Method of Singularities for Supersonic Flow

In Sec. 4-5-2, the method of cone-symmetric flow was applied to the computations of flows about wings in supersonic incident flow. This method is limited to the treatment of special cases, such as wings without twist and with straight edges. Wings of arbitrary planform with twist cannot be treated using this method. For them, the method of singularities is available.

Figure 4-72 Application of the super­position principle to the inclined rec­tangular flat plate, (a) Given wing, (b) Basic wing (infinitely wide plate), (c), (d) Compensation wings 1 and 2. (e) Procedure for determination of the pressure distribution.

A detailed presentation of this method and of its applications is found in Jones and Cohen [39] and Heaslet and Lomax [30]; see also the basic contribution of Keune and Burg [42].

The basic features of the method of singularities for incompressible flow have been explained in Secs. 3-2 and 3-6. An analogous procedure has been developed for supersonic flows. The equation for the velocity potential of three-dimensional incompressible flow Ф(х, у, z) is given for Max > 1 in Eq. (4-8).

Vortex distribution It has been shown in Sec. 3-2-2 that a solution of the potential equation for a wing with lift in incompressible flow can be obtained by means of a vortex distribution in the xy plane. By designating the vortex element at station xy (Fig. 3-17) by k{x, y), Eqs. (346) and (347) yield for the contribution of this element to the velocity potential

By applying the supersonic similarity rule Eq. (4-10) with Eq. (4-12), the corresponding solution for supersonic incident flow becomes

Г = У(* – Х’У – (Mai ~ 1) [(У – y’f + z2]

The analogous formula for a source distribution is Eq, (4-101).

For the transition to the potential of supersonic flow, the term in the incom­pressible equation that is formed by multiplication with the 1 in the brackets must be eliminated because it is real in the entire space and, therefore, physically impossible in supersonic flow. The term with 1 jr in the potential equation of incompressible flow becomes, in the potential equation of the supersonic flow, a term that is real only within the Mach cone. Because a point P is affected by two disturbances in supersonic flow but by only one in subsonic flow, as demonstrated in Fig. 4-73І, the factor before the vortex element к has, for supersonic flow, twice the value of that for incompressible flow.

In order to obtain now the total potential at a point x, y, z, the contributions of the vortex elements have to be integrated in the xy plane. Here, only the downstream cones of the vortex elements are taken into account; the upstream cones remain unused. Hence, the potential of the vortex distribution, see Eq. (346), becomes

Figure 4-73 The effect of a sound point source at subsonic and supersonic velocities.

with the kernel function

(?(ж, y, z; y’) = 2

-у(У)

In Eq. (4-94), the integration has to be conducted over the width of the upstream cone in span direction (see Fig. 4-74). Integration of Eq. (4-95) has to be conducted over x in the upstream cone of the point x, y, z from the leading edge to the Mach cone х0(У), given by

xo(y’) = x — І(Ма4 — 1) [(y — y’)2 + z2] (4-96)

Corresponding to Eq. (3-45), the velocity components in the x and z directions at the wing location z = 0 are obtained from Eqs. (4-94) and (4-95) [compare also Eqs. (3-37) and (3-41)] as

The equation for the vortex density k(x, y) is obtained from the kinematic flow condition, which for the wing without twist with z = 0 and су = a is given from Eq. (340) by

U^a – j – w(x, у) = 0

Figure 4-74 The integration range for the veloc­ity potential of a wing at supersonic incident flow velocity from Eqs. (4-94) and (4-95).

By introducing Eq. (4-98) into Eq. (4-100), the latter equation becomes an integral equation for the determination of the vortex density k(x, y) in which the wing shape must be given. Solution of this integral equation is quite difficult, as in the incompressible case; see [15, 18, 30].

From the velocity component u, the pressure difference between the lower and upper sides of the wing is obtained in the form of the pressure coefficient from Eq. (344).

A relatively simple solution for the method of singularities was outlined in the early days of aerodynamics in a few examples by Prandtl [75] and Schlichting [80].

Source distribution It has been shown in Sec. 3-6-2 that the potential equation for incompressible flow with Маж = 0 can be solved through a source distribution on the wing surface. The method of source distributions for supersonic flow has been developed into a computational procedure by Eward [18]; see also Puckett [76].

The source element q(x’,y’) at the station x’,y’ contributes, from Eq. (3-174), to the perturbation velocity potential the amount

d20(x, y, zx’,y’) = — ^ q^X ’^y ^ dx’ dy’ (Max = 0)

where, again,

r = y/(x—x’)2 + (y —y’)2 + z2

The corresponding solution at supersonic incident flow becomes, with Eq. (4-12) and the supersonic similarity rule, Eq. (4-10),

d2<${x, y,z-,x, y’)= — j – 2q(yXdx’ dy’ (4-101)

where r is given by Eq. (4-93). It can be proved that this expression is a solution of the potential equation, Eq. (4-8). The square root has real values only within the two Mach cones of the point x , y’, z — 0 (upstream and downstream cones, see Fig. 4-57) with the apex semiangle д, where tan д = 1 ЫМаЬ — 1. For physical reasons, however, the source element produces a contribution to the potential of only the points x, y, z that lie in the downstream cone of the source element. Equation (4-101) contains an additional factor of 2, however, for reasons that were explained for Eq. (4-93).

