Inclined Wing in Supersonic Flow

Before reporting on a general computational procedure for the determination of the lift distribution on wings of finite span in supersonic incident flow, first two particularly simple wing shapes will be treated, namely, the rectangular wing and the triangular wing (delta wing). Fundamentally, these two wings can be computed by the relatively simple method of cone-symmetric flow of Sec. 4-5-2. For arbitrary wing shapes, however, the method of singularities discussed in Sec. 4-5-3 must be used.

Rectangular wing The simple rectangular wing is obtained by setting 7 = rr/2 in Fig. 4-70. Thus, from Eq. (4-81), m = °°. During transition from the swept-back leading edge of Fig. 4-70a to the unswept leading edge of Fig. 4-7Ob, the Mach line originating at point A disappears because point A is no longer a center of disturbance. Hence, range II of constant pressure distribution now embraces the entire surface outside of range IV. The solution for the edge zone of the rectangular wing (range IV) is obtained from Table 4-5 for m -*■ 00 as

— — arccos (1 – f – 2t) (4-111)

cp pi 71

with t from Eq. (4-92). This pressure distribution is shown in Fig. 4-76. It was first investigated by Schlichting [80]. From Fig. 4-76 it can be seen that the lift of the edge zone is only half as high as that of an area of the same size in plane flow. This solution allows a simple determination of the total lift of a rectangular wing. The lift slope becomes

dcL ^ 4 ^_________ 1

dec J/M4 _ 1 2/11/Jf<4 – 1

This formula is applicable as long as the two edge zones do not overlap, that is, for As/Mai, — 1 > 2 (Fig. 4-lla). They overlap for 1 < A/MaL — I < 2 (Fig. 4-77/?). The Mach lines from the upstream corners intersect the wing trailing edge. For А/1 —Mab < 1, they intersect the side edges and are reflected from them as shown in Fig. 4-llc. The pressure distribution in the ranges affected by two Mach cones (simple overlapping) may be gained by superposition (see Sec. 4-5-2).

Figure 4-76 Inclined rectangular plate at super­sonic incident flow, (a) Planform. (b) Pressure distribution at the wing edge, from Eq. (4-111).

The lift slope of the rectangular wing is seen in Fig. 4-78a, where Eq. (4-112) is valid even up to AsjMaL — 1 = 1. A detailed explanation thereof will be omitted here. In Fig. 4-786 and c, the neutral-point positions and the drag coefficients are also shown. Finally, the pressure distribution over the wing chord and the lift distribution over the span are given in Fig. 4-79 for a rectangular wing of aspect ratio /1=2.5; in Fig. 4-79tf the Mach number Max — 1.89, and in Fig. 4-796


Figure 4-77 Inclined rectangular plate of finite span at supersonic incident flow for several Mach num – bers. (a) A Mala — 1 > 2. (b) 1 < J y/Mal, – 1 < 2. (с) a s/Malo -1 < 1.

Figure 4-78 Aerodynamic forces on inclined rectangular wings of various aspect ratios at supersonic incident flow, (a) Lift slope. (b) Neutral-point position. (c) Drag coefficient.

Maoo = 1.13. It can be shown easily that a wing with А у/Mai* — 1 = 1, as at Масс = 1, has an elliptic circulation distribution. The influence of the profile thickness of an inclined rectangular wing has been investigated, in the sense of a second-order theory, by Bonney [8]; compare also Leslie [50].

Delta wing As a further example, the delta wing will be discussed. This includes wings with subsonic and supersonic leading edges, depending on the Mach number (Figs. 4-67 and 4-69).

Wings with subsonic trailing edges are entirely described by range I, as can be

concluded from Fig. 4-66a, The corresponding pressure distribution has already been given in Table 4-5 and in Fig. 4-67. In terms of the mean value of the pressure over the span from Eq. (4-87), the total lift is obtained by integration over the wing area as

where ЛСрр =Сррц —Cppiu is the mean pressure difference between the lower and upper surfaces of the unswept plate. With Acpv = Aafy/MaL — 1, the lift slope of the delta wing with subsonic leading edge becomes

dc£ m 2 л

dec E'(m) y’jlfalo – 1

= tany (0 < m < 1) (4-113b)

E (?n)

for MaBо > 1 and 0 < m < 1.

