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This turbulence closure assumes the velocity increment skewness is the same throughout the equilibrium range, with its value fixed by the Kolmogorov two-thirds law and Kolmogorov four-fifths law. It yields a simple crossover between differentiable and inertial-scale increments, but suppresses the scale dependence of skewness and has no independent spectral-transfer dynamics.

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.

The concern is that the rapid-decay assumptions need not be preserved by the dynamics. In a constant-density incompressible flow, the divergence of the Navier-Stokes equation gives the pressure Poisson equation for incompressible flow, with $\pi =p/\rho$:For a localized eddy, inversion with the free-space Green function and two integrations by parts yieldAt large separation the differentiated kernel has leading form $(3n_i{n}_j-{\delta }_{ij})/r^3$. Thus an anisotropic stress moment of an individual eddy produces a quadrupolar fluid pressure of order $r^{-3}$, even if the velocity source is spatially localized. Correlating that fluid pressure with local $u_x^2$ allows $⟨u_x^2{p}^{\prime }⟩=O(r^{-3})$. Isotropy may cancel a mean multipole but does not automatically cancel such a mixed fourth-order statistic.

The triple-correlation evolution contains a pressure-gradient term of the form $-⟨u_x^2{\partial }_{x^{\prime }}{\pi }^{\prime }⟩$, so these long-range pressure correlations in turbulence can generate a longitudinal triple-correlation tail $K\sim a{r}^{-4}$. With $L=0$, part (ii) then gives $\dot{I}=8\pi u^{3}a$, which need not vanish. This is the substance of the objection to deducing exact conservation from initially short velocity correlations. Batchelor and Proudman's original long-range calculation concerned anisotropic turbulence; extending it to isotropic decay requires the appropriate higher-order correlations and allows cancellations. A permitted tail is not a proof that its coefficient stays appreciable in fully developed turbulence.

This possibility appealed to workers constructing heuristic turbulence closures. A closure that generated a changing low-wavenumber $k^4$ coefficient and a decay exponent different from $10/7$ could then be viewed as representing a pressure-mediated effect, rather than as automatically violating a universal invariant. The closure's far-field coefficient still needs physical validation; closure-generated nonconservation does not itself prove the true dynamics nonconserving.

For evidence available before this exam, the 2006 simulations of Ishida, Davidson and Kaneda found that $I$ approached an approximately constant value after a transient when the box was much larger than the integral scale of turbulence and the Reynolds number was large. The energy decay was correspondingly close to $u^2\sim t^{-10/7}$. The observed conclusion was approximate late-time conservation in that regime, indicating weak net long-range triple-correlation transfer, rather than exact conservation for every initial condition, box size or Reynolds number.

The quadratic nonlinear term of the Navier-Stokes equation couples three Fourier modes whose wave vectors close to a triangle. These spectral triad interactions transfer kinetic energy between wave numbers. A turbulence closure can model their triple correlations while retaining more scale information than a prescribed local power law.