Analysis of Universal Structure and Development of Statistical Theory of Inhomogeneous Anisotropic Turbulence

本研究的目的是了解湍流的普遍结构，这对于发展非均匀各向异性湍流的统计理论是必不可少的。本文基于湍流统计理论和大尺度直接数值模拟（DNS）的方法，阐明了以下问题：(A)湍流的DNS和数据分析：通过对网格点数高达4096^3、自由度大于10^11、非线性相互作用的世界最大的湍流DNS数据的分析，阐明了(i)各种统计量的Reynolds（Re）依赖性和（ii）高Re极限下的渐近通约性。我们还对（iii）高Re下Kolmogorov的相似性假设和（iv）大涡模拟中的涡流-黏性假设进行了定量检验。(B)各向异性湍流的通用结构和建模：我们在分析拉格朗日谱闭合方程的基础上推导了均匀湍流剪切流的小尺度各向异性谱表达式。我们还推导了层状湍流中各向异性谱的一般表达式，并通过DNS证实了其形式，并证明了类似于非平衡系统线性响应理论的思想的有效性。(C)壁面有界湍流的DNS和数据分析：平面泊泽维尔湍流是边界层湍流或固体壁面湍流中最基本的湍流之一。为了实现DNS，我们分别基于Chebyshev-tau方法和组合压缩方案编写了代码，并验证了它们的性能。利用DNS，通过分析墙-法向速度相关谱，明确了涡旋尺度分布对墙-法向距离和Re的依赖关系。

The aim of this study is to understand the universal structure of turbulence, which is indispensable for developing statistical theory of inhomogeneous anisotropic turbulence. On the basis of the method of statistical theory of turbulence and large scale Direct Numerical Simulation(DNS), the study has clarified the followings :(A)DNS of Turbulence and Data-Analysis :By analyzing the data of the world largest DNS of turbulence with the number of grid points up to 4096^3, more than 10^11 degrees of freedom and 10^22 nonlinear interactions, we clarified some of (i)Reynolds(Re)-dependence of various statistics, and (ii)asymptotic universality in the high Re-limit. We also performed quantitative examination of (iii)Kolmogorov's similarity hypotheses, and (iv)the eddy-viscosity hypothesis in Large Eddy Simulation, at high Re.(B)Universal Structure and Modeling of Anisotropic Turbulence :We derived an expression for the anisotropic spectrum at small scales of homogeneous turbulent shear flow on the basis of the analysis of a Lagrangian spectral closure equation. We also derived a general expression for the anisotropic spectrum in stratified turbulence, confirmed its form by DNS, and showed the validity of an idea similar to that of the linear responses theory for non-equilibrium system.(C)DNS of Wall Bounded Turbulence and Data-Analysis :Plane Poiseuille turbulence is one of the most basic turbulences among boundary layer turbulences or turbulences with solid wall. In order to perform the DNS, we made codes based on the Chebyshev-tau method and a Combined Compact Scheme, respectively, and validated their performance. By the DNS, we clarified the dependence on the wall normal distance and Re of eddy-scale distribution, by analyzing the wall-normal velocity correlation spectrum.