Eulerian and Lagrangian renormalization in turbulence theory
Eulerian and Lagrangian renormalization in turbulence theory
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湍流理论中的欧拉和拉格朗日重正化
DOI：
10.1017/s0022112077001232
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发表时间：
1977
影响因子：
3.7
通讯作者：
R. Kraichnan
中科院分区：
文献类型：
--
作者：
R. Kraichnan
构造了湍流和湍流对流在随机伽利略变换下各阶不变的系统重归一化摄动展开式。发展了两种类型的展开式，其最低截断分别给出了拉格朗日历史直接相互作用近似和简化拉格朗日历史直接相互作用近似。这些近似以前是作为欧拉直接相互作用近似的启发式修正推导出来的（Kraichnan 1965）。所使用的技术包括对广义速度场u（x, t/s）的原始扰动展开的反演，定义为在时间t时通过x的流体单元在时间s时测量的速度。新的展开通过应用于随机线性振荡器，随机速度的被动标量对流和拉格朗日速度协方差来说明。被动标量展开的最低项给出了Taylor（1921）关于流体单元色散的精确结果，而较高的项描述了粒子位移分布与高斯形式的偏差。在所有的应用程序中，假定的基础统计比高斯统计更普遍，这是一种特殊情况。
Systematic renormalized perturbation expansions for turbulence and turbulent convection are constructed which are invariant at each order under random Galilean transformations. Two types of expansion are developed whose lowest truncations give, respectively, the Lagrangian-history direct-interaction approximation and the abridged Lagrangian-history direct-interaction approximation. These approximations previously were derived as heuristic modifications of the Eulerian direct-interaction approximation (Kraichnan 1965). The techniques used involve reversion of primitive perturbation expansions for the generalized velocity field u(x, t/s), defined as the velocity measured at time s in the fluid element which passes through x at time t. The new expansions are illustrated by application to a random linear oscillator, to passive-scalar convection by a random velocity and to the Lagrangian velocity covariance. The lowest term of the expansion for the passive scalar gives Taylor's (1921) exact result for dispersion of fluid elements, and higher terms describe the deviations of the particle-displacement distribution from Gaussian form. In all the applications the assumed underlying statistics are more general than Gaussian statistics, which appear as a special case.