雷诺方程、脉动运动方程及雷诺应力输运方程的推导

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

ϕ

\phi

ϕ具有不规则性，常常将其分解为平均量与脉动量之和，即

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\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

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\left<\phi\right>=\int_{-\infty}^{\infty}\phi p\left(\phi\right)\rm d\phi.

⟨ϕ⟩=∫−∞∞​ϕp(ϕ)dϕ.其中

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p\left(\phi\right)

p(ϕ)是概率密度。时间平均：

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\Phi=\overline{\phi}=\frac{1}{\Delta t}\int_0^{\Delta t}\phi {\rm{d}}t.

Φ=ϕ​=Δt1​∫0Δt​ϕdt.

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：

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\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}

f​​gf​​g+f​f′​fg​∂x∂f​​∂t∂f​​g′f​​​=f​=g​f​=g​+f​=0=f​g​+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：

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\left\right>=0、 \left<\frac{\partial f}{\partial x}\right>=\frac{\partial \left}{\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：

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\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′f​​fg​​=g′​⋅f​​=g′​⋅f​=0=(f​+f′)(g​+g′)​=f​g​+f​g′+f′g​+f′g′​=f​g​​+f​g′​+f′g​​+f′g′​=f​g​+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

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\frac{\partial u_i}{\partial x_i}=0 \tag{1}

∂xi​∂ui​​=0(1)

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(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

f

i

f_i

fi​，即只考虑

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(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

(

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(1)

(1)、

(

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(2)

(2)作平均运算(系综平均)：

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(3)

\left<\frac{\partial u_i}{\partial x_i}\right>=0 \tag{3}

⟨∂xi​∂ui​​⟩=0(3)

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(4)

\left<\frac{\partial u_i}{\partial t}\right>+ \left = \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：

<

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(5)

\left<\frac{\partial u_i}{\partial x_i}\right>=\frac{\partial \left}{\partial x_i}=0 \tag{5}

⟨∂xi​∂ui​​⟩=∂xi​∂⟨ui​⟩​=0(5)同理：

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\begin{aligned}\left<\frac{\partial u_i}{\partial t}\right>&=\frac{\partial \left}{\partial t}\\ \left<-\frac{1}{\rho}\frac{\partial p}{\partial x_i}\right>&=-\frac{1}{\rho}\frac{\partial \left}{\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

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⟨uj​∂xj​∂ui​​⟩则有：

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\begin{aligned} \left&=\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\\ &=\left<\frac{\partial u_iu_j}{\partial x_j}\right>\\ &=\frac{\partial \left}{\partial x_j}\\ &=\frac{\partial \left\left}{\partial x_j}+\frac{\partial \left}{\partial x_j}\\ &=\left\frac{\partial \left}{\partial x_j}+\frac{\partial \left}{\partial x_j} \end{aligned}

⟨uj​∂xj​∂ui​​⟩​=⟨∂xj​∂ui​uj​​−ui​∂xj​∂uj​​⟩=⟨∂xj​∂ui​uj​​⟩−⟨ui​∂xj​∂uj​​⟩=⟨∂xj​∂ui​uj​​⟩=∂xj​∂⟨ui​uj​⟩​=∂xj​∂⟨ui​⟩⟨uj​⟩​+∂xj​∂⟨ui′​uj′​⟩​=⟨uj​⟩∂xj​∂⟨ui​⟩​+∂xj​∂⟨ui′​uj′​⟩​​上面的推导用到了

∂

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u

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>

∂

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=

0

\frac{\partial \left}{\partial x_j}=0

∂xj​∂⟨uj​⟩​=0、

∂

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0

\frac{\partial u_j}{\partial x_j}=0

∂xj​∂uj​​=0、

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\left= \left\left+\left

⟨ui​uj​⟩=⟨ui​⟩⟨uj​⟩+⟨ui′​uj′​⟩。整理以上各项可得：

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\frac{\partial \left}{\partial t}+ \left\frac{\partial \left}{\partial x_j}+ \frac{\partial \left}{\partial x_j}= -\frac{1}{\rho}\frac{\partial \left}{\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

