Method of Cone-Symmetric Supersonic Flow

Fundamentals Before the general theory of the three-dimensional wing in supersonic incident flow is treated in the following sections, a simple special case will be discussed first that has great significance, particularly for wings of finite span. Consider the flow about a triangular plane surface. In Fig. 4-63, two Mach lines originate at the apex A0 of the triangle, where, in this example, the right-hand edge of the triangle is a subsonic edge, the left-hand edge a supersonic edge. Further, the flow conditions are studied on a ray originating at the triangle apex. The flow conditions at point At of this ray are determined exclusively by the area that is cut out of the triangle by the upstream cone of A1} supplemented—if applicable—by the area between the Mach line M. L. and the wing leading edge (influence range of Ai). The flow conditions at Аг likewise are determined exclusively by the influence range of A7. The two influence ranges of Ax and A2 are geometrically similar, and the flow conditions in and A2 must be equal. It follows that the flow properties

(pressure, density, velocity, and temperature) are constant on the whole ray through A0. This statement is valid for any ray through A0. The flow field thus described is called a cone-symmetric (conical) flow field, according to Busemann. It is a requirement for the above considerations that the edges of the triangular area be straight lines; they are two special rays of the cone-symmetric flow field.

Figure 4-63 Cone-symmetric flow over a triangular flat plate at supersonic flow.

A few examples of the application of such cone-symmetric flows are given in Fig. 4-64. Figure 4-64a shows a delta wing with a double-wedge profile in sections normal to the incident flow direction. This is an example of a wing of finite thickness at zero lift. Figure 4-64b depicts the triangular flat plate with angle of attack (lift problem). The flow over the side edge of an inclined rectangular plate is seen in Fig. 4-64c. In the triangular part of the plate surface, limited by the Mach line M. L., the flow conditions are constant on each of the rays through the corner A0. On the remaining part of. the surface, the flow field is constant because here, in sections normal to the plate leading edge, the flow is two-dimensional and supersonic (see Fig. 4-22Z?).

For the cone-symmetric flow just discussed, the three-dimensional potential equation, Eq. (4-8), assumes a simplified form. By choosing for the cone-symmetric flow the coordinate system according to Fig. 4-65, the perturbation potential

Ф {x, y, z) = xf (rj, f) (4-83<z)

Figure 4-65 Cone-symmetric flow at supersonic velocity.

satisfies the condition that the velocity components from Eq. (4-6) are constant on the rays through the cone apex A. By introducing Eqs. (4-83<z) and (4-836) into Eq, (4-8), the following differential equation of second order for f(rj, f) is obtained,

where tan ]± = 1 IvMctL — 1. This equation for the new function / depends only on the two space variables 77 and £ in the plane normal to the incident flow direction (x direction) (see Fig. 4-65). In the lateral planes (x = const), the v and w components form a quasi-plane flow. Application of the cone-symmetric supersonic flow was restricted at first to wings with straight edges. Later it was extended to “quasi-cone-symmetric” flows, see [30].

Qassification of ranges The application of this method will be demonstrated for one wing at various Mach numbers by means of Fig. 4-66. The chosen example, a pointed swept-back wing without twist, is shown in Fig. 4-66. In Fig. 4-66a, it has subsonic leading edges only, in Fig. 4-666 only supersonic leading edges. In range I of Fig. 4-66я, the flow is cone-symmetric with the wing apex A as the cone center. In the remaining crosshatched zones, no cone symmetry exists with reference to the centers В and C, since on the Mach lines through В and C the pressure cannot be constant because of range I. In Fig. 4-666, the pressure is constant over the entire range II, as will be shown later. In range III, there is cone-symmetric flow, the cone

Figure 4-66 Flow types of inclined wings of finite span at supersonic incident flow; example of a tapered swept-back wing. M. L. = Mach line. (a) Wing with subsonic leading edge, ;i > 7. (b) Wing with supersonic leading edge, p < 7. With­out hatching = pressure is constant. Single hatch­ing = pressure distribution is cone-symmetric. Cross-hatching = pressure distribution is not cone – symmetric.

