一、雷诺分解

对于湍流而言，其流动变量ϕ\phiϕ具有不规则性，常常将其分解为平均量与脉动量之和，即ϕ=ϕ‾+ϕ′=Φ+ϕ′ϕ=<ϕ>+ϕ′\begin{aligned}\phi&=\overline{\phi}+\phi'=\Phi+\phi '\\\phi&=\left<\phi\right>+\phi'\end{aligned}ϕϕ=ϕ+ϕ′=Φ+ϕ′=⟨ϕ⟩+ϕ′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<ϕ>=∫−∞∞ϕp(ϕ)dϕ.\left<\phi\right>=\int_{-\infty}^{\infty}\phi p\left(\phi\right)\rm d\phi.⟨ϕ⟩=∫−∞∞ϕp(ϕ)dϕ.其中p(ϕ)p\left(\phi\right)p(ϕ)是概率密度。
时间平均：Φ=ϕ‾=1Δt∫0Δtϕdt.\Phi=\overline{\phi}=\frac{1}{\Delta t}\int_0^{\Delta t}\phi {\rm{d}}t.Φ=ϕ=Δt1∫0Δtϕdt.

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：f‾‾=f‾gf‾‾=g‾f‾g+f‾=g‾+f‾f′‾=0fg‾=f‾g‾+f′g′‾∂f∂x‾=∂f‾∂x∂f∂t‾=∂f‾∂tg′f‾‾=0\begin{aligned}\overline{\overline{f}}&=\overline{f}\\ \overline{g\overline{f}}&=\overline{g}\overline{f}\\\overline{g+f}&=\overline{g}+\overline{f}\\ \overline{f'}&=0 \\ \overline{fg}&=\overline{f}\overline{g}+\overline{f'g'}\\ \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'\overline{f}}&=0 \end{aligned}fgfg+ff′fg∂x∂f∂t∂fg′f=f=gf=g+f=0=fg+f′g′=∂x∂f=∂t∂f=0上面这些性质对于系综平均是同样适用的，例如：<g′<f>>=0、<∂f∂x>=∂<f>∂x\left<g'\left<f\right>\right>=0、 \left<\frac{\partial f}{\partial x}\right>=\frac{\partial \left<f\right>}{\partial x}⟨g′⟨f⟩⟩=0、⟨∂x∂f⟩=∂x∂⟨f⟩。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：g′f‾‾=g′‾⋅f‾‾=g′‾⋅f‾=0fg‾=(f‾+f′)(g‾+g′)‾=f‾g‾+f‾g′+f′g‾+f′g′‾=f‾g‾‾+f‾g′‾+f′g‾‾+f′g′‾=f‾g‾+f′g′‾\begin{aligned} \overline{g'\overline{f}}&=\overline{g'}\cdot\overline{{\overline{f}}}\\ &=\overline{g'}\cdot\overline{f}\\ &=0\\ \overline{fg}&=\overline{\left(\overline{f}+f'\right)\left(\overline{g}+g'\right)}\\ &=\overline{\overline{f}\overline{g}+\overline{f}g'+f'\overline{g}+f'g'} \\ &=\overline{\overline{f}\overline{g}}+\overline{\overline{f}g'}+\overline{f'\overline{g}}+\overline{f'g'}\\ &=\overline{f}\overline{g}+\overline{f'g'} \end{aligned}g′ffg=g′⋅f=g′⋅f=0=(f+f′)(g+g′)=fg+fg′+f′g+f′g′=fg+fg′+f′g+f′g′=fg+f′g′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∂ui∂xi=0(1)\frac{\partial u_i}{\partial x_i}=0 \tag{1}∂xi∂ui=0(1)∂ui∂t+uj∂ui∂xj=−1ρ∂p∂xi+ν∂2ui∂xj∂xj+fi(2)\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 \tag{2} ∂t∂ui+uj∂xj∂ui=−ρ1∂xi∂p+ν∂xj∂xj∂2ui+fi(2)在下面的推导中我们暂时不考虑体积力项fif_ifi，即只考虑∂ui∂t+uj∂ui∂xj=−1ρ∂p∂xi+ν∂2ui∂xj∂xj(2)\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} \tag{2} ∂t∂ui+uj∂xj∂ui=−ρ1∂xi∂p+ν∂xj∂xj∂2ui(2)对公式(1)(1)(1)、(2)(2)(2)作平均运算(系综平均)：<∂ui∂xi>=0(3)\left<\frac{\partial u_i}{\partial x_i}\right>=0 \tag{3}⟨∂xi∂ui⟩=0(3) <∂ui∂t>+<uj∂ui∂xj>=<−1ρ∂p∂xi>+<ν∂2ui∂xj∂xj>(4)\left<\frac{\partial