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EDQNM closure models the evolution of the turbulent energy spectrum through spectral triad interactions. Fourth-order correlations are approximated using products of second-order correlations, an eddy damping limits the resulting triple correlations, and their memory integrals are replaced by expressions using current spectra. The modeled damping times are additional physical input. This allows evolving spectra and scale-dependent transfer, but does not automatically recover coherent structures or internal intermittency. Bertoglio, Squires and Ferziger's simulation comparison explains the damping assumptions and tests their limitations.

A fixed observation point alternates between turbulent and non-turbulent fluid because a fluctuating interface passes it. This is distinct from internal intermittency, which concerns variations within fluid that remains turbulent.

Internal intermittency is the fluctuating concentration of gradients, transfer and viscous dissipation inside turbulent fluid. It is distinct from an interface between turbulent and non-turbulent regions intermittently passing an observer. Its scale dependence is reflected in non-Gaussian increments and coarse-grained viscous dissipation moments.

Kolmogorov 1941 theory assumes locally universal small-scale velocity statistics set by mean viscous dissipation and kinematic viscosity at large Reynolds number. In its inertial range dimensional form, $S_p(r)\sim {ϵ}^{p/3}{r}^{p/3}$. Large-scale modulation and internal intermittency require qualifying this unconditional scaling.

Small-scale joint velocity increment statistics become approximately invariant under rotations of the observation configuration, even when the energy-containing flow is anisotropic. This is a scale-restricted hypothesis; it does not make the entire velocity field isotropic or remove internal intermittency.

Assume statistically steady, three-dimensional turbulence at large Reynolds number, with local statistical homogeneity and statistical isotropy at scales small compared with the forcing scale $L$. The Kolmogorov 1941 theory additionally assumes a local energy cascade, constant mean energy flux per unit mass $ϵ$, and negligible viscous dissipation within the inertial range. Its dimensional form neglects corrections from internal intermittency.

At separation $l$, let $\delta u_l$ be a typical velocity increment. The eddy turnover time is ${\tau }_l\sim l/\delta u_l$. Constant transfer rate then gives $ϵ\sim \delta u_l^2/{\tau }_l\sim \delta u_l^3/l$, henceHere $E(k)$ is the one-dimensional turbulent energy spectrum, obtained from $\delta u_l^2\sim kE(k)$ with $k\sim l^{-1}$. The scaling holds only forThe Kolmogorov length scale follows by setting the scale-dependent Reynolds number $\delta u_{l}l/\nu$ to one. The associated Kolmogorov microscales give ${\tau }_{\eta }=(\nu /ϵ{)}^{1/2}$. At larger scales the forcing matters; at smaller scales kinematic viscosity matters.

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.

For a Burgers vortex, integrating the axial vorticity gives the azimuthal velocity:The swirl contribution to the viscous dissipation per unit length and per unit density isFor the equality, expand the two squares using ${\omega }_z=u_{\theta }^{\prime }+u_{\theta }/r$; their integrated difference is proportional to $[u_{\theta }^2{]}_0^{\infty }$, which vanishes. Substitution of the Gaussian vorticity profile givesThis excess dissipation of a Burgers vortex is independent of kinematic viscosity at fixed strain $\alpha$ and circulation $\Gamma$. Multiply by the mass density if a dimensional power per length is wanted.

The qualification "excess" matters. The imposed uniform strain itself has viscous dissipation density $3\nu {\alpha }^2$ per unit mass, so its integral over an infinite cross-section diverges. One must subtract that background or restrict to a finite cross-section. In a core-sized area, its contribution is of order ${\nu }^2\alpha$, smaller than the swirl contribution by order $(\nu /\Gamma {)}^2$ as $\Gamma /\nu \to \infty$.

Meanwhile $\delta \propto {\nu }^{1/2}$ shrinks and the central vorticity $\Gamma /(\pi {\delta }^2)$ grows. The finite dissipation becomes concentrated in a narrow tube: a useful local model of internal intermittency and the turbulent dissipation anomaly, though not a proof that an entire turbulent flow consists of such vortices.