‘*1

tip of which is the wing apex A0, since the pressure is constant on the Mach lines from point A because of range II. Also, range IV is covered by cone-symmetric flow with reference to point B. In the crosshatched zones, however, the flow is not cone-symmetric. Now, some information will be given on the pressure coefficients in the various ranges (Table 4-5). The values are referred to the constant pressure coefficient of the inclined flat plate, according to Eq. (4-43):

Range	Center	m	cplcp pi
Swept-back leading edge	Unswept leading edge (m <*>)
I	A	0 < m < 1	m 1 f E'{m) ]ji _ щг	—
II	—	m > 1	7П Ут2 — 1	1
III	A	m > 1	2 m I 1 -12 я Ут2- 1 у m-~ 17	1
IV	В	m > 1	————– arc cos {1 4- 2t———— j 71 ]/m2 — 1 [ m—l/_	— arc cos (1 + 21) 71

Table 4-5 Basic solutions for the pressure distribution of the inclined flat surface in supersonic incident flow (cone-symmetric flow) for ranges I, II, III, and IV of Fig. 4-66*

Figure 4-67 Inclined wing with subsonic leading edge (0 < m < 1). (a) Wing plan – form (triangular wing). (b) Pressure distri­bution on a section normal to the flow direction, m — 0.6.

The index pi designates the plane problem. The upper sign will be used for the upper side, the lower sign for the lower side.

Wing with subsonic leading edge Without going into the details, the computed pressure distributions in sections through the wing, normal to the incident flow direction (0 < fj < 1), are tabulated in Table 4-5; see [20, 77] for a wing with subsonic leading edge (range I in Fig. 4-66<z). In the present case, m assumes the values 0 <m < 1. From Fig. 4-6la, the following relation applies to

Range I: v = — cot у = tan (4-86)

x tan у

On the wing, fj runs from —1 to +1, where f? = —1 and 77 = +1 are the leading edges.

In Fig. 4-67b, the pressure distribution is shown. On the two edges, cp is infinitely large, as would be expected for flow around a sharp subsonic leading edge (see Figs. 4-6 Їй and 4-62c and b). The mean value of the pressure over the width (span) is

Wing with supersonic leading edge The simplest case of a wing with supersonic leading edge is the inclined flat plate in incident flow normal to the leading edge.

This problem has been treated before in Sec. 4-3-3 as a plane problem [see Fi

4-22b and Eq. (4-85)].

The pressure distribution of the swept-back flat plate, the leading edge of which forms the angle у with the incident flow direction (Fig. 4-68) is obtained by considering that only the component of the incident flow velocity normal to the leading edge, that is, (Jx sin j, is affecting the lift (see Fig. 3-45). In the section normal to the leading edge, the plate angle of attack a* — a/sin j. Here a is the angle of attack in the plane of the velocity £/«. Consequently, the pressure distribution of the swept-back inclined flat plate becomes

The swept-back plate, like the unswept plate, has a constant pressure distribution over the wing chord. The ratio of the pressure coefficients of swept-back and unswept plates becomes, with Eqs. (4-81) and (4-85),

____ m

CP pi ]/m2 – 1

where m > 1, according to the assumptions made. It is noteworthy that cp/cpv > 1, which signifies that the swept-back plate produces a higher lift per unit area than the unswept plate, presupposing that the angles of attack, measured in the incident flow direction, are equal. For 7 = 7t/2, that is, m — ®°, cplcppi = 1, as would be expected. For j — n, that is, m ~ 1, cpjcppi = co. In this case, the Mach line falls on the leading edge, and thus the incident flow component normal to the leading edge is equal to the speed of sound. Linear supersonic theory therefore fails.

These results for two-dimensional flow about a swept-back flat plate can be applied to the wing of finite span. To that end, an inclined delta wing with

Figuie 4-68 Swept-back flat plate with super­sonic leading edge.

