Nice idea.
First, I imagine you have to recognize that surfaces are
already fractals in a sense.
You
can use mathematical fractal surfaces to model real surfaces. As you "become" smaller (in your imagination, of course) the surface of an object becomes larger. I'll give two examples (none of them in aerodynamics, but in asphalts and soils, which are "my" subject):
Fractal grip
I've wrote a dozen times about the theory of tyre grip by Bo Persson. It's based loosely in kekekeke concept: you have to estimate or calculate the interlocking forces between tyres and asphalt at ever decreasing "characteristical lengths", down to the molecular level.
That theory states that the relationship between load and friction force, which since the XVIIth century has been considered lineal, it's not.
What happens is that as you increase the load, the rubber starts to creep between the valleys and hills in the asphalt. The amount of "fractal" surface you have for interlocking is enormous, compared with the effective area of the patch. That's the real reason why load and friction force are approximately related in linear fashion: because the rubber inserts itself in the "fractal surface" as the load increases.
If someone is interested I can post the links, but I don't want to post them for the sixth or seventh time by myself.
Fractal clays
Clay is a material with extremely small grains. It behaves very different from sands or silts, where grains are submillimeter sized. Clay grains are so small that gravitation is not important: the relationship between their area and their weight is very large. Thus, clays are binded by the electric forces among grains. When you model clay, you're modelling a lump of soil held in place by electric forces.
Now, the size of pores in clays can be studied by their "fractal length". The size of pores is very important in consolidations studies because it determines how fast the water will seep out when under load and thus, how fast the soil will compress.
A very smooth pore has a fractal length (Ds) close to 2, while a very rough one has larger values (the fractal length is the relationship between the length of a "flat" surface and the length of the actual surface). Electron microscopy has proved that the consolidation of clay grains follow predictable fractal shapes.
Pores in clay under the electron microscope
Fractal powders
Powders have been studied also (for aerosols) based in their fractal dimension.
"Surfaces of most materials, including natural and synthetic, porous and non-porous, amorphous and crystalline are fractal on a molecular scale [1, 2]. Surface roughness plays an important role by increasing the drag force of a particle as it settles and therefore reduces the settling velocity. The degree of surface corrugation is commonly quantified by a surface fractal dimension (DS) with values varying from 2 (smooth) to 3 (very rough)."
Two different surfaces used in aerodynamic studies. The smoothness of the spheres on the left is apparent: if you get closer, its surface will reveal itself to be a fractal (altough smaller in characteristical length than the one on the right)
What catches my eye in that article is this:
"
Conclusion. Powders with varying degrees of corrugation were successfully obtained by spray drying with their surface roughness quantified by fractal analysis. It was shown that
only a relatively small degree of surface corrugation was sufficient to accomplish a considerable improvement in the aerosol performance of the powder."
By aerosol performance, the writer means that the powder floats more in air.
Fractal heat exchange
I won't give examples of that. As mentioned, it has been explored since a proposal by Hylde van der Vyver (I'm not sure about the name, I'm writing from memory).
End of examples
I suppose the same thing happens in real life with aerodynamic surfaces, with a caveat: the molecules are moving at high speeds, so I imagine it's hard for them to "feel" the fractal "innards".
So, even if I still think your idea is remarkable and can lead to some new ideas about new aerodynamic shapes, I think that the smooth planes you see on the wings of Formula One cars are
not smooth: if you were small enough (perhaps the size of Bernie Ecclestone) you could realize how rough they are for a molecule of air.
The fact is that if you want to explore the idea, you only need rough surfaces: they're "fractal" already.
Anyway, I predict you're onto something... any drawings?
