本系列博文如下：

$k-\varepsilon$ 模型通用形式

$k-\varepsilon$ 模型有多个不同版本，对于不同版本的模型，$k$ 和 $\varepsilon$ 的输运方程都可以写成以下通用形式：

$$ \begin{equation} \frac{\mathrm{D} (\rho k)}{\mathrm{D} t} = P_k - \Phi_k + D_k^\mu + D_k^T \label{eq:keqn-g} \end{equation} $$

$$ \begin{equation} \frac{\mathrm{D} (\rho \varepsilon)}{\mathrm{D} t} = P_\varepsilon - \Phi_\varepsilon + D_\varepsilon^\mu + D_\varepsilon^T \label{eq:omegaeqn-g} \end{equation} $$

其中，$P$ 是产生项（Production term），$\Phi$ 是毁灭项（Destruction term），$D^\mu$ 是粘性耗散项（Viscous dissipation term ，$D^T$ 是湍流耗散项（Turbulent dissipation term）。

标准 $k-\varepsilon$ 模型

标准 $\displaystyle k-\varepsilon$ 模型1是最常用的一种形式，在这个模型中，$k$ 方程右端各项的定义如下

$P_k$ $\Phi_k$ $D_k^\mu$ $D_k^T$
$P$ $\rho \varepsilon$ $\displaystyle \frac{\partial }{\partial x_j} \left( \mu \frac{\partial k}{\partial x_j} \right)$ $\displaystyle \frac{\partial }{\partial x_j} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial x_j} \right)$

$\omega$ 方程右端各项的定义如下

$P_\varepsilon$ $\Phi_\varepsilon$ $D_k^\mu$ $D_k^T$
$\displaystyle C_{1\varepsilon} \frac{\varepsilon}{k} P$ $\displaystyle C_{2\varepsilon} \frac{\rho \varepsilon^2}{k}$ $\displaystyle \frac{\partial }{\partial x_j} \left( \mu \frac{\partial \varepsilon}{\partial x_j} \right)$ $\displaystyle \frac{\partial }{\partial x_j} \left( \frac{\mu_t}{\sigma_\varepsilon} \frac{\partial \varepsilon}{\partial x_j} \right)$

将以上各项代入湍流输运方程 $\eqref{eq:keqn-g}$ 和 $\eqref{eq:omegaeqn-g}$，整理后得到文献中最常见的形式：

$$ \begin{equation} \frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_j} = P - \rho \varepsilon + \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j} \right] \end{equation} $$

$$ \begin{equation} \frac{\partial(\rho \varepsilon)}{\partial t} + \frac{\partial(\rho u_j \varepsilon)}{\partial x_j} = C_{1\varepsilon}\frac{\varepsilon}{k} P - C_{2\varepsilon}\frac{\rho \varepsilon^2}{k} + \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_\varepsilon}\right) \frac{\partial \varepsilon}{\partial x_j} \right] \end{equation} $$

从 $\varepsilon$ 方程到 $\omega$ 方程

下面推导如何从 $\varepsilon$ 方程变换到 $\omega$ 方程。

$\omega$ 的定义如下：

$$ \begin{equation} \omega \equiv \frac{1}{C_\mu} \frac{\varepsilon}{k} \end{equation} $$ 其中，$C_\mu = 0.09$ ，和 $k-\omega$ 模型中的常数 $\beta^\ast$ 意义相同。

对 $\rho \omega$ 取全微分，根据链式求导法则，有

$$ \begin{equation} \begin{aligned} \frac{\mathrm{D}(\rho \omega)}{\mathrm{D} t} &= \frac{1}{C_\mu}\frac{\mathrm{D}(\rho \varepsilon/k)}{\mathrm{D} t} \newline &= \frac{1}{C_\mu} \left[ \frac{1}{k}\frac{\mathrm{D}(\rho \varepsilon)}{\mathrm{D} t} - \frac{\varepsilon}{k^2} \frac{\mathrm{D}(\rho k)}{\mathrm{D} t} \right] \newline &= \frac{1}{C_\mu k} \frac{\mathrm{D}(\rho \varepsilon)}{\mathrm{D} t} - \frac{\omega}{k} \frac{\mathrm{D}(\rho k)}{\mathrm{D} t} \newline &= \left( \frac{1}{C_\mu k} P_\varepsilon - \frac{\omega}{k} P_k \right) - \left( \frac{1}{C_\mu k} \Phi_\varepsilon - \frac{\omega}{k} \Phi_k \right) + \left( \frac{1}{C_\mu k} D_\varepsilon^\mu - \frac{\omega}{k} D_k^\mu \right) + \left( \frac{1}{C_\mu k} D_\varepsilon^T - \frac{\omega}{k} D_k^T \right) \end{aligned} \end{equation} $$