Figure 4-69 Inclined wing with supersonic leading edge {m > 1). (a) Wing planform (triangular wing). (The hatched area A’ is explained on page 293.) (b) Pressure distri­bution on a section normal to the flow direction, m = 1.5.

supersonic leading edge {m > 1), according to Fig. 4-69a, may be considered. Here, m is given by Eq. (4-81), and the following relationships apply to

Ranges II and III: t — tan y – = — cot и = — V’MaL — 1 (4-90)

tan ц x x

The straight lines t = const are rays through the wing apex, where t runs from 0 to m > 1; t = ±1 represents the Mach line, t = ±m the leading edge. On this wing, the ranges II and III of Fig. 4-66b must be distinguished. The pressure is constant and is given by Eq. (4-89), between the Mach line and the leading edge, that is, in range II (1 <t<m). Details of the computation for range III (0 < f < 1) will not be given here.

In Table 4-5, formulas are listed for the basic solutions in ranges II and III at cone-symmetric supersonic incident flow. Figure 4-69b gives the pressure distribu­tion in a section normal to the flow direction. Note that the pressures on the portions of the surface that lie before the Mach lines originating at the apex are larger than in the case of a leading edge normal to the incident flow. Conversely, the pressures are considerably smaller behind these Mach lines. The mean value of cp over the span is

Wing with a supersonic leading edge and supersonic side edge So far, the wing with a supersonic leading edge has been treated. Now, for a further basic solution, the wing with a supersonic leading edge and a supersonic side edge will be discussed. A side edge is defined as an edge that is parallel to the incident flow in the planform (Fig. 4-70). From point В of the side edge, a wedge-shaped range IV of cone-symmetric flow is formed rearward (see Fig. 4-666). This range is bounded by the side edge of the wing and the two Mach lines issuing from A and B. The boundary conditions for the pressure distribution in range IV are cp = 0 on the side edge and cp = срц = const on the Mach line. By using the coordinate system x, у of Fig. 4-70<z, the following relationship applies to

Range IV: t = -§• cotju = ^ ІMa^ — 1 (4-92)

X X

where t = 0 represents the side edge and t — — the Mach line. The relationship for the pressure coefficient is given in Table 4-5.

A particularly comprehensive compilation of basic solutions is found in Jones and Cohen [39].

Superposition principle Determination of the lift distributions at supersonic flow over an arbitrary wing shape is not yet possible by means of the basic solutions of

Figure 4-70 Inclined wing with super­sonic leading edge and side edge, (a) Swept-back wing. (b) Rectangular wing.

Table 4-5. In those ranges of the wing that are covered by the Mach cones of several disturbance sources, for example, the crosshatched zones in Fig. 4-66, the basic solutions cannot be immediately applied. For these areas, a solution can be found, however, with the help of a simple superposition procedure, which will be sketched briefly.

Sought is the lift distribution of a tapered swept-back wing without twist, ABCD in Fig. 4-71. To this end, the wing is complemented to a wing with a sharp tip AED for which the basic solution of the lift distribution is known from Table 4-5. To obtain the given wing ABCD from this initial wing AED, a disturbance source is thought to be placed on point B. Two Mach lines under the angle ji with the side edge BC issue from this point. The left-hand Mach line intercepts the trailing edge of the given wing at point F. In the range ABFD of the given wing, no change in lift distribution is caused by the disturbance source B. Now, the following solution has to be added to the solution of the wing AED to obtain the solution for the given wing ABCD: For the range BEF, a solution is to be found with the following characteristics (so-called compensation wing). In the partial range ВЕС, the lift of the compensation wing has to be equal but opposite to that of the wing AED so that the total lift disappears in the former after superposition (lift extinction). In the partial range BCF, the compensation wing must not have a normal velocity component to keep the angle of attack, of this range unchanged after superposition. The details for the computation of such compensation wings cannot be discussed here. A comprehensive listing of the most important compensation wings and their velocity distributions is found, however, in Jones and Cohen [39]. For the fundamentals of the theory, compare also Mirels [62]. The above method may be applied to a simple example like that given by Fig. 4-72.