It is widely believed that at high Reynolds number (Re) all turbulent flows approach a limiting state of “fully developed turbulence" in which the statistics of the velocity fluctuations are independent of Re. Nevertheless, direct measurements of the velocity fluctuations have failed to yield firm empirical evidence that even the second-order structure function becomes independent of Re at high Re, let alone structure functions of higher order. Here we relate the friction coefficient of rough-pipe flows to the second-order structure function. Then we show that in light of experimental measurements of our results yield unequivocal evidence that the second-order structure function becomes independent of Re at high Re, compatible with the existence of fully developed turbulence.
A similar expression holds for the structure function of order , .
“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 is independent of Re for all . Here we concern ourselves only with .
For example, the form of the structure function should not change if we used in lieu of a global Re.
Barenblatt and Goldenfeld also assumed a general type of incomplete similarity with respect to in which the intermittency exponent, , may depend on Re (Ref. 7). Then, in keeping with the principle of asymptotic covariance, they wrote , and pointed out that the experimental data of Praskovsky and Oncley (Ref. 2) favor (or at high Re). Since is of slight concern here, we continue to represent it as a constant. Nevertheless, in all our equations may be substituted by .
Italics in the original.