目录

• 一、雷诺分解
• 二、平均运算的性质
• 三、雷诺方程的推导
• 四、脉动运动方程的推导
• 五、雷诺应力输运方程的推导
• 六、参考资料

一、雷诺分解

对于湍流而言，其流动变量$\varphi$具有不规则性，常常将其分解为平均量与脉动量之和，即$$\begin{array}{rl}\varphi & =\overline{\varphi }+{\varphi }^{\prime }=\Phi +{\varphi }^{\prime }\\ \varphi & =⟨\varphi ⟩+{\varphi }^{\prime }\end{array}$$平均量常常有系综平均、时间平均和空间平均三种形式，这里不展开介绍其中的细节，只给出前两种的表达式：

• 系综平均$$⟨\varphi ⟩={\int }_{-\infty }^{\infty }\varphi p\left(\varphi \right)\mathrm{d}\varphi .$$其中$p\left(\varphi \right)$是概率密度。
• 时间平均：$$\Phi =\overline{\varphi }=\frac{1}{\Delta t}{\int }_0^{\Delta t}\varphi \mathrm{d}t.$$

二、平均运算的性质

这里简要列举平均运算(系综平均、时间平均、空间平均)的性质，其对于推导雷诺方程及雷诺应力输运方程至关重要：$$\begin{array}{rl}\overline{\overline{f}}& =\overline{f}\\ \overline{g\overline{f}}& =\overline{g}\overline{f}\\ \overline{g+f}& =\overline{g}+\overline{f}\\ \overline{f^{\prime }}& =0\\ \overline{fg}& =\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}\\ \overline{\frac{\partial f}{\partial x}}& =\frac{\partial \overline{f}}{\partial x}\\ \overline{\frac{\partial f}{\partial t}}& =\frac{\partial \overline{f}}{\partial t}\\ \overline{g^{\prime }\overline{f}}& =0\end{array}$$上面这些性质对于系综平均是同样适用的，例如：$⟨g^{\prime }⟨f⟩⟩=0、⟨\frac{\partial f}{\partial x}⟩=\frac{\partial ⟨f⟩}{\partial x}$。这些性质是简单的积分运算，推导相对容易，这里只证明其中两项：$$\begin{array}{rl}\overline{g^{\prime }\overline{f}}& =\overline{g^{\prime }}\cdot \overline{\overline{f}}\\ & =\overline{g^{\prime }}\cdot \overline{f}\\ & =0\\ \overline{fg}& =\overline{\left(\overline{f}+f^{\prime }\right)\left(\overline{g}+g^{\prime }\right)}\\ & =\overline{\overline{f}\overline{g}+\overline{f}{g}^{\prime }+f^{\prime }\overline{g}+f^{\prime }{g}^{\prime }}\\ & =\overline{\overline{f}\overline{g}}+\overline{\overline{f}{g}^{\prime }}+\overline{f^{\prime }\overline{g}}+\overline{f^{\prime }{g}^{\prime }}\\ & =\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}\end{array}$$更为详细的推导可以参考博文：湍流模型(2)——雷诺平均方程。

