Doublet-Lattice Method

The doublet lattice method is based on a linear theory using a numerical approach to study the subsonic three dimensional flows past complex lifting surfaces (Albano and Rodden 1969). In this method as lifting surfaces, the wing, the horizontal and vertical tails are subdivided into trapezoidal surfaces (the parallel sides of the trapezoids are in line with the free stream direction) to discretize the flow domain. In addition, if there is a tank or a store as an external type body, its surface is also subdivided into trapezoids during discretization. In order to dis­cretize a surface on each panel, a doublet line is placed at quarter chord of the panel and a control point is assigned at three quarter chord point as shown in Fig. 5.6. On the doublet line the unknown but constant value of the doublet strength is considered and at the control point known downwash value is assigned. This way the Kutta condition is satisfied numerically. As seen in Fig. 5.6, the wing surface is divided into n panels where the information about locations of the doublet line and the control points are used to obtain the algebraic equations from Eq. 5.37 using numerical integration.

The downwash, wi, induced on the control point of the ith panel by the doublet lines of the panels j = 1, 2, 3,…,n, shown in Fig. 5.6, can be expressed as follows

n

w = jj (5-51)

j=i

Here, Dij is written in terms of the numerical integrals over the Kernel function. In order to perform the numerical integrals for each penal we have to transform the local coordinates (П, g, f), into the global coordinates (x, y, z), as shown in Fig. 5.7.

I

According to Fig. 5.7 tangent angle ys reads as

g = y cos ys + z sin ys 1 = —y sin ys + z cos ys

Using the geometry of the panel shown in Fig. 5.7, we can obtain the coefficient matrix Dij of Eq. 5.51 as follows. First we determine the coordinates of the control point, R = (xR, yR, zR), then the doublet line coordinates as inner left, si, middle: sm, outer right, so, coordinates are considered. If there is a spanwise curvature for the wing the local coordinates from the global ones read as

g0 = (yR — ySm) cos ys + (zr — ZSm) sin Js

10 = -(yR — ysm) sin ys + (zr — ZSm) COs ~fs

r = g 0 + l0

If the doublet line length is j and the angle between the doublet line and the span is kj then let us define e = 1/2 lj coskj. The value of the Kernel function in terms of new coordinates becomes

j = rK and Km = j(R, Sm), Ki = k(R, Si), K0 = k(R, S0).

If the panel length is small enough the parabolic change of the Kernel function will give us sufficiently accurate approximation. Accordingly, along the doublet line we have the following approximate integral to represent the Kernel integration with respect to di

Here, A, B and C respectively read as

A =(Ki — 2 Km + K0)/2e2, B =(K0 — Ki)=2e, C = Km

The resulting

With all these, we have given the necessary information for the ‘doublet-lattice’ method to be applied numerically. This method is applicable to wing-tail, wing – external store and wing-fuselage interaction problems as well as the wings with curved spans.