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

An Eulerian time average of an incompressible constant-density Navier-Stokes equation gives the displayed mean momentum equation. Decompose velocity into its mean and a zero-mean fluctuation. In conservative form, the averaged momentum flux is $\overline{u_i{u}_j}=U_i{U}_j+\overline{u_i^{\prime }{u}_j^{\prime }}$; mean incompressibility then turns its first term into mean advection. The fluctuation covariance is the Reynolds stress divided by density, using the positive covariance convention. Closing this extra unknown requires a turbulence model.