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

1.通过大规模并行数值计算，获得了R_λ=460时的定常各向同性湍流，空间分辨率为N=10243。在湍流数值模拟的历史上，首次观测到小而有限宽度的动能惯性范围。Kolmogorov常数为1.64，与实验数据符合较好，并计算了各种速度增量的结构函数。计算了结构函数的标度指数，发现与唯象理论计算的标度指数一致。研究还发现，高于4阶的横向结构函数的标度指数小于纵向结构函数的标度指数。2002年3月，在圣达菲举行的统计流体动力学国际研讨会上宣读了这些研究结果。与Bifela…教授合作从那时起，对结构函数各向异性的SO(3)分析开始了更多的研究，至今仍在继续。到目前为止所获得的结果是非常积极的，支持我们之前的发现。此外，还得到了关于各向异性的新发现，但还需要理论分析。已有许多研究将Tsallis统计应用于湍流。Tsallis的统计数据是熵的不可扩展的性质，预计将揭示湍流的统计性质。速度增量的标度指数、速度增量和拉格朗日加速度的概率密度函数是理论应用的对象。Kraichnan博士和我对这些应用程序进行了严格的检查。目前我们的结论对这些研究是否定的，因为湍流具有多个自由度之间的强耦合，这与Tsallis统计量的非可加性是不一致的，并且因为湍流的本质-能量级联在理论上没有得到适当的描述。这种合作仍在进行中，预计会有进一步的发展。为了得到NS方程各项之间更定量的关系，我们研究了由NS方程导出的高阶结构函数方程。它包含需要闭合的压力-速度关联项。我们用条件平均值研究了压力的贡献。结果表明，在速度增量下，条件平均值为二次函数。提出了一个基于Bernoulli定理的理论模型来解释这一现象，该理论的含义是同一阶的一些标度指数是相同的，这与DNS的观测结果不同。与国际合作的进一步研究目前正在进行中。较少

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