u_i}{\partial t}\right>+ \left<u_j\frac{\partial u_i}{\partial x_j}\right> = \left<-\frac{1}{\rho}\frac{\partial p}{\partial x_i}\right>+ \left<\nu \frac{\partial^2 u_i}{\partial x_j\partial x_j}\right> \tag{4} ⟨∂t∂ui⟩+⟨uj∂xj∂ui⟩=⟨−ρ1∂xi∂p⟩+⟨ν∂xj∂xj∂2ui⟩(4)由平均运算的性质有：<∂ui∂xi>=∂<ui>∂xi=0(5)\left<\frac{\partial u_i}{\partial x_i}\right>=\frac{\partial \left<u_i\right>}{\partial x_i}=0 \tag{5}⟨∂xi∂ui⟩=∂xi∂⟨ui⟩=0(5)同理：
<∂ui∂t>=∂<ui>∂t<−1ρ∂p∂xi>=−1ρ∂∂xi<ν∂2ui∂xj∂xj>=ν∂2<ui>∂xj∂xj\begin{aligned}\left<\frac{\partial u_i}{\partial t}\right>&=\frac{\partial \left<u_i\right>}{\partial t}\\ \left<-\frac{1}{\rho}\frac{\partial p}{\partial x_i}\right>&=-\frac{1}{\rho}\frac{\partial \left<p\right>}{\partial x_i}\\ \left<\nu \frac{\partial^2 u_i}{\partial x_j\partial x_j}\right>&=\nu \frac{\partial^2\left< u_i\right>}{\partial x_j\partial x_j}\\ \end{aligned}⟨∂t∂ui⟩⟨−ρ1∂xi∂p⟩⟨ν∂xj∂xj∂2ui⟩=∂t∂⟨ui⟩=−ρ1∂xi∂⟨p⟩=ν∂xj∂xj∂2⟨ui⟩对于<uj∂ui∂xj>\left<u_j\frac{\partial u_i}{\partial x_j}\right>⟨uj∂xj∂ui⟩则有：<uj∂ui∂xj>=<∂uiuj∂xj−ui∂uj∂xj>=<∂uiuj∂xj>−<ui∂uj∂xj>=<∂uiuj∂xj>=∂<uiuj>∂xj=∂<ui><uj>∂xj+∂<ui′uj′>∂xj=<uj>∂<ui>∂xj+∂<ui′uj′>∂xj\begin{aligned} \left<u_j\frac{\partial u_i}{\partial x_j}\right>&=\left<\frac{\partial u_iu_j}{\partial x_j}-u_i\frac{\partial u_j}{\partial x_j}\right>\\ &=\left<\frac{\partial u_iu_j}{\partial x_j}\right>-\left<u_i\frac{\partial u_j}{\partial x_j}\right>\\ &=\left<\frac{\partial u_iu_j}{\partial x_j}\right>\\ &=\frac{\partial \left<u_iu_j\right>}{\partial x_j}\\ &=\frac{\partial \left<u_i\right>\left<u_j\right>}{\partial x_j}+\frac{\partial \left<u_i'u_j'\right>}{\partial x_j}\\ &=\left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}+\frac{\partial \left<u_i'u_j'\right>}{\partial x_j} \end{aligned}⟨uj∂xj∂ui⟩=⟨∂xj∂uiuj−ui∂xj∂uj⟩=⟨∂xj∂uiuj⟩−⟨ui∂xj∂uj⟩=⟨∂xj∂uiuj⟩=∂xj∂⟨uiuj⟩=∂xj∂⟨ui⟩⟨uj⟩+∂xj∂⟨ui′uj′⟩=⟨uj⟩∂xj∂⟨ui⟩+∂xj∂⟨ui′uj′⟩上面的推导用到了∂<uj>∂xj=0\frac{\partial \left<u_j\right>}{\partial x_j}=0∂xj∂⟨uj⟩=0、∂uj∂xj=0\frac{\partial u_j}{\partial x_j}=0∂xj∂uj=0、<uiuj>=<ui><uj>+<ui′uj′>\left<u_iu_j\right>= \left<u_i\right>\left<u_j\right>+\left<u_i'u_j'\right>⟨uiuj⟩=⟨ui⟩⟨uj⟩+⟨ui′uj′⟩。