methods will be found in Ashley, Widnall, and Landahl (1965).
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