According to Kolmogorov, the energy spectrum function of a turbulent fluid is given as,
E(k)=Cϵ23k−53
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 τ∼rv(r)∼1kv, the energy density for scale r is ∼v(r)2, so you get an energy flux rate ϵ∼v2/τ∼v3k which is assumed constant. You then estimate she spectral energy density as E∼v2/k∼ϵ2/3k−2/3−1=ϵ2/3k−5/3. I hope that is a little better than pure dimensional analysis.
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