 Solution

Let $C(r)=⟨\mathbf{u}(\mathbf{x})\cdot \mathbf{u}(\mathbf{x}+\mathbf{r})⟩$ in zero-mean isotropic turbulence. Define the scalar Saffman integral $L={\int }_{{\mathbb{R}}^3}C(r)d^{3}r=4\pi {\int }_0^{\infty }{r}^{2}C(r)dr$. The spectral transform isFor an integrable $r^{2}C(r)$, dominated convergence applied to $\sin (kr)/(kr)$ givesA further $O(k^4)$ expansion requires an additional correlation moment and is not automatic from finite $L$. The normalization agrees with $\int Edk=⟨|\mathbf{u}{|}^2⟩/2$.

For a control volume $V$ put ${\mathbf{P}}_V={\int }_V\mathbf{u}dV$. Statistical homogeneity gives the exact finite-volume identityFor geometrically similar volumes increasing without bound, the overlap fraction $|V\cap (V-\mathbf{r})|/|V|$ tends to one at every fixed separation. If $C$ is absolutely integrable, dominated convergence givesThe printed finite-large-volume expression is the corresponding asymptotic approximation, not an exact identity for an arbitrary finite volume. For instance, positive nonzero $C$ with a finite correlation length gives an overlap fraction smaller than one and hence a strictly smaller finite-volume value. The limit also shows $L\ge 0$; a nonzero Saffman integral is positive.