6- 1-1 General Remarks on the Interactions

among Parts of the Airplane

The aerodynamic coefficients of the major components of the airplane—wing, fuselage, empennage—are quite well established through theory and systematic measurements. The aerodynamics of the wing was treated thoroughly in Chaps. 2-4. The findings established there apply accordingly to the empennage (vertical stabilizer and rudder, and horizontal stabilizer and elevator; see Chap. 7). The aerodynamics of the fuselage was the subject of Chap. 5. When these individual parts are assembled into a complete airplane, however, their interaction (inter­ference) plays a very important role in the formation of aerodynamic forces. In many cases these interference effects are of the same order of magnitude as the contributions of the individual parts to the aerodynamic forces of the airplane as a whole. For this reason, consideration of these interactions is indispensible to the study of the aerodynamics of the airplane. The physical processes behind the aerodynamics of the interactions, are, of course, much harder to conceive than those of the aerodynamics of the individual parts. Consequently, the theoretical study of the interference problem has been attacked much later and is, even today, not yet established to the extent of that of the individual parts. The theory of interference aerodynamics is available to a large extent for inviscid flow only.

Most important of the numerous interference effects among the various airplane components are the interactions between the wing and the fuselage and between the wing and the empennage. The interference between the wing and the fuselage is felt mainly in a changed lift distribution over these parts. The effect of the wing on the empennage, on the other hand, lies mainly in a changed incident flow direction of the empennage caused by the induced velocity field of the wing.

A further important interference effect is the so-called ground effect, which is created during flight near the ground. Hereby for equal lift, the lift slope is increased and the induced drag is usually reduced. This problem has been treated in detail in theory and experiment; see the references cited in Sec. 3-3-1.

In this chapter, only the interaction between wing and fuselage will be investigated. The interference problems related to the empennage will be treated in Chap. 7 together with the aerodynamics of the empennage.

The Fuselage in Subsonic Incident Flow

Computational procedures In Sec. 5-3-1 it was shown that at Mach numbers Maaо < 1, the computation of the flow about a fuselage may be reduced to the determination of the incompressible flow for a fuselage that is suitably transformed. The computation of the incompressible flow over a fuselage was discussed in detail in Sec. 5-2. The starting point for further consideration is the subsonic similarity rule. By assigning the index “inc” to the reference fuselage that corresponds to the given fuselage at a given Mach number, the transformation formulas for the geometric data of the fuselage become, from Eqs. (540), (543), and (544),


xinc = x fine = r Vl —Mat, djnc = $


Fuselage radius:



Fuselage length:

ІР’тс ~ іF


Thickness ratio:

^Finc = 6^7 v/l МаЪ>


Angle of attack:

Qinc = a Vl —MaL

(5-52 d)

The transformation formula for the pressure coefficient is, from Eq. (5-45),


This computation procedure will now be applied to fuselages in axial and inclined incident flow at subsonic velocities.

The fuselage at axial incident flow The pressure coefficient for incompressible flow from Eq. (5-10д) can be given in the form

[/"(-*•) "b<?(-*0 hi 5/г|ПС] 6^7jnc (cp)jvf(

where the functions f(x) and g(x) are independent of the thickness ratio of the fuselage. Introducing Eqs. (5-52c) and (5-53) into the above equation yields


Ap — nR2 as the fuselage cross section. From Eq. (5-54), it can be seen that the influence of compressibility on the pressure distribution is taken into account by a term additive to the pressure distribution at incompressible flow. It is proportional to the second derivative of the distribution of the fuselage cross section. Since, in general, this derivative is negative, the additive term represents an increase in the negative perturbation pressure. The similarity rule of Sec. 5-3-1 is thus confirmed, namely, that the computation of subsonic flow of arbitrary Mach number, 0 <M(2oc < 1 may be reduced to the computation for Маж = 0.

The pressure distributions for the paraboloid of revolution of thickness ratio 8p = 0.1 are shown in Fig. 5-22 for several Mach numbers. Marked changes of the pressure distribution because of the compressibility effect are found only near the fuselage center section (see Krause [29]).

Drag-critical Mach number The critical Mach number of the incident flowM2ooCr at which the velocity of sound is reached locally on the body is obtained, from Eq. (4-53b), from the lowest pressure on the body cpm-in. In Fig. 5-23, determination of the drag-critical Mach number for paraboloids of revolution is demonstrated for several thickness ratios SF. As the figure shows, the intersections of the curves cPrnin versus Маж of the various paraboloids of revolution from Eq. (5-54) with the curve from Eq. (4-53b) have to be established. For comparison see also Fig. 4-28. The critical Mach number, determined in this way, is plotted in Fig. 5-24 against the fuselage thickness ratio. The critical Mach numbers of ellipsoids of revolution are included in this figure. They are somewhat larger than those of the paraboloids. Comparison of these critical Mach numbers of bodies of revolution with those of wing profiles of Fig. 4-29 shows that, for the same thickness ratio (5p = S), the critical Mach number for three-dimensional flow is considerably larger than for plane flow.

The drag-critical Mach number is of significance for the drag rise at high subsonic Mach numbers; compare Fig. 4-14 for wing profiles. Finally, the drag

Figure 5-22 Pressure distribution on a paraboloid of revolution of thickness ratio 527 = 0.1 in axial flow for several Mach numbers.

coefficients at axial incident flow, from Gothert [15] are presented in Fig. 5-25 for a few relatively thick fuselages as a function of the Mach number (see Krauss [30]). These measurements show that the drag rise for fuselages lies at higher Mach numbers than for wing profiles of the same thickness, as would be expected from theory.

The fuselage in asymmetric flow The pressure distribution due to the angle of attack is given for incompressible flow by Eq. (5-25) when the index inc is added to all quantities. Introducing Eqs. (5-51)-(5-53) into this equation yields the pressure distribution at compressible flow as

cv(x, 0) = -2 ~ [*(*) №(*)] (5-55)

Figure 5-23 Determination of the drag-critical Mach number Archer of paraboloids of revolution of thickness ratio 5p at axial incident flow. Curve 1 from Eq. (5-54) and Fig. 5-6. Curve 2 from Eq. (4-536).

Figure 5-24 The drag-critical Mach number MaXQT of paraboloids and ellipsoids of revolution vs. thickness ratio bp and axial incident flow.

By comparison with Eq. (5-25д), it is apparent that the pressure distribution due to the angle of attack is independent of the Mach number. It follows that the relationships of Sec. 5-2-3 for the lift distribution, the lift, and the moment in incompressible flow apply directly to compressible subsonic flow.*

Studies of the computation of the pressure distribution on fuselages of arbitrary cross section shapes, for both subsonic and supersonic flows, have been conducted, for example, by Hummel [23]. A nonlinear second-order theory is given by Revell [42].

5- 3-3 The Fuselage in Supersonic Flow

Fundamentals The essential difference between subsonic and supersonic flows has already been explained by Fig. 1-9. Furthermore, the specific problems of the wing

Figure 5-25 Drag coefficients of bodies of revolution in axial incident flow vs. Mach number, from measurements of Gothert. Cjjp refers to the frontal area.

of finite span at supersonic velocities were discussed in Sec. 4-5. The essential physical difference between flows of subsonic and supersonic velocities lies in the fact that, at the latter, a given point can affect only the space enclosed by the downstream cone. This point itself can be affected only by disturbances within the upstream cone. The application of these fundamental facts of supersonic flow to a fuselage is explained in Fig. 5-26. The flow at a station x, r can be influenced only by the crosshatched range cut out of the fuselage by the upstream cone of apex semiangle fj.. The Mach angle /і is related to the approach Mach number Max by Eq. (4-80). The upstream cone to the point (x, r) intersects the fuselage axis (x axis) at the point


In the following discussions, the length x0 will be termed “influence length.” The Mach cone generated by the fuselage nose is also sketched in Fig. 5-26. The supersonic flow about a circular cone (fuselage nose tip) in axial incident flow represents the simplest case of a cone-symmetric supersonic flow, which has been discussed previously in Sec. 4-5.

