# Reynolds-averaged Navier-Stokes equations

The Reynolds-averaged Navier-Stokes equations are time-averaged equations of motion for fluid flow.

For open-channel flow, the incompressible Reynolds Momentum Equations are:

[itex]{ \partial \bar{u_i} \over \partial t} + \bar{u_j} { \partial \bar{u_i} \over \partial x_j} = g_i - { 1 \over \rho}{ \partial \bar{p} \over \partial x_i } - {\partial \overline{u_i^\prime u_j^\prime} \over \partial x_j} + \upsilon {\partial^2 \bar{u_i} \over \partial x_j^2}[itex]

Essentially a detailed version of Newton's second law for fluid flow, the left hand side of the equation represents the change in momentum of the flow, caused by the forces acting on it - the driving force of gravity, the pressure gradient, turbulent diffusion or Reynolds stresses, and viscous diffusion.

Applicable to steady flows, these equations are obtained by separating flow velocity into mean (or time-averaged) and fluctuating parts

[itex] u = \bar{u} + u^\prime [itex]

u = instantaneous velocity
ū = time-averaged velocity
u′ = velocity fluctuation

This is known as Reynolds decomposition.

This averaging results in new terms in the equations of motion, for example u′u′ or u′w′, known as Reynolds stresses — a measure of flow turbulence.

These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate averaged solutions to the Navier-Stokes equations of motion. Be aware that ensemble averaging is not the same as Reynolds averaging, and confusing the two concepts can give nonsensical results.

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