Category AERODYNAMICS OF THE AIRPLANE

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

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)

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].

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.

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):

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.

4- 5-1 Fundamentals of Wing Theory at Supersonic Flow

Mach cone (influence range) There is an essential physical difference between flows of subsonic and supersonic velocities, namely, that the disturbances of a sound point source in the former flow propagate in all directions, but in the latter flow only within a cone that lies downstream of the sound source (Fig. 1-9b and d). This so-called Mach cone has the apex semiangle fi, which, by Eq. (1-33), is given by

sin« = ~— and tan ц = ■■■ ■ ———- (4-80)

I/Ma*, – 1

with Маж = Uoojaoo. The state of affairs just discussed may also be interpreted (see Fig. 4-57) that a given point in a supersonic flow, U„ >#«, can influence only the space within the downstream cone, whereas it can itself be influenced only from the space within the upstream cone. Application of this basic fact of supersonic flow on a wing of finite span is demonstrated in Fig. 4-58. The flow conditions at a point x,

Figure 4-56 Profile drag coefficients vs. Mach number for an unswept and a swept-back wing Op = 45°), t/c = 0.12, a = 4.

y, z = 0 on the wing can be influenced only from the crosshatched area A’ of the wing that is cut out of the wing by the upstream cone. When the Mach line M. L. lies before the wing leading edge, as in Fig. 4-58, the area between this Mach line and the leading edge also contributes to the influence on point x, y, z — 0. Downstream, the influence range is bounded by the two Mach lines through the point x, y, z = 0.

Subsonic and supersonic edge The conditions of Fig. 4-57 find an important application in oblique incident flow on a wing edge. If, as in Fig. 4-59a, a Mach line lies before the wing edge, the component vn of the incident flow velocity £/«, normal to the edge is smaller than the speed of sound аю. Such an edge is termed subsonic edge. Conversely, if, as in Fig. 4-596, the Mach line lies behind the wing edge, then vn is larger than а*,. In this case, the edge is termed supersonic edge. With /г as the Mach angle and 7 as the angle of the edge with the incident flow direction (Fig. 4-59), the expression

m = – = tan у Ma^ — 1 (4-81)

Figure 4-57 Upstream cone and downstream cone of a point in supersonic flow. fj. = Mach angle.

Figure 4-58 Wing in supersonic incident flow. A’ = influence range.

allows one to determine whether the edge is subsonic or supersonic. Thus the edges are characterized as follows.

Subsonic edge: vn < a« /<« > 7 m < 1 ■ (4-82a)

Supersonic edge: ип>ак, іл<.ут>1 (4-82 b)

The special case 7 = 0° (m = 1) is a subsonic edge for all supersonic Mach numbers, and the case 7 = 90° (m — °°) is a supersonic edge. The concept of subsonic and supersonic edges is of significance not only for the leading edge, but also for the trailing and side edges. This fact is explained in Fig. 4-60. Here, the subsonic edges are drawn as dashed fines, the supersonic edges as solid fines. For the same wing planform, the Mach fines for three different Mach numbers are drawn. At the lowest

WINGS IN COMPRESSIBLE FLOW 279

Figure 4-60 Example for the explanation of sub­sonic and supersonic edges of swept-back wings. Dashed lines: subsonic edges; solid lines: super­sonic edges, (a) Subsonic leading edge and sub­sonic trailing edge. (6) Subsonic leading edge and supersonic trailing edge, (c) Supersonic leading edge and supersonic trading edge.

Mach number (Fig. 4-60a), all edges are subsonic, at the highest Mach number (Fig.

4- 60c), the leading and trailing edges are supersonic, but the side edges are still subsonic. Distinction between subsonic and supersonic edges is conditioned by the difference in flow patterns in the vicinity of the edges. In Fig. 4-61, the various types of flow patterns are sketched, which are the sections normal to the leading and trailing edges, respectively. In close vicinity to the section plane, the flow may be considered to be approximately two-dimensional. The basically different character of subsonic and supersonic flows over an inclined flat plate was demonstrated in Fig. 4-22. Based on this figure, Fig. 4-61 shows the subsonic leading edge, at which flow around the leading edge is incompressible accord­ing to Fig. 2-9a. An essential characteristic of this flow is the formation of an upstream-directed suction force on the nose (see Fig. 4-22a). Figure 4-61 b shows the subsonic trailing edge with smooth flow-off according to the Kutta condition (see Sec. 2-2-2). At such a trailing edge, the pressure difference between the lower and upper surfaces is equal to zero (Fig. 4-22г). Complete pressure equalization between the lower and upper surfaces is achieved. In Fig. 4-6lc and d, the supersonic leading edge and the supersonic trailing edge, respectively, are shown. In both cases, neither flow around the edge nor smooth flow-off is achieved, but Mach lines originate at the edges along which the flow quantities change unsteadily. Between the lower and upper surfaces, a finite pressure difference exists (see Fig.

4- 22b).

Finally, the pressure distributions over a wing section are shown schematically for the three different cases of Fig. 4-60. For the section with subsonic leading and

Figure 4-61 Typical flow patterns at subsonic and supersonic edges (see Fig. 4-59). (a) Subsonic leading edge, vn < ctoo, flow around edge. (6) Subsonic trailing edge, ип<аю, smooth flow-off (Kutta condition), (c) Supersonic leading edge, vn > with Mach lines, (d) Supersonic trailing edge, vn > a«, with Mach lines.

trailing edges, Fig. 4-62^, the pressure distribution is similar to that of incompressible flow, as would be expected. The rear Mach line, however, causes a break in the pressure distribution. In the case of the section with supersonic leading and trailing edges (Fig. 4-6 2c), the pressures at the leading and trailing edges have finite values. The front Mach line again produces a break in the pressure distribution.

Pressure distribution In this section, the wing of finite span at incident flow of subsonic velocities will be investigated, at zero lift (displacement problem). The pressure distribution of such a wing of finite thickness is of particular interest with regard to the determination of the drag-critical Mach number at high subsonic incident flow. The concept of critical Mach number has already been explained in Sec. 4-3-4. The incident flow velocity of the critical Mach number is the lower limit for the formation of the shock waves, which change the entire flow pattern considerably and, in particular, lead to a strong drag rise (see Figs. 4-14 and 4-15).

The pressure distribution of a three-dimensional wing with symmetric wing profiles at subsonic incident flow is obtained from that of the transformed wing of Eq. (4-69) as

where cp jnc is the pressure distribution of the transformed wing for which the pressure distribution of incompressible flow is to be computed. The computational

Figure 4-51 Lift slope of delta wings of various thicknesses; aspect ratio л = 3, from [16]. Comparison with linear theory.

method was given in Sec. 3-6. The transformation of the wing pianform follows Eqs. (4-66)-(4-68); the thickness ratio 8 = t/c remains unchanged (version II of the subsonic similarity rule of Sec. 4-3-2).

Drag-critical Mach number On three-dimensional wings, contrary to the plane problem, frequently the wing leading or trailing edges are not perpendicular to the incident flow direction. The simplest cast of that kind is the swept-back wing of constant chord and infinite span. This case has been treated previously for incompressible flow in Sec. 3-6-3. The sweepback has a significant influence on the magnitude of the critical Mach number, because only the velocity component normal to the leading edge determines the maximum perturbation velocity on the contour of such wings of finite thickness. From Eq. (4-53д), the critical pressure pCT of the wing in incident flow normal to its leading edge is obtained after multiplication with Oc«i7«cr/2 and with Mr», cr = as

By introducing now, in agreement with the above statement, U=<,CTcosy as the effective velocity instead of I/»cr, and again adopting the dimensionless notation, the critical pressure coefficient of the swept-back wing becomes

Pci~Poc 2 1 —MaloCT cos2 ф

JTi 7+1 Mai

– C/cocr

Here, as in Eq. (4-76), the pressure coefficient of the swept-back wing is referred to the dynamic pressure of the incident flow. The relation cpCT{Ma<xct’) is shown in Fig. 4-52 for у = 0° (see curve 1 of Fig. 4-28) and for <p= 45°. To determine the critical Mach number of the incident flow Ma«,at the curve cpmin is drawn in Fig.

4- 52 up to its intersection with the curve cpcx (see Fig. 4-28).

