Computational Models
Panel Methods
Many computational models and analysis methods are based on linear three-dimensional
potential flow theory. These are discussed in the overview of
panel methods in an earlier chapter.
In this section we take a look at the simplest panel method in more detail.
Weissinger Method
Weissinger theory or extended lifting line theory differs from lifting line
theory in several respects. It is really a simple panel method (a vortex
lattice method with only one chordwise panel), not a corrected strip theory
method as is lifting line theory. This model works for wings with sweep
and converges to the correct solution in both the high and low aspect ratio
limits.
The version of this model used in the Wing Design program is actually a
variant of Weissinger's method: it uses discrete skewed horseshoe vortices
as shown.
Each horseshoe vortex consists of a bound vortex leg and two trailing vortices.
This arrangement automatically satisfies the Helmholtz requirement that
no vortex line ends in the flow. (The trailing vortices extend to infinity
behind the wing.)
The basic concept is to compute the strengths of each of the "bound"
vortices required to keep the flow tangent to the wing surface at a set
of control points.
If the vortex of unit strength at station j produces a downwash velocity
of AICij at station i, then the linear system of equations representing
the boundary conditions may be written:
where {alpha} represents the angle of incidence of the sections along the
span (assumed a flat plates). If the section has camber, the local angle
of attack is taken as the angle from the zero lift line of the section.
The linear system of equations to be solved may also be written in terms
of the angle of attack at the wing root and the twist amplitude. For wings
with a linear distribution of twist (washout):
where:
{ar} is a vector containing the root angle of attack as each element
{y} is the spanwise coordinate, varying from 0 at the root to b/2.
{q} is the total twist (washout) in the wing from root to tip
Thus, the wing circulation distribution can be written as the sum of two
distributions:
Since the section lift (lift per unit length along the span) is related
to the circulation by:
The lift distribution can be expressed as:
where l1 and l2 are independent of the incidence angles and depend only
on the planform shape of the wing.
Since the lift coefficient of the wing, CL, is linearly related to the
angle of attack we can also write the lift distribution in the following
form:
The first term is known as the additional lift distribution and the second
term is called the basic lift distribution. They scale linearly with the
wing lift coefficient and the twist angle respectively. Additional information
on basic and additional lift distributions is available in the section on
wing design.
An interactive computation
based on this idea is available on the internet. Use it to investigate
the effect of wing shape on lift distributions or to design wings as discussed
in the following sections. The source
code is available as well.