雷诺平均纳维-斯托克斯方程

雷诺平均纳维－斯托克斯方程（英語：Reynolds-averaged Navier–Stokes equations，简称RANS）是流体力学中一种用来描述湍流的时均纳维－斯托克斯方程。其思想是将湍流运动看作时间平均与瞬时脉动两种流动的叠加，即任一物理量${\displaystyle \phi }$满足：

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

其中，${\displaystyle {\bar {\phi }}}$为时均值，${\displaystyle \phi '}$为脉动值。时均值可定义为：

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

如果不考虑密度脉动的影响，对纳维－斯托克斯方程中的物理量按上述方法取时间平均，可得到可压缩流体平均流动的控制方程（即雷诺平均方程）：^{[注 1]}

${\displaystyle {\frac {\partial \rho }{\partial t}}+\operatorname {div} (\rho \mathbf {u} )=0}$

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

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

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

如果使用张量中的指标符号，则又可表示为：

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x_{i}}}(\rho u_{i})=0}$

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

上式中的${\displaystyle -{\overline {u_{i}'u_{j}'}}}$被称作雷诺应力，即：

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

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.