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).

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