A note on third order structure functions in turbulence

Starting from the Navier-Stokes equation, we rigorously prove that a modi ed third order structure function ~ S 3 (r) asymptotically equals 4 r=3 ( is the dissipation rate) in the large Reynolds number limit of Navier-Stokes equation in an inertial regime. From this result, we rigorously con rm the Kolmogorov fourfth law, without the Kolmogorov assumption on isotropy. Our de nition of the structure function involves a solid angle averaging over all possible orientation of the displacement vector y, besides space-time averaging. Direct numerical simulation for a highly symmetric ow for a Taylor Reynolds number of 155 shows that the ow remains signi cantly anisotropic and that without solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of r = jyj for some orientation of y, but not for another.