In the VLM, the finite wing again is represented by N flat panels as defined by a preliminary grid generation. In the following discussion, it is assumed that thickness effects are negligible. What follows, then, is a so-called lifting-surface theory, although the VLM may be applied to wings with nonzero thickness. The wing is represented by a camber surface and the tangency-boundary condition is applied on the camber surface rather than on the wing surface. The wing may have arbitrary camber and planform. The angle of attack is assumed to be small; therefore, this inviscid theory describes a thin wing with negligibly small regions of separation.
Again, vortex singularities are distributed over a surface. However, in contrast to the VPM described herein, in the VLM discussion, each panel is assigned a horseshoe vortex rather than a vortex ring. The placement of the transverse element of the horseshoe vortex and the control point is suggested by the following: Consider a wing panel as a flat plate in an effectively two-dimensional flow. Such a flat plate of chord c is shown at the angle of attack in Fig. 6.22.
Represent the lift on this plate by a combined vortex of strength Г located at c/4, which is the center of pressure for an airfoil according to thin-airfoil theory. From the Biot-Savart Law, at some distance, h, along the plate, the vortex induces a velocity V=r/2nh. Now, we recall from Eq. 5.14 that the circulation around a two-dimensional flat-plate airfoil is given by Г = nmVTC. Finally, the tangency-boundary condition (i. e., no flow through the surface) requires that V must be equal and opposite to a component of freestream velocity given by VM(sin a) = V^a. Appealing to tangency:
or, solving for h:
Figure. 6.22. Two-dimensional flat plate at the angle of attack.
Thus, with the transverse-bound vortex on a panel at c/4, the tangency-boundary condition is satisfied at one point, c/4 + h = 3c/4. We choose to locate the control point on each panel at this location and we apply the tangency-boundary condition at this control point. If the wing leading edge is swept, the bound vortex is skewed at the sweep angle relative to the у-axis. The bound vortices have different (i. e., unknown) strengths that vary in both a spanwise and chordwise direction.
Two free-vortex filaments always trail downstream from the ends of the transverse element on the panel (Fig. 6.23). Each pair of (free) trailing-vortex elements must have the same strength as the bound vortex from which it originates (i. e., Helmholtz). The free-vortex elements cannot support any pressure difference, so they must trail off of the bound vortex and away from the wing in a direction parallel to the local streamlines; that is, they must follow a curved path. However, because the wing angle of attack is small the free vortices may be assumed to follow a straight line at the freestream or another convenient direction. If desired, the trailing vortices may be divided into straight-line segments so as to better model the physical reality and, if desired, may follow the wing surface until the trailing edge is reached. As in the lifting-line theory, the influence of the starting vortex portion of each horseshoe at the wing is ignored as being negligible.
Carefully compare Figs. 6.20 and 6.23 to understand better the difference between the vortex-panel and the vortex-lattice formulation of the finite-wing problem. In particular, contrast the modeling of the trailing vortices. In the VPM, the trailing-vortex pairs behind the wing have the same strength as the bound vortex on the trailing-edge panel from which they originated. In the VLM, the trailing-vortex pairs have the same strength as the bound vortex on the individual chordwise panel from which they originated.
A control point is located at the three-quarter chord of each panel and the Biot-Savart Law is used to calculate the velocity induced at each control point by all of the other horseshoe vortices. The bound vortex located on the panel that contains the particular control point in question contributes to the induced velocity at that control point because the point of interest and the bound vortex do not coincide, as in the lifting-line theory. Thus, each horseshoe vortex contributes three velocity
Figure 6.23. VLM.
components at each control point, and the induced velocities may be computed by using Eq. 6.2. The induced-velocity components are normal to a plane containing the control point and the vortex element, so that all of the induced velocities finally may be assembled into the resultant velocity, Vp, which is perpendicular to the panel in question. Thus, the resultant normal velocity, induced at any panel, s, due to all of the horseshoe vortices located on that panel and on every other panel is given by:
4 = Vi| )sn. m (<5.66)
where each horseshoe vortex makes three contributions; one due to the bound vortex and two due to the free vortices.
The resultant induced velocity at each panel control point must be equal and opposite to the normal component of freestream velocity at each control point if the tangency-boundary condition (i. e., no flowthrough) is to be satisfied. The result is a set of N simultaneous linear-algebraic equations, where N is the number of panels. There is only one unknown vortex strength associated with each panel because the free vortices trailing from each panel have the same strength as the bound vortex associated with that panel. The array of N simultaneous equations then can be solved for the N unknown values of Г.
As in the VPM there is significant geometry for the computer code to handle; however a general code may be written to describe the location of the control points and the horseshoe vortices, and the calculations are repetitive. With the bound-vortex
Under this assumption, the wake-vortex sheet has a small pressure difference across it (not physically correct because the sheet does not have the correct physical shape). However, under this assumption, the wake sheet contributes zero drag because the wake vorticity is parallel to the freestream. In this case, the induced drag of the wing from which the wake originates can be evaluated by integration of the wake properties in the two-dimensional flow occurring in a cross plane far downstream of the wing and perpendicular to the wake sheet. This plane is called the Trefftz plane.
A detailed numerical example of the VLM as applied to a swept wing with zero thickness and camber (i. e., a flat plate) is in Thomas, 1976. Close study of this example is of great assistance if the student must write a VLM program. Results of the VLM as applied to a rectangular wing and a comparison with experiment, are shown in Fig. 6.24. In this figure, the wing is not taken to be of zero thickness, as discussed, but rather is modeled by a combination of sources and vortex lattices. The predicted surface-pressure distribution on the wing at two spanwise stations and the predicted induced drag of the wing shows excellent agreement when compared with experimental data.
(b) Source and vortex-lattice pressure
coefficients on a wing
(c) Induced drag for Lockheed ATT-95 aircraft
Figure 6.24. VLM compared with experimental data (Thomas, 1976).
The panel methods described herein are valid only for inviscid, incompressible flow. Furthermore, the solutions are found only on the wing or camber surface. If flow details away from the wing are desired within these two assumptions, then the methods can be extended to compute the flow induced by the flow singularities and the freestream.