Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 87 / 3 / iv / Solution

The motivation for Kolmogorov refined similarity hypothesis is internal intermittency: instantaneous energy dissipation is spatially uneven, so scaling every velocity increment by the global mean dissipation can mix regions of very different activity. Define the coarse-grained energy dissipation over a region of size $r$ byIts mean is $ϵ$ in homogeneous turbulence. The refined hypothesis asserts that, at high local Reynolds number and within the inertial range, the normalized incrementhas universal conditional statistics independent of ${ϵ}_r$ and of $r$. In particular, $⟨V_r^p\mid {ϵ}_r⟩={\beta }_p$ for finite moments. Taking a conditional expectation and then averaging givesHere ${ϵ}_r$ is the paper's ${ϵ}_{AV}(r)$. Signed odd moments need not have positive coefficients; the third-order forward-cascade coefficient is negative. The signed first moment is zero in homogeneous flow, so one does not assign it a nonzero scaling law. Absolute moments are often used when discussing general positive-order scaling exponents.

The refined hypothesis alone does not specify the distribution of ${ϵ}_r$. Kolmogorov's 1962 estimate supplements it with a lognormal intermittency model. Let $Y_r=\ln ({ϵ}_r/ϵ)$ be Gaussian with variance ${\sigma }_r^2=\mu \ln (\ell /r)+\mathrm{constant}$. The condition $⟨{ϵ}_r⟩=ϵ$ fixes its mean at $-{\sigma }_r^2/2$. The Gaussian exponential moment then gives, with $q=p/3$,Combining with the refined moment formula yields, for the nonzero moments described by that model,The intermittency parameter $\mu$ must be supplied by dissipation statistics; it is not determined by dimensional analysis. The formula preserves ${\zeta }_3=1$. Its extrapolation to arbitrarily high orders is not justified: the quadratic correction eventually predicts pathological exponents and is one limitation of the lognormal model, distinct from the refined hypothesis itself.