Friday, June 7, 2019

turbulence - Physical Explanation for "Kolmogorov -5/3 spectrum" in Fluid Mechanics


According to Kolmogorov, the energy spectrum function of a turbulent fluid is given as,



E(k)=Cϵ23k53


where ϵ is the energy flux and k=2πr where r is the length scale.


The normal explanation I see in most physics texts and articles is that the -5/3 exponent is found purely through dimensional analysis. Obviously dimensional analysis is very useful, but often times there is a more physical explanation as well. Does anyone have any insights?



Answer



The standard explanation is that there is a constant flux of energy from large eddies to smaller eddies. The time scale for an eddy of scale r to turn over is τrv(r)1kv, the energy density for scale r is v(r)2, so you get an energy flux rate ϵv2/τv3k which is assumed constant. You then estimate she spectral energy density as Ev2/kϵ2/3k2/31=ϵ2/3k5/3. I hope that is a little better than pure dimensional analysis.


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