(у -+ 0, A -*■ 0)

(4-114а) (4-114*)

One result of Eq. (4-113b) should be emphasized: For very slender wings (7 very small), m approaches zero for any Mach number, and because E) = 1,

with tan 7 = AI A. Thus, the lift slope of very slender wings is independent of the Mach number when Маж > 1. The same result was found in Eq. (4-75a) for Моею < 1. This is the so-called slender-body theory of Jones [37]. For Max = 1, again m — 0, and in this case Eq. (4-113) is also valid, in agreement with Eq. (4-75c). Thus it has been shown that the lift slope at Ma„ = 1 has the value dcLjda — я A12 for arbitrary aspect ratios A, whether Ma^ — 1 is reached from the subsonic or from the supersonic range (see Fig. 4-51).

The neutral point lies in the surface center of gravity because the pressure differences, averaged in the lateral direction, are constant in the longitudinal direction. Thus, the neutral point lies at

*jv = 2 cr 3

The drag of a wing with subsonic leading edge is composed of the partial force La, which depends on the pressure distribution on the wing, and the suction force S, which is produced by the flow around the leading edge (see Sec. 3-4-3). Thus, the drag is given by

D = La-S (4-116)

The contribution La is known from the above discussion. The suction force S can be determined from the vortex density k(x, у) in the vicinity of the leading edge. This relationship for plane incompressible flow is given in Eq. (2-76). Determination of the suction force for compressible flow with subsonic leading edge is treated, for example, in [37] and [77]. For a delta wing with subsonic leading edge (m < 1), the drag coefficient without suction force cD = c^a becomes

Here it has been taken into account that, from Eq. (4-81),

The coefficient of the suction force is determined from [77] as

According to Eq. (4-116), this quantity is to be subtracted from the drag coefficient from Eq. (4-117) to obtain the total drag (reduced drag + wave drag + suction force). Hence

= [2 W{m) – ]/l –

The wing with supersonic leading edge is composed of ranges II and III of Fig. 4-666 only. The corresponding pressure distributions have been given previously in Table 4-5 and in Fig. 4-696.

By taking the mean value of the pressure over the width from Eq. (4-91), the lift slope of a delta wing with supersonic leading edge becomes

^ = * . (m > 1) (4-121)

Hence, the lift slope of a delta wing with supersonic leading edge is equal to that of the plane problem (Table 4-2).

Likewise, the neutral point of a delta wing with supersonic leading edge lies in the area center of gravity because the pressure difference, averaged laterally, is constant in the longitudinal direction. Thus the neutral-point position is the same as that of a delta wing with subsonic leading edge, namely, xNjcr = f [Eq. (4-115)]. The total drag (induced + wave drag) is


Since there is no flow around the leading edge, no suction force is created. The drag coefficient becomes, therefore,





I ч


(4-122 a)

= f І Mai – 1


— зі m

71 A.


in agreement with the flat plate of infinite span (see Table 4-2). Equation (4-122c) is obtained with Eq. (4-118),

The ratio of the lift slopes of a delta wing from Eqs. (4-113) and (4-121) and that of an inclined flat plate of Table 4-2, with

jdcL 4

doi)oо y/MaL ~ 1

is plotted in Fig. 4-80 against m [Eq. (4-118)]. The lift slope of a delta wing with a subsonic leading edge (0 <m < 1) is considerably smaller than that of a delta wing with a supersonic leading edge (m > 1). The theoretical results for the lift slope of delta wings in the entire Mach number range are compiled in Fig. 4-82a. The values for Масо < 1 have been established from the linear theory of subsonic incident flow (Sec. 44-2), those for supersonic incident flow from the above formulas, which are also linear. The curve for A = 00 corresponds to the plane problem in Fig. 4-2Gc.

The neutral-point positions of a delta wing for the entire Mach number range

Figure 4-80 Lift slope of a delta wing at supersonic incident flow. Subsonic leading edge: 0 < лп < 1. Supersonic leading edge: m > 1.

are presented in Fig. 4-826 for several aspect ratios A. The curve for /1 = °° corresponds to the plane problem in Fig. 4-206.

The drag coefficient of delta wings is given in Fig. 4-81, where curve la represents the case Mz<l without suction force, Eq. (4-117), and curve lb the case with suction force. Curve 2 is the case m > 1 from Eq. (4-122c).

