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
满足:

其中,
为时均值,
为脉动值。时均值可定义为:

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 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}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ecd351ee66ba2e0e11d6793bb4d56b2c0d30ae59)
![{\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}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f9886eb0222028cc5d8b427f82bd8761d3a5cd0b)
![{\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}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0e5acac5e3e302c285c19fe1f9e51aaeb4bd3746)
如果使用张量中的指标符号,则又可表示为:


上式中的
被称作雷诺应力,即:
