Small-Scale Turbulence Is Not Gaussian: Intermittency and the Failure of Local Isotropy — Epoche C1
The number that starts the argument Take two points in a turbulent flow separated by a small distance $r$, measure the difference between the velocity components along the line joining them, repeat many millions of times, and compute the fourth moment of that difference divided by the square of its second moment. If the velocity difference were a Gaussian random variable this ratio — the kurtosis, or flatness — would equal exactly $3$, since for a zero-mean Gaussian $\langle x^{2n}\rangle = (2n-1)!!\,\sigma^{2n}$ and so $\langle x^4 \rangle = 3\sigma^4$. In laboratory turbulence at small separations the measured value is far larger, and it grows without apparent bound as the Reynolds number is raised. That single fact is what this note is about, and it has two distinct consequences for Kolmogorov's 1941 theory which the original version of this note ran together. Separating them is the main work here. What K41 asserts, in a form that can be tested Kolmogorov's theory of 1941 rests on three hypotheses. The first, local isotropy , says that at sufficiently high Reynolds number the statistics of velocity differences over small separations become independent of direction, so that whatever anisotropy the large-scale forcing imposes is forgotten by the time the energy has cascaded down. The second says that at small $r$ those statistics are determined entirely by two quantities: the kinematic viscosity $\nu$ and $\epsilon$, the mean rate at which the flow dissipates kinetic energy per unit mass. The third says that in the inertial range — separations much larger than the scale at which viscosity acts but much smaller than the scale at which energy is injected — even $\nu$ drops out, leaving only $\epsilon$ and $r$. Everything then follows by dimensional analysis, and the exponents are worth deriving rather than quoting. The viscosity has dimensions $\mathrm{L^2\,T^{-1}}$ and the dissipation rate $\mathrm{L^2\,T^{-3}}$. A length built from them, $\eta = \nu^a \epsilon^b$, requires $2a + 2b = 1$ for the length and $-a - 3b = 0$ for the time; the second gives $a = -3b$, and substituting yields $-4b = 1$. Hence $b = -\tfrac14$, $a = \tfrac34$, and $$\eta = \left(\frac{\nu^3}{\epsilon}\right)^{1/4},$$ the Kolmogorov scale, below which viscosity smooths the field. In the inertial range the only combination of $\epsilon$ and $r$ with the dimensions of a squared velocity is $(\epsilon r)^{2/3}$, so the second-order structure function — the mean squared velocity difference — must obey $S_2(r) = \langle [\delta u_L(r)]^2 \rangle \sim (\epsilon r)^{2/3}$, where $\delta u_L$ is the component of the velocity difference along the separation vector. Extending to arbitrary order gives the prediction that anchors the whole debate: $S_p(r) = \langle |\delta u_L(r)|^p \rangle \sim r^{\zeta_p}$ with $$\zeta_p = \frac{p}{3}.$$ Note what this asserts. Linearity of $\zeta_p$ in $p$ is equivalent to the statement that the probability distribution of $\delta u_L(r)$, once rescaled by $r^{1/3}$, is the same at every scale. K41 is a self-similarity hypothesis, and it entails in particular that the kurtosis is independent of $r$ — which is precisely what the measurement in the opening paragraph contradicts. The one exact result Before the failures, the success. Kolmogorov also derived, not from dimensional analysis but from the Navier-Stokes equations themselves, an exact relation for the third moment of the longitudinal increment in homogeneous isotropic turbulence at infinite Reynolds number: $$\langle [\delta u_L(r)]^3 \rangle = -\frac{4}{5}\,\epsilon\, r .$$ This is the four-fifths law, and it is the only quantitative result in the theory of turbulence with the status of a theorem: given homogeneity, isotropy, stationarity and the limit of vanishing viscosity, it follows. Its consequence for what comes later is that $\zeta_3 = 1$ exactly, so no model of intermittency may predict otherwise. Two cautions belong with it. First, the law concerns the signed third moment, whereas the structure functions defined above use absolute values; there is no exact result for $\langle|\delta u_L|^3\rangle$, and the widespread practice of anchoring measurements at $\zeta_3 = 1$ borrows an exact result for one quantity to calibrate another. Second, the negative sign is itself physical: it encodes the direction of the energy cascade, from large scales to small. What is measured The classic measurements are those of Anselmet, Gagne, Hopfinger and Antonia, who obtained structure functions up to order eighteen from hot-wire anemometry in a turbulent jet and a duct flow. Their exponents, with the K41 prediction beside them, are as follows. $p$ K41: $p/3$ measured $\zeta_p$ log-normal, $\mu = 0.25$ She-Leveque 2 0.667 0.70 0.694 0.696 3 1.000 1.00 1.000 1.000 4 1.333 1.28 1.278 1.280 6 2.000 1.78 1.750 1.778 8 2.667 2.17 2.111 2.211 10 3.333 2.54 2.361 2.594 Three features of this table matter. The departure begins below $p = 4$, not at it: $\zeta_2 \approx 0.70$ already exceeds $2/3$, and the sign of the deviation reverses at $p = 3$, with $\zeta_p > p/3$ below and $\zeta_p < p/3$ above. The gap widens with order, reaching almost a quarter of the predicted value by $p = 10$. And the high-order entries carry the largest uncertainty for a reason that is not incidental: estimating $\langle|\delta u|^{10}\rangle$ requires resolving the far tail of the distribution, so the very rarity that makes intermittency interesting also makes it hard to measure, and reported exponents differ between experiments by several per cent. The physical picture behind the numbers is that dissipation is not spread evenly through the fluid but concentrated in sparse, intense structures — vortex filaments and sheets — occupying a small and scale-dependent fraction of the volume. The distribution of velocity gradients accordingly has tails far heavier than Gaussian, closer to exponential or stretched exponential, and the flatness of the longitudinal derivative is obse