Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 73 / 2 / ii

For the Landau intermittency counterexample, combine two locally homogeneous turbulent populations, with the same sufficiently wide inertial interval but different viscous dissipation rates ${ϵ}_1$ and ${ϵ}_2$. This can mean an ensemble randomly selected between two intensities, or large turbulent patches whose small-scale increments are sampled away from the patch boundaries. Suppose each population individually obeys the proposed universal coefficient ${\beta }_p$. If their proportions are $w$ and $1-w$, the combined longitudinal structure function isIts effective coefficient referenced to the global mean viscous dissipation is thereforeFor a concrete example let $w=1/2$ and ${ϵ}_{1,2}=ϵ(1\pm d)$, with $0<d<1$. ThenFor $p=6$ this is $1+d^2$, explicitly dependent on the strength of the large-scale modulation. For $p=2$ it is less than one by concavity. Thus coefficients based solely on mean viscous dissipation cannot be universal under arbitrary integral-scale intermittency. The common inertial interval can be maintained by taking both local Reynolds numbers sufficiently large.

At $p=3$ the ratio is exactly one. This exception is consistent with the signed third-order energy-flux law and must not be used to rescue universality of all other orders. For noninteger orders one uses absolute increments; the even-order examples already establish the counterexample without a sign convention ambiguity. The issue is the difference between a moment of the local viscous dissipation and the same power of its mean, not a failure of averaging itself.