Combining the Kolmogorov two-thirds law with the Kolmogorov four-fifths law gives the velocity increment skewness in the inertial range:The constant-skewness turbulence closure assumes this same value throughout the universal equilibrium range, including the dissipative crossover. This is an extra modeling assumption; neither similarity law establishes it near $r=0$.

Write $A=\beta (15\beta {)}^{1/2}$, $B=(15\beta {)}^{3/4}$, $S_2=A{v}^{2}h$ and $r=B\eta x$. Since $\nu =v\eta$ and $ϵ\eta =v^3$, substituting $S_3=-4S_2^{3/2}/(5{\beta }^{3/2})$ into the Kolmogorov equation for structure functions givesDividing by $2B/5$ yields the normalized constant-skewness structure-function equation:At small $x$, $h\sim x^2$, recovering $S_2\sim ϵr^2/(15\nu )$ for differentiable velocity increments. At large $x$, the dominant balance gives $h\sim x^{2/3}$, recovering the assumed Kolmogorov two-thirds law.

This turbulence closure is inexpensive: a single ordinary differential equation interpolates between the two regimes while respecting the local third-order balance. Its prescribed skewness cannot independently predict scale-dependent derivative statistics, transient spectral evolution or internal intermittency.

EDQNM closure instead evolves the turbulent energy spectrum through modeled spectral triad interactions. Its quasi-normal fourth-order approximation is supplemented by eddy damping and a Markovian treatment of correlation memory. It retains more information about scale-dependent transfer and evolving spectra, at the cost of spectral integrals and modeled damping times. It is still approximate and does not automatically describe coherent structures or internal intermittency. The simple constant-skewness model is therefore useful as an equilibrium interpolation; EDQNM closure addresses a broader spectral evolution problem.

Let $u^2$ denote one-component velocity variance, $F=u^{2}f$ the longitudinal velocity correlation, and $S_p=⟨(\Delta v{)}^p⟩$ the signed longitudinal structure functions. Homogeneity gives $F=u^2-S_2/2$, while the third-order convention here is $u^{3}K=S_3/6$. Thus the Kármán-Howarth equation becomesThe mean kinetic energy per unit mass is $3u^2/2$, so its decay gives ${\partial }_t{u}^2=-2ϵ/3$. In the universal equilibrium range, the small-scale variation ${\partial }_t{S}_2$ is negligible at leading order, hence ${\partial }_{t}F\simeq -2ϵ/3$. Integrating from zero and using regularity givesThe Kolmogorov equation for structure functions is thereforeFor finite local unsteadiness the right side has the additional term $-3r^{-4}{\int }_0^r{s}^4{\partial }_t{S}_2(s,t)ds$; dropping it is the local-equilibrium approximation used here.

In the inertial range the viscous term is also negligible, leaving the Kolmogorov four-fifths law, $S_3(r)=-4ϵr/5$. Its coefficient and sign follow from the exact energy balance and Kármán-Howarth equation, rather than from dimensional similarity. It supplies a firm third-order constraint on Kolmogorov 1941 theory and measures the forward energy cascade, while leaving the second- and higher-order exponents open to internal intermittency corrections.

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.

Internal intermittency is the uneven distribution of activity within an already turbulent fluid: intense gradients and viscous dissipation occupy relatively small, fluctuating regions separated by much quieter regions. It concerns the internal field, rather than a turbulent/non-turbulent interface intermittently passing a probe.

Integral-scale intermittency is the variability of the energy-containing eddies and their local or time-dependent energy supply. Its modulation occurs on lengths and times of order $\ell$ and $\ell /u$, can depend strongly on forcing or initial conditions, and changes the background intensity experienced by smaller eddies.

Equilibrium-range intermittency concerns the increasing fluctuations of local transfer and viscous dissipation as the averaging scale is reduced below the integral scale of turbulence. It appears as non-Gaussian velocity-gradient statistics, fluctuating coarse-grained viscous dissipation and anomalous longitudinal structure function exponents. It includes the inertial range and its continuation into the dissipation range, where kinematic viscosity ultimately limits gradients. Large-scale modulation varies the local energy supply; small-scale intermittency concentrates and redistributes that supply. These effects can coexist, so they should not be identified simply because both broaden a distribution.