Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 87 / 3

In this question $\Delta v$ is the velocity increment in the direction of the separation, so $S_2(r)=⟨(\Delta v{)}^2⟩$ is a longitudinal velocity structure function. This interpretation is necessary for the kernel in part (iii); the full vector increment has a different kernel.

A velocity component that varies very little over distance $r$ is almost the same at the two sample points, so its common part cancels from their difference. A Fourier component acquires the difference factor $e^{i\mathbf{k}\cdot \mathbf{r}}-1$. Its squared magnitude is $2[1-\cos (\mathbf{k}\cdot \mathbf{r})]$, of order $(kr{)}^2$ when $kr\ll 1$ and of order one after averaging rapid phases when $kr\gg 1$. Thus a longitudinal structure function attenuates large eddies and retains smaller eddies, but is a smooth filter rather than an exact scale cutoff.

The heuristic cutoff $k_c=\pi /r$ marks wavelengths of order $2r$. If motions below that cutoff are treated as perfectly coherent and those above it as uncorrelated at the two points, isotropy allocates one third of their velocity variance to the longitudinal component. Since the turbulent energy spectrum integrates to one half of total velocity variance, the uncorrelated high-wavenumber component contributesorThe factor $3/4$ follows from the component and energy normalizations. The approximation discards the finite gradient of eddies larger than $r$; part (iii) quantifies that missing contribution.

The first similarity hypothesis of Kolmogorov 1941 theory states that at sufficiently large Reynolds number, the small-scale statistics in the equilibrium range are locally universal and determined by the mean turbulent kinetic energy dissipation rate $ϵ$ and kinematic viscosity $\nu$, together with the separation or wavenumber being measured. Large-scale forcing details are excluded only at scales well below the integral scale of turbulence.

These parameters define the Kolmogorov microscalesDimensional analysis gives universal functions $F$ and $G$ withThe dimensional factors follow because $S_2$ has units of velocity squared, whereas $E$ has units of velocity squared times length. The universal equilibrium forms include viscous effects; they do not impose a power law through the dissipation range.

The second similarity hypothesis asserts independence of kinematic viscosity at scales large compared with $\eta$ but still small compared with the integral scale. In this inertial range, $S_2$ can depend only on $ϵ$ and $r$. Balancing time and length dimensions forces powers $2/3$ and $2/3$. The same calculation for $E$, using $k$ instead of $r$, forces powers $2/3$ and $-5/3$. HenceThe constants $C_2$ and $C_K$ are universal within the hypotheses and the stated normalization. They are related by the exact structure-function spectral filter, rather than by assigning all energy to one sharp scale.

For isotropic turbulence with incompressible flow, use the spectral velocity correlation tensorChoose the polar axis along $\mathbf{r}$ and put $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{r}}$. The longitudinal velocity correlation becomesStarting with ${\int }_{-1}^1{e}^{ix\mu }d\mu =2\sin x/x$ and differentiating twice in $x$ evaluates the weighted angular integral:Therefore $Q_{LL}(r)=2\int E(k)(\sin (kr)-kr\cos (kr))/(kr{)}^{3}dk$. At zero separation $Q_{LL}(0)=(2/3)\int E(k)dk$. Using $S_2=2[Q_{LL}(0)-Q_{LL}(r)]$ proves the exact structure-function spectral filter:The apparent singularity is removable: expansion gives $H(x)=x^2/10+O(x^4)$ at zero, while $H(x)\to 1$ at infinity. Substitute the given approximate filter and split the integral at $k_c=\pi /r$. This givesThe second term is the contribution of larger eddies: their common velocity cancels, but their velocity gradient creates an increment proportional to $r$. Their squared gradient contributes the $k^2$ weight. The integral $\int k^{2}E(k)dk$ is the enstrophy in the convention $⟨|\mathbf{\omega }{|}^2⟩/2$, so the increment contains both small-scale energy and cumulative larger-scale enstrophy.

This makes the longitudinal structure function a poor diagnostic of a finite inertial range. With a limited range of $k^{-5/3}$ spectrum, the two integrals retain outer-scale and dissipation-scale contributions, leading schematically to a mixture $a_0+a_2{r}^2+a_{2/3}{r}^{2/3}$. The local logarithmic slope need not then equal $2/3$, even if the spectral inertial interval exhibits a clearer $-5/3$ slope. This finite-range effect was demonstrated in Davidson and Krogstad's 2008 controlled-field study. It challenges interpreting $S_2(r)$ as energy exclusively at scale $r$; it does not refute the formal infinite-inertial-range two-thirds law, since inserting an ideal $k^{-5/3}$ spectrum into the exact transform does produce $r^{2/3}$ scaling.

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.

Separate a smooth large-scale velocity $U_L$ from a remaining smaller-scale component $w_L$. If the large-scale variation length is much greater than $r$, their longitudinal increment has the formFor a positive integer $p$, the leading expansion of its structure function isThus even a velocity component at scales much larger than $r$ enters through its gradients and through mixed moments with the finer field. It is not generally legitimate to discard those mixed moments: large-scale strain can organize and modulate the finer motions. For example, at fourth order the expansion containsin addition to the small-scale fourth moment and the generally present odd mixed terms. Even if the components were independent and symmetric, the displayed mixed second-moment term would survive. This explicitly exhibits large-scale contributions to structure functions beyond second order.

The implication for K62 refined similarity is that an observed structure function need not measure a purely scale-local velocity fluctuation. Dividing it by $({ϵ}_{r}r{)}^{1/3}$ may still leave dependence on outer-scale gradients, correlations or finite-range cutoffs. The universal conditional coefficients ${\beta }_p$ and the identification of a measured exponent with the lognormal intermittency model therefore require additional locality and scale-separation tests. Finite-range mixed contributions can imitate or obscure anomalous scaling. This is an objection to treating the 1962 dimensional argument as an unconditional law for measured increments; it is not, by itself, a proof that an asymptotic refined similarity regime cannot exist.