Swept-back airfoil of infinite span For the determination of the pressure difference (p — Р») of a swept-back wing, it should be observed that (p — pM) is proportional to the dynamic pressure of the effective velocity (goo/2)£/2, cos2 </>. It is also proportional to the thickness ratio or the angle of attack, respectively, determined in the plane of the effective incident flow; that is, it is proportional to (t/c) cos <p. It follows that in incompressible flow,

Pine P°° ~~(Pinc P°°)<p= 0 COS

Referred to the dynamic pressure of the incident flow (q*,I2)U1, the relation between the pressure coefficients becomes (cpmin)inc = cos йпс(сртіп)тс,<р=о – With Eqs. (4-76) and (4-68c) it is

COS, v. –

cpmin “ min)inc, =0 With COS —

By substitution, finally,

The above-explained procedure has been applied to an example in Fig. 4-52. Chosen were two airfoils of infinite span, one unswept and one with a sweepback angle of 45°. For the unswept airfoil, (сртт)іпс, у?=о =“0.2 has been assumed, resulting in a critical Mach number = 0-83. The effect of the sweepback is seen in

a shift of the critical Mach number to a considerably larger value of (Ма„ост)^-45 — 1.13. This shift is caused by three effects. First, the curve cpCT is shifted to the right because of the sweepback; second, by the sweepback, cpmin at Ma<* = 0 is

Figure 4-52 Determination of drag – critical Mach number Ma^ cr for an unswept and a swept-back airfoil of infinite span. (cp minV = o, Mace =o = -0.2.

Figure 4-53 Drag-critical Mach number of the incident flow of swept-back airfoils of infinite span, (a) Effect of pressure coefficient, (b) Effect of thickness ratio (biconvex parabolic shape).

reduced; and third, the rise of cpTnin with Mach number is much weaker for a swept-back wing than for an unswept wing.

An extension of Fig. 4-52 is given in Fig. 4-53a, where the critical Mach numbers of swept-back airfoils of infinite span are presented relative to (£pmm)inc)¥> = 0- For a biconvex parabolic profile,(сртіГі)іпс^=0 = -2(итах/С/о«,)тс = —(8/7г)(//с). Corresponding to the example shown in Fig. 4-53c, the sweepback angle has been evaluated in Fig. 4-53& as a function of the critical Mach number and for several thickness ratios. For 5 = tjc = 0, this function is

Ma„ cr =

Thus, sweepback may raise the drag-critical Mach number of very thin profiles considerably above unity.

Middle (root) section of the swept-back wing The discussions about the effect of wing sweepback presented so far are valid only for the straight airfoil of infinite span (see Fig. 4-52). For folded wings (Fig. 3-74), the favorable sweepback effect (raising of the drag-critical Mach number) is not realized fully in the vicinity of the root section. The middle portion of the wing performs somewhat as if it were unswept. For the computation of the critical Mach number of the middle section of the folded swept-back wing, the following procedure has to be applied: For incompressible flow, the velocity distribution over the root section is given by Eq. (3-187). The maximum velocity over the root section produces the largest underpressure (Cpmin)inc 2(Ищах/^°°)іпс* The Value of (^тпах/^°°)іпс ®f ^ parabolic profile is plotted in Fig. 3-76 against the sweepback angle ipinc. Conversion of (Opmin)inc into Cpmin f°r the various Mach numbers is given by Eq. (4-76), where the sweepback angle also has to be transformed according to Eq. (4-68c). The critical Mach number is then obtained as the intersection of the curves cpmjn and Cpcr °f Fig – 4-52, where for the root section the curve cpcT for ip = 0 has to be taken. The result of this computation is presented in Fig. 4-54, for sweepback angles t/?=0, 45, and —45° and for several relative thickness positions Xt. The dashed curve for <p = ±45° shows the values for the straight swept-back wing. They are valid for sections of the folded wing at large distances from the root. It is clearly seen that the swept-back wing (9?=+45°) has the most favorable critical Mach number of the root section for relative thickness positions of about 30%,

whereas the swept-forward wing (<p = —45°) is most favorable for relative thickness ratios of about 70%. These results show that the critical Mach number of the middle section of folded swept-back wings is, in general, considerably lower than that of the tip section. It follows that the favorable sweepback effect of the straight swept-back wing cannot be fully realized by folded wings.

Investigations of the drag-critical Mach number of folded swept-back wings were made by Neumark [64]. He also studied the influence of finite aspect ratios on the critical Mach number, but no marked differences with the airfoil of infinite span were found; see Fig. 3-71.

Experimental results Raising of the drag-critical Mach number by sweepback has found practical applications of great importance for airplane design. As has previously been shown in Sec. 4-3-2, increasing the critical Mach number produces a shift of the compressibility-caused drag rise to higher Mach numbers (Fig. 4-1 Ad). It must be expected, therefore, that sweepback causes a shift to higher Mach numbers of the strong rise of the profile-drag coefficients with Mach number,

This fact was first realized by Betz in 1939 and has been checked experimentally by Ludwieg [57]. A few of his measurements are plotted in Fig. 4-55, The polars for an unswept and for a swept-back trapezoidal wing (ip = 45°) show the following: The profile drag (cL = 0) of the unswept wing is several times larger at Ma„ = 0.9 than at Ma„ = 0.7. Thus the drag-critical Mach number of this wing lies between Ma* = 0.7 and Mr» = 0.9. For the swept-back wing, however, the profile drag at Ma* = 0.9 is only insignificantly higher than at Mz„ = 0.7. In other words, the critical Mach number of this wing lies above Ma„ = 0.9. Another example of this important swept-back wing effect is demonstrated in Fig. 4-56. Here, from [71], cDp is shown versus Ma„=> for an unswept and a swept-back wing (<p = 45°). The sweepback effect is manifested by a shift of the onset of the drag rise from about Max — 0.8-0.95. This favorable sweepback effect has been exploited by airplane designers since World War II. The presentation of Fig. 34c, namely, sweepback angle versus flight Mach number, shows very clearly that the sweepback angle of airplanes actually built increases markedly when Mach number Mz» = 1 is approached.

Thick wing at sonic incident flow The subsonic similarity rule of Sec. 4-4-3 leads to useful results in computing the lift for incident sonic flow (Маж = 1). It fads, however, in the computation of the displacement effect of a finitely thick wing at sonic incident flow. The reason is that the pressures on the wing become infinitely high. Compare, for example, [70] for an account of this difference between the lift problem and the thickness problem in the limiting case Ma„ -+ 1. To obtain useful information on the thickness problem at Mam = 1, nonlinear approximation methods have to be applied. The transonic similarity rule (see Sec. 4-3-4) is particularly well suited for classification and systematic presentation of test results on wings of finite span; see Spreiter [103]. Further information on the theory of transonic flow of wings is found in publications by Keune [43] and Pearcey [69] and in reference [68] on the equivalence theorem of wings of small span in transonic flow of zero incidence.

It has been shown in Sec. 4-34 that the aerodynamic coefficients of a wing profile undergo strong changes during transition from subsonic to supersonic flow, that is, at transonic flow. The linear approximation methods for incident flows of subsonic and supersonic velocities for the airfoil of infinite span fail when sonic velocity, /kfoco 1, is approached (see Fig. 4-33). For wings of finite span, however, physically plausible results may be obtained at = 1. In this case, the same limiting values are obtained for the lift-related coefficients (see, e. g., Fig. 4-82), by approaching Max = 1 both from subsonic and from supersonic incident flow.

Now, for the lift problem at Max = 1, a few results will be presented that have been obtained according to the method of Truckenbrodt [95]; compare also the publications of Mangier [58], Mangier and Randall [58], and Spreiter [85].

For tapered swept-back wings, the lift slope and the neutral-point position are shown in Fig. 449 as functions of the geometric parameter cyfa for several values of crla0. The wing geometry is seen in Fig. 449a, the lift slope in Fig. 4-49b, and the neutral-point position in Fig. 449c. It is noteworthy that for crja > 1 [i. e., when the trailing edge of the inner (root) section lies farther back than the leading edge of the outer (tip) section], the lift slope is equal to 7тЛ/2 for all wing shapes in agreement with Eq. (4-75c). For cr/a < 1 (i. e., when the trailing edge of the root section lies farther upstream than the leading edge of the tip section), the lift slope is smaller than тгЛ/2. The neutral point for cr/a> 1 lies at xNja = (see Fig.

4- 49c). For delta wings (a0 = a = cr), xNjcr= f. For crja< 1, the neutral point

shifts upstream. The linear theory for Ma„ = 1 also allows computation of the pressure distribution on the wing surface. Here, for uncambered wings, wing areas of which the local span remains constant in the chord direction (Fig. 4-50a), or decreases (Fig. 4-50h), do not contribute to the lift (Д cp = 0).

