Conference on Turbulence and Interactions TI2006 May June Porquerolles France
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Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France Statistics at Small Scales in DNS of Turbulence – Dependence on Reynolds Number and Length Scale – Y. Kaneda Department of Computational Science and Engineering, Nagoya University, Nagoya, 464-8603, Japan ?Email: ABSTRACT A review is given on results of data-analysis based on recent high-resolution direct numerical simulations (DNSs) of incompressible turbulence in a periodic box with the number of grid points up to 40963. Emphasis is put on the possible universality of small-scale statistics in turbulence at high Reynolds number, as well as on the dependence of the statistics on the Reynolds number and the length scale. 1 INTRODUCTION Turbulence consists of the so-called ‘eddies' with a wide range of length scale. The statistics of turbulence depends on the length scale as well as the Reynolds number Re. According to the celebrated Kolmogorov hypotheses [1], one may assume that at sufficiently high Re 1 and at sufficiently small scale r such that r L, the statistics is universal in the sense that it is insen- sitive to the detail of boundary conditions and forcing at large scale ? L, where L is the charac- teristic length scale of energy containing eddies. This assumption plays a key role in most modern theories and modeling of turbulence.
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Conference on Turbulence and Interactions TI2006, May 29 June 2, 2006, Porquerolles, France
Statistics at Small Scales in DNS of Turbulence – Dependence on Reynolds Number and Length Scale –
Y. Kaneda Department of Computational Science and Engineering, Nagoya University, Nagoya, 4648603, Japan ∗ Email: kaneda@cse.nagoyau.ac.jp
ABSTRACT A review is given on results of dataanalysis based on recent highresolution direct numerical simulations 3 (DNSs) of incompressible turbulence in a periodic box with the number of grid points up to4096. Emphasis is put on the possible universality of smallscale statistics in turbulence at high Reynolds number, as well as on the dependence of the statistics on the Reynolds number and the length scale.
1 INTRODUCTIONdirect numerical simulations (DNSs) with em phasis on theRe and scale dependence of the statistics. We use two kinds of DNS data. One is Turbulence consists of the socalled ‘eddies’ withfrom DNSs of almost isotropic turbulence (Sec a wide range of length scale. The statistics oftions 2 and 3), and the other is from DNSs of turbulence depends on the length scale as wellanisotropic turbulence (Section 4). as the Reynolds numberRe. According to the celebrated Kolmogorov hypotheses [1], one may 2ReDEPENDENCE OFSTATISTICS assume that at sufficiently highRe1and at sufficiently small scalersuch thatrL, the statistics is universal in the sense that it is insen Two series of DNSs of incompressible homoge sitive to the detail of boundary conditions and neous turbulence in a periodic box with the num forcing at large scale∼L, whereLis the charac3 3 ber of grid pointsNup to4096were performed teristic length scale of energy containing eddies. on the Earth Simulator. One is withkmaxη∼1 This assumption plays a key role in most modern and the other is withkmaxη∼2, wherekmax theories and modeling of turbulence. is the highest wavenumber in each simulation, andηis the Kolmogorov length scale [2,3]. The However, little seems known about the implica DNSs are based on a spectral method free from tion of the conditionRe1norrL. For alias error. The total kinetic energy is kept al example, one may ask “AreRe= 10,000and most timeindependent by introducing external L/r= 100large enough or not for the observa forcing in the form of negative viscosity at a low tion of the universality ?” What is lacking here wavenumber range. is the quantitative understanding of theRe and scale dependence of the statistics. The DNS data agree well with the well known scaling, We consider here the possible universality of the 3/4 3/4 1/2 statistics under the light of recent high resolutionN∝Re ,L/η∝Re ,Rλ∝Re ,