分别对右端各项进行整理。

产生项

$$ \begin{equation} \begin{aligned} P_\omega = \frac{1}{C_\mu k} \left( C_{1\varepsilon}\frac{\varepsilon}{k} P \right) - \frac{\omega}{k} P = (C_{1\varepsilon} - 1) \frac{\omega}{k} P \end{aligned} \end{equation} $$

毁灭项

$$ \begin{equation} \begin{aligned} \Phi_\omega = \frac{1}{C_\mu k} \left( C_{2\varepsilon}\frac{\rho \varepsilon^2}{k} \right) - \frac{\omega}{k} \rho \varepsilon = (C_{2\varepsilon} - 1) C_\mu \rho \omega^2 \end{aligned} \end{equation} $$

粘性耗散项

$$ \begin{equation} \begin{aligned} D_\omega^\mu &= \frac{1}{C_\mu k} \left[ \frac{\partial}{\partial x_j} \left(\mu \frac{\partial \varepsilon}{\partial x_j} \right) \right] - \frac{\omega}{k} \left[ \frac{\partial}{\partial x_j} \left(\mu \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{k} \frac{\partial}{\partial x_j} \left[ \mu \left( k \frac{\partial \omega}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \right) \right] - \frac{\omega}{k} \left[ \frac{\partial}{\partial x_j} \left(\mu \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{\mu}{k} \left[ \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} + k \frac{\partial \omega}{\partial x_j} \frac{\partial \omega}{\partial x_j} + \frac{\partial \omega}{\partial x_j} \frac{\partial k}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \frac{\partial k}{\partial x_j} \right] - \frac{\mu \omega}{k} \frac{\partial^2 k}{\partial x_j^2} \newline &= \frac{2 \mu}{k} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} + \mu \frac{\partial^2 \omega}{\partial x_j^2} \end{aligned} \end{equation} $$

湍流耗散项

$$ \begin{equation} \begin{aligned} D_\omega^T &= \frac{1}{C_\mu k} \left[ \frac{\partial}{\partial x_j} \left(\frac{\mu_t}{\sigma_\varepsilon} \frac{\partial \varepsilon}{\partial x_j} \right) \right] - \frac{\omega}{k} \left[ \frac{\partial}{\partial x_j} \left(\frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{\sigma_\varepsilon k} \frac{\partial}{\partial x_j} \left[ \mu_t \left( k \frac{\partial \omega}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \right) \right] - \frac{\omega}{\sigma_k k} \left[ \frac{\partial}{\partial x_j} \left(\mu_t \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{\sigma_\varepsilon k} \left[ \frac{\partial \mu_t}{\partial x_j} \left( k \frac{\partial \omega}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \right) + \mu_t \frac{\partial}{\partial x_j} \left( k \frac{\partial \omega}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \right) \right] - \frac{\omega}{\sigma_k k} \left[ \frac{\partial}{\partial x_j} \left(\mu_t \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{\sigma_\varepsilon k} \left[ \frac{\partial \mu_t}{\partial x_j} \left( k \frac{\partial \omega}{\partial x_j} + \omega \frac{\partial k}{\partial x_j} \right) + \mu_t \left( 2\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} + k \frac{\partial^2 \omega}{\partial x_j^2} + \omega \frac{\partial^2 k}{\partial x_j^2} \right) \right] - \frac{\omega}{\sigma_k k} \left[ \frac{\partial}{\partial x_j} \left(\mu_t \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{\sigma_\varepsilon k} 2\mu_t \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} + \frac{1}{\sigma_\varepsilon k} \left[ k \left( \frac{\partial \mu_t}{\partial x_j} \frac{\partial \omega}{\partial x_j} + \mu_t \frac{\partial^2 \omega}{\partial x_j^2} \right) + \omega \left( \frac{\partial \mu_t}{\partial x_j} \frac{\partial k}{\partial x_j} + \mu_t \frac{\partial^2 k}{\partial x_j^2} \right) \right] - \frac{\omega}{\sigma_k k} \left[ \frac{\partial}{\partial x_j} \left(\mu_t \frac{\partial k}{\partial x_j} \right) \right] \newline &= \frac{1}{\sigma_\varepsilon k} 2\mu_t \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} + \frac{1}{\sigma_\varepsilon} \frac{\partial}{\partial x_j} \left( \mu_t \frac{\partial \omega}{\partial x_j} \right) + \left( \frac{1}{\sigma_\varepsilon} - \frac{1}{\sigma_k} \right) \frac{\omega}{k} \frac{\partial}{\partial x_j} \left( \mu_t \frac{\partial k}{\partial x_j} \right) \end{aligned} \end{equation} $$

