On the decay of isotropic turbulence

Abstract

We investigate the decay of freely evolving isotropic turbulence. There are two canonical cases: $E(k \,{\to}\, 0)\,{\sim}\, Lk^2$ and $E(k \,{\to}\, 0)\,{\sim}\, Ik^4,$$L$ and $I$ being the Saffman and Loitsyansky integrals respectively. We focus on the second of these. Numerical simulations are performed in a periodic domain whose dimensions, $l_{box}$, are much larger than the integral scale of the turbulence, $l$. We find that, provided that $l_{box} \,{\gg}\, l$ and $\hbox{\textit{Re}}\,{\gg}\, 1$, the turbulence evolves to a state in which $I$ is approximately constant and Kolmogorov's classical decay law, $u^2\,{\sim}\, t^ {{- 10} / {7}}$, holds true. The approximate conservation of $I$ in fully developed turbulence implies that the long-range interactions between remote eddies, as measured by the triple correlations, are very weak. This finding seems to be at odds with the non-local nature of the Biot-Savart law.