The minimum energy decay rate in quasi-isotropic grid turbulence

We consider high Reynolds number, freely-decaying, isotropic turbulence in which the large scales evolve in a self-similar manner when normalized by the integral scales, u and $ℓ$⁠. As it is well known, a range of possible behaviors may be observed depending on the form of the longitudinal velocity correlation at large separation, $u2f∞=u2f(r→∞)$⁠. We consider the cases $u2f∞=cmr-m,2≤m≤6$⁠, whose spectral counterpart is $E(k→0)~cmkm-1$ for $m<6$⁠, with or without a $lnk$ correction, and $E(k→0)~Ik4$ for $m=6$⁠. (I is Loitsyansky’s integral.) It has long been known that the $cm$ are invariants for $m<6$⁠, which demands $u2ℓm=constant$ during the decay. This, in turn, sets the energy decay rate as $u2~t-(1-p)2m/(m+2)$⁠, where p is the power-law exponent for the normalized dissipation rate, $ɛℓ/ɛℓu3u3~t-p$⁠, observed empirically to be a small positive number in grid turbulence. We systematically explore the properties of these different classes of turbulence and arrive at the following conclusions. (i) The invariance of $cm$ is a direct consequence of linear momentum conservation for $m≤4$⁠, and angular momentum conservation for $m=5$⁠. (ii) The classical spectra of Saffman, $E(k→0)~c3k2$⁠, and Batchelor, $E(k→0)~Ik4$⁠, are robust in the sense that they emerge from a broad class of initial conditions. In particular, it is necessary only that $〈ωiω'j〉∞≤O(r-8)$ at $t=0$⁠. The non-classical spectra (⁠$m=2,4,5$⁠), on the other hand, require very specific initial conditions in order to be realized, of the form $〈ωiω'j〉∞=O(r-(m+2))$⁠. (Note the equality rather than the inequality here.) This makes the non-classical spectra less likely to be observed in practice. (iii) The case of $m=2$⁠, which is usually associated with the $u2~t-1$ decay law, is pathological in a number of respects. For example, its spectral tensor diverges as $k→0$⁠, and the long-range correlations $〈uiu'j〉∞=O(r-2)$ are too strong to be a consequence of the Biot-Savart law. (It is the Biot-Savart law that lies behind the long-range correlations in the classical spectra.) This suggests that $m=2$ spectra are unlikely to manifest themselves in grid turbulence. A detailed review of the available experimental data is consistent with this assertion. We conclude, therefore, that the minimum energy decay rate in homogeneous grid turbulence is most probably that of Saffman turbulence.