Optimization of Lift-Curve Slope for Wing-Fuselage Combination

The paper presents results obtained by the author for wing-body interference. The lift-curve slopes of the wing-body combinations are considered. A 2D potential model for cross-flow around the fuselage and a discrete vortex method (DVM) are used. Flat wings of various forms and the circular and elliptical cross sections of the fuselage are considered. It was found that the value of the lift-curve slopes of the wing-body combinations may exceed the same value for an isolated wing. An experimental and theoretical data obtained by other authors earlier confirm this result. Investigations to optimize the wing-body combination were carried within the framework of the proposed model. It was revealed that the maximums of the lift-curve slopes for the optimal midwing configuration with elliptical cross-section body had a sufficiently large relative width (more than 30% of the span wing). The advantage of the wing-fuselage combination with a circular cross section over an isolated wing for wing aspect ratio greater than 6 can reach 7.5% at the relative diameter of fuselage equal to approximately 0.2.


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][2][3][4][5][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][11][12][13][14][15][16][17][18][19][20][21] or the strip method [14,15,22], and on the application of the velocity potential [23][24][25][26] or the stream function [27] written in the Trefftz's plane. The application of

The calculation method of the interference for wing-fuselage combination
The calculation method of the interference for the wing-fuselage combination [42] includes two methods: (1) a discrete vortex method (DVM) for the surface of the wing and (2) 2D potential model of the flow for cross-flow around fuselage [41].
The original three-dimensional problem ( Figure 1) is divided into two parts. First part is the two-dimensional problem of the flow around the cross section of the fuselage (Figure 2), and the second part is a three-dimensional problem for the isolated wing. In the 2D problem, the flow around the cross section of the fuselage adds a pair of discrete point vortices. The added vortices are the consequence of lift on the wing. According to Zhukovsky's theory about the lift of the wing, any lifting surface can be replaced by an equivalent Π-shaped vortex; free vortices at low angles of attack lie in the plane of the wing and extend to infinity. In our model, it is 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 present of the pair vortices is shown in Figure 3.
In this formulation, the problem is reduced to solving the following system of algebraic linear equations: where L is the number of control points (collocation points) equal to the number of attached vortices on the right-wing console, n j is the unit normal vector to the jth 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 j is a column vector of the velocity induced in the jth the control point on the wing surface by incoming flow and the flow from the cross section of the fuselage that includes  For small angles of attack of the wing-body combination α ≪ 1 ð Þand small wing deflection angle (angle of inclination wing), δ ≪ 1 ð Þcan use the linear formulation, and then a solution can be written as a linear function of the angle of attack and wing deflection angle: Right parts of the system of algebraic linear equations Eq. (1) can be represented also as In Eqs. (2) and (3), Γ α i , Γ δ i , F α j , F δ j are derivatives of the Γ i , F j on the angle of attack α and wing deflection angle δ, respectively.
For calculating the right parts of the system Eq. (1) for the problem with the fuselage of an arbitrary cross section of the body, the panel method that leads to the solution system of algebraic linear equations (4) is proposed: where σ is a column vector Let us give the final formula for the components of the induced velocity, for example, for the case of the circular cross section of the fuselage in jth control point of the wing panel: where y j , z j ,ỹ vz v À Á are coordinates of the control point and the inversion vortex point (see Figure 1), respectively, and V nj is a normal component of the velocity to the surface at the jth control point wing panel induced velocity component along the OZ-axis of the cylinder in cross-flow (see Figure 1). Coordinate inversion vortices are defined by Milne-Thomson's theorem about the circle [43]. The coordinate y t and intensity Γ t of the free vortex can be found on the connection equations [41,42,44]. So the task of the wing-body interference is reduced to the solution Eq. (1) with the right part Eq. (6) or right-hand parts, obtained by solving the system (4) that provides the solution of the problem for the potential flow around an arbitrary contour of the panel method. The method of the successive iterations provides an agreement of the velocity field on the surface wing and the surface fuselage. Each iteration is reduced to the solution of systems of linear algebraic equations (1) with corrected right part Eq. (6). The zero iteration can select the solution for the isolated wing. For small angles of attack and wing deflection angle, the proposed model or the linear formulation allows to get the solution of two problems at once, which can be called αα-problem (fuselage and wing have the same angle of attack, angle of the wing deflection angle equal to zero) and δ0-problem (the fuselage has a zero angle of attack, and the wing has deflection angle δ not equal to zero). For this linear case, formulas for the coefficients of the normal forces of the wing and the body are of the form Eq. (7) where values C L δ W B ð Þ , C L δ B W ð Þ , C L α W B ð Þ , C L α B W ð Þ are obtained from the solution δ0and αα-problem, respectively.

