Dissipation and enstrophy statistics in turbulence : are the simulations and mathematics converging?

Since the advent of cluster computing over 10 years ago there has been a steady output of new and better direct numerical simulation of homogeneous, isotropic turbulence with spectra and lower-order statistics converging to experiments and many phenomenological models. The next step is to directly compare these simulations to new models and new mathematics, employing the simulated data sets in novel ways, especially when experimental results do not exist or are poorly converged. For example, many of the higher-order moments predicted by the models converge slowly in experiments. The solution with a simulation is to do what an experiment cannot. The calculation and analysis of Yeung, Donzis & Sreenivasan (J. Fluid Mech., this issue, vol. 700, 2012, pp. 5–15) represents the vanguard of new simulations and new numerical analysis that will fill this gap. Where individual higher-order moments of the vorticity squared (the enstrophy) and kinetic energy dissipation might be converging slowly, they have focused upon ratios between different moments that have better convergence properties. This allows them to more fully explore the statistical distributions that eventually must be modelled. This approach is consistent with recent mathematics that focuses upon temporal intermittency rather than spatial intermittency. The principle is that when the flow is nearly singular, during ‘bad’ phases, when global properties can go up and down by many orders of magnitude, if appropriate ratios are taken, convergence rates should improve. Furthermore, in future analysis it might be possible to use these ratios to gain new insights into the intermittency and regularity properties of the underlying equations

Bounds for Euler from vorticity moments and line divergence

The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments obey Dm ≥ Dm+1, the reverse of the usual Ωm+1 ≥ Ωm Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with D2 m → � sup ӏωӏ ~ Am(Tc-t)-1 where the Am are nearly independent of m. In the second phase, the new Dm ordering breaks down as the Ωm converge towards the same super-exponential growth for all m. The transition is identified using new inequalities for the upper bounds for the -dD-2m/dt that are based solely upon the ratios Dm+1/Dm, and the convergent super-exponential growth is shown by plotting log(d log Ωm/dt). Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth

A farewell to Neo-Punic: Tac-Caghqi revisited

The alleged inscriptions in the south-east hypogeum of Tac-Caghqi (within the premises of St Nicholas College in Rabat, Malta) were discussed in depth by Mons. Benedetto Rocco in 1972. Rocco interpreted the glyphs as Neo-Punic, with a long 'Inscription' supposedly consisting of a prayer to appease the deceased through the offering of a gift, and an alleged minor 'Inscription' addressing the dead to 'rise'. These readings were discussed against the notion of possible libation rites that may have been a custom within the hypogeum, as suggested by the tomb furniture in situ. Rocco based his readings of the script and types of letters on his previous study of further alleged Punic and Neo-Punic inscriptions within cave sites in Palermo and Favignana (Sicily), in combination with semantic analysis of Biblical Hebrew. Nevertheless, the supposed Tac-Caghqi inscriptions come across as ambiguous sets of glyphs that are illegible, and actually cannot be deemed Punic or Neo-Punic script.peer-reviewe