In incompressible flow it is customary to designate the contribution cD = cjtA, caused by the velocity field induced behind the wing, as induced drag. Such a contribution is made to the drag in compressible flow, too, and it is logical to call it induced drag also (Fig. 4-81, curve 3). Subtracting this drag from the total drag at supersonic velocities, the wave drag (Fig. 4-81) is obtained. For practical purposes, separate determination of the induced drag has no particular significance. Only the sum of induced drag and wave drag is required; see Schlichting [80].

The drag coefficient of delta wings without twist at various aspect ratios A is shown in Fig. 4-82c versus the Mach number. The curve for Л = 00 corresponds to

Figure 4-81 Drag of delta wings at supersonic incident flow vs. m from Eq. (4-118). in < 1: subsonic leading edge, m > 1: supersonic leading edge. Curve la, from Eq, (4-117). Curve lb, from Eq. (4-120). Curve 2, from Eq. (4-122). Curve 3, induced drag from Eq. (3-134).

the plane problem in Fig. 4-20c. Note that the aspect ratio has a strong effect on the lift-related drag at subsonic incident flow. Conversely, this effect is negligible for supersonic incident flow.

For airplane design, wing forms with large aspect ratio do not offer an advantage at supersonic flight velocities (see Fig. 3-4b).

A survey of the pressure distributions over the wing chord and the lift distributions over the span is found in Fig. 4-83 for delta wings with subsonic and supersonic leading edges. The lift distributions (c? c) are illustrated in Fig. 4-84 for several values of m. It is noteworthy that the lift distributions over the span are elliptic for all wings with subsonic leading edge, 0 < m < 1. For wings with supersonic leading edge, m> 1, the lift distribution approaches a triangular form at very high Mach numbers

Systematic measurements to check the three-dimensional wing theory at supersonic incident flow have been published by Love [56] for delta wings with rounded and sharp-edged noses. In these measurements the aspect ratio A lies

v* I

Figure 4-83 Pressure distribution over the wing chord and lift distribution over the wing span of delta wings at supersonic incident flow, (a) Subsonic leading edge, 0<m<l. (b) Super­sonic leading edge, m > 1.

Figure 4-84 Lift distributions over the span of delta wings at supersonic flow for several values of m from Eq. (4-118). 0<m<l: subsonic leading edge, m >1: supersonic leading edge.

between 0.7 and 4, the profile thickness is 5 = tfc = 0.08, and the relative thickness position Xt = xtjc = 0.18; the Mach numbers are MaQ„ = 1.62, 1.92, and 2.40.

The results for the lift slope are given in Fig. 4-85. As the abscissa, the parameter m was chosen. The ordinate for the range of subsonic leading edges (m < 1) is the quantity cot у (dcL/dot) = (4/A)(dcjJda) for the range of supersonic leading edges (m > 1), the quantity (dcL I da) у/МаЬ — 1 is the ordinate. Test results for the 22 wings at the 3 different Mach numbers lie quite close to one curve, confirming the validity of the supersonic similarity rule of Sec. 4-2-3. The measured curve follows the theoretical curve fairly well. The deviations between theory and measurements at m = 0 and m = 1 are understandable, because m ~ 0 means transonic flow (Me» «1), and m = 1 signifies transition from a subsonic to a supersonic leading edge.

The analogous plotting of the drag coefficients is given in Fig. 4-86. Only the values for rounded noses are included. Here also, the measured drag coefficients He near one single curve, again confirming the supersonic similarity rule. In the range of subsonic leading edges the curve of the measured drag coefficients lies, at the lower values of m, between the theoretical curves with and without suction force.

Finally, in Fig. 4-87, the measured neutral-point positions are plotted. Here, too, the supersonic similarity rule finds a satisfactory confirmation. The neutral points of wings with rounded noses lie somewhat more upstream than those with


Figure 4-86 Measured drag coefficients due to lift of delta wings in supersonic incident flow, from Love. 0 < m < 1: subsonic leading edge, m > 1: supersonic leading edge.

sharp-edged noses. The measured neutral-point position moves slightly upstream and increases with Mach number, although, from the linear theory, it should be independent of Mach number.

