We use data from well-resolved direct numerical simulations at Taylor-scale Reynolds numbers from 140 to 1000 to study the statistics of energy dissipation rate and enstrophy density (i.e. the square of local vorticity). Despite substantial variability in each of these variables, their extreme events not only scale in a similar manner but also progressively tend to occur spatially together as the Reynolds number increases. Though they possess non-Gaussian tails of enormous amplitudes, ratios of some characteristic properties can be closely linked to those of isotropic Gaussian random fields. We present results also on statistics of the pressure Laplacian and conditional mean pressure given both dissipation and enstrophy. At low Reynolds number intense negative pressure fluctuations are preferentially associated with rotation-dominated regions but at high Reynolds number both high dissipation and high enstrophy have similar effects.

As computing power has grown and turbulent flows at increasing Reynolds numbers are being computed by direct numerical simulations, conventional assumptions on adequate spatial and temporal resolutions are being continually challenged. We perform a systematic study of the resolution effects via numerical simulations at various spatial and temporal resolutions to clarify the proper scaling of dissipation and enstrophy (vorticity squared). Results show that inadequate resolution in space and/or time leads to overestimation of the likelihood and intensity of extreme fluctuations in dissipation and enstrophy. In order to capture rare events accurately, for instance, one needs to have not only grid resolutions that are increasingly smaller fractions of the Kolmogorov scale as the Reynolds number increases but also the Courant number that becomes increasingly smaller than that assumed to be adequate previously. Some comparisons are made with results obtained from an alternative approach where a stricter criterion for truncation in wave-number space allows aliasing errors to be removed completely. In contrast to prior work, the present data do not support the notion that dissipation and enstrophy probability density functions (PDFs) approach each other in the far tails at high Reynolds number. However the two PDFs are remarkably similar in form, being well described by stretched exponentials.

As the US Department of Energy (DOE) computing facilities began deploying petascale systems in 2008, DOE was already setting its sights on exascale. In that year, DARPA published a report on the feasibility of reaching exascale. The report authors identified several key challenges in the pursuit of exascale including power, memory, concurrency, and resiliency. That report informed the DOE's computing strategy for reaching exascale. With the deployment of Oak Ridge National Laboratory's Frontier supercomputer, we have officially entered the exascale era. In this paper, we discuss Frontier's architecture, how it addresses those challenges, and describe some early application results from Oak Ridge Leadership Computing Facility's Center of Excellence and the Exascale Computing Project.

Sustained and rapid advances in computing have enabled the conduct of direct numerical simulations (DNS) at increasing problem sizes and higher levels of physical realism, which have in turn contributed to many advances in understanding turbulence. However, continuing and future success at the “extreme-scale” level will likely require new algorithms adapted to emerging heterogeneous architectures, and even then, long simulations at extreme problem sizes are probably still too costly. In this paper we first describe the essential elements of an asynchronous parallel algorithm for DNS of incompressible isotropic turbulence, which scales effectively up to $18{432}^3$ resolution (more than 6 trillion grid points) on a world-class IBM-NVIDIA CPU-GPU machine. We then propose a simulation paradigm, built on the idea that, for physical quantities of short timescales, sampling over well-separated snapshots in a long simulation at high resolution can be replaced by sampling over short simulation segments with a high degree of statistical independence evolved from snapshots at modest or even low resolution. The total computational cost is now counted in Kolmogorov timescales instead of large-eddy timescales, leading to tremendous savings at high Reynolds number. This “Multiple Resolution Independent Simulations” (MRIS) approach is validated through a series of comparisons, and subsequently applied to obtain results on fine-scale intermittency, at Taylor-scale Reynolds numbers 390 to 1300, with grid spacing smaller than the Kolmogorov length scale. The results show conclusively that extreme fluctuations of the dissipation rate are usually accompanied by extreme enstrophy, while extreme enstrophy is usually accompanied by less-intense dissipation. Statistics of the locally averaged dissipation and enstrophy suggest these two variables scale together at inertial-range scale sizes (but not in the dissipation range). Finally, brief remarks are made concerning perspectives on likely major challenges in an exascale future, and several other topics of study where the MRIS approach may be useful.

Statistics of the trajectories of molecules diffusing via Brownian motion in a turbulent flow are extracted from simulations of stationary isotropic turbulence, using a postprocessing approach applicable in both forward and backward reference frames. Detailed results are obtained for Schmidt numbers ( $Sc$ ) from 0.001 to 1000 at Taylor-scale Reynolds numbers up to 1000. The statistics of displacements of single molecules compare well with the earlier theoretical work of Saffman (J. Fluid Mech. vol. 8, 1960, pp. 273–283) except for the scaling of the integral time scale of the fluid velocity following the molecular trajectories. For molecular pairs we extend Saffman’s theory to include pairs of small but finite initial separation, which is in excellent agreement with numerical results provided that data are collected at sufficiently small times. At intermediate times the separation statistics of molecular pairs exhibit a more robust Richardson scaling behaviour than for the fluid particles. The forward scaling constant is very close to 0.55, whereas the backward constant is approximately 1.53–1.57, with a weak Schmidt number dependence, although no scaling exists if $Sc\ll 1$ at the Reynolds numbers presently accessible. An important innovation in this work is to demonstrate explicitly the practical utility of a Lagrangian description of turbulent mixing, where molecular displacements and separations in the limit of small backward initial separation can be used to calculate the evolution of scalar fluctuations resulting from a known source function in space. Lagrangian calculations of the production and dissipation rates of the scalar fluctuations are shown to agree very well with Eulerian results for the case of passive scalars driven by a uniform mean gradient. Although the Eulerian–Lagrangian comparisons are made only for $Sc\sim O(1)$ , the Lagrangian approach is more easily extended to both very low and very high Schmidt numbers. The well-known scalar dissipation anomaly is accordingly also addressed in a Lagrangian context.