整理以上各项可得：∂<ui>∂t+<uj>∂<ui>∂xj+∂<ui′uj′>∂xj=−1ρ∂∂xi+ν∂2<ui>∂xj∂xj\frac{\partial \left<u_i\right>}{\partial t}+ \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}+ \frac{\partial \left<u_i'u_j'\right>}{\partial x_j}= -\frac{1}{\rho}\frac{\partial \left<p\right>}{\partial x_i}+ \nu \frac{\partial^2\left< u_i\right>}{\partial x_j\partial x_j}∂t∂⟨ui⟩+⟨uj⟩∂xj∂⟨ui⟩+∂xj∂⟨ui′uj′⟩=−ρ1∂xi∂⟨p⟩+ν∂xj∂xj∂2⟨ui⟩将∂<ui′uj′>∂xj\frac{\partial \left<u_i'u_j'\right>}{\partial x_j}∂xj∂⟨ui′uj′⟩移到右边即有雷诺方程：∂<ui>∂t+<uj>∂<ui>∂xj=−1ρ∂∂xi+ν∂2<ui>∂xj∂xj−∂<ui′uj′>∂xj(6)\frac{\partial \left<u_i\right>}{\partial t}+ \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}= -\frac{1}{\rho}\frac{\partial \left<p\right>}{\partial x_i}+ \nu \frac{\partial^2\left< u_i\right>}{\partial x_j\partial x_j} -\frac{\partial \left<u_i'u_j'\right>}{\partial x_j}\tag{6}∂t∂⟨ui⟩+⟨uj⟩∂xj∂⟨ui⟩=−ρ1∂xi∂⟨p⟩+ν∂xj∂xj∂2⟨ui⟩−∂xj∂⟨ui′uj′⟩(6)式中的−<ui′uj′>-\left<u_i'u_j'\right>−⟨ui′uj′⟩ 乘上密度ρ\rhoρ便是雷诺应力−ρ<ui′uj′>-\rho\left<u_i'u_j'\right>−ρ⟨ui′uj′⟩，可以写成张量形式：(−ρ<u′2>−ρ<u′v′>−ρ<u′w′>−ρ<u′v′>−ρ<v′2>−ρ<v′w′>−ρ<u′w′>−ρ<v′w′>−ρ<w′2>).\begin{pmatrix} -\rho \left<{u^{\prime 2}} \right>& -\rho \left<{u^{\prime }v^{\prime}} \right>& -\rho \left<{u^{\prime }w^{\prime}}\right>\\ -\rho \left<{u^{\prime }v^{\prime}} \right>& -\rho \left<{v^{\prime 2}}\right> & -\rho \left<{v^{\prime }w^{\prime}}\right> \\ -\rho \left<{u^{\prime }w^{\prime}} \right>& -\rho \left<{v^{\prime }w^{\prime}} \right>& -\rho \left<{w^{\prime 2}}\right> \end{pmatrix}. \quad⎝⎛−ρ⟨u′2⟩−ρ⟨u′v′⟩−ρ⟨u′w′⟩−ρ⟨u′v′⟩−ρ⟨v′2⟩−ρ⟨v′w′⟩−ρ⟨u′w′⟩−ρ⟨v′w′⟩−ρ⟨w′2⟩⎠⎞.

四、脉动运动方程的推导

将NS方程(1)(1)(1)、(2)(2)(2)减去雷诺方程(5)(5)(5)、(6)(6)(6)，并进行一定的整理即可得到脉动运动方程：∂ui′∂xi=0(7)\frac{\partial u_i'}{\partial x_i}=0 \tag{7}∂xi∂ui′=0(7)∂ui′∂t+<uj>∂ui′∂xj+uj′∂<ui>∂xj=−1ρ∂p′∂xi+ν∂2ui′∂xj∂xj−∂∂xj(ui′uj′−<ui′uj′>)(8)\frac{\partial u_i'}{\partial t}+ \left<u_j\right>\frac{\partial u_i'}{\partial x_j} + u_j'\frac{\partial \left<u_i\right>}{\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}{\partial x_j}\left(u_i'u_j'-\left<u_i'u_j'\right>\right)\tag{8} ∂t∂ui′+⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩=−ρ1∂xi∂p′+ν∂xj∂xj∂2ui′−∂xj∂(ui′uj′−⟨ui′uj′⟩)(8)下面逐项进行推导：∂ui∂xi−∂<ui>∂xi=∂(ui−<ui>)∂xi=∂ui′∂xi\frac{\partial u_i}{\partial x_i}- \frac{\partial \left<u_i\right>}{\partial x_i}= \frac{\partial \left(u_i-\left<u_i\right>\right)}{\partial x_i}= \frac{\partial u_i'}{\partial x_i}∂xi∂ui−∂xi∂⟨ui⟩=∂xi∂(ui−⟨ui⟩)=∂xi∂ui′故有：∂ui′∂xi=0(9)\frac{\partial u_i'}{\partial x_i}=0\tag{9}∂xi∂ui′=0(9)同理：∂ui∂t−∂<ui>∂t=∂(ui−<ui>)∂t=∂ui′∂t\frac{\partial u_i}{\partial t}- \frac{\partial \left<u_i\right>}{\partial t}= \frac{\partial \left(u_i-\left<u_i\right>\right)}{\partial t}= \frac{\partial u_i'}{\partial t}∂t∂ui−∂t∂⟨ui⟩=∂t∂(ui−⟨ui⟩)=∂t∂ui′−1ρ∂p∂xi+1ρ∂∂xi=−1ρ∂(p−)∂xi=−1ρ∂p′∂xi-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+ \frac{1}{\rho}\frac{\partial \left<p\right>}{\partial x_i}= -\frac{1}{\rho}\frac{\partial \left(p -\left<p\right>\right)}{\partial x_i}= -\frac{1}{\rho}\frac{\partial p'}{\partial x_i}−ρ1∂xi∂p+ρ1∂xi∂⟨p⟩=−ρ1∂xi∂(p−⟨p⟩)=−ρ1∂xi∂p′ν∂2ui∂xj∂xj−ν∂2<ui>∂xj∂xj=ν∂2(ui−<ui>)∂xj∂xj=ν∂2ui′∂xj∂xj\nu \frac{\partial^2 u_i}{\partial x_j\partial x_j}- \nu \frac{\partial^2\left< u_i\right>}{\partial x_j\partial x_j}= \nu \frac{\partial^2\left(u_i-\left< u_i\right>\right)}{\partial x_j\partial x_j}= \nu \frac{\partial^2 u_i'}{\partial x_j\partial x_j}ν∂xj∂xj∂2ui−ν∂xj∂xj∂2⟨ui⟩=ν∂xj∂xj∂2(ui−⟨ui⟩)=ν∂xj∂xj∂2ui′另外：uj∂ui∂xj−<uj>∂<ui>∂xj=(<uj>+uj′)∂(<ui>+ui′)∂xj−<uj>∂<ui>∂xj=<uj>∂<ui>∂xj+<uj>∂ui′∂xj+uj′∂<ui>∂xj+uj′∂ui′∂xj−<uj>∂<ui>∂xj=<uj>∂ui′∂xj+uj′∂<ui>∂xj+uj′∂ui′∂xj=<uj>∂ui′∂xj+uj′∂<ui>∂xj+∂ui′uj′∂xj−ui′∂uj′∂xj=<uj>∂ui′∂xj+uj′∂<ui>∂xj+∂ui′uj′∂xj\begin{aligned} u_j\frac{\partial u_i}{\partial x_j}- \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}&= \left(\left<u_j\right>+u_j'\right)\frac{\partial \left(\left<u_i\right>+u_i'\right)}{\partial x_j}- \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}\\&= \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}+ \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\partial x_j}+ u_j'\frac{\partial u_i'}{\partial x_j}- \left<u_j\right>\frac{\partial \left<u_i\right>}{\partial x_j}\\&= \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\partial x_j}+ u_j'\frac{\partial u_i'}{\partial x_j}\\&= \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\partial x_j}+ \frac{\partial u_i'u_j'}{\partial x_j}- u_i'\frac{\partial u_j'}{\partial x_j}\\&= \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\partial x_j}+ \frac{\partial u_i'u_j'}{\partial x_j} \end{aligned}uj∂xj∂ui−⟨uj⟩∂xj∂⟨ui⟩=(⟨uj⟩+uj′)∂xj∂(⟨ui⟩+ui′)−⟨uj⟩∂xj∂⟨ui⟩=⟨uj⟩∂xj∂⟨ui⟩+⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩+uj′∂xj∂ui′−⟨uj⟩∂xj∂⟨ui⟩=⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩+uj′∂xj∂ui′=⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩+∂xj∂ui′uj′−ui′∂xj∂uj′=⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩+∂xj∂ui′uj′上面的推导用到了∂uj′∂xj=0\frac{\partial u_j'}{\partial x_j}=0∂xj∂uj′=0。