6- 3-1 The Wing-Fuselage System in Subsonic Incident Flow

Fundamentals The following discussions on the flow about a wing-fuselage system at subsonic velocities will be limited to the case of straight flight. The effect of

Figure 6-28 Side force due to sideslip vs. lift coefficient. Measurements from Moller (system as in Fig. 6-27).

compressibility on the flow about a wing has been explained by means of the Prandtl-Glauert-Gothert rule in Sec. 4-4-2 for wings, and for fuselages in Sec. 5-3-2. This rule renders feasible the determination of a subsonic flow {Маж < 1) about wings and fuselages by means of a transformation to incompressible flow. By this means the incompressible flow will be computed for a transformed wing and a transformed fuselage. The transformation of the geometric quantities for wing and fuselage is given by Eqs. (4-66), (4-67a), (4-67Z?), (4-68д)-(4-68с), (5-51), (5-52<z), and (5-52b), where the quantities for incompressible flow are marked by the index “inc” and those for the compressible flow are given without the index. These quantities are as follows:

By computing the incompressible flow for the transformed wing-fuselage system at the angle of attack of the compressible flow, the transformation of the pressure coefficient from Eq. (4-69) becomes

cp = i-^—f-Cpinc C«inc=«) (6-32)[30]

Through an analogous transformation, the lift coefficients and the pitching-moment coefficients of wing-fuselage systems are obtained.

The discussions about the incompressible flow over wing-fuselage systems of Sec. 6-2-2 led to the conclusion that the lift slope of the wing-fuselage system is little different from that of the wing alone if the relative width of the fuselage is small to moderately large. Consequently, the relationship Eq. (4-74) for the wing alone applies directly to the wing-fuselage system, or

‘dcL_____________ 2rtA_______

У(1 – Mat)A~ + 44-2

The dependence of the neutral-point position on the Mach number of a wing-fuselage system follows immediately from the relationships just stated, because xN/c = —dcM/dcL, as

XN(W+F) XN(W+F) inc

~c " c (6’34)

Here xn(w+F) inc is the neutral-point position of the wing-fuselage system at incompressible flow as transformed according to Eqs. (6-29)-(6-31). As an example, the lift slopes and neutral-point positions are presented in Fig. 6-29 against the Mach number. These measurements of Schneider [42] are compared with theoretical results. The lift slope is little affected by the fuselage; the neutral-point displacement, however, shows a considerable fuselage influence. Results for wing-fuselage systems with a rectangular and a delta wing are also available in [42].

Investigation of the wing-fuselage system by means of the panel method As a result of utilizing efficient computers, methods have become more useful that are based on singularities distributed on the body surface, thus satisfying exactly the kinematic boundary conditions. Such generalizations have the advantage that geometric restrictions in the body shape are essentially eliminated. Based on the computational procedure for the displacement flow of Smith and Hess [13], the simultaneous treatment of the displacement flow and the lift flow of wing-fuselage systems has been presented independently by Kraus and Sacher [22] and Labrujere et al. [25]. In the method of Kraus and Sacher, the displacement flow is generated through a

Figure 6-29 Lift slope (a) and neutral-point shift (b) due to the fuselage effect vs. Mach number for axisymmetric fuselage with swept-back wing. Measurements from Schneider; theory for

incompressible flow from Sec. 6-2-2. (o———– o) Wing alone, л =2.15, A = 0.5, ^=52.4°.

(?——– v) Wing and fuselage, Ipjd = 12.5, ejd = 7.25, b/d = 5.0. — ■ — • — ■ a) Wing and

fuselage, lp/d = 10, ejd = 6.5, bid = 3.33.

source-sink distribution on the surface of the body, the abrupt change of the potential of the circulation flow through a vortex distribution within the body. The total potential results from the superposition of the individual contributions. The defining equations for the as yet unknown singularities are established by means of the kinematic boundary condition, to be satisfied on the surface. The flow conditions for the displacement problem are expressed by the requirement that no flow is to penetrate the body surface, that is, that the velocity normal to the body surface is zero. For the lift problem (circulation flow), this condition requires smooth flow-off at the trailing edges of the lift-producing surfaces. The distribution of the singularities and the perturbation potentials are obtained from the solutions of the equations defining the singularities. Thus the total potential and the velocities and pressure coefficients are obtained in the entire flow field and on the body surface. The pressure coefficients are computed with all three of the components of the perturbation velocity.

A suitable approach to the solution of the defining equations is found in the panel method. The singularities, first assumed to be distributed continuously, are now assumed to be constant on small flat surfaces (panels) and thus are accessible to analytic integration of their defining equations. For the displacement flow, panels covered with a constant source-sink density as in Fig. 6-30a are distributed on the surface. For the circulation flow, panels on the inner surface are used, and on the edges of these panels a vortex filament is laid of constant vortex strength. This leads, as in Fig. 6-3Ob, to the well-known picture of lifting surfaces consisting of vortex ladders composed of individual horseshoe vortices forming elementary wings.

On each surface panel, carrying a singularity density assumed to be constant, a control point lies in its center of gravity at which the kinematic flow condition is to be satisfied. Hence there exist as many control points as surface panels with singularity densities individually assumed to be constant but as yet unknown in magnitude. To each vortex ladder (consisting of several inner panels with a vortex filament of constant circulation strength on their edges) a control point is assigned at the trailing edge of the wing in which the Kutta condition is satisfied. Thus there are as many Kutta control points as vortex ladders with unknown total circulation strength (the circulation distribution within each of the vortex ladders on the individual panels is assumed a priori to be a Bimbaum distribution).

The requirement of the defining equation that the kinematic boundary condition has to be satisfied at all control points for all perturbation potentials of all panels leads to a system of linear equations of a form similar to that derived for the lifting-surface method (see [22]). This, in most cases very extensive, system of linear equations is solved through iteration by means of a Gauss-Seidel procedure. By this means the singularity strength and thus the velocity and the pressure distribution at the control points are obtained. A typical result of a computation of a wing-fuselage system is shown in Fig. 6-31 by means of the pressure distributions on a few selected fuselages and wing sections in comparison with measurements by Schneider [42]. The effect of compressibility has been taken into account through the Gothert rule (see [22]). Further methods of general validity for the computation of the aerodynamics of wing-fuselage systems at subsonic flow have been developed by Woodward [52], Giesing et al. [9], and Korner [21].

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