kOmega 模型

kOmega (k-ω) 模型是 CFD 计算中常见的一种湍流模型。是作为雷诺平均 NS 方程 (RANS) 的封闭模型。

$$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \rho P - \beta^* \rho \omega k + \frac{\partial}{\partial x_j} \left[\left(\mu + \sigma_k \frac{\rho k}{\omega} \right)\frac{\partial k}{\partial x_j}\right], \qquad \text{with } P = \tau_{ij} \frac{\partial u_i}{\partial x_j}$$

$$\displaystyle \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\gamma \omega}{k} P - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_{\omega} \frac{\rho k}{\omega} \right) \frac{\partial \omega}{\partial x_j} \right] + \frac{\rho \sigma_d}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$$

其中，$v_t$ 涡粘度定义为 $k/\omega$，$k$ 为湍动能，$\omega$ 为比耗散率