Finally, a few test results [16] are given in Fig. 4-51 for the lift of delta wings at Mach numbers close to unity. The lift slopes dcildot are plotted against the parameter Л2(Мгі — 1), which results from the similarity transformation of compressible flow [see Eq. (4-26)]. The pronounced peak in the theoretical curve of dci/da at Maw = 1 is not fully confirmed through measurements. In the subsonic and supersonic range, theory is well represented by the measurements. Further experimental results on wings in transonic flow are found in Frick [24].

General formulas The local lift coefficient of a wing section is obtained through integration of the pressure distribution over the wing chord according to Eq, (2-Ю). By taking into account Eq. (4-69), the transformation formulas for the local lift coefficient and, accordingly, for the local pitching-moment coefficient are thus given as

For incompressible flow, the wing theory of Sec. 3-3-5 produces the dimensionless lift distribution 7inc (i? inc) and the dimensionless pitching-moment distribution from Eqs. (3-115fl) and (3-115b). By introducing Eqs. (4-67д), (4-672?), (4-70/z), and (4-702?), the dimensionless distributions for subsonic flow become

ctc

7 ~ 2b~ 7inc

_ cmc___

22? ~Minc

These equations show that the dimensionless lift and moment distributions remain unchanged during transition from incompressible to compressible flow. It should be noted, however, that the distributions 7 and 7inc, and fi and qinc belong to different planforms (Fig. 4-44).

The transformation of the coefficients of total lift and pitching moment, taking into account Eqs. (4-66), (4-67й), (4-672?), and (4-69), results in

The coefficient of induced drag in incompressible flow for elliptic lift distribution is, from Eq. (3-312?), cDi-mc =сь-1ПС1пЛ[пс. Introducing cL inc and /linc into the above transformation formulas yields the relationship

Hence, the formula for the coefficient of induced drag in relation to the lift coefficient is independent of the Mach number. The transformation formulas for the remaining aerodynamic coefficients are compiled in Table 4-4.

Elliptic wing Simple closed formulas for the lift slope as a function of the Mach number can be established for wings with elliptic planform. For incompressible flows, computations follow Eq. (3-98) of the extended lifting-line theory. Applying the subsonic similarity rule yields

dc^ 2 пЛ

У(1 – Mai,) Л2 + 4 + 2

Figure 445 Ratio of lift slopes at subsonic and incompressible flow fox elliptic wings of various aspect ratios л vs. Mach number of incident flow according to Eq. (4-74).

from which the limiting values

^=~Л (Л->0) (4-750

da 2 v ‘

= … 2?… — (Л -> oo) (4-75b)

da yi _ Mai,

are obtained. Equation (4-75д) is identical to Eq. (3-101b). For very small aspect ratios, the dependence of the lift slope on the Mach number thus disappears. Equation (4-75b) is identical to the expression of the plane problem from Table

4- 1.

For the case Ma„ — 1, the lift slope becomes

^ = v Л (Л&. = 1) (4-75c)

Contrary to the airfoil of infinite span (A =°°), for which (dcL/da)00 = °°, the lift slope of wings of finite span has finite values. The significance of this result will be investigated more closely in Sec. 4-44.

The ratio of the lift slopes for Маж Ф 0 and Ma„ = 0 is shown in Fig. 445 for several aspect ratios against the Mach number. This figure shows that the compressibility influence on the lift slope becomes smaller when the aspect ratio is reduced. This fact was first pointed out by Gothert [28].

Wings without twist The aerodynamic coefficients will be computed for the same wings for which the lift distribution was determined in Sec. 3-3. These were a trapezoidal, a swept-back, and a delta wing, with aspect ratios between A = 2 and /1=3. These three given wings are depicted in the upper boxes of Fig. 446. The geometric data for the wings are compiled in Table 34. The second and third rows of boxes show the wings transformed with the subsonic similarity rule for Ma„ = 0.4 and 0.8, respectively. The lift distributions of these wings have been

computed according to the wing theory for incompressible flow of Sec. 3-3-5.

The results of these computations for the lift distribution of the wing without twist (a= 1) are presented in Fig. 4-47. The lower figures give the dimensionless lift distributions 7 according to Eq. (4-71 a) for Mach numbers Max = 0 and Маж = 0.8. The curves for Ma<* = 0 are identical to curve 3 of the distributions in Fig. 3-33. In the upper figures, the local neutral points and the total neutral points N are plotted on the wing planform. At the upper part of Fig. 4-48, the lift slopes are plotted against the Mach number; at the lower part, the neutral-point displacements with respect to the geometric neutral point. The points for Маж = 1, shown as open circles, are theoretical values of an approximation method that will be explained in Sec. 4-4-4. They agree with Eq. (4-75c) for trapezoidal and delta wings. In addition, in all six diagrams, measurements by Becker and Wedemeyer [5] are included. The measured lift slopes agree well with theory in all cases. In general, the dependence on Mach number of the neutral-point positions is given satisfactorily by theory.

Certain discrepancies between theory and experiment of the neutral-point positions can be explained mainly by the effect of the finite profile thickness disregarded in the theory; compare also Fig. 4-13b. It is noteworthy that the neutral point of the trapezoidal wing shifts considerably upstream under the compressibility influence. However, this theoretical result is only partially confirmed by measurements, because shock waves form when the drag-critical Mach number is exceeded. On the two other wings, the swept-back and the delta wings, the neutral points are displaced toward the rear.

No more detailed statements are needed on the induced drag, since, as shown by Eq. (4-73), the quotient С£ц1с is independent of Mach number and thus equal to that of incompressible flow (see Table 34).

Further results on the aerodynamic coefficients of delta wings of various aspect ratios are compiled in Fig. 4-82, together with results for supersonic incident flow.

Data for the compressibility effects on the flight mechanical coefficients at subsonic incident flow, for example, of the rolling, pitching, and yawing wing, are found in Kowalke [5] and Krause [5].

4- 4-1 Application of the Subsonic Similarity Rule

It has been shown in Sec. 4-2-3 that the computation of flow about a wing of finite span with incident flow Mach number Mam < 1 can be reduced to the determina­tion of the incompressible flow for a wing of finite span by means of the subsonic similarity rule (Prandtl, Glauert, Gothert). The corresponding problem for the airfoil of infinite span (profile theory) was discussed in Sec. 4-3-2. Computation of incompressible flows was treated in detail in Chap. 2 for the airfoil of infinite span and in Chap. 3 for the wing of finite span. The methods of wing theory for incompressible flow therefore have a significance that reaches far beyond the area of incompressible flow.

The second version of the subsonic similarity rule of Sec. 4-2-3 is the starting point for further discussions. In what follows, the reference wing in incompressible flow that is coordinated to the given wing at given Mach number will be designated by the index “inc.” Thus, the transformation formulas for the wing planform according to Eqs. (4-10) and (4-15) are

The geometric transformation for a trapezoidal swept-back wing in straight flight and in yawed flight for Mach number Ma„ = 0.8 is presented in Fig. 4-44.

For unchanged profile (h/cnc=h/c, (t/c)inc = t/c, and unchanged angles of attack ainc = a, the pressure coefficient of the given wing cp is obtained according to Eq. (4-23) from that of the transformed wing cp-mc as

cP = —===== (version II) (4-69)

У 1 — Ma%>

Compare Figs. 4-8 and 4-9 for the Mach number range 0 < 1. In the case of

airfoils of infinite span, the subsonic similarity rule is no longer valid for Маж = 1 (see Sec. 4-3-2). Approximately, however, it may be applied to Ma„ = 1 in the case of wings of finite span. More details will be given later. Attention should he drawn to the panel method of Kraus and Sacher [44], which includes the influence of compressibility.

By taking into account the sirmlarity rules of Sec. 4-2-3, specific profile theories have been developed for flow about wing profiles (slender bodies) that depend on

the values of the incident flow Mach number. For Mam < 1 the subsonic flow is described in Sec. 4-3-2, for > 1 the supersonic flow in Sec. 4-3-3, and for Маж – 1 the transonic flow in Sec. 4-3-4. For very high Mach numbers of incident flow, that is, Мйоп > 1, the theory of supersonic flow does not lead to satisfactory results. For this case of incident flow with hypersonic velocity {Max > 4), a few statements on a profile theory of hypersonic flow will be made. First, the following considerations will be based on a slender profile, pointed in front.

