Recovering isotropic statistics in turbulence simulations: The Kolmogorov 4/5th law

Abstract

One of the main benchmarks in direct numerical simulations of three-dimensional turbulence is the Kolmogorov 1941 prediction for third-order structure functions with homogeneous and isotropic statistics in the infinite-Reynolds number limit. Previous DNS techniques to obtain isotropic statistics have relied on time-averaging structure functions in a few directions over many eddy turnover times, using forcing schemes carefully constructed to generate isotropic data. Motivated by recent theoretical work which removes isotropy requirements by spherically averaging structure functions over all directions, we will present results which supplement long-time averaging by angleaveraging over up to 73 directions from a single flow snapshot. The directions are among those natural to a square computational grid, and are weighted to approximate the spherical average.The averaging process is cheap, and for the Kolmogorov 1941 4/5ths law, reasonable results can be obtained from a single snapshot of data. This procedure may be used to investigate the isotropic statistics of any quantity of interest.

The Lagrangian exploration module: An apparatus for the study of statistically homogeneous and isotropic turbulence

“…The currently accepted value for C 2 = 2.1 was used. 30 Although these scaling relations are strictly applicable only to statistically homogeneous and isotropic turbulence, experience 31 shows that they work well even in other situation. As we will find, the LEM can produce statistically homogeneous and isotropic turbulence.…”

Energy flux fluctuations in a finite volume of turbulent flow

“…The recent theoretical [9,10,11] and numerical work [12] leads to a generalisation of Eq. 1, which is local in space and time.…”

“…The spherical average in Eq. 2 permits the recovery of the isotropic sector from an arbitrary (anisotropic) flow [12].…”

Power laws and inverse motion modelling: application to turbulence measurements from satellite images

“…Besides providing a closure for motion estimation, power law 2 The hypothesis of isotropy can be relaxed; see, e.g., Taylor et al (2003). The method to recover isotropic statistics described there motivates the use of different directions for the displacements in Section 3.1.…”