Examples
The following links bring up tables of boundary points that
can be copied and pasted into the input table of the Vortex Panel Method.
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Detailed NACA0012 paneling. Solution
window for 8 deg. angle of attack. This table contains data for a 201
panel representation of the NACA0012 airfoil. Compare the results obtained
with this data with that obtained from the (much lower resolution) default
NACA0012 paneling available in the applet. The smoother results are much
more suitable for input into further calculations such as boundary layer
methods.
-
Detailed NACA4412 paneling. Solution
window for 8 deg. angle of attack. A 201
panel representation an airfoil with the same thickness distribution as a NACA 0012, but with 4% maximum camber at the 40% chord location. Compare the results obtained
with this data with that obtained for the NACA0012. The less peaked pressure distribution results in a better behaved boundary layer and improved stall characteristics.
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NACA 23012 paneling. Solution
window for 8 deg. angle of attack. A standard airfoil with camber, 60 panels. (Courtesy: Dan Shafer and http://www.aae.uiuc.edu/m-selig/ads/coord_database.html)
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Flat plate paneling. Solution
window for 8 deg. angle of attack. A 35 panel representation of a flat
plate. (Actually a plate with a thickness of 0.02% chord - a completely
flat plate would place the control points for the airfoil upper surface
exactly on top of those for the lower surface, resulting the program trying
to invert a singular matrix). The interesting feature of the flat plate
airfoil is that its flow can be computed exactly using a technique known
as the Joukowski mapping. The Joukowski mapping is implemented by the Ideal
Flow Machine applet. Try using this applet to compute the same flat
plate flow, and compare the answers (you will need some background in conformal
mapping to do this). Alternativelu, compare the answers you get with the
results of thin-airfoil theory - a simple approximate method for computing
aerodynamic characteristics of airfoils.
-
Clark Y paneling. Solution
window for 10 deg. angle of attack. A classic section represented
by 69 panels (smoothed to eliminate pressure oscillations near the leading
edge).
Exercises
The following are some suggestions for more extensive uses of
the applet, designed to develop understanding of airfoil aerodynamics.
-
Copy the detailed NACA0012 paneling into Excel.
Use Excel's functions to multiply the y-coordinates by a constant
factor. Copy and paste the resulting point pairs back into the applet and
recompute. Repeat this process with different thicknesses until you have
a clear picture of the effects of airfoil thickness on surface pressure
distribution, lift and moment coefficient. Note that the NACA 0012 airfoil
is initially 12% thick (i.e its maximum thickness is 12% of its chord).
Document your calculation by copying the results back into Excel and plotting
them.
-
Copy the detailed NACA0012 paneling into Excel.
Use Excel's functions to add a circular arc to the y-coordinates of the
panel points (the equation sqrt(r2-(x-0.5)2)
- sqrt(r2-0.25) gives the necessary
y-increments in terms of the chordwise position x and the
radius of curvature r (>0.5), both in chordlengths. Repeat this
process with different radii of curvature until you have a clear picture
of the effects of airfoil camber on surface pressure distribution, lift
and moment coefficient and zero lift angle of attack. Note that the NACA
0012 airfoil is initially uncambered. Document your calculation by copying
the results back into Excel and plotting them.
-
Borrow a copy of Abbott, I.H., and von Doenhoff, A.E., Theory of Airfoil
Sections, Dover, New York, 1959. Select an airfoil section for which
they present experimental data. Use the applet to compute the aerodynamic
characteristics of the airfoil. Compare your results with the experimental
measurements. Comment on the differences.
-
Write an Excel program to generate the panel coordinates for a standard
series of NACA airfoil sections (e.g. the 4-digit series, see for example
Abbott, I.H., and von Doenhoff, A.E. for the necessary equations). Use
the applet to contrast the different characteristics of the airfoils that
can be generated from this formulation.
-
Derive or find out (e.g. from A. M. Kuethe and C.-Y. Chow, Foundations
of Aerodynamics, Fourth Edition, Wiley, 1985) the general equation
for the velocity field generated by a vortex panel of linearly varying
strength. For one of the above examples, copy the 'Panel Strength' output
into Excel or a similar program and use it, along with your equation, to
plot the flowfield (velocity and/or pressure coefficient) at points away
from the airfoil surface.
-
Perform a calculation using the detailed NACA0012
paneling of the pressure distribution around the NACA 0012 at a small
angle of attack. Copy the results into Excel. Use the Prandtl-Glauert compressibility
correction to then estimate the pressure distribution (and lift coefficient
generated by this airfoil) as a function of Mach number M. The Prandtl
Glauert compressibility correction (see for example Anderson, John D.,
Modern Compressible Flow, McGraw Hill, 1990) relates the pressure
coefficient generated by an airfoil at non-zero Mach number M (say
Cp|M) to the pressure coefficient calculated
or measured for incompressible flow (Cp) through the relation
Cp|M = Cp / sqrt(1 - M2).
Note that this correction can only be used at Mach numbers up to about
0.7 for which the flow over the airfoil is almost entirely subsonic.
-
Repeat the previous exercise, using a typical supercritical airfoil section
(use the library to find coordinates for this), and compare the answers.
-
(For more experienced users only.) Use the panel method applet in conjunction
with the boundary
layer applets to compute the drag on the NACA 0012 airfoil, or any
other foil you choose. In principle, by trial and error, you may also be
able to use the boundary layer applets to estimate the angle of attack
at which the airfoil stalls.
Current Applet Version 1.0. Last HTML/Applet
update 8/28/98. Questions or comments please contact William J. Devenport