Theory of small deflections in hypersonic flow Through a concave deflection by the angle d > 0, a compression flow is produced that can be computed according to the theory of the oblique shock. Conversely, an expansion flow is formed behind a convex deflection by the angle d < 0 that can be treated as a Prandtl-Meyer corner flow. The fluid mechanical quantities before and behind the deflection will be marked by the indices 1 and 2, respectively. The deflection angle is assumed to be small!#!<!, which means that the velocities before and behind the deflection differ only by a small perturbation velocity. The range of Mach numbers of the hypersonic flow considered here is Ma{ > 1 and Ma2 > 1. The pressure coefficients cp = Apjqx of the pressure change Ap – p7 —pb relative to the dynamic pressure before the deflection qx = (Qi/2)Ui, are obtained as [53]

d – <0 (d<0) (4-58Z?)

In either case, the pressure coefficient at small deflections of a hypersonic flow is given as

Cp = $2f(Ma1 її) (4-59)

where Max її is the similarity parameter of hypersonic flow. The parameter will be discussed later in more detail in connection with the hypersonic similarity rule.

For large values of Max її > 1, the expressions

cp = (7 + 1)тЯ (Max ~* °°) (4-6Qa)

= – /2- (-Магїї>—^) (4-60b)

у(Махїї)2 7-І)

are valid. The latter formula indicates that after deflection, vacuum (p2 = 0) is obtained for values of —Max її> 2/(y — 1). In Fig. 4-40, the pressure coefficient in relation to the square of the deflection angle ср/її2 is plotted as a function of the hypersonic similarity parameter Max її by curves 1 and 2. For comparison, the supersonic approximation of Eq. (443a) for high Mach numbers is

shown as curve 3. This approximation agrees better with the expansion flow than with the compression flow. The deviations are too large, however, to adopt this approximation as the pressure equation for hypersonic flow with small deflections.

Inclined flat plate in hypersonic flow By setting її = ±a in Eqs. (4-58a) and (4-58h), a being the angle of attack, the pressure distributions on the lower and upper surfaces of an inclined flat plate in hypersonic flow can be easily computed. They are constant over the chord. The lift is then obtained from the resultant pressure distribution of the lower and upper surfaces. The lift coefficient is obtained as

cL = a2F(Maaa a) (4-62a)

Figure 440 Pressure coefficients at hypersonic flow (7 = 1.4). (1) Expansion: lower sign, from Eq. (4-58b). (2) Compression: upper sign, from Eq. (4-58c). (3) Supersonic approximation from Eq. (4-61).

cl — (7 + l)**2 (Mdoo °°) (4-62b)

In Fig. 441, this result is presented for various Mach numbers of the incident flow Max = Mcioo according to Linnel [53]. It can be seen that the lift coefficient for a fixed angle of attack decreases sharply with increasing Mach number and that the hypersonic theory deviates from the supersonic theory. The curves for Маж = 0 (incompressible flow) and Max = 00 mark the limiting cases.

Hypersonic similarity rule Specific similarity rules were established in Sec. 4-2-3 for subsonic, transonic, and supersonic flows. With their help, flows about geometrically similar bodies can be related to each other. Such a similarity rule also exists for hyper­sonic flow. It was first presented by Tsien [98] and proved to be completely general

by Hayes [98]. The relation between pressure coefficient and deflection angle and Mach number is expressed in Eq. (4-59). For symmetric incident flow, the deflection angle is proportional to the thickness ratio t/c. In this case the Mach number Mat becomes the incident flow Mach number Max. Hence, in analogy to Eqs. (4-35) and (4-36), the following expressions are obtained for the pressure and drag coefficients:

cp = 62/ ^5Mzo=,^ (4-63)

cD = 53F(5 Mcioo) (4-64)

Hypersonic flow over a blunt profile The flow pattern in the vicinity of the nose of a body in hypersonic incident flow is sketched in Fig. 4-42. Keeping in mind the

Figure 441 Lift coefficient of the flat plate vs. angle of attack a for various Mach numbers (y = 1.4). Hypersonic theory for small angles of

attack according to Linnell. (——– ) Hypersonic

theory, Eq. (4-62a), Mam -*■ с/, = (7 + l)a2.

(——- ) Theory based on Eq. (446), Ma^ -*0:

cl — 27га.

important fact that the leading edge of every body is somewhat—even if very little—rounded, it is obvious that a stagnation point always exists on the nose, and therefore a detached shock wave is formed upstream of the stagnation point in which the approaching hypersonic flow is abruptly reduced to subsonic flow. As a result, extremely high temperatures are produced near the stagnation point, which may lead to dissociation and ionization of the gas and thus to deviations from the properties of ideal gases. The thermic equation of state [Eq. (1-1)] is no longer valid, for instance, and the specific heat capacity cp does not stay constant either.

The dependence of the temperature rise that occurs near the stagnation point after passage of the shock wave on the Mach number is presented in Fig. 4-43 for air. The dashed line is valid for the ideal gas (see Fig. 4-2b) and the solid curves for a

real gas at several values of the static pressure of the incident flow. Because of dissociation, the temperature rise at high Mach numbers is considerably smaller for real gases than for ideal gases.

At larger distances from the stagnation point the shock wave closely approaches the body contour. It is strongly curved, therefore, particularly near the stagnation point (Fig. 4-42). On the body contour itself, a friction (boundary) layer (range A) forms because of the viscosity, the thickness of which is now of the same order of magnitude, however, as the distance between shock wave and the outer edge of the boundary layer (range B). The formation of the boundary layer is governed by the pressure distribution on the body, which, at hypersonic incident flow, is determined mainly by the shape of the shock wave. This, in turn, depends on the body contour and its boundary layer. There prevails, consequently, a very strong interaction between friction layer and shock wave in hypersonic flow.

Another difficulty contributes to the problem. Since the shock wave is curved, the entropy increases in the shock wave are different for each streamline. These increases depend on the shock-wave inclination at the respective stations. Therefore, the flow behind the curved shock is no longer isentropic. This means that the flow behind the shock is no longer irrotational and that the separation into a rotational friction layer and an irrotational outer flow, customary in boundary-layer theory, is no longer possible. On the contrary, the total flow field between shock wave and body contour is now rotational. The friction effects, however, are of significance only in the zone next to the wall, zone A of Fig. 4-42, whereas zone В represents an inviscid, but not irrotational, layer. An important characteristic of hypersonic flow is its small lateral extent. Therefore the flow quantities vary strongly in the lateral direction, whereas they vary only little in direction of the incident flow.[26]

The computations of the flow about a body with a blunt leading edge, and particularly the computation of the shock-wave shape and of the pressure distribution. on the body, are very difficult, even when friction is disregarded, because the flow field contains, side by side, zones of hypersonic, supersonic, and subsonic flow.

In the special case (Mar»-*00, у 1), the incident flow would remain undisturbed up to the body contour and then be deflected in direction of the contour. Thereby a portion of the horizontal momentum would be transmitted to the body wall and thus produce the body drag. This special case is termed Newtonian flow because Newton based his theory for the drag of arbitrary bodies on this concept. It leads to the following expression for the pressure coefficient:

cp = 2 sin2 # (Newtonian approximation) (4-65)

with # being the deflection angled This relationship serves as a rough approxima­tion for the front portion of the body, whereas the above momentum consideration does not give an answer for the rear body portion. In this context the expression aerodynamic shadow is used.

The methods for the exact computation of hypersonic flows are very lengthy and can be handled only with modern electronic computers. Investigations in this field are still in progress, and many aerodynamic problems—particularly those including the deviations from the properties of ideal gases—are not yet completely solved.

Monographs in book form on hypersonic flow are listed in Section II of the Bibliography. Compare also Schneider [82].

Both approximation theories for subsonic and supersonic flows discussed in Secs.

4- 3-2 and 4-3-3 fail when the incident flow velocity approaches the speed of sound. In this case the flow becomes of the mixed type; that is, both subsonic and supersonic velocities exist in the flow field. At certain points the flow therefore passes the speed of sound. In transonic flow fields of this kind, shock waves are formed in most cases, and theoretical treatment is made much more difficult.

