Re: Whitcomb supercritical airfoil wings in a Mathematica model

• To: mathgroup at smc.vnet.net
• Subject: [mg102600] Re: Whitcomb supercritical airfoil wings in a Mathematica model
• From: Roger Bagula <roger.bagula at gmail.com>
• Date: Sun, 16 Aug 2009 06:41:22 -0400 (EDT)
• References: <h65vdq\$eoa\$1@smc.vnet.net>

```I have been looking at aerodynamic surfaces.
Are you aware that dimpling on golf balls was developed to reduce their drag?
http://www.aerospaceweb.org/question/aerodynamics/q0215.shtml
Roughness on a sphere can also reduce fluid drag by
reducing surface effects.

My first idea was that  you could make flat surfaces with one side
raised slightly in a self-similar dimpling to make a new kind of
fractal wing. Dimpling on wings is actually not very new.
The idea of fractal roughness in self-similar terms
to decrease  friction and increase lift seems to be new.
Something like a Fresnel lens in fractal terms.
They have Photoshop brushes called "fractal wings":
http://zatstyles.blogspot.com/2009/04/fractal-wings-brushe-by-flashtu...
but that has noting to do with vortex effects by self-similar
fractals.
I found this paper online:
http://www.tc.gc.ca/innovation/tdc/projects/air/f/9117.htm
that seems to confirm that some relationship might exist.

At the least it is an idea worth exploring.
A profile that might be possible for a wing roughness model is this
tri-farey function
with limited height:
http://www.flickr.com/photos/fractalmusic/3820841066/
Tri-Farey Function working: ( the less than or equal to sign comes out
badly on news groups)
f[x_] := (9*x/(8 - 18*x)) /; 0 =E2=89=A4 x <= 2/9
f[x_] := 9*x/4 /; 2/9 < x <= 4/9
f[x_] := ((4/5)(1 - x)/(x)) /; 4/9 < x =E2=89=A4 1
:two curved parts and one straight/linear.

I call the Mandelbrot cartoon function or Besicovitch - Ursell
function
as biscuit function.
I attached the fractal that results which is not symmetrical.
This resulting curve is more like the kind of responses seen in
nature.

In continuum mechanics they call that kind of response curve a
Voigt solid/ model which corresponds to a force model of how a solid
responds to force.
F=a*x+b*dx/dt

The tri-Farey function, then is more like quadratic
nature force system.

Models from Mathematica of aircraft with fractal
wings ( fractal function as cylinder on the top of the wings);

http://www.flickr.com/photos/fractalmusic/3821840478/
http://www.flickr.com/photos/fractalmusic/3821033855/

Getting the wings into my ray trace program was the hard part.
The DXF files came out 31mb for 120 points on wing;
5.8 for 50 points
and finally which worked 3.3 for 38 points.
The Mathematica picture is the 120 point wings...
"Real' wings of this type should probably have 1000 points at least.

I don't know if the fractal surface should be on the top of the wing
or the bottom:
but it would show in pictures on the bottom so I used the top.
Something like a von Koch surface might be a better choice,
but I was working on trifurcation processes.

```

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