LES一个方程

李东岳

一个朋友要我退一下这个方程，就是这个 $S^{2}$ 怎么来的，我先发在这里，谁能看看不？

李东岳

简单推了一下没想到这个是个坑这么复杂文章里面一笔带过我这个用的2维的推了一下后来发现有个乘以2那个忘记乘了中间还需要调用连续性方程有空我在电脑上重新写一下

李东岳

整理一下

%(#ff0000)[问题：证明均一湍流有 $S^{2} = \frac{1}{2} {(\frac{\partial u_{i}}{\partial x_{j}})}^{2}$ ]

推导：

为方便讨论，考虑二维情况，
$\begin{matrix} (1) & S^{2} = S_{i j} S_{i j} \end{matrix}$
$\begin{matrix} (2) & S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) \end{matrix}$
将方程 $(2)$ 代入到 $(1)$ 有
$\begin{matrix} (3) & S^{2} = S_{i j} S_{i j} = S_{11} S_{11} + S_{12} S_{12} + S_{21} S_{21} + S_{22} S_{22} \end{matrix}$
$\begin{matrix} (4) & = {(\frac{\partial u_{1}}{\partial x_{1}})}^{2} + \frac{1}{4} {(\frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}})}^{2} + \frac{1}{4} {(\frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}})}^{2} + {(\frac{\partial u_{2}}{\partial x_{2}})}^{2} \end{matrix}$
$\begin{matrix} (5) & = {(\frac{\partial u_{1}}{\partial x_{1}})}^{2} + \frac{1}{2} {(\frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}})}^{2} + {(\frac{\partial u_{2}}{\partial x_{2}})}^{2} \end{matrix}$
$\begin{matrix} (6) & = {(\frac{\partial u_{1}}{\partial x_{1}})}^{2} + \frac{1}{2} ({(\frac{\partial u_{1}}{\partial x_{2}})}^{2} + 2 \frac{\partial u_{1}}{\partial x_{2}} \frac{\partial u_{2}}{\partial x_{1}} + {(\frac{\partial u_{2}}{\partial x_{1}})}^{2}) + {(\frac{\partial u_{2}}{\partial x_{2}})}^{2} \end{matrix}$
$\begin{matrix} (7) & = {(\frac{\partial u_{1}}{\partial x_{1}})}^{2} + \frac{1}{2} {(\frac{\partial u_{1}}{\partial x_{2}})}^{2} + \frac{\partial u_{1}}{\partial x_{2}} \frac{\partial u_{2}}{\partial x_{1}} + \frac{1}{2} {(\frac{\partial u_{2}}{\partial x_{1}})}^{2} + {(\frac{\partial u_{2}}{\partial x_{2}})}^{2} \end{matrix}$

李东岳

这个跟湍流动能耗散率和形变率的关系有关，应该不仅仅是LES的

李东岳

此问题同类于湍流动能耗散率的定义。湍流动能耗散率的定义为：
$\begin{matrix} (8) & ε = 2 ν_{t} \bar{S_{i j} S_{i j}} \end{matrix}$
经过各向同性假定后有：
$\begin{matrix} (9) & ε = ν_{t} \bar{\frac{\partial u_{i}^{'}}{\partial x_{k}} \frac{\partial u_{i}^{'}}{\partial x_{k}}} \end{matrix}$

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LES一个方程