 Loitsyansky integral

The Loitsyansky integral is the second spatial moment of the trace velocity correlation tensor, with the displayed sign convention. If correlations decay sufficiently rapidly, it is the coefficient of $k^4/(24{\pi }^2)$ in the turbulent energy spectrum. In terms of the normalized longitudinal velocity correlation it is $8\pi u^2{\int }_0^{\infty }{r}^{4}f(r)dr$ when the boundary terms vanish; conventions omitting $8\pi$ also occur. Its evolution is $\dot{I}=8\pi [r^4{u}^{3}K{]}_{\infty }-12\nu L$, where $L$ is the Saffman integral. Thus it is invariant when $L=0$ and the longitudinal triple correlation is $o(r^{-4})$, not under arbitrary turbulent conditions.