**1. Introduction**

An analysis of a lift-curve slope for wing-fuselage combinations currently plays an important role in studies of aerodynamics and the preliminary design of a modern aircraft.

Since the aircraft occurrence aircraft designers have been interested in the problems of the wing-body interference in aviation and missile technology. Initially, research is focused on the experimental study of specific wing-body combinations [1–6]. First mathematical models of the wing-fuselage interference were offered later. The solution of the linearized problem of the ideal incompressible flow around arbitrary shape wings in the presence of the fuselage is a difficult task since it is necessary to solve the three-dimensional Laplace equation for the velocity potential which satisfies the boundary conditions on the surface of the wing-body combination and the boundary conditions at infinity. One of the few exact solutions was obtained by Golubinsky in the article [7]. The first theoretical calculations were based on the inversion of discrete vortices inside the cross-section body [8], on the solution of integral equations [9], on the application the thin body theory [10–21] or the strip method [14, 15, 22], and on the application of the velocity potential [23–26] or the stream function [27] written in the Trefftz's plane. The application of the velocity potential in the Trefftz's plane [22–26, 28–30] gives the opportunity to get the distribution of lift along the wingspan. Let us pay attention to one important result that was first theoretically obtained by Multhopp and presented in a review by Ferrari [22]. It is important to note that this fact was experimentally confirmed by Jacobs and Ward [1]. This result shows that the value of the lift-curve slope of the wing-body combination at a certain relative value diameter of the fuselage *D* ¼ *df =b* (*b* is the wingspan; *df* is the diameter of the fuselage) exceeds the same value for the isolated wing of the same geometry which is used in the wing-fuselage configuration. Let us give some examples confirming this fact. For the scheme of midwing monoplane with cylindrical fuselage *D* ¼ 0*:*14 and trapezoidal wing with aspect ratio *AR* ¼ 4*:*83 and taper ratio *λ* ¼ 2*,* 38, the value of the lift-curve slope of the wing-body combination *CLα*W*,* <sup>B</sup> exceeds the same value *CLα<sup>W</sup>* for the isolated wing of the same geometry which is used in the wing-body configuration by approximately 5% (4.75%, an experiment; 4.92%, theory) [22]. For the wing-body combination No. 13 in the experimental Jacobs and Ward's paper [1], the relative increase in the value of the lift-curve slope of the wing-fuselage combination was slightly greater (≈6.49%) than the isolated wing of the same geometry, which was obtained also. In Korner's book [6], it was noted that the value of the lift-curve slope of the wing-body combination is approximately 5% higher than the same value for the wing alone. In the theoretical papers [31, 32], the excess value of the lift-curve slope of the wing-body combination above the same value for the isolated wing of the same geometry was also noted. In paper [32] the midwing monoplane scheme has received an increase in the value of the lift-curve slope of the wing-fuselage combination compared with the isolated wing with the same geometry approximately 19%. Theoretical results of the calculation value of the lift-curve slope of the wing-body combination are devoted also in papers [33–35] and book [36]. Woodward in papers [37, 38] investigated the aerodynamic characteristics of wing-fuselage combinations using the panel method. The same panel method was used in the paper [39]. An experimental study of wingbody-tail combinations was performed in the work [40].

This chapter by no means covers all papers on the interference of the wing and fuselage. Author's book [41] and paper [42] contain more detailed bibliography on the problems of the lift of the wing-body combination.

proposed each console part of the wing replaces one Π-shaped vortex lying in the plane of the wing. The Π-shaped vortex in the left-wing console is shown in **Figure 1**. The coordinate of the free vortex and its intensity can be found from the bond equation; after the lift-isolated wing by DVM will be defined. The inversion method (**Figure 2**) can be used to satisfy the boundary conditions of impermeability on the surface of the body cross section for the canonical body, and for the arbitrary two-dimensional cross section can use the panel method. An example solution for the potential flow around the elliptical cross section of the fuselage in

*The mathematical model of the potential flow around the elliptical cross section of the fuselage in the present*

In this formulation, the problem is reduced to solving the following system of

where *L* is the number of control points (collocation points) equal to the number of

� �*, j* <sup>¼</sup> <sup>1</sup>*,* …*L,* (1)

� � ¼ � **<sup>F</sup>***<sup>j</sup>* � **<sup>n</sup>***<sup>j</sup>*

attached vortices on the right-wing console, **n***<sup>j</sup>* is the unit normal vector to the *j*th control point on the surface of the wing, A*ij* is the matrix of the aerodynamic influence or the matrix of the induced velocities at the control points of the wing surface from all system of horseshoe vortices (left and right consoles for the isolated wing), and **F***<sup>j</sup>* is a column vector of the velocity induced in the *j*th the control point on the wing surface by incoming flow and the flow from the cross section of the fuselage that includes

the present of the pair vortices is shown in **Figure 3**.

**Γ***<sup>i</sup>* A*ij* � **n***<sup>j</sup>*

X *L*

*i*¼1

algebraic linear equations:

**Figure 1.**

**Figure 2.**

**83**

*pair of vortices.*

*The mathematical model of wing-body interference.*

*Optimization of Lift-Curve Slope for Wing-Fuselage Combination*

*DOI: http://dx.doi.org/10.5772/intechopen.89056*

The main purpose of this paper is to give results of solving optimization problems for the values of the lift wing-body configurations and to demonstrate the conformity of computational author's results with the known experimental and theoretical results of other authors.