最后雷诺应力项−∂<ui′uj′>∂xj-\frac{\partial \left<u_i'u_j'\right>}{\partial x_j}−∂xj∂⟨ui′uj′⟩符号变为正号，整理各项可得：∂ui′∂t+<uj>∂ui′∂xj+uj′∂<ui>∂xj+∂ui′uj′∂xj=−1ρ∂p′∂xi+ν∂2ui′∂xj∂xj+∂<ui′uj′>∂xj\frac{\partial u_i'}{\partial t}+ \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\partial x_j}+ \frac{\partial u_i'u_j'}{\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 \left<u_i'u_j'\right>}{\partial x_j}∂t∂ui′+⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩+∂xj∂ui′uj′=−ρ1∂xi∂p′+ν∂xj∂xj∂2ui′+∂xj∂⟨ui′uj′⟩将∂ui′uj′∂xj\frac{\partial u_i'u_j'}{\partial x_j}∂xj∂ui′uj′移到右边可得∂ui′∂t+<uj>∂ui′∂xj+uj′∂<ui>∂xj=−1ρ∂p′∂xi+ν∂2ui′∂xj∂xj−∂∂xj(ui′uj′−<ui′uj′>)(10)\frac{\partial u_i'}{\partial t}+ \left<u_j\right>\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left<u_i\right>}{\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}{\partial x_j}\left(u_i'u_j'-\left<u_i'u_j'\right>\right)\tag{10}∂t∂ui′+⟨uj⟩∂xj∂ui′+uj′∂xj∂⟨ui⟩=−ρ1∂xi∂p′+ν∂xj∂xj∂2ui′−∂xj∂(ui′uj′−⟨ui′uj′⟩)(10)

五、雷诺应力输运方程的推导

从脉动运动方程(10)(10)(10)出发，在ui′u_i'ui′脉动方程上乘以uj′u_j'uj′，uj′u_j'uj′脉动方程上乘以ui′u_i'ui′，两式相加后作平均运算，得到雷诺应力输运方程：∂<ui′uj′>∂t+<uk>∂<ui′uj′>∂xk=−<ui′uk′>∂<uj>∂xk−<uj′uk′>∂<ui>∂xk−1ρ(<uj′∂p′∂xi>+<ui′∂p′∂xj>)+ν<uj′∂2ui′∂xk∂xk+ui′∂2uj′∂xk∂xk>−∂∂xk<ui′uj′uk′>\begin{aligned} \frac{\partial\left<u_i'u_j'\right>}{\partial t}+ \left<u_k\right>\frac{\partial\left<u_i'u_j'\right>}{\partial x_k}=& -\left<u_i'u_k'\right>\frac{\partial\left<u_j\right>}{\partial x_k} -\left<u_j'u_k'\right>\frac{\partial\left<u_i\right>}{\partial x_k} -\frac{1}{\rho}\left(\left<u_j'\frac{\partial p'}{\partial x_i}\right>+\left<u_i'\frac{\partial p'}{\partial x_j}\right>\right)\\&+ \nu\left<u_j'\frac{\partial^2 u_i'}{\partial x_k\partial x_k}+u_i'\frac{\partial^2 u_j'}{\partial x_k\partial x_k}\right>- \frac{\partial }{\partial x_k}\left<u_i'u_j'u_k'\right> \end{aligned}∂t∂⟨ui′uj′⟩+⟨uk⟩∂xk∂⟨ui′uj′⟩=−⟨ui′uk′⟩∂xk∂⟨uj⟩−⟨uj′uk′⟩∂xk∂⟨ui⟩−ρ1(⟨uj′∂xi∂p′⟩+⟨ui′∂xj∂p′⟩)+ν⟨uj′∂xk∂xk∂2ui′+ui′∂xk∂xk∂2uj′⟩−∂xk∂⟨ui′uj′uk′⟩下面逐项进行推导：

(1)<uj′∂ui′∂t+ui′∂uj′∂t>=<∂ui′uj′∂t>=∂<ui′uj′>∂t\left<u_j'\frac{\partial u_i'}{\partial t} +u_i'\frac{\partial u_j'}{\partial t}\right>= \left<\frac{\partial u_i'u_j'}{\partial t}\right>= \frac{\partial \left<u_i'u_j'\right>}{\partial t}⟨uj′∂t∂ui′+ui′∂t∂uj′⟩=⟨∂t∂ui′uj′⟩=∂t∂⟨ui′uj′⟩
(2)下面已将原式的jjj替换为kkk以符合爱因斯坦求和约定
uj′<uk>∂ui′∂xk+ui′<uk>∂uj′∂xk=<uk>∂ui′uj′∂xku_j'\left<u_k\right>\frac{\partial u_i'}{\partial x_k} +u_i'\left<u_k\right>\frac{\partial u_j'}{\partial x_k} =\left<u_k\right>\frac{\partial u_i'u_j'}{\partial x_k}uj′⟨uk⟩∂xk∂ui′+ui′⟨uk⟩∂xk∂uj′=⟨uk⟩∂xk∂ui′uj′取时间平均运算有：
<<uk>∂ui′uj′∂xk>=<uk><∂ui′uj′∂xk>=<uk>∂<ui′uj′>∂xk\left<\left<u_k\right>\frac{\partial u_i'u_j'}{\partial x_k}\right> =\left<u_k\right>\left<\frac{\partial u_i'u_j'}{\partial x_k}\right> =\left<u_k\right>\frac{\partial \left<u_i'u_j'\right>}{\partial x_k}⟨⟨uk⟩∂xk∂ui′uj′⟩=⟨uk⟩⟨∂xk∂ui′uj′⟩=⟨uk⟩∂xk∂⟨ui′uj′⟩
