A note on third–order structure functions in turbulence

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Abstract
Starting from the Navier–Stokes equation, we rigorously prove that a modified third–order structure function, Stilde3 (r) asymptotically equals –4 over 3 ∈r (∈ is the dissipation rate) in an inertial regime. From this result, we rigorously confirm the Kolmogorov four–fifths law, without the Kolmogorov assumption on isotropy. Our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector y, besides space–time averaging. Direct numerical simulation for a highly symmetric flow for a Taylor Reynolds number of up to 155, shows that the flow remains significantly anisotropic and that, without solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of r = lyl for some orientation of y, but not for another.