# 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

## 1 Answer

The statement that "fluids are fractal" is not quite correct (or at the very least is not precise). Instead what really happens is that energy in fluids transitions to higher and higher frequencies via a recursive formula which looks slightly fractal-like (called the "Selection rule"). This is one of the most famous results in fluid mechanics and is due to Kolmogorov. Here is a slightly mathematical, but very eloquent exposition of this beautiful discovery:

http://www.sjsu.edu/faculty/watkins/kolmo.htm

Note however that this only tells us the approximate structure of a fluid flow; ie its power spectrum. It does not give us enough information to say exactly what any given fluid will do at fine scales, or even if a solution exists! These latter two problems remain unsolved even today and are the subject of one of the famous millennium problems:

http://www.claymath.org/millennium/Navier-Stokes_Equations/

• Do you have a somewhat better reference for the selection rules and the spectral form of the navier-stokes equation? I looked through the webpage and its nice but very general. – user346 Mar 23 '11 at 8:08
• Kolmogorov spectrum model works only for certain areas of energy spectrum - called inertial interval, where transport of energy is mainly inertial and flow is from bigger vertexes which decays to smaller ones. But outside this area, picture is quite different. For larger scales flow is non universal and is governed by global geometry of boundary. For smaller scales than inertial, flow is governed by decay down to thermal scales. Turbulence is strongly nonlinear and very complicated phenomenon and selfsimilarity describes it only in partial way. – kakaz Mar 23 '11 at 11:18
• This is not correct--- Kolmogorov's power spectrum for turbulence is precisely and exactly fractal. There is an exact scale invariance in the solution! The "selection rule" terminology is a bit misleading--- that wavenumbers are produced by addition is only a selection rule in a very technical sense. – Ron Maimon Nov 3 '11 at 7:02
• It has been well said that in turbulence we can give precise answers to vague questions but only vague answers to precise questions – Philip Roe May 30 '17 at 20:51