Drag-critical Mach number First, the limiting Mach number should be established up to which the theory of subsonic flow of Sec. 4-3-2 is still valid. In the case of a wing profile at subsonic incident flow velocity {Маж < 1), Fig. 4-13a demonstrated that the lift slope can no longer be described by the linear theory at higher subsonic Mach numbers. The results on the neutral-point position of Fig. 4-136, and in particular those on the drag coefficients of Figs. 4-14 and 4-15, confirm this fact, which is caused by flow separation on the profile. Depending on the profile shape (thickness ratio, camber ratio, nose ratio) and the angle of attack, a critical Mach number Ma„cr can be established up to which no significant flow separation occurs. This will be designated as the drag-critical Mach number. It can be defined, for
instance, as the Mach number at which the drag coefficient cD is higher by ЛcD = 0.02 than at Маж = 0.6.

The physical reason for flow separation at higher subsonic Mach numbers is that shock waves are formed when sonic velocity is reached locally on the profile and exceeded over a certain range. The critical Mach number Ma„CT is understood, therefore, to be the Mach number of the incident flow at which sonic velocity is reached locally on the profile. The critical pressure coefficient at the critical Mach number Мажст is cpCT. The critical Mach number Mz«cr is obtained by setting for cpcr the highest underpressure сртїп that occurs at the body. For slender bodies, cpmin is small and МаЖСТ is close to unity. In this case, based on streamline theory of compressible flow, neglecting higher-order terms, cpCT becomes

2 1 – Mai

7 + 1 Mai, cr

Cp min

From Eq. (4-38), cDmin is a function of Mach number. Introducing Eq. (4-38) into Eqs. (4-53a) and (4-53b) yields

In Fig. 4-28, cpcr from Eq. (4-53a) is shown versus as curve 1. For a given

wing profile, Maeccr is determined by the intersection of curve 1 according to Eq. (4-53&) with curve 3 according to Eq. (4-38); see also Fig. 4-16. More simply, MzooCr can be obtained by starting from Eq. (4-54). This relationship is given as curve 2. .

The value of Cpmin depends strongly on the profile shape and the angle of attack. It is obtained from the velocity distribution of potential flow with Cpmin =—2wmax/£/«,. The maximum pressures for various profiles in incompressible flow are plotted in Fig. 2-34 against the thickness ratio. The critical Mach numbers for chord-parallel flow are shown in Fig. 4-29 for several profiles as functions of

Figure 4-28 Illustration of determination of drag-critical Mach number MaXCI of a wing profile; 7= 1.4. Curve 1 from Eq. (4-53), curve 2 from Eq. (4-54), curve 3 from Eq. (4-38).

profile thickness 5 = t/c and relative thickness position Xt = xt/c. As would be expected, the critical Mach number decreases sharply with increasing thickness ratio for all profiles.

Physical behavior of transonic profile flow When a wing profile is exposed to an incident flow velocity high enough to form areas of local supersonic velocity in its vicinity, shock waves are formed in the ranges where the velocity is reverted from supersonic to subsonic. In these shock waves, pressure, density, and temperature change very strongly. The strong pressure rise in the shock wave frequently leads to flow separation and consequently to a complete change of the flow pattern. This effect causes a strong increase in the drag (pressure drag).

To demonstrate these processes, the pressure distribution on a wing profile is given in Fig. 4-30д for various Mach numbers from measurements in reference [89]. The pressure distribution is steady for Mach numbers at which the maximum velocity on the profile contour is everywhere smaller than the local sound speed, wc<a. In the present case, this holds up to Маж ~ 0.6. Up to Mam «0.6 the pressure rise at the rear end of the profile is as steady as the pressure drop is in front. For higher Mach numbers, Маж > 0.7, at which the sonic velocity is exceeded locally, wc > a, the pressure rise behind the pressure minimum occurs unsteadily in a shock wave. The height of the pressure jump increases with Mach number. This abrupt pressure rise is very undesirable with respect to the boundary layer, which tends to separate even at a steady pressure rise. In most cases, the shock wave causes separation of the flow from the wall and thus a strong drag rise, as is obvious from the curve of the drag coefficient versus Mach number of Fig. 4-30b see also Figs. 4-14 and 4-15.

In Fig. 4-31, a Schlieren picture and an interferometer photograph from Holder [33] are shown of a wing of angle of attack a — 8° in a flow field of Mam = 0.9. The formation of the shock wave and a strong separation immediately behind the shock are clearly noticeable.

The flow pattern in the transonic velocity range, which is, in general, quite complicated, is displayed schematically in Fig. 4-32 for a biconvex profile in symmetric incident flow. Pressure distributions and streamline patterns are given over a range of increasing Mach number. Figure 4-32a represents the incompressible case, Fig. 4-32b the subsonic case in which the “sonic limit” has not yet been exceeded anywhere. Figure 4-32c-e demonstrates the formation of the shock wave after the “sonic limit” of the pressure distribution (critical pressure) has been passed. Figure 4-32f and g represents the typical pressure distribution of supersonic flow that was previously shown in Fig. 4-24.

The formation of shock waves in the transonic range also has a strong effect on the lift. This is demonstrated schematically in Fig. 4-33, in which the solid curve represents a typical measurement of the relation between lift coefficient and Mach number, whereas the dashed line corresponds to the linear theory according to Fig.

4- 20a. For a better understanding of the measured lift curve, the positions of the shock wave and the velocity distributions on the profile for the points А, В, C, D, and E are shown in Fig. 4-34. At Mach number Ma^ — 0.75 (point A), a shock wave does not yet form because the velocity of sound has not been exceeded

Figure 4-30 Measurements on a wing profile at subsonic incident flow from [89], angle of attack a = 0°. (a) Pressure distribution at various Mach numbers, (b) Drag coefficient vs. Mach number.

Figure 4-33 Lift coefficient of wing vs. Mach number. Solid curve: typi­cal trend of measurements. Dashed curve: theory according to Fig,

4-20c.

significantly on either side of the profile. Up to this Mach number, the flow is subsonic and the lift follows the linear subsonic theory (Prandtl, Glauert). At Ma„ = 0.81 (point B), the velocity of sound has been exceeded significantly on the front portion of the profile upper surface. A shock wave at the 70% chord is the result. The lower surface is still covered everywhere by subsonic flow. Up to point В, the lift increases with Mach number. At Mach number 0.89 (point C), the velocity of sound is also exceeded over a large portion of the lower surface. A shock wave therefore forms on the lower surface near the trailing edge. This changes the velocity distribution over the profile considerably, resulting in a marked lift reduction. At Mach number Ma^ = 0.98 (point D), the two shock waves on the upper and lower surfaces are considerably weaker than at Mz» = 0.89 and are located at the trailing edge. The lift, therefore, is again larger than at point C. Finally, at Маж = 1.4 (point E), pure supersonic flow has been established with a velocity distribution typical for supersonic flow. The magnitude of the lift now corresponds to the linear supersonic theory (Ackeret).

All tests indicate that the processes in the shock wave are markedly affected by the friction layer. This interaction between shock wave and boundary layer is, besides other effects, particularly complicated because the behavior of the boundary layer changes with Reynolds number, but on the other hand, the shock wave depends strongly on the Mach number. Above a certain shock strength, the pressure rise in the shock causes boundary-layer separation which, in addition to the drag rise already discussed, leads to strong vibrations as a result of the nonsteady character of this flow. This phenomenon is also called “buffeting” in aeronautics; see, for example, Wood [109]. Both the Mach numbers of sudden drag rise and of buffeting are influenced by the profile shape and the angle of attack a (see Fig.

4- 35). The so-called buffeting limit restricts the Mach number range for safe airplane operation. By increasing the incident flow Mach number to supersonic velocities, the shock moves to the wing trailing edge and the buffeting effects disappear again. For very thin and slightly inclined profiles, this state can be reached without the shock’s gaming sufficient strength to excite buffeting while it is moving over the profile. The individual phases of the flow in Fig. 4-3 5a are explained by the pressure distributions of Fig. 4-356.

Because of the complicated flow processes above the critical Mach number, a strictly theoretical determination of the buffeting limit is not possible. However, Thomas and Redeker [109] developed a semiempirical method for the determina­tion of the buffeting limit; see Sinnott [84]. A comprehensive experimental investigation of this problem, which is most important for aeronautics, has been reported in detail by Pearcey [69] and Holder [33].

