Fractal nature of turbulence

Someone described to me the difficulty of numerically simulating turbulence as that as you look at smaller length scales you see more structure like you do in a fractal. Searching on google for 'fractal turbulence' does seem to bring up quite a few hits. But since fractals are self repeating as you go to smaller and smaller scales, doesn't this mean that we would already know all about the turbulence if we calculate it at one scale, and then add in the all the other scales assuming they are self similar?

I am not a physicist, so please answer with simple language and not overly mathematical.

• I can't answer your question as I am not an expert on turbulence but your statement "we would already know all" is certainly wrong. Most of the fractals (and in particular ones found in nature) aren't completely self-identical (as Sierpinsky's carpet or Koch's snowflake) but instead possess infinite variety between scales (think about Mandelbrot's set). So while self-similarity lets you answer some questions (e.g. determining Hausdorff dimension) it definitely doesn't give you everything. – Marek Mar 22 '11 at 8:31
• In addition to @Marek's comment above, I'd point out that fluids in real life do not have arbitrarily small features. Eventually the length scale becomes that of the mean-free path of the constituent molecules, and one must use kinetic theory to compute further. One must, when reading the literature, keep in mind that mathematical literature usually completely forgets this point, because it is interested in the Navier-Stokes equation from a mathematical point of view. In computational settings, one never discretises that far anyway, and constitutive relations are sufficient. – genneth Mar 22 '11 at 9:27
• @Marek Right, and self-similarity of turbulent fluids ends when approaching molecular (or mean free path) dimensions. – Georg Mar 22 '11 at 9:30
• @Ginsberg your intuition is partially correct. Modulo the caveats mentioned in other comments, the (approximate) scale-invariance of turbulent flow suggests that there should be an effective description of turbulent flow in terms of a conformal field theory for a wide range of scales. CFT is the modern tool used to study scale-invariant phenomena. Despite the self-similarity there still remain non-trivial aspects of such phenomena which tools such as CFT help us understand. – user346 Mar 22 '11 at 10:18
• @Deepak: could you give some references? Naturally I know about lots of applications of CFT to condensed matter and statistical mechanics systems but I am afraid I am completely ignorant when it comes to fluid dynamics. – Marek Mar 22 '11 at 10:27