(3)下面也已将原式的jjj替换为kkk
<uj′uk′∂<ui>∂xk>+<ui′uk′∂<uj>∂xk>=<uj′uk′><∂<ui>∂xk>+<ui′uk′><∂<uj>∂xk>=<uj′uk′>∂<ui>∂xk+<ui′uk′>∂<uj>∂xk\begin{aligned} \left<u_j'u_k'\frac{\partial \left<u_i\right>}{\partial x_k}\right>+ \left<u_i'u_k'\frac{\partial \left<u_j\right>}{\partial x_k}\right>&= \left<u_j'u_k'\right>\left<\frac{\partial \left<u_i\right>}{\partial x_k}\right>+ \left<u_i'u_k'\right>\left<\frac{\partial \left<u_j\right>}{\partial x_k}\right>\\&= \left<u_j'u_k'\right>\frac{\partial \left<u_i\right>}{\partial x_k}+ \left<u_i'u_k'\right>\frac{\partial \left<u_j\right>}{\partial x_k} \end{aligned}⟨uj′uk′∂xk∂⟨ui⟩⟩+⟨ui′uk′∂xk∂⟨uj⟩⟩=⟨uj′uk′⟩⟨∂xk∂⟨ui⟩⟩+⟨ui′uk′⟩⟨∂xk∂⟨uj⟩⟩=⟨uj′uk′⟩∂xk∂⟨ui⟩+⟨ui′uk′⟩∂xk∂⟨uj⟩
(4)
<−uj′ρ∂p′∂xi>+<−ui′ρ∂p′∂xj>=−1ρ(<uj′∂p′∂xi>+<ui′∂p′∂xj>)\left<-\frac{u_j'}{\rho}\frac{\partial p'}{\partial x_i}\right>+ \left<-\frac{u_i'}{\rho}\frac{\partial p'}{\partial x_j}\right>= -\frac{1}{\rho}\left(\left<u_j'\frac{\partial p'}{\partial x_i}\right>+\left<u_i'\frac{\partial p'}{\partial x_j}\right>\right)⟨−ρuj′∂xi∂p′⟩+⟨−ρui′∂xj∂p′⟩=−ρ1(⟨uj′∂xi∂p′⟩+⟨ui′∂xj∂p′⟩)
(5)
<νuj′∂2ui′∂xk∂xk>+<νui′∂2uj′∂xk∂xk>=ν<uj′∂2ui′∂xk∂xk+ui′∂2uj′∂xk∂xk>\left<\nu u_j'\frac{\partial^2 u_i'}{\partial x_k\partial x_k}\right>+ \left<\nu u_i'\frac{\partial^2 u_j'}{\partial x_k\partial x_k}\right> =\nu\left<u_j'\frac{\partial^2 u_i'}{\partial x_k\partial x_k}+u_i'\frac{\partial^2 u_j'}{\partial x_k\partial x_k}\right>⟨νuj′∂xk∂xk∂2ui′⟩+⟨νui′∂xk∂xk∂2uj′⟩=ν⟨uj′∂xk∂xk∂2ui′+ui′∂xk∂xk∂2uj′⟩
(6)下面已将原式的jjj替换为kkk
∂ui′uj′∂xj=ui′∂uj′∂xj+uj′∂ui′∂xj=uj′∂ui′∂xj=uk′∂ui′∂xk\frac{\partial u_i'u_j'}{\partial x_j}= u_i'\frac{\partial u_j'}{\partial x_j}+ u_j'\frac{\partial u_i'}{\partial x_j}= u_j'\frac{\partial u_i'}{\partial x_j}= u_k'\frac{\partial u_i'}{\partial x_k}∂xj∂ui′uj′=ui′∂xj∂uj′+uj′∂xj∂ui′=uj′∂xj∂ui′=uk′∂xk∂ui′故
uj′uk′∂ui′∂xk+ui′uk′∂uj′∂xk=uj′uk′∂ui′∂xk+ui′uk′∂uj′∂xk+ui′uj′∂uk′∂xk=uj′uk′∂ui′∂xk+ui′∂uj′uk′∂xk=∂ui′uj′uk′∂xk\begin{aligned} u_j'u_k'\frac{\partial u_i'}{\partial x_k}+ u_i'u_k'\frac{\partial u_j'}{\partial x_k}&= u_j'u_k'\frac{\partial u_i'}{\partial x_k}+ u_i'u_k'\frac{\partial u_j'}{\partial x_k}+ u_i'u_j'\frac{\partial u_k'}{\partial x_k}\\&= u_j'u_k'\frac{\partial u_i'}{\partial x_k}+ u_i'\frac{\partial u_j'u_k'}{\partial x_k}\\&= \frac{\partial u_i'u_j'u_k'}{\partial x_k} \end{aligned}uj′uk′∂xk∂ui′+ui′uk′∂xk∂uj′=uj′uk′∂xk∂ui′+ui′uk′∂xk∂uj′+ui′uj′∂xk∂uk′=uj′uk′∂xk∂ui′+ui′∂xk∂uj′uk′=∂xk∂ui′uj′uk′取时间平均运算得<∂ui′uj′uk′∂xk>=∂<ui′uj′uk′>∂xk\left<\frac{\partial u_i'u_j'u_k'}{\partial x_k}\right>= \frac{\partial \left<u_i'u_j'u_k'\right>}{\partial x_k}⟨∂xk∂ui′uj′uk′⟩=∂xk∂⟨ui′uj′uk′⟩上面的推导应用了∂uj′∂xj=0\frac{\partial u_j'}{\partial x_j}=0∂xj∂uj′=0、∂uk′∂xk=0\frac{\partial u_k'}{\partial x_k}=0∂xk∂uk′=0，即式(9)(9)(9)。
