雷诺平均纳维-斯托克斯方程

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雷诺平均纳维-斯托克斯方程英语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\phi满足:

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

其中,\bar{\phi}为时均值,\phi'为脉动值。时均值可定义为:

\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]

\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

如果使用张量中的指标符号,则又可表示为:

\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

上式中的-\overline{u_i'u_j'}被称作雷诺应力,即:

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

注释[编辑]

  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[编辑]

  • 王福军. 《计算流体动力学分析》. 清华大学出版社.