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),

Coordinates:	xinc = x fine = r Vl —Mat, djnc = $	(5-51)
Fuselage radius:	Riric=R/l-Mal	(5-52c)
Fuselage length:	ІР’тс ~ іF	(5-52h)
Thickness ratio:	^Finc = 6^7 v/l МаЪ>	(5-52c)
Angle of attack:	Qinc = a Vl —MaL	(5-52 d)

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

(5-53)

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

(5-54)

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

(5-56)

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)

о

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)

(5-64)

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.

(5-65)[27]

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

1 277*

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.

(5-72)*

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

(5-73)

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.