We review studies of the statistics of isotropic turbulence in an incompressible fluid at high Reynolds numbers using direct numerical simulation (DNS) from the viewpoint of fundamental physics. The Reynolds number achieved by the largest DNS, with 40963 grid points, is comparable with the largest Reynolds number in laboratory experiments. The high-quality DNS data in the inertial subrange and the dissipative range enable the examination of detailed statistics at small scales, such as the normalized energy-dissipation rate, energy and energy-flux spectra, the intermittency of the velocity gradients and increments, scaling exponents, and flow-field structure. We emphasize basic questions of turbulence, universality in the sense of Kolmogorov's theory, and the dependence of the statistics on the Reynolds number and scale.

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The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from Re λ = 34.6 up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, η. The vorticity stretching motions scale with the Taylor length scale, λ T , while the flow outside the shear layer scales with the integral length scale, L. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is 120η in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of 4λ T shows that transitions in flow structure occur where Re λ ≈ 45 and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is 4λ T in width and height, which is consistent with observations in high Reynolds number flow of a 4λ T wide instantaneous shear layer with many η-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895-929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain † Email address for correspondence: g.e.elsinga@tudelft.nl ‡ Present address: Graduate School of Environmental and Life Science, Okayama University, Okayama 700-8530, Japan. 32 G. E. Elsinga, T. Ishihara, M. V. Goudar, C. B. da Silva and J. C. R. Hunt at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

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