(7)下面已将原式的jjj替换为kkk
<uj′∂<ui′uk′>∂xk>+<ui′∂<uj′uk′>∂xk>=<uj′><∂<ui′uk′>∂xk>+<ui′><∂<uj′uk′>∂xk>=0\begin{aligned} \left<u_j'\frac{\partial \left<u_i'u_k'\right>}{\partial x_k}\right> +\left<u_i'\frac{\partial \left<u_j'u_k'\right>}{\partial x_k}\right>&= \left<u_j'\right>\left<\frac{\partial \left<u_i'u_k'\right>}{\partial x_k}\right> +\left<u_i'\right>\left<\frac{\partial \left<u_j'u_k'\right>}{\partial x_k}\right> =0 \end{aligned}⟨uj′∂xk∂⟨ui′uk′⟩⟩+⟨ui′∂xk∂⟨uj′uk′⟩⟩=⟨uj′⟩⟨∂xk∂⟨ui′uk′⟩⟩+⟨ui′⟩⟨∂xk∂⟨uj′uk′⟩⟩=0整理以上各项便可以得到雷诺应力输运方程∂<ui′uj′>∂t+<uk>∂<ui′uj′>∂xk=−<ui′uk′>∂<uj>∂xk−<uj′uk′>∂<ui>∂xk−1ρ(<uj′∂p′∂xi>+<ui′∂p′∂xj>)+ν<uj′∂2ui′∂xk∂xk+ui′∂2uj′∂xk∂xk>−∂∂xk<ui′uj′uk′>(11)\begin{aligned} \frac{\partial\left<u_i'u_j'\right>}{\partial t}+ \left<u_k\right>\frac{\partial\left<u_i'u_j'\right>}{\partial x_k}=& -\left<u_i'u_k'\right>\frac{\partial\left<u_j\right>}{\partial x_k} -\left<u_j'u_k'\right>\frac{\partial\left<u_i\right>}{\partial x_k} -\frac{1}{\rho}\left(\left<u_j'\frac{\partial p'}{\partial x_i}\right>+\left<u_i'\frac{\partial p'}{\partial x_j}\right>\right)\\&+ \nu\left<u_j'\frac{\partial^2 u_i'}{\partial x_k\partial x_k}+u_i'\frac{\partial^2 u_j'}{\partial x_k\partial x_k}\right>- \frac{\partial }{\partial x_k}\left<u_i'u_j'u_k'\right> \end{aligned}\tag{11}∂t∂⟨ui′uj′⟩+⟨uk⟩∂xk∂⟨ui′uj′⟩=−⟨ui′uk′⟩∂xk∂⟨uj⟩−⟨uj′uk′⟩∂xk∂⟨ui⟩−ρ1(⟨uj′∂xi∂p′⟩+⟨ui′∂xj∂p′⟩)+ν⟨uj′∂xk∂xk∂2ui′+ui′∂xk∂xk∂2uj′⟩−∂xk∂⟨ui′uj′uk′⟩(11)

进一步整理
<uj′∂p′∂xi>+<ui′∂p′∂xj>=<∂uj′p′∂xi−p′∂uj′∂xi>+<∂ui′p′∂xj−p′∂ui′∂xj>=<∂uj′p′∂xi>−<p′∂uj′∂xi>+<∂ui′p′∂xj>−<p′∂ui′∂xj>=(∂<uj′p′>∂xi+∂<ui′p′>∂xj)−<p′(∂uj′∂xi+∂ui′∂xj)>ν<uj′∂2ui′∂xk∂xk+ui′∂2uj′∂xk∂xk>=ν<∂∂xk(ui′∂uj′∂xk)+∂∂xk(uj′∂ui′∂xk)>−2ν<∂ui′∂xk∂uj′∂xk>=ν∂2<ui′uj′>∂xk∂xk−2ν<∂ui′∂xk∂uj′∂xk>\begin{aligned} \left<u_j'\frac{\partial p'}{\partial x_i}\right> +\left<u_i'\frac{\partial p'}{\partial x_j}\right>&= \left<\frac{\partial u_j'p'}{\partial x_i}-p'\frac{\partial u_j'}{\partial x_i}\right>+ \left<\frac{\partial u_i'p'}{\partial x_j}-p'\frac{\partial u_i'}{\partial x_j}\right>\\&= \left<\frac{\partial u_j'p'}{\partial x_i}\right> -\left<p'\frac{\partial u_j'}{\partial x_i}\right> +\left<\frac{\partial u_i'p'}{\partial