三、雷诺方程的推导

不可压缩牛顿型流体的NS方程为$$\frac{\partial u_i}{\partial x_i}=0\text{\hspace{1em}(1)}$$$$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}+f_i\text{\hspace{1em}(2)}$$在下面的推导中我们暂时不考虑体积力项$f_i$，即只考虑$$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}\text{\hspace{1em}(2)}$$对公式$(1)$、$(2)$作平均运算(系综平均)：$$⟨\frac{\partial u_i}{\partial x_i}⟩=0\text{\hspace{1em}(3)}$$ $$⟨\frac{\partial u_i}{\partial t}⟩+⟨u_j\frac{\partial u_i}{\partial x_j}⟩=⟨-\frac{1}{\rho }\frac{\partial p}{\partial x_i}⟩+⟨\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}⟩\text{\hspace{1em}(4)}$$由平均运算的性质有：$$⟨\frac{\partial u_i}{\partial x_i}⟩=\frac{\partial ⟨u_i⟩}{\partial x_i}=0\text{\hspace{1em}(5)}$$同理：
$$\begin{array}{rl}⟨\frac{\partial u_i}{\partial t}⟩& =\frac{\partial ⟨u_i⟩}{\partial t}\\ ⟨-\frac{1}{\rho }\frac{\partial p}{\partial x_i}⟩& =-\frac{1}{\rho }\frac{\partial ⟨p⟩}{\partial x_i}\\ ⟨\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}⟩& =\nu \frac{{\partial }^2⟨u_i⟩}{\partial x_j\partial x_j}\end{array}$$对于$⟨u_j\frac{\partial u_i}{\partial x_j}⟩$则有：$$\begin{array}{rl}⟨u_j\frac{\partial u_i}{\partial x_j}⟩& =⟨\frac{\partial u_i{u}_j}{\partial x_j}-u_i\frac{\partial u_j}{\partial x_j}⟩\\ & =⟨\frac{\partial u_i{u}_j}{\partial x_j}⟩-⟨u_i\frac{\partial u_j}{\partial x_j}⟩\\ & =⟨\frac{\partial u_i{u}_j}{\partial x_j}⟩\\ & =\frac{\partial ⟨u_i{u}_j⟩}{\partial x_j}\\ & =\frac{\partial ⟨u_i⟩⟨u_j⟩}{\partial x_j}+\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}\\ & =⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}+\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}\end{array}$$上面的推导用到了$\frac{\partial ⟨u_j⟩}{\partial x_j}=0$、$\frac{\partial u_j}{\partial x_j}=0$、$⟨u_i{u}_j⟩=⟨u_i⟩⟨u_j⟩+⟨u_i^{\prime }{u}_j^{\prime }⟩$。整理以上各项可得：$$\frac{\partial ⟨u_i⟩}{\partial t}+⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}+\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}=-\frac{1}{\rho }\frac{\partial ⟨p⟩}{\partial x_i}+\nu \frac{{\partial }^2⟨u_i⟩}{\partial x_j\partial x_j}$$将$\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}$移到右边即有雷诺方程：$$\frac{\partial ⟨u_i⟩}{\partial t}+⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}=-\frac{1}{\rho }\frac{\partial ⟨p⟩}{\partial x_i}+\nu \frac{{\partial }^2⟨u_i⟩}{\partial x_j\partial x_j}-\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}\text{\hspace{1em}(6)}$$式中的$-⟨u_i^{\prime }{u}_j^{\prime }⟩$ 乘上密度$\rho$便是雷诺应力$-\rho ⟨u_i^{\prime }{u}_j^{\prime }⟩$，可以写成张量形式：$$\left(\begin{array}{ccc}-\rho ⟨u^{\prime 2}⟩& -\rho ⟨u^{\prime }{v}^{\prime }⟩& -\rho ⟨u^{\prime }{w}^{\prime }⟩\\ -\rho ⟨u^{\prime }{v}^{\prime }⟩& -\rho ⟨v^{\prime 2}⟩& -\rho ⟨v^{\prime }{w}^{\prime }⟩\\ -\rho ⟨u^{\prime }{w}^{\prime }⟩& -\rho ⟨v^{\prime }{w}^{\prime }⟩& -\rho ⟨w^{\prime 2}⟩\end{array}\right).$$

