A note on third-order structure functions in turbulence

Starting from the Navier-Stokes equation, we rigorously prove that a modifi ed third-order structure function, (S) over tilde(3)(r), asymptotically equ als -4/3 epsilon r (epsilon is the dissipation rate) in an inertial regime. From this result, we rigorously confirm the Kolmogorov four-fifths law, wi thout the Kolmogorov assumption on isotropy. Our definition of the structur e function involves a solid angle averaging over all possible orientations of the displacement vector y, besides space-time-averaging. Direct:numerica l simulation for a highly symmetric flow for a Taylor Reynolds number of up to 155 shows that the flow remains significantly anisotropic and that, wit hout solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of r = \y\ for some orient ation of y, but not for another.