Similarity rule for transonic profile flow So far, analytical determinations of transonic flows with shock waves have succeeded only in a few cases. In some cases, however, a steady transition through the sonic velocity (without shock waves) has also been observed. In this latter case, transonic flows can be treated theoretically by means of an approximation method. They lead to similarity rules for pressure distribution and drag coefficient (Sec. 4-2-3) that are in quite good agreement with

measurements. It can be shown that the transonic similarity rule remains valid even when the flow includes weak shock waves.

Between pressure distribution and drag coefficient of wing profiles of various thickness ratios tjc and at various transonic Mach numbers of the incident flow (Mfloo 1), the following expressions are valid according to reference [103], and extend Eqs. (4-35) and (4-36):

Here, cp is called the reduced pressure coefficient, and cq is the reduced drag coefficient. For the special case Маж = 1 (sonic incident flow), тж — 0 from Eq. (4-57). From this it follows immediately that the pressure coefficient cp is proportional to (t)cf’z in this case and the drag coefficient proportional to {tfcf/3 [see Eqs. (4-35) and (4-36), respectively].

Malavard [103] checked the similarity rules, Eqs. (4-55)-(4-57), in compre­hensive experiments. He clearly verified the transonic similarity rule for pressure distribution and drag coefficient of symmetric biconvex profiles of thickness ratios i/c = 0.06-0.12 at chord-parallel flow of incident Mach numbers of Маж = 0.775-1.00. Plotting of the drag coefficient cD against the Mach number in Fig.

4- 36a shows the well-known strong drag rise near Ma«, = 1 and, moreover, the strong increase of this rise with the thickness ratio tjc.

Theories for the computation of transonic profile flows The transonic profile flow with shock waves can be treated only by nonlinear theory, in contrast to the linear theories of subsonic and supersonic profile flows. There exist numerous trials and methods for the solution of this task. A survey of the more recent status of understanding of theory and experiment for transonic flow is given by Zierep [111]. So far, the hodograph method, the integral equation method, the parabolic method, and the method of characteristics have been applied to computations. Guderley uses mainly the hodograph method, Oswatitsch generally prefers the integral equation method. The many publications quoted in [63, 66, 79, 84-87, 111] show that no generally valid solution has been found for the computation of the pressure distribution of wings on which shock waves form at transonic incident flow. More recent progress has been discussed at the two Symposia Transsonica [67].

Supercritical profiles For wing profiles operating at high subsonic flight velocities, the drag-critical Mach number Мажсr according to Figs. 4-14a and 4-29 can be shifted to higher values by reducing the profile thickness ratio or by lowering

the profile lift coefficient.* Profiles at which the critical pressure coefficient cp cr from Eq. (4-53a) has not yet been exceeded or has just been reached on the suction side (profile upper side) are termed subcritical profiles. On them no shock waves form, and therefore no shock-induced flow separation occurs. Through suitable profile design, local areas of supersonic flow can be created on the profile in which recompression to subsonic flow occurs steadily or in weak shock waves only. On these profiles the pressure rise in the recompression zone is gradual and therefore does not cause flow separation. Transonic profiles designed according to the stated criterion are termed supercritical profiles.

A few more statements should be made about the evolution from subcritical to supercritical wing profiles. In many designs the product of lift-to-drag ratio and Mach number must be optimized. This request may roughly be transferred to the aim to achieve for a given profile thickness ratio at the design Mach number the highest possible lift at fully attached flow conditions. By starting with the pressure distribution la in Fig. 4-37 found on the suction side of the conventional NACA 64A010 profile a gain in lift first may be obtained by further upstream and downstream extension of the minimum suction pressure just along its critical value

according to curve 2a. Such profiles are called “roof-top profiles.” In the range of the profile nose, a strong acceleration of the flow is required, which is accomplished by increasing the nose radius. The onset of the recompression needed to match the pressure at the profile trailing edge (pressure at the rear stagnation point in inviscid flow) must be chosen to allow establishment of a pressure gradient over the rear portion of the profile that does not cause flow separation. Chordwise linear recompression according to curve 2a has been found to be good in practical applications. A further marked increase in lift is obtained by admitting a local supersonic flow field on the profile suction side, which means choosing pressure distributions exceeding the critical pressure coefficient. That kind of flow implies a further increase in nose radius, and, in addition, a flattening of the upper surface. In this case, an essentially shock-free or weak shock pressure distribution along the profile chord, allowing recompression without separation, curve 3a, is of decisive importance. The pressure distribution over the rear portion of the pressure side of conventional profiles is little different from that on the suction side (curves la and lb). Thus, the rear portion of such profiles contributes little to the lift. A larger difference in the pressure distribution of upper and lower side, curves la and 3b, is obtained through changing the profile lower contour between the range of maximum thickness and the trailing edge such that a reduced local thickness is obtained. This change means, according to Fig. 4-38, the establishment of a corresponding profile camber. Measures of that kind are known as “rear loading.” At such profile designs, caution is necessary to avoid flow separation in the recompression region, precisely as it was required on the suction side.

A comparison of the geometries of a subcritical and supercritical profile with “rear loading” and thickness ratios tjc = 0.12 is shown in Fig. 4-38. Systematic investigations on profiles with shock-free recompression from subsonic to supersonic flow have been made by Pearcy [69]. The first design intended to produce shock-free supercritical profiles, so-called quasi-elliptic profiles, was conducted by Niewland [65] and confirmed in the wind tunnel (Fig. 4-39). Since then, a number of generally applicable design methods for supercritical profiles have been developed, and profile families have been checked out successfully in the wind tunnel [4, 54, 55].

4- 3-1 Survey

Now that a basic understanding of the compressible flow over wings (slender bodies) has been established in Sec. 4-2, the airfoil of infmite span will be discussed. On the basis of the similarity rules of Sec. 4-2-3, it turns out to be expedient to study pure subsonic and supersonic flows (linear theory) first, that is, flows with subsonic and supersonic approach velocities (Ma„ ^ 1), Secs. 4-3-2 and 4-3-3. The validity range of linear theory for Маж < 1 is limited by the critical Mach number Maxcr for the drag of Sec. 4-3-4. Later, transonic flow (nonlinear theory) will be discussed, at which the incident flow of the wing profile has sonic velocity {Маю & 1). Lastly, in Sec. 4-3-5, a brief account of hypersonic flow will be given, characterized by incident flow velocities much higher than the speed of sound {Маж >1).

4-3-2 Profile Theory of Subsonic Flow

Linear theory (Prandtl, Glauert) The exact theory of inviscid compressible flow leads to a nonlinear differential equation for the velocity potential for which it is
quite difficult to establish numerical solutions in the case of arbitrary body shapes. For slender bodies, however, particularly for wing profiles, this equation can be linearized in good approximation, Eq. (4-8). For such body shapes, explicit solutions are therefore feasible. In these cases, the physical condition has to be satisfied that the perturbation velocities caused by the body are small compared with the incident flow velocity. This condition is satisfied for wing profiles at small and moderate angles of attack. Linear theory of compressible flow at subsonic velocities leads to the Prandtl-Glauert rule. It allows the determination of compressible flows through computation of a subsonic reference flow. As discussed in Sec. 4-2-3, this subsonic similarity rule (version II) consists essentially of the following.

For equal body shapes and equal incident flow conditions, the pressure differences in the compressible flow are greater by the ratio l/Vl —Mai, than those in the incompressible reference flow. Here, Mam = ихаж is the Mach number, with Uoc the incident flow velocity and the speed of sound. Hence, the pressure distribution over the body contour from Eq. (4-23) becomes

P(x) ~ Poo = ■ …… -1 = [PincOO – P°° 3 (4’37)

Vl – Ma-^

Here the quantities of compressible flow are left without index, those of the incompressible reference flow have the index “inc.”

For the dimensionless pressure coefficient, the formula of the translation from incompressible to subsonic flow is obtained as

Cp = : cpinc (version II) (4-38)

?oo у l — Ma2^

Here it has been assumed that profile contours and angles of attack of compressible flow and of the incompressible reference flow are equal; that is,

ZincW = Z{X) (4-3 9 д)

«inc = a (4-396)

where X = xjc and Z = zjc are the dimensionless profile coordinates according to Eq. (2-2).

An experimental check of Eq. (4-38) is given in Fig. 4-11 for the simple case of a symmetric profile of 12% thickness in chord-parallel flow. Agreement between theory and experiment is very good in the lower Mach number range. At higher Mach numbers some differences are found. In Fig. 4-11, the values of the local sonic speed (Ma =1) are included, showing that sonic speed is first reached locally at. Мдет = 0.73.