Now the. slender body of revolution at flows without (axial) and with a small angle of incidence will be treated. Either case can be computed approximately with the method of singularities (source-sink and dipole distributions, respectively). This method has been presented previously in Sec. 5-2 for incompressible flow. Another possibility is the application of the method of characteristics. Besides the linear theory of supersonic flow over fuselages, which will be presented below in detail, nonlinear theories of higher order have been developed by, for example, van Dyke [51] and Lighthill [33]. Comprehensive presentations concerning the fundamentals of the aerodynamics of fuselages in compressible flow are found in the pertinent publications on gas dynamics, listed in, Section II of the Bibliography.

The fuselage at axial incident flow The axisymmetric fuselage in axial incident flow of supersonic velocity can be treated by means of the source-sink method in a way similar to that which has been explained for incompressible flow (Sec. 5-2-2). This

Figure 5-26 Fuselage theory at supersonic incident flow.

method was developed by von Karman and Moore [55]. The relationship between the source distribution q(x) and the fuselage contour R(x) can be established through the same considerations as in the case of incompressible flow; that is, here, too, Eq. (5-4b) is valid. The procedure for translating the source-sink method of incompressible flow into that of supersonic flows has been treated in detail for the wing in Sec. 4-5-3 and can be applied to the fuselage.

The potential Ф (x, r) of the flow induced by the linear source distribution #(.x) on the x axis is given [see also Eq. (4-102)] as

Ф (x, r)


Here x0 is the influence length from Eq. (5-56). The velocity components are obtained for the entire space in the well-known way as

8Ф дФ

и ——– w~ =———-

д x Ъ г

In executing these differentiations it should be noted that the upper limit x = x0 of the integral in Eq. (5-57) depends on x and r, and that for x = x0, that is, on the Mach cone, the denominator of the integrand vanishes.

To determine the velocity distribution on the fuselage surface, the values of the induced velocities are needed for small radial distances r; see Eqs. (5-6a) and (5-6b). Equations (5-57) and (5-58) yield

1 #(Д3)

2 dx*

The final form of the induced velocity components is obtained by introducing Eq. (.5-4b) into Eqs. (5-59a) and (5-59b) as

To determine the pressure distribution from the induced velocities, the formulas of the incompressible flow are directly applicable, that is, Eq. (5-8) for the first approximation and Eq. (5-9) for the second approximation. In analogy to Eq.

(5-1 Oa), the dependence of the pressure distribution on the body thickness ratio SF can be found for fuselages in axial incident flow. By observing Eqs. (5-60) and (5-61), this dependence is obtained as

cp(x) = [/i(x) + gx{x) In (8p fMaL — l)]SF (5-62)

The functions fi(x) and g(x) depend on the fuselage geometry. They do not depend, however, on the thickness ratio of the fuselage. Consequently, Eq. (5-62) for the pressure distribution may be written in the following form:

Cp = (<ь)ыа„-п -715TIllV*r<&-l (5-63)

This equation is analogous to Eq. (5-54) for subsonic incident flow.

It may be seen from Eq. (5-63) that, at supersonic incident flow, the compressibility effect on the pressure distribution is given by a term additive to the pressure distribution at Маж = /2. This confirms the similarity rule of Sec. 5-3-1, stating that the computation of a supersonic flow of arbitrary Mach number can be reduced to the computation at MaM = y/l. The above computational procedure for the pressure distribution of fuselages in supersonic axial flow will be explained now by means of a few examples. The supersonic flow over the nose tip of a cohe-shaped body was treated early by Taylor and Maccoll [46], Tsien [48], and Busemann [6]. Results for a blunt-body nose in supersonic incident flow have been published by Holder and Chinneck [21] and van Dyke [52].

In Fig. 5-27, the pressure coefficient for Ma„ =y/2 (second approximation) is presented for the paraboloid of revolution of thickness ratio 5^ = 0.1. The functions fi and gt of Eq. (5-62) become in this case

UX) = —4(22 Jf2 – 16X +■ 1) —8(6X2 —6X + l)ln(l – X)


9i(X) — —8(6X2 — 6X – F 1)

with X = x/lp. For comparison, the pressure coefficient from the method of characteristics is also shown. Agreement of these two computational methods is very good. Furthermore, the pressure distribution for incompressible flow (Маж — 0) from Fig. 5-5 is added. It is noteworthy that, in supersonic flow, the pressure minimum lies behind the middle of a body that is symmetric to X= 0.5. Furthermore, it should be noted that for the same shape of the body cross sections, the pressure distribution in the axisymmetric case shows a completely different character than in the plane case, as may be verified by comparison with Fig. 4-23д.

As in the case of a wing, the pressure distribution over the total surface of an axisymmetric fuselage in supersonic incident flow results in a force in flow direction that is different from zero. As in the case of the wing, this force is termed wave drag. It is caused by the Mach waves originating at the body. Computation of the wave drag may be done either with the help of the momentum law or through direct integration of the pressure distribution over the surface. Only the latter computational procedure will be described below.

Integration of the pressure distribution over the surface (component of the

Figure 5-27 Pressure distribution on a paraboloid of revolution of thickness ratio 5p = 0.1 at = sj2 and M?« = 0 in axial flow. Curve 1, singular­ities method; second approximation from Eqs. (5-62) and (5-64). Curve 2, linear method of characteristics.


pressure force in the x direction) yields the wave drag of the body of revolution in axial incident flow as



To establish the effect of the Mach number on the wave drag, Eq. (5-63) is substituted for cp in Eq. (5-65). Integration by parts yields

Here the “number” depends on the fuselage geometry, but not on the thickness ratio. Consequently, the coefficient of the wave drag, referred to the frontal area, is
proportional to 8p* Evaluation of the above equations for the paraboloid results in

cdf = T &F— Ю.675p (paraboloid) (5-68)

where cp(x) of Eq. (5-62) has been substituted, using the expressions of Eq. (5-64). The coefficients of the wave drag of truncated paraboloids of Wegener and Kowalke [11] are given in Fig. 5-28. For the paraboloid cut off in the middle (Ip(Ipо = г)> it becomes

cdf = у = 4.675p (paraboloid tip) (5-69)

This drag coefficient does not include the contribution made by the suction pressure on the blunt tail surface (so-called base pressure). For paraboloids of thickness ratios 5^ = 0.1 and 0.2, the drag coefficients as determined by the method of source distributions are compared in Fig. 5-29 with those from the linear method of characteristics. The deviations of the coefficients from the two methods are very small for 8p = 0.1. They are no longer negligible for 5^ = 0.2, however. By substituting in Eq. (5-65) the expression for cp(x) of Eq. (5-62), a formula for the wave drag of a general pointed body of revolution is obtained that depends on the

*In this connection it should be remembered that the wave-drag coefficient of the wing of finite span, referred to the planform area, is likewise proportional to the thickness ratio (Sec. 4-3-3).

Figure 5-28 Coefficients of wave drag for truncated paraboloids of revolution vs. thick­ness ratio bp = dpmax/lp and Mach number, from Wegener and Kowalke.

Figure 5-29 Coefficients of wave drag for paraboloids of revolution of thick­ness ratios 6j7 = 0.1 and 0.2. Compari­son of the singularities method (1), from Eq. (5-68), and the method of characteristics (2).

body geometry, von Karman and Moore [55] and Ward [57] established the following equation; see the derivation [5]

{ (f

Df=^UI^< A’f{If) I Af Ы dx

~ j J AF(x’)AF(x) In X і dx’ dx

о о

where A’p = dAF/dx and A’p = d2AFjdx2, with AF(x) being the fuselage cross- sectional area. With this formula, the wave drag at given body geometry may be determined through relatively simple quadratures (see [9]).

Das [8] discusses some basic questions about the connection between the various theores for the computation of the wave drag of fuselages, about the ranges of their applicability and the limitations in their accuracies. Both the various theories (linear, nonlinear) and the test results are compared. The summary report on wave drag of fuselages by Morris [39] and the investigations on the base drag of bodies with blunt tails of Tanner [45] should be mentioned here. The computation of the friction drag of bodies of revolution in supersonic flow has been treated by Young [60].