四、脉动运动方程的推导

将NS方程$(1)$、$(2)$减去雷诺方程$(5)$、$(6)$，并进行一定的整理即可得到脉动运动方程：$$\frac{\partial u_i^{\prime }}{\partial x_i}=0\text{\hspace{1em}(7)}$$$$\frac{\partial u_i^{\prime }}{\partial t}+⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p^{\prime }}{\partial x_i}+\nu \frac{{\partial }^2{u}_i^{\prime }}{\partial x_j\partial x_j}-\frac{\partial }{\partial x_j}\left(u_i^{\prime }{u}_j^{\prime }-⟨u_i^{\prime }{u}_j^{\prime }⟩\right)\text{\hspace{1em}(8)}$$下面逐项进行推导：$$\frac{\partial u_i}{\partial x_i}-\frac{\partial ⟨u_i⟩}{\partial x_i}=\frac{\partial \left(u_i-⟨u_i⟩\right)}{\partial x_i}=\frac{\partial u_i^{\prime }}{\partial x_i}$$故有：$$\frac{\partial u_i^{\prime }}{\partial x_i}=0\text{\hspace{1em}(9)}$$同理：$$\frac{\partial u_i}{\partial t}-\frac{\partial ⟨u_i⟩}{\partial t}=\frac{\partial \left(u_i-⟨u_i⟩\right)}{\partial t}=\frac{\partial u_i^{\prime }}{\partial t}$$$$-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{1}{\rho }\frac{\partial ⟨p⟩}{\partial x_i}=-\frac{1}{\rho }\frac{\partial \left(p-⟨p⟩\right)}{\partial x_i}=-\frac{1}{\rho }\frac{\partial p^{\prime }}{\partial x_i}$$$$\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}-\nu \frac{{\partial }^2⟨u_i⟩}{\partial x_j\partial x_j}=\nu \frac{{\partial }^2\left(u_i-⟨u_i⟩\right)}{\partial x_j\partial x_j}=\nu \frac{{\partial }^2{u}_i^{\prime }}{\partial x_j\partial x_j}$$另外：$$\begin{array}{rl}{u}_j\frac{\partial u_i}{\partial x_j}-⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}& =\left(⟨u_j⟩+u_j^{\prime }\right)\frac{\partial \left(⟨u_i⟩+u_i^{\prime }\right)}{\partial x_j}-⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}\\ & =⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}+⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}+u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}-⟨u_j⟩\frac{\partial ⟨u_i⟩}{\partial x_j}\\ & =⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}+u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}\\ & =⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}+\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_j}-u_i^{\prime }\frac{\partial u_j^{\prime }}{\partial x_j}\\ & =⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}+\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_j}\end{array}$$上面的推导用到了$\frac{\partial u_j^{\prime }}{\partial x_j}=0$。最后雷诺应力项$-\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}$符号变为正号，整理各项可得：$$\frac{\partial u_i^{\prime }}{\partial t}+⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}+\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p^{\prime }}{\partial x_i}+\nu \frac{{\partial }^2{u}_i^{\prime }}{\partial x_j\partial x_j}+\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_j}$$将$\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_j}$移到右边可得$$\frac{\partial u_i^{\prime }}{\partial t}+⟨u_j⟩\frac{\partial u_i^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p^{\prime }}{\partial x_i}+\nu \frac{{\partial }^2{u}_i^{\prime }}{\partial x_j\partial x_j}-\frac{\partial }{\partial x_j}\left(u_i^{\prime }{u}_j^{\prime }-⟨u_i^{\prime }{u}_j^{\prime }⟩\right)\text{\hspace{1em}(10)}$$

五、雷诺应力输运方程的推导

从脉动运动方程$(10)$出发，在$u_i^{\prime }$脉动方程上乘以$u_j^{\prime }$，$u_j^{\prime }$脉动方程上乘以$u_i^{\prime }$，两式相加后作平均运算，得到雷诺应力输运方程：$$\begin{array}{rl}\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial t}+⟨u_k⟩\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k}=& -⟨u_i^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_j⟩}{\partial x_k}-⟨u_j^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_i⟩}{\partial x_k}-\frac{1}{\rho }\left(⟨u_j^{\prime }\frac{\partial p^{\prime }}{\partial x_i}⟩+⟨u_i^{\prime }\frac{\partial p^{\prime }}{\partial x_j}⟩\right)\\ & +\nu ⟨u_j^{\prime }\frac{{\partial }^2{u}_i^{\prime }}{\partial x_k\partial x_k}+u_i^{\prime }\frac{{\partial }^2{u}_j^{\prime }}{\partial x_k\partial x_k}⟩-\frac{\partial }{\partial x_k}⟨u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }⟩\end{array}$$下面逐项进行推导：

• (1)$$⟨u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial t}+u_i^{\prime }\frac{\partial u_j^{\prime }}{\partial t}⟩=⟨\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial t}⟩=\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial t}$$

• (2)下面已将原式的$j$替换为$k$以符合爱因斯坦求和约定
  $$u_j^{\prime }⟨u_k⟩\frac{\partial u_i^{\prime }}{\partial x_k}+u_i^{\prime }⟨u_k⟩\frac{\partial u_j^{\prime }}{\partial x_k}=⟨u_k⟩\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_k}$$取时间平均运算有：
  $$⟨⟨u_k⟩\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_k}⟩=⟨u_k⟩⟨\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_k}⟩=⟨u_k⟩\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k}$$

