Rough-pipe flows and the existence of fully developed turbulence

A similar expression holds for the structure function of order $n$⁠, $(ul)n¯=Pn[Re,l∕L](εl)n∕3$⁠.

“The mathematical expression of the assumption of a limiting state of fully developed turbulence is that statistical averages of the flow exhibit complete similarity with respect to Re” (Ref. 4). Therefore, in fully developed turbulence the leading term of $(ul)n¯$ is independent of Re for all $n⩾2$⁠. Here we concern ourselves only with $n=2$⁠.

For example, the form of the structure function should not change if we used $Reλ$ in lieu of a global Re.

Barenblatt and Goldenfeld also assumed a general type of incomplete similarity with respect to $l∕L$ in which the intermittency exponent, $α$⁠, may depend on Re (Ref. 7). Then, in keeping with the principle of asymptotic covariance, they wrote $α(lnRe)=α0+α1∕lnRe+o(1∕lnRe)$⁠, and pointed out that the experimental data of Praskovsky and Oncley (Ref. 2) favor $α0=0$ (or $α→0$ at high Re). Since $α$ is of slight concern here, we continue to represent it as a constant. Nevertheless, in all our equations $α$ may be substituted by $α0+α1∕lnRe+o(1∕lnRe)$⁠.

Italics in the original.