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)

d*(R2)

d2 (B2) dx2

‘d?(R2) 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)

U CO

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)

where

2

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].