Reynolds-averaged momentum equation (${\partial }_t{U}_i+U_j{\partial }_j{U}_i=-{\rho }^{-1}{\partial }_i\bar{p}+\nu \Delta U_i-{\partial }_j\overline{u_i^{\prime }{u}_j^{\prime }}$)

For an incompressible flow of constant mass density $\rho$ and kinematic viscosity $\nu$, Reynolds averaging of the Navier-Stokes equation, without a body force, givesHere $\mathbf{u}=\mathbf{U}+{\mathbf{u}}^{\prime }$ and $\overline{{\mathbf{u}}^{\prime }}=0$, with averaging assumed to commute with the displayed derivatives. The mean acceleration ${\partial }_t{U}_i$ vanishes only when the averaged velocity field is stationary; taking an average does not by itself remove it.

To derive the extra term, average the conservative momentum flux: $\overline{u_i{u}_j}=U_i{U}_j+\overline{u_i^{\prime }{u}_j^{\prime }}$. Mean incompressibility turns the divergence of the first product into $U_j{\partial }_j{U}_i$. The second product is the Reynolds stress divided by mass density, using the positive-covariance convention. It describes transport of mean momentum by fluctuating velocities. A turbulence closure is needed because that covariance is an additional unknown, rather than a specified molecular transport coefficient.