The lift, obtained by integration of the pressure distribution over the profile chord, increases with the transition from incompressible to compressible flow as l/Vl — Mai, because of Eq. (4-38).

The expression for the lift coefficient is given in Table 4-1, which also contains the transformation formulas for the other lift-related aerodynamic coefficients. For

incompressible flow, the determination of neutral-point position, zero-lift angle, zero-moment coefficient, and angle of attack and angle of smooth leading-edge flow has been discussed in Sec. 24-2. For lift slope and neutral-point position of the skeleton profile, the values found for the inclined flat plate are valid, namely, {dcLldanc — 2/Г and (xNlc)inc = -, respectively.

In Fig. 4-12, the theoretical lift slopes are plotted against the incident flow Mach number.

Since, according to Eq. (4-37), the pressure distributions over a body at various Mach numbers are affine to the incompressible pressure distribution, it follows immediately that the position of the resultant aerodynamic force in the subsonic range (as long as no shock waves are formed) is equal to that in incompressible flow. Also, the drag in the subsonic range is determined by the same processes as in incompressible inviscid flow; that is, it is equal to zero.

Comparison with test results In Fig. 4-13, the most important results of the subsonic similarity rule are compared with measurements of Gothert [88]. For 5 symmetric

wing profiles of thickness ratios tjc — 0.06, 0.09, 0.12, 0.15, and 0.18, lift slopes are plotted in Fig. 4-13a and neutral-point positions in Fig. 4-1 Зі?, both against the Mach number of the incident flow. For comparison, the theory with (dcLlda)inc = 5.71 is drawn as a straight line in Fig. 4-1 За.*

In the lower Mach number range, agreement between theory and measurement is very good, with the exception of the profile of 18% thickness. The theoretical curve follows the experimental data up to a certain Mach number, which shifts toward Mao, = 1 with decreasing profile thickness. The differences between theory and experiment beyond this Mach number are caused by strong flow separation. This fact can also be seen in the presentation of the drag coefficients of the same profiles in Fig. 4-14a.

According to the present linear theory for very thin profiles, the neutral-point position should be independent of Mach number. The experimental results of the profiles of Fig. 4-13& show, however, a considerable dependence of the neutral – point position on the Mach number when the profile thickness increases.

For the same symmetric profiles that have just been discussed with regard to lift slope and neutral-point position, the dependence of the drag (= profile drag) on the angle of attack a and on the Mach number of the incident flow Max is demonstrated in Fig. 4-14. The behavior of the curves for the drag coefficient CDp(Ma^), with tjc as the parameter, is characterized by the near independence of cDp from the Mach number in the lower Mach number range, whereas a very steep

”’Presented in double-logarithmic scale is dcj^jda vs. (1 —Malo).

drag rise occurs when approaching Ma„ = 1. This drag rise results from flow separation, caused by a shock wave that originates at the profile station at which the speed of sound is locally exceeded. The associated incident flow Mach number is designated as drag-critical Mach number Maxcr. In the case of chord-parallel incident flow (a = 0) the drag rise and, therefore, Mz«,cr, occur closer to Ma„ — 1 for thin profiles than for thick ones (Fig. 4-14a). For a profile with angle of attack (а Ф 0), the profile thickness has a negligible influence on the drag rise, as seen in Fig. 4-14b. As would be expected, the drag rise shifts to smaller Mach numbers with increasing angle of attack of the profile. The effect of the geometric profile parameters of relative thickness ratio, nose radius, and camber on the trend of the curves cDp{Maao) is shown in Fig. 4-15.

Attention should be called to the test results reported by Abbott and von Doenhoff, Chap. 2 [1], and by Riegels, Chap. 2 [50].

In summary, it can be concluded from the comparison of theory and experiment that the subsonic similarity rule (Prandtl-Glauert rule) is always in good agreement with measurements before sound velocity has been reached locally on the profile, that is, when no shock waves and corresponding separation of the flow can occur. Since these two effects are not covered by linear theory, the drag-critical Mach number is at the same time the validity limit of linear profile theory. Determination and significance of the critical Mach number Ma«, cr will be discussed in detail in Sec. 4-3-4.

Higher-order approximations (von Karman-Tsien, Krahn) From the derivation of the linear theory (Prandtl, Glauert), it can be concluded that the deviations of this approximate solution from the exact solution are increasing when the Mach number approaches Ma — l. The same is shown in the pressure-distribution measurements of

Fig. 4-11. Several efforts have been made, therefore, to improve the Prandtl-Glauert approximation. Steps in this direction have been reported by von Karman and Tsien [96], Betz and Krahn [7], van Dyke [99], and Gretler [29]. By the von Karman-Tsien formula, the computation of a compressible flow about a given profile is reduced to the determination of an incompressible flow about the same profile. The result is given here without derivation:

It can be seen immediately that this equation becomes the Prandtl-Glauert formula [Eq. (4-38)] for small values of cp*пс. According to von Karman-Tsien, the underpressures assume larger values and the overpressures smaller values than according to Prandtl-Glauert. In Fig. 4-16, the von Karman-Tsien rule and the Prandtl-Glauert rule are compared with measurements on the profile NACA 4412. Obviously, for the higher Mach numbers the von Karman-Tsien rule is in markedly better agreement with experiment than the Prandtl-Glauert rule.

At the stagnation point of a profile, both theories give the pressure coefficients too high, whereas the Krahn theory, which will not be discussed here, describes the behavior at this point accurately. Also, for Маж 1, Eqs. (4-38) and (440) lose validity, as would be expected from the assumptions made in their derivation. The relationship for the critical pressure coefficient cpcT (Mam) is shown in Fig. 4-16 as a limiting curve (see Sec. 4-34, Fig. 4-28).

4- 3-3 Profile Theory of Supersonic Flow

When a slender body with a sharp leading edge is placed into a supersonic flow field streaming in the direction of the body’s longitudinal axis (Fig. 4-17), the leading edge of this body assumes the role of a sound source in the sense of Fig. 1-9d. As a consequence, Mach lines originate at the sharp leading edge, upstream of which the incident parallel flow remains undisturbed. Only downstream of these Mach lines is the flow disturbed by the body. As an example of this behavior, the flow pattern about a convex profile in supersonic incident flow is shown in Fig. 4-18. The Mach lines, at which the pressure changes abruptly, have been made visible by the Schlieren method. The incident flow velocity can be determined quite accurately, with Eq. (1-33), from the angle of the Mach lines that originate at the profile leading edge.

Linear theory (Ackeret) In analogy to the case of subsonic incident flow of Sec.

4- 3-2, inviscid compressible flow about slender bodies (wing profiles) can be

Figure 4-17 Supersonic flow over a sharp-edged wedge.

computed by a linear approximation theory in the case of supersonic incident flow as well. The linearized potential equation, Eq. (4-8), is valid both for subsonic and supersonic flows. It was Ackeret [1 ] who laid the foundation for this linear theory of supersonic flow. The essential concept of this linear theory is expressed by the requirement that the perturbation velocity и in the x direction is a function only of the inclination of the profile contour area elements with respect to the incident flow direction, of the velocity Um, and of the Mach number Ma„:

u{x) =—– — – Uoc with w(x) = &{x)Ux (441)

Maio — 1

according to the kinematic flow condition (£ > 0: concave; # < 0: convex).

The inclinations of the contour on the upper and lower surfaces against the incident flow direction, &u and fy, respectively, are given for slender profiles of finite thickness and pointed nose (see Fig. 4-19) as

Here the upper sign applies to the upper surface of the profile, the lower sign to the lower surface. Equation (443) confirms the supersonic similarity rule (version II) as

Figure 4-19 Geometry and incident flow vector used in the profile theory at supersonic velocities.

derived in Sec. 4-2-3 [see Eq. (4-25)]. For the further evaluation of Eq. (443), it is expedient to separate the profile contours again, as in the case of the incompressible flow in Chap. 2, into the profile teardrop and the mean camber (skeleton) line [see Eq. (2-1)].

Z = – = Z<s> ±Z<f) and X=— (4-44)

c c

Here, as previously in Eq. (2-2), the coordinates have been made dimensionless with the profile chord c. Again, the upper sign applies to the upper surface of the profile, the lower sign to the lower surface.

