= Loitsyansky integral
{c}
{title2=$I=-\int r^2\langle\boldsymbol u\cdot\boldsymbol u'\rangle\,d^3r$}
= Loitsyansky's integral
{c}
{synonym}
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 $8\pi u^2\int_0^\infty r^4f(r)dr$ when the boundary terms vanish; conventions omitting $8\pi$ also occur. Its evolution is $\dot I=8\pi[r^4u^3K]_\infty-12\nu L$, 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.
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