Calculation results
The comparison of the calculation results obtained from the above theoretical model with calculations by the DVM for case αα-problem [34][35][36] is shown in Figures 4-9. The rectangular, triangular, and swept wings were considered. It may be noted is enough good agreement of calculated data.
The changing of coordinates of the aerodynamic center x AC /c measured from the beginning of the mean aerodynamic chord is also shown in Figures 4, 6, and 8. The coefficient of the interference K Σ in these figures is defined by the formula where value C L α W is a lift-curve slope for an isolated wing composed of two consoles of this wing.
The comparison of the calculation results obtained from the above theoretical model with calculations by the numerical method of singularities for case δ0-problem [31] is presented in Figure 10.
The comparison of calculated data for the mathematical model described above and the calculated and experimental data of other researchers [45][46][47][48][49] is shown in Figures 11-13.    Figure 14 shows an influence of compressibility on the values of theoretical liftcurve slopes for case midwing monoplane combination with the rectangular and delta-shaped wing. Figure 15 also shows an influence of compressibility on values of theoretical lift-curve slopes for case high-wing monoplane combination with the rectangular and delta-shaped wing [33].   Of particular interest is the comparison of calculated and experimental data to prove that the lift-curve slope for the wing-body combination exceeds this value for an isolated wing. Figure 16 shows this comparison.
The area shown in color in Figure 16 indicates the advantage of the lift-curve slopes of the wing-body combinations over an isolated wing. Calculations and experiments show that with an increasing aspect ratio of the wing, this advantage will increase. This circumstance is important since the modern development of the aircraft industry tends to increase the aspect ratio of the wing. Another conclusion is that the maximum of the lift-curve slopes with a wing aspect ratio of 6 is achieved at relative fuselage diameter of approximately 0.2. Such a relative diameter of the fuselage allows the design of modern aircraft with a wide fuselage. Numerical studies have shown that with increasing aspect ratio of the wing and the ratio of the width to the height of the fuselage elliptical cross sections, the advantage of liftcurve slopes of the wing-body combinations over isolated wings becomes larger. The noted facts allow us to formulate and solve an optimization problem.

The formulation of the optimization problem
Note that in some theoretical and experimental papers devoted to the wing-body interference revealed a maximum dependence ∂C L =∂α ¼ f d f =b À Á : Our calculations on the above mathematical model also confirm this fact. It was found that the maximums of lift-curve slopes for a wing-body combination depends on the shape of the wing and the cross-section shape of the fuselage. The paper presents solutions to the optimization problem for the wing-body combinations with unswept trapezoidal wings and circular or elliptical cross sections.  We will use the formulation of the optimization problem as a nonlinear programming problem as follows: where X ¼ x 1 , x 2 , 0 ½ T is a vector of the project parameters connected with geometrical characteristics of the wing-body configuration by formulates where x 1 x 2 are auxiliary variables. The problems Eq. (9) and (Eq. (10) are a problem of unconditional optimization, for which there are D ∈ 0;1 ½ , 1=λ ð Þ∈ 0;1 ½ and x 1 ∈ À∞; þ∞ ½ , x 2 ∈ À∞; þ∞ ½ .

5.
Results of the optimization problem for lift-curve slope for midwing-body monoplane configuration Figures 17 and 18 show results of the optimization problem for lift-curve slope for midwing-body monoplane configuration with circular cross-section fuselage vs. the aspect ratio of the rectangle wing. In Figure 16, the notation is used: where C L α W, B is the lift-curve slope of the wing-body combination the same as in Eq. (9) and C L α W is a lift-curve slope of the isolated wing in which it is included part of the occupied fuselage. Figure 18 shows the results of the solution of the optimization problem for liftcurve slopes for midwing-body monoplane configuration with elliptical crosssection fuselage. Maximum values of the lift-curve slopes depend on the aspect ratio of the rectangular wing and the ratio of the axes of the ellipse. Figure 18 shows that the advantage of the wing-fuselage combination over an isolated wing is enhanced with increasing the aspect ratio of the rectangular wing and with increasing the ratio of the axes of the cross-section fuselage. The optimal ratio of the width of the body to the span of the wing can reach 30% and more! Figure 19a shows the effect of the compressibility and the statistics for modern aircraft also (Figure 19b). Red color point shows the project of fifth-generation aircraft (project M-60, Russia). The feature of the project M-60 is a wide fuselage. As can be seen from Figure 19b, with the aspect ratio wing equal to 15, the optimal ratio of the width of the circular cross section to the wingspan can reach 20%!

Conclusions
The paper presents results obtained by the author for wing-body interference. The lift-curve slopes of the wing-body combinations are considered. A 2D potential Figure 17. The optimal relative diameter of the fuselage with circular cross-section body for the midwing configuration vs. the aspect ratio of the rectangular wing. Figure 18. Maximums of the lift-curve slopes for the optimal midwing configuration with elliptical cross-section body vs. the aspect ratio of the rectangular wing. model for cross-flow around the fuselage and the discrete vortex method for the wing were used. Flat wings of various forms and the circular and elliptical cross sections of the fuselage are considered. It was found that the value of the lift-curve slopes of the wing-body combinations may exceed the same value for an isolated wing. An experimental and theoretical data obtained by other authors earlier also confirms this result. Investigations to optimize the wing-body combination were carried within the framework of the proposed model. The proposed mathematical model for the solution optimization problem for the wing-body combination allows selecting the optimal geometric parameters for configuration to maximize the values of the lift-curve slopes of the wing-body combination.
It was revealed that the maximums of the lift-curve slopes for the optimal midwing configuration with elliptical cross-section body reach their values at sufficiently large relative width of the body (more than 30% of the span wing!). The advantage of the wing-fuselage combination with a circular cross section over an isolated wing at the wing aspect ratio greater than 6 can reach 7.5% at the relative diameter of fuselage equal to approximately 0.2. The advantage of the wingfuselage combination with the elliptical cross section with the ratio of axes of the body equal to 2.5 over an isolated wing with aspect ratio equal to 12 is that it can reach 29% at relative width of fuselage equal approximately to 0.35!