The evaluation of drag measurements for the determination of the wave drag includes considerable uncertainties, because the measured total drag is composed of friction drag and, if the tail is blunt, base drag, besides the wave drag. Measurements in which these three contributions were determined individually have been conducted by Chapman and Perkins [10] and Evans [10]. In Fig. 5-30, the test results of [10] for a truncated paraboloid are plotted as drag coefficients against the Mach number. The comparison of these measurements with theory was accomplished by adding to the measured base drag the theoretical friction drag from Fig. 4-5 and the wave drag from Fig. 5-28. Agreement of the drag coefficients computed in this way with the measurements is quite good. It should be mentioned, however, that there are cases of larger differences between measurements and

Figure 5-30 Measured drag coeffi­cients of a truncated paraboloid of revolution in axial flow (dpmax/ Ip = 0.07) at supersonic velocities, from Evans. Comparison with theory. Curve 1, base drag. Curve 2, base and friction drag. Curve 3, total drag.

Figure 5-31 Drag coefficients (pressure drag without base drag) of slender fuse­lages vs. Mach number Ma«, from mea­surements of [3] (body contour shown with increased ordinates). (1) Optimum body, from Haack and Sears, dpmax/lp =

0. 086. (2) Paraboloid of revolution,

dpmaxdF — 0.091. (3) Cylindrical body, dpmaxllp – 0.08. (4) Cylindrical body

with contraction, dpmax/lp = 0.08.

theory. Additional test results are given in Fig. 5-31, namely, the coefficients of the pressure drag cDp of four slender fuselages in axial incident flow plotted against the Mach number Ml». These drag coefficients do not include the base drag. Fuselage 1 is a body of minimum wave drag for a given volume and a given length, from Haack [43] and Sears [43]. Fuselage 2 is a paraboloid of revolution. Fuselages 3 and 4 have cylindrical tad sections. For fuselages 2 and 3, the theoretical values of Eq. (5-70) are also shown.

Another optimum fuselage configuration with pointed nose and blunt tail was specified by von Karman [55]. Also, Das [7] concerned himself with the determination of optimum shapes of a fuselage with regard to its drag at supersonic flow. A compilation of additional test results and of comparisons with theory is found in Fiecke [11]. Miles [38] derived a linear theory for the computation of the wave drag of fuselages at supersonic incident flow.

The flow picture of Fig. 5-32 gives a more profound insight into the flow about a fuselage in supersonic flow of axial incidence. In particular, it shows clearly the bow wave and the tail wave at a Mach number of Маж = 3.5.

Body of revolution with a blunt nose in hypersonic incident flow In Sec. 4-3-5, the profile with a blunt nose in hypersonic incident flow was treated. For the computation of the pressure distribution on the body surface, Newton’s approxima­tion, Eq. (4-65), was furnished as the simplest expression. This relationship, which was established for plane flow, can be applied likewise to axisymmetric flow as present in the case of fuselages. The pressure distribution on a half-body consisting of a cylinder with a matching spherical nose pertaining to such a hypersonic flow is plotted in Fig. 5-33. According to Newton’s concept of momentum transfer from the flow particles to the body, the pressure distribution would be given by Eq. (4-65). The real flow does not correspond to this concept, and Eq. (4-65) cannot properly represent the pressure distribution. Nevertheless, a very good approxima-

Figuie 5-33 Pressure distribution of a half-body with spherical nose, from Lees, (о) Масо = 5.8, Re = UcoR/v^ = 1.2 ■ 10s. (Д) Маж = 3.8, Re = UaoRlvx =

1.4 • 10s. (——– ) Modified Newtonian

approximation, from Eq. (5-71).

tion for the pressure distribution is obtained, at least near the stagnation point, by substituting in Eq. (4-65) the actual value at the stagnation point for the factor 2. Thus, the so-called modified Newton formula is obtained:

c,> = Cpmajcsinz#

This relationship is also given in Fig. 5-33, showing very good agreement. It should be emphasized, however, that Eq. (5-71) is an empirical relationship.


The fuselage in asymmetric incident flow The fuselage in asymmetric incident flow of supersonic velocity can be treated by means of a dipole distribution on the body axis, similar to the method presented in Sec. 5-2-3 for incompressible flows. The adaptation of the dipole distribution of incompressible flow to supersonic flow follows the rules explained for the axial incident flow. The potential Ф (x, r, £) of a line distribution of three-dimensional dipoles m(x) on the x axis becomes, in analogy to Eq. (5-20д),

Here, xQ is the influence length from Eq. (5-56) and Fig. 5-26. The expansion of Ф (x, r, $) for small radial distances r yields


in agreement with Eq. (5-206) for incompressible flow. Consequently, the velocity components determined from Eq. (5-21) for supersonic flow are identical to those for incompressible flow. Furthermore, the kinematic flow condition of Eq. (5-22), and hence the determining equation for the dipole distribution Eq. (5-24), applies directly to supersonic flows. Finally, it follows that the formula for the pressure distribution at incompressible flow, Eq. (5-25a), is also valid for any supersonic Mach number of incident flow. Since it has been found that Eq. (5-55) for the pressure distribution at subsonic incident flow is identical to Eq. (5-25a), the remarkable result is obtained that, over the entire Mach number range, the pressure distribution due to the angle of attack of the fuselage, and the lift distribution, the lift, and the moment, can be determined from the formulas for incompressible flow. For instance, the lift of a fuselage, truncated in the rear, at supersonic incident Полу is, from Eq. (5-29a),

Lp — ‘Іокр ж A pi

where Apt is the cross-sectional area of the fuselage tail.

The sign £ signifies, according to Hadamard, that only the finite part of this integral has

to be taken.

Figure 5-34 Lift coefficient c^p = Lp! Apmaxqoo of a slender body of revolu­tion with blunt tail vs. angle of attack a, from [53]. Body thickness ratio dpm3iX/ Ip = 0.10, Mach number Mam = 1.97, Reynolds number Re = UoJ. plv » 106, linear theory from Eq. (5-74).

All computational methods for the lift of fuselages treated so far lead to a linear dependence of the fuselage lift on the angle of attack. At larger angles of attack, however, the lift increases more than linearly with angle of attack. As an example, in Fig. 5-34 the lift coefficient cLF of a slender body of revolution with a blunt tail is plotted against the angle of attack for Mach number Ma„ « 2. Compare also Fig. 5-3 for the case of incompressible flow. This lift characteristic much resembles that of a wing of extremely small aspect ratio (see Sec. 3-3-6). The nonlinearity is caused by viscosity effects. At larger angles of attack, the flow separates on the upper and lower surfaces of the fuselage because of cross flow over the body. Subsequently, the flow rolls up and, as in the case of the flow over the side edges of a wing of small aspect ratio, free vortices form that are shed from the body under an angle different from zero (see Fig. 3-50a). The formation of the vortex sheet on slender bodies at large angles of attack is sketched in Fig. 5-35 for a rectangular wing and for a delta wing of small aspect ratio, and for a slender fuselage. Details of the flow about slender bodies at large angles of attack and the theoretical determination of the nonlinear lift characteristic are treated in [2, 24, 35, 37, 51].

For transonic flow about bodies of revolution, generally valid solutions are not yet available. However, the investigations of Keune and Oswatitsch [25, 27], Spreiter [44], Fink [12], and Krupp and Murman [31] must be mentioned here.


5- 3-1 Similarity Rules for Fuselage Theory of Compressible Flow

Velocity potential (linearization) For slender fuselages under a small angle of incidence, the magnitude and direction of the local velocities are only a little different from the velocity of the incident flow иж. It is expedient, therefore, to split up the total flow into a basic, undisturbed flow and a superimposed perturbation flow:

U — – j- и Wr = wr W& = wi& (5-35)

where u, wr, w# are the perturbation velocities with

и < Ux wr < иж w# < U00

Here Ma = Ufa is the local Mach number. Equation (5-36) applies to subsonic, transonic, and supersonic flows. The components of the perturbation velocity become

The relationship between the local Mach number Ma and the Mach number of the incident flow Ma« = U^ja^ is given by Eq. (4-7).

For purely subsonic and purely supersonic flows, Ma can be replaced approximately by Маж. Hence, the following linear differential equation for the potential is obtained in analogy to Eq, (4-8):

In analogy to Eq. (4-9), the equation for transonic flow becomes

Contrary to Eq. (5-38), this differential equation for the potential is nonlinear. In analogy to the case of the wing of finite span, the potential equations derived above, Eqs. (5-38) and (5-39), will now be applied to the development of similarity rules for subsonic, transonic, and supersonic flows.

