Page 217 - Aerodynamics for Engineering Students

200 Aerodynamics for Engineering Students

4.10 Computational (panel) methods
for two-dimensional lifting flows

The extension of the computational method, described in Section 3.5, to two-
dimensional lifting flows is described in this section. The basic panel method was
developed by Hess and Smith at Douglas Aircraft Co. in the late 1950s and early
1960s. The method appears to have been first extended to lifting flows by Rubbert*
at Boeing. The two-dimensional version of the method can be applied to aerofoil
sections of any thickness or camber. In essence, in order to generate the circulation
necessary for the production of lift, vorticity in some form must be introduced into
the modelling of the flow.
It is assumed in the present section that the reader is familiar with the panel
method for non-lifting bodies as described in Section 3.5. In a similar way to the
computational method in the non-lifting case, the aerofoil section must be model-
led by panels in the form of straight-line segments - see Section 3.5 (Fig. 3.37).
The required vorticity can either be distributed over internal panels, as suggested by
Fig. 4.22a, or on the panels that model the aerofoil contour itself, as shown in
Fig. 4.22b.
The central problem of extending the panel method to lifting flows is how to satisfy
the Kutta condition (see Section 4.1.1). It is not possible with a computational
scheme to satisfy the Kutta condition directly, instead the aim is to satisfy some of
the implied conditions namely:

(a) The streamline leaves the trailing edge with a direction along the bisector of the
trailing-edge angle.
(b) As the trailing edge is approached the magnitudes of the velocities on the upper
and lower surfaces approach the same limiting value.

(a 1 Internal vortex panels

'
( b ) Surface vortex panels
Fig. 4.22 Vortex panels: (a) internal; (b) surface

* P.E. Rubbert (1964) Theoretical Characteristics of Arbitrary Wings by a Nonplanar Vortex Lattice Method
D6-9244, The Boeing Co.