$\omega$ 方程的最终形式

最终得到如下形式的 $\omega$ 方程：

$$ \begin{equation} \begin{aligned} \frac{\partial(\rho \omega)}{\partial t} + \frac{\partial(\rho u_j \omega)}{\partial x_j} =& \underbrace{% (C_{1\varepsilon} - 1) \frac{\omega}{k} P \vphantom{\frac{\partial^2 \omega}{\partial x_j^2}} % underbrace at same level height }_ {P_\omega} - \underbrace{% (C_{2\varepsilon} - 1) C_\mu \rho \omega^2 \vphantom{\frac{\partial^2 \omega}{\partial x_j^2}} % underbrace at same level height }_ {\Phi_\omega} + \underbrace{% {\color{blue}\frac{2 \mu}{k} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}} + \mu \frac{\partial^2 \omega}{\partial x_j^2} }_ {D_\omega^\mu} \newline &+ \underbrace{% {\color{red}\frac{1}{\sigma_\varepsilon k} 2\mu_t \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}} + \frac{1}{\sigma_\varepsilon} \frac{\partial}{\partial x_j} \left( \mu_t \frac{\partial \omega}{\partial x_j} \right) + {\color{blue}\left( \frac{1}{\sigma_\varepsilon} - \frac{1}{\sigma_k} \right) \frac{\omega}{k} \frac{\partial}{\partial x_j} \left( \mu_t \frac{\partial k}{\partial x_j} \right)} }_ {D_\omega^T} \end{aligned} \end{equation} $$

Menter 的文献2中提到了变换后的 $\omega$ 方程忽略了一项耗散项：

A small additional diffusion term is neglected in the transformation. It is shown in Ref. 9 that the term has virtually no effect on the solutions

经对比后发现实际忽略了两项，即上式中用蓝色标出的两项。

注意 $\mu_t$ 与 $k$ 和 $\omega$ 之间有如下关系：

$$ \begin{equation} \mu_t = \frac{\rho k}{\omega} \end{equation} $$

因此上式中红色的交叉耗散项可进一步写作：

$$ \begin{equation} {\color{red}\frac{1}{\sigma_\varepsilon k} 2\mu_t \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}} = \frac{2\rho}{\sigma_\varepsilon} \frac{1}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} \end{equation} $$

于是得到文献中的形式：

$$ \begin{equation} \begin{aligned} \frac{\partial(\rho \omega)}{\partial t} + \frac{\partial(\rho u_j \omega)}{\partial x_j} =& (C_{1\varepsilon} - 1) \frac{\omega}{k} P - (C_{2\varepsilon} - 1) C_\mu \rho \omega^2 \newline &+ \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_\varepsilon}\right) \frac{\partial \omega}{\partial x_j} \right] + \frac{2\rho}{\sigma_\varepsilon} \frac{1}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} \end{aligned} \end{equation} $$

  1. Launder, B. E., & Sharma, B. (1974). Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and Mass Transfer, 1(2), 131–137. https://doi.org/10.1016/0094-4548(74)90150-7 ↩︎

  2. Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8), 1598–1605. https://doi.org/10.2514/3.12149 ↩︎