∂

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\frac{\partial \left}{\partial x_j}

∂xj​∂⟨ui′​uj′​⟩​移到右边即有雷诺方程：

∂

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(6)

\frac{\partial \left}{\partial t}+ \left\frac{\partial \left}{\partial x_j}= -\frac{1}{\rho}\frac{\partial \left}{\partial x_i}+ \nu \frac{\partial^2\left< u_i\right>}{\partial x_j\partial x_j} -\frac{\partial \left}{\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

−

<

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-\left

−⟨ui′​uj′​⟩ 乘上密度

ρ

\rho

ρ便是雷诺应力

−

ρ

<

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-\rho\left

−ρ⟨ui′​uj′​⟩，可以写成张量形式：

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\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

(

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(1)

(1)、

(

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(2)

(2)减去雷诺方程

(

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(5)

(5)、

(

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(6)

(6)，并进行一定的整理即可得到脉动运动方程：

∂

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′

∂

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0

(7)

\frac{\partial u_i'}{\partial x_i}=0 \tag{7}

∂xi​∂ui′​​=0(7)

∂

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′

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<

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>

∂

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∂

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∂

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(

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(8)

\frac{\partial u_i'}{\partial t}+ \left\frac{\partial u_i'}{\partial x_j} + u_j'\frac{\partial \left}{\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\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：

∂

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∂

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\frac{\partial u_i}{\partial x_i}- \frac{\partial \left}{\partial x_i}= \frac{\partial \left(u_i-\left\right)}{\partial x_i}= \frac{\partial u_i'}{\partial x_i}

∂xi​∂ui​​−∂xi​∂⟨ui​⟩​=∂xi​∂(ui​−⟨ui​⟩)​=∂xi​∂ui′​​故有：

∂

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′

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(9)

\frac{\partial u_i'}{\partial x_i}=0\tag{9}

∂xi​∂ui′​​=0(9)同理：

∂

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∂

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\frac{\partial u_i}{\partial t}- \frac{\partial \left}{\partial t}= \frac{\partial \left(u_i-\left\right)}{\partial t}= \frac{\partial u_i'}{\partial t}

∂t∂ui​​−∂t∂⟨ui​⟩​=∂t∂(ui​−⟨ui​⟩)​=∂t∂ui′​​

−

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-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+ \frac{1}{\rho}\frac{\partial \left}{\partial x_i}= -\frac{1}{\rho}\frac{\partial \left(p -\left\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

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\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：

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\begin{aligned} u_j\frac{\partial u_i}{\partial x_j}- \left\frac{\partial \left}{\partial x_j}&= \left(\left+u_j'\right)\frac{\partial \left(\left+u_i'\right)}{\partial x_j}- \left\frac{\partial \left}{\partial x_j}\\&= \left\frac{\partial \left}{\partial x_j}+ \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\partial x_j}+ u_j'\frac{\partial u_i'}{\partial x_j}- \left\frac{\partial \left}{\partial x_j}\\&= \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\partial x_j}+ u_j'\frac{\partial u_i'}{\partial x_j}\\&= \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\partial x_j}+ \frac{\partial u_i'u_j'}{\partial x_j}- u_i'\frac{\partial u_j'}{\partial x_j}\\&= \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\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

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\frac{\partial u_j'}{\partial x_j}=0

∂xj​∂uj′​​=0。最后雷诺应力项

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-\frac{\partial \left}{\partial x_j}

−∂xj​∂⟨ui′​uj′​⟩​符号变为正号，整理各项可得：

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\frac{\partial u_i'}{\partial t}+ \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\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}{\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

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\frac{\partial u_i'u_j'}{\partial x_j}

∂xj​∂ui′​uj′​​移到右边可得

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)

(10)

\frac{\partial u_i'}{\partial t}+ \left\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \left}{\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\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

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ui′​脉动方程上乘以

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u_j'

uj′​，

u

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u_j'

uj′​脉动方程上乘以

u

i

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u_i'

ui′​，两式相加后作平均运算，得到雷诺应力输运方程：

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\begin{aligned} \frac{\partial\left}{\partial t}+ \left\frac{\partial\left}{\partial x_k}=& -\left\frac{\partial\left}{\partial x_k} -\left\frac{\partial\left}{\partial x_k} -\frac{1}{\rho}\left(\left+\left\right)\\&+ \nu\left- \frac{\partial }{\partial x_k}\left \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)

