# 雷诺平均纳维－斯托克斯方程

$\phi=\bar{\phi}+\phi'.$

$\bar{\phi}=\frac{1}{\Delta t} \int_{t}^{t+\Delta t} \phi(t) dt.$

$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho \mathbf{u}) = 0$
$\frac{\partial (\rho u)}{\partial t} + \operatorname{div}(\rho u\mathbf{u}) = \operatorname{div}(\mu\ \operatorname{grad}u) - \frac{\partial p}{\partial x} + \left[ -\frac{\partial (\rho \overline{u'^2})}{\partial x}-\frac{\partial (\rho \overline{u'v'})}{\partial y}-\frac{\partial (\rho \overline{u'w'})}{\partial z} \right] + S_u$
$\frac{\partial (\rho v)}{\partial t} + \operatorname{div}(\rho v\mathbf{u}) = \operatorname{div}(\mu\ \operatorname{grad}v) - \frac{\partial p}{\partial x} + \left[ -\frac{\partial (\rho \overline{u'v'})}{\partial x}-\frac{\partial (\rho \overline{v'^2})}{\partial y}-\frac{\partial (\rho \overline{v'w'})}{\partial z} \right] + S_v$
$\frac{\partial (\rho w)}{\partial t} + \operatorname{div}(\rho w\mathbf{u}) = \operatorname{div}(\mu\ \operatorname{grad}w) - \frac{\partial p}{\partial x} + \left[ -\frac{\partial (\rho \overline{u'w'})}{\partial x}-\frac{\partial (\rho \overline{v'w'})}{\partial y}-\frac{\partial (\rho \overline{w'^2})}{\partial z} \right] + S_w$

$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}(\rho u_i) = 0$
$\frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j}(\rho u_i u_j) = -\frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}(\mu \frac{\partial u_i}{\partial x_j} - \rho \overline{u_i'u_j'}) + S_i$

$\tau_{ij} = \overline{u_i'u_j'}$

## 注释

1. ^ 式中为方便起见，对于非脉动值的时均值，使用去掉上划线的$\phi$代替含上划线的$\bar{\phi}$

## 参考资料

• 王福军. 《计算流体动力学分析》. 清华大学出版社.