 Solution

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.