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6.6 Applications of the Panel Method 197

6.6 Applications of the Panel Method

To demonstrate the use of the computer program of Section 6.5, subsection 6.6.1
presents the calculation of the lift and pitching moment coefficients of an airfoil
for a range of angles of attack. To demonstrate the application of the panel
method to geometries other than an airfoil, in subsection 6.6.2, the calculation
of the surface pressure and external velocity distributions on a circular cylinder
discussed in Section 6.2 with a finite-difference method is presented and the
modifications required to the panel program are described.
The extension of the panel program to multielement airfoils is described in
subsection 6.6.3. The modifications for this case are discussed in detail and
the resulting program is applied to two-element and three-element airfoils to
compute their pressure distributions and lift and pitching moment coefficients.

6.6.1 Flowfield and Section Characteristics of a N A C A 0012 Airfoil

Consider a NACA 0012 airfoil that is symmetrical with a maximum thickness
of 0.12c [3]; the pressure distribution and external velocity distribution on its
upper and lower surface is computed and its section characteristics determined
with the panel program. Sometimes the airfoil coordinates are given analytically
but often in tabular form similar to those given in Appendix B, Chapter 6.
The calculations for this airfoil are performed for angles of attack from 0°
to 20° with 6a increments of 4°. The x/c and y/c values are read in starting
on the lower surface trailing edge (TE), traversing clockwise around the nose to
the upper surface TE.
In identifying the upper and lower surfaces of the airfoil, it is necessary to
t
determine the x/c-location where V /V OQ = 0. This location, called the stag-
1
nation point, is easy to determine since the V jVoo values are positive for the
upper surface and negative for the lower surface. In general it is sufficient to
take the stagnation point to be the x/c-location where the change of sign in
t
V /V OQ occurs. For higher accuracy, if desired, the stagnation point can be de-
t
termined by interpolation between the negative and positive values of V /V OQ
as a function of the surface distance along the airfoil.
Figures 6.8 and 6.9 show the computed pressure coefficients, C p , on the lower
and upper surfaces and dimensionless external velocity distributions, V/VOQ, on
the upper surface of the airfoil at three angles of attack starting from 0°. As
expected, the results show that the pressure and external velocity distributions
on both surfaces are identical to each other at a = 0°. With increasing incidence
angle, the pressure peak moves upstream on the upper surface and downstream
on the lower surface. In the former case, with the pressure peak increasing in
magnitude with a, the extent of the flow deceleration increases on the upper
surface and, as shall be shown later in Chapter 7, it increases the regions of flow
separation on the airfoil and in the wake. On the lower surface, on the other