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

概率密度函数的数学结构和湍流的间歇性以及大规模并行数值计算。

基本信息

批准号：
12640118
负责人：
GOTOH Toshiyuki
金额：
$ 2.3万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

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

1. 通过大规模并行数值计算，得到了R_λ=460处的稳定均匀各向同性湍流，空间分辨率N=1024^3。从DNS数据库中收集了各种统计数据，并与湍流理论进行了比较。在湍流动力学史上首次观测到小而有限宽度的动能惯性范围。得到了与实验数据相符的Kolmogorov常数为1.64，并计算了速度增量的各种结构函数。计算了结构函数的标度指数，发现与现象学理论计算的结果一致。横向结构函数在4阶以上的标度指数小于纵向结构函数。这些发现发表在2002年3月在圣达菲举行的统计流体力学国际研讨会上。更多关于结构功能各向异性的SO(3)分析从那时开始，并一直持续到现在。到目前为止得到的结果非常积极地支持了我们以前的发现。在各向异性方面也有新的发现，但需要进行理论分析。已有许多研究将Tsallis统计应用于湍流。Tsallis统计是熵的非泛化性质，并期望对湍流的统计性质有所启示。速度增量的标度指数和速度增量和拉格朗日加速度的概率密度函数是该理论的应用对象。kraichnan博士和我对这些应用进行了严格的研究。目前我们对这些研究的结论是否定的，因为湍流在多个自由度之间具有很强的耦合性，这与Tsallis统计的非加性不一致，而且湍流的本质——能量级联在理论中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Navier-Stokes （NS）方程各项之间更为定量的关系，我们研究了由NS方程导出的高阶结构函数方程。它包含需要封闭的压力-速度相关项。我们已经研究了压强对条件平均的贡献。结果表明，条件平均是速度增量的二次函数。提出了一个基于伯努利定理的理论模型来解释它，该理论的含义是同一阶的一些缩放指数是相同的，与DNS观测不同。目前正在国际合作下进行进一步研究。少