Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 79 / 1

The zeroth law of turbulence states that at fixed outer velocity $u$ and integral scale of turbulence $\ell$, the mean viscous dissipation per unit mass approaches a nonzero finite value as the Reynolds number $\mathrm{Re}=u\ell /\nu$ becomes large. In an established forward energy cascade, this gives $ϵ\sim C_{ϵ}{u}^3/\ell$, with $C_{ϵ}$ of order one. The small-scale velocity gradients become large enough to offset decreasing kinematic viscosity; this is the turbulent dissipation anomaly.

At the Kolmogorov microscales, the local Reynolds number is of order one, $v\eta /\nu \sim 1$, and the viscous dissipation is $ϵ\sim \nu (v/\eta {)}^2$. Solving these balances givesSubstituting the outer-scale estimate, and omitting order-one constants, givesThe associated eddy turnover time is $\eta /v=(\nu /ϵ{)}^{1/2}$.

Local isotropy of turbulence means that joint small-scale velocity increment statistics are approximately unchanged by rotations, even when the large-scale flow is anisotropic. The universal equilibrium range contains separations much smaller than the integral scale of turbulence: their adjustment times are short compared with the outer eddy turnover time, and their normalized statistics are assumed independent of the detailed large-scale forcing. It includes the inertial range and the dissipation range.

The Kolmogorov first similarity hypothesis says that these locally isotropic statistics depend only on $r$, mean viscous dissipation $ϵ$ and kinematic viscosity $\nu$. Dimensional analysis therefore expresses the second-order longitudinal structure function in terms of the Kolmogorov microscales:The universality of $F$ is an assumption of the theory, not a consequence of dimensional analysis alone.

The Kolmogorov second similarity hypothesis removes dependence on kinematic viscosity when $\eta \ll r\ll \ell$. Only $ϵ$ and $r$ remain, so the Kolmogorov two-thirds law isThe same reasoning gives the predicted order-$P$ longitudinal structure function:These are signed moments for integer $P$; for noninteger orders, use $⟨|\Delta v{|}^P⟩$. In particular, the signed third-order coefficient is negative, ${\beta }_3=-4/5$, in the Kolmogorov four-fifths law.

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.

A passive scalar $\theta$ is transported by the velocity field without feeding back on it:Take its mean to be zero, and define the scalar dissipation rate by the half-scalar variance convention $\chi =\kappa ⟨|\nabla \theta {|}^2⟩$. In a statistically equilibrated energy cascade of scalar fluctuations, this is also the flux of half-scalar variance toward small scales.

The Kolmogorov two-thirds law gives a typical velocity increment $\delta u_r\sim (ϵr{)}^{1/3}$ and hence eddy turnover time ${\tau }_r\sim r/\delta u_r\sim {ϵ}^{-1/3}{r}^{2/3}$. Assuming local scalar transfer on this same time scale, $\chi \sim ⟨(\Delta \theta {)}^2⟩/{\tau }_r$. The Obukhov-Corrsin theory therefore gives the scalar analogue:This applies to separations at which both direct forcing and molecular diffusion are negligible and the transporting velocity increments lie in the inertial range. For very large or very small ratios $\nu /\kappa$, the scalar and velocity cutoff scales differ, so this common range must be checked. Defining $\chi$ as dissipation of the full scalar variance instead simply changes the convention for $C_{\theta }$.