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Coarse-grained energy dissipation is a spatial average of the local rate $2\nu S_{ij}{S}_{ij}$ over a region of size $r$. Its random value ${ϵ}_r$ has mean $ϵ$ in homogeneous turbulence. Refined similarity uses this local averaged viscous dissipation to scale velocity increments.

Refined similarity writes $\Delta v=(r{ϵ}_r{)}^{1/3}V$ with universal normalized conditional statistics at high local Reynolds number. Then $S_p=C_p{r}^{p/3}⟨{ϵ}_r^{p/3}⟩$. The distribution of coarse-grained energy dissipation need not be universal. Refined similarity itself does not prescribe that distribution.

External intermittency is the alternation of turbulent and non-turbulent fluid at a point as a fluctuating interface passes it. Internal intermittency in the equilibrium range instead describes uneven, bursty concentrations of small-scale velocity gradients and viscous dissipation within turbulent fluid. Integral-scale intermittency is modulation of the energy-containing motions and their energy supply. It is non-universal because it retains information about forcing, boundary conditions and the history of the large-scale flow.

The Landau intermittency counterexample challenges the unconditional universality asserted by Kolmogorov 1941 theory. Suppose turbulent populations have different conditional mean viscous dissipation rates ${ϵ}_a$. Even if each obeys the same conditional scaling, averaging over populations giveswhich generally differs from ${\beta }_P{r}^{P/3}⟨{ϵ}_a{⟩}^{P/3}$. Thus a coefficient normalized by the global mean depends on the large-scale intensity distribution. The third-order case escapes this particular objection because the power of ${ϵ}_a$ is one.

The Kolmogorov refined similarity hypothesis replaces global mean viscous dissipation by coarse-grained energy dissipation ${ϵ}_{\mathrm{AV}}(r)$. It assumes thathas universal conditional statistics at sufficiently large local Reynolds number. If the conditional $P$th moment is ${\beta }_P$, the law of total expectation givesThe refined similarity hypothesis does not itself prescribe the distribution of coarse-grained energy dissipation. Scale dependence of those moments can change the predicted exponents. As before, noninteger orders require absolute velocity increments.

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.

Let ${ϵ}_{\mathrm{loc}}(\mathbf{x})=2\nu S_{ij}{S}_{ij}$ and define a coarse-grained energy dissipation at scale $r$, for example by a ball average,This is the random variable denoted ${ϵ}_{\mathrm{AV}}(r)$. The refined similarity hypothesis uses the local supply of energy available to motions of size $r$, rather than replacing it everywhere by a global mean. In its inertial range form,At high local Reynolds number, $V$ has a universal conditional distribution independent of ${ϵ}_r$, forcing details and $r$ within the inertial range. This givesFor signed odd orders the coefficients carry the appropriate sign; absolute moments use their own coefficients. Universality is retained in the normalized conditional velocity statistics, while the distribution of ${ϵ}_r$ can retain integral-scale information. This distinction resolves the particular averaging objection: its nonuniversal viscous dissipation moments are now explicitly present.

The quoted second viscous dissipation moment does not by itself determine every longitudinal structure function exponent. The extra lognormal intermittency model assumes $Y_r=\log ({ϵ}_r/ϵ)$ is Gaussian. Write its variance as ${\sigma }_r^2$. Normalization of the mean requires $⟨Y_r⟩=-{\sigma }_r^2/2$, soThe second-moment estimate gives $\exp ({\sigma }_r^2)=B(\ell /r{)}^{\mu }$. Hence, with $q=p/3$,In particular ${\zeta }_3=1$, ${\zeta }_2=2/3+\mu /9$ and ${\zeta }_6=2-\mu$.

The most problematic step is the extrapolation of a lognormal viscous dissipation law to rare events and all moment orders. It is not entailed by a measured second moment. Its quadratic exponents eventually decrease with $p$ and become negative for $p>3+6/\mu$. If the velocity has finite uniformly bounded moments, the bound $⟨|\Delta v{|}^p⟩\le 2^p⟨|u_x{|}^p⟩$ prohibits such divergence as $r\to 0$. Thus the model cannot be an unrestricted high-order inertial range law. This is a criticism of the lognormal closure, rather than an algebraic contradiction in refined similarity itself.