The total potential at the point x, y, z is obtained by integration over the contributions of source elements in the xy plane, considering only the downstream cones of the source elements. The upstream cones are not considered. Hence

Here, A’ is the influence range (integration range) of the point x, y, z. It is shown for z = 0 in Fig. 4-58. For z^0, the influence range is bounded by a hyperbola (see Fig. 4-74).

The velocity components in the x and z directions at the wing location z = 0 are obtained from Eqs. (345) and (4-102) as

и {x, у) ==	1 …3 f f __. _	q{x’, у’) dx’d у’	(4-103)
271 tix J J V(® — a U’)	У)2 _ (i¥f4 _ 1) {y – y’f
w(x, y) =	±{з(*. У)		(4-104)

where the upper sign is valid for z > 0 and the lower for z < 0. The partial differentiation with respect to x in Eq. (4-103) requires particular precautions because the integrand goes to infinity on the boundaries of the integration ranges formed by the Mach lines, and these boundaries depend on x and y. Those integrals are best solved by the method of finite constituents of divergent integrals of Hadamard.*

The pressure coefficient of supersonic flow becomes the same as in incom­pressible and subsonic flow [Eq. (4-18)]:

cpix> V) = (4-105)

O’CO

Equation (4-103) is suitable immediately in the given form for the computation of the velocity distribution on a wing of finite thickness at supersonic flow (displacement problem) (see Sec. 4-5-5 for a specific discussion).

The method of source distribution will now be applied to the inclined wing at supersonic flow (lift problem); the inclined wing with subsonic leading edge cannot be treated by the discussed method of source distribution without complications, because in this case flow around the leading edge is present. Instead of the source distribution, the dipole distribution according to [30] and a vortex distribution of the kind described above are therefore preferable. A method will be given later, however, by which a wing with subsonic leading edge can be computed after all by the source method. A simple application of the source distribution method is the computation of the inclined wing with supersonic leading edge. Since the incident flow component normal to the leading edge is larger than the speed of sound and, consequently, there is no flow around the leading edge (Fig. 4-61c), the solution for the lower and upper sides of a wedge profile with linearly growing thickness is at the same time the solution for the inclined flat surface (see Fig. 4-64z and b). The starting point for further consideration is the velocity potential of the source distribution of Eq. (4-102). For an inclined wing, source distributions of different signs have to be arranged in the wing plane on the upper and lower wing surfaces. Thus, a pressure discontinuity is produced at the wing that results in lift. Further discussion needs to be conducted for the upper half-space, z ^ 0, only. The upper source distribution corresponds to the potential Ф(х, у, z). Then, the velocity components of the perturbation flow are computed with Eq. (345). The source strength from Eq. (4-104) is

q{x, у) = 2w(x, у) (4-106)

*Translator’s note: See M. A. Heaslet and H. Lomax in W. R. Sears (ed.), “General Theory of High Speed Aerodynamics,” Princeton University Press, Princeton, N. I., 1954, for a discussion of Hadamaid’s method.

WINGS IN COMPRESSIBLE FLOW 295

For the solution of the problem the following conditions must be satisfied: For the supersonic leading edge, the flow in the range before the wing is undisturbed. For the wing with subsonic leading edge, the flow is undisturbed before the Mach lines. Thus, in these two ranges Ф = 0.

On the wing, the kinematic flow condition must be satisfied, namely,

U^txix, y) – f w{x, y) = 0 (4-107)

where a(x, у) is the angle-of-attack distribution. Thus, from Eq. (4-106), the source distribution of the wing becomes

q (x, y) = — 2 a (x, у) (4-108)

For the wing with subsonic leading edge, an up wash range with the local streamline inclination X(x, y) lies between the Mach lines and the wing leading edge. In analogy to Eq. (4-108), it follows that

q(x, y) = —2UxX{x, у) (4-109)

У<* – x’f – {MaU – 1) [{у – уУ + г*]

(*«)

In this upwash range, no pressure discontinuity can exist in the z direction, however, requiring that u(x, y) — v(x, y) = 0. Introducing Eqs. (4-108) and (4-109) into Eq. (4-102) yields

Here, Rw is the integration range on the wing and Ru that of the upwash zone. These ranges may be explained now through three examples: In Fig. 4-69, a delta wing with two supersonic leading edges is shown. In this case, the range Ru does not exist, whereas the range Rw is identical to the hatched range A’. In Fig. 4-75, a wing with a supersonic and a subsonic leading edge is sketched. As has been shown

by Eward [18], only the integral over the range R’w is left for the potential at the point P(x, y, 0), because the integrals over the ranges Ru and R’w just cancel each other. The wing with two subsonic leading edges is shown in Fig. 4-75/?. In this case, the above Eward theorem, applied twice, leads to the conclusion that, approximately, only the hatched ranges R’w contribute to the integral Eq. (4-110); see Etkin and Woodward [17], Hancock [18], and Zierep [18]. Application of the Eward procedure is always feasible for wings with supersonic trailing edges. The flows with subsonic trailing edges, however, require consideration of the vortex sheet behind the wing. A contribution to the solution of this problem was made by Friedel [25].