Mathematical Structure of the Probability Density Function and Intermittency in Turbulence and Massive Parallel Numerical Computation.

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1. Steady homogeneous isotropic turbulence at R_λ=460 was obtained by using massive parallel numerical computation with the spatial resolution N=1024^3. Various statistical data were gathered from the DNS data base and compared with turbulence theories. The inertial range of the kinetic energy with small but finite width was observed for the first time in the history of DNS of turbulence. The Kolmogorov constant was found to be 1.64 in agreement with experimental data, and various kinds of structure functions for the velocity increments were also computed. The scaling exponents of the structure functions were computed and found to be in agreement with those computed by phenomenological theories. It was also found that the scaling exponents of the transverse structure functions at the order higher than 4 are smaller than those of the longitudinal ones. These findings was read at International Workshop on Statistical Hydrodynamics at Santa Fe, March 2002. A collaboration with Prof.Bifela … More re on the SO(3) analysis for the anisotropy of the structure functions has begun since then, and been continuing by now. The results so far obtained are very positive to support our previous findings. Also new findings regarding to the anisotropy are obtained, but need theoretical analysis.2. There have been many studies which apply the Tsallis statistics to turbulence. The Tsallis statistics is nonextensitve nature for the Entropy, and expected to shed some lights on the statistical nature of the Turbulence. Scaling exponents of the velocity increments and probability density functions of the velocity increments and the Lagrangian acceleration are the objects for the theory to be applied. Dr.Kraichnan and myself have critically examined those applications. Currently our conclusion is negative to those studies because the turbulence has strong coupling among the many degrees of freedom which is inconsistent with the non-additiveness of the Tsallis statistics, and because the energy cascade, the essence of turbulence, is not properly described in the theory. This collaboration is still under way and further development will be expected.3. In order to obtain more quantitative relation among various terms of Navier-Stokes (NS) equation, we have examined the equation of higher order structure functions derived from the NS equation. It contains the pressure-velocity correlation term which needs a closure. We have studied the pressure contributions in terms of the conditional average. It was found that the conditional average is of the quadratic function in the velocity increments. A theoretical model based on the Bernoulli theorem was proposed to explain it, The implication of this theory is that some of the scaling exponents at the same order are identical, differing from the DNS observation. Further study with international collaboration is now under way. Less