= Solution
In this question $\Delta v$ is the <velocity> increment in the direction of the separation, so $S_2(r)=\langle(\Delta v)^2\rangle$ 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 $e^{i\boldsymbol k\cdot\boldsymbol r}-1$. Its squared magnitude is $2[1-\cos(\boldsymbol k\cdot\boldsymbol r)]$, of order $(kr)^2$ when $kr\ll1$ and of order one after averaging rapid phases when $kr\gg1$. 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 contributes
$$
S_2(r)\simeq2\left(\frac13\,2\int_{\pi/r}^\infty E(k)dk\right),
$$
or
$$
\boxed{\frac34S_2(r)\sim\int_{\pi/r}^\infty E(k)dk.}
$$
The 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.
Back to article page