For the pressure difference between the lower and upper surfaces of the profile (load distribution), Eq. (4-43Й) yields with Eq. (444):

The aerodynamic coefficients are easily obtained from the pressure distribution through integration. The lift coefficient is, from Eq. (2-54a),

і

cL = f Acp{X)dX = …… 4 – [25] (446)

J ІМа% – 1

О

It is a remarkable result that the lift coefficient depends only on the angle of attack a and not at all on the profile shape; that is, the zero-lift direction coincides with the profile chord (x axis). The moment coefficient, referred to the profile leading edge (nose up = positive), becomes, from Eq. (2-55u)*:

The lift-related aerodynamic coefficients are compiled in Table 4-2. They include the lift slope dcLfda and the neutral-point position xN/c = —dcMldcL, of which the dependence on the incident Mach number Max > 1 is demonstrated in Fig. 4-20tf and b. For comparison, the dependencies for the skeleton profile in subsonic incident flow, Ma^ <1, are also shown (see Table 4-1). These results are identical to those of the inclined flat plate. For Ma„ 1, both linear theories presented here fail, because the assumptions made are no longer valid. This is true particularly for

the lift slope, as can be seen from Fig. 4-33. The location of the neutral point is at xNjc = for subsonic flow and at xNfc = for supersonic flow. This marked shift toward the rear when the flow changes from subsonic to supersonic velocities should be emphasized.

In addition to lift, drag is produced in supersonic frictionless flow. It is called wave drag. The two forces are expressed by

C

D=b {Apfti± A pu$u) dx

о

where Api{x) — Рі(х)—рж and Apu(x)= Pu(x)~P°° are the pressures on the lower and upper surfaces of the profile, respectively, and дг and are the profile inclinations from Eq. (4-42). By using the pressure coefficients from Eq. (4-436) and evaluating the integrals under the, assumption that the profiles are closed in front and in the rear, the lift coefficient cL is obtained as in Eq. (446), and the drag coefficient cD becomes*

*Note that, also in subsonic flow, the wing of finite span has a drag that is proportional to the square of the lift (induced drag, see Sec. 34-2).

Figure 4-20 Aerodynamic forces of the inclined flat plate at subsonic and supersonic flows, (a) Lift slope dc^lda. (b) Position of the resultant of the aerodynamic forces хдг. (c) Drag coefficient Cjj.

Replacing a by as in Eq. (446), and by Z^ and as in Eq. (444), results in

It should be noted that the total wave drag is composed of three additive contributions. The first contribution is proportional to c and independent of the profile geometry. It is plotted in. Fig. 4-20c against the incident flow Mach number/ The second and third contributions are independent of the lift coefficient and proportional to the square of the relative camber and the relative thickness, respectively. Consequently, it can be seen directly that the flat plate is the so-called best supersonic profile, because the second and third contributions are equal to zero in this case.

The formulas for the drag rise dcjjfdc2L and for the zero drag cD at <?£, = 0 have been listed once more separately in Table 4-2. A simple explanation of the wave drag will be given for the subsequently discussed case of the inclined flat plate.

Results of linear theory The physical understanding of the last section was applied for the first time by Ackeret [1] to a quite simple computation of the flow over a flat plate in a flow of supersonic velocity Um at a small angle of incidence a. According to Fig, 4-21, the streamline incident on the plate leading edge forms with the plate a corner of angle a that is concave on the lower side of the plate and convex on the upper side. Consequently, an expansion Mach line originates on the upper side and a compression Mach line on the lower side. At the trailing edge, the compression line is above, the expansion line below the plate. Behind the plate the velocity is again equal to £/« and the pressure equal to рж, as it is ahead of the plate. Consequently, there is a constant underpressure pu on the entire upper surface and a constant overpressure pi on the lower surface. The pressure coefficient cp(x) = const follows from Eq. (4-436) with аФ 0 and z(x) = 0. The characteristic difference in the pressure distributions for supersonic and subsonic incident flow is explained in Fig. 4-22. From Fig. 4-22<z, at subsonic velocity the pressure distribution produces a force-resultant N normal to the plate, and in addition, the flow around the sharp leading edge produces a suction force S directed upstream along the plate (see Sec. 3-4-3). The resultant of the normal force N and the suction force S is the lift L, which acts normal to the incident flow direction £/«,. The resultant aerodynamic force has no component parallel to the incident flow direction; in other words, the drag in the frictionless subsonic flow is equal to zero.

For the case of supersonic flow, Fig. 4-226, the force N resulting from the pressure distribution also acts normal to the plate. However, because there is no flow around the leading edge, no suction force parallel to the plate exists here. The normal force N in inviscid flow therefore represents the total force. Separation info components normal and parallel to the incident flow direction establishes the lift L = TV cos a. and the wave drag D = N sin a « La. There is another physical explanation for the existence of drag at supersonic incident flow, namely, that for the production of the pressure waves (Mach lines) originating at the body during its motion, energy is expended continuously.

As a further example of the pressure distribution on profiles in supersonic flow, a biconvex parabolic profile and an infinitely thin cambered parabolic profile, given by the equations

Z(t) = – X)

Figure 4-2! Inclined plate in supersonic incident flow.

Z<*) = 4^X(1 – X)

are compared in Fig. 4-23. Both profiles are in chord-parallel incident flow, a = 0°. Consequently, from Eq. (446), cL = 0 for either profile. The pressure distributions, as computed from Eq. (443), are given in Fig. 4-23. The zero moment of the teardrop profile is equal to zero, whereas that of the skeleton profile is turning the leading edge down (nose-loaded). The lift-independent share of the wave drag is obtained from Eq. (448b) as

_______ 16 (iY

Cdo~ з Ужитті c/

_______ 64 jhy

~ З У Mai, – 1 N c /

These expressions show that the zero-drag coefficients are proportional to the squares of the thickness ratio t/c and the camber h/c, respectively. In Fig. 4-24, the

Figure 4-23 Pressure distribution at supersonic incident flow for para­bolic profiles at chord-parallel inci­dent flow, (a) Biconvex teardrop profile. (b) Skeleton profile.

pressure distributions of an inclined flat plate (Fig. 4-24я), a parabolic skeleton (Fig.

4- 24Z?), a symmetric biconvex profile, and a circular-arc profile at angle of attack a = 0° (Fig. 4-24c and d), as well as at а Ф 0° (Fig. 4-24e and/), are compared.

Further, a few data should be given about the dependence of wave drag on the relative thickness position for double-wedge profiles and parabolic profiles. The

geometry of parabolic profiles was given by Eq. (2-6). In Table 4-3 the results are compiled, and in Fig. 4-25 the contribution to the wave drag that is independent of cL is plotted against the relative thickness position. For a relative thickness position xt = 0.5, the wave drag of the double-wedge profile is

(4-51)

Thus, the drag of this double-wedge profile is lower by a factor f than that of the parabolic profile (Xt= 0.5). The double-wedge profile (Xt = 0.5) is the profile of lowest wave drag for a given thickness. Data on additional profile shapes are found in Wegener and Kowalke [21].

Information on the remaining aerodynamic coefficients, namely, zero-lift angle and zero moment, is compiled in Fig. 4-26 for skeleton profiles of all possible relative camber positions. The geometric data of the skeleton line were given in Eq. (2-6). For comparison, the coefficients for subsonic velocities are also shown. The zero-lift angle and the zero moment are plotted against the relative camber position in Fig. 4-2бй and b, respectively. In either case the basically different trends at subsonic and supersonic velocities are obvious.

Higher-order approximations (Busemann) The above-stated linear profile theory for supersonic flow, characterized by a local pressure difference (p—pcX) proportional to the local profile inclination & was later extended by Busemann [10] to a higher-order theory by adding terms of $2 and г}3. The pressure coefficient of the extended theory changes Eq. (4-43a) into

(4-52л)

Table 4-3 Wave drag at supersonic incident flow for double-wedge profiles and parabolic profiles (see Fig. 4-25)

Figure 4-25 Wave drag at supersonic flow vs, relative thickness position for double-wedge profile (1) and parabolic profile (2), from [21] (see Table 4-3).

with

The aerodynamic coefficients can be determined from Eq. (4-52), but no details will be given here. For the lift-independent contribution, an additional term is obtained that is proportional to (t/c)3 for symmetric profiles. Theoretical drag values, computed using this theory of second-order approximation, are compared in Fig. 4-27 with measurements by Busemann and Walchner [10]. Good agreement is obtained.

With greater accuracy than by the above-illustrated theory of second-order approximation, the supersonic flow about thin profiles can be determined by the method of characteristics. Compare, for instance, the publications of Lighthill [51, 52].