Subsonic and supersonic similarity rules The similarity rules for subsonic and supersonic flows are obtained through a transformation of the potential equation [Eq. (5-38)]. To this end, the given compressible flow is transformed into a flow, the potential equation of which no longer contains the Mach number. This transformation is accomplished, in analogy to Eq. (4-10), by setting

x’ = x r’ = cxr #’ = # Ф — сгФ’ U’cz = ZJoq (5-40)

where the primes signify the transformed quantities. The factor cx is determined in such a way that the Mach number Ma„ no longer appears in the transformed potential equation. The factor c2 is obtained by applying the streamline analogy (kinematic flow condition). These factors cx and c2 are given by the expressions Eqs. (4-12) and (4-21) derived earlier. The transformed potential equations are, in analogy to Eqs. (4-13) and (4-14),

The transformed potential equation for subsonic flow is identical to the potential equation (Max = 0). The transformed potential equation for supersonic flow is identical to the linear potential equation Eq. (5-38) for Max = /2. These transformations show that the computation of subsonic flows of any Mach number can be reduced to the computation of the flow at Маж = 0 and the computation of supersonic flows of any Mach number to that at Max = /2. This is the Prandtl-Glauert-Gothert-Ackeret rule for fuselages. It can be formulated in the following way, corresponding to version I for wings of finite span (Sec. 4-2-3).

From the given fuselage and the given Mach number, a transformed fuselage is obtained by a distortion of its dimensions in the у and z directions and of its angle of attack by the factor cx = %/|l —Mat|. Its dimensions in the x direction remain unchanged. For the fuselage, transformed in this way, the incompressible flow has to be computed if the given Mach number is subsonic. If the given Mach number is supersonic, however, the compressible flow for Ma„ = y/2 has to be computed.

The transformation formulas of the geometric quantities of the fuselage are

Thickness ratio: = Vll —MaL – p (5-43я)

lF if

Camber ratio: mr = Vil —MaL~ (543b)

lF if

Angle of attack: a = л/і 1 —Mo’Ll a (544)

Hence, when the velocities of the incident flow of the given and the transformed fuselages are equal, the pressure coefficients are related by

The geometric transformation of Eq. (5-43д) is illustrated in Fig. 5-21, in which the transformed thickness ratio is plotted against the Mach number. The hatched body is the given body the flow over which is computed for different Mach numbers. The transformed bodies belonging to the given Mach numbers are drawn without hatches. The flow about these transformed bodies has to be computed as incompressible flow when Ma„ < 1, and as flow at Mam = y/2 when Маж > 1. Applications of this rule will be discussed in Secs. 5-3-2 and 5-3-3.

Transonic similarity rule The similarity rules explained above apply only to subsonic and supersonic flows. Now, a similarity rule for fuselages at transonic flow {Масо = 1) of axial incidence will be given. This similarity rule was first formulated by von Karman [56]. A more detailed presentation of this similarity rule was later given by Keune and Oswatitsch [27]. The following simplified derivation should be sufficient.

By starting with the nonlinear potential equation, Eq. (5-39), the problem may be formulated as follows: Given is an axisymmetric fuselage of revolution at Ma„ = 1. Then, what is the pressure distribution over an affine reference fuselage at the same incident flow Mach number Max =1? In analogy to Eq. (4-28), the following transformation is introduced:

ж’ = x r’ = czr Ф — c&’ С/’,» = и,» (546)

*The validity of this transformation formula for the pressure distribution reaches beyond the framework of the first approximation of Eq. (5-8), as has been shown, e. g., by Truckenbrodt [49]. It applies to the second approximation of Eq. (5-9) as well.


Figure 5-21 The application of the Prandtl-Glauert-Ackeret rule to fuse­lages. Thickness ratio 5’p of the trans­formed fuselage vs. Mach number.

Again, the quantities with primes signify the reference fuselage, those without primes the given fuselage. Substituting Eq. (5-46) in Eq. (5-39) yields, in analogy to Eq. (4-29), c — c4. To establish another relationship between the constants c3 and c4, the radial velocity component wr is derived from the boundary condition Eq. (5-6c):

lim (rwr) = UcoR^- lim (/ w’r) =UOQR’^7 (547)

r—»0 dx r’—>0

Because of the affinity of the two fuselages, R’ = (8’Fj8F)R, with and 8p being the fuselage thickness ratios. With xvr = дФ/дг and w’r = дФ’Ідг’,

Finally, the relationship between the pressure distributions cp and cp of the two fuselages remains to be determined. Because cp = — 2u! Uao = — (2/{Уоо)ЭФ/Эх, this relationship is obtained immediately as

cp — C^Cj! — ^i~j Cp (549)

This is the well-known von Karman similarity rule for bodies of revolution at transonic incident flow.

As was first shown by Oswatitsch, a correction term to this formula can be determined, leading to

cp = cp Qrj + 2g(x)5p In (5-50)

Here, g(x) is given by Eq. (5-1 Oh).

The Fuselage in Asymmetric Incident Flow

General remarks Now the asymmetric inviscid flow about an inclined fuselage as in Fig. 5-12 will be considered.

First, it is important to state that, in inviscid flow, only a moment, not a resultant force, is acting on the – inclined fuselage. This is caused by the underpressures on the upper side of the body nose and the lower side of the tail and, conversely, the overpressures on the lower side of the nose and the upper side of the tail. This pressure distribution results in a moment Mp that attempts to turn the fuselage nose up (unstable moment). At small angles of attack a, this moment is proportional to the angle of attack.

The fuselage-wing interaction changes the magnitude of this moment greatly (see Chap. 6). However, the moment of the fuselage alone will be treated here, first in inviscid flow and later with consideration of friction. It should be mentioned that

Figure 5-12 Inviscid flow about an inclined fuselage.

Figure 5-13 Coefficient k for the computation of the moment of an inclined general ellipsoid of Eq. (5-186), from Vandrey.

the effect of friction on the aerodynamic properties of the fuselage is considerable. The moment in inviscid flow can be obtained from simple momentum considera­tions. Computation of the pressure distribution on the fuselage surface requires application of potential theory. As in the case of the fuselage in axial incident flow, exact solutions and approximate solutions to the singularities method are known. Finally, the effect of friction can be determined with the help of boundary-layer theory.

Fuselage moment by the momentum method of Munk An early account of the computation of the moment of an inclined fuselage was given by Munk [41]. It is based on an application of the momentum law. The momentum far behind a body moving at constant velocity in an inviscid fluid remains unchanged and no resultant force acts on the body, but this does not exclude the existence of a free force couple. According to the Munk theory, lift and pitching moment (free force couple) of a fuselage at an angle of incidence a and at free stream velocity Ux are

Lf = 0 (5-1 So)

MF = 2kqaoVFa (5-186)

Here, = (q{2)UL is the dynamic pressure of the incident flow, VF is the body volume, and A: is a factor describing the ratio of the volume of the fluid quantity moving with the body to the body volume. Values of к for general ellipsoids have been given by Zahm [36] and presented graphically by Vandrey [40]. The coefficients к for general ellipsoids of volume VF = mbc are given in Fig. 5-13 as a function of the axis ratios c/а and 6/c. Accordingly, the coefficient к for slender ellipsoids of revolution (b = c and c/a < 0.2) differs little from unity. Thus, from

Eq. (5-186), the moment of slender bodies of revolution is obtained from the

simple approximation formula

MF = 2qaaVFa (5-19)

Note that the unstable moment of the aerodynamic forces acting on an inclined slender fuselage of revolution is proportional to the angle of attack a and the body volume VF [Eq. (5-la)].

Pressure distribution by the method of dipole distribution The flow field of an inclined body of revolution can be computed by the singularities method. In the simplest approach, a distribution of spatial dipoles as in Fig. 5-14 is arranged on the body axis.* The axes of the dipoles are parallel to the z axis. The potential of the dipole distribution is

cos # m{x) 2л r

where m(x) stands for the dipole strength. The second relationship results from the expansion of the potential for small distances r from the axis, as required for slender bodies. The velocity components in axial, radial, and circumferential directions, respectively, are obtained from Eq. (5-206) as

и =—–



cos $ dm(x)

(5-21 а)


2 л

г dx


Щ = а

1 cos# . .