• (3)下面也已将原式的$j$替换为$k$
  $$\begin{array}{rl}⟨u_j^{\prime }{u}_k^{\prime }\frac{\partial ⟨u_i⟩}{\partial x_k}⟩+⟨u_i^{\prime }{u}_k^{\prime }\frac{\partial ⟨u_j⟩}{\partial x_k}⟩& =⟨u_j^{\prime }{u}_k^{\prime }⟩⟨\frac{\partial ⟨u_i⟩}{\partial x_k}⟩+⟨u_i^{\prime }{u}_k^{\prime }⟩⟨\frac{\partial ⟨u_j⟩}{\partial x_k}⟩\\ & =⟨u_j^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_i⟩}{\partial x_k}+⟨u_i^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_j⟩}{\partial x_k}\end{array}$$

• (4)
  $$⟨-\frac{u_j^{\prime }}{\rho }\frac{\partial p^{\prime }}{\partial x_i}⟩+⟨-\frac{u_i^{\prime }}{\rho }\frac{\partial p^{\prime }}{\partial x_j}⟩=-\frac{1}{\rho }\left(⟨u_j^{\prime }\frac{\partial p^{\prime }}{\partial x_i}⟩+⟨u_i^{\prime }\frac{\partial p^{\prime }}{\partial x_j}⟩\right)$$

• (5)
  $$⟨\nu u_j^{\prime }\frac{{\partial }^2{u}_i^{\prime }}{\partial x_k\partial x_k}⟩+⟨\nu u_i^{\prime }\frac{{\partial }^2{u}_j^{\prime }}{\partial x_k\partial x_k}⟩=\nu ⟨u_j^{\prime }\frac{{\partial }^2{u}_i^{\prime }}{\partial x_k\partial x_k}+u_i^{\prime }\frac{{\partial }^2{u}_j^{\prime }}{\partial x_k\partial x_k}⟩$$

• (6)下面已将原式的$j$替换为$k$
  $$\frac{\partial u_i^{\prime }{u}_j^{\prime }}{\partial x_j}=u_i^{\prime }\frac{\partial u_j^{\prime }}{\partial x_j}+u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}=u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}=u_k^{\prime }\frac{\partial u_i^{\prime }}{\partial x_k}$$故
  $$\begin{array}{rl}{u}_j^{\prime }{u}_k^{\prime }\frac{\partial u_i^{\prime }}{\partial x_k}+u_i^{\prime }{u}_k^{\prime }\frac{\partial u_j^{\prime }}{\partial x_k}& =u_j^{\prime }{u}_k^{\prime }\frac{\partial u_i^{\prime }}{\partial x_k}+u_i^{\prime }{u}_k^{\prime }\frac{\partial u_j^{\prime }}{\partial x_k}+u_i^{\prime }{u}_j^{\prime }\frac{\partial u_k^{\prime }}{\partial x_k}\\ & =u_j^{\prime }{u}_k^{\prime }\frac{\partial u_i^{\prime }}{\partial x_k}+u_i^{\prime }\frac{\partial u_j^{\prime }{u}_k^{\prime }}{\partial x_k}\\ & =\frac{\partial u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }}{\partial x_k}\end{array}$$取时间平均运算得$$⟨\frac{\partial u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }}{\partial x_k}⟩=\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }⟩}{\partial x_k}$$上面的推导应用了$\frac{\partial u_j^{\prime }}{\partial x_j}=0$、$\frac{\partial u_k^{\prime }}{\partial x_k}=0$，即式$(9)$。

• (7)下面已将原式的$j$替换为$k$
  $$\begin{array}{rl}⟨u_j^{\prime }\frac{\partial ⟨u_i^{\prime }{u}_k^{\prime }⟩}{\partial x_k}⟩+⟨u_i^{\prime }\frac{\partial ⟨u_j^{\prime }{u}_k^{\prime }⟩}{\partial x_k}⟩& =⟨u_j^{\prime }⟩⟨\frac{\partial ⟨u_i^{\prime }{u}_k^{\prime }⟩}{\partial x_k}⟩+⟨u_i^{\prime }⟩⟨\frac{\partial ⟨u_j^{\prime }{u}_k^{\prime }⟩}{\partial x_k}⟩=0\end{array}$$整理以上各项便可以得到雷诺应力输运方程$$\begin{array}{rl}\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial t}+⟨u_k⟩\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k}=& -⟨u_i^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_j⟩}{\partial x_k}-⟨u_j^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_i⟩}{\partial x_k}-\frac{1}{\rho }\left(⟨u_j^{\prime }\frac{\partial p^{\prime }}{\partial x_i}⟩+⟨u_i^{\prime }\frac{\partial p^{\prime }}{\partial x_j}⟩\right)\\ & +\nu ⟨u_j^{\prime }\frac{{\partial }^2{u}_i^{\prime }}{\partial x_k\partial x_k}+u_i^{\prime }\frac{{\partial }^2{u}_j^{\prime }}{\partial x_k\partial x_k}⟩-\frac{\partial }{\partial x_k}⟨u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }⟩\end{array}\text{\hspace{1em}(11)}$$

