Development and verification of analytical and statistical theory of turbulence

本文的研究目的是发展既适用于各向同性湍流又适用于各向异性湍流的湍流分析和统计理论，并通过数值模拟加以验证，主要研究结果如下：（1）发展了强分层湍流中湍流扩散的分析和统计理论。它是基于一个线性近似（快速畸变理论）和Corrsin的猜想，并阐明了强烈的层结抑制垂直湍流扩散的机制。它还表明，一个异常的湍流扩散，不能解释现有的湍流模型，可能会发生在湍流与简单的剪切运动的平均流量。这些理论结果得到了数值模拟的证实。(2)应用拉格朗日重整化近似（LRA），证明了在二维湍流中可能存在一种新的能谱，这种能谱不能用Kraichnan-Batchelor-利斯（KBL）理论来解释，但仍具有KBL所预言的k^3能谱。这是一个众所周知的模型，用于旋转球体如行星上的流体运动，分析了由于波和湍流之间的相互作用而引起的相移。理论计算结果与直接数值模拟结果吻合较好。(3)本文提出了一种新的计算方法，可以有效地估计在湍流传质和传热研究中起关键作用的拉格朗日两时间关联。该方法使用时间幂的泰勒展开式和相应的帕德近似。计算结果表明，无论是对单粒子关联还是对两粒子关联，无论是对各向同性还是各向异性轴对称湍流，该方法都能很好地工作，其结果与DNS方法符合得很好。

The purpose of this study is to develop analytical and statistical theories of turbulence that are practically applicable not only to isotropic, but also to anisotropic turbulence, and to verify them by numerical simulations.The results of the study include the followings :(1) An analytical and statistical theory is developed for turbulent diffusion in strongly stratified turbulence. It is based on a linearized approximation (Rapid Distortion Theory) and Corrsin's conjecture, and clarifies an mechanism of suppression of vertical turbulent diffusion by strong stratification. It is also shown that an anomalous turbulent diffusion that could not be explained by existing turbulence models, may occur in turbulence with a mean flow of simple shearing motion. These theoretical results were confirmed by numerical simulations.(2) By applying the Lagrangian Renormalized Approximation (LRA) , it is shown that a new kind of energy spectrum in two-dimensional turbulence may exist, which is not explained by the theory of Kraichnan-Batchelor-Leith (KBL) , but still exhibits the k^3 spectrum as predicted by KBL.By constructing a Lagrangian statistical theory for the beta-plane turbulence, which is a well known model for the fluid motion on a rotating sphere such as a planet, an analysis is made of the phase shift due to the interaction between waves and turbulence. The theoretical results were shown to be in good agreement with direct numerical simulations (DNS).(3) A new computational method is proposed for efficiently estimating Lagrangian two-time correlations that play the key roles in the study of mass and heat transfer in turbulence. The method uses Taylor expansions in powers of time and corresponding Pade approximations. It is shown thet the method works well and its results are in good agreement with DNS, not only for single but also two-particle correlation, and not only for isotropic but also anisotropic axisymmetric turbulence.