„ , т{х)


8 г

2 л rs


гщ —




1 sin# , . 2л тіХ)

(5-21 с)

The dipole strength is determined from the kinematic flow condition, which demands that, on the body, the velocity component normal to the surface is zero.

For a body with a cambered skeleton line as in Fig. 5-15, which is a generalization of Fig. 5-14, the kinematic flow condition becomes’

tx(x) U„з cos# – j – wr(it) = 0 for r = R (5-22)

Here a(x) is the local angle of attack of the skeleton line referred to the incident flow direction of U„ as given by

where a is the angle of attack of the fuselage axis and zF(x) is the skeleton line of the fuselage. Introduction of wr from Eq. (5-21 b) into Eq. (5-22) yields, for the dipole distribution,

m(x) = 2 я Ucq a (x) R2 {x) — 2 U^ a (x) AF(x) (5-24)

where Af(x) is the cross-sectional area of the fuselage.

Pressure distribution The inclination of the fuselage causes a pressure distribution on the body surface that, from Eq. (5-8), is given in first approximation as cp(x, $) = —2u(x, #)/f/oo. Introducing Eq. (5-24) into Eq. (5-2Їй) yields

cp (X, 0) = – 2 — £ [«(*) R – (*)] (5-25e)

If the angle of attack is constant along the fuselage axis, this equation takes the simpler form


cJx, #) = —4a cos $ —-— [a(x) = const] (5-25b)

и CiX

An example of these pressure distributions is given in Fig. 5-16 by means ol ellipsoids of revolution of thickness ratios SF = dFmaxllF = 0.1 and 0.2 and angle

"Here, the dipole distribution can be left on the body axis, as in the case of the plane skeleton theory (see Sec. 2-4-2).

Figure 5-16 Pressure distribution result­ing from asymmetric incident flow on ellipsoids of revolution of thickness ratios 5р = 0.1 and 0.2 from Eq. (5-26). Exact solution of Eq. (5-33).

of attack a= const. The following expression for the pressure distribution is easily found:

cp = — 2<x cos$ Sp (5-26)

fX(l – X)

Lift distribution The lift distribution is obtained from the pressure distribution by integration. A fuselage portion of length dx is supported by the lift force dbF of magnitude

dLp = — qcoR(x)dx J cpcos&d& (5-27)


Observing Eq. (5-25a) and integrating over i3 yields the lift distribution,

j>И R2(ж)] (5-28)

CL tC G/2C

This relationship has been derived by Multhopp [40] from momentum considera­tions.

Equation (5-28) shows directly that the total lift force of a closed body vanishes, because

if i?(x) = 0 at the nose and tail of the body [see Eq. (5-18a)].

Pitching moment The pitching moment of the fuselage at constant angle of attack a(x) = const is obtained from Eq. (5-28) through integration by parts as

where VF is the fuselage volume from Eq. (5-1/2). In this way, the Munk approximation formula for the moment of slender bodies of revolution has also been obtained by means of the singularities method. Because LF = 0, the fuselage moment is independent of the location of the reference axis. It is a so-called free moment.

As an example, Fig. 5-17 illustrates, for theory and experiment, the lift distribution from [16] of an inclined ellipsoid of revolution of thickness ratio 8p=^. The theoretical lift distribution is obtained from Eq. (5-28) with a(x) — const as


Tills approximation is included in Fig. 5-17 as the solid line (line 1). The measurements agree well with theory in the front portion of the fuselage, but some deviations are found for the rear portion. For comparison, see Sec. 5-2-2.

The above. discussions apply to bodies of revolution. To determine the moments of bodies of noncircular cross sections at constant angle of attack, it should be realized that, essentially, only the fuselage width distribution bp(x) determines the

moment caused by the inclination. Equation (5-29b) can therefore be applied to fuselages of noncircular cross sections by substituting Dpi2 for R and introducing a correction factor k*. This leads to



Here Vp is the volume of a body of revolution that has the body width for its diameter. The correction factor can be determined by comparing Eq. (5-31 я) with the exact equation Eq. (5-18b) for general ellipsoids. Because Vp = (bjc)VF, we have k* = kc/b, where к is given in Fig. 5-13. The values of k* thus computed are presented in Fig. 5-18 as functions of fuselage width ratio 5p = bpm&x/lp and the cross-section ratio of the fuselage XF = hpmaxlbFmzx (see Fig. 5-1). It follows, therefore, that the factor k* is almost unity for slender fuselages of all practical cross-section ratios Лр. Thus, the above discussion has shown that for the computation of the moment of slender fuselages of noncircular ctoss sections, Eq. (5-29b) may be used in good approximation if the radius R is replaced by the semiwidth bpl2.

The moment of the fuselage of variable angle of attack a(x) is obtained by using the semiwidth bp/2 in Eq. (5-28) instead of R. Hence, integration over the fuselage length yields for the pitching moment, from Eq. (5-29b),


MF – Qoo^ j a(x)bp(x) dx (5-32)


This equation is applicable to the fuselage with cambered skeleton line from Eq. (5-23) and to fuselages in curved flow as encountered during rotation about the lateral axis [see Eq. (3-147)]. Furthermore, this relationship is important for the

Figure 5-18 Coefficient k* for the computation of the moment of an inclined fuselage of noncirculai cross sections, from Eq. (5-3la).

computation of the fuselage moment when a wing is attached to the fuselage (see Chap. 6).

The above considerations on the lift distribution and on the moment furnish, accordingly, the side force distribution and the yawing moment due to sideslip for a yawed fuselage.

Exact solutions A few data will now be given on the exact solutions for inclined ellipsoids of revolution. Maruhn [36] determined the pressure distribution of the inclined ellipsoid of revolution at small angles of attack as


Here, b/a = bp is the fuselage thickness ratio. The quantity В is defined as

where a0 is given by Eq. (5-16). The angle-of-attack-dependent pressure distribution for this exact solution is shown in Fig. 5-16 for 8F =0.2. In the vicinity of the nose and the tail, the exact solution gives somewhat smaller values of the pressure coefficient than the approximate solution by the method of singularities. This means that the correction factor k* for the moment in Eq. (5-3 la) is somewhat smaller than unity. In Fig. 5-17, too, the exact solution for the lift distribution is included as curve 2. Near the nose and the tail, the exact solution deviates somewhat from the approximation solution. In the vicinity of the nose, the measurements agree quite well with the exact solution. Larger differences remain, however, near the tail. They are caused by viscosity effects to be treated in the next section.

The values of the moment from the exact solution have already been given in Eq. (5-182?). From Eq. (5-3la), the theoretical moment coefficient cMF = Mpjq^Vp is obtained as

cMf — 2k*a

This theoretical value, with k* — 0.95, is compared in Fig. 5-3 with a measurement. The moment slope dcMFida from this theory is considerably steeper than that of the measurement. This difference is caused by viscosity effects. Viscosity effects are also responsible for the deviation of the measured lift from zero, as seen in Fig. 5-3.

Viscosity effects Qualitatively, viscosity affects the flow over the inclined fuselage (Fig. 5-12) in such a way that the pressure on the tail section is reduced, because the inviscid outer flow is forced outward by the boundary layer. Consequently, the negative lift of the tail section is somewhat smaller than the positive lift of the nose section. Overall, therefore, viscosity effects cause a positive lift, which is also termed friction lift. This fact may be seen in Fig. 5-17 for the lift distribution. The friction
lift changes the moment; specifically, it creates an additive nose-down moment with reference to the lateral axis through the fuselage center. Hafer [16] has described a method for the approximate computation of the viscosity effect by means of boundary-layer theory. Accordingly, the boundary-layer displacement thickness 6г is determined along the fuselage surface at axial incident flow. A rather strong growth of the boundary-layer thickness near the tail is found, as sketched in Fig. 5-19. Consequently, to compute the pressure distribution, the local fuselage radius R(x) in Eq. (5-25a) must be replaced by the radius [i?(x) 4- 5i(x)], that is, the radius R(x) enlarged by the displacement thickness. The lift distribution is obtained from the pressure distribution corrected for viscosity by integration. In Fig. 5-17, the lift distribution, computed in this way from [16], is also shown. Through the correction for viscosity, better agreement is reached with the measurements, particularly in the vicinity of the tad. Lift, pitching moment, and neutral-point position are determined through further integrations. In Fig. 5-20, the lift slope dcLp/da, the moment slope dcMF/da, and the neutral-point position xNFllF are plotted against the inverse fuselage thickness ratio lFldFmax for several axisymmetric fuselages. These measurements were taken by Truckenbrodt and Gersten [50]. Curves 1 are those from the inviscid theory, curves 2 from the viscous theory of Hafer [16]. The latter theory agrees quite well with measurements.

