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

Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box [0,L]3 is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t)=(ϖ−10Ωm)αm for 1≤m≤∞ where αm=2m/(4m−3) and [Ωm(t)]2m=L−3∫V|ω|2mdV with ϖ0=νL−2. All four simulations unexpectedly show that the Dm are ordered for m=1,…,9 such that Dm+1<Dm. Moreover, the Dm squeeze together such that Dm+1/Dm↗1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 40963

Vortex stretching as a mechanism for quantum kinetic energy decay

A pair of perturbed anti-parallel quantum vortices, simulated using the three-dimensional Gross-Pitaevskii equations, is shown to be unstable to vortex stretching. This results in kinetic energy K∇ψ being converted into interaction energy EI and eventually local kinetic energy depletion that is similar to energy decay in a classical uid, even though the governing equations are Hamiltonian and energy conserving. The intermediate stages include: the generation of vortex waves, their deepening, multiple reconnections, the emission of vortex rings and phonons and the creation of an approximately -5/3 kinetic energy spectrum at high wavenumbers. All the wave generation and reconnection steps follow from interactions between the two original vortices, unlike the self- interactions in vortex wave models. A four vortex example is given to demonstrate that some of these steps might be general

The growth of vorticity moments in the Euler equations

A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced profile for the initial magnitude of vorticity along with a new algorithm for the initial vorticity direction. The new analysis uses a new adaptation to the Euler equations of a rescaling of the vorticity moments developed for Navier-Stokes analysis. All rescaled moments grow in time, with the lower-order moments bounding the higher-order moments from above, consistent with new results from several Navier-Stokes calculations. Furthermore, if, as an inviscid flow evolves, this ordering is assumed to hold, then a singular upper bound on the growth of these moments can be used to provide a prediction of power law growth to compare against. There is a significant period where the growth of the highest moments converges to these singular bounds, demonstrating a tie between the strongest nonlinear growth and how the rescaled vorticity moments are ordered. The logarithmic growth of all the moments are calculated directly and the estimated singular times for the different Dm converge to a common value for the simulation in the best domai

Swirling, turbulent vortex rings formed from a chain reaction of reconnection events

Long, straight, anti-parallel vortex tubes, with balanced profiles and a local perturbation, are simulated using the Navier-Stokes equations and evolve into a chain of spiral vortex rings with the characteristics of three-dimensional turbulence. This includes evidence of a cascade of energy to high wavenumbers, the formation of a k −5/3 inertial subrange and a new hierarchy of rescaled vorticity moments where, against expectations, the lower-order moments bound the higher-order moments. This order holds for all times even as the individual moments fluctuate significantly and could explain the observation that ratios converge faster than the individual moments in very high Reynolds number forced turbulence simulations. The transformation of the original pair of vortices into turbulent swirling vortex rings is outlined by describing first the twists and turns of the first two reconnection steps in detail, next how these create the first set of vortex rings, and finally the formation of the additional reconnections and stretched, swirling rings that lead to turbulence. The k −5/3 spectrum is interpreted in terms of a model of stretched, spiral vortices similar to those seen in these simulations

Helicity generation in three-dimensional Euler and turbulence

Helicity produced by nearly singular vortex interactions is shown to play a role in the ensuing development of turbulence. This might provide a link between turbulence and the dynamics of the three-dimensional Euler equations. where numerical evidence has suggested that there might be a singularity. Interactions between regions of oppositely signed helicity in both physical and Fourier space are shown to be associated with the transfer of energy to small scales and the formation of vortex tubes, both being properties of fully, developed turbulence

The ever-elusive blowup in the mathematical description of fluids

This contribution introduces you to the Euler equations of ideal fluids and the Navier–Stokes equations which govern fully developed turbulent flows. We describe some of the unresolved mathematical issues, including the “Navier–Stokes millennium problem”, and the role numerical simulations play in developing this field

Evidence for a mid-latitude, mesoscale downscale energy cascade from the marine boundary layer

The third-order longitudinal velocity structure function SLLL and the longitudinal and transverse second order structure functions SLL (r) and ST T (r) are determined for the marine boundary layer over the Pacific Ocean. The velocities are the 10 meter equivalent neutral winds inferred from measurements of radar backscatter by the scatterometer on the QuikSCAT satellite. The sign of SLLL is used to infer the direction of the energy cascade, with SLLL 0 indicating an upscale energy cascade. This is analogous to how SLLL would behave in three- and two- dimensional turbulence respectively, but does not imply that either of these dynamical systems is completely applicable. In order to distinguish different geophysical effects, the analysis of structure functions is divided into latitudinal and longitudinal boxes. In this paper we concentrate on mid-latitude and mid-longitudes over the North Pacific Ocean, because these boxes were found to be least influenced by the equatorial waveguide and large sea surface temperature gradients. The overall conclusion is that for scales r less than 1000 km SLLL 0 is favored. In this sense the results are consistent with upper tropospheric aircraft measurements and recent numerical results for the developing theory of stratified, rotating tur- bulence