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.

A modified nonlinear sub-grid scale model for large eddy simulation with application to rotating turbulent channel flows Phys. Fluids 24, 075113 (2012) Scaling range of velocity and passive scalar spectra in grid turbulence Phys. Fluids 24, 075101 (2012) On Lagrangian single-particle statistics Phys. Fluids 24, 055102 (2012) The length distribution of streamline segments in homogeneous isotropic decaying turbulence Phys. Fluids 24, 045104 (2012) Conditional vorticity budget of coherent and incoherent flow contributions in fully developed homogeneous isotropic turbulence Phys. Fluids 24, 035108 (2012) Additional information on Phys. Fluids

A method of renormalized expansions in theory of turbulence is developed with the use of the Lagrangian position function. The introduction of this function makes it easy to express the Lagrangian development of the velocity field. A simple truncation of a set of renormalized expansions is shown to lead to an approximation which is compatible with Kolmogorov's inertial range energy spectrum.

No abstract

A dynamical analysis is made of the Lagrangian and Eulerian two-time velocity correlations QL and QE for small time difference in turbulence at very high Reynolds number. The short-time analysis yields a refinement of the so-called random sweeping model approximation for the Eulerian correlation QE. An estimate of the Lagrangian frequency spectrum of the Lagrangian correlation QL in the inertial subrange is derived on the basis of a deductive Lagrangian renormalized approximation (LRA), which is consistent with the short-time analysis. Estimates of some nondimensional constants associated with the Eulerian and Lagrangian time microscales are also obtained by making use of the energy spectrum predicted by the LRA.