The Fuselage in Axial Flow

Pressure distribution by the method of source-sink distribution The method of source-sink distribution for bodies of revolution in axial flow was first presented in detail by Fuhrmann [13]. The flow over such a body can be represented, as in Fig.

5- 4, by a distribution q(x) of three-dimensional sources on the body axis that is superimposed by a translational flow £/«. Compare the discussions of the plane problem (profile teardrop) of Sec. 2-4-3.

The connection between the source distribution q(x) and the fuselage contour R(x) can be established intuitively through application of the continuity equation to the volume element ABCD of Fig. 5-4:

(*7oo + u) nR2 qdx — (um + « + ~ dx^ n {e – f ~ dxj

Hence, it follows the source distribution

?(*) =7lJ^ К*- + u) Я2]

(54 b)


d2 (B2)


t dx~“ j x-tQ

direction of the velocity vector on the surface is tangent to the surface.

Pressure distribution From the Bernoulli equation, the pressure’ distribution on the surface of the body is obtained as

where Wc = (Uoo + u)2 + wj is the velocity on the fuselage contour.

As in wing profile theory, the quadratic terms of the induced velocities may be disregarded. Thus, the first approximation of the equation becomes

Equations (5-8) and (5-9), together with Eq. (5-la), allow the determination of the relationship between the pressure coefficient and the fuselage thickness ratio 8p = dFmax/lp. This relationship is found as

cp(x) = [f(x) + Я(х) bSF]6F (5-10fl)

where the functions f(x) and g(x) depend only on the fuselage form but not the thickness ratio.

Examples A few examples of this method of source-sink distribution will now be discussed. The induced velocity u(x) of an ellipsoid of revolution of thickness ratio dp = dpmaxjlp is obtained from Eq. (5-7a) with X = x/lp as

The pressure distribution of an ellipsoid of revolution of thickness ratio 6F = 0.1 is given in Fig. 5-5. Both the first approximation from Eq. (5-8) and the second approximation from Eq. (5-9) are shown. For comparison, the exact solution is given, and will be discussed in the next section. The second approximation agrees

well with the exact solution over the entire contour. The first approximation deviates at the front and rear portions.

For the maximum perturbation velocity at the ellipsoid of revolution that occurs at station X = , Eq. (5-1 Ід) yields

= — /і In*-^ Sp (ellipsoid) (5-Ий)

Uoo 2 /

This value is plotted against the fuselage thickness ratio in Fig. 5-6. Here, too, the exact solution is shown for comparison. At large thickness ratios the values of the

Figure 5-6 Maximum perturbation velocity of bodies of revolution in axial flow vs. thickness ratio Ьр. (1) Exact solution from Eq. (5-15) or Lessing, respectively. (2) Approximation from Eq. (5-Ий) or (5-12b), respectively.

exact solution are larger than those of the approximation solution of the source-sink method with the source distribution on the axis. Also included in Fig. 5-6 is the perturbation velocity for the plane problem of the elliptic profile as in Fig. 2-34. In this case UjnzJUoo – 8 (=8p). The comparison of the curves for the elliptic profile and the ellipsoid demonstrates by how much the maximum perturbation velocity at the body of revolution is smaller than that at the wing profile of the same thickness ratio.

For the induced velocity of a paraboloid of rotation [see Eqs. (2-6) and (2-7a)], Eq. (5-7a) yields

^ = 2 [1 – 6X(1 – X)][3 – f lnAr(l — X) + 2)n8p] 8% (5-12a)


The corresponding pressure distribution (second approximation) for 8p =0.1 is shown as curve 2 of Fig, 5-5. The maximum perturbation velocity, lying again at X= is obtained from Eq. (5-12a) as

= _ (з 4- 2 ln^f) 8p (paraboloid) (5-126)

This value is represented by curve 2 of Fig. 5-6.

The computations discussed so far are based on source distributions on the fuselage axis. Results of Lessing [32] for distributions of source rings on the body surface are included in Figs. 5-5 and 5-6 as curves 1. These results can be considered to be “exact.” The considerable improvement of the theory based on surface distribution over that based on axial distribution is obvious in Fig. 5-6.

The pressure distribution for a body of revolution, composed of a half-ellipsoid of revolution and a matching infinitely long cylinder, is given in Fig. 5-7. For evaluation of Eq, (5-la) at the station of the curvature discontinuity, x/lp = the

Figure 5-7 Pressure distribution on an axisymmetric half-body (dpmzх/7р = 0.1) in axial flow (source distribution on the axis).

Figure 5-8 Pressure distribution on a body of revolution with cylindrical center section. Sp = dpmax/lp = 0.09 (source distribution on the axis).

footnote to this equation must be observed. In this way, the specifically marked value of cp is obtained.* Finally, Fig. 5-8 shows the pressure distribution of a body of revolution composed of a frontal half-ellipsoid of revolution, a rear half-paraboloid of revolution, and a matching cylindrical center section. For the marked points at the stations of curvature discontinuity, the comment that was made for Fig. 5-7 applies.

A body of revolution of the airship kind has been studied particularly by Fuhrmann [13]. The flow pattern produced by a slender body of revolution (so-called streamlined body) is illustrated in Fig. 5-9. Its generating source-sink distribution is indicated in the upper picture. The theoretical pressure distribution is in excellent agreement with measurements.

Exact solutions A few more data will be given on the exact solutions for fuselages in axial incident flow. Such exact solutions of the spatial potential equation can be found in closed form for a few cases only.

The general ellipsoid in axial incident flow was first investigated by Tucker – mann [36] and Zahm [36] and later, more explicitly, by Maruhn [36]. The pressure distribution on the surface of the ellipsoid, Fig. 5-10, in incident flow parallel to the x axis is given in [36] as

where a, b, and c are the semiaxes of the ellipsoid. The origin of the coordinates lies at the center of the ellipsoid. The quantity d is a function of the two axis ratios а/с and b/c; it is presented in Fig. 5-11, from [36].

The special case of an ellipsoid of revolution in axial incident flow is obtained from Eq. (5-13) for b = c as

Figure 5-10 Geometry of a general ellipsoid.

Figure 5-11 Coefficient A for the determina­tion of the pressure distribution on a general ellipsoid in axial incident flow, from Eq. (5-13), vs. the two axis ratios а/с and b/c.

Here, b/a =5/7 is the thickness ratio of the body of revolution. The evaluation of Eq. (5-14) is included in Fig. 5-5 as the exact solution. The pressure minimum Cpmin = 1 ~A1 lies at jc = 0. Hence, the maximum perturbation velocity becomes

^ = A – 1 (5-15)



A = —with я„ = – – ■F.. (tanh’1 Vl -<Sj. – Уі – dp) (5-16)

A ~~ ао у 1 — <5p

The relation between umzx! Uoo and bp is shown in Fig. 5-6 as the exact solution for the ellipsoid of revolution. For small values of SF, the three equations above yield the relationship Eq. (5-1 lb) that was derived by means of the singularities method.

Effect of viscosity So far in this chapter, the fluid has been assumed to be inviscid and incompressible. The effect of compressibility on the aerodynamic properties of a fuselage will be treated in the following sections. First, a few data will be given on the effect of viscosity in incompressible flow (effect of Reynolds number). At moderately large Reynolds numbers (Re > 10s), the effect of viscosity on the pressure distribution on bodies of revolution in axial incident flow is quite small. This can be seen, for instance, from Fig. 5-9, in which the pressure distribution computed for inviscid flow is compared with measurements. The slight deviations of the pressure distribution as obtained from potential theory from that found in viscous flow is responsible for the pressure drag of the body of revolution. In addition there is a friction drag, which is produced by the wall shear stress.