x_j}\right> -\left<p'\frac{\partial u_i'}{\partial x_j}\right>\\&=\left(\frac{\partial \left<u_j'p'\right>}{\partial x_i}+\frac{\partial \left<u_i'p'\right>}{\partial x_j}\right) -\left<p'\left(\frac{\partial u_j'}{\partial x_i}+\frac{\partial u_i'}{\partial x_j}\right)\right>\\ \nu\left<u_j'\frac{\partial^2 u_i'}{\partial x_k\partial x_k}+u_i'\frac{\partial^2 u_j'}{\partial x_k\partial x_k}\right>&= \nu\left<\frac{\partial}{\partial x_k}\left(u_i'\frac{\partial u_j'}{\partial x_k}\right)+\frac{\partial}{\partial x_k}\left(u_j'\frac{\partial u_i'}{\partial x_k}\right)\right> -2\nu\left<\frac{\partial u_i'}{\partial x_k}\frac{\partial u_j'}{\partial x_k}\right>\\&= \nu\frac{\partial^2\left<u_i'u_j'\right>}{\partial x_k\partial x_k} -2\nu\left<\frac{\partial u_i'}{\partial x_k}\frac{\partial u_j'}{\partial x_k}\right> \end{aligned}⟨uj′∂xi∂p′⟩+⟨ui′∂xj∂p′⟩ν⟨uj′∂xk∂xk∂2ui′+ui′∂xk∂xk∂2uj′⟩=⟨∂xi∂uj′p′−p′∂xi∂uj′⟩+⟨∂xj∂ui′p′−p′∂xj∂ui′⟩=⟨∂xi∂uj′p′⟩−⟨p′∂xi∂uj′⟩+⟨∂xj∂ui′p′⟩−⟨p′∂xj∂ui′⟩=(∂xi∂⟨uj′p′⟩+∂xj∂⟨ui′p′⟩)−⟨p′(∂xi∂uj′+∂xj∂ui′)⟩=ν⟨∂xk∂(ui′∂xk∂uj′)+∂xk∂(uj′∂xk∂ui′)⟩−2ν⟨∂xk∂ui′∂xk∂uj′⟩=ν∂xk∂xk∂2⟨ui′uj′⟩−2ν⟨∂xk∂ui′∂xk∂uj′⟩
得：
∂<ui′uj′>∂t+<uk>∂<ui′uj′>∂xk⏟Cij=−<ui′uk′>∂<uj>∂xk−<uj′uk′>∂<ui>∂xk⏟Pij+<p′ρ(∂uj′∂xi+∂ui′∂xj)>⏟Φij−∂∂xk(<p′ui′>ρδjk+<p′uj′>ρδik+<ui′uj′uk′>−ν∂<ui′uj′>∂xk)⏟Dij−2ν<∂ui′∂xk∂uj′∂xk>⏟Eij\begin{aligned} &\underset{C_{ij}}{\underbrace{\frac{\partial\left<u_i'u_j'\right>}{\partial t}+ \left<u_k\right>\frac{\partial\left<u_i'u_j'\right>}{\partial x_k}} }= \underset{P_{ij}}{\underbrace{-\left<u_i'u_k'\right>\frac{\partial\left<u_j\right>}{\partial x_k} -\left<u_j'u_k'\right>\frac{\partial\left<u_i\right>}{\partial x_k}}} + \underset{\Phi_{ij}}{\underbrace{\left<\frac{p'}{\rho}\left(\frac{\partial u_j'}{\partial x_i}+\frac{\partial u_i'}{\partial x_j}\right)\right>}} \\& -\underset{D_{ij}}{\underbrace{\frac{\partial}{\partial x_k} \left( \frac{\left<p'u_i'\right>}{\rho}\delta_{jk}+ \frac{\left<p'u_j'\right>}{\rho}\delta_{ik}+ \left<u_i'u_j'u_k'\right>- \nu\frac{\partial \left<u_i'u_j'\right>}{\partial x_k} \right) }} -\underset{E_{ij}}{\underbrace{2\nu\left<\frac{\partial u_i'}{\partial x_k}\frac{\partial u_j'}{\partial x_k}\right>}} \end{aligned}Cij∂t∂⟨ui′uj′⟩+⟨uk⟩∂xk∂⟨ui′uj′⟩=Pij−⟨ui′uk′⟩∂xk∂⟨uj⟩−⟨uj′uk′⟩∂xk∂⟨ui⟩+Φij⟨ρp′(∂xi∂uj′+∂xj∂ui′)⟩−Dij∂xk∂(ρ⟨p′ui′⟩δjk+ρ⟨p′uj′⟩δik+⟨ui′uj′uk′⟩−ν∂xk∂⟨ui′uj′⟩)−Eij2ν⟨∂xk∂ui′∂xk∂uj′⟩

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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