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\left= \left<\frac{\partial u_i'u_j'}{\partial t}\right>= \frac{\partial \left}{\partial t}

⟨uj′​∂t∂ui′​​+ui′​∂t∂uj′​​⟩=⟨∂t∂ui′​uj′​​⟩=∂t∂⟨ui′​uj′​⟩​

(2)下面已将原式的

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k

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k以符合爱因斯坦求和约定

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u_j'\left\frac{\partial u_i'}{\partial x_k} +u_i'\left\frac{\partial u_j'}{\partial x_k} =\left\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：

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\left<\left\frac{\partial u_i'u_j'}{\partial x_k}\right> =\left\left<\frac{\partial u_i'u_j'}{\partial x_k}\right> =\left\frac{\partial \left}{\partial x_k}

⟨⟨uk​⟩∂xk​∂ui′​uj′​​⟩=⟨uk​⟩⟨∂xk​∂ui′​uj′​​⟩=⟨uk​⟩∂xk​∂⟨ui′​uj′​⟩​

(3)下面也已将原式的

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\begin{aligned} \left}{\partial x_k}\right>+ \left}{\partial x_k}\right>&= \left\left<\frac{\partial \left}{\partial x_k}\right>+ \left\left<\frac{\partial \left}{\partial x_k}\right>\\&= \left\frac{\partial \left}{\partial x_k}+ \left\frac{\partial \left}{\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)

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\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+\left\right)

⟨−ρuj′​​∂xi​∂p′​⟩+⟨−ρui′​​∂xj​∂p′​⟩=−ρ1​(⟨uj′​∂xi​∂p′​⟩+⟨ui′​∂xj​∂p′​⟩)

(5)

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\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

⟨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

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\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

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\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

<

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\left<\frac{\partial u_i'u_j'u_k'}{\partial x_k}\right>= \frac{\partial \left}{\partial x_k}

⟨∂xk​∂ui′​uj′​uk′​​⟩=∂xk​∂⟨ui′​uj′​uk′​⟩​上面的推导应用了

∂

u

j

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∂

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0

\frac{\partial u_j'}{\partial x_j}=0

∂xj​∂uj′​​=0、

∂

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∂

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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

j

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k

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<

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=

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\begin{aligned} \left}{\partial x_k}\right> +\left}{\partial x_k}\right>&= \left\left<\frac{\partial \left}{\partial x_k}\right> +\left\left<\frac{\partial \left}{\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

∂

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(11)

\begin{aligned} \frac{\partial\left}{\partial t}+ \left\frac{\partial\left}{\partial x_k}=& -\left\frac{\partial\left}{\partial x_k} -\left\frac{\partial\left}{\partial x_k} -\frac{1}{\rho}\left(\left+\left\right)\\&+ \nu\left- \frac{\partial }{\partial x_k}\left \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

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\begin{aligned} \left +\left&= \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 +\left<\frac{\partial u_i'p'}{\partial x_j}\right> -\left\\&=\left(\frac{\partial \left}{\partial x_i}+\frac{\partial \left}{\partial x_j}\right) -\left\\ \nu\left&= \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}{\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：

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\begin{aligned} &\underset{C_{ij}}{\underbrace{\frac{\partial\left}{\partial t}+ \left\frac{\partial\left}{\partial x_k}} }= \underset{P_{ij}}{\underbrace{-\left\frac{\partial\left}{\partial x_k} -\left\frac{\partial\left}{\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}{\rho}\delta_{jk}+ \frac{\left}{\rho}\delta_{ik}+ \left- \nu\frac{\partial \left}{\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′​⟩​)​​−Eij​

2ν⟨∂xk​∂ui′​​∂xk​∂uj′​​⟩​​​

六、参考资料

《湍流理论与模拟》第二版

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⋅张兆顺、崔桂香、许春晓、黄伟希