The effect of friction on the flow about fuselages is determined from

boundary-layer computations, quite similar to the case of wing profiles. In the latter case the boundary layers are two-dimensional, whereas in the case of fuselages with circular cross sections in axial incident flow, the boundary layers are axisymmetric. The computational procedures for the latter are very similar to those for the two-dimensional boundary layers, both laminar and turbulent.

These boundary-layer computations for a given body produce distributions of boundary-layer thicknesses (momentum thickness and displacement thickness) and of a form parameter of the boundary-layer profiles over the contour. They determine drag and position of the separation point. Young [59] and Scholz [59] conducted comprehensive computations of the drag of bodies of revolution. They found that the contribution of the wall shear stress to the drag of bodies of revolution is, in general, equal to that of the flat plate in parallel incident flow of equal surface area and equal Reynolds number with reference to the body length. For fully turbulent flow, the body drag due to friction Dpf may be obtained approximately from the flat plate drag Dp from the formula

DFf=Dp(l+c8F) (5-17)

with c « 0.5. Here Dp is the drag of the flat plate in parallel incident flow of the same surface area Sp and the same length Ip as those of the body of revolution. Hence, Dp — CfSpqoo, where the coefficient Cf for smooth surfaces can be taken from Fig. 248. Further data on fuselage drag are found in Hoerner [19].


5- 2-1 General Remarks

Now that some experimental results have been given, the theory of flow over fuselages will be presented. Fuselage theory can be established, similar to profile theory, by two different approaches.

The first approach consists of the establishment of exact solutions of the three-dimensional potential equation, which can be done successfully in only a few cases. The second approach is the so-called method of singularities, in which the flow pattern about the fuselage is formed by arranging sources, sinks and, if necessary,
dipoles on the fuselage axis. This procedure is fairly simple for bodies of revolution (see von Karman [54] and Keune and Burg [26]). An extention of this method for the computation of the flow over fuselages consists of arranging ring-shaped source distributions on the body surface (see Lotz [34], Riegels [32], and Hess [IB]). By this method, body shapes can be treated whose cross sections deviate somewhat from circles.

First, the fuselage in axial flow will be discussed, then the fuselage in oblique flow.

Forces and Moments on the Fuselage

The following sections will be devoted to a detailed discussion of fuselage aerodynamics. To give a feeling for the magnitudes of the forces and moments acting on the fuselage, a typical measurement on a fuselage will be presented first. In Fig. 5-3, some results of a three-component measurement on an axisymmetric fuselage by Truckenbrodt and Gersten [50] are plotted. Here, the following dimensionless coefficients have been introduced for the components of the resultant force (lift and drag) and for the pitching moment:

Lift: Lp — Cip Ур! г q ж

Drag: DF = cDFV^qco (5-3)*

Pitching moment: MF = cMFVFqoo

where =(p/2)i/i is the dynamic pressure of the incident-flow velocity £/«, and

VF is the fuselage volume. Figure 5-3 shows the lift coefficient cLF, the drag

coefficient CjjF, and the pitching-moment coefficient c^f plotted against the angle of attack a. The position of the axis of reference for the pitching moment is indicated in Fig. 5-3. In the range near a = 0, the lift coefficient changes linearly with angle of attack a. At larger angles of attack, cLF grows more than linearly. This lift characteristic cL(a) is very similar to that of a wing of very small aspect ratio (see Fig. 3-49). The drag coefficient cDF is approximately proportional to the square of the angle of attack, similar to that of the wing. In the range of large angles of attack, the pitching-moment coefficient depends almost linearly on the angle of attack.

Forces and moments, in addition to those discussed above, act on the fuselage

Fuselage volume is introduced in this case as a quantity of reference in compliance with the theory of fuselages (see Sec. 5-2-3). The drag coefficient is frequently referred to the surface SF or the frontal area Apmax of the fuselage.

Figure 5-3 Three-component mea­surements Cip, Cdf, an^ CMF vs. angle of attack on an axisymmetric fuselage. Reynolds number Re = 3 • 106. Theory for cj/fp from Eq. (5-34).

as a result of the turning and sideslip motions of the airplane, as has been discussed for the wing in Sec. 3-5.

The summary reports of Munk [41], Wieselsberger [58], Goldstein [14], Thwaites [47], Howarth [22], Heaslet and Lomax [17], Brown [5], Ashley and Landahl [4], Hess and Smith [18], and Krasnov [28] deal with the questions of flow over a fuselage in incompressible, and, to some extent also in compressible flow. Also, the survey of Adams and Sears [1] must be mentioned. Furthermore, the comprehensive compilations of experimental data on the aerodynamics of drag and lift of fuselages of Hoerner [19] and Hoerner and Borst [20] should be pointed out.



___________________________________________ FIVE



5- 1-1 Geometry of the Fuselage

Whereas the main function of the airplane wing is the formation of lift, it is the main function of the fuselage to provide space for the net load (payload). It is required, therefore, that the wing at given lift and the fuselage at given volume have the least possible drag. Consequently, the fuselage has, in general, the geometric shape of a long, spindle-shaped body, of which one dimension (length) is very large in comparison with the other two (height and width). The latter two dimensions are of the same order of magnitude. In Fig. 5-1, a number of idealized fuselage shapes are compared. In general, the plane of symmetry of the fuselage coincides with that of the airplane. The cross sections of the fuselage in the plane of symmetry and normal to the plane of symmetry (planform) have slender, profilelike shapes. The most important geometric parameters of the fuselage that are of significance for aerodynamic performance will now be discussed.

In analogy to the description of wing geometry, a fuselage-fixed rectangular coordinate system as in Fig. 5-1 will be used, with

x axis: fuselage longitudinal axis, positive in rearward direction у axis: fuselage lateral axis, positive toward the right when looking in flight direction

z axis: fuselage vertical axis, positive in upward direction

Figure 5-1 Geometric nomenclature for fuselages, (a) General fuselage shape, (b), (c) Fuselage teardrop with noncircular cross sections, (d) Fuselage teardrop with circular cross sections (axisymmetric fuselage). (<?) Fuselage mean camber (skeleton) line.

In general, it is expedient to place the origin of the coordinates on the fuselage nose. For axisymmetric fuselages, utilization of cylinder coordinates as in Fig. 5-Id is frequently preferable, where r stands for the radius and # for the polar angle.

— 8p

— bp

_ s**

— Op

The main dimensions of the fuselage are the fuselage length lF, the maximum fuselage width bFmax, and the maximum fuselage height hFmax (Fig. 5-1). The fuselage cross sections in the yz plane are usually oval-shaped (Fig. 5-lb and c). The simplest case is the fuselage with circular cross sections as in Fig. 5-1 d, with ^Fmax = ^Fmax=^FmMJ where dFmzx is the maximum fuselage diameter. From these four main dimensions, the following relative quantities can be formed:

The first three quantities are measures of the slenderness or fineness ratio of the fuselage. For the fuselage of circular cross section, 8F = 8p = 8p* and p = 1.

A more detailed description of fuselage geometry can be given by introducing the fuselage mean camber line. As shown in Fig. 5-1 a, this line is defined as the connection of the centers of gravity of the cross-sectional areas Ap{x). The line connecting the front and rear endpoints of the skeleton line is designated as the fuselage axis; it should coincide with the x axis. The fuselage skeleton line zF(x) as shown in Fig. 5-1 e lies in the fuselage symmetry plane. The largest distance of the skeleton line from the fuselage axis is designated as fp.

In analogy to the wing shape, Sec. 2-1, a general fuselage shape as shown in Fig. 5-Ій can be thought of as being composed of a skeleton line zF(x) on which cross sections Л^(х) are distributed. The body with this cross-section distribution is also termed a fuselage teardrop. In the case of noncircular cross sections of the fuselage, fuselage teardrops are characterized by the distributions hp{x) and bF(x) as in Fig. 5-1 b and c. In the case of circular fuselage cross sections, the fuselage teardrop is determined uniquely by the distribution of the radii R(x) (Fig. 5-1 d). The geometric parameters of a wing (teardrop and skeleton) can be selected first for the required aerodynamic performance. For fuselages this procedure is possible only to a very limited degree, because the fuselages must satisfy important requirements that may not be compatible with the aerodynamic considerations. For theoretical investigations on the aerodynamic properties of fuselages, the profile teardrops discussed in Sec. 2-1 are well suited.