进一步整理
$$\begin{array}{rl}⟨u_j^{\prime }\frac{\partial p^{\prime }}{\partial x_i}⟩+⟨u_i^{\prime }\frac{\partial p^{\prime }}{\partial x_j}⟩& =⟨\frac{\partial u_j^{\prime }{p}^{\prime }}{\partial x_i}-p^{\prime }\frac{\partial u_j^{\prime }}{\partial x_i}⟩+⟨\frac{\partial u_i^{\prime }{p}^{\prime }}{\partial x_j}-p^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}⟩\\ & =⟨\frac{\partial u_j^{\prime }{p}^{\prime }}{\partial x_i}⟩-⟨p^{\prime }\frac{\partial u_j^{\prime }}{\partial x_i}⟩+⟨\frac{\partial u_i^{\prime }{p}^{\prime }}{\partial x_j}⟩-⟨p^{\prime }\frac{\partial u_i^{\prime }}{\partial x_j}⟩\\ & =\left(\frac{\partial ⟨u_j^{\prime }{p}^{\prime }⟩}{\partial x_i}+\frac{\partial ⟨u_i^{\prime }{p}^{\prime }⟩}{\partial x_j}\right)-⟨p^{\prime }\left(\frac{\partial u_j^{\prime }}{\partial x_i}+\frac{\partial u_i^{\prime }}{\partial x_j}\right)⟩\\ \nu ⟨u_j^{\prime }\frac{{\partial }^2{u}_i^{\prime }}{\partial x_k\partial x_k}+u_i^{\prime }\frac{{\partial }^2{u}_j^{\prime }}{\partial x_k\partial x_k}⟩& =\nu ⟨\frac{\partial }{\partial x_k}\left(u_i^{\prime }\frac{\partial u_j^{\prime }}{\partial x_k}\right)+\frac{\partial }{\partial x_k}\left(u_j^{\prime }\frac{\partial u_i^{\prime }}{\partial x_k}\right)⟩-2\nu ⟨\frac{\partial u_i^{\prime }}{\partial x_k}\frac{\partial u_j^{\prime }}{\partial x_k}⟩\\ & =\nu \frac{{\partial }^2⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k\partial x_k}-2\nu ⟨\frac{\partial u_i^{\prime }}{\partial x_k}\frac{\partial u_j^{\prime }}{\partial x_k}⟩\end{array}$$
得：
$$\begin{array}{rl}& \underset{C_{ij}}{\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial t}+⟨u_k⟩\frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k}}=\underset{P_{ij}}{-⟨u_i^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_j⟩}{\partial x_k}-⟨u_j^{\prime }{u}_k^{\prime }⟩\frac{\partial ⟨u_i⟩}{\partial x_k}}+\underset{{\Phi }_{ij}}{⟨\frac{p^{\prime }}{\rho }\left(\frac{\partial u_j^{\prime }}{\partial x_i}+\frac{\partial u_i^{\prime }}{\partial x_j}\right)⟩}\\ & -\underset{D_{ij}}{\frac{\partial }{\partial x_k}\left(\frac{⟨p^{\prime }{u}_i^{\prime }⟩}{\rho }{\delta }_{jk}+\frac{⟨p^{\prime }{u}_j^{\prime }⟩}{\rho }{\delta }_{ik}+⟨u_i^{\prime }{u}_j^{\prime }{u}_k^{\prime }⟩-\nu \frac{\partial ⟨u_i^{\prime }{u}_j^{\prime }⟩}{\partial x_k}\right)}-\underset{E_{ij}}{2\nu ⟨\frac{\partial u_i^{\prime }}{\partial x_k}\frac{\partial u_j^{\prime }}{\partial x_k}⟩}\end{array}$$

六、参考资料

《湍流理论与模拟》第二版$\cdot$张兆顺、崔桂香、许春晓、黄伟希

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