Swept-back wing Lift slopes of swept-back wings with constant wing chord (taper X = 1) are given in Fig. 4-88 with A cot 7 as the parameter (A = aspect ratio, 7 = sweepback angle measured from the wing longitudinal axis). The lift slope is referred to that of the plane problem (dcLldot)cc = 4/y/MaL — 1 and depends on the parameter m = tan 7/tan д = tan 7 yjMaL — 1 and on the purely geometric quantity A cot 7, and may be written as

The fact that the lift slopes depend only on these three parameters can be realized by setting tan = cot 7 in the supersonic similarity rule Eq. (4-26) and observing that /1 ]MaL — IjA tan tp = tan 7/tan д = m [see Eq. (4-81)]. Under flow conditions rendering the leading edge of the present wing shapes subsonic, the lift slopes—in a way similar to that shown for delta wings (Fig. 4-80)—deviate considerably from those of the plane problem. Conversely, when the leading edge of the present wing shapes is supersonic, the lift slopes are almost equal to those of

Figure 4-88 Lift slope of swept-Ъаск wings (taper X = 1) at supersonic incident flow, from [55]. 0 < m < 1: subsonic leading edge, m > 1: supersonic leading edge.

the plane problem. For a better illustration, the wing planforms are sketched in Fig. 4-88 for /1 = 3. However, the diagram applies to other values of A, too. The figure does not include rectangular wings, because the chosen presentation is not applicable to the case of у = тг/2. The lift slopes of the rectangular wing were given earlier in Fig. 4-7 8д.

Arbitrary wing planforms So far, results have been presented for the linear wing meory at supersonic incident flow for the unswept rectangular wing, the delta (triangular) wing, and the swept-back wing. In this section, a few results will be given for a trapezoidal wing, a swept-back wing, and a delta wing; see Fiecke [21]. The theoretical lift slopes of these three wings are given in Fig. 4-89 for the Mach number range from Max = 0 to Max = 2.5. For the same Mach number range, the drag coefficients of these three wings are presented in Fig. 4-90. Two curves each apply to the subsonic range and to the supersonic range with subsonic leading edge. The dashed curve applies to the values with suction force, the solid curve to those without. The former are described by the well-known formula for the induced drag C£, = cl/тіA. The drag without suction force is found from cD = cLa = c2L(daldcL), where the values of dcLjda are taken from Fig. 4-89. It can be expected that the suction force is fully effective on a well-rounded profile nose and that the dashed lines represent the drag coefficients. Conversely, the suction force is negligible on thin profiles with sharp noses, as used in most cases on supersonic airplanes, and thus the upper curve applies. In Fig. 4-91, the neutral-point positions of these three wings are shown schematically against the Mach number. The typical behavior during transition from subsonic to supersonic velocities is seen, namely, that the neutral point moves considerably downstream when a Mach number of unity is

Figure 4-90 Drag coefficient due to lift vs. Mach number for a trapezoidal, a swept – back, and a delta wing of aspect ratio /■1 = 3, from [21]. Dashed curve: with suc­tion force. Solid curve: without suction force.

Figure 4-91 Neutral-point position vs. Mach number for a trapezoidal, a swept-back, and a delta wing, from [21]. (o) Neutral-point posi­tion for Ma oo < 1 – {•) Neutral- point position for Ma oo > 1.

exceeded. This means an increase in longitudinal stability of the airplane during transition from subsonic to supersonic flight.

Finally, a brief account will be given of the experimental confirmation of linear wing theory. In Fig. 4-92, the lift slopes dc^jda are plotted over the Mach number for four different wings (rectangular, trapezoidal, triangular, and swept-back). For the subsonic range, the theoretical curves were determined according to Sec. 4-4-2, for the supersonic range, from Friedel [25]. The measured lift slopes are in good agreement with theory, except for the immediate vicinity of Маю = 1. Additional details of a three-component measurement in the subsonic and supersonic ranges of the trapezoidal wing of Fig. 4-92Z? are illustrated in Fig. 4-93. The curves cL(a) of Fig. 4-93йг show clearly that the linear range and the coefficient of maximum lift cL are considerably larger in supersonic than in subsonic flow. Also, the pitching – moment curves cL{cM) in Fig. 4-93c confirm that the linear range is markedly larger for Mao* > 1 than for < 1. In this connection, the publications [59, 76, 90] are noted; they are concerned with the computation of twisted wings and flight mechanical coefficients of wings at supersonic velocities.

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