The ellipsoid of revolution of Fig. 5-2a is a simple fuselage configuration for subsonic velocities. Another simple fuselage of axial symmetry that is used particularly for supersonic flight velocities is the paraboloid of revolution with a pointed nose as shown in Fig. 5-2b* To accommodate jet engines, fuselage configurations with blunt tails may be chosen. Among the design parameters not only fuselage length and diameter play an important role, but also fuselage volume and surface area. Volume and surface area of axisymmetric fuselages are given by


Figure 5-2 Special axisymmetric fuselages, (a) Ellipsoid of revolution. (b) Paraboloid of revolution.


SF = 2n J R(x) ds (5-1 b)


where s is the path length along the fuselage contour and Ip is the associated length of a meridional section measured on the fuselage contour.

Finally, a few data are given here for the volume of the ellipsoid of rotation and the paraboloid of rotation (Ip = !p0) of Fig. 5-2, respectively:

VF=llFAFmax (ellipsoid) (5-2 a)

Vp-Ysh^Fxmx (paraboloid) (5-26)

Here, lF is the fuselage length and Apm^x is the maximum fuselage cross-sectional area, also called the frontal area.

Wing of Finite Thickness. in Supersonic Flow

General statements In the previous sections, the inclined wing of finite span in supersonic flow was treated (lift problem). Now, the special case of a wing of finite
thickness with zero lift (displacement problem) will be discussed in more detail. Of interest here are the pressure distribution over the wing contour and the resulting wave drag. The latter is a strong function of the profile thickness, as was discussed for the plane problem in Sec. 4-3-3. The most general method of determining the pressure distribution of wings of finite thickness at zero lift is the source-sink method of von Karman [100]. The fundamentals of this method for the wing with supersonic incident flow were furnished in Sec. 4-5-3. The basic concept of this method is to cover the planform area of the given wing with a source distribution q(x, y) in the xy plane. From this, the x component of the velocity on the wing surface u(x, y) is obtained from Eq. (4-103) and the z component w(x, y) from Eq. (4-104). By describing the wing contour by z^x, y) = z(x, y), the kinematic flow condition is expressed by Eq. (3-173b). Introducing this into Eq. (4-104) yields the source distribution Eq. (3-176). Introducing this result into Eq. (4-103) furnishes the pressure. coefficient cp = —2и/иж as

~ [x’, 7/0 dx dy’ _______________ dx

І(х – –1[Ma^ – ЇМ*/ – УТ

Here, A’ is the influence range of the point x, y, as indicated in Fig. 4-58 by cross-hatching. The pressure distribution for a given wing contour z(x, y) can thus be determined.

Wave drag The coefficient of wave drag of the wing at zero lift is obtained through integration of the pressure distribution over the wing area A as


This formula is applicable to sharp-edged profiles only. The dependence of the drag coefficient on profile thickness ratio, taper, aspect ratio, sweepback angle, and Mach number of the incident flow is given according to the supersonic similarity rule by Eq. (4-27). This relationship is of great value for a systematic presentation of theoretical and experimental results.

Rectangular wing For the wing of rectangular planform and spanwise constant profile z(x, y) = z(x), introducing Eq. (4-124) into Eq. (4-125) and integrating twice yield (see Dorfner [15])


Note that, for A’ = As/MaL — 1 >1, the drag formula for the rectangular wing of finite span is identical to that of the rectangular wing of infinite span (see Table 4-2).

For a convex parabolic profile Z = z/c = 2bX( — X) with X = x/c, the integration yields

= 1 (A’ > 1)

(4-1276) where cDo* is given by Eq. (4-50a). The numerical evaluation is given in Fig, 4-94.

Delta wing A few results will be added for delta (triangular) wings. Delta wings with double-wedge profiles have been computed ‘ by Puckett [76], those with biconvex parabolic profiles by Beane [76]. Coefficients of the wave drag at zero lift for double-wedge and biconvex parabolic profiles of 50% relative thickness position are shown in Fig, 4-95 as a function of the parameter m = /MaL — Ы/4. For the double-wedge profile, cDQO* is expressed by Eq. (4-51). For supersonic leading edges

Figure 4-94 Drag coefficient (wave drag) at zero lift for rectangular wings at super­sonic incident flow vs. Mach number. Biconvex parabolic profile cd0«, from Eq. (4-5 Oc).

(m> 1), cD0/cDQoo is almost independent of Mach number, whereas it changes strongly with Mach number for subsonic leading edges (m < 1). Both curves have pronounced breaks at m = 1, that is, when the Mach line coincides with the leading edge. The curve for the double-wedge profile has another break at m = , that is, when the Mach line is parallel to the line of greatest thickness.

In Fig. 4-96, a number of measurements on delta wings with double-wedge profiles and 19% relative thickness position are plotted from [56]. Similar to Fig. 4-86, different representations have been chosen for m < 1 and m> 1. At the kind of presentation chosen here, these measurements on 11 wings at Mach numbers Ma„ = 1.62, 1.92, and 2.40 fall very well on a single curve. Hence, the supersonic similarity rule of Eq. (4-27) has been confirmed again. The theoretical curve from




Puckett [76] for the relative thickness position X*=0.18 shows a high peak at m = 1 that is not confirmed by measurements, as would be expected because the incident flow velocity at the leading edge is just sonic. Comparison between theory and experiment suffers from the uncertainty in the determination of the friction drag, which has to be subtracted from the measured values.

The treatment of the thickness problem of a delta wing with sonic leading edge has been compared with transonic flow theory by Sun [93].

Swept-back wing The wave drag coefficients of swept-back wings of constant chord are illustrated in Fig. 4-97. The corresponding information for the lift slope was given in Fig. 4-88. The wing has a double-wedge profile, of which the drag coefficient in plane flow cDo is obtained from Eq. (4-51). The curves show a pronounced break at m = 1, that is, when the Mach line and leading edge fall together. It should be noted that, according to [15],

-02- = —— for m > 1 + —(4-128)

Cdqoo iml — 1 /loot у

is obtained in the range of the supersonic leading edge if the Mach line originating at the apex (line g) intersects the trailing edge.

Arbitrary wing planforms To conclude this discussion, the total drag coefficient at zero lift (wave drag + friction drag) of the three wings (trapezoidal, swept-back, and delta) treated earlier (Figs. 4-89-4-91) is plotted in Fig. 4-98 against the Mach number. These three wings have double-wedge profiles with a thickness ratio t/c = 0.05 and an aspect ratio A = 3. Within the Mach number range presented, the

Figure 4-98 Total drag coefficient (wave drag 4- friction drag) vs. Mach number for a trapezoidal, a swept-back, and a delta wing of aspect ratio л = 3. Double-wedge profile tjc = 0.05, xtjc =

0. 50, from [21].

wave drag is two to three times larger than the friction drag. The latter has been determined from Fig. 4-4 for Reynolds numbers Re ^ 107. Since the wave drag at supersonic incident flow is proportional to (t/c)2, this contribution, and thus the total wing drag at zero lift, can be reduced considerably by keeping t/c small. This fact is taken into account in airplane design by choosing extremely small thickness ratios for supersonic airplanes; compare Fig. 3-4a.

Concluding remarks In addition to the references included in the text, attention should be directed toward summary reports and reports dealing with various theories on the aerodynamics of the wing in supersonic flow [6, 11, 19, 22, 23, 40, 51, 92, 105-107] – The special case of the aerodynamics of the wing of small aspect ratios, first studied by Jones [37], has been investigated comprehensively as the “slender-body theory” for both lift and drag problems [2, 13, 14, 41, 108]. The aerodynamics of slender bodies is treated in Sec. 6-4. The influence of vortex shedding at the lateral wing edges of rectangular wings, and the leading-edge separation on swept-back and delta wings at supersonic flow, are treated in [12, 72, 91], based on the understanding of incompressible flow. Based on a suggestion of Jones, questions concerning the minimum wing drag have been investigated by several authors [36, 61, 97, 110]. In this connection, the investigations on the design aerodynamics of wings at high flight velocities, promoted mainly by Kuchemann, play an important role [9, 38, 46, 60].

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.