Kármán–Howarth equation

Loitsianskii's invariant

L.G. Loitsianskii derived an integral invariant for the decay of the turbulence by taking the fourth moment of the Kármán–Howarth equation in 1939, i.e.,

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(u^{\prime 2}{\int }_0^{\mathrm{\infty }}{r}^{4}fdr\right)={\left[2\nu u^{\prime 2}{r}^4\frac{\mathrm{\partial }f}{\mathrm{\partial }r}+u^{\prime 3}{r}^{4}h\right]}_0^{\mathrm{\infty }}.}$

If ${\displaystyle f(r)}$ decays faster than ${\displaystyle r^{-3}}$ as ${\displaystyle r\to \mathrm{\infty }}$ and also in this limit, if we assume that ${\displaystyle r^{4}h}$ vanishes, we have the quantity,

${\displaystyle \mathrm{\Lambda }=u^{\prime 2}{\int }_0^{\mathrm{\infty }}{r}^{4}fdr=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$

which is invariant. Lev Landau and Evgeny Lifshitz showed that this invariant is equivalent to conservation of angular momentum. However, Ian Proudman and W.H. Reid showed that this invariant does not hold always since ${\displaystyle \underset{r\to \mathrm{\infty }}{lim}(r^{4}h)}$ is not in general zero, at least, in the initial period of the decay. In 1967, Philip Saffman showed that this integral depends on the initial conditions and the integral can diverge under certain conditions.

Decay of turbulence

For the viscosity dominated flows, during the decay of turbulence, the Kármán–Howarth equation reduces to a heat equation once the triple correlation tensor is neglected, i.e.,

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(u^{\prime 2}f)=\frac{2\nu u^{\prime 2}}{r^4}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^4\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right).}$

With suitable boundary conditions, the solution to above equation is given by

${\displaystyle f(r,t)=e^{-r^2/8\nu t},u^{\prime 2}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\times (\nu t{)}^{-5/2}}$

so that,

${\displaystyle R_{ij}(r,t)\sim (\nu t{)}^{-5/2}{e}^{-r^2/8\nu t}.}$