The Local Nature of Dissipative Scales in High Reynolds Number Turbulence — Epoche C1
What universality claims, and how precise the claim is Hold a hot-wire probe in the exhaust of a jet engine, in the wake of a bridge pier and a metre above a wheat field on a windy afternoon, and Kolmogorov's 1941 theory predicts that the statistics of the velocity fluctuations at the smallest resolvable separations will be the same in all three, once each is expressed in the right units. That is the claim this essay examines. The units in question are set by two quantities and no others: the kinematic viscosity $\nu$, a property of the fluid measured in $\mathrm{m^2\,s^{-1}}$, and $\varepsilon$, the mean rate at which the flow converts kinetic energy into heat per unit mass, measured in $\mathrm{m^2\,s^{-3}}$. Andrei Kolmogorov's hypotheses of 1941 — hereafter K41 — assert first that at separations far below the scale of the energy-containing motions, the statistics of velocity differences depend only on $\nu$ and $\varepsilon$, and second that in the range of separations that are also far above the scale at which viscosity acts, they depend on $\varepsilon$ alone. The picture behind them is a cascade: energy is injected at a large scale $L$ by whatever stirs the flow, is transferred by inertial interactions to smaller and smaller eddies without loss, and is removed by viscosity only when the eddies are small enough for viscous forces to compete with inertial ones. The scale at which that happens can be derived rather than postulated, and deriving it fixes the exponents that otherwise look arbitrary. Seek a length $\eta = \nu^{a}\varepsilon^{b}$. Matching the powers of length gives $2a + 2b = 1$; matching the powers of time gives $-a - 3b = 0$, hence $a = -3b$. Substituting, $-6b + 2b = 1$, so $b = -1/4$ and $a = 3/4$: $$\eta = \left(\frac{\nu^{3}}{\varepsilon}\right)^{1/4}.$$ The same procedure gives a velocity $u_\eta = (\nu\varepsilon)^{1/4}$ and a time $\tau_\eta = (\nu/\varepsilon)^{1/2}$. The reason $\eta$ deserves to be called the dissipation scale then follows in one line: the Reynolds number formed from these, $u_\eta \eta/\nu = (\nu\varepsilon)^{1/4}(\nu^{3}/\varepsilon)^{1/4}/\nu = (\nu^{4})^{1/4}/\nu = 1$. At $\eta$, and only there, inertia and viscosity are comparable. One consequence is worth recording because it explains why the subject is hard. Estimating the injection rate at the large scale as $\varepsilon \sim u'^{3}/L$, where $u'$ is the typical large-scale velocity fluctuation, gives $L/\eta = (u'L/\nu)^{3/4} = \mathrm{Re}^{3/4}$, so the number of independent degrees of freedom in a three-dimensional volume scales as $(L/\eta)^{3} \sim \mathrm{Re}^{9/4}$. At the Reynolds numbers of atmospheric flow this is why direct simulation is out of reach, and why the argument has to be conducted with statistics rather than with solutions. The one exact result, and what it forbids Before examining the failures of K41 it is necessary to record what is not in doubt, because the earlier version of this essay made a claim about small-scale statistics that this result contradicts. Define the longitudinal velocity increment $\delta u_r = [\mathbf{u}(\mathbf{x} + \mathbf{r}) - \mathbf{u}(\mathbf{x})]\cdot \hat{\mathbf{r}}$, the difference in velocity along the line joining two points a distance $r$ apart. For homogeneous isotropic turbulence, Kolmogorov derived from the Navier–Stokes equations themselves, in the limit of vanishing viscosity, the four-fifths law: $$\langle (\delta u_r)^{3} \rangle = -\tfrac{4}{5}\,\varepsilon\, r .$$ This is an exact consequence of the equations of motion together with the assumption of a finite non-zero dissipation rate, not a hypothesis, and it is essentially the only such result in the theory. Two things follow. First, the third-order exponent is exactly one, which any candidate theory of the higher orders must reproduce. Second, and this is the correction: the earlier claim that K41 implicitly predicts Gaussian statistics at small scales is wrong, and the four-fifths law is what shows it. A Gaussian distribution is symmetric, so all its odd moments vanish; if $\delta u_r$ were Gaussian, the left-hand side would be zero and the cascade would carry no energy. The negative sign encodes the direction of the transfer — energy flows to small scales — and requires a skewed distribution. Measured longitudinal velocity-derivative skewness is accordingly negative, of order $-0.5$, and grows slowly in magnitude with Reynolds number. Non-Gaussianity is not evidence against K41; it is required by it. Why the scaling formula in the earlier version cannot be right What intermittency does challenge is the scaling of higher moments, and here the earlier text wrote something that repays close reading. Intermittency is the observed fact that dissipation is not spread evenly through the fluid but is concentrated in a small fraction of the volume, so that a probe registers long quiet intervals broken by violent bursts. The consequence is that structure functions of high order are dominated by rare events and depart from K41's prediction $\langle (\delta u_r)^{n}\rangle \sim (\varepsilon r)^{n/3}$. The earlier version expressed this departure as $$\langle (\delta u_r)^{n} \rangle \sim \varepsilon^{n/3} r^{\zeta_n}.$$ Check the dimensions. The left side has dimensions of velocity to the $n$, that is $\mathrm{m^{n}\,s^{-n}}$. The right side has $(\mathrm{m^{2}s^{-3}})^{n/3}\,\mathrm{m}^{\zeta_n} = \mathrm{m}^{2n/3 + \zeta_n}\,\mathrm{s}^{-n}$. Equality requires $2n/3 + \zeta_n = n$, that is $\zeta_n = n/3$ — the very thing the formula was written to deny. As stated it is not merely imprecise but self-refuting. The repair is instructive, and it turns out to establish the essay's thesis rather than to weaken it. A dimensionally consistent form with anomalous exponents must import a second length, and the only one available is the scale $L$ at which the flow is stirred: $$\langle (\delta u_r)^{n} \rangle = C_n\,(\varepsilon L)^{n/3}\left(\frac{r}{L}\right)^{\zeta_n}.$$ Now t