# 雷诺平均纳维－斯托克斯方程

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

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

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

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x_{i}}}(\rho u_{i})=0}$
${\displaystyle {\frac {\partial }{\partial t}}(\rho u_{i})+{\frac {\partial }{\partial x_{j}}}(\rho u_{i}u_{j})=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}(\mu {\frac {\partial u_{i}}{\partial x_{j}}}-\rho {\overline {u_{i}'u_{j}'}})+S_{i}}$

${\displaystyle \tau _{ij}=-{\overline {u_{i}'u_{j}'}}}$

## 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${\displaystyle \phi }$ 代替含上划线的${\displaystyle {\bar {\phi }}}$

## 参考资料

• 王